GO a w o 3 SUPER- CONDUCTIVITY w Ernest A. Lynton METHUENS MONOGRAPHS ON PHYSICAL SUBJECTS Superconductivity ERNEST A. LYNTON Although the fascinating phenomenon of superconductivity has been known for fifty years, it is largely through the concentrated experimental and theoretical work of the last decade that a basic (though at present very incomplete) understanding of the effect has been reached. This monograph is a largely descriptive introduction to superconduc- tivity, requiring little more than an under- graduate physics background. It is written to serve two functions ; first as a stepping stone towards more intensive study for those who intend to work in the field of research and development of superconductivity and its applications and, secondly, as a basic refer- ence on the present state of the subject of superconductivity. The book contains a description of the principal characteristics of a superconductor, together with a detailed discussion of the most useful phenomenological models which have been applied to superconductors. The second part of the monograph describes the funda- mental microscopic properties in terms of the theory of Bardeen, Cooper and Schrieffer. It is shown how remarkably successful this theory has been in explaining the behaviour of an idealized superconductor. There is a chapter on superconducting devices, a sub- ject index and a bibliography of more than 330 books and articles. SECOND EDITION LONDON : METHUEN & CO. LTD NEW YORK JOHN WILEY & SONS INC methuen's monographs on physical subjects General Editor: B. L. WORSNOP, b.sc, ph.d. SUPERCONDUCTIVITY Superconductivity E. A. Lynton Professor of Physics Rutgers, The State University New Brunswick, N.J., U.S.A. Powder patterns of the intermediate stale, showing thesbrink- ing of the superconducting (dark) regions as /; takes on the values (left to right, top to bottom) 0, 008, 027, 053, 0-79, and 0-90. (After Faber, 1958. Reproduced by kind permission of the Royal Society and the author.) Proc. Roy. Soc. A248 464, plate 25. LONDON: METHUEN & CO LTD NEW YORK: JOHN WILEY & SONS INC First published in 1962 Second edition 1964 © 1962 and 1964 by E. A. Lynton Printed in Great Britain by Spottiswoode, Ballantyne & Co Ltd London & Colchester Catalogue No. Methuen 12/4081/66 2-1 For Carla CHRJS. : i Acknowledgements This book has grown, beyond recognition, from a set of lecture notes written and used during my stay at the Institut Fourier of the Univer- sity of Grenoble in 1959-60. I should like once again to thank my hosts, Professors Neel and Weil and Dr Goodman, for a stimulating and pleasant year. I am very grateful to a large number of people who have helped me with written or oral comments, with news of their un- published work, with preprints, and with copies of graphs. In par- ticular I thank Drs Coles, Collins, Cooper, Douglass, Faber, Gar- funkel, Goodman, Masuda, Olsen, Pippard, Schrieffer, Shapiro, Swihart, Tinkham, Toxen, and Waldram. My colleagues Lindenfeld, McLean, and Weiss provided much helpful discussion. Above all my gratitude is due to Bernard Serin, from whose guidance and friendship I have profited for many years. He found the time to read the entire first draft of the manuscript and suggested many improvements, not all of which I have been wise enough to incorporate. September 1961 E. A. LYNTON Preface to the Second Edition This edition contains revisions and additions which bring the mono- graph essentially up to date, as of the end of June 1964. The treatment of superconductors of the second kind has been considerably ampli- fied, a discussion of the Josephson effect has been added, and a num- ber of other changes have been made. Many of these were also incor- porated in the excellent French translation of Mme Nozieres, which was published early this year. I am very grateful to her, as well as to Dr Nozieres, for his valuable comments and help. A Russian trans- lation, edited with many illuminating footnotes by Dr. Gor'kov, unfortunately reached me too late for these comments to be included in the present edition. August 1964 E. A. LYNTON vi Contents Introduction page I n m IV Basic Characteristics 1 . 1 Perfect conductivity and the critical magnetic field 1.2 Superconducting elements and compounds 1.3 The Meissner effect 1.4 The specific heat 1.5 Theoretical treatments Phenomcnological Thermodynamic Treatment 2.1 The phase transition 2.2 Thermodynamics of mechanical effects 2.3 Interrelation between magnetic and thermal properties 2.4 The Gorter-Casimir two-fluid model Static Field Description 3.1 Perfect diamagnetism 3.2 Influence of geometry and the intermediate state 3.3 Trapped flux 3.4 The perfect conductor 3.5 The London equations for a superconductor 3.6 Quantized flux The Pippard Non-local Theory 4.1 The penetration depth, A 4.2 The dependence of A on temperature and field 4.3 The range of coherence 4.4 The Pippard non-local relations The Ginzburg-Landau Phenomcnological Theory vii 3 3 5 7 9 11 13 13 16 17 19 22 22 23 26 27 28 32 34 34 36 40 41 48 Vlll VI vn Vffl IX Contents The Surface Energy page 55 6.1 The surface energy and the range of coherence 55 6.2 The surface energy and the intermediate state 59 6.3 Phase nucleation and propagation 61 6.4 Supercooling in ideal specimens 65 6.5 Superconductors of the second kind 67 6.6 The mixed state or Shubnikov phase 71 6.7 Surface Superconductivity 74 Low Frequency Magnetic Behaviour of Small Specimens 75 7.1 Increase in the critical field 75 7.2 High field threads and superconducting magnets 77 7.3 Variation of the order parameter and the energy gap with magnetic field 78 The Isotope Effect 81 8.1 Discovery and theoretical considerations 81 8.2 Precise threshold field measurements 83 Thermal Conductivity 87 9.1 Low temperature thermal conductivity 87 9.2 Electronic conduction 89 9.3 Lattice conduction 93 9.4 Thermal conductivity in the intermediate state 94 The Energy Gap 95 10.1 Introduction 95 10.2 The specific heat 96 10.3 Electromagnetic absorption in the far infrared 98 10.4 Microwave absorption 100 10.5 Nuclear spin relaxation 104 10.6 The tunnel effect 106 10.7 Far infrared transmission through thin films 109 10.8 The Ferrell-Glover sum rule 1 14 ,. XII XIII Contents Microscopic Theory of Superconductivity 11.1 Introduction 11.2 The electron-phonon interaction 1 1 .3 The Cooper pairs 1 1 .4 The ground state energy 11.5 The energy gap at 0°K 1 1.6 The superconductor at finite temperatures 11.7 Experimental verification of predicted thermal properties 11.8 The specific heat 1 1.9 Coherence properties and ultrasonic attenuation 11.10 Electromagnetic properties Superconducting Alloys and Compounds 12.1 Introduction 12.2 Dilute solid solutions with non-magnetic im- purities 12.3 Compounds with magnetic impurities 1 2.4 Superimposed metals Superconducting Devices 13.1 Research devices 13.2 Superconducting magnets 1 3.3 Superconducting computer elements Bibliography Index IX page 116 116 117 118 120 126 127 129 134 136 139 141 141 141 145 150 153 153 154 157 165 183 Introduction Although the fascinating phenomenon of superconductivity has been known for fifty years, it is largely through the concentrated experi- mental and theoretical work of the past decade that a basic (though as yet very incomplete) understanding of the effect has been reached. Far from being an oddity of little physical interest it has been shown to be a co-operative phenomenon of basic importance and with close analogies in a number of fields. At the present time one important period in the development of the subject has been completed, and the next is already well under way, with much effort in theory and experi- ment to carry our understanding from the general to the particular, from the idealized superconductor to the specific metal. Somewhat coincidentally, there now also is great interest in possible practical applications of superconductivity. This monograph is a largely descriptive introduction to super- conductivity, requiring no more than an undergraduate physics back- ground, and written to serve two functions. It can be a first survey and a stepping stone toward more intensive study for those who intend to become actively engaged in the further development of superconduc- tivity, be it in basic research or in technical applications. Such readers will benefit from the extensive bibliography, listing more than 450 books and articles. At the same time the book is sufficiently complete in its description both of experimental details and of theoretical approaches to be a basic reference for those who wish to be acquainted with the present state of superconductivity. It will enable them to follow further developments as they appear in the scientific and technical literature. The contents of the book can be grouped into a number of sections which treat the subject of superconductivity in successive layers with increasing resolution of detail. The first three chapters introduce the reader to the principal characteristics of bulk superconductors, and treat these in terms of the basic phenomenological models of London and of Gorter-Casimir. With this section the reader thus acquires a broad outline and a general understanding of the thermodynamic and 1 2 Superconductivity the static electromagnetic behaviour of idealized, bulk superconduc- tors. The treatment of the subject is then pursued in greater detail along two essentially parallel directions. In the section comprising Chapters IV-VII are discussed those aspects of the behaviour of superconductors which lead to the non-local treatments of Pippard and of Ginzburg and Landau. These more sophisticated phenomeno- logical models account for an interphase surface energy, in terms of which the later chapters of this section describe the intermediate state, phase nucleation, propagation, and supercooling, superconductors of the second kind, and the magnetic behaviour of specimens of small dimensions. Chapters VIII-X can be read without a study of the preceding section (IV-VII) and describe in much detail those characteristics of a superconductor which during the past decade have indicated the microscopic nature of superconductivity, and have led to the theory of Bardeen, Cooper, and Schrieffer. The fundamental aspects of this theory are presented with a minimum of mathematics. The book closes with a chapter on the behaviour of alloys and com- pounds, and with one on superconducting devices. In describing the principal empirical characteristics of supercon- ductors I have tried to include only the key experiments through which the phenomenon in question was established, as well as more recent work which gives the most detailed or the most precise informa- tion. It is both unnecessary and impossible in a monograph of this small size to be encyclopaedic either in the enumeration of all per- tinent experiments, or in the description of superconducting be- haviour in minute detail. My selection of what aspects of the latter to emphasize may appear arbitrary, especially to those whose work has been slighted. The choice was not a judgement of the scientific value of such work, but rather of its didactic usefulness in illuminating the elementary characteristics of superconductors. CHAPTER I Basic Characteristics 1.1. Perfect conductivity and critical magnetic field The behaviour of electrical resistivity was among the first problems investigated by Kamerlingh Onnes after he had achieved the lique- faction of helium. In 1911, measuring the resistance of a mercury sample as a function of temperature, he found that at about 4°K the resistance falls abruptly to a value which Onnes' best efforts could not distinguish from zero. This extraordinary phenomemon he called superconductivity, and the temperature at which it appears the critical temperature, T c (Kamerlingh Onnes, 1913). When a metallic ring is exposed to a changing magnetic field, a current will be induced which attempts to maintain the magnetic flux through the ring at a constant value. For a body of resistance R and self-inductance L, this induced current will decay as /(/) = 7(0)exp(-i?//L). (LI) /(/) can be measured with great precision, for example, by observing the torque exerted by the ring upon another, concentric one which carries a known current. This allows the detection of much smaller resistance than any potentiometric method. A long series of such measurements on superconducting rings and coils by Kamerlingh Onnes and Tuyn (1924), Grassman (1936), and others recently cul- minated in an experiment by Collins (1956), in which a superconduct- ing ring carrying an induced current was kept below T c for about two and a half years. The absence of any detectable decay of the current during this period allowed Collins to place an upper limit of 10 -21 ohm-cm on the resistivity of the superconductor.! This can be com- pared to the value of 10 -9 ohm-cm for the low temperature resistivity of the purest copper. There is, therefore, little doubt that a superconductor is indeed a t Quinn and Ittner (1962) have lowered this upper limit to 10" 23 ohm-cm by looking for the time decay of a current circulating in a thin film tube. 3 4 Superconductivity perfect conductor, in the interior of which any slowly varying electric field vanishes. A current induced in a superconducting ring will persist indefinitely without dissipation. Below T c , the superconducting behaviour can be quenched and normal conductivity restored by the application of an external mag- netic field. This field, H c , is called the critical or threshold magnetic field, and, as shown in Figure 1 , it varies approximately as H c *H [l-(lJ], (1.2) Normal T Temperature T Fig. 1 where H = H c at T= 0°K. It is convenient to introduce reduced co- ordinates / ■ T/T c , and h(t) = H C (T)/H , in terms of which ft» l-/ 2 . (I.2a) The actual temperature variation of h is more accurately represented by a polynomial in which the coefficient of the t 2 term differs from unity by a few per cent. The superconductivity of a wire or film carrying a current can be quenched when this reaches a critical value. For specimens sufficiently Basic characteristics 5 thick so that surface effects can be ignored, the critical current is that which creates at the surface of the specimen a field equal to H c . Smaller samples remain superconducting with much higher currents than those calculated from this criterion, which is called Silsbee's rule (Silsbee, 1916). 1.2. Superconducting elements and compounds Table I lists all presently known superconducting elements and their characteristic H and T c . In addition there have been found by many investigators, in particular by Matthias and co-workers, by Alekseevskii and co-workers, and by Zhdanov and Zhuravlev (see Table I Element T C (°K) H (gauss) Aluminium 119 99 Cadmium 0-56 30 Gallium 109 51 Indium 3-407 283 Iridium 014 ~20 Lanthanum-a ~5 Lanthanum-/* 5-95 1600 Lead 718 803 Mercury-a 4153 411 Mercury-/? 3-95 340 Molybdenum 10 — Niobium 9-46 1944 Osmium 0-7 65-82 Rhenium 1-70 201 Ruthenium 0-49 66 Tantalum 4-482 830 Technetium 11-2 _ Thallium 2-39 171 Thorium 1-37 162 Tin 3-722 306 Titanium 0-40 100 Tungsten ~001 Uranium-a 0-6 ~2000 Uranium-y 1-80 Vanadium 5-30 1310 Zinc 0-92 53 Zirconium 0-75 47 (cf. Roberts (1963) for most references) 6 Superconductivity Matthias, 1957; Roberts, 1961), a very large number of alloys and compounds which also become superconducting. Some of these compounds consist of metals, only one of which by itself becomes superconducting, some have constituents of which neither by itself is superconducting, and some even are semiconductors. The possibility of superconductivity in semiconductors and semimetals has been discussed by M. L. Cohen (1964), and both GeTe (Hein et al., 1964) and SrTi0 3 (Schooley et al., 1964) have been found to be supercon- ducting at very low temperatures. 2 4 6 8 10 No. valence electrons/atom Fio.2 The critical temperatures of superconductors range from very low values up to 181°K for Nb 3 Sn (Matthias et al., 1954). Matthias (1957) has pointed out a number of regularities in the appearance of superconductivity and in the values of T c , the principal of which are the following : (1) Superconductivity has been observed only for metallic sub- stances for which the number of valence electrons Z lies between about 2 and 8. (2) In all cases involving transition metals, the variation of T c with the number of valence electrons shows sharp maxima for Z = 3, 5, and 7, as shown in Figure 2. (3) For a given value of Z, certain crystal structures seem more favourable than others, and in addition T c increases with a high power of the atomic volume and inversely as the atomic mass. Basic characteristics 7 1.3. The Meissncr effect, and the reversibility of the S.C. transition If a perfect conductor were placed in an external magnetic field, no magnetic flux could penetrate the specimen. Induced surface currents would maintain the internal flux, and would persist indefinitely. By the same token, if a normal conductor were in an external field before it became perfectly conducting, the internal flux would be locked in by induced persistent currents even if the external field were removed. o o o A:H e =0, B%0. C:0<H e <H o B:H e =0 T>T C . T<T C . T<T C . T<T C . (a.) (b) (c) (d) Fig. 3 C:0<Hp<H e n c. T<T C B: H e =0, T<T C . (d) Fig. 4 Because of this, the transition of a merely perfectly conducting speci- men from the normal to the superconducting state would not be reversible, and the final state of the specimen would depend on the path of the transition. As an example, Figures 3 and 4 show the flux configuration for a perfectly conducting sphere taken from point A in Figure 1 to point C by the different paths ABC and ADC, respectively. The final field distribution at C, as well as that at B, depends on whether one pro- ceeded via Bov via D, and the irreversibility of the transition is evident. Careful measurements of the field distribution around a spherical 8 Superconductivity specimen by Meissner and Ochsenfeld (1933), however, indicated that regardless of the path of transition the situation at point C is always that shown in Figure 3c : the magnetic flux is expelled from the interior of the superconductor and the magnetic induction B vanishes. This is called the Meissner effect, and shows that the superconducting transi- tion is reversible. Figure 5 illustrates this by showing B vs. H e curves both for a perfect conductor and for a superconductor, taking the case of long cylin- drical specimens with axes parallel to the applied field. H e is a uniform, He cO 3 "a c >' perfect conductor /> «= super- conductor Applied Field He Fro. 5 external field. In increasing field both specimens have 5=0 until H e = H c , when they become normal and B = H e . If the field is now again decreased, the induction inside the perfect conductor is kept at its threshold value B = H c by surface currents, and in zero field the specimen is left with a net magnetic moment, as is illustrated in Figure 4d. The superconductor, however, expels the flux at the transition and returns reversibly to its initial state with B = for < H e < H c . The vanishing of the magnetic induction, corresponding to the ex- pulsion of the magnetic flux, is the basic characteristic of all ideal superconducting material with dimensions large compared to a basic length which will be mentioned later. It is quite independent of the Basic characteristics 9 connectivity of the body, so that if one has a superconductor with a hole, the Meissner effect occurs in the metal and only the hole may be threaded by magnetic flux. The magnetic properties of such a super- conducting ring are thus essentially determined by the relative size of the diameter of the ring to the diameter of the hole. 1.4. The specific heat The specific heat of a superconductor consists, like that of a normal metal, of the contribution of the electrons (C e ) and that of the lattice (C g ). For a normal metal at low temperatures *^n Wntt, (1.3) , gn = yT+A(T/0) 3 . y is the Sommerfeld constant, which is proportional to the density of electronic states at the Fermi surface, is the Debye temperature, and A a numerical constant for all metals. Experimentally the two contri- butions to C n can be separated by plotting CJTvs. T 2 , so that the slope of the resulting curve is A/0 3 , and the intercept is y. In the superconducting phase C| = C es +C gs . Figure 6 shows values of both C s (H=0) and C„ (H> H c ) for tin as measured by Corak and Satterthwaite (1954), displaying the charac- teristic features of a sharp discontinuity in C,of the order of 2y7 c at T c , and a rapid decrease of C s to values below C n varying about as T 3 . It is customary to attribute the difference between C s and C„ entirely to changes in C e , on the assumption that C g is the same in both phases. This seems reasonable in view of the electronic nature of the super- conducting phenomenon, and is supported by the absence of any observable change in the lattice parameters (Keesom and Kamerlingh Onnes, 1924), and by the detection of only minimal changes in the elastic properties (see, for instance, Alers and Waldorf, 1961). On this assumption ^s *-n — f-'es ^e (1.4) which allows one to determine C es from measured values of the specific heat difference after C e „ = y7"has been determined separately. There has recently been some evidence that the lattice contributions to the specific heat in the two phases are not quite equal in the case of 10 Superconductivity Basic characteristics 11 (deg.Kf) Fig. 6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1/t Fig. 7 indium (Bryant and Keesom, 1960; O'Neal et al., 1964), so that 1.4 may not be exact for this element and possibly other superconductors as well. Ferrell (1961) has suggested that this is due to a shift in the phonon frequency spectrum. However, the superconducting ele- ments for which reliable values of C„ exist are those with a relatively high Debye temperature for which C g < C e in both phases down to very low temperatures. For these elements possible small differences in C g therefore do not much affect the validity of 1.4. Figure 7 displays C cs for tin calculated on the basis of 1.4 from the results in Figure 6, plotted logarithmically in units of l/yT c vs. 1//. This shows that for l/t>2, one can represent C es by the equation CJyT c = aexp(-b/t). (1.5) A subsequent chapter will discuss that this is an indication of the existence of a finite gap in the energy spectrum of the electrons separating the ground state from the lowest excited state. The number of electrons thermally excited across this gap varies exponentially with the reciprocal of the temperature. In recent years it has become apparent that such an energy gap determines the thermal properties as well as the high frequency electromagnetic response of all super- conductors, and that it must indeed be one of the principal features of a microscopic explanation of superconductivity. 1.5. Theoretical treatments The macroscopic characteristics of a superconductor have been the subject of a number of phenomenological treatments of which the principal ones will be discussed in subsequent chapters. F. and H. London (1935a, b) developed a model for the low frequency electro- magnetic behaviour which is based on a point by point relation between the current density and the vector potential associated with a magnetic field. This implies wave functions of the superconducting electrons which even in the presence of such a field extend rigidly to the limits of the superconducting material and then vanish abruptly. A thermodynamic treatment and an associated two-fluid model based on essentially equivalent simplifications were worked out by Gorter and Casimir (1934a, b). These complementary theories provide highly successful and useful tools in the semi-quantitative analysis of many problems involving superconductors. Their limitations become apparent principally in situations in which size and surface effects are important. Pippard (1950, 1951) has shown that such effects become tractable when one takes into account the finite coherence of the superconduct- ing wave functions which is such as to allow them to vary only slowly over a finite distance. This leads (Pippard, 1953) to a non-local 12 Superconductivity integral relation between the current density at a point and the vector potential in a region surrounding the point. The equation has only been solved for a few special cases. In many instances, however, it reduces to a modified version of the London equation, so that the much simpler London formalism can then be used with the Pippard modifications (Tinkham, 1958). Ginzburg and Landau (1950) have developed on a thermodynamic basis an alternate method of treating the coherence of the super- conducting wave functions. Their treatment is compatible with Pippard's electromagnetic approach, and forms a highly useful com- plement to it. A successful microscopic theory of superconductivity has recently been developed by Bardeen, Cooper, and Schrieffer (1957). It is based on the fact, established by Cooper (1956), that in the presence of an attractive interaction the electrons in the neighbourhood of the Fermi surface condense into a state of lower energy in which each electron is paired with one of opposite momentum and spin. Bardeen, Cooper, and Schrieffer (BCS) have been able to show that a finite energy gap separates the state with the largest possible number of Cooper pairs from the state with one pair less. This leads to the correct thermal and electromagnetic properties to display superconductivity. The attraction between electrons necessary to form Cooper pairs can in principle be due to any suitable kind of interaction. The dis- covery (Maxwell, 1950; Reynolds et al., 1950) that for many super- conducting elements the critical temperature depends on the isotopic mass showed that for these substances the attractive interaction is one between the electrons and the lattice. The BCS theory and its exten- sions have been worked out on this basis. However, the isotope effect is apparently absent or considerably reduced in some transition metals and their compounds (see section 8.1). Furthermore the effect of pressure in transition metals does not correlate with the Debye temperatures as it does in non-transition superconductors (Bucher and Olsen, 1964). Kondo (1962) and Garland (1963a, b) have attri- buted these anomalies to the existence of overlapping bands in the electronic energy spectrum at the Fermi surface. However, there is also a hypothesis that in transition metals the attractive interaction responsible for pairing may be a magnetic one (Matthias, 1960). CHAPTER II Phenomenological Thermodynamic Treatment 2.1. The phase transition Long before the determination of the reversibility of the supercon- ducting transition by the discovery of the Meissner effect, attempts had been made to apply thermodynamics to it by Keesom (1924), by Rutgers (Ehrenfest, 1933), and in particular by Gorter (1933), who virtually predicted the Meissner effect by pointing out that the success of these early thermodynamic treatments strongly suggested the reversibility of the transition. The discovery of the Meissner effect finally enabled Gorter and Casimir (1934a) to develop a full treatment of the superconducting phase transition in a manner analogous to that of other phase transi- tions. They start with the fact that two phases are in equilibrium with one another when their Gibbs free energies (G) are equal. The free energy of a superconductor is most easily expressed by a diamagnetic description developed in Chapter III, which attributes to the super- conductor a magnetization M (H e ) in the presence of an external field H e . Then V He G,(H e ) = G/0)- j dvj M(H e )dH e . (H.l) For an ellipsoid, M(H £ ) is uniform, and He G s (H e ) = G s (0)- VJ M(H e )dH e . (11.10 o The last term in this expression gives the work done on the specimen by the magnetic field. As the magnetization is diamagnetic, that is, negative, the field raises the energy of the superconducting specimen. It will be shown in Chapter III that only for a quasi-infinite cylinder parallel to the external field does the superconducting phase change into the normal one at a sharply defined value of H e . For all other 13 14 Superconductivity shapes, there is an intermediate state consisting of a mixture of normal and superconducting regions. Even under these circumstances, how- ever, any magnetic work is done solely on the superconducting por- tions, and for any shape of specimen this always equals, per unit volume, He JM(H e )dH e = -H 2 /Stt. (II.2) o Thus one can write for any specimen : G,(/r c ) = (7,(0)+WJ c 2 /87r. (H.3) In the normal state the susceptibility is generally vanishingly small, so that G n (H c ) = G„(0). Since the condition of equilibrium defining H C (T) is that one has G n (H c ) = G S {H C ), 01.4) This is the basic equation of the thermodynamic treatment de- veloped by Gorter and Casimir. As S = - (dG/dT) Pi H , differentiation of 11.4 yields S n (0) - Sffl = - (VHJAtt) (dHJdT). (II.5) At T= T c , H c = 0, and S n = S s . At any lower temperature, H c > 0, and furthermore Figure 1 shows that for < T< T c , dHJdT < 0. Hence the entropies of the two phases are equal at the critical temperature in zero field; at any lower, finite temperature the entropy of the super- conducting phase is lower than that of the normal one, indicating that the former is the state of higher order. This ordering will later be shown to follow from a condensation of electrons in momentum space. It follows from Nernst's principle that S n = S s at T= 0, so that in this limit the slope of the threshold field curve must vanish. As the entropies of the two phases are also equal at T= T c , their difference must pass through a maximum at some intermediate temperature. Phenomenological thermodynamic treatment 15 Equation II.5 also shows that the latent heat Q = T(S n — S s ) is zero at the transition in zero field, and is positive when H c > 0. Thus there is an absorption of heat in an isothermal superconducting-to-normal transition, and a corresponding cooling of the specimen when this takes place adiabatically. The resulting possibility of cooling by adiabatic magnetization of a superconductor was suggested by Mendelssohn (Mendelssohn and Moore, 1934) and has been used by Yaqub (1 960) for low temperature specific heat measurements of tin. A further differentiation of II.4 yields, upon multiplication by T: C s -C n = (VT/47r)[H c (d 2 H c /dT 2 ) + (dHJdT) 2 ). (II.6) Atr=r o ^ c = 0,and C,- C n = (*T/4tt) {dHJdTfj^ Te > 0, (H.60 so that the thermodynamic treatment predicts the observed dis- continuity in the specific heat. As the entropy difference between the tv vo phases passes through an extremum at some temperature below T c , the specific heats of the two phases at that temperature must be 16 Superconductivity equal, and at even lower temperatures C s is smaller than C„. Both of course tend toward zero at T= 0°. The variation of C s - C„ as a func- tion of temperature, as well as that of S s -S„, are shown in Figure 8. 2.2. Thermodynamics of mechanical effects The thermodynamic treatment developed thus far has ignored any changes in the volume at the transition, as well as any dependence of H c on pressure as well as on temperature. In taking these into account one should begin by considering possible magnetostrictive field effects on the volume in going from II. 1 to II. 1'. Ignoring this, however, and noting (see Figure 1 1) that for the special case of a quasi-infinite cylinder parallel to the external field the area under the magnetization curve up to any field value H e < H c is equal to H 2 j%tt, one can write QABQ-Gjm = (VJ*ir)Hl 01.7) Differentiating this with respect top in order to obtain V= (8GI8p) T H yields V 5 {H e )- Vjm = (H}l87r)(dV s !8p) T . (n.8) Similar differentiation of II.3 and II.4 leads to V n {H c )-VM = *mv,EftT,p)l*n\ V n {H c )-V s {0) = {H}l%ir)(dVJBp) T +{V s HJAn)(dHcl*P)T- 01.9) Comparing II.9 with II.8 shows that the first term on the right-hand side of the former is just the magnetostriction of the superconductor upon changing the field from zero to the critical value. It is the second term which gives the actual volume change at the transition : V„(H C )-V S (H C ) - (V s H c /47r)(dH c l8p) T . (11.10) This term exceeds the magnetostrictive one by more than an order of magnitude. The derivatives of 11.10 with respect to T and to p yield expressions for the changes at the transition of the coefficient of thermal expansion et=(l/V)(SV/dT) t and of the bulk modulus K = - V{8pj8V). AtT= T c , H c = 0, this yields and «„-«, = (U47T)(8H c ldT)(dH c l8p\ (11.11) *„-« = (K 2 /47r)(dH c /8p) 2 . (IU2) Phenomenological thermodynamic treatment 17 There has been extensive experimental work on pressure effects on the critical field. This has been reviewed by Swenson (1960) and sum- marized most recently by Olsen and Rohrer (1960). These latter authors (1957) and, independently, also Cody (1958), have succeeded in refining earlier work of Lazarev and Sudovstov (1949), and have obtained for different superconducting elements empirical values of the length change of a long rod at the transition. (Andres et at., 1 962). Differences in the behaviour of transition and non-transition metals have been pointed out by Bucher and Olsen (1964). The magnitudes of the several mechanical effects are exceedingly small. Typical values for 8HJ8p are of the order of 10~ 8 -10~ 9 gauss/dyne-cm -2 , and the fractional length change of a long rod is a few parts in 10 -8 . Using the above thermodynamic relations this yields a difference in the thermal expansion coefficient of about 10 -7 per degree, and a fractional change in compressibility of one part in 10 5 . 2.3. The interrelation between magnetic and thermal properties One of the most remarkable features of the thermodynamic treatment outlined in the preceding sections is the manner in which it links the magnetic and the thermal properties of a superconductor. Equation II. 5, for example, indicates that quite independently of the detailed shape of the magnetic threshold field curve, its negative slope indi- cates that the superconducting phase has a lower entropy than the normal one. The quantitative verification of an equation such as II. 6', called Rutgers' relation, provides the best available confirmation of the basic reversibility of the superconducting transition. The following table, taken from Mapother (1962), compares for a few particularly favourable elements the specific heat discontinuity measured calori- metrically, with its value calculated with II.6' from measured thres- hold field curves. The agreement is seen to be excellent: Element Indium Tin Tantalum (millijoules/°mole) 9-75 9-62 10-6 10-56 41-5 41-6 18 Superconductivity The relations between the thermal properties and the threshold field curve of course also imply that if a specific temperature variation is either assumed or empirically determined for one of the former, this uniquely specifies the temperature variation of the latter. Kok (1 934), for example, showed that if one substitutes into equation II.6 a para- bolic variation of H c , as given by equation 1.2, one obtains a cubic temperature variation of C cs . It was mentioned in Chapter I that both of these are only fair approximations to the actual temperature dependence of these quantities, and that in fact the threshold field can be represented more accurately by a polynomial which in reduced co-ordinates has the form h(t) = 1- 2 a n t\ (11.13) The first coefficient a, must vanish, as otherwise S s —S„ would not vanish at T= (see equation II.5), and 2>„ = 1 to make //(l) = 0. If n this polynomial is substituted into equation II.6, and one continues to neglect any changes in the lattice specific heat, it follows that ~^f=(MWHHT^-a 2 t 2 ...)x x(2a2) + (-2<7 2 /-...)