CURRENT STATE OF THE THEORY OF SUPERCONDUCTIVITY

V. L. Ginzburg

I. MACROSCOPIC THEORY

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
§ 1. Basic experimental facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
§ 2. The old phenomenological theory . . . . . . . . . . . . . . . . . . . . 180
§ 3. The new phenomenological theory. Basic relations . . . . . . . 198
§ 4. The new phenomenological theory. Concrete results . . . . . . 204
§ 5. Normal current in superconductors. § 6. Normal and anomalous skin effect in metals. References (see the following issue)

INTRODUCTION

The phenomenon of superconductivity was discovered by Kamerlingh Onnes almost 40 years ago (in 1911). Nevertheless, to this day superconductivity continues to appear in all books and reviews as a rather mysterious phenomenon and, indeed, remains comparatively little studied both experimentally and theoretically. Thus, up to the present time the fundamental question of the depth of penetration of a magnetic field into a superconductor remains insufficiently clarified. Work on the intermediate state, on normal conductivity in superconductors, and so forth is also far from complete. In the field of theory, beginning in 1935, a certain success was achieved, connected above all with the construction of a phenomenological electrodynamics of superconductors \(^{1-3}\). However, this theory suffers from a number of substantial shortcomings that make it impossible to use it precisely in the most interesting and important cases. A more general phenomenological theory of superconductivity, free from these shortcomings, was developed only

very recently⁴ and has not yet been subjected to special experimental verification. As for the microscopic theory of superconductivity, here only individual considerations and ideas have been expressed, but no even purely approximate, yet at least somewhat coherent, theory (for example, such as the existing quantum theory of ferromagnetism) has yet been constructed. The attempts undertaken in this direction in recent times⁶–¹² do not alter this assessment, since, in our opinion, they cannot be considered successful. As a result, the microscopic theory of superconductivity is now the chief problem, unresolved even in principle, in the physics of metals and, indeed, in the whole of atomic and molecular physics.

Indeed, if one does not speak of nuclear physics, cosmic-ray physics, and astrophysics, then in all other fields the understanding of essential phenomena has advanced considerably farther than can be said of superconductivity. The latter also applies to the phenomenon of superfluidity of helium II, which is related to superconductivity. Despite the fact that superfluidity was discovered by Kapitza only in 1938, i.e. considerably later than superconductivity, superfluidity has at present been studied better than the latter, mainly thanks to the theoretical and experimental investigations carried out at the Institute for Physical Problems of the Academy of Sciences of the USSR in Moscow (see the reviews¹³–¹⁴; for some unresolved questions see also¹⁵). The successes achieved in the study of superfluidity—a phenomenon simpler and at the same time analogous to superconductivity—the general progress in the technique of physical experiment, and other factors have contributed to the fact that in recent years the study of superconductivity has been proceeding on an ever broader front. One may therefore think that many experimental questions which remain unclear here up to now will soon be clarified. Such a situation cannot but be accompanied by an increase in the never-extinguished interest in the theory of superconductivity. And this is indeed taking place—it is enough to say that, beginning in 1945, about 40 corresponding papers have appeared (true, in the overwhelming majority of cases not representing any special interest).

All that has been said justifies, as it seems to us, the appearance of the present article, whose aim is to illuminate the contemporary state of the phenomenological and microscopic theory of superconductivity. In doing so, the author considered it possible and proper to dwell in detail only on those questions and works which seem to him essential, and did not seek to illuminate in detail the attempts to create a theory of superconductivity which do not seem promising. In the present article not

it would also have been appropriate to dwell on a number of particular concrete theoretical questions already considered in the review literature\(^1\)–\(^3\). Of course, such an inevitably somewhat subjective approach to the selection of material has its drawbacks. In order, to a certain extent, to neutralize possible omissions in this respect and, in general, to help in finding one’s bearings among the new works, the bibliography has been made as complete as possible.

The entire article is divided into two parts, devoted respectively to macroscopic and microscopic theory. At the same time, in the present case it would be both impossible and inexpedient to carry out such a division into micro- and macro-theory with any strictness; for example, it proved convenient to include in the first part published below the question of the anomalous skin effect, which is connected rather with the microscopic (i.e., so-called electronic) theory of metals. The second part of the article, devoted mainly to microscopic theory, the author hopes to complete in the near future.

§ 1. BASIC EXPERIMENTAL FACTS

Before turning to the discussion of the theory, let us recall the basic experimental data concerning superconductivity (for more details see\(^2\) and the literature cited there; references to new works are given below). The phenomenon of superconductivity was discovered and received its name as a result of the discovery of the fact that certain metals at low temperature completely lose their electrical resistance. In this case the loss of resistance occurs discontinuously at a certain critical temperature \(T_k\), characteristic of the given metal. For all superconducting elements \(T_k < 10\) degrees Kelvin. The temperature dependence of the resistance \(R\) on the temperature \(T\) for superconductors and nonsuperconductors is clear from the schematic Fig. 1. In experiment, of course, the fall of the resistance occurs over a certain finite temperature interval \(\Delta T\). This interval for

Fig. 1.

alloys and superconducting elements with impurities may be very broad, reaching several degrees. However, for sufficiently pure, single-crystal, and linearly stressed metals the value \(\Delta T\) decreases to \(\sim 10^{-3}\) degrees (see, however, \(^{16}\)). Therefore one may assume that, under ideal conditions, the loss of resistance indeed occurs discontinuously. In what follows we shall suppose that we are dealing only with such ideal superconductors, and shall ignore phenomena—such as, for example, hysteresis—which may be regarded as due to the nonideality of the specimen under investigation. Let us also note that there is no indication that in the superconducting state, i.e. at \(T<T_k\), the resistance of the metal, though extremely small, is nevertheless different from zero. On the contrary, all available data argue in favor of the fact that in a superconductor electrical resistance, in the usual sense of this concept, disappears completely; i.e., for example, an electric current flowing in a closed circuit does not decay at all.

The second special feature of superconductors, apart from the loss of resistance, is their characteristic behavior in a magnetic field, which amounts, roughly speaking, to the fact that the magnetic field never penetrates into the bulk of a superconducting metal. This property of superconductors (established in 1933 and sometimes called the Meissner effect) is often called ideal diamagnetism, since in a diamagnetic body with magnetic permeability \(\mu=0\) (i.e. susceptibility

\[ \chi=\frac{\mu-1}{4\pi}=-\frac{1}{4\pi} \]

) the mean microscopic magnetic field \(B\) (magnetic induction) is likewise equal to zero. However, the analogy between a superconductor and an ideal diamagnetic body has a limited range of applicability. Thus, in a diamagnetic body, as in any magnetic material, at the boundary with vacuum the tangential component of the induction vector \(B_t\) changes discontinuously, so that

\[ B_{t(\text{vacuum})}=\frac{B_{t(\text{metal})}}{\mu}. \]

From the microscopic point of view this means that the transition layer in which the magnetic field changes has a thickness of the order of atomic dimensions, i.e. \(\sim 10^{-8}\div 10^{-7}\ \text{cm}\). From experimental data, however, it is known that the depth of penetration of the magnetic field into the bulk of a superconductor is \(\delta\sim 10^{-5}\ \text{cm}\), and near \(T_k\) is even larger. Therefore, for superconductors of small dimensions (films, wires), whose properties in the absence of a field are still entirely identical with those of a massive metal, one cannot speak of the nonpenetration of the field. In this case the field inside the specimen is only somewhat weaker than at its surface. The character of the penetration of a magnetic field into a superconductor has, as will be clear from what follows, extremely great significance. At the same time it is clear that in all cases when we

although we wish to consider this question, a superconductor cannot be regarded as an ideal diamagnet, i.e., characterized simply by the permeability \(\mu = 0\). This can be done only in such problems as, for example, the problem of the distribution of the magnetic field around a massive superconductor (i.e., if the dimensions of the superconductor \(L \gg \delta\)). In such cases, evidently, inside the superconductor \(B=0\), while outside it the field is the same as in the case of an ideal diamagnet of the same shape. The distribution of a field parallel and perpendicular to the axis of a cylinder in the superconducting or normal states is shown, for illustration, in Fig. 2. Since none of the known superconductors is

Superconductor  Non-superconductor  Superconductor  Non-superconductor

Fig. 2.

a ferromagnet in the normal state (for \(T > T_k\)), the magnetic field in a superconductor is practically the same as in vacuum (since \(|\chi| \leq 3 \cdot 10^{-5}\)). Below, we shall nowhere distinguish the induction \(\mathbf{B}\) from the field \(\mathbf{H}\).

