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

List of Figures

Chapter 1
Introduction

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-T_c 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 .

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 

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.

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, V_X, V_X¢, V_Y, V_Y¢, and V_Z. The electrode configuration is shown in fig .

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.

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

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

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

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 q_x = [(2dx)/ L] and q_y = [(2dy)/ L], so the extra displacements are:

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:

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%.

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:

Writing the electrode voltage differences as:

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.

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:

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:

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, f_I, and that of the STM, f_S, which are the frequencies most likely to cause problems, the transfer function is nearly constant,

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 .

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 T_c 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, H_c(T) (see section ).

These two properties of a superconductor may be summed up as follows: the resistivity of the superconductor r = 0, provided T < T_c, and the magnetic field inside the superconductor B = 0, provided also that H < H_c(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 T_c. Observed values of T_c 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 T_c for an elemental superconductor is 9.26 K for niobium, which corresponds to a thermal energy k_BT_c of 0.8 meV, very much lower than the energy levels which are normally important in the theory of metals (E_F ~ 5 eV; (^h/_2p) w_D ~ 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 T_c. Above T_c 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 T_c, 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 T_c: for T < 0.5T_c, D(T)/D₀ > 0.95. The behaviour of D(T)/D₀ as a function of the reduced temperature t = T/T_c is shown in fig ; data are taken from Mühlschlegel (1959).

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

  Type II superconductors

A type II superconductor has two critical fields H_c1 and H_c2 associated with it. These tend to zero as T increases towards T_c.

When the applied field is below H_c1 the flux is excluded. Above H_c2 the flux penetrates totally, and the specimen reverts to normal. Between H_c1 and H_c2 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:

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 < T_c a fraction n_s(T)/n of the total number of conduction electrons are capable of flowing as a supercurrent. The density of superconducting electrons n_s(T) drops to zero as T approaches T_c, and approaches n as T falls well below T_c. 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

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,

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 kT_c of the Fermi energy are involved in superconductivity, and the momentum range Dp » kT_C/v_F, so Dx ~ (^h/_2p)/Dp » (^h/_2p) v_F/kT_c, hence the coherence length is defined as:

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:

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 @ w_D^-1 @ 10^-13 s, the distance travelled by a Fermi surface electron will be of order l @ v_F t @ 1000 Å, taking v_F 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 k_F, so these scattering processes must all occur in a shell of width (^h/_2p) w_D 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 w_D, and zero for higher frequencies. This implies the condition that, for an attractive interaction to occur,

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:

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

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

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 c_{k ­}^* and c_{k ­} 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:

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

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:

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

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

Minimising the total energy áY_G | H | Y_G ñ with respect to the v_k gives (Tinkham, 1975, pp 25-28):

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

A trivial solution to this equation is for D_k = 0, when v_k = 1 for all e_k < 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) w_D (see equation 52), a solution exists with:

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:

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

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:

  Critical temperature T_c

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):

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

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 c_k­^* acting on the BCS ground wavefunction (equation 58):

The required excitation energy is just E_k, 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¢, e_k¢ minus the expected KE if the state were not occupied, i.e.  the probability of occupation |v_k¢|² times the pair energy 2e_k¢:

For any given excitation energy E > D, there exist two k-states with appropriate energy, conventionally known as k _< and k _> , below and above k_F 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 N_S(E) is given by:

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

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 u_k and v_k to be real for simplicity):

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 |u_k|²), or the annihilation of an electron in state -k (if the state (k­, -k¯) is occupied, with probability |v_k|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 E_F. 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 E_F to above E_F, 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 E_F are electron-like, and below E_F, 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 n_S or |y|², or where the local density of superconducting electrons, n_S, varies with position, for which a few examples follow:

• normal metal-superconductor (N-S) interfaces

• the surface of a superconductor

• intermediate state of type I superconductors

• mixed state of type II superconductors

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 n_S. It is valid for temperatures close to T_c, 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:

The Ñy (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:

There are two cases to consider:

• If a > 0 the minimum free energy occurs when |y|² = 0, corresponding to the normal state, with zero density of superconducting electrons.

• If a < 0 the minimum occurs at:

  |y|2 = - a b

  (93)

  so there is a finite density of superconducting electrons, and the material is superconducting.

The critical field is given by:

3.5.2  The Ginzburg-Landau differential equations

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

3.5.3  The temperature-dependent coherence length

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

From equations 96 and 101,

3.5.4  Results from Ginzburg-Landau theory

Combining the BCS results that

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

Typically, for clean superconductors, k << 1. It can be shown (Solymar, 1972, pp. 8-9) that type I superconductors have k < 1/Ö2, and type II, k > 1/Ö2, 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 H_c2, but below a critical field H_c3 (Saint-James et al. , 1969, ch. 4). The value of H_c3 can be above or below H_c2 and/or H_c; this effect is exhibited by type II superconductors if H_c3 > H_c2, and by type I superconductors of H_c3 > H_c, 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 H_c2) and by resistivity techniques (giving H_c3, since the surface superconducting sheath short-circuits the current). The ratio H_c3/H_c2 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:

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)):

For the clean case, where the coherence length in the N-metal is small compared with the electron mean free path, x_N < l_N, the probability F of finding an electron pair a distance x from the interface (on the N side) is:

F = f(x) exp(-K|x|)

(112)

where f(x) is a slowly-varying function, |x| is large, and the penetration length K^-1 is given by:

K-1 = (h/2p) vN 2pkT

(113)

and v_N is the Fermi velocity in the N-metal.

Equation (113) implies that the penetration length becomes very large as T® 0. In this case, the form of F is:

F µ 1 x+x1

(114)

where x₁ is a constant.

For a dirty N-metal, x_N > l_N, the pairs diffuse from the superconductor into the normal metal. Equation (112) holds for large |x|, but K^-1 is now given by:

K-1 = æ ç è (h/2p) vN lN 6pkT ö ÷ ø 1/2   .

(115)

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-T_c Superconductors

The high-T_c 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.  YBa₂Cu₃O_7-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-T_c superconductors are discussed, along with some of the experimental difficulties posed by these features.

3.7.1  High-T_c Properties

The high-T_c 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-T_c 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-T_c 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-T_c 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 d_x²-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 d_x²-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-T_c cuprate superconductors.

The mechanism of high-T_c 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:

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.

  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 E_I, the work function on either side is effectively reduced by an amount E_I. The effective barrier heights f₁ = Y₁ - E_I and f₂ = Y₂ - E_I 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):

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 ).

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

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:

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 .

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

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 .

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 D^1/2.

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 e^ikx, 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:

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

  Steady state approach

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

. 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 v_x = (^h/_2p)^-1¶E/¶k_x, 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(E_x). 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):