Electrons in Solids

Contemporary topics

Richard A Dunlap

Department of Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia, Canada

Morgan & Claypool Publishers

Copyright © 2019 Morgan & Claypool Publishers

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organizations.

Certain images in this publication have been obtained by the author from the Wikipedia/Wikimedia website, where they were made available under a Creative Commons licence or stated to be in the public domain. Please see individual figure captions in this publication for details. To the extent that the law allows, IOP Publishing and Morgan & Claypool Publishers disclaim any liability that any person may suffer as a result of accessing, using or forwarding the images. Any reuse rights should be checked and permission should be sought if necessary from Wikipedia/Wikimedia and/or the copyright owner (as appropriate) before using or forwarding the images.

Rights & Permissions

To obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, please contact info@morganclaypool.com.

ISBN 978-1-64327-690-8 (ebook)

ISBN 978-1-64327-687-8 (print)

ISBN 978-1-64327-688-5 (mobi)

DOI 10.1088/2053-2571/ab2f2c

Version: 20190901

IOP Concise Physics

ISSN 2053-2571 (online)

ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise Physics

Published by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA, 94901, USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

Contents

Preface

Acknowledgements

Author biography

Part I The quantum Hall effect

1 The normal Hall effect

1.1 Introduction

1.2 The basic physics of the Hall effect

1.3 The Hall effect and holes

1.4 The effective mass tensor

1.5 Applications of the Hall effect

References and suggestions for further reading

2 The integer quantum Hall effect

2.1 Introduction

2.2 Discovery of the integer quantum Hall effect

References and suggestions for further reading

3 High magnetic field technology

3.1 Introduction

3.2 Solenoids

3.3 Magnetic core electromagnets

3.4 Superconducting magnets

3.5 Bitter solenoids

References and suggestions for further reading

4 Integer quantum Hall effect theory

4.1 Introduction

4.2 Quantum harmonic oscillator

4.3 Landau levels

4.4 Landau level degeneracy

4.5 Lifting of the Landau level degeneracy

4.6 Explanation of the plateaus

References and suggestions for further reading

5 The fractional quantum Hall effect

5.1 Introduction

5.2 Discovery of the fractional quantum Hall effect

5.3 Theory of the fractional quantum Hall effect

References and suggestions for further reading

6 Applications of the quantum Hall effect

6.1 Introduction

6.2 The resistance standard

6.3 Relationship of the quantum Hall effect to physical constants

References and suggestions for further reading

Part II Superconductivity

7 Superconductivity and magnetism

7.1 Introduction

7.2 The discovery of superconductivity

7.3 The effect of magnetic fields on superconductors

7.4 Diamagnetism and the Meissner effect

References and suggestions for further reading

8 Other properties of superconductors

8.1 Introduction

8.2 The London penetration depth

8.3 Flux quantization

8.4 The isotope effect

8.5 Specific heat

References and suggestions for further reading

9 BCS theory

9.1 Introduction

9.2 Cooper pair formation

9.3 The superconducting energy gap

9.4 The coherence length

References and suggestions for further reading

10 High temperature superconductivity

10.1 Introduction

10.2 The discovery of high temperature superconductors

10.3 Properties of high temperature superconductors

10.4 The theory of high temperature superconductivity

References and suggestions for further reading

11 The Josephson effect

11.1 Introduction

11.2 The Josephson effect

References and suggestions for further reading

12 Superconductor applications

12.1 Introduction

12.2 Power transmission

12.3 Particle physics

12.4 Fusion research

12.5 SQUIDs

References and suggestions for further reading

Preface

The manifestations of electric charge have been known since antiquity in the form of lightning, static electricity and electric potential produced by certain species of fish. The properties of electric currents in metals was studied extensively in the 19th century, although the details of the nature of these currents remained unknown until 1987 when J J Thomson discovered the electron. The development of quantum theory in the 20th century provided a means of describing the electronic band structure of solids and led to an understanding of the importance of both electrons and holes in the conduction process in solids and the differences between insulators, semiconductors and conductors. The present volume deals with two cases where the charge transport in a solid is not described by the normal band structure picture of the solid. These cases are related to the phenomena of the quantum Hall effect and superconductivity. Part I of this book deals with the quantum Hall effect, which is a consequence of the behavior of electrons in solids when they are constrained to move in two dimensions. Part II of the present volume describes the behavior of superconductors where electrons are bound together in Cooper pairs and travel through a material without resistance.

Acknowledgements

I am grateful to Nicki Dennis for her support and encouragement in her role as Commissioning Editor and Karen Donnison for all her work as Permissions Editor. I am also grateful to Chris Benson at IOP Publishing for his careful and efficient editing of the manuscript, and Melanie Carlson and Brent Beckley at Morgan & Claypool for handling the production of the printed version of the book.

Author biography

Richard A Dunlap

Richard A Dunlap received a BS in Physics from Worcester Polytechnic Institute in 1974, an AM in Physics from Dartmouth College in 1976 and a PhD in Physics from Clark University in 1981. Since receiving his PhD, he has been on the Faculty at Dalhousie University. He was appointed Faculty of Science Killam Research Professor in Physics from 2001 to 2006 and served as Director of the Dalhousie University Institute for Research in Materials from 2009 to 2015. He currently holds an appointment as Research Professor in the Department of Physics and Atmospheric Science. Professor Dunlap has published more than 300 refereed research papers and his research interests have included: magnetic materials, amorphous alloys, critical phenomena, hydrogen storage, quasicrystals, superconductivity and materials for advanced batteries. Much of his work involves the application of nuclear spectroscopic techniques to the investigation of solid-state properties. His previous books include: Experimental Physics: Modern Methods (Oxford 1988), The Golden Ratio and Fibonacci Numbers (World Scientific 1997), An Introduction to the Physics of Nuclei and Particles (Brooks/Cole 2004), Sustainable Energy (Cengage, 1st ed. 2015, 2nd ed. 2019), Novel Microstructures for Solids (Morgan & Claypool 2018), Particle Physics (Morgan & Claypool 2018) and The Mössbauer Effect (Morgan & Claypool 2019).

Part I

The quantum Hall effect

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 1

The normal Hall effect

1.1 Introduction

The experimental study of the influence of an applied magnetic field on the electrical properties of a material dates back to the 1850s when Lord Kelvin investigated changes of the resistance of a material by the application of a magnetic field. This behavior is now known as magnetoresistance. In the 1870s Edwin Hall tried to determine if the force exerted on a conductor carrying a current by an applied magnetic field acted on the conductor as a whole or only on the charge carriers in the conductor. His observation of an electric potential normal to the flow of current is now called the Hall effect. While the phenomena of magnetoresistance and the Hall effect can be described in terms of the classical behavior of charge carriers in a solid, more recently, quantum mechanical manifestations of these behaviors have been observed. Specifically, giant magnetoresistance (GMR), and later colossal magnetoresistance (CMR) and extraordinary magnetoresistance (EMR) have been discovered. These show an effect that can be more than 10^5 times as large as the conventional magnetoresistance. Under specific conditions, at low temperatures and in large applied magnetic fields, quantized Hall effect behavior has also been observed. In part I of the present volume, these quantum effects are contrasted to the well known classical behavior that has been observed for the electrical properties of solids in externally applied fields. In this chapter, we begin with an overview of these classical effects.

1.2 The basic physics of the Hall effect

The electrical conductivity of a material can be considered for a sample with geometry, as shown in figure 1.1. Here, the length, x_0, is much greater than the width, y_0, which is much greater than the thickness, z_0. An electric field is applied along the x-direction by closing a switch connected to a voltage source. This voltage creates an electric field

along the x-direction. For the present, we will consider that the current in the sample may be represented by the flow of electrons as shown in the figure. The acceleration, ax, of the electrons in response to the applied electric field will be

where me is the mass of the electron. A time t after the switch is closed, the velocity of the electrons will be

Current is defined as the charge transported per unit time and may be written as

where A is the cross sectional area of the sample normal to the current flow, i.e. A = y_0z_0. The current density, jx, is defined as

and this equation can be combined with equations (1.3) and (1.4) to give

It is customary to assume that the mean free time of the charge carriers, τe, is limited by scattering from other charges and phonons in the material giving

where the conductivity, σe, resulting from the current carried by the electrons is defined as

and the resistivity is defined as

Equation (1.7) is the intrinsic form of Ohm’s law, V = IR, which depends only on material properties, and not on sample dimensions.

Figure 1.1. Geometry of a current carrying sample, showing the sample dimensions and the coordinate axes.

Now consider an experiment where a magnetic field, B, is applied along the z-direction. The electrons flowing through the material will experience a Lorentz force

The magnetic field, B, sometimes called the magnetic induction, is related to the magnetic field strength, H, and the magnetization, M, as

In the case where a material has no magnetization, B and H are the same except for a numerical factor, the permeability of free space, μ_0.

The above situation is illustrated by the geometry shown in figure 1.2. The applied electric field is in the x-direction, Ex and this causes a current to flow in the x-direction, which is comprised of negatively charged electrons traveling in the negative x-direction as illustrated in figure 1.3(a). The Lorentz force on these electrons as given by equation (1.10) is shown in figure 1.3(b), where the cross product is a vector pointing in the y-direction and the force on the electrons (because of their negative charge) is in the negative y-direction. This force will cause the electrons to be deflected, as shown in figure 1.3(c). Because there can be no net flow of current out of the sample in the y-direction, there must be a force acting on the electrons to prevent them from moving continually in this direction. This force is provided by an electric field that is established in the negative y-direction to counteract the electron flow from the Lorentz force, as shown in figure 1.3(d). One way of looking at this is to realize that if electrons flow in the negative y-direction, then this will represent a redistribution of charge in the sample and will give rise to an electric field which will oppose this flow.

Figure 1.2. Geometry for the Hall effect experiment.

Figure 1.3. View of the sample in the xy plane. (a) Direction of current and electron flow showing the coordinate system and the direction of the magnetic field (out of the page), (b) direction of the cross product and the resulting force on the negatively charged electron, (c) trajectory of electron subject to the Lorentz force and (d) electric field which opposes flow of electrons in the y-direction.

The above situation can be expressed mathematically by writing down the y-component of the Lorentz force from equation (1.10) as

The reason for the minus sign in front of the Ey term can easily be seen by an examination of the directions illustrated in figure 1.3. Since, in equilibrium, the net force in the y-direction must be zero, equation (1.12) gives the electric field in the y-direction as

From equation (1.3) we can write the x-component of the equilibrium velocity in terms of the electron mean free time as

Combining equation (1.13) with equation (1.14) gives

Combining equations (1.5) and (1.14) gives the current density as

Solving equation (1.16) for eExτe/me and substituting into equation (1.15) gives

where RH is the Hall coefficient which is defined as

The Hall resistivity, Rxy, that is the resistance measured in the y-direction when the current is in the x-direction, is written as

This resistance is in contrast to the normal (or longitudinal) resistance, Rxx = Vx/Ix, which is the resistance in the x-direction when the current is in the x-direction. There is sometimes some ambiguity about the terminology used for the Hall resistance. It is sometimes referred to as RH, which can be confused with the terminology for the Hall coefficient. It is sometimes referred to as ρH or ρxy, which suggests that it is a resistivity (although in two dimensions resistance has the same dimensions as resistivity in three dimensions). We, therefore, adopt the terminology as in equation (1.19) to avoid confusion.

The experimental measurement of the Hall coefficient provides three pieces of very important information about the electrical properties of the material;

- (1) The sign of the charge carriers, in this case negative, because they were assumed to be electrons.

- (2) The charge per charge carrier, assumed to be one unit of electronic charge in the above.

- (3) The number density of charge carriers.

The experimental measurement of the Hall coefficient follows from a consideration of the right-hand side of equation (1.18). The electric field in the y-direction can be written in terms of the potential difference across the sample in the y-direction, Vy, as

and the current density in the x-direction follows from equation (1.5) as

These two equations give

The quantities on the right-hand side of equation (1.21) are all readily measurable as illustrated by the sample geometry in figure 1.1 and by the experimental schematic shown in figure 1.2. It is important not to extract current along the y-direction in the sample during the measurement of the potential difference across the sample, so a high input impedance voltmeter should be used.

A simple picture of electrical conduction in metals would suggest that the current is carried by electrons in the conduction band with a charge of one fundamental unit of electronic charge. The number density of charge carriers can be estimated on the basis of the assumed atomic valence state and the atomic density. Some results of Hall coefficients, as calculated by equation (1.18) (i.e. −1/ene), and experimental values, as determined by Hall voltage measurements according to equation (1.21), for some ‘simple’ monovalent metals are shown in table 1.1. The experimental results are in reasonable agreement with calculated values. They certainly show the proper sign for the electron charge and are consistent with a single electron per atom contributing to the conduction. However, Hall voltage measurements are subject to a number of uncertainties due to the experimental method and more accurate measurements of the Hall coefficient by methods such as helicon wave techniques yield results that are typically in much closer agreement with the calculated values. This result shows, at least for the materials in the table, that the assumptions used for the calculation of the Hall coefficient are quite reasonable.

Table 1.1. Calculated and experimental measurements of the Hall coefficient for some simple metals as discussed in the text. (Experimental values are from Hurd 1972 and references therein.)

1.3 The Hall effect and holes

The discussion in the previous section shows that a simple approach that deals with charge carriers as free electrons in metals provides a good description of the results obtained from Hall effect measurements. We know, however, that this simple approach is not suitable for all materials. There are two factors that this approach does not consider; firstly, that electrons, even in a metal, are not necessarily free and are constrained by the details of the band structure of the material and, secondly, in some materials, notably some semiconductors, the current can also be carried by holes. The derivation in the previous section can be repeated by assuming that the charge carriers are positively charged holes rather than negatively charged electrons. In the case of holes, the charge will be +e (rather than −e) and the number density, mass and mean free time will have values appropriate for the hole carriers, nh, mh and τh, respectively. This leads to a Hall field, analogous to equation (1.15), of

and a Hall coefficient, analogous to equation (1.18), of

Measurements of normal resistivity or conductivity cannot distinguish between currents carried by electrons moving in one direction and currents carried by holes moving in the other direction. However, a simple measurement of the Hall voltage can readily tell the difference between these two situations. The change of sign of the Hall coefficient is reflected, from an experimental standpoint, by a change in the sign of the Hall field and, hence, the Hall voltage.

Experimental studies of p-type semiconductors, as opposed to n-type semiconductors, clearly show positive Hall coefficients indicative of the existence of hole states carrying the current. Hall effect measurements also show some surprising, and highly informative, results for some metals that are clearly related to band structure effects. Table 1.2 gives the calculated and measured Hall coefficients for some ‘not-so-simple’ metals. The electronic structure of aluminum is [Ne]3s^23p and for indium it is [Kr]5s^25p. Thus, in a simple model, we would expect one conduction electron per atom for both Al and In as a result of the unpaired outer p electron. However, both Al and In show a positive Hall coefficient with a value consistent with one hole carrier per atom. This situation is the result of overlapping bands giving rise to the formation of hole states.

Table 1.2. Calculated and measured (by helicon wave methods) Hall coefficients of metals with currents carried by holes. Calculated values assume one hole carrier per atom. (Experimental values are from Kittel 2005 and references therein.)

Above absolute zero temperature we expect that all semiconducting materials will have both electrons and holes present. The majority carriers, which are the result of impurities, as well as thermal excitations, are the dominant carriers, and the minority carriers, which are thermally excited, occurs with a lower density. In the case where both electron and hole carriers are present, a thorough evaluation of the Hall coefficient requires a more detailed treatment rather than merely adding the negative (for electrons) and positive (for holes) contributions from equations (1.18) and (1.24) to the coefficient. This is because the relative contribution from each charge carrier to the total Hall field is related to its ability to respond to the applied magnetic field and this involves carrier specific characteristics, such as mass and mean free time. The question of mass will be discussed further in the next section, but for present we will consider values of me and mh for the electron and hole, as was used above. These factors, which appear, for example in equation (1.16), can be taken into account by introducing the carrier mobility, which is defined for electrons and holes as

and

for the electrons and holes, respectively. Including both electron and hole carriers in the derivation of the Hall coefficient yields the result

Since the mobilities are related to mass, as in equations (1.25) and (1.26), it is important to consider the meaning of carrier mass in detail.

1.4 The effective mass tensor

Although the mass of the charge carrier does not appear explicitly in the expressions for the Hall coefficients, equations (1.18) and (1.24), it is an important parameter in the description of charge transport in solids. For example, mass appears explicitly through the mobilities in equation (1.27) for systems in which both carrier polarities are present. It is also an important factor in the description of electron behavior in the presence of very high magnetic fields. In the case illustrated in figure 1.2 where there is no electric field in the x-direction and the magnetic field is sufficiently high, then the electrons will assume circular orbits in the material. This will be an important feature of the quantum Hall effect described below. In this case, the Lorentz force given by equation (1.12) with Ey = 0, can be set equal to the centripetal acceleration of the electron in a circular orbit of radius r;

The angular frequency of the electron in the orbit is related to its velocity by

Equations (1.28) and (1.29) can be combined to yield the angular frequency as

This frequency is known as the cyclotron frequency.

Although it may be unclear what the mass of the hole is, one might expect that the mass of the electron which has appeared in the equations above is the mass of a free electron. This is not the case for an electron in a solid as the behavior of the electron is governed by the band structure of the material. The mass in the above discussion is, therefore, not the free particle mass but an effective mass that appears in the equations of motion. Let us look at a simple one-dimensional example.

The dispersion relation for free electrons relates the energy, E, to the momentum, p (or equivalently wave vector, k), by the relationship

Taking the second derivative of this expression gives

from which the inverse mass may be defined as

If this approach is extended to electrons which are not free electrons (i.e. they are on bands which are not parabolic) then the mass in equation (1.33) is the effective mass, m*;

If this picture is extended to three dimensions, then we find that the effective mass can be (and usually is) different in different directions. This is obvious because the shape of the bands (that is the energy as a function of wave vector) is different in different directions and thus, the second derivative in equation (1.34) will be different in different directions. It is, therefore, necessary in a full treatment of this problem, to express mass as a tensor. This tensor will include terms representing (for example) the response of a particle in the x-direction to a force exerted in the x-direction but can also include terms representing the response of a particle in the x-directions to a force exerted in the y-direction. We can consider a simple two-dimensional classical analog of this situation to clarify this concept.

