Scanning Tunnelling Spectroscopy of Superconductors and Coulomb Blockade Effects

Dominic Simon Wilson
Girton College, Cambridge



Dissertation submitted for the Degree of Doctor of Philosophy
at the University of Cambridge

January, 1996

Contents

1  Introduction
    1.1  Scanning Probe Spectroscopy
    1.2  Plan of the Thesis
2  Experimental Design
    2.1  The STM design and specifications
    2.2  The STM Head
        2.2.1  The sample mounting and carriage
        2.2.2  The piezo-electric tubes
        2.2.3  Calculation of piezo tube motion
        2.2.4  Stick-slip approach mechanism
        2.2.5  The scanning piezo tube
        2.2.6  The cryostats and inserts
            The continuous flow cryostat
            The superinsulated cryostat
    2.3  System performance
        2.3.1  Current sensitivity
        2.3.2  Vibration Isolation
            Frequency response of the suspension
    2.4  The STM Control and Electronics
        2.4.1  The current amplifier and feedback loop
        2.4.2  Surface scanning
        2.4.3  Current-voltage characteristics
        2.4.4  Analogue-to-digital conversion
        2.4.5  Numerical differentiation
        2.4.6  Current imaging tunnelling spectroscopy (CITS)
        2.4.7  Modifications for point-contact spectroscopy
    2.5  Experimental procedure
        2.5.1  Tip preparation
        2.5.2  Procedure for scanning tunnelling microscopy
        2.5.3  Procedure for point-contact spectroscopy
3  Theory of Superconductivity
    3.1  General Features of Superconductivity
        3.1.1  Fundamental properties of superconductors
        3.1.2  Observed properties of superconductors
            Critical temperature
            Electrical resistance
            Thermoelectric properties
        3.1.3  Critical field: type I and type II superconductors
            Type I superconductors
            Type II superconductors
            Temperature dependence of the critical field
    3.2  A Microscopic Theory of Superconductivity
        3.2.1  The London equation
        3.2.2  Nonlocal theory and the Pippard coherence length
    3.3  BCS Theory
        3.3.1  Development and justification of the BCS theory
        3.3.2  Formation of Cooper pairs
        3.3.3  The BCS ground state
        3.3.4  Finite temperature effects
            Critical temperature Tc
    3.4  Excitations of Superconductors
        3.4.1  Single-particle excitations
        3.4.2  Derivation of the superconducting energy gap
        3.4.3  Quasiparticle creation and annihilation operators
    3.5  Ginzburg-Landau Theory
        3.5.1  Ginzburg-Landau free energy
        3.5.2  The Ginzburg-Landau differential equations
        3.5.3  The temperature-dependent coherence length
        3.5.4  Results from Ginzburg-Landau theory
            Superconducting sheath effect
    3.6  The Proximity Effect
        3.6.1  Standard results for the proximity effect
    3.7  High-Tc Superconductors
        3.7.1  High-Tc Properties
4  Theory of Tunnelling
    4.1  Introduction to Tunnelling
        4.1.1  Historical development
    4.2  Theoretical Models for Tunnel Junctions
        4.2.1  Theoretical treatment of the barrier
            Work function effects
            Barrier models
        4.2.2  Corrections to the image potential
            Local density approximation
        4.2.3  Theoretical treatment of the electrodes
            Allowed elastic tunnelling events
            Finite temperature effects
        4.2.4  Indirect and higher order tunnelling processes
            Inelastic processes
            Resonant tunnelling
            Two-step and multiple-step tunnelling
            Surface states
            Image states and Gundlach oscillations
            Tamm states
            Coulomb blockade effects
    4.3  Methods of Tunnel Current Calculation
        4.3.1  The WKB method
            Steady state approach
            Angular dependence of the tunnel current
            Sharp boundary approximation
        4.3.2  Time-dependent perturbation theory approach
    4.4  Metal-Insulator-Metal (MIM) Junctions
        4.4.1  Electron tunnelling picture
        4.4.2  Numerical calculations for asymmetrical barriers
        4.4.3  Tunnelling spectroscopy
    4.5  NIS Junctions
        4.5.1  Excitation description of tunnelling events
        4.5.2  Other factors influencing tunnelling data
            Lifetime effects
            Phonon effects
    4.6  SIS Junctions
        4.6.1  Semiconductor model
        4.6.2  Multiple electron effects
    4.7  Proximity Effect (NINS) Tunnelling
        4.7.1  Experimental observation
        4.7.2  Theory of NINS tunnelling
            The pair potential
            Normal metal with V > 0
            Normal metal with V = 0 (or V < 0)
            Summary of expected results from NS tunnelling
        4.7.3  Other processes at NS interfaces
5  Theory of Point-Contact Spectroscopy
    5.1  NcN Point Contacts
        5.1.1  Sharvin results
            Interpolation between ballistic and thermal (Maxwell) regimes
        5.1.2  Phonon spectroscopy in the ballistic regime
            Diffusive regime versus ballistic regime
        5.1.3  Experimental results
            Transverse electron focussing
    5.2  NcS Point Contacts and Andreev Reflection
        5.2.1  The BTK model
        5.2.2  BTK results
            Theoretical conductance curves
            Determination of barrier strength Z from experimental data
            Meaning of the barrier-strength parameter Z
            Finite temperature effects
            Experimental observations of Andreev reflection
            Experimental fit to BTK model
        5.2.3  Extension of the BTK model to include the proximity effect
6  Scanning Tunnelling Microscopy and Spectroscopy
    6.1  Introduction to the Scanning Tunnelling Microscope
        6.1.1  Scanning tunnelling spectroscopy (STS)
    6.2  STM Tunnel Current Calculations
        6.2.1  Treatment of the barrier in STM
        6.2.2  Apparent barrier height
            Theoretical calculations of apparent barrier height
            Behaviour of the apparent barrier height for small tip/sample separations
            Effect of atomic forces between the electrodes
            Transition from tunnelling to ballistic transport
        6.2.3  The modified Bardeen approach (MBA)
        6.2.4  Green's function approach
    6.3  STM as a Topographic and Spectroscopic Tool
        6.3.1  Topographic techniques
            Constant-current imaging
            Spectroscopic considerations
            Constant-height imaging
            Other modes of operation of STM
        6.3.2  Spectroscopic techniques
        6.3.3  Related scanning microscopies
    6.4  Theories of Topographic Imaging in STM
        6.4.1  Perturbation theory and plane-wave transmission approaches
        6.4.2  Tip atom considerations
            Derivative rule
        6.4.3  Imaging of graphite
        6.4.4  Transition to point-contact regime
    6.5  Topographic Imaging: Preliminary Tests of the STM
        6.5.1  Atomic resolution with graphite
            Liquid helium-temperature atomic resolution
        6.5.2  Gold films
7  Coulomb Blockade Effects: Theory and Results
    7.1  Theory of Coulomb Gap and Coulomb Staircase Effects
        7.1.1  Coulomb gap: Zeller & Giaever's model
        7.1.2  Highly asymmetric junctions
        7.1.3  More general case: symmetric and asymmetric junctions
        7.1.4  Multiple junctions: ensemble effects
        7.1.5  Superconducting particles
        7.1.6  Coulomb staircase
        7.1.7  Experimental detection of Coulomb effects: planar junctions
        7.1.8  Experiments using low-temperature STM
    7.2  Results
        7.2.1  Coulomb staircase effects
        7.2.2  Coulomb gap
        7.2.3  Discussion
8  Tunnelling Spectroscopy of Superconductors: Results and Discussion
    8.1  Tunnelling into Elemental Superconductors
        8.1.1  Pb films
            Sample preparation
            Pb results
            Comparison with theoretical curve
        8.1.2  Nb films
            Sample preparation
            Nb results
    8.2  Tunnelling into Niobium Nitride (NbN) and NbN/Au Films
            NbN film preparation
        8.2.1  Bare NbN films
        8.2.2  NbN/Au bilayers
    8.3  High-Tc Superconductors
        8.3.1  Bare YBCO films
            Measurement of the reduced gap in YBCO
        8.3.2  YBCO/Ag films
        8.3.3  BSCCO crystals
            Tunnelling measurements of bare BSCCO crystals
            Other factors affecting energy gap detection
            Measurement of the reduced gap of BSCCO
            Symmetry of the BSCCO gap: d-wave or s-wave?
            Gap broadening and linear background
        8.3.4  BSCCO/Au bilayers
9  Point-Contact Spectroscopy of Superconductors: Results and Discussion
    9.1  Bare NbN films
    9.2  Comparison with Results for Nb
    9.3  NbN/Au Bilayers
    9.4  Summary of Point-Contact Spectroscopy Results
10  Conclusions and Suggestions for Further Work
    10.1  The STM System
    10.2  Coulomb Blockade Effects
    10.3  Tunnelling Experiments on Superconductors
        10.3.1  Bare superconductors
        10.3.2  NINS tunnelling
    10.4  Point-Contact Spectroscopy of Superconductors
        10.4.1  Bare superconductors
        10.4.2  NcNS structures
    10.5  Concluding Remarks
11  References

List of Figures

    2.1  The STM head. Scale shown is approximately four times actual size.
    2.2  The sample carriage. The aluminium bar is pushed against the glass rods by the phosphor-bronze clip, so the carriage is held up by static friction. The screw adjustment, which is screwed into the main body of the carriage, is used to optimise the friction for the stick-slip mechanism. Scale is approximately five times actual size.
    2.3  The scanning tube electrode configuration.
    2.4  The curved piezo tube. The piezo tube may be considered to curve along an arc of a circle. The coefficients x and y represent the increase in height of the centres of the X- and Y-electrodes owing to the tilt of the top face of the piezo, which is assumed to be an undistorted circle. In this diagram, y = 0, and x is negative, so if -x is the distance by which the centre of the X-electrode is lowered, the part of the tube's top surface at an angle f from here is lowered by a distance -x \mathgroup \symoperators cos
\nolimits f.

    2.5  The curved piezo tube (cross-section). The relation of the x- and y-displacements of the tip to the tilt of the top surface of the piezo tube is shown, for a tube curved along the arc of a circle. The x-z plane cross-section is displayed.
    2.6  Electrode configurations for single- and double-ended amplifiers.
    2.7  Point-contact spectroscopy circuit. The circuit was capable of generating and measuring currents in the range 30 nA-480 mA.
    3.1  Temperature dependence of the energy gap.
    3.2  Temperature dependence of the critical field Hc(T).
    3.3  The pair occupation fraction (solid line) for a superconductor at T = 0 compared to the Fermi function (dotted line) for a metal at T = Tc.
    3.4  Energy Ek of quasiparticle excitations in a superconductor (solid line) and a normal metal (dotted line) plotted against corresponding free-electron energy ek.
    3.5  The theoretical normalised BCS superconducting density of states for tunnelling experiments, nS(E). The abscissa is the ratio of superconducting and normal state densities of states.
    3.6  Ginzburg-Landau free energy functions. If a > 0 the minimum occurs at y = 0. If a < 0 the minimum occurs at a finite value, y = y, and has a value [ 1/ 2]m0 Hc2.
    3.7  Variation of the pair potential near an NS interface for (a) attractive interaction between electrons in the normal metal; (b) repulsive interaction; (c) Ginzburg-Landau approximation.
    4.1  Trapezoidal barrier between planar metallic electrodes.
    4.2  Infinite line of image charges generated by two parallel planar conductors.
    4.3  Barrier shape for planar junction showing the effect of image charges. The effect of the image charges is to reduce the effective height and width of the barrier. I is the energy of the insulator below the vacuum energy, and f1, f2 are the effective work functions, Y1-I and Y2-I. \mathaccent "7016\relax f is the mean barrier height as seen by an electron in the left electrode, and the junction is biased with a voltage +V applied to the right electrode. s1 and s2 are the classical turning points, between which the particle may be considered to tunnel.
    4.4  Elastic tunnelling events at zero temperature. Elastic tunnelling events correspond to a horizontal transition from left to right. The right hand electrode is biased to a voltage +V. Electrons can only tunnel across from an occupied state to an unoccupied one, so transitions can occur only in an energy range eV. The probability of electron tunnelling is proportional to the density of occupied states on the left hand side, multiplied by the density of unoccupied states at the same energy on the right.
    4.5  Elastic tunnelling processes at finite temperature. Elastic tunnelling processes are horizontal transitions. Electron tunnelling probability is weighted by the probability of occupancy (the Fermi function, f(E)) on the left hand side multiplied by the probability of vacancy (1-f(E+eV)) at the same energy on the right hand side. Because both electrodes have both filled and unfilled states at all energies, electron tunnelling can occur in either direction.
    4.6  Elastic (E) and inelastic (I) tunnelling processes. The inelastic process can only occur when the bias exceeds a voltage \mathchar 26\mkern -9muhw/e, where w is the frequency of the phonon involved in this process.
    4.7  Whole-system wavefunction used in WKB (steady-state) approach.
    4.8  One-dimensional rectangular barrier. This is the simple barrier shape used for comparison of the WKB approximation with the exact result for transmission through a square barrier.
    4.9  Wavefunctions used in the Bardeen approach to tunnelling theory (transfer Hamiltonian method). The wavefunction of each electrode is a standing wave plus exponentially decaying tail. The tails overlap in the barrier region only, and the tunnel current is calculated from the overlap using the Fermi Golden Rule.
    4.10  NIS tunnelling events at finite temperature.
    4.11  Theoretical conductance curves for an NIS junction at various temperatures. S=Pb: Tc = 7.2 K, 2D/kT = 4.67. Curves calculated for T = 0 (dashed); T = 1 K, 3 K, 4.2 K, 6 K and 7 K.
    4.12  Tunnelling from a normal metal into a superconductor in the excitation description. The excitation in the superconductor may be created in either branch of the E-k diagram.
    4.13  Lifetime effects for different values of G/D. The effect of G is to smear the curve, and reduce the conductance peaks at eV = D. The zero-bias conductance is always zero. Curves are shown for (with decreasing peak heights) G/D = 0.05, 0.2, 0.3. The BCS curve (dashed) is also shown.
    4.14  Excitation diagram for SIS tunnelling at T = 0. Single electron tunnelling can only occur with the splitting up of a Cooper pair from the condensate, after which one of the electron can cross the barrier.
    4.15  Density of states tunnelling diagram for SIS junction at T = 0. At eV = D1+D2, a singularity of filled states on the left is opposite a singularity of empty states on the right, leading to a very sharp increase in the tunnel current as the voltage is increased past this level.
    4.16  Tunnelling diagram for SIS junction at finite temperature. At finite temperatures, some electrons in the right hand electrode are excited above the gap, enabling electrons to tunnel from the left to the right, into the few empty states. At eV = |D1-D2| the two singularities are opposite, as shown, giving a sharp feature in the tunnelling characteristic, considerably smaller than the feature at D1+D2.
    4.17  Pair potential over an NINS tunnelling structure, with V positive in the normal metal. F and V are both positive in the N layer, so D = FV is positive. An energy gap exists at the surface, of size Dsurface.
    4.18  Pair potential and pair amplitude (dashed) over an NINS tunnelling structure, with V = 0 in the normal metal (e.g.  Au). F is positive in the N layer, but V = 0, so D = FV-0. No energy gap exists in the normal metal, but effects may be detected by tunnelling in the ballistic limit, since electrons entering the normal metal from the superconductor are all of energy E > D, and peaked at E = D.
    5.1  Theoretical conductance curves for several values of barrier strength, Z. RN dI/dV is the ratio of junction conductance to normal-state conductance.
    5.2  Theoretical I-V  curves for several values of barrier strength, Z.
    5.3  Pair potential used by van Son for calculation of Andreev reflection probability.
    6.1  Surface atomic arrangement in cleaved graphite. (a) shows all the surface atoms; (b) shows those atoms visible to STM as filled circles; atoms represented by open circles are invisible.
    6.2  STM image of graphite surface, 10 nm×10 nm in area. (a) shows the untransformed image; (b) shows the transformed image obtained by anticlockwise rotation of 90, vertical stretch by 10% and shearing by 10 anticlockwise. The expected atomic positions are shown as filled circles.
    6.3  4 nm×4 nm images of graphite showing surface contaminant. The images were taken in the order (a), (b), (c), and the surface feature appears larger and smoother in later images.
    6.4  4 nm×4 nm image of graphite taken at T = 4 K. Atomic positions are unclear, although lines of atoms can be distinguished.
    6.5  Images of gold deposited on glass. (a) shows a 200 nm×200 nm image taken at T = 5 K; (b) shows a 500 nm×500 nm image taken at room temperature; (c) shows a 1.5 mm×1.5 mm image taken at room temperature.
    7.1  Coulomb gap conductance curves. (a) Tunnelling through a single particle of capacitance C (dashed line: dQ = 0; solid line: dQ = 0.3e); (b) Ensemble of particles of capacitance C; (c) Ensemble of particles with inverse capacitances uniformly distributed between 1/2C and 2/C; (d) Ensemble of particles with inverse capacitances normally distributed, mean C, s.d. 0.3C. R is the junction resistance (numerical calculations performed by the author).
    7.2  Theoretical I-V  characteristics for Coulomb effects. The limiting cases of the Coulomb gap (dashed) and the Coulomb staircase (solid) are shown.
    7.3  Selected Area Channelling Pattern for thin Au on mica film deposited at 400C. The six-fold symmetric pattern indicates that this film is epitaxial, and (111)-aligned.
    7.4  Room temperature 250 nm×250 nm image of the gold on mica film. Image (a) uses a grey-scale to represent heights; (b) displays the same data in 3D projection. The surface is smooth and continuous.
    7.5  Coulomb staircase observed tunnelling into Au on mica film. Conductance curves (c) and (d) correspond to I-V  characteristics (a) and (b) respectively. Step width in each case is 82 mV. The equal step widths, as a function of voltage, obtained for different scan rates, confirm that the current oscillations are a function of voltage (and not time), as expected for the Coulomb staircase effect.
    7.6  Coulomb staircases with different step widths, observed with Au on mica film. Conductance curves (c) and (d) correspond to I-V  characteristics (a) and (b) respectively.
    7.7  Coulomb gap obtained tunnelling into Au on mica film. Graphs (c), (d) are the numerical derivatives of the I-V  graphs (a), (b) respectively.
    8.1  Experimental conductance curves for Pb films on glass at various temperatures. The superconducting energy gap in the density of states is shown.
    8.2  Comparison of measured (solid) and theoretical (dashed) conductance curves for Pb/Au junction. The theoretical curve is calculated for Tc = 7.2 K, T = 5.3 K, 2D/kTc=4.67.
    8.3  Measured (solid) and Theoretical (dashed) conductance curves for Nb/Au junction, GNN = 2 mS. Theoretical parameters are Tc = 9.2 K, T = 2.2 K, 2D/kTc=3.89.
    8.4  Measured (solid) and Theoretical (dashed) conductance curves for Nb/Au junction, GNN = 230 nS. Theoretical parameters are Tc = 9.2 K, T = 2.2 K, 2D/kTc=3.89.
    8.5  Conductance curve for bare NbN on MgO film at 5 K. Parameters used to calculate fit were Tc = 13.7 K; T = 5.0 K; 2D/kTc
    8.6  Conductance curve for bare NbN on sapphire film. Parameters used for fit (dashed) were: Tc = 16 K; T = 3.5 K; 2D/kTc
    8.7  Topographic (a) and CITS (b) 250 nm×250 nm images of NbN/Au film at T = 4 K. The darker areas of (b) indicate places where the current is depressed at V = 0.5 mV; these areas are outlined in the topographic image (a), showing correlation between the topographic and spectroscopic data. In the bottom right hand area outlined in (a), the effect is much smaller.
    8.8  Current-voltage characteristics with NbN/Au sample. Graphs (a) and (b) were taken at different positions in the darker regions of the CITS image; (c) was taken very close to the edge of the region, and (d) within the light (linear I-V ) region of the image.
    8.9  Energy gap in YBCO resolved by STM at 5 K.
    8.10  Fit of experimental I-V  for STM junction with YBCO to BCS curve. Curve (a) shows the numerically differentiated conductance curve (solid line) and linear background (dashed); (b) shows the curve with the asymmetric linear background subtracted out; (c) shows the fit of experimental data (solid line) to a BCS curve with 2D/kTc=3.5. Tc = 92 K. BCS curves are shown for (short dashes) T = 5 K (actual temperature) and (long dashes) T = 40 K (to cause smearing of the BCS curve).
    8.11  150 nm×150 nm topographic (a) and CITS (b) image of (103)-oriented YBCO/Ag bilayer. The lighter areas show regions of depressed current at a bias voltage V = -1 mV. The outline of these regions is superimposed on the topographic image.
    8.12  Conductance dips obtained tunnelling into depressed current region of YBCO/Ag bilayer. High-Tc energy gap corresponds to 2D/kTc=6.5.
    8.13  Sample of conductance curves for Au tip/BSCCO tunnel junction showing conductance background. The horizontal scale in (d)-(f), which show a flattening-off of the linear background, is double that for (a)-(c).
    8.14  Sample of conductance curves for Au tip/BSCCO tunnel junction showing dip around zero bias.
    8.15  Conductance curve for Au tip/BSCCO tunnel junction showing possible BCS energy gap. Curve (a) shows the conductance data; (b) shows the linear conductance fit (dashed line); (c) shows the curve with the linear background subtracted out; (d) shows the curve fitted to the BCS curve for 2D/kTc=6 (Tc = 85 K; T = 5 K).
    8.16  Poor fit of BSCCO gap to lifetime broadening model; G = 2 meV.
    8.17  Other conductance curves obtained with Au tip/BSCCO tunnel junction. Curve (a) shows a BCS-like characteristic for positive bias only; (b) shows dips outside the gap region.
    8.18  Conductance curves for tunnelling from Au tip to Au/BSCCO structure (NINS). A zero-bias conductance dip can be discerned in curves (a)-(c), of full width 20 mV. Curve (d) exhibits only a conductance background.
    8.19  150 nm×150 nm topographic image of BSCCO/Au taken at room temperature. The gold appears to be continuous.
    9.1  Point-contact spectroscopy conductance curves for bare NbN film at several temperatures. Curves measured with junction resistance r = 0.2 W (G = 5 S). Successive curves are offset vertically by 5 S.
    9.2  PCS conductance curve for NbN film at T = 3.8 K.
    9.3  PCS conductance curve for NbN film at T = 14.8 K.
    9.4  Comparison of PCS conductance curve for NbN film at T = 3.8 K with thermally smeared BTK model. Parameters used were: Tc = 17 K; T = 3.8 K; 2D/kTc=4; Z = 0.84.
    9.5  Behaviour of normalised conductance dip position with temperature for bare NbN film. The behaviour of D(T) is also shown, for Tc = 17 K. The variation of the dip position follows the variation of D(T) fairly well, except for a small deviation near Tc.
    9.6  Conductance curves for NbN over a range of junction conductances. Successive curves are offset vertically by 1 unit. Normalised conductance is calculated as (differential conductance)/(absolute conductance at V = 20 mV); the absolute conductance is given to the right of each curve.
    9.7  Conductance curves for bare NbN sample at various junction conductances. Curves are offset vertically by 1 unit.
    9.8  Conductance curves for bare NbN sample at various junction conductances. Curves are offset vertically by 2 units. Negative conductance regions appear in the top two graphs.
    9.9  Conductance curve for NbN film at G = 6 S.
    9.10  Conductance curves for Nb sample at G = 400 mS. Some similarity with BTK predictions is shown, but peaks are insufficiently high, and narrow conductance dips either side of central peak are not predicted by BTK.
    9.11  I-V  and s-V curves for point contacts with NbN/Au. Graphs (a) and (c) are the I-V  and s-V curves for I = 300 mA at 20 mV; (b) and (d) for I = 30 mA at 20 mV.
    9.12  Current-voltage curves for NbN/Au film using point-contact spectroscopy at temperatures between 4.8 K and 14.2 K. Successive curves are vertically offset by 0.1 A for clarity.
    9.13  Conductance curves for NbN/Au film using point-contact spectroscopy at temperatures between 4.8 K and 14.2 K. Successive curves are vertically offset by 10 S for clarity. The flat central increased conductance region has exactly twice the background conductance (at temperatures up to 12 K), as predicted by BTK, but the plateau widths are very much larger than expected, and extremely asymmetric.
    9.14  Dependence of normalised conduction dip position on temperature for NbN/Au film. The behaviour of D(T) is also shown, calculated for Tc = 16 K.
    9.15  I-V  curves for NbN/Au film at several temperatures.
    9.16  Conductance curves for NbN/Au film at several temperatures.
    9.17  I-V  (a) and s-V (b) curves for point contact junction with NbN/Au at 2.6 K.
    9.18  Comparison of plateau width for NbN/Au conductance curves with D(T)/D0.
    9.19  Conductance curve taken at T = 16.8 K for point contact junction with NbN/Au film. The two measurements were made before and after driving the sample normal.
    9.20  Conductance curve taken at T = 4.5 K for point contact junction with NbN/Au film.
    9.21  Conductance curve for point contact with NbN/Au at 2.0 K.
    9.22  Normalised conductance dip positions plotted against temperature.

Chapter 1
Introduction

This dissertation describes the construction of a low temperature scanning tunnelling microscope, and applications of the STM to the study of superconducting structures. The STM was designed to be easily portable between different cryostats, by making its overall dimensions small, and using piezoelectric tubes to control both coarse and fine tip positioning, enabling only electrical feedthroughs to be used, rather than mechanical ones.

1.1  Scanning Probe Spectroscopy

In the STM, a metallic tip is positioned close enough to the sample's surface for quantum-mechanical tunnelling to occur. For topographic imaging a feedback system was employed, to move the tip in such a way as to keep the tunnel current constant. Moving the tip over the surface laterally allows topographic data to be taken, and used to build up an image, whose lateral dimension is between a few nm and a few mm.

The STM can probe the sample's electronic density of states as a function of energy by measuring the I-V  characteristic of the tip/sample junction (scanning tunnelling spectroscopy). The resolution of this technique is limited mainly by the temperature; by cooling to liquid helium temperatures (down to 1.3 K), a resolution (FWHM) of 0.4 meV may be obtained (compared to 90 meV at room temperature).

The advantage of using an STM, rather than fabricating a planar junction, is that the results obtained are based on the local density of states, rather than on a spatial average over the sample surface. The differential conductance at a voltage V is proportional to the density of states of the sample, at an energy E = eV (smeared by a thermal smearing function), provided that the tip density of states is constant. By imaging topographic and spectroscopic information simultaneously (applying a set of voltages to each point of the image, and measuring the corresponding tunnel currents), the electronic and physical characteristics of different regions of the sample may be correlated.

Tunnelling spectroscopy of superconductors and normal/superconductor (NS) bilayer structures (both conventional and high-Tc superconductors were used in this study) can reveal the superconducting density of states at the surface of the sample. For tunnelling from a normal tip into a superconducting sample, the experimental conductance curves are expected to reflect the superconducting density of states. In NS bilayers, the superconducting density of states affects the electron energy distribution in the normal layer.

Coulomb blockade effects, in which charging of a small particle requires a non-negligible voltage to be applied, can affect the experimental conductance curves obtained by scanning tunnelling spectroscopy. In the Coulomb gap effect, no current can flow until the voltage is sufficiently high that the energy required to charge the particle is provided; the current then rises linearly with voltage. In the Coulomb staircase effect, incremental charging of the particle occurs, giving rise to current steps at regular voltages.

Point-contact spectroscopy (PCS) experiments were also carried out during the course of this study. In PCS, the STM tip comes into contact with the sample, and a much (up to 8 orders of magnitude) higher current flows. The analysis of these data is more complicated than for the tunnelling case, owing to the additional processes of Andreev reflection (of an electron as a hole, with precisely reversed momentum); branch-crossing transmission, and phonon processes which depend on the junction size. Heating effects can also affect the junction behaviour. In the simplest model, in the limit of a good contact, Andreev reflection at the surface of the superconductor is expected to double the differential conductance at voltages lower than the superconducting energy gap voltage, D/e.

1.2  Plan of the Thesis

Chapter 2 describes the experimental apparatus and techniques employed in this study, in particular the design of the low-temperature scanning tunnelling microscope. Chapter 3 describes the properties of superconductors, and the development of the BCS theory. The Ginzburg-Landau theory for spatially-varying pair density is described, and used to explain the behaviour of NS bilayers. The properties peculiar to high-temperature superconductors are also considered. Chapter 4 reviews the theory of tunnel junctions, and its application to NIS and NINS junctions. In chapter 5, point-contact spectroscopy is briefly reviewed. The BTK theory for NS point contacts, and the predictions of this model, are discussed. Chapter 6 describes theories of STM imaging and spectroscopy, and the results of the preliminary tests of the STM system, imaging Au films and graphite. Chapter 7 describes the theory of Coulomb blockade effects, and the experimental observation of Coulomb staircase and gap effects by STM. Chapter 8 presents and discusses the results of tunnelling into superconductors and NS bilayers. The reduced gap values for NbN, c-oriented YBCO and BSCCO are estimated, and compared with previous reported values. Tunnelling experiments with proximity structures are discussed. Chapter 9 describes the results obtained by point-contact spectroscopy of NbN and NbN/Au films. Conclusions drawn from the work described in this thesis are presented in chapter 10.

Chapter 2
Experimental Design

2.1  The STM design and specifications

A portable cryogenic STM, and associated electronics, computer control, and vibration-isolated cryostats, were the major pieces of apparatus used for this study. This equipment was continually modified throughout the course of the experiments, as the required specifications changed, although the basic design, and the control system, remained the same throughout.

The central piece of equipment used in these experiments was the STM head. This was the mechanism which brought together the tip and sample, to a distance of approximately 1 nm, so that electron tunnelling could occur. The tip could then be scanned over the surface whilst maintaining a constant tunnel current, in order to obtain a topographic image of the surface, or held at a constant distance above a fixed point on the surface, whilst the current-voltage (I-V ) characteristic of the tip/sample junction was measured, in order to obtain spectroscopic data. The extreme mechanical stability (few×10-12 m) and current sensitivity (few×10-12 A) required necessitated very careful design to minimise vibration and electrical interference.

The STM head had to be small enough to fit into the sample spaces, and able to function over the temperature range from room temperature down to the base temperature of the cryostats. The STM had to be constructed as a transferrable unit, so that it could be moved from one cryostat to another. This, combined with the need to reduce sensitivity to vibrations, was the reason why the STM head and inserts were designed with no mechanical connections to the top of the cryostat.

Initially, various STM head designs were considered. All designs included piezo-electric control over both scanning and coarse approach mechanisms, to eliminate the need for mechanical connections to the top of the cryostat; an extremely rigid structure for resistance to vibrations; well-shielded signal wires, to enable current sensitivity of a few pA; and a very compact design to allow the STM to fit into cryostat sample spaces. The final design is shown in Figure .


Figure 2.1: The STM head. Scale shown is approximately four times actual size.

2.2  The STM Head

The STM head was mounted on a machineable ceramic1 base, of diameter 25.4 mm and thickness 6.35 mm. The scanning and approach piezo tubes were attached to the base with Araldite epoxy. Stainless steel (low-temperature) coaxial cable was used for the connections to the tip and sample, which had to be well-shielded, and varnished copper wire for the connections to the piezos. The wires ran through holes drilled into the ceramic base, which was also inset to hold the piezo tubes firmly. The wires were attached to the piezo tubes using silver paint2, which was also used to connect the sample wire to the surface of the sample, and to attach the sample to the aluminium carriage (described in section ).

The central piezo tube moved the STM tip in all three dimensions, while the outer two moved the sample carriage towards and away from the tip using a stick-slip mechanism. The STM head was built to be as rigid as possible, to reduce problems caused by vibrations.

The electrical connections to tip and sample were carefully shielded to prevent electromagnetic interference. The sample wire ran through a steel feedthrough, with a PTFE sheath separating the sample wire from the outer earth. The tip wire, which ran to the high-sensitivity current amplifier, passed through the centre of the scanning piezo tube, the inner electrode of which was earthed, greatly reducing capacitative effects caused by the high voltages (up to 185 V) applied to the outer electrodes. The inner and outer electrodes of the scanning piezo were separated from the tip holder by a ceramic spacer.

Because the coaxial cable for the tip ran inside the inner piezo tube, the whole length of the tip wire was shielded, from the inside of the piezo tube, to the amplifier, reducing electrical interference.

2.2.1  The sample mounting and carriage

The sample carriage, shown in fig 


Figure 2.2: The sample carriage. The aluminium bar is pushed against the glass rods by the phosphor-bronze clip, so the carriage is held up by static friction. The screw adjustment, which is screwed into the main body of the carriage, is used to optimise the friction for the stick-slip mechanism. Scale is approximately five times actual size.

was made of aluminium. The first version had a strong magnet mounted within it, which attracted a steel rod on the other side of the glass tubes, providing the correct amount of force required to hold it against the glass, while the piezo tubes were moved slowly, but allow it to slip, while the glass tubes were sharply accelerated by the piezos.

The final version of the carriage used a phosphor-bronze clip which pushed the aluminium bar against the glass tubes. This had the advantages that the force pushing the carriage against the tubes could be optimised using a screw adjustment, and that the sample was not automatically exposed to a large magnetic field by the presence of the carriage magnets.

The stick-slip mechanism relied on the frictional force between the carriage and tubes being of the correct magnitude. The static friction had to be sufficiently large for the carriage to be held firmly in position, to atomic precision, when the stick-slip mechanism was not enabled, but sufficiently small to be overcome, when the stick-slip mechanism was used to move the carriage up or down the glass tubes, by application of a sawtooth waveform to the approach piezos. The frictional force holding the carriage in place was directly related to the normal force applied by the phosphor bronze clip. The best results were achieved when the carriage was held quite firmly, requiring a moderate force to manually push the carriage up or down the tubes. The stick-slip mechanism was found to work very reliably and smoothly in either direction, often at voltages as low as 30-50 V (the lowest which could be applied by the sawtooth generator).

The sample was mounted to the carriage using silver paint, which was also used to attach the sample wire. When carrying out four-terminal measurements (required for point-contact spectroscopy), two wires were attached to the sample, and a second wire to the tip.

2.2.2  The piezo-electric tubes

Piezo-electric tubes were used for all the moving parts required in the STM head. This had the advantage that there were no mechanical connections to the top of the cryostat.

The piezo tubes used were of two types, which were made of different forms of lead zirconate titanate. The scanning piezo in the first model of the low-temperature STM, and the approach mechanism piezos in both models, were cylinders of type I, while the later model had a scanning piezo of type II. Table  shows a comparison of the two tube types.

Piezo material Length Outer diameter Wall thickness
Type I PZT-5A 12.7 mm 6.35 mm 0.51 mm
Type II PZT-5H 17.78 mm 6.35 mm 0.51 mm

Table 2.1: Comparison of the two piezo tubes used for the scanning tube.

The inner and outer curved surfaces were coated with a thin metal layer to form the two electrodes. The central scanning piezo had its outer electrode segmented into four parts, causing the piezo tube to bend under application of unequal voltages to the four outer electrodes. This allowed the tip to move in all three directions.

The greater tube length, and greater sensitivity of PZT-5H, gave the final version of the STM head a considerably greater sensitivity and maximum displacement. The theoretical increase in sensitivity (displacement per unit applied voltage) was a factor of 3.1, but the experimentally determined factor was nearly 5. This may have been due to partial depoling of the earlier tube, before its sensitivity was experimentally determined.

The sensitivity of the piezo tube is determined by the value of d31, the component of the tensor, relating applied voltage to induced strain, which corresponds to a radial voltage, and an axial strain. For a cylindrical tubular geometry, with a voltage V applied between the inner and outer electrodes, where L is the tube length, and t the wall thickness, the extension DL is given by DL = |d31|LV/t. The value of d31 for PZT-5A is -171×10-12 m/V. For PZT-5H, the value of d31 is -274×10-12 m/V. This gives an extension of 43 Å/V for PZT-5A, and 96 Å/V for PZT-5H. The calculation of the lateral motion for a segmented tube with different voltages applied to its outer electrodes is given in section .

2.2.3  Calculation of piezo tube motion

In this section, the displacement in x-,y- and z-directions for a piezo tube with an arbitrary set of voltages applied to its five electrodes is calculated.

The voltages applied to the X, X, Y, Y and Z electrodes are, respectively, VX, VX, VY, VY, and VZ. The electrode configuration is shown in fig .


Figure 2.3: The scanning tube electrode configuration.

The piezo tube is attached at both ends to a rigid circular disc, and the voltages applied do not vary along its length (except at the tip end, where the electrodes are milled off within 0.5 mm of the end). The tube can therefore be modelled as a section of a torus, i.e.  it has circular cross-section, and is in the shape of a very slightly curved cylinder, as shown in fig . Because the ends are rigidly attached to discs, they are constrained to be both planar and circular, which makes this approximation more realistic than the finite element analysis calculations of Carr (1988), which allow the tip end to assume any shape.


Figure 2.4: The curved piezo tube. The piezo tube may be considered to curve along an arc of a circle. The coefficients x and y represent the increase in height of the centres of the X- and Y-electrodes owing to the tilt of the top face of the piezo, which is assumed to be an undistorted circle. In this diagram, y = 0, and x is negative, so if -x is the distance by which the centre of the X-electrode is lowered, the part of the tube's top surface at an angle f from here is lowered by a distance -x cosf.

The extension of the tube at an angle f from the centre of the x electrode is given by:

DL(f) = DL + xcosf+ ysinf
(1)
where x, y describe the tilt of the top of the piezo.

At a given angle f, the equilibrium value of DL(f) (which would be attained if the tube were infinitely elastic) is given by:

DLeq(f) = |d31|L
t
(Vz - V(f)).
(2)
V(f), the voltage applied to the outside of the tube at an angle f, is:
V(f) =











Vx
| f| < p
4
Vy
p
4
< f < 3p
4
Vy
- 3p
4
< f < - p
4
Vx
| f| > 3p
4
.
(3)
The total strain energy of the tube is then given by:
E = k
p

-p 
(DL(f) - DLeq(f))2 df.
(4)
In this equation, k is some undetermined spring constant. Writing K = |d31|L/t, and substituting equations 1 and 2 into the above equation, gives the energy as:
E = k
p

-p 
((DL + xcosf+ ysinf) - K(Vz - V(f)))2 df.
(5)
Expanding V(f), this gives:
E
=
k
[(p)/ 4]

-[(p)/ 4] 
((DL + xcosf+ ysinf)- K(Vz - Vx))2 df+
k
[(3p)/ 4]

[(p)/ 4] 
((DL + xcosf+ ysinf)- K(Vz - Vy))2 df+
k
-[(p)/ 4]

-[(3p)/ 4] 
((DL + xcosf+ ysinf)- K(Vz - Vy))2 df+
k
-[(3p)/ 4]

-p 
((DL + xcosf+ ysinf)- K(Vz - Vx))2 df+
k
p

[(3p)/ 4] 
((DL + xcosf+ ysinf)- K(Vz - Vx))2 df.
(6)
This expression integrates to:
E
=
2p(DL - K(Vz - 1
4
(Vx + Vx + Vy + Vy)))2 +
22Kx(Vx - Vx) + 22Ky(Vy - Vy) +
p(x2 + y2).
(7)
In order to calculate the equilibrium values for DL, x and y, this expression for the total energy is differentiated with respect to each, and the derivatives set equal to zero, giving a minimum energy:
dE
dDL
=
2p(DL - K(Vz - 1
4
(Vx + Vx + Vy + Vy)).
(8)
Setting this equal to zero gives:
DL
=
K(Vz - 1
4
(Vx + Vx + Vy + Vy))
=
|d31|L
t
(Vz - Vouter ).
(9)
For x and y,
dE
dx
=
22K(Vx - Vx) + 2px
(10)
dE
dy
=
22K(Vy - Vy) + 2py.
(11)
Setting these equal to zero gives:
x
=
- 2
p
|d31|L
t
(Vx - Vx)
(12)
y
=
- 2
p
|d31|L
t
(Vy - Vy).
(13)
DL is simply the z-displacement, but the geometry must be considered in order to calculate the x- and y-displacements from the values of x and y. Figure  shows the x-z plane cross-section of the tube. Here, R is the radius of curvature of the tube, and q is the angle of the top of the tube in the x-z plane.


