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D. Shoenberg, Moscow

Chapter V. Thermodynamics of Superconductivity

The idea of applying thermodynamics to the transition from the superconducting to the normal state was first put forward by Keesom[^2], and then by Rutgers[^3], and was developed in detail by Gorter[^4]. At that time, however, it was believed that the transition in a magnetic field is essentially irreversible, since the superconductor was regarded as an ideal conductor (in the sense indicated in Chapter II), so that the surface currents associated with the field, upon the destruction of superconductivity, would disappear with the liberation of Joule heat. This presumed irreversibility made Gorter’s treatment doubtful, and it seemed surprising that his results agreed well with experiment. The reason for this agreement became clear subsequently, after the discovery of the Meissner effect, which showed that the disappearance of surface currents in a pure metal is in fact not connected with any irreversible transformations of energy (somewhat analogously to the disappearance of surface currents in a ferromagnet at the Curie point), so that Gorter’s basic assumption of reversibility proves to be correct.

The thermodynamics of the superconducting transformation in a magnetic field is entirely analogous to that of any other phase transition, and all the results can be obtained most simply by equating the free energies of the two phases. In order to determine the free energy of the superconducting phase in a magnetic field, it is most convenient to consider a long wire, since in this case the field at the surface of the superconductor is equal to the external field. If \(G_s\) is the free energy per unit volume in the absence of a field2, then in a field \(H\) it is equal to \(G_s - \frac{1}{2}HI\), where \(-\frac{1}{2}HI\) is the energy of magnetization. Since \(I = -\frac{H}{4\pi}\), the free energy

is \(G_s+\dfrac{H^2}{8\pi}\). The free energy \(G_n\) of the normal state is not changed in a magnetic field, since there is no magnetization in a magnetic field. When the field is made equal to \(H_c\), the free energies of both phases become equal, and the two phases can coexist in equilibrium with one another; at smaller fields there exists only the superconducting state, as corresponding to the smaller free energy, and at larger fields—only the normal phase. Thus the condition of phase equilibrium is

\[ G_n = G_s + \frac{H_c^2}{8\pi}. \tag{1} \]

Although we derived this condition by considering a long wire, it is in fact quite general. If a body has a demagnetizing coefficient different from zero, then the transition of the whole body from the superconducting to the normal state is spread over an entire interval of external fields; however, as we saw in Chap. III, during this transition the specimen breaks up into a large number of superconducting and normal regions, and the field at their boundaries is always equal to \(H_c\). A more detailed consideration, into which we shall not enter here, shows that equation (1) is the condition of phase equilibrium between superconducting and normal regions also in such a gradual transition.

From equation (1) we can derive the differences between all thermodynamic quantities of the two phases. Thus, differentiating with respect to temperature, we find the difference between the entropies of the normal and superconducting phases, since \(S=-\dfrac{\partial G}{\partial T}\). We have

\[ S_n - S_s = -\frac{H_c}{4\pi}\frac{dH_c}{dT}. \tag{2} \]

Since \(\dfrac{dH_c}{dT}\) is always negative, we see that the entropy of the superconducting state is always less than (or equal to, if \(H_c=0\)) the entropy of the normal state; in other words, the superconducting state corresponds to a more ordered state of the metal. The entropy difference becomes zero at the normal transition temperature \(T_0\), i.e. in the absence of a magnetic field; according to Nernst’s theorem it must also vanish at absolute zero; this means that at \(T=0\) one must have \(\dfrac{dH_c}{dT}=0\), which is apparently in agreement with experimental results (the measurements, however, have not been carried out down to temperatures sufficiently low to make this agreement quite definite). Since the entropy difference vanishes both at \(T=T_0\) and at \(T=0\), it must pass somewhere through a maximum (Fig. 17).

From (2) we also see that, in the isothermal destruction of superconductivity by a magnetic field, heat is absorbed, and if

if the transition takes place adiabatically, then the temperature is lowered (we shall consider the magnitude of this cooling below). The heat absorbed in the transition of a unit volume from the superconducting to the normal state at temperature \(T\) is equal to

\[ Q = T(S_n - S_s) = -\frac{T H_c}{4\pi}\frac{dH_c}{dT}^{\,1)} . \tag{3} \]

This result is quite analogous to the well-known Clapeyron–Clausius equation, where the variables \(p\) and \(V\) have been replaced respectively by \(\dfrac{H_c}{4\pi}\) and \(B\). We see that this heat is absent at the normal transition temperature \(T_0\). Thus, in the absence of a magnetic field there is no heat of transition. This has already been mentioned in Chap. I as an experimental fact.

Differentiating (2) with respect to \(T\), we find the difference of the heat capacities (per unit volume)

\[ \Delta c = c_s - c_n = \frac{T}{4\pi}H_c\frac{d^2H_c}{dT^2} + \frac{T}{4\pi}\left(\frac{dH_c}{dT}\right)^2 . \tag{4} \]

In particular, in the absence of a magnetic field (i.e. at \(T = T_0\)), we have

\[ \Delta c = \frac{T}{4\pi}\left(\frac{dH_c}{dT}\right)^2 . \tag{5} \]

This formula is known as Rutgers’ formula and shows that if a metal is cooled or heated in the absence of a magnetic field, then the heat capacity undergoes a jump at the transition point. In the absence of a magnetic field the heat capacity of the superconducting phase is, obviously, always greater than that of the normal phase, but at lower temperatures (i.e. when superconductivity is destroyed by a magnetic field) the sign of \(c_s - c_n\) must change in accordance with the fact that \(S_s - S_n\) passes through a minimum. The jump in the heat capacity was experimentally discovered by Keesom and Kokom\(^4\) for tin and thallium and agrees very well with Rutgers’ formula, thereby confirming that the reversibility assumption underlying the thermodynamic treatment is indeed valid.

Formulas (4) and (3) determine the heat of transition and the heat capacity for the complete transition between the superconducting and normal phases. However, as we have already noted, if the specimen has a nonzero demagnetization coefficient, then this transition is in reality spread over a whole interval of external fields at constant temperature (the field at the superconducting boundaries is all the time equal to \(H_c\)) or over an interval of temperatures, if the external field is constant (in this case the field at the superconducting boundaries is also always equal to \(H_c\), but this \(H_c\) changes with temperature). Thus, if the transition takes place at constant temperature in an ellipsoid with demagnetization coefficient \(4\pi n\), then the transition will be

\(^1\) This result was first obtained by Keesom\(^1\) in 1924.

occurs gradually as the external field changes from \((1-n)H_c\) to \(H_c\), and correspondingly this heat \(Q\) (1) will be absorbed gradually. Similarly, (4) gives only the difference of the heat capacities before the beginning of the transition and after its end, but says nothing about how the heat capacity changes during the transition. In order to determine how the heat capacity changes during the transition, let us define the amount of heat that must be supplied to the ellipsoid in order to raise its temperature by \(dT\) at a constant value \(H\) of the external field lying in the intermediate region. In Chapter III we saw that in the intermediate state we are dealing with a fine mixture of normal and superconducting phases, the quantities of which are respectively proportional to \(x\) and \(1-x\), where

\[ x=\frac{B}{H_c}. \tag{6} \]