Clarifying the question of the depth of penetration of the field into a superconductor is, experimentally, very difficult and is still far from complete, despite the appearance of a number of new works in this direction \(^{17-20a}\). The problem of field penetration into a superconductor is closely connected with the theory of superconductivity, and we shall have to touch upon it again below. For the moment we shall point out that, according to all data, the penetration depth \(\delta\) (defined, for example, as the distance from the surface of the metal at which the field decreases by a factor of \(e = 2.72\)) at \(T \to 0\) is of the order of \(10^{-5}\) cm and increases on approaching the critical temperature \(T_k\) in such a way that, for \(T \to T_k\), \(\delta \to \infty\). Apparently, for \(T \sim T_k\),

\[ \delta^2 = \frac{\mathrm{const}}{T_k - T}. \tag{1,1} \]

The dependence of \(\delta\) on \(T\) for mercury, according to various data which we shall discuss in § 4, is shown in Fig. 3. In this case along the axis

on the ordinate not \(\delta\), but \(\delta(T)-\delta(2.5^\circ)\). For curve \(II\), obtained in \({}^{4}\) on the basis of the measurements of \({}^{21}\), \(\delta(2.5^\circ)=0.66\cdot 10^{-5}\) cm (curves \(I\) and \(III\) were constructed from the data of works \({}^{17}\) and \({}^{19}\), in which precisely the difference \(\delta(T)-\delta(2.5^\circ)\) was determined directly).

The superconducting and normal states of a given metal can and must be regarded as two of its phases in the usual thermodynamic sense of the word. In this case, in the absence of a magnetic

Fig. 3.

field, the transition from the normal state to the superconducting state, occurring at the temperature \(T_k\), is a phase transition of the 2nd kind, i.e. it is accompanied only by a jump in the heat capacity, and not by the liberation of latent heat, as would occur in the case of a transition of the 1st kind. In the presence of a magnetic field, the transition to the superconducting state already occurs at a temperature lower than \(T_k\), and is a transition of the 1st kind. At each given temperature \(T<T_k\), the superconducting phase exists or, more precisely, is thermodynamically stable only in a field \(H<H_k\), where \(H_k(T)\) is the critical magnetic field. At \(T=T_k\), \(H_k=0\). The state diagram of a superconductor in a magnetic field

is represented in Fig. 4*). The critical field at \(T=0\) is maximal and reaches several hundred oersteds (for example, for mercury \(H_k(0)=412\) oersteds). At \(H>H_k\) the metal passes into the normal state, which is not qualitatively different in any respect from the normal state at \(T>T_k\). If, in agreement with experiment, one assumes that the metal at \(T<T_k\) can be in a normal state which, in the absence of a field, is not thermodynamically stable, then the very fact of the existence of the critical field \(H_k\) is quite understandable. The point is that, owing to the nonpenetration of the field into the superconductor, as a simple thermodynamic consideration shows (see § 2), in a sufficiently strong field the existence of the normal phase, in which the magnetic field is the same as in vacuum, will already be more advantageous. In this case, for massive specimens,

Fig. 4.

\[ F_n-F_s=\frac{H_k^2}{8\pi}, \tag{1,2} \]

where \(F_n\) and \(F_s\) are the free energies of unit volume, respectively, of the normal and superconducting phases in the absence of a field. It follows from experiment that near \(T_k\)

\[ H_k=\left(-\frac{dH}{dT}\right)_{T_k}(T-T_k), \tag{1,2a} \]

where

\[ \left(\frac{dH_k}{dT}\right)_{T_k}<0 \]

and, for example, for mercury

\[ \left|\frac{dH_k}{dT}\right|_{T_k}\simeq 190\,\frac{\text{oersted}}{\text{degree}}. \]

Using (1,2), one can express through \(H_k\), \(\dfrac{dH_k}{dT}\), and \(\dfrac{d^2H_k}{dT^2}\) the entropy difference \(\Delta S=S_1-S_n\) and the heat-capacity difference \(\Delta c=c_s-c_n\) in the superconducting and normal states. The corresponding formulas are given in § 2; here we only wish to point out that the jump in heat capacity \(\Delta c\) at \(T\leq T_k\) can be determined with the aid of magnetic measurements.

The heat capacity of a metal is composed of the heat capacity of the lattice \(c^r\) and the electronic heat capacity \(c^e\), and for a nonsuperconducting—

*) The transition from the superconducting state to the normal state in a magnetic field is sharp only if the field is homogeneous, as is the case for a cylinder in a field parallel to its axis. We shall always assume that we are dealing precisely with this case.

conductors and superconductors in the normal state at low temperatures

\[ c_n=c^0+c_n^e;\quad c^0=aT^3,\quad c_n^e=\gamma T, \tag{1,3} \]

where \(a\) and \(\gamma\) are constants independent of \(T\) \(\left(a=\dfrac{464.4}{\theta^3}\ \dfrac{\text{calories}}{\text{mole}\cdot\text{degree}}\right)\), where \(\theta\) is the Debye temperature, equal, for example, for mercury to \(\theta=90^\circ\); for mercury \(\gamma=4.5\cdot10^{-4}\ \dfrac{\text{calories}}{\text{mole}\cdot\text{degree}^2}\). The state of the lattice in the superconducting and normal states, apparently, may with a high degree of accuracy be regarded as identical. This is indicated both by the complete identity of the X-ray structure of the two phases and by the extreme smallness of the change in the volume of the metal upon the destruction of superconductivity in a magnetic field \(\left(\dfrac{\Delta V}{V}\sim 10^{-7}\right.\) at \(H_k\sim 100\) oersteds, at \(\left.T=T_k\ \Delta V=0\right)\). Therefore one may think that the heat capacity of the lattice in the superconducting state is the same as in the normal state and is equal to \(c^0\). The electronic heat—

Fig. 5.

Fig. 6.

capacity in the superconducting state \(c_s^e\) differs strongly from \(c_n^e\) and, as is clear from the above,

\[ c_s^e=\Delta c+c_n^e=\Delta c+\gamma T, \tag{1,4} \]

where \(\Delta c=c_s-c_n\) is the jump in heat capacity, which can be determined from magnetic or from calorimetric measurements. The values \(c_s=c^0+c_s^e\) and \(c_n=c^0+c_n^e\) can be determined, and in fact in a number of cases have been determined, from calorimetric measurements.

measurements. The available experimental data fully confirm the law (1.3) for \(c_n^e\), make it possible for most metals to find \(\gamma\) (see\(^ {33}\)), and indicate that in a number of cases \(c_s^e \sim T^3\) (see Part II). The values \(\Delta c = c_s^e - c_n^e = c_s - c_n\) and \(c_s^e\) for mercury, indium, and thallium are given in Figs. 5 and 6.

Superconductivity is destroyed not only by an external magnetic field, but also by a current flowing through the superconductor. Here, however, we do not have two different phenomena, but essentially one and the same phenomenon, since the current is associated with a magnetic field. In the case of a massive superconductor, say a circular cylinder through which a current \(I\) flows, the critical value of the current is \(I_k = 2cRH_k\), where \(R\) is the radius of the cylinder and \(c\) is the speed of light. Thus, in this case, and in general for massive specimens, the destruction of superconductivity by a current begins when the current field at the surface reaches the critical value \(H_k\) (as is known, the current field in the case of a cylinder is just equal to \(H = \dfrac{I}{2cR}\)).

For thin specimens (i.e., when the film thickness \(2d\) or the wire radius \(R\) is of the order of, or less than, the penetration depth \(\delta\)), the situation changes. In this case the critical magnetic field \(H_k\) is greater than the critical field \(H_{km}\) for a massive superconductor (in (1.2) the field \(H_{km}\) appears, but earlier we simply omitted the subscript “\(m\)”). Experimentally, for sufficiently thin films \(H_k\) is greater than \(H_{km}\) by tens of times\(^ {21}\). Conversely, the critical current \(I_k\) for thin specimens is smaller than in the case of thick ones. The calculation of \(H_k\) and \(I_k\) is a problem of the theory and will be discussed below.

Fig. 7.