Figure 2.5: The curved piezo tube (cross-section). The relation of the x- and y-displacements of the tip to the tilt of the top surface of the piezo tube is shown, for a tube curved along the arc of a circle. The x-z plane cross-section is displayed.

The displacement of the tube in the x-direction is:

dx = R(1-cosq).
(14)
Since L Rq and (1-cosq) 1/2q2, this gives:
dx 1
2
Lq.
(15)
The vertical displacement, -x, of the right hand edge of the tube end, with respect to the centre of the tube end (see fig 2.5), is:
-x = D
2
q
(16)
where D is the tube diameter. Hence dx [Lx/ D], and similarly, dy [Ly/ D]. This gives:
dx
=
2
p
|d31|L2
Dt
(Vx - Vx)
(17)
dy
=
2
p
|d31|L2
Dt
(Vy - Vy)
(18)
dz
=
|d31|L
t
(Vz - Vouter ).
(19)

There is an additional displacement in the x- and y-directions, because the tip projects from the end of the tube at the tilt angle of the end of the tube. This term is equal to Tsinq Tq, where T is the length of the tip. The z-displacement is unaffected by the tip length. The tilt angles in the x-z and y-z planes respectively are qx = [(2dx)/ L] and qy = [(2dy)/ L], so the extra displacements are:

dxtip
=
2T
L
dx = 2
p


2LT
Dt


|d31|(Vx - Vx)
(20)
dytip
=
2T
L
dy = 2
p


2LT
Dt


|d31|(Vy - Vy).
(21)
The sensitivities of the piezos (displacement per volt) are therefore:
Sz
=
|d31|L
t
(22)
Sx,y
=
2
p
|d31|L(L+2T)
Dt
.
(23)
The value of the constant [(2)/( p)] is approximately 0.45. This compares with the value 0.5, which arises from simpler considerations.

It is possible to take into account the finite thickness of the piezo tubes, by integrating over the width of the tube (still assumed to curve in a circular arc). If the tube thickness is t, the x,y-sensitivity is modified to:

Sx,y = 2
p


3-6r+4r2
3(1-3r+4r2-2r3)


|d31|L(L+2T)
Dt
.
(24)
where r = t/D << 1. Because of the inherent inaccuracies, such as measurement of tip length and tube thickness, and the exact shape of the tube, this may be approximated as:
Sx,y = 2
p


1+ t
D


|d31|L(L+2T)
Dt
.
(25)
This result is equivalent to replacing D in equation 23 by the average of the inner and outer diameters (D-t), an intuitively reasonable result. The factor [(2)/( p)] (1+t/D) is approximately equal to 0.486 for the 12.7 mm PZT-5A tubes, and 0.490 for the 17.78 mm PZT-5H tubes. The calculated theoretical sensitivities in Å/V for the tubes are shown in table . The tip length correction is the additional sensitivity in the X and Y directions per mm of tip length.

Tube type Z-sensitivity X,Y-sensitivity X,Y tip length correction
PZT-5A 12.7 mm 42.6 Å/V 41.4 Å/V 6.5 Å/V/mm
PZT=5H 17.78 mm 87.2 Å/V 119.6 Å/V 13.5 Å/V/mm

Table 2.2: Theoretical sensitivities for the two types of scanning piezo. The tip length correction is the extra sensitivity per mm of tip length.

These figures may be used to compare the theoretical displacements with the experimentally determined values, for the two types of tube, as shown in table . The calculations assume a tip length of 3 mm. Agreement between theoretical and measured values agree to within 25% in all cases. In particular, the values for the x,y-sensitivity of the PZT-5H tube (which determine the scale in the images obtained after construction of the final STM head design) agree to within 7%.

PZT-5A, 12.7 mm PZT-5H, 17.78 mm
theoretical measured theoretical measured
z-displacement 43 Å/V 36 Å/V 87 Å/V 96 Å/V
x,y-displacement 61 Å/V 76 Å/V 160 Å/V 170 Å/V

Table 2.3: Comparison between theoretical and experimental piezo sensitivities. Theoretical and measured sensitivities are in reasonable agreement, the largest difference being about 25%. The tip length is assumed to be 3 mm for the theoretical calculations.

2.2.4  Stick-slip approach mechanism

The STM head used a stick-slip approach mechanism to bring the tip and sample together from a macroscopic distance to a distance of the order of 1 nm, required for electron tunnelling. Various arrangements were tried until the most reliable performance was achieved.

The two piezo tubes, one on either side of the central, scanning piezo tube, were supplied with a sawtooth signal, which caused them to move slowly in one direction, but extremely fast in the opposite direction. The polarity, amplitude, and frequency of the sawtooth could be adjusted to achieve the desired direction and speed of movement of the sample carriage. The carriage was held by friction on two glass tubes attached to the ends of the piezo tubes. Slipping of the carriage against the glass tubes occurred only in the direction of fast piezo movement, so a net motion in the opposite direction resulted.

This mechanism was used both for the manual approach of tip and sample from a few mm apart, to a distance as close as could be achieved visually using a microscope, and also for the slower approach to tunnelling distance, which cut off automatically when a current flowed between the tip and sample.

In the initial, coarse approach, the sawtooth generator operated at several hundred Hz. The final approach was made once the STM insert was in the cryostat, in a Helium atmosphere. For the fine approach, the sawtooth generator operated at a few Hz.

Because differential thermal contraction caused the tip and carriage to move apart as they cooled, the carriage had to be moved back towards the tip once the required low temperature was attained.

The sawtooth amplitude needed to be at least 30-50 V for the mechanism to operate at room temperature, but at Helium temperature, or lower, the sensitivities of the piezo tubes were reduced by a factor of approximately five, and sawtooth amplitudes at least five times larger had to be used. The maximum sawtooth voltage supplied was 290 V.

2.2.5  The scanning piezo tube

Once the tip and sample were sufficiently close together for tunnelling to occur, a feedback system attempted to keep the tunnel current equal to a preset value, by altering the length of the piezo tube with an appropriate applied voltage.

Figure 2.3 shows the geometry of the scanning tube's five electrodes. Since there are five inputs (applied voltages) and only three outputs (X-,Y- and Z-co-ordinates of the tip), there are two degrees of freedom in the choice of applied voltages, given the desired co-ordinates.

As shown earlier, the co-ordinates of the tip are given by:

X
=
Sxy(VX - VX)
(26)
Y
=
Sxy(VY - VY)
(27)
Z
=
Sz(VZ - Vouter ).
(28)
The values of [(VX+VX)-(VY+VY)] and [VZ + Vouter] do not affect the position of the tip.

Writing the electrode voltage differences as:

DVXX
=
VX - VX
DVYY
=
VY - VY
DVZo
=
VZ - Vouter .
(29)
The voltages supplied to the five piezo electrodes must satisfy, for desired displacements X, Y and Z,
DVXX
=
Sxy-1 X
DVYY
=
Sxy-1 Y
DVZo
=
Sz-1 Z.
(30)

Initially, the amplifiers used to drive the piezos were only capable of single-ended output, with three outputs whose voltages corresponded to the desired X,Y and Z-displacements fed directly to the X, Y and Z electrodes, with the X and Y electrodes held at earth potential, i.e.

VX
=
DVXX
VX
=
0
VY
=
DVYY
VY
=
0
VZ
=
DVZo.
(31)
This was undesirable, because it coupled the lateral motion to the longitudinal motion of the tip: a change to the X or Y voltage also moved the tip in the Z-direction.

The amplifiers were later replaced with double-ended output amplifiers, with four outputs, each corresponding to one of the outer electrodes. The inner electrode was held at earth, and the voltages applied symmetrically to the outer electrodes:

VX
=
1
2
DVXX - DVZo
VX
=
- 1
2
DVXX - DVZo
VY
=
1
2
DVYY - DVZo
VY
=
- 1
2
DVYY - DVZo
VZ
=
0.
(32)
The single- and double-ended amplifier electrode configurations are shown in fig .


Figure 2.6: Electrode configurations for single- and double-ended amplifiers.

The double-ended amplifier arrangement had two advantages over the previous arrangement. It removed the coupling between lateral and longitudinal deflections of the tip (except for small, second-order effects), and it reduced by nearly an order of magnitude the capacitance between the tip and the Z-signal, which had previously been large enough to cause oscillations. The capacitance was measured by applying an a.c. voltage signal to the tube electrodes, and monitoring the a.c. current.

With the inner electrode of the piezo tube at earth, the signal wire from the tip could be passed down the centre of the piezo tube to provide good electromagnetic shielding, both from external signals, and from the signals applied to the four outer electrodes.

The STM head was calibrated using a freshly cleaved graphite surface, so the X- and Y-displacements could be measured in terms of the interatomic spacing in the graphite array. By tilting the surface, and measuring the angle of tilt, and the ratio of Z-displacement to X- and Y-displacement for several angles of tilt, it was possible to calibrate for motion in the Z-direction. The X- and Y-direction displacements depend upon the length of the tip, so this was unchanged between these sets of measurements. The Z-direction calibration is unaffected by tip length.

2.2.6  The cryostats and inserts

The STM head and connections were designed to allow easy transfer of the STM head from one cryostat to another. The piezo-driven stick-slip mechanism had only electrical connections to the top of the cryostat, allowing for easy portability. The small size of the STM head also made it easily transferable from one cryostat to the other, as well as improving the rigidity and vibration characteristics.

The STM was operated in two different cryostats. The first was a continuous-flow cryostat, with PID temperature control, which allowed it to stabilise at any given temperature between 4 K and room temperature. The base temperature was measured, using a resistance thermometer, as 3.7 K. This was achieved using the maximum helium flow rate, but a temperature of 4 K was routinely achieved3. The time taken for the cryostat to cool from room temperature to 4 K was approximately 90 minutes.

A superinsulated cryostat was used where a lower base temperature was needed, and was only adapted to take the STM at a late stage in the course of this work. The insert was modified to allow this cryostat to be used for four-terminal measurements on point-contact junctions, as well as for point-tunnelling work.

  The continuous flow cryostat

The continuous flow cryostat was used for the initial tests of the low-temperature STM, and for most of the subsequent work. It was suspended by elastic cords from a rigid frame, in order to minimise vibrations entering the system through the support. The time constant of this suspension was approximately 2 seconds. Wires from the electronics to the cryostat were also able to be suspended by elastic, to reduce vibrations entering the cryostat along the signal wires.

Vibrations could also enter the continuous flow cryostat through the connection to the helium dewar. These could be reduced by standing the dewar on a rubber mat, but it is not believed that vibrations entering through the dewar and transfer tube were the largest source of mechanical vibrations entering the system. (The transfer tube was made of flexible stainless steel, which would absorb some of the vibrations from the dewar.)

Vibrations from the pumping system, and from the circulating liquid helium, could not easily be reduced. However, these did not seriously affect the operation of the STM. The other potential source of vibrations was from sound waves striking the smooth metal surface of the cryostat. No signal was detected when the cryostat was exposed to typical laboratory sounds (such as talking and footsteps; clapping, however, did cause a distinct ringing to occur in the tunnel current signal).

  The superinsulated cryostat

The superinsulated cryostat had a base temperature of around 1.3 K, and could remain at this temperature for several hours. Another possible advantage of using the superinsulated cryostat was that vibrations from circulating liquid helium would not be present. Also, once the temperature fell below 2.17 K, the lambda point of helium, the liquid helium in the cryostat stopped boiling, and merely evaporated from the surface, thus reducing the vibration. Once cooling had started, the STM tip was not brought into contact with the sample until the helium temperature had fallen below the lambda point. The cooling process took approximately 3-4 hours.

The superinsulated cryostat was mounted on a semi-inflated car inner tube, placed on a rigid framework. The time constant of oscillations of the cryostat on the inner tube was about 1 second. The rigid frame had four 25 kg blocks of lead resting on its base, and was placed on a rubber mat, to further reduce vibrations from the floor.

The connection to the pumping line was through a long, flexible stainless steel tube, and wires to the electronics could be suspended as with the continuous flow cryostat. The pumping line was probably the largest source of vibration, but it was possible to close the pumping line and switch off the pump, leaving the cryostat to warm up slowly as measurements were taken.

2.3  System performance

Initially, the system was designed only for tunnelling operation, and the range of the current amplifier was approximately 5 pA to 50 nA. The most important system requirements were very high current sensitivity (demanding very low current noise levels and electromagnetic interference) and resistance to vibrations.

At a later stage, a facility for point-contact spectroscopy measurements was added, which used much higher currents. This necessitated modifying the superinsulated cryostat insert to perform four-terminal measurements. The new amplifier allowed measurements over the range 5 nA to 0.5 A. Thus I-V  measurements could be performed over a range of eleven orders of magnitude of current. In the case of the work on superconducting films, and measurement of the energy gap, where the applied voltage was typically in the range -20 to +20 mV, this corresponds to a range of resistances from 4 GW down to 0.04 W. In practice, measurements of I-V characteristics were in the range 50 MW to 0.1 W.

In tunnelling mode, the STM was capable of obtaining topographic images of conducting surfaces, at temperatures between 1.3 K and 300 K. Atomic resolution images of freshly-cleaved highly-oriented pyrolitic graphite, a form of graphite with an atomically smooth surface, were taken both at room temperature and at 4.2 K.

At room temperature the maximum scan size of the final model of STM head was approximately 3 mm, but the scan size was decreased to approximately 0.6 mm when the head was cooled to 4.2 K. The earlier model of the STM head had a smaller scan size, of around 1.5 mm at room temperature, because it used a less sensitive scanning piezo. The voltages applied to the earlier model were also larger, because the amplifiers used to drive the piezo tubes were changed at the same time as the STM head. The newer amplifiers provided a double-ended output, at a maximum voltage of 225 V4 whereas the output voltage of the older, single-ended amplifiers was approximately 600 V.

2.3.1  Current sensitivity

The STM operated with tunnel currents of 1 nA, 0.1 nA, or even 10 pA, and the current sensitivity and current noise level had to be at least an order of magnitude smaller for tunnelling spectroscopy measurements to be performed. The effect of current noise could be reduced by performing several (up to 256) I-V  measurements, then averaging the results. However, a noise level of a few pA was still required, if the tunnel current was 0.1 nA or less.

The noise level of the amplifier was around 2 pA, whereas the noise generated by the earlier model of the STM head was over 10 pA. However, with the final STM head design , with better electrical shielding of the tip and tip wire, the total noise level increased only slightly when the STM head was connected to the amplifier: the noise level remained at around 2 pA.

2.3.2  Vibration Isolation

In tunnelling mode, the tunnel current depends very sensitively on the tip/sample separation distance. Roughly, a change in separation of around 1 Å will alter the tunnel current by an order of magnitude. The stability required for tunnelling spectroscopy is even greater than that required for scanning tunnelling microscopy, because a movement of 10-12 m will change the tunnel current by approximately 2%. Motions slightly larger than this may be tolerated, since a gradual drift of several per cent will not affect the visibility of any features, and vibrations causing fluctuations in the tunnel current may be reduced by averaging several I-V  measurements.

In order to achieve this mechanical stability, the STM head itself had to be built very rigidly, and precautions had to be taken to eliminate as much vibration as possible from reaching the STM head. Several methods were tried, before the final arrangement was reached. The continuous flow cryostat was suspended from a wall-mounted frame by elastic cords. Connections to the helium return system were made using flexible stainless steel tubes. Vibrations entering through the dewar and transfer tube could be reduced by standing the dewar on a rubber mat. An important precaution was the use of the main pumping line to suck the helium through the cryostat, rather than a diaphragm pump in the same room.

The superinsulated cryostat was mounted on a large metal slab, resting on an inflated car inner tube. This inner tube rested on a very rigid frame, with four 25 kg lead weights on the bottom of the frame. The frame itself stood on a rubber mat. The main pumping line was connected to the cryostat via a long, flexible stainless steel tube.

For both cryostats, signal leads could be suspended from elastic cords to further improve vibration isolation.

The lowest vibrational frequency of the STM head was slightly over 4 kHz. Vibration at this frequency was initially a problem, and was prevented by improvements to the tip shielding, and by the use of double-ended output amplifiers, so that the inner electrode of the scanning piezo tube could be kept at earth potential. This reduced the capacitance between the piezo electrodes and the tip by nearly an order of magnitude, so interference between the Z-signal and the current amplifier was greatly reduced. The maximum feedback loop gain which could be used without oscillations occurring was increased in this way, also by an order of magnitude.

  Frequency response of the suspension

The continuous flow cryostat suspension consisted of a single spring. The resonant frequency of the mass-spring system formed by the cryostat and the suspension is given by:

f0 = 1
2p
  


g
DL
 
(33)
where f0 is in Hz, and DL is the extension of the spring due to the addition of the mass.

For the STM suspension, DL = 1 m and g = 10 ms-2, so the predicted natural frequency is f0 = 0.5 Hz. This is in agreement with the observed free oscillatory frequency.

For the superinsulated cryostat, the suspension system had higher resonant frequencies. The cryostat swung sideways with a resonant frequency of 2 Hz, and oscillated vertically with a resonant frequency of 5 Hz. The higher frequencies were due to the smaller displacement of the suspension system (the car inner tube), although this was partially counteracted by the higher mass of the cryostat.

For frequencies, f, intermediate between the frequency of the isolation system, fI, and that of the STM, fS, which are the frequencies most likely to cause problems, the transfer function is nearly constant,

K(f) =

f
fS


2

 


fI
f


2

 
=

fI
fS


2

 
.
(34)
Taking values of 5 kHz for fS, and 0.5 Hz, 5 Hz for the two isolation systems (continuous flow cryostat and superinsulated cryostat), the transfer constants are, respectively, 10-8 and 10-6, corresponding to respective reductions in power of 160 dB and 120 dB.

This analysis neglects the effect of interaction with acoustic vibrations, which could couple directly with the STM, bypassing the vibration isolation. These were minimised by performing the experiments at times when there was as little acoustic interference as possible.

2.4  The STM Control and Electronics

2.4.1  The current amplifier and feedback loop

The STM electronics used a very sensitive current amplifier to amplify the tunnelling current at the tip to a measurable voltage, which was fed into the main electronics box. The feedback electronics was designed to operate best when the input voltage, produced by the amplifier, was 0.1 V.

The amplifier was in two parts, so that the first stage amplifier, the head amplifier, could be placed as close as possible to the STM tip. It connected directly to the top of each of the cryostats.

The amplifier pair could be set to amplify by a factor of 107, 108, 109, or 1010 V/A. When the feedback stabilisation voltage was set to its optimal value of 0.1 V, these amplifier gain levels corresponded to current stabilisation values of 10 nA, 1 nA, 0.1 nA and 10 pA respectively. It was also possible to change the stabilisation voltage, but the feedback functioned optimally when the voltage was within a factor of about three of 0.1 V.

The feedback loop controlled the Z-output to the scanning piezo tube in order to maintain the tunnel current at the preset value. This was achieved by keeping the input voltage from the current amplifier at a constant value, usually within a factor of three of 0.1 V. The current amplifier was a logarithmic amplifier, so a wide range of tunnel currents could be used, and the current range altered by factors of ten, by changing a feedback resistor. This allowed the feedback to operate in its most efficient range for any current between 3 pA and 30 nA.

The current noise level of the amplifier pair was 2 pA rms. A higher current resolution could be obtained when carrying out I-V  measurements by repeating the measurements several times, and averaging the results. This method was used to give current sensitivities of approximately 1 pA.

2.4.2  Surface scanning

The usual method of taking an image of the surface was to scan the tip back and forth over the surface in a raster pattern, controlling the Z-displacement of the tip with the feedback loop, to maintain a constant tunnel current, while recording the Z-displacement of the tip at each point. The constant current image obtained in this way is a topographic representation of the sample's surface.

It was also possible to set the ADC to collect current data, rather than Z-displacement data, and to disable the feedback loop, so that a constant tip displacement is maintained. This method could only be used over very small areas and very smooth surfaces, to avoid crashing the tip into the surface.

The raster motion of the tip could be set to scan in different directions, to confirm the presence of real surface features. Data was collected as the tip moved along in the set direction, after which the tip moved back along the same path, without collecting data. It was also possible to set the electronics to scan the same area repeatedly, allowing small changes in the image to be seen, and allowing the scan to be terminated when the clearest image was visible.

2.4.3  Current-voltage characteristics

One of the most important features of the system was the ability to measure the current-voltage (I-V ) characteristic of the tunnel junction which the tip made with the surface.

Before an I-V  measurement, the feedback loop holds the tip above the surface, at the correct distance to give the preset tunnel current. Both the bias voltage applied to the sample, and the desired tunnel current could be selected, giving control over the resistance of the junction, and the current through it.

During the I-V  measurement, the feedback loop was disabled, and the voltage was moved over the desired range, while the current was measured. The voltage was stepped over a discrete number of voltage points (up to 512). The tip was held at each voltage point for a short time, before the current measurement was made. The amount of time for which the tip was held at a constant voltage is selectable.

Each I-V  characteristic could be measured several times, and averaged, to reduce the effect of random noise. The number of scans could be set to any power of two up to 256. The feedback loop was re-enabled for a short time between scans, to re-establish the correct tip/sample displacement, giving the set current.

Current-voltage characteristics taken by the electronics were numerically differentiated by the computer to obtain the differential conductance as a function of applied voltage across the junction. The use of digital-to-analogue and analogue-to-digital converters (DACs and ADCs) to pass voltage and current data between the computer and electronics had certain consequences upon the I-V  characteristics obtained in this way, and upon the numerical differentiation of these graphs. These are discussed in section .

2.4.4  Analogue-to-digital conversion

Because a computer was used to control much of the operation of the STM electronics, and to store the collected data, digital-to-analogue and analogue-to digital converters (DACs and ADCs) were used. Many of the operational constants, such as the gain of the feedback loop, were stored as digital values.

The DACs and ADCs used in the STM electronics were 12-bit converters for the voltage, and 16-bit converters for the current, however a software error (corrected in 1995) reduced the effective current sensitivity to roughly the equivalent of 12-bit (Czorniy, 1996). Constants such as the loop gain, the scan size, and the x- and y-displacements of the scan area were passed from the computer to the electronics as 12-bit binary.

The bias voltage, applied to the sample, was output as a 12-bit number, implying that the minimum voltage increment was 1/4096 of the maximum voltage which could be applied. There were several voltage ranges which could be used: for the 2 V range, the minimum voltage increment was 0.5 mV. This was insufficient for the measurement of features whose size was of the order of 1 mV, so a voltage attenuator was used to divide the voltage down by a factor of either 10 or 100. Voltage step sizes of 50 mV or 5 mV were sufficiently small to resolve these features.

For measurement of current without digitization effects, it was necessary to set the ADC to its most sensitive range, where the maximum measurable current was approximately three times the current corresponding to a current amplifier output voltage of 0.1 V. For the 1 nA range, the minimum detectable current change was about 1.2 pA. Considering that the noise level of the amplifier was 2 pA, this level of quantization was acceptable on the 1 nA and smaller current ranges. For the 10 nA range, the minimum detectable current change was 12 pA.

The effects of current and voltage quantization are similar. For either case, the I-V  characteristic would consist of a series of current plateaus, separated by steps, where the current changes rapidly. The difference between the two situations is that, where voltage quantization is the problem, the actual voltage applied across the junction, and therefore the current tunnelling through the junction, is quantized, but where current quantization is the problem, it is only the measurement of the current which is quantized. In the latter case, the `plateaus' would be completely flat, while in the former case, there will be small deviations in the current. Digitization effects were essentially eliminated, by appropriate choice of ADC and DAC gain.

2.4.5  Numerical differentiation

Numerical differentiation was used to calculate the differential conductance from the measured current-voltage characteristics. Conductance curves obtained in this way could show up small changes in conductance which were not visible in the raw I-V  data.

The algorithm used to calculate the derivative of the current data was a least-squares method. The number of current points, n, used in the calculation of each conductance point was user-selectable; typically, n = 5.

For each set of n adjacent points in the dataset, the gradient of the best-fit line through these points was calculated, and used as the conductance datum. With a total N points in the current dataset, this resulted in a set of conductance data comprising N-n+1 points.

Using too many points caused excess smoothing of features in the processed data, decreasing the effective voltage resolution, while using too few points to calculate the derivative produced a noisy, jagged conductance curve. The number of points, n, used in calculation of the derivative, varied between 3 and 20, with higher values used for noisy data. Most of the data was obtained using n = 5, or 5 < n 10.

2.4.6  Current imaging tunnelling spectroscopy (CITS)

In current imaging tunnelling spectroscopy (CITS), current data were collected at each point of the image. During the scan, the tip stopped at each point where current data was to be collected, and was held stationary for a short time to allow the feedback loop to stabilise the current at the preset value. The sample bias voltage then ran through a set of up to 16 preset values, and the tunnel current was measured for each of these voltages. This allowed a number of current images to be obtained, in addition to the topographic image. In each of these images, one corresponding to each of the preset voltages, the current flowing through the tunnel junction at each point was recorded.

When the applied bias voltage is equal to the scanning bias voltage the tunnel current should be equal to the preset stabilisation current. If the I-V  characteristic is the same at each point in the image, then all the points on any given current image should show the same current. Any structure in the images reveals variation of the I-V  characteristic of the point tunnel junction with position of the tip over the surface, e.g.  a surface containing regions exhibiting a superconducting energy gap will exhibit a reduced current in those regions of the image, at voltages inside the gap region (and increased current at voltages corresponding to the conductance peaks). CITS enables location of areas for further measurement of I-V  characteristics.

2.4.7  Modifications for point-contact spectroscopy

In point-contact spectroscopy, the tip comes into contact with the surface, forming a junction with a very much lower resistance. Instead of the typical STM tunnelling resistance of 107 W or greater, the junction resistance may be in the range 0.04-100 W, a factor of 105 to 2.5×108 times smaller. The current passing through the junction is correspondingly higher: instead of the usual 10 pA-10 nA, the current may be as much as several hundred mA. The design of the electronics for point contact spectroscopy had to take these factors into account.

Because the resistance of the low-temperature stainless steel wires which carried the tip and sample signals was large compared to the junction resistance, four-terminal measurements of the junction I-V  characteristics were made. (The resistances of the tip and sample leads which ran down the insert were 24 W at room temperature, falling to 20 W at liquid helium temperatures. Four-contact measurements enabled junction resistances as low as 0.1 W to be attained.)

The electronics used to measure the I-V  characteristics had to ensure that the voltage applied at the bottom of the cryostat was the correct bias voltage, using a feedback loop, while keeping the tip end of the junction at earth potential, using a second feedback loop, and to measure the current flowing through the junction. The circuitry had to be able to withstand currents of up to several hundred mA.

Since it was desirable to be able to vary the junction resistance continuously from the tunnelling range through to the point-contact range, the electronics was designed to be able to produce and measure currents in the range 30 nA to 480 mA - a range of over seven orders of magnitude. This meant that, using either the sensitive head amplifier, or the point-contact spectroscopy amplifier, currents in the range 10 pA to 480 mA could be measured (a range covering almost eleven orders of magnitude).

The point-contact spectroscopy circuit was designed to output a voltage to the main STM electronics box, in place of the output from the usual current amplifier. The current range could be set to one of eleven values between 100 nA and 300 mA. This selected current range determined the size of the current which gave an output of 0.1 V to the main STM electronics box. Changing the current setting on the computer allowed the STM feedback system to stabilise this voltage at a level other than 0.1 V, thereby allowing any current in the range 30 nA to 480 mA to be selected. The circuit is shown in figure .


Figure 2.7: Point-contact spectroscopy circuit. The circuit was capable of generating and measuring currents in the range 30 nA-480 mA.

The amplifiers used in the final version of the circuit were OPA111 (very high input impedance) amplifiers, and EI2008 buffer amplifiers (capable of a current output of up to 1 A). The buffer amplifiers were attached to the wall of the electronics box to provide adequate heat-sinking.

The circuit was tested using a network of resistors to simulate the resistances of the junction and the wires going down the cryostat. The measured voltages at either end of the sample were found to be correct to within 0.1% for sample resistances of 1 W and above, on current ranges 100 mA and below. With lower resistances or higher currents, the applied bias was within 1% of the desired value. This resistance network was also used to check the calibration of the current amplifier output voltage, which was found to be accurate to better than 2% on all ranges.

2.5  Experimental procedure

2.5.1  Tip preparation

Two methods were used for the preparation of sharp tips for STM work. In most of the experiments, the cut tip technique was used, in which a 0.5 mm gold wire was mounted in the tip holder, using a specially designed mounting tool, then cut using a very clean, sharp scalpel blade, at an angle of about 45 to the tip axis. The tip holder was then screwed into the end of the scanning piezo tube. This method could also be employed for some other commonly used metal wire tips, such as platinum-iridium.

The second method was the etched tip technique, in which a hard metal tip (e.g.  tantalum or tungsten) was electrochemically etched, by mounting the tip vertically, partially immersed in 2M KOH solution, and applying a voltage (7 V) between the tip (positive) and another strip of the same metal, also immersed in the etchant. After several minutes (typically 15 minutes) the immersed end of the tip dropped into the solution, and the remaining part was cleaned by use of sequential dips in distilled water, acetone and propanol. This method was only used in trial experiments. Production of tips by etching can produce extremely warped tips, when a substantial length of tip is submerged in the etchant, such that, in some cases, the sharp point is bent round more than 90 degrees, and tunnelling occurs from some other point. Nonetheless, such tips may still obtain high-quality images (Nicolaides et al. , 1988).

The choice of a tip for spectroscopic measurements is governed by the likelihood of the tip states affecting the measured characteristics. The use of electrochemically etched tips made from W or Ta wires is more likely to result in nonlinearities in the I-V  characteristic owing to the tip states than the use of mechanically cut tips made from Au or PtIr wires (Wilkins et al. , 1990). For this reason, Au tips were consistently used.

It is generally the case that the most reliable spectroscopic data are obtained by using relatively blunt tips, while sharp tips, capable of obtaining atomic resolution, are unreliable for the purpose of spectroscopy (Feenstra et al. , 1987; Klitsner et al. , 1990).

2.5.2  Procedure for scanning tunnelling microscopy

The sample wire was attached to the sample, and the sample mounted onto the carriage, using silver paint.

The sample carriage was moved towards the tip using the stick-slip coarse approach mechanism while observing the junction under a low power microscope. The approach was manually stopped when the sample was close to the tip.

The shielding was fixed around the STM head, and the insert placed into the cryostat. The sample space was pumped out, and back-filled with one atmosphere of helium gas. The carriage was moved in by steps towards the STM carriage, until a tunnel current flowed between the sample and the tip. The sample was then retracted by several steps, to prevent damage when cooling started.

The cryostat was cooled to its base temperature. Measurements at temperatures above the base temperature of the cryostat were performed after the STM had operated at the base temperature. The tip was then moved towards the sample using a higher stepping voltage (owing to the decreased sensitivity of the piezo tube at liquid helium temperatures). Once a stable tunnel current was attained at this temperature, data could be taken.

2.5.3  Procedure for point-contact spectroscopy

The preparations for point-contact spectroscopy were similar to those for scanning tunnelling microscopy. The superinsulated cryostat was used for point-contact spectroscopy, because the continuous flow cryostat did not have enough coaxial feedthroughs.

The tip and sample each required two connections, as four-terminal measurements were necessary to eliminate the wire resistances. These extra connections were attached with silver paint, before the coarse approach.

With the extra connections in place, both scanning tunnelling and point-contact spectroscopy could be performed. For the latter, the usual high-sensitivity current amplifier was replaced by the point-contact spectroscopy amplifier.

For the approach, it was possible to set the point-contact spectroscopy amplifier to two-terminal mode. This prevented damage to the amplifier if the tip and sample touched, forming a junction with an extremely low resistance. There was also an intermediate setting, which afforded protection to the amplifier, and gave valid results at lower junction resistances than the simple two-terminal mode, by connecting the pairs of terminals through 1 kW resistors, rather than shorting them together.

The temperature of the STM head was monitored when very high currents were flowing, because of heating effects, which became large with junction currents above about 100 mA.

Chapter 3
Theory of Superconductivity

This chapter describes the fundamental properties of superconductors, and the development of the BCS microscopic theory of superconductivity. This theory is applied to derive the elementary excitations of a superconductor, which determine (chapter ) the behaviour of NIS and NINS (proximity effect) tunnel junctions and NcS and NcNS point contact structures. Ginzburg-Landau theory is described, for application to tunnelling into structures with NS interfaces (NINS and NcNS).

3.1  General Features of Superconductivity

3.1.1  Fundamental properties of superconductors

The phenomenon of superconductivity was discovered in 1911, when Kammerlingh Onnes (1911) found that the resistance of a sample of mercury dropped by more than four orders of magnitude, becoming immeasurably small, when the temperature fell below a critical value (about 4.2 K for Hg).

The phenomenon of the disappearance of electrical resistance below a critical temperature Tc has since been found to occur in several elemental metals, alloys, and other compounds.

With the discovery of the Meissner effect (Meissner & Ochsenfeld, 1933) it was realised that magnetic flux is totally excluded from a superconductor, except for a thin ( ~ 0.1 mm) penetration layer at the surface, provided the applied field H does not exceed a characteristic critical value, Hc(T) (see section ).

These two properties of a superconductor may be summed up as follows: the resistivity of the superconductor r = 0, provided T < Tc, and the magnetic field inside the superconductor B = 0, provided also that H < Hc(T).

There is a third important property of a superconductor, which is that the wavefunction of the superconductor exhibits phase coherence over macroscopic distances owing to the formation of a superfluid condensate of electron pairs. This is a less easily measured property, but is the most fundamental property of a superconductor, as it leads to all the other properties.

3.1.2  Observed properties of superconductors

  Critical temperature

The transition to the superconducting state is a sharp one in bulk samples, with properties changing abruptly from normal to superconducting as the temperature falls below the critical value Tc. Observed values of Tc for superconductors range from a few mK for some elemental superconductors and heavy fermion systems, to over 100 K for some cuprate high-temperature superconductors. The highest value of Tc for an elemental superconductor is 9.26 K for niobium, which corresponds to a thermal energy kBTc of 0.8 meV, very much lower than the energy levels which are normally important in the theory of metals (EF ~ 5 eV; (h/2p) wD ~ 0.1 eV) (Ashcroft & Mermin, 1976).

  Electrical resistance

As mentioned above, the resistance of a superconductor drops abruptly to zero as the temperature falls below the transition temperature Tc. Above Tc the resistivity has the usual form for a metal, r(T) = r0 +B T5, with the two terms coming respectively from impurity scattering and from phonon scattering. In a superconductor below Tc, these mechanisms are no longer able to degrade a current. Currents can flow in a superconductor indefinitely with no discernible dissipation of energy.

Superconductivity is destroyed by a sufficiently large magnetic field (section ). If the current in a superconductor exceeds a critical value, then the field at the surface will exceed the critical field, and the material will become normal.

AC currents can also pass through a superconductor without dissipation, provided that the frequency is not too high. The response becomes dissipative as the frequency exceeds approximately D/(h/2p), where D is the energy gap.

At finite temperature, the magnitude of the energy gap D(T) (the temperature-dependent energy gap) is reduced below its zero temperature value D0. The effect is small unless T is close to Tc: for T < 0.5Tc, D(T)/D0 > 0.95. The behaviour of D(T)/D0 as a function of the reduced temperature t = T/Tc is shown in fig ; data are taken from Mühlschlegel (1959).


Figure 3.1: Temperature dependence of the energy gap.

  Thermoelectric properties

Superconductors are poor thermal conductors. This is contrary to the independent electron approximation, which predicts that good electrical conductors will also be good thermal conductors. Superconductors also exhibit no Peltier effect (thermal current caused by an electrical current).

3.1.3  Critical field: type I and type II superconductors

When a magnetic field is applied to a superconductor, surface currents flow in order to cancel this field out inside the superconductor. The magnetic fields associated with these currents take energy to create, and if the applied field is sufficiently large, it will be energetically favourable for the superconductor to revert to a normal state. There are two types of behaviour which can occur when a a sufficiently large field is applied, and superconductors can be divided into two types correspondingly:

  Type I superconductors

Below a critical field Hc(T), which decreases with increasing temperature, falling to zero at T = Tc, flux does not penetrate the superconductor. Above the critical field Hc, the entire specimen reverts to normal behaviour, with perfect flux penetration.

  Type II superconductors

A type II superconductor has two critical fields Hc1 and Hc2 associated with it. These tend to zero as T increases towards Tc.

When the applied field is below Hc1 the flux is excluded. Above Hc2 the flux penetrates totally, and the specimen reverts to normal. Between Hc1 and Hc2 there is partial flux penetration, and the sample consists of both normal and superconducting regions, a state known as the mixed state. In this state, the flux penetrates in the form of flux vortices. Where the vortices penetrate the specimen, the material is normal, but it is superconducting elsewhere.

The upper critical field for some `hard' type II superconductors can be as high as 10 T (and may be much higher for high-temperature superconductors), compared with a critical field of around 10 mT for type I superconductors.

  Temperature dependence of the critical field

The critical field may be determined by equating the energy 1/2m0 H2 (associated with holding the external field out) with the condensation energy of the superconductor, equal to the Helmholtz free energy difference per unit volume between the normal and superconducting states:

1
2
m0 H2 = fn(T) - fs(T).
(35)
The critical field attains its maximum value Hc(0) at T = 0. As the temperature is increased towards Tc, the value of Hc(T) falls towards zero at T = Tc. Writing the reduced temperature as t = T/Tc, Hc(T) is well approximated by a parabolic law,
Hc(T) = Hc(0)[1-t2]
(36)
which is illustrated in fig .


Figure 3.2: Temperature dependence of the critical field Hc(T).

3.2  A Microscopic Theory of Superconductivity

Prior to 1957, and the BCS theory of superconductivity, the reason for the abrupt change to a superconducting state was not understood, although some progress was made by taking the observed properties of superconductors, in particular those of zero electrical resistance and magnetic flux exclusion, as a starting point.

3.2.1  The London equation

According to the Gorter and Casimir two-fluid model (London, 1954), in a superconductor at a temperature T < Tc a fraction ns(T)/n of the total number of conduction electrons are capable of flowing as a supercurrent. The density of superconducting electrons ns(T) drops to zero as T approaches Tc, and approaches n as T falls well below Tc. Since the supercurrent flows with no resistance, it carries the entire current induced by any applied electric field, so the normal electron fluid can be ignored in the following discussion.

If an electric field E appears in the superconductor, the superconducting electrons are accelerated without dissipation, so

m vs
t
= -e E.
(37)
The current density carried by these electrons is j = ns e vs, so the current density j obeys

t
 j = ns e2
m
E.
(38)
It is convenient to define a quantity L by:
L = m
ns e2
(39)
so that
E =
t
(L J ).
(40)
Substituting this equation into Faraday's law of induction,
×E = - B
t
(41)
gives the relation between current density and magnetic field:

t


×j + nse2
m
B

= 0.
(42)
This relation, and the Maxwell relation,
×B = m0 j
(43)
determine the possible configurations of current density and magnetic field for a perfect conductor.

