Some facts about superconductivity

Gorter, C. J. 1953

Physica XIX 745-754 Lorentz

Kamerlingh Onnes Conference

SOME FACTS ABOUT SUPERCONDUCTIVITY

by C. J. GORTER

Suppl. No. 1058 of the Communications from the Kamerlingh Onnes Laboratorium, ieiden, Nederland

1. It has been remarked repeatedly that superconductivity presents at least a number of formal analogies with ferromagnetism. In reviewing the facts and theories of ferromagnetism one may focus attention upon three different groups of data and theories.

First one may be interested in ferromagnetism at the absolute zero of temperature. One may ask which metals are ferromagnetic at that temperature and may note the magnitude of the spontaneous magnetization of elements and compounds. The theoretician may investigate the nature of the interaction between electronic spins which leads to their partial parallelism and he may propose explana- tions for the magnitude of interaction energies and spontaneous magnetization.

Secondly one may study ferromagnetism as a function of temperature. The spontaneous magnetization decreases with rising temperature and finally vanishes at the Curie point, and this decrease is accompanied by an extra specific heat which is quite pronounced just below the Curie temperature. In theory we have a problem of statistical mechanics which presents an analogy to the order-disorder problem in binary alloys. An approximate but remarkably success- ful solution of the statistical problem was given by W e i s s long before any understanding of the more fundamental problems of group one had been obtained.

Thirdly one has the various problems of the technical magnetization curve. The part played by crystalline anisotropy, inhomogeneities and strains is important while also the behaviour of thin films and fine powders as well as ferromagnetism in alternating fields may be included in this third group. The theory considers Weiss domains and Bloch walls and the interplay of the energies due to anisotropy, inhomogeneities, magnetostriction, sample boundaries, etc.

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Physica XIX 47.

746 C. J. GORTER

2. Our knowledge on superconductivity, which is as yet much more superficial, may similarly be divided into three chapters.

As far as we know superconductivity occurs in the neighbourhood of T = 0°K in 21 pure metals, a number of their alloys and in some compounds containing Bi, W, Cu or MO. Monovalent metals and ferromagnetics are not represented among them and it has been remarked that metals with a high valency electron density and nevertheless a small electrical conductivity at high temperatures are particularly apt to become superconductive l).

Upon application of a magnetic field higher than the threshold field, H,hr.o, a superconductive metal goes over into the normal phase. According to Nernst’s law the heat of transformation is zero and the magnetic work H&r,o V/81-c is equal to the energy difference AU, between the normal and the superconductive phase 2). This energy difference is about lo5 times smaller than the corresponding energy in the ferromagnetic case. The fact that superconductivity persists at temperatures about lo3 times AU,/R (see next section) indicates that only a small fraction of the valency electrons is engaged 3, 4). The study of different isotopic mixtures has revealed that A U, NM -‘, where M is the average atomic weight of the mixture “).

The ohmic resistance is certainly less than 10-i’ times that in the normal state, hence it seems reasonable to consider E = 0 as cha- racteristic for the steady situation in a superconductor.

The Meissner effect 6, would indicate that also B = 0 ‘). It is clear, however, that in order to screen off an external magnetic field a superconductive current of finite thickness is required. If we sup- pose this is due to ?z frictionless electrons per unit volume and thus

dJ/dt = (ne”/m) E, (1) where J denotes the local current density and na, e and c have the usual meaning, the thickness of the layer is 8)

1 = (mc/4n ne2)$. (2)

Thus a rapid change of the conditions must occur near the boundary of a superconductive region. F. and H. L o n d o n 9) proposed considering

curl J = - (ne2/m) B (3)

as the fundamental equation which replaces Ohm’s law in superconductors. Taking (1) into account, the statement that B = 0 at a

SOME FACTS ABOUT SUPERCONDUCTIVITY 747

distance much larger than 1 from the’boundaries of a superconductive region is, of course, equivalent with (3). Since 1 is of the order of low5 cm it is difficult to verify (3) accurately 10). Experimentally in particular the situation in single crystals is not yet clear. F. L o n- d o n 11) has pointed out that (2) would follow from the supposition of a long range order of the momentum vector p = VW + (e/c) A. Different kinds of interaction between electrons have been proposed, without startling success. The suggestion of Fr ii hlic h. and B a r d e e n l), however, that elastic lattice waves are responsible for the interaction concerned, is supported by the remarkable pro- portionality of AU, and iPi-‘.