When the temperature is raised by \(dT\), \(H_c\) decreases somewhat, and \(x\) increases, i.e. the amount of the normal phase increases. This conversion of a fraction \(dx\) of the specimen from the superconducting to the normal state, however, requires an amount of heat \(Qdx\), so that the rise in temperature will require a larger amount of heat than would be needed for each of the pure phases separately. In other words, the heat of transition, instead of being detected directly, appears as an anomaly—as an increase of the heat capacity. Disregarding the heat of transition, we may write for the heat capacity of the intermediate state \(xc_n+(1-x)c_s\), and therefore the actually observed heat capacity will be

\[ c=xc_n+(1-x)c_s+Q\frac{dx}{dT}. \tag{7} \]

Substituting the value of \(x\) from (6), the value of \(B\) from Chapter III, i.e.

\[ B=H_c-\frac{(H_c-H)}{n}, \]

and the value of \(Q\) from (3), we find for the heat capacity of a metal in the intermediate state\(^1\)

\[ c=c_n\left[1-\frac{1}{n}\left(1-\frac{H}{H_c}\right)\right]+c_s\left[\frac{1}{n}\left(1-\frac{H}{H_c}\right)\right]+\frac{T}{4\pi n}\frac{H}{H_c}\left(\frac{dH_c}{dT}\right)^2. \tag{7a} \]

We see that at \(H=(1-n)H_c\) the heat capacity abruptly increases from the value \(c_s\) to

\[ c_s+\frac{T(1-n)}{4\pi n}\left(\frac{dH_c}{dT}\right)^2, \]

and then continues to change (increasing or decreasing, depending on the value of the temperature), until the external field reaches \(H_c\); at this moment the heat capacity abruptly drops from

\[ c_n+\frac{T}{4\pi n}\left(\frac{dH_c}{dT}\right)^2 \]

to \(c_n\). If, instead of the heat capacity, one measures only the absorption of heat in the destruction of superconductivity when the magnetic field is increased, we find that when the field is increased by \(dH\), heat \(Qdx\) must be absorbed, where \(dx\) is the fraction of the metal that has passed from the super-

\(^1\) This result was first obtained by Peierls\(^5\).

conducting state into the normal state as the field is increased. It is easy to see that this heat is equal to \(\dfrac{T}{4\pi n}\left(\dfrac{dH_c}{dT}\right)dH\), i.e., the heat is absorbed uniformly (it does not depend on the magnitude of \(H\)). The change of heat capacity with temperature in a constant external field is also determined by equation (7a), where, instead of \(H\), \(T\) now varies, and hence so does \(H_c\). In this case there will again be two jumps at the temperatures at which the constant external field is equal, respectively, to \((1-n)H_c\) and \(H_c\). Let us note that (7a) predicts infinite jumps in the case \(n=0\) (an infinite cylinder or a thin disk parallel to the field), but this should not trouble us, since in this case both jumps are infinitely close to each other, and their presence simply shows that all the heat of transition is absorbed at one definite temperature or field. Since in this limiting case the transition takes place quite abruptly, there is no need to discuss a smeared transition, and formulas (3) and (4) are fully applicable.

Fig. 15.

These predictions have not yet been experimentally confirmed for an ellipsoid, but Keesom and Kok \(^{6}\) have measured the heat capacity of a piece of thallium of irregular shape in a magnetic field \(^{1}\), which gives qualitative confirmation of the theory. As we have already indicated, the field distribution around an irregularly shaped specimen is also irregular, and, in contrast to an ellipsoid, the specimen cannot pass entirely into the intermediate state at a definite value of the external field. The result of this is, generally speaking, that the penetration of the magnetic field into the specimen becomes more gradual, so that, for example, the magnetization curve has no sharp corners. Consequently, discontinuities in the heat-capacity curve are smoothed out, and the heat of transition appears only as a smooth “hump” on the heat-capacity curve. Figure 15 shows the experimental curve of Keesom and Kok, as well as the (schematic) curve that one would have to expect for an ellipsoid comparable with their specimen (we have taken \(n=0.12\)). Keesom and Kok point out that if the temperature interval in which the “hump” occurs is relatively small (as in Fig. 15), then the complete latent heat of transition can be derived from the heat-capacity curve. Thus, measurements show that during the transition

\(^{1}\) Owing to the somewhat incomplete Meissner effect, the results obtained were somewhat more complicated when the magnetic field was applied before the specimen became superconducting. The quoted results refer only to the case of a field applied after cooling in the absence of a field.

the last term on the right-hand side of (7) is larger than the other two, so that we shall not make a large error if we assume that \(x\) varies linearly with temperature. If this assumption is made, it is evident from (7) that the heat of transition \(Q\) is determined by the area of the region shaded in Fig. 15 (more precisely, by the average over the temperature region under consideration, \(Q\)). In this way they determined the values of \(Q\), which were in good agreement with (3), thereby again confirming the reversibility of the superconducting transition. Conversely, knowing the value of \(Q\), one can determine the magnitude \(x\) at every instant of the transition as the ratio of the area of the shaded region up to the considered value of the external field to \(Q\)¹). In this way the curve in Fig. 16 was obtained, showing the variation of \(x\) with \(T\) (the curve for the ellipsoid is drawn with a dotted line; it is almost straight, since only a small temperature interval is being considered).

Fig. 16.

Let us note once more that discontinuities in the heat capacity are smoothed out owing to the rounding of the corners of this curve²).

The detailed form of the temperature dependence of the heat capacity (and entropy) was measured by Keesom and Van Laer⁴ᵃ for tin down to a temperature of \(1^\circ,2K\). Their measurements are of interest in that the heat capacity in the superconducting state is approximately proportional to the cube of the temperature. In magnitude this heat capacity is of the same order as the heat capacity due to lattice vibrations in a normal metal. Unfortunately, the lattice heat capacity cannot be determined exactly, because even at temperatures sufficiently high to neglect the electronic heat capacity and sufficiently low in comparison with the Debye \(\Theta\), the lattice heat capacity is not exactly proportional to \(T^3\). Therefore the heat capacity of the superconducting electrons alone cannot be found exactly. If one assumes that the lattice heat capacity below \(3^\circ,7\) is already proportional to \(T^3\), it follows from this that the heat capacity of the electrons in the superconducting state is also proportional to \(T^3\). Measurements with \(Tl\) indicate, although less definitely, the same result. The question of whether the heat capacity of all superconductors depends on temperature as \(T^3\) can be settled by further experiments. If this result proves to be general,

¹) This is, of course, only a first approximation; a better result will be obtained if one uses the first approximation to correct [in the first two terms in (7)] the assumption of a linear variation of \(x\).

²) The rounding of the corner at the end of the curve corresponding to higher temperatures (and analogously on the heat-capacity curve, Fig. 15) is caused, possibly, not by an incorrect form of the specimen, but by small contaminations in it; an incorrect form usually causes only a gradual beginning, not the end, of the curve.

then it will be very essential for the construction of the theory of superconductivity.

Daunt and Mendelssohn1 drew attention to the circumstance that from purely magnetic measurements one can derive a lower limit for the entropy of the normal state, since the difference of the entropies of the normal and superconducting states

\[ \left(-\frac{H_c}{4\pi}\cdot\frac{dH_c}{dT}\right) \]

obviously cannot be greater than the entropy of the normal state. On the basis of exact measurements of the \(H_c—T\) curves they constructed curves for the entropy differences as a function of temperature for a number of metals (Fig. 17). These curves show that at temperatures low in comparison with the normal transition temperature, the entropy difference varies linearly with temperature (the measurements, however, were not carried out down to temperatures low enough to make this conclusion definite), and we may conclude that the entropy, and therefore also the heat capacity of the metal in the normal state, also varies linearly at sufficiently low temperatures. By extrapolating their experimental results, Daunt and Mendelssohn obtained the coefficient of the limiting linear law of the temperature dependence of the entropy difference, which, as mentioned above, is a lower limit of the corresponding coefficient in the limiting linear law of variation of the heat capacity of the normal state. They found that this linear term is in all cases larger than is predicted by Sommerfeld’s formula for the electron gas[^2] (for tantalum and niobium—several times larger). This discrepancy is not surprising, since the electron gas is a very crude description of a metal, and, in any case, the theory on which this formula is based cannot explain the appearance of superconductivity.