Equation 42 implies that any time-independent B may exist within the perfect conductor, since equation 43 gives the required (static) current density j to support any given static field B, and equation 42 is satisfied by any pair of static functions. However, superconductors permit no magnetic fields in their interior, so the required time-independent value in equation 42 must be zero, and hence the London equation,

×j = - ns e2
m
B.
(44)
Combining this equation with equation 43, gives:

2 B
=
m0 ns e2
m
 B
(45)
2 j
=
m0 ns e2
m
 j.
(46)
These equations imply that currents and magnetic fields decay exponentially into the superconductor, and only exist within a layer of thickness lL(0) of the surface, where lL(0), the London penetration depth, is given by:
lL(0) =

m
m0 ns e2


1/2

 
(47)
and is typically of magnitude 100-1000 Å, for T well below Tc.

3.2.2  Nonlocal theory and the Pippard coherence length

Pippard (1953) proposed a nonlocal generalization of the London equation, by letting the current at a point depend on the vector potential A(r) throughout a region of space whose size was determined by uncertainty arguments: electrons within kTc of the Fermi energy are involved in superconductivity, and the momentum range Dp kTC/vF, so Dx ~ (h/2p)/Dp (h/2p) vF/kTc, hence the coherence length is defined as:

x0 = a (h/2p) vF
kTc
(48)
where a is some dimensionless constant.

The effect of this nonlocal behaviour is that the response of the currents to vector potentials varying on a scale smaller than x0 is reduced, and the current is now given by:

J(r) = - 3e0
x0 L

R[R ·A(r)]
R4
e-R/x dr
(49)
where R = r-r and x is related to the coherence length and mean free path l by:
1
x
= 1
x0
+ 1
l
.
(50)
Pippard found experimentally that a = 0.15, and BCS theory (see section ) later gave a theoretical value of a = 0.18.

The relative values of the two characteristic lengths l and x0 are important in determining the behaviour of the superconductor, as discussed in section . The analysis is often simplified by considering the two cases of the clean limit, where l >> x0, and scattering is relatively unimportant, and the dirty limit, where l << x0, and scattering is more important than coherence length effects.

3.3  BCS Theory

In this section, the microscopic theory of superconductivity of Bardeen, Cooper and Schrieffer (1957) is discussed in order to apply the results to tunnelling and point contact work in which one of the electrodes is superconducting.

The main conclusions of BCS theory are that the Fermi sea is no longer the ground state, and that a lower energy state is realised by electrons forming non-localised pairs with opposite momenta and spin.

The excitations of this ground state have a minimum energy, D, i.e.  the superconductor acquires an energy gap below its transition temperature.

BCS theory also allows the derivation of an effective superconducting density of states, and an E-k relationship.

3.3.1  Development and justification of the BCS theory

BCS Theory is based on the idea that an attractive interaction can take place between two electrons near the Fermi surface in a metal. Cooper (1956) showed that this interaction, however weak, causes the Fermi sea to be unstable against the formation of at least one electron pair. BCS (1957) showed that the superconducting ground state consists purely of paired electrons, and leads to a superconducting state, with greatly altered properties from the normal (`Fermi sea') ground state. These properties have been described in section 3.1, and include zero resistance to electrical current, exclusion of magnetic flux, and changes to the thermal conductivity and heat capacity of the metal.

In most superconducting metals, the origin of the attractive interaction is believed to be electron-phonon coupling. The idea is that an electron moving through a lattice of positive ions may polarise the ions to such an extent that the lattice is overscreened, resulting in a region of net positive charge, which can then attract a second electron. This model also suggests a reason why the electron separation in a pair is so large ( ~ 1000 Å). In the time for the overscreening to develop, t @ wD-1 @ 10-13 s, the distance travelled by a Fermi surface electron will be of order l @ vF t @ 1000 Å, taking vF as 106 m/s. This distance is similar to a typical BCS coherence length for a metal.

The above process may be considered as the emission of a virtual phonon by one electron, and its subsequent absorption by a second electron. This is a scattering process in which the electron is scattered from a full state to an empty state. The maximum change in the wavevector k of the electron is equal to the wavevector q of the exchanged phonon. This is much smaller than the Fermi wavevector kF, so these scattering processes must all occur in a shell of width (h/2p) wD around the Fermi surface.

The actual phonon-mediated interaction is quite complex, and would be very difficult to deal with, so in BCS theory a very simple approximation to the actual potential is used: the attraction between electrons is assumed to be constant up to a cut-off frequency equal to the Debye frequency wD, and zero for higher frequencies. This implies the condition that, for an attractive interaction to occur,

|Ek - Ek| < (h/2p) wD
(51)
where the electron is scattered from a state of momentum k into one of momentum k. Since the scattering can only occur from a full to an empty state, and thus within energy (h/2p) wD of the Fermi surface, the interaction may also be restricted to states where |ek| and |ek| (the energies of the states as measured with respect to the Fermi surface) are both individually less than (h/2p) wD.

Since a very crude approximation to the actual attractive interaction between electrons is used, it is clear that detailed predictions may be inaccurate, especially those which depend on the parameters of the attractive potential. However, it turns out that the theory produces some results which appear to be independent of the actual interaction; moreover, it can be shown that any attractive interaction, however weak, will cause the Fermi sea to be unstable with respect to the formation of electron pairs, and thus that the Fermi sea can no longer be the ground state.

3.3.2  Formation of Cooper pairs

The scattering matrix element for the potential described in section 3.3.1 is defined by:

Vkk =

-V
|ek| and |ek| < (h/2p) wD
0
otherwise.
(52)
Cooper (1956) considered adding two electrons to a Fermi sea at T = 0, with the electrons interacting with each other via this potential, and interacting with the Fermi sea only through the fact that they must obey the Pauli exclusion principle. The electrons were considered to have equal and opposite momenta, k and -k, since this state, with zero total momentum, is the most energetically favourable situation, and therefore be the ground state of the two added electrons.

The two-particle wavefunction of the added electrons is a function of the distance r1-r2 between the electrons, and is given by:

Y(r1,r2) = Y(r1-r2) =

k 
gk eik.r1 e-ik.r2 =

k 
gk eik.(r1-r2).
(53)
Since the overall state must be antisymmetric, and the spatial part is symmetric owing to the angular symmetry, the electrons must be in an antisymmetric, singlet spin state, [1/( 2)]( - ), and the electron pair is represented by (k,-k ).

Taking this singlet state and substituting it into the Schrödinger equation gives an equation for the coefficients gk and energy eigenvalue E,

(E-2ek)gk =

k > kF 
Vkk gk.
(54)
With Cooper's approximation that the interaction V is constant below a cut-off energy (h/2p) wD from EF, and using the weak-coupling approximation, that N(0)V << 1, this leads (Tinkham, 1975, p 18) to an energy eigenvalue which is below the Fermi energy by an amount
e = -2(h/2p) wD e-2/N(0)V
(55)
where N(0) is the density of single electron states of a given spin at the Fermi level. This negative energy (with respect to the Fermi surface) state is formed from electrons with k > kF, and hence with kinetic energy greater than EF. However, the reduction in potential energy caused by the potential V outweighs the increase in kinetic energy.

Since the total energy change is negative for any V > 0, the formation of a bound state between two electrons is energetically favourable, and that the Fermi sea is unstable to the formation of electron pairs, given the existence of an arbitrarily weak electron-electron attraction. This formation will continue until the system state is no longer adequately described by a Fermi sea, and hence the Fermi sea cannot be the true ground state of the system.

3.3.3  The BCS ground state

From the arguments in the previous section, it is clear that the true ground state will be the state which is reached by the repeated formation of electron pairs from an initial `Fermi sea' state, until formation of pairs is no longer favourable. It will be seen that the ground state consists only of paired electrons, all of which are described by the same wavefunction, so the superconductor exhibits phase coherence over macroscopic distances.

In order to describe the BCS wavefunction, it is convenient to use the terminology of second quantization, in which creation and annihilation operators ck * and ck are used to create or remove an electron with momentum k and spin up. The singlet wavefunction for the electron pair previously discussed then becomes:

|Y =

k > kF 
gk ck * c-k * |F
(56)
where |F represents the Fermi sea with all states up to kF filled.

In order to describe an N-electron wavefunction consisting of Cooper pairs, the most general wavefunction is given by:

|YN =
g(ki, , kl) cki* c-ki * ckl* c-kl * |0
(57)
where |0 represents the vacuum state, and ki, , kl represent the k-values included in the sum.

To simplify this sum, BCS used a self-consistent field method, in which the occupancy of each state k is taken to depend only upon the average occupancy of the other states. They took as their ground state:

|YG =

k = k1,,kM 
(uk +vk ck * c-k *) |0
(58)
where the probability that the state (k,-k ) is occupied is hk = |vk|2, and empty, |uk|2 = 1-|vk|2.

To find the ground state of the system, the vk are adjusted so as to minimise the total energy. The pairing Hamiltonian for the system is:

H =

k,s 
ek nk s +

k,k 
Vkk ck * c-k *c-k ck
(59)
where nk s is the particle-number operator for particles of spin s, i.e.   nk s = ck s* ck s. Note that, as before, e is measured from the Fermi energy, so there is no need to take into account the fact that particle number is conserved, as particles at the Fermi energy will not contribute to the sum. The first term represents the total kinetic energy, while the second represents the scattering between states with momenta (k,-k) and (k,-k), and is the potential energy term.

Since the probability of occupation of the state (k,-k) is hk = |vk|2, the KE term is:

KE = 2

k 
ek |vk|2.
(60)
The scattering term describes scattering from a occupied state (k,-k) to an unoccupied state (k,-k); after the scattering, the (k,-k) state will be unoccupied and the (k,-k) state occupied. The probability amplitudes for the initial and final states are respectively ukvk and v*ku*k so the PE term is:
PE =

kk 
Vkk uk v*ku*k vk.
(61)
Note that the uk and vk are in general complex, but for simplicity may be taken to be real.

Minimising the total energy YG | H | YG with respect to the vk gives (Tinkham, 1975, pp 25-28):

vk2 = 1
2


1- ek
Ek


(62)
where the quantity Ek is defined by:
Ek2 = (ek2 + Dk2 )
(63)
and Dk is defined by:
Dk = -

k 
Vkk uk v*k.
(64)
It will become apparent that Ek is just the excitation energy, and Dk the magnitude of the superconducting energy gap.

Taking these equations (62-64) together leads to the self-consistent BCS gap equation for Dk:

Dk = - 1
2


k 
Dk
Ek
Vkk = - 1
2


k 
Dk
(Dk2 + ek2)1/2
Vkk.
(65)

A trivial solution to this equation is for Dk = 0, when vk = 1 for all ek < 0, and we just obtain a Fermi sea at T = 0. However, using the BCS model in which the potential is constant up to a cut-off at (h/2p) wD (see equation 52), a solution exists with:

Dk =

D
|ek| < (h/2p) wD
0
|ek| > (h/2p) wD.
(66)
Substituting this expression into equation 65 gives:
D = 1
2


k 
D
(D2 + ek2)1/2
V
(67)
where the sum is over k states satisfying |ek| < (h/2p) wD.

Cancelling the D, and using the fact that N(0) is the density of states at the Fermi level, dn/de, to convert the sum to an integral over energy, this equation becomes:

1 = N(0)V
(h/2p) wD

0 
1
(D2 + ek2)1/2
dek
(68)
and performing the integral,
1 = N(0)V sinh-1 (h/2p) wD
D
(69)
i.e.
D = (h/2p) wD
sinh(1/N(0)V).
(70)
Using the weak-coupling approximation N(0)V << 1, gives:
D @ 2(h/2p) wD e-1/N(0)V.
(71)

With D constant (with respect to k), the dependence of the pair occupation function vk on ek is now:

vk2 = 1
2


1- ek
(D2 +ek2 )1/2


.
(72)
Note that this is in contrast to a normal metal at T = 0, where only those states below the Fermi energy are occupied. In a superconductor at T = 0, the pair occupation fraction resembles the electron occupancy (Fermi) function for a normal metal at T = Tc. This is shown (using D = 1.76kT) in figure .


Figure 3.3: The pair occupation fraction (solid line) for a superconductor at T = 0 compared to the Fermi function (dotted line) for a metal at T = Tc.

3.3.4  Finite temperature effects

The previous discussion has all been for T = 0. To generalize the BCS gap equation (65) for finite T, a factor [1-2f(E)] = tanh(E/2kT) must be included5, to allow for the fact that occupation of a state k above the Fermi level excludes the occupation of the state (k,-k) by a pair of electrons. The integral form (68) then becomes:

1 = N(0)V
(h/2p) wD

0 
1
(D2 + ek2)1/2
[1-2f(E)] dek.
(73)
Since E = {D2 + ek2}, it is convenient to convert this integral to one over E, using de = (E/e)dE, and D << (h/2p) wD, to obtain:
D = N(0)V
(h/2p) wD

D 
D
  


E2 -D2
 
[1-2f(E)] dE.
(74)

  Critical temperature Tc

The critical temperature of the superconductor is found by letting D tend to zero in (74), and dividing each side by D, which yields (Wolf, 1985 p100):

1
N(0)V
=
(h/2p) wD/kT

0 
tanhx
x
dx
(75)
which gives in the weak-coupling case, 1/N(0)V << 1,
kTc @ 1.13 (h/2p) wD e-1/N(0)V.
(76)
Substituting into ( 71) gives:
2D(0) = 3.53kTc
(77)
which is the usual expression relating the superconducting gap to the transition temperature in the weak-coupling case. In practice, measured values of the ratio 2D(0)/kTc, known as the reduced energy gap, usually fall between 3.0 and 4.5, with many near the BCS value of 3.5 (Tinkham, 1975 p. 34). Strong coupling superconductors such as Pb and Hg are described by modification of the theory. Reduced energy gaps of 4.3-4.67 have been reported for Pb (Wolf, 1985, p. 524), and 4.60.1 for Hg (Meservey & Schwartz, 1969, p. 141).

It is found that D(T) varies rapidly with temperature near Tc:

D(T) @ 1.8D(0)

1- T
Tc


1/2

 
.
(78)
The corresponding results in the strong coupling case are:
2D(0) = 4.0kTc
(79)
and, for T close to Tc,
D(T) @ 1.73D(0)

1- T
Tc


1/2

 
.
(80)

3.4  Excitations of Superconductors

In considering the physical situation of electrons tunnelling or moving ballistically between two metals, at least one of which is superconducting, an understanding of the possible excitations of a superconductor is required. One effect of the minimum energy D for single particle excitations, as discussed in the previous section, is to create an effective superconducting density of states with a gap of magnitude D(T) about the Fermi energy, which may be detected by tunnelling experiments.

3.4.1  Single-particle excitations

We consider the situation where a superconductor has just one single-electron excitation or wavevector k, and the rest of the electrons are paired. The wavefunction for this state is given by the electron creation operator ck* acting on the BCS ground wavefunction (equation 58):

Y = ck*

k = k1,,kM 
(uk + vk ck * c-k*) |0 .
(81)
The single electron occupies the state k, while the state -k is unoccupied. This prevents the state (k,-k) from being occupied by any pair of electrons. This state can no longer contribute to the potential energy of all the pairs of electrons through the scattering potential Vkk, and therefore slightly raises the energy of every pair of electrons. For this reason, a finite energy is required to inject a lone electron into a superconductor.

The required excitation energy is just Ek, which has a minimum value (at k = 0) of D, the energy gap. It is therefore impossible to inject an electron (or hole) into the superconductor with energy less than D.

3.4.2  Derivation of the superconducting energy gap

If we consider the increase in energy of the superconductor resulting from the change in the expressions for KE and PE derived earlier (equations 60 and 61), the expected change in KE is given by the kinetic energy of the electron in state k, ek minus the expected KE if the state were not occupied, i.e.  the probability of occupation |vk|2 times the pair energy 2ek:

(Change in KE) = (1-2|vk|2)ek.
(82)
The expected change in PE is given by the terms no longer present in the potential energy calculation, since scattering into state (k,-k) cannot occur:
Change in PE) = -2 u*k vk

k 
Vkk uk v*k.
(83)
From the definition of D (equation 64) the PE term is equal to -2u*k vk Dk, hence the change in total energy of the system due to the addition of a single electron is:
Change in total E) = (1-2|vk|2)ek -2 u*k vk Dk.
(84)
Substituting for uk and vk in terms of Ek, Dk and ek using (62) reduces this expression to just Ek, showing that Ek does indeed represent the quasiparticle excitation energy. Moreover, (63) shows that D is the smallest possible value of Ek, and hence that there is an energy gap of magnitude D either side of the Fermi energy. This contrasts with the case of a normal metal, where the excitation energy is simply the absolute value of ek, its sign determining whether the excitation is an electron or a hole. This dependence of Ek, the quasiparticle energy on ek, the corresponding free-electron energy is shown in fig .


Figure 3.4: Energy Ek of quasiparticle excitations in a superconductor (solid line) and a normal metal (dotted line) plotted against corresponding free-electron energy ek.

For any given excitation energy E > D, there exist two k-states with appropriate energy, conventionally known as k < and k > , below and above kF respectively. Since the states of momentum k and -k are paired, this structure may be considered to be reflected in the y-axis, giving four available k-states for each permissible energy E (neglecting the further twofold spin degeneracy).

The BCS superconducting density of states NS(E) is given by:

NS(E) = dn
dE
= dn
de
de
dE
(85)
(omitting the k subscripts for clarity).

The quantity dn/de is just the normal metal density of states NN(E), which may be taken to be constant near EF, and e in terms of E is e = (E2-D2). The BCS superconducting density of states (as would be measured in a tunnelling experiment) is therefore:

NS(E) = NN(E) d
dE
[(E2 - D2)1/2] =







NN(E) E
  


E2 - D2
 
Ek > D
0
Ek < D
(86)
which may also be written as:
NS(E) = Re |E|
  


E2-D2
 
NN(E).
(87)
Writing nS(E) = NS(E)/NN(E), the normalised superconducting density of states,
nS(E) = Re |E|
  


E2-D2
 
=







|E|
  


E2 - D2
 
|Ek| > D
0
|Ek| < D.
(88)
This function is plotted in fig . Note that the function is zero between D - the superconducting energy gap .


Figure 3.5: The theoretical normalised BCS superconducting density of states for tunnelling experiments, nS(E). The abscissa is the ratio of superconducting and normal state densities of states.

3.4.3  Quasiparticle creation and annihilation operators

The method used so far to consider the creation of a single electron excitation to a BCS ground state is not easily generalized to further excitations. A better approach, discovered independently by Bogoliubov (1958) and by Valatin, is to introduce quasiparticle creation and annihilation operators (Bogoliubov operators) which are defined by (taking the uk and vk to be real for simplicity):

g*k 0
=
uk c*k - vk c-k
g*k 1
=
uk c*-k + vk ck
gk 0
=
uk ck - vk c*-k
gk 1
=
ukc-k + vk c*k .
(89)
With these definitions, the single-particle creation and annihilation operators may be expressed as a linear combination of the Bogoliubov operators:
ck
=
uk gk0 + vkg*k1
c*k
=
uk g*k0 + vkgk1
c-k
=
-vk gk0 +uk g*k1
c*-k
=
-vk g*k0 +uk gk1.
(90)
The Bogoliubov operators have the property that the states created by repeated application of these operators are orthogonal. These operators therefore properly describe the elementary excitations of a superconductor, which are of mixed hole and electron character.

It is straightforward to modify these operators by the addition of an electron pair creation or annihilation operator to one of the two terms, in order to form operators which definitely add (or remove) one electron to (or from) the system (Bardeen, 1961).

The operators have this form (equation 89) because the presence of an unpaired electron in a given momentum and spin state, say k, (and therefore a hole in the opposite momentum state) prevents the occupation of the (k, -k) state either by an electron pair or by a hole pair. To create such a state requires either the creation of an electron in state k (if the state (k,-k) is unoccupied, with probability |uk|2), or the annihilation of an electron in state -k (if the state (k, -k) is occupied, with probability |vk|2). In either case, the net change in spin is +, and momentum, +k. However, the change in the total number of electrons depends on the values of u and v, and hence the energy of the state above EF. The above discussion describes the operator g*k0, corresponding to putting an unpaired electron in state k. The operator g*k1 corresponds to putting an unpaired electron in state -k. The two annihilation operators can be analogously defined.

For states well above the Fermi energy, the ground state gives a very high probability of the momentum state being unoccupied. Putting an unpaired electron into the state will therefore tend to increase the number of electrons in the (k,-k) state by +1. For states well below the Fermi energy, the state has a very high probability of being occupied (by 2 electrons). Consequently, putting an unpaired electron into this state is better thought of as putting an unpaired hole into the opposite momentum state. The net change in number of electrons occupying the (k,-k) state is therefore -1.

For states near the Fermi energy, the ground state probability of occupation by a pair of electrons is intermediate, and as the energy of the state being considered is changed from below EF to above EF, the net change in number of electrons occupying the state when it acquires an unpaired electron changes continuously from a negative value to a positive one. The character of the excitation can be considered to undergo a gradual change from hole-like well below the Fermi energy to electron-like excitations well above the Fermi energy, although in all cases the excitations have a mixture of both electron and hole character. This is in contrast to the excitations of a normal metal, where the electron-like and hole-like excitations are completely separate. In a normal metal at T = 0 the states below the Fermi energy are all occupied, and those above all empty. All excitations above EF are electron-like, and below EF, hole-like, and there is no minimum excitation energy.

3.5  Ginzburg-Landau Theory

Up to this point, BCS theory has been used to describe the ground state and excitations of homogeneous superconductors, where the wavefunction and energy gap of the superconductor are not functions of position. Ginzburg-Landau theory (Ginzburg & Landau, 1950) is a phenomenological theory for cases where fields are strong enough to change nS or |y|2, or where the local density of superconducting electrons, nS, varies with position, for which a few examples follow:

A pseudowavefunction y(r) is introduced as a complex order parameter, where |y(r)|2 represents the local density of superconducting electrons, nS(r). This results in a generalization of the London theory (section 3.2.1) to situations where nS is a function of position.

Originally derived using phenomenological, macroscopic arguments, G-L theory was shown (Gor'kov, 1959) to be a limiting case of BCS theory, generalized to allow for a varying nS. It is valid for temperatures close to Tc, and slowly-varying y(r) and A(r) (the vector potential).

3.5.1  Ginzburg-Landau free energy

G-L theory postulates that the free energy density f may be expanded (if y is small and slowly-varying) in a series of the form:

f = fn0 + a|y|2 + 1
2
b|y|4 + 1
2m*




(h/2p)
i
- e*
c
A

y

2

 
+ 1
2
m0 h2
(91)
where h is the local field. For y = 0, this reduces to the normal state free energy fn0 + 1/2m0 h2, as required.

The y (fourth) term on the right hand side of equation 91 makes it energetically unfavourable to have rapid changes in the order parameter y, and is a crucial point, which leads to the existence of type II superconductors (section ) and the proximity effect (section ).

In the case of a uniform |y|, the superconducting free energy deficit is:

fs-fn = a|y|2 + 1
2
b|y|4
(92)
which is a series expansion neglecting higher powers of |y|.

There are two cases to consider:

The value of y in equation 93 is conventionally known as |y|2 because it occurs infinitely deep inside a bulk superconductor. The situation is shown in fig .


Figure 3.6: Ginzburg-Landau free energy functions. If a > 0 the minimum occurs at y = 0. If a < 0 the minimum occurs at a finite value, y = y, and has a value 1/2m0 Hc2.

The critical field is given by:

Hc2 = a2
m0b.
(94)
Taking the effective charge and mass in (91) as e* = 2e and m* = 2m, according to experimental data, and that ns* = 1/2ns, we find (Tinkham, 1975 p. 109) that:
|y|2
=
mc2
8pe2 leff2
(95)
a(T)
=
- 2e2
mc2
Hc2(T)leff2(T)
(96)
b(T)
=
16pe4
m2c4
Hc2(T) leff4(T)
(97)
where leff is the effective penetration depth, including impurity and geometric effects.

3.5.2  The Ginzburg-Landau differential equations

Considering T close to Tc and minimising the free energy (possibly in the presence of applied fields and currents) gives the Ginzburg-Landau differential equations:

ay+ b|y|2 y+ 1
2m*


(h/2p)
i
- e*
c
A

2

 
y = 0
(98)
and
J = e*(h/2p)
2m*i
(y* y- y y* ) - e*2
m*c
y*yA
(99)
which is the usual expression for current.

3.5.3  The temperature-dependent coherence length

In the absence of fields, and in one dimension, equation 98 reduces to:

g =

2
x2(T)


g
(100)
where g = 1+y/y, and x is given by:
x2(T) = (h/2p)2
2m*|a(T)|
(101)
which is proportional to (1-t)-1 and hence diverges as T Tc. A disturbance of the value of y away from y will decay on a length scale of order x(T), the temperature-dependent coherence length.

From equations 96 and 101,

x(T) = F0
22 pHc(T) leff(T)
(102)
where F0 is the flux quantum, F0 = hc/2e.

3.5.4  Results from Ginzburg-Landau theory

Combining the BCS results that

x0 = (h/2p) vf
pD(0)
(103)
and, equating magnetic and condensation energies,
1
2
m0 Hc2 = 1
2
N(0)D2(0)
(104)
we find that
F0 =

2
3


1/2

 
p2 x0 lL(0) Hc(0).
(105)

Combining this with equation 102 gives the result that, near Tc,

x(T)
=
0.74 x0
(1-t)1/2
clean limit
(106)
x(T)
=
0.855 (x0l)1/2
(1-t)1/2
dirty limit.
(107)
The dimensionless Ginzburg-Landau parameter, k, given by
k = leff(T)
x(T)
(108)
is found, in the clean and dirty limits respectively, to be:
k
=
0.96 lL(0)
x0
clean limit
(109)
k
=
0.715 lL(0)
l
dirty limit.
(110)

Typically, for clean superconductors, k << 1. It can be shown (Solymar, 1972, pp. 8-9) that type I superconductors have k < 1/2, and type II, k > 1/2, since this determines whether the domain wall energy between regions with different magnetic fields is positive and negative. In the latter case, it is energetically favourable for many domain walls to form, until the flux penetrates the superconductor in cores, each containing one flux quantum, F0.

  Superconducting sheath effect

The previous discussion has been for an infinite bulk superconductor. In the case of a finite superconductor, it is found that superconductivity can exist in a surface layer at fields above Hc2, but below a critical field Hc3 (Saint-James et al. , 1969, ch. 4). The value of Hc3 can be above or below Hc2 and/or Hc; this effect is exhibited by type II superconductors if Hc3 > Hc2, and by type I superconductors of Hc3 > Hc, and occurs when the field is nearly parallel to the surface. The existence of this effect has been verified by tunnelling experiments, and is the reason for discrepancies between measurements of the critical field by magnetization techniques (giving Hc2) and by resistivity techniques (giving Hc3, since the surface superconducting sheath short-circuits the current). The ratio Hc3/Hc2 is as large as 2 in some materials.

The above discussion is no longer true if the superconductor is coated with a conductor, since the `no perpendicular current' boundary condition no longer holds. This is considered in section .

3.6  The Proximity Effect

The arguments of the previous section imply that the superconducting order parameter y cannot change over a scale smaller than x(T). At an interface between a normal metal and a superconductor, the superconducting order parameter (pair potential) will therefore decay into the metal with a decay length of order x(T), rather than suffering an abrupt drop to zero. The order parameter in the superconductor will also be somewhat depressed close to the interface with the normal metal. This is the proximity effect.

The behaviour of the pair potential D(r) at the NS interface depends on the interaction between electrons in the normal metal:

D(r) = F(r)V(r)
(111)
where F(r) is the pair amplitude (|F(r)|2 is the pair density, proportional to the probability density of finding a pair at position r), and V the electron interaction (positive if the interaction is attractive). This is shown for attractive (a) and repulsive (b) interactions in fig . The approximation used in the Ginzburg-Landau scheme is to model the pair potential as a linear variation near the boundary, shown in fig c. The pair potential falls to zero smoothly over a distance b on the normal side.


Figure 3.7: Variation of the pair potential near an NS interface for (a) attractive interaction between electrons in the normal metal; (b) repulsive interaction; (c) Ginzburg-Landau approximation.

In order for the proximity effect to occur, the contact between the two metals in the NS junction must be very good, and metals must be chosen which do not form intermetallic compounds or solid solutions, otherwise interdiffusion can take place extremely rapidly at the interface, to a depth of hundreds of Å ngströms in a few minutes at room temperature (Weaver & Brown, 1962; Caswell, 1964).

3.6.1  Standard results for the proximity effect

In the case where the normal (N) side of the interface is normal for all temperatures (i.e.  not a superconductor above its transition temperature), and the interaction between electrons is negligible, the following results apply (Deutscher & de Gennes, 1969; Wolf, 1985, p. 184)):

When the N-metal is a superconductor above its transition temperature TcN, the form (114) applies as T TcN.

The main consequence of the non-zero order parameter in the normal metal is that, provided the normal layer is sufficiently thin, a supercurrent can flow through a normal region between two superconductors (see e.g.  de Gennes & Guyon, 1963; de Gennes, 1964; Clarke, 1969). The maximum supercurrent (critical current) decreases exponentially with the normal metal thickness, increases with the mean free path of the normal metal, and increases rapidly with decreasing temperature.

For an NS sandwich consisting of a thin normal metal layer backed by a superconducting slab, the pair density, F, decays exponentially into the normal metal, and, provided the normal layer is sufficiently thin, will be non-zero at the surface. If the metal has non-zero electron-electron attractive potential, V, the pair potential, D = FV, will be non-zero at the surface, so electron states below this energy will not exist, giving rise to a energy gap in the density of states. If the metal has V = 0, then D = 0 at the surface (i.e.  there is no energy gap) although tunnelling characteristics are affected by the altered energy distribution of the electrons. Tunnelling in NINS junctions is further discussed in section .

3.7  High-Tc Superconductors

The high-Tc superconductors are a class of ceramic cuprates, some of which have abnormally large transition temperatures. The first of these to be discovered, Ba-doped lanthanum copper oxide was discovered in 1986 by Bednorz & Müller (1986, 1988). These superconductors have in common a structure consisting of stacked CuO2 planes. The superconductivity arises from doping the insulating material, e.g.  by adding oxygen to YBa2Cu3O6. The stoichiometry of the material thus formed is often oxygen-deficient, e.g.  YBa2Cu3O7-d. The structure consists of n closely-spaced copper-oxide planes, with larger gaps between, where n depends on the material, and is equal to 2 for YBCO and BSCCO, and generally between 1 and 4.

The highest transition temperature observed in a cuprate superconductor is 133.5 K for HgBa2Ca2Cu3O9, and transition temperatures for the superconductors used in this study are 92 K for YBa2Cu3O7 (YBCO) and 85 K for Bi2Sr2CaCu2O8 (BSCCO).

In this section, the properties peculiar to high-Tc superconductors are discussed, along with some of the experimental difficulties posed by these features.

3.7.1  High-Tc Properties

The high-Tc superconductors such as YBCO, BSCCO etc. , are layered materials, all of which contain planes of Cu and O atoms, in which the superconductivity is believed to flow. These cuprate planes occur with different separations and local environments in the different high-Tc materials, which gives rise to the different superconductive properties measured. They are type II superconductors, with very large upper critical fields.

The layered nature of these materials results in an anisotropic superconducting behaviour, in which the supercurrent flows within the cuprate planes, but not across them. The orientation of a single crystal sample is therefore of great importance in any experiment, particularly one, like STM, which probes the surface of the material. In scanning tunnelling spectroscopy the current is injected into the superconductor over a range of angles, so the results obtained from any such experiment are dependent upon the superconductivity of the material in these directions. With polycrystalline samples, e.g.  polycrystalline YBCO films, there are also preferential orientations for the crystallites. The existence of mis-oriented crystallites may also be of importance.

Another property shared by most of these materials is their unstable surface. YBCO degrades easily, losing oxygen from the surface in vacuum, and hence altering the stoichiometry. It also reacts with water, and degrades fairly rapidly in air at room temperature, so precautions must be taken, and contact with the atmosphere minimised. BSCCO has considerably better surface stability.

The high-Tc materials have an extremely short coherence length, of the order of just a few Å (compared to a few hundred Å for conventional superconductors). The existence of a degraded surface layer in these materials, even if only a few atoms thick, can therefore destroy any properties observed at the surface. This is particularly so because the degraded layer is typically insulating rather than conducting (the superconductors also become insulating above the transition temperature). Hence, for STM work, it has often been necessary to dig the tip into the surface in order to observe any superconducting features. In any case, the existence of a thin insulating layer complicates the behaviour of the junction, since the barrier is no longer a vacuum barrier. In addition, the poor surface characteristics may give rise to conducting or superconducting inclusions in an insulating matrix, leading to Coulomb effects (see chapter )

Many authors (e.g.  Monthoux et al. , 1993; Tsuei et al. , 1994; Tanaka & Kashiwaya, 1995; Ichimura et al. , 1995) have proposed that the high-Tc superconductors exhibit d-wave superconductivity (in which the paired electrons are in a state with angular momentum m = 2, rather than the s-wave (m = 0) state of conventional superconductors). A summary of evidence for dx2-y2 symmetry is given by Van Harlingen (1995). There is a solid body of evidence for strong anisotropy in D(k), consistent with the full nodes characteristic of dx2-y2 symmetry. For a d-wave superconductor, the tunnelling probability depends on the angle of the electron wavevector to the c-axis, so the conductance curve no longer directly represents the local density of states. However, there is also a large amount of evidence in favour of s-wave symmetry (Burns, 1992), from Josephson and conventional tunnelling experiments. Most of the evidence for d-wave symmetry has been obtained using YBCO, and it is unknown, if YBCO is a d-wave superconductor, whether this is generally true of the high-Tc cuprate superconductors.

The mechanism of high-Tc superconductivity is still hotly-debated (Mott, 1996; Anderson, 1990,1995,6), and it is unclear whether the electron pairing is mediated by phonons, by magnons, or by some other interaction. Specific heat measurements imply the presence of states in the gap, suggesting that there is no gap in the conventional sense. Currently, the answers to these questions are still unknown.

Chapter 4
Theory of Tunnelling

4.1  Introduction to Tunnelling

Quantum-mechanical tunnelling is the process by which a particle may pass through a classically forbidden region (where the potential energy of the particle is greater than its total energy). A classical particle of the same energy would be unable to enter such a region. However, a quantum-mechanical (i.e. realistic) particle has a finite probability of passing through such a region. The total energy of the particle is lower than the potential energy, and the wavevector of the particle in the direction of propagation becomes imaginary. The probability wave which describes the particle's motion is therefore an evanescent wave (which decays exponentially in the direction of propagation). The non-zero amplitude of the wavefunction implies that the particle has a finite probability of traversing the barrier. The reason for this behaviour is the inherently smooth behaviour of the probability density y*y even at points of finite discontinuity in the potential function, U(x), such as the interface between a conductor and an insulator or vacuum.

4.1.1  Historical development

Electron tunnelling was used by Gamow (1928) to explain a-decay from heavy nuclei, by the tunnelling of a-particles through the potential barrier formed by the repulsive electrostatic force and the attractive strong nuclear force. Oppenheimer (1928) explained the field ionization of hydrogen by means of electrons tunnelling from localised states near the nucleus into spatially extended states. Fowler & Nordheim (1928) explained field emission of electrons from metals using steady-state tunnelling theory, with electrons tunnelling from the surface of the metal to a point in the vacuum where the total energy became positive due to the applied field.

Electron tunnelling between two metallic planar electrodes separated by a vacuum was considered by Frenkel in 1930, and the situation in which an insulator formed the barrier was considered by Sommerfeld & Bethe (1933). The first convincing experimental verification of tunnelling in such a system was made by Giaever (1960a) and Fisher & Giaever (1961).

4.2  Theoretical Models for Tunnel Junctions

The simplest tunnelling structure consists of two normal-state electrodes separated by a thin potential barrier. Junctions of this type are usually fabricated by allowing a thin, natural oxide barrier to grow on an evaporated film, then evaporating a second, crossing counterelectrode film (Fischer & Giaever, 1961). In such a junction, tunnelling has been demonstrated to account for greater than 99.9% ot the total current (Giaever, 1960a,b). The tunnel current through the barrier can be calculated either by the steady state approach of Gamow (1928), or the time-dependent perturbation theory (Transfer Hamiltonian) approach of Oppenheimer (1928), generalised to tunnel junctions by Bardeen (1961). The latter approach explicitly involves the density of states, and allows theoretical treatment of junctions where one or both electrodes are superconducting.

4.2.1  Theoretical treatment of the barrier

The barrier is the crucial element in any tunnelling experiment, as it is this that decouples the two electrodes, so that electrons moving from one electrode to the other are injected at a specific energy. For appreciable tunnelling to occur, the width of the barrier must be no more than a few nm thick.

The simplest treatment of the barrier is to consider a 1-D rectangular potential barrier, of constant height and width. The (imaginary) wavevector of a given electron in the gap region is then constant. At low voltages, the current density through such a barrier, of width d and height f, is:

J =

3e2
8p2(h/2p)


V
  


2mf
 

(h/2p) w
exp(-2
  


2mf
 

(h/2p)
w)
(116)
where V is the applied bias across the junction, and m the electron mass (Simmons, 1963).

With a bias voltage applied, the barrier becomes trapezoidal, as shown in fig . For an electron with a given energy (relative to the Fermi energy of the left hand electrode) moving from left to right, this lowers the average barrier height by an amount eV/2, where e is the electronic charge, and V the applied bias, increasing the tunnelling probability.


Figure 4.1: Trapezoidal barrier between planar metallic electrodes.

  Work function effects

The barrier is also affected by the difference in the work functions of the electrodes on either side. Biasing the right electrode to a voltage +V lowers its Fermi energy by an amount eV with respect to the left electrode. However, the height of the barrier at either electrode is equal to the work function of the electrode material. Therefore, if the junction is unbiased, but has electrodes composed of metals with different work functions on either side, the barrier is still trapezoidal. If the work functions of the left and right hand electrodes are, respectively, Y1 and Y2, the height of the barrier on the left hand side is greater than that on the right by an energy eV+Y1-Y2.

If the barrier is formed from an insulator rather than the vacuum, the height of the barrier is reduced by an amount which depends on the material used. If the energy of an electron in the insulator is lower than that of one in the vacuum by an energy EI, the work function on either side is effectively reduced by an amount EI. The effective barrier heights f1 = Y1 - EI and f2 = Y2 - EI will be used for the rest of this chapter.

  Barrier models

For a barrier of width d, the energy of the barrier at a distance x from the left hand electrode, measured with respect to the Fermi energy of the left hand electrode (which will be the convention generally used in this chapter), is given by (see fig 4.1):

f(x,V) = f1+ x
d
(f2-eV-f1 ).
(117)
For a more accurate treatment, the WKB (Wentzel-Kramers-Brillouin) approximation may be used (see section ). This takes account of a slowly-varying barrier height by integrating the decay constant of the evanescent wavefunction, k = (2m(V(x)-Ex)/(h/2p)2) over the width of the barrier. The WKB approximation is only valid when the potential in which the particle moves varies slowly with position across the barrier, on the scale of the particle's wavelength. In this case, the decay constant may also be considered to vary slowly with position across the barrier.

A further refinement to the barrier potential is to include the effect of image charges. Two parallel conducting electrodes create an infinite line of image charges with alternating sign (see fig ).


Figure 4.2: Infinite line of image charges generated by two parallel planar conductors.