Consider a mass, m, on a frictionless inclined plane at an angle of θ with respect to the x-axis as illustrated in figure 1.4. To simplify the analysis the effect of gravity will be ignored. If an external force is applied in the x-direction to the mass as shown in figure 1.4(a) then the acceleration of the mass will be as illustrated. The movement of the mass is constrained by its interaction with the surface of the plane, just as the movement of electrons (or holes) in a solid is constrained by the features of the band structure. The acceleration as illustrated in figure 1.4(a) has both x- and y-components which may be written as

and

where aij represents the acceleration in the i-direction as a result of a force in the j-direction. For figure 1.4(b) the force is along the y-axis and the accelerations will be

and

Figure 1.4. Classical analog of the effective mass tensor; (a) force applied in the x-direction and (b) force applied in the y-direction.

On the basis of equations (1.35) to (1.38) the simple scalar form of Newton’s law, a = F/m may be written as a tensor equation

where the vector components of the acceleration are related to the vector components of the force by the inverse effective mass tensor. This may be written in simple component form as

This approach is analogous to the situation in three dimensions for electrons in a solid where the components of the 3 × 3 inverse effective mass tensor may be written in terms of second derivatives of the energy with respect to the spatial components of the wave vector as

1.5 Applications of the Hall effect

All applications of the Hall effect are related to the sensitivity of the electronic transport properties of materials to the presence of an external magnetic field. In general, practical devices that make use of the Hall effect are designed with materials which will optimize the effect of magnetic fields. Devices fall basically into two categories, those designed for measurement and those designed for control.

The Hall effect is one of the simplest accurate methods of measuring magnetic fields. If a Hall effect device as illustrated in figure 1.2 is constructed from a material with a known Hall coefficient, then the magnetic field can be obtained from a measurement of the Hall voltage according to equation (1.22). Rewriting this equation to give the magnetic field, we find

If the dimensions of the sample (i.e. z_0) and the current in the x-direction (i.e. Ix), then a simple measurement of the Hall voltage will provide the magnetic field.

The ability of a Hall effect sensor to detect magnetic fields makes it useful for a number of control applications. A simple proximity detector can be constructed using a Hall sensor and a permanent magnet. If the spatial variation of the field produced by the permanent magnet is known, then the relative locations of the magnet and the Hall sensor can be determined.

A Hall sensor can be used to measure the frequency of rotation of a shaft may be measured by placing a permanent magnet on the side of the shaft. This approach can be used with a feedback circuit to control rotational frequency.

References and suggestions for further reading

Ashcroft N W and Mermin N D 1976 Solid State Physics (Boston, MA: Cengage)

Hurd C M 1972 The Hall Effect in Metals and Alloys (New York: Plenum)

Kittel C 2005 Introduction to Solid State Physics 8th edn (Hoboken, NJ: Wiley)

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 2

The integer quantum Hall effect

2.1 Introduction

The quantum Hall effect, sometimes called the quantized Hall effect, was discovered in 1980, almost exactly a century after Hall’s discovery of the normal Hall effect, by Klaus von Klitzing of the University of Wurzburg. In 1985, von Klitzing, who had since moved to the Max Planck Institute in Stuttgart, won the Nobel Prize in Physics for his discovery. The quantum Hall effect is an extension of the normal Hall effect and describes the behavior of charge carriers in very large applied magnetic fields and at very low temperatures. It is only seen in systems in which the charge carriers are confined to move in two dimensions.

2.2 Discovery of the integer quantum Hall effect

In 1980, Klaus von Klitzing reported the existence of the quantum Hall effect in a MOSFET. The MOSFET (Metal Oxide Semiconductor Field Effect Transistor) is a device in which, under the appropriate conditions, the movement of the charge carriers (electrons in this case) is two-dimensional.

The cross section of a typical MOSFET is shown in figure 2.1. The MOSFET name comes from the structure of the device as shown in the illustration; a layer of conductive metal, a layer of insulating oxide and a layer of semiconductor. The ‘field effect’ term comes from the fact that the device is controlled by an applied electric field (rather than a current, as is the case for a conventional bipolar transistor). The device in the illustration is an n-channel enhancement MOSFET. ‘n-channel’ meaning that it is negative charge carriers (i.e. electrons) that conduct the current through the device and ‘enhancement’ meaning that the current is enhanced (rather than depleted) by the application of an appropriate electric field.

Figure 2.1. Cross sectional view of an n-channel enhancement MOSFET. S = source, G = gate, D = drain. The ‘+’ in ‘n+’ refers to the fact that this n-type semiconducting region is heavily doped with donor impurities. (Sergey Merkulov / Shutterstock.com / https://www.shutterstock.com/image-vector/mosfet-structure-531373156?src=e7jc8dItHkXGHqUteh6nnA-1-2.)

The operation of the n-channel enhancement MOSFET is illustrated in figure 2.2. Normally, when there is no voltage between the gate and the body, there will be no conduction between the source and drain. This is also true for voltages of either polarity applied to the source and drain, as in either case one of the junctions between the n+ source or drain and the p-type body will be reversed biased and the applied electric field will inhibit charge flow across the junction. When a positive voltage is applied to the gate (relative to the body), as shown in the figure, the metal-oxide-semiconducting structure acts like a parallel plate capacitor and positive charge collects on the metal electrode and negative charge collects on the semiconductor. The negative charges (electrons) are minority carriers in the p-type body region and the application of the voltage between the gate and body will cause the accumulation of these minority carriers and the expulsion of majority carriers from the region in the body immediately adjacent to the oxide layer. This region, referred to as an inversion layer, now acts as an n-type material and allows current to flow through the channel between the source and drain, in either direction, as there is now no junction of dis-similar semiconducting materials for the charge carriers to cross. In properly designed MOSFETs under appropriate conditions, the charge carrying channel between the source and drain can be made very thin and the charge carriers will be confined to move in a two-dimensional plane.

Figure 2.2. Formation of a minority electron carrier channel in the n-channel enhancement MOSFET.

Over the years, there has been a move to make electronic devices smaller and smaller, or equivalently, to incorporate more components into the same size device. This requires making each circuit element smaller. There is, however, a limit to how small electronic devices can be made. Early MOSFETs had channel lengths of a few μm, while modern devices have channels of a few tens of nm in length. One of the limiting factors to the reduction of MOSFET dimensions has to do with the properties of the oxide layer. When the oxide layer is of the order of 1 nm thick (corresponding to about 4–5 atomic spacings) then charge carriers can tunnel across this layer between the gate and the body. The appearance of tunneling reduces the ability to form an inversion layer comprised of minority carriers for the current to flow between the source and drain and reduces the ability of the MOSFET to effectively turn the source–drain current on and off by controlling the gate voltage.

The quantum Hall effect was discovered when von Klitzing was studying the Hall behavior of some Si MOSFETs (made by Gerhardt Parda at the Siemens Research Laboratory in Munich and Michael Pepper at the University of Cambridge) in applied fields near 15 T and at a temperature of about 1.5 K. The basic experimental setup for observing the Hall effect in a MOSFET is illustrated in figure 2.3. As in the normal Hall effect experiment described previously, the current flows in the x-direction (in this case between the source and drain) and the applied magnetic field is applied along the z-direction. The Hall voltage is measured perpendicular to both the current and the magnetic field along the y-direction as shown in the figure.

Figure 2.3. Configuration of n-channel enhancement MOSFET for quantum Hall effect experiment.

Typical results for this experiment are illustrated in figure 2.4. The figure shows the Hall resistance, defined in terms of measured quantities as Rxy = Vy/Ix and the longitudinal resistance defined as Rxx = Vx/Ix. In a normal Hall effect experiment we would expect that the Hall resistance, as given in equation (1.19), would show a linear increase with increasing applied magnetic field. We would also expect that the longitudinal resistance would be constant and independent of the applied field.

Figure 2.4. Hall resistance, RH = Rxy, and longitudinal resistance, Rx = Rxx, for a GaAs heterostructure as a function of applied magnetic field at temperature of 0.3 K. Note: in equation (2.1). (© 2012 IEEE. Reprinted, with permission, from Schurr et al 2012.)

The results of the quantum Hall effect experiment show results that are distinctly different than those for a normal Hall effect experiment. Specifically, we observe that the Hall resistance increases with increasing magnetic field but shows a series of well-defined plateaus. The longitudinal resistance is not constant but shows a series of maxima and minima where the minima occur for the same applied magnetic fields as the plateaus in the Hall resistance. Upon analysis of these results it is found that the plateaus correspond to values of the Hall resistance of the form

where h is Planck’s constant and ν is an integer. It is easy to see that the values of the plateaus in the Hall resistance in figure 2.4, which is plotted in units of h/e^2, occur at values of 1/1, 1/2, 1/3, 1/4, … . This quantization of the Hall resistance is now known as the integral quantum Hall effect because the plateaus occur at values of h/e^2 divided by an integer. The quantity h/e^2 has units of resistance and has a value of 25812.807557(18) Ω (Note: (18) is the uncertainty in the two least significant digits of the value). This quantity is known quite accurately, as both Planck’s constant and the electronic charge have been measured with great accuracy. However, the ratio of these two fundamental quantities is known more accurately (by an order of magnitude or more) from quantum Hall effect measurements than it is from the direct measurement of the fundamental quantities themselves. It is quite interesting that these values do not vary from one semiconducting device to another. This is an unusual feature in semiconductor physics where it is customary that properties vary greatly from one semiconducting sample to another as a result of even minor differences in composition. In fact, quantum Hall effect experiments require devices made from ‘dirty’ semiconducting materials; that is, semiconducting materials that are disordered and contain at least some impurities of different valence states. This is discussed further below.

The ‘zero’ in the longitudinal resistance are not regions of actual zero resistance, as we would expect for a superconducting material (see part II of the present book). As well, the value of the longitudinal resistance corresponding to the Hall resistance plateaus, does depend on experimental factors such as applied magnetic field and temperature. However, these values are ‘vanishingly small’ compared to the measured values for normal electrically conducting materials and experimentally can have values that are as much as 13 orders of magnitude smaller than those observed between the Hall resistance plateaus.

The experimental requirements for the observation of the quantum Hall effect are the use of an appropriate device in which the electrons are confined to move in two dimensions, as described above. The quantum Hall effect has been observed in, for example, MOSFET devices, gallium-arsenide heterostructures and, more recently in graphene. Here we concentrate primarily on the use of MOSFETs for the observation of the quantum Hall effect. The technology for the production of a suitable MOSFET device are described below. These are the techniques used for the manufacture of commercial semiconducting devices of this type. In addition, the observation of the quantum Hall effect requires appropriately high magnetic fields and suitable temperatures. The technology for the production of high magnetic fields is discussed in the next chapter. Although most quantum Hall effect experiments require low temperatures, these quantum effects have also been observed in graphene at room temperature.

References and suggestions for further reading

Ibach H and Luth H 1993 Solid State Physics: An Introduction to Theory and Experiment (Berlin: Springer)

Prange R E and Girvin S M (ed) 1990 The Quantum Hall Effect 2nd edn (Berlin: Springer)

Schurr J, Ahlers F and Kibble B P 2012 The ac quantum Hall resistance as an electrical impedance and its role in the SI Meas. Sci. Technol. 23 124009

Streetman B G and Banerjee S 2000 Solid State Electronic Devices 5th edn (New York: Prentice-Hall)

Yennie D R 1987 Integral quantum Hall effect for nonspecialists Rev. Mod. Phys. 59 781–824

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 3

High magnetic field technology

3.1 Introduction

As noted in the previous chapter, the observation of the quantum Hall effect requires the use of large applied magnetic fields. The production of magnetic fields that can be controlled experimentally requires the use of electric currents in a conductor to generate the magnetic field. Two approaches can be taken to maximize the field that is produced by an electromagnet; maximizing the current through the conductor and using the properties of ferromagnetic materials to augment the field. These two approaches are discussed in the present chapter.

3.2 Solenoids

The simplest arrangement for an electromagnet is an air core coil. A solenoid, as illustrated in figure 3.1, is most appropriate for the production of a large magnetic field. If the length to diameter ratio is large, then the axial field at the center of the solenoid can be quite uniform. It is easy to calculate the value of this field from Ampère’s circuital law. The integral form of Ampère’s law is written as

where the integral on the left-hand side of the equation is the integral of the magnetic field around a closed path and the integral on the right-hand side of the equation is the integral of the current density over the surface enclosed by the closed path. For the analysis of the solenoid, we construct a closed path as illustrated in the figure. The contribution to the first integral from the component of the path along the axis of the solenoid will be BL. The contributions to the integral from the sides of the path will be zero because the dot product will be zero. We can make the other end of the path far from the solenoid, so the intensity of the field will be zero. The right-hand side of equation (3.1) will be equal to the current, I, in the conductor times the number of coils, N, within a length L. We write

where n is the number of turns per unit length. Thus, equation (3.1) becomes

or

The magnitude of the magnetic field inside the solenoid can be increased by increasing the number of turns in the coil per unit length or by increasing the current in the wires (or both). There is a limit to both of these factors. Unfortunately, increasing the number of turns per unit length is most easily accomplished by decreasing the diameter of the wire, while increasing the current requires increasing the diameter of the wire. In either case an effective means of dispersing heat generated by Ohmic heating is needed.

Figure 3.1. A simple solenoid for the production of a region of magnetic field. The path for calculating the internal field using Ampere’s law is illustrated. (Reproduced from HyperPhysics by Rod Nave, Georgia State University.)

3.3 Magnetic core electromagnets

In order to optimize the magnetic field that can be produced by a coil of a particular size, it is convenient to utilize the magnetization provided by a ferromagnetic material. According to equation (1.11), the net magnetic field will be the applied field plus the field associated with the magnetization of the medium. Ferromagnetic materials, below the Curie temperature, have a net non-zero magnetization due to the preferential alignment of the magnetic moments associated with the atoms in the material. However, in most cases, the net bulk magnetization of a ferromagnet will be zero, as illustrated in figure 3.2(a). The figure shows that the magnetic moments are preferentially oriented within a region of the sample, called a domain, but the magnetization of the domains is oriented in random directions within the sample. Thus, although each domain has a net magnetization that is the vector sum of the individual atomic moments, the bulk material has zero net magnetization because the vector sum of the domain magnetizations is zero. The domains in a typical ferromagnet are of the order of a few 10 s of μm on a side, although the exact size depends on the composition of the material, the geometry of the sample, the method by which it was prepared and any externally applied stresses.

Figure 3.2. (a) Ferromagnet with randomly oriented domains and (b) magnetized ferromagnet.

Figure 3.2(b) shows a ferromagnet that has been magnetized. This is most easily accomplished by applying an external magnetic field to the sample. The interaction of the external field and the magnetization of the domains causes the domains to align with the magnetic field. In this case the sample has a net non-zero magnetization that contributes to the total field according to equation (1.11).

The reason for the formation of domains deals with minimizing the magnetic energy of the system. Figure 3.3 illustrates this principle. In figure 3.3(a) a ferromagnetic sample has a single preferred direction of its magnetization. This situation produces a macroscopic magnetic field in space outside the sample. There is considerable magnetic energy stored in this field and this configuration does not represent the most energetically favorable situation. In figure 3.3(b) the sample has been divided into two oppositely oriented domains. This configuration largely cancels the external magnetic field around the sample, thereby reducing the energy associated with the external field. Figure 3.3(c) shows an even more energetically favorable situation where the sample is divided into four domains with an internal field that circulates around the sample, virtually eliminating any external field and giving rise to a net zero magnetization. Another way of looking at this behavior is to realize that there is energy associated with the discontinuity in the magnetic field in going from inside a ferromagnetic material to free space and this energy is minimized when the field lines are parallel to the surface as seen in figure 3.3(c).

Figure 3.3. Explanation for domain formation based on magnetic energy; (a) single domain, (b) two domains and (c) four domains showing closure domains. External magnetic field lines are shown by the broken lines.

The alignment of the magnetic domains requires energy, so once the domains are aligned it requires energy to misalign them. When the applied field is removed from a magnetized sample, the domains will (at least partially) remain aligned in the direction of the field. It actually requires the application of an applied magnetic field in the opposite direction to return the magnetization to zero. This behavior results in the so-called magnetic hysteresis loop as illustrated in figure 3.4. Here the magnetic induction (see equation (1.11)) is plotted as a function of the applied magnetic field; this form of the hysteresis is sometimes called the B–H loop. Alternately, the magnetization may be plotted as a function of the applied field. When a sample with zero magnetization (resulting from a net zero vector sum of the magnetization of the domains) is subject to an applied magnetic field the induction follows the initial magnetization curve as shown in the figure. When the field is sufficiently large, all of the domains are aligned in the direction of the magnetic field and saturation induction (or saturation magnetization) is achieved. When the applied field is removed, then the residual magnetization of the partially aligned domains is called the retentivity (or the remanence). An applied field in the opposite direction (the coercivity) is required to reduce the magnetization to zero, as shown in the figure. Cycling the applied field between a large value in one direction and a large value in the opposite direction will cause the induction to follow the hysteresis loop as shown in the figure.

Figure 3.4. Magnetic hysteresis loop as described in the text. Bs = saturation induction, Br = remanence, Hc = coercivity. Hysteresis loops for (a) a soft magnetic material and (b) a hard magnetic material are shown. (Reprinted from Roozeboom et al 1998, with permission from Elsevier.)

Different materials, as a result of their composition and/or processing, show different hysteresis. Materials in which the domains are easy to align (and subsequently easy to misalign) are referred to as soft magnetic materials and have thin hysteresis loops as shown in figure 3.4. Materials in which the domains are more difficult to align and misalign have broad hysteresis loops as shown in the figure and are referred to as hard magnetic material.