Fig. 17.

Let us note that the magnitude of the entropy difference per gram-atom is only about \(10^{-3}R\), which is very small in comparison with the difference

entropies for an ordinary transition (\(\sim R\))\(^{1}\); this shows that either the redistribution of electrons in the superconducting transition is very slight, or that only a small fraction of all electrons participates in this transition. Quite analogously, the differ-

Fig. 18. Entropy–temperature curves for tin

ence of the energies (per unit volume) of the superconducting and normal phases is determined as

\[ \Delta E = E_n - E_s = \Delta G - T \frac{d\Delta G}{dT} \]

or

\[ \Delta E = \frac{H_c^2}{8\pi} - \frac{T H_c}{4\pi}\frac{dH_c}{dT}. \tag{8} \]

This quantity is also of the order of \(10^{-3} RT\) (per gram-atom), instead of \(RT\) for ordinary phase transitions. The difference of the energies, of course, vanishes at the normal limiting temperature \(T_0\).

In order to calculate the magnetocaloric effect mentioned above, it is necessary to know the exact form of the entropy curves as functions of temperature for the superconducting and nor-

\(^{1}\) It is easy to see that if the entropy difference were of the order of \(R\), this would mean that the entropy of the normal state is very large; and since the entropy must go to zero at \(T = 0\), the heat capacity of the normal state would have to have a very high maximum at some temperature below \(T_0\), as in the case of paramagnetic salts.

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of the normal state of the metal; these curves are shown for tin in Fig. 18 (computed from the data of Keesom and Van Laer\(^{4a}\)). Figure 18 shows the curves at constant external field \(H\) for the case of a spherical specimen. So long as the temperature is so low that \(H < \frac{2}{3} H_c\), the sphere is wholly superconducting, and the entropy \(S_s\) is determined by the lower curve. For \(H_c > H > \frac{2}{3} H_c\) the sphere is in an intermediate state with entropy \(xS_n + (1 - x)S_s\), where \(x = \frac{3H}{H_c} - 2\). Finally, for \(H_c < H\) the specimen is wholly in the normal state, and the entropy \(S_n\) is determined by the upper curve.

With the aid of Fig. 18 it is easy to determine the magnetocaloric cooling. Thus, if a sphere is thermally insulated at temperature \(A\) (\(2^\circ.6\mathrm{K}\)) and a magnetic field greater than \(\frac{2}{3}H_c\) is applied (say, 65 gauss), then the sphere must cool to \(B\) (\(2^\circ.25\mathrm{K}\)) so that the entropy remains constant. Obviously, for the given initial temperature the sought temperature will be lowest if the applied field is greater than the critical field at this temperature. For example, starting from \(A\), the lowest temperature that could be obtained by applying a field greater than 200 gauss is \(C\) (\(2^\circ.0\mathrm{K}\)). Let us note that if the applied field is not greater than the critical field at the final temperature, then the latter will be smallest for a specimen with a large demagnetizing coefficient, since in this case the entropy curve in the intermediate state is more elongated (or, in other words, at the given external field a larger part of the specimen will be in the normal state). Thus, starting from the lowest temperature attainable by ordinary methods, say \(1^\circ\mathrm{K}\), for tin one can in the manner described reach about \(0^\circ.05\mathrm{K}\). Still lower temperatures could be obtained by means of a cyclic process, in which one superconductor cools another that is not in a magnetic field; this second superconductor is then separated from the first and magnetized, starting already from a lower temperature, and so on.

The idea of using this cooling effect to obtain very low temperatures was first put forward by Mendelssohn and Moore\(^{9}\). But although they, as well as Keesom and Kok\(^{6}\), showed that a lowering of temperature can indeed be achieved, this method has not yet been developed for practical use\(^{1}\). Let us mention that it has certain disadvantages in comparison with the Debye–Giauque method of adiabatic demagnetization of paramagnetic salts. The most important of these is the very low absolute value of the heat capacity of the metal,

\(^{1}\) Daunt and Mendelssohn\(^{7}\) pointed out that niobium and tantalum may be very suitable substances for obtaining low temperatures, since the entropy differences for these metals are comparatively large (owing to their relatively large critical fields).

Thus, if it is used for cooling some other substance with a comparable or greater heat capacity, the resulting lowering of the temperature will be greatly reduced. This can be seen directly from Fig. 18; if to the metal one adds some other substance with a larger heat capacity, then the entropy curves of the whole system become much steeper, so that the difference of the temperatures at which the metal is superconducting or normal (for the same entropy of the whole system) will be much smaller. Thus this method would be useless for cooling any appreciable amount of liquid helium or paramagnetic salt, and would be of little productivity even for cooling any other nonsuperconducting metals. Its chief application would be the cooling of the superconductor itself, for the purpose of studying its properties (for example, the dependence of entropy itself on temperature) at low temperatures. In addition, this method has the obvious advantage that it requires only comparatively weak magnetic fields. Let us note one more feature of the method which decreases its productivity: in order for the process to be adiabatic, the magnetic field must be applied infinitely slowly. Indeed, during the cooling process the specimen is in an intermediate state, so that the change of the field induces Foucault currents in the normal regions, the damping of which is accompanied by the liberation of Joule heat. Consequently, the lowering of the temperature will be smaller than that calculated on the assumption of constant entropy, unless the increase of the field is very slow.

Up to now we have considered the thermodynamics of the superconducting transition at constant pressure \(p\) and volume \(V\) of the metal. However, as we saw in Chapter I, there is a weak dependence of the critical field on stresses in the metal, and we shall now turn to an investigation of the thermodynamic consequences of this dependence. We shall deal only with homogeneous pressures; owing to the paucity of experimental data, it is not possible to consider more complicated deformations.

Differentiating the basic equilibrium condition (1) with respect to \(p\), and remembering that \(\dfrac{\partial G}{\partial p}=V\), we find that in a magnetic field there occurs a change of volume equal to (per unit volume)¹

\[ V_n - V_s = \frac{H_c}{4\pi}\left(\frac{\partial H_c}{\partial p}\right)_T \tag{9} \]

(if we substitute \(-\left(\dfrac{\partial H_c}{\partial T}\right)\left(\dfrac{\partial T}{\partial p}\right)_{H_c}\) in place of \(\left(\dfrac{\partial H_c}{\partial p}\right)_T\), and recall that \(-\dfrac{T H_c}{4\pi}\left(\dfrac{\partial H_c}{\partial T}\right)_p\) is the latent heat \(Q\), we shall see that (9)

¹ In these formulas, \(G\) refers, strictly speaking, to unit mass; but since the changes of volume are very small, one may retain \(G\) referred to unit volume (strictly speaking, \(G\) refers to the mass of a unit volume of one of the phases).

is the ordinary Clapeyron–Clausius equation). The only data on \(\dfrac{\partial H_c}{\partial p}\) are those of de Haas, Sizoo, and Kamerlingh Onnes\(^{10}\) for tin and indium, according to which \(\dfrac{\partial H_c}{\partial p}\sim 10^{-10}\) abs. units. Taking \(H_c\sim 100\) gauss, we find that the destruction of superconductivity by a magnetic field must be accompanied by an increase of volume of order \(10^{-9}\) per unit volume. MacLennan, Allen, and Wilhelm looked for a change in the dimensions of a lead wire upon the destruction of superconductivity, but found no change in length greater than \(10^{-8}\) per 1 cm, so that there was no change in volume greater than \(3\cdot 10^{-8}\) per unit volume. If one assumes that \(\dfrac{\partial H_c}{\partial p}\) for lead is of the same order of magnitude as for tin, this negative experimental result is in agreement with equation (9). Let us note that in the absence of a magnetic field there would be no change in volume at all.