Measurements of the thermal conductivity of a metal in the normal and superconducting states showed that at \(T_k\) there is no jump in thermal conductivity, but only a change in its dependence on temperature (see Fig. 7, which shows the thermal resistance of lead; the dashed line indicates the resistance at \(H > H_k\), when lead is in the normal state; recent works on thermal conductivity\(^ {23,24,25}\)). Meanwhile, if at \(T = T_k\) the electronic part of the thermal conductivity were to become zero, then the thermal conduct-

of the metal, which is determined, in addition to the electronic part of the thermal conductivity, by the thermal conductivity of the lattice, would undergo a discontinuity. Thus, it follows from experiment that the electronic part of the thermal conductivity does not undergo a discontinuity. Further, the superconducting current, as follows from experiment, is completely unrelated to the transfer of heat and, therefore, if, roughly speaking, at \(T = T_k\) all the conduction electrons passed into some superconducting state, then a discontinuity of the thermal conductivity would necessarily take place. Hence one may arrive at the conclusion, supported by a whole series of other facts and considerations, that in the superconducting state only part of the electrons can participate in the transfer of the superconducting current, while the other part, vanishing as \(T \to 0\), gives rise to transport processes of the same type as in the normal state. It follows from this that, in addition to electronic thermal conductivity, there must also be observed in superconductors a “normal conductivity,” i.e., in addition to the superconducting current there may flow a “normal” current associated with the release of Joule heat. Under ordinary conditions this current cannot be observed, since it is masked by the superconducting current. More precisely, under stationary conditions the electric field in a superconductor is equal to zero and the normal current, obviously, must be absent. However, in an alternating field in a metal there is inevitably present not only a magnetic but also an electric field, and Joule losses must occur. This is indeed observed experimentally. Recently the normal conductivity in superconductors, observed in a high-frequency field, has been the subject of a number of works\(^{26—29b}\), which will be discussed in § 6. Let us note here only that at very high frequencies in the infrared part of the spectrum (frequency \(\omega \sim 5 \cdot 10^{14}\)) the properties of superconductors and non-superconductors are already practically identical, i.e., in other words, superconductivity disappears. At the highest radio frequencies achieved

\[ \left( \frac{\omega}{2\pi} = 2.4 \cdot 10^{10},\ \lambda = \frac{2\pi c}{\omega} = 1.25\ \text{cm} \right), \]

on the contrary, superconductivity is still observed\(^{29}\). The study of the intermediate frequency region, where superconductivity must disappear, has not yet been carried out, although the difficulties of work at such frequencies are by no means insurmountable.

We shall not dwell here on the rather curious thermoelectric properties of superconductors (see § 5 and\(^{2}\); new works\(^{30,31}\)). We shall also confine ourselves merely to mentioning new works devoted to the influence of pressure\(^{32—34}\), to the properties of the intermediate state\(^{35—40}\), and to other questions\(^{41—65б}\).

In conclusion of this brief survey of experimental data, let us recall that the phenomenon of superconductivity is not universal and, in any case, such metals as Ag, Au,

Cu, Bi, Pt, and some others are not superconductors down to temperatures of \(0.05 \div 0.1^\circ\). The elements indicated in Table 1 and a number of alloys are superconducting. Certain empirically established features of superconducting elements (their place in the Mendeleev table, the magnitude of the atomic volume, etc.) are discussed in \({}^{2,64}\).

Table 1

Superconducting elements

Element	\(T^\circ_{\mathrm{k}}\)	Element	\(T^\circ_{\mathrm{k}}\)
1. Niobium Nb (Columbium Cb)	9.22	10. Thorium Th	\(\sim 1.35\)
2. Lead Pb	7.26	11. Uranium U	1.30
3. Lanthanum La	4.7	12. Aluminum Al	1.17
4. Tantalum Ta	4.38	13. Gallium Ga	1.07
5. Vanadium V	4.3	14. Zinc Zn	0.95
6. Mercury Hg	4.16	15. Rhenium Re	\(\sim 0.9\)
7. Tin (white) Sn	3.71	16. Zirconium Zr	\(\sim 0.7\)
8. Indium In	3.37	17. Cadmium Cd	\(\sim 0.6\)
9. Thallium Tl	2.38	18. Titanium Ti	0.53
9. Thallium Tl	2.38	19. Hafnium Hf	\(\sim 0.3\)

In connection with the data given in Table 1, it should be borne in mind that the temperature \(T_{\mathrm{k}}\) depends rather strongly on the presence of impurities and, in general, varies from sample to sample (the corresponding changes in \(T_{\mathrm{k}}\) even for good samples sometimes reach \(\sim 0.1^\circ\)). In addition, when determining \(T_{\mathrm{k}}\) from resistance measurements, in some cases completely false values of \(T_{\mathrm{k}}\) are obtained, apparently owing to the formation in the metal of superconducting filaments whose properties are determined by impurities (see, for example, the cases of titanium and uranium \({}^{2,43,51,58,58a}\)).

Recently, in superconductors a considerable isotope effect has been discovered \({}^{102—104}\), consisting in a shift, upon passing from isotope to isotope, of the critical temperature \(T_{\mathrm{k}}\) and of the entire curve \(H_{\mathrm{k}}(T)\). Thus, for example, for the isotope \(\mathrm{Hg}^{198}\), \(T_{\mathrm{k}} = 4.177^\circ\); for natural mercury \((A = 200.6)\), \(T_{\mathrm{k}} = 4.156^\circ\), and for a Hg sample with atomic weight \(A = 203.4\), \(T_{\mathrm{k}} = 4.137^\circ\). The changes in \(T_{\mathrm{k}}\) are such that approximately

\[ A^{1/2}T_{\mathrm{k}} = \mathrm{const}, \]

that is,

\[ \frac{T_{\mathrm{k}}}{\theta} = \mathrm{const}, \]

where \(\theta\) is the Debye temperature, which is just inversely proportional to the square root of the mass of the atom.

§ 2. THE OLD PHENOMENOLOGICAL THEORY

The theory of superconductivity can naturally be divided into a phenomenological and a microscopic one. The task of the first consists in establishing equations for the superconducting current and, in general, a complete system of electrodynamic equations for the case of superconductors. The microscopic theory, on the other hand, must lead to a justification of the accepted equations, i.e. to their derivation as a result of considering the motion of electrons in a metal, and also to an order-of-magnitude estimate of the quantities and to the establishment of the temperature dependence of the coefficients entering the phenomenological equations. Of course, drawing a sharp boundary between phenomenological theory and microscopic theory is not always expedient and even, in a certain sense, not always possible. However, in the case of superconductivity, whose mechanism still remains unclear in many respects, it is especially important, insofar as possible, to separate clearly the phenomenological scheme from various model representations. We shall proceed precisely along this path and begin with the phenomenological theory*).

Electrodynamic processes in any system of bodies, some of which may be superconductors, are described by Maxwell’s equations

\[ \begin{aligned} \operatorname{rot}\mathbf{H} &= \frac{4\pi}{c}\mathbf{j} + \frac{1}{c}\frac{\partial \mathbf{D}}{\partial t}, \qquad \operatorname{div}\mathbf{D} = 4\pi\rho,\\ \operatorname{rot}\mathbf{E} &= -\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}, \qquad \operatorname{div}\mathbf{B} = 0, \end{aligned} \tag{2,1} \]

where \(\mathbf{j}\) is the current density, \(\rho\) the charge density, \(\mathbf{E}\) and \(\mathbf{H}\) the intensities of the electric and magnetic fields, and \(\mathbf{D}\) and \(\mathbf{B}\) the inductions of these fields.

Equations (2,1) acquire definite content only if the relations between \(\mathbf{B}\), \(\mathbf{D}\), \(\mathbf{j}\), \(\mathbf{E}\), and \(\mathbf{H}\) are specified. In stationary isotropic bodies it is usually assumed, in agreement with experiment, that

\[ \mathbf{D} = \varepsilon \mathbf{E}, \qquad \mathbf{B} = \mu \mathbf{H}, \tag{2,2} \]

\[ \mathbf{j} = \sigma \mathbf{E}, \tag{2,3} \]

where \(\varepsilon\), \(\mu\), and \(\sigma\) do not depend on the field vectors.

In superconductors there is no reason to abandon the relations (2,2). It need only be noted that the dielectric constant \(\varepsilon\) may, in principle, turn out to be superconducting.

*) In order not to encumber the exposition, we shall not, as a rule, refer to the older theoretical works, and we refer the reader, for the corresponding references, to reviews \(^{1-3}\).

state is quite different than in the normal one (see about this in § 6); in addition, in both states the permeability \(\mu\), apparently, is very close to unity (in the case of the normal state this is firmly established; in the case of the superconducting state there is no indication that \(\mu\) differs appreciably from 1). Therefore, in what follows we shall always assume that \(\mu=1\).