}. (11.14) Of the two terms on the left-hand side, the second just equals the Sommerfeld constant y. The first is subject to the following general argument: As shown by equation II.5, S n > S s , and since S„ varies linearly with T, S s must approach zero with some power of T greater than unity. Hence one can write o: T l+X ,x > 0, so that and C„oc T l+X , CJTozT*. It follows, therefore, that no matter what the precise temperature dependence of C es is, CJT^Q as T-+0°. Applying this limit to equation 11.14 thus yields y = (l/27T)a 2 (Hl/Tl). (11.15) Phenomenological thermodynamic treatment 19 An equivalent expression results from applying the above argument directly to equation II.6, and recalling that as T-+0, dH c /dT-+0. One then obtains y = -(l^Tr^Hl/T^ihd^rldr 2 )^. (11.16) Both of these last equations are exact expressions which permit the evaluation of the Sommerfeld constant from a detailed knowledge of the threshold field curve. Mapother (1959, 1962) has carried out a searching analysis of the extent to which magnetic and thermal data can actually be correlated in practice without introducing excessive errors due to extrapolation; Serin (1955) and Swenson (1962) have also discussed the relation between the two types of data. 2.4. The Gorter-Casimir two-fluid model The so-called phenomenological two-fluid models of superconduc- tivity have in common two general assumptions: (1) The system exhibiting superconductivity possesses an ordered or condensed state, the total energy of which is characterized by an order parameter. This parameter is generally taken to vary from zero at T= T c to unity at T= 0°K, and can thus be taken to indicate that fraction of the total system which finds itself in the superconducting state. (2) The entire entropy of the system is due to the disorder of non- condensed individual excited particles, the behaviour of which is taken to be similar to that of the equivalent particles in the normal state. In particular, two-fluid models make the conceptually useful assumption that in the superconducting phase a fraction #" of the conduction electrons are 'superconducting' electrons condensed into an ordered state, while the remaining fraction 1 — #" remain 'normal'. The artificiality of this division cannot be overemphasized; its use- fulness will presently appear. The free energy per unit volume of the ' normal ' electrons continues to be the same as that of electrons in a normal metal, that is g n (T) = -\yT 2 (11.17) where y is the Sommerfeld constant. For the 'superconducting' elec- trons g s (T) is taken to be a condensation energy relative to the normal CHRIST'S COLLEGE LIBRARY 20 Superconductivity phase, and the considerations of the first section of this chapter show this to be g s (T) = -HllSrr. (H.1J The total free energy per unit volume of the superconducting phase containing a fraction #" of V electrons and 1 - ^ of l n ' electrons is therefore G s (ir, T) = a(\ - fT)g n {T) + b(iT)8 s (T). (11.19) The simplest choice of a(l-iT) = 1-iT; b(iT)=ir, makes G&if, T) a linear function of 1P, so that the equilibrium condition (3C/air)r = can be satisfied for only one value of Tat which iT can assume any value between and l . This would mean that for any value of W the normal and superconducting phases can be in equilibrium at only that one temperature, which is not the case. Thus it is necessary to choose a(l - iT) and b(iT) with more care. Gorter and Casimir (1934b) chose a(l - #") = (l - ir)\ b{iT) = Hr, (11.20) so that G 3 (ir,T) = -\{\-Hr)*yT 1 -inill%iT. (11.21) Applying the equilibrium condition yields a(l - *0"~ ' m HllA-n yT 2 , (IL22) which at T c , with 1T - 0, reduces to y = (WttxHHIIT 2 .). Substituting 11.23 back into 11.22: (\-1T)«- 1 = (TJT) 2 = r 2 , Phenometwlogical thermodynamic treatment 21 so that Hr = |_^/(i~«0 (11.24) Oi.: Putting this back into 11.21 and differentiating to obtain other thermal quantities yields and s s (ir,T) = yT(\-iry = y r c r (1+0[)/(1 - a) , C s (iT,T) = [(l + a)/(l-a)]yr c / (l+a)/(, - a \ The value of a must be chosen so as to give a reasonable fit to experi- mental data. With a = £ one obtains QOT.n = 3yT c t\ S S (T) = yT c t\ and iT(J) = l-/ 4 , y = (WirKHlfT?). (11.270 (11.260 (11.250 (11.230 Here again is the cubic temperature variation of the specific heat which is only an approximation. Clearly no value of a will change equation 11.27 into an exponential expression. A comparison of equation 11.23' with equation 11.15 also shows that the choice a = £ makes a 2 =\, and reduces the polynomial representation of h(t) to a parabolic form. This not only indicates once again the interrelation between the magnetic and thermal properties, but also points up that the Gorter-Casimir model can be used at best only semi-quantita- tively. Within this limitation, however, the concept of the two inter- penetrating 'fluids' of condensed and uncondensed electrons is very useful in obtaining a semi-quantitative understanding of many super- conducting phenomena, and will be used repeatedly in subsequent chapters. There have been a number of attempts (see [5], p. 280) to improve the quantitative aspects of the Gorter-Casimir model so as to yield more nearly the correct exponential variation of C es and the corre- sponding non-parabolic dependence of h(t). These modifications have either tried different functional forms for a(l - 1T) and b(iT) in equation TJ.19, or have introduced additional adjustable parameters. Some of these variations do yield considerably better equations for the thermal and magnetic superconducting properties. However, the principal virtue of a two-fluid model is to provide a conceptual tool of primarily qualitative nature, and the various suggested improve- ments rarely add much to the basic physical picture of the two groups °f electrons. CHAPTER III Static Field Description 3.1. Perfect diamagnetism Even in the absence of a microscopic explanation of the phenomenon of superconductivity, it is reasonable to assume that the vanishing of the magnetic induction at the interior of a superconductor is due to induced surface currents.! In the presence of an external magnetic field, the magnitude and distribution of this current is just such as to create an opposing interior field cancelling out the applied one. A formal description of a macroscopic superconductor in the presence of an external field H e is, therefore, the following: in the interior: B, = H,- = M,- = 0, where M,- is the magnetization per unit volume; at the surface: 3 S ^ 0, where 3 S is the surface current density; and outside: B e = U e +H s , where H s is the field due to the sur- face currents. It is this field which causes the distorted field distribution near a super- conductor as shown in Figure 3c. Although this description is formally correct, it is much more con- venient to replace it by an equivalent one which treats the supercon- ductor in the presence of an external field as a magnetic body with an interior field and magnetization such that in the interior: B, = 0, H, ^ 0, M, ^ 0; at the surface: 3 S = 0; and outside: B e = H e + H s , where now H, is the field due to the magnetization of the sample. t That electron currents and not, for example, spins are responsible for the diamagnetism of a superconductor is demonstrated by its gyromagnetic ratio which is found to have the value of -e\2mc (Kikoin and Goobar, 1940; cf. [1], p. 50 and p. 193 ; [2], p. 83). 22 As Static field description B = H+4ttM, 23 this description is equivalent to attributing to the superconductor a magnetization per unit volume M/= -(1/4tt)H, (HI.1) which means that the superconductor has the ideal diamagnetic susceptibility of — 1/47T. 3.2. The influence of geometry and the intermediate state The great convenience of the diamagnetic mode of description is illustrated by considering an ellipsoidal superconducting specimen in an external field H e which is parallel to the major axis. The conven- tional proof, that inside a uniform ellipsoid B, H, and M are all con- stant and parallel to H e , is independent of susceptibility and therefore applies to the superconductor. Further standard treatments show that (with vector notation now unnecessary) : H, = B.-AirDMu (IH.2) where D is the demagnetization coefficient of the specimen. For an ellipsoid of revolution this is given by »-(HMS-'> a and b are, respectively, the semi-major and semi-minor axes, and e - (1 -b 2 /a 2 ) 112 . For an infinite cylinder with its axis parallel to H e , D = 0; for an infinite cylinder transverse to the field, D = \, and for a sphere, D-l/3. Combining III.l and III.2 yields : and M,= -H e /47r(l-D) H, = HJ{l-D). (IU.3) (IH.4) In the neighbourhood of the superconductor, the external field is distorted by the magnetization of the specimen. It follows from the 3 24 Superconductivity continuity of the normal component of B and of the tangential com- ponent of //that for an ellipsoidal specimen the exterior field distribu- tion is as shown in Figure 9. At the equator of the specimen Static field description 25 and at the pole H~ = H,= H e l(l-D), H p = Bi = 0. (III.5) (IH.6) For the longitudinal infinite cylinder with axis parallel to H e ,D = and H eq = H e . The exterior field at the surface of the specimen is, therefore, everywhere the same, and the cylinder remains entirely superconducting until the applied field becomes equal to the critical Fig. 9 value H c . The entire body then becomes normal. The magnetization curve for such a specimen is shown in Figure 10, in which for con- venience -A-nM is plotted against H e . For all other ellipsoidal shapes, D ^ 0, and the non-uniformity of the field distribution around the superconductor raises the question of what happens when H cq = H c > H e . To assume that a portion of the specimen near the equator then becomes normal, as shown in Figure 11, would lead to a contradiction: the boundary between the superconducting and normal regions occurs where H = H C) but in the now normal region the field would equal H e <H c \ There is, in fact, no simple, large-scale division of such a specimen into normal and superconducting regions, which allows a field distribution such that H>H C in the former, H < H c in the latter, and H=H C at the boundaries. He i (a) transverse , H cu !i nder (aX \p) sphere (c) longitudinal cylinder \\ Superconducting state \\ Intermediate state H c /2 2H c /3 Applied Field H e Fio. 10 Fig. 11 Instead one must postulate, as was first done by Peierls (1936) and b y F. London (1936), that once H e > (1 - D)H C , the entire specimen is subdivided into a small-scale arrangement of alternating normal and superconducting regions, with B = H c in the normal regions, and ** = in the others. The distribution of these regions varies in such a 26 Superconductivity way that the total magnetization per unit volume changes linearly from Mi = -HJ47r(l - D) = - BJ4*r at H e = H c {\ - D), to M ( = at H e m H c . Hence, for (1 -D)H c <H e < H c , M,= -(l/4irZ»(/f c -fQ, (III.7) Hi = H e -A-nDMi = H c , (UI.8) Bi = H c -(MD)(H C -H e ). (III.9) Magnetization curves for a transverse cylinder (D = I) and for a sphere (D = 1/3) are also shown in Figure 10. Note that the area under each of the curves is given by j MidH e = -H?I8tt. (111.10) This is just the magnetic work done on the specimen in raising the field from zero to H c , as cited in equation II.2. In the region (1 — D) H c =5 H e ^ H c , in which the specimen is neither entirely normal nor entirely superconducting, it is said to be in the intermediate state. The detailed structure of this state will be discussed in Chapter VI ; at this time it is only necessary to emphasize that this intermediate state exists, in some field interval, for any geometry other than that of a quasi-infinite cylindrical sample parallel to the external field. 3.3. Trapped flux It is important to distinguish the reasons and conditions for the inter- mediate state from those giving rise to the phenomenon of trapped flux or the incomplete Meissner effect. As mentioned earlier, the magnetic flux threading a multiply connected superconductor is trapped by an indefinitely persisting current, and cannot change unless the superconductivity of the specimen is quenched. A similar situation can arise in a simply-connected but non-homogeneous superconductor. Strains, concentration gradients, and other imper- fections can create inside a superconductor regions with anomalously Static field description 27 high critical fields. Thus if such a non-ideal specimen is placed in a magnetic field sufficiently high to make it entirely normal, and the field is then reduced, the anomalous regions will become supercon- ducting before the bulk of the specimen. Should some of these regions be multiply-connected, then the flux threading them at the moment of their transition into superconductivity can no longer escape, and is trapped even when the external field is reduced to zero, for as long as the specimen remains superconducting. Applied Field H e Fig. 12 As a result, after it has once been normal in an external field, such an imperfect specimen is less than perfectly diamagnetic in an external field H< H c , and retains a paramagnetic moment in zero field. This »s shown in Figure 12 for a long cylinder parallel to the field, using the same units as in Figure 10. The ratio of m to -H c is called the fraction of trapped flux. 3.4. The perfect conductor To emphasize once again the difference between a perfect conductor a nd a superconductor, it is useful to outline an electromagnetic treat- ment of the former, as developed by Becker et al. (1933) just before the discovery of the Meissner effect. 28 Superconductivity In a perfect conductor, the equation of motion for an electron of mass m and charge e in the presence of an electric field E does not contain a retarding term and would simply be mv = eE. (IU.11) In terms of the current density J = nes, where n is the number density of the electrons, one can write III.l 1 in the form E = (47rA 2 /c 2 ) J, 011.12) where A 2 ■ mc 2 /4wne 2 . (IE. 13) The parameter A has the dimensions of length, and for a density of electrons corresponding to one electron per atom it has a value of the order of 10 -6 cm. Using Maxwell's equation curlE = -H/c, one finds that (4ttA 2 /c) curl J + H = 0, (UI. 14) and applying another Maxwell equation curlH = 4tt3jc yields for the perfect conductor the equation V 2 H = H/A 2 . (111.15) Von Laue (1949) showed that the solution of III. 15 for any specimen geometry yields a value of H which decreases exponentially as one enters the specimen. For a semi-infinite slab extending in the x- direction from the plane x = 0, the appropriate solution is H(x) = H(0) exp ( - x/X). (UI. 1 6) Clearly, for x P A, H(x) « 0. Thus equation III. 1 6 confirms that in the interior of a perfect conductor the magnetic field cannot change in time from the value it had when the specimen became perfectly conducting. 3.5. The London equations for a superconductor The incorrectness of IU.16 was demonstrated by the discovery of Meissner and Ochsenfeld (1933) that regardless of the magnetic his- tory of the specimen, the field inside a superconductor always vanishes. F. and H. London (1935a, b; see [2]) therefore proposed to Static field description 29 add to Maxwell's equations the following two relations in order to treat the electromagnetic properties of a superconductor: and E = (^AVJ), (47rA 2 /c)curlJ+H = 0. (A) (B) Replacing the field by a vector potential curl A = H and choosing a gauge such that div A = 0, (B) reduces to 4ttA 2 J + A = 0. (BO Note that (A) is identical to UI.2, and thus describes the property of perfect conductivity, but that the difference between (B) and III.4 is the important one that application of Maxwell's equations now leads to V 2 H = H/A 2 . OH. 17) Solution of this for any geometry now shows that H, and not only H, decays exponentially upon penetrating into a superconducting speci- men. For the semi-infinite slab described above, the solution of III. 17 is H(x) = H(0)exp(-x/A), (IU. 18) which shows that for x > A, H(x) « 0, in accordance with the Meissner effect. Clearly the London equations (A) and (B) do not, in fact, yield the complete exclusion of a magnetic field from the interior of a super- conductor. Instead, the*y predict the penetration of a field such that it decays to 1 /e of its value at the surface in a distance A. This is called the London penetration length. Its existence has been fully confirmed experimentally, although empirical values are consistently higher than those predicted by the defining equation ni.13, as will be dis- cussed in a later chapter. The existence of this slight penetration of an exterior field must be taken into consideration in the discussion of superconducting thin films, wires, or colloidal particles, and in a detailed treatment of the intermediate state. 30 Superconductivity Applying curlE = — H/c to equation (B), one obtains curl[E-(47rA 2 /c 2 ) J] = 0, showing that E— (47rA 2 /c 2 ) J = grad<£, where <f> is a scalar. In the most general case of a multiply connected superconductor or a superconducting portion of a current-carrying circuit, one cannot prove that <£ vanishes. Hence (A) does not always follow from (B) and the perfect conductivity implicit in (A) and the perfect diamagnetism in (B) must be considered as independent postulates. In a system of N particles of charge q described by the wave function ^ / (r l ,r 2 ,...,r N ), the mean current density at a point R in the presence of a magnetic field H(r a ) = curlA(r a ) (III. 19) is given by N 1 N s 2 r U(R-r. - ^-AOrJ V * f 8(R- ra )</r, . . .dr N . (111.20) tnc J In the absence of a field, A(r a ) ■ 0, W=W Q , and the current density vanishes, so that > [■■■ fl^tnv^o-^VamscR-rjx Q = l , N xdr t ...dr N = 0. (IH.21) If, therefore, one assumes that the wave function W is perfectly rigid under the application of a magnetic field, that is, that W= X F () always, then it follows that N 2 J(R)= -2J ...J ^- c A(r )W*V8Ql.-r x )dr l ...dr N . (111.22) Static field description 3 1 By defining a particle density /i(R) = S j ...j x P*Y8(R.-r a )dr l ...dr N , QSU8) 1 N equation 111.22 can be written as J(R) = -n(R) — A(R). mc (111.24) But if the particle density n(R) is a sufficiently smooth function so that one can replace it by a constant n, then in view of the defining equation III. 13, 111.24 is seen to be identical to (BO- Thus the London equation (B) or (B') implies that the magnetic properties of a superconductor are due to a complete rigidity of the wave functions of the superconducting carriers. In F. London's own words ([2], p. 150): '...superconductivity would result if the eigen- functions of a fraction of the electrons were not disturbed at all when the system is brought into a magnetic field (H< i/ c ).' A possible explanation of this is contained in the London equations themselves. The mean local value of the carriers momentum in the presence of a field is given by p = W v + (9/c)A, which can be rewritten as P = (^)[(4ttA 2 /c)J + A]. (111.25) In the same gauge as that leading to (BO, 111.25 for a simply connected superconductor reduces to p, = 0. (IIL26) The London equation thus implies that superconductivity is due to a condensation of a number of carriers into a lowest momentum state P s = 0. By the uncertainty principle this requires the essentially un- limited spatial extension of the appropriate wave functions, and makes it impossible for them to be affected by local field variations. It also follows from 111.26 that v* = -(qlmc)A, (IH.27) 32 Superconductivity showing that in a simply connected superconductor the charge flow is entirely determined by the externally applied field, and exists only in its presence. 3.6. Quantized flux F. London already observed ([2], p. 151] that the unlimited extension of the wave function of the superconducting charge carriers has a very fundamental consequence in a multiply connected superconductor. Consider, for example, a superconductor containing a hole. The wave functions must then be single valued along any closed path enclosing the hole. By analogy to the electronic wave functions in an atomic orbit one can then apply the Bohr-Sommerfeld quantization rules and require that for the superconducting charge carriers (J> p-dl = nh, (IU.28) along any path enclosing the hole. According to 111.25 this then means that jMttA 2 m r he (b J-dl+&>A-dl = n- (m.29) Since H = curlA, the contour integral of A is equal to the surface integral of H over the area enclosed by the contour, and this in turn equals the magnetic flux <P threading the contour: Thus <J)A-rfl= f f H-c/S = 0. r * he J<fl+0 = n— • Q (111.30) (111.31) London called the left-hand side of this equation a. fluxoid, and we see that according to 111.31 such a fluxoid is quantized in integral multiples of he & » - ' (H1.32) Note that if the contour is taken at a distance from the hole large compared to the penetration depth A, the current density vanishes, Static field description 33 and the fluxoid is just equal to the total flux associated with the hole. This flux is thus seen to be quantized. The quantization of flux was verified experimentally by Doll and Nabauer (1961) and by Deaver and Fairbank (1961). These experi- ments have shown that the quantum of flux is given by A - kC 2x10 7 gauss-cm 2 . This shows that q = 2e, that is, that the superconducting charge carriers are pairs of electrons. It has already been mentioned that indeed this is the fundamental premise of the microscopic theory. A number of authors (Byers and Yang, 1961; Onsager, 1961; Bardeen, 1961b; Keller and Zumino, 1961; Brenig, 1961) have extended the London argument for flux quantization in a rigorous fashion. In particular Byers and Yang as well as Brenig have shown explicitly that the quantization is due to a periodicity of the free energy of the superconductor as a function of flux. The free energy of the normal phase is essentially independent of flux, and there must therefore occur a corresponding periodic variation of the critical temperature at which the free energies of the two phases are equal. This flux periodicity of T c has been observed by Little and Parks (1961). CHAPTER IV The Pippard Non-local Theory 4.1. The penetration depth A The London equations lead to an exponential penetration of an externally applied magnetic field into a superconductor, so that the penetration can be characterized by the depth A at which the field has fallen to 1/e of its value at the surface. Quite in general, and inde- pendently of any particular set of electromagnetic equations for the superconductor, one can define the penetration depth for an infinitely thick specimen by ■±J H{x)dx. (1V.1) This would apply equally well to an exponentially decaying field as to one, improbable though it may be, which remains constant to a certain depth and then vanishes suddenly. Shoenberg ([1], p. 140) has pointed out that in this way one can treat problems involving either very thick specimens (thickness a > A) or very thin ones {a <^ A) independently of a detailed knowledge of the appropriate electromagnetic equations. Using IV. 1 to calculate the ratio of the magnetic susceptibility \ of a sample into which the applied field has penetrated, to the susceptibility xo of an identical sample from which the field is entirely excluded, he finds equations of which the following are applicable to a plate of thickness la in a uniform field parallel to its surface : x/Xo = 1 - A/a for a > A, x/xo = aa 2 /A 2 for a < A. (IV.2) (IV.3) The detailed form of the field penetration does not enter at all into IV.2 and all other equations for large specimens of other shapes, and does so only through the numerical parameter a in the equations for 34 The Pippard non-local theory 35 small specimens. As a consequence it is impossible to test the validity of any particular penetration law, such as, for example, the London relation III. 17, by measurements on large specimens; with very small specimens this can only be done if one can determine absolute values of a and of A, which is very difficult. On the other hand, one can measure the variation of the susceptibility of large or of small speci- mens with any parameter affecting only A: temperature, external field, impurity content, etc. One can then deduce directly the variation of A with the parameter in question, without having to make any assumptions about the true penetration law. The results of such measurements can therefore help to choose between different electro- magnetic theories if these predict different parametric dependences of the penetration depth. The oldest method of measuring the penetration depth consists of determining the magnetic susceptibility of samples with large surface to volume ratios to make the penetration effects appreciable. Shoenberg (1940) measured the temperature dependence of the sus- ceptibility of a mercury colloid containing particles of diameter between 100 and 1000 A. Desirant and Shoenberg (1948) used com- posite specimens consisting of about 100 thin mercury wires of diameter about 10" 3 cm, and Lock (1951) carried out extensive measurements of the magnetic behaviour of thin films of tin, indium, and lead. Casimir (1940) suggested a method using macroscopic specimens in which the mutual inductance was measured between two coils closely wound around a cylinder of superconducting material. It was applied successfully by Laurmann and Shoenberg (1947, 1949), by Shalnikov and Sharvin (1948), and most recently with certain refinements by Schawlow and Devlin (1959), and by McLean (1960). A superconductor has finite surface impedance at high frequency, and this impedance is limited in the superconducting phase by the penetration depth A, as in the normal phase it is limited by the skin depth S. Pippard (1947a) was the first to use this as a means of measuring A, and he and his collaborators have carried out a large number of experiments at different frequencies and varying experi- mental conditions (see Pippard, 1960). Basically all these measure- ments involve observing the change in the resonant frequency of a 36 Superconductivity cavity containing the specimen when the specimen passes from the normal to the superconducting phase. At T< T c , where \<8, these changes are proportional to 8 - A. If 8 is independent of temperature, as is the case for a metal in the residual resistivity range, then any temperature variation of the observed changes must be due to the temperature variation of A. Dresselhaus et al. (1964) use instead of a cavity a rutile resonator to which the specimen is coupled, and also observe changes in the resonant frequency. Schawlow (1958), Jaggi and Sommerhalder (1959, 1960), and most recently Erlbach et al. (1960) have measured the penetration of a magnetic field through a thin cylindrical film of thickness less than the penetration depth. 4.2. The dependence of A on temperature and field According to the London theory, an external magnetic field pene- trates into a superconductor to a depth characterized by (see equa- tion HI. 13) A = (mc 2 l4irn s e 2 ) 112 , where n s is the number density of the superconducting electrons. It is reasonable to expect this to be the only temperature-dependent factor in this defining equation, and in fact the Gorter-Casimir two-fluid model assumes that n s (t) = #xq*m av.4) where i^it) is the order parameter. The temperature dependence of W is given by 11.25', so that sub- stituting this and IV.4 into the defining equation for the penetration depth one obtains A(/) = A(0)/(l-/ 4 ) ,/2 , (IV.5) where A(0) = (mc 2 /47r/;/0)e 2 ) ,/2 (IV.6) is the penetration depth at T= 0°K. Very near T c , IV.5 can be written as *»-£Ww. (IV.7) The Pippard non-local theory 37 sented to a very high degree of approximation by IV.5. This is a striking success of the phenomenological theories discussed in the preceding chapters. Close inspection of the recent very precise measurements, however, shows a small deviation from IV.5 at / < 0-8, which becomes particularly pronounced at low temperatures. This deviation is barely discernible in the normal plot of A(r) vs. y(t), where y(.t) = (1-f 4 ) -1 ' 2 , but is displayed strikingly in Figure 13, which shows for Schawlow's results (1958) the variation of the slope, 1400p 1200 >■ 1000 d y 600 400 200 a BCS THEORY 1 Daunt et al. (1948) were the first to point out that the empirical temperature variation of the penetration depth can indeed be repre- 1.0 1.5 2.0 25 3.0 3.5 4.0 4.5 5.0 Fig. 13 dXjdy, with y(t). The solid line indicates the values of dX/dy calculated by Miller (1960) on the basis of the BCS theory; the experimental results appear to deviate less from IV.5 than is predicted by theory. Furthermore it appears that in impure specimens no deviation from IV.5 can be found at all (Waldram, 1961). The slope of A(r) plotted as a function of y(t), as well as the inter- cept, yield values for A(0) if one ignores the small deviations from IV.5. Appropriate empirical values for pure bulk samples are shown in Table II. They exceed by a factor of about five what one would expect from the London definition IV.6, unless one makes rather 38 Superconductivity unlikely assumptions of low densities of superconducting electrons or of a large effective mass. Experiments on very small samples, and measurements on impure metals, yield even higher values of A(0), although none of the factors in IV.6 appear to depend on size or purity. This failure of the London theory will be discussed in the following sections of this chapter. Table II lement A(0)(A) Reference Al 500 Faber and Pippard, 1955a Cd 1300 Khaikin, 1958 Hg 380-450* Laurmann and Shoenberg, 1949 In 640 Lock, 1951 Nb 440 McLean and Maxfield (1964) Pb 390 Lock, 1951 Sn 510 Pippard, 1947; Laurmann an Shoenberg, 1949; Lock, 1951 470-600* Schawlow and Devlin, 1959 Tl 920 Zavaritskii, 1952 * Anisotropy. Pippard ( 1 950) investigated the change of the penetration of a small, r.f. field (9-4 kMc/s) at a given temperature as an external d.c. field H e is raised from zero to the critical value. This change, divided by the penetration depth in zero field, is plotted against temperature in Figure 14. There are clearly two effects: one at low temperatures (which Bardeen (1952, 1954) has shown to follow from an extension of the London equations to include non-linear terms), and one near T c . This latter involves a change in A with H in just that region in which A varies appreciably with temperature. By a thermodynamic derivation Pippard has shown that this temperature variation of A leads to a dependence of the superconducting entropy on field. He finds that S(H e )-S(0) = ^^V/U ~/ 4 )] av>8) for a superconductor of total surface A, assuming IV.5 to hold. Near T c this change contributes as much as one-fourth of the total entropy The Pippard non-local theory 39 difference between the normal and superconducting phases, which is quite considerable. Pippard pointed out that to assume that the entire entropy change takes place in the thin layer into which the field penetrates would, therefore, result in an unreasonably high entropy density in this layer. Yet this is just what one is led to believe by the London model, accord- ing to which the superconducting wave functions or, in two-fluid language, the corresponding order parameter #", remains rigidly unchanged by the application of an external field. Any change in the Z0 25 3.0 T(°K) Fig. 14 35 T C thermodynamic functions with field must therefore be confined to the thin layer into which the field penetrates. Recent measurements of the field dependence of the penetration depth at 1 and 3 kMc/s (Spicwak, 1959; Richards, 1960, 1962; Pip- pard, 1960; Dresselhaus et al., 1964) have shown that this effect has certain unexpectedly complicated features. Not only is the magnitude of the change frequency dependent, but even its sign can change under certain conditions. In particular there can be an increase in the pene- tration depth when the applied field is parallel to the specimen surface, and at the same frequency a decrease when the field is perpendicular. Bardeen (1958) and Pippard (1960) have suggested that these com- plexities may be due to field induced deviations of the superconducting and normal electron densities from their equilibrium values. 4 40 Superconductivity 4.3. The range of coherence The unreasonably high entropy density in the surface layer led Pippard (1950) to propose a basic modification of the London model, according to which the order parameter changes gradually over a certain length f , which he calls the range of coherence of the super- conducting wave functions. In terms of the microscopic theory this distance can be considered as the typical size of the Cooper pairs. Any change in the thermodynamic functions of course extends over as wide a region as the change in the order parameter, and thus a value £ > A would correspond to a more reasonably small value of the field- induced entropy density. Pippard (1950) obtained an estimate of the range of coherence of the order parameter by minimizing the Gibbs free energy of the super- conductor in the presence of an external field. The resulting relation between the fractional change of the penetration depth and the ratio A(0)/£ allows him to estimate from his experimental data on the field effect on A that £ M 20A(0) « 10 -4 cm. Such a distance is much larger than the smallest colloidal specimen or the thinnest films in which superconductivity is still known to exist, and it is therefore of great importance to realize that, to quote Pippard (1950, p. 220) : ' the range of order must therefore not be regarded as a minimum range necessary for the setting up of an ordered state, but rather as the range to which order will extend in the bulk material'. Strong support is given to the existence of this range of coherence by the extreme sharpness of the superconducting transition under suitable conditions. De Haas and Voogd (1931) have observed resistive transitions in single crystals of tin taking place within a range of one millidegree, and a sharpness approaching this value has come to be the criterion for the quality of a specimen of suitable shape and orientation. Applying a simple statistical argument, Pippard shows that fluctuations would create a broader transition unless the super- conductivity of a bulk sample can be created or destroyed only over an entire domain of diameter M 10A(0). In the next section as well as in 6.5 it will be shown that the range of coherence is much smaller than 10 ~ 4 cm in low mean free path alloys as well as in certain pure metals. For such materials one would expect The Pippard non-local theory 41 a broadened transition even for ideally homogeneous samples. Goodman (1962c) has discussed this in some detail. 4.4. The Pippard non-local relations In 1953 Pippard measured the penetration depth in a series of dilute alloys of indium in tin, and found that the decrease in the normal electronic mean free path of the metal was accompanied by an 10- E o .L 10 20 )0 6 4 40 30. (cm) Fig. 15 50 appreciable rise in the value of A(0). This has been confirmed by Chambers (1956), and by Waldram (1961), whose results are shown in Figure 15. Such a dependence of A(0) on the mean free path is quite incompatible with the London model, for clearly none of the para- meters in the defining equation IV.6 varies appreciably with the electronic mean free path. This experimental result, added to the previous questions which had been raised about the correctness of the London phenomenological treatment, led Pippard (1953) to develop a fundamental modification of this model, based on the concept of the range of coherence of the 42 Superconductivity superconducting phase. The basic London equations, it will be remembered, lead to the relation J(R)= - 4-nXl A(R), where A^, = mc 2 /4ime 2 , so that one can also write this as „2 IW~ J(R) = A(R). (IV. 10) (IV.ll) One way to introduce a dependence of the penetration depth on the electronic mean free path is to write (IV.P1) where £ is a constant of the superconductor in question, and £(/) a parameter depending on the mean free path /. It is evident from the analysis in Chapter HI that IV.P1 leads to an expression for the field penetration into a semi-infinite slab which has the London form: but where now H{x) = H e exp(-x/X), X = X L (TV. 12) As experimentally A is found to increase with decreasing /, it is clear that £(/) must decrease as / decreases. As a first step toward a modification of the London model, Pippard identifies £ with the range of coherence of the pure superconductor, and assumes that £(/) tends toward this value as /->• a>, but that £(/) ->/as /->0. This is the case if, for example, 1 1 1 (IV. 13) where a is a constant of order unity. £(/) is thus an effective range of coherence which has a size (w 10 " 4 cm) characteristic of the metal in a pure superconductor, but which becomes limited by the normal electronic mean free path as the latter becomes much smaller than 10- 4 cm. The Pippard non-local theory 43 These equations satisfactorily explain the onset of a mean free path effect on the penetration depth at a critical value of /, as found by Pippard and others, as well as the very large penetration depth values obtained from experiments where / is limited by boundary scattering. They do not, however, satisfactorily explain the finding that A(0) in pure, bulk superconductors exceeds the London value IV.6 by a factor of four to five. According to Pippard this is because IV.P1 does not correctly describe the relation between current density and the vector potential in such a case. PI still implies, as does equation III.B', the basic London idea of a wave function which is completely rigid under the application of an external field because the electronic momenta are ordered or correlated over an infinite distance. Thus the distance over which H and A vary is quite immaterial ; the same kind of relation would hold if the field varied very slowly as if it varied very rapidly. But according to Pippard the range of momentum coherence is not infinite but only about 10~ 4 cm, so that the electromagnetic response of the superconductor should be affected profoundly if the field varies rapidly over this distance. A relation like PI could apply only if the field varied slowly over a distance of the order off. The situation is somewhat analogous to the problem of electrical conductivity in a normal conductor, for which the relation J(R) = a(/)E(R) (IV. 14) is valid only if E(R) varies slowly over a distance of the order of/. An applied alternating field penetrates only a finite distance, S, which varies inversely as the square root of the frequency. At sufficiently low temperatures and high frequencies, the electronic mean free path in the normal metal may be longer than this skin depth, so that electrons may spend only part of the time between collisions in the field pene- trated region. Pippard (1947a) showed how this makes the electrons less effective as carriers of current and leads to a higher surface resistance, as observed by H. London (1940) and Chambers (1952). Under these conditions Ohm's law (IV. 14) can no longer be a valid approximation; the current at a point must be determined by the integrated effect of the field over distances of the order of the mean free path (see Pippard, 1954). The details of this so-called anomalous 44 Superconductivity skin effect were worked out by Reuter and Sondheimer (1948), who derived that 3a CR(R-E)e- R "dT JCR)_ 4ir/J F (IV. 15) where a is the d.c. conductivity and / the mean free path. The form of this equation ensures that in the case of a rapidly varying field the current density at a point R is determined by the integral of the field over a distance comparable to the mean free path /. In a superconductor of range of coherence £, the current density at a point in the case of a rapidly varying field should also be deter- mined by an integral of the field over a distance of the order of g, and not, as is implicit in the London equation as well as in PI, by the field variation over a quasi-infinite distance. Because of this analogy to the anomalous conduction in a normal metal, and because some special solutions of equation IV. 15 were already known, Pippard (1953) proposed as the basic relation for the electromagnetic response of a pure superconductor the equation J(R)= - 3ne 2 C R(R-A)e~ R ltdr 4n$ m S R* (IV.P2) Somewhat misleadingly, as this erroneously implies that the basic London equation is a truly local relation, P2 is called the Pippard non-local relation. This relation leads to a reversal of the phase of the magnetic field penetrating into a superconductor (Pippard, 1953). Drangeid and Sommerhalder (1962) have observed this effect. The validity of P2 is strongly supported by Bardeen's proof ([5], pp. 303 ff.) that an energy gap in the single electron spectrum requires a non-local relation between current density and vector potential. In fact the BCS theory leads to a relation entirely equivalent to P2 if one assumes & = nv /7re(0), (IV. 16) where 2e(0) is the energy gap at 0°K. Substituting the BCS value 2e(0) = 3-52k B T c , IV.16 becomes g, = 0l8hv /k B T c . (IV.17) The Pippard non-local theory 45 This is just the expression IV.9 for the range of coherence derived from an uncertainty principle argument, with a = 018. From P2, the penetration depth A as defined by IV. 1 can be evaluated explicity in two limiting cases: A = V(Zolt)*L for 5< A, (London limit), (IV. 18a) A.