Differentiation of (9) with respect to temperature or pressure shows that the thermal expansion and compressibility must undergo a jump at the superconducting transition even in the absence of a magnetic field. We have

\[ \Delta \left(\frac{\partial V}{\partial T}\right) = \frac{\partial V_n}{\partial T} - \frac{\partial V_s}{\partial T} = \frac{1}{4\pi}\frac{\partial H_c}{\partial T}\frac{\partial H_c}{\partial p} + \frac{H_c}{4\pi}\frac{\partial^2 H_c}{\partial p \partial T} \tag{10} \]

and

\[ \Delta \left(\frac{\partial V}{\partial p}\right) = \frac{\partial V_n}{\partial p} - \frac{\partial V_s}{\partial p} = \frac{1}{4\pi}\left(\frac{\partial H_c}{\partial p}\right)^2 + \frac{H_c}{4\pi}\frac{\partial^2 H_c}{\partial p^2}. \tag{11} \]

Since we have no data on the second derivative of \(H_c\) with respect to pressure and since, moreover, we are interested only in the order of magnitude, we shall consider only the transition at the normal transition temperature, i.e. in the absence of a magnetic field (the order of magnitude may perhaps not change under this restriction). Introducing the coefficient of thermal expansion \(\alpha=\dfrac{1}{V}\dfrac{dV}{dT}\) and the modulus of compression \(\chi=-V\dfrac{dp}{dV}\), we have from (10) and (11), for the case \(H_c=0\) (putting \(V=1\), since all the equations below refer to unit volume),

\[ \frac{\Delta \alpha}{\alpha} = \frac{1}{4\pi\alpha}\frac{\partial H_c}{\partial T}\frac{\partial H_c}{\partial p}; \qquad \frac{\Delta \chi}{\chi} = \frac{\chi}{4\pi}\frac{\partial H_c}{\partial p}. \tag{12} \]

Substituting rough numerical data \(\left(\dfrac{\partial H_c}{\partial p}\sim 10^{-10},\ \dfrac{\partial H_c}{\partial T}\sim 10^{2},\ \alpha\sim 10^{-7},\ \chi\sim 10^{12}\right)\), we find

\[ \frac{\Delta \alpha}{\alpha}\sim 10^{-2}; \qquad \frac{\Delta \chi}{\chi}\sim 10^{-9}. \]

The change in compressibility is too small to be observed, and this is confirmed by the negative results of the experiments of de Haas and Kinoshita. Although the relative change in \(\alpha\) is not so

small, but $\alpha$ itself at low temperatures is so small that its change may be regarded as lying below the limits of measurement accuracy. In the experiments of MacLennan, Allen, and Wilhelm $^{11}$ the thermal expansion of lead could have been detected only precisely at the transition temperature, so that it was impossible to say whether there is a jump of the predicted order of magnitude.

Chapter VI. Superconducting Alloys

In addition to pure superconducting elements, there exists a large number of alloys which become superconducting at low temperatures $^{12}$. Roughly speaking, these alloys may be divided into two classes.

1. Alloys of superconducting elements with one another or with elements that are at present known as non-superconducting. These alloys may, generally speaking, be superconducting over a large range of concentrations, so that superconductivity is not a property inherent in a definite composition, say, a chemical compound. There is no universal rule concerning the dependence of the transition temperature on composition, but in many cases the transition temperature of a pure superconducting element is raised by the addition of a non-superconducting component.

2. Alloys of elements (both superconducting and non-superconducting) in definite ratios, i.e. chemical compounds; an example of this type is Au$_2$Bi $^{13}$. Although the experimental data on this subject are meager, it may nevertheless be thought$^{*)}$ that if such an alloy were prepared in pure form (i.e. with an exact stoichiometric composition), it would behave like a pure superconducting element. In the case of these alloys it appears that superconductivity is characteristic of a definite arrangement of atoms corresponding to the lattice of a chemical compound. We should also mention that there exists a series of alloys of this type which, above their transition temperature, are poor conductors $^{14}$; such are many carbides, nitrides, borides, and silicides, as well as CuS. Apart from the fact of their superconductivity, however, very little is known about them. It is interesting to note that in all cases of such superconducting alloys in which none of the components is itself superconducting, at least one of the components lies immediately adjacent to one of the two groups of superconducting elements in the periodic system of the elements. This possibly confirms (Chapter I) that the grouping in the periodic system is not accidental.

In this chapter we shall study the properties of alloys predominantly of the first type, which differ from pure—

$^{*)}$ We have in mind the low critical field of Au$_2$Bi, which is of the same order of magnitude as the critical field of pure metals, and not of alloys of type 1.

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of these superconducting elements in almost all essential respects. The existence of another type of alloys may prove significant for the development of the theory of superconductivity, but since they probably do not differ in their general properties from ordinary superconductors, there is no need to consider them separately. Therefore, in what follows it should be remembered that when we speak of alloys, we have in mind alloys of the first type.

A characteristic difference between an alloy and a superconducting element is, first of all, the earlier restoration of resistance by a magnetic field. The resistance curves as a function of the field for alloys, generally speaking, are of the type shown in Fig. 19, and differ sharply from the corresponding curves for a pure superconductor. Especially significant is the very large value of the field[^15] required for the appearance of the first traces of resistance. This field varies over wide limits in different alloys (the figure takes the case of the Pb—Bi eutectic, which has one of the largest known critical fields), but is in general of the order of a thousand gauss instead of several hundred in most superconducting elements.

Fig. 19. Dependence of resistance on field for the Pb—Bi eutectic (at 4.2 K)

This critical field is very sensitive to an increase in the current strength in the alloy, rapidly decreasing for currents that produce only a very weak field at the surface—this is connected with the invalidity of Silsbee’s hypothesis for alloys (see below). Another important feature is that the resistance is not restored at once, but over a comparatively large interval of fields (even if the magnetic field is parallel to the current in the alloy wire). Owing to this diffuseness of the transition, there is a much smaller difference between the curves in transverse and longitudinal fields than in pure superconductors. If, after the restoration of resistance, the magnetic field is again decreased, a noticeable hysteresis is usually observed (Fig. 19; the return curve is not drawn completely because of the absence of experimental data).

This hysteresis is possibly connected with hysteresis of the magnetic properties (i.e. the absence of the Meissner effect), which we shall consider below. We have, however, too little experimental data that would allow more detailed conclusions to be drawn about the nature of this connection; unfortunately, most experiments with alloys have been devoted to some single feature, and there is no

not a single alloy for which the various properties were studied on one and the same specimen.