3. At the lowest temperatures the specific heat of the superconductive phase is much smaller that that of the normal phase, yT, but it rises much more steeply. At T M 1.1 (A U,/r)” the two specific heats are equal, while at the transition temperature T a 2(4U,/~)~ the specific heat *) in zero field falls discontinuously from about 2.7 yT, to yT, l2 ) 13). It . is remarkable how similarly many superconducting metals behave in these respects 14). If an external field H is introduced, the difference of the free energies, AF, of the two phases is decreased by H2V/8n; the transition temperature is accordingly decreased, and, since AF then becomes zero with a negative tangent, there is a heat of transformation 2). Fig. 1 gives the difference in free energy, AF, the difference in entropy, AS, and the threshold field as functions of the temperature. The most striking point of the thermal behaviour is that at T, both the energy and the entropy of the superconductive phase approach the corresponding quantities of the normal phase, suggesting that just below T, the superconductive phase differs very little t) from the normal phase 4).

A somewhat similar situation is known to occur in ferromagnetics in which at the Curie temperature the last trace of spontaneous magnetization disappears as well as in certain binary alloys with an order-disorder transformation point. More or less in analogy with spontaneous magnetization and the degree of order, an internal parameter has been introduced which only has a physical meaning

*) These specific heats, of course, do not include the Debije specific heat of the lattice. 7) The thermal conductivity, which will be discussed separately by Dr. 11 e II d e I s-

s o h n, is lower in the superconductive than in the normal phase, having also no jump

at T,

748 C. J. GORTER

in a well-defined interval of values. This internal parameter then is supposed to adjust itself at a given temperature so as to minimize the free energy. In 1934 the following expression 4, was proposed for the free energy :

F = xAU, - $2 yT2,

where the internal parameter x should lie between 0 and 1.

AF

T

-T Fig. 1.

This expression leads, among other things, to

x - (T/T,)‘,

(4)

(5) c, = 3yT3/T$ (6)

which gives AC, = 0 at T = 0.58 T,, where

T, = 2(A U&J)“, (7) and to a parabolic threshold curve. All this is in approximate agreement with the actual caloric data. 1 --x apparently-indicates the degree of superconductivity. As long as 1 -x > 0 the superconductive electrons would short-circuit the normal electrons whose


number would be proportional to x. These early suggestions were remarkably confirmed by the observation that 1 increases rapidly when T approaches T, while n in (2) even appears to be proportional to 1 -x as given by (5) 15). On the other hand, the presence of normal electrons particularly just below T, was confirmed by H. L o n d o n’s 16) observation that an alternating electric field in the surface layer gives rise to dissipation of Joule heat. Besides 1, the mean free path of the normal electrons and the normal skin dkpth must be taken in consideration and so far no quantitative explanation has been proposed for P i p p a r d’s data in this latter field 17).

There are no thermo-electric forces in a superconductive circuit 18). According to Kelvin’s formulae therefore also the Peltier effect between two superconductors and the difference between the Thom- son coefficients of two superconductors should be zero. An experiment carried out by D au n t and M e n d e 1 s s o h n 19) indicates that also the Thomson coefficient in any superconductor is zero. Early Leiden observations of thermoelectric anomalies in the normal phase in the neighbourhood 20) of the threshold curve have not been confirmed by recent Washington 21) and Cambridge 22) d8ta. If these latter results are correct, the thermoelectric force per degree and the Thomson coefficient are approximately proportional to T in the normal phase and, of course, zero in the superconductive phase.

The only theory which describes the gradual disappeirance of the degree of superconductivity with increasing temperature is the treatment of Heisenberg and Koppess). It is assumed that interactions between electrons may lead to strong perturbations of the normal free electron states near the Fermi limit. This could lead to a separating out of a regular electron structure at the expense of the normal free-electron wave functions. The number of electrons in that structure which would be responsible for superconductivity is considered as an internal parameter, which in fact is proportional to our 1 - x, and the free energy is minimized with respect to variations of that parameter. A rather simple statistical treatment which has no direct connection with the nature of the interaction between the electrons, leads to x N T4 at relatively high temperatures, but to a much more rapid variation of x with T below about 0.5 T, 12).

4. In the preceding sections we have supposed the existence of a sharp magnetic threshold curve. In practice 24) we only encounter

750 C. J. GORTER

that in a few favourable cases, viz. sufficiently thick samples of very pure Sn, i$-Ig or a few other metals having a negligeable demagnetization factor in the direction of the outside field. Such very pure samples will carry superconductive currents as long as the magnetic field at the surface is.lower than the threshold field (Silsbee’s rule).