The infinite series expression for the resulting image potential Vi is modelled well by a hyperbolic approximation6:

Vi
=
- e2
8pe


1
2x
+

n = 1 


nd
[ (nd)2 - x2 ]
- 1
nd




(118)
- 0.575 ld2
x(d-x)
(119)
where s is the distance between the electrodes, x is the distance of the electron from one electrode, and l = e2 ln2/8pe0er d (Simmons, 1963). The effect of the image charges is to round off the corners of the barrier, making it both lower and narrower, and hence increasing the tunnelling probability. The effect of adding the image potential to a trapezoidal barrier is shown in fig .


Figure 4.3: Barrier shape for planar junction showing the effect of image charges. The effect of the image charges is to reduce the effective height and width of the barrier. I is the energy of the insulator below the vacuum energy, and f1, f2 are the effective work functions, Y1-I and Y2-I. [`(f)] is the mean barrier height as seen by an electron in the left electrode, and the junction is biased with a voltage +V applied to the right electrode. s1 and s2 are the classical turning points, between which the particle may be considered to tunnel.

The image potential causes the wavevector of the electron to become real some distance away from each electrode, at the points where the total energy is zero.

Adding the approximate image potential to the trapezoidal barrier potential gives the barrier potential:

f(x,V) = f1 + x
d
(f2-eV-f1)- 1.15ld2
x(d-x)
.
(120)

4.2.2  Corrections to the image potential

In addition to the image potential, other corrections may be made to the simple models already described, in order to take account of many-body effects, i.e.  interactions between the electrons and other charge carriers, in both the surface and the bulk material.

  Local density approximation

In this widely-used approximation, the electron-electron interaction is modelled by the exchange-correlation energy, which acts as a one-electron potential for tunnelling electrons.

Lang & Kohn (1970) modelled tunnelling from a jellium metal (i.e.  one where the effect of the positive ion cores is approximated by a uniform positive charge density), using the local density approximation to calculate the one-electron potential, which was found to be accurate close to the surface, but that at larger distances the model failed, since it does not include the image potential.

The image charge takes a finite time to build up, since it may be considered as an interaction with surface plasmons. Thus the image potential must be reduced if tunnelling processes occur on a shorter time than the inverse plasmon frequency (van de Leemput & van Kempen, 1992, section 2.3). Thus, the tunnelling time effectively influences the barrier shape. Since the barrier shape also influences the tunnel time, this is a problem which requires a self-consistent method of calculation (see section ).

4.2.3  Theoretical treatment of the electrodes

  Allowed elastic tunnelling events

In a junction with two normal metal electrodes at zero temperature, the electron states in each electrode are full at energies below the Fermi level, and empty above this level. Increasing the bias voltage shifts the Fermi level in the right electrode with respect to the Fermi level in the left electrode, and increases the number of electrons in the left electrode which can tunnel across the barrier into an empty state of the same energy on the other side (elastic tunnelling). This situation is shown in fig .


Figure 4.4: Elastic tunnelling events at zero temperature. Elastic tunnelling events correspond to a horizontal transition from left to right. The right hand electrode is biased to a voltage +V. Electrons can only tunnel across from an occupied state to an unoccupied one, so transitions can occur only in an energy range eV. The probability of electron tunnelling is proportional to the density of occupied states on the left hand side, multiplied by the density of unoccupied states at the same energy on the right.

The rate of change of tunnel current with applied bias is therefore proportional to the integral of the product of the densities of states, between the two Fermi levels, on each side of the barrier. This assumes that the voltage change is small enough not to affect greatly the tunnelling probability. If the density of states is only slowly-varying with energy in one electrode, the graph of dI/dV against V reflects the density of states in the other electrode.

If a large voltage (more than about 0.5-1 V) is applied, it is the densities of states at and opposite the higher Fermi level which dominate, because the barrier is lower for electrons tunnelling across at this level than for electrons at the lower Fermi level. In this case, tunnelling is occurring predominantly from the higher energy Fermi level (the Fermi level of the negatively-biased electrode). As this level is moved up and down with respect to the Fermi level of the positively-biased electrode, by changing the bias, it is the density of states in this electrode (the positively-biased one) which plays more of a role in determining the tunnel current. The consequences of this effect for scanning tunnelling microscopy are discussed in section .

For bias voltages V which are sufficiently small that the electron density of states is fairly constant within an energy range eV of the Fermi energy on either side of the barrier, and that the tunnelling probability does not change greatly over this energy range, this model implies that the tunnel current should be proportional to the applied voltage. At a higher applied bias, the lowering of the average barrier height will increase the tunnelling probability, so the current increases more quickly than the voltage. If there are variations in the electron density of states in either electrode over this bias range, this will also have an effect on the tunnel current.

  Finite temperature effects

At a finite temperature, the occupancy of electron states does not change abruptly from 1 to 0 at the Fermi energy, because electrons have a finite probability of being in an excited state due to thermal excitations. The effect of this is to smear out the step function into the Fermi function, giving an occupancy f(E) at an energy E above the Fermi level7, where

f(E) = 1
1 + exp(E/kT)
.
(121)
This function differs substantially from the zero temperature equivalent (where the occupancy changes abruptly from 1 to 0, as E goes above the Fermi level, E = 0) only at energies within a region of order kT from the Fermi level. The physical effect of a finite temperature is that electrons may tunnel from states just above the Fermi level of the left hand electrode (which have a finite probability of occupation), or into states just below the Fermi level of the right hand electrode (which have a finite probability of being vacant), and also that there is a finite probability of electron tunnelling from right to left (since both electrodes possess both filled and unfilled states at all energies), although this probability becomes vanishingly small when the bias is large, eV >> kT. This situation is shown in fig ,


Figure 4.5: Elastic tunnelling processes at finite temperature. Elastic tunnelling processes are horizontal transitions. Electron tunnelling probability is weighted by the probability of occupancy (the Fermi function, f(E)) on the left hand side multiplied by the probability of vacancy (1-f(E+eV)) at the same energy on the right hand side. Because both electrodes have both filled and unfilled states at all energies, electron tunnelling can occur in either direction.

and is treated mathematically by integrating over electrons of all energies, with a weighting equal to the Fermi function, f(E), corresponding to the left electrode, and the complement of the Fermi function, 1-f(E-eV), corresponding to the right electrode, for electrons moving from left to right, and similarly for electrons moving from right to left. The effect of this is that the graph of dI/dV against V shows the density of states, smeared out by a function whose width is approximately 3.5kT (see section ).

4.2.4  Indirect and higher order tunnelling processes

Tunnelling processes can involve an interaction between the tunnelling electron and phonons in the barrier material, or an interaction with states in the barrier between the two electrodes (resonant and two-step, or multi-step, tunnelling).

  Inelastic processes

In addition to the elastic process described in section 2.3.1, it is possible for an electron to lose energy via a phonon as it tunnels across. This process is possible as long as there is an empty state to tunnel into, at an appropriate energy. At zero temperature, as the bias is gradually increased, the conductance due to inelastic processes will increase by a step each time the bias V passes a voltage (h/2p) w/e, allowing another inelastic tunnelling channel to open. This process is illustrated in fig .


Figure 4.6: Elastic (E) and inelastic (I) tunnelling processes. The inelastic process can only occur when the bias exceeds a voltage (h/2p) w/e, where w is the frequency of the phonon involved in this process.

Generally, both elastic and inelastic processes contribute to the tunnel current across a junction.

Inelastic electron tunnelling spectroscopy (IETS) is a well-established technique which gives information about the resonant frequencies of the barrier, or of adsorbed molecules deposited on the inner surfaces of the electrodes (Jaklevic & Lambe, 1966). However, the required sensitivity and stability are extremely high, since it is the second derivative of the I-V  curves which gives the spectra. Prospects for detection of adsorbed atoms by STM are discussed by Baratoff & Persson (1988), who conclude that IETS by scanning tunnelling microscopy is an achievable goal.

At a finite temperature, the smearing out of the electron occupancies by the Fermi function smears out the steps in the conductance due to inelastic processes, by an amount of order kT.

  Resonant tunnelling

Resonant tunnelling occurs when the barrier contains internal resonant levels. This may be due to a potential well, inside the barrier, which could be caused by an embedded metallic impurity.

In the case of resonant tunnelling, this level is not a true bound state, because any electron placed in the resonant level would quickly tunnel its way out to either side of the barrier. The presence of the resonant state can increase the tunnelling probability for electrons with matching energy by several orders of magnitude, giving a conductance peak at a particular voltage bias (Duke, 1969). With a finite lifetime effect such as this, a Lorentzian energy peak is expected in a one-dimensional situation. The resonant state can increase the junction conductance at voltages above some cut-off value(s) determined by the energy of the resonant level(s) (see e.g.  Gadzuk & Plummer, 1973; Knauer et al. , 1977).

Halbritter (1982) pointed out that the reason resonant tunnelling does not dominate the tunnel current in thick junctions is the Coulomb energy (see chapter ), which suppresses tunnelling via the resonant state, if it is caused by an embedded particle.

  Two-step and multiple-step tunnelling

Two step tunnelling occurs when a localised state, or `trap', is present inside the barrier. It may occur when a degraded, insulating surface of a metal acquires a conducting island on its surface.

The effect of the localised state is that, once the energy of the electrons is sufficient to enter the localised state, tunnelling across the barrier may occur in two short steps, rather than a single long step, greatly increasing the tunnelling probability. Since the WKB transmission factor depends on a path integral across the barrier, the effect of splitting up the path into two steps (taken as being of equal length) is that the transmission factor is changed from D to 1/2 D1/2.

The effect of such tunnelling processes is to render spectroscopy impossible, because energy is lost by the tunnelling electron into the localised state. This is an inelastic process, with a step increase in conductance, as well as the increase in slope owing to the increased WKB transmission factor.

Such processes may also occur with more than one localised state in the barrier (multi-step processes), and the treatment is equivalent.

  Surface states

Surface states are states localised at the surface, which lie inside a band gap, and would therefore be forbidden in the bulk (Zangwill, 1988). A surface state can mix with a degenerate extended state, forming a surface resonance, which has a large surface amplitude and can lead to resonant tunnelling (van de Leemput & van Kempen, 1992). The lifetime of the state is governed by coupling (e.g.  via phonons) between the state and a bulk state (Garcia et al. , 1987), so if the decay time is large, this becomes the rate-determining step in the tunnelling process, leading to current saturation at some level.

  Image states and Gundlach oscillations

An image state is a localised surface state formed by means of the attractive image potential interaction. If the energy of the state is within the band gap, there is also a repulsive force from the surface, so an electron can be trapped in this state, just outside the surface. If the energy is not within the band gap, the state can hold the electron for a finite time before it escapes.

If a sufficiently large voltage is applied that the part of the barrier near the surface becomes classically allowed (i.e.  several volts), interference between transmitted and reflected electrons can lead to conductance oscillations (Gundlach, 1966). Coombs & Gimzewski (1988) demonstrated that this could give rise to fine structure in the conductance curves obtained by STM, owing to the effect of different parts of the tip, with slightly different work functions, causing beats to occur.

  Tamm states

Surface states which are localised just inside the surface of the metal can also exist, as demonstrated by Tamm (1932) in a 1-D model using a Krönig-Penney potential8 with a boundary consisting of a constant repulsive potential.

  Coulomb blockade effects

When a localised state which may take part in two-step tunnelling consists of a very small particle, its capacitance C may be sufficiently low for the electrostatic energy required to place an electronic charge e on the particle, e2/2C to be important. The condition for the tunnelling channel to be blocked is that e2/2C > eV. This is known as a Coulomb blockade, and below the threshold voltage V = e/2C, the tunnelling conductance reduction is known as a Coulomb gap.

If the capacitances between the particle and the two electrodes on either side are both considered, the different values of capacitances and resistances in the theoretical model may produce either a Coulomb gap, with a linear I-V  characteristic apart from a central symmetrical zero conductance region, or a Coulomb staircase, with a series of steps corresponding to placing charges e, 2e, 3e,... onto the central particle. These effects are reviewed in chapter . It is important to bear these effects in mind to avoid possible misinterpretation of gap-like features for BCS superconducting gaps (section 3.4.2) which have distinct features.

4.3  Methods of Tunnel Current Calculation

4.3.1  The WKB method

The Wentzel-Kramers-Brillouin (WKB) approximation can often be used to estimate the tunnelling rate across a barrier. The requirement for the approximation to be valid is that the potential U(x) through which the electron moves is only a slowly varying function of position. The WKB approximation yields the fraction D of the incident probability current which is transmitted across the barrier.

For a plane wave eikx, the probability current is (h/2p) k/m. This is the rate at which electrons arrive at the barrier. In general, the probability current in the x-direction is defined for a wave y as:

j =   < y|j|y >   = i(h/2p)
2m


y y*
x
- y* y
x


.
(122)

The fraction of transmitted probability current, D, in the WKB approximation is given by:

D = exp(-2K)
(123)
where
K =
x2(Ex)

x1(Ex) 
k(x,Ex) dx
(124)
and
k(x,Ex) =

2m*[V(x)-Ex ]
(h/2p)2


1/2

 
.
(125)
In these equations, Ex is the kinetic energy related to the motion in the x-direction, and x1, x2 are the classical turning points, between which the potential V(X) exceeds the energy Ex. The WKB approximation is valid when the potential in which the electrons are moving is slowly varying compared to the electron wavelength.

  Steady state approach

In this approach a single wavefunction is taken to extend over the whole tunnelling structure (fig )


Figure 4.7: Whole-system wavefunction used in WKB (steady-state) approach.

. If the variation in the potential is taken to be sufficiently smooth that the WKB method can be employed, then the net current density across the barrier from metal 1 to metal 2 is obtained by integrating over available k-states in metal 1, weighting each by the group velocity vx = (h/2p)-1E/kx, and multiplying by f(E)[1-f(E+eV)] to ensure that the initial state 1 is occupied and the final state 2 empty, and by the transmission factor D(Ex). The expression for electrons tunnelling from metal 2 to metal 1 is obtained similarly, and the net current density J = J12 - J21 is given by (Wolf, 1985):

J12
=
2e
(2p)3



d2 kt dkx

1
(h/2p)
E
kx


Df(E)[1-f(E+eV)]
(126)
J21
=
2e
(2p)3



d2 kt dkx

1
(h/2p)
E
kx


Df(E+eV)[1-f(E)]
(127)
J
=
2e
(2p)2 h



0 
dEx[f(E)-f(E+eV)]

d2 kt D(Ex,V).
(128)

The difference between the WKB results and those obtained by directly solving the time-independent Schrödinger equation and matching the wavefunctions at the boundaries can be illustrated by considering the case of a 1-dimensional rectangular barrier of height V0, extending from -a to +a (fig ).


Figure 4.8: One-dimensional rectangular barrier. This is the simple barrier shape used for comparison of the WKB approximation with the exact result for transmission through a square barrier.

The transmission factor T for an electron of energy E approaching the barrier from the left, in a state eikx, is

T = 2ik
(k2-k2)sinh(2ka) +2ikk cosh(2ka)
(129)
where
k =

2m
(h/2p)2
(V0 - E)

1/2

 
.
(130)
The tunnelling probability |T|2 (corresponding to the factor D in the WKB approximation) is
|T|2 = 4k2 k2
(k2+k2)2sinh2(2ka)+4k2 k2
.
(131)
In the case of a high, wide barrier, the sinh2 term may be approximated by an exponential:
|T|2 = 16k2 k2 exp(-4ka)
(k2 + k2)2
.
(132)
This is valid for 2ka >> 1. Typically, 2ka ~ 10 for planar junctions.

In the WKB approximation applied to the same barrier, for an electron of energy E approaching the barrier, k(x,Ex) = (2m(V0-E)/(h/2p)2) = k. K = 2a (2m(V0-E)/(h/2p)2) = 2ka, and

D = exp(-4ka).
(133)
The loss of the prefactor is typical of WKB solutions, but the general behaviour of the tunnel current is the same as for the exact case.

  Angular dependence of the tunnel current

A consequence of the dependence of D upon kt, is that tunnelling is favoured for electrons with a minimum kt. Writing f as the angle between the wavevector of a Fermi surface electron in metal 1 and the normal to the barrier, E = m1 (the chemical potential in the left electrode) and Ex = E cos2 f m1(1 - f2). For a barrier of constant height U0 = m1+UB, (UB is the barrier height above the chemical potential in the left electrode) and width d, the WKB approximation gives:

D = exp(-2K)
(134)
where
K =

2m*d(U0-Ex)
(h/2p)2


1/2

 
.
(135)
Writing k = (2mUB/(h/2p)2),
D = exp(-2kd(1+ m1 f2
UB
)1/2).
(136)
The angle fe at which the tunnelling probability is reduced by a factor e is then given by:
fe2 = UB
m1
( 1
kd
+ 1
4 k2 d2
).
(137)
Using typical values for a planar junction, of UB = 1 eV, d = 25 Å, m1 = 5 eV and m = me = 9.1 ×10-31 kg, one obtains:
k = 5.15 ×109 m-1
(138)
and
fe = 7.2.
(139)
In an STM junction, with typical values UB = 5 eV, d = 5 Å, m1 = 5 eV and m = me, one obtains:
k = 1.15 ×1010 m-1
(140)
and
fe = 24.4.
(141)
This indicates that electrons tunnel from an STM tip over a greater range of angles. The physical reason for this is that the shorter tunnelling distance leads to a slower fall-off of tunnelling probability with angle, which dominates the effect of the higher barrier, which would tend to have the opposite effect. Tunnelling by STM through an oxide barrier, such as the degraded surface of a high-temperature superconductor, would give values nearer the planar junction case.

For most junctions, the value of kd is approximately 10, since the tunnel current depends exponentially on this factor. This means that the term 1/4k2 d2 can be ignored, and that the tunnelling rate depends on f simply as D exp(bf2), where b = kd m1/UB.

  Sharp boundary approximation

As an alternative to using the WKB approximation, which assumes that the potential is a slowly varying function with position, the potential may be assumed to change sharply at boundaries between the conductors and the insulator. In this case, the wavefunction must be matched at the boundaries, so that y(x) and dy/dx are continuous. Generally, although there are differences in the theoretically calculated junction conductances between this method and the WKB method, the overall behaviour is similar (Brinkman et al. , 1970).

4.3.2  Time-dependent perturbation theory approach

In this approach (also known as the Bardeen Approach), developed by Bardeen (1961) from Oppenheimer's (1928) transfer Hamiltonian method, the electrodes are considered to be very weakly coupled by the barrier, as borne out by the near-unity probability of reflection from the barrier.

The barrier divides the system into two nearly independent portions, in each of which the approximate wavefunction decays exponentially into the barrier and beyond (see fig ).


Figure 4.9: Wavefunctions used in the Bardeen approach to tunnelling theory (transfer Hamiltonian method). The wavefunction of each electrode is a standing wave plus exponentially decaying tail. The tails overlap in the barrier region only, and the tunnel current is calculated from the overlap using the Fermi Golden Rule.

The weak coupling by the barrier is treated as a perturbing Hamiltonian, HT which allows electrons in states corresponding to one side of the barrier to undergo a transition to a state corresponding to the other side of the barrier:

H = H1 + H2 + HT.
(142)

The two systems on either side of the barrier are first considered separately, to obtain the approximate wavefunctions. The WKB wavefunctions for the system are (Bardeen, 1961; Harrison, 1961):

y1 =







w-1/2 exp[i(kyy+kzz)] cos(kxx+d),
x < 0
1
2
w-1/2 exp[i(kyy+kzz)]g exp(-kx),
0 < x < d
0,
x > d
(143)
for the left electrode, and a similar expression for the right electrode9. This is a standing wave inside the electrode, matched to an exponentially decaying wave in the barrier region. The amplitude is taken to be zero in the other electrode, which is valid only for the case of small tunnelling probability.

Substituting an initial time-dependent wavefunction into the time-dependent Schrödinger equation and using the fact that the wavefunctions overlap only in the barrier region, the matrix element is given by the Bardeen (1961) integral, which is taken over a plane surface separating the two electrodes:

M = (h/2p)
2m



z = z0 


yl* yr
z
- yr* yl
z


dS
(144)
where yl, yr are the wavefunctions of the two electrodes, and S is any plane surface between them. In this case the integral is given for tunnelling in the z-direction. This integral is easily modified for the non-planar case (e.g.  STM) by taking the integral over an arbitrary (in general, non-planar) surface between the electrodes. The matrix element is then (van de Leemput & van Kempen, 1992, p. 1171):
M = (h/2p)
2m



S 
(yl*yr -yr*yl ) ·dS
(145)
where the choice of the surface S is determined by the geometry.

To deal properly with STM junctions, it is necessary to use the Modified Bardeen Approach (MBA) (see section ), which allows for the thinness of the vacuum barrier in scanning tunnelling microscopy by modifying the wavefunctions in the two electrodes (Chen, 1993, p. 70). This is necessary because the electrodes are no longer independent when the barrier width is very small (van de Leemput & van Kempen, 1992, pp. 1173, 1175). Also, the barrier height in an STM junction is greatly reduced by the image potential. The barrier collapses completely for sufficiently small tip/sample separation, so the usual Bardeen integral is unsatisfactory. These areas are further discussed in chapter .

The transition rate (probability of tunnelling per unit time), w, for transitions from a given state l on the left of the barrier, to a set of states of equal energy, and of density r(Er) on the right, is given by Fermi's Golden Rule:

w = 2p
(h/2p)
|M|2 r(Er) d(Er-El).
(146)
The net current density may be calculated by integrating over all the states on the left from which tunnelling can take place to calculate the current density from the left electrode to the right, and subtracting a similar expression for tunnelling from right to left. Since tunnelling events are always from a full state to an empty state, the expressions involve Fermi factors f(E)[1-f(E+eV)] and f(E+eV)[1-f(E)] respectively: the difference between these expressions is [f(E)-f(E+eV)]. Taking into account the densities of states on either side of the barrier, we obtain a result equivalent to the steady state calculation for a planar barrier, where the WKB transmission factor D(Ex,V) in equations 126-128 is replaced by the matrix element |M|2 (Harrison, 1961):

J = 2e
h



- 
[f(E)-f(E+eV)] rl(E) rr(E+eV) |M|2 dE.
(147)

4.4  Metal-Insulator-Metal (MIM) Junctions

4.4.1  Electron tunnelling picture

From the results of section 4.3.2, the tunnel current in an MIM junction is given by the integral (147), inserting the appropriate densities of states for the two electrodes.

If |M| is fairly constant over the energy interval of interest, and the finite temperature effects are neglected, so f(E)-f(E+eV) is only non-zero for E between 0 and eV, an acceptable approximation if the energy resolution required is no better than kT, the tunnelling current is just proportional to the convolution of the densities of states in the two electrodes:

J
eV

0 
rl(E) rr(E+eV) dE
(148)
where E is the energy measured from the Fermi level of the left electrode.

In a normal metal, as a first approximation, the densities of states are constant at the Fermi energy, so we obtain the simple result that for MIM junctions at small bias (eV << EF),

I V
(149)
and it will be convenient to define a quantity GNN, the normal state tunnelling conductance, to that for small applied voltages,
I = GNNV.
(150)

With reference to fig 4.4, which shows electron tunnelling between two normal metallic electrodes, it can be seen that increasing the applied bias V by a small amount will slightly raise the energy of the left hand electrode with respect to the right, and bring a few more unoccupied states on the right opposite to occupied states on the left. This situation described in this diagram is exactly equivalent to a convolution of the two densities of states, as previously noted.

A more careful consideration of the symmetrical barrier case leads to a cubic correction term in the current, i.e.  

I = aV + bV3
(151)
with a conductance
s = a+ 3bV2.
(152)
The conductance has the form of a parabola with minimum conductance occurring at V = 0.

4.4.2  Numerical calculations for asymmetrical barriers

The trapezoidal potential modified by image charges (equation 120; see fig 4.3) was used by Brinkman, Dynes and Rowell (1970) to obtain a theoretical conductance curve for an MIM junction of thickness 20 Å, with f1 = 1 eV; f2 = 3 eV, which shows a marked asymmetry, the minimum conductance occurring at V = 40 mV. The effect of the image charges is to increase the overall transmission, slightly reduce the asymmetry of the conductance, and increase the rate of increase of conductance with voltage, but the authors concluded that the effect was relatively minor, and it has therefore been neglected by later researchers. The increased transmission due to the rounding of the barrier edges can be taken account of by means of a reduced effective barrier height.

4.4.3  Tunnelling spectroscopy

For spectroscopy, a free electron metal electrode should be used, so that the only DOS features affecting the tunnel current are those in the unknown electrode. In this case, again considering fig 4.4,

dI
dV
= rr(eV)
(153)
i.e.  the density of states of the unknown electrode is proportional to the junction conductance.

4.5  NIS Junctions

In considering NIS junctions, it is convenient to write the tunnel current in terms of the normal-state junction conductance,

s = (dJ/dV)S
(dJ/dV)N
.
(154)
We may rewrite the expression for the tunnel current given in equation 147, as:
I(V) = GNN


- 
[f(E)-f(E+eV)] rl(E) rr(E+eV) |M2| dE
(155)
where GNN is the normal-state junction conductance, as defined in equation 150.

In this expression, taking the right hand electrode as the superconducting one, rl(E) is the normal metallic density of states, and almost constant near EF, while rr(E) = NS(E), the superconducting density of states given by equation 87.

We can therefore write that rl(E) @ rl(0), taking the Fermi energy of the left electrode as zero, and the conductivity is given by:



dI
dV




S 
= GNN


- 


-
V
f(E+eV)

rl(0) NS(E) dE.
(156)
The corresponding normal conductance contains NN(E), which is taken as constant, so the value of s = (dI/dV)S/(dI/dV)N is:
s =


- 


-
V
f(E+eV)

nS(E) dE.
(157)
This is a convolution of the normalised superconducting density of states function nS(E) (88) with a thermal broadening function, -f(eV)/V:
- f(eV)
V
= e
kT
exp(eV/kT)
(1+exp(eV/kT))2
= e
4kT
sech2

eV
2kT


.
(158)
This is a bell-shaped function whose full width at half height is 3.52kT/e. In order to make theoretical predictions about tunnelling experiments at finite temperatures, it may be necessary to consider the temperature variation of the superconducting gap, D(T) (section 3.3.4), and the thermal broadening caused by this convolution function.

The NIS tunnelling process is represented in fig .


Figure 4.10: NIS tunnelling events at finite temperature.

This shows the origin of the convolution of the superconducting DOS with the thermal broadening function, owing to the finite temperature of the normal metal electrode. If D >> kT, there will be few excitations in the superconductor, as the minimum excitation energy is D.

At T = 0, the conductance of the NIS junction is zero for |V| < D/e. As the bias V is increased, the Fermi level of the metal is moved up in energy with respect to the superconductor, but there are no occupied states in the normal electrode opposite unoccupied states in the superconductor until V = D/e, when the differential conductance becomes very large. The current increases very rapidly at V = D/e, owing to the singularity in nS(E), and tends to linear behaviour for large V.

In order to determine the density of states in the superconducting electrode, it is best to use as low a temperature as possible, since the T = 0 limit gives a conductance curve which is simply the superconducting density of states. However, finite temperature experimental data may be deconvolved with the thermal smearing function to obtain the density of states. The superconducting energy gap is also reduced in size for T > 0, so the variation of the observed conductance curve with temperature depends upon both these factors. Theoretical conductance curves for various temperatures for an NIS junction whose superconducting electrode is lead (Tc = 7.2 K) are shown in fig , which takes 2D/kT = 4.67 for Pb.


Figure 4.11: Theoretical conductance curves for an NIS junction at various temperatures. S=Pb: Tc = 7.2 K, 2D/kT = 4.67. Curves calculated for T = 0 (dashed); T = 1 K, 3 K, 4.2 K, 6 K and 7 K.

4.5.1  Excitation description of tunnelling events

It is useful to consider tunnelling in terms of the creation of excitations in the two electrodes. This method is equally applicable to MIM or NIS (or indeed SIS) junctions, and instead of describing the tunnelling event as an electron traversing a barrier, it is considered as the creation of excitations in both electrodes; a hole in the left hand electrode and an electron in the right hand electrode (for tunnelling from left to right).

To clarify the situation, an MIM junction is considered, with applied bias V, where an electron is tunnelling across from an energy 0.3eV below EF on the left, gains an energy eV, and arrives on the right with energy 0.7eV.

The same situation may be considered as the creation of a hole excitation of energy 0.3 eV in the left electrode, and an electron excitation of energy 0.7 eV in the right electrode. The total energy eV required for the two excitations comes from the electron tunnelling across the barrier.

In general, tunnelling from a state of energy -E1 in the left electrode to a state of energy E2 in the right (each measured with respect to the Fermi energy of the corresponding electrode), may be considered as the creation of a hole excitation of energy E1 in the left electrode, and an electron excitation of energy E2 on the right. In either description, energy conservation demands E1 + E2 = eV. One advantage of the excitation description is that it can handle tunnelling in the reverse direction, simple by considering E1 to be negative. It is also very well suited to describing tunnelling where one or both electrodes are superconducting, and the quasiparticle excitations in the superconducting electrode(s) are of mixed electron/hole character.

For an electron tunnelling from a normal metal into a superconductor, the appropriate excitation diagram is shown in fig .


Figure 4.12: Tunnelling from a normal metal into a superconductor in the excitation description. The excitation in the superconductor may be created in either branch of the E-k diagram.

There are two possible final states for the electron arriving in the superconductor, because the two branches of the E-k diagram may both receive electrons, although for E1 >> D it is very likely to enter the electron-like branch. Tunnelling cannot occur for eV < D.

Combining the probabilities of tunnelling into a hole-like state and tunnelling into an electron-like state gives a factor of |uk|2+|vk|2 = 1 (uk and vk are defined in section 3.3.3), so the values of u and v need not be considered. The combination of these two channels is equivalent to considering the superconductor as an ordinary semiconductor with an energy gap 2D. This is known as the semiconductor model, and may be used provided that there is no interference between the two tunnelling channels. This approximation is not valid for point contacts, since Andreev reflection may occur. In this case, the two channels must be considered separately, e.g.  BTK theory for NcS contacts, which is considered in detail in section . The excitation description is most useful in those cases where more than one elastic tunnelling event is possible, since it provides a systematic way of dealing with each channel, and allows inclusion of processes involving pairs from the superconducting condensate.

4.5.2  Other factors influencing tunnelling data

  Lifetime effects

If the lifetime of the state into (or from) which an electron is tunnelling is short (e.g.  tunnelling into a superconductor with a finite quasiparticle recombination time), lifetime broadening will smear out the features observed (Dynes et al. , 1978). This conductance curve obtained in this case may be described by adding an imaginary part to the energy, replacing E by E-iG in equation 88 for the superconducting density of states:

s(V) Re |E-iG|
  


(E-iG)2-D2
 
(159)
where G represents the intrinsic lifetime broadening.

The effect on the theoretical conductance curve of using different values of G/D is shown in fig .


Figure 4.13: Lifetime effects for different values of G/D. The effect of G is to smear the curve, and reduce the conductance peaks at eV = D. The zero-bias conductance is always zero. Curves are shown for (with decreasing peak heights) G/D = 0.05, 0.2, 0.3. The BCS curve (dashed) is also shown.

  Phonon effects

Features appear in the I-V  characteristics of NIS junctions at energies corresponding to phonon energies (particularly in strong-coupling superconductors) (McMillan & Rowell, 1965, 1969; Eliashberg, 1960). These are manifested as step-like features in the conductance curve at well-defined voltages above the energy gap.

4.6  SIS Junctions

For completeness, the behaviour of SIS junctions, where both electrodes are superconducting (either the same or different superconductors) is considered.

At T = 0, the superconductors are in the ground state, with all the electrons paired, so single electron tunnelling may only occur by splitting up a Cooper pair in one of the electrodes, and placing one of the resulting single electrons in each electrode (fig ), hence requiring an energy D1+D2 to be supplied. This implies that, at T = 0, there is no current flow for an applied bias eV < D1+D2. At finite temperatures the tunnelling probability is non-zero owing to the presence of thermally excited electrons.


Figure 4.14: Excitation diagram for SIS tunnelling at T = 0. Single electron tunnelling can only occur with the splitting up of a Cooper pair from the condensate, after which one of the electron can cross the barrier.

4.6.1  Semiconductor model

As before, the semiconductor model may be used, in which tunnelling events into (or from) electron-like and hole-like states are combined, in order to eliminate the u and v factors in the tunnelling probabilities. The current-voltage characteristic of the tunnel junction is then dependent upon the densities of states of the two electrodes (Giaever, 1960b).

The density of states diagram is shown in fig .


Figure 4.15: Density of states tunnelling diagram for SIS junction at T = 0. At eV = D1+D2, a singularity of filled states on the left is opposite a singularity of empty states on the right, leading to a very sharp increase in the tunnel current as the voltage is increased past this level.

For the general situation of different superconductors (SIS), with energy gaps D1 and D2, a sharp jump in the tunnel current appears at eV = D1+D2 (or at 2D if the superconductors are identical), when the singularities of the density of states functions of the two electrodes are moved past each other. The singularity on the left consists almost entirely of full states, while the singularity on the right consists of empty states, leading to a very large increase in tunnel current at this bias.

At finite temperatures, there is also a feature at the difference energy, |D1-D2|, caused by tunnelling of thermally excited electrons or holes across the junction (fig ).


Figure 4.16: Tunnelling diagram for SIS junction at finite temperature. At finite temperatures, some electrons in the right hand electrode are excited above the gap, enabling electrons to tunnel from the left to the right, into the few empty states. At eV = |D1-D2| the two singularities are opposite, as shown, giving a sharp feature in the tunnelling characteristic, considerably smaller than the feature at D1+D2.

At this bias, singularities from each superconducting density of states are again opposite, but in this situation the states are either both almost all full or both almost all empty. The number of possible tunnelling events is considerably smaller, as is the resulting feature in the conductance curve. The tunnel current is non-zero for T > 0 because of the existence of some full states in the singularities above the Fermi energy, and some empty states in the singularities below. The sum and difference features may be used to calculate the energy gaps of the two electrodes rather accurately.

The I-V  characteristics obtained from an SIS junction can exhibit a negative differential conductance region between |D1-D2| and D1+D2 if the junction is at a finite temperature (for more detail on SIS and SIS I-V  characteristics see e.g.  Giaever, 1960b; Solymar, 1972, pp. 34-37). As the voltage is increased past |D1-D2|, the singularities in the two densities of states no longer match up, so the tunnelling probability decreases, until the full and empty singularities approach at V = D1+D2.

4.6.2  Multiple electron effects

Other effects can arise from simultaneous tunnelling of two or more electrons across the junction. Adkins (1963) observed a step in the conductance of Pb/Al2O3/Pb junctions at 1.2 K, occurring at a voltage D/e, where D is the energy gap in Pb. (Similar effects were also seen with Sn/Al2O3/Pb junctions.) As shown in section 4.6.1, the single electron process occurs at twice this voltage, since a voltage of 2D/e brings the full and empty singularities in the densities of states opposite. The two-electron processes which can occur at D/e involve either the breaking or the formation of a condensed pair, with energy conservation obeyed due to electrons tunnelling across the barrier appropriately. The feature occurs as a conductance step, since these events can occur at all voltages above D/e.

For completeness, it should be noted that Josephson tunnelling can also occur (see also Josephson, 1962; Waldram, 1976), in which a condensed pair tunnel across the barrier into the condensate of the other electrode, without breaking up. This process occurs at zero bias, and results in a tunnelling supercurrent (i.e.  current with no applied voltage): the dc Josephson effect.

Processes involving larger numbers of electrons are also possible (Adkins, 1964), but for an n-electron process (giving rise to a conductance step at V = (D1+D2)/n), the probability is proportional to |M|2n, where M is the matrix element for the transition, so higher-order processes become rapidly less likely. The observation of such processes to orders as high as 12 (Rowell, 1964), the independence on temperature and the dependence on the barrier suggest that the features are a result of Josephson radiation (Giaever & Zeller, 1970) rather than multiparticle tunnelling, although the latter may be possible for small values of n if the barrier is very thin in places. Solymar (1972, p. 55) suggests that, although multiparticle tunnelling is theoretically possible, it may never have been observed, and that all subharmonic structure, including n = 2, may be due to Josephson radiation processes.

4.7  Proximity Effect (NINS) Tunnelling

Electron tunnelling into a proximity effect structure (NINS) reveals information about the local electron density of states. This allows verification of the behaviour of the superconducting parameters with different thicknesses of the superconducting and normal metal layers. In particular, the superconducting gap parameter, D, is expected to decay exponentially into the normal metal (see section 3.6).

4.7.1  Experimental observation

Adkins & Kington (1966) performed tunnelling experiments with planar Al/Al2O3/Cu/Pb junctions, where the Pb was thick, and the Cu of variable thickness, to obtain conductance curves which showed the existence of a superconducting gap in the density of states of the Cu. The width and depth of the gap decreased with increasing Cu thickness, i.e.  the superconducting gap was found to decay into the Cu, with a decay length of approximately 400 Å (for thin films - for thicker films the decay was slower), similar to the estimated coherence length. They also found that the measured density of states for tunnelling into the Cu side of the Cu/Pb sandwich was enhanced at energies just above the gap energy, but suppressed at greater energies ( ~ 2D). This was explained by localization of low-energy excitations in the copper. Conversely, tunnelling into the Pb side of the sandwich resulted in a suppression of the states just above the gap energy. Phonon effects were observed at energies up to 10D.

4.7.2  Theory of NINS tunnelling

  The pair potential

In a superconductor, the pair potential D(r) is equal to the product of the interaction potential V, and the pair density F (see equation 111). D is the potential experienced by quasiparticles, as a result of the presence of the Cooper pairs.

As discussed in section 3.6, the pair potential varies smoothly in the region of an NS interface, if the normal metal has a positive interaction potential, V. This is because the pair density |F|2 (and hence F) cannot change on a scale smaller than x(T), the temperature-dependent coherence length (the coherence lengths in the superconductor, xS(T), and the normal metal, xN(T), are, in general, different). Since F and V are both positive in the normal region, so is D.

  Normal metal with V > 0

For tunnelling in an NINS structure where the normal metal has a positive V, the behaviour of the pair potential is similar to that shown in fig .


Figure 4.17: Pair potential over an NINS tunnelling structure, with V positive in the normal metal. F and V are both positive in the N layer, so D = FV is positive. An energy gap exists at the surface, of size Dsurface.

The pair potential is positive at the surface of the normal metal, so quasiparticles cannot be injected into the normal metal (of the NS bilayer) at energies below the value Dsurface at the surface of the metal. The density of states detected by tunnelling has an energy gap of size Dsurface, and the ideal conductance is zero for eV < Dsurface.

  Normal metal with V = 0 (or V < 0)

In the situation where the normal metal has a zero (or negative) interaction potential V (e.g.  V = 0 for Au, Ag), the pair amplitude F still decays into the normal metal, but the pair potential D = FV is zero (or negative) (see fig ). Hence, quasiparticles of all energies are allowed in the normal metal, and there is no strict energy gap (i.e.  no energies for which the density of states is zero).


Figure 4.18: Pair potential and pair amplitude (dashed) over an NINS tunnelling structure, with V = 0 in the normal metal (e.g.  Au). F is positive in the N layer, but V = 0, so D = FV-0. No energy gap exists in the normal metal, but effects may be detected by tunnelling in the ballistic limit, since electrons entering the normal metal from the superconductor are all of energy E > D, and peaked at E = D.

However, although there is no completely forbidden energy range, conductance curves obtained from tunnelling in this structure will have a peak corresponding to eV D, because, provided electrons cross the normal metal ballistically, the peak at E = D in the superconductor layer will result in more electrons with that energy travelling into the normal metal layer. Similarly, there will be fewer electrons of energy E < D, since no electrons travelling into the normal metal from the superconductor will have energies below D.