Soft magnetic materials are suitable for applications such as cores for transformers or electromagnets. This is because the bulk magnetization induced by a relatively small applied magnetic field adds to the externally applied field as indicated in equation (1.11). Hard magnetic materials are suitable for use as permanent magnets since, once magnetized, they retain considerable domain alignment and magnetization even when the applied field is removed.

A region of high magnetic field can be produced in an air gap incorporated into a magnetic circuit comprised of a ferromagnet inside a solenoid, as illustrated in figure 3.5. The maximum field which can be produced in this way is limited by the saturation magnetization of the ferromagnetic materials used in the device. For practical materials (e.g. iron based ferromagnetic materials) the maximum achievable field is around 2 T.

Figure 3.5. Solenoid with a ferromagnetic flux path incorporating air gaps for regions of uniform high magnetic field. (https://en.wikipedia.org/wiki/Electromagnet#/media/File:Electromagnet_with_gap.svg)

3.4 Superconducting magnets

One approach to producing higher magnetic fields in the laboratory is to go back to the air core solenoid design and to eliminate the problem of resistive heating by using superconducting wire. Superconductors are discussed in detail in part II of this book. The use of a superconductor not only eliminates the problem of removing excess heat from the magnet, which is caused by the wire’s resistance, but once the current is injected into the coil it will circulate indefinitely without loss, thereby producing a magnetic field without the need for continuously supplying current to compensate for resistance in the wire.

The disadvantage of superconducting magnets is that they must be cooled to low temperatures, requiring the use of a low temperature cryogen or a suitable refrigeration system. Nevertheless, superconducting magnets are in common use for a variety of applications. One of the most common applications of this technology is in magnetic resonance imaging (MRI) devices used for medical diagnostics. Typical MRI instruments, as shown in figure 3.6, use magnetic fields in the range of about 3 T.

Figure 3.6. Superconducting magnet used in a magnetic resonance imaging (MRI) system. (abramsdesign / Shutterstock.com / https://www.shutterstock.com/image-illustration/3d-illustration-mri-machine-391]261144?src=zwu2KaVfISI8bdNmm1uwBA-1–12.)

Laboratory superconducting magnets commonly produce 10–12 T but there is a limit to the field that they can produce. The maximum field that has been achieved with a superconducting magnet is 26.8 T, although it is expected that fields up to about 30 T might be feasible with present technology. The limiting factor for superconducting magnets is the relationship between superconductivity and magnetism as discussed in detail in chapter 7. A current through a superconductor produces a magnetic field and a magnetic field (if it is sufficiently high) will destroy the superconducting state. Thus, there is a limit to the magnetic field which a superconducting magnet can produce, or more specifically there is a limit to the current which a superconductor can carry and still remain superconducting.

3.5 Bitter solenoids

Both iron core magnets and superconducting magnets have limitations that are based on the properties of known materials. One way of getting around these limitations is to go back to the simple air core solenoid and figure out how to make the current large enough to fulfil the requirements for a quantum Hall effect experiment. The solution to this problem is the Bitter solenoid, named after its developer, Francis Bitter. The solenoid does not consist of coils of wire but of cylindrical copper plates with holes in the center as illustrated in figure 3.7. Each plate is cut along a radial direction and the plate is bent so as to offset the edges of the cut line as shown in the figure. Each plate represents a portion of a helix and the bottom edge of the cut from one plate is connected to the top edge of the cut from the plate below it. Insulating disks are alternated with the copper plates to insulate them from one another. Each plate has holes which align with holes in the other plates and the insulating disks. These holes allow for cooling water to flow through the device. Figure 3.8 shows the details of a Bitter plate and an image of an assembled Bitter magnet.

Figure 3.7. Exploded view of the geometry of (a) conducting plates and (b) insulating spacers in a Bitter solenoid. (Reproduced from Bird 2004 with permission from IOP Publishing.)

Figure 3.8. (a) Details of Bitter plate showing holes for cooling water circulation and (b)-(d) the assembly of Bitter plates into a complete magnet. (Reproduced from Menard et al 2016 with permission).

The large cross sectional area of the conductor, combined with the effective water cooling, allows the Bitter solenoid to produce high magnetic fields. These are sometimes used in conjunction with a superconducting solenoid. The largest static fields available, 45 T, are produced by such an arrangement at the National Magnet Facility in Tallahassee, Florida (see figure 3.9).

Figure 3.9. Hybrid superconducting-Bitter magnet system at the National Magnet Facility in Tallahassee, Florida. (Larkablueyes/wikipedia commons / CC-BY-SA 3.0 / https://creativecommons.org/licenses/by-sa/3.0/deed.en.)

References and suggestions for further reading

Bird M D 2004 Resistive magnet technology for hybrid inserts Supercond. Sci. Technol. 17 R19–R33

Menard J E et al 2016 Fusion nuclear science facilities and pilot plants based on the spherical tokamak Nuclear Fusion 56 106023

Roozeboom F, Abedrabbo S, Ravindra N M, Walk H and Falter M 1998 Rapid thermal magnetic annealing as an emerging technology in field-annealing of thin magnetic films for recording heads Mater. Sci. Semicond. Process. 1 303–15

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 4

Integer quantum Hall effect theory

4.1 Introduction

The explanation for the integral quantum Hall effect is based on two factors; the presence of a large magnetic field and the fact that the electrons are confined to move in two dimensions. We begin with a closer look at the behavior of electrons in a magnetic field. In the normal Hall effect, the curved paths followed by the electrons are, in fact, portions of circles. In the absence of an electric field and if the applied magnetic field is sufficiently strong, then when the dimensions of the sample in the x–y plane are large enough, the electrons will travel in circular orbits. The angular frequency of the electrons in these orbits will be the cyclotron frequency, ωc;

where m* is the effective electron mass. If the motion of the electron is constrained to the x–y plane, then there will be no motion in the z-direction and the electron’s total energy will be given by the energy of its orbit. It is important to understand the conditions under which the electron can be considered to be moving in two dimensions. The spatial extent of the electron is determined by its de Broglie wavelength. This wavelength is related to the electron energy by

Thus, if the electrons are constrained within a plane of thickness equal to (or less than) the de Broglie wavelength, then their behavior can be considered as two-dimensional.

Classically, we expect that the energy of an electron in a cyclotron orbit could take on a continuous range of values. This is not true in the quantum case, where the energy becomes quantized and electrons can only exist in discrete energy states, referred to as Landau levels. In order to understand the quantization of the Landau levels, it is appropriate to begin with a brief overview of the quantum harmonic oscillator.

4.2 Quantum harmonic oscillator

Classically, the harmonic oscillator describes, for example, the motion of a mass on a spring where, in one dimension, the force acting on the mass is given by Hooke’s law as

where k is the spring constant. The potential that corresponds to Hooke’s law is given as

The quantum harmonic oscillator is described by the Schrödinger equation for the wavefunction of a particle with the potential as given by equation (4.4);

The solution to this differential equation will give the energy eigenvalues and the corresponding wave functions of the particle. Mathematically this is somewhat tedious, so we will not get involved in the details. The energy eigenvalues that are obtained from the solution to equation (4.5) give the quantized energy levels of the harmonic oscillator as

where the principle quantum number, n, can take on integer values of n = 0, 1, 2, 3… The oscillation frequency is given by the classical frequency of a mass on a spring as

4.3 Landau levels

In the presence of an applied magnetic field along the z-direction, the Schrödinger equation can be written as

where m* is the effective electron mass and ky is the y-component of the wavevector.

Mathematically equation (4.8) looks like equation (4.5) with the minimum in the potential shifted by along the x-axis. The energy eigenvalues obtained from equation (4.8) are, therefore,

where the cyclotron frequency is given by equation (4.1). The allowed quantum numbers are, as for the harmonic oscillator, n = 0, 1, 2, 3… These eigenvalues are illustrated in figure 4.1(a). The energy levels are discrete levels with the energy defined by the principle quantum number n. The Schrödinger equation also contains the quantity ky, that is the y-component of the electron wavevector. However, the energy eigenvalues are not dependent on ky. Therefore, the energy levels are degenerate in ky. The figure shows a plot of the density of states, that is the number of states per unit energy, on the horizontal axis, as a function of energy, on the vertical axis. In the next section, the degeneracy of the Landau levels, as shown in figure 4.1(b), will be considered quantitatively.

Figure 4.1. (a) Discrete Landau levels for electrons confined to two dimensions in the presence of an applied magnetic field. (b) Smeared Landau levels in a disordered material showing extended and localized states.

4.4 Landau level degeneracy

Because of the degeneracy in ky the number of allowed states per Landau level, N, can be viewed in a somewhat phenomenological way as follows. The number of levels is equal to the number of allowed cyclotron orbits within the sample. We write this as

where ϕ_0 is the fundamental magnetic flux quantum and A is the area of the sample in the x–y plane. Therefore, the quantity B/ϕ_0 is the number of flux quanta per unit area. The single electron flux quantum is defined as

and this will be discussed further in chapter 8. The number of allowed orbits per unit area, and hence the Landau level degeneracy, will be

4.5 Lifting of the Landau level degeneracy

The discussion above concerning the formation of Landau levels and their properties was based on the assumption that there is no interaction, other than that described by the Lorentz force, between the moving charge carrier and the applied magnetic field. In a real material, particularly one containing impurities, as are obviously present in the conduction channel in a MOSFET, the electrons will interact with these impurity atoms, as well as with any crystallographic defects in the material. These interactions will influence the energy of the electrons and will lift the degeneracy of the Landau levels. This will smear the Landau level energy, as illustrated in figure 4.1(b). The amount of smearing, ΔE, may be estimated on the basis of the Heisenberg uncertainty principle as

where τ is a measure of the mean electron scattering time.

Another way of looking at this is through a consideration of how an electron in a Landau level will interact with an impurity atom in the material. As indicated previously, it is important for the observation of the quantum Hall effect for the semiconducting material, specifically the body of the MOSFET, to contain both acceptor and donor impurities. The acceptor impurities, which are the majority impurities in an n-channel enhancement MOSFET, become negatively charged ions when they accept an additional electron. The donor impurities, on the other hand, become positively charged when they give up an electron. Thus, the electrons in the Landau levels will interact with both negatively and positively charged impurity ions.

Electrons in Landau levels which interact with negatively charged ions will have their energy decreased and electrons which interact with positively charged ions will have their energy increased. Electrons which avoid interactions (more or less) with impurities will have energies near the unperturbed Landau level value. The interactions with impurity ions (as well as other types of defects in the semiconductor) will lift the degeneracy in ky and will spread out the energy of the Landau levels, as shown in figure 4.1(b).

Electrons which generally avoid interactions with impurities are relatively free to travel about in the material in response to (say) an applied electric field. Electrons which interact substantially with impurities become trapped (at least at low temperature) and cannot travel throughout the material. This behavior gives rise to the formation of a band of extended states centered on the Landau energy value and regions of localized states on either side as illustrated in figure 4.1(b). It is important to realize in this picture that the total number of allowed states for the electrons remains fixed, it is only the degeneracy that is lifted by the interactions with impurities.

In a system of fermions, it is important to consider which energy levels will be occupied and which will be vacant. The Fermi energy is calculated by a consideration of the occupancy of the energy levels when the system is in its ground state such that the Pauli principle is obeyed. Ideally this refers to the situation when the temperature is at absolute zero. If the temperature is above absolute zero, then the occupancy of the levels will get smeared out in the vicinity of the Fermi energy. This means that there will be unoccupied states in the localized regions and there may be electrons from localized states that move into extended states. Thermal effects can, therefore, allow electrons from localized states to move thorough the material and the existence of such behavior will eliminate the features of the quantum Hall effect. It is, therefore, important to undertake quantum Hall effect experiments at temperatures that are sufficiently low that the thermal energy is small compared to the Landau level spacing. Specifically, this requires that

This condition is best satisfied in the case of low temperatures and high cyclotron frequencies (corresponding to high applied magnetic fields).

4.6 Explanation of the plateaus

A simple phenomenological explanation of the plateaus that are observed in the quantum Hall resistance (and the corresponding minima in the longitudinal resistance) follows from an investigation of equations (4.1), (4.9) and (4.12) and figure 4.1(b). As seen in figure 2.4, quantum Hall effect measurements are made as a function of applied magnetic field. Equation (4.1) shows that as the field increases the cyclotron frequency increases and equation (4.9) shows that this means that the spacing of the Landau levels also increases. Equation (4.12) shows that as the field increases, the degeneracy of each Landau level increases. Figure 4.2 shows what happens to the filling of the Landau levels as the applied magnetic field is increased. The diagram shows the filled levels in the ground state (T → 0 K) and the uppermost filled level represents the Fermi energy. It is the electrons that are at or near the Fermi energy that determine the conductivity through the material. As the applied magnetic field is increased the energy and degeneracy of the Landau levels changes as indicated in equations (4.1), (4.9) and (4.12). Figure 4.2 shows that as the field increases the Fermi energy drops, alternately passing through regions of localized and extended electron states.

Figure 4.2. Filling of the energy levels in the ground state up to the Fermi level for the n = 0 and n = 1 landau levels as a function of increasing applied magnetic field.

When the Fermi energy passes through a region of extended states, the electrons contribute to conduction in the same way that charge carriers are responsible for the normal Hall effect. In this case, the Hall resistance increases more or less linearly as a function of the applied magnetic field. As well, the longitudinal resistance shows a finite value similar to that exhibited in normal Hall effect experiments.

When the Fermi energy passes through regions of localized states the electrical properties do not change, as changing the magnetic field does not change the population of electrons in extended states. That is, the changes in the state occupancy that occurs as the field is increased in this region, do not change the ability of the charge carriers to contribute to current flow in the material and the Hall resistance remains constant, thus forming the observed plateaus. The corresponding longitudinal resistance becomes vanishingly small in the limit of T → 0 K. This behavior may seem counterintuitive but basically arises because when the electrons are confined to localized states there is no scattering and the electrons will drift without resistance in the y-direction in response to the small applied electric field.

This problem can be considered in a more quantitative manner in order to understand the details of the experimental results. When an integer number of Landau levels, ν, is filled, the total number of occupied states per unit area will be given in terms of equation (4.12) as

At this point the Fermi energy will be half way between the energies of two unperturbed Landau levels and will, therefore, be at the center of the localized states. This situation, therefore, corresponds to a plateau in the Hall resistance as a function of applied magnetic field. Following the discussion in section 1.2 for the normal Hall effect in three-dimensional samples, we define the Hall resistance in two dimensions as

where the Hall coefficient is given as

The sign is usually not included here as it only affects the polarity and not the magnitude of the Hall voltage. From equations (4.16) and (4.17), the Hall resistance is found to be

Substituting equation (4.15) for n_0 into equation (4.18) gives the quantization condition as

as observed for the plateaus in the experimental results.

References and suggestions for further reading

Ibach H and Luth H 1993 Solid State Physics: An Introduction to Theory and Experiment (Berlin: Springer)

Prange R E and Girvin S M (ed) 1990 The Quantum Hall Effect 2nd edn (Berlin: Springer)

Yennie D R 1987 Integral quantum Hall effect for nonspecialists Rev. Mod. Phys. 59 781–824

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 5

The fractional quantum Hall effect

5.1 Introduction

In 1982, David Tsui, Horst Stormer and Arthur Gossard of Bell Laboratories were investigating the quantum Hall effect in some heterojunction devices. These devices are similar to MOSFETs in the sense that the current is confined to travel in a thin plane. The results that they observed were similar to those obtained in previous quantum Hall effect experiments, except that additional plateaus that corresponded to values of the Hall resistance as given in equation (4.19) with non-integer values of the parameter ν were seen. The results of these observations, as well as a possible explanation for their features, are described in the present chapter.

5.2 Discovery of the fractional quantum Hall effect

The devices used by Tsui and co-workers had a lower level of disorder (i.e. impurities, defects, etc) than devices that are typically used for integer quantum Hall effect experiments, and this feature is important for the behavior described below. Typical results illustrating the fractional quantum Hall effect are shown in figure 5.1. Plateaus in the Hall resistance occur for simple fractional values of ν in equation (4.19), although these plateaus are not as well defined as those which occur at integer values of ν. Corresponding to these plateaus are minima, although not ‘zeros’, in the longitudinal resistance. This phenomenon is known as the fractional quantum Hall effect. For a number of years, quantized states were observed only for fractional values of ν with odd denominators. In 1986, Robert Clark and co-workers at Oxford University found suggestions of a state with a Hall resistance of 2h/5e^2, i.e. ν = 5/2, and this state was confirmed in 1987 by Tsui and co-workers. Other even denominator plateaus are now known, although by far the majority of plateaus correspond to odd denominators and theories have been most successful at describing the presence of these odd denominator plateaus. To date more than 70 different fractional plateaus have been observed.

Figure 5.1. The fractional Hall effect showing the Hall resistance and the normal resistance. (Reproduced from Willett et al 1987, Copyright (1987) by the American Physical Society.

5.3 Theory of the fractional quantum Hall effect

While the description in the previous chapter provides a suitable explanation for the integral quantum Hall effect, it does not explain the appearance of less well-defined steps in the Hall resistance at fractional multiples of h/e^2. We can see some general features of the fractional quantum Hall effect by returning to the description of the integer quantum Hall effect and looking carefully at the reasons for this behavior. It is easy to see from a direct comparison of equations (4.12) and (4.15) that the quantum number ν is merely the total number of occupied states divided by the number of states per Landau level. This is the number of filled Landau levels and the quantum number

is referred to as the Landau level filling factor. Obviously, for the integer quantum Hall effect the filling factor is an integer. Integer values of ν correspond to filled extended states, i.e. the Fermi level is within the next higher region of localized states. The following, phenomenological approach explains the presence of the odd denominator fractional plateaus. Fractional values of ν correspond to the uppermost region of extended states being partially filled. The presence of a plateau for a specific fraction of the extended states being occupied, i.e. 1/i of the states, may be explained if the charge carriers are quasiparticles with an effective charge of e* = e/i. Obviously, charge conservation must be considered in this approach and it is appropriate to view the quantum of charge, e, to be divided into i spatially distinct fractional charges e/i. Returning to equation (4.11), we define a flux

where i is an integer. This leads to

When the total number of occupied states is an integer, j, multiple of equation (5.3) a plateau will appear in the Hall resistance. Thus,

and

which corresponds to equation (4.19) with a fractional value of the quantum number

Therefore, the basic integer quantum Hall effect approach is preserved with the difference being in the characteristics of the fermions themselves. Theoretically fractional charges are defined such that

giving rise to the odd denominator plateaus that are observed experimentally.