- feM 1/3 for g> A, (Pippard limit). (IV. 1 8b) The second of these is the one applicable to the case of an infinite mean free path, and correctly predicts a penetration depth into very pure superconductors which is much larger than the London value. IV. 18a is identical to IV. 12 obtained directly from PI. This is of course a reflection of the fact that PI is the limiting form for £ <^ A of the more general equation P2. Equations IV. 18 show that the range of coherence of a super- conductor can be calculated from absolute values of the penetration depth. Faber and Pippard (1955a) have in this way obtained values of 2-1 x 10 -5 cm for tin, and 12-3 x 10 -5 cm for aluminium. These values differ very much, but when substituted into equation IV.9 together with known values of T c and v (from anomalous skin effect data [Chambers, 1952]), both correspond to a = 015. This is in striking agreement with the BCS value of 0-18, as cited in IV.17. A later chapter will mention how measurements of the transmission of infrared radiation through thin superconducting films lends further strong support to this value. Peter (1959) has solved the Pippard non-local relation P2 for the case of cylindrical superconducting films of thickness d < A and radius r. He finds that an external field H e penetrates through the film to a value Hi such that HJH^Vrd^faFitld). (IV. 19) £o 's the range of coherence in a specimen of unlimited mean free path, and can be calculated from IV.17; £ is the actual range of coherence in the film, and A should be the London penetration depth as calcu- lated from IV.5 and IV.6. Schawlow (1958), however, has shown that 46 Superconductivity good agreement with his measurement on tin films can be found by substituting for A the empirical value for bulk samples (510 A) and considering £ as being determined by the size-limited mean free path of the electrons in the films. A similar analysis has been used by Sommerhalder (1960). It is now generally accepted that whenever one applies the equation of the Pippard theory (or those of the Ginzburg-Landau treatment to be discussed presently) to the case of small or impure specimens, one obtains good agreement by using for the ideal penetration depth in a bulk sample, not the London value A^. but rather the depth deter- mined experimentally. For example, the results of Whitehead (1956) on the magnetic properties of mercury colloids were shown by Tinkham (1958) to be in excellent agreement with the prediction of the London limit of the Pippard theory if one modifies equation IV.18b and writes ■\W (IV.20) where X b is now the empirical penetration depth for a bulk sample and takes the place of the London value A L . The mean free path / is limited by boundary as well as by impurity scattering. Ittner (1960a) has similarly found that such a modification of the Pippard equations adequately predicts the results of the observations by Blumberg (1 962) of the critical field of moderately thin films. In analysing the magnetic behaviour of small (or very impure) specimens, for which £ * / <^ A, it is thus in general possible to obtain adequate precision without attempting to solve the difficult relation IV. P2. Instead one can use IV.20 to calculate the penetration depth, and then substitute this value of A into the London equation IV.10. In discussing the mean free path dependence of the coherence length one must remember that it is related to the behaviour of a superconductor in two subtly different ways. One of these, as men- tioned in Section 4.3, is the distance over which the order parameter of the superconducting phase varies. It is this aspect which, for example, in Chapter VI will enter into the discussions of the width of a boundary between the normal and superconducting phases. It follows from Gor'kov's analysis of the influence of impurities The Pippard non-local theory 47 (Gor'kov 1959b) that the mean free path dependence of this aspect of the range of coherence is given by £ = £oX- 1/2 (0 (IV.21) x(/) is a function of the mean free path shown graphically by Gor'kov and approximated to within about 20 per cent by the simple expression (Douglass and Falicov, 1964) x(/) (s4 In the limit / <^ £ , IV.21 thus reduces to i = V(t o 0- (IV.22) (IV.23) The relatively slow variation of this aspect of the coherence length with mean free path is essentially due to the fact that not every electronic collision destroys the superconducting coherence. The other aspect of the range of coherence is that it determines the distance over which the magnetic field or the vector potential at a given point influences the current density. This is expressed by the Pippard equation IV.P2. What is important in this application is the actual mean distance between electron collision, so that now equa- tion IV. 13 applies. This means that $ « / (IV.24) for / <g £ . It is this mean free path dependence which enters, for example, into equation IV.20. CHAPTER V The Ginzburg-Landau Phenomenological Theory In 1950 Ginzburg and Landau (G-L) introduced a phenomenological approach to superconductivity which, like that of Pippard, modifies the absolute rigidity of the superconducting order parameter or wave function which is implicit in the London model. Although the theory was originally formulated so as to reduce always to the 'local' London equations in zero field, Bardeen (1954) has shown that it can be modified so as to be compatible with a non-local equation of the Pippard type. Furthermore, Gorkov (1959, 1960) has derived the G-L equations, under certain conditions, from his formulation of the BCS theory. G-L introduce an order parameter >p which they normalize so as to make \*fi\ 2 = n s , where n s is the density of the superconducting elec- trons, ip is thus a kind of 'effective' wave function of the supercon- ducting electrons. According to the general Landau-Lifshitz theory of phase transitions (1958), the free energy of the superconductor depends only on \ifj\ 2 and can be expanded in series form for tem- peratures near T c . In the absence of an external field, the supercon- ducting free energy (per unit volume, as are all equations listed) is then (7,(0) = G B (O) + o#| 2 +(|8/2)|0| 4 . (V.l) Minimizing the free energy with respect to \<p\ 2 yields the zero field equilibrium value M>l 2 =-«/A (V.2) from which (7,(0) - (7„(0) = -<x 2 /20. (V.3) In the immediate vicinity of T c one can assume that the coefficients a and /? have the simple form *(j) = (r c -r)(sa/ar) r=re and jB(D = p(T c ) - fi e 48 (V.4) The Ginzburg-Landau phenomenological theory 49 With these one then finds from V.3, remembering that the free energy difference between the phases equals the magnetic energy, that 47ra 2 47r(r c -r) 2/fl - x2 2 _ Ht = \8T/ T=Te (V.5) P Pc Near T c , H c indeed is known to vary linearly with (T c -T), so that the correctness of equation V.5 justifies the assumptions V.4. All further thermodynamic manipulations are now possible, but they and all other conclusions drawn using V.4 are restricted to tempera- tures very near T c . Both Bardeen (1954) and Ginzburg (1956a) have considered extensions of the model to the full superconducting range by introducing different forms for a(T) and @{T), the former using expressions based on the Gorter-Casimir two-fluid model. The outstanding contribution of the G-L model in any temperature range arises from its ability to treat the superconductor in an external field H e « H c . The free energy G s (H e ) is now increased not only by the usual volume term H*/9v, but also by an extra term connected with the appearance of a gradient of i/j, as ifj is not completely rigid in the presence of H e . Such a gradient would contribute to the energy in analogy to the kinetic energy density in quantum mechanics which depends on the square of the gradient of the wave function. Intro- ducing this extra energy is equivalent to requiring that */> not change too abruptly. One is thus led to a concept of gradual, extended varia- tions of the superconducting order parameter quite analogous to Pippard's model of the range of coherence. In order to preserve gauge-invariance, G-L assume the extra energy term to be J-L/^-^aJ , (V.6) where A is the vector potential of the applied field, and e* a charge which, as stated in the original version of the theory, 'there is no reason to consider as different from the electronic charge'. Modifica- tions of this view will be discussed presently. G-L thus write t2 1 r * 12 (V.7) G s (H e ) = GM + fr+^A -W*~A Uc m ~M - 50 Superconductivity One must now minimize this with respect to both tp and to A, which leads to the two equilibrium equations: 2m\ c J r dtfj* V 2 A = -^J, = t^'w^-W^ c mc (V.G-L1) + • Aire*"' mc I0I 2 A. (V.G-L2) In a very weak field, HxO, the function iff remains practically con- stant (that is, rigid), V0 = 0, 4> » 0o> and G-L2 reduces to V 2 A A-ne* 2 mc' l^ol 2 A = Aire* 1 « C A. (V.8) </ 2 dz 2 -tN**w^*-t^-* H ^4 A 2 mc 2 (V.10) Here, as in V.2, the subscript denotes the zero field value. This of course is just London's expression (B). Non-local versions of the G-L treatment are obtained by substituting an integral expression for the second term in V.6. In its present local form the G-L treatment is restricted to temperatures near T c for two reasons: in the first place because of the simple forms assumed for the functions a(T) and fi(T), and secondly because only near T c is A §> £ , and can the non-local electromagnetic character of superconductivity be ignored. The set G-Ll and G-L2 of coupled non-linear equations in ifj and A have been solved for essentially one-dimensional problems. Taking the z-axis to be normal to the infinite superconducting boundary, the field H along the>--axis, and the current J s and potential A along the x-axis, one obtains (using V.l) The Ginzburg-Landau phenomenological theory 5 1 Note that with this geometry H = curlA = ~ dz The meaning of these equations becomes clearer by introducing a dimensionless parameter k defined by 2e* 2 KT := .-, .,/ZcAq, h 2 < where Ag- m c 2 4ne* 2 ifa (V.ll) (V.12) The subscript again denotes zero field, k, A , and H c are the three parameters of the G-L theory which are to be determined experi- mentally, and in terms of which various field and size effects can be expressed. H c is the bulk critical field. A is the empirical penetra- tion depth of a superconductor in the weak field limit, and is the quantity which through equation V.12 determines the zero field equilibrium value of the order parameter i/jq. For a bulk sample con- taining impurities A increases, as was discussed in the previous chapter, and this in turn affects both O and k. k can be determined in a number of ways, two of which follow directly from the defining equation V. 1 1 . In the immediate vicinity of T ct the experimental variation of A (/) can be expressed by IV.7: Also one can write H c = dH r so that 8/jV dT dH r xAT, T=T e dT T=T„ xT 2 xX 4 (0). (V.l 3) Thus k is seen to be temperature independent, at least for T& T c . 52 Superconductivity One can also use the expression for the penetration depth derived from the BCS theory to be, very near T c : m = x -^ (l _„-./! _y?/ik- J/2 V2 V2 \ATj so that ,2 _ ,*2 4/i 2 , dH„ dT t=t c x7?xAf(0). (V.14) where now A L (0) is the London penetration depth calculated from 111.13 using the actual free electron density n. This can be calculated from the value of the normal state anomalous skin resistance (Chambers, 1952). Another method of calculating k for a given superconductor is to use results on supercooling, as will be discussed in a subsequent sec- tion. Ginzburg (1955) pointed out already before the formulation of the BCS theory that values as calculated from V.13 and V.14 could be made to agree very well with those obtained from supercooling data by taking e* = 2 or 3e. More recently Gor'kov (1958) has formu- lated the electromagnetic equation of the BCS microscopic theory in terms of Green's functions, and was able to show (1959, 1960) that the G-L equations G-Ll and G-L2 are identical to his expressions near T c when ip is taken to be proportional to the energy gap, and when one takes e* = 2e. This again is an indication that the current carriers in superconductivity are the doubly charged Cooper pairs. With this value of e*, V.13 and V.14 yield and *c= 108 xlO 7 k = 216xl0 7 dH r dT dH c T=T r tMo), dT T=T e T c X 2 L (0). (V.130 (V.140 For tin, the first of these yields k = 0-158, the second 0-149, two values which are in excellent agreement. For indium, however, the respective values are 01 12 and 0051 (Davies, 1960; Faber, 1961). For aluminium, the equations yield 005 and 001 (Davies, 1960). This lack of agreement may be in part due to errors in anomalous The Ginzburg-Landau phenomenological theory 53 skin-effect measurements used to evaluate A L (0), and in part, particu- larly in the case of aluminium, due to the large value of £ > because of which non-local conditions set in very close to T c . The values of k calculated from supercooling are probably the most reliable. In terms of the parameters k, Aq, and H c , equation (V.9) reduces Far from the phase boundary, for z-> °o, tp 2 = ipl, and dz At the boundary, z = 0, V.9' is satisfied in the absence of an external field {A = 0) by ip 2 = ipl; difi/dz = 0. In other words, the presence of the phase boundary as such has no influence on the function tp, which has the same value tp everywhere. In the presence of an external field H e , however, this solution no longer applies, and one must integrate V.9' and V.10 with the boundary condition tp 2 = ipl for z-»-co, and the condition H = dA/dz = H e , and difi/dz = for z = 0. This integra- tion cannot be carried out exactly. Neglecting higher order terms, however, one finds equations for i/j and for A as functions of z. At z = 0, the value of tp is ^o 4(«+V2) CV ' 13) With values of k m 0- 1 , this equation predicts a decrease of ip by only about 2-3 per cent when H e = H c . It is not surprising, therefore, that the change in penetration depth with field is also very small. This can be calculated formally by using the defining equation IV. 1 from which one finds that, with a weak measuring field normal to H e : 1 + An 1 + 8(k+V2) 2 # ( 2 k H 2 For a measuring field parallel to H e , the effect is tripled. (V.16) 54 Superconductivity It is evident that in the limit k-*0, the effect of the external field on ifj and on A vanishes, so that one returns to a situation formally equivalent to the London picture. It must however be noted that even for k = 0, j/tq is deduced from the empirical value of A . As a result one can in certain cases, such as, for example, the treatment of very thin films, allow k to vanish without necessarily reducing the G-L treatment to the London one. CHAPTER VI The Surface Energy 6.1. The surface energy and the range of coherence Closely tied to the range of coherence of the superconducting wave functions is the existence of an appreciable surface energy on a boundary between the superconducting and normal phases. H. London (1935) already pointed out that the total exclusion of an external field does not lead to a state of lowest energy for a super- conductor unless such a boundary energy exists. In the presence of an excluded external field, H e , the energy of a superconductor in- creases by Hg/Sir per unit volume. It would, therefore, be energetically more favourable for a suitably shaped superconductor to divide up into a very large number of alternately normal and superconducting layers such that the width of the latter is less than A, and that of the former very much smaller than that. The resulting penetration of the external magnetic field into the superconducting layers much reduces the magnetic energy of the sample, while the extreme narrowness of the normal layers keeps negligible their contribution to the total free energy. This situation is made energetically unfavourable by the existence of a surface energy. To make each superconducting layer narrower than A, a slab of thickness c/must have d/X such layers. This is avoided by an interphase surface energy cc„, per unit surface whose contribution exceeds the gain in magnetic energy, that is: 2d Hid 8tt ' (VI.l) where the energies have been calculated for a volume of slab of unit surface area. Hence XH[ > 28tt 55 (WW) 56 Superconductivity It is convenient and customary to express the surface energy in terms of a parameter A ' of dimensions of length, such that Thus one sees that 8tt A'> (VI.2) (VI.3) Position Fio. 16 is the condition for the diamagnetic behaviour of superconductors.! Empirical values of A ' for pure superconductors turn out to be an order of magnitude larger than the penetration depth. The surface energy is intimately related to the Pippard range of co- herence. Figure 16 shows the variation of the order parameter iV and of the externally applied field H e along a direction perpendicular to the s-n interphase boundary. One can define two effective bound- aries, indicated by M and C. M is the magnetic boundary defined so that if inside the superconductor B = H c up to M and then dropped off sharply to zero, the total magnetic energy would equal the actual value, given by the integral of BH/Stt over the entire superconductor. t F. London ([2], pp. 125-130) has shown that taking into account the detailed field penetration leads to the condition A' > A. The surface energy 57 Similarly C is the configurational boundary such that if #" dropped sharply to zero at C after being constant up to that point, one would have the same superconducting free energy as the actual amount. The free energy per unit volume of the superconductor is lower than that in the normal state by an amount Hc/Stt. A configuration boundary as shown on the inside of the magnetic boundary is essentially equiva- lent to a reduction of the superconducting volume and hence an increase in the total free energy by an amount equal to HcI%t times the distance C-M per unit area of interphase boundary. The intro- duction of the Pippard range of coherence thus leads to a configur- ational boundary surface energy A ' w £. From this one must subtract the decrease in energy due to the penetration of the field. Figure 16 indicates that the distance C-M corresponds to the resulting net surface energy parameter A m £-A. (VIA) The condition for the Meissner effect is that f > A, i.e. that A > 0. The Ginzburg-Landau theory was formulated so as to lead ex- plicitly to the existence of a surface energy, which arises as in the Pippard approach from the gradual variation of the order parameter *p over a finite distance, from the zero value in the normal region to its full equilibrium value in the superconducting domain. Again the surface energy is that amount which is needed to equate the energies of the two phases in equilibrium, with H e = H c . In the supercon- ducting phase the increase in the free energy in the region where is changing is given by V.6; in addition there is a reduction of the energy due to the penetration of the field equal to H(z)-H c -MH„ = - 4tt If,- (VI.5) where H(z) is the value of the penetrated field at any point inside the superconducting region. Thus the surface energy is given by the integral of the difference between the superconducting and normal free energies over the entire superconducting half-space : 00 G s {H,z)- H(z)H c . H\ H: 4tt +-P-G /I (0)--^ (VI.6) 58 which gives for A : Superconductivity — CO where A is the empirical penetration depth into a bulk supercon- ductor, and k the dimensionless parameter defined in the previous chapter. This equation requires numerical integration. For k <^ 1 it reduces to A = 1-89-° ■ K (VI.8) The thickness of the transition layer is thus, according to the G-L theory, of the order of A /k « 10A for most pure elements. The intimate relation between the G-L model and Pippard's range of coherence is shown by Gorkov's derivation of G-Ll and G-L2 from first principles. He finds an expression for the G-L parameter k in terms of the critical temperature and the Fermi momentum and velocity of the metal. Using equations IV. 17 and 111.13, this simplifies to 0-96 A ?- 50 (VI.9) Comparing VI.8 and VI.9 shows that, as expected, the Pippard range of coherence g and the surface energy parameter A as derived from the G-L theory are of comparable size. In short, both approaches necessarily lead to a positive surface energy because both require that the characteristic superconducting order parameter vary over a finite distance. Both, therefore, obtain a net surface energy parameter of length comparable to the difference between this distance and the penetration depth of an external magnetic field. It therefore also follows from both theories that the surface energy must decrease and may even become negative when the range of co- herence decreases and the penetration depth increases. Equations IV.12 and IV.21 show that this is just what happens to A and to £ when the mean free path of the superconductor decreases. In alloys one would, therefore, expect A to decrease with increasing impurity con- The surface energy 59 tent, and ultimately to become negative. This has indeed been inferred by Pippard (1955) and by Doidge (1956) from their studies of flux trapping and the superconducting transition in dilute solid solutions of indium in tin. Direct measurements of J in such alloys by Davies (1960) has demonstrated its decrease with shortening /, and Wipf (1961) has traced this decrease to actual negative values. All the work cited indicates that A becomes negative at a critical concentration of approximately 2-5 atomic per cent of indium in tin. Changes of A with decreasing mean free path also follow from the numerical integration of VI.7, which yields that A < for k > 1/V2. (VI. 10) This prediction is in good agreement with the work on tin-indium alloys just cited. Chambers (1956) found that the addition of 2-5 atomic per cent of indium to tin about doubles the penetration depth as compared to pure tin, so that according to the defining equation k should be increased by a factor of approximately four. This would make k « 0-6, which is close to the theoretical value of 0-707. 6.2. The surface energy and the intermediate state Chapter II mentioned that a superconducting specimen with a demag- netization coefficient D is in the intermediate state when the external magnetic field H e satisfies the inequality (1 -D)H C <H C < H c . All experiments on the detailed structure of this state have generally sub- stantiated the suggestion of Landau (1937, 1943) of a laminar struc- ture of alternating normal and superconducting layers. The thickness of the normal layer grows at the expense of the superconducting one as the external field approaches H c . Landau further suggested that in the normal layers B = H c , while B = in the superconducting ones. Clearly the width of the laminae is strongly influenced by the mag- nitude of the interphase surface energy A. Indeed Landau finds that for an infinite plate of thickness L oriented perpendicularly to the field (£> = 1), the sum a of the thickness of the superconducting layer, a s , and that of the normal one, a„, is given by LA (VI.11) 60 Superconductivity where W is a complicated function of the ratio of the external to the critical field H e jH c . Numerical values for Y(HJH£ have been calcu- lated by Lifshitz and Sharvin (1951). A typical result is a value a « 1 -4 mm for L = 1 cm and HJH C = 0-8. A similar equation has also been derived by Kuper (1951), who predicts numerical values which are smaller by a factor of two or three. Typical experimental results fall in between these predictions. These results have been obtained by a variety of methods, all making use of the fact that in the intermediate state lines of flux pass only through the normal laminae, and emerge from the specimen wherever these laminae end on the surface. A number of authors (Meshkovskii and Shalnikov, 1947; Shiffman, 1960, 1961) have e2& i\,\t\ && i\i\i\i\ Fig. 17 passed very fine bismuth wire probes across the surface of a specimen, and observed the magnetoresistive fluctuations in the probe resistance when passing from the end of a normal lamina to that of a super- conducting one. Others have spread on the surface of a flat specimen fine powder, superconducting (Schawlow et al, 1954; Schawlow, 1956; Faber, 1958; Haenssler and Rinderer, 1960) or ferromagnetic (Balashova and Sharvin, 1956; Sharvin, 1960). The former will shun flux and cluster on the ends of the superconducting laminae, as shown schematically in Figure 17; the latter will be attracted by flux and move onto the ends of the normal laminae. The resulting powder patterns can be easily seen and photographed. Another optical method consists of placing a thin sheet of magneto-optic glass (for example, cerium phosphate glass) on the specimen surface, and observing the reflection of polarized light (P. B. Alers, 1957, 1959; De Sorbo, 1960, 1961). The surface energy 61 The frontispiece shows a series of photographs obtained by Faber (1958) with superconducting tin powder on an aluminium plate, taken with increasing external field oriented perpendicularly. The dark areas are covered with powder and are therefore the ends of the superconducting laminae. The gradual shrinking of these areas with increasing field and the corresponding growth of the light, normal regions is clearly visible. The domains show a peculiar type of corrugation, not predicted by the Landau model, and adding to the surface to volume ratio of the laminae. 6.3. Phase nucleation and propagation H. London (1935) pointed out that the existence of a positive surface energy at the interphase boundary must under suitable conditions give rise to phenomena analogous to superheating and supercooling in the more familiar phase transitions. In fact a stable nucleus for the phase transition cannot exist at all if the surface energy is everywhere positive. Indeed there are many experimental observations that when a specimen is placed in a greater than critical magnetic field which is then reduced, the normal phase persists in fields less than H c . This is the superconducting equivalent of supercooling. A typical magnetiza- tion curve illustrating this is shown in Figure 18. The degree of this 'supercooling' is characterized by the parameter Si = H t IH c , or by the parameter <j>, m 1-af - {Hl-Hf)IHl (VI.12) For tin, 5/ is commonly of the order of 0-9; in aluminium the degree of supercooling is usually much larger, and values of S t as low as 002 have been observed. Superheating is the name given to the persistence of the supercon- ducting phase at fields above H c . This is very rarely observed. Garfunkel and Serin (1952) have shown that this is so because the ends of any conventional specimen cannot resist the initiation of the normal phase, probably because of large local field values resulting from demagnetization effects. Centre portions of long tin rods could be made to superheat to S t = 1-17. Much information on the nucleation of the superconducting phase and on its relation to the surface energy has been obtained by Faber 62 Superconductivity (1952, 1955, 1957) in a series of measurements on supercooling in tin and aluminium. His technique consisted of winding on a long cylin- drical specimen several small, spaced coils the field of which could be made to add or to subtract from a field produced by a large solenoid surrounding the entire sample. With the sample normal, the field of the solenoid could be lowered to some value between H, and H c , and the field could then be lowered locally by a suitably directed current through one of the smaller coils. The superconducting phase then nucleated in the portion of the sample under the coil, and spread H, H c Applied Field H e Fig. 18 rapidly throughout the sample. In this fashion supercooling could be studied at different portions of the sample. The transition was de- tected by pick-up coils distributed along the specimen. At a given temperature the degree of supercooling varied con- siderably from point to point in a given specimen but at a given point frequently remained reproducible even when in between measure- ments the specimen was warmed to room temperature. This indicated that nucleation must occur at particular spots, some of which pro- mote nucleation more effectively than others. As the surface energy can be lowered and may even become negative due to strain, it is reasonable to assume that the spots favouring nucleation are regions The surface energy 63 of local strain, some of which exist in even the purest specimens. This is supported by Faber's finding that any handling of the specimens between measurements could change the location and effectiveness of the nucleation centres. Strained regions probably contain a high density of dislocations. By correlating the size of the nucleating field H t with the time it took to be effective, Faber could deduce the depth of the nucleating flaw below the sample surface, and found this always to be between 10 -4 and 10 -3 cm. Etching down the surface to this depth would uncover further flaws extending to a similar depth. It is thus reason- able to take 10 -4 -10~ 3 cm as being the approximate size of the nucleating flaws. At temperatures well below T c , this length is con- siderably bigger than the width of the interphase boundary, and one can therefore imagine such a flaw to consist of a region of negative surface energy surrounded by a shell across which the surface energy increases to the normal positive value of the bulk material. Faber (1952) has shown that there is a potential barrier against the further growth of this nucleus until one has reached a degree of supercooling such that , A <f>, « -+n, (VI. 13) where A is the surface energy parameter, r a length of the order of the flaw size, and // a small constant determined by the flaw's shape and demagnetization factor. The measurements in fact show that the tem- perature variation of </>, is very much like that of A, as determined from other experiments. Both Faber (see Faber and Pippard, 1955b) and Cochran et al. (1958) found that supercooling was much enhanced after a specimen had been placed temporarily in a field much higher than the bulk critical value. This shows that certain superconducting nuclei can be quenched only by such a high field and supports much other evidence that in a non-ideal specimen there can exist small regions of high strain which remain superconducting in very high fields. By means of a series of pick-up coils along his specimens, Faber (1954) was able to observe the propagation of the superconducting phase once the transition had been initiated at some nucleus. From CHRIST'S COLLEGE I IDDADV 64 Superconductivity his results he infers that the growth of the superconducting phase occurs in a series of distinct stages. The nucleus, which is always near the surface, first expands to form an annular sheath around the speci- men. This sheath then spreads along the entire length of the specimen with a velocity of the order of 10 cm/sec, and finally the supercon- ducting phase spreads inwards to fill the entire sample. The growth of a superconducting region is limited principally by the interphase surface energy on the one hand, and by eddy current damping on the other. If there were no surface energy, the super- conducting phase could propagate by means of very thin filaments which displace no magnetic flux and therefore create no retarding induced currents. For a sheath of finite thickness, on the other hand, which propagates in the presence of an external field H c , eddy currents are generated, and the magnetic energy gained in the phase transition is balanced by the unfavourable surface energy as well as the eddy current joule heating. Faber (1954, 1955) has shown that the resulting velocity of propagation for very pure specimens under optimum con- ditions is given by v = A(I/o)A- 2 (lH c -H e ]/H c y (VI. 14) where / is the electronic mean free path in the normal phase, a the normal electrical conductivity, and A is a constant of the specimen. By measuring the temperature variation of v, Faber has used this equation to obtain the temperature variation of A for tin and for aluminium. The values of A(T) obtained in this way by Faber, as well as those measured in different ways by Davies (1960), Sharvin (1960), and Shiffman (1 960), can be fitted by a number of empirical functions of temperature. According to the G-L theory, the surface energy should have the same temperature variation as A , at least very near T c , where k is independent of temperature. Hence one would expect A(t) = J(0)(l-/ 4 )- ,/2 , which can also be written j»-«o-«-w for/ « 1. (VI.15) (VI.150 The surface energy 65 The second of these functions appears to give a good fit to various results for tin over a rather wide range of temperature, but Faber's aluminium data can be represented only by the first of these. There seems to be a definite difference in the temperature dependence of the surface energy for these two metals which is at present not under- stood. The uncertainty in the temperature dependence of A of course introduces a degree of doubt about the extrapolated value at 0°K. The table below lists the best available values of ^(0) for a number of metals, from a comparison of all available experimental data. Also listed are values of £ > th e range of coherence, as calculated from equation IV. 17, as well as empirical values for A (0). Element Aluminium Indium Tin 10 5 <d(0) (cm) 18 3-4 2-3 10 5 £ (cm) 16 4-4 2-3 10 5 Ao(0) (cm) 0-50 0-64 0-51 6.4. Supercooling in ideal specimens Near T c , A becomes large, and the flaws lose their effectiveness as nucleation centres. Measuring H t in this region can, therefore, give some information on supercooling in ideal, unflawed material. Faber (1957) has found for aluminium, S, = 0036, for In 016, and for Sn 0-23 ; values of Cochran et al. (1958) for aluminium are in reasonable agreement. These results can be compared with theoretical predic- tions arising from the G-L model. Equation V.9' has an interesting consequence with regard to the normal phase. One would expect that with H e S* H c , the half-space described by the equation would be entirely normal, with </< = 0. This is indeed a solution, but the equation is also satisfied by a second solution with ifj # 0. Assuming that for this solution #/> <^ 1, so that H(z) M H e everywhere, and remem- bering that in the geometry chosen A(z) = H(z)z, the equation becomes 7ES 66 Superconductivity This has the form of a wave equation for a harmonic oscillator, which is known to have periodic solutions tp which vanish for z= ±co (which is the required boundary condition for the normal phase) if «= V2^(/»+i),« = 0,1,2,... tic In other words, for any value of k, the normal phase of the super- conductor becomes unstable with regard to the formation of laminae of superconducting material when HJH C = K /(n + J) V2, of which the highest value, with n = 0, is H c2 IH c = V(2)k. (VI. 17) (VI. 18) A distinction must now be made as to whether k < 1 / V2 or k> 1 /V2. In the former case, which is that of most pure superconductors, H c2 < B and the field H c2 is then the lowest field to which the normal phase can persist in a metastable fashion. H c2 is thus the lower limit to which an ideal superconductor can be supercooled, and therefore in the region very near T c one would expect the experi- mental value of S t to equal \/(2) k (Ginzburg, 1956, 1958a; Gor'kov, 1959b, c). The values of k calculated in this fashion from Faber's measure- ments of S t are: 01 64 for tin, 0112 for indium, and 0026 for alumi- nium. The first two of these agree very well with k values deduced from experimental penetration depths. In aluminium the lack of agreement is probably due to the appearance of non-local effects very close to T c . Ginzburg (1958b) has noted that this is more likely to invalidate calculations involving the penetration depth than those regarding the surface energy and supercooling. Non-local effects become important for the former when £ > A ; for the latter only when K Thus K-values calculated from supercooling data are probably the most reliable, except for the effect discussed in section 6.7. The surface energy 67 For superconductors with a dimension small compared to X /k « £ , the order parameter is essentially constant throughout and one can solve the G-L equation with the simplifying assumption k a 0. The critical fields of supercooling are then given by H C 2 = V6-/Z, for a slab of thickness la, H c2 = 2V5^H C a for a sphere of radius a, and A H c2 = V8jH c for a cylinder of radius r (Ginzburg, 1958a). For all these geometries H c2 decreases with increasing specimen size, approaching mono- tonically the value given by VI. 18, which depends only on the value of k characteristic of the material. The compatibility of the G-L theory with the Pippard range of co- herence under those circumstances of temperature, size, or mean free path which eliminate the need for a non-local electromagnetic formu- lation is brought out once again by the similarity of VI. 18 with the corresponding expression derived by Pippard (1955). He finds, also by minimizing the free energy, that H - 2 V 3A o„ n c2 — jn c - (VI. 19) This differs from VI. 18 only by a numerical factor of order unity since «■« A /£. 