The transition temperature of alloys is of the same order of magnitude as that of elements, but the transition between the normal and superconducting states is smeared over a comparatively large temperature interval (of the order of \(1^\circ K\)) even in the absence of a field. In the literature it is usually customary to define the transition temperature as that temperature at which the resistance is one-half of its total value, but it must be remembered that this field is sensitive to the strength of the measuring current.

Until the magnetic properties of alloys had been investigated, the high value of their critical field suggested certain interesting possibilities. Thus, it seemed possible, with the aid of such an alloy, to realize the old idea of a superconducting solenoid for fields of several thousand gauss, since smaller external fields did not restore any trace of resistance. Further, if \(H_c\) and \(\frac{dH_c}{dT}\) were as large as follows from resistance measurements, all caloric effects would be hundreds of times greater than in pure metals, so that upon the destruction of superconductivity in an alloy there should occur a colossal jump in heat capacity and colossal cooling. In practice, however, it turned out that none of these possibilities could be realized, perhaps because the critical field given by resistance measurements is characteristic only of a very small part of the volume of the alloy. We shall now briefly describe the experimental data which point to such an interpretation.

De Haas and Casimir-Jonker \(^{16}\) showed, using the method with a bismuth wire, that the magnetic field in fact penetrated into the alloy long before it became large enough to begin restoring the resistance, and that the penetration was almost complete already at fields of the same order of magnitude as for pure elements. Similarly, Mendelssohn and Moore \(^{17}\), measuring the \(B—H\) curve of a long wire made of a superconducting alloy, found that \(B\) became different from zero and began to approach the value \(H\) at fields considerably smaller than those needed to restore the first traces of resistance (Fig. 20). These circumstances immediately show that the true critical field in the greater part of the alloy specimen is of the same order of magnitude as in pure elements, but that as the field is increased a very small part of the specimen may remain superconducting by virtue of its larger critical field, and thus a superconducting “path” remains for the measuring current.

The incorrectness of Silsbee’s hypothesis for alloys is also in qualitative agreement with the explanation given. Thus, Keesom \(^{18}\), attempting to use an alloy to obtain strong magnetic fields, found that the resistance of a wire made of the alloy

is restored by considerably weaker currents than those which would create, at the surface of the wire, a field of the order of those critical fields that are determined from resistance measurements. In some experiments it was shown that the field at the surface of the wire, produced by the current at which resistance begins to appear, does not depend on the diameter of the wire and has the same order of magnitude (more precisely, about 60% of it) as the longitudinal field which begins to penetrate into the specimen. According to the hypothesis stated above, this may mean that the field of the current which begins to restore the resistance is equal to the smallest critical field of the alloy (assuming that, owing to inhomogeneity, there exists a continuous distribution of critical fields). When this current is exceeded, superconductivity is destroyed throughout the whole wire. The current cannot flow along “filaments” with a high critical field, perhaps because their radii are too small, i.e., roughly speaking, because the ratio of the radius of such a filament to the radius of the wire is smaller than the ratio of the critical field of the bulk of the wire to the greatest critical field. This qualitative explanation of the inapplicability of Silsbee’s hypothesis means that in reality this inapplicability is only apparent, and that the hypothesis is still valid if by critical field one understands not the field which restores the resistance, but that which begins to penetrate into the wire, i.e., of the order of the critical field of the bulk of the specimen.

Fig. 20. The \(B—H\) curve for \(\mathrm{Pb}+2\%\ \mathrm{In}\) at \(1^\circ.95\ \mathrm{K}\)

Although the explanation given is plausible, it is only very qualitative and at present has no theoretical foundation. Let us enumerate the known facts from a purely experimental point of view, without emphasizing this hypothetical interpretation. We may say that the alloy has three critical fields: 1) the field \(H_1\), produced at the surface of the wire by the current which just begins to restore the resistance; 2) the field \(H_2\), at which the lines of force begin to penetrate into the specimen; 3) the field \(H_3\), necessary for the restoration of the resistance, at which only a weak current flows. The variation of these fields with temperature is shown in Fig. 21 for one particular case (the field \(H_1\) was determined only for this case).

In a pure metal these three critical fields are identical, whereas in an alloy \(H_3\) is much greater than \(H_2\), and \(H_1\) somewhat less than \(H_2\) (in the present case by 30%). In our interpretation of the inapplicability of Silsbee’s hypothesis we assumed that there is in fact no difference between \(H_1\) and \(H_2\), and it is possible that the field at which penetration of the field begins is in fact somewhat smaller, i.e. the first penetration is too small to be detected by the experimental method. It is possible, however, that a difference between \(H_1\) and \(H_2\) does exist—at present this question must be left open. In Fig. 21 the shaded region is the region in which the penetration of the field becomes more and more complete, but usually the penetration is already almost complete soon after \(H_2\) is reached. The field \(H_3\), as already mentioned, is very sensitive with respect to the measuring current, decreasing rather rapidly as the current strength is increased. Let us also mention that for complete restoration of the resistance a field is required that considerably exceeds \(H_3\) (by approximately 50%).

Fig. 21.

Fig. 20 also shows another essential difference between alloys and elements, namely that in superconducting alloys the Meissner effect is not observed. Indeed, we see that the \(B—H\) curve exhibits noticeable hysteresis when the magnetic field is decreased, beginning with the value at which it appreciably penetrates into the superconductor; the greater part of the flux that has penetrated into the specimen remains “frozen” in it. The exact form of the \(B—H\) curve depends on the particular alloy and also on the temperature; curves for different alloys or for the same alloy at different temperatures cannot be made to coincide by a simple change of scale1.

This absence of the Meissner effect in alloys can be interpreted in general terms in terms of the hypothesis of the existence—

... in alloys consisting of regions with an abnormally large critical field, if one makes the further hypothesis that these regions are multiply connected. Mendelssohn^20 proposed that the structure of the alloy resembles a sponge made of a substance with a smaller critical field, containing regions of a substance with a larger critical field. Although this assumption is rather a convenient working hypothesis than an established fact, we shall see that it makes it possible to give a qualitative explanation of the observed magnetic properties. When the magnetic field is increased above the value corresponding to the critical field of the bulk of the alloy, it begins to penetrate into the specimen, but the penetration is still delayed by the superconducting regions in the “holes of the sponge,” which tend to keep the flux through the specimen constant. We have seen, however, in Chapter IV, that a multiply connected superconductor cannot maintain a constant flux in the material it encloses when the external field increases too much, since such a superconductor cannot carry a current greater than a certain strength; moreover, this limiting value of the external field is the smaller, the thinner the superconducting sections. Thus, despite the high value of the critical field of the “holes of the sponge,” because of their thinness they are not able to prevent the increase of the flux through the specimen as the external field is increased, and, finally, the penetration of the external field becomes almost complete, despite the fact that the holes (at least some of them) are still superconducting and can carry a small current without any resistance. When the field is again decreased, starting from some large value, the “holes” are at first not able to retain the whole initial flux through the specimen, since this would require excessively strong induced currents in the superconducting regions. But when the field reaches the critical value for the bulk of the alloy, the superconducting regions can become thicker at the expense of the surrounding substance, which was initially in the normal state and is now in a field smaller than its critical one. As their thickness increases, the superconducting regions can carry a larger current and thus oppose the decrease of the flux through the alloy as the external field is reduced to zero, while the main mass of the alloy (through which the flux passes) remains in the normal state.