  Summary of expected results from NS tunnelling

To summarise, energy gap structure may be detected in the V = 0 metal, if some ballistic transport occurs, not because the states in the normal metal are forbidden, but simply because fewer electrons of this energy, and more electrons of energy E D, enter the normal layer from the superconductor.

If the normal metal is too thick or too dirty, the effect will be completely smeared out by scattering, it is important to use normal metal films which are thin compared to their mean free path if the effect is to be observed.

Adkins & Kington (1966) observed an energy gap (with non-zero conductance at zero bias) reduced in width compared to that in Pb, tunnelling into the Cu side of a Cu/Pb bilayer. The conductance curve obtained using a V = 0 normal metal layer is not expected to have such narrow features, suggesting that, for Cu, V > 0, although superconductivity in Cu has never been experimentally observed (Deutscher & de Gennes, 1969, p. 1019).

4.7.3  Other processes at NS interfaces

An important process which can occur at the NS interface is Andreev reflection (Andreev, 1964), whereby an electron incident on the N side of the boundary causes transmission of an electron pair into the superconductor, with reflection of a hole, back along the original path of the electron. This occurs with probability P, where

P =

E-(E2-D2)1/2
E+(E2-D2)1/2


(160)
which is equal to 1 for E < D and falls as D2/4E2 for E >> D (Wolf, 1985, p. 190).

The consequences of the possibility of Andreev reflection in NS point contacts are discussed in section .

De Gennes & Saint-James (1963) showed that the process of Andreev reflection implied the existence of at least one quasiparticle bound state, of energy no more than D, confirmed experimentally by Rowell (1973) by tunnelling in an SINS structure, where Pb was used for both superconductors, and Zn (2 mm and 3.5 mm) for the N-metal.

Wolf et al.  (1980a,b; Arnold et al. , 1980) coated Nb with an extremely thin layer of Al ( 100 Å) to enable the growth of a reliable oxide layer for tunnelling experiments to determine the phonon spectrum. A similar approach was used to obtain the phonon spectrum for vanadium (Zasadzinski et al. , 1982). In these experiments, where the normal metal is thin compared to the coherence length in the superconductor, the Ginzburg-Landau theory cannot be applied. Arnold (1978) assumed spatially constant pair-potentials in the N and S metals, and neglected scattering at the NS boundary, to calculate the N-metal surface tunnelling density of states, which is highly structured, and dependent on the electron-phonon coupling in the N layer. He also showed that a very narrow gap in the density of states occurred at E = D at the surface of the normal metal.

Arnold & Wolf (1982) showed that the sampling depth into the superconductor can be enhanced by the presence of a normal Al layer, by as much as a factor of 6 (for Nb), owing to the small Fermi velocity in the transition metals.

Chapter 5
Theory of Point-Contact Spectroscopy

In point-contact spectroscopy, two electrodes are brought to mechanical contact at a point, greatly reducing the resistance of the junction. The geometry may be in the form of two crossed edges, a pinhole in an oxide barrier, or a `spear and anvil' junction (a sharp point on a planar surface). The barrier between the two electrodes is replaced by a microconstriction, which complicates the transport process, by allowing retroreflection of electrons back through the contact by phonon scattering. The behaviour of such contacts depends on the relative size of the orifice and the electron mean free path; these contacts have been investigated in detail by previous authors, since they are a valuable tool, e.g.  for phonon spectroscopy (section ).

A `spear and anvil' junction can be varied continuously between tunnelling and contact by adjusting the tip/sample separation (see section ). The physics of point-contact (NcN and NcS) junctions is considered in this chapter.

For NcS junctions (i.e.  superconducting and normal-state electrodes separated by a constriction), intermediate cases between tunnelling and contact have been treated by means of a barrier of adjustable height separating the two electrodes (BTK (Blonder, Tinkham & Klapwijk, 1982) theory: see section ). A barrier strength parameter, Z, determines the behaviour of the junction as its character changes between the two extremes.

5.1  NcN Point Contacts

Reviews of theoretical and experimental work on normal state point-contact junctions up to 1989 are to be found in Duif et al.  (1989) and Jansen et al.  (1980). Sharvin (1965) considered electron transport across a point contact or microconstriction between two bulk conductors, and realised that the behaviour of the junction depended on the relative sizes of the electron mean free path l, and the contact dimension r. The contact was modelled as a circular orifice of radius r.

5.1.1  Sharvin results

If l >> r (the ballistic regime) the junction has characteristics in common with a tunnel junction, since there is no scattering close to the orifice, and the electron is simply injected from one side of the junction into the other, gaining energy eV.

The net current across the junction is simply:

I = neAdv
(161)
where dv is the increase in velocity for an electron at the Fermi energy moving across the barrier and A is the contact area.

Since the gain in kinetic energy dE is given by

eV = dE = mvF dv
(162)
for an electron on the Fermi surface, we have
dv = eV
mvF
(163)
and hence,
R = V
I
= mvF
ne2A
.
(164)
Substituting into this equation the formula for resistivity, r = mvF/ne2l, and using A = 3pr2/4 as the effective contact area, we find (Sharvin, 1965):
RS = 4rl
3 pr2
.
(165)

If l << r (the Maxwell, thermal, or dirty regime) almost all the energy of the electron is lost in the region of the contact, heating the contact region. Maxwell (1904) calculated this resistance RM by solving Poisson's equation, to find:

RM = r
2r
.
(166)

In the Maxwell regime, collisions cause the electrons to lose the energy gained by traversing the microconstriction within a small distance of the contact region, causing local heating. This is unrevealing for spectroscopy.

If the elastic scattering length le << r, but the diffusion length L = (lile)1/2 >> r, (the diffusive regime), where le and li are respectively the elastic and inelastic scattering lengths, elastic scattering occurs within the contact region, altering the electrons' momenta but not their energies. Energy is only lost in inelastic processes, which occur some distance from the contact. This situation may therefore provide useful spectroscopic information.

  Interpolation between ballistic and thermal (Maxwell) regimes

A ballistic point-contact junction is very similar to a tunnel junction, since the methods for calculating tunnel current may be modified to calculate the current across such a junction. It is shown in section  that, in an STM junction, the distinction between ballistic transport and tunnelling no longer exists when the tip and sample are sufficiently close, and both processes may be described using a general theory of quantum transport.

For the intermediate regime between ballistic and thermal transport, an interpolation between the limiting cases may be used (Wexler, 1966):

R = 4rl
3pr2
+ G(K) r
2r
= 4rl
3pr2


1 + 3p
8
G(K) r
l


.
(167)

5.1.2  Phonon spectroscopy in the ballistic regime

Experiments on point-contact junctions show a clear relationship between the measured d2V/dI2 and the phonon density of states F(e) as measured by neutron scattering experiments. The explanation of this is that the current across the junction is reduced by scattering of some electrons back across the junction (scattering also prevents some electrons from traversing the contact at all). By considering the current reduction due to scattering processes occurring within an effective volume Weff, which is approximately a sphere of radius r about the centre of the contact, an expression may be obtained for the differential conductivity:

dI
dV
= 1
RS
- 2
3
e2 r3 N(0) 1
t(eV)
(168)
where N(0) is the Fermi level density of states, and t(eV) is the energy-dependent scattering time for electrons with energy eV above the Fermi level. The inverse scattering time is related to a spectral function S(e) (which is an integral over initial and final states for the interaction combined with an angular efficiency function) by:
1
t(eV)
= 2p
(h/2p)

eV

0 
S(e) de.
(169)
The second derivative d2I/dV2 is then directly proportional to S(e):
d2I
dV2
= - 2pe3
(h/2p)
WeffN(0)S(eV).
(170)
If the scattering process is an electron-phonon interaction, this spectral function is the Eliashberg function a2 F(e) modified by the angular efficiency function mentioned above.

In measurements of d2I/dV2 for point-contact junctions, it is also found that there is a background signal, slowly-varying with e, and given by:

B(eV) = K
eV

0 
a2 F(e)
e
de
(171)
which may be subtracted out to determine a2 F(e) from experimental data.

  Diffusive regime versus ballistic regime

Lysykh et al.  (1980) added Ni to one electrode of a Cu-Cu junction to change the behaviour towards that of the diffusive regime. The features in the second derivative curves become very much less distinct: the peaks in the signal are (a) broadened, and (b) the amplitude reduced. These phenomena may be explained respectively by (a) the less stringent momentum conservation condition, owing to the elastic scattering, and (b) the reduction in the effective volume over which scattering occurs, owing to the elastic scattering occurring within a distance le of the contact, resulting in a disc-shaped scattering region of width le and radius r, rather than a spherical one of radius r, hence reducing the current derivative by a factor le/r.

5.1.3  Experimental results

Yanson (1974) showed that d2I/dV2 was proportional to the Eliashberg function a2F(w) (the product of the phonon density of states, F(w) at frequency w with the scattering matrix element squared, a2) for high-resistance junctions with strong-coupling metals (Pb, Sn), using a oxide layer pinhole contact. Jansen et al.  (1977) used a `spear and anvil' contact, which allowed the junction resistance to be altered by changing the applied pressure, to obtain the phonon spectra of Cu, Ag and Au, which were in general agreement with the results from inelastic neutron scattering, and from pinhole junctions formed by electric breakdown of an oxide barrier (Yanson & Shalov, 1976). The two methods have also been used to obtain phonon spectra for In, Zn, Cd, Fe, Co, Ni, K, Na and Li (Jansen et al. , 1980, p. 6101), and scattering by other processes, such as magnetic impurities (Jansen et al. , 1980, p. 6112).

For investigations of NcS point contacts, a third type of junction, formed between two crossed wedges, has been successfully used (e.g.  Reinertson et al. , 1992), and has the advantage of higher mechanical stability.

  Transverse electron focussing

If two point contacts are made close together on the surface, application of a magnetic field allows electrons emitted from one contact to be focussed into the other. The resulting voltage signal at the second contact, as a function of the voltage at the first contact, gives information on the energy distribution of the electron-phonon interaction strength for specific Fermi surface orbits, and also the anisotropy of the interaction (van Son et al. , 1987). This method has been used to obtain surface reflectivities for electrons incident from below the surface (Benistant et al. , 1986).

5.2  NcS Point Contacts and Andreev Reflection

As previously mentioned, there are similarities between tunnelling and ballistic transport across a point contact. However, the behaviour of an NcS point contact allows a wider range of behaviour than a corresponding NIS tunnel junction, owing to the possibility of Andreev reflection, in which an electron approaching an NS interface from the normal side is reflected as a hole, with a pair of electrons injected into the superconductor.

5.2.1  The BTK model

Blonder, Tinkham & Klapwijk (BTK, 1982; see also García et al. , 1988) considered the case of a one-dimensional NS interface, with the two regions separated by a delta-function barrier of variable height H. They used the Bogoliubov operators (see section 89), suitably modified to conserve particle number by the addition of S* and S operators, which respectively add or remove a pair of electrons to or from the superconducting condensate, i.e.

g*ek0 = uk c*k - vkS* c-k
(172)
g*hk0 = uk S c*k - vk c-k = S g*ek0
(173)
and similarly for the remaining operators.

They considered all the transitions resulting in a net single electronic charge transfer across the interface, or equivalently, the possible transitions which could occur when an electron was incident on the barrier from the normal side. These are: normal reflection of the electron (B), Andreev reflection, in which a hole is reflected, and a Cooper pair transmitted (A), normal (non-branch-crossing) transmission of the electron (C), and branch-crossing transmission of the electron into a hole-like excitation, with creation of a Cooper pair (D). The probabilities A,B,C and D are functions of the initial electron energy E, measured with respect to the Fermi surface.

The barrier was modelled by a d-function, Hd(x), and the model was simplified by ignoring proximity effects (i.e.  the energy gap rises sharply to its full value, on a distance scale smaller than the coherence length x). The calculation was performed for T = 0.

A dimensionless barrier strength Z was defined as:

Z = kFH
2eF
= H
(h/2p) vF
.
(174)
BTK solved the Bogoliubov equations,
i(h/2p) f
t
=

- (h/2p)22
2m
-m(x)+V(x)

f(x,t) + D(x)g(x,t)
(175)
and
i(h/2p) g
t
= -

- (h/2p)22
2m
-m(x)+V(x)

g(x,t) +D(x)f(x,t)
(176)
to obtain energy-dependent values for A(E), B(E), C(E) and D(E).

The Bogoliubov equations are essentially Shrödinger equations for the electrons and holes, modified by the additional term containing D(x), which describes a coupling together of electrons and holes. This leads to the formation of quasiparticle excitations of mixed electron/hole character, which have a minimum energy D, the energy gap of standard BCS theory.

5.2.2  BTK results

The results of the BTK calculations for the A,B,C and D factors are given in table  (after Blonder et al. , 1982). Note that in the NcN case the ordinary transmission and reflection coefficients, C and B respectively, are related to Z by the familiar transmission and reflection formulae. Note also that, for energies within the gap, C = D = 0 always, so A+B+C+D = 1 implies that B = 1-A.

A B C D
NcN junction 0 [(Z2)/( 1+Z2)] [1/( 1+Z2)] 0
General form
E < D [(D2)/( E2+(D2-E2)(1+2Z2)2)] 1-A 0 0
E > D [(u02v02)/( g2)] [((u02-v02)2Z2(1+Z2))/( g2)] [(u02(u02-v02)(1+Z2))/( g2)] [(v02(u02-v02)Z2)/( g2)]
No barrier
E < D 1 0 0 0
E > D [(v02)/( u02)] 0 1-A 0
Strong barrier
E < D [(D2)/( 4Z2(D2-E2))] 1-A 0 0
E > D [(u02v02)/( Z4(u02-v02)2)] 1-[1/( Z2(u02-v02))] [(u02)/( Z2(u02-v02))] [(v02)/( Z2(u02-v02))]

Table 5.1: BTK Transmission and Reflection Coefficients. A is probability of Andreev reflection; B of ordinary reflection; C of non-branch-crossing transmission; and D of branch-crossing transmission. The `no barrier' case corresponds to Z = 0, and the `strong barrier' case corresponds to Z2(u2-v2) >> 1. g2 = [u02+Z2(u02-v02)], and u02 = 1-v02 = 1/2{1+[(E2-D2)/E2]1/2}.

The process of Andreev reflection causes a charge 2e to flow across the junction, while normal reflection of an electron causes no charge to flow across. Increasing the junction bias from V to V+dV allows electrons of energy eV to e(V+dV) to traverse the junction, as described in section 4.2, so the differential conductance dI/dV is proportional to 2A+C+D, or equivalently, 1+A-B. This quantity tends to 1/(1+Z2) at high bias, so the normalised junction conductance GNS/GNN is given by:

GNS
GNN
= RN

dI
dV




S 
= (1+A+B)(1+Z2)
(177)
where RN is the normal-state (or high-bias) junction resistance, and is related to the Sharvin resistance RS for a perfect contact by RN = (1+Z2)RS.

  Theoretical conductance curves

For large values of Z, this function approaches the BCS function for the density of states, nS(E) (equation 88), as measured in tunnelling experiments (section 4.5). It can be seen that this approach describes both classical tunnel junctions (Z = ) and perfect contacts (Z = 0), along with all intermediate junctions, and that the predicted transmission and reflection coefficients, and junction current, evolve smoothly as a function of Z. Examples of the theoretical conductance curves for several values of Z are shown in fig .


Figure 5.1: Theoretical conductance curves for several values of barrier strength, Z. RN dI/dV is the ratio of junction conductance to normal-state conductance.

The theoretical conductance curves may be integrated numerically to obtain theoretical I-V  relationships (fig ). These all become linear for high V, but, unlike the tunnelling case (Z = ), the asymptotes for finite Z do not pass through the origin.


Figure 5.2: Theoretical I-V  curves for several values of barrier strength, Z.

  Determination of barrier strength Z from experimental data

Writing Iexc as the excess current, Iexc = INS(V)-INN(V), the vertical displacement of the asymptotes is equal to limV Iexc, which is a function of Z increasing from 0 to 4/3 as Z decreases from Z = (tunnelling) to Z = 0 (perfect contact). This quantity is useful, because it can be measured from experimental data, allowing the corresponding value of Z to be found.

The zero-voltage normalised conductance GS(0)/GN(0) is also a useful quantity, which rises from 0 to 2 as Z decreases from Z = to Z = 0. The values of Z deduced in these two ways may be compared to check the exactness of the fit to measured data.

Alternatively, one can determine the theoretical curve which best fits the experimental data, giving the most likely value of Z for the junction concerned.

For datasets taken over finite temperature ranges which are not small compared to Tc, Smith et al.  (1993) describe methods for obtaining Z using theoretical graphs of normalised zero-bias conductance RN(dI/dV)V = 0 against reduced temperature t = T/Tc.

  Meaning of the barrier-strength parameter Z

The theoretical model above defines Z according to the height of the d-function barrier between the two electrodes. However, Z is also related to the junction resistance, since the normal-state or high-bias resistance of the junction is a factor 1+Z2 higher than the resistance of a pure contact (which is the reason for the appearance of this factor in equation 177). Z may therefore be considered as an experimental parameter describing the effect of several scattering mechanisms in the junction.

The effect of a Fermi velocity mismatch between the two electrodes is to cause some (extra) normal reflection to occur, which increases the effective value of Z. This effect is exactly quantifiable, and the effective value, Zeff is given by (Blonder & Tinkham, 1983):

Zeff =

Z2 + (1-r)2
4r


1/2

 
(178)
where r is the ratio of Fermi velocities, r = v1/v2.

Experiments by Hass et al.  (1992) utilised this phenomenon for determination of the Fermi velocity in YBCO by point-contact spectroscopy. They obtained Zeff 0.3 and hence r 2, giving an in-plane Fermi velocity for YBCO of at least 7×105 m/s. This is five times higher than the expected value, which the authors explain by the effect of a singularity in the electron velocity at kF, and the van Hove singularity, on the 2D electron gas in the Cu-O planes. The authors also obtained values for the gap of 18-20 mV (2D/kTc=4.5-5) for the in-plane direction, and subgap structure at 12 mV corresponding to 2D/kTc=3 for the across-plane direction, in agreement with measurements by other groups.

Blonder & Tinkham (1983) found that the fit of experimental data from Cu-Nb point contacts (using a sharpened Nb tip and polished Cu surface) to the theoretical I-V  curves is very good for junctions resistance R > W, but that for R < W the characteristics are distorted by junction heating effects.

  Finite temperature effects

The above results are easily generalised to finite temperature (noting that D, and hence A, B, C and D are temperature-dependent), by multiplying by the appropriate Fermi occupation factors and integrating:

I = (1+Z2)
eRN



 
[1+A(E)+B(E)][f(E-eV)-f(E)]dE.
(179)
For T << Tc and for small voltages, V << kT/e, the difference of Fermi functions may be replaced by a derivative:
Y(Z,T) = RN

dI
dV


= (1+Z2)


- 


- f
E


[1-A(E)-B(E)]dE.
(180)

  Experimental observations of Andreev reflection

Andreev reflection has been observed in electron-focussing experiments using double point contacts by Bozhko et al.  (1982) and Benistant et al.  (1983). Benistant et al.  (1985) measured Andreev reflection between an Ag single crystal (200 mm thick, compared to the 700 mm mean free path) and Pb film, by injecting electrons from a point contact. The electrons spread out from the point, and retroreflected (Andreev reflected) holes can re-enter the point contact, enhancing the current. Normally reflected electrons are not retroreflected, so dissipate. An enhanced current was detected, but by a much smaller amount than would be expected assuming perfect retroreflection. This was explained by considering that Andreev reflection occurred over a range of angles, within a cone of angle 10-4 rad about the incoming direction, owing either to non-ideal retroreflection, or scattering by dislocations, the required density being 106-107 cm-2.

Van Son et al.  (1988a) performed similar experiments with a thinner (20 mm) Ag slab backed by 0.8 mm Pb, and estimate that a dislocation density of10 107-108 cm-2 is sufficient to cause scattering over this range of angles. For small applied magnetic fields, the deviations from the ballistic model are reduced, since the deflections of the electron and hole orbits by the scattering can cancel out. This lends support to the theory that scattering by dislocations is the cause of the range of angles of the retroreflected electrons.

  Experimental fit to BTK model

Reinertson et al.  (1992) measured conductance curves of Ag/Nb and Ag/Nd1.85Ce0.15CuO4-d point contacts at several temperatures. The point contacts were formed by two crossed wedges. They found results for the Ag/Nb junction in good agreement with the BTK predictions, with barrier strengths between 0.3 and 0.8 (methods for determination of Z are discussed in Smith et al. , 1993). For the Ag/Nd1.85Ce0.15CuO4-d junction, barrier strengths ranged between 0.8 and 1.0, owing to Fermi velocity and lattice mismatches. The fit to the BTK model for behaviour of the zero-bias resistance was good, but the observed conductance dip width was constant with temperature, contrary to the BTK predictions, but in agreement with results obtained by Ekino & Akimitsu (1989).

5.2.3  Extension of the BTK model to include the proximity effect

Van Son et al.  (1988b) extended the work of Blonder et al.  (1982) to NS interfaces where the pair potential varies gradually (Blonder et al.  considered the case where the pair potential is a step function) using a numerical solution of the Bogoliubov equations (equations 175 and 176). As with the Blonder et al.  (1982) calculation, a variable-height d-function repulsive potential separates the layers. The pair potential is depressed in the superconductor near the interface, and positive in the normal metal near the interface, owing to the proximity effect (see section 3.6), which Blonder et al.  ignored.

Rather than evaluating the pair potential D(x) self-consistently, various forms of D(x) were assumed. If D(x) is slowly-varying compared with the coherence length x0, the BTK results are reproduced. The form of D(x) used was parabolic, with zero slope at x = -xN and x = xS (where x = 0 at the interface), reaching values of D- and D+ on the N and S sides of the d-function barrier respectively (see fig ).


Figure 5.3: Pair potential used by van Son for calculation of Andreev reflection probability.

The exact choice of potential did not greatly affect the results - the important characteristics were the values of D- and D+ and the scale over which D(x) changed. Taking the Z = 0.3 case, the conductance curves obtained for trial values of these parameters were similar to the Blonder et al.  results, but in all cases the Andreev reflection probability was increased for E < D (the bulk gap) and decreased for E > D. The sharp peak at E = D was rounded off, and the maximum (still with near-unity probability of Andreev reflection) shifted to slightly lower E if D+ < D.

Van Son et al.  (1988b) also considered the situation of tunnelling into the normal side of an NS bilayer exhibiting the proximity effect, by shifting the position of the d-function barrier from x = 0 to x = -xT < -xN (see fig 5.3). The junction was taken to be a pure tunnel junction, with Z infinite, and the barrier at x = 0 omitted for simplicity. They calculated the behaviour of the normalised transmission coefficient (normalised by dividing by the normal-state transmission factor, 1/(1+Z2)) of the whole structure with energy for various values of D- and D+, and found that oscillations as a function of energy occurred above D, with an amplitude proportional to the gap discontinuity at the NS boundary. There were also two peaks in the transmission for E < D, whose position depended upon the values D- and D+, but were near E = 0.3D and E = 0.9D.

Considering the differential conductance of the junction as measuring the excitation density of states in the N layer, for energies less than D, the Andreev reflection probability is 1, and bound states in the normal layer exist. For energies above D, maxima in the density of states result from `quasibound states'.

Van Son et al.  (1988b) also considered the effect of a negative pair potential in the N-metal, and showed that the discontinuity of D (not |D|) at the boundary is the important quantity in determining the general behaviour.

Chapter 6
Scanning Tunnelling Microscopy and Spectroscopy

6.1  Introduction to the Scanning Tunnelling Microscope

The scanning tunnelling microscope (STM) was developed by Binnig & Rohrer (1982). STMs utilise the phenomenon of tunnelling by measuring the tunnel current caused by electrons tunnelling between a very fine tip and a conducting sample, separated by a vacuum gap. (STMs are often operated in an atmosphere of air, or, as in the present study, helium gas, and have also been used in liquids such as corrosive solutions (van de Leemput & van Kempen, 1992).) If the width of the gap is sufficiently small, enough electrons tunnel across the gap for a measurable current to flow.

Since the wavefunction in the vacuum decays with distance on a scale comparable with the size of an atom, the tip-sample distance must be only a few atomic diameters for a measurable tunnel current to flow. A change in the tip/sample distance of approximately 0.1 nm can cause an order of magnitude change in the tunnel current, so sub-Ångstrom resolution is theoretically possible. Hence, the STM is sensitive to atomic-scale variations in the surface (exactly which feature of the surface is being imaged will be discussed in section ).

If a feedback loop is used to adjust the relative position of the tip and sample in such a way as to keep the tunnel current constant as the tip is moved across the surface (the usual mode of operation of an STM), the tip must follow the contours of the surface very accurately indeed. A computer is used to form an image in this way, by correlating the motion of the tip towards and away from the sample with its lateral motion across the sample.

An STM requires the ability to hold the tip/sample junction stable to within 0.1 Å or better, and to detect and measure tunnel currents in the range 0.01-10 nA. Scanning tunnelling spectroscopy requires even greater stability and sensitivity (section ).

An STM tip/sample junction may be treated similarly to a planar tunnel junction for the purposes of theoretical analysis, although there are several important differences: the most important of these are the localised nature of the junction; the reduced barrier width (and the effect of this on the overall shape of the barrier); and the three-dimensional nature of the geometry.

It is instructive to compare the parameters of a typical STM junction with those of a planar tunnel junction, and this comparison is shown in table . A planar junction is usually formed by natural growth of an aluminium oxide layer on the bottom (Al) electrode, followed by deposition of the second electrode. The barrier height is a function of the energy gap in the insulating oxide. Current-voltage characteristics are an average over the area of the junction. In scanning tunnelling microscopy, the barrier is usually a vacuum or air gap, with a barrier height equal to the work function of the electrodes, reduced by the image charge potential (which can make the barrier height very low, or even negative, for small tip/sample separations).

Planar junction STM junction
Barrier width/nm 2-3 0.4-0.7
Barrier height/eV 1 -2-5
Junction resistance/W 10-1000 107-1010
Junction current/A 10-5-10-3 10-10-10-8
Junction area/mm2 1-10 10-14-10-13
Current density/Amm-2 10-5-10-3 103-106

Table 6.1: Comparison of planar and STM tunnel junction parameters. The barrier height can be negative in STM.

An STM junction has a much thinner, higher barrier than a planar junction; one of the effects of this is an increased range of tunnelling angles in scanning tunnelling microscopy (see section 3.1.2). The angle at which the tunnelling probability has dropped by a factor e compared to tunnelling straight across is about 24 in scanning tunnelling microscopy, compared to about 7 in a planar oxide junction.

6.1.1  Scanning tunnelling spectroscopy (STS)

In scanning tunnelling spectroscopy, the variation in the electron tunnelling current with the applied voltage across the junction (I-V  characteristic) is measured. This gives information on the probability of electron tunnelling, and hence the sample's electronic density of states, as a function of the energy range of the electrons involved in the process (described by the model in section ). For many purposes, the conductance curve obtained by differentiation of the I-V  characteristic is proportional to the sample's density of states.

Planar tunnel junctions have been used for many years to investigate the electron density of states of various systems, but it is necessary to form the material to be investigated into a planar electrode. By the nature of this process, any measurements obtained are an average over the whole of the electrode's surface. In contrast, with scanning tunnelling microscopy and spectroscopy the electrodes are in the form of a sharp tip and a sample (often, but not necessarily, planar), separated by a vacuum gap, typically of order 1 nm. The only requirement is that the sample is a conductor. The tip/sample separation is usually held constant for topographic and spectroscopic measurements.

This geometry has the advantage that the sample to be measured does not need to be formed into a planar electrode. In addition, scanning tunnelling spectroscopy is the only tool which can measure the local electron density of states of the surface of a material to a resolution of a few Ångstroms. However, the necessary stability of the junction, and the size of the variations in tunnel current which must be measured, pose even more stringent requirements than those for scanning tunnelling microscopy, necessitating an elaborate vibration-damping system, and an extremely sensitive, well-shielded current amplifier. To detect small changes in a current of order 0.1-10 nA requires a current sensitivity of the order of pA. Since the tunnel current is extremely sensitive to the tip/sample separation (a movement of 0.1 nm can change the current by an order of magnitude), for the mechanically-induced current variations to be less than 5% of the total current, the junction must be held stable to an accuracy of about 5×10-12 m or better.

6.2  STM Tunnel Current Calculations

The transfer Hamiltonian, or Bardeen integral method for calculating the tunnel current across a (generally non-planar) barrier has been described in section 4.3.2, in which the matrix element describing transitions between the two electrodes is given by the Bardeen integral, (145), and the tunnel current may be calculated using Fermi's Golden Rule, (146).

The overlap integral of the two wavefunctions is taken over a surface which divides the two systems, the exact choice of surface not being critical.

This approach is satisfactory for cases where the electrodes are sufficiently distant, but in scanning tunnelling microscopy, the tip/sample separation distance may be as little as 0.4 nm, in which case the Bardeen integral calculation is no longer sufficient, and the nature of the tip/sample interaction must be considered more carefully.

6.2.1  Treatment of the barrier in STM

Simultaneous measurements of the atomic force gradient between the tip and the sample, and the conductance of the STM junction, with varying tip/sample distance have shown (Dürig et al. , 1988) that the point of equilibrium (i.e.  zero force gradient) corresponds to a junction resistance of 105 W. The tunnelling conductance increased by about a factor of ten per 0.1 nm, so junction resistances in the range 1 MW to 1 GW, typical for scanning tunnelling microscopy, correspond to distances between 0.1 nm to 0.4 nm from the equilibrium point (mechanical contact).

Dürig et al. also integrated the measured force gradient to find that the tip/sample interaction force increased from 0.1 nN to 1 nN in a smooth near-exponential fashion, as the gap width was reduced from 4 Å to 1 Å. For these experiments, an iridium tip and polycrystalline iridium sample were used.

The point of equilibrium is roughly equal to the atomic spacing in the sample, typically 0.3 nm, corresponding to tip/sample distances in scanning tunnelling microscopy of 0.4-0.7 nm, as previously quoted.

Since in scanning tunnelling microscopy the barrier is very thin, the image potential caused by the two electrodes is important in determining the barrier height. Simmons (1963) considered the potential between two electrodes as the sum of the trapezoidal electrostatic potential with a hyperbolic image potential. However, the classical image potential diverges at the barrier edges in this model, while the proper, quantum-mechanical expression for the image potential (Appelbaum & Hamann, 1972) joins smoothly with the local exchange and correlation potential of the electrodes (Inkson, 1971; Payne & Inkson, 1985). Also, the image planes are not precisely at the surface of the electrodes, but slightly outside them if the charge is a few Å or more from the electrodes (Appelbaum & Hamann, 1972; Payne & Inkson, 1985). For a charge very close to a conductor, the image plane moves back inside the surface of the electrode. A typical value for the distance from the electrode surface (using the jellium model) to the image planes is 0.075 nm, so that the image plane spacing is 0.15 nm less than the electrode spacing (Binnig et al. , 1984; Wiesendanger, 1994, p. 132).

6.2.2  Apparent barrier height

According to the WKB approximation (section 4.3) the tunnel current I varies with barrier width W as:

I exp(-2kW)
(181)
where the decay constant k is given by:
k =   


2mf
(h/2p)2
 
.
(182)
Here, f is the barrier height, measured with respect to the Fermi level of the left electrode. In the case of a wide vacuum gap this would just be equal to the work function of the metal, but it is reduced by the image charge potential. These equations may be used to define an apparent barrier height fapp, by considering the rate at which the tunnel current increases with decreasing barrier width. From equation 181, lnI/W = -2k, and hence, from equation 182, an expression for fapp may be obtained:
fapp = (h/2p)2 k2
2m
= (h/2p)2
8m


lnI
W


2

 
.
(183)
This forms the basis of the experimental technique for measuring the barrier height: measurement of the squared logarithmic derivative of the current with gap separation give the apparent barrier height, and by plotting this function against gap separation, the variation of the apparent barrier height with tip/sample spacing may be found. A reduction of the current by one order of magnitude per Ångstrom corresponds to an apparent barrier height of 5 eV, roughly the work function of gold. Measurements of local variations in the apparent barrier height may be used to obtain surface images with atomic resolution on metal surfaces (e.g.  Marchon et al. , 1988).

From the results of Simmons (1963), the apparent barrier height measured in this way might be expected to decrease as the tip/sample separation was reduced, owing to the reduction in the barrier height due to the image potential. However, both experimental results (Teague, 1978; Schuster et al. , 1992) and theoretical calculations (Teague, 1978; Binnig et al. , 1984; Coombs et al. , 1988, Payne & Inkson, 1985) show that this is not the case, and that the apparent barrier height is constant for tip/sample spacings almost down to mechanical contact. The apparent barrier height falls rapidly towards zero only at spacings below about 0.4 nm (Lang, 1987b, 1988; Pitarke et al. , 1989).

Schuster et al. (1992) measured I and dI/dz for smooth Cu(110) surfaces, and found that d(lnI)/dz was, to the experimental accuracy, constant with varying current (and hence varying gap width). The image potential did not cause the measured barrier to collapse11.

This observed behaviour may be explained by the fact that, as the tip/sample separation decreases, the decreasing barrier height causes the tunnel current to increase more quickly than would otherwise be expected. The apparent barrier height is therefore higher than the actual barrier height.

  Theoretical calculations of apparent barrier height

Binnig et al. (1984) calculated the tunnel barrier potential for a W-Au junction from the electrostatic potential calculated from Poisson's equation, using the electron density profile of Smith (1969), and an approximate expression for the image potential:

Vim(z) @ e2
d


(ln2 -1)+

z
d


2

 
+ 1
1-4(z/d)2


(184)
where d, the distance between the two image planes, is given by d = s-1.5 Å, s being the interelectrode distance12, and z is the distance measured from the centre of the barrier.

The barrier potential was modelled by fitting equation 184 to the calculated potential at the centre of the barrier (electrostatic + image potentials) and at the edges of the barrier (taken as equal to the bulk exchange and correlation potential13). They approximated the resulting potential by a rectangular barrier. For this barrier, it was found that the term in the apparent barrier height which was first order in 1/d disappeared. Graphs of the theoretical apparent barrier height with barrier width were linear, indicating that the apparent barrier height is constant as the tip/sample separation is reduced.

Coombs et al.  (1988) used the more accurate image potential approximation of Simmons (1963), also using a mean barrier height approximation, to show that the first-order term again disappeared, and that the effective barrier height was almost constant, and equal to the work function of the electrode, changing by only 3% while the actual barrier height fell by a factor of 3.

Payne & Inkson (1985) used the corrected14 image charge potential of Simmons (1963) to obtain similar results, and noted that the use of the correct quantum mechanical expression for image charge (Appelbaum & Hamann, 1972), which joins into the bulk potential of the electrodes rather than diverging at the edges of the barrier, and hence has a smaller gradient near the barrier edges, would result in an even greater difference between the actual and apparent barrier heights.

Teague (1978) performed numerical calculations to show that the apparent barrier height is equal to the nominal barrier height at tip/sample separations all the way down from 2 nm to 0.3 nm (almost mechanical contact). The barrier height becomes negative (i.e.  a microconstriction, rather than a tunnelling barrier) for a gap 0.2 nm across, so the model used became inaccurate.

  Behaviour of the apparent barrier height for small tip/sample separations

Lang (1987b, 1988) calculated the behaviour of the apparent barrier height between two parallel planar electrodes, and where one electrode had an adsorbed sodium atom (adatom-on-jellium model), for tip/sample separations below 0.7 nm. He ignored the effects of image charges, because, as previously discussed, for very small separations the classical image potential is no longer valid, and for larger separations the image potential has no effect on the apparent barrier height because of the cancellation of the first-order term in 1/d. However, he found that the exponential decay of electron density into the barrier led to a slow, roughly exponential decay of the exchange and correlation potential into the barrier, and that the barrier height is therefore depressed within 0.2-0.3 nm of the surface. The apparent barrier height was therefore substantially decreased as the barrier width was reduced, particularly in the situation where one electrode had an adsorbed sodium atom tip, where the apparent barrier height fell off sharply for separations below 0.5 nm, decreasing below 0.5 eV for a separation of 0.3 nm.

Pitarke et al.  (1989) used numerical integration to find the exact result for the current density. Using the Simmons (1963) potential, they found that the apparent barrier height was constant, and almost equal to the work function, down to about 0.4 nm, and fell to almost zero at 0.2 nm. Using a more sophisticated model for the barrier, the apparent height was found to be about 0.5 eV below the work function for large separations (1 nm), again falling off most rapidly between 0.4 nm-0.2 nm, and reaching zero at zero separation.

Persson & Baratoff (1992) used a numerical self-consistent calculation (see section 4.2.2; the barrier height and tunnelling time are interdependent), to estimate that the classical expression of Simmons (1963) could overestimate the image potential by 20-30% in STM barriers.

  Effect of atomic forces between the electrodes

Teague's (1978) experimental measurements showed that, for very small spacings, the attractive atomic forces between the electrodes distorted them, so that the measured barrier height actually increased as the barrier width was reduced. Further reductions in the spacing led to a repulsive force, causing the apparent barrier to fall to zero. These effects were a result of the electrode distortion due to the atomic forces causing the actual electrode spacing to differ from the value calculated. Chen & Hamers (1991) obtained similar results, with the apparent barrier height first increasing from 3.5 eV (large separation) to 4.8 eV, then falling to below 0.3 eV within a fraction of an Å ngström, 1.5 AA from mechanical contact. These results were reproduced by a simple theoretical model of an elastically deformable tip.

  Transition from tunnelling to ballistic transport

Chen (1993, p. 62) showed that, although generally the behaviour of a `barrier' which is classically allowed (i.e.  the barrier height is negative) is quite different from that of a classically forbidden one15, in the case where the barrier width is sufficiently thin (less than half the de Broglie wavelength of the electron), the transmission coefficient varies smoothly as the barrier height is varied from positive to negative values. This behaviour may be justified by the uncertainty principle, since the energy uncertainty of a typical tunnelling electron, of kinetic energy E-UB ~ 2 eV, calculated from its tunnelling time across a 3 Å wide barrier, exceeds its kinetic energy, so the distinction between the cases E < UB and E > UB is lost (Chen, 1993, p. 64).

For this reason, for a sufficiently thin barrier, the distinction between the tunnelling and ballistic electron transmission is lost.

For a very thin barrier, where the barrier height is very low or negative, the WKB method fails, since it will predict 100% transmission if f < 0. Similarly, the Bardeen integral approach is invalid, since the electrodes can no longer be considered as near-separate systems coupled by a weak interaction, and predicts zero transmission for f < 0 (Chen, 1993, p. 61). For accurate calculation of the tunnel current over the whole range of barrier heights and junction conductances relevant to scanning tunnelling microscopy, a different approach is required: this is the modified Bardeen approach.

6.2.3  The modified Bardeen approach (MBA)

In order to apply the transfer Hamiltonian to STM junctions, the tip and sample wavefunctions must be modified to allow for the presence of interactions between the two electrodes. The electrode wavefunctions are both altered from their independent forms by the presence of the other electrode. Attractive electron exchange forces (short distance, 3-10 Å) and van der Waals forces (long-distance, 10-100 Å), and very short range ( < 3 Å) repulsive forces, can occur between the tip and sample.