6.5. Superconductors of the second kind According to equation VI. 10, the surface energy becomes negative when k > l/\/2. A similar conclusion follows from the Pippard non- local model when A > $ (equation VI. 4: seeDoidge, 1956). The exist- ence of a positive surface energy was shown to be necessary for much 68 Superconductivity of the magnetic behaviour usually found in superconductors. It is, therefore, not surprising that superconductors in which this energy is negative display quite different characteristics. They are accordingly called superconductors of the second kind. For a bulk specimen of such a superconductor the volume free energy in the superconducting phase remains lower than that of the normal one in external fields up to the thermodynamic value H c defined by equation II.4. The negative surface energy, however, makes it energetically favourable for interphase boundaries to appear at field lower than H c , and for superconducting regions to persist to fields higher than H c . Goodman (1961) has shown that this can already be deduced from the London model by the single addition of a negative surface energy term. The details of the behaviour of superconductors of the second kind can be deduced from the G-L theory, which is equally valid for k > \/\/2 as for k < 1 /\/2. In particular, the analysis of the preceding section still holds; that is, the normal phase has a stability limit at a field H c2 given by equation VI. 18, which shows that for k > 1/V2, H c2 > H c . Abrikosov (1957) has used the G-L equations to analyze in some detail the magnetic behaviour of superconductors of the second kind, and finds the features indicated in the magnetization curve shown in Figure 19 for a cylindrical sample parallel to an external field H c . There is a complete Meissner effect only up to H e = H ci < H c , at which point the magnetization changes with infinite slope. For values of k not much larger than 1 /V2, Abrikosov predicts in fact a discontinuity. At somewhat higher field, the magneti- zation approaches zero linearly, with a slope -4tt^= 1-18/(2^-1), and vanishes entirely at BL = H c2 = V(2)kH c . (VI.20) (VI. 18) The magnetization curve should be fully reversible. Abrikosov can- not solve the equations determining H cl for all values of k; for k > 1 he obtains V(2)kH c i/H c = In k+ 0-08 (VI.21) The surface energy 69 In the limit k = 1/V2, H cl =H C = H c2 . Goodman (1962a) has inter- polated between the latter value and those given by VI.21 to get a graphical representation of H cl /H c for all k . A numerical solution has been obtained by Harden and Arp (1963). The magnetization curve predicted by the Ginzburg-Landau- Abrikosov (G-L- A) model can be compared with experiment, as it is possible to determine the value of k for a specimen by independent measurement. Gor'kov (1959) has derived an expression for k valid when the electronic mean free path is much smaller than the intrinsic Hcl H c Applied Field H e Fig. 19 H c2 coherence length £ . This was shown by Goodman (1962a) to have the convenient form k = K +7-5xloy /2 p. (VT.22) k o is the parameter for the pure substance, y the Sommerfeld specific heat constant, in erg cm -3 deg -2 , and p the residual resistivity in ohm-cm. For tin this predicts quite closely the resistivity at which the surface energy becomes negative (Chambers, 1956). Using this equation, Goodman (1962a) has shown that the G-L-A model satisfactorily explains the magnetic behaviour of substances 70 Superconductivity such as Ta-Nb alloys investigated by Calverley and Rose-Innes (1960) and his own U-Mo alloys (Goodman et ai, 1960). Further- more, recent magnetization measurements on Pb-Tl single crystals (Bon Mardion et ai, 1962), indicate a considerable degree of rever- sibility. Detailed quantitative verification of the G-L-A magnetiza- tion curves, as is possible only near T c , was provided by Kinsel et al. (1962), who used In-Bi specimens to compare values of k calculated from equations VI. 18, VI.20, VI.22, and from Harden and Arp's values of H ci /H c . The different values of k for a given specimen agree to within a few per cent. Similar agreement can also be deduced from the results of Stout and Guttman (1952) on In-Tl alloys. The G-L-A model is thus well substantiated. The negative surface energy need not be due to a short mean free path. In principle, it is possible for the coherence length to be shorter than the penetration depth, even in a pure superconductor; this is most likely in superconductors with a high T c (cf. equationIV.17). Indeed, Stromberg and Swenson (1962) have found that the magneti- zation curve of very highly purified niobium is that of a supercon- ductor of the second kind, with a value of H cl and H c2 corresponding to k ~ 1 • 1 . This conclusion is consistent also with the results of Autler (1962) as well as of Goedemoed et al. (1963). Kinsel et al. (1963) have found with their In-Bi alloys that the effective value of k as defined by equation VI. 18 rises gradually as the temperature decreases below T c , increasing by about 25 per cent as T approaches 0°K. This agrees with the calculations of Gor'kov (1960). The experiments further show that at any temperature k = 1/V2 continues to be the critical value for the onset of super- conductivity. Thus a specimen with k ~ 0-65 at its transition tem- perature is there a superconductor of the first kind, but becomes one of the second kind at that temperature at which k reaches the critical value. The temperature dependent increase in k leads to a corresponding decrease of the surface energy. Specimens for which k goes through the value 1/V2 at some temperature are those for which at that tem- perature the surface energy changes from being positive to being negative, as has been observed for indium alloys by Kinsel et al. ( 1 964). There is reason to believe that neither a negative surface energy The surface energy 71 nor the size effects to be discussed in the next chapter can increase the critical field of a superconductor without limit. Both Clogston (1962) and Chandrasekhar (1962) have pointed out independently that in sufficiently high fields it is no longer correct to assume that the free energy of the normal phase is independent of field. With a finite paramagnetic susceptibility X p (which was ignored in deriving equa- tion II.4), this free energy is, in fact, lowered by an amount \X.H 2 . Thus, in sufficiently high fields, this alone could already bring about a transition from the superconducting to the normal phase. The limit on the critical field imposed by this mechanism is estimated to be two or three hundred K gauss, and this is consistent with the results of Berlincourt and Hake (1962). 6.6. The mixed state or Shubnikov phase The magnetization curve of type II superconductors clearly shows that for H cX < H e < H cl , the material is neither in the usual super- conducting nor in the normal phase. Abrikosov (1957) has called this region the mixed state, and De Gennes ([14]) has suggested naming it the Shubnikov phase, honouring the scientist who first suggested the fundamental nature of type II superconductivity (Shubnikov et al., 1937). It is evident from the importance of the negative surface energy that in the mixed state the specimen must contain as large an area of interphase surfaces as is compatible with a minimum of normal volume. This could be brought about by a division of the material into a large number of very thin normal and superconducting sheets or laminae (Goodman, 1961, 1964; Gorter, 1964). According to the G-L-A theory, however, the mixed state consists of a regular array of normal filaments of negligible thickness which are arranged parallel to the external field and are surrounded by superconducting material. At the normal filaments the superconducting order parameter vanishes, and then rises from these linearly with distance. It reaches its maximum value as quickly as possible, that is over a distance of the order of £. The magnetic field has a maximum value at the normal filaments, and falls off over a distance of the order of A > £. This means that the field decreases to zero only if the filaments are spaced at distances at least of the order of A. This mixed state structure can 6 72 Superconductivity be shown to have a lower energy than any laminar arrangement ([14], p. 111.81). One can thus think of the mixed state as if the superconducting material were pierced by a number of infinitesimally thin filamentary holes, regularly spaced parallel to the external field and thus each containing magnetic flux. From the discussion in Chapter II it there- fore follows that the total flux associated with each normal thread is quantized in units of <f> . This flux does not penetrate far into the superconducting material because of superconducting currents circu- lating in planes perpendicular to the filament. This creates a vortex line of superconducting pairs along each normal thread, in striking analogy to the vortices existing in liquid Helium II (Rayfield and Rcif, 1964). The flux and the currents associated with an isolated vortex line extend over a distance of about A. The interaction between two vortex lines can thus be appreciable only at distances less than A. This means that when the formation of vortex lines becomes energeti- cally favourable at H = H ci , they can essentially immediately achieve a density corresponding to a separation of about A without creating much interaction energy ([14], pp. III. 74ff.). This causes the abrupt decrease of the magnetization at H cl predicted by Abrikosov and verified experimentally. It is not certain, however, whether the magnetization actually decreases discontinuously at this field or whether it merely drops with an infinite slope. The former would correspond to a first order transition with a latent heat, the latter only to an infinity in the specific heat. With the external field increasing beyond H cX , more and more vortex lines are formed until their spacing approaches •Mh e as //nears H c2 (Abrikosov, 1 957). According to Abrikosov, the vortex lines form a square array at all fields except very near H cU but Kleiner et al. (1 964) and Matricon (1 964) have shown that a triangular array has a somewhat lower energy throughout the mixed state. This changes the coefficient in equation VI.20 from 118 to 116. The surface energy 73 A fundamental feature of the vortex structure of the mixed state is that the order parameter W is everywhere finite except along the centre of the vortices, which are normal filaments of negligible volume. Thus the material can still be considered as entirely super- conducting. Abrikosov (1957) showed in fact that in the mixed state one can characterize the material by a mean square order parameter y* 2 , and that near H c2 this varied linearly with the magnetization. The correctness of this and therefore the validity of the vortex structure has been substantiated by measurements of the specific heat (Morin, et al., 1962; Goodman, 1962b; Hake, 1964; Hake and Brammer, 1964) and of the thermal conductivity (Dubeck et al., 1962, 1964). De Gennes and his collaborators (cf. [14], Vol. II) have studied the properties of an isolated vortex line, as well as the interaction between such lines. This leads to possible collective vibrational modes (De Gennes and Matricon, 1 964), as well as to a surface barrier inhibiting the motion of lines into or out of the superconducting material (Bean and Livingston, 1963; [14], p. 111.85), De Gennes and Matricon (1964) have also suggested the possibility of investigating the vortex line structure of the mixed state by slow neutron diffraction. Prelimi- nary results have recently been reported (Cribier et al., 1 964). In an ideal type II superconductor, homogeneous and devoid of lattice imperfection, the vortex lines would be pushed out of the material by the Lorentz force if the specimen carried any current at right angles to the field (Gorter, 1962a, b). In any actual material, the motion of the lines is inhibited by defects and inhomogeneities which form potential barriers by which lines the are pinned. Anderson (1963) has investigated the thermally activated 'creep' of lines at low current densities, and has shown that on a local scale the density of lines tends to remain uniform, so that bundles of lines move together. This vortex or flux creep has been further discussed by Friedel et al. (1963). With increasing current densities to creep changes into a viscous flow of the lines, giving rise to resistive phenomena (Anderson and Kim, 1964). Extensive experimental work on this has been done by Kim era/. (1963, 1964). Tinkham (1963, 1964) has shown that a quantized vortex structure like that of the mixed state occurs even in pure films if they are very 74 Superconductivity thin and placed in a perpendicular external field. This is in agreement with magnetization measurements on such films by Chang et al. ( 1 963) and penetration depth and critical field data of Mercereau and Crane (1963). Guyon et al (1963) have investigated the dependence of the critical field on thickness. For thin films so narrow as to contain only a single row of vortices Parks and Mochel (1964) calculated that the free energy should have a minimum at values of the perpendicular external field at which the vortex diameter just equals the film width. At T c this should result in a corresponding minimum of the film resistance. They have observed such minima and take this as direct evidence for the existence of quantized vortices. 6.7. Surface Superconductivity As mentioned in Chapter V, the boundary condition applicable to the G-L order parameter W is that its derivative vanish. Saint-James and De Gennes (1963) have shown that in an external field parallel to the surface this leads to the persistence of an outer superconducting sheath up to a field H c3 = 1-695 H c2 The thickness of this sheath is of the order of £. Its existence, explicitly verified by many experiments (see, for example, Hempstead and Kim, 1963; Tomasch and Joseph, 1963), explains what had often been a puzzling discrepancy between magnetic and resistive transitions. The surface sheath exists also in Type I superconductors, but can be detected only if H c2 > H c . As H c2 = \/2kH c , it follows that H c3 = 2-40kH c , so that H c3 > H c for k > 0-42. Under this condition, a measurement of H c3 /H c is in fact a way of obtaining k for Type I materials (Strongin et al. 1964; Rosenblum and Cardona, 1964). The existence of the surface sheath in Type I superconductors means that if supercooling experiments are carried out on cylindrical samples in a longitudinal field, as is usually the case, the ideal lower limit for super cooling is H c2 rather than H c2 (cf. section 6.4). Thus the experimental value of S, should be set equal to 1-695V(2)k and the values of k thus calculated are therefore correspondingly reduced. CHAPTER VII The Low Frequency Magnetic Behaviour of Small Specimens 7.1. Increase in critical field When one of the dimensions of a superconducting specimen becomes comparable to the penetration depth, its critical magnetic field be- comes much higher than that of a bulk sample of the same material at the same temperature. This follows already from the basic Gorter- Casimir thermodynamic description, according to which the free energy difference per unit volume between the superconducting and normal phases is Hi G n (0)-GM = ■£• (VII.l) In an external field H e a superconductor acquires an effective mag- netization M(H e ) and becomes normal when lie M{H e )dH e = -*■ 077 (VII.2) The integral is the area under the magnetization curve, and it was pointed out in Chapter ni that for any ellipsoidal specimens VU.2 was satisfied when H e = H c . Actually this is true only when one neglects the penetration of the external field into the sample, which lowers the effective magnetization and the susceptibility of the sample, as shown by equations IV.2 and IV.3. The susceptibility determines the initial slope of the magnetization curve; a lower x means that the curve has to go to a higher critical field to satisfy equation VTI.2. Clearly, assuming this curve to remain linear with slope x right up to a critical field H s \ "I Hi Xo X (VII.3) 75 76 and Superconductivity 7T = v^3 — for a <^ Aq, CVII.4) (VII.5) using the London equation to evaluate a = $ in IV.3 (Ginzburg, 1 945 ; [11 p. 172). Similar expressions can be derived for spherical and cylindrical samples. The resulting equations agree well with the fre- quently observed enhancement of the critical field in small specimens when one uses for the penetration depth A the appropriate Pippard value as calculated from IV.20. This is a good example of how ex- pressions derived from the London model can be used with the modi- fied value of A (Tinkham, 1958). The field enhancement calculated from the Ginzburg-Landau theory leads to nearly identical results. The essential difference is that because of the additional terms V.6 in the G-L free energy of the superconductor, the penetration depth increases in the presence of an external field (see equation V.16), so that the critical field for small samples becomes even higher. For thin films of thickness 2a the critical field is si r la (VII.6) where /(*) is the same function of k which appears in equation V. 1 6, and is very small for small values of k. For very thin films, a < A , G-L find that which for very small k reduces to H s A ti c a (VII.8) This is the same expression which for thicker films gives the super- cooling field H c2 . Expressions similar to VII.6 and VII.8 have also been derived for spheres and wires (Silin, 1951; Ginzburg, 1958a; Low frequency magnetic behaviour of small specimens 11 Hauser and Helfand, 1962), and have been used by Lutes (1957) in the interpretation of his measurements of the critical field enhancement in tin whiskers. It is possible to relate the thin film critical field to the basic super- conducting parameters £ anti K of the bulk material. The penetration depth appearing in VII.8 should be given by 1V.20, in which £(/) is determined by IV. 13 with the film thickness taken as the effective mean free path (Tinkham, 1958). In the limit a < £ this yields (Alloy* (VII.8 ') (Douglass and Blumberg, 1962). The use of the thin film susceptibility as derived by Schrieffer (1957) with non-local electrodynamics leads to 20-40 per cent higher values of the numerical constant (Ferrell and Glick, 1962;Toxen, 1962). 7.2. High field threads and superconducting magnets The size effect on the critical field is particularly striking in experi- ments using extremely thin evaporated films. In their experiment on the Knight shift in tin, Androes and Knight (1961) used films of thick- ness a 100 A and found H c (0) « 25 kgauss. Ginsberg and Tinkham (1960) saw no effect on the superconducting properties of their 10-20 A lead film in a field of 8 kgauss. The equivalent of small superconducting specimens can exist also in bulk material. In an inhomogeneous specimen there will be local variation of the surface energy due to varying strain or to varying electronic mean free path. If locally the surface energy is sufficiently lower than the value elsewhere, it may be energetically favourable for this region to remain superconducting in the presence of an external field even when the surrounding material has become normal (Pippard, 1 955). Under these conditions one can thus have a situation quite analogous to that of small specimens: small superconducting regions exist in a matrix of normal material (Gorter, 1935; Mendels- sohn, 1935; Shaw and Mapother, 1960). If their dimensions are small compared to the penetration depth, the critical field of these regions will be correspondingly raised, and it is known (Faber and Pippard, 1955b; Cochran et al, 1958) that such regions can persist in 78 Superconductivity high fields. In many instances these regions are threads which can form continuous superconducting paths from one end of the speci- men to the other, resulting in a resistive transition much broader and extending to much higher fields than the magnetic one (Doidge, 1 956). The threads are, of course, likely to touch each other in many places, resulting in what Mendelssohn (1935) called a superconducting sponge. The multiple connectivity of such a structure generally leads to highly irreversible magnetic transition with almost total flux trap- ping. Bean (1962) has used a simplified model with which to calculate the magnetization curve of such a sponge. He has confirmed some features of this model with an artificial filamentary superconductor made by forcing mercury into the pores of Vycor glass (Bean et a/., 1962). The possible relevance of this to superconducting magnet wire will be discussed in Chapter XIII. 7.3. Variation of the order parameter and the energy gap with magnetic field From equations V.G-L1 and V.G-L2 one can also calculate the varia- tion of the order parameter *fi inside the thin films. For thicknesses 2a very small compared to the width of the transition layer Xq/k, or in the equivalent Pippard terms for 2a < g , if; can be considered con- stant, and one can take k K 0. This leads to (Ginzburg, 1958a) X ) 3oUc7\V H "6 L \"c/ OA 6J/ L \ A o/ 30 \H For very thin films VII.8 applies, so that (VII.9) <l<i For such films, therefore, i/j(H s ) = 0, which means that the transition into the normal state is of second order, without a latent heat and with a discontinuity only in the specific heat, and not in the entropy. There can be no supercooling, and therefore, no hysteresis. For thicker films and bulk samples the transition in an external field, as discussed in Chapter II, is always of first order. The critical thickness below which there is a second order transition is 2a = V(5)A , Low frequency magnetic behaviour of small specimens 79 which has been verified by Zavaritskii (1951, 1952). Note that as the penetration depth is inversely proportional to 0, A(//) for thin films is much larger than A even in fairly small fields (Douglass, 1961c). Douglass (1961a) has pointed out that because of the proportion- ality of the energy gap to «/r, as derived by Gorkov (1959, 1960), equations VII.9 and VII. 10 represent the field dependence of the energy gap in sufficiently thin films. Thus one can write e\H s ) 4 = for 2a < V(5)A . (VII. 11) For thicker films, VII.9 and VII. 10 do not apply, and G-Ll and G-L2 must be solved numerically. The resulting variation of the energy gap at H e = H s asa function of film thickness has been calcu- lated by Douglass (1961a). It is displayed by the curve in Figure 20. The points are gap values which Douglass (1961b) obtained from tun- nelling experiments (see Section 10.6). Similar results have been found by Giaever and Megerle (1961), also by means of the tunnel effect, as well as by Morris and Tinkham (1961) with thermal conductivity measurements. With H e « H s , the empirical variation of the energy gap with field closely agrees with the Ginzburg-Landau-Gorkov pre- dictions even at temperatures well below T c . In such high fields the 80 Superconductivity order parameter is then small enough to make tenable the basic G-L assumptions as well as Gorkov's identification of the energy gap with W, Tinkham (1962) has proposed ways in which the G-L equations can be extended to give agreement also with low field results over a wide range of temperature. The limitations of these equations in this region have been discussed by Meservey and Douglass (1964). Bardeen (1962) has calculated the critical field and critical current for thin films on the basis of the BCS theory. At higher temperatures his results generally confirm the predictions of the Ginzburg-Landau theory, including the vanishing of the energy gap and a resulting second-order transition at the critical field in sufficiently thin films. At much lower temperatures, however, below about TJ3, Bardeen finds that for any thickness the energy gap remains finite and the transition a first-order one. However, Maki (1963) as well as Nambu and Tuan (1963) predict that the phase transition should be of the second order at all temperatures. Merservey and Douglass (1964) verify this down to t = 0- 14. CHAPTER VIII The Isotope Effect 8.1. Discovery and theoretical considerations The various phenomenological treatments based on the empirical characteristics of a superconductor provide an astonishingly com- plete macroscopic description of the superconducting phase. How- ever, they do not give any clear indications as to the microscopic nature of the phenomenon. One of the first such clues arose through the simultaneous and inde- pendent discovery, in 1950, by Maxwell, and by Reynolds et al., that the critical temperature of mercury isotopes depends on the isotopic mass by the relation T c M a = constant, (vm.i) r- 4.00 Ave. mass no.: x 199.5 N 200.7 (nat) 202.0 203.3 J I I u J ! I I 4.20 Fig. 21 81 82 Superconductivity where Mis the isotopic mass and a m £. This is illustrated in Figure 21, showing the variation of threshold field near T c for different isotopes. The effect has since also been established in a number of other ele- ments. The following table contains the most reliable experimental values of the exponent a, together with quoted probable errors. Reference Olsen, 1963 Reynolds et al., 1951 Matthias et al., 1963 Hein and Gibson, 1964 Shaw etal., 1961 Hake et al., 1958 Maxwell and Strongin, 1964 Gcballee/o/., 1961 Finnemoreand Mapother, 1962 Maxwell, 1952a Serin et al., 1952 Lock et al., 1951 Maxwell, 1952b Alekseevskii, 1953 Geballeand Matthias, 1964 In all the non-transition metals, with the exception of molybdenum, the results are consistent with a = 1/2. However, small mass differ- ences and the possibility of impurity and strain effects limit the experi- mental reliability, as is made evident by the variations between differ- ent measurements on the same element. Thus one cannot rule out deviations from the ideal value of a = £ which may be as high as 20 per cent in some cases. In view of recent theoretical work to be dis- cussed in Chapter XI, it is significant that the trend of the published deviations from a = \ is toward lower values. The situation in the transition metals ruthenium and osmium, however, appears to be different. This will be further discussed in Section 1 1.5. The inference to be drawn from the dependence of T c on the isotopic mass is startling. A relation between the onset of superconductivity, which is quite certainly an electronic process, and the isotopic mass, which affects only the phonon spectrum of the lattice, must mean that superconductivity is very largely due to a strong interaction between the electrons and the lattice. Thus the discovery of the isotope effect Element a Cd 0-51 ±010 Hg 0-504 Mo 0-33 Os 0-21 Pb 0-461 ±0.025 0-501 ±001 3 Rh 0-4 Ru <01 <005 Sn 0-505 ±0019 0-46 ±002 0-462 ±001 4 Tl 0-50 ±005 0-62 ±01 Zn 0-5 The isotope effect 83 clearly pointed out the direction in which a microscopic explanation of the phenomenon had to be sought. In fact, Frohlich (1950) had independently suggested just such a mechanism without knowing of the experimental work. However, it took several more years until the subtle nature of the pertinent electron-lattice interaction was recog- nized and a valid microscopic theory began to be developed. 8.2. Precise threshold field measurements The variation of critical temperature with isotopic mass was estab- lished by measuring the critical field H c as a function of temperature, and then extrapolating this to zero field. Magnetic measurements of course make use of the perfect diamagnetism of a superconductor, and can be made in one of two ways : either the change in flux through the sample at the transition induces an e.m.f. in a pick-up coil which is connected to a suitable galvanometer, or the changing susceptibility of the sample is reflected in the change of the mutual inductance of coaxial coils of which the sample forms part of the core. Either of these methods can be applied with great accuracy in spite of simple apparatus, and has the further advantage of measuring a bulk property virtually unaffected by the possible presence of small regions with different superconducting characteristics. By providing a 84 Superconductivity misleading short-circuiting path, such minor flaws can lead to very erratic results when T c is measured by observing the variation of electrical resistance. The careful determination of critical field curves which arose as almost a by-product of the work on the isotope effect established a 4-LEAD Fig. 23 number of interesting characteristics. Figure 22 shows the variation of the reduced critical field h m H c /H as function of t 2 m T 2 /T 2 C for a number of tin isotopes measured by Serin et al. (1952). It is evident, as was indicated earlier, that equation I.2a is only an approximation] and that a better representation for h is a polynomial N W) = i - S of, (vni.2) n=2 The isotope effect 85 A polynomial which fits the data for all tin isotopes as found by Lock et al. (1951) to within one-half of a per cent is // = l-10720/ 2 -0-0944/ 4 + 0-3325/ 6 -01660/ 8 . (VIII.3) All measurements to date have indicated that to within the available precision all isotopes of a given element follow the same critical field polynomial. One also finds that H has the same mass dependence as T c . This means that the superconducting condensation energy Hq/Btt varies proportionally to the isotopic mass, and also that, as shown by equations 11.15 or 11.16, the value of y is independent of isotopic mass. It has also been found that in going from one element to another, the reduced threshold field curves show small but definite variations. For all elements there are deviations from a strictly parabolic varia- tion, generally by a similar small amount in one direction, but in the case of lead and mercury by an amount in the opposite direction. Figure 23 shows these deviations as a function of reduced tempera- ture. It is important to emphasize the smallness of these deviations. 86 Superconductivity so as not to allow them to obscure the basic similarity of the super- conducting behaviour of all elements in terms of reduced co-ordinates. This not only sanctions the continuing discussion of superconduc- tivity in general terms with only occasional references to specific elements, but also allows one to look for a microscopic explanation of superconductivity, which in first approximation need not concern itself with the distinctive characteristics of individual elements, but only takes account of general and common features. The deviations of the measured threshold fields from a simple parabolic variation must be, according to the thermodynamic treatment developed in Chapter n, correlated with the empirical deviations of the specific heat from the corresponding change as the cube of the temperature Serin (1955) showed this strikingly by plotting both these deviations on the same graph, using the best available data for tin. This is shown in Figure 24. Mapother (1959) has since established the correlation between the experimental non-parabolic threshold fields and the exponential variation of the specific heat. CHAPTER IX Thermal Conductivity 9.1. Low temperature thermal conductivity In normal metals, heat is carried both by the conduction electrons and by the quantized lattice vibrations, the phonons. The total thermal conductivity consists of the sum of these two contributions : Km — k en -r k B (IX.1) where e and g denote the electrons and the lattice, respectively. The electronic conductivity is limited by two scattering mechanisms : the phonons and the lattice imperfections, and one can write at T< ©: \\k cn = aT 2 + Po /LT. (1X.2) The first of the terms on the right gives the resistivity due to the electron scattering by phonons, and predominates at higher tem- peratures; the second that due to scattering by imperfections, which becomes important below the temperature at which k en has a maximum: TL* = Po/2aL. (IX.3) In these equations p is the residual electrical resistivity, L the Lorentz number (2-44 xlO -8 watt-ohm/deg 2 ) and a is a constant of the material which is inversely proportional to @ 2 . Note that for a given material the addition of impurities increases p and thus raises T max . In pure metals and dilute alloys, k en > k g „; it is only in metals con- taining as much as several per cent impurities that the two contri- butions are of the same order of magnitude. The two-fluid model allows one to predict qualitatively what hap- pens to the thermal conductivity of a metal when it becomes super- conducting (see Mendelssohn, 1955; Klemens, 1956). The condensed 'superconducting' electrons cannot carry thermal energy nor can they scatter phonons. With decreasing temperature their number increases, and that of the 'normal' electrons correspondingly 7 87 88 Superconductivity decreases, which will result in a rapid decrease of the electronic heat conduction. At the same time the conduction by phonons will be enhanced, as these are no longer scattered as much by electrons. In pure specimens, the decrease in k es will usually exceed any gain in k gs , and the total conductivity in the superconducting phase will then be much smaller than in the normal phase. This is illustrated, for example, by the results of Hulm (1950) on pure Hg shown in Fig. 25 Figure 25. There exist, however, pure materials in which the normal conductivity is not very high but which are very free of grain boun- daries and other lattice defects. In the superconducting phase of such substances at very low reduced temperatures the phonons are then hardly scattered by anything except the specimen boundaries, result- ing in a large value of k gs . This has been observed, for example, by Calverley et al. (1961) in tantalum and niobium. Suppressing the electronic conduction in the normal phase by adding impurities decreases the effect of condensing electrons out of Thermal conductivity 89 the thermal circuit. For moderately impure specimens the super- conducting conductivity will then not be very different from the corresponding normal one. This is shown, for example, by the results of Hulm (1950) on a Hg-In alloy, also displayed in Figure 25. The results of Lindenfeld (1 96 1 ) on lead alloys shown in Figure 26 indicates what happens with increasing inpurity content: as the phonon con- tribution to the normal conductivity becomes more appreciable, the 0.30 K (watt/cm-deg) Pb + 6%Bi •— Pb+3*h -Pb+6%ln 0.20- 0.10- gain in k^ increasingly outweighs the decrease in k es , and the conduc- tivity in the superconducting phase becomes much larger than that in the normal one. 9.2. Electronic conduction If the effect on thermal conductivity by the superconducting transition is indeed due to the disappearance of electrons from the conduction process, then one should be able to write IX.2 for a superconductor as !