However, a less simple behaviour is found in the following cases: a) samples that are not very thick compared to I; we then find an

increase of the threshold field, combined with a decrease of the field corresponding to the threshold current ;

b) samples that do not have a negligeable demagnetization factor in the direction of the field. The field then starts to penetrate into the sample at a value lower than the threshold field “) g5) ;

c) samples of a number of other metals (Th, Ta, V, Ti, Cb) and many impure metals and alloys 26). Magnetic fields may penetrate into samples at relatively weak fields, but the last trace of superconductivity disappears at much stronger fields; moreover, upon redu- cing a strong field, magnetic flux may be frozen in.

Though the cases b) and c) differ in several respects we apparently have in both cases to do with a fine division of superconductive and normal regions. A division, analogous to that of the so-called inter- mediate state encountered in b), for that matter also occurs if a wire carries a current which is higher than the value indicated by Silsbee’s rule.

Just as in the case of ferromagnetism the boundaries between the regions appear to depend on inhomogeneities, strains, the outer shape and other secondary circumstances. With the surface energies this forms a complicated array of factors which is still quite con- fusing.

From the absence of large influence of an outside field on the high frequency losses in the boundary layer of a superconductor P i p- p a r d 27) has concluded that the gradient of the internal parameter x cannot be very high. It is, however, not excluded that considerable gradient variations of x inside the superconductive regions may occur so that the boundaries between normal and superconductive regions are less sharp than is usually assumed. Gradual variations of x may also be of importance for the behaviour of thin layers and wires (case a) in a magnetic field.

5. Summing up, we may say that superconductivity confronts us


still with a number of unsolved problems, It seems clear that some interaction between the electrons in the Fermi tail must be responsible. But it has not yet been proved that one of the interactions proposed may lead to equations (1) and (3). Perhaps we have to wait until that first fundamental problem is solved before the gradual decrease of the degree of superconductivity with rising temperature is understood and thus the internal parameter is interpreted.

The actual behaviour of samples and compounds also presents a number of complicated problems which have to do with fine ‘sub- divisions into superconductive and normal regions. It seems possible to attack those problems without fundamental knowledge on the nature of superconductivity, while on the other hand it is not excluded that the conclusions may elucidate certain of the more fundamental problems.

Received 9-5-53. REFEREKCES

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( 1952) 597. 13) W o r 1 e y, R. D., Z e m a n s k y, M. W., and B o o r s e, H. A., Phys. Rev. (2)

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752 C. J. GORTER

16) L o n d o II, H., Proc. roy. Sot. London A 170 (1940) 522. I 7) P i p p a r d, A. B., Proc. roy. Sot. London A 101 ( 1947) 370, 385, 399. i8) M e i s s n e r, W., Z. ges. Kllte. Ind. 34 (1927) 197. B o r e 1 i u s, G., K e e s o m,

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21) Steele, ~I.C.,Phys.Rev.(2)88(1951)262. Webber, R.T.,and Steele, M. C., Phys. Rev. (2) 70 (1950)\1028.

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Physik (6) 1 (1947) 405. 24) D e H a a s, W. J., G u i n a u, 0. A., and C a s i m i r-J o n k e r, J. M., Commun.

Suppl. No. 82d; Rapp. Commun. Lab. Kamerlingh Onnes September Congr. int. Froid No. 26 (1936); Act. 7e Congr. int. Froid 2 (1936) 236. S b o e n b e r g, D., Superconductivity (Cambridge 1938). V o n L a u e, M., Theorie der Suprnleitung (Berlin 1949). Lo n do II, F., Superfluids I (New York, 1950).

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26) M e n d e 1 s s o h II, Ii., and hI o o r c, J. R., Proc. ray. Sot. London A IsI (I 935) 334. G o r t e r, C. J., Physica 5 (1935) 449.

27) Pip p ar d, A. B., Proc. roy. Sot. London A 203 (1950) 210; Phil. Mag. (7) 41 (1950) 243 and 43 (1952) 273.

SIMON: In evaluating the isotope effect one has to bear in mind that a change in Debye 0 involves a change in volume. In the pressure effect also we have to consider the influence of the change of volume proper and of the corresponding change of 0. As volume and 0 depend on each other in a different way in the effects mentioned, it is possible to calculate the magnitude of the two components. It turns out that they both contribute in comparable degree to the pressure effect, while the component due.to the volume change affects the isotope effect only slightly (a few percent). Nevertheless this correction has to be applied if one wants to check the validity of F r 6 h 1 i c h’s relation between the mass of the isotope and the transition temperature.