By modifying the wavefunctions of the tip and sample (using Green's functions to approximate the wavefunction distortion), then putting the results into the standard surface integral (equation 145), Chen (1993, pp. 65-72) showed that the standard transfer Hamiltonian result is modified by a (separation-dependent) multiplicative factor. The results obtained using this method (calculated with the tip/sample separation at the centre of the barrier, and also for a separation surface 1/3 of the way across the barrier) were compared with the exact results for a square barrier. The standard transfer Hamiltonian result for the same barrier was also calculated. The latter approach was found to fail when the barrier height maximum fell below 2 eV, but the MBA gave a good approximation to the exact result at all energies. The transmission coefficient, as a function of barrier height, calculated using the MBA, was almost independent of the exact choice of separation surface, for all barrier heights from +4 eV down to -2 eV. In the case of the midpoint separation surface, the agreement was accurate to within a few percent over all barrier energies.

6.2.4  Green's function approach

Noguera (1988) calculated the Green's function of the whole electrode/barrier/electrode system by matching the individual Green's functions at the boundaries. This approach is (unlike the Bardeen (1961) transfer Hamiltonian integral) not limited to thick barriers. It was found that the derivatives of the density of states are the important quantities, and that the van Hove singularities, where the density of states becomes infinite, do not contribute to any current (see also Noguera, 1990), owing to their zero group velocity. Similarly, d-states are less important than s- or p-states. It was also shown that the Bardeen tunnelling theory is invalid for energies corresponding to the energy of surface states, and that the effect of these cannot therefore be determined by Bardeen's theory. It was noted that these conclusions differ from experimental findings, in which surface states are indeed imaged (Stroscio et al. , 1987; van de Leemput & van Kempen, 1992), and it was suggested that this may be due to a current parallel to the surface (which would not suffer the exponential decay which eliminates the effect of the surface states), or electron-phonon coupling to bulk (propagating) states, at finite temperatures (see section 4.2.4).

Sacks & Noguera (1988) used the method of Noguera (1988) to calculate the tunnel current in specific 1-D cases. For a square barrier, they found a result identical with the Bardeen theory, where the transmission coefficient is proportional to the product of the local densities of states of the two electrodes at the centre of the barrier. For a square barrier containing a resonant state (modelled by an attractive d-function potential which could contain a single bound state), they found that the resonance depended on both electrodes, unlike the Bardeen result, and that the latter tended to overestimate the resonant tunnelling effect, and was invalid near resonance, and near the top of the barrier.

6.3  STM as a Topographic and Spectroscopic Tool

There are several modes of operation of STM, which in general reveal different information. These are considered in the current section.

6.3.1  Topographic techniques

In the simplest picture of STM operation, there are two different modes of topographic imaging: constant current or constant height scans.

  Constant-current imaging

In the constant current scan, the simple picture is that the tip is held at a constant distance from the sample surface, by means of keeping the tip/sample tunnel current constant. The tunnel current is exponentially dependent upon the tip/sample spacing, with a decay constant of about 0.5 Å, hence (provided the sample and tip mountings are sufficiently stable and vibration-isolated, the current amplifier sufficiently sensitive and well-shielded from interference, and the feedback loop has a sufficiently high bandwidth to drive the tip at the surface corrugation16 frequency) the tip accurately follows the contours of the surface to atomic resolution as it is scanned across it, and surface features of lateral dimensions of the order of Å ngströms and vertical displacements of a fraction of an Å ngström, can be detected and imaged. There are many situations in which this simple picture is true to a first approximation, and STM imaging can indeed allow straightforward imaging in this manner, however, in general a more sophisticated model is required to describe tunnelling in the STM.

It has been suggested in the previous paragraph that the STM tip follows contours in the surface, but it is clear that for atomic-scale objects, the concept of surface contours is an equivocal one, since the surface is comprised of continuous matter, but of (for a metal) discrete positive ions surrounded by a sea of electrons. The simplest picture which takes this into account is one in which the tip follows contours of constant electron density (this assumes that the tunnel current is directly related to the electron density at the end of the tip). Atomic resolution then occurs when the tip detects the corrugations in the surface electron density caused by the presence of the atomic sites.

The electron density model explains why only half the atoms are detected in a scan of the surface of graphite: it is only those atoms which have upward-pointing dangling bonds (for a more accurate picture, see section ) which give a large contribution to the electron density corrugations, and are imaged.

A drawback of STM imaging is that the shape of the tip may affect the observed image, since surface features smaller than the end of the tip cannot be resolved. Atomic resolution is therefore highly sensitive on the positioning of, potentially, a single atom at the end of the tip. If the tip is insufficiently slender, surface asperities on a rough surface may scan the tip, resulting in an image consisting of a superimposed set of images of the end of the tip, where only the large-scale structure describes the sample topography. For images of no better than nanometre resolution, deconvolutions procedures may be employed to remove tip effects, with the assumption that all the current from the tip flows to the nearest point on the surface (Chicon et al. , 1987), but at higher resolutions, this model breaks down, since the current has a spatial distribution, and the behaviour must be described quantum mechanically (see section ).

  Spectroscopic considerations

A further consideration is that the STM is an inherently spectroscopic tool. From the Fermi exclusion principle, it may be seen that only those electrons with energies between the Fermi levels of the tip and the sample may tunnel (i.e. , if the sample is at a positive bias V with respect to the tip, only those electrons in the tip in an energy range eV below EF may tunnel across to the sample). Hence, it is not the electron density per se , but the electron density with energies in this range, which is the crucial parameter.

The spectroscopic sensitivity implies that images obtained using different bias voltages or polarities may differ considerably; this has been observed in studies where different applied bias voltages can reveal the locations of different atoms (either different elements, or atoms of the same element in crystallographically different positions) in the surface (Lang, 1987a). This technique has been applied particularly successfully to the study of semiconductor surfaces (Feenstra, 1990). The combination of spectroscopy with extremely high spatial resolution is unique to the scanning tunnelling microscope.

  Constant-height imaging

In constant-height imaging, the STM tip is held at a constant displacement by deactivating the feedback loop before the scan, thus holding the voltage applied to the scan tube(s), and the tip's vertical position, constant (Hansma & Tersoff, 1987). A variation of this method is simply to scan the tip faster than the feedback loop can respond, thus allowing the feedback loop to stabilise the tip's average distance from the surface, and to detect the high-frequency variations in the tunnel current (variable-current imaging, Bryant et al. , 1986a,b). The tunnel current variations may be used to map the surface if the current is taken to depend exponentially on the tip/sample distance.

This method detects surface features by their effect on the tunnel current at a fixed displacement above the average surface position. This method has several advantages. The scan rate is not limited by the bandwidth of the feedback loop, allowing STM images to be collected at very much faster rates (including real time video rates: Wiesendanger, 1994, p. 128). It is also more sensitive, being less susceptible to noise and mechanical vibrations, because of the fast scan rate. However, this mode is limited to surfaces which are sufficiently flat that there are no high-frequency variations (including steps) larger than the tip/sample spacing, otherwise the tip will crash into the surface. Another drawback is the difficulty in obtaining absolute vertical heights with this method, but this may be overcome by combining the constant-current and constant-height techniques over the same sample.

  Other modes of operation of STM

Variants on the standard STM techniques have been employed for several reasons. The zero-bias noise level has been used (Möller et al. , 1989) for topographic imaging to avoid application of high electric fields to the sample. To extend STM to insulating samples, a.c. voltages at GHz frequencies have been used (Kochanski, 1989), enabling the high-frequency tunnelling of electrons back and forth across the barrier to be detected, since the tunnelling of, say, ten electrons onto the surface of the insulator, and back, every half-cycle at 1 GHz corresponds to an a.c. current of order 1 nA. The third harmonic of the current was used, to eliminate the current due to the junction capacitance. Abraham et al.  (1988a,b) vibrated the STM tip laterally, and measured both the topographic image and the I/x image, improving the signal-to-noise ratio.

6.3.2  Spectroscopic techniques

Tunnelling across a barrier, and the mechanism by which the conductance curve of a tunnel junction may reveal information about the density of states in the sample, have been discussed in chapter 4, and it has been stated (section 2.1.2) that the decay constant of the wavefunction is proportional to {f}, where f is the barrier height.

Electrons with higher energies experience a lower barrier height (f = V-E where E is the electron energy, V the absolute barrier height), but for situations where the applied bias is small, the range of energies allowable for tunnelling electrons is also small, and the tunnelling probability is relatively unaffected by this small decrease in barrier height. Features which appear in the I-V  characteristic are dependent mainly upon variations in the density of states of the sample, provided that the density of states of the tip is constant over this range. This situation is satisfied very well for superconductive tunnelling, where the density of states of the superconductor is highly energy-dependent, and the energy range involved is just a few meV.

If higher biases are used, the variation of effective barrier height with energy means that higher energy electrons (i.e.  those nearer the Fermi level) are the most likely to tunnel. This process typically leads to an order of magnitude increase in tunnel current per volt applied. Thus, for small applied bias the junctions are nearly ohmic in the absence of features in the density of states (e.g.  for an MIM junction), but for higher applied bias a smoothly varying background appears, on which the density of states information is superimposed.

The consequence of the disproportionate contribution to the tunnel current from states near the Fermi level of the more negative electrode is that the conductance curve depends on the density of states of the more positive electrode. The sample bias should therefore be positive, to obtain information relevant to the sample density of states (Klitsner et al. , 1990). Griffith & Kochanski (1990) calculated the conductance curve for tunnelling between a tip and sample with different frequency oscillations in their densities of states, and showed that for sample bias above about 0.7 V (below -0.5 V) only the sample (tip) density of states was apparent. In cases where the bias voltage cannot be considered small compared to the barrier height, the dependence of the transmission probability on the bias voltage can have a large effect on the measured conductance curve. Theoretical calculations by Chandler (1993, section 4.2) using the WKB method showed that density-of-states features were still present, but distorted, and an asymmetric V-shaped background appeared.

6.3.3  Related scanning microscopies

A wide range of related techniques exists, including spin-sensitive imaging and the use of photon- and phonon-assisted processes to obtain the image. Good reviews of related techniques are van de Leemput & van Kempen (1992, section 6) and Wiesendanger (1994, pp. 158-204).

6.4  Theories of Topographic Imaging in STM

STMs are capable of achieving atomic resolution on many surfaces, including close-packed metal surfaces such as Au(111) (Hallmark et al. , 1987); Al(111) (Wintterlin et al. , 1989); Au(110), Ag(111) and Cu(111) (Berndt et al. , 1992). In this section, theoretical studies of the origin of the atomic resolution are presented.

6.4.1  Perturbation theory and plane-wave transmission approaches

García et al.  (1983) used a square-barrier approximation and considered s-wave tunnelling. The current density was calculated from electrons within a shell of width 0.01 eV (the bias energy) of EF. They considered a periodic array of tips (sufficiently separated to decouple them) to enable preiodic bounadary conditions to be applied, and used numerical techniques to solve for various types of corrugated surface. They found that the tunnel current was proportional to the density of states, and to a geometric function G which is a monotonically increasing function of the effective radius of curvature of the junction, Reff, where

1
Reff2
=

1
R1t
+ 1
R1s




1
R2t
+ 1
R2s


(185)
where R1,2t and R1,2s are the two radii of curvature of the tip and sample respectively. Thus a larger tip (i.e.  radius of curvature) gives a larger current, as might be expected.

The lateral resolution, Leff, was found by considering the distance over which the sample was involved in tunnelling, calculated from the tunnelling angles. Leff was defined by considering the diameter which a constant cylindrical tunnel current would require to produce a given total current I and maximum density j0, as:

p(Leff/2)2 = I/j0.
(186)
Leff was found to increase monotonically with the effective radius of curvature, Reff, as expected, but only fairly slowly. It was also found to increase if the barrier height was reduced; again, this result is intuitively expected since a lower barrier height implies that electrons can tunnel further. Values of 4-8 Å were found for the lateral resolution in this way, depending upon the parameters.

Tersoff & Hamann (1983) used the Bardeen approach (independent electrode approximation), with the tip modelled by a spherical potential well. Using an s-wave tip wavefunction, in the low voltage, low temperature limit, they found that the lateral resolution was approximately [2(R+d)/k]1/2 (where k = {2mf/(h/2p)2}, and R, d are the tip radius of curvature and minimum distance from the surface respectively), since higher spatial frequencies in the surface wavefunction are suppressed. The implication is that sharper tips, and shorter tip/sample separations, improve the resolution, but only as a square root function. The distance of the tip centre of curvature from the surface is the important quantity: Tersoff & Hamann (1985) stated that (again, in the low V, T limit) the conductance is proportional to the Fermi-level local density of states (LDOS) evaluated at the tip. The surface wavefunctions in the barrier were expanded in terms of a sum over surface reciprocal lattice vectors, and the tip wavefunctions by an s-wave model. Substitution of these wavefunctions into the Bardeen integral (equation 145) gave the current:

I = 32p3 e2
(h/2p)
Vf2Nt(EF)R2e(2kR)r(rt,EF)
(187)
where Nt(EF) is the Fermi-level LDOS and r(rt,EF) is the Fermi-level electron density at the tip, rt.

Substitution of parameters17 (f = 5 eV; R = 9 Å; d = 6 Å) into the theory gives a resolution of 5 Å, but this could be improved upon by using an atomically fine tip.

Tersoff & Hamann's result (equation 187) is relatively simple to apply, giving reasonable results, e.g.  for the Si (7×7) reconstruction (Tromp et al. , 1986). Sacks et al.  (1987) applied this result to a free-electron metal with a sharp boundary. For large separations, R+d >> k-1, the image is a 2-D convolution of the surface with a Gaussian distribution, of width (FWHM) 2[ln2(R+d)/k]1/2, giving a lateral resolution of a few Å.

The electron density of states decays into the barrier as exp[-2k(R+d)]/(R+d), and the current as:

I R2
R+d
e-2kd.
(188)

Stoll et al.  (1984; see also Stoll, 1984) considered the transmission and reflection of waves by a 1-D sinusoidally corrugated interface, representing the sample surface, and, like García, used a periodic array of tips. Using a three-dimensional numerical technique to treat diffraction by a corrugated wall, they found that for tip/sample separations less than 6 Å, deviations from the exponential dependence of the tunnel current occurred. The ratio Dd/hs of the observable and real corrugation amplitudes did however depend exponentially on the separation except at very small separations, so that the constant-current contours flattened out as the tip/sample separation was increased:

Dd/hs exp(-p2(d+R)/a2k)
(189)
where a is the metal corrugation wavelength, and d, R, k have the same meanings as in the Tersoff & Hamann calculation.

6.4.2  Tip atom considerations

Lang (1985) used the adatom-on-jellium model to calculate the tunnel current density in the barrier, using a self-consistent approach. Maps of current density were obtained, showing a large, sharp enhancement to the current in the region of the sodium adatom. A calcium adatom (chosen because of its similar equilibrium metal-adatom separation) has four times the density of states, and gives rise to a fourfold more sharply-peaked current distribution, with most of the current flowing within an atomic radius of the centre of the adatom. Partially filled p-shell atoms cause the additional current to undergo a sign change, causing a 20-30% current density decrease near the adatom.

Lang (1986) used a similar model where each surface had an adatom to compute the behaviour of tip displacement vs. lateral separation of the adatoms. Using a sodium atom for the tip, and a sodium, sulphur or helium atom for the sample, he found that the vertical corrugation observed for Na was considerably higher than that for S (1.5 Å compared to 0.5 Å), whereas He gave a negative corrugation of -0.3 Å. The FWHM feature widths observed were, for Na, S, He respectively, 5 Å, 2.5 Å, 2.5 Å. The observed vertical heights depend on the adatom's effect on the Fermi level density of states. It is proposed that some adatoms could have virtually no effect on the tunnel current. Doyen et al.  (1988) performed similar calculations for a single-atom tungsten tip chemisorbed onto a W(110) surface, obtaining similar results, i.e.  that for tip/sample distances less than 6 Å, the tip did not follow contours of constant Fermi-level charge density, and that (for small tip/sample separations only) an adsorbed oxygen atom showed up as a dip rather than a peak in the tip/sample distance.

Lucas et al.  (1988) modelled the tip as a hemispherical protrusion on one of two parallel planar free-electron metal electrodes. They used the classical image potential approach, with corrections where it becomes invalid near the boundaries. Using three-dimensional scattering theory over a discretized finite grid, they found that the zero angular momentum state contributed 90% of the tunnel current. They also found that most of the current was contained within a small range of angles of the symmetry axis, and that atomic corrugations of the size of the tip radius would be detectable. However, an unphysically small (2 Å) tip/sample separation was used to expedite the computer calculations.

  Derivative rule

Chen (1988; 1993, chapter 3) showed that the matrix elements determining the tunnelling contribution from a tip state depends on the angular dependence of the tip state, so that for an s-state, the relevant quantity is y; for a p-state the quantity proportional to the matrix element is y/x for px etc. ; for a d-states the second derivatives are important. This result may be visualised by considering the effect of lobes of the tip state on the tunnel current; e.g.  for a px state, the two lobes have opposite phase, so a uniform sample wavefunction would give zero net tunnel current - the wavefunction must have a non-zero gradient in the x-direction for a net current to flow, i.e.  finite y/x. Chen (1988) also considered the effects of different states in both tip and sample.

Chen (1990b) suggested that atomic resolution on close-packed metal surfaces could be attained by using a tip with a dangling pz or dz2 state, e.g.  the tip should be made out of a d-band metal. A theoretical calculation showed the resolution should be better by an order of magnitude. Explicit expressions were derived (Chen, 1991) for the conductance distributions corresponding to different tip and sample states, and numerical factors found for the relative corrugations corresponding to different tip or sample states, these being s (1), p (2.73) and d (12.9), the factors for tip and sample combining multiplicatively. Assumption of a dz2 tip state explained the observation of corrugations on even the smoothest close-packed metal surfaces (section 6.4).

Wintterlin et al.  (1989) observed atomic resolution using an Al sample, in UHV, with a corrugation height of 0.1-0.4 Å. The calculated value for a tip 3 Å distant is 0.02 Å, and an elastic deformation of the tip by an adhesive tip/sample interaction is proposed to explain the observed corrugation. It was also demonstrated that a high voltage applied to the tip could form a sharp metallic cluster on the tip.

6.4.3  Imaging of graphite

STM images of graphite have often shown unphysically large corrugations (Soler et al. , 1986) of up to 10 Å (compared to the 2 Å width of the unit cell). These were initially attributed to a mechanical interaction between the tip and surface, leading to deformation of the sample surface. Colton et al.  (1988) obtained images (using a graphite-coated tungsten tip) which matched theoretical images calculated for multiple-atom tips. Mamin et al.  (1986) suggested that contaminants between the tip and sample were mediating the interaction, which would cause the current to vary less rapidly with tip/sample separation (Coombs & Pethica, 1986). The contaminant model was strengthened by measurements in UHV showing a corrugation of 0.3 Å (Hamers, 1993). Tersoff (1993) has suggested that the `dirt', such as metal oxide, can slide over the graphite surface. Alternatively, the contaminant could be water. The effect of a contaminant-mediated interaction is simply a vertical scaling (Coombs & Pethica, 1986).

It should be noted that graphite, unlike metals, has nodes in its density of states, which would give unusually high experimentally-determined corrugations heights, of up to 2 Å, owing to the feedback system pushing the tip forward to maintain constant current. The tip would not in fact be driven into the material because of contributions to the current from non-zero angular momentum states or tip atom vibrations (Leavens, 1988). The effect of a singularity in the LDOS of graphite is the basis of Tersoff's (1986) anomalous corrugation theory - the theoretical corrugation with an s-wave tip is logarithmically divergent, i.e.  infinite, but Lawumni & Payne (1990) showed that even a very small amount of state mixing with a non-zero angular momentum state would eliminate the effect.

There are other peculiarities associated with imaging of graphite: multiple tips, or the transfer of a graphite flake onto the tip, may give rise to artefacts with appropriate symmetry (Mizes et al. , 1987). The imaging of only half the atoms occurs because the wavefunctions of those atoms directly above atoms in the layer below couple to form bands of width ~ 1 eV around EF, while those which are not directly above an atom have wavefunctions sharply peaked at EF. The STM images the Fermi-level LDOS, hence the latter atoms cause a much larger contribution to the corrugation than the former (Batra & Ciraci, 1988).

Since graphite easily flakes off onto the tip, Tersoff & Lang (1990) considered the effect of various tip atoms, including carbon, on the STM image. They used Lang's adatom-on-jellium model to calculate the expected corrugation heights and the contributions of different angular momentum states to the total current, for tip atoms of Na, Ca, Si, Mo and C. Large variations were found: Na closely obeyed the s-wave tip model, with almost all (97%) of the current from the m = 0 state, and a corrugation of 0.9 Å. For Ca, Si and Mo the contribution of non-zero angular momentum states became progressively higher (for Mo: 46% from m = 0; 28% from m = 1; 26% from m = 2), and the corrugations progressively lower (0.1 Å for Mo), while for a C atom tip, the m = 1 contribution to the current was 99%, and the corrugation negative (-0.2 Å). This behaviour is dependent on the sample lattice, but it is clear that, in general, the image at atomic resolution, particularly that of graphite, is highly dependent upon the tip atom.

6.4.4  Transition to point-contact regime

Measurements by Gimzewski & Möller (1987) of the variation in the tunnel current between a clean Ir tip and Ag surface at a voltage bias of 20 mV, as the tip/sample separation was decreased from the point where I = 1 nA (R = 20 MW), showed that the tunnel current initially increased exponentially, by a factor of about 8 per Å for the first 3 Å of motion, before reaching a plateau of nearly-constant resistance of 35 kW, over a range of 2.5 Å. The current then jumped by an order of magnitude at a well-defined position. That this jump corresponds to contact, and to large adhesive forces between the tip and sample, is strongly suggested by the appearance of hysteresis in the current-distance characteristic after this point is reached. Lang (1987b) used the adatom-on-jellium model, keeping the adatom-nearest surface distance constant, with a self-consistent density calculation (the transfer Hamiltonian method is invalid when the tip/sample separation is small, and hence the wavefunction overlap is large). The resistance was exponentially dependent upon the separation at large separations, but at smaller separations (less than about 3 Å) levelled out to a plateau with a resistance of 32 kW for a Na tip atom (18 kW for Ca), in good agreement with the results of Gimzewski & Möller. This compares with a theoretical resistance of p(h/2p)/e2 = 12.9 kW for an ideal 1-D conduction channel (Landauer, 1987).

6.5  Topographic Imaging: Preliminary Tests of the STM

6.5.1  Atomic resolution with graphite

Highly-oriented pyrolitic graphite (HOPG) has often been used as a test of the ability of an STM to achieve atomic resolution. The material consists of planes of atoms, with the separation between planes almost 2.4 times the nearest-neighbour distance within planes. For this reason, graphite cleaves easily along the a-b planes, forming an atomically flat surface. The atoms are arranged in interlocking hexagons, as shown in fig (a). However, only one out of each two adjacent atoms has an upward-pointing atomic bond, so imaging by STM, which may be considered to image the Fermi-level electron density, usually only reveals half of the atoms (see section 6.4.3). The resulting image will then show only the atoms represented by filled-in circles in fig (b).


Figure 6.1: Surface atomic arrangement in cleaved graphite. (a) shows all the surface atoms; (b) shows those atoms visible to STM as filled circles; atoms represented by open circles are invisible.

The nearest-neighbour distance is a = 0.2456 nm, and the parameters visible to STM are the atomic spacing within observed rows, 3a = 0.4254 nm, and the inter-row spacing, 3a/2 = 0.3684 nm. Measurement of these parameters by STM was used to calibrate the lateral sensitivity of the scan piezo. (The calculated sensitivity was confirmed by Czorniy (1996) with measurements of a diffraction grating.) Measurement of the sensitivity in the z-direction was achieved by obtaining images of tilted graphite surfaces at known angles.

The parameters used for imaging the surface of graphite were typically Vbias = 35 mV and I = 1 or 10 nA. Fig a shows a 10 nm×10 nm scan of the surface of graphite, with the atomic array clearly visible. The features are somewhat distorted, but the distortion may be corrected by applying an appropriate linear transformation to the image. After rotating the image by 90, then applying a linear stretch in the vertical direction by 10%, and a horizontal shear by 10 anticlockwise, the resulting image fits the expected structure accurately (fig b).


Figure 6.2: STM image of graphite surface, 10 nm×10 nm in area. (a) shows the untransformed image; (b) shows the transformed image obtained by anticlockwise rotation of 90, vertical stretch by 10% and shearing by 10 anticlockwise. The expected atomic positions are shown as filled circles.

It has also proved possible to image contaminants lying on the surface of the graphite. Fig  shows three consecutive images (4 nm×4 nm) of an area adjacent to fig 6.2. In these images it is possible to discern the atomic array, but the main feature is a 1.5 nm wide protrusion near the centre of the images. The large height of this feature relative to the atomic corrugation height renders the latter hard to distinguish. After each image scan, the peak appears to become larger and smoother, suggesting that gold atoms are migrating from the tip onto the surface asperity, or that the asperity is physically affected by the forces applied by the tip moving across it. The asperity is likely to be a contaminant molecule or group of atoms, and may be a small group of graphite atoms left behind by the cleaving process. (Ganz et al.  (1988) have observed stable clusters and single atoms of Au, Ag and Al on a graphite surface by UHV STM.)


Figure 6.3: 4 nm×4 nm images of graphite showing surface contaminant. The images were taken in the order (a), (b), (c), and the surface feature appears larger and smoother in later images.

  Liquid helium-temperature atomic resolution

The images referred to above were all taken at room temperature. Using the improved model of the STM head and amplifier arrangement, images of graphite have been taken with the STM mounted in the continuous flow cryostat, at a temperature of approximately 5 K18 (fig ). The lines of atoms can be seen in this image, but the atomic positions are unclear, possibly due to vibrations caused by the flow of liquid helium through the cryostat. It has not been possible to obtain completely clear images of graphite with atomic resolution, probably for this reason. Another effect seen at low temperature is that the images are composed of short lines in the scan direction, believed to be caused by a few monolayers of water on the surface of the graphite, which freeze, causing the tip repeatedly to stick, as it moves across the surface (Czorniy, 1996).


Figure 6.4: 4 nm×4 nm image of graphite taken at T = 4 K. Atomic positions are unclear, although lines of atoms can be distinguished.

6.5.2  Gold films

As a further test of the ability of the STM to perform topographic imaging at room temperature and liquid helium temperature, images of gold films deposited on glass were taken. The expected morphology of these films consists of circular islands of gold, as a result of the deposition process. Figure  shows three images of the surface of such a film. The first image is of an area 200 nm×200 nm, and was taken at T = 5 K. The other two images were taken at room temperature, and show areas of 500 nm×500 nm and 1.5 mm×1.5 mm respectively. From these images it may be seen that the STM is capable of imaging surfaces on scales ranging from a few microns, down to a few nanometres across. The agreement of the size of gold `hills' in fig a-c indicates that the factor of 5 reduction in sensitivity between room temperature and helium temperature, used in scale calculations, is correct.


Figure 6.5: Images of gold deposited on glass. (a) shows a 200 nm×200 nm image taken at T = 5 K; (b) shows a 500 nm×500 nm image taken at room temperature; (c) shows a 1.5 mm×1.5 mm image taken at room temperature.

Chapter 7
Coulomb Blockade Effects: Theory and Results

Coulomb blockade (single-electron charging) effects occur when the energy required to move a single electron across a junction is larger than kBT. This can result from tunnelling into an isolated particle of conducting material separated from the main bulk of the sample by an insulating layer, thus forming a double tunnel junction. The effect occurs when the capacitances between the central and outer electrodes, and the self-capacitance of the isolated particle, are small ( ~ 1-10 aF). It may also result from tunnelling into bulk material, because the junction is effectively current-biased on short timescales (Mullen et al. , 1988a), owing to the finite bandwidth of the control circuitry used to voltage-bias the junction.

Convincing data showing Coulomb blockade effects were obtained in this study, using evaporated thin gold films and NbN/Au films. Both Coulomb gap and Coulomb staircase effects were observed.

7.1  Theory of Coulomb Gap and Coulomb Staircase Effects

7.1.1  Coulomb gap: Zeller & Giaever's model

Giaever & Zeller (1968; Zeller & Giaever 1969) studied the behaviour of planar tunnel junctions fabricated with small Sn particles, which became superconducting at a sufficiently low temperature19, embedded in the oxide barrier.

Initially considering normal state inclusions, they realised that the electrostatic energy required to place a charge e on a sphere of radius r is e2/2r. For a sphere radius 5 nm, this energy is equal to 130 meV, but would in practice be an order of magnitude smaller owing to the dielectric effect of the oxide.

For liquid helium temperatures, kT ~ 0.36 meV, so the number of electrons on a small sphere of the size described would be fixed, at constant junction bias. The Coulomb effects can only occur if this condition is satisfied, i.e.  if e2/2C >> kBT. The number of electrons will be such that the resulting electrostatic potential on the sphere is as near as possible to the Fermi level, although equality cannot in general be reached, since the sphere's potential is effectively quantized in steps of e/C, where C is the capacitance of the sphere, C = 4pe0 er r. There will then in general be a voltage difference VD between sphere and the nearby electrode, where -e/2C < VD < e/2C. Adding (removing) an electron to (from) the sphere requires an energy, supplied by the bias voltage, of

E = 1
2


e
C
VD

2

 
- 1
2
VD2 C
(190)
where + corresponds to addition of an electron to the particle, and - to removal of an electron.

The capacitance C of a particle is just the sum of the capacitances between the particle and each of the two electrodes, C = C1+C2. (C1 is the capacitance of the left junction, and C2 that of the right). This is because, to the particle, the two capacitances are effectively in parallel, since the two electrodes differ in potential only by a constant bias.

7.1.2  Highly asymmetric junctions

The following discussion corresponds to the case where the particle is much closer to the right electrode, so C C2 >> C1.

If the value of VD were zero, the tunnelling conductance through the inclusion would be zero for |V| < e/2C, and constant otherwise, i.e.  it would be an inverted top-hat function centred at V = 0. At lower applied voltages, the bias voltage supplies insufficient energy to place an electron onto the particle (or remove one). The effect of the finite VD is to displace the zero conductance region by VD along the voltage axis, since now the zero bias potential of the particle is displaced by VD from the Fermi level of the electrode. Thus, if a single spherical particle is present in the tunnelling barrier, which is feasible for STM work, the conductance will be just such an inverted top-hat function, and the I-V  characteristic will be zero in this region, but linear elsewhere.

7.1.3  More general case: symmetric and asymmetric junctions

Initially considering tunnelling from left to right (positive bias), if an electron tunnels across from the left electrode to the central electrode, the charge on this electrode is changed from dQ = CVD to dQ-e (c.f. 190). The charge on the left capacitor is changed by -e and the source must supply a charge of -eC2/(C1+C2) from the right to the left electrode, to keep the total voltage constant. The total energy change both from moving the charge through the voltage source, and placing a charge on the central electrode is:

dE = e
C1+C2


e
2
-dQ-C2V

.
(191)
This is negative, and hence the charge movement can occur, if V > (e/2-dQ)/C2. Considering the trade-off between the energy supplied by the source and the electrostatic energy required to charge the central electrode, expressions may also be derived for the minimum voltage for which tunnelling can occur in the case of initially removing an electron from the central electrode, and placing it on the right electrode, and also for the two processes corresponding to tunnelling of electrons from right to left (negative bias). The tunnelling cannot occur, and a Coulomb gap exists, when the applied voltage is insufficient either to add an electron to, or to remove an electron from the central electrode:
min

-e/2+dQ
C1
, -e/2-dQ
C2


< V < min

e/2+dQ
C1
, e/2-dQ
C2


(192)
i.e.  the overlap of two gaps of full width e/C1 and e/C2 whose centres are displaced from the origin by +VD and -VD respectively, and which correspond to an initial step of changing the number of electrons on the central electrode by -1 or +1 respectively.

In the case where C = C2 >> C1, this reduces to a gap of width e/C:

-e/2+dQ
C
< V < e/2+dQ
C
(193)
which is the inverted top-hat function previously derived for the asymmetric case, of full width e/C and displaced by dQ/C = VD from the origin (see fig a).

7.1.4  Multiple junctions: ensemble effects

If an ensemble of spherical particles with very similar sizes, and hence capacitances, is present, the conductance observed will be the average of the conductance characteristics for each particle between the electrodes. Since the capacitances are all similar, the size of the gap in each conductance curve will be almost the same, but because a small change in capacitance will shift the spacing of the electrostatic levels, the values of VD will take all possible values. Hence the total conductance will be the sum of a large number of inverted top hat functions, centred at various voltages between -e/2C and +e/2C, i.e.  the convolution of an inverted top-hat function, of full width e/C with a top-hat function of full width e/C, producing an inverted triangular function, which has zero conductance at V = 0, reaching the background conductance at V = e/C (see fig b).

In the case where there are large variations in the sizes, and hence capacitances, of the particles, the sum will include gaps of different widths, so the conductance will be the sum of a set of inverted triangular functions of full width 2e/C. The weighting of the function of full width e/C will be proportional to the particle size distribution function, as shown in fig c,d, calculated numerically (by the author) for a uniform distribution of inverse capacitances between 1/2C and 2/C (c), and for a normal distribution of inverse capacitances, with mean 1/C and standard deviation 0.3/C (d).


Figure 7.1: Coulomb gap conductance curves. (a) Tunnelling through a single particle of capacitance C (dashed line: dQ = 0; solid line: dQ = 0.3e); (b) Ensemble of particles of capacitance C; (c) Ensemble of particles with inverse capacitances uniformly distributed between 1/2C and 2/C; (d) Ensemble of particles with inverse capacitances normally distributed, mean C, s.d. 0.3C. R is the junction resistance (numerical calculations performed by the author).

7.1.5  Superconducting particles

Zeller & Giaever also considered the effect of superconducting particles included in the oxide layer. The results they obtained, again assuming an asymmetric barrier with C2 >> C1, were that, for tunnelling through a single such particle, the conductance is BCS-like, but displaced outward from the origin by an amount equal to the Coulomb gap size in the normal particle case.

7.1.6  Coulomb staircase

Experiments by Lambe & Jaklevic (1969), using a similar arrangement to the Zeller & Giaever junction, but with one of the oxide layers too thick for tunnelling to occur, showed incremental charging effects, where the total charge on the central electrode increased by single quanta as the voltage across the junction was increased. The equilibrium charge on the central electrode can be calculated simply. A modification of this model, in which the smaller capacitance junction can transmit a tunnel current, but very much more slowly than the other junction, gives an insight as to how the Coulomb staircase effect (in which the conductance increases in a quantized fashion with bias voltage) may be observed in an asymmetric junction. The central electrode is charged quickly to the equilibrium level by tunnelling through one junction, but can only lose charge through the other junction slowly. The successive charging of the central electrode by an electron opens new tunnelling channels, increasing the conductance of the junction by a constant amount, at regular voltage steps, as described later in this section.

Mullen et al.  (1988a,b) investigated the theoretical behaviour of voltage-biased series double tunnel junctions. The voltage of the central electrode is not fixed, so fluctuations in this voltage occur when the number of electrons in the central region is changed, which affects the probability of further electrons traversing each of the junctions. Mullen et al. (1988a) modelled this process stochastically, and found that the qualitative nature of the I-V  characteristics depended on the relative resistances and capacitances of the two junctions. Taking these resistances and capacitances as R1, C1 for the left electrode; R2, C2 for the right electrode, the following behaviour is predicted: when R1 = R2 and C1 = C2 a Coulomb gap, consisting of a zero conductance region around zero bias, should be seen. When the junctions are non-identical, a Coulomb staircase, consisting of a series of current steps at regular voltages, may be seen. This is due to incremental charging of the particle. The size and sharpness of the steps, and the gradient of the current between steps (i.e.  the background conductance between conductance peaks) depends on the exact parameters used.

The theoretical I-V  characteristics in the two limiting cases of the Coulomb gap and Coulomb staircase regimes are shown in fig .


Figure 7.2: Theoretical I-V  characteristics for Coulomb effects. The limiting cases of the Coulomb gap (dashed) and the Coulomb staircase (solid) are shown.

Laikhtman (1991) used the quasi-classical description of Mullen et al.  (1988a,b) to derive and solve the transport equation for the system of two series tunnel junctions, giving an exact analytical expression for the Coulomb staircase. In the realistic case where the junctions have greatly differing resistances, the I-V  characteristic is determined by the junction with the larger resistance, and the steps are periodic:

I = 1
R1+R2


V- e
C1+C2


+DI
(194)
where
DI = e(1-d)
R1 (C1+C2)
.
(195)
Here, R1 is the larger of the two junction resistances, and d-1/2 represents the `fractional charge' on the corresponding capacitor, in units of e, so 0 d < 1, i.e.
d = Fractional part of

C1 V
e
+ 1
2


.
(196)

As the voltage is increased, each time the value of C1 V/e+1/2 in (196) passes an integer, the value of d drops from 1 to 0, resulting in a step increase in DI in expression (195), of magnitude e/[R1(C1+C2)]. These steps occur at regular voltage spacings of e/C1. There is also a background current gradient (conductance) of 1/(R1+R2), from (194).

For junctions where the resistances of the two junctions are comparable the expressions are considerably more complex, and less intuitive.

Laikhtman (1991) also used second-order perturbation theory to explain why a non-negligible conductance may be seen in the gap region, owing to processes in which the middle electrode is only charged for a very short time (the process being allowed by the uncertainty principle, DEDt > (h/2p)/2). The conductance is linear in V for small voltages (i.e.  the I-V  characteristic in quadratic in V).

Ruggiero & Barner (1991; see also Barner et al. , 1989) used planar junctions of the form Al/Al2O3/particles/oxide/Ag, where the particles were Pb, Cu or Ag, and the oxide either naturally-grown PbO or sputtered Al2O3. They used scanning transmission electron microscopy (STEM) to study the Ag particle size distribution, and found that it was closely related to the Fourier transform of the conductance, because the peak spacing is related to the particle sizes which are present.

Ben-Jacob & Gefen (1985) and Averin & Likharev (1986) showed that in a current-biased junction, where the Coulomb energy dominates the thermal energy, the electrons will tunnel across at regular intervals, with a frequency n = I/e (`SET oscillations'). Even for voltage-biased junctions, the tunnelling events can occur sufficiently rapidly that the voltage across the junction varies, owing to the finite bandwidth of the circuit. The junction behaves as if it is current-biased, the circuit only fixing the time-averaged voltage across the junction (van Bentum et al. , 1988c). An electron tunnelling across the junction will then cause the Fermi level of the electrode it enters to increase, so a finite bias is required to ensure it can arrive in an unoccupied state, otherwise the tunnelling process cannot occur. In this situation, a Coulomb gap may be seen even with a single junction into the bulk material.

7.1.7  Experimental detection of Coulomb effects: planar junctions

Giaever & Zeller (1968) reported large zero-bias resistance peaks (Coulomb gap) tunnelling into small tin particles separated from an Al film by the oxide layer.

Fulton & Dolan (1987) observed Coulomb blockade effects in planar film junctions, and the Coulomb staircase was seen by Barner & Ruggiero (1987), with a granular film of Ag particles embedded in an oxide layer between planar electrodes. The ensemble of junctions containing particles of different size (and hence conductance) led to an averaging of characteristics. Kuz'min & Likharev (1987) manufactured a geometry with a planar junction, in which most of the tunnel current flowed through a single embedded particle.