Further confirmation of the general ideas described is provided by calorimetric data. As might have been expected on the basis of the facts described, no jump in heat capacity of the magnitude previously expected on the basis of the value of the critical field obtained from resistance measurements was found (Mendelssohn and Moore^21). The reason for this is that the main mass of the metal, as we have seen, has quite ordinary critical fields, so that the heat capacity of the whole specimen must behave almost as does the heat capacity of the pure element. The experiments were calculated for the large jump that had been expected...

heat capacities and were insufficiently accurate to detect small changes in heat capacity. It is possible that more accurate experiments will indeed reveal not a sharp jump on the heat-capacity curve, but only its bend in that temperature region in which the transition between the superconducting and normal states of the alloy takes place.

The calorimetric method can be used to show that after a sufficiently strong magnetic field has been applied and then switched off (and the specimen has been left with “frozen-in” flux), the bulk of the alloy is in fact in the normal state. Thus, Mendelssohn and Moore^20 measured the temperature dependence of the heat capacity of a loaded tin specimen (which, with respect to the absence of the Meissner effect, behaved like an alloy), first on cooling in the absence of a field, and then after a strong field had been applied and switched off at the very lowest temperature. Their results are shown schematically in Fig. 22 and show that in the second case the heat capacity follows the curve corresponding to the normal state without any noticeable jump, i.e., that the “frozen-in” flux prevents almost the entire mass of the specimen from becoming superconducting again. With more accurate measurements this method can probably be used to determine the magnitude of the superconducting part of the volume of an alloy when, in the absence of a field, it contains “frozen-in” flux.

Fig. 22.

Although, owing to the inevitable inhomogeneity of alloys of the type described, it is natural to expect that they possess a whole interval of critical fields (corresponding to the different composition in different parts of the alloy), nevertheless at present there is no acceptable explanation of why some small regions have such an abnormally large value of the critical field that they remain superconducting in fields of several thousand gauss. At first glance it may seem that these large critical fields are characteristic of some definite composition, which exists only in small regions of the specimen. This idea, however,

is unlikely to be correct, since, quite apart from the fact that such a high critical field has never been found for the bulk of an alloy, it seems improbable that a superconductor in its bulk could have such a critical field, because the entropy in the normal state would then have to be extraordinarily large in comparison with the entropy in the superconducting state (of course, in the presence of a magnetic field), and consequently the substance in the normal state would have to possess a much greater heat capacity than ordinary metals. It is more likely that the large critical field is connected with the small dimensions of regions that remain superconducting in a strong magnetic field. There are, indeed, indications that the critical field begins to increase when the dimensions of the superconductor are reduced below \(10^{-4}\) cm (see Chapter VIII). Although we are inclined to think that the large field values required to restore the resistance of a superconducting alloy are due to a cause of this kind, we have no explanation for why the same mechanism does not occur in pure superconductors, i.e. why very thin superconducting filaments do not remain in them after the usual critical field has been exceeded. We must suppose that in this respect there is some essential difference between alloys and elements; this difference may in some way be connected with the inhomogeneity of the alloy, but this connection is at present inexplicable1.

In conclusion let us mention one more circumstance, of more practical and historical than theoretical interest. The behavior of alloys described above is in practice almost always observed in simple superconductors, unless they are extremely pure and free of mechanical stresses. Since the discovery of the Meissner effect, a large number of experiments have been devoted to separating the properties that we attributed in the preceding chapters to pure superconductors from those effects that are caused by slight impurities; these works have indeed shown that even very small impurities are sufficient to produce some of the effects that we described as characteristic of alloys. Thus, for most superconducting elements it has so far not been possible to obtain a specimen that would exhibit the full Meissner effect (i.e. absence of hysteresis for any magnetization), in which the critical field determined from resistance measurements would not be somewhat greater than the field determined by magnetic measurements, and in which the transition in a long cylinder would occur not over a small interval of fields or temperatures, but at one definite point (elements only weakly contam—

... which, however, never exhibit such large critical fields as are characteristic of alloys1. At the present time it is still unclear exactly which impurities are effective in causing the properties of a superconductor to deviate from the “ideal”; possibly they are dissolved gases (usually not taken into account in determining purity), or else internal stresses, which are responsible for the “non-ideal” behavior of many apparently very pure superconductors.

Chapter VII. Superconducting Thin Films

We have seen that a characteristic property of a superconductor is \(B = 0\), and, consequently, that any current must be a surface current. It is obvious, however, that the current cannot be entirely superficial, i.e. there must be some definite depth of penetration of the current, which at the same time is the depth of penetration of the magnetic field into the superconductor. Since the description with \(B = 0\) is suitable for specimens of macroscopic dimensions (\(\sim 1\) cm), this depth of penetration must evidently be very small in comparison with 1 cm, and we come to the conclusion that effects for which this depth is significant can occur only in specimens of very small dimensions. Besides the possibility of studying these effects, experiments with specimens of small dimensions are also of interest in that they can give an idea of the minimum size characteristic of superconductivity. Like ordinary conductivity, superconductivity is a “cooperative” phenomenon, requiring a large number of atoms, so that it is natural to expect the existence of minimum dimensions below which superconductivity cannot occur.

In working with specimens small in all their dimensions, there are a number of obvious difficulties. Up to the present time the only attempt in this direction was made by Tarr and Wilhelm[^24], who investigated the magnetic properties of an emulsion of mercury droplets with an average diameter of \(10^{-4}\) cm. The experiments showed that the magnetic permeability of these droplets was in any case less than unity and that the Meissner effect occurred, but the absolute value of \(B\) was not determined, so that the experiments proved only that the penetration depth could not be appreciably greater than \(10^{-4}\) cm. Pontius[^25] investigated the properties of thin lead wires and found that the critical field for the restoration of resistance begins to increase when the diameter of the wire is less than \(2 \cdot 10^{-3}\) cm. The increase for the thinnest wire used (\(5 \cdot 10^{-4}\) cm) reached, however, only 4%, so that the results must be accepted with caution, all the more so since

that the transition to normal conductivity was always very gradual. And here the only definite conclusion is that the penetration depth is no more than \(10^{-4}\) cm.

The only experiments that have unquestionably revealed a dependence of superconducting properties on dimensions are experiments on specimens in which only one dimension is small, i.e., on thin films. Misener and others\(^{26}\) worked in Toronto with thin films of lead and tin deposited electrolytically on cylindrical metallic wires and tubes (usually of constantan or nichrome). They found that the transition temperature of these films began to decrease rapidly when the thickness was made less than \(10^{-4}\) cm, and that films thinner than about \(2 \cdot 10^{-5}\) cm did not become superconducting at all down to temperatures of about \(2^\circ\)K\(^{1}\). We shall see below that this particular result concerning the dependence on dimensions was not confirmed by subsequent experiments and is possibly connected with the method used to obtain thin films. More interesting are the magnetic properties of these films, which are perhaps due to a substantial influence of dimensions, since they were confirmed by subsequent experiments carried out by an entirely different method.

It was found that, as in alloys, these films have three distinct critical fields (only tin films were studied). Namely:

1. The field \(h_1\), created by the current necessary for the reappearance of the first traces of resistance. It was considerably smaller than the critical field for massive tin (about 30% for the thickest films used—\(12 \cdot 10^{-4}\) cm—and about 3% for the thinnest—\(3 \cdot 10^{-5}\) cm). No simple dependence was found between the field and the film thickness. In contrast to massive specimens of a pure metal, these films did not show a jump when resistance was restored by a current—the restoration of resistance was spread over a large interval of currents. It was shown that the current necessary for the restoration of resistance is proportional to the radius of the cylinder on which the film is deposited (for films of the same thickness). Therefore it is permissible to connect the restoration of resistance with the magnetic field of the current at the surface of the film.