/*„ = x(ir)aT 2 +y(^)p Q ILT, (IX.4) 90 Superconductivity where x(if) and y(ir) are functions only of the order parameter or which indicates the fraction of condensed electrons. Equation 11.25 shows that IT is a function only of / = T/T c , so that one can write instead Uk„ = aT 2 /g(t) + Po LT/f(t). (IX.4) The equation has been written in this form to agree with the nomen- clature introduced by Hulm (1950). He pointed out that if one chooses a sample in which the electronic heat conduction is predominantly limited by one or the other of the two scattering mechanisms, the measured ratio k e Jk en then equals the appropriate ratio function g(t) or /(r). To a first approximation, at least, these functions should be universal functions for all superconductors and be related to the microscopic nature of the phenomenon. For a specimen for which T max < T c , as is the case for reasonably pure Hg and Pb, and for extremely pure Sn and In, the heat conduc- tion just below T c is by electrons limited by phonon scattering. For such samples KJKn * sit). (IX.5) All pertinent measurements show the same qualitative features: g(t) at / = 1 breaks away sharply from unity with a discontinuous slope, and decreases as a power of / which is about 2 for Sn and In (Jones and Toxen, 1960: Guenault, 1960), and 4 to 5 for Pb and Hg (Watson and Graham, 1963; see also Klemens, 1956). Calculations by Kada- noff and Martin (1961), by Kresin (1959) and by Tewordt (1962, 1963a) appear to explain the experimental results for Sn and In, but not for Hg and Pb. For specimens for which T max ^ T c , the electronic conduction in the superconducting phase is at all temperatures limited by impurity scattering, so that for these *«/*«, * /CO. (IX.6) Several investigations (see Klemens, 1956) have shown that at /= 1 /(0 approaches unity smoothly with a continuous slope, and that at lower temperatures it decreases more slowly than git). The results are in reasonable agreement with expressions for /(/) derived by Bardeen etal. (BRT, 1959) and by Geilikman and Kresin (1959) on Thermal conductivity 91 the basis of the BCS theory. The gradual change from a phonon- scattered to an impurity-scattered electronic conduction in the same material of increasing impurity is particularly well illustrated by the recent results of Guenault (1960) on a series of monocrystalline tin specimens. When thermal conductivity measurements on superconductors are extended to small values of t, as was first done by Heer and Daunt (1949) and later by Goodman (1953), /(/) is found to decrease very rapidly. Goodman pointed out that this could be represented by an equation of the form At) = aexp(-*/0, (IX.7) and suggested that this implied the existence of an energy gap between the ground state and the lowest excited state available to the assembly of superconducting electrons. This conclusion can be inferred from thermal conductivity results in the following manner. Simple transport theory shows that k e = (1/3) lv C e , (IX.8) where / is the mean free path, v tne average velocity, and C c the specific heat of the electrons. Assuming that v , the Fermi velocity in the normal metal, remains the same for the uncondensed 'normal' electrons in the superconducting phase, and that in both phases the mean free paths (which may differ in magnitude) vary only slowly with temperature, then the temperature variation of k e Jk e „ must be due entirely to that of the specific heats. In other words f(t) * kjk„ M CJC e (IX.9) C en is known to vary linearly with temperature, so that IX.7 implies that C es = a'T c texpi-b/t). (IX.10) That such a temperature variation of the specific heat corresponds to an energy gap in the electronic spectrum can be shown as follows: If a gap of width 2e lies below the lowest available excited state, the number of thermally excited electrons will be proportional to expi-2e/2k B T), where k B is the Boltzmann constant, and the factor 2 arises because every excitation creates two independent particles, 92 Superconductivity an electron and a hole. Thus the free energy of the superconducting phase is equal to the condensation energy per particle multiplied by the exponential factor, which remains unchanged, through two dif- ferentiations with respect to temperature, to appear in the specific heat. The parameter b in IX. 10 is thus seen to equal 2e/2k B T c . 1.0 1 - .6 K en Aluminium (.Zavaritskii) .4- {Aluminium(Satterthwaite) ITheory (BRT) .4 t Fig. 27 .6 .8 1.0 According to the microscopic theory to be discussed in Chapter XI, the energy gap is a function of temperature. The parameter b can therefore be written as b = <T) *(0) x e(D k B T c k B T c " € (0) where € (0) is the gap value at 0°K. The detailed dependence of kjk en on b has been calculated by BRT, and the function e(DMO), calculated from the BCS theory, has been tabulated by Miihlschlegei (1959). Measurements oikJk e „ can thus be used to infer the value of <0)lk B T c . The appropriate temperature dependence of kjk en has been observed in a number of metals. The results for aluminium by Satterthwaite (1960) and by Zavaritskii (1958a) are shown in Thermal conductivity 93 Figure 27, together with a theoretical curve calculated with a gap equal to 3-50 k B T c . The agreement is somewhat deceptive, since there is good evidence that the gap width for aluminium is only 3 40 k B T c . From an observed anisotropy in the temperature dependence of k„ at very low temperatures Zavaritskii (1959, 1960a, b) has been able to infer a corresponding anisotropy in the width of the energy gap in the spectrum of the superconducting electrons in the case of cadmium, tin, gallium, and zinc. To the last he could apply theoretical expres- sions due to Khalatnikov (1959), from which he deduced a gap aniso- tropy of about 30 per cent. A similar result holds for cadmium. The measurements of Zavaritskii also show that the gap anisotropy can have different forms : in the case of gallium the value of the gap can be approximated by an ellipsoid compressed along the axis of rota- tion; for zinc and cadmium this ellipsoid is stretched out along the axis of rotation. In cases where the energy gap is a function of the magnetic field, measurements of k c Jk cn can be used to infer this field dependence. This technique has been used by Morris and Tinkham (1961) for thin films (see Section 7.3), and by Dubeck et al. (1962, 1964) for type II superconductors in the mixed state (see Section 6.6). 9.3. Lattice conduction Far below T c the fraction of 'normal' electrons becomes so small as to make k cs <^ k gs . At the very lowest temperatures, the phonons are primarily scattered by crystal boundaries in a manner which is the same in the superconducting as in the normal phase. The charac- teristic T 3 dependence in this limit (Casimir, 1938) has been well established experimentally (Mendelssohn and Renton, 1 955 ; Graham, 1958). In the normal state there occurs at these temperatures still appre- ciable heat conduction by electrons, limited only by impurity scat- tering and varying linearly with temperature (see equation IX.2). Thus in this range kjk s = aT-\ (IX.11) where a is a constant of the material which can have values as high as several hundred. For example, a suitable lead wire can have 94 Superconductivity k n /k s « 10 5 at 0-1 °K. A number of authors (see Mendelssohn, 1955) suggested using such wires in ultra-low temperature experiments as thermal switches which would be 'open', i.e. non-conducting, in the superconducting phase, and 'closed' when the superconductivity is quenched by means of a suitable magnetic field. Such heat switches are now widely used (see, for instance, Reese and Steyert, 1962). At somewhat higher temperatures, at which the phonons begin to be scattered by the 'normal' electrons even in the superconducting phase, there is necessarily a concurrent rise of the electronic con- duction. Experimentally it is very difficult to separate the conduction mechanisms. Where this has been possible (Conolly and Mendels- sohn, 1962; Lindenfeld and Rohrer, 1963) the results have been consistent with the pertinent calculations by BRT and by Geilikman and Kresin (1958, 1959). 9.4. The thermal conductivity in the intermediate state A number of experiments, in particular those of Mendelssohn and co- workers (Mendelssohn and Pontius, 1937; Mendelssohn and Olsen, 1950; Mendelssohn and Shiftman, 1 959), have shown that the thermal conductivity of a superconductor in the intermediate state generally does not change linearly from its value in the one phase to its value in the other when at a given temperature the external field is varied. Instead there appears an extra thermal resistance, which in some cases can be very large, and which is attributed to the scattering of the predominant heat carriers (electrons or phonons) at the boundaries between the superconducting and normal laminae which make up the intermediate state. For materials in which phonon conduction domi- nates this has been analyzed by Cornish and Olsen (1953) and by Laredo and Pippard (1955). Strassler and Wyder (1963) have devel- oped a treatment for very pure specimens in which the conduction is mostly by electrons. Experiments on the thermal conductivity in the intermediate state thus yield strong confirmation that the laminar structure, observed by various techniques at the surface of a specimen, actually persists throughout a bulk sample. CHAPTER X The Energy Gap 10.1. Introduction Ever since the initial discovery of superconductivity it had been known but barely noted that the striking electromagnetic behaviour of a superconductor at low frequencies is not accompanied by any corre- sponding changes in its optical properties: there is no visible change at T c , although the reflectivity of a metal at any frequency is related to its conductivity at that frequency. Thus at the very high optical frequencies the resistance of a superconductor is a constant, inde- pendent of temperature, and equal to that of the normal metal. At about the time of the discovery of the isotope effect steadily im- proving high frequency techniques had shown that atO°K the normal resistance persisted down to frequencies of the order of 10 13 c/sec, but that it remained zero up to frequencies of the order 10 10 c/sec. In 1952 already Shoenberg ([1], p. 202) concluded from this that at some fre- quency between these two limits '.. . quantum processes set in which could raise electrons from the condensed to the uncondensed state and thus cause energy absorption'. As shown in the previous chapter, Goodman (1953) very shortly after this inferred from his thermal conductivity results the existence of an energy gap in the single electron energy spectrum. A similar conclusion had been deduced a few years earlier by Daunt and Mendelssohn (1946) from the absence of any Thomson heat in the superconducting state. This indicated to them that the supercon- ducting electrons remain effectively at 0°K up to T= T c by being in low-lying energy states separated from all excited states by an energy gap of the order of k B T c . In the years which followed, the existence of such a gap was firmly established by a large number of experiments, and this, together with the electron-phonon interaction indicated by the isotope effect, pro- vided the keystones of a microscopic theory. This chapter will 95 96 Superconductivity describe a few experiments which indicate the energy gap most clearly and directly. The subject has been reviewed by Biondi et al. (1958) and recently by Douglass and Falicov (1964). 10.2. The specific heat After the resurgence of interest in specific heat measurements as a result of the suggestive results of precise threshold field measurements, of Goodman's thermal conductivity results, and of the first clear experimental verification of a deviation from a T 3 law by Brown et al (1953) on niobium, there have been in recent years a number of measurements which clearly indicate the exponential variation of C„ corresponding to an energy gap. The first of these were the results of Corak et al. (1954) on vanadium and by Corak and Satterthwaite (1 954) on tin ; and since then the exponential variation of C es has been established in a number of elements. The appropriate column in Table III lists the energy gap values of these elements deduced from the specific heat measurements. Note that in units of k B T c these gaps are of very similar size for widely varying superconductors. This again bears out the basic similarity of all superconductors in terms of re- duced co-ordinates. It is perhaps useful to consider briefly the difficulty of obtaining good values for C es . What is measured, of course, in both the super- conducting and in the normal phase, is the total specific heat. It is then necessary to separate the electronic from the lattice contribution in the normal phase in order to be able to subtract the latter from the total specific heat in the superconducting phase. Unfortunately, even at low temperatures, C ga is small compared to C„ only for metals with large Debye temperatures. These are just the hard, high-melting point metals which are difficult to obtain with high purity, without which superconducting measurements are misleading. The softer and lower melting point metals, on the other hand, have a very unfavour- able ratio of electronic to lattice specific heat. Measurements by Goodman (1957, 1958), Zavaritskii (1958b), and Phillips (1959) on aluminium have shown at very low temperatures (/ < 0-2) a deviation of C es from a simple exponential law (Boorse, 1959). Cooper (1959) has pointed out that this can be a consequence of anisotropy in the energy gap. At the lowest temperatures most The energy gap 97 electron excitations would be expected to occur across the narrower portions of the gap, and this would be reflected in an upward curva- ture of C es when plotted semi-logarithmically against 1/r. Figure 28 compares a number of measurements which show this curvature with BCS . 3CWT) 3 Fig. 28 the exponential law expected from the BCS theory. According to a theory of Anderson (1 959) (see Section 12.2) the gap anisotropy of an element diminishes with the addition of impurities. Indeed Geiser and Goodman (1963) have found in aluminium specimens of different purity that the deviation of C cs from an exponential form decreases with increasing impurity. 98 Superconductivity 10.3. Electromagnetic absorption in the far infrared The magnitude of the energy gap 2e(0) can be characterized by a fre- quency i/ ? such that hv ? = 2e(0). It is at this frequency that one would expect the change from the characteristically superconducting re- sponse to low frequency radiation, to the normal resistance main- The energy gap 99 The measurements of the transmission of such radiation through superconducting films will be discussed in a later section. Richards and Tinkham (1960), Richards (1 961), and Ginsberg and Leslie (1962) have observed directly the absorption edge at the gap frequency in bulk superconductors. Radiation from a quartz mercury arc infrared J-VL 10 15 20 25 30 35 40 45 50 FREQUENCY (cm -1 ) Fig. 29 tained at high frequencies. Unfortunately, the frequencies corre- sponding to gap widths inferred from the specific heat measurements are 10 u -10 12 c-sec~ \ which is an experimentally awkward range at the upper limit of klystron-excited frequencies, yet very low for mercury arc ones. Only recently have Tinkham and collaborators developed the techniques needed to detect the very low radiation intensities available in this far infrared region. Table III Element Energy gap (2<Q)lk B T c ) A B C D E F Aluminium 316 2-9 3-37 b , 3-43 c 3-3 d 3-5 Cadmium ... ... 3-3 Gallium ... ... ... 3-5 Indium 4-1 3-9 3-9 3-63 a , 3-45 b 3-5 Lanthanum 2-85 ... 3-7 Lead 414 4-0 ... 4-33 a , 4-26 b 418 d 3-9 Mercury 4-6 ... ... ... 3-7 Niobium 2-8 ... 4-4 3-84 e ,3-6*, 3-59 8 3-7 Rhenium ... 3-3 Ruthenium 31 Tantalum < 30 3-6 3-60°, 3-5', 3-65 u 3-6 Thallium 3-2 2-8 Thorium ... 3-5 Tin 3-6 3-3 3-5 3-6 3-46 a , 3-47 b 3-65 d 3-6 Vanadium 3-4 ... 3-6 3-4' 3-6 Zinc ... ... ... 2-5 ... 3-4 A — from infrared absorption (lead: Ginsberg and Leslie, 1962; lanthanum: Leslie et al., 1964; all others Richards and Tink- ham, 1960). B — from infrared transmission (Ginsberg and Tinkham, 1960). C — from microwave absorption (aluminium: Biondi and Garfunkel, 1959; tin: Biondi et ai., 1957). D — by fitting specific heat data to exponential (Goodman, 1 959). E — from tunneling ("Giaevcr and Megerle, 1961; b Zavaritskii, 1961; c Douglass, 1962; d Douglass and Merservey, 1964; Townsend and Sutton. 1962; 'Giaever, 1962; "Sherrill and Edwards, 1962; "Dietrich, 1962). F— calculated from XI.32 (Goodman, 1959). 100 Superconductivity monochromator was fed by means of a light pipe into a cavity made of the superconducting material under investigation. The cavity con- tained a carbon resistance bolometer, and was shaped so that the incident radiation would make many reflections before striking this detector. For frequencies lower than v g , the superconducting walls of the cavity do not absorb, and much radiation reaches the bolometer. At v g , absorption by the walls sets in, and the signal from the bolo- meter decreases sharply. Figure 29 shows normalized curves of the fractional change in the power absorbed by the bolometer, in arbi- trary units, plotted against frequency for all the metals investigated by Tinkham and Richards. The gap values obtained are listed in Table III. The absorption edges for Pb and Hg show a certain struc- ture, which has also been found in the same elements in infrared transmission measurements (Ginsberg et al., 1959). Ginsberg and Leslie (1962) have shown that this structure persists even in a lead alloy containing 10 atomic per cent of thallium, so that it is probably not due to gap anisotropy. The effect may be due to states of collective excitations lying in the gap (Tsuneto, 1960) which have not been taken into account in the BCS theory. However, calculations of Maki and Tsuneto (1962) lead one to expect that the energy of collective excita- tions should be drastically shifted by impurity scattering. Richards (1961) has reported measurements on single crystals of pure tin and of tin containing 01 atomic per cent indium. His results show that the position of the absorption edge varies with crystal orientation, which clearly indicates the anisotropy of the gap. Further- more this anisotropy decreases with increasing impurity, which strongly supports Anderson's suggestion (1959) that the anisotropy becomes smoothed out in impure samples. The absorption edges observed by Richards have a structure which, unlike that seen in Pb and in Hg, occurs for frequencies greater than v g . These postcursor peaks do not seem to change with impurity, and have not yet found an explanation. 10.4. Microwave absorption Although the resistivity of a superconductor vanishes at 0°K for fre- quencies up to v g , there is a finite resistance at higher temperatures even at lower frequencies (H. London, 1940). One can understand The energy gap 101 this from a simple two-fluid picture, according to which at any finite temperature a fraction of the electrons remains ' normal'. H. London pointed out that in the presence of an alternating electric field these electrons absorb energy as they would in a normal metal, and that such a field is needed to sustain an alternating current even in a super- conductor because of the inertia of the superconducting electrons. Into a normal metal an alternating field penetrates to a skin depth 8, which leads to anomalous results if the mean free path l> 8, as is the case at high frequencies and low temperatures (see p. 42). In the superconducting phase, the theory of the anomalous skin effect still applies in principle, but has to be modified both because for high fre- quencies the superconducting penetration depth A is much smaller than the skin depth 8 (except very near T c ) and decreases very rapidly with decreasing temperature, and because the number of 'normal' electrons also drops sharply below T c . Both of these lead to a reduc- tion of the resistance in the superconducting phase as compared to that in the normal one : the ratio of the resistances decreases rapidly below T c , changes more gradually at lower temperatures where both A and the order parameter are fairly constant, and finally vanishes at 0°K where there are no more 'normal' electrons. Unpublished calculations of the variation of RJR„ with tempera- ture and with frequency have been made by Serber and by Holstein on the basis of the Reuter-Sondheimer equations, the London theory, and the two-fluid model. Typical results are the solid curve labelled 0-65Ar B r c and the dashed one labelled 2-37 k B T c in Figure 30. With frequencies up to 8 x 10 10 c/sec there is general experimental agree- ment with these calculations, as shown, for example, by the recent results of Khaikin (1958) on cadmium and of Kaplan et al. (1959) on tin. Their temperature dependence for a given frequency can be represented by an empirical function, suggested by Pippard (1948): #0 = / 4 (l-/ 2 )0-/ 4 r 2 . (X.1) The frequency dependence is as v 4/3 at low frequencies, tending to- ward a constant value at higher frequencies. However, surface impedance measurements at frequencies con- siderably higher than 8 x 10 10 c/sec show appreciable deviations from the predictions of the simple two-fluid model. Figure 30 shows the Superconductivity 102 ratio RJR n as a function of reduced temperature for aluminium as measured at three microwave frequencies by Biondie/A/ (1957) The frequences are given in units of k B TJh. For 0-65, the results agree well with the temperature variation calculated without regard to an energy gap. For 2-37, however, such calculations would give the dashed curve, and it is evident that for / > 0-7, the measured ratio considerably exceeds the predicted one. The same is true for hv - 304k B T c , except that in this case the deviation already begins at O.^r^k ' ' ' J ' 1 t I 0.6 07 0.8 0.9 1.0 U Fio. 30 Clearly an additional absorption mechanism occurs for these fre- quencies, and of course this is due to the boosting of condensed elec- trons across the energy gap. If this gap had a constant width at all t< 1 the appearance of this extra absorption would depend only on the frequency. Its temperature dependence, however, clearly shows that the energy gap varies with temperature, tending toward zero as /-M. As a result, photons of energy 2-37k B T c , for example, are not sufficient to bridge the gap at / = 0, but become effective at that tem- perature at which the gap has shrunk to a width of 2-37 k B T A series of measurements of the resistance ratio as a function both of frequency and of temperature thus serves to map out the temperature variation no™? u aP ° f 3ny g,Ven ^Perconductor. Biondi and Garfunkel (1959) have obtained values of the resistance ratio by measuring The energy gap 103 calorimetrically the amount of energy absorbed by an aluminium wave guide, over a range of frequencies ranging from 0-65k B T c (1-5 x 10 10 c/sec) to 3-91^7^ (10 x 10 10 c/sec) at temperatures down 1.0 1.5 2.0 2.5 3.0 3.5 Energy (in units of kT c ) Fig. 31 4.0 to 0-35°K. The accuracy of the measurements was such that the ab- sorption of 10 " 9 watt could be detected. Their results give a tempera- ture variation of the gap which is in close agreement with the predic- tions of the BCS theory. Mattis and Bardeen (1958) and Abrikosov et ul. (1958) have de- veloped a theory of the anomalous skin effect in superconductors on the basis of the BCS theory. Miller (1960) used the work of the former to calculate the surface impedance for many different frequencies and temperatures. The close agreement between his results and the 8 104 Superconductivity measurements of Biondi and Garfunkel is shown in Figure 31, in which points calculated by Miller are superimposed on smooth curves representing the empirical values. The theoretical treatments are equally successful in the lower frequency range in which there are no gap effects. 10.5. Nuclear spin relaxation When the nuclear spins of a substance are aligned by the application of an external field, they again relax to their equilibrium distribution predominantly by interaction with the conduction electrons. In this interaction, a nucleus flips its spin one way as the electron spin flips the other way so as to conserve the total spin. The electron can do this only if there is available an empty final state of correct energy and spin direction, and the nuclear relaxation rate in a normal metal depends therefore both on the number of conduction electrons (itself proportional to the product of the density of states and the energy derivative of the Fermi function) and on the density of states in the vicinity of the Fermi surface. To predict the temperature variation of this relaxation process in the superconducting phase one is tempted to use again a simple two- fluid model, according to which the number of 'normal' electrons available decreases rapidly below T c . To this should, therefore, corre- spond a decrease in the relaxation rate as compared to that in the normal phase. But the energy gap severely modifies the density of states available to the interacting electrons. In the gap there are, by definition, no available states at all, and the missing states are 'piled up ' on either side. The presence of the energy derivative of the Fermi function in the relaxation rate expression makes this rate essentially proportional to the square of the density of states evaluated over a range k B T on either side of the Fermi energy. At temperatures near 7;., the gap is still very narrow and < k B T, so that the pile-up of states on either side results in an appreciable increase of the rate over that in the normal phase. At lower temperatures, the gap becomes wider than k B T, and the relaxation rate rapidly diminishes, approach- ing zero as T-*-0. The measurements of Hebel and Slichter (1959), of Redfield (1959).. and of Masuda and Redfield (1960a) fully confirm this consequence The energy gap 105 of the energy gap. In particular, a detailed analysis by Hebel (1959) has shown that the empirical results are compatible with the manner of piling up predicted by the microscopic theory. Hebel's results and some empirical values are given in Figure 32, in which the ratio of the relaxation rate in the superconducting phase to that in the normal one is plotted against temperature. The temperature variation of what can be considered as the attenuation of the nuclear alignment is markedly different from the corresponding change in the attenuation of an '0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 t Fig. 32 ultrasonic elastic wave in a superconductor. As this difference is one of the most striking consequences of the BCS theory, its discussion and the general description of ultrasonic attenuation in supercon- ductors is postponed until a later chapter. In his calculations, Hebel avoids singularities on either side of the gap by introducing a parameter r which represents a smearing of the density of states over an energy interval small compared to the width of the gap. It is possible to interpret this in terms of an anisotropy of the gap, since the relaxation process samples the gap over all direc- tions simultaneously. With this interpretation, the data on aluminium 106 Superconductivity of Masuda and Redfield (1960a, 1962) indicate an anisotropy of the order of 1/10 of the gap width, and recent measurement by the same authors (Masuda and Redfield, 1960b; Masuda, 1962b) indicate that this anisotropy decreases in impure aluminium. Anisotropy of mag- nitude similar to that in aluminium has been found by Masuda (1962a) in cadmium. 10.6. The tunnel effect The most recent and the most direct measurement of the energy gap has been provided by the work of Giaever (1960a), who essentially superconductor normal metal insulator Fig. 33 measured the width of the gap with a voltmeter. He accomplished this by observing the tunneling of electrons between a superconducting film and a normal one across a thin insulating barrier. Quantum- mechanically, an electron on one side of such a barrier has a finite probability of tunneling through it if there is an allowed state of equal or smaller energy available for it on the other side. Figure 33 shows the density of states function in energy space for a sandwich consist- ing, from left to right, of a superconductor, an insulator, and a normal The energy gap 107 metal, all at 0°K. In the last of these, electrons fill all available states up to the Fermi level E F ; in the superconductor, there is a gap of half- width e(0), and states up to E F — e(0) are filled. With such conditions there can be no tunneling either way, as on neither side of the barrier are there any available states. A potential difference applied between the two metals will shift the energy levels of one with respect to the other. It is evident from Figure 33 that tunneling will abruptly become possible when the applied voltage equals e(0). The subsequent variation of tunneling current with applied voltage of course depends on the details of the density of states curve of the superconductor on either side of the gap. At first, there is a very rapid rise of current with voltage due to the €(0) Voltage Fig. 34 large density of piled-up stages; for voltages much exceeding e(0), the tunnelling samples the density of states well beyond the gap, and the variation of I vs. V approaches the purely ohmic character of a junc- tion of two normal metals. This is summarized in Figure 34, which gives with the solid line the current- voltage characteristic of the superconducting-normal junction at 0°K. The dotted line indicates the behaviour at < T< T c , the modification being due to the fact that at finite temperatures on both sides of the junction some electrons are excited across the gap or the Fermi level, respectively. The dashed line shows the behaviour at T > T c , i.e. for a junction of normal metals. Nicol et al. (1960) and Giaever (1960b) have extended such experi- ments to cases where both metals of the junction are superconductors, but with very different critical temperatures, such as Al (T c = 1-2°K) and Pb (T c = 7-2°K). The gaps of the two will be correspondingly '08 Superconductivity different, and for such a junction the density of states function at 0°K is shown in Figure 35. A tunneling current will begin to flow when the potential difference between the two metals is €(0) Pb +e(0) AI . In this case, however, the modification due to finite temperature is more significant than with an s-n junction. Imagining the density of states curve of Figure 35 with a few excited electrons beyond both gaps, and a few available states remaining below both, one recognizes that now superconductor 1 superconductor insulator Fig. 35 the current / at first increases with increasing potential V, then de- creases for e(0) Pb -e(0) A1 < K<€(0) Pb + € (0) A1 , and then increases again. Figure 36 shows the current-voltage characteristics in this case; the limits of the negative resistance region are very sharp. Thus the current-potential characteristics yield the energy gap values at a given temperature for both metals. The energy gap values obtained by this method for several super- conductors are listed in Table III, and can probably be considered as the most reliable of all experimental determinations. Measurements as a function of temperature closely support the thermal variation of the energy gap predicted by the BSC theory. The films used are thin The energy gap 109 compared to the penetration depths, and because of their size their critical fields are very high. This and its use in investigating the varia- tion of the energy gap with magnetic field was discussed in Chapter VI I. Recent tunneling studies have verified other aspects of the energy gap, in particular its relationship to the phonon spectrum of the superconducting lattice. This will be summarized in Chapter XI. Simultaneous tunneling of two electrons has been observed by Taylor and Burstein (1962), in agreement with the calculations of 6,-e 2 e,+e 2 Voltage Fig. 36 Schrieffer and Wilkins (1962). This is not to be confused with the tunneling of Cooper pairs, as predicted by Josephson (1962), which will be discussed in Section 1 1.7. The results of Taylor and Burstein also indicate the possibility of tunneling assisted by the simultaneous absorption of a phonon. Theoretical aspects of this have been dis- cussed by Kleinman (1963) and Fibich (1964). 10.7. Far infrared transmission through thin films In a series of experiments, Tinkham, Glover, and Ginsberg (Glover and Tinkham, 1957; Ginsberg and Tinkham, 1960) have measured 110 Superconductivity the transmission through thin superconducting films of electromag- netic radiation in the far-infrared range of wavelengths between 0-1 and 6 mm. Their results lend themselves to an ingenious analysis leading to a number of very fundamental conclusions about the inter- relation of the energy gap, the response to high frequency radiation, Fig. 37 and the existence of perfect conductivity and of the Meissner effect in the limit of zero frequency (see Tinkham [10], pp. 168-176). In Figure 37, the curve labelled TJT n is one which can be drawn through the empirical values of the ratio of the transmissivity in the superconducting phase, T s , to the normal value, T„, all suitably normalized for film resistance and substrate refraction, and plotted against frequency. The transmissivity of a substance is related to its conductivity. One can approximate the conductivity of the film in the normal state by a real number, a n , which to a good approximation is The energy gap 1 1 1 independent of frequency in the range under investigation. The super- conducting conductivity can be written as the complex quantity It then follows from general electromagnetic theory that (X.2) £ = ( T^Hi-rt^) +[( I -^> ,/2 ^]~}~ • (X- 3 > Microwave work on bulk superconductors, such as the measurements of Biondi and Garfunkel (1959), have shown that at T< T c and ho < k B T c , the surface resistance vanishes. It follows from this that the real, lossy part of the conductivity must also vanish in this range, or a, « 0, so that the low frequency measurements of T s /T„ can be used to evaluate the corresponding values of a 2 /a„. For a number of samples of tin and lead with widely varying normal conductivity, all the data of Glover and Tinkham fit a universal curve represented by °il° n = (\la)(k B TJtiw), a = 0-27. As a n is independent of frequency, X.4 implies that a 2 cc 1/cu. (X.4) (X.5) This is just the frequency dependence which follows from the London equation curlJ + -_- 2 H = 0, (X.6) since this with Maxwell's equation curlE = -H c leads to c 2 1 a 2 = 4ttA 2 , (X.7) An imaginary conductivity which is inversely proportional to the frequency thus corresponds to the consequences of X.6 : the Meissner CHRIST'S COLLEGE I mm nw 112 Superconductivity effect and a finite penetration depth A. However, the magnitude of A calculated from the experimental transmission results with the aid of X.7 exceeds the London value \ L = mc 2 jATT 2 ne 2 by at least a factor often. Furthermore, there is nothing in the London theory to explain why a 2 /cr„ for different superconductors should satisfy a universal equation like X.4. On the other hand the Pippard treatment predicts for these films, in which £ as l< A, that (see equation IV. 18a) : where (equation IV.9) : Hence A 2 = (lo/OAl, £ = afiv /k B T c . CT 2 On ne 2 I 1 1 X— X— X — m £ to o„ For a normal metal mvQ and hence the Pippard theory leads to Z 2 O n i k B T t a htxi for all superconductors. This is strikingly verified by the results of Glover and Tinkham. The real and imaginary parts of any linear response function, such as the electrical conductivity, are related by a pair of integral trans- forms known as the Kramers-Kronig (K-K) relations. In terms of the conductivity these take the form: i<«) + 00 co x a 2 (co x )dco\ t0 2 — (0 2 + 00 o 2 (to) ■-. co 2 — to 2 (X.8) The energy gap 1 1 3 Substituting X.7 into the first of these two relations shows that the imaginary conductivity o 2 must be accompanied by a real conduc- tivity which takes the form of a delta-function at the origin: a, (to) = (c 2 /8A 2 ) 8(co - 0). (X.9) Similarly, in terms of the empirical value X.4 for o 2 /o„ one would have o x _ TT\ k B T r o~ 2a h S(o)-O). (X.10) Such an infinite real conductivity at zero frequency of course does not introduce losses. 