7.1.8  Experiments using low-temperature STM

McGreer et al.  (1989) observed a Coulomb staircase I-V  characteristic tunnelling into a Pb film using an STM with a tungsten tip. The tip was allowed to touch the surface of the film, and became coated in Pb. Sharp steps were observed (and hence sharp peaks in the conductance curve), at a constant spacing, except that, within 4D of zero bias, the step spacing was increased by 8D (where D is the superconducting gap). This is compatible with theory, which predicts a displacement from the origin, on each side, of between D and 4D, depending on the junction resistances and capacitances. It was also observed that moving the tip too close or too far away from the surface caused the sharp steps to become smeared out, owing respectively to breakdown of the conditions R1 >> R2 and C1 >> C2 respectively (R1, C1 correspond to the tip side of the double junction).

Van Bentum et al.  (1988a,b,c; 1989b) measured I-V  characteristics between a tungsten tip and a wide range of samples, including tungsten, aluminium and YBCO, and detected charging effects independent of the sample material or microstructure, which were in agreement with Monte Carlo simulations of the process. Experimentally, they found that current steps could occur either at a set of odd multiples of e/2C, or a set of even multiples. Capacitances obtained were between 5-15 aF, yielding current steps at a spacing of between 10-30 meV. Monte Carlo simulations of SET oscillations showed that the linewidth depends on the bias impedance, disappearing for a pure voltage source (van Bentum et al. , 1989b). Measurements of the resonance using a 48 GHz microwave signal showed only a broad, irreproducible signal, rather than a narrow peak. Van Bentum et al.  (1989a) make the point that the obseved I-V  curve produced by SET effects (in particular, Coulomb gap) resembles that for NIS tunnelling; the latter can be distinguished by the presence of conductance peaks either side of zero bias.

Wilkins et al.  observed Coulomb staircase effects with a film consisting of 100 Å diameter indium droplets deposited on the oxide layer on an Al film (Wilkins et al. , 1989), and also with bulk metal samples of tungsten (Wilkins et al. , 1990), due to metal-oxide impurities. In some cases, the positive bias differential conductance was found to be about twice as high as the negative bias value.

Medina et al.  (1990) observed a linear conductance background tunnelling into high-Tc superconductors, a feature commonly reported in the literature, and attributed this to the presence of a broad distribution of particle sizes. Taking the effective potential of an electron tunnelling across the junction to be V+e/2C or V-e/2C depending on its direction, to take account of the Fermi-level shifts induced by the tunnelling electrons, they found a conductance proportional to Vcoth(eV/2kT), which is linear in V for eV >> 2kT, tending to |V| for large V and a finite constant value for small V.

Berthe & Halbritter (1991) observed single-electron effects up to room temperature, and concluded that these were related to resonant tunnelling occurring through tip adsorbates.

7.2  Results

Both Coulomb blockade and Coulomb staircase effects were observed, and were consistent with the results and theoretical models previously discussed. The tip was in all cases produced by cutting an Au wire, as described in section 2.5.1. The samples with which these effects were observed were evaporated Au on mica, and Au/NbN films.

The Au on mica film was deposited onto a heated mica substrate at 400C. The selected area channelling pattern transmission electron microscope image (fig ) shows six-fold symmetry, indicating that the film was epitaxial, and that the surface was a (111) face. The thickness was approximately 210 nm.


Figure 7.3: Selected Area Channelling Pattern for thin Au on mica film deposited at 400C. The six-fold symmetric pattern indicates that this film is epitaxial, and (111)-aligned.

A room temperature image of the surface, shown in fig , indicates that the surface is smooth and continuous.


Figure 7.4: Room temperature 250 nm×250 nm image of the gold on mica film. Image (a) uses a grey-scale to represent heights; (b) displays the same data in 3D projection. The surface is smooth and continuous.

7.2.1  Coulomb staircase effects

To check that the observed oscillations of tunnel current as a function of voltage were real, and not caused by STM vibrations, I-V  characteristics were obtained over different voltage scales, as shown in fig . (The current steps are more clearly distinguished in the conductance curves, obtained by numerical differentiation of the I-V  characteristics.) If the measured oscillations in voltage were due to some mechanical vibration or electromagnetic interference, the number of cycles observed in the scan would be the same, corresponding to different apparent step widths.

The results shown in fig  indicate that the steps appear at the same voltages in each of the scans, taken at the same point of the sample. This is strong evidence that the observed effect is a real oscillation of the tunnel current with applied voltage. The step width of approximately 82 mV20 corresponds to a capacitance C given by DV = e/C, implying that the capacitance is just 2.0 aF.


Figure 7.5: Coulomb staircase observed tunnelling into Au on mica film. Conductance curves (c) and (d) correspond to I-V  characteristics (a) and (b) respectively. Step width in each case is 82 mV. The equal step widths, as a function of voltage, obtained for different scan rates, confirm that the current oscillations are a function of voltage (and not time), as expected for the Coulomb staircase effect.

2 aF is the self-capacitance of a sphere of radius 18 nm in vacuum. (This makes the assumption that there is no dielectric contamination.) If the observed I-V  characteristic is caused by tunnelling from a gold (or other) particle, isolated from the bulk sample and tip, this sets an upper limit on the particle size. The particle could be much smaller, since the capacitance also depends on the tip/particle and particle/surface separations. Self-capacitance is not generally the main contribution to the observed capacitance, since the tip/particle capacitance is likely to be larger. Approximating this capacitance as a parallel plate capacitor, C = e0A/d, where A is the effective tip area contributing to the capacitance, and d is the tip/particle separation. Taking d = 7 Å gives an effective tip area of 1.6×10-16 m2, i.e.  an effective tip diameter of 14 nm. This is the diameter of the region of the tip contributing to the capacitance; the region from which tunnelling occurs may be much smaller.

The observation of Coulomb staircases (more clearly shown in the data of fig ) implies that the two junctions are unequal, suggesting that the tip/particle junction has a larger resistance and a larger capacitance than the particle/surface junction. This was also found by McGreer et al.  (1989) (discussed in section 7.1.8). The shape of the conductance curves is very similar to that obtained by van Bentum et al.  (1988b,c), tunnelling into a granular Al sample, except that the voltage steps obtained in the experiments in this study were a factor of 5 larger.

The conductance peaks appear at voltage spacings of 87 mV and 57 mV for figs c and d, indicating capacitances of 1.8 aF and 2.8 aF respectively. The step heights seen in fig b vary between 25 pA (for large negative bias) to 50 pA (large positive bias). Equating this value with e/RC, where C = 2.8 aF, yields a junction resistance of between 2 GW (large negative bias) and 1 GW (large positive bias) for the larger of the two junction resistances.


Figure 7.6: Coulomb staircases with different step widths, observed with Au on mica film. Conductance curves (c) and (d) correspond to I-V  characteristics (a) and (b) respectively.

Conductance curves with the spacing shown in fig 7.6c were repeatedly seen as the tip was moved to different positions over the surface. After a short time, the step width changed to that shown in fig 7.6d, which was also seen several times. This implies that the Coulomb staircase was due to a particle attached to the tip. It is proposed that moving the tip over the sample surface eventually caused the particle to alter its position, which increased the capacitance, decreasing the observed voltage spacing.

Another feature shown in fig 7.6b is an asymmetry in the I-V  characteristic, the locally-averaged differential conductance increasing almost uniformly from 1 nS (R = 1 GW) for V = -0.5 V to 2 nS (R = 0.5 GW) for V = +0.5 V. This effect has also been reported by Wilkins et al.  (1990), tunnelling into etched tungsten with a platinum tip, where the observed asymmetry was also as much as a factor of 2. The values obtained from the step heights in fig 7.6b for the larger junction resistance are in agreement with the magnitude (apart from a constant factor of 2) and variation of the measured junction resistance across the voltage range. The reason for the factor of two is that the I-V  characteristics do not show an ideal Coulomb staircase, probably due to breakdown of R1 >> R2 and/or C1 >> C2 for the tip/particle and particle/surface junctions respectively. This produces the observed conductance background, reducing the apparent step height.

A more detailed approach to the data analysis would be to match the data obtained here to data obtained from numerical simulations of the double junction system for different parameters (Mullen et al. , 1988a). This approach, however, would be time-consuming, and the previous analysis was deemed satisfactory for confirmation of the theory, so this simulation was not carried out.

The data of fig 7.6a,c were taken at a higher tunnel current than those of fig 7.6b,d by a factor of 10. This implies that the tip/particle separation should be about 1 Å smaller. This does not account for the change in the I-V  characteristic, since a reduction in the distance would increase the tip/particle capacitance, C1, decreasing the voltage spacing. The step width difference cannot be a result of this change, since reducing the tip/particle separation would increase C1 and decrease DV, but the opposite change is seen. It is therefore believed that the change results from a movement of the particle on the tip.

7.2.2  Coulomb gap

After the tip had been further moved around the sample, the I-V  characteristics changed abruptly, exhibiting a Coulomb gap, the conductance dropping very rapidly to zero at the gap edges (analysis of the raw current data shows it to be consistent with a discontinuity in the gradient, and certainly the change occurs over a voltage of less than 10 mV). In all cases, the gap occurred at voltages between -1726 mV and +1358 mV. Representative data are shown in fig .


Figure 7.7: Coulomb gap obtained tunnelling into Au on mica film. Graphs (c), (d) are the numerical derivatives of the I-V  graphs (a), (b) respectively.

For the Coulomb gap effect, the total gap width is equal to e/C, where C is the larger junction capacitance. The average gap width of 307 mV implies a capacitance of 0.52 aF. Van Bentum et al.  (1988b) reported a gap corresponding to a capacitance of 1.3 aF, tunnelling into stainless steel with a tungsten tip, so the magnitude of the capacitance observed here does not seem unreasonable. Approximating the tip/particle capacitance C as a parallel plate capacitor and taking the tip/particle distance d to be 7 Å (a estimate of the upper limit of d, based on the fact that tunnelling was occurring) gives an effective tip area (contributing to the capacitance) of A = dC/e0 = 4.1×10-17 m2, i.e.  a particle diameter of 7.2 nm.

7.2.3  Discussion

It is clear from these results that the parameters C1, C2, R1 and R2, and quantities derived from these, vary from one junction to another. It is proposed that the reduction of the tip/particle capacitance from ~ 2 aF to ~ 0.5 aF causes the observed I-V  characteristics to change from Coulomb staircase to Coulomb gap behaviour, since C1 >> C2 is no longer satisfied. This also implies that the capacitance between the particle and the surface is less than about 0.5 aF, constraining the geometry. A greater capacitance would cause C1 >> C2 to break down for C1 2 aF, while a much smaller particle/surface capacitance would satisfy this condition for C1 0.5 aF. As an example, a particle of diameter 10 nm, a distance of 2 nm from the bulk material, and 0.4 nm from the tip, would give C2 = 0.35 aF and C1 = 1.8 aF, compatible with the data.

A current-biased junction (which the STM junction may effectively be, owing to the limited bandwidth of the control system) can give rise to a Coulomb gap (Mullen et al. , 1988a) but not to a Coulomb staircase, so the results strongly imply the presence of either isolated particles over the surface, or a particle on the tip, giving rise to double tunnel junctions. The parameters calculated from the data on the basis of the theory are all physically believable, and comparable with results obtained by other authors, supporting this interpretation of the data.

It is proposed that the particle was attached to the tip, which explains why Coulomb blockade effects were observed at several points over the surface, and provides an explanation (alteration of the tip particle) for the abrupt changes in the I-V  characteristics; a decrease in the Coulomb staircase step width, followed by a change to Coulomb gap characteristics. The surface is expected to be smooth and continuous (confirmed by a topographic image (fig 7.4) taken at room temperature), so it is unlikely that several particles are attached to the surface.

It is concluded that Coulomb blockade effects can occur in situations where they are not expected with STM, and it is necessary to take this into account in analysis of data, particularly where similar I-V  features, such as superconducting (NIS) energy gap, are expected. (The Coulomb gap effect can occur over a similar voltage range if the parameters are correct.) If the hypothesis that the particle is attached to the tip is correct, Coulomb blockade characteristics could be observed with almost any sample.

Chapter 8
Tunnelling Spectroscopy of Superconductors: Results and Discussion

8.1  Tunnelling into Elemental Superconductors

In order to check that the STM system was indeed capable of providing spectroscopic information on the superconducting density of states, I-V  characteristics of STM tunnel junctions with samples of the elemental superconductors Pb and Nb were measured. These samples were both in the form of thin films.

8.1.1  Pb films

  Sample preparation

The Pb films were deposited on glass using an evaporator with a base pressure of around 5×10-6 mbar, at room temperature. The film thickness was 200 nm. The films were transferred as quickly as possible from the evaporator into the STM, so that they spent only a short time exposed to the atmosphere, to minimise the formation of surface layers such as lead hydroxide.

  Pb results

Current-voltage characteristics were measured at the base temperature of the continuous-flow cryostat. By using the heater in the cryostat, measurements were also taken as the temperature of the sample increased above Tc for Pb, and then decreased below Tc again. Each current-voltage graph obtained was the average of a set of 16 individual measurements, each consisting of 256 data points between -10 and +10 mV. The voltage and current settings for the feedback loop were 10 mV and 1 nA respectively, corresponding to a junction conductance of 100 nS. The 16 measurements each took 13 ms, with a pause of 100 ms between each measurement, so the complete process took under 2 seconds. This process was repeated at intervals of about 30 seconds, with the group of 25 taking a total of 757 seconds. The I-V  characteristics were differentiated numerically to obtain the conductance curves. A selection of the results is shown in fig .


Figure 8.1: Experimental conductance curves for Pb films on glass at various temperatures. The superconducting energy gap in the density of states is shown.

As the temperature was changed during these measurements, the I-V  characteristics became linear when the temperature was sufficiently high, and reverted to their previous form when the temperature dropped back below Tc. The conductance is proportional to the superconducting density of states, convolved with a thermal smearing factor, as shown in fig 4.11 (plus additional structure due to other processes such as phonon scattering peaks and multiple electron processes). These results are rather noisy, but clearly show the superconducting energy gap, thermally smeared, at a number of different temperatures21 The temperature at which each curve was measured is not shown, because the temperature was continually changing, and thermometry was too far from the sample, had too large a time constant, and was insufficiently accurate (in the continuous flow cryostat) to give reliable temperature readings. All temperatures were in the range 5-8 K.

  Comparison with theoretical curve

In the curve taken at the lowest temperature, the maximum conductance was 120% of the normal value, and the minimum 35%. The peak occurred at a bias of 2.5 mV. For Pb, Tc = 7.2 K. Taking a value of 4.67 for 2D/kTc, the value of the energy gap is expected to be D = 1.45 meV. The effect of thermal smearing at finite temperatures is that the maximum conductance appears at a somewhat higher bias voltage (the energy gap is also somewhat lower than the zero-temperature gap, D0, but this effect is small unless T is close to Tc.) At a temperature of T = 5.3 K, the maximum conductance (120% of normal value) would be expected to occur at a bias of 2.0 mV. The experimental and theoretical curves22 are compared in fig . The curves agree only moderately well; the experimental results show a wider gap with broader `shoulders' than predicted by theory, although the noisy conditions could be the cause.


Figure 8.2: Comparison of measured (solid) and theoretical (dashed) conductance curves for Pb/Au junction. The theoretical curve is calculated for Tc = 7.2 K, T = 5.3 K, 2D/kTc=4.67.

The temperature of best fit, 5.3 K, is higher than the temperature indicated by the readout of the temperature controller. This is probably a result of the location of the thermometer, which was not actually inside the sample space, combined with possible thermometer inaccuracy.

8.1.2  Nb films

Nb was used for further experiments to measure the density of states. There wer several reasons for choosing Nb. It has a higher Tc than Pb, enabling sharper features to be measured, and thus produces a better test of the energy resolution achieved by STM. It forms a hard surface oxide, then undergoes no further reaction (as opposed to Pb, which continues to degrade with time, forming a hydroxide layer), so the surface is better characterised. It was also used to ensure that the STM was capable of energy gap spectroscopy with a range of superconducting materials.

  Sample preparation

The Nb films were prepared by sputtering onto a sapphire substrate. They were subsequently kept in vacuum in a dessicator until required, but necessarily exposed to the atmosphere before mounting in the STM.

  Nb results

A sample of results obtained tunnelling into Nb films is shown in figs  and , along with the theoretical conductance curves, calculated using Tc = 9.2 K, T = 2.2 K and 2D/kTc=3.89. The thermometer reading during these measurements was 1.5 K. Possible reasons for this discrepancy included a temperature difference of 0.7 K between the thermometer and the sample, or the existence of some additional smearing effect. In the superinsulated cryostat the thermometer was located in the sample space, about 20 mm away from the sample, so the temperature measurements ware expected to be accurate. Lifetime broadening was expected to be negligible. A further possibility is that the Nb had a lower Tc than 9.2 K, an effect frequently observed in thin superconducting films, mainly due to the presence of impurities (Tarte, 1996). A combination of small errors in the measured temperature and the transition temperature could account for the discrepancy.

The data shown in fig  were obtained using a tunnel current of 20 nA, for 10 mV applied bias, corresponding to a junction conductance of 2 mS. The data is the average of 16 scans of 512 data points between -10 and +10 mV, differentiated numerically.


Figure 8.3: Measured (solid) and Theoretical (dashed) conductance curves for Nb/Au junction, GNN = 2 mS. Theoretical parameters are Tc = 9.2 K, T = 2.2 K, 2D/kTc=3.89.

Fig  shows data obtained from the same sample, but with a tunnel current of 2.3 nA for an applied voltage of 10 mV, and hence a junction conductance of 230 nS. All other parameters were unchanged.


Figure 8.4: Measured (solid) and Theoretical (dashed) conductance curves for Nb/Au junction, GNN = 230 nS. Theoretical parameters are Tc = 9.2 K, T = 2.2 K, 2D/kTc=3.89.

In these measurements there is a consistent deviation from the theoretical curve, with the measured energy gap slightly smaller than the expected one. However, the agreement between theory and experiment is sufficiently good to give confidence in the ability of the STM to perform spectroscopic measurements of sufficient resolution to detect superconducting energy gaps.

8.2  Tunnelling into Niobium Nitride (NbN) and NbN/Au Films

To perform a systematic study on the behaviour of proximity effect structures, it was decided to use the superconductor NbN, because it has a highly stable, smooth, unreactive surface, is well-characterised, and has a fairly high transition temperature, allowing measurements to be taken down to T 0.1Tc.

  NbN film preparation

The films used in this study included bare NbN films deposited on sapphire, MgO or silica. Some of the films were coated with an overlayer of gold, of various thicknesses. The NbN films were produced by D.C. (in some cases r.f.) sputtering onto the substrate. With some of the films, acetylene (C2H2) was added during sputtering to produce NbCN.

The transition temperatures of the films ranged between 13.7 K for carbon-free r.f. sputtered films on both MgO and c-plane sapphire, to 17.6 K for NbCN deposited on m-plane sapphire. Typical values were around 16 K for films on all substrates. The thickness of the NbN ranged between 60-120 nm, but was typically around 100 nm.

A gold overlayer was deposited in situ onto a selection of the NbN films, and was typically 30 nm thick. In one run, a shutter was used to partially obscure some of the substrates during the gold deposition, producing a set of films with varying thicknesses of gold.

8.2.1  Bare NbN films

Given the stable surface characteristics of NbN, it was expected that good tunnelling conditions and repeatable I-V  characteristics would be obtained from tunnelling into bare NbN using a gold tip.

Data obtained from tunnelling experiments was fitted to theoretical BCS curves (thermally smeared) for different values of 2D/kTc. It was found that a value of 4.2 gave a good fit of experimental data with the theoretical model.

Fig  shows the conductance curve obtained from a bare NbN on MgO film with a measured Tc of 13.7 K. The sample was mounted in the continuous flow cryostat. The voltage range used was -16 mV to +16 mV, and the tunnel current set at 1 nA for a bias of 100 mV (R = 100 MW). The theoretical conductance curve (dashed) was calculated using T = 5 K (typical for the continuous-flow cryostat) and 2D/kTc, this value giving the best fit to the data.


Figure 8.5: Conductance curve for bare NbN on MgO film at 5 K. Parameters used to calculate fit were Tc = 13.7 K; T = 5.0 K; 2D/kTc

The fit in fig 8.5 is good, except for the region just outside the gap, where the conductance falls from its peak value rather faster with increasing bias voltage than is predicted by BCS theory. This phenomenon appears for both positive and negative bias, and may be related to phonon modes in the NbN.

Fig  shows the conductance curve obtained from a sample of bare NbN on sapphire using the superinsulated cryostat. This sample had a transition temperature of approximately 16 K. The measured temperature was below 1.5 K, but theoretical conductance curves were calculated both for T = 1.5 K and T = 3.5 K. The latter temperature gave an extremely good fit to the experimental data, as shown in the figure. The value of 2D/kTc used for the calculation was 4.2, as before.


Figure 8.6: Conductance curve for bare NbN on sapphire film. Parameters used for fit (dashed) were: Tc = 16 K; T = 3.5 K; 2D/kTc

The value of 2D/kTc=4.2 found in the present study is in general agreement with other reported data, although this value is near the high end of experimental values obtained. Barber et al.  (1995), who provided the films used for the present work, obtained results of D = 2.5 meV; Tc = 16 K, corresponding to 2D/kTc 3.6. LeDuc et al.  (1987, 1989) obtained values for D of 2.58 meV and 2.7 meV, giving 2D/kTc 3.75-3.9, while other reported values range from D = 2.4 meV (Beasley & Kircher, 1981) to D = 3.09 meV (Gurvitch et al. , 1985), corresponding to values of 2D/kTc between approximately 3.5 and 4.5.

From the good fit of data to BCS tunnelling theory, it may be concluded that the STM is capable of direct measurement of the tunnelling density of states, as described in section 3.4.2 (equation 88; fig 3.5).

8.2.2  NbN/Au bilayers

NbN films with a thin overlayer of gold were used to attempt to detect the proximity effect by STM. Samples on different substrates and with different thicknesses of gold were used, but in no case did the I-V  characteristic show any sign of any BCS energy gap. Experiments and theory of Kashiwaya & Koyanagi (1994) suggest that observation of an energy gap, of reduced width, is expected by tunnelling spectroscopy. They used NbC0.3N0.7/Au films, and took D = 0 in Au layer. Degradation of the interface was modelled with a d-function potential, and the energy gap width depended on the reflection coefficient for electrons approaching the interface. The conductance was not expected to fall to zero, since states of all energies exist in the normal layer, because D = 0. However, the measured gap width decayed into the normal layer, at a rate depending strongly on the interface quality.

The fact that the BCS energy gap was consistently observed with bare NbN films, and never observed with NbN/Au films, implies that the surface electron density of states of the structure does indeed have no energy gap.

Many of the I-V  characteristics obtained were linear, but one sample did contain certain areas which had repeatable non-linear I-V  characteristics consistent with a Coulomb gap. A CITS scan (see section 2.4.6) over a 250 nm square area revealed which sites had a reduced current at certain bias voltages, compared to those with a linear I-V  characteristic. The topographic and CITS images are shown in fig ; the darker areas in fig b are those with a depressed current (compared to areas with a linear I-V  characteristic) at a bias of 0.5 mV (and at 0.2 mV, 0.4 mV). The outlines of these depressed current regions are superimposed on the topographic image (fig a), showing a correlation between the topographic and spectroscopic data: the outlines of regions which exhibit a Coulomb gap encircle `hills' on the gold surface.


Figure 8.7: Topographic (a) and CITS (b) 250 nm×250 nm images of NbN/Au film at T = 4 K. The darker areas of (b) indicate places where the current is depressed at V = 0.5 mV; these areas are outlined in the topographic image (a), showing correlation between the topographic and spectroscopic data. In the bottom right hand area outlined in (a), the effect is much smaller.

The I-V  characteristics obtained at different points on the sample are shown in fig .


Figure 8.8: Current-voltage characteristics with NbN/Au sample. Graphs (a) and (b) were taken at different positions in the darker regions of the CITS image; (c) was taken very close to the edge of the region, and (d) within the light (linear I-V ) region of the image.

In the light areas, the I-V  characteristics were all linear, while in the darker areas, the I-V  characteristics flattened out around V = 0. Approximately 60 I-V  characteristics were measured at different positions on the surface, and the results obtained all followed the same pattern. There was variation in the exact form of the non-linearity, particularly near the edges of the darker regions in the CITS image.

The behaviour described has also been seen with a sample of YBCO/Ag (section ), in which a localised area had I-V  characteristics with a conductance dip around zero bias. In each case the regions affected were of the order of 100 nm across.

The size of the reduced conductance region in fig 8.8 suggests that the origin of the reduced conductance region is not the superconducting gap, since the gap in NbN is typically ~ 2.5 mV, while the observed reduced conductance region in fig 8.8a lies at |V| < 6 mV. The characteristics for tunnelling into the gold film, if it were sufficiently thin, would be expected to yield conductance peaks, and a dip similar to that of the bare superconductor, but smeared and reduced in magnitude, as a result of scattering of the electrons from the superconductor, traversing the normal region, into lower energy states. Conservation of states implies that conductance peaks will be present, as well as the zero-bias dip, but these were not seen.

It is proposed that the observed nonlinearity is the result of a Coulomb gap effect, since the two branches of the nonlinear I-V  curves are relatively displaced. Coulomb gaps (and staircases) have also been observed by tunnelling into a gold on mica film (section 7.2). The I-V  curve of fig 8.8a becomes asymptotically linear at high (positive and negative) voltages. The total gap width is about 12 mV, but a better measure of the Coulomb gap is found by taking the voltage displacement of the two asymptotes to the curve, giving 16 mV, corresponding to a capacitance of 10 aC. If the tip/sample separation is take to be 7 Å, the junction area can then be estimated as 8×10-16 m2, corresponding to a junction diameter of 320 Å. The mechanism by which the Coulomb gap is observed over some regions of the film, but not over others, is unclear. Comparison of the topographic and CITS images of fig 8.7 shows that the regions above which a gap is detected correspond to distinct topographic features, and that individual gold `hills' exhibit the gap either over their whole area, or not at all.

If the observed conductance dips were caused by the superconducting energy gap, it would be expected that the effect would be observed more clearly in the lower (thinner) regions of the Au film, whereas the regions where it is observed correlate with higher regions of the film.

If these conductance features are indeed caused by a Coulomb blockade effect, the correlation of regions where this effect is observed with `hills' in the topographic image may be due to a isolated particle attached to the tip. The Coulomb gap would then be observed only where tunnelling occurred through the isolated particle. The topography of the surface may cause different parts of the tip to contribute the major part of the tunnel current as the tip is moved over the surface. If part of the tip is covered by the attached particle, tunnelling from this part will produce a Coulomb gap conductance characteristic, whereas tunnelling from other parts of the tip will yield a linear I-V  characteristic.

8.3  High-Tc Superconductors

The high-Tc superconductor samples used in the tunnelling spectroscopy experiments were of the following types:

8.3.1  Bare YBCO films

Current-voltage characteristics of tunnel junctions with bare c-oriented YBCO (YBa2Ca3O7-d) films were measured initially, in order to determine whether the energy gap could be measured by STM, despite the formation of degraded surface layers.

In some of the tunnel junctions formed using STM, the conductance curve appeared to exhibit a BCS-like gap with `shoulders'. Examples of this energy gap are shown in fig .


Figure 8.9: Energy gap in YBCO resolved by STM at 5 K.

The fit to BCS theory was poor, as found by other authors: the conductance did not reach zero despite the fact that T 0.06Tc, and the curves exhibited the well-known linear conductance background. The gap features obtained were highly variable, disappearing completely in some cases. The gap shape was also extremely non-ideal. Estimation of the energy gap from such conductance curves is inaccurate, and dependent upon the method chosen, since various different data analysis techniques may be employed (e.g.  Srikanth, 1992, sec. 1.5.1; van Bentum et al.  1989a). The conductance curves shown in fig 8.9 all have different, and non-ideal, shapes, but in each case the energy gap estimated roughly by the position of the conductance maxima is approximately 20 meV, corresponding to (with Tc = 92 K) 2D/kTc 5. However, smearing of the BCS density of states, whether thermal or by some other mechanism, produces a curve with maxima higher than D/e, since the BCS density of states drops sharply to zero (instantaneously for T = 0) for V < D/e. The graphs shown are therefore compatible with values of 2D/kTc as low as the ideal BCS value of 3.5.

  Measurement of the reduced gap in YBCO

Another effect which tends to move the maxima to higher voltages is the linear conductance background usually measured by tunnelling experiments with YBCO. As an illustration of these effects, fig  shows the process of obtaining an estimate of 2D/kTc: fig a shows the raw conductance data, in which the curve has conductance maxima at approximately -21.5 2 mV and +202 mV. Subtraction of the linear conductance background on each side (fig b) yields a curve with maxima at -201 mV and +181 mV (fig c). Comparison with the BCS curve (2D/kTc=3.5; T = 92 K; D = 13.9 mV), for T = 40 K (purely in order to smear the BCS curve23) shows (fig d) that 2D/kTc=3.5 is a reasonable estimate.


Figure 8.10: Fit of experimental I-V  for STM junction with YBCO to BCS curve. Curve (a) shows the numerically differentiated conductance curve (solid line) and linear background (dashed); (b) shows the curve with the asymmetric linear background subtracted out; (c) shows the fit of experimental data (solid line) to a BCS curve with 2D/kTc=3.5. Tc = 92 K. BCS curves are shown for (short dashes) T = 5 K (actual temperature) and (long dashes) T = 40 K (to cause smearing of the BCS curve).

Lifetime broadening (Dynes et al. , 1978) can require anomalously large (as much as D/2) values of G (see equation 159) to fit the data (Srikanth, 1992, section 1.5.1). Alternatively, broadening of the gap may be accounted for by a distribution of gap energies (Kirtley, 1990) to account for inhomogeneities (such as variations in stoichiometry), which can cause the local gap energy to vary over the scale of the coherence length (if high-Tc superconductivity is d-wave, anisotropy in the gap could cause a similar effect). The shortness of the coherence length allows D(r) to vary over the scale sampled by the tip, so the tip can sample a range of gap energies.

The value of 3.5 tentatively obtained for 2D/kTc is in agreement with other results from tunnelling into c-oriented YBCO. Chandler (1993) reports a mean value for YBCO of 2D/kTc=5 (with standard deviation s = 2.2) from a sample of 70 results. However, tunnelling into c-axis YBCO typically yields values nearer the BCS value of 3.5, while tunnelling in the ab-planes usually gives larger values, in the range 4-6 (Burns, 1992, ch. 5) or as much as 8 (Kirtley, 1990). In some cases tunnelling experiments can sample both c-axis and ab-plane gaps, giving a double-humped conductance curve.

The zero-bias conductance is over 50% of the conductance just outside the gap `shoulders'. This may be due to some leakage current, or may be a true reflection of the YBCO density of states. If YBCO is a d-wave superconductor, the conductance curve would not be expected to have an almost flat zero conductance region near zero bias, since all electron energies would be permitted, for some k (see section  for further discussion and BSCCO results). The poor surface qualities of the YBCO, particularly after exposure to air, mean that tunnelling is occurring through degraded layers of altered-stoichiometry YBCO, rather than vacuum. The presence of this insulating layer will influence the tunnelling characteristics obtained, as will the nature of the superconducting region of the YBCO into which the tunnelling occurs, which may itself have non-ideal stoichiometry.

Other I-V  characteristics obtained by direct tunnelling into bare YBCO yielded similar results to those detailed in fig 8.9. Various irreproducible gap shapes were obtained, with conductance maxima typically occurring at between 15-30 mV.

8.3.2  YBCO/Ag films

As decribed in section 4.7.2, the Ag layer in an NS bilayer will not exhibit an energy gap, although the superconducting energy gap will affect the measured conductance by means of ballistic transport. There are, however, several reasons why Ag was used as an overlayer for the experiments, rather than a metal with a nonzero attractive electron potential, such as Al. The surface of Ag (and also Au, used for the BSCCO overlayer) is unreactive, and does not develop an oxide layer. In addition, Ag and Au have been found to cause much less surface disruption to the surfaces of YBCO and BSCCO than other metals (Weaver, 1993, section 7.2.2).

Gijs et al.  (1990) measured the proximity effect by coating YBCO with Ag (thus minimising surfce disruption), followed by Al, which enabled an oxide layer to be grown before deposition of a counterelectrode. Although the Ag layer exhibits no energy gap, since the attractive electron potential V = 0, so D = FV = 0, the pair density F is finite, decaying through the Ag, and into the Al layer. Hence, in the Al layer, both F and V are finite, yielding an energy gap. This idea could be utilised in future STM experiments to investigate the proximity effect.

Tunnelling characteristics of the junction between the Au STM tip and YBCO/Ag bilayers, with between a few hundred and a thousand Å ngströms of normal metal overlayer, were measured for several films, with both (100)- and (103)-oriented YBCO, in order to determine whether any superconducting gap was detectable by means of the proximity effect. No such effect was ever seen with this system, because the poor surface characteristics of the YBCO prevent the proximity effect from occurring between the YBCO and Ag layers. For the proximity effect to occur, a very good contact is required between the superconductor and the normal metal, which will not be the case if a thin layer of (insulating) degraded YBCO is present at the interface.

A large sample of tunnelling characteristics were measured, with different YBCO orientations and gold thicknesses, but results were inconsistent. Many of the conductance curves were linear, but a large number showed conductance dips around zero bias. The sizes of dips measured varied between 10 mV and 80 mV (full width), and the shapes of the dips varied enormously also. The reduction in conductance at zero bias varied between 10% and 70%, but the values were hard to estimate owing to the often asymmetric gap characteristics obtained.

For one (103)-oriented YBCO/Ag film, a CITS image revealed areas with a depressed current at V = -1 mV. In these regions, conductance dips were consistently seen at zero bias, whereas in the other regions, linear I-V  characteristics were obtained. The topographic (a) and CITS (b) images, 150 nm×150 nm, are shown in fig , in which depressed current regions appear lighter. The outlines of the depressed current regions are superimposed on the topographic image, indicating a correlation between the two images. This is particularly clear in the lower middle part of the topographic image, where the outline of the depressed current region curves around the silver `hill'. The regions of depressed current correlate with the lower (i.e.  thinner silver) regions on the topograph, evidence that the observed effect is related to the effect of the underlying superconductor, as described in section 4.7.2. The ballistic nature of the effect requires that the normal layer be thin for observation of any effect.


Figure 8.11: 150 nm×150 nm topographic (a) and CITS (b) image of (103)-oriented YBCO/Ag bilayer. The lighter areas show regions of depressed current at a bias voltage V = -1 mV. The outline of these regions is superimposed on the topographic image.

Experiments by Koyanagi et al.  (1991, 1992) with a-, c- and (103)-oriented YBCO/Au films, only the (103)-oriented films (and only a small percentage of those) were observed to display BCS-like conductance curves. This is unsurprising, considering the poor quality of the YBCO surface.

A sample of the conductance curves obtained in the depressed current regions is shown in fig . The conductance dips reached down to 10% of the background.


Figure 8.12: Conductance dips obtained tunnelling into depressed current region of YBCO/Ag bilayer. High-Tc energy gap corresponds to 2D/kTc=6.5.

The gap widths are approximately 50 mV (full width), implying D ~ 25 meV, and 2D/kTc ~ 6.5. This is in line with reported values for 2D/kTc for the a,b-planes of YBCO (2D/kTc ~ 4-8). (The use of (103)-oriented YBCO allows contact of the silver with the a,b-planes.) The measured gap width has not been greatly decreased by the normal layer, in contrast to the results of Kashiwaya & Koyanagi (1994), who found (with NbC0.3N0.7) that the experimental gap width reduced rapidly with thickness of the normal layer, particularly if the NS interface was poor. The zero-bias conductance was 15% of the conductance background. No conductance peaks were observed, but this may be due to the non-ideal nature of the NS interface.

8.3.3  BSCCO crystals

BSCCO (Bi2Sr2CaCu2O8) was chosen for further tunnelling experiments because it has a better surface stability than YBCO, and was obtainable in the form of single crystals (of dimensions approximately 1 mm×1 mm×0.1 mm). The surface stability was also important, so that good contacts could be made between the superconducting BSCCO and the Au overlayer, without an intervening degraded insulating layer of BSCCO.

  Tunnelling measurements of bare BSCCO crystals

Tunnelling into bare BSCCO crystals (using an Au STM tip) yielded many conductance curves with a linear conductance background. A sample of these characteristics are shown in fig .


Figure 8.13: Sample of conductance curves for Au tip/BSCCO tunnel junction showing conductance background. The horizontal scale in (d)-(f), which show a flattening-off of the linear background, is double that for (a)-(c).

The behaviour of the curve near zero bias is variable, but the asymmetry between positive and begative bias is a general feature. The background flattened out at higher bias, as shown in fig 8.13d-f. Other characteristics occasionally obtained showed a dip around zero bias (without BCS conductance peaks either side) as shown in fig .


Figure 8.14: Sample of conductance curves for Au tip/BSCCO tunnel junction showing dip around zero bias.

The zero bias conductance dips are rather wide (about 40-50 mV) to be due to the BCS energy gap, since this would require 2D/kTc 11-14 (for BSCCO, Tc=85 K). Chandler (1993) also obtained data with BSCCO indicating a large reduced gap, which fitted well to a Gaussian distribution of gap energies of mean 37.7 meV (corresponding to 2D/kTc=10.3) and width 24 meV. However, with the data shown in fig 8.14 there are no conductance peaks, so the nonlinearities cannot be attributed with any certainty to the superconducting gap. The more ideal characteristics obtained by some other authors using BSCCO (e.g.  Matsumoto et al. , 1992) suggest that the crystals used in the current experiments were degraded on contact with air. Contact with the atmosphere for a period of at least ten minutes is unavoidable with the STM system used, so without redesigning the system completely, samples which degrade on contact with air cannot be investigated in a satisfactory state.

  Other factors affecting energy gap detection

There are several mechanisms which may lead to an overestimate of the gap size. Formation of an SIS junction by contact of the tip with an isolated BSCCO island, from which tunnelling occurs into the bulk, would give an apparent gap twice as large as the actual value. Charging effects can displace the conductance curves to higher voltages, increasing the apparent gap by as much as 4D (McGreer et al. , 1989; see also section 7.1.8).

A possible explanation for the observed linear conductance background is that the surface degradation produces an insulating layer with embedded superconducting, normal or semiconducting islands, which can undergo charging effects. A broad distribution of particle sizes with capacitances as small as ~ 1 aF, corresponding to particles ~ 100-200 Å across (taking the tip/particle distance to be ~ 1 nm), is required to explain the linear background, which appears to flatten off at around 200 mV. Medina et al.  (1990) found that this model gave a conductance proportional to Vcoth(eV/2kT), which is linear for large V and rounded off for small V.

It is possible that some of the zero-bias conductance dips may be essentially the same effect as the linear background, but occurring at a smaller voltage scale, owing to larger tip/particle capacitances. The capacitances involved would need to be ~ 3 aF. It is therefore important to obtain more BCS-like features in order to be confident that the energy gap is responsible for the measured conductance curves.

  Measurement of the reduced gap of BSCCO

In a few of the junctions, convincing BCS energy gap-like features were observed. The conductance curve in fig a shows a BCS energy gap superimposed on a linear conductance background. The conductance at V = 0 falls to just 6% of the value at the `shoulders'. Fitting straight lines to the asymmetric linear background (fig b) and subtracting this background out24 yields a curve (fig c) with conductance peaks at -26 mV and +20 mV. The points between the conductance peaks where the conductance reaches the same value as the background level, which are often used to estimate the value of D, occur at -19 mV and +12 mV. Fig d shows this curve with a superimposed BCS curve calculated for Tc = 85 K; T = 5 K; 2D/kTc=6.