Equation (2,3) is Ohm’s law written in differential form. Precisely this relation (2,3) does not hold for the superconducting current, which meets with no resistance. The task of the theory, first of all, consists in replacing Ohm’s law (2,3) by another one, without which it is impossible to use equations (2,1). In doing so it is expedient to take into account at once that in a superconductor, besides the superconducting current, a normal current \(\mathbf{j}_n\) may flow. The total current is

\[ \mathbf{j}=\mathbf{j}_s+\mathbf{j}_n. \tag{2,4} \]

For the normal current it is natural to assume the validity of Ohm’s law

\[ \mathbf{j}_n=\sigma\mathbf{E}, \tag{2,5} \]

where \(\sigma\) is the “normal” conductivity in the superconducting state. As has become clear recently, the direct use of equation (2,5) proves in a number of cases to be inadmissible, since the depth of penetration of the electromagnetic field into a superconductor may be smaller than the mean free path of the “normal” electrons in the metal. In this case the density of the normal current at a given point is determined not only by the field \(\mathbf{E}\) at the same point, but also by the field in an entire region of the metal. Under such conditions, both in superconductors and in normal metals, the so-called anomalous skin effect is observed. We shall dwell on this circle of questions in § 6. For the time being, however, for definiteness, we shall use relation (2,5) for \(\mathbf{j}_n\). Regardless of whether this relation is correct or not, the normal current is not equal to zero only if the electric field \(\mathbf{E}\) is not equal to zero (we are now disregarding currents arising in the presence of a temperature gradient; see also the note to p. 183). The electric field in a superconductor, however, is nonzero only in the nonstationary case, since an electric field constant in time and not equal to zero would lead to an increase of the superconducting current. Thus, the possibility of the presence of a normal current must be taken into account only in an alternating field (and also in the presence of a temperature gradient, see § 5). Meanwhile, of primary interest is the case of a magnetic field constant in time, in the presence of which a stationary superconducting current flows in the superconductor.

It is precisely this case that we shall mainly consider below, with the exception of §§ 5 and 6.

Thus, setting aside the question of the normal current, it is necessary to establish equations determining the connection with the field of the superconducting current \(\mathbf{j}_s\). Within the framework of a phenomenological theory this can be done either by relying directly on experimental data, as is the case for Ohm’s law (2.3), or by proceeding from certain assumptions, justified by experiment, and from general invariance requirements. In the stationary case, as follows from experiment, the current \(\mathbf{j}_s\) is uniquely connected with the magnetic field \(\mathbf{H}\) at the same point or in its neighborhood. Assuming that the connection between \(\mathbf{j}_s\) and \(\mathbf{H}\) is linear, it is easy to see that the simplest equation determining this connection has the form:

\[ \operatorname{rot}\Lambda \mathbf{j}_s=-\frac{1}{c}\mathbf{H}, \tag{2.6} \]

where \(\Lambda\) is a certain new constant, which may depend on temperature, \(c\) is the speed of light, and the equation has at once been written in the generally accepted form. That equation (2.6) is the simplest possible one is not difficult to verify, taking into account that \(\mathbf{j}_s\) is a polar vector, while \(\mathbf{H}\) is an axial vector. Therefore the relation \(\mathbf{j}_s=\operatorname{const}\cdot\mathbf{H}\) is non-invariant and, consequently, inadmissible. Further, two equations containing only first-order derivatives are possible: \(\operatorname{rot}\mathbf{H}=\operatorname{const}\cdot\mathbf{j}_s\) and \(\operatorname{rot}\mathbf{j}_s=\operatorname{const}\cdot\mathbf{H}\). The first of these expressions, with the corresponding choice of the constant, coincides with one of Maxwell’s equations, while the second expression coincides with relation (2.6). In this case, as already stated, the constant is chosen in such a way (in the form \(\operatorname{const}=-c\Lambda\)) that the equation in appearance coincides with that obtained by another method, indicated below.

Assuming that equation (2.6) is also valid in a variable field, we differentiate both sides of this equality with respect to time and take into account that

\[ -\frac{1}{c}\frac{\partial\mathbf{H}}{\partial t}=\operatorname{rot}\mathbf{E}. \]

It follows that

\[ \operatorname{rot}\left(\frac{\partial\Lambda\mathbf{j}_s}{\partial t}-\mathbf{E}\right)=0 \]

and

\[ \frac{\partial\Lambda\mathbf{j}_s}{\partial t}=\mathbf{E}+\nabla\chi, \tag{2.7} \]

where the function \(\chi\) is for the time being arbitrary \((\nabla\chi=\operatorname{grad}\chi)\). A number of considerations (see §§ 13, 16 and 23, paper \(^{66}\), and below, equation (2.18)) speak in favor of the fact that \(\chi=\operatorname{const}\cdot j_s^2\). It can be shown that the presence of this term should, generally speaking, be unimportant \(^{2,66}\). From the experimental point of view, too, there are no indications testi-

attesting to the necessity of assuming that in (2.7) \(\nabla\chi\ne0\)*). Therefore it is usually assumed that \(\nabla\chi=0\) and the equation

\[ \frac{\partial \Lambda \mathbf{j}_s}{\partial t}=\mathbf{E}. \tag{2.8} \]

is used.

As is clear from what has been said, equation (2.8) cannot be regarded simply as a consequence of equation (2.6), and is in a certain respect independent. It is essential only in the nonstationary case.

Equations (2.6) and (2.8), together with the first pair of Maxwell’s equations and the relations (2.2), form a complete system of electrodynamic equations (the equations \(\operatorname{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{H}}{\partial t}\) and \(\operatorname{div}\mathbf{H}=0\), as is easy to see, are consequences of equations (2.6) and (2.8)). The electrodynamics of superconductors, based on equations (2.6) and (2.8), was developed by F. and H. London in 1935 and later; the solution, on the basis of these equations, of concrete problems was carried out chiefly by Laue\(^3\). The generalization to the case of anisotropic, moving, and nonuniformly heated superconductors, as well as the allowance for possible nonlinearity of the equations, see in \({}^{2,67-71}\). The phenomenological theory of superconductivity based on equation (2.6) is, in a certain respect, internally closed, and its creation was undoubtedly a substantial step forward. However, as we shall see below, this theory, which for brevity we shall call the “old” one, has a limited range of applicability and cannot be used for considering the boundary between the normal and superconducting phases, the process of destruction of superconductivity in thin superconductors by field and current, and, in general, is not valid in strong fields, i.e. when \(H\sim H_k\). We shall discuss in greater detail the shortcomings of the old phenomenological theory shortly, and then, in §§ 3, 4, set forth a new, more general theory\(^4\). For the present, however, bearing in mind that the “old” theory is applicable in a number of cases and, moreover, until very recently was the only one considered, we shall devote some further attention to it.

Equations (2.6) and (2.8) were originally established not in the manner done above, but by another, less

*) It should be noted that if \(\nabla\chi\ne0\), then in the stationary case, when \(\partial\Lambda\mathbf{j}_s/\partial t=0\), \(\mathbf{E}=-\nabla\chi\ne0\). Therefore, if \(\nabla\chi\ne0\), equation (2.5) must be written in the form \(\mathbf{j}_n=\sigma(\mathbf{E}+\nabla\chi)\), since otherwise in the stationary state dissipation of energy would be observed, which contradicts experiment. Above, when we indicated that in the stationary case \(\mathbf{E}=0\), it was implicitly assumed that \(\nabla\chi=0\). It is easy to see that if \(\nabla\chi\ne0\), all our argument remains essentially unchanged, and the normal current is different from zero only under nonstationary conditions.

formal. In view of the importance of the question, we shall give this derivation, although, strictly speaking, what is involved is not a derivation but certain suggestive considerations. Since the superconducting current meets no resistance, the necessity of taking into account the inertia of the electrons carrying this current is evident. If one starts from the classical model of a free electron gas in a metal, then the equation of motion of an electron in the absence of friction has the form \(\dfrac{m d\mathbf v}{dt}=e\mathbf E\), where \(e\) and \(m\) are the charge and mass of the electron, and \(\mathbf v\) is its velocity. Multiplying both sides of this equation by the charge \(e\) and the concentration of “superconducting” electrons \(n_s\), and introducing the current density \(\mathbf j_s=e n_s\mathbf v\), we obtain equation (2.8), where

\[ \Lambda=\frac{m}{e^2 n_s}. \tag{2.9} \]

This derivation of equation (2.8), even within the framework of the chosen crude model, is by no means rigorous, since in the initial equation of motion we, on the one hand, did not take into account the influence of the magnetic field, i.e. the Lorentz force \(\dfrac{e}{c}[\mathbf v\mathbf H]\), and, on the other hand, instead of the total derivative \(\dfrac{d\mathbf v}{dt}=\dfrac{\partial\mathbf v}{\partial t}+(\mathbf v\nabla)\mathbf v\), in passing to (2.8) we retained only the partial derivative \(\dfrac{\partial\mathbf v}{\partial t}\). The meaning of these assumptions and their well-known justification will be clear from what follows.