It is frequently observed that a series of fractional plateaus will occur for values of the quantum number

for integer n = 1, 2, 3, 4, 5, …, and

for integer n = 2, 3, 4, 5, …. These states correspond to

and

respectively. The relevance of these sequences of values is readily seen by the sequential minima in the longitudinal resistance curve shown in figure 5.1.

A more rigorous theoretical approach views the charge carriers in the system as composite fermions comprised of electrons coupled to an even number of flux quanta (although this picture should not be taken too literally). The composite fermion must contain an even number of flux quanta (in addition to the electron) so that it remains a fermion (and does not become a boson).

References and suggestions for further reading

Eisenstein J P and Stormer H L 1990 The fractional quantum Hall effect Science 248 1510–6

Willett R, Eisenstein J P, Störmer H L, Tsui D C, Gossard A C and English J H 1987 Observation of an even-denominator quantum number in the fractional quantum Hall effect Phys. Rev. Lett. 59 1776–9

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 6

Applications of the quantum Hall effect

6.1 Introduction

The quantum Hall effect does not show any features that are likely to be of any great direct economic use. Added to this is the fact that its observation requires a combination of very low temperatures and very high magnetic fields which, at present, is only obtainable in research laboratories. However, the quantum Hall effect, both the integer effect and the fractional effect, have made some important contributions to science.

The integral quantum Hall effect provides some interesting information about the basic physics of electrons in two-dimensional semiconducting materials. The fractional quantum Hall effect emphasizes the complexity of the behavior of electrons in two-dimensional systems and suggests the need to expand our basic thinking about the properties of electrons in solids. In both cases, the interest in the quantum Hall effect has led to renewed interest in semiconductor research and may have indirectly been responsible for some of the progress in the development of semiconducting devices made during the past two and a half decades.

6.2 The resistance standard

The most direct use of the results of quantum Hall effect experiments, however, has been in the field of metrology, that is, the science of measurements. In the SI system (Système international d’unités) there are seven base units. In addition to the meter, kilogram and second, used for distance, mass and time, there are units for amount of substance (mole), temperature (K), luminous intensity (candela) and electric current (ampere). The ampere is defined as the current flowing through two parallel conductors that results in a specific force between them. Specifically, one ampere is the current flowing through each of two parallel conductors one meter apart in vacuum that will produce a force between the conductors of 2 × 10−7 newtons (N). All other electromagnetic units are derived from the ampere and other fundamental units. For example, the unit of power, the watt, is defined as one joule/second or kg m^2 s−3. From Ohm’s law, the watt is

so that

The ohm is obtained from

The standard by which the ohm is defined has changed over the years since it was first used in the mid-1800s. Early definitions of the ohm were often based on the electrical resistivity of a given length of a particular material (typically iron or copper) with a specified diameter or cross section. However, the actual resistance of such a standard was only as accurate as the dimensions of the material and depended, as well, on the purity and thermal treatment of the material. In 1863 standard coils of wire with a resistance defined as one ohm were deposited in the Kew Observatory in the United Kingdom. However, even the resistance of these standard coils is subject to fluctuations due to changes in temperature, humidity and atmospheric pressure.

Since 1990, the quantum Hall effect has been used to define the ohm. The von Klitzing constant, RK, is defined as

and has been measured by quantum Hall effect experiments to be

It is interesting to note from a quantum mechanical standpoint that the von Klitzing constant is merely the ratio of the fundamental flux quantum to the fundamental charge quantum. The measurement of the von Klitzing constant by means of the quantum Hall effect, therefore, provides a definition of the unit of electrical resistance, the ohm.

6.3 Relationship of the quantum Hall effect to physical constants

The von Klitzing constant may also be related to the fine structure constant, α. The fine structure constant is a dimensionless constant that was first defined by Arnold Sommerfeld in 1912 as the ratio of the classical velocity of an electron in the first Bohr orbit of a hydrogen atom to the velocity of light. We can understand the physical significance of this quantity by writing the classical equation of motion for an electron in a hydrogen atom

Bohr assumed that the angular momentum of the electron in its orbit was quantized in units of Planck’s constant as

Combining equations (6.6) and (6.7) gives

For the n = 1 level, the fine structure constant is, therefore, defined as

While Sommerfeld’s original definition of the fine structure constant involved the Bohr orbit velocity and the speed of light, it is currently viewed in the context of its definition in equation (6.9) as a dimensionless coupling constant that quantifies the electromagnetic interactions between charges and photons.

The relationship of the quantum Hall effect and the fine structure constant may be viewed in the following way. The permittivity of free space, ε_0, and the permeability of free space, μ_0, are related to the speed of light, c, as

In the SI system, the speed of light is defined to be exactly 2.997 924 58 × 10^8 m s−1. The speed of light is an exact quantity in the SI system because (in 1983) the unit of length, the meter, was defined in terms of this value for the speed of light. Since in the SI system, μ_0 is also a defined quantity, μ_0 = 4π × 10^7 m kg s−2 A−2, then ε_0 is also defined exactly. Thus, the definition of the fine structure constant from equation (6.9) may be written in terms of a single measurable quantity, the von Klitzing constant, as

Since the von Klitzing constant is known very accurately, equation (6.11) serves as alternative means to equation (6.9) for determining the numerical value of the fine structure constant. Results of such measurements are typically presented for values of α−1 rather than α, as α−1 is very close to the integer 137 and allows for a more convenient comparison of values. Figure 6.1 shows values of the fine structure constant as obtained by different methods. These include the calculation of the fine structure constant by equation (6.11) on the basis of the measured von Klitzing constant. The value based on RK−90 is shown by the solid vertical line and the limits of uncertainty are given by the broken vertical lines. The value based on the most recent value of the von Klitzing constant, RK, from equation (6.5) is shown with the appropriate uncertainty indicated. Other values as obtained by methods described in the figure caption are also shown. All measured values fall within the limits set by the RK−90 value, that is the known value of the von Klitzing constant at the time the standard definition of the ohm was established, 1990. However, all values are not consistent with each other in the context of their uncertainties. The value obtained from RK, is consistent with all except the proton gyromagnetic ratio measurement. The precise determination of the von Klitzing constant by means of quantum Hall effect measurements serves as a potential standard for establishing the value of the fine structure constant.

Figure 6.1. Values of α−1 (referenced to a value of 137.03) as obtained by different methods;

- h/mn = ratio of Planck’s constant to the neutron mass

- RK = quantum Hall effect using value of the von Klitzing constant as in equation (6.11)

- RK−90 = quantum Hall effect using the value of the von litzing constant from the 1990 standard

- ae = anomalous magnetic moment of the electron

- Δνmu = hyperfine splitting of the muon ground state

- Γ’90 = gyromagnetic ratio of the shielded proton

The value based on the 1990 value of the von Klitzing constant is shown by the vertical solid line and the uncertainty in this value is indicated by the vertical broken lines. (Reproduced from Jeckelmann and Jeanneret 2003 with permission.)

References and suggestions for further reading

Jeanneret B and Benz S P 2009 Application of the Josephson effect in electrical metrology Eur. Phys. J. Special Topics 172 181–206

Jeckelmann B and Jeanneret B 2003 The quantum Hall effect as an electrical resistance standard Meas. Sci. Technol. 14 1229–36

Part II

Superconductivity

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 7

Superconductivity and magnetism

7.1 Introduction

Superconductivity is a phenomenon in which the electrical resistivity of a material becomes zero below a given temperature (the critical temperature, Tc). The first observation of superconductivity was made in mercury by Heike Kamerlingh Onnes in 1911 as described below. The fact that the resistivity of most metals decreases as the temperature is decreased has been known since the 19th century. It was also known that the resistivity of a metal, particularly at low temperatures, depended upon the amount of impurities present and it was, therefore, believed that if a metal could be made pure enough its resistance would go continuously to zero as the temperature was lowered to absolute zero. Therefore, Kamerlingh Onnes’ observation of superconductivity, as described below, was totally unexpected.

7.2 The discovery of superconductivity

In 1908, Kamerlingh Onnes succeeded in liquefying helium, which has a boiling point of 4.2 K, and had available the facilities to make measurements down to temperatures that had previously been inaccessible. In fact, by controlling the vapor pressure of the liquid helium, he was able to make measurements down to about 1.5 K. Using this new technology, he embarked on an investigation of the resistance of metals at low temperatures. One of the metals he studied was mercury, Hg, because it could be purified to a very high degree. The results of his original experiment on Hg are illustrated in figure 7.1. The resistance below about 4.2 K fell to a value that was lower than Kamerlingh Onnes could measure. This behavior, known as superconductivity, was subsequently observed in more than twenty other elements, although, in many cases the superconducting transition temperature was sufficiently low (below about 1.5 K) that the discovery had to wait for the development of more sophisticated refrigeration methods. Table 7.1 gives experimental results for the elements that are known to show a superconducting transition. Additional elements are known that show superconducting at elevated pressure and some transuranic elements have been reported to show superconductivity.

Figure 7.1. Temperature dependence of the electrical resistance of a sample of Hg at low temperature. Data are from Kamerlingh Onnes (1911).

Table 7.1. The superconducting elements (at ambient pressure). α and β refer to different crystalline phases of the same element.

It is obvious that these materials have enormous possibilities for commercial application. There are, however, two basic obstacles to the use of these materials: the need for very low temperatures and the fact that superconductivity is destroyed by the application of a magnetic field. Before we see how progress has been made towards overcoming these difficulties, let us look at some of the basic physical properties of superconductors.

7.3 The effect of magnetic fields on superconductors

Even during the early stages of superconductivity research, the relationship between superconductivity and magnetism, as illustrated in figure 7.2, was known. This relationship shows that at a given temperature there is a critical magnetic field, Bc, at which the superconductivity is destroyed. At Tc, the figure shows that Bc(Tc) = 0. At absolute zero the material remains superconducting up to some critical field, Bc(T = 0), at which point it becomes normal (i.e. it has electrical resistance). Values of Bc(T = 0) are shown in table 7.1 for elemental superconductors. The temperature dependence of the critical field is well approximated by the relationship

which allows for the extrapolation of the critical field to zero temperature based on measurements at the temperatures that are accessible below Tc.

Figure 7.2. Relationship between the critical magnetic field, Bc, and the temperature for some elemental superconductors.

At this point it is appropriate to comment on the use of magnetic units in the literature and the distinction between B and H. From the relationship between B, H and M;

it is clear that H and M have the same units and these units differ from those of B (in the SI system) by the units of the permeability of free space, μ_0. Table 7.2 shows the SI and cgs-Gaussian units of B, H and M and the terminology that is used to designate these three quantities. It is obvious that the terminology can be ambiguous or even misleading as the term ‘magnetic field’ is used to denote both B and H. In the present text, we use the term ‘magnetic field’ to denote B while the term ‘magnetic field strength’ is used for H. Although these quantities are often used interchangeably, it is interesting to consider the origin of these terms. The magnetic field, B, is viewed as the field that is responsible for the force which acts upon moving charges as described by the Lorentz force;

On the other hand, H, is viewed in terms of the response of magnetic moments in a system that is responsible for the magnetic susceptibility of the material;

Table 7.2. SI units for the magnetic quantities B, H and M, and the terminology used to designate these quantities.

While the present text uses SI units, it is important to note, for a comparison of the present results with magnetic results from the literature, the relationship of SI quantities to cgs-Gausian units as shown in the table. Quantities in the SI system can be converted to cgs-Gaussian units using the conversions 1 T = 10^4 G and 1 A m−1 = 4π × 10−3 Oe. The data reported here for the critical magnetic field are values of B in Tesla. It is, however, common to find critical field values in the literature given as Hc in Tesla where the quoted values are actually μ_0Hc.

The discovery of superconductivity brought the short-lived hope that large currents could be carried in very small wires without energy loss. Unfortunately, when current flows through a wire (even if its resistance is zero), a magnetic field is produced and if this magnetic field is greater than Bc then the material will become normal. This means that there is not only a critical field for superconductivity, but a critical current, as well. In elemental superconductors, the critical current is sufficiently small that these materials are relatively useless as practical superconductors.

In the 1950s it was discovered that a second class of superconductor existed. These materials were called type-II superconductors, while the previously discovered type of superconductor became known as type-I. To understand the difference between type-I and type-II superconductors we need to look in detail at how magnetic fields affect superconductors.

In the case of a normal (non-superconducting) material, the magnetic field lines from an applied magnetic field penetrate the material, as illustrated in figure 7.3(a). A type-I superconductor that is placed in a magnetic field will distort the magnetic field lines in such a way (figure 7.3(b)) that they do not pass through the material. This phenomenon, now known as the Meissner effect, was first studied in the 1930s by the German physicists Walther Meissner and Robert Ochsenfeld. If the field is sufficiently strong (i.e. greater than Bc), then the magnetic field lines will enter the material and cause it to become normal.

Figure 7.3. Penetration of magnetic field lines in a type-I superconductor. (https://commons.wikimedia.org/wiki/File:EfektMeisnera.svg.)

The situation in a type-II superconductor is slightly different. For small applied magnetic fields, the field lines are expelled as for the type-I superconductor. For larger fields, the field lines begin to penetrate the material (as shown in figure 7.4), but unlike the case for the type-I superconductor, only part of the sample becomes normal. The magnetic field lines penetrate the normal regions of the sample and the current, which, follows the path of least resistance, will flow through the superconducting regions. As the magnetic field is increased, the proportion of the material which is normal will increase, and at some point, the superconductivity will disappear. The phase diagrams of type-I and a type-II superconduictors are shown in figure 7.5. In the mixed state the material can still conduct as a superconductor because the electric current will follow paths through portions of the sample that remain superconducting and it is only above the critical field Bc2 that superconductivity is completely elimiated. This state, as shown in figure 7.4, is referred to as a mixed state or vortex state and superconducting currents will circulate in vortices around the normal regions. Interactions between the vortices cause them to take on a regular (lattice-like) pattern in the superconductor. This pattern is called an Abrikosov lattice, as illustrated in figure 7.6.

Figure 7.4. Magnetic flux penetration in a type-II superconductor for T < Tc and B < Bc1. (Adwaele / Wikimedia Commons / CC-BY-SA 3.0 / https://creativecommons.org/licenses/by-sa/3.0/deed.en.)

Figure 7.5. Phase diagram of (a) type-I superconductor and (b) type-II superconductor. (Reproduced from Dunlap 2019.)

Figure 7.6. Magneto-optical images of an Abrikoisov lattice on the surface of a NbSe_2 type-II superconductor at 4.0 K after cooling in the Earth’s field (a) and in an 8 Oe applied field (b). The scalebar represents 10 μm. (Reproduced from Goa et al 2001 with permission.)

In type-II superconductors Bc2 can be much larger that Bc in type-I superconductors. For this reason, virtually all superconductors that have practical applications are type-II materials. Most elemental superconductors are type-I superconductors, the exceptions (as shown in table 7.1) are niobium, vanadium and technetium. Niobium shows the highest critical temperature of all elemental superconductors and has typically been a component in conventional (non-high temperature) superconductors that have found practical applications. The transition temperatures and critical fields of some conventional type-II superconductors are given in table 7.3. MgB_2, a well-known compound, was only discovered to be superconducting in 2001 and has been used in some applications to replace the traditionally used Nb-compounds. Figure 7.7 shows the phase diagrams of some potentially useful type-II superconductors.

Table 7.3. Properties of some useful conventional type-II superconductors.

Material

Tc (K)

Bc2 (Tesla) at T = 0 K

NbTi

9.3

15

PbMoS

14.4

60

V_3Ga

14.8

20.8

NbN

15.7

15.3

V_3Si

16.9

23.5

Nb_3Sn

18.3

30

Nb_3Al

18.7

32.4

Nb_3Ge

23.2

37

MgB_2

40

74

Figure 7.7. Phase diagram of some type-II superconductors.

Superconductors exhibit diamagnetism, that is, when subject to an externally applied magnetic field, they show a negative magnetic susceptibility. In the next section we develop a simple theoretical description of diamagnetism.

7.4 Diamagnetism and the Meissner effect

The Langevin theory of diamagnetism provides a basic physical picture of the magnetic behavior of diamagnetic materials and provides a description of the Meissner effect, as illustrated in figures 7.3 and 7.4.

In paramagnetic, ferromagnetic, antiferromagnetic and ferrimagnetic materials the atoms (or in some cases the conduction electrons) carry a magnetic moment and these magnetic moments can be aligned by placing the material in an externally applied magnetic field. As well, the magnetic moments of ferromagnets, antiferromagnets and ferrimagnets can align as a result of their interaction with one another. Typically, diamagnetic materials have atoms that possess no magnetic moment, but a magnetic moment may be induced by an externally applied magnetic field and this moment is antiparallel to the applied field. The physics of this behavior is described below.

We begin with the classical magnetic dipole of a current loop where the charge is defined to be the electronic charge;

Here ω is the angular frequency of the charge and r is the radius of the loop. The negative sign comes from the negative electron charge.

If this current loop is placed in a magnetic field where the magnetic flux density is given by B, then there will be a torque on the dipole moment;

The geometry of the magnetic moment arising from a current loop and an applied magnetic field is shown in figure 7.8. From the geometric relationships shown in this figure we can express the magnitude of the torque as

Classically, the torque is related to the time rate of change of the angular momentum, L, as

The direction of this change is illustrated in figure 7.9.

Figure 7.8. Geometry of a magnetic moment arising from a current loop and an applied magnetic field.