Figure 8.15: Conductance curve for Au tip/BSCCO tunnel junction showing possible BCS energy gap. Curve (a) shows the conductance data; (b) shows the linear conductance fit (dashed line); (c) shows the curve with the linear background subtracted out; (d) shows the curve fitted to the BCS curve for 2D/kTc=6 (Tc = 85 K; T = 5 K).

The value of 2D/kTc=6 obtained from this data compares well with values measured by other authors. Chandler (1993) reviewed 59 measurements of 2D/kTc, and found that the mean value was 6.7 (with standard deviation s = 2.2). Hasegawa et al.  (1994) found a value for 2D/kTc of 9-10, and also obtained a very low conductance at V = 0, and similarly-shaped conductance curves to those in the current study. Other measurements of the energy gap in high-Tc superconductors by tunnelling spectroscopy (planar junction, break junction, STM and point-contact spectroscopy) are reviewed in Kirtley (1990) and Hasegawa (1992).

  Symmetry of the BSCCO gap: d-wave or s-wave?

The low values of zero-bias conductance are evidence for s-wave rather than d-wave superconductivity, since in the latter case D(k) contains nodes in some directions. This implies that there exist allowed electron states at all energies (although only at some momenta), so tunnelling into the material should sample some of these states, resulting in an elevated conductance around zero bias. Ichimura et al.  (1995) considered the effect of d-wave superconducting gap anisotropy of the form:

D(k) = D0 cos(2f)
(197)
where f is the azimuthal angle. The resulting theoretical conductance curve reached zero at V = 0, but had a discontinuity in gradient (i.e.  it was V-shaped). This shape arises because as E = eV is reduced, it falls below D in more and more directions, until, at E = 0, only electron states with momenta along the nodel lines exist. The predictions of this model do not fit the data of fig 8.15, or of Hasegawa et al.  (1994) or Ichimura et al.  (1995).

While the weight of evidence for YBCO is in favour of d-wave symmetry (van Harlingen, 1995), it is possible that this is not generally true for the high-Tc cuprates.

  Gap broadening and linear background

The broadening apparent in the curves of fig 8.15 was fit to the lifetime broadening model of Dynes et al.  (1978), the best fit occurring for G = 2 meV), shown in fig . The fit is very poor, indicating that lifetime scattering is not the predominant source of broadening. Higher values of G give peaks which are too small, while lower values produce gap sides which are too steep.


Figure 8.16: Poor fit of BSCCO gap to lifetime broadening model; G = 2 meV.

The linear background was observed by most other authors, and indicates that the sample is non-ideal. The results may be influenced by inhomogeneities of the particular sample, and broadening may be due to local variations in D(r).

Other conductance curves obtained similarly yielded a BCS-like characteristic only for positive bias (fig a), and dips outside the gap region (fig b). Both of these characteristics were obtained in several immediately repeated measurements.


Figure 8.17: Other conductance curves obtained with Au tip/BSCCO tunnel junction. Curve (a) shows a BCS-like characteristic for positive bias only; (b) shows dips outside the gap region.

The conductance peak in fig 8.17a would imply 2D/kTc 2.5, far lower than reported values, and the extremely non-ideal nature of these data means that no reliable conclusions can be drawn from this dataset.

8.3.4  BSCCO/Au bilayers

In order to form a clean BSCCO/Au interface, the BSCCO crystals were cleaved in vacuum, whilst Au was being evaporated, in order that the freshly-exposed surface was immediately coated with Au, to a thickness of 30-50 nm. This avoided the problem of exposure of the BSCCO crystal to the atmosphere, since the crystal was only exposed once it had been coated with a protective Au layer. The small size of crystal used in this procedure (1mm ×1 mm×0.1 mm) made this procedure difficult, since the BSCCO could separate in such a way that only a few grains were left on one surface. The sign of a good cleave was a shiny crystal face on the uncoated part, and an equally-sized mound under the gold on the coated part. Positioning of the STM tip was also difficult to achieve with the system used, since the tip width was about the same as the crystal size, and the tip point could be at different positions on the circumference of the tip, depending on the orientation of the cut.

Conductance curves may be expected to be highly non-ideal, since the crystal temperature exceeds room temperature during preparation, owing to the proximity of the hot gold boat below it in the evaporator. The exposure of the crystal at high temperature to vacuum may also cause oxygen depletion of the BSCCO surface.

Conductance dips were measured using tunnelling into the BSCCO/Au structure prepared as described: a sample of the conductance curves obtained tunnelling from the Au tip into the Au/BSCCO structure is shown in fig .


Figure 8.18: Conductance curves for tunnelling from Au tip to Au/BSCCO structure (NINS). A zero-bias conductance dip can be discerned in curves (a)-(c), of full width 20 mV. Curve (d) exhibits only a conductance background.

In curves (a)-(c), the conductance almost reaches zero at bias V = 0. Such a low conductance is not expected for tunnelling into an NS bilayer where the normal metal has zero attractive electron interaction (V = 0, where V is the interaction potential), because in this case there is, strictly speaking, no gap in the normal metal (see section 4.7.2): all energies of electrons are allowed, and the features measured in tunnelling conductance curves are due to ballistic travel of electrons from the superconductor through the normal metal. Scattering in the normal metal smears out the conductance curve, by scattering the electrons to lower energies, obscuring the gap.

The width of the observed features can only be used to provide a rough magnitude for D, owing to the extremely nonideal conductance curve. If D ~ 10 meV, 2D/kTc ~ 2.5. The low value may be explained by phonon processes in the normal metal layer, which scatter some electrons from above D to lower energies.

A Coulomb gap effect does not appear to be the origin of the feature, because the linear background suggests that the BSCCO is affecting the conductance curve: the I-V  characteristics obtained with NbN/Au were almost ideal Coulomb gap candidates, while those with BSCCO/Au are not.

A 150 nm×150 nm topographic image of the surface taken at room temperature is shown in fig . The gold appears to be continuous, and consists of `hills' approximately 20 nm across. The continuity of the gold film confirms that the effects do not result from tunnelling directly into the BSCCO layer but are due to the proximity effect.


Figure 8.19: 150 nm×150 nm topographic image of BSCCO/Au taken at room temperature. The gold appears to be continuous.

Chapter 9
Point-Contact Spectroscopy of Superconductors: Results and Discussion

The important effect allowed by point-contact spectroscopy (PCS) is the detection of Andreev reflection, in which an electron reaching the NS interface from the N side can pair up with another electron and enter the superconducting condensate. The result is that a hole propagates along the path traversed by the electron, with precisely reversed momentum.

In an ideal (Z = 0) situation with a normal metal tip and bare superconducting sample, Andreev reflection would be expected always to result in a hole entering the tip, causing a twofold enhancement of the current for |eV| < D, since Andreev reflection then has a probability of 1. The theoretical I-V  graphs are shown in section 5.2 for different values of Z, but in all cases, the current should be enhanced near eV = D.

If the superconductor is coated with a normal layer, Andreev reflection can still enhance the current if the hole is retroreflected back into the tip. The effect is reduced by the presence of a magnetic field, which would curve the trajectories of the electron and hole away from the point contact, and also by phonon scattering in the normal layer (this effect is influenced by the normal layer thickness and mean free path).

The experimental set-up allowed the junction type to be varied continuously between tunnelling and contact, enabling results to be gathered as a function of junction resistance, temperature, and position in the sample (although it was not possible to move to another point of the sample while the tip was in contact with the surface, without causing damage to both).

The junction characteristics used for these experiments varied from typical STM values (I 0.01-1 nA; R 107-109 W) to point-contact regime measurements, with currents as high as I 100 mA (R 0.2 W). The current (and hence junction resistance) could be set at any level between these extremes, allowing the I-V  characteristics to be monitored as the junction type was gradually altered from tunnelling to point-contact.

9.1  Bare NbN films

It was found that the I-V  characteristic of bare NbN films changed completely as the junction resistance was lowered. The BCS energy gap filled in gradually as the resistance was decreased, until the zero bias conductance exceeded the normal state conductance, but new features also appeared when the junction resistance became sufficiently small.

As the temperature was increased towards Tc, the features in the I-V  characteristic altered, disappearing above T = Tc. It is concluded from this that the observed features are a phenomenon of the superconductivity of the NbN film. The conductance curves taken at several temperatures are shown in fig . The NbN film from which these data were obtained was 107 nm thick. The I-V  characteristics were measured between -60 and +60 mV, with the current set to 100 mA at V=20 mV, corresponding to a junction resistance of r = 0.2 W, (GNN = 5 S). The conductance curves for T = 3.8 K and T = 14.8 K are shown in more detail, in figs  and .


Figure 9.1: Point-contact spectroscopy conductance curves for bare NbN film at several temperatures. Curves measured with junction resistance r = 0.2 W (G = 5 S). Successive curves are offset vertically by 5 S.


Figure 9.2: PCS conductance curve for NbN film at T = 3.8 K.


Figure 9.3: PCS conductance curve for NbN film at T = 14.8 K.

The central features of these conductance curves are compatible with thermally-smeared BTK curves (discussed in section 5.2) for different values of Z. As the parameter Z is decreased from (corresponding to the BCS tunnelling behaviour) to 0 (perfect contact), the central dip reduces in depth and width, disappearing completely for Z = 0 (at higher values of Z for thermally smeared BTK curves). For the T = 3.8 K curve, fig 9.2, the positions of the conductance maxima are in broad agreement with the prediction of BTK, shown for Z = 0.84 in fig .


Figure 9.4: Comparison of PCS conductance curve for NbN film at T = 3.8 K with thermally smeared BTK model. Parameters used were: Tc = 17 K; T = 3.8 K; 2D/kTc=4; Z = 0.84.

This model accounts for the conductance maxima at about D, and the variable depth of the central dip, but the experimental peaks are broader than predicted. The broad conductance maximum within 20 mV of zero bias (with some effect extending as far out as 40 mV), and the sharp conductance minima are not accounted for by this simple model.

As the temperature increases towards Tc, the central dip in the conductance fills in, and the peak surrounding it increases, until for T close to Tc the feature becomes a zero bias conductance peak (predicted qualitatively by the BTK model, although the quantitative fit is very poor). In addition, as the temperature is raised, the conductance dips at higher voltages move towards lower voltages, in approximately the same way as D(T), except for a small deviation near Tc. The dependence of the dip positions on temperature is shown in fig , in which the bias voltages at which the conductance dips appear are normalised with respect to the T = 3.8 K values.


Figure 9.5: Behaviour of normalised conductance dip position with temperature for bare NbN film. The behaviour of D(T) is also shown, for Tc = 17 K. The variation of the dip position follows the variation of D(T) fairly well, except for a small deviation near Tc.

The data imply that the positions of the conductance dips are not simply related to the magnitude of the superconducting energy gap. However, the reduction in the voltage bias at which these features occur as T approached Tc, which followed the same general trend as D(T), and their disappearance at temperatures above Tc, indicate that they are an effect of the superconductivity of the NbN. The difference in the behaviour of the conductance dip voltages, and of D(T) is not believed to be due to a temperature difference between the NbN film and the thermometer, since the thermometer is located inside the STM shield25, very close to the STM head. It is not believed that a temperature gradient in the vacuum can could reasonably account for the discrepancy (as much as 2.5 K near Tc) between the two graphs, particularly since measurements of the transition temperature, using the thermometer inside the STM shield, gave the expected figure, from which the curves of D(T) were calculated.

The variation in the conductance curves obtained as the junction resistance is changed from tunnelling to point-contact character are shown in fig . As the junction conductance increases from 0.5 mS to 4 S, the zero bias conductance increases from virtually zero, for pure tunnelling, to a value slightly exceeding the background conductance, when the junction is a point contact with G = 4 S. In this point-contact conductance curve, new features appear in the conductance curve; these are conductance dips at V = 16 mV. These features are also shown in fig 9.1.


Figure 9.6: Conductance curves for NbN over a range of junction conductances. Successive curves are offset vertically by 1 unit. Normalised conductance is calculated as (differential conductance)/(absolute conductance at V = 20 mV); the absolute conductance is given to the right of each curve.

The main changes which occur when the junction character is changed from tunnelling to point-contact behaviour may be summarised as follows: the zero bias conductance dip due to the BCS energy gap `fills in' with increasing conductance, and the peaks either side become sharper, but their position does not change. This behaviour is substantially different from that predicted by the BTK model (section 5.2), in which the normalised zero bias conductance is expected to increase as the junction conductance increases, and hence the barrier strength, Z, decreases. However, the conductance peaks either side are expected to decrease in size, so that as the normalised differential conductance at zero bias increases towards 2, the peak height decreases towards 2. Although the conductance dip `fills in' as expected, the behaviour of the conductance peaks is the reverse of that expected, leading to a much higher than predicted excess current.

This excess current may be seen even more clearly in the curves of fig 9.1 for T close to Tc. The decreased value of D(T) and the increased thermal smearing cause the gap feature to be completely obscured by the large conductance peaks. It is, however, unexpected that the observed features are so clear for T so close26 to Tc.

Further sets of results obtained from the NbN film are shown in figs  and . These results were obtained from the same film, but the tip was moved between taking the two sets of data.


Figure 9.7: Conductance curves for bare NbN sample at various junction conductances. Curves are offset vertically by 1 unit.


Figure 9.8: Conductance curves for bare NbN sample at various junction conductances. Curves are offset vertically by 2 units. Negative conductance regions appear in the top two graphs.

The data in fig 9.7 show the BCS energy gap filling in, and the conductance peaks either side becoming higher, as the conductance of the junction is increased from 10 mS to 0.67 S, as before. Further increasing the conductance to 4.67 S causes the peaks to become much wider and higher, and at 5.33 S, dips in the conductance at certain well-defined voltages again appear.

Fig 9.8 shows strikingly different behaviour; the G = 67 mS graph exhibits a zero-bias conductance peak, which acquires a central dip as the conductance is increased to 2 S. The top graphs, for both of which G = 6 S, show regions of negative differential conductance either side of the central feature, with sharp dips at V = 6 mV and V = 7.5 mV. This is shown more clearly in fig .


Figure 9.9: Conductance curve for NbN film at G = 6 S.

9.2  Comparison with Results for Nb

Similar behaviour to that described above (fig 9.9) has also been seen with pure Nb films, although it was not possible to reach such a low value of T/Tc, owing to the lower Tc, nor such a low junction resistance, probably owing to the existence of a surface oxide layer. The data have shown similar features to those obtained using NbN. Data obtained for a junction conductance (absolute) of 400 mS (at V = 10 mV) are shown in fig . The results are compatible with the predictions of BTK theory, except for the conductance dips just outside the gap region. The heights of the conductance peaks are lower than predicted by the ideal theory (1.5 times the high voltage conductance, rather than double), which may be attributed to poor surface characteristics. The gap width is also less than expected (for Nb, with Tc = 9.3 and 2D/kTc=3.89, D0 = 1.56 mV). Experimental difficulties in obtaining a wide range of conductances for junctions with Nb suggest that NbN is a superior material to use for this purpose, as predicted.


Figure 9.10: Conductance curves for Nb sample at G = 400 mS. Some similarity with BTK predictions is shown, but peaks are insufficiently high, and narrow conductance dips either side of central peak are not predicted by BTK.

9.3  NbN/Au Bilayers

In the tunnelling experiments, the NbN/Au films showed no features related to the underlying NbN layer. The I-V  characteristics obtained were either linear, or showed features incompatible with a BCS energy gap; a Coulomb gap27 was seen in some cases (see section 7.2). It was concluded that the results obtained by tunnelling spectroscopy were unrelated to the underlying superconductor.

The low-resistance conductance curves obtained by point-contact spectroscopy were similar to those observed with the bare NbN films. In addition, the features seen with NbN/Au films also disappeared above T ~ 16 K, indicating that they were indeed due to the superconductivity of the NbN.

The conductance curves obtained using different junction conductances did not vary in a systematic way with the conductance, but instead changed rapidly as the conductance was increased past a certain value (which varied as the contact was altered), by pushing the tip into the sample. Current-voltage and conductance curves for junctions with currents, at V = 20 mV, of 30 mA and 300 mA are shown in fig .


Figure 9.11: I-V  and s-V curves for point contacts with NbN/Au. Graphs (a) and (c) are the I-V  and s-V curves for I = 300 mA at 20 mV; (b) and (d) for I = 30 mA at 20 mV.

The width of the feature is larger than D by at least 50% in each case. Current-voltage characteristics of the form of fig 9.11a were observed by Pankove (1966) with NS contacts between crossed Al and Nb wires, who claimed the effect was caused by exceeding the critical current in some region of the superconductor. The negative conductance region observed in some junctions was hysteretic, and not explained. The critical current hypothesis does not account for the results of fig 9.11a, since the negative differential conductance region also occurs for decreasing current magnitude.

The temperature variation, with a junction conductance of G = 3 S (the highest achievable with this sample and tip), of the I-V  characteristics obtained with NbN/Au by point-contact spectroscopy, is illustrated in fig , and the corresponding conductance curves are shown in fig . The central conductance region has double the background conductance, as predicted by BTK theory, but the high-conductance plateaus are far wider than expected, and extremely asymmetric.


Figure 9.12: Current-voltage curves for NbN/Au film using point-contact spectroscopy at temperatures between 4.8 K and 14.2 K. Successive curves are vertically offset by 0.1 A for clarity.


Figure 9.13: Conductance curves for NbN/Au film using point-contact spectroscopy at temperatures between 4.8 K and 14.2 K. Successive curves are vertically offset by 10 S for clarity. The flat central increased conductance region has exactly twice the background conductance (at temperatures up to 12 K), as predicted by BTK, but the plateau widths are very much larger than expected, and extremely asymmetric.

The decrease in the width of the central conductance plateau with temperature is unexpectedly rapid compared to the variation of D(T) with T well below Tc. The dependence of the positions of the conductance minima at positive and negative bias (which occur at both ends of the central conductance plateau) with temperature is shown in fig , in which the dip voltage is normalised with respect to the T = 4.8 K dip voltage. The behaviour of D(T) is also shown, calculated for Tc = 16 K, for comparison.


Figure 9.14: Dependence of normalised conduction dip position on temperature for NbN/Au film. The behaviour of D(T) is also shown, calculated for Tc = 16 K.

The more rapid decrease of the conductance dip voltages with temperature in the case where the NbN film has an Au overlayer suggests that the superconducting order parameter decays into the Au layer more rapidly when the temperature is closer to Tc, as predicted by theory. However, the other features in the conductance curves give no support to this hypothesis. The central conductance plateau is far wider than expected, extending from -7 mV to +30 mV in the lowest temperature measurements. According to BTK theory, the normalised conductance for Z = 0 is 2 for eV < D, but D/e < 3 mV for NbN. The features would be expected to be smaller still for tunnelling into an NS structure, particularly with a normal metal thickness as high as 1650 Å. Indeed, for T close to Tc it is surprising that any deviation from NcN characteristics is shown, since the proximity effect is usually only measurable with thin N-metal layers and low temperatures. The conductance variations would be expected to be lower in a proximity film, particularly at higher temperatures and thicknesses. It is clear that some additional mechanism is required to explain the results obtained.

Current-voltage (fig ) and conductance graphs (fig )


Figure 9.15: I-V  curves for NbN/Au film at several temperatures.


Figure 9.16: Conductance curves for NbN/Au film at several temperatures.

for a different NbN/Au film, with a 1000 Å Au overlayer (200 Å deposited immediately after the NbN; 800 Å several days later) are also shown. For clarity, typical low temperature (T = 2.6 K) I-V  and conductance curves are shown to a larger scale in fig .


Figure 9.17: I-V  (a) and s-V (b) curves for point contact junction with NbN/Au at 2.6 K.

As before, a low resistance junction was formed before increasing the temperature, and performing consecutive I-V  measurements. The transition temperature was found to be 16.80.2 K, from observation of the current as the temperature of the chamber was increased and decreased past this value. This is in agreement with the highest temperature at which non-linear I-V  relations were obtained, namely 16.8 K. The narrowing of the central conductance plateau is clearly seen in these curves, and is plotted in fig 


Figure 9.18: Comparison of plateau width for NbN/Au conductance curves with D(T)/D0.

, in which it is seen that the normalised width of the plateau follows D(T)/D0 fairly closely. The main deviation from the trend of the points is caused by the T = 10.6 K point, which, along with the T = 6.6 K point, also deviates from the trend in the form of its I-V  characteristic.

The absolute plateau width is also in good agreement with the value of D for NbN (D ~ 3 eV, compared to a plateau half-width of 3.7 K in the widest case). There are also conductance dips (in some cases negative), as before, at higher bias voltages. Again, the width of the observed features would be expected to be lower owing to scattering in the normal metal layer by phonons, particularly at higher energies (van Bentum et al. , 1988d).

A feature which is unclear from figs 9.15 and 9.16 is the sharpness of the zero bias conductance peak for T close to Tc. The I-V  characteristic at T = 16.8 K was measured both before and after increasing the temperature to 17 K, which made the sample go normal, and the I-V  characteristic become linear (only the first of these is shown in fig 9.16). These two conductance curves are shown in fig , which also indicates the repeatability of the measurements.


Figure 9.19: Conductance curve taken at T = 16.8 K for point contact junction with NbN/Au film. The two measurements were made before and after driving the sample normal.

The large conductance ratio is surprising for tunnelling into an NS bilayer so close to Tc. The form of the conductance curve is believed to be due to the reduction of D(T) with increasing T, which causes the conductance peaks shown in fig  (T = 4.5 K), typical of the conductance curves obtained at lower temperatures, to become closer, merging together into a single peak owing to thermal smearing, and obscuring the zero bias dip.


Figure 9.20: Conductance curve taken at T = 4.5 K for point contact junction with NbN/Au film.

Similar experiments were performed with a NbN/Au film where the Au thickness was between 2000 Å and 3000 Å (a failure in the crystal counter led to this uncertainty in the thickness). I-V  characteristics with conductance dips either side of zero bias were again obtained. A typical reproducible conductance curve taken at T = 2.0 K is shown in fig .


Figure 9.21: Conductance curve for point contact with NbN/Au at 2.0 K.

In addition, variously-shaped zero bias conductance peaks occurred with some of the point contacts formed, with widths generally of the order of 20 mV FWHM, and heights between 15% and 50% of the background conductance.

Asymmetrical conductance dips appeared at all temperatures, as previously seen with bare NbN. The conductance dip positions in the T = 2.5 K case were -39.0 mV and +49.6 mV, and the normalised (with respect to the 2.5 K values) dip positions plotted against temperature in figure .


Figure 9.22: Normalised conductance dip positions plotted against temperature.

The asymmetry in the conductance dip positions remains fairly constant as the sample is heated.

From the NbN and NbN/Au results, a wide range of effects are observed by point-contact spectroscopy. The interpretation of the results obtained is far less straightforward than for the tunnelling case, since other physical processes are allowed once the materials are in physical contact, such as Andreev reflection and phonon scattering of electrons back through the contact28. The far higher currents employed in point-contact spectroscopy can also give rise to large local heating effects.

9.4  Summary of Point-Contact Spectroscopy Results

The features which were reproducibly seen in the conductance curves obtained by PCS are summarised in this section.

For bare NbN films, sharp conductance dips occur at well-defined positive and negative bias voltages, at positions between 2D/e and 20D/e, moving to lower voltages as the temperature approaches Tc, and disappearing above Tc. The conductance dips at positive bias often occurred at voltages 15-25% higher than the negative bias conductance dips, and were also more pronounced.

The temperature dependence of the conductance dip positions is similar to that of D(T). In some cases the conductance dips are adjacent to the central conductance peak. The central conductance peak can be as much as 6 times the background conductance, and extend out as far as 2D/e. (For the Nb films, the central conductance peak was also flanked by sharp conductance dips, but these occurred at voltages less than D/e.)

The origin of the central maximum appears to be that, as the junction conductance is increased, the peaks at D/e broaden out, and the central minimum `fills in', progressively leading to a broad central maximum with (usually) a small conductance dip centred at zero bias. The broadening as the conductance is increased can produce a central broad maximum of width 2D, with increased conductance as far out as 6D.

For some of the contacts with NbN/Au, the conductance dips at positive and negative bias exhibited very large asymmetries in both their magnitudes and their positions (by factors of 6 and 3.5 respectively, in fig 9.13). As before, the positive bias conductance dips occurred at higher biases, and were larger. Between the conductance dips, a very wide plateau of almost constant conductance, double the background conductance, was obtained. However, other contacts showed symmetrical conductance curves (fig 9.17), with sharply-defined conductance peaks at D, and a shallow zero-bias conductance dip. In these cases, the conductance peaks were approximately 6 times the high-voltage conductance (as for some NbN I-V  curves), and the negative differential conductance regions occurred immediately adjacent to these. Apart from the anomalously large conductance peaks, and the adjacent negative conductance regions, the shape of the some of the curves (figs 9.17 and 9.20) is very similar to that predicted by BTK (Blonder et al. , 1982).

The temperature dependence of the conductance dip positions observed with NbN/Au films appeared to fall off more rapidly with increasing temperature than D(T).

The wide variation in the results obtained suggests a dependence on uncontrollable aspects of the point contact. However, some of the features, such as the appearance of conductance dips at sharply-defined voltages, and the broadening and increase in magnitude of the conductance peaks near zero bias were universally observed, in both NbN and NbN/Au films. The narrow conductance dips at V ~ 30-50 mV are reproducible, but have not been reported by other authors for proximity effect systems29.

A suggested explanation for the conductance dips given by Chandler (1993, section 7.2.1) and Speakman (1992, section 4.5.4.2) is that they occur when the critical current is exceeded in some region of the superconductor. (This model was proposed for junctions containing YBCO, and a general decrease in the voltage positions with temperature was found, and attributed to the decrease in critical current with temperature). The voltages at which the conductance dips due to this effect occur would depend upon the exact geometry of the NS interface. The problem with this explanation is that the junctions are voltage-biased rather than current-biased, so this model cannot be correct: when the voltage is decreasing, the current increases over the voltage range of the negative differential conductance region. This effect cannot be attributed to a critical current phenomenon.

Narrow conductance dips just outside the conductance peaks flanking the gap region have been observed in planar proximity-effect junctions (Adkins & Kington, 1969). Arnold (1978) showed that this dip was caused by a narrow dip in the density of states around E = D. Work by Wolf et al.  (1980a,b) and Arnold et al.  (1980) on Nr-Zb/Al junctions supported this conclusion. The mechanism by which a symmetrical negative conductance region appears is uncertain (the symmetry suggests that hysteresis caused by heating effects is not the origin of this effect). The appearance of these conductance dips just outside the gap region with a Nb films suggests either a thin normal metal contaminant layer on the surface, or that the end of the STM tip in contact with the Nb was affected by the proximity effect. However, the observation of conductance dips at positions just outside the central peak, when this peak is as wide as 2D, is not readily explained by a gap in the density of states at E = D.

Chapter 10
Conclusions and Suggestions for Further Work

10.1  The STM System

The STM system constructed for this project was capable of achieving atomic resolution with graphite at both room temperature and Helium temperatures, and of imaging local variations in the I-V  characteristics of the surface, enabling measurement of the energy gap in superconducting samples, at a small region of the surface. Conductance curves obtained by numerical differentiation of the I-V  characteristic fit well with theoretical models for the sample density of states.

There are several changes which could be made to the experimental procedure, which would improve the prospects of obtaining more accurate spectroscopic and topographic data. These include:

For the point-contact measurements (as opposed to tunnelling), the effects of electrical noise and vibrations are both less important, owing to the far higher currents involved, and the more stable geometry.

10.2  Coulomb Blockade Effects

Tunnelling into a thin gold film has yielded good `Coulomb staircase' and `Coulomb gap' I-V  characteristics, in agreement with theory and with other reported results. The capacitances involved have been as low as ~ 2 aF for the Coulomb staircase effect, and ~ 0.5 aF for the Coulomb gap effect, and it is proposed that the reduction in the tip/particle capacitance is responsible for the change from the `staircase' to the gap effect, as predicted by theory.

Coulomb blockade effects have also been detected in some of the tunnelling experiments with bare and coated superconductors, and are, in general, an important factor to consider in interpretation of spectroscopic data obtained by STM.

10.3  Tunnelling Experiments on Superconductors

10.3.1  Bare superconductors

Close fits of the conductance curves with the BCS density of states have been obtained for tunnelling experiments with Pb, Nb and NbN films, at a range of temperatures, and the measured energy gaps have been compatible with previous results.

Measurements of the energy gap in bare YBa2Cu3O7-d and Bi2Sr2CaCu2O8 have also been made: the values of 2D/kTc=3.5 for c-axis YBa2Cu3O7-d , and 2D/kTc=6 for Bi2Sr2CaCu2O8, are both near the middle of the range of reported values for these substances. Both materials exhibited linear backgrounds, and subtraction of these backgrounds showed an energy gap feature with conductance peaks. In the case of Bi2Sr2CaCu2O8, the conductance fell to just 6% of that outside the gap, which is possible evidence for s-wave superconductivity.

10.3.2  NINS tunnelling

A CITS image on NbN/Au has revealed areas with a conductance gap at zero bias (i.e.  dI/dV 0), but these have been attributed to a Coulomb gap effect, rather than to the electron distribution in the overlayer.

Scanning tunnelling spectroscopy of high-Tc NS bilayers has yielded energy gap-like features, with 2D/kTc 6.5 for (103)-oriented YBa2Cu3O7-d/Ag, and 2.5 for Bi2Sr2CaCu2O8/Au. The value for YBa2Cu3O7-d/Ag is in line with reported value of the ab-plane reduced energy gap, 2D/kTc ~ 4-8. The features were observed in the thinnest areas of the silver (shown by current-imaging tunnelling spectroscopy), as expected, and the gap feature was prominent (the zero-bias conductance was just 15%).

The low value of 2D/kTc obtained for Bi2Sr2CaCu2O8/Au may be attributed to phonon scattering of electrons into lower energy states in the normal layer reducing the apparent gap below the actual value (measured as 2D/kTc ~ 6). The gap features seen with Bi2Sr2CaCu2O8/Au were unclear, and estimation of the gap width rather arbitrary, owing to the poor shape, so calculations based on this estimate are unreliable.

For further work, it is suggested the Bi2Sr2CaCu2O8 film is coated first with a thin layer of Ag or Au, to minimise surface disruption to the superconductor, then with a layer of a superconducting metal (to be kept above its transition temperature), such as Al. In this way, an energy gap should be induced in the top layer, providing sharper features than could be observed with an Ag or Au film alone, in which ballistic transport effects are required for detection of the gap. It is believed that this has not previously been attempted by scanning tunnelling microscopy: this could improve upon the indistinct tunnel junction results by probing the surface with CITS to find an area where the proximity effect is more apparent. The existence of a true energy gap should also enable more accurate study of the decay of the pair density, F, in the Ag (or Au) and Al layers, by variation of the film thicknesses.

10.4  Point-Contact Spectroscopy of Superconductors

10.4.1  Bare superconductors

Point-contact spectroscopy of NbN films revealed a variety of conductance curve shapes. Conductance peaks were observed around zero bias, and there was generally agreement with BTK theory, but with other effects also occurring. Features, not predicted by BTK, which were observed included:

The mechanism for these additional features is unclear, but all features disappeared sharply at T = Tc, confirming that these were superconductive effects. Some of the curves obtained were similar to the predictions of BTK (except for the conductance dips), with conductance peaking around V = D/e, and dipping slightly near V = 0. Heating effects are likely to have an effect on low-resistance junctions.

10.4.2  NcNS structures

With NbN/Au films, many features similar to those for bare NbN films were observed. Large asymmetries between positive and negative bias features were measured. Conductance peaks appeared at V = D/e, with a shallow conductance peak around V = 0. Conductance peaks of 6 times the background conductance were seen.

The variation of the conductance dip positions with temperature was markedly faster for the NbN/Au films than for the NbN films (where the graph of normalised voltage against reduced temperature closely followed D(T)).

10.5  Concluding Remarks

It has been shown that the STM head designed and constructed for this work was capable of obtaining measurements of the density of states in normal and superconducting samples. Measurements of the energy gap in several superconductors have been made, yielding 2D/kTc values of 4.2 for NbN, 3.5 for c-plane YBa2Cu3O7-d, and 6 for Bi2Sr2CaCu2O8, all of which are in line with previous reports. The low (6%) zero-bias conductance observed with Bi2Sr2CaCu2O8 is evidence that it is an s-wave superconductor.

Current-imaging tunnelling spectroscopy of YBa2Cu3O7-d/Ag revealed that the thinner regions of the silver exhibited an energy gap of width 2D/kTc=6.5, in line with reported values for the ab-plane gap. Spectroscopy of Bi2Sr2CaCu2O8/Au yielded ambiguous data.

Point-contact spectroscopy of superconductors and NS bilayers yielded similar conductance curves, which exhibited features, not yet fully understood, which decreased in voltage as T was increased. In some NbN/Au curves, similarities to the predictions of Blonder, Tinkham & Klapwijk (1982) were observed, but large differences (anomalously high conductance peaks and negative conductance regions) existed. All features disappeared above T = Tc, proving the superconducting origin of the features.

Coulomb blockade (gap and staircase) effects were observed, with a smooth gold film, owing, it is believed, to the presence of a conducting particle attached to the tip (but not connected electrically). Results are in agreement with previous reports.

While the tunnelling experiments have performed largely as expected, and yielded useful data, the results of the point-contact spectroscopy experiments are not completely understood, and further work is required to study the effects of increasing the junction conductance. In particular, those features seen, which were not predicted by BTK, should be investigated further.

Chapter 11
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Where multiple papers with the same first author are listed, I follow the scheme recommended by Butcher (1992, pp. 253-5) for lists with large numbers of `et al. ' references, in which the references are listed in the following sequence:

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Footnotes:

1 Ceramics such as Macor are prone to microscopic hairline fractures which may cause STM instability after a few temperature cycles, if sharp inside edges exist (Adler et al. , 1991).

2 The silver paint was a suspension of silver particles in a volatile liquid, which evaporated to leave a silver residue, forming a good electrical and mechanical connection.

3 By closing the valve to the helium dewar, and continuing to pump on the helium in the cryostat, a temperature of 3.55 K was achieved.

4 The maximum voltage signal for lateral motion was 80 V, and that for motion in the z-direction was 185 V, but the total voltage applied to any segment could not exceed 225 V.

5 The Fermi function f(E) = 1/[1+exp(E/kT)]

6 The original expression of Simmons was a factor of two too large (van de Leemput and van Kempen, 1992). The corrected expression is given here.

7 The chemical potential, m, is the correct value, but this differs from the Fermi energy EF at finite temperatures by an amount of order 10-4EF at room temperature, and less at lower temperatures.

8 The Krönig-Penney potential consists of a semi-infinite row of attractive d-functions (representing ion cores).

9 Note that it is conventional to choose axes for a planar barrier such that tunnelling occurs in the X-direction, while in an STM junction it is more natural to consider the tunnelling to take place in the Z-direction, in which case this expression can be suitably modified.

10 The figure given is actually 10-7-10-8 cm-2, but clearly the indices should be positive.

11 For small tip/sample separations with a corrugated sample, (1×2) missing-row reconstructed Au(110), measured barrier heights were found to vary from 2 eV above topographic maxima to 0.1 eV for topographic minima. It was concluded that, for highly-corrugated metal surfaces, this technique for measuring the apparent barrier height could fail close to the surface.

12 The image planes are not exactly located at the surface of a jellium metal, but slightly outside it, for charges more than a few Bohr radii away, and retreating inside the surface for very small separations (Appelbaum & Hamann, 1972; Payne & Inkson, 1985)

13 The quantum-mechanical image potential, unlike the classical image potential, does not become infinite at the edges of the barrier, but smoothly connects to the bulk potential (Inkson, 1971; Payne & Inkson, 1985).

14 The expression of Simmons (1963) is too large by a factor of 2.

15 In the case where the height is negative, the electron energy exceeds the barrier potential, E > UB, and the transmission probability oscillates with `barrier' width, and is equal to unity for kW = np, k being the wavenumber corresponding to the negative barrier height, k = {2m(E-UB)/(h/2p)2}, and W the barrier width; in the case where a classically forbidden barrier exists, the probability decays as exp(-kW), where k is the decay constant corresponding to the barrier height, k = {2m(UB-E)/(h/2p)2}.

16 i.e.  for a given speed of tip motion over the surface, the maximum frequency present in its vibrations when it follows the contours in the surface sufficiently accurately to allow atomic corrugations to be followed.

17 The tip/sample separation and tip radius are arrived at by comparison of experimentally-observed corrugation in Au(110) with calculated charge-density contours at different heights above the surface.

18 The temperature readout registers temperatures as low as 3.7 K, but it is unclear how accurate these are, owing to the separation between the thermometer and the sample space. The figure of 5 K is suggested by the results of tunnelling experiments performed in this cryostat (section )

19 For bulk Sn, the transition temperature is 3.7 K, but the transition temperature increased with decreasing particle size, reaching 4.2 K for particles of radius 7 nm.

20 The measured voltage between peaks, using the raw data, exhibited small variations in the step widths, due to the background noise slightly shifting the apparent conductance peaks.

21 For all the conductance curves shown, the data have been obtained by numerical differentiation of the current data using a least-squares fit (typically using 5 points to calculate the gradient; see section 2.4.5). This has the effect of an additional smearing out of the data, and is equivalent to convolving the data which would be obtained from taking direct differences by the dataset: {...0, 0, 0.2, 0.3, 0.3, 0.2, 0, 0...}. The effect of convolving the theoretical conductance curves with this dataset (or with the equivalent 5-point version) is negligible even for data where the voltage range used is relatively large compared to D (e.g.  Vmax-Vmin = 14D), where the difference would be greatest. For this reason, the smearing effect of the differentiation routine may safely be ignored in all future discussion.

22 Theoretical conductance curves for finite-temperature NIS junctions were calculated by convolution (using FFT routines given in Press et al. , 1987) of the BCS density of states (value of D(T) calculated using a look-up table and interpolation with data from Mühlschlegel, 1959) with the bell-shaped thermal broadening function -f(eV)/V of FWHM 3.52kT/e, given in (158).

23 The mechanism of the smearing is not thermal, since T/Tc ~ 0.055, neither it is (predominantly) due to lifetime effects, since the curves resulting from this model have reduced or nonexistent conductance maxima.

24 The fitted linear background is linear down to V = 0, which artificially distorts the bottom of the resulting curve: using a background which is linear at high V, but smoothly curved at low V, such as Vcoth(eV/2kT) (Medina et al. , 1990) would give a more ideal curve shape.

25 The STM shield was a metal can, to reduced electromagnetic pickup, with a hole drilled through it to allow free flow of helium inside the can, preventing the formation of an internal vacuum.

26 T = 17.7 K would appear to exceed Tc = 17 K, but the sample temperature must indeed be lower than Tc; the discrepancy, of less than 1 K, is believed to have resulted from the previous recent heating of the sample. No discrepancies in the measured temperature of greater than 1 K have been seen to occur.

27 Both the Coulomb gap and the superconducting energy gap produce a depressed (ideally zero) conductivity at zero bias, but the energy gap must have conductance peaks (`shoulders') either side, to conserve states, whereas the Coulomb gap does not, resulting in a vertical relative displacement of the two halves of the I-V  characteristic.

28 Phonon effects are also seen in the I-V  characteristics of NIS tunnel junctions for strong-coupling superconductors such as Pb, but the interpretation of these effects is relatively straightforward, allowing deduction of the Eliashberg function, a2F(w).

29 Nguyen et al.  (1992) reported broad conductance dips in quantum well structures with Nb electrodes, which were due to a combination of Andreev reflection and multiple normal reflection, but the dips were far broader and shallower, and the geometry quite different.


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