6 8 10 12 -h<y/k B T c Fig. 38 Turning now to the high frequency far infra-red transmissivity data, the peak and subsequent decrease of T s /T„ indicates that at a fre- quency roughly corresponding to the peak, a real, lossy component ct, of the superconducting conductivity must appear. In the absence of such a component T s /T„ would continue to rise. The appearance of a real component of conductivity at or near some critical frequency is, of course, highly suggestive of an energy gap. Taken by themselves, the data of Tinkham et al. do not determine the gap quite unam- biguously (see Forrester, 1958). However, accepting the existence of a gap from other experiments allows a fully consistent interpretation of the transmission results from which the magnitude of the gap as well as other interesting quantities can be derived. 1 14 Superconductivity The calculations of Miller (1960) of the variation of a x \a n are shown in Figure 37, the ordinate being scaled in units of 2e(0) where 2c(0) is the width of the gap at 0°K. An energy gap implies that, as for a normal metal, the imaginary part of the conductivity vanishes for frequencies beyond the gap. Using X.3 one can then calculate cr,/<x n from the measured values of TJT„ to a first approxi- mation, and then apply an iterative procedure using the K-K relations as well as the sum rule about to be mentioned to obtain final values of ajcr,,. Figure 38 gives the result thus obtained by Ginsberg and Tinkham for lead, showing the precursor peak also found for mer- cury. One ignores this in deriving energy gap values from the limit <y i/ CT /.- > 0- The resulting gap widths are listed in Table III. 10.8. The Ferrell-Glover sum rule The intimate connection between the experimentally verified decrease of aj/a„ near co g , corresponding to the existence of a gap, and the low frequency London-type imaginary conductivity a \ <x 1/eo, corre- sponding to infinite conductivity and the Meissner effect at zero frequency, was first pointed out by Ferrell and Glover (1958) and further elaborated by Tinkham and Ferrell (1959). The first of these papers pointed out that at extremely high frequencies, such that hut far exceeds any of the binding energies of an electron in the metal, the real part of the conductivity vanishes. The appropriate K-K rela- tion for the imaginary conductivity then becomes, since a t is an even function, a 2 (o>) « — TTOi I 9f(a>i)dh>|. CX.11) At these very high frequencies all electrons are free in both the normal and the superconducting phases, and one would thus expect cr 2 (co) and, therefore, the integral in X.l 1 to have the same value in both phases. In other words, there exists the sum rule that this integral remains unchanged under the superconducting transition. From this The energy gap 1 1 5 it follows that any area A removed from under the o-,(o>) curve by the energy gap must reappear somewhere else, and it can do so only at the origin in the form of a delta function of strength A. This being the case, one can then again apply the K-K relations to show that associated with such a delta function a,(oi) = A8(oj-0) (X.12) must be a contribution to the imaginary conductivity of magnitude aiiai) = IA/tho. (X.13) The argument has now come through a full circle. An energy gap corresponds to a disappearance of a^w) in the superconducting phase over some frequency range in which this conductivity is finite in the normal metal. This, according to the Ferrel-Glover sum rule, must lead to the appearance of a delta function X.12. In turn this leads to a London-type imaginary conductivity ff 2 ccl/£o, which was seen to correspond to the Meissner effect and infinite conductivity. One sees further that in terms of the parameter a of X.4, one can write A\o n = (7Tl2)(k B T c lh)(Ma). (X.14) Determining A/a„ from their transmission data and using this rela- tion, Ginsberg and Tinkham obtain values for a of 0-23 for lead, 0-26 for tin, and 0- 19 for thallium. These, as well as Glover and Tinkham's value of 0-27 for both tin and lead, are in remarkable agreement both with the Faber-Pippard data (0-15 for tin and indium) and with the BCS prediction for all metals (0-18). The agreement is particularly convincing if one considers the simplifications of the theory on the one hand, and the wide variety and considerable difficulty of the experiments on the other. From X.13 and X.7 it is evident that A 2 = c 2 I8A. (X.15) Thus the Ferrell-Glover sum-rule leads to an inverse proportionality between the square of the penetration depth and the energy gap. Such a relation is implicit in the Pippard model and the Bardeen theory, and appears explicitly in the Ginzburg-Landau treatment as extended by Gor'kov. CHAPTER XI Microscopic Theory of Superconductivity 11.1. Introduction In reviewing the contents of the preceding chapters, which give an empirical description of superconductivity, perhaps the most striking feature to be noticed is how much quantitative information can be given about superconductivity in general without speaking about the specific properties of any one of the many superconducting elements. The astonishing degree of similarity in the superconducting behaviour of metals with widely varying crystallographic and atomic properties indicates that the explanation for superconductivity should be in- herent in a general, idealized model of a metal which ignores the com- plicated features characterizing any individual metallic element. It should, therefore, be possible to find in the simple model of the ideal metal the possibility of an interaction mechanism leading to the super- conducting state, and to derive from this at least qualitatively the properties of an ideal superconductor. One would judge from this that an explanation for superconduc- tivity should be fairly easy, until he realizes the extreme smallness of the energy involved. A superconductor can be made normal by the application of a magnetic field H c which at absolute zero is of the order of a few hundred gauss. The energy difference between the superconducting and the normal phase at absolute zero, which is given by Hq/Stt, thus is of the order of 10~ 8 e.v. per atom. How very small this is can best be judged by remembering that the Fermi energy of the conduction electrons in a normal metal is of the order of 10-20 e.v. The simple model of Bloch and Sommerfeld gives a reasonably accurate description of the basic characteristics of a metal although it completely ignores, among other things, the correlation energy of the conduction electrons due to their Coulomb interaction. This energy is of the order of 1 e.v. ! As a further difficulty in arriving at a microscopic theory of super- conductivity one must add the extreme sharpness of the phase 116 Microscopic theory of superconductivity 117 transition under suitable conditions. The absence of statistical fluctu- ations shows that the superconducting state is a highly correlated one involving a very large number of electrons. Thus it is necessary to find inherent in the basic properties common to all metals an interaction correlating a large number of electrons in such a way that the energy of the system relative to the normal metal is lowered by a very small amount. The discovery of the isotope effect in a number of super- conducting elements clearly indicated that in these the interaction in question must be one between the electrons and the vibrating crystal lattice, and indeed Frohlich (1950) had suggested such a mechanism independently of the simultaneous experimental results. 11.2. The electron -phonon interaction Frohlich and, a little later, Bardeen (1 950) pointed out that an electron moving through a crystal lattice has a self energy by being 'clothed' with virtual phonons. What this means is that an electron moving through the lattice distorts the lattice, and the lattice in turn acts on the electron by virtue of the electrostatic forces between them. The oscillatory distortion of the lattice is quantized in terms of phonons, and so one can think of the interaction between lattice and electron as the constant emission and reabsorption of phonons by the latter. These are called 'virtual' phonons because as a consequence of the uncertainty principle their very short lifetime renders it unnecessary to conserve energy in the process. Thus one can think of the electron moving through the lattice as being accompanied or 'clothed', even at 0°K, by a cloud of virtual phonons. This contributes to the electron an amount of self-energy which, as was pointed out by Frohlich and by Bardeen, is proportional to the square of the average phonon energy. In turn this is inversely proportional to the lattice mass, so that a condensation energy equal to this self-energy would have the correct mass dependence indicated by the isotope effect. Unfortu- nately, however, the size turns out to be three to four orders of magnitude too large. It was only seven years later that Bardeen, Cooper, and Schrieffer (BCS, 1957) succeeded in showing that the basic interaction respon- sible for superconductivity appears to be that of a pair of electrons by means of an interchange of virtual phonons. In the simple terms 118 Superconductivity used above this means that the lattice is distorted by a moving elec- tron, this distortion giving rise to a phonon. A second electron some distance away is in turn affected when it is reached by the propagating fluctuation in the lattice charge distribution. In other words, as shown in Figure 39, an electron of wave vector k emits a virtual phonon q which is absorbed by an electron k'. This scatters k into k — q and k' into k' + q. The process being a virtual one, energy need not be conserved, and in fact the nature of the resulting electron-electron interaction depends on the relative magnitudes of the electronic energy change and the phonon energy fico q . If this latter exceeds the Fig. 39 former, the interaction is attractive — the charge fluctuation of the lattice is then such as to surround one of the electrons by a positive screening charge greater than the electronic one, so that the second electron sees and is attracted by a net positive charge. The fundamental postulate of the BCS theory is that supercon- ductivity occurs when such an attractive interaction between two electrons by means of phonon exchange dominates the usual repulsive screened Coulomb interaction. 11.3. The Cooper pairs Shortly before the formulation of the BCS theory, Cooper (1956) had been able to show that if there is a net attraction, however weak, Microscopic theory of superconductivity 1 19 between a pair of electrons just above the Fermi surface, these elec- trons can form a bound state. The electrons for which this can occur as a result of the phonon interaction lie in a thin shell of width ^ hu) q , where hco q is of the order of the average phonon energy of the metal. If one looks at the matrix elements for all possible interactions which take a pair of electrons from any two k values in this shell to any two others, he finds that because of the Fermi statistics of the electron these matrix elements alternate in sign and, being all of roughly equal magnitude, give a negligible total interaction energy, that is, a vanishingly small total lowering of the energy relative to the normal situation of unpaired electrons. One can, however, restrict oneself to matrix elements of a single sign by associating all possible k values in pairs, kj and k 2 , and requiring that either both or neither member of a pair be occupied. As the lowest energy is obtained by having the largest number of possible transitions, each represented by a matrix element all of the same sign, one wants to choose these pairs in such a way that from any one set of values (kj, k^, transitions are possible into all other pairs (k|", k£. As momentum must be conserved, this means that one must require that k 1 + k 2 = k,' + k^ = K (XI.1) that is, that all bound pairs should have the same total momentum K. (See, for example, Cooper, 1960.) To find the possible value of kj and k 2 which satisfy XI. 1 and at the same time lie in a narrow shell straddling the Fermi surface k F one can construct the d iagram shown in Figure 40, d rawing concentric circles of radii k F - 8 and k F + 8 from two points separated by K. It is clear that all possible values of k, and k 2 satisfying XI. 1 are restricted to the two shaded regions. This shows that the volume of phase space available for what has become known as Cooper pairs has a very sharp maximum for K = 0. Thus the largest number of possible transitions yielding the most appreciable lowering of energy is obtained by pairing all possible states such that their total momentum vanishes. It is also possible to show that exchange terms tend to reduce the interaction energy for pairs of parallel spin, so that it is ener- getically most favourable to restrict the pairs to those of opposite spin. One can, therefore, summarize the basic hypothesis of the BCS 9 120 Superconductivity theory as follows: At 0°K the superconducting ground state is a highly correlated one in which in momentum space the normal electron states in a thin shell near the Fermi surface are to the fullest extent possible occupied by pairs of opposite spin and momentum. The most direct verification of the existence of these pairs arises from the flux quanti- zation measurements mentioned in Chapter III. The energy of this state is lower than that of the normal metal by a finite amount which is the condensation energy of the superconducting state and which at 0°K must equal Hi/Sir per unit volume. Further- more, this state has the all-important property that it takes a finite quantity of energy to excite even a single ' normal ', unpaired electron. For not only does this require the very small amount of energy needed Fig. 40 to break up a bound pair, but more importantly the occupation of a single k state by an unpaired electron removes from the system a large number of pairs which could have interacted so as to occupy k and — k. Hence the total energy difference between having all paired electrons and having a single excited electron is finite and equal to a large multiple of the single pair correlation energy. In terms of the single electron spectrum, therefore, theBCS theory correctly yields an energy gap. It has already been shown that such an energy gap not only leads to the observed variation of the specific heat, the thermal conductivity, and the absorption of high frequency electromagnetic radiation, but also that it is correlated with the existence of perfect diamagnetism and perfect conductivity in the low frequency limit. 11.4. The ground state energy The recognition of the basic electron interaction mechanism respon- sible for superconductivity does not remove the major difficulty Microscopic theory of superconductivity 121 mentioned earlier, namely that the correlation energy in question is so very much smaller than almost any other contribution to the total electronic energy. BCS therefore take the bold step of assuming that all interactions except the crucial one are the same for the supercon- ducting as for the normal ground state at 0°K. Taking as the zero of energy the normal ground state energy and including in this all normal state correlations and even the self energy of the electrons due to virtual phonon emission and reabsorption, BCS proceed to calcu- late the superconducting ground state energy as being due uniquely to the correlation between Cooper pairs of electrons of opposite spin and momentum by phonon and screened Coulomb interaction. The interaction leading to the transition of a pair of electrons from the state (k t , -k | ) to (k' t , -k' I ) is characterized by a matrix element, -]^ fc/ = 2(-k'i,k'tl#int|-k!»M), (XI.2) where i/ int is the truncated Hamiltonian from which all terms com- mon to the normal and superconducting phases have been removed. V kk - is the difference between one term describing the interaction between the two electrons by means of a phonon, and a second one giving their screened Coulomb interaction. The basic similarity of the superconducting characteristics of widely different metals implies that the responsible interaction cannot crucially depend on details charac- teristic of individual substances. BCS therefore make the further simplifying assumption that V kk - is isotropic and constant for all electrons in a narrow shell, straddling the Fermi surface, of thickness (in units of energy) less than the average energy of the lattice, and that V kk - vanishes elsewhere. Measuring electron energy from the Fermi surface, and calling e k the energy of an electron in state k, one can state this formally by the equations: and V kk .= V forhfcl.M «&»„ V kk - = elsewhere. (XI.4) The basic BCS criterion for superconductivity is equivalent to the condition V< 0. 122 Superconductivity It is well to note clearly at this point that this simplification of the interaction parameter F necessarily leads to what can be called a law of corresponding states for all superconductors, that is, virtually identical predictions for the magnitudes of all characteristic quantities in terms of reduced co-ordinates. Any empirical deviation from such complete similarity is, therefore, no invalidation of the basic premise of the BCS theory, but merely an indication of the oversimplification inherent in XI.4. (See footnote, page 1 30.) Let h k be the probability that states k and — k are occupied by a pair of electrons, and (\—h k ) the corresponding probability that the states are empty. W(Q), the ground state energy of the superconducting state at 0°K as compared to the energy of the normal metal, is then given by ^(0)= S^ArSW^l-MMl-^}" 2 . (XI.5) k kk" The summation is over all those k-values for which V kk - 9* 0» so that using XI.4 one can simplify to wm - S 2e k h k - v 2 {h k (i -MMi -h)) m - (Xi.50 k kK The first term gives the difference of kinetic energy between the super- conducting and normal phases at 0°K. The factor 2 arises because for every electron in state k of energy e k there is with an isotropic Fermi surface another electron of the same energy in — k. This first term can be either positive or negative, and is smaller than the second term which gives the correlation energy for all possible transitions from a pair state (k, — k) to another (k', — k'). For such a transition to be possible, k must initially be occupied and k' empty. The simultaneous probability of this is given by h k {\ — h k >). The final state must have k empty and k' occupied, and this has probability h k -{\ — h k ). The square root of the product of these probabilities multiplied by the matrix element for the transition and summed over all possible values of k and k' gives the total correlation energy. W(0) must of course be negative for the superconducting phase to Microscopic theory of superconductivity 123 exist, and to see whether this is possible XI.5' can be minimized with respect to h k . This leads to [h k (l-h k )} 1 ' 2 = v w l-2h. 2e k By defining equation XI.6 simplifies to e(0)= KStMl-Ml kf 1/2 =*-!)• h,=- where E k = [4+e 2 (0)] 1/2 (XI.6) (XI.7) (XI.8) (XI.9) Substituting XI.8 back into XI.7 one obtains a non-linear relation for€(0): Ky g(0) e{0) "2Z[ £ I + eW 2 ' (XI. 10) This can be treated most readily by changing the summation to an integration and transforming the variable of integration from k to e. Assuming symmetry of states on either side of the Fermi surface ( c = 0), and introducing the density of single electron states of one spin in the normal state at e = 0: M0), XI. 10 becomes JlCOq M0) [e 2 +€ 2 (0)] ,/2 The limit of integration is the phonon energy above which, according to XI.4, V=0. The solution of XI. 11 is e(0) = /mysinMl/MO) V\. (XI.12) Putting this back into XI.9 and XI.7 and finally into XI.5', one finds that the ground state energy of the superconducting state is given by W(0) = - 2MQ)(W 2 exp[2/M0)F]-l (XI. 13) 124 Superconductivity The numerator of this quantity follows from dimensional reasoning from any theory which postulates an interaction between electrons and phonons and allows this interaction to be cut off at some average phonon energy hw q « k B 6, beyond which the interaction becomes repulsive. A term like this had been contained in the earlier attempts of Frohlich and of Bardeen, and, as mentioned before, is much too large. The success of the BCS theory lies in the appearance of the exponential denominator which reduces W(0) by many orders of magnitude. Although a precise calculation of the average interaction parameter V for a specific metal continues to be among the most important questions still to be solved, various estimates (Pines, 1958 ; Morel, 1959; Morel and Anderson, 1962) indicate that the values of N(0) Vx 0-3, derived from a knowledge of H Q , are reasonable. Thus the denominator has a value of about e 7 . The isotope effect follows from the numerator of XI. 1 3, as it would from any theory involving electron-phonon interaction with a cut- off frequency related to the Debye 6 and hence to the isotopic mass. Equation XI. 13 shows that H 2 ~ = ^(0) cc {hu> q ) 2 (k D Q) 2 cc Mfol (XI. 14) For a group of isotopes, one finds H cc T c , so that T c « Mr* 12 . (XI.15) Any appreciable deviation of the isotope effect exponent from the value 0-5 could indicate that the simplifying BCS assumption of a cut- off for both Coulomb and phonon interaction at hw q has to be modi- fied (Tolmachev, 1958; Swihart, 1959, 1962). Bardeen (1959) has pointed out that the cut-off may be determined by the lifetime of the 'normal' electrons which can be excited across the gap. These elec- trons are not the bare, non-interacting electrons of the simple Bloch- Sommerfeld model. Instead they are so-called quasi-particles 'clothed' by their interactions with each other and with the lattice (see \1], pp. 184-95). As a result the wave functions describing them are not eigenfunctions of the system, so that the particles have a finite lifetime. The effect of this on the pair interaction has been further Microscopic theory of superconductivity 125 discussed by Ehashberg(1961),Bardeen[9],Schrieffer(1961),Betbeder- Matibet and Nozieres (1961), and Bardasis and Schrieffer (1961). The damping of the quasi-particles is found to be very small even up to energies well beyond the Fermi energy. This is in contradiction to the BCS assumption embodied in XI .4, as the justification of the cut- off of the Coulomb interaction at hu) q is essentially that quasi-particles of larger energy are so strongly damped as not to be available for pair formation. It is thus necessary to modify the BCS cut-off by taking into account the existence of the repulsive interaction for e k > hw q . This does not appreciably affect the gap at the Fermi surface (e k = 0), but will result in its variation with e k , as will be further discussed in Section 11.7. With a compound tunnelling arrangement in which electrons are injected into a layer of superconducting lead and then have the possi- bility of tunnelling through a second junction into normal metal, Ginsberg (1962) was recently able to place an upper limit on the life- time of the quasi-particles in a superconductor. According to his pre- liminary result this upper bound is 2-2 x 10~ 7 sec, which is only about five times as large as the average time calculated by Schrieffer and Ginsberg (1962) for quasi-particle recombination into pairs by means of phonon emission. This has also been calculated by Rothwarf and Cohen (1963). Swihart (1962) as well as Morel and Anderson (1962) have studied the isotope effect for different forms of the energy dependence of the electron-electron interaction. They find that the exponent of the iso- topic mass in equation XI. 1 5 is less than the ideal value of one half by amounts of 10-30 per cent which increase with decreasing TJ6. However, the isotope effects in ruthenium (Geballe et ol., 1961, Finnemore and Mapother, 1962), osmium (Hein and Gibson, 1964) and perhaps also in molybdenum (Matthias et a I., 1963) appear to be too small to be explained by these calculations. This raises questions about the origin of the attractive interaction responsible for the formation of Cooper pairs in these as well as perhaps in other metals. Matthias (see for example, 1960) has repeat- edly suggested that in all transition metals there exists an attractive magnetic interaction responsible for superconductivity. However, both Kondo (1963) and Garland (1963a) have tried to explain the 126 Superconductivity apparently anomalous superconducting behaviour of the transition metals as a consequence of the overlap at the Fermi energy of the .v and d bands of the electronic spectrum, and not because of a magnetic interaction. Rondo assumes a larger interband interaction; Garland, on the other hand, believes that the electrons of high effective mass in the d-band tend not to follow the motion of the s-electrons. This results in 'anti-shielding' the interactions between ^-electrons, leading to an attractive screened Coulomb interaction between them being added to the usual attractive interaction by exchange of virtual phonons. Garland (1962b) calculated the magnitude of the isotope effect for all superconducting elements and obtains results which agree closely with all available experimental results, including in particular the reduced effect in transition metals. This also results, at least quali- tatively, from Rondo's calculations. Garland was also able to explain the anomalous pressure effect in transition metals (Bucher and Olsen, 1964). 11.5. The energy gap at 0°K From XI.5' one can see that the contribution of a single pair state (k, -k) to this total condensation energy is W k = 2* k h k -2Vj: {0-MM ,/2 . (XI.16) The first term represents the kinetic energy of both electrons in the pair state k, and the second term the total interaction energy due to all possible transitions into or out of the state. At 0°R the lowest excited state of the superconductor must corre- spond to breaking up a single pair by transferring an electron from a state k to another, leaving an unpaired electron in - k. The condensa- tion energy is then reduced by W k . The first term of this can be made arbitrarily small, and is analogous to the excitation energy in a normal metal, for which there is a quasi-continuous energy spectrum above the ground state. The second term of W k , however, is finite for all values of k, which is why in the superconducting phase the lowest excited state is separated from the ground state by an energy gap. Microscopic theory of superconductivity 127 Comparing XI.16 with XI.7 one sees that this energy gap has the value 2e(0), which according to XI. 12 equals 2c(0) = 2^^/sinh [1 /N(0) V\. (XI. 1 7) As 1/N(0) V& 3-4, this can be approximated by 2 € (0) = 4Aco 9 exp [- 1 /tf(0) V\. (XI. 1 8) 11.6. The superconductor at finite temperatures As the temperature of the superconductor is raised above 0°R, an increasing number of electrons find themselves thermally excited into single quasi-particle states. These excitations behave like those of a normal metal; they are readily scattered and can gain or lose further energy in arbitrarily small quantities. In what follows they are simply called normal electrons. At the same time there continues to exist the configuration of all electrons still correlated into Cooper pairs, and displaying superconducting properties, being very difficult to scatter or to excite. One is thus led again to a two-fluid point of view. As at 0°R, one can write down an analytic expression for the ground state energy W{J) containing a kinetic energy term and an interaction term. In both, the presence of the normal electrons must be accounted for, which is done by introducing a suitable probability factor f k . Letting f k = probability of occupation of k or of -k by a single normal electron, then 1 -2f k = probability that neither k nor -k is occupied by a normal electron. This leads to a kinetic energy term Wn] K . E , = 2£ M[f k + V-2f k )h k ], k (XI. 19) where the summation is over the same range as at 0°R, and h k retains the same definition, though no longer the same value. The second term in the brackets clearly gives the probability that the pair state (k, — k) not be occupied by normal electrons but by a correlated pair. The correlation energy at a finite temperature is kk' x(l-2/*)(l-2/*0. (XI.20) 128 Superconductivity The last two terms ensure that the correlated pair states not be occu- pied by normal electrons. It is obvious that the presence of these terms decreases the pairing energy. The thermal properties of the superconductors can now be found quite readily by writing down the free energy of the system and requiring this to be at a minimum. The free energy is G = Wm-TS = [W(T)\ K . B MW(Tj\ C0 „-TS, (XI.21) where J is the temperature and S the entropy. This last is due entirely to the normal electrons; the electrons which are still paired are in a state of highest possible order and do not contribute at all. Thus the entropy is given by the usual expression for particles obeying Fermi- Dirac statistics : TS = -2k B T-Z {A.lnA. + (l-/*)m(l-A)}. (XI.22) k Substituting XI. 19, XI.20 and XI.22 into XI.21, and minim ising this free energy with respect to h k , one now obtains Ml-hM«* S[Mi-M]" 2 (i-2A0 l-2h k This time one defines = V 2e 4 <T)= F£[M1-/'a<)] ,/2 (1-2/,0, and again obtains hi -Hi (XI.23) (XI.24) (XI.25) where E k is now defined as E k m [e k + e 2 (!T)] ,/2 . One sees that, as at 0°K, 2e(T) represents the contribution of a single pair state to the total correlation energy, and that to break up one such pair at any finite temperature removes from the supercon- ducting energy at least this amount. In other words, the supercon- ducting state continues to contain an energy gap 2e(T) separating the lowest energy configuration at any given temperature from that with one less correlated pair. Microscopic theory of superconductivity 129 To evaluate the magnitude of the energy gap one must first find an expression for f k , which one obtains by minimizing the free energy with respect to f k . This yields f k = [ C xp(E k /k B T)+])- 1 . (XI.26) XI.26, XI. 18, and XI.24 yield for e(T) sl non-linear relation which, changing as before from a summation over k to an integration over e, becomes Jiuia [e2+ 7^^(-^H' <XL27) o Wv = J The critical temperature T c is reached when all pair states are broken up so that e(T c ) = 0. Hence 1 f de . € W = J 7 tan W c o (XI.28) As long as k B T c < hw q , the solution of this can be written as k B T c = M4/K^exp[- 1/W(0) V\. (XI.29) The exponential dependence of the transition temperature has been verified by Olsen et at, (1964) by means of measurements of its varia- tion with pressure in aluminium. 11.7. Experimental verification of predicted thermal properties Combining equations XL 18 and XI.29 yields for the width of the energy gap at 0°K 2e(0) = 3-52 k B T c . (XI.30) This is in remarkable quantitative agreement with empirical values obtained from the wide variety of measurements mentioned in Chap- ter X. Table HI shows that for the most widely different elements the energy gap does not appear to deviate from this idealized value by more than about 20 per cent. The theoretical temperature variation of the gap width is displayed in Figure41 ; this has also been well con- firmed by a number of experiments. 130 Superconductivity Muhlschlegel (1959) has tabulated values of the energy gap, the entropy, the critical magnetic field, the penetration depth, and the specific heat, all in reduced coordinates, as functions of the reduced temperature. All these are in close agreement with experimental results. These agreements clearly vindicate the basic BCS approach , accord- ing to which the similarities between superconductors outweigh their differences, so that an approximate law of corresponding states should 1.0 OB em o.6 02 0.2 0.4 0.6 t Fig. 41 0.8 t.O hold.f This similarity principle had of course emerged from much pre- vious experimental evidence. However, differences between metals and anisotropics in a given metal do exist, and the experimental evi- dence for gap variations from one metal to another, as well as for gap anisotropics, clearly indicates the need to refine the details of the BCS calculations. For one thing it is of course desirable to take into account t Deviations from such a law can occur even with the BCS assumption of constant Kif in solving equations XI.27 and XI.28 one takes into account higher order terms in k B T e lhw q (Muhlschlegel, 1959). The resulting correc- tion factors appearing in equations XI. 30, XI.35, and XI.36 are, however, too small to explain the empirical deviations from similarity discussed in this section. Thouless (I960) has shown that in the BCS formulation the energy gap at 0°K is only 4-0 k B T c even in the non-physical limit Microscopic theory of superconductivity 131 the dependence of the interaction parameter Fon k and k', so as to be able to calculate directional effects. Even more challenging are the previously mentioned attempts to relax, even in an isotropic model, the assumption XT.4 that Fis strictly constant for e k < hoj q and is then cut off abruptly. A better knowledge of the variation of Fwith e k in turn would allow the more precise calculation of the corresponding dependence of the energy gap on e k . The actual form of this variation undoubtedly more nearly resembles the solid line in Figure 42 rather than the dotted line which corresponds to the simple BCS assumption. Usually one is interested in excitation energies of the order of k B T c and the BCS assumption is then fully applicable as long as k B T c < hu) q , which is called the weak coupling limit. As hw q &k B ®, where 6 is the Debye temperature, this requires that T c < e. For a number of superconducting elements, in particular for Pb and Hg, this condition does not hold. Swihart (1962, 1963) as well as Morel and Anderson (1962) have investigated the consequences of an energy dependence of the inter- action Kmore realistic than that assumed by BCS. In particular they take into account that, as was mentioned earlier, lifetime effects are too small to justify cutting off the Coulomb repulsion at hm q . There- fore these authors include in the interaction a repulsive part (V> 0) for energies e k > hw q . The resulting variation of the energy gap at 0°K as a function of e k has been shown by Morel and Anderson to have the form represented schematically in Figure 42. Swihart found 132 Superconductivity that a rise of this gap function on moving from the Fermi surface leads to the correct specific heat jump for lead at T c . The relation between calorimetric and magnetic properties indicates that such a gap variation is also consistent with the observed critical field curve for lead and probably also with that for mercury. For even higher quasi particle energies the energy gap continues to change sign periodically at multiples offiw q . This is consistent with the observations of Rowell et al. (1962), who found maxima in the tunneling conductance with that periodicity. Excitation of high energy quasi particles involve multi-phonon interactions. A precise calculation of the energy gap variation with e k cannot content itself with assigning to the phonons an average energy hcu q . Instead it must take into account the details of the phonon spectrum, as determined, for example, by neutron diffraction. In particular it is necessary to recognize the different frequency distributions for the longitudinal and transverse phonons. Such a calculation was carried out for lead by Culler et al. (1962), using an on-line computer facility. The relation between the phonon spectrum and the tunneling characteristics has been fully discussed by Scalapino and Anderson (1964). In considering an energy gap which changes sign as a function of e k , it must be remembered that in an experiment involving thermal or electromagnetic absorption across the gap, the quantity actually observed is the energy E k , denned by XI. 9. This involves only the square of the gap, and is therefore always positive. However, the details of the variation of the variation of the gap with e k can be verified by tunneling experiments, in which the conductance dl/dV is directly proportional to the density of states in the superconductor (Bardeen, 1961a, 1 962a; Cohen et al. 1962).Schrieffer<?