2. If the magnetic field was applied across the tube on which the film had been deposited, penetration of the lines of force through the tube occurred only when the external field reached the value \(h_2\). With a further increase of the external field, the penetration became complete only gradually, and even in fields more than 40 times greater than \(h_2\), the penetration amounted to only 99%. The value of \(h_2\) was usually several times greater than \(h_1\), but always less than,

\(^{1}\) The results for lead and tin differed only in details, while the order of magnitude of the critical thickness was one and the same.

than \(H_c\) for massive tin. It is easy to see that if \(h_2\) did not depend on the dimensions, it would have to be equal to \(\frac{1}{2}H_c\), and this was roughly true for the thickest films \((12\cdot 10^{-4}\ \mathrm{cm})\), although the penetration was not complete even at \(10H_c\); in the thinnest films \((6\cdot 10^{-5}\ \mathrm{cm})\), \(h_2\) was of the order of \(0.3H_c\). These results are very reminiscent of the penetration of the field into alloys, but, in contrast to alloys, no hysteresis was observed here when the field was decreased, i.e., the field could not “freeze” into the tube.

1. The field \(h_3\), which restored the first traces of resistance, was greater than the critical field of massive tin, as would also be the case if the massive specimen were contaminated. But, in contrast to the behavior of contaminated tin, in the direction transverse to the measuring current a larger field was required than in the direction parallel to the current. In all cases the restoration of resistance appeared to be spread over an interval of fields comparable with \(h_3\) itself, and since \(h_3\) was only slightly greater than \(H_c\) of massive tin, it might have seemed that the resistance was completely restored before the magnetic field had fully penetrated into the film. It is possible, however, that this property was due simply to an insufficiently accurate determination of the field at which the restoration of resistance ended. This explanation is plausible because, when an appreciable resistance of the film has already been restored, the resistance of the metal on which the film is deposited is much smaller than the resistance of the film itself, so that the measured resistance is mainly the resistance of this metal, and it would be very difficult to detect any incomplete restoration of the film resistance. Thus these results are not incompatible with the hypothesis that the restoration of resistance and the penetration of the field end simultaneously at fields greatly exceeding \(H_c\). The values of \(h_3\) did not depend noticeably on the thickness, but, as in alloys, were very sensitive to the magnitude of the measuring current, with \(h_3\) decreasing as the current increased. The curves of the dependence of resistance on the field always exhibited noticeable hysteresis.

We see that the behavior of these electrolytically deposited films is in many respects analogous to the behavior of alloys, and therefore it seemed possible to suppose that some of the anomalies described are in fact due to the formation of an alloy from the film and the metal on which it is deposited. It was found that if a layer of a nonsuperconducting metal, say copper, was deposited on a superconducting film, then the film acquired properties corresponding to a much thinner film without the covering layer; thus, the transition temperature began to fall with thickness for considerably thicker films (approximately 10 times thicker). If the lowering of the transition temperature were indeed due to the formation of an alloy, the latter result would be quite natural, since a copper-covered film has two surfaces-

…surfaces on which an alloy may form, instead of one. The formation of an alloy cannot, however, explain why \(h_1\) and \(h_2\) are considerably smaller than for massive tin, and this property is undoubtedly a primary size effect.

In order to exclude the possibility of complications caused by alloy formation, Shal’nikov\(^{27}\) carried out experiments with very thin tin and lead films deposited on glass by evaporation, with very strict observance of purity conditions. At present there are only preliminary results, but even they are of great interest and, apparently, confirm our point of view that the effects observed by Meissner are only partly due to the influence of size. First of all, Shal’nikov found that a film of lead and tin \(5 \cdot 10^{-7}\) cm thick is still definitely superconducting. The transition temperature in the case of tin films was about \(4.7^\circ\) K, i.e. \(1^\circ\) higher than for the solid metal. In the case of lead films the transition temperature could not be determined accurately in these experiments; however, it appeared that it too is higher than for the solid metal. Thus, in contrast to Meissner’s results, it appears that superconductivity can still exist even in films only about 15 atomic layers thick.

For technical reasons Shal’nikov has so far worked only with films deposited on a plane surface, and therefore his results concerning the destruction of superconductivity by a current are not so easy to interpret in terms of the magnetic field of the current as Meissner’s results with cylindrical films; for this reason only a qualitative comparison of the two results is possible. It was found that superconductivity is destroyed by very weak currents. A rough calculation shows that the maximum field produced by these currents (2 mA for a lead film \(5 \cdot 10^{-7}\) cm thick and 3 mm wide at \(4.2^\circ\) K) at the surface of the film—and specifically at its edge—is only about 10 gauss, which is only about \(2\%\) of the usual critical field of lead. The critical current increases with the thickness of the film, but in a rather complicated way. An important feature of these experiments is that the resistance is restored very abruptly, and with a further increase of the current above its critical value only a very slight further increase of the resistance is observed, again in contrast to the smeared-out transitions found by Meissner. This confirms that the smearing of the transition is a secondary effect, not caused by the influence of size. Qualitatively we may say that Shal’nikov’s work confirms the result that the critical field \(h_3\) for a current is considerably smaller than the critical field of a massive metal.

The most remarkable property of films obtained by evaporation is perhaps the large magnitude of the external magnetic field (both transverse and longitudinal) which must be applied to the film in order to restore its resis-

tion. The magnetic field necessary for restoring the resistance decreases as the measuring current is increased, and by a rough extrapolation to zero current, for the field of a lead film of thickness \(6\cdot 10^{-6}\) cm a value of 15,000 gauss was found. For the thinnest films it was not possible to make a reliable extrapolation, but it was evident that \(h_3\) is still considerably higher, i.e. of the order of \(10^5\) gauss. The question of the penetration of magnetic fields into the film has not yet been studied, and therefore at present it is impossible to say whether these films have an \(h_2\) distinct from \(h_1\) and \(h_3\).

All the properties described belonged to films deposited at \(4.2^\circ\)K. If these films were then warmed to room temperature and cooled again, they showed a clearly pronounced annealing, since the normal resistance after this proved to be noticeably smaller than the initial one. This annealing also had a strong influence on the superconducting properties of the films; thus, for tin films the transition temperature fell from an anomalously high value to approximately \(3.7^\circ\)K (which would correspond to the transition temperature of the solid metal).

For all films the other anomalous properties became less pronounced: the critical currents became higher, and the values of the field \(h_3\) decreased substantially. Nevertheless, they still differed essentially from the values characteristic of solid metals. Thus one may assert that, with the exception of the transition temperature, such annealing does not significantly change the qualitative characteristics of the superconducting properties of thin films.

It is obvious that Shalnikov’s results must to some extent depend on the structure of the films, which can change upon annealing. However, it is hardly possible to reduce the observed phenomena to the influence of structure alone, and one may consider that to one degree or another they are determined by the small thickness of the metal in the film.

We see that, in order to clarify the properties of thin films, a very large amount of experimental work still remains to be done. But, summarizing the results of Meissner and Shalnikov, we may say that the following facts have already been established almost definitely:

1) superconductivity can exist in films only 15 atomic layers thick;

2) as in alloys, in thin films there exist different critical fields for the restoration of resistance by a current, for the penetration of a magnetic field, and for the restoration of resistance by an external field. In particular, the field \(h_1\) for restoring the resistance by a current is much smaller, while the field \(h_3\) is much larger, than in a massive superconductor.