Figure 7.9. Relationship of the angular momentum of an electron in a current loop and the torque caused by an applied magnetic field.

From figure 7.9, some simple geometry gives

Using the angular momentum of a particle of mass m in an orbit;

and substituting into equations (7.8) and (7.9) gives

From equation (7.7) this gives;

The torque causes the angular momentum vector to precess around the magnetic field direction at the Larmour frequency, as illustrated in figure 7.9;

This precession represents a current defined by

and results in an induced magnetic moment of

where A is the area of the loop defined by the precession, as shown in figure 7.9. The position vector in spherical coordinates is expressed in terms of its Cartesian coordinates as

Quantum mechanically the electron is represented by its probability distribution and in the case of a spherical distribution the mean values of the Cartesian coordinates may be written as

Thus, the area of the current loop in the x–y plane, as shown in figure 7.9, is

Combining equations (7.14), (7.15) and (7.18) gives

This is the magnetic moment associated with one electron from one atom. In a material consisting of N atoms per unit volume, where each atom has Z unpaired electrons, then the magnetization per unit volume will be

This gives the Langevin (or diamagnetic susceptibility) per unit volume as

Note that is quantity is negative, that is the induced magnetization is antiparallel to the direction of the applied magnetic field, as is the case for a diamagnetic material. Note also that a calculation of the quantity on the right-hand side of equation (7.21) will show that this value is small compared to typical values of the paramagnetic susceptibility of paramagnetic materials. While the negative diamagnetic contribution to the susceptibility is present for paramagnetic materials (and ferromagnetic materials as well), it is much smaller than the positive contribution and has no practical affect on the experimental measurement of the susceptibility in these materials. In a diamagnetic material, however, these is no positive paramagnetic contribution to the susceptibility and the negative diamagnetic contribution (however small) will dominate.

Superconducting materials represent a unique class of diamagnetic materials. Superconductors are referred to as perfect diamagnets and have a diamagnetic susceptibility of −1. That is, from equation (7.21)

This means that the induced magnetization in the superconductor exactly cancels the magnetic flux density from the applied magnetic field leading to a net zero magnetic field inside the superconductor. This is the origin of the Meissner effect and the expulsion of the magnetic flux lines within the material when it is in the superconducting state, as shown in figure 7.3. The magnetization curve of a type-I superconductor is shown in figure 7.10. The perfect diamagnetic behavior is seen for applied fields up to the critical field after which the diamagnetic susceptibility reverts to its normal, very small negative value, which appears as zero on the scale of the graph.

Figure 7.10. Magnetization curve of a type-I superconductor showing perfect diamagnetic behavior for applied magnetic fields up to the critical field value. Note the negative scale on the vertical axis.

The magnetic behavior of a type-II superconductor is illustrated in figure 7.11. Perfect diamagnetic behavior is seen for applied magnetic fields up to Bc1. In this region, the material is in the superconducting state and the Meissner effect expels all magnetic flux lines with an induced magnetization that exactly cancels the applied magnetic field. Between Bc1 and Bc2 the material is in the vortex state and flux penetrates the sample in vortices while leaving the remainder of the sample in the superconducting state. The superconducting diamagnetic susceptibility decreases with increasing applied field as the density of flux lines in the samples increases up to an applied field of Bc2. For fields above Bc2 all superconductivity disappears and the material is in the normal state.

Figure 7.11. Magnetization curve of a type-II superconductor showing the behavior at Bc1 and Bc2.

References and suggestions for further reading

Ashcroft N W and Mermin N D 1976 Solid State Physics (Boston, MA: Cengage)

Dunlap R D 2019 Renewable Energy Storage (San Rafael, CA: Morgan & Claypool)

Goa P E, Hauglin H, Baziljevich M, Il’yashenko E, Gammel P L and Johansen T H 2001 Real-time magneto-optical imaging of vortices in superconducting NbSe_2 Supercond. Sci. Technol. 14 729–31

Kamerlingh Onnes H 1911 The superconductivity of mercury Comm. Phys. Lab. Univ. Leiden Nos. 122 and 124

Kleiner R, Buckel W and Huebener R 2016 Superconductivity: An Introduction 3rd edn (Berlin: Wiley)

Tinkham M 2004 Introduction to Superconductivity 2nd edn (New York: Dover Books)

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 8

Other properties of superconductors

8.1 Introduction

The previous chapter showed the close relationship between superconductivity and magnetism. The present chapter describes, in more detail, the interaction between magnetic fields and superconducting materials. It is also shown that quantum mechanics plays an important role in the description of superconductivity. Finally, the thermal properties of superconducting materials are discussed.

8.2 The London penetration depth

If we think about the laws of electromagnetism carefully, the cancelation of the internal flux in a superconductor in the superconducting state is somewhat counter-intuitive. We begin with Faraday’s law of induction which states

where the quantity on the left-hand side is the line integral of the electric field around any closed loop and the right-hand side is the time derivative of the magnetic flux inside the loop. The magnetic flux (in weber) is given from the definition of the magnetic field, B, which is the magnetic flux density in tesla (or weber per m^2) as,

where A is the area of the loop. Since the resistance of a superconductor is zero, then from Ohm’s law the voltage drop across a superconductor is zero and the electric field inside the superconductor is zero. Thus, from equations (8.1) and (8.2) the rate of change of the magnetic field inside the superconductor must be zero. That is, the magnetic field inside the superconductor must be constant. If we observe the behavior of the type-I superconductor that is cooled through the critical temperature, as illustrated in figure 7.3(a) and 7.3(b), then the Meissner effect would appear to violate Faraday’s law.

There is a straightforward explanation for this apparent violation of Faraday’s law; when the superconductor is lowered through its critical temperature, the entire sample does not become superconducting. There is a thin layer on the surface of the superconductor that remains normal. This region has a non-zero resistance and, therefore, is a region in which the electric field can be non-zero. This normal region contains currents, as seen in figure 8.1, which produce a magnetic field inside the superconductor that exactly cancel the applied field.

Figure 8.1. Formation of surface currents on a superconductor below Tc that cancel the externally applied field inside the sample.

The properties of these surface current can be described, in a somewhat phenomenological manner, in terms of the London equations. These equations were developed in 1935 by the brothers Fritz and Heinz London and consider the behavior of superconducting electrons in a solid if an electric field momentarily exists.

We begin with Newton’s equation for superconducting electrons in an electric field,

where m is the electron mass, is the vector velocity, e is the charge and E is the electric field. The superconducting current density, j, for the superconducting electrons is given by

where n is the superconducting electron density. Combining equations (8.3) and (8.4) gives,

This is known as the first London equation and relates changes in current density to the electric field.

We take the curl of both sides of equation (8.5) to obtain

From Faraday’s law we write

and substituting into equation (8.6) gives

Taking the time integral of both sides yields the second London equation;

The two London equations describe the behavior of the electric and magnetic fields in the material. We can relate this behavior to the formation of surface currents in the superconductor and the penetration of the external magnetic field into the normal outer layer of the superconductor by combining the second London equation with Ampere’s law;

Taking the curl of both sides of equation (8.10) gives

Using the basic vector identity

and equation (8.9) allows us to write

If we look carefully at the coefficient in front of the magnetic field on the right-hand side of equation (8.13), we will see that this quantity has the units of inverse length squared. Thus, we define the London penetration depth as

The solution of equation (8.13) gives the depth dependence of the magnetic field in the superconductor;

where x is the depth measured from the surface and B_0 is the magnetic field outside of the superconductor. Equation (8.15) shows that the magnetic field decreases exponentially beneath the surface of the superconductor with a characteristic length scale of the London penetration depth. That is the penetration depth is defined as the 1/eth length of the exponential decay. This behavior is illustrated in figure 8.2. It is in the region where the magnetic field penetrates the superconductor that the material is in the normal state and, as such, an electric field can be present, and currents exist that will exactly shield the superconducting region from the external magnetic field. It is interesting to note that if the external magnetic field is removed, there is no field to penetrate into the superconductor and hence there is no shielding current.

Figure 8.2. The exponential decay of the magnetic field inside a superconductor illustrating the London penetration depth, λ.

It is of interest to consider the temperature dependence of the London penetration depth. If the penetration depth at zero temperature is λL0 then the temperature dependent penetration depth is given by the empirical relationship

This expression shows that as the temperature approaches the critical temperature, the penetration depth diverges and becomes infinite at Tc when the entire sample becomes normal. At low temperatures, i.e. well below Tc, typical London penetration depths are of the order of 1–100 nm. The exact value of λL0 in a particular material depends on the electronic properties of the material in its normal state, specifically, the normal electron mean free path.

8.3 Flux quantization

While the London approach provides a description of the Meissner effect and explains, in a phenomenological way, the behavior of the superconductor at the critical temperature, it is a macroscopic theory. It does not explain the microscopic behavior of the charge carriers that are responsible for the superconducting state nor does it address their quantum mechanical properties. The BCS theory developed by John Bardeen, Leon Cooper and J Robert Schrieffer, as described in the next chapter, was the first successful microscopic theory of superconductivity. Before we overview this theoretical approach, it is important to look at some additional properties of superconductors that help to reveal the importance of quantum effects.

If a superconducting ring is placed in an applied magnetic field when the temperature is above Tc then the field will penetrate the material and, as well, the space inside the ring, as illustrated in figure 8.3(a). If the superconductor’s temperature is now lowered below the critical temperature, the magnetic flux will be expelled from the superconducting material as a result of the Meissner effect. However, the magnetic field will still exist in the region outside the superconducting ring and in the region inside the ring. If the applied magnetic field is now turned off, then the field outside the ring will disappear but the field inside the ring will be trapped by the superconducting material, as illustrated in figure 8.3(b), and a persistent current will be induced in the ring, as in figure 8.3(c). This current circulates without resistance and thus without loss and will continue to produce the flux trapped inside the ring indefinitely.

Figure 8.3. (a) Magnetic flux penetration in a superconducting ring above Tc and (b) trapped flux in a superconducting ring below Tc, and (c) persistent current in superconducting ring. (Reproduced from Fagaly 2006, with the permission of AIP Publishing.)

By the early 1950s it was clear that superconductivity was fundamentally a quantum mechanical phenomenon and, as a result, Fritz London hypothesized that the superconducting current and hence the magnetic flux in figure 8.3(b) would be quantized. This is analogous to the situation seen for the quantum Hall effect, as described by equation (4.11), which gives the fundamental flux quantum as h/e. However, it was determined experimentally that the flux inside a superconducting ring was quantized in units of

indicating that the superconducting currents in a superconductor were basically different than currents in a normal conductor and the charge carriers appeared to have a charge of twice the fundamental electronic charge.

8.4 The isotope effect

The electronic properties of a material are normally the result of the properties of the electrons in the material, which are related to the electronic structure of the atoms and the details of the interactions between the atoms. These properties are not directly related to the properties of the nuclei of the atoms, and hence electronic properties of different isotopes of the same element are typically the same. However, the properties of superconductors have been found to vary between different isotopes of the same element. This is illustrated in figure 8.4 where the critical temperatures of various tin isotopes are plotted as a function of atomic mass. Tin is an interesting example to illustrate this property as tin is not only superconducting but has the largest number of stable isotopes of any element. The reasons for this behavior will become apparent in the next chapter.

Figure 8.4. Superconducting transition temperature as a function of atomic mass for different isotopes of tin. The fit is to equation (8.18). Data are from Maxwell (1952) and Serin et al (1952).

The data shown in figure 8.4 are plotted on a log–log graph and the figure shows a linear relationship between the logarithms of the critical temperature, Tc, and the atomic mass, M. This indicates an empirical relationship between these two quantities which can be expressed as a power law. An analysis of the data in the figure shows a relationship of the form

8.5 Specific heat

Another physical property that shows unusual properties in superconductors is the specific heat. When a normal metal is exposed to a source of heat, the metal will absorb thermal energy. This energy can be absorbed by the lattice, in which case it will excite phonons, or it can be absorbed by the conduction electrons, in which case it will increase their kinetic energy. At very low temperatures there is also a contribution from nuclei, but this is not relevant to the present discussion. The specific heat is defined as the quantity of thermal energy required to raise the temperature of a material by a specific amount, i.e. in the SI system the molar specific heat is the thermal energy in joules needed to raise the temperature of one mole of substance by one degree Kelvin.

The specific heat of a superconductor can be studied as a function of temperature in both the normal state and in the superconducting state by making measurements without an applied field, in which case the sample will be superconducting below Tc, and in an applied field greater than Bc, in which case the sample will remain normal to low temperature. Figure 8.5 shows the behavior that is typically seen in superconducting materials.

Figure 8.5. Specific heat of a superconducting material (solid line) and a normal metal (broken line) as a function of temperature.

In the normal state, the specific heat, C, can be shown to be well approximated by the expression

where the term that is linear in temperature arises from the contribution of the conduction electrons, while the term that is cubic in temperature arises from the lattice (or phonon) contribution. At low temperature, the more-or-less linear behavior shown in figure 8.5 corresponds to the dominant electronic contribution.

The behavior of the specific heat in the superconducting phase shows two differences from that of the normal state. Firstly, there is a discontinuity at the critical temperature where the specific heat above Tc follows normal behavior, as expected, and secondly, below Tc the behavior of the specific heat is noticeably non-linear. We can understand the importance of the electronic component of the specific heat in the superconducting regime by plotting this component of the specific heat as a function of inverse temperature on a semi-logarithmic graph. This plot, as seen in figure 8.6 for gallium in its superconducting state, shows a linear relationship. This means that the superconducting specific heat is well defined by a relationship of the form

Figure 8.6. Electronic component of the specific heat of gallium in the superconducting regime plotted on a semi-logarithmic scale as a function of inverse temperature. The specific heat is scaled to the critical temperature and the constant γ = 6.0 × 10−4 mJ mol−1 K−2. (Reprinted figure with permission from Phillips 1964, Copyright (1965) by the American Physical Society.)

The behavior shown in figure 8.6 and by equation (8.20) is typical of a system in which there is an energy gap in the density of states and indicates the importance of the quantum behavior of the charge carriers in the system. This is discussed further in the next chapter.

The three properties of superconductors, as described above, flux quantization, the isotope effect and the specific heat, provide insight into three important features of the superconducting state, specifically, the nature of the charge carriers, the importance of phonons and the formation of an energy gap, respectively. The BCS theory, as overviewed in the next chapter, was the first microscopic description of the quantum mechanical nature of superconductivity and provided a physical understanding of these properties.

References and suggestions for further reading

Ashcroft N W and Mermin N D 1976 Solid State Physics (Boston, MA: Cengage)

Fagaly R L 2006 Superconducting quantum interference device instruments and applications Rev. Sci. Instrum. 77 101101

Goa P E, Hauglin H, Baziljevich M, Il’yashenko E, Gammel P L and Johansen T H 2001 Real-time magneto-optical imaging of vortices in superconducting NbSe_2 Supercond. Sci. Technol. 14 729–31

Kittel C 2005 Introduction to Solid State Physics 8th edn (Hoboken, NJ: Wiley)

Kleiner R, Buckel W and Huebener R 2016 Superconductivity: An Introduction 3rd edn (Berlin: Wiley)

Maxwell E 1952 Superconductivity of the isotopes of tin Phys. Rev. 86 235–42

Phillips N E 1964 Low-temperature heat capacities of gallium, cadmium, and copper Phys. Rev. 134 A385–91

Serin B, Reynolds C A and Lohman C 1952 The isotope effect in superconductivity. II. Tin and lead Phys. Rev. B 86 162–4

Tinkham M 2004 Introduction to Superconductivity 2nd edn (New York: Dover Books)

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 9

BCS theory

9.1 Introduction

From the time of Kamerlingh Onnes’ discovery, the theoretical basis of superconductivity remained a mystery for 46 years. In 1957, John Bardeen, Leon Cooper and J Robert Schrieffer, then at the University of Illinois, presented a theory of superconductivity known as the Bardeen–Cooper–Schrieffer (BCS) theory that has since gained universal acceptance, at least for the behavior of the superconducting materials observed up to that time. The present chapter provides an overview of this theory beginning with a description of the formation of Cooper pairs.

9.2 Cooper pair formation

In a normal (non-superconducting) metal, electrical current is carried by individual electrons or holes. When an electron interacts with an atom in the metal there is an opposition to the flow of electrons, and this contributes to the resistance. BCS theory states that electrons in a superconductor form tightly bound pairs (known as Cooper pairs) which travel together through the material. If an electron interacts with an atom in the material, this interaction is seen by the second electron of the Cooper pair through the coupling between them. The second electron responds in such a way as to compensate for the interaction of the first electron with the atom. In this way, the paired electrons can travel through the material without interference from the atoms and, therefore, without resistance. It is the pairing of electron in the superconductor that explains the observed quantization of magnetic flux as described by equation (8.17). All electrons in a material, or even all conduction electrons, do not participate in superconductivity, but it is those electrons that do form Cooper pairs that can travel through the material without resistive loss and can contribute to the persistent current shown in figure 8.3(c). Thus, the flux quantum is defined by charge carriers with a charge of 2e as given by equation (8.17).

To understand the properties of superconductors it is important to understand how electrons in a metal can form Cooper pairs. Two negatively charged electrons will experience a repulsive Coulombic interaction but the formation of a bound pair requires an attractive interaction between the electrons. This net attractive interaction is the result of interactions that involve the crystalline lattice that are sufficient to overcome the repulsive Coulombic interaction. As a conduction electron travels through the lattice it experiences an attractive Coulombic interaction with the positively charged ions on the lattice sites. This interaction can displace the ions from their equilibrium lattice positions as illustrated in figure 9.1.

Figure 9.1. Displacement of ions by the attractive Coulombic interaction with a conduction electron.

The displaced ions produce a region of increased positive charge density and this positive charge distribution is an attractive potential for other electrons. As figure 9.1 illustrates a second electron can be attracted to the electron that caused the ion displacement and forms a bound Cooper pair. Two important questions need to be considered in order to quantify this process; how are the details of the lattice distortions described and which electrons can become bound to one another?