Equation (2.8) corresponds to the case of an “ideal conductor.” To describe the properties of a superconductor this equation is insufficient. Indeed, applying to (2.8) the operation \(\operatorname{rot}\) and taking into account the equation \(\operatorname{rot}\mathbf E=-\dfrac{1}{c}\dfrac{\partial\mathbf H}{\partial t}\), we have

\[ \frac{\partial}{\partial t}\left(\operatorname{rot}\Lambda\mathbf j_s+\frac{1}{c}\mathbf H\right)=0. \tag{2.10} \]

It follows from this that \(\operatorname{rot}\Lambda\mathbf j_s=-\dfrac{1}{c}\mathbf H-\dfrac{1}{c}\mathbf H_0\), where \(\mathbf H_0\) is a certain time-independent magnetic field determined by the initial conditions of the problem. The field \(\mathbf H_0\) may be different from zero also in the interior of the superconductor, where it must, however, remain constant in time. Indeed, neglecting in the first equation (2.1) the displacement current and the normal current, which is always possible at sufficiently low frequencies, we have

\[ \operatorname{rot}\mathbf H=\frac{4\pi}{c}\mathbf j_s. \tag{2.11} \]

Hence \(\operatorname{rot}\operatorname{rot}\mathbf H=-\Delta\mathbf H=\dfrac{4\pi}{c}\operatorname{rot}\mathbf j_s\), and, taking (2.10) into account,

\[ \frac{\partial}{\partial t}\left(\Delta\mathbf H-\frac{4\pi}{c^2\Lambda}\mathbf H\right)=0. \tag{2.12} \]

From (2.12) it follows that, for example, in the case of a plane surface of a superconductor parallel to the external field, inside the metal

\[ \frac{\partial \mathbf H}{\partial t} = \left(\frac{\partial \mathbf H}{\partial t}\right)_0 e^{-z/\delta}, \tag{2.13} \]

where \(\left(\dfrac{\partial \mathbf H}{\partial t}\right)_0\) is the value of \(\dfrac{\partial \mathbf H}{\partial t}\) at the interface, \(z\) is the distance from this boundary into the body, and \(\delta\) is the penetration depth

\[ \delta^2=\frac{\Lambda c^2}{4\pi}=\frac{mc^2}{4\pi e^2 n_s}. \tag{2.14} \]

In the limit, if \(\Lambda \to 0\), as is clear from (2.8) and (2.13), in a superconductor

\[ \mathbf E=0,\qquad \frac{\partial \mathbf H}{\partial t}=0. \tag{2.15} \]

The equalities (2.15) correspond to the conditions which are regarded as holding in ideal conductors, i.e. conductors with infinitely large conductivity. Taking account of the inertia of the electrons leads to the replacement of the conditions (2.15) by the relations (2.8) and (2.13), which is important only in the transition layer and, for \(z\gg \delta\), plays no role. Thus, if a superconductor could be considered simply as an ideal conductor, the magnetic field in its interior need not be zero. It would seem that precisely this case should occur if a superconductor heated above \(T_k\) is placed in a constant magnetic field and then cooled below \(T_k\). Meanwhile, the experimental facts (the Meissner effect) show that in all cases, including the experiment just mentioned, in the bulk of a sufficiently pure superconducting metal one always has \(\mathbf H=0\). Hence it is clear that the equation (2.8) is insufficient, and that some additional condition must be imposed, selecting only a part of the solutions of this equation, namely those solutions for which \(\mathbf H=0\) in the bulk of the metal. This selection is equivalent to the requirement that the field \(\mathbf H_0\) introduced above, after formula (2.10), always be equal to zero. In this way one obtains precisely equation (2.6). In this case, instead of (2.13), we obtain the solution \(\mathbf H=\mathbf H_0 e^{-z/\delta}\), which will be discussed further below (see (2.26)). Such, approximately, is the path which led to the formulation of the old phenomenological theory based on equations (2.6) and (2.8).

These equations also have a clear hydrodynamic meaning; to elucidate it, let us consider the motion of a certain charged liquid\({}^{66}\). In a known approximation, such a liquid may be taken to be the aggregate of the conduction electrons in a metal. The Euler equation for a charged liquid evidently has the form

\[ \frac{d\mathbf v}{dt} = \frac{\partial \mathbf v}{\partial t} + (\mathbf v\nabla)\mathbf v = \frac{\partial \mathbf v}{\partial t} + \frac{1}{2}\nabla \mathbf v^2 - [\mathbf v,\operatorname{rot}\mathbf v] = \]

\[ = -\frac{\nabla p}{\rho_m} + \frac{e}{m} \left( \mathbf E+\frac{1}{c}[\mathbf v\mathbf H] \right), \tag{2.16} \]

where \(p\) is the pressure, \(\rho_m=mn\) is the mass density, and \(\dfrac{e}{m}\) is the ratio of the charge density of the liquid \(en_s\) to its density \(\rho_m=mn_s\). (Thus \(n_s\) is the concentration of the particles forming the liquid, and \(e\) and \(m\) are their charge and mass.) Equation (2,6) is a consequence of the requirement that in (2,16) the terms containing the vector product vanish. Indeed, this requirement, which may be called the generalized condition for the absence of vortices, is evidently as follows:

\[ \operatorname{rot}\mathbf{v}=-\frac{e}{mc}\mathbf{H}. \tag{2,17} \]

If we introduce the current \(\mathbf{j}_s=en_s\mathbf{v}\) and the constant \(\Lambda=\dfrac{m}{e^2n_s}\) (see (2,9)), then (2,17) goes over into (2,6). The remaining part of equation (2,16) after introducing the current \(\mathbf{j}_s\) is

\[ -\frac{\partial\Lambda\mathbf{j}_s}{\partial t} = \mathbf{E} -\frac{1}{en_s}\nabla\left(\frac{\Lambda j_s^2}{2}+p\right). \tag{2,18} \]

This equation has the form (2,7), and, if the term with the gradient is discarded, becomes (2,8). Thus equations (2,6) and (2,8) are the equations of motion of an ideal charged liquid under condition (2,17). This condition, however, is by no means obligatory in hydrodynamics, and for its justification one must go beyond its limits. Moreover, equations (2,17) and (2,6) obviously cannot be obtained as the result of a consistently purely classical treatment. The point is that, as is well known, according to classical theory the magnetic moment of any body in a state of thermodynamic equilibrium is zero (see, for example, \(^{72}\S 29\)). To explain magnetization it is therefore necessary to use quantum theory, or to introduce into classical theory the requirement, alien to it, of the existence of discrete electronic orbits, etc. Thus, to explain the diamagnetism of atoms in such a semiclassical treatment it is assumed that there are electronic orbits which, in a magnetic field, precess about the field with the Larmor angular velocity

\[ \omega_L=-\frac{eH}{2mc} \tag{2,19} \]

(in the case of an electron the charge \(e<0\)).

If the mean radius of the orbit is \(a\) and the distribution of the orbits in space is isotropic, then for the susceptibility of the gas, taking (2,19) into account, one obtains the well-known formula (\(Z\) is the number of electrons in the atom, \(N_a\) the concentration of atoms):

\[ \chi=-\frac{e^2ZN_a}{6mc^2}\,a^2. \tag{2,20} \]

The quantum nature of magnetization is reflected in (2,20) also in the fact that the orbit radius \(a\) is determined by Planck’s constant \(\hbar=\)

\[ =1.05\cdot 10^{-27}\left(\text{for the hydrogen atom in the ground state } a=\frac{\hbar^2}{me^2}\right). \]

In our case it is easy to see that, if the field \(\mathbf H\) is homogeneous\(^*\), then condition (2.17) will be satisfied if the liquid rotates around the field as a whole, i.e. as a solid body, with frequency (2.19). This also shows that condition (2.17) has the meaning of imposing on the system certain rigid constraints. A system of free electrons, of course, will not move in this way; suffice it to say that a free charge rotates around the field with frequency

\[ \omega_H=2\omega_L=-\frac{eH}{mc}, \]

and not with frequency \(\omega_L\). The derivation of equation (2.6) on the basis of considering the motion of electrons in a metal constitutes a problem of microscopic theory and will be discussed in the second part of the article.

Equations (2.6) and (2.8), or (2.17) and (2.18), can be obtained from a certain variational principle\(^{73-75}\), but this adds nothing essential to the matter.