SIMON: Prof.G.0. Jones andMrP.F. Chester ofQueenMary College, London University, have recently shewn that bismuth becomes superconducting at high pressures. They subjected bismuth to approximately hydrostatic pressure at room temperature and then cooled, under pressure, to liquid helium temperatures. At all pressures tried, between 20 000 and 40 000 atm, a superconductive transition was observed at about 7°K by the change in mutual inductance between two coils mounted round the specimen. At pressures below 20 000 atm no transition was observed down to 2.1”K. B r i d g m a n has shown that bismuth at room temperature shows the following polymorphic trahsitions: I-II at 25 000 atm and II-III at 27 000 atm. The crystal structure of the high pressure modific-


ation is not known. As there exists a slight pressure gradient in the sample no exact conclusions can be drawn, but it would appear that modification II and perhaps III are superconductive.

LAMB: Does T, usually depend on pressure to an appreciable extent? SHOENBERG: The coefficient dT,/d# is sometimes negative and some-

times positive. A pressure of 20 000 atm could cause a change of T, of the order of a few degrees. The occurrence of superconductivity in bismuth at high pressures discovered by J o n e s is however probably due to a modification as he suggests.

BROER: The example of the pressure shift of the freezing point of water shows that it is not safe to extrapolate even the sign of pressure coefficients to pressures of the order of 20 000 atm.

PIPPARD: H i 1 s c h has found that bismuth films deposited at 4°K are superconductive; superconductivity disappears on annealing.

FR~HLICH: What is the experimental accuracy of the &Y/M-law in the isotope effect?

PIPPARD: S h o e n b e r g , L o c k and I found the exponent of n/I in the isotope effect to be between 0.92 and 0.96, we did not feel that it could reasonably be pushed as high as 1. On the other hand the American workers believe it to be very close to unity.

CASIMIR: For which elements have measurements on the isotope effect been carried out so far? Has it been explained why experiments with lead from radioactive sources did not yield results ?

GORTER: The isotope effect has recently been studied in tin, mercury and thallium. The shift which could have been found in the early attempts to discover the effect are of the order of the experimental errors in those times.

FR~HLICH : What is the accuracy for the T3-law for the specific heat ? GORTER: Early investigations of K e e s o m and V a n L a e r on tin

showed small systematic deviations from the T3-law. Recent measurements on niobium carried out in Columbia University at relatively lower temperatures lead to larger deviations in the sense predicted by K o p p e’s formula (Cf. ref. l*) and I”)).

LONDON: I never understood the formula for the free energy

F = xAUo-&a+yT2.

I could understand that the power xi is adjusted to fit the facts. But y, being the Sommerfeld specific heat, would be then expected to be proportional to (1 - a) f. Instead, here y has been assumed to be constant.

CASIMIR: One may write also the equation as

F = y2AU,,-+yyT2

and say that y is the fraction of the Fermi surface that is still normal. The energy then is proportional to the square of this parameter. Perhaps this is slightly more plausible. Of course the theory remains essentially pheno- menological.

HEISENBERG: As to the use of the normal y, K o p p e’s idea was simply

Physica XIX 48

754 SOME FACTS ABOUT SUPERCONDUCTIVITY

to assume that in the superconductive state part of the surface of the Fermi sphere is covered with the condensed phase; therefore the number of free states and accordingly y is reduced. Straightforward calculation then yields a law for the specific heat very near that of G o r t e r’s theory. The difficulty of K o p p e’s hypothesis is the assumption that the condensation starts just at the top of the Fermi-distribution for T = 0. Even for higher temperatures a satisfactory explanation of this assumption has so far not been given, but it seems essential to ensure agreement with experiment.

HEISENBERG: To what accuracy does one know at present that the number of free electrons as derived from the first L o n d o n equation is identical with the number of free electrons as derived from the normal state (e.g. from the Hall effect)?

PEIERLS : In order to estimate the penetration depth in the L o n d o n formula as H e i s e n b e r g suggests, one should use the values of the specific heat coefficient which contains 12 and 112 and one other datum in addition.

LONDON : The &,,p8,C0P,d might be different from WZ,,~,~,~~. HEISENBERG: Isn’t it being too sceptical, to assume a difference between

the effective mass of the electron in the superconducting and the normal state ? The condensation changes so little in the general behaviour of the electrons, the number of electrons that condense on the surface of the Fermi sphere is so small, that I would not like a change of the effective mass.

SHOENBERG : The value of n/nz in the superconducting state at T = 0°K is about 3 times less than in the normal state.

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Some facts about superconductivity