The details of the ion movements in a lattice are described by the phonon modes of the lattice. In the Debye model the Debye frequency, ωD, is the frequency of the most energetic phonons in the phonon spectrum. It is related to the Debye temperature, θD, by the expression

This model gives the Debye frequency as

where n is the number density of primitive cells in the lattice and is the velocity of sound in the material. The sound velocity may be expressed as

where K is a measure of the stiffness of the lattice (the isentropic bulk modulus) and ρ is the material density. The density may be written in terms of the atomic mass of the atom, M, and the number density of primitive cells (which, for a monatomic basis, contain one atom) as

Combining equations (9.2) to (9.4) gives

The BSC theory predicts a relationship between the critical superconducting temperature and the Debye frequency;

where φ is the electron-phonon interaction strength and N(EF) is the electron density of states at the Fermi energy.

It is seen from equation (9.6) that the critical temperature is a linear function of the Debye frequency. It is also clear from equation (9.5) that the critical temperature is related to the atomic mass by equation (8.18) and follows the trend shown in figure 8.4.

A more detailed investigation of isotope effects in elemental superconductors shows that a more general form of equation (8.18) may be written as

where the value of the exponent in equation (8.18) is α = 1/2. Table 9.1 gives experimentally observed values of α for some elemental (type-I) superconductors. A value of α = 1/2 is an accurate description of the isotope effect in many elemental superconductors. However, some elemental superconductors show anomalously low values, e.g α = 0.37 and 0.21 for Mo and Os, respectively, and results for Zr and Ru show that α ≈ 0, indicating that there is no isotope effect in these elements. The simple form of BCS theory which predicts the relationship in equation (8.18) does not account for the Coulombic repulsion between electrons or electronic band structure effects. More detailed calculations that include these effects show that the basic approach of BCS theory is valid and provide an explanation for these anomalous cases. Overall the results of isotope effect measurements indicate the importance of atomic mass on superconductivity and demonstrate the existence of a Cooper pair formation mechanism based on electron–lattice interactions.

Table 9.1. Experimental observed exponents from equation (9.7) for some elemental superconductors. Z = atomic number and <Tc> = critical temperature of naturally occurring element.

The second question posed above dealt with the identity of the electrons that form the Cooper pairs. The conduction electrons in a metal have energies that are near the Fermi energy. There is a distribution of energies that results from thermal effects and the energies of the electrons that can bind together to form a Cooper pair must be the same within a factor comparable to the electron–lattice interaction energy. This interaction energy is what excites phonons in the lattice and is of the order of the Debye energy as given by

It is interesting to consider the relationship between the Debye energy and the Fermi energy for some type-I superconductors. This comparison is shown in table 9.2. Typically, Fermi energies are two to three orders of magnitude larger than the Debye energy. As the Fermi energy depends on electronic structure and the Debye energy depends on bonding characteristics, there is not a direct correlation between these values. It is clear, however, from the table that it is only electrons in states that are very close to the Fermi energy that can form Cooper pairs.

Table 9.2. Fermi energy and Debye energy for some type-I elemental superconductors.

The electrons which form Cooper pairs also have constraints on their momentum and spin. Figure 9.2 shows the electrons that form a Cooper pair. These electrons have both spin and momentum vectors antiparallel. The result of this arrangement is that; (1) the net momentum of the Cooper pair is near zero and (2) the total spin of the Cooper pair is zero.

Figure 9.2. Representation of the momentum and spin vectors of the electrons that form a Cooper pair.

In the case of the momentum, the net momentum may be identically zero, in which case no current flows through the material. Alternately there may be a small difference in the magnitudes of the momenta and a net current (i.e. a superconducting current) will flow one way or the other.

The net spin of the Cooper pair presents an interesting situation. As the total spin of the bound pair is zero, the Cooper pair is a boson. Hence, the usual constraint on electrons, which are fermions, that they must obey the Pauli exclusion principle, is no longer relevant. The ground state of a system of Cooper pairs is one in which all the Cooper pairs occupy the same state. The ground state of a collection of Cooper pairs can be represented quantum mechanically by a single macroscopic wavefunction. BCS theory explains the lack of electrical resistance in a superconductor in the context of this wavefunction. All Cooper pairs act as a single quantum mechanical entity and scattering of any Cooper pair is only possible if all Cooper pairs are simultaneously scattered. In normal metals, resistance results from electron scattering from (for example) phonons. It is somewhat ironic that the very mechanism (or at least one of them) that is responsible for resistivity in normal metals is also responsible for the formation of Cooper pairs in superconductors and the total elimination of resistance. This situation is reflected by the fact that in most cases metals that are very good conductors in the normal state (e.g. copper, silver and gold) are not superconductors while metals that have a relatively high resistance in the normal state (e.g. lead, tin and mercury) are superconducting at low temperature.

9.3 The superconducting energy gap

The energy gap in a semiconductor arises because of the interaction of electrons with the periodic potential of the ions in the crystal lattice. This gap represents the energy that an electron in the valence band would need to acquire in order to occupy a state in the conduction band. In a superconductor, the energy gap, as inferred from the behavior of the specific heat in the superconducting regime, arises for a completely different reason. The energy gap represents the energy that is needed to break the bond between two electrons in a Cooper pair and arises because of the phonon mediated electron–electron interaction. A comparison of occupied and vacant states at T = 0 for a normal metal and a superconductor is shown in figure 9.3. The superconducting energy gap can be obtained directly from measurements of the superconducting density of states as shown in figure 9.4.

Figure 9.3. (a) Fermi energy for conduction band of a normal metal showing the occupied states below EF and the vacant states above EF at T = 0. Energy gap between filled and vacant states. (b) Energy gap between occupied and vacant states in a superconductor at T = 0.

Figure 9.4. (Insert) Observation of the superconducting energy gap of NbN using a low temperature scanning tunneling microscopy. An energy gap of 2.38 mV is obtained from an analysis of the data. (Main graph) Temperature dependence of the measured superconducting energy gap of NbN. (©Deutsche Physikalische Gesellschaft. Reproduced by permission of IOP Publishing. CC BY-NC-SA, from Romestain et al 2004.)

The zero temperature energy gaps for some elemental superconductors are given in table 9.3. An inspection of the table shows that large energy gaps correspond to large values of the critical temperature. This is expected as larger energy gaps correspond to materials in which it is more difficult to break Cooper pairs. BCS theory predicts a relationship between Eg and Tc (at T = 0) of the form

This has been shown to be valid for a number of type-I superconductors. The energy gap also appears as a parameter in equation (8.20) for the specific heat and measurements of this quantity are a convenient means for determining the energy gap.

Table 9.3. Critical temperatures and energy gaps (at T = 0) in some elemental superconductors.

Element

Tc (K)

Eg (meV)

aluminum

1.19

0.34

gallium

1.083

0.33

indium

3.40

1.05

α-lanthanum

4.9

1.9

lead

7.19

2.73

α-mercury

4.15

1.65

niobium

9.26

3.05

tantalum

4.482

1.4

β-tin

3.722

1.15

zinc

0.875

0.24

The energy gap disappears as the critical temperature is approached. That is, at T = Tc the energy gap becomes zero and Cooper pairs spontaneously break. BCS theory predicts a mean-field-like temperature dependence of the energy gap below Tc as

This general behavior is seen for the temperature dependence of the superconducting energy gap in figure 9.4. The temperature dependence of the energy gap for some elemental superconductors is seen in figure 9.5 along with an analysis based on the predictions of BCS theory.

Figure 9.5. Energy gap of tin (triangles), tantalum (circles) and niobium (squares) normalized to the T = 0 value and plotted as a function of temperature scaled to Tc. The predicted value from BCS theory is shown by the solid line. (Reprinted figure with permission from Townsend and Sutton 1962, Copyright (1962) by the American Physical Society.)

9.4 The coherence length

The formation of Cooper pairs provides a means of defining a second length scale for a superconductor (along with the London penetration depth), referred to as the coherence length. There are various definitions that have been used for the coherence length that in some way measure the spatial correlations between superconducting Cooper pairs. These different definitions are, more-or-less, equivalent well below Tc, so we will not concern ourselves with these differences. The most appropriate definition of the coherence length is the spatial extent of the Cooper pair wavefunction. This is distinct from, and can be appreciably different than, the actual physical separation between the electrons of the Cooper pair and represents the minimum distance which is necessary for a substantial change in the Cooper pair density to occur.

We can derive a simple expression for the coherence length from Heisenberg’s uncertainty principle. The uncertainty in energy is given by the energy gap as

That is, the Cooper pair energy can have any value that is less than the energy gap and the electrons will remain bound together. Heisenberg’s uncertainty principle gives the uncertainty in position in terms of the uncertainty in momentum and this can be approximated to the coherence length, ξ_0, at T = 0;

Substituting equation (9.11) into (9.12) gives

From the relationship between energy and momentum this expression gives

The relationship between the London penetration depth (at T = 0) and the coherence length (at T = 0) provides valuable information about the behavior of the superconducting state. Some typical values of λL0 and ξ_0 along with their ratio defined as κ = λL0/ξ_0 are given in table 9.4.

Table 9.4. Zero temperature London penetration depths and coherence lengths and their ratio κ = λL0/ξ_0 for some elemental superconductors. (http://hyperphysics.phy-astr.gsu.edu/hbase/solids/chrlen.html)

Phenomenologically, if the London penetration depth is less than the coherence length, or more precisely κ < 1/√2, then the superconductor shows type-I behavior. If κ > 1/√2, then the superconductor shows type-II behavior. Table 9.4 shows that Nb falls into the category of a type-II superconductor, while the other elements shown in the table are of type-I. Figure 9.6 shows the typical relationship between the London penetration depth and the coherence length in type-I and type-II superconductors. Here the magnetic field below the surface of a superconductor shows the characteristic penetration depth in terms of the exponential decay of the field. The Cooper pair density, i.e. the square of the Cooper pair wavefunction, shows the characteristic scale of the coherence length. The relationship of the London penetration depth and the coherence length as shown in the figure can describe, in a phenomenological manner, the differences in the behavior described for the type-I superconductor in figure 7.3 and the for the type-II superconductor in figure 7.4.

Figure 9.6. (a) Relationship of the London penetration depth (λL0) and the coherence length (ξ_0) for a type-I superconductor. For the London penetration depth, the magnetic field as a function of depth below the superconductor surface is shown (solid line) and for the coherence length, the Cooper pair wavefunction as a function of depth is shown (broken line). (b) London penetration depth and coherence length as in (a) for a type-II superconductor.

The importance of the relationship between the penetration depth and the coherence length on determining the type of superconductivity can be understood by considering the minimization of the total energy of the system. The magnetic flux penetration into the superconductor represents an increase in magnetic energy for the system (compared to the case where there is no magnetic field in the sample). On the other hand, since the Cooper pair density must increase slowly over some distance, there is a decrease in the energy associated with the condensation of bosons into their ground state. These two factors compete with one another and when the coherence length becomes appreciably smaller than the penetration depth it becomes energetically favorable for the regions of superconductivity to be interspersed with normal regions where the flux can penetrate.

The importance of impurities in a superconductor is apparent in terms of the relationship between the characteristic superconducting length scales and the normal electron mean free path. Impurities will have the obvious effect of reducing the normal electron mean free path by increasing the scattering of conduction electrons. This decrease in mean free path will result in an increase in the penetration depth and a decrease in the coherence length. Thus, the presence of impurities can change a type-I superconductor into a type-II superconductor by changing the situation in figure 9.6(a) into the situation in figure 9.6(b). This behavior is nicely illustrated by the superconducting properties of lead with indium impurities in figure 9.7, where the magnetization as a function of applied field is plotted for alloys with different impurity concentrations. It is clearly seen that the type-I behavior of pure lead with a single well-defined critical field becomes characteristic of type-II behavior where the magnetization returns slowly to zero as a function of applied field. While the addition of indium impurities in lead has a relatively small effect on the value of the critical temperature, the figure shows that the value of the upper critical field, Bc2, increases substantially with increasing impurity concentration. Thus, the addition of impurities may increase potential applications of some superconducting materials by increasing the critical current that they can carry.

Figure 9.7. Effects of indium impurities on the superconducting properties of lead; (a) pure Pb, (B) Pb+2.08 wt% In, (C) Pb+8.23 wt% In and (D) Pb+20.4 wt% In. (Note 1 Oersted ≡ 0.0001 Tesla.) (Reprinted figure with permission from Livingston 1963, Copyright (1963) by the American Physical Society.)

BCS theory provides an understanding of many of the basic properties of conventional superconductors. It describes the superconducting charge carriers as Cooper pairs that are formed by phonon mediated electron–electron interactions. The superconducting state is described in terms of a condensation of Cooper pairs at low temperature and the effect of phonons provides an explanation for the observed isotope effect. The theory also provides a description of the specific heat in the context of a superconducting energy gap and an explanation for the Meissner effect.

References and suggestions for further reading

Ashcroft N W and Mermin N D 1976 Solid State Physics (Boston, MA: Cengage)

Kittel C 2005 Introduction to Solid State Physics 8th edn (Hoboken, NJ: Wiley)

Livingston J D 1963 Magnetic properties of superconducting lead-base alloys Phys. Rev. 129 1943–9

Romestain R, Delaet B, Renaud-Goud P, Wang I, Jore C, Villegier J C and Poizat J P 2004 Fabrication of a superconducting niobium nitride hot electron bolometer for single-photon counting New J. Phys. 6 129

Townsend P and Sutton J 1962 Investigation by electron tunneling of the superconducting energy gaps in Nb, Ta, Sn and Pb Phys. Rev. 128 591–5

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 10

High temperature superconductivity

10.1 Introduction

Superconductors have a number of potential applications as described later in this chapter. These applications fall roughly into two categories, those that depend on the unique nature of the charge carriers in superconductors and their unusual magnetic properties and those that are related to superconductor’s ability to carry large currents with no resistive loss. In the first case applications include sensors and in the second case applications include power transmission and superconducting magnets. In the second case, in particular, the requirement for superconductors to be operated at low temperature is an inconvenience and an additional expense. In fact, superconductors that are designed to carry large currents must be cooled sufficiently below the critical temperature to have a critical field, and hence a critical current, that is sufficient for the desired application. The discovery of superconductors which functioned at room temperature would eliminate the need for expensive cryogenic systems. Even superconductors which could be used with conventional refrigeration methods or liquid nitrogen rather than liquid helium would be advantageous.

Elemental type-I superconductors have sufficiently low critical field values that their practical applications are very limited. The discovery of type-II superconducting metallic compounds with higher values of Tc and much higher values of the critical field Bc2, as indicated in table 7.3, substantially increased the potential uses for superconductors. The highest temperature conventional superconductor, Nb_3Ge with a critical temperature of 23.2 K and a critical field of 37 T at 0 K, was discovered 1973. Even this material required the use of liquid helium to cool the sample to superconducting temperatures.

In the 1960s the approach to superconductivity research began to change and the idea that materials which were not good conductors at room temperature could become superconducting at low temperature. This research direction led to the discovery of high temperature superconductors as discussed in the present chapter.

10.2 The discovery of high temperature superconductors

Early research on superconductors concentrated on metals, as it was believed that a material should be conductive in the normal state in order to become superconducting at low temperatures. However, in 1964 James F Schooley and co-workers at the National Bureau of Standards showed that SrTiO_3, which was a very poor normal conductor, became superconducting at 0.3 K. The compound SrTiO_3 is a ceramic which has a crystal structure known as a perovskite. The perovskites derive their name from the naturally occurring material known as perovskite (CaTiO_3). Natural perovskites are some of the Earth’s most abundant minerals and have been widely studied by geologists. Synthetic perovskites are important ceramic materials which are used principally in the electronics industry.

In 1973 David Johnston of the University of California at San Diego found superconductivity in LiTi_2O_4 at 13.7 K, however, these materials were difficult to prepare, and it was not possible to isolate the superconducting phase. Arthur Sleight of DuPont Research prepared Ba–Pb–Bi–O as a single phase in 1975, and showed that it was superconducting at 13 K. However, it was in 1986 when Johannes Georg Bednorz and Karl Alexander Müller discovered superconductivity at above 30 K in the barium-lanthanum-copper-oxide system (see figure 10.1) that widespread interest in superconductivity in non-metals really began. Figure 10.1 shows that the resistivity decreases more-or-less linearly with decreasing temperature down to around 100 K. It then increases logarithmically as a result of electron localization. At around 30 K the resistivity drops by more than three orders of magnitude, indicating a transition to a superconducting state.

Figure 10.1. Superconducting transition in BaxLa5−xCu_5O5(3−y) for x = 1 (upper curves and left scale) and x = 0.75 (lower curve and right scale). The upper curves also show the affect of different current densities. (Reprinted by permission from Springer Nature: Springer Verlag, 1986. Bednorz and Müller 1986.)

About a year after the discovery of superconductivity in Ba–La–Cu–O, superconductivity in YBa_2Cu3O7−δ was observed at 93 K as illustrated in figure 10.2. Although the critcal field at T = 0 is too high to measure with current technology, an extrapolation from lower field data suggests that it is in the range of 160 T. This was the first observation of superconductivity above the temperature of liquid nitrogen.

Figure 10.2. Temperature dependence of the resistance of YBa_2Cu3O7−δ showing the superconducting critical temperature at 93 K. (Reprinted figure with permission from Wu et al 1987, Copyright (1987) by the American Physical Society.)

10.3 Properties of high temperature superconductors

Y–Ba–Cu–O and many other high temperature superconductors have the layered perovskite structure. This structure is illustrated in figure 10.3 where the layers of Y, Ba and Cu–O are shown. The Y–Ba–Cu–O structure is orthorhombic where the lattice parameters in the x and y directions are different (a ≠ b ≠ c). The stoichiometric composition would be YBa_2Cu_3O_7, however, when the material is heated in an oxygen containing atmosphere some of the oxygen sites are normally vacant. Oxygen stoichiometry is very important to the properties of this material, and other perovskites, as the compound with δ = 1 shows antiferromagnetic order with a Néel temperature above 500 K due to a localized magnetic moment on the Cu2+ ions. As δ is decreased the Néel temperature drops drastically and for optimal oxygen doping, the material is a superconductor with Tc = 93 K. It has been found that the critical current in the x–y plane (i.e. in the Cu–O planes) is very high, while the critical current in the z direction is much lower. Samples for applications are often prepared as thin tapes which have the z-axis preferentially oriented perpendicular to the tape surface in order to optimize current carrying capacity in the plane of the tape. This is discussed further in section 12.2.