/a/. (1963) have however pointed out that for tunneling one cannot use the standard expression for the quasi-particle density of states. This is because when an electron tunnels from one side of the barrier to the other, the initial and final states are not quasi-particle eigenstates of the individual metals making up the tunnel. Instead the appropriate density of states to use is E k N ™ = m) Hvv^} Microscopic theory of superconductivity 133 in which the energy gap e(0) is taken to vary with e k . Rowell et al. (1963) have closely verified the expected structure by tunneling experi- ments with lead, tin, and aluminium. Indications of this structure had been seen earlier by Giaever et al. (1962) in lead and by Adler and Rogers (1963) in indium. The tunneling discussed thus far in this section and in Section 10.6 involves the passage of one or more quasi-particles. Josephson (1962) has predicted an additional tunneling current when both sides of the tunnel are superconducting. This current can be considered 10 r4 4.0 _ Pb ■ /sla 3.6 SaTh V S + n V^g 26(0) Al| kT c 3.2 1 Re Ru In IS, - f Tl Zr 2.4 1 i 10' 3 10' 2 T c /€) Fig. 43 10" as being due to the direct passage of coherent Cooper pairs from one side of the insulating barrier to the other. As has been elaborated by Anderson (1963) and by Josephson (1964), the relative phase of the superconducting wave functions on either side of the barrier has physical meaning because it is a quantity conjugate to the number of electrons on each of the two sides and because this number is not constant, that is, not fully determined. As a result the energy of the system depends on this phase difference, and in turn this gives rise to a flow of pairs across the barrier in the absence of an applied potential difference. The Josephson current is very difficult to detect because it is quen- ched by a magnetic field of a few tenths of a gauss. It was first observed 1 34 Superconductivity by Anderson and Rowell (1963) and by Shapiro (1963). The tempera- ture dependence has been studied by Fiske (1964), following calcu- lations by Ambegeokar and BaratofF (1963). Ferrell and Prange (1963) have discussed the self-limitation of the Josephson current by the magnetic field it generates itself, and De Gennes (1963) has derived an expression for the current from the Ginzburg-Landau-Gor'kov equations. A more realistic cut-off can also yield theoretical justification for the apparent correlation of e(0) with TJ@. Such a correlation was suggested by Goodman (1958), whose plot of energy gap values against TJ0 for 17 different superconductors is shown in Figure 43. Goodman used gap values deduced from empirical values of y, H , and T c by combining XI. 1 3, XT. 1 8, and XI.20, and remembering that y = ffl^jAftO). This yields k B T c V3 Appropriate values of 2e(0)/(£ B r c ) are listed in Table III. (XI.32) 11.8. The specific heat One can obtain the electronic specific heat in the superconducting phase by twice differentiating with respect to temperature the free energy expression XI.21. At sufficiently low reduced temperatures, for which 2e(7") > k D T c , this yields yTc where K { and Kj, are first and third order modified Bessel functions of the second kind. This simplifies in the temperature regions indicated to the following exponential expressions: — « 8-5 exp(-l -44TJT), yT c 2-5 < TJT < 6, 26 exp (- 1 -62TJT), 7 < TJT < 1 1 . (XT.34) Experimental data at this time exist only in the first of these two regions where they are in good agreement with the BCS values, Microscopic theory of superconductivity 135 except for the upward deviation at the lowest temperature which was mentioned earlier (see Figure 28). Further numerical predictions of the BCS theory include yT c = 2 43. (XI.35) The following table is taken from [7] (p. 212) and shows how closely Element Lead Mercury Niobium Tin Aluminium Tantalum Vanadium Zinc Thallium CJTJ yT c 3-65 318 307 2-60 2-60 2-58 2-57 2-25 215 most experimental values agree with this. The theory also yields that (XI.36) ^ = 0170, Hi from which one can calculate (see equation 11.15) that the predicted coefficient of t 2 in the polynomial expansion of the threshold field is a 2 m 107. (XI.37) This agrees exactly with the experimental value for tin (VIII.3) and closely with that for several other elements. In terms of the deviation of the threshold field curve from a strictly parabolic variation as dis- played in Figure 23, any value of a 2 greater than unity corresponds to a curve below the abscissa; only mercury and lead are seen to have deviations corresponding to values of a 2 smaller than unity. The BCS calculations are based on an isotropic model, in which the interaction parameter Kdoes not depend on the direction of A: and k ' . Pokrovskii (1961) and Pokrovskii and Ryvkin (1962) have investi- gated the effects of anisotropy on thermal and magnetic properties. 10 136 Superconductivity They find that in anisotropic superconductors the specific heat ratio in Xl.35 should be smaller than 2-43, the quantity in XI. 36 larger than 01 70, and therefore the coefficient a 2 larger than 107. In the second of the papers cited these results are compared with extensive experi- mental data. The thermal conductivity in superconductors has been calculated on the basis of the BCS theory for several of the pertinent mechanisms. Bardeen et al. (BRT, 1959) and Geilikman (1958) have derived the ratio of the electronic conductivity in the superconducting phase to that in the normal one when this is primarily limited by impurity scattering (equation IX.6). Their results have been well confirmed experimentally, as was discussed in Chapter IX. The derivation of BRTforthecaseofelectronicconductionlimitedbyphononscattering (equation IX.5) does not, however, lead to the empirical behaviour. Calculations by Kadanoff and Martin (1961) and by Kresin (1959) are in better agreement, but further theoretical work is needed for this conduction mechanism, in which quasi-particle life times may again be important (see [7], pp. 272 ff.). According to calculations of Tewordt (1962, 1963), however, these appear to have little effect on this conduction mechanism. BRT as well as Geilikman and Kresin (1958, 1959) have derived the lattice conductivity limited by electron scattering. Experimentally it is very difficult to separate out this part of the heat transport. Where this has been possible (Connolly and Mendelssohn, 1962; Lindenfeld and Rohrer, 1963) the results have been in general agreement with the theoretical predictions. 11.9. Coherence properties and ultrasonic attenuation One of the most striking predictions of the BCS theory arises as a direct consequence of the pairing concept, and experimental verifica- tion of this point is thus of particular importance. In a normal metal the scattering of an electron from state k t to state k' t is entirely independent from the scattering of an electron from - k | to -k' ]• or of any other transition. The coherence of the paired electrons in the kf and -k j states in the superconducting phase, however, makes these two transitions interdependent. The details of the theory (see [7], pp. 212-24) show that the contribution of the two possible Microscopic theory of superconductivity 137 transitions interfere either constructively or destructively depending on the type of scattering phenomenon involved. There is constructive interference in the case of electromagnetic interaction, such as the absorption of electromagnetic radiation, and the hyperfine inter- action which determines the nuclear relaxation rate. The experi- mental results expected in these two cases are therefore qualitatively those which follow from a two-fluid model consideration of the total number of electrons available as well as from the density of available states. It has already been mentioned how this explains the observed rise in the nuclear relaxation rate just below the critical temperature (Figure 32). On the other hand, the contributions of the two transitions inter- fere destructively in the case of the absorption of phonons, such as occurs in the attenuation of ultrasonic waves. This destructive inter- ference so decreases the probability of absorption that the effect of the increase in density of states on either side of the gap is completely wiped out, and the absorption just below T c drops very sharply. For low frequency phonons, ha> <^ 2e(0), the ratio of attenuation coeffi- cient in the superconducting and normal phase o.J<x„ drops below T c with an infinite slope, and is given by -? = 2/{l+exp[2e(T)lk B T]}. a„ (XI.38) This function is shown in Figure 44, which includes experimental points on both tin and indium by Morse and Bohm (1957). It should be contrasted with the theoretical prediction for nuclear relaxation rate, shown in Figure 32. Measurements of the ultrasonic attenuation in single crystals of tin in different crystal directions has yielded very convincing demonstra- tion of the anisotropy of the energy gap. When an electron absorbs a phonon, energy and momentum can both be conserved only if the component of the electronic velocity parallel to the direction of sound propagation is equal to the phonon velocity, which is the velocity of sound S. Since, however, the Fermi velocity of the electrons, v , is several orders of magnitude larger than S, this is possible only for electrons which move almost at right angles to the direction of sound 138 Superconductivity propagation. Thus a measurement of the attenuation of sound propa- gated in a particular crystalline direction involves only those electrons whose velocity directions lie in a thin disk at right angles to this direc- tion. The value of the energy gap appearing in equation XI. 38 i > thus one averaged over this particular disk. Such measurements have been 1.0 r 0.8 0.6 0C n 0.4 02 • TIN x INDIUM performed on variously oriented tin single crystals both by Morse et al. (1959) and by Bezuglyi et al. (1959). Their results are in good agree- ment and are summarized in the following table: Wave vector q 2e(0)/k B T c parallel to [001] 3-2 ±01 parallel to [1 10] 4-3 ±0-2 perpendicular to [001] and 18° from [100] 3-5 ± 01 Microscopic theory of superconductivity 139 11.10. Electromagnetic properties To describe the many-particle wave function of the superconducting state in the presence of an external field, BCS treat the electromagnetic interaction as a perturbation, and obtain an expansion in terms of the spectrum of excited states in the absence of the field. This wave func- tion is then substituted into an equation of the form III. 20 to calculate the current density. Mattis and Bardeen (1958) and also Abrikosov et al. (1958) have expanded this to treat fields of arbitrary frequency. The result of the former has been used by Miller (1960) to calculate values of cri/o n and of a 2 la n over a wide range of temperatures and frequencies. His calculations are in excellent agreement with all the experimental results using weak fields at high frequencies, described in Chapter X, if the energy gap is taken as a parameter to be adjusted to its empirical value. The treatment of a magnetic field as a perturbation in the BCS formulation makes it very difficult to extend it to high field values (H m H c ). This can be done more readily from a representation of the BCS ideas in terms of Green's functions which has been developed by Gor'kov (1958). A simplified version of this method has been pre- sented by Anderson (1960). The electromagnetic equations occurring in this formulation were shown by Gorkov (1959, 1960) to be equiva- lent to the Ginzburg-Landau expressions in the region near T c and under circumstances where A > £. As was pointed out in Chapters V and VII, Gor'kov showed that the energy gap is proportional to the G-L order parameter, so that the dependence of the latter on tem- perature, magnetic field, and co-ordinates, also applies to the former. The successful application of these ideas to a number of experimental results has been mentioned in Chapter VII. An apparent shortcoming of the original BCS treatment is its lack of gauge invariance. It was suggested by Bardeen (1957) and worked out by various authors that this can be remedied by taking into account the existence of collective excitations. A discussion of this with full references is given in [7] (pp. 252 ff.). There exists as yet no fully satisfactory explanation that the Knight shift in superconductors does not vanish in any of the elements in which is has thus far been studied : mercury (Reif, 1 957), tin (Androes and Knight, 1961), vanadium (Noer and Knight, 1964) and aluminium '40 Superconductivity (Hammond and Kelly, 1964). The Knight shift is defined as the frac- tional difference in the magnetic resonance frequency of a nucleus in a free ion and the same nucleus in a metallic medium. It is due to the field at the nucleus created by the free electrons, and is usually taken to be proportional to the electronic spin susceptibility. A literal interpretation of the Cooper pairs of opposite spin would lead one to expect that in a superconductor this susceptibility and hence the Knight shift should vanish at 0°K. A number of authors (see [7], pp. 261-263; Anderson, 1960; Suhl, 1962; Cooper, 1962) have suggested why this may actually not be the case, and although none of these explanations appears fully adequate, they have shown that the Knight shift offers no fundamental disagreement with the idea of the BCS theory. It is, furthermore, possible that the Knight shift in some of these elements is not primarily due to spin paramagnetism. Clogston et al. (1962, 1964) deduce from the temperature variation of the Knight shift in vanadium that in the superconducting state the dominant contribution due to the d-electron spin does vanish, as the simple theory would predict. This, however, leaves a finite Knight shift due to orbital paramagnetism which involves electrons too far from the Fermi surface to be involved in pairing. Thus this contribution to the Knight shift in vanadium is not affected by the superconducting transition, and perhaps the orbital part is the dominant one in tin and mercury. CHAPTER XII Superconducting Alloys and Compounds 12.1. Introduction Ever since the discovery of superconductivity there have been many searches for new superconducting materials. Roberts (1961) has recently listed more than 450 alloys and compounds with critical temperatures ranging from 016° up to 18-2°K. In the appearance of superconductivity among these substances there exist certain regu- larities which were discovered by Matthias (1957) and to which reference was made in Chapter I. One might consider as an ultimate goal of any complete microscopic theory the ability to derive these Matthias rules from first principles. This would be equivalent to being able to calculate with some precision the actual critical temperature of any superconductor. At the moment our understanding of super- conducting and of normal metals is still very far from such achievements. One of the many ways of increasing this understanding is a sys- tematic study of superconducting alloy systems in which solvent or solute are used as controlled parameters. This has been done in a number of experiments. 12.2. Dilute solid solutions with non-magnetic impurities Serin, Lynton, and collaborators (Lynton et al., 1957; Chanin et al., 1959) have investigated the superconducting properties of dilute alloys of various solutes into tin, indium, and aluminium, up to the limit of solid solubility. For low impurity concentrations, of the order of a few tenths of an atomic per cent, T c decreases linearly with the reciprocal electronic mean free path, independently of the nature of the solute. When plotted against the reduced co-ordinate £ //, where | is the coherence length of the pure solvent, the fractional change in T c is the same for elements as different as Sn and Al (Serin, 1960). This is shown in the initial portions of both curves in Figure 45. The existence and the magnitude of this seemingly general effect lend 141 N 142 Superconductivity strong support to Anderson's model of impure superconductors (Anderson, 1959). He suggested that the energy gap anisotropy is smoothed out by impurity scattering and disappears when the elec- tronic mean free path is comparable to fiv MO) ~ p - so- +.02 +.01 -.01 -02 % -04 -05- ELECTRONEGATIVE 5 6 7 Aoll Fig. 45 This should then result in a lowering of T c by an amount approxi- mately equal to the square of the fractional anisotropy. Nuclear resonance in aluminium (Masuda and Redfield, 1960a, 1962) and ultra- sonic and infrared absorption in tin (Morse et al., 1959; Bezuglyi,e/ al., 1959; Richards, 1961) have shown that the gap in these elements varies by about 10 per cent from its average value, so that T c should be lowered by about 1 per cent when / « P . The measurements of T c confirm this very well. Recently Caroli et al. (1962), Markowitz and Kadanoff (1963), and Tsuneto (1962) have shown in terms of the microscopic theory that Anderson's idea of the smoothing of an Superconducting alloys and compounds 143 anisotropic energy gap indeed leads to a lowering of T c of the observed magnitude. Hohenberg (1963) has calculated the dependence of T c , the energy gap, and the density of states on the concentration of impurities. This general mean free path effect on T c has also been found in tantalum by Budnick (1960). It has been verified by using a number of different ways of scattering the electrons : by mechanical deforma- tion and cold work in aluminium (Joiner and Serin, 1961), by size effects in indium (Lynton and McLachlan, 1962), by quenching (De Sorbo, 1959), electron irradiation (Compton, 1959), neutron bombardment (Blanc et al., 1960), and by using isoelectronic ternary compounds (Wipf and Coles, 1959) in tin. Figure 45 shows that for P /l > 1 , the effect on T c deviates from the initial linear decrease in a way which depends on whether the solute is electropositive (valence smaller than that of solvent) or electro- negative (valence larger). Chiou et al. (1961) have extended such measurements to higher concentrations. They found that for both types of impurities T c ultimately rises to values above that of the solvent, and were able to repissent the variation of T c with impurity concentration in all cases by an empirical relation containing two parameters adjusted according to the particular solvent-solute combination. According to the BCS theory (equation XI.29), T c depends on three parameters : an average phonon frequency m q (which is proportional to the Debye temperature ©), the density of normal electron states at the Fermi surface, N(0) (which is proportional to the Sommerfeld y), and the BCS interaction parameter V. Specific heat measurements on tin alloys have recently enabled Gayley et al. (1962) to find the effects of the addition of indium, bismuth, and indium antimonide on the values of y and of © for tin. One can use equation XT.29 to calculate the corresponding change in T c . This seems to account for most of the difference in the behaviour of electropositive and electronegative solutes, at least in the case of indium and bismuth, but not for the increase in T c at high solute concentrations. One concludes that this increase is mainly due to effects of alloying on the interaction energy V. Any attempt to calculate Kin the presence of impurities has to take 144 Superconductivity into account that with scattering the wave vectors k are no longer good quantum numbers. Hence the question arises of the criterion for pairing of the electrons. Abrahams and Weiss (1959) and Anderson (1959) have pointed out that in impure superconductors Cooper pairs are formed of two electrons the wave functions of which are identical except for the reversal of the time co-ordinate, and which have the same energy. The former authors have used this to deduce Applied Field He Fig. 46 several impurity effects, the magnitude of which is difficult to esti- mate. Anderson (1959, 1960) has discussed the general implications of the use of time-reversed wave-functions. Detailed microscopic calculations of the impurity effects on the superconducting parameters have been attempted by Caroli et al. (1962) and by Markowitz and Kadanoff(1963). It is interesting to note that the work on carefully homogenized and annealed solid solutions has shown these to be 'well-behaved' super- conductors according to several criteria. Transitions occur within a few millidegrees, and very little flux remains in suitably oriented cylindrical samples after an external field has been removed (Budnick Superconducting alloys and compounds 145 et al., 1956). Also the absorption edge of infrared radiation at the gap frequency can be very sharp (Ginsberg and Leslie, 1962). Detailed magnetization curves (Lynton and Serin, 1958) however, show that the transitions for such alloys are nevertheless not fully reversible, as shown in Figure 46 for 311 per cent In-Sn cylinders transverse to an external field. In decreasing field the magnetic moment does not attain its full diamagnetic value until H vanishes. This indicates that flux is initially trapped, but then leaks out as suggested by Faber and Pippard (1955b). 12.3. Compounds with magnetic impurities Matthias and collaborators have traced the occurrence of super- conductivity in a large number of compounds containing para- magnetic and ferromagnetic impurities (Matthias, 1960). Their results can be summarized as follows: Ferromagnetic transition elements with 3d electrons (Cr, Mn, Fe, Co, and Ni) put into fourth column superconductors (Ti, Zr) raise T c more than do corresponding amounts of transition elements with Ad electrons (Re, Rh, Ru, etc.) (Matthias and Corenzwit, 1955; Matthias et al., 1959b). At the same time magnetic measurements on Ti-Fe and Ti-Co alloys indicated the absence of localized moments. The effect of the 4d electrons can be attributed to the increase in the number of valence electrons per atom toward five, a number particularly favourable for superconductivity. The extra rise with 3d electrons is attributed to a magnetic electron-electron inter- action favouring superconductivity. For the same reason adding Fe (3d electrons) to a Ti . 6 V . 4 compound lowers T c less than does an equal amount of Ru (4d electrons): in both cases T c is decreased because the number of valence electrons per atom rises beyond five, but with Fe the apparent magnetic interaction counteracts this in part. It must be pointed out, however, that ferromagnetic transition elements with 3d electrons put into fifth column superconductors (Nb, V) lower T c in approximate agreement with the expected effect due to the change in valence electrons per atom (MUller, 1959). There does not appear to be any added effect due to the magnetic nature of the impurities. Why such effects should appear with fourth column metals but not with fifth column ones is far from clear, as in neither 146 Superconductivity case are there any localized magnetic moments associated with the 3d solute atom. Quite recently Cape (1963) has measured the electrical and mag- netic properties of very carefully prepared alloys of Ti containing 0.2 to 4 at % Mn. Depending on the method of preparation these specimens are either in a single hexagonal close packed (hep) phase, or contain an admixture of a second, body centred cubic (bec) phase. Localized moments exist only in the hep phase, which however is not superconducting. This is consistent with the usual suppression of superconductivity by impurities retaining localized moments (see below). The non-magnetic bec phase, on the other hand, has a tran- sition temperature which is raised above that for pure Ti by an amount commensurate with the increase in the number of valence electrons. Hake et al. (1962) had earlier deduced from their measurements of transport properties that the hep phase of Ti-Cr, Ti-Fe, and Ti-Co also carried localized magnetic moments. In addition there is calori- metric evidence (Cape and Hake, 1963) that in Ti-Fe samples only a small fraction of the volume is superconducting. These results throw considerable doubt on Matthias' speculation that iron-group im- purities which do not carry a localized magnetic moment enhance superconductivity by means of a magnetic interaction between electrons. While 3d impurities in fifth column metals (for example Nb) do not show any evidence for a localized moment, they do when put into sixth column metals (for example, Mo), and in fact Matthias et al. (1960) found that the change in behaviour occurs in Nb-Mo solutions at a concentration of about 60 per cent Mo. One would therefore expect some special effects on T c to appear in 3d compounds with sixth column metals. Until the recent discovery of the superconduc- tivity of Mo, no such metal was known to be superconducting. For that reason, this effect was studied on superconducting Mo . 8 Re . 2 , and indeed small amounts of 3d impurities lower T c far more than one would expect from valence effects. It is in fact this which made the dis- covery of the superconductivity of Mo so difficult: a few parts per million of iron are enough to depress T c below the measurable range (Geballe et al, 1962). A less abrupt decrease in T c is obtained when rare earth elements with 4/electrons are put into lanthanum (Matthias Superconducting alloys and compounds 147 et al., 1 958b, 1959a). The magnitude of this decrease, for each per cent of rare earth impurity, is correlated with the spin rather than with the effective magnetic moment of the solute. This is shown in Figure 47 in which - AT C for each per cent, the spin, and the effective moment /x eff are plotted for the different rare earths. A higher effective moment, in fact, appears to tend to raise T c , perhaps for the same reason as in the case of the 3d impurities in fourth column metals : erbium, with spin 3/2 and large moment lowers T e less than does an equal per- centage of neodynium, which has the same spin but a smaller moment. Fig. 47 All these compounds containing 4/electrons show ferromagnetic behaviour at somewhat higher concentrations of the rare earth solutes, with the Curie temperature rising with increasing number of 4/ electrons. Such dilute ferromagnetism has not been observed for compounds with 3d electrons which indicates that the s-f magnetic interaction is rather long range, while the d-d one is a short-range inter-action effective only through nearest neighbours, which is impossible in dilute solutions (Matthias, 1960). Interesting analogies in the variation of the Curie temperature and the superconducting critical temperature are found by investigating the magnetic characteristics of so-called Laves compounds AB 2 , where B is germanium or a noble metal (Ru, Os, Ir, Pt) and A is either 148 Superconductivity a rare earth with 4/ electrons (A') or one of the group Y, Sc, Lu, or La {A"), none of which contain 4/ electrons (Suhl et al., 1959). A'B 2 is always ferromagnetic, A"B 2 always superconducting. Comparing the Curie temperatures of the former with the critical temperatures of the latter one finds a similar dependence on spin and on the number of valence electrons per atom. This is but one of a number of interest- ing correspondences which Matthias has found between supercon- ductivity and ferromagnetism. There are, for example, several groups of isomorphous compounds which are either superconducting or ferromagnetic (see, for example, Matthias et al., 1 958a ; Compton and Matthias, 1959; Matthias et al., 1962). Also, the appearance of locali- zed moments when a ferromagnetic impurity is put into a non-mag- netic transition element seems to depend on the number of valence electrons in a manner similar to the criterion for the appearance of superconductivity (Matthias, 1962). Matthias hasfrequentlysuggested that an electron configuration favourable to superconductivity may also be favourable to ferromagnetism. The possible coexistence of superconductivity and ferromagnetism in the same substance has been investigated in lanthanum-rare earth binary compounds (Matthias et al., 1958b) and in Laves compound mixtures (A'i^ x A%)B 2 (Matthias et al., 1958c; Suhl et a/.,1959). Both magnetic (Bozorth et al., 1960) and calorimetric measurements (Phillips and Matthias, 1960) have shown that ferromagnetism and superconductivity occur in the same sample, but the evidence is not entirely conclusive in ruling out the possibility that these two phe- nomena merely exist side by side in different portions of the specimen. Anderson and Suhl (1959) have shown that the actual coexistence of ferromagnetism and superconductivity on a microscopic scale is energetically possible if the ferromagnetic alignment occurs in the form of extremely small domains probably of the order of 50 A. They call this 'cryptoferromagnetic' alignment. Suhl and Matthias (1959) have treated the general problem of the lowering of T c due to the presence of magnetic impurities by extending an argument of Herring (1958), according to which the polarization due to the coupling of the conduction electrons with the spins of the paramagnetic impurity ions lowers the free energy in both the normal and in the superconducting phases. The free energy is lowered by each Superconducting alloys and compounds 149 electron-spin scattering interaction by an amount proportional to the reciprocal energy difference between the initial and final electron state. In the normal state this difference can be arbitrarily small, in the superconducting case this difference cannot be smaller than the energy gap. As a result the free energy of the normal phase is lowered more than that of the superconducting one, and the onset of super- conductivity therefore occurs at a lower temperature. Suhl and Matthias ignore the small changes in the interaction matrix element V, and as a result their prediction (dTJdc -* 0) for very small magnetic impurity concentrations is probably wrong. Abrikosov and Gor'kov (1960) show that magnetic impurity effects on V initially lowers T c linearly with impurity concentrations. Atmuch higher concentrations Suhl and Matthias find that S7y3c->co, which is supported by experiment (Hein et al., 1959). Similar calculations have been carried out by Baltensperger (1959). Suhl (1962) has recently reviewed this and similar work. Abrikosov and Gor'kov (1960) as well as De Gennes and Sarma (1963) show that magnetic impurities will lower the energy gap more rapidly than the transition temperature. There should thus be a range of concentration for which the alloy has a finite critical tem- perature at which its DC resistance disappears without the existence of an energy gap. Indeed Reif and Woolf ( 1 962) have found this para- doxical behaviour to exist. They measured the electrical resistance as well as the tunneling characteristics of a number of lead and indium film containing magnetic impurities. The gap decreased twice as rapidly as the transition temperature, and an indium film containing 1 at % Fe, for example, had no resistance below 3°K but a perfectly ohmic tunneling conductance. Phillips (1963) has pointed out that such gapless superconductivity does not violate any fundamental principle. The spin-flipping scatter- ing of the conduction electrons by the magnetic ions gives the former a very short lifetime. This broadens the electron states, particularly those nearest the gap, so much as to spread the states into the gap. At a certain impurity concentration states will have spread throughout the width of the gap, making it disappear. The density of states, however, will still have maxima at what used to be the edge of the gap, and as long as this exists the material will have zero DC resistance. 1 50 Superconductivity The details of this have been worked out by Skalski et al. (1963). In terms of the Ferrell-Glover rule and the frequency dependence of the conductivities shown in Figure 37, the situation can be described by saying that at some concentration there is a finite real conductivity a i at all frequencies. For some further range of impurity concentration, however, o-j will still be less than a N for cu/w g < 1 , resulting in a reduced delta function at the origin. At even higher concentrations, ct, x a N for all frequencies. The sum rule is now satisfied without an infinite DC conductivity, so that the metal is normal in every respect. 12.4. Superimposed metals A recent series of experiments by Meissner (see Meissner, 1960 for full references) has revived interest in the question whether thin layers of superconducting material deposited on a normal metal would them- selves become normal, and whether conversely sufficiently thin layers of normally non-superconducting metal would become superconduct- ing when in contact with a superconductor. Such superimposed metals differ from the sandwiches used in the tunnelling experiments by the absence of an insulating layer. Parmenter (1960a) has constructed a theory for such direct metallic contacts in which he attempts to introduce directly into the BCS formulation a dependence of the energy gap on position by postulating a spatial variation of the parameter h k appearing in equation XI.5. This adds to the kinetic energy portion of the ground state energy (the first term in equation XI.5) terms involving the square of the gradient of h k 12 and of (1 -h k ) 112 , broadly analogous to the extra energy term V.6 in the Ginzburg-Landau theory. Near the boundary of a super- conductor this leads to a significant variation of the energy gap over distances which are of the order of 10 -6 cm, that is, two orders of magnitude smaller than the coherence length £ . The configurational surface energy resulting from this gap variation turns out in this theory to be about 10 -5 cm (Parmenter, 1960b) which is an order of magnitude smaller than the generally accepted value. To investigate the behaviour of superimposed metallic layers under this theory, Parmenter postulates a set of plausible boundary con- ditions involving the continuity of h k and of the normal component Superconducting alloys and compounds 1 5 1 of the gradient of h k , which, however, have yet to be justified from more fundamental considerations. From these he concludes that a normal layer sandwiched between two superconductors will itself be supreconducting if it is no thicker than about 10 -5 cm. A supercon- ducting layer between two normal metals will remain normal up to some similar critical thickness. Cooper (196 1 ) has given a more intuitive argument for the possibil- ity that the superconducting properties of thin metallic films may be strongly affected by direct contact with other metals. He emphasized that in the BSC theory one must clearly distinguish between the range of the attractive interaction between electrons, and the distance over which as a result of this interaction the electrons are correlated into Cooper pairs. The range of the interaction is very short (10~ 8 cm) ; the 'size' of the wave packets of the pairs, on the other hand, is of the order of the coherence length, that is, 10 -4 cm. This, as Cooper points out, is analogous to the difference between the range of the nuclear interaction and the much larger size of the resulting deuteron wave packet. Because of this long coherence length the Cooper pairs can extend a considerable distance into a region in which the interaction between electrons is not attractive. Thus when a thin layer of super- conducting material is in contact with a layer of normal metal, the zero-momentum pairs formed because of the attractive interaction in the superconductor extend into both layers. As a result the ground state energy of this thin bimetallic layer is characterized by some aver- age over both metals of the parameter N(0) V, which in turn deter- mines the energy gap of the layer and its transition temperature, according to equations XI. 1 8 and XI.29. The form of this average of course depends on the nature of the boundary between the two metals ; the better the contact, the more effective is a superimposed layer in changing the properties of the substrate. Regardless of how one accounts for this, one would expect the average to depend also in some manner on the relative thickness of the two layers. The thicker the normal layer, the smaller the average interaction, and the more the energy gap width and the transition temperature are decreased from the values they would have if only the superconductor were present. Similarly a combination of two superconductors would be expected to have a T e somewhere between the T c values of the two materials, 11 1 52 Superconductivity varying from one extreme to another as the relative thickness of the two layers is varied. These qualitative conclusions presuppose that both layers of the bimetallic film are sufficiently thin so that the coherent electron pairs extend over the entire volume. One expects the critical thickness for this to be of the order of the coherence length, although it is not clear whether this should be the ideal value £ , or the mean free path limited value £(f). If one of the two superimposed metals is much thicker than whatever critical length is appropriate, then presumably the average interaction is determined by the ratio of the smaller thickness to the critical length. The experiments of Smith et al. (1 96 1 ) with lead films of about 500 A on or between silver films varying from 100 to 7000 A indicate a de- crease of the transition temperature of the lead with increasing silver thickness, supporting earlier work of Misener and Wilhelm ( 1 935) and of Meissner (1960). Similar results have been obtained by Hilsch and Hilsch (1961 ) with combinations of copper and lead films. These agree with the calculations of De Gennes and Guyon (1962) and the more extensive treatment of Werthamer (1964). However, the work of Rose-lnnes and Serin (1961) has shown that results can be strongly influenced by varying evaporation procedures, even under condi- tions which quite preclude ordinary bulk intermetallic diffusion. Interpenetration of metals seems to occur quite readily with super- imposed layers, possibly by the mechanisms of surface and defect diffusion. Because of this the experimental situation is at this time far from clear. CHAPTER XIII Superconducting Devices 13.1. Research devices The characteristics of superconductors have for a long time already been put to use in many low temperature experiments. It is very com- mon to use niobium wires, for example, in electrical connections to samples which one wishes to isolate thermally as well as possible. Such wires are superconducting with a high critical field throughout the entire liquid helium temperature range, and combine low thermal transport with perfect electrical conductivity. The use of lead wires as heat switches at temperatures below 0T°K has been mentioned in Chapter DC. More specialized research devices using superconducting com- ponents of varying complexity have been suggested or used frequently, and it is possible in this cursory survey to mention only a few of these. A number of such devices have been developed to detect very small potential differences, as occur, for instance, in studies of thermo- electric powers. Pippard and Pullan (1952) improved earlier designs by Grayson Smith and co-workers (Grayson Smith and Tarr, 1935; Grayson Smith et al., 1936) by using a single turn of superconducting wire to construct a galvanometer capable of detecting e.m.f.s of 10" ,2 V. With a resistance as low as 10~ 7 ohm this required a current sensitivity of only 10~ 5 amp ; the time constant L/R was kept short by the single turn design which reduced the effective inductance. A super- conducting magnetic shield made possible controlling fields as low as 001 gauss. A different approach to the measurement of very small potentials was suggested by Templeton (1955b) and by De Vroomen (De Vroomen, 1955; De Vroomen and Van Baarle, 1957). These authors designed 'chopper' amplifiers in which the small d.c. signal is con- verted into an alternating one by passing through a superconducting wire which is modulated into and out of the normal state by being 153 154 Superconductivity placed in an alternating magnetic field. The resulting oscillating potential across the wire is then amplified in a conventional manner. These devices can operate stably with a noise level at about 10~ ' ' V. Templeton (1955a) has also designed a superconducting reversing switch to suppress undesirable thermal voltages in measurements of potential differences of the order of about 10~ 6 V. Many low temperature experiments as well as superconducting magnets require rather high direct currents at very low voltages. To avoid the use of thick electrical leads which would bring too much heat into the helium dewar, Olsen (1958) has designed a superconducting rectifier and amplifier which, together with a low temperature transformer, allows one to feed in a low alternating current through thin leads. The rectification occurs as the current flows through a superconducting wire placed in an external, nearly critical field, such that the field due to the current in one direction is sufficient to make the wire normal during about one-half of each cycle. D. H. Andrews et al. (1946) made use of the change in resistivity at the superconducting transition in designing a bolometer. A different superconducting radiation detector has been suggested by Burstein et al. (1961) who pointed out that a tunnelling device (Chapter X) suitably biased would respond to absorption of electromagnetic radiation in the microwave and submillimetre range. RF detection with a tunneling device has been achieved by Shapiro and Janus (1964). For work at high frequencies superconducting metals may also be used to construct resonant cavities of extremely high Q. This has been discussed by Maxwell (1960), and preliminary experiments have been reported by Fairbank et al. (1964) as well as by Ruefenacht and Rinderer (1964). Thought is also being given to the use of superconducting cavities in high energy proton linear accelerators (Parkinson, 1962; Fairbank et al., 1964). Many of the devices listed in this section as well as others have recently been discussed by Parkinson (1964). 