The question of whether the analogy between the behavior of alloys and thin films is accidental, or whether it has a deeper significance (as was speculatively assumed in Ch. VI), must remain open until some theory of superconductivity has been developed.

The result that the critical field \(h_1\) is smaller than in a massive

of a superconductor, is quite natural, since in the limiting case of infinitely thin films it is obvious that a film cannot carry any appreciable current, even if it were to remain superconducting. It is natural to expect that the thickness at which the critical field \(h_1\) begins to decrease should be comparable with the depth in which the current flows (“penetration depth”); but without special additional assumptions it is impossible to predict how \(h_1\) should depend on the thickness. From the experiments we may conclude that the penetration depth perhaps lies between \(10^{-5}\) and \(10^{-4}\) cm, but even this conclusion is speculative until further data are obtained (thus, if Misener’s results were complicated by the formation of an alloy, then it is possible that the effective thickness of the superconducting parts of his films was less than the total thickness, so that from the anomalous behavior of films \(10^{-3}\) thick one still cannot conclude that the penetration depth is of the same order of magnitude).

Perhaps the most inexplicable feature of the results obtained is that the critical field \(h_3\) is larger than in the bulk metal. It is also very difficult to understand how a magnetic field can penetrate through a thin film at a time when the resistance of the film is still zero, since (as we saw in Chap. II) the vanishing of the resistance of a bulk specimen is only a consequence of the fact that \(B\) vanishes somewhere inside. A possible way out of this difficulty is the assumption that penetration of the field occurs (as in alloys) only in some parts of the film, while the other parts remain superconducting and form a path through the film with zero resistance; such speculations, however, have little value as long as there are no further experimental data.

Chapter VIII. Conclusion

In this concluding chapter we shall briefly summarize the data set forth in our review of the phenomena of superconductivity.

It appears that an essential property of a pure bulk superconductor is that a magnetic field cannot penetrate into it deeper than some very small depth, i.e. a superconductor of macroscopic dimensions has permeability equal to zero. A consequence of this is that a current may flow over the surface of a superconductor in the absence of an electric field, so that it behaves as a body with zero resistance.

A large amount of recent work has been devoted to investigating the transition between the superconducting and normal states caused by a magnetic field. This has led to the idea of an “intermediate state,” which is a special kind of fine mixture of superconducting and normal phases. With the help of this idea one can explain fairly well, at least in simple cases, most of the features of these transitions. However, some properties relating to the electrical resis—

of the metal’s presence in an intermediate state. They are possibly connected with the fact that, on the surface of the specimen, the division into superconducting and normal regions is such that it is no longer possible to speak of the two phases as such, and we are dealing with a special kind of “mixed” phase, the properties of which are not at present entirely clear. The investigation of the mixed phase and of the size of the layers in the intermediate state are the principal unresolved problems concerning transitions in a magnetic field. Apart from these two questions, closely connected with the properties of superconductors of small dimensions, we may say that the explanation of the transition process follows from the property that the magnetic permeability of a superconductor is equal to zero and does not require a deeper theory of the origin of this property.

Similarly, the application of thermodynamics to the superconducting transition has shown that the jump in the heat capacity, the latent heat of transition (which vanishes in the absence of a magnetic field), and the more complicated caloric properties are all consequences of the magnetic properties of the superconductor. From the weak dependence of the critical field on pressure we have also deduced the fact that the change in volume, the coefficient of thermal expansion, and the elastic properties at the superconducting transition are so small as to lie beyond the limits of observation.

When we turn to superconducting alloys, the situation is no longer so clear; on general grounds it seems unlikely that there should exist two different kinds of superconductivity, and therefore it is natural to try to explain the anomalous behavior of alloys as due to some incidental cause, say, inhomogeneity of composition. The assumption that an alloy is a mixture of superconductors with different critical fields (each of which has all the properties of a pure superconductor) can indeed apparently explain the smearing and irreversibility of the superconducting transition in alloys, as well as the existence of different critical fields for the destruction of superconductivity by current, by an external magnetic field, and for the penetration of the magnetic field. This interpretation, however, is only a very crude one and cannot explain why the critical field for the restoration of resistance is so much larger than the other two. Therefore at present it is not quite clear to what extent the behavior of alloys is a consequence of the properties of pure superconductors, and to what extent a deeper theory of superconductivity is necessary for explaining these properties.

The properties of superconductors whose dimensions are comparable with, or smaller than, the depth of penetration of the magnetic field (studied up to now in the form of thin films) evidently cannot be explained from the equality to zero of the magnetic permeability of massive superconductors, since this property is valid only when the penetration depth may be neglected.

Further investigation of the influence of dimensions is apparently an especially promising path, since it can

provide a basis for a deeper theory of superconductivity, i.e., to explain the origin of the vanishing of the magnetic permeability of massive conductors. The analogy between the properties of thin films and of alloys shows that they may be related to one another, and therefore an explanation of the former may also lead to an explanation of the latter.

The discovery of the reversibility of the magnetic properties and the applicability of thermodynamic consequences from it have shown that the superconducting transition is an ordinary phase transition (in the absence of a magnetic field—a transition of the second kind), and the circumstance noted in Chapter I, that many properties of the metal in the normal and superconducting states differ little from one another, finds its explanation in the low temperature at which the transition occurs. Indeed, by analogy with other phase transformations we may expect that, for different phases, only those properties will differ appreciably which are connected with energies comparable to, or smaller than, \(kT\) (\(T\)—the transition temperature). Thus, for example, the energy of a light quantum in the visible region is of the order of \(10^4 kT\), and we could hardly expect any appreciable difference in the reflection and absorption of visible light by a super- and a nonsuperconductor. Ordinary low-frequency electromagnetic waves are not absorbed in the superconducting phase, and therefore it is natural to expect the existence of a critical frequency of the order of

\[ \frac{kT}{h}, \]

at which absorption should begin. This corresponds to a wavelength of about \(1\ \mathrm{cm}\), which, unfortunately, is technically difficult to obtain. At present it is known from experiment only that the critical wavelength is greater than \(10^{-2}\ \mathrm{cm}\) and less than \(10^3\ \mathrm{cm}\); this is in agreement with the considerations given above.

At present there is still no electronic theory of superconductivity. A substantial difficulty is that the modern electronic theory of metals treats the electrons as a gas, and since, as is known, phase transitions cannot occur in a gas, it is not surprising that this theory cannot explain the appearance of superconductivity. We may hope for the appearance of a theory of superconductivity only after the development of methods that make it possible to take into account the strong interaction of the electrons in a metal with one another. Such a theory might also explain the properties of the thermal conductivity of superconductors and give the dependence of the heat capacity on temperature at low temperatures. It is possible, however, that a more phenomenological theory will first have to be developed. Just as in the theory of ferromagnetism the magnetic properties were first explained by means of domains of magnetization, and only later were the causes of this magnetization explained, so here as well the first step may be the elucidation of the mechanism by which the magnetic permeability becomes equal to zero, and only then will an electronic theory of superconductivity be developed.

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1. Let us recall that very pure tin and mercury are metals which exhibit the most “ideal” behavior. By contrast, tantalum[^14] and niobium are examples of metals that behave like alloys, even when they are apparently very pure. ↩↩↩↩↩

2. The free energy considered here is, strictly speaking, the Gibbs free energy, often called the thermodynamic potential. ↩