In (2.6) and (2.8) it is assumed, just as in (2.2), (2.3), and (2.5), that the superconductor is electrically isotropic. In non-cubic metals it is necessary, generally speaking, to take into account the anisotropy of the normal conductivity and of the depth of penetration of the magnetic field into the superconductor. Since, on the basis of experiment, the anisotropy of the penetration depth has not yet been firmly established\(^{19}\), we shall not dwell here on the quite obvious generalization of the entire scheme to the anisotropic case\(^{67,72,68,76}\). It is also not difficult to consider the case of moving superconductors\(^{69}\). Equations (2.6)—(2.8) may further be generalized to the case of a nonlinear dependence of \(\mathbf j_s\) on \(\mathbf H\), for which there are certain grounds\(^{2,70,71}\). In the simplest nonlinear theory in (2.6)—(2.8) one may assume that \(\Lambda\) depends on \(\mathbf j_s\) or \(\mathbf H\). If one starts from equations (2.17)—(2.18), the nonlinear generalization is obtained in a natural way if one assumes that the number of superconducting electrons \(n_s\) depends on \(v^2\) or \(H^2\) (\(\Lambda\) and \(n_s\) may depend only on even powers of \(v\) or \(H\), since they cannot change when the sign of \(v\) or \(H\) is changed)\(^{2,70}\). The nonlinearity of the equations for the superconducting current is obtained automatically in the new phenomenological theory, which we shall take up below. Therefore we shall not dwell on a formal nonlinear generalization of the old theory. Finally, let us note that equations (2.6)—(2.8) must be replaced by more general ones also in the presence of a temperature gradient. This case is important for

\(^*\) In a superconductor the field \(\mathbf H\) is in fact always inhomogeneous, but this, of course, does not change the essence of the matter. Moreover, for example, for a sphere or cylinder of radius \(R\ll\delta\) the field is approximately homogeneous.

consideration of thermoelectric phenomena and the thermal conductivity of superconductors\(^{2,67}\); we shall touch on it in § 5.

Leaving aside for the moment all these directions, in which the simplest variant of the old phenomenological theory may be generalized, let us turn to some consequences of the established equations:

\[ \operatorname{rot}\Lambda \mathbf{j}_s=-\frac{1}{c}\mathbf{H}, \tag{2,6} \]

\[ \frac{\partial \Lambda \mathbf{j}_s}{\partial t}=\mathbf{E}, \tag{2,8} \]

\[ \mathbf{j}_n=\sigma \mathbf{E}. \tag{2,5} \]

The equation of energy conservation in electrodynamics

\[ \frac{\partial}{\partial t}\left(\frac{\mu \mathbf{H}^{2}+\varepsilon \mathbf{E}^{2}}{8\pi}\right)+\frac{c}{4\pi}\operatorname{div}[\mathbf{E}\mathbf{H}]=-\mathbf{j}\mathbf{E} \]

in the case of superconductors, according to (2,4), (2,5), and (2,8), takes the form:

\[ \frac{\partial}{\partial t}\left(\frac{\mathbf{H}^{2}+\varepsilon \mathbf{E}^{2}}{8\pi}+\frac{\Lambda \mathbf{j}_s^{2}}{2}\right)+\frac{c}{4\pi}\operatorname{div}[\mathbf{E}\mathbf{H}] =-\mathbf{j}_n\mathbf{E}=-\sigma \mathbf{E}^{2}. \tag{2,21} \]

Thus, the energy density in a superconductor is

\[ w_s=\frac{\mathbf{H}^{2}+\varepsilon \mathbf{E}^{2}}{8\pi}+\frac{\Lambda \mathbf{j}_s^{2}}{2}, \tag{2,22} \]

where the last term takes into account the energy of the current and represents, as is clear from (2,9), simply the kinetic energy of the superconducting electrons \(\frac{mn_s v^{2}}{2}\).

In the absence of a magnetic field, the transition from the normal to the superconducting state occurs continuously (it is a second-order transition). In accordance with this, as \(T\to T_k\), \(\delta\to \infty\), and equations (2,5), (2,6), (2,8) also continuously pass into the single equation (2,3), determining the current in the normal state. Indeed, for \(T\to T_k\)

\[ \Lambda=\frac{4\pi\delta^{2}}{c^{2}}\to\infty \tag{2,23} \]

and, consequently, as is clear from (2,6), the superconducting current must vanish, i.e. for \(T\to T_k\)

\[ \mathbf{j}_s\to 0. \tag{2,24} \]

If, however, \(\Lambda\to\infty\) and \(\mathbf{j}_s\to 0\), then equations (2,6), (2,8) impose no conditions on the fields \(\mathbf{E}\) and \(\mathbf{H}\), while \(\mathbf{j}=\mathbf{j}_n=\sigma \mathbf{E}\). Hence it also follows that the normal conductivity in a superconductor as \(T\to T_k\) must tend to the value of the conductivity of the normal metal at \(T=T_k\). This conclusion, already clear from the fact that

that at \(T=T_k\) the phase transition is a second-order transition, is fully confirmed experimentally (see § 6).

Let us now turn to finding the distribution of the field and current in superconductors placed in an external magnetic field.

In the stationary case, using (2.6) and (2.11), we have

\[ \Delta \mathbf H-\frac{1}{\delta^2}\mathbf H=0,\qquad \delta^2=\frac{\Lambda c^2}{4\pi}=\frac{mc^2}{4\pi e^2 n_s}. \tag{2.25} \]

At the boundary between a superconductor and vacuum, as always, the tangential components of the vector \(\mathbf H\) must be continuous. Therefore, in the case of a superconducting half-space, i.e. in fact for a sufficiently thick plate, the solution of equation (2.25) is as follows:

\[ H=H_0 e^{-z/\delta}, \]

\[ j_s=j_{s0}e^{-z/\delta} =-\frac{cH_0}{4\pi\delta}e^{-z/\delta} =-\frac{c}{4\pi\delta}H, \tag{2.26} \]

where the coordinate \(z\) is measured from the boundary separating the metal from the vacuum, and \(H_0\) is the field at \(z=0\) (i.e. the external field); the field \(\mathbf H\) and the current \(\mathbf j_s\) are mutually perpendicular and, by assumption, lie in the \(xy\) plane.

In the case of a superconducting plate of thickness \(2d\), placed in a field \(\mathbf H_0\) parallel to it, the solution evidently has the form:

\[ H=H_0\frac{\operatorname{ch} z/\delta}{\operatorname{ch} d/\delta}, \qquad j_s=-\frac{cH_0}{4\pi\delta} \frac{\operatorname{sh} z/\delta}{\operatorname{ch} d/\delta}, \tag{2.27} \]

where the origin of coordinates is placed at the middle of the plate, the field \(\mathbf H\) is directed along the \(y\)-axis, and the current has a component only along the \(x\)-axis (see Fig. 8).

Fig. 8.

If a circular cylinder of radius \(R\) is placed entirely in a homogeneous magnetic field \(\mathbf H_0\) parallel to it, then the field is everywhere directed along the axis of the cylinder \(z\), and the current lines form circles, with:

\[ H=H_0 \frac{I_0\!\left(\dfrac{r}{\delta}\right)} {I_0\!\left(\dfrac{R}{\delta}\right)}, \]

\[ j_s=\frac{c}{4\pi}\operatorname{rot}_{\varphi}H =-\frac{c\,\partial H_z}{4\pi\,\partial r} =-\frac{cH_0}{4\pi\delta}\, \frac{I_1\!\left(\dfrac{r}{\delta}\right)} {I_0\!\left(\dfrac{R}{\delta}\right)}. \tag{2.28} \]

where \(I_n(x)=i^{-n}J_n(ix)\), \(J_n(x)\) is the Bessel function of the first kind of order \(n\) (recall that \(\dfrac{dI_0(x)}{dx}=I_1(x)\), \(I_0(x)=1+\dfrac{x^2}{4}+\dfrac{x^4}{64}+\cdots;\) for \(x\gg 1\), \(I_0(x)\simeq \dfrac{e^x}{\sqrt{2\pi x}}\)).