13.2. Superconducting magnets As early as 1931, De Haas and Voogd found critical fields as high as 15 kgauss in some lead-bismuth alloy wire. Other instances of rela- Supei conducting devices 155 tively large values of the critical field have been observed for many alloys and for strained or impure samples of the superconducting elements. For niobium published values of the critical field at 0°K vary from about 1950 to 8200 gauss. Quite recently Kunzler et al. (1 961 b) discovered Nb s Sn to have a critical field of about 200 kgauss, and similar critical fields have since been found in other substances. These seem to be either intermetallic compounds of the /3-wolfram structure, or body centered cubic alloys. When suitably prepared these materials remain superconducting while carrying current den- sities as high as 5 x 10 4 amp/cm 2 in fields almost up to the critical value. The critical fields of these materials are much too high to be the thermodynamic critical fields H c as defined by equation II.4. How- ever, earlier chapters have discussed two reasons why superconduc- tivity can persist in a given specimen to fields higher than H c . One possibility is that the material is sufficiently inhomogeneous, so as to display the characteristics of a 'Mendelssohn sponge', as discussed in Section 7.2. In such a specimen superconductivity persists in a filamentary structure, the dimensions of which are much smaller than the penetration depth. As a result, the filaments remain super- conducting to a field H s > H c , as given by equation VII.8. On the other hand, a quite different mechanism for high field superconduc- tivity was discussed in Chapter VI, where it was shown that super- conductors of the second kind remain in a superconducting mixed state up to U c2 > H c (Abrikosov, 1957). Superconductors of the second kind are materials which may be quite homogeneous and which have a negative surface energy, generally because of their very short electronic mean free path. Goodman (1961) was the first to suggest the possible relevance of this mechanism to explain the high critical field, found by Kunzler and others, and there is convincing evidence that this is indeed the case (see for example, Berlincourt and Hake, 1963). The specific heat results of Morin et al. (1962) on V 3 G a are consistent with the behaviour expected for superconductors of the second kind (Goodman, 1963b), and so are the critical fields observed by Berlincourt and Hake (1962, 1963) in the low current limit for a number of high field alloys and compounds. Hauser (1962) as well as Swartz (1962) have further shown that the 156 Superconductivity magnetization curves of suitably prepared specimens of various compounds are consistent with the identification of these materials as superconductors of the second kind. A systematic study of the role of defects on the magnetization curve has been carried out by Livingston (1963, 1964). However, Gorter (1962a, b) has pointed out that a homogeneous superconductor with uniform negative surface energy cannot in the presence of a transverse magnetic field carry the high current densities which are actually observed in most of the compounds and alloys under discussion. This can best be understood in terms of the vortex structure of the mixed state which is created by the external field (see Section 6.6). When a current passes through the specimen at right angles to the vortices, it interacts with the latter so as to push them out of the specimen. This, as was mentioned in Section 6.6, can be prevented only if the vortices are pinned down by local variation of the surface energy, as would be present if the specimen were inhomo- geneous. Indeed, there is much evidence that the high current carrying capacity is associated with the presence of dislocation in cold-worked specimens (Hauser and Buehler, 1962). Annealed samples may still have a very high critical field while carrying a low current density, but turn normal when the latter is increased. Rose-Innes and Heaton (1963) have used Ta-Nb wire to show very strikingly how sample treatment can change the current carrying capacity without changing the critical field. Thus the present picture of high field superconductors is that basically they are materials characterized by a negative surface energy. They are further able to carry high current densities in high fields if through cold work they are made to contain a high density of dislocations which pin down the current carrying regions. A nearly uniform distribution of these dislocations explains why the critical current increases as the cross-sectional area of the specimen (Lock, 1961a; Hauser and Buehler, 1962). The ability of some superconductors to carry high current densities in high fields, of course, suggests their use in the winding of magnets. Yntema (1955) described a superconducting solenoid wound with niobium wire and producing up to 7 kgauss, but this received little attention. In 1960 Autler wound a niobium solenoid creating a field Superconducting devices 157 of 4-3 kgauss, and since then the interest in the subject has grown explosively, with much scientific and technical activity in a large number of laboratories. Kunzler?/ al. (196 la) and others used Mo 3 Re to wind solenoids producing up to 1 5 kgauss; much higher fields were achieved soon thereafter as a result of work with Nb 3 Sn (Kunzler et al. 1961b), Nb 2 Zr (Kunzler, 1961; Berlincourt et al., 1961) and NbTi (Coffey et. al., 1964). Solenoids wound of these materials have produced fields up to 100 kgauss, and both suitable superconducting wire as well as entire solenoid assemblies have become commercially available. At the moment, the size of these is still measured in inches, but large-scale superconducting coils producing fields well in excess of 100 kgauss seem quite feasible. Kropschot and Arp(1961) have recently reviewed the subject of super- conducting magnets, and have discussed the considerable technical and economic advantages of such devices. Much information can also be found in [11] as well as in Berlincourt (1963). 13.3. Superconducting computer elements Much research and development work is currently being devoted to attempts to use superconductors both as switching devices and as memory storage elements in electronic computers. The basic idea for a superconducting switching element originated with Buck (1 956) who invented the cryotron. This consists of a layer of thin (0003 in.) niobium wire wound on to a thicker (0009 in.) tantalum wire. A sufficiently large current through the former, called the control winding, can quench the superconductivity of the latter, called the gate. The two materials are chosen because the convenient operating temperature of 4-2°K is only a little below the critical temperature of Ta, but much lower than that of Nb, so that a control current sufficient to 'open the gate' is still much less than the critical cur- rent of the control. The diameter of the gate is furthermore kept large so as to maximize the amount of gate current, I g , which can be controlled by the control current, I c . Calling H c the critical field of the tantalum gate at the operating temperature, and D its diameter, then (/,)m« = H c ttD, (xni.i) 158 and Superconductivity *c t n (XIII.2) where n = number of turns/unit length of control winding. Thus (XIII.3) This is the 'gain' of the cryotron, which must be kept at a value greater than unity in order that the gate current of one cryotron can be used to control another. .+ ;-! Fig. 48 A great variety of logical circuits can be built up by making use of this reciprocal control of a number of cryotrons. Most of these cir- cuits contain the basic flip-flop or bistable element, shown in Figure 48. Current through this element can flow in either one or the other branch and, once established in one, will flow in it indefinitely since it makes the other one resistive. The choice of branch can be dictated by placing a further cryotron gate in series with each branch, and con- trolling this by an outside signal, which can 'open the gate', making the corresponding branch resistive and forcing the current into the other path. This is shown in Figure 49, which also indicates that if each branch also controls the gate of a read-out cryotron, the position of the bi-stable element can be read. Figure 50 shows other basic logical circuits using cryotrons; the current through the heavy line Superconducting devices 1 59 flows only if: (a) cryotron A or B is open, (b) cryotrons A and B are open, (c) neither A nor B are open. More complicated logical circuits are discussed by Buck (1956) as well as in review articles by Young (1959), by Haynes (1960), and by Lock (1961b). Basically all these cryotron circuits consist of a number of parallel superconducting paths between which the current can be switched by the insertion of a resistance into the non-desired branches. Under steady-state conditions the power dissipation is zero as long as there is always at least one path which remains superconducting. The speed READ "ZERO" INPUT "ZERO" 37 INPUT 1 "ONE" § * V ± Fig. 49 READ "ONE" with which the resistance can be inserted, that is, the speed with which a given gate can be made normal, depends on the basic phase transi- tion time and is small enough ( as 10~ 10 sec) not to be a limiting factor at this time (see, for instance, Nethercot, 1961 ; Feucht and Woodford, 1961). On the other hand, the switching time from one current path to another is determined by the ratio L/R, where L is the inductance of the superconducting loop made up of the current paths, and R the resistance introduced by an opened gate. The usefulness of wire- wound cryotrons is severely limited by the fact that this time is no less than 10~ 5 sec, even if the gate consists of a tantalum film evaporated on to an insulating cylinder. Because of this all current research and development effort is directed toward making thin film cryotrons 160 Superconductivity consisting of crossed or parallel gate and control films separated by insulating layers, and placed between additional superconducting shielding films called ground planes. The resistance of the thin film gates is comparable to that of a wire gate, but the ground planes con- fine magnetic flux to a very small region and thus result in L/R values of the order of 10" 8 -10~ 10 sec. Cryogenic loops with a time constant of 2x 10 -9 sec have been operated (Ittner, 1960b). An account of many of the design considerations governing such thin film cryotrons can be found in several papers in [9]. INPUTS a\or IB _<ztXp xr5=-fo (neither B T T A c\ a — cnto QXlo — t nor B -^^Zpr*^ Fig. 50 Suggestions for superconducting memory devices were advanced simultaneously by Buckingham (1958), Crittenden (1958), and Crowe (1958). Their devices are basically quite similar and make use of the fact that a current induced in a superconducting ring will persist in- definitely. Since the current can circulate either way one has the possibility of a two-state memory storing one bit of information with no dissipation of power other than that required to maintain the low temperature. Of the three suggestions it is that of Crowe on which in recent years most attention has been concentrated and which will be briefly described here. Before doing so it might be noted that per- sistent current memory devices have in common with switching cryotrons that a current in one superconducting circuit quenches the Superconducting devices 161 superconductivity in another. There is, however, no need for a greater-than-unity gain, as the controlled current is not in turn used to drive another unit. One therefore often calls the memory elements low gain cryotrons. The Crowe cell basically consists of a thin film of superconducting material (for example, lead) with a small hole, a few millimetres in diameter, which has a narrow cross-bar running across it. This is shown schematically in Figure 51. A drive 'wire' in the form of a second narrow strip lies just above the cross-bar, separated only by a thin insulating layer. As long as the entire configuration remains superconducting, the magnetic flux threading the hole must retain its SENSE WIRE. Fio. 51 original value, which we shall take to be zero. Therefore if a current is passed through the drive wire, it will induce currents in the cross-bar and the remainder of the film. The direction of this induced circu- lating current will be such as to keep the flux from penetrating, and results in a flux distribution indicated in Figure 52a, which shows a schematic cross section of the cell. The cross-bar is very thin and narrow and therefore has a low critical current. When the induced current exceeds this critical value, the cross-bar becomes normal. The flux now changes to the configuration shown in Figure 52b, as the remainder of the film remains superconducting. If finally the drive current is again removed, the superconductivity of the cross-bar is restored, and now the flux threading the hole is trapped, as long as the cross-bar remains superconducting, by a persistent current which is in the opposite direction of the originally induced flow. Even when 162 Superconductivity the drive wire current is now removed, the flux distribution remains that of Figure 52c. The idealized operation of a Crowe cell (Garwin, 1957) is indicated in Figure 53, which shows on equal time scales, but arbitrary vertical scales, the drive current I d , and the cross-bar current / c . Pulse 1 is too small to induce a critical value of I c . Pulse 2 results in I c > I cril ; the cross-bar becomes momentarily normal, and after the drive pulse is removed a persistent current l v is stored. Pulse 3 is now a 'read- out' pulse which has no effect since it induces a current in a direction opposite to that of the persistent current. With pulse 4, however, the persistent current is reversed, storing the other possi bility of the two- state memory, and now read-out pulse 5 succeeds in driving the cross-bar well beyond the critical value. Note that this is a destructive read-out. The memory is sensed by means of a wire below the cross-bar, also very close to it but electrically insulated. A current pulse will be in- duced in the sense wire because of its proximity whenever the flux linking the cross-bar changes, that is, whenever the cross-bar becomes normal. Thus we note on Figure 53 that the sense wire response I s to pulse 3 is nothing, which can be taken as ' Read 0', while its response to 5 is a pulse which can be taken as 'Read 1 '. Superconducting devices 163 The operation of the Crowe cell is rendered more complicated than is indicated in the preceding simplified account because the cross-bar heats up through joule heat when it becomes normal, and the thermal recovery time may be appreciable. Crowe (1957), Rhoderick (1959), Von Ballmoos (1961), and several papers in [9] discuss the resulting complications. Id _ STORE READ STORE READ "0" "1" STORED STORED "0" "0 - V v-- STORED "I" k READY K READO" IV Fig. 53 Crowe cells can be arranged into a two-dimensional matrix of memory elements with the drive wire forming part both of an x- and a y-circuit, as indicated in Figure 51 . Driving pulses I x , I y are then so chosen that either alone is not sufficient to activate the device, but that both together do. The reader is again referred to [9] for a number of papers on superconducting memories built up of such matrices. Rose-Innes (1959) has estimated the consumption of liquid helium required to keep a memory like that cold, and finds this to be of the order of two litres per hour for an array of one million cells. This is well within the capacity of closed cycle helium refrigerators such as the one described by McMahon and Gifford (1960). Bibliography General References [I] shoenberg, d., Superconductivity, Cambridge University Press, 1952. 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Index Adiabatic Magnetization, 15 Anisotropy of energy gap and BCS theory, 131 decrease with impurity, 97, 100, 106, 142 deduction from infrared absorption, 100 nuclear spin relaxation, 106 specific heat, 96 thermal conductivity, 92 ultrasonic attenuation (table), 138 Anomalous skin effect, 44, 101 , 103 Anomaly in lattice specific heat, 9 Atomic mass, effect on T c , 6 {See also isotope effect) Atomic volume, effect on T c , 6 Bardeen-Cooper-Schrieffer (BCS) theory, 12, 117-40, 150 (See also energy gap, interac- tion parameter V, quasi particles) and G-L theory, 52, 139 Pippard non-local relations, 44 anomalous skin effect, 103 basic hypothesis, 120 coherence effects, 136-8 collective excitations, 100 critical current and field in thin films, 80 critical field, 130 critical temperature, 129, 143 electromagnetic properties, 103, 114, 139-40 electron -phonon interaction, 118-20, 125 ground state energy, 120-6 183 Bardeen-Cooper-Schrieffer (BCS) theory - cont. high frequency conductivity, 103, 114 Knight shift, 139^10 nuclear relaxation rate, 104-6, 137 penetration depth, 37, 52, 130 range of coherence, 48, 115 similarity principle, 122, 130 specific heat, 97, 130, 134-6 thermal conductivity, 91, 92, 136 thermal properties, 127-36 weak coupling limit, 131 Bulk modulus, 16 Coefficient of thermal expansion, 17 Coherence : see range of coherence Coherence effects, 136-8 Cold work, 143 Collective excitations, 100 Colloidal particles, 29, 35, 46 Compressibility, 17 Condensation energy, 19, 85, 91, 117, 126 Condensation of electrons in mo- mentum space, 12, 14, 19, 21, 31, 41 Cooper pairs, 12, 32, 52, 118-21, 125, 140, 144, 151 Critical current, 4, 80, 155, 161 Critical magnetic field, 4 in BCS theory, 130 in G-L theory, 51 of lead and mercury, 132 of small specimens, 46, 58, 75-8, 80, 109, 155 of superconducting elements (table), 5 of thin films, 46, 76-7, 80, 109 184 Index Critical magnetic field - cont. precise measurements, 83-5 pressure effects, 16—17 relation of thin film value to A 6 and lo, 77 relation to thermal properties, 13-19,21,86, 132, 135 similarity of reduced field curves, 4, 85, 135 temperature dependence, 4, 18, 84-6, 130, 135 very high values, 27, 70-1, 109, 155 Critical field for supercooling, 66, 74,76 for surface superconductivity, 74 Critical temperature, 3 {See also isotope effect) dependence on atomic mass, 6 atomic volume, 6 discontinuity of specific heat, 9, 15, 17, 132, 135 effect of magnetic impurities, 145-50 effect of non-magnetic impurities, 141-4 in BCS theory, 129-43 Matthias' rules, 6, 141, 148 of superconducting elements (table), 5 Crowe cell, 161-3 Cryotron, 157-63 Cylindrical specimens, 13, 16, 24 {See also thin wires) Debye temperature, 10, 87, 124-5, 131, 134, 143 Demagnetization coefficient, 23 Density of electron states, 104-8, 123, 127, 129, 134, 143 Diffusion, 152 Dilute alloys critical temperature, 141-4 magnetization curve, 144-5 Dilute alloys - cont. specific heat, 143 thermal conductivity, 87-8, 90 variation of surface energy, 59, 70 Effective charge, 49, 52 Elastic properties, 9 Electron irradiation, 143 Electron-electron interaction, 12, 82-3, 95, 117-25 Energy Gap {See also anisotropy of energy gap) correlation with TJ9, 134 deduction from infrared absorption, 98-100 infrared transmission, 113-14 microwave absorption, 100-4 nuclear spin relaxation, 104-6 specific heat, 11,91,96-7 thermal conductivity, 92, 95 tunnelling, 79, 106-9 ultrasonic attenuation, 137-8 dependence on field, 79-80, 92, 109, 139 phonon spectrum, 132 position, 150-2 quasi-particle energy, 131-3 size, 79, 139 temperature, 90, 103, 108, 129, 139 in BCS theory, 120, 126-34, 139 in thin films, 79-80 of superconducting elements (table), 99 relation to G-L order parameter, 52, 79, 139 Meissner effect and perfect conductivity, 110, 114-15, 120 penetration depth, 1 1 5 range of coherence, 44 Thomson heat, 95 Entropy, 14, 18,38,78,128 Index 185 Ferrell-Glover sum rule, 114-15, 150 Ferromagnetism, 145-8 Flux creep, 73 Flux quantization, 32-3, 72, 120 Free energy, 13-4, 19-20, 47-8, 57, 75, 93, 128, 148-9 Gapless superconductivity, 149 Gauge invariance in BCS theory, 139 in G-L theory, 49 Geometry, influence of, 23-6 Ginzburg-Landau (G-L) theory, 12, 48-54, 150 basic equations, 50, 139 critical field of small specimens, 75-7 extension to lower temperatures, 49,80 free energy, 48-9, 57, 75 limitations, 49, 52-3, 66, 80 non-local modifications, 48, 50 range of coherence, 58 relation to BCS theory, 48, 52, 139 London equations, 50, 54 small specimens, 76-80 superconductors of second kind. 67-72 supercooling, 65-7 surface energy, 57-9 G-L order parameter, 48 and free energy, 48-50, 57 effect of magnetic field, 53, 78, 139 gradual spatial variation, 49-50, 57-8, 73 proportionality to energy gap, 52, 79, 139 relation to penetration depth, 49, 53^, 78 G-L parameter «-, 5 1 -3, 58, 66, 68-70 and range of coherence, 58 G-L parameter k - cont. critical value for negative surface energy, 59, 66, 67, 70 deduction from penetration depth, 51-3 supercooling, 51-3, 66 in thin films, 54, 78 relation to normal conductivity and specific heat constant, 69 surface energy, 58 temperature dependence, 70 Gorter-Casimir thermodynamic treatment, 11, 13-19 Gorter-Casimir two-fluid model, 11, 19-21 {See also two-fluid model; two-fluid order parameter) application to G-L theory, 49 relation to penetration depth, 36 Gyromagnetic ratio, 22 Impurity effects : see mean free path effects Infrared absorption, 98-100 Infrared transmission, 45, 99, 109- 114, 115 Interaction parameter V, 121 anisotropy, 131 BCS cut-off, 121, 124-5, 131 effect of non- magnetic impurities, 1 43-4 magnetic impurities, 149 influence on isotope effect, 124-6 quasi-particle lifetime effects, 124-5, 131 variation with quasi-particle energy, 131 Intermediate state, 14, 23-6, 29, 59-61,94 Isotope effect, 12, 81-3, 95, 117, 124-6 absence in transition metals, 12, 82, 125 186 Index Isotope effect - cont. effect of quasi-particle life time, 124-6 in the BCS theory, 124 table of values, 82 Josephson effect, 109, 133-4 Knight shift, 77, 139^K) Kramers-Kronig relations, 112, 114 Latent heat, 1 5, 78 Lattice parameters, 10 Laves compounds, 147-8 Lifetime effects, 124-6, 131 Localized magnetic moment, 145-6 London theory, 1 1 , 28-32, 36, 41-2 basic equations, 29, 42, 44,46, 1 1 1 incorrect values of penetration depth, 29, 37-8, 43 microscopic implications, 11, 31, 43 non-linear extension, 38 prediction of penetration depth, 29,36 Low frequency behaviour diamagnetic description, 22-6 influence of geometry, 23-6 relation to high frequency re- sponse and energy gap, 1 14-15 small specimens, 75-80 Magnetic field distribution, 7-8, 22-4 Magnetic field dependence of energy gap, 79-80 entropy, 37 free energy, 13-14, 49 G-L order parameter, 53, 78 penetration depth, 35, 38-9, 53, 76 Magnetic field penetration: see penetration depth Magnetic susceptibility, 14, 23, 34-5, 75, 77, 83 Magnetic impurities, 145-50 Magnetization area under magnetization curve, 16, 26, 75 dilute alloys, 144 filamentary superconductors, 78 ideal superconductors, 13-14, 24-6,68 small specimens, 75 superconductors of second kind, 68-9, 156 Magnetostriction, 16 Matthias' rules, 6, 141, 148 Mean free path effects on anisotropy of energy gap, 97, 100 106, 142 critical temperature, 141-4 G-L parameter k, 69 infrared absorption, 100 nuclear relaxation rate, 106 penetration depth, 35, 38, 41-3, 45-6,58, 112 range of coherence, 42-3, 45-7, 152 surface energy, 58-9, 70, 77 Mechanical effects, 16-17, 143 Mendelssohn 'sponge', 78, 155 Microwave absorption, 100-4 Mixed state, 71-74 Neutron bombardment, 143 Nuclear spin relaxation, 104-6 Nuclcation of superconducting phase, 61-3 Order in the superconducting phase, 14, 19 Order parameter, see G-L order parameter; two-fluid order para- meter Penetration depth, 29, 58 defining equations, 28, 34, 36 Index 187 Penetration depth - cont. dependence on field direction, 39, 53 frequency, 39 magnetic field, 35, 38-9, 53, 76 mean free path, 35, 38, 41-3, 45-6,58,112 range of coherence, 43, 46, 112 size, 38, 46, 79 temperature, 35-8, 51-3, 130 in BCS theory, 37, 52, 130 in Pippard theory, 45-6, 112 incorrectness of London values, 29,37-8,43, 112 methods of measurement, 35-6 relation to energy gap, 115 entropy, 38 frequency variation of con- ductivity, 111-2 G-L order parameter, 51, 78 surface energy, 55-7 susceptibility, 34-5 thin film critical field, 77 values in superconducting ele- ments (tables), 38, 65 Perfect conductivity of supercon- ductors, 4, 29-30, 114-15 Perfect conductor, 4, 6-8, 27-8 Persistent current, 3, 6, 26, 158-60 Phase propagation, 63-5 Phonon spectrum, 132 Pippard non-local theory, 12, 41-6 (See also range of coherence) basic equations, 42, 44, 45 critical field in thin films, 46 field penetration through thin films, 45 penetration depth, 42-3, 45-6 reduction to local form (London limit) 12, 45-6 relation to energy gap and BCS theory, 44-5 susceptibility of thin films, 77 Pressure effects, 12, 16-17 Quantized flux, 32-3, 72, 120 Quasi-particles, 124-5, 132 Quenching, 143 Range of coherence, 1 1 and superimposed metals, 150-1 dependence on mean free path, 42, 45-7, 152 in BCS theory, 44 in G-L theory, 49, 58 relation to energy gap, 44 field dependence of penetra- tion depth, 40 mean free path effect on T c , 142 penetration depth, 40, 44-6 sharpness of transition 40-1 surface energy, 57 uncertainty principle 40, 45 values for Al, In, Sn (table), 65 Relation between magnetic and thermal properties, 13-21, 86, 96, 132, 135 Rutgers' relation, 15, 17 Semiconductors, superconducting, 6 Silsbee's rule, 5 Similarity, 85-6, 96, 116, 121-2, 130 Size effect on critical field, 46, 76-7, 155 critical supercooling field, 67 critical temperature, 143 energy gap, 79-80 magnetic susceptibility, 34 penetration depth, 38, 46, 79 range of coherence, 46 Skin depth, 35-6, 43 Small specimens critical field, 46, 75-8 in G-L theory, 46, 54, 67, 75-80 in Pippard theory, 46 low frequency behaviour, 75-80 188 Index Small specimens - cont. penetration depth, 35-8, 43, 45-6, 79 range of coherence, 45-6 Sommerfeld specific heat constant, 9 in dilute alloys, 143 independence of isotopicmass,85 relation to critical field, 19-21,135 G-L parameter *, 69 Specific heat of the electrons comparison of magnetic and calorimetric data, 17, 19-21, 86, 134, 135 dependence on temperature, 9, 11,18,20-1, 86,91,96-7,134 discontinuity at T c , 9, 15, 17, 78, 132, 135 in BCS theory, 130, 135 relation to critical field, 15, 17, 21,86, 96 energy gap, 11, 90, 96-7, 132 thermal conductivity, 91, 95 Rutgers' relation, 15, 17 Specific heat of the lattice, 9-10, 18 Spherical specimens critical field of small spheres, 76 magnetization, 8, 26 penetration depth of small spheres, 29, 35, 46 supercooling in small spheres, 67 Spin, effect on T c , 147 Strain, 26-7, 62, 77, 155 Superconducting alloys and com- pounds, 5-6 dilute alloys, 58-9, 87-90, 141-5 ferromagnetism, 145-8 high critical fields, 154-6 Laves compounds, 147-8 magnetic impurities, 145-50 Matthias' rules, 6, 141 non-magnetic impurities, 141-5 rare earth and transition metal solutes, 145-9 thermal conductivity, 88-9 Superconducting devices cavities, 154 computer elements, 157-64 d.c. amplifiers, 153-4 galvanometers, 153 heat switches, 94, 1 53 leads, 153 magnets, 78, 154-7 memory devices, 1 60-4 radiation detectors, 154 rectifiers, 154 reversing switches, 154 Superconducting elements (table), 5 Superconducting filaments, 78, 155 Superconducting ring, 3, 8, 26, 30, 32 Superconducting transition contrast with perfect conductor, 6-8 discontinuity of specific heat, 9, 15, 17, 78, 134 entropy, 14 free energy, 13-14,48 in dilute alloys, 144 in thin films, 78-81 length and volume changes, 16-17 order, 72, 78-81 reversibility, 7, 13, 17, 144 speed, 159 Superconductors of second kind, 67-73, 155-6 Supercooling, 52-3, 61-3, 65-7, 74 Superheating, 61 Superimposed metals, 150-2 Surface currents, 22 Surface energy, 55-74, 150, 156 dependence on temperature, 64-5 effect of strain, 62, 77, 156 in G-L theory, 57-8 in inhomogeneous specimens, 77, 156 Index 189 Surface energy - cont. in Pippard theory, 56-7 mean free path effect, 58-9, 67, 77 negative values, 58-9, 62, 67, 69, 70, 157 relation to intermediate state, 59 phase nucleation and propa- gation, 61-6 range of coherence, 56-8 values for AI, In, Sn (table), 65 Surface impedance, 35-6, 52-3, 95, 101-4, 111, 139 Table of critical fields and temperatures, 5 energy gap values, 99 energy gap anisotropy, 138 isotope effect exponents, 82 penetration depth values, 38, 65 ranges of coherence, 65 specific heat discontinuities, 17, 135 superconducting elements, 5 surface energies, 65 Temperature dependence of critical field, 4, 18, 84-6, 130, 135 energy gap, 90, 103, 108, 130 G-L parameter k, 70 penetration depth, 35-8, 51-2, 130 specific heat, 9, 11, 18, 20-1, 86, 91,96-7, 130, 134 surface energy, 64-5, 70 surface impedance, 101-3, 139 thermal conductivity, 87-94, 136 two-fluid order parameter, 19-21, 36 Thermal conductivity, 87-94 in BCS theory, 92, 136 of thin films, 79, 93 Thermal conductivity - cont. relation to energy gap, 91-3, 95 gap anisotropy, 92 specific heat, 91 Thermal expansion coefficient, 17 Thermodynamics of superconduc- tors BCS theory, 128 G-L theory, 48-9 Gorter-Casimir treatment, 11, 13-19 relation between magnetic and thermal properties, 13-21, 86, 96, 132, 135 Thin films (See also superimposed metals) critical current, 80 critical field, 47, 75-80 critical thickness for second order transition, 78 cryotrons, 157-63 energy gap, 79, 93 infrared transmission, 46, 109-15 in G-L theory, 54, 75-80 in perpendicular field, 73 magnetic behaviour, 33, 73-4, 75-80 penetration depth, 29, 35-6, 46, 79 relation of critical field to X b and &»77 second order transition, 78 supercooling, 67 susceptibility, 34-5, 76, 77 thermal conductivity, 79, 93 total field penetration, 36, 45 variation of G-L order para- meter, 78 Thin wires, 29, 35, 67, 76 Thomson heat, 95 Threshold magnetic field: see criti- cal magnetic field Time-reversed wave functions, 144 190 Index Transition metals absence of isotope effect, 12, 82, 125-6 effect on T c , 145-50 Trapped flux, 3, 26-7, 32, 120, 144, 161 Tunnelling, 106-9, 132-4 Two-fluid model (See also Gorter-Casimir two- fluid model) and BCS theory, 127 extension of G-L theory, 49 relation to nuclear spin relaxation, 104 penetration depth, 36 thermal conductivity, 87 Two-fluid order parameter, 19-21, 36 gradual spatial variation, 40, 56-7 relation to penetration depth, 36 surface energy, 56-7 thermal conductivity, 89 rigidity in London theory, 39 Ultrasonic attenuation, 105, 136-8 Uncertainty principle, 31, 40, 45 Valence elections, effect on T c . 143, 145 Vortex lines, 72-4 6, s> LIVERPOOL turn this book to th 2 the last date si Monographs on Physical Subjects — continued IONIZATION AND BREAKDOWN IN GASBS F. Llewellyn Jones LOW TEMPERATURE PHYSICS L. C. JaCKSOl magnetic amplifiers George M. EtringC magnetic materials F. Brailsford MASERS AND LASERS G. J. F. TrOUp THE MEASUREMENT OF RADIO ISOTOPBS Taylor MBCHANICAL AND ELECTRICAL VIBRATION J. R. Barker mercury arcs F. J. Teago THE METHOD OF DIMENSIONS Alfred W. P MICROWAVE LBNSBS J. Brown microwavb spectroscopy M. W. P. Stran MOLECULAR BEAMS K. F. Smith NUCLEAR RADIATION DBTBCTORS J. Sharp the nuclear reactor Alan Salmon optical masers O. S. Heavens ORDER-DISORDER PHENOMENA E. W. ElCQ PHOTONS AND ELECTRONS K. H. Spring PHYSICAL CONSTANTS W. H. J. Childs physical formulae T. S. E. Thomas THE PHYSICAL PRINCIPLES OF WIRBLBSS J. A. Ratcliffe PRINCIPLES OF APPLIED GEOPHYSICS D. Parasnis relativity physics W. H. McCrea sbismology K. E. Bullen SBMI-CONDUCTORS D.A.Wright SHOCK TUBES J. K. Wright THB SPECIAL THBORY OF RELATIVITY M< Dingle superconductivity Ernest A. Lyman THE THBORY OF GAMBS AND LINEAR gramming S. Vajda THERMIONIC VACUUM TUBBS W. H. Aldoi Sir Edward Appleton thermodynamics Alfred W. Porter wave filtbrs L. C. Jackson wave guides H. R. L. Lamont WAVB MBCHANICS H. T. Flint x-ray crystallography R. W. James x-ray optics A. J. C. Wilson Printed in Great Britain Monographs on Physical Subjects o ALTERNATING CURRENT MEASUREMENTS David Owen APPLICATIONS OF INTERFEROMETRY W. Ewait Williams CQ APPLICATIONS OF THERMOELECTRICITY H. J. Goldsmid ^J atmospheric electricity B. F. J. Schonland ATMOSPHERIC TURBULENCE O. G. Sutton ^3 atomic spectra R. C. Johnson ^J cartesian tensors George Temple classical mechanics J. W. Leech THB CONDUCTION OF ELBCTRICITY THROUGH GASES K. G. Emeteus the cosmic radiation J. E. Hooper and M. Scharff dielectric aerials D. G. Kiely dipole moments R. J. W. le Fevre the earth's magnetism Sydney Chapman ,—4 ELASTICITY, FRACTURE AND FLOW J. C. Jaegar THB electric arc J. M. Somerville C^ elements OF pulse circuits F. J. M. Farley ^3 BLBMENTS OF TENSOR CALCULUS H. Lichnerowicz ^_{ fluid dynamics G. H. A. Cole ^t frequbncy modulation L. B. Arguimbau and R. D. Stuart FRICTION AND lubrication F. P. Bowden and D. Tabor FUNDAMENTAL OF DISCHARGE TUBE CIRCUITS V. J. Francis gbnbral circuit theory Gordon Newstead THE GENERAL PRINCIPLES OF QUANTUM THEORY G. Temple glass G. O. Jones heaviside's electric circuit theory H. J. Josephs high energy nuclear physics W. Owen Lock HI-GH FREQUENCY TRANSMISSION LINES Willis Jackson INTEGRAL TRANSFORMS IN MATHEMATICAL PHYSICS C. J. Tranter AN INTRODUCTION TO ELECTRONIC ANALOGUE COMPUTERS M. G. Hartley AN INTRODUCTION TO ELECTRON OPTICS L. Jacob AN INTRODUCTION TO FOURIER ANALYSIS R. D. Stuart AN INTRODUCTION TO THE LAPLACE TRANSFORMATION J. C. Jaegar AN INTRODUCTION TO SERVOMECHANISMS A. Porter AN INTRODUCTION TO VECTOR ANALYSIS B. Hague AN INTRODUCTION TO TENSOR CALCULUS AND RBLATIVITY Derek F. Lawden AN INTRODUCTION TO PHASE-INTEGRAL METHODS J. Heading [continued on back flap] W