Figure 10.3. Layered perovskite structures of YBa_2Cu_3O7−δ. (Gadolinist / Wikimedia Commons / CC-BY-SA.3.0 / https://creativecommons.org/licenses/by-sa/3.0/deed.en.)

Subsequent to the discovery of superconductivity in layered perovskites above the temperature of liquid nitrogen, intense research focused on other perovskite type materials (particularly copper oxide based ones) as potential superconductors. A number of other perovskite-type materials have since been found which exhibit superconductivity, some with critical temperatures higher than that of Y–Ba–Cu–O. Figure 10.4 shows the maximum superconducting critical temperature as a function of year from 1910 to 2015. A number of different classes of materials have been found to exhibit superconductivity. These materials include carbon nanotubes, doped fullerenes and actinide compounds. Although these typically have lower critical temperatures than the perovskites, they are of considerable significance for understanding the basic physics of superconductivity. Of particular interest are superconducting Fe-containing compounds, which were first discovered in 2006. Most of these materials can be referred to as ferropnictides, that is compounds of iron with a group 15 element such as arsenic. Some iron chalcogenide compounds, such as FeSe, have also exhibited superconductivity. Superconductivity in these compounds has been observed at up to about 100 K and these materials show promise for future research.

Figure 10.4. Maximum experimentally measured values of Tc for superconductors. ● = metals; ♦ = perovskites; ★ = Ce compounds and actinides; ▼ = fullerenes; ■ = Fe compounds; ▲ = carbon nanotubes. (Reproduced from Ray 2015.)

The highest temperature superconductors (up until 2015) were Cu–O based perovskites (or cuprates) and it has been shown that there is a more-or-less direct relationship between the number of copper oxide layers in the unit cell and the critical temperature. This relationship is illustrated in figure 10.5 where critical temperatures are plotted as a function of the number of Cu–O layers in the unit cell for three series of perovskites. Although there was hope that higher critical temperatures might be achieved in cuprates with more Cu–O layers, evidence has suggested that beyond three layers, the critical temperature decreases.

Figure 10.5. Superconducting critical temperature as a function of the number of Cu–O planes in the perovskite structure for Bi-based materials (triangles); Tl-based materials (circles) and Hg-based materials (squares). The Bi-series is Bi_2Sr_2CuO_6, Bi_2Sr_2CaCu_2O_8, Bi_2Sr_2Ca_2Cu_3O_10; the Tl-series is Tl_2Ba_2CuO_6, Tl_2Ba_2CaCu_2O_8, Tl_2Ba_2Ca_2Cu_3O_10; and the Hg-series is HgBa_2CuO_4, HgBa_2CaCu_2O_6, HgBa_2Ca_2Cu_3O_8; for 1, 2 and 3 Cu–O layers, respectively.

In 1993 HgBa_2Ca_2Cu_3O_8 was found to have a superconducting transition temperature of 134 K and about a year later it was reported that this material had a critical temperature of 164 K when subject to a pressure of about 30 GPa. The measured pressure dependence of Tc for this material is shown in figure 10.6. Most elemental superconductors show a decrease in Tc when subject to a moderate applied pressure. It is generally believed that this change is the result of the effect of pressure on the density of electron states at the Fermi energy, N(EF). In larger applied pressures, changes in the electrical properties of materials can occur (including changes in their superconductive properties) because the applied pressure can induce structural changes (i.e. changes from one crystalline structure to another). In fact, at ambient pressure there are 29 elements that are known to exhibit superconductivity. If applied pressure is included as an experimental parameter, that number jumps to 53. Perovskite superconductors typically show an increase in critical temperature with applied pressure. In some cases, a maximum is observed followed by a decrease as shown in figure 10.6. The reasons for this behavior are complex but are generally believed to be related to changes in carrier density with applied pressure.

Figure 10.6. Pressure dependence of the critical temperature of HgBa_2Ca_2Cu_3O_8 (Hg-1223). (Reprinted figure with permission from Gao et al 1994, Copyright (1994) by the American Physical Society.)

Despite extensive studies of perovskites and related materials, including measurements with externally applied pressure, the maximum observed superconducting transition temperature remained at 164 K for more than 20 years. In 2015, however, superconductivity was observed at 203 K in hydrogen sulfide (H_2S) under an applied pressure of 155 GPa. Figure 10.7 shows typical results from this investigation. The figure also shows the results for D_2S and illustrates the substantial isotope effect that exists in this material.

Figure 10.7. Superconducting transition in H_2S at an applied pressure of 141 GPa. The result for D_2S (deuterated H_2S) at 155 GPa is shown for comparison to illustrate the isotope effect. (Reprinted by permission from Springer Nature: Drozdov et al 2015 (3 September 2015).)

At standard temperature and pressure, H_2S, is a colorless gas with a strong odor of rotten eggs. In sufficient concentrations it is very toxic. It is also flammable and corrosive. At 1 atmosphere it has a boiling point of 213 K and a melting point of 191 K. At low pressure, solid H_2S has a cubic crystal structure. The phase diagram as a function of temperature and pressure is quite complex but it is generally known that the material partially dissociates at a pressure of about 95 GPa into H_3S and S. At about the same pressure it undergoes a transition from insulating behavior to metallic behavior. It is not known how H_3S and S each contribute to this insulator to metal transition. It is speculated that the superconducting behavior at higher pressure is related to the H_3S phase. The observation of superconductivity at high pressure in a material, which is gaseous at room temperature and that is an insulator at low pressure in its solid state, opens the possibility of searching for superconductivity in classes of materials that have not previously been considered. Hydrogen based materials seem to be promising candidates and it has been suggested by Neil Ashcroft that hydrogen itself may be superconducting at near room temperature when subject to sufficient pressure. These possibilities are discussed further in the next section.

10.4 The theory of high temperature superconductivity

The BCS theory provides a good explanation of the properties of what are generally referred to as conventional superconductors. These are traditionally the superconducting materials that were known up until the mid-1980s and which have superconducting transition temperatures up to 23.2 K. However, the recently discovered non-perovskite superconductors such as H_2S may also fall into this category. These conventional superconducting materials are all metals, and hence are reasonably good conductors in the normal state. BCS theory describes the superconductivity as resulting from phonon mediated electron–electron interactions that give rise to the formation of Cooper pairs which have s-state wavefunctions. The theory provides an accurate description of the Meissner effect, the isotope effect, the formation of the superconducting energy gap and the anomalous behavior of the specific heat, as well as a number of other physical properties of the superconducting state.

When superconducting perovskites were first discovered in the mid-1980s, superconductors were categorized as either ‘conventional superconductors’, meaning metals with Tc < 23.2 K or ‘high temperature superconductors”, meaning cuprates with Tc > 30 K. This is a very well-defined distinction which allows superconductors to be separated into two categories merely by drawing a horizontal line somewhere between about 20 K and 30 K on figure 10.4, where those superconductors below the line were conventional materials that were well described by BCS theory and those above the line were high temperature materials where the validity of BCS was unclear. This distinction may be useful from a practical standpoint in terms of superconducting applications, but it is an overly simplistic approach to understanding the physics of superconductors. This is because some classes of superconductors would cross the line between conventional and high temperature. This is seen, for example, for the ferropnictnides and for the metallic superconductors, as well. The latter is the result of the discovery of superconductivity in MgB_2 (magnesium dibromide) at 39 K in 2001 and the recent results concerning H_2S as discussed above. Therefore, from a fundamental physical standpoint, it is probably most relevant to describe the types of superconductors in terms of the theoretical description of the basic superconductivity mechanism.

The cuprate superconductors have been the most widely studied of all the non-conventional classes of superconductor and there is some consensus concerning the ways in which the superconductivity in these materials differs from that in conventional superconductors, although there is no detailed theory of this behavior. In many other classes of superconductors, the situation is even less clear. In the present section we concentrate on cuprate superconductors and how their superconducting properties differ from those described by BCS theory and on the way in which high temperature conventional superconductors like H_2S can be explained.

Experimental evidence thus far indicates that all cuprate superconductors are type-II superconductors. In fact, as indicated above, they have exceptionally high Bc2 and are, therefore, suitable, at least in terms of critical field values, for many superconductor applications. This will be discussed further below. Two characteristics of the cuprate superconductors that cannot be justified in the context of BCS theory are the nature of the Cooper pair wavefunction and the observed isotope effect.

In the mid-1990s Chang C Tsuei and John R Kirtley at the IBM T J Watson Research Center conducted a series of experiments on flux quantization in cuprate superconductors. They measured the spontaneous magnetic flux in superconducting rings which contained either two or three Josephson junctions (see chapter 12). They found that rings with two Josephson junctions (see chapter 11) contained spontaneous magnetic flux equal to an integer number of fundamental flux quanta as defined by equation (8.17), i.e.

while rings with three Josephson junctions contained spontaneous magnetic flux equal to a half integer of the fundamental flux quanta, i.e.

These results provided strong evidence that antiferromagnetic ordering in these cuprates played an important role in the formation of Cooper pairs. More importantly, it provided fairly conclusive evidence that the Cooper pair wavefunction had d-state symmetry. Further experiments have confirmed that this is the case. This is in contrast with the assumption of BCS theory that the Cooper pair wavefunction has s-state symmetry.

Experimental studies of the isotope effect in cuprate superconductors has also provided results that are difficult to reconcile with the predictions of BCS theory. Extensive studies of cuprate superconductors with two different isotopes of oxygen, 16O and 18O have been undertaken. A few studies have also reported the isotope effect for copper using 63Cu and 65Cu. The isotope effect on the critical temperature is defined by the exponent, α, as indicated in equation (9.7). In the case of superconductors with multiple elemental components, a partial isotope exponent may be defined for each element as

where ΔTc(i) is the shift in Tc that results from the change in isotopic mass ΔM(i) for element i. Typical results for measurements of α(i) in cuprate superconductors is shown in figure 10.8. These data show the exponent, α(i) for i = O, Cu, as a function of critical temperature for a series of cuprates with differing compositions. It is readily seen that the exponent decreases as a function of critical temperature and has a value near zero for the cuprates with the highest Tc. The composition of a series of cuprates that yields the highest critical temperature is referred to as optimal doping; see comments concerning doping in section 7.7. It has generally been found that cuprate superconductors with optimal doping have isotope exponents near zero. This effect is difficult to explain in the context of BCS theory, especially when more detailed treatments of Coulombic repulsion and band structure are included.

Figure 10.8. Isotope exponents for oxygen isotopes and copper isotopes in a series of Y–Pr–Ba–Cu–O cuprate superconductors of different compositions as a function of critical temperature. (Reproduced from Zhao et al 2001 with permission of IOP Publishing.)

A number of other properties of superconducting materials (in addition to Tc) are known to be influenced by the isotope effect. Among these are the effective charge carrier mass, the antiferromagnetic ordering temperature and the temperature dependence of the energy gap. Overall, observed isotope effects in cuprate superconductors are difficult to explain using BCS theory.

Even after more than 30 years of investigation, the details of the superconductivity mechanism in cuprate perovskites is not clear. The exact role of phonons, if any, is still uncertain. It has been speculated that Cooper pair formation in these materials may be mediated by spin density waves, that is, periodic modulations of the electron spins, and that the antiferromagnetic coupling of Cu spins may play an important role in this behavior.

It is seen in figure 10.4 that H_2S is identified as a conventional metallic superconductor. In fact, it is known, as mentioned above, that H_2S undergoes a transition to a metallic state at a pressure around 95 GPa, although the details of this transition are not entirely clear. When high temperature superconducting perovskites were first discovered, one of the reasons for looking at superconducting mechanisms other than BCS theory was a general feeling that BCS type superconductivity was limited to critical temperatures around 30 K. However, there is, in principle, no temperature limit associated with BCS type superconductivity. A conventional superconductor merely requires high frequency phonons, a strong electron–phonon coupling and a sufficient density of electron states at the Fermi energy. This is clear from equation (9.6) which summarizes the importance of these parameters on Tc in the context of BCS theory. Atomic mass is a key factor for determining Tc, as reducing atomic mass increases the phonon frequency and gives rise to a corresponding increase in critical temperature. This effect is obvious in the BCS expression that describes the isotope effect (equation (9.7)). Extending the isotope concept to very light atoms, suggests that this approach might lead to high values of the critical temperature.

Hydrogen, being the lightest atom, is of particular interest. In 1935 Eugene Wigner and Hillard Bell Huntington predicted that hydrogen would exhibit an insulator to metal transition at high pressure. Experimental evidence concerning this transition is still not definitive. However, pressure induced insulator to metal transitions are well known in other systems and a conductive high-pressure phase of hydrogen (probably a semimetal) was reported in 2011. In 1968 Neil Ashcroft suggested that metallic hydrogen could become superconductive at around room temperature under a pressure of about 500 GPa. He also later suggested that hydrogen-rich compounds might also benefit from the high Debye frequency of the light hydrogen atom and exhibit conventional superconductivity at high temperatures. The report of superconductivity in H_2S at above 200 K is the first evidence of conventional superconductors that results from the high frequency phonon modes of hydrogen. As well, the observed isotope effect, as shown in figure 10.7, is consistent with predictions of a conventional superconducting phase.

While the search for high temperature superconducting cuprates has shown no progress in more than 20 years, the recent discovery of conventional superconductivity in hydrogen-based materials under pressure has given new promise to the possibility of room temperature superconductivity.

References and suggestions for further reading

Bednorz J G and Müller K A 1986 Possible high Tc superconductivity in the Ba–La–Ca–O system Z. Phys. B 64 189–93

Burns G 1991 High-Temperature Superconductivity: An Introduction 1st edn (New York: Academic)

Drozdov A P, Eremets M I, Troyan I A, Ksenofontov V and Shylin S I 2015 Conventional superconductivity at 203 K in the sulfur hydride system Nature 525 73–7

Gao L, Xue Y Y, Chen F, Xiong Q, Meng R L, Ramirez D, Chu C W, Eggert J H and Mao H K 1994 Superconductivity up to 164 K in HgBa_2Cam−1CumO2m+2+δ ( m=1, 2, and 3) under quasihydrostatic pressures Phys. Rev. B 50 4260–3

Ray P J 2015 Structural investigation of La2−xSrxCuO4+y – Following staging as a function of temperature Master’s Thesis University of Copenhagen

Uchida S 2015 High Temperature Superconductivity: The Road to Higher Critical Temperature (New York: Springer)

Wu M K, Ashburn J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang Y Q and Chu C W 1987 Superconductivity at 93 K in a new mixed-phase Y–Ba–Cu–O compound system at ambient pressure Phys. Rev. Lett. 58 908–10

Zhao G M, Keller H and Conder K 2001 Unconventional isotope effects in high-temperature cuprate superconductors J. Phys. Condens. Matter 13 R569–87

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 11

The Josephson effect

11.1 Introduction

Quantum mechanical tunneling is a well know phenomenon that clearly demonstrates the difference between classical physics and quantum mechanics. Macroscopic manifestations of quantum tunneling are observed in a wide variety of phenomena such as nuclear fusion in stars, radioactive alpha decay, photosynthesis and the operation of many electronic components. These processes typically involve the tunneling of electrons, or sometimes protons, through potential barriers. In 1962 Brian David Josephson predicted the tunneling of Cooper pairs across a weak link between two superconductors. The weak link between the two superconductors can be either an insulating layer or a normally conducting layer (where the Cooper pairs cannot exist as real charge carriers). Devices based on this behavior have a number of important applications, as described later in this chapter and in chapter 12. We begin with the basic physics of Cooper pair tunneling in Josephson junctions.

11.2 The Josephson effect

As noted above, Cooper pair tunneling between superconductors separated by a weak link comprised of an insulator or normal conductor was predicted by Brian Josephson. A simple example of the construction of a Josephson junction is illustrated in figure 11.1. The equations that govern the time dependent voltage, V(t), and current, I(t), characteristics of this device were developed by Josephson and are

and

where Ic is the critical current and ϕ(t) is the time dependent phase shift across the junction. These are known as the first and second Josephson equations, respectively. The electrical properties of the Josephson junction depend on the details of the applied voltage as described below.

Figure 11.1. Geometry of a Josephson junction. (Reprinted from Hamilton 2000, with the permission of AIP Publishing.)

When there is no external electromagnetic field applied to the Josephson junction then V(t) in equation (11.2) is zero and the phase shift is a constant. Thus, there will be a constant current across the junction given by equation (11.1) where ϕ is not time dependent. In this case the constant may take on values between −Ic and +Ic, depending on the value of ϕ.

If a constant voltage is applied across the Josephson junction then equation (11.2) may be integrated, with V as a constant, to give

Substituting this into equation (11.1) gives

From this it clearly follows that an alternating current will flow across the junction and the frequency of this current will be

This same relation can be obtained more directly without the need for the Josephson equations. A Cooper pair with charge 2e tunnels across the junction, and through a potential V, thereby gaining an energy 2eV. Since the Cooper pair does not exist as a real particle inside the junction, the energy that it gains must be radiated as a photon with energy

Equation (11.5) follows directly from equation (11.6).

The coefficient on the right-hand side of equation (11.5) is called the Josephson constant, KJ, and has a value of

It is easy to see that the Josephson constant is merely the inverse of the fundamental flux quantum given in equation (8.17).

Equation (11.5) shows a fundamental relationship between two physical quantities, voltage and frequency, where the proportionality constant is dependent only upon the fundamental constants e and h, and not on any device dependent parameters such as dimensions, materials, temperature, etc. This fundamental relationship allows for a basic definition of a voltage standard as discussed below.