For a thick cylinder,

\[ R\gg \delta:\quad H\simeq H_0\sqrt{\frac{R}{r}}\, e^{-\frac{(R-r)}{\delta}}, \]

\[ j_s\simeq -\,\frac{cH_0}{4\pi\delta}\sqrt{\frac{R}{r}}\, e^{-\frac{(R-r)}{\delta}}. \tag{2,28a} \]

In the case of a sufficiently thin cylinder,

\[ R\ll \delta:\quad H\simeq H_0\, \frac{1+\left(\dfrac{r}{2\delta}\right)^2} {1+\left(\dfrac{R}{2\delta}\right)^2}; \qquad j_s\simeq -\,\frac{cH_0}{4\pi\delta}\, \frac{\dfrac{r}{2\delta}} {1+\left(\dfrac{R}{2\delta}\right)^2}. \tag{2,28b} \]

If a superconducting sphere of radius \(R\) is placed in a uniform magnetic field, then outside it the field is the sum of the external field \(H_0\) and the field of a magnetic dipole with moment

\[ \left. \begin{aligned} \vec{\mu} &= -\frac{1}{2} \left(1-\frac{3\delta}{R}\operatorname{cth}\frac{R}{\delta} +\frac{3\delta^2}{R^2}\right)R^3\mathbf{H}_0,\\ R\gg\delta:\quad \vec{\mu} &\simeq -\,\frac{R^3}{2}\mathbf{H}_0,\\ R\ll\delta:\quad \vec{\mu} &\simeq -\,\frac{R^5}{30\delta^2}\mathbf{H}_0. \end{aligned} \right\} \tag{2,29} \]

We shall not give the formulas for the distribution of the field and current in the sphere itself; they may be found in \(^{1,3}\); there too are given solutions of a number of other problems (a cylinder in a perpendicular field, a cylinder in a parallel field along which a current flows, etc.; in addition to \(^{1,3}\) we point to new works in the field of the phenomenological theory \(^{77-83}\)). One of the experimental methods for determining \(\delta\) is based on measuring the magnetic moment of a superconductor of small dimensions in a magnetic field \(^{14}\). In the case of superconducting spheres, the magnetic susceptibility referred to one sphere is determined by formula (2,29). For any cylinder in a field \(H_0\) parallel to its axis, the magnetic moment referred to unit length is equal to

\[ \psi=\int \frac{H(s)-H_0}{4\pi}\,dS, \tag{2,30} \]

where \(H(s)\) is the field in the cylinder and \(dS\) is an element of the cross-sectional area of the cylinder. The vectors \(\vec{\mu}\), \(\mathbf{H}(s)\), and \(\mathbf{H}_0\), of course, are collinear, and, since \(H(s)<H_0\), the moment \(\vec{\mu}\) is antiparallel to the field \(\mathbf{H}_0\). To formula

(2.30) we arrive, taking into account that the field in a superconductor plays the role of the induction $\mathbf{B}$ in a magnetic material. Moreover, if our cylinder were made of a magnetic material, then the magnetic field $\mathbf{H}$ in it would be equal to the external field $\mathbf{H}_0$ (since the tangential components of $\mathbf{H}$ are continuous). It follows from this that in a superconductor the role of the magnetic field of the phenomenological theory of magnetics is played by the field $\mathbf{H}_0$ and, consequently, in the superconductor the magnetization

\[ \mathbf{M}=\frac{\mathbf{B}-\mathbf{H}}{4\pi} \]

is precisely equal to

\[ \frac{\mathbf{H}-\mathbf{H}_0}{4\pi}. \]

For a plate, using (2.27), we have

\[ \mu=-\left(1-\frac{\delta}{d}\operatorname{th}\frac{d}{\delta}\right)\frac{d\cdot H_0}{2\pi}; \tag{2.31} \]

where the moment is referred to a unit surface area of the plate.

In the case of a circular cylinder, the moment per unit length is equal to

\[ \left. \begin{aligned} \mu&=-\left[1-\frac{2\delta}{R}\frac{I_1\left(\dfrac{R}{\delta}\right)}{I_0\left(\dfrac{R}{\delta}\right)}\right]\frac{R^2}{4}H_0,\\[6pt] R\gg\delta:\quad \mu&\simeq-\left(1-\frac{2\delta}{R}\right)\frac{R^2}{4}H_0, \end{aligned} \right\} \tag{2.32} \]

where, in substituting the solution (2.28) into (2.30), it has been taken into account that

\[ \int_0^x I_0(x)\,x\,dx=xI_1(x). \]

Let us now dwell on the question of the destruction of superconductivity by a magnetic field. To find the critical field at which the transition to the normal state occurs, consider a cylindrical specimen in a magnetic field parallel to its axis. Suppose that initially the external magnetic field $H_0$ is greater than the field $H_k$ corresponding to the given temperature. In this case the whole specimen is in the normal state. When the field is decreased to the value $H_k$, a phase transition begins which, if the temperature is kept constant, will end with the complete transition of the cylinder into the superconducting state. Since this transition, when carried out sufficiently slowly, may be regarded as reversible, the change in the free energy of the body upon the transformation is equal to the work $A_{12}$ performed on the body by the electromagnetic field,

\[ F_2-F_1=A_{12}, \tag{2.33} \]

where all free energies and the work $A_{12}$ are referred to a unit volume of the cylinder,

\[ F_1=F_{n0}+\frac{H_k^2}{8\pi} \]

is the free energy of the body ($F_{n0}$) and of the field

\[ \left(\frac{H_k^2}{8\pi}\right) \]

before the transition, i.e. in the normal state, and $F_2=F_{s0}+W$ is the free energy after the transition, where $W$—

energy associated with the field and current in the superconductor. The quantities \(F_{n0}\) and \(F_{s0}\) are the free energies per unit volume of the normal and superconducting phases in the absence of a field. \(F_{n0}\) and \(F_{s0}\) are considered unchanged also in the presence of a field, since the magnetic permeability \(\mu=1\). Thus, the influence of the field in \(F_1\) and \(F_2\) is taken into account only by the terms \(\dfrac{H_k^2}{8\pi}\) and \(W\). As is clear from (2,22),

\[ W=\frac{1}{V}\int\left(\frac{H_{(s)}^2}{8\pi}+\frac{\Lambda j_s^2}{2}\right)dV =\frac{1}{S}\int\left(\frac{H_{(s)}^2}{8\pi}+\frac{\Lambda j_s^2}{2}\right)dS, \]

where \(V\) is the volume of the cylinder, and \(S\) the area of its cross section. The work \(A_{12}\) is equal to the change in the energy of the specimen in the external magnetic field. In the normal state this energy is equal to zero, while in the superconducting state it is equal to \(-\mu(H_k)\cdot H_k\); thus, according to (2,30),

\[ A_{12}=\mu(H_k)\cdot H_k=-\frac{H_k^2}{4\pi} +\frac{H_k}{4\pi S}\int H(s)\,dS . \tag{2,34} \]

This expression can also be obtained by direct calculation of the electromagnetic energy flowing during the transition through the surface of the cylinder under consideration (see \(^{2}\)).

Taking what has been said into account, we finally obtain

\[ F_{n0}-F_{s0} = \frac{H_k^2}{8\pi} +\frac{1}{S}\int \left( \frac{H_{(s)}^2}{8\pi} +\frac{\Lambda j_s^2}{2} -\frac{H_k H(s)}{4\pi} \right)dS . \tag{2,35} \]

For a massive cylinder the integral term in (2,35) is small, since the field \(H(s)\) and the current \(j_s\) differ from zero, or, more precisely, are not too small, only in a surface layer of thickness of order \(\delta\). Therefore, for a massive superconductor

\[ F_{n0}-F_{s0}=\frac{H_{km}^2}{8\pi}, \tag{2,36} \]

where \(H_{km}\) is the critical field for massive specimens. This formula has already been given in § 1 (see (1,2)), where the indices 0 and \(M\) were omitted*).

\[ \text{*)} \]

In view of the importance of relation (2,36), we point out that it follows at once from (2,33), since for a massive metal \(W=0\),

\[ A_{12}=\mu H_{km}=-\frac{H_{km}^2}{4\pi} \quad\text{and}\quad F_2-F_1=F_{s0}-F_{n0}-\frac{H_{km}^2}{8\pi} =\frac{-H_{km}^2}{4\pi}. \]

Let us also note that, in order to obtain all the formulas, one need not speak of the work of the field forces, but may simply equate, at \(H=H_k\), the total energies of the superconductor and of the normal metal in the magnetic field, taking into account the free energy of the superconductor in the external field, equal to \(-\mu\cdot H_k\). Indeed, in this case

\[ F_{n0}+\frac{H_k^2}{8\pi}=F_{s0}+W-\mu\cdot H_k, \]

which is equivalent to (2,33), since \(A_{12}=\mu\cdot H_k\).