Equation (11.4) is a simplified version of a more general expression that takes into account junction properties such as capacitance and resistance, which will influence current flow across the junction by normal electrons. An equivalent circuit diagram for the real Josephson junction is shown in figure 11.2.

Figure 11.2. Equivalent circuit for a real Josephson junction. The ‘×’ symbol represents the Josephson junction.

This circuit is described by a generalization of equation (11.4) as

where the first term on the right-hand side represents Cooper pair tunneling, the second term represents the normal electron current through a resistive element and the third term represents the normal electron current through a circuit element with capacitive impedance and takes into account time varying potentials.

The characteristics of the solution to equation (11.8) depends on several factors, including the relative values of C, R and Ic as well as the form of the applied voltage. Here we show a couple of representative examples. In figure 11.3 the I–V curve of a typical Josephson junction with DC applied voltage is shown. When there is no applied voltage a supercurrent up to the critical value, Ic, can be carried across the junction. When a DC voltage is applied across the junction, the tunneling current is negligible up to a point (for both positive and negative directions) where the current suddenly becomes large. This voltage, as shown in the figure, is Eg/e, that is the voltage corresponding to the superconducting energy gap. At this point the superconducting Cooper pairs break and normal electron current flows.

Figure 11.3. I–V characteristics for a Josephson junction showing the applied voltage necessary to break superconducting Cooper pair bonds.

Figure 11.4 shows the I–V characteristics of a Josephson junction exposed to microwave radiation. Here the external radiation causes a quantization of the energy (hence voltage) values of the Cooper pair tunneling current where

Figure 11.4. Quantized voltage levels in a Josephson junction exposed to microwave radiation. (Reprinted by permission from Springer Nature: Jeanneret and Benz 2009.)

Equation (11.9) provides a clear relationship between voltage and frequency in experimental data such as shown in figure 11.4. As the Josephson constant is based on well established fundamental constants and frequency and is fairly easy to measure to a high degree of accuracy, this equation provides a convenient means of defining the volt. In fact, the voltage standard, as adopted in 1990 by the National Institute of Standards and Technology in the United States, uses the standard definition of the volt based on the Josephson junction. This is now used extensively worldwide.

The difficulty with adopting a standard definition of the volt on the basis of the Josephson effect results from the value of the Josephson constant in equation (11.9) and the magnitude of the quantized voltages in a Josephson junction. The voltage levels shown in figure 11.4 for a frequency of 96 GHz are separated by about 0.2 mV. In order to obtain a measurement of a voltage around one volt, a large number of Josephson junctions are constructed in series. In practice, combined series and parallel combinations of Josephson junctions are sometimes used and a device developed by the United States National Institute of Standards and Technology is shown in figure 11.5.

Figure 11.5. Integrated circuit comprised of 67 410 Josephson junctions used as a 2.6 V standard. This device was developed by the United States National Institute of Standards and Technology in 2005. (Credit: Benz/NIST.)

References and suggestions for further reading

Barone A and Paterno G 1982 Physics and Applications of the Josephson Effect (Hoboken, NJ: Wiley)

Chong Y, Burroughs C J, Dresselhaus P D, Hadacek N, Yamamori H and Benz S P 2005 2.6-V high-resolution programmable Josephson voltage standard circuits using double-stacked MoSi_2-barrier junctions IEEE Trans. Instrum. Meas. 54 616–9

Gallop J C 1990 SQUIDS, the Josephson Effects and Superconducting Electronics (Boca Raton, FL: CRC Press)

Hamilton C A 2000 Josephson voltage standards Rev. Sci. Instrum. 71 3611–23

Jeanneret B and Benz S P 2009 Application of the Josephson effect in electrical metrology Eur. Phys. J. Special Topics 172 181–206

IOP Concise Physics

Electrons in Solids

Contemporary topics

Richard A Dunlap

Chapter 12

Superconductor applications

12.1 Introduction

Superconductors, both conventional and cuprate, have found numerous applications over the years. Some applications have the potential to be of great commercial importance while other are of greater scientific significance. In this, and the next two sections, we review a few of the applications of superconductors. This treatment is far from exhaustive and does not cover some of the applications that are more speculative, or which have not found the success that was anticipated. We do, however, cover the more important applications where the use of superconducting materials has proven to fulfil applications which would otherwise not be feasible or where the use of superconductors has the potential for increased efficiency.

12.2 Power transmission

When superconductivity was first discovered by Kamerlingh Onnes, it was envisioned that superconducting materials would revolutionize many aspects of technology that involve the conduction of large currents. Unfortunately, this did not turn out as anticipated because of the low critical fields of the superconductors that were known in the early years of research in this field. It was not until the 1950s and 1960s with the discovery of materials such as Nb_3Sn and Nb_3Ge, which had substantial critical fields at achievable temperatures, that superconductors found an appreciable number of applications.

With the discovery of superconducting cuprates with critical temperatures above the temperature of liquid nitrogen, it was hoped that the use of superconductors would become widespread and that these materials would be able to fulfil requirements that had been difficult in the past. Unfortunately, once more, the application of superconductors was not as straightforward as hoped. While it was true that superconducting cuprates such as Y–Ba–Cu–O had extremely high critical fields at T = 0, a comparison of Y–Ba–Cu–O at 77K with Nb-based conventional superconductors at liquid helium temperature, showed that the differences were not all that significant. Figure 12.1 shows a comparison of the critical field of some superconductors from different classes. While it is true that there are clear advantages to operating a superconducting device at liquid nitrogen temperate rather than liquid helium temperature in terms of the complexity and cost of equipment as well as cost and ease of use in dealing with the cryogen, particularly for more commercial applications, the use of superconducting cuprates is not without its own difficulties. Firstly, these materials are quite brittle (as they are ceramics and not metals) and the fabrication of a current carrying superconductor in the form of a wire is very challenging. Secondly, because these materials are highly anisotropic the electronic properties can be considerably different in the a–b plane and in the c-direction. Finally, the grain structure of the material can have a substantial effect on its overall properties. While intragranular properties may be excellent, intergranular may be poor, as the nature of the contact between grains can adversely affect the overall transport of current. It is clear that, although the basic superconducting properties of cuprates may be encouraging, the materials preparation aspects of their application could be challenging.

Figure 12.1. Upper critical field phase diagram of several superconductors having potential for commercialization in high field magnets. (Reprinted by permission from Springer Nature: Larbalestier et al 2001.)

Perhaps the most obvious application of superconducting materials is for the transmission of electrical power. The ability to use liquid nitrogen as a cryogen is of significant importance for this applications because of the large scale that is involved. The first superconducting cuprate ‘wires’ were made from Bi–Sr–Ca–Cu–O because of its high critical temperature. However, these were not practical because of cost and materials fabrication issues. Second generation superconducting cuprate wires have been made from Y–Ba–Cu–O. To fabricate a suitable power transmission cable from this material, a 0.1 mm thick tape is prepared by sandwiching 1 μm of the high temperature superconductor between layers of copper and nickel. The superconducting ribbons are made into cables suitable for power distribution by winding the ribbons around a copper core as shown in figure 12.2. The cable is surrounded by an insulated jacket that carries liquid cryogen to maintain the superconductor below its critical temperature.

Figure 12.2. Construction of a high temperate cuprate superconductor power transmission cable. (Credit: energy.gov.)

The first commercial power distribution cable that uses superconducting technology was installed in Lake Ronkonkoma, New York in 2008 (see figure 12.3). This 600 m cable is installed underground and is operated by the Long Island Power Authority (LIPA). It is part of a 138 kVpower distribution grid that supplies electricity to about 300 000 homes.

Figure 12.3. The first superconducting power distribution cable installed in New York in 2008. The connection to the underground superconducting cables from the distribution facility is seen in the photograph. (Credit: energy.gov.)

Since that time, a number of other similar cables have been installed worldwide. The ability of using liquid nitrogen rather than liquid helium as a cryogen is particularly beneficial for this type of application because economic factors, as well as system complexity and reliability, are important. While systems of this type have met with success, it is unclear if long range power distribution networks utilizing superconducting technology would be practical at present. Urban networks, where considerable power is distributed, but individual lines are of limited length, might be the most appropriate application of this technology in the near future.

12.3 Particle physics

The Biot–Savart law gives the magnetic field at a distance r from a long straight conductor carrying a current I as

Clearly, in the absence of magnetic materials, the ability to produce large magnetic fields, amounts to the ability to carry large currents in a conductor. The approach to producing large magnetic fields has been discussed in chapter 3 where it was shown that the largest fields can be produced by a combination of a superconducting coil and a Bitter solenoid. This approach is appropriate for some scientific applications where very strong fields in relatively small volumes are needed and economical operation is not a major consideration. For many other applications a strong magnetic field is required over a substantial volume and minimizing power requirements is beneficial. In such cases superconducting magnets have found widespread application. The use of superconducting magnets has several advantages over the use of conventional electromagnets, either with or without a magnetic core. These include:

- A large magnetic field can be maintained over a substantial volume.

- Magnet volume can be minimized as large currents can be carried in superconductors with minimal cross section.

- Power requirements are minimized, as once energized, superconducting magnets carry persistent currents without the need for a power source.

- As there are no resistive losses in the superconductor, there is no Ohmic heating and no need for a cooling system.

For virtually all magnet applications conventional superconductors using liquid helium cooling have been most successful. In such cases, the most common superconductor is a Nb–Ti alloy, although some systems utilize the high temperature conventional superconductor MgB_2. Implementation of perovskite superconductors is in its early stages. In some systems continuous flow helium refrigerators, which eliminate the need for constantly replenishing (expensive) liquid helium, have been incorporated.

The use of superconducting magnets is widespread for medical imaging systems (MRI scanners) as mentioned in chapter 3. In the present section we look at some other applications where superconducting magnets have been successfully utilized. These include, particle accelerators, magnetic confinement fusion reactors and magnetically levitated trains.

Particle accelerators utilize a large number of magnets to direct the path of charged subatomic particles. These magnets are used for two purposes. Dipole magnets (i.e. bending magnets) are used to direct the beam of charged particles around corners. The charged particles are directed in a circular trajectory according to the Lorentz force. Magnets are also used to focus the beam as shown in the figure. The magnetic fields required for large accelerators necessitate the use of superconducting magnets and these have been in common use worldwide since the 1970s when Nb–Ti superconductors became available. The Large Hadron Collider (LHC) operated by CERN (Conseil Européen pour la Recherche Nucléaire) near Geneva (see figure 12.4) is the world’s largest accelerator and utilizes 1232 dipole magnets, 15 meters in length, for bending the beam and 392 quadrupole magnets, 5 to 7 meters in length, for focusing the beam. The liquid helium costs at such a facility are typically a small fraction of the overall operating costs and changing extensive existing and well-proven infrastructure to high temperature superconductor technology is not economical or wise.

Figure 12.4. Beamline at the LHC operated by CERN. (Credit: Cern.)

12.4 Fusion research

Research on controlled fusion reactors has taken a variety of approaches in recent years. The most successful of these has been inertial confinement fusion (i.e. laser fusion) and magnetic confinement fusion. The latter approach is an important application of superconducting magnets. The most common geometry for magnetic confinement fusion reactors is toroidal and the tokamak (from the Russian for toroidal chamber with magnetic coils) shown in figure 12.5 is probably the best known of these machines. Magnetic fields confine the charged particles of the plasma which travel in inside a toroidal region surrounded by magnetic coils. Superconducting coils, typically utilizing Nb–Ti superconductors, are in common use to produce magnetic fields that are sufficient for this purpose.

Figure 12.5. Typical design of a tokamak or toroidal magnetic confinement fusion reactor. Superconducting magnets are shown in red and the toroidal plasma chamber is shown in yellow. (Vergara Pina Fernando Wla / Shutterstock.com.)

While Tokamaks have shown some success in achieving conditions that are appropriate for nuclear fusion reactions, they have limitations and fall short of reaching conditions necessary for net energy production. The particles which travel in paths around the inside edge of the torus experience a different magnetic field than those which travel around the outside surface of the torus. This situation results in the separation of the positive and negative charges in the plasma and leads to instabilities. One way of dealing with this problem is to twist one end of the torus relative to the other so that particles which travel along the inside edge of the torus on one side of the device travel along the outside edge of the torus on the other side of the device; and vice versa. This type of device is called a stellarator and was one of the earliest types of magnetic confinement fusion reactors. There has been renewed interest in this design now that tokamaks seem to have reached their limitations. Figure 12.6 shows a diagram of the magnetic coil arrangement in the Wendelstein X-7, a recently constructed stellarator operated by the Max Planck Institute of Plasma Physics (IPP) in Greifswald, Germany. The unusual shape of the magnet coils is the result of extensive computer modeling of the plasma.

Figure 12.6. Diagram of the arrangement of the superconducting magnets in the Wendelstein X-7 fusion reactor. There are 50 non-planar inner superconducting magnets (shown in blue) and 20 planar outer superconducting magnets (shown in brown). (Graphic: IPP, Dr. Christian Brandt)

12.5 SQUIDs

The Josephson effect plays a key role in the operation of several devices (particularly sensors) that are in common use. In this section we discuss one of the widely used types of Josephson sensors; the SQUID. The name SQUID is an acronym for ‘Superconducting QUantum Interference Device’. The device is basically a very sensitive magnetic field sensor that operates on the principle of the Josephson junction. It is not only a very sensitive magnetic field detector but it approaches the quantum limit of sensitivity and is, therefore, close to being as sensitive as any detector of anything can possibly be.

There are fundamentally two different types of SQUIDs, dc SQUIDs and rf (or ac) SQUIDs. Although the rf SQUID is, at least in principle, simpler to construct, the dc SQUID is more sensitive and more commonly utilized. We confine our discussion here to the dc SQUID, which is illustrated in figure 12.7. This device consists of a superconducting ring with two weak links or Josephson junctions, one on either side of the ring. A current is injected into the device, on the left in the figure, and is split between the two sides of the loop, thereby passing through (by tunneling) the two Josephson junctions. The currents from the two paths are rejoined on the right side of the diagram. If magnetic flux lines are present inside the loop, then a phase difference is introduced across the junctions and this gives rise to a voltage across the loop. As this voltage is proportional to the magnetic flux enclosed in the loop, then a simple measurement of the voltage across the loop constitutes a sensitive measurement of the magnetic field.

Figure 12.7. Diagram of a dc SQUID magnetometer showing two Josephson junctions. (Miraceti / Wikimedia Commons / CC-BY-SA 3.0 / https://creativecommons.org/licenses/by-sa/3.0/deed.en.)

The equivalent circuit of the dc SQUID is shown in figure 12.8. Each real Josephson junction consists of an equivalent circuit that contains a resistor and a capacitor in parallel. In a real SQUID there may be an actual resistor in the circuit in addition to the resistance of the weak link between the superconductors. The voltage drop across the SQUID, Vs, for a given bias current, IB, is related to the phase difference introduced by the two junctions. This phase difference is, in turn, a function of the magnetic flux through the loop and the bias current. In general, we can write the total phase difference across the SQUID, δϕ, as

where δϕB is the phase difference introduced by the magnetic flux and δϕI is the phase difference introduced by each of the junctions. The magnetic flux through the loop is given by

where ΦA is the flux from the external magnetic field and Φ_0 is the fundamental flux quantum. The bias current is related to the critical current of the weak links, Ic, and the phase shift as given in equation (12.2) that is introduced by this current;

From equations (12.2) to (12.4) it can be shown that the total phase shift across the loop will be an integer multiple of 2π when

By providing a small modulating magnetic flux, Φm < Φ_0/2, then a voltage drop is observed across the loop and this is proportional to the linked flux.

Figure 12.8. Equivalent circuit diagram of a dc SQUID.

The dc SQUID is, by far, the most sensitive method of measuring magnetic fields and can have a field sensitivity on the order of 1 × 10−15 T. This can be compared to some typical magnetic fields;

- Earth’s magnetic field: 5 × 10−5 T

- Magnetic field produced by a human heart: 5 × 10−11 T

- Magnetic field produced by a human brain: a few × 10−15 T.

The Earth’s magnetic field is easily measured by magnetometers that are far simpler (and less sensitive) than the SQUID. Even fluctuations in the Earth’s field caused by auroral activity are of the order of 5 × 10−7 T and are readily observed by conventional methods. Although SQUIDs have been used in many fields, the sensitivity, as shown above, makes them suitable for several medical and biophysics applications. The use of SQUID magnetometry in conjunction with traditional electroencephalography (measurement of the electrical signals from the brain) has been an important technique for the diagnosis of a number of neurological disorders.

The ability to utilize high temperature superconductors in SQUIDs would make these devices simpler and more economical to use and experimental SQUIDs have been constructed using Y–Ba–Cu–O cuprates. These devices tend to show a much greater level of noise than SQUIDs based on traditional superconducting materials, thereby reducing their sensitivity to around 3 × 10−14 T. Although this may not be sufficient for the most demanding SQUID applications, it may be suitable for a number of measurements and progress in this area may, in the future, make high temperature SQUIDs competitive with Nb-based SQUIDs.

References and suggestions for further reading

Barone A and Paterno G 1982 Physics and Applications of the Josephson Effect (Hoboken, NJ: Wiley)

Burns G 1991 High-Temperature Superconductivity: An Introduction 1st edn (New York: Academic)

Clarke J and Braginski A I (ed) 2006 The SQUID Handbook: Applications of SQUIDs and SQUID Systems vol 2 (Weinheim: Wiley)

Gallop J C 1990 SQUIDS: The Josephson Effects and Superconducting Electronics (Boca Raton, FL: CRC Press)

Kleiner R, Buckel W and Huebener R 2016 Superconductivity: An Introduction 3rd edn (Berlin: Wiley)

Larbalestier D, Gurevich A, Feldmann D M and Polyanskii A 2001 High-Tc superconducting materials for electric power applications Nature 141 368–77

Uchida S 2015 High Temperature Superconductivity: The Road to Higher Critical Temperature (New York: Springer)