From (2.36) the relation between the heat of transition \(Q\) and the jump in heat capacity \(\Delta c\) with \(H_{\mathrm{cm}}\) is immediately clear. Namely, since the entropy \(S=-\dfrac{\partial F}{\partial T}\) and the heat capacity \(c=T\dfrac{\partial S}{\partial T}\),

\[ Q=T(S_n-S_s)=-\frac{T H_{\mathrm{cm}}}{4\pi}\frac{dH_{\mathrm{cm}}}{dT}, \]

\[ \Delta c=c_s-c_n=\frac{T}{4\pi}H_{\mathrm{cm}}\frac{d^2H_{\mathrm{cm}}}{dT^2} +\frac{T}{4\pi}\left(\frac{dH_{\mathrm{cm}}}{dT}\right)^2 . \tag{2.37} \]

For

\[ T=T_k:\quad H_k=0,\quad Q=0\quad \text{and}\quad \Delta c=\frac{T_k}{4\pi}\left(\frac{dH_{\mathrm{cm}}}{dT}\right)^2_{T_k}. \]

Above it was assumed that the transition from the normal state to the superconducting one occurs not only at constant temperature, but also at constant volume. Usually this assumption is quite legitimate, but in order to determine the effect of pressure on \(H_{\mathrm{cm}}\), to find the change of the specific volume in the transition \(\Delta V=V_n-V_s\), etc., it is necessary, of course, to consider the transition at constant pressure, and not at constant volume. In this case the difference of the thermodynamic potentials \(\Phi=F\cdot V+pV\), referred to unit mass, is equal to \((V\simeq V_n\simeq V_s\) is the specific volume)

\[ \Phi_n-\Phi_s=\frac{H_{\mathrm{cm}}^2}{8\pi}\cdot V . \tag{2.38} \]

Since \(V=\left(\dfrac{\partial \Phi}{\partial p}\right)_T\), the change of the specific volume upon destruction of superconductivity is

\[ \Delta V=V_n-V_s=\frac{H_{\mathrm{cm}}}{4\pi} \left(\frac{\partial H_{\mathrm{cm}}}{\partial p}\right)_T\cdot V, \tag{2.39} \]

and in (2.38) \(V\) is not differentiated with respect to \(p\), since the term thereby obtained,
\(\dfrac{H_{\mathrm{cm}}^2}{8\pi}\left(\dfrac{\partial V}{\partial p}\right)_T\), takes into account the change of the volume of the superconducting phase under the influence of the electromagnetic pressure \(\dfrac{H_{\mathrm{cm}}^2}{8\pi}\); if, however, we are interested in the change of volume \(\Delta V\) in the field \(H_{\mathrm{cm}}\), associated only with the destruction of superconductivity, then the change of the volume \(V_s\) under the influence of the magnetic field need not be taken into account*). The change \(\Delta V\) is very small and can usually be safely neglected

\[ \left(\frac{\Delta V}{V}\sim 10^{-7}\ \text{for}\ H_{\mathrm{cm}}\sim 100\ \text{and}\ \frac{\partial H_{\mathrm{cm}}}{\partial p}\sim 10^{-8}\right). \]

*) This circumstance, as I. M. Lifshitz kindly pointed out to the author, was not taken into account in \({}^{2}\); therefore formula (4.18) in § 19 of \({}^{3}\) determines the change of volume associated both with the destruction of superconductivity and with the compression (striction) of the superconductor in a magnetic field.

Differentiating (2.39) with respect to pressure and temperature, one can determine the change in the compressibility \(\beta\) and the coefficient of thermal expansion \(\alpha\) associated with the destruction of superconductivity (see [2]; new experimental works \(^{32-35,47}\)).

The phenomenological theory, whose main aspects we have elucidated, is, of course, in agreement with the experimental fact that a magnetic field does not penetrate into the bulk of superconductors. But for testing this theory, this is still not enough, since one can propose a large number of other equations leading to the same qualitative result (for example, the same equation (2.6) leads to it, but with a quantity \(\Lambda\) depending on \(H\) and \(j_s\); true, this dependence cannot be arbitrary, but the relation is not uniquely determined). A genuine test of the equations obtained could primarily be based on an experimental determination of the law of attenuation of the field in a superconductor, i.e. on comparison with experiment of formulas (2.26)—(2.32) and similar ones. However, measurement of the magnetic moment of thin specimens, such as colloidal particles, thin wires, etc., is very difficult. Therefore, although this method has also been successfully used to determine changes of the penetration depth \(\delta\) with temperature, it has not yet been used for sufficiently precise absolute measurements.

Besides measuring the moment, the theory can be tested by another method, which is also important in itself and is based on relatively easily performed experiments. The point is that the critical field \(H_k\) depends on the dimensions of the superconductor; moreover, this dependence, as is clear from (2.35), is determined by the character of the distribution of the field and current in the specimen. Thus, in the case of a plate (film) of thickness \(2d\), substituting the solution (2.27) into (2.35), we have

\[ F_{n0}-F_{s0}=\frac{H_{km}^{2}}{8\pi}=\frac{H_k^2}{8\pi}+\frac{\sigma}{d}, \]

\[ \sigma=\int_0^d \left( \frac{H^2(z)}{8\pi} +\frac{\lambda j_s^2(z)}{2} -\frac{H_k H(z)}{4\pi} \right)\,dz = -\frac{H_k^2}{8\pi}\,\delta\,\operatorname{th}\frac{d}{\delta}, \tag{2.40} \]

where the quantity \(\sigma\) may be interpreted as the free energy associated with a unit surface. Such terminology is justified by the fact that, in the absence of boundaries,

\[ \frac{H_k^2}{8\pi}=\frac{H_{km}^2}{8\pi}, \]

and the term \(\sigma/d\) is precisely equal to the energy per unit volume caused by the presence of a surface. The well-known conventionality of interpreting \(\sigma\) as surface energy is clear from the fact that \(\sigma<0\), whereas the surface energy in its usual introduction is always positive.

relative. Independently of the term used, according to (2.35) and (2.40) the change in \(H_k\) for thin superconductors may be regarded as the result of the existence of a certain energy \(\sigma\), connected with the presence of a superconducting current and the partial penetration of a magnetic field into the metal. From (2.40) it is immediately clear that

\[ \left(\frac{H_k}{H_{km}}\right)^2 = \frac{1}{1-\frac{\delta}{d}\operatorname{th}\frac{d}{\delta}} . \tag{2.41} \]

This relation, however, is not in agreement with experiment \(^{21}\). Thus, if at a given temperature one determines, from the measured values \(\dfrac{H_k}{H_{km}}\) for different \(d\), the constant \(\delta\) by formula (2.41), then this “constant” \(\delta\) changes strongly with changing \(d\) (for example, at \(T=4^\circ\) and \(d=0.3\cdot10^{-5}\), \(\delta=3.4\cdot10^{-5}\), while at \(d=1.2\cdot10^{-5}\), \(\delta=2.0\cdot10^{-5}\)). In \(^{86}\) it was pointed out that the situation can be corrected if one takes into account the change in the surface energy at the boundary between the metal and the vacuum according as the metal is in the superconducting or in the normal state. In this case, instead of (2.41), one obtains

\[ \left. \begin{aligned} \left(\frac{H_k}{H_{km}}\right)^2 &= \frac{1+\frac{\beta}{d}} {1-\frac{\delta}{d}\operatorname{th}\frac{d}{\delta}}, \\[6pt] \beta &= \frac{\sigma'_n-\sigma'_s}{\dfrac{H_{km}^{2}}{8\pi}}, \end{aligned} \right\} \tag{2.42} \]

where \(\sigma'_n\) and \(\sigma'_s\) are the surface energies, unrelated to the magnetic field, at the boundary of the vacuum with the metal when it is respectively in the normal and superconducting states. The “total surface energy” \(\sigma\) itself, now appearing in (2.40), has the form

\[ \sigma = -(\sigma'_n-\sigma'_s) - \frac{H_k^2}{8\pi}\,\delta\,\operatorname{th}\frac{d}{\delta}. \tag{2.43} \]

If formula (2.42) is adopted, then comparison with experiment implies \(^{86}\) that \(\beta>0\) (i.e. \(\sigma'_n>\sigma'_s\)) and, moreover, \(\beta\sim\delta\sim10^{-5}\ \mathrm{cm}\). However, the introduction of the surface energy

\[ \sigma'_n-\sigma'_s = \beta\,\frac{H_{km}^{2}}{8\pi} \sim \delta\,\frac{H_{km}^{2}}{8\pi} \]

appears, in the present case, to be at least strained. Indeed, the surface energy is usually, in order of magnitude, equal to the volume free energy multiplied by a length of the order of atomic distances; thus, in our case, where the difference of the volume free energies is equal to \(\dfrac{H_{km}^{2}}{8\pi}\), one might expect the pres...

the differences of surface energies of order \((10^{-8} \div 10^{-7})\,\dfrac{H_{km}^{2}}{8\pi}\), but not of order \(10^{-5}\cdot\dfrac{H_{km}^{2}}{8\pi}\). This contradiction appears still more vividly in the case of the interface between the superconducting and normal phases of a metal. In this case, as is clear at least from (2,43), if one does not take into account some additional surface energy,

\[ \sigma=\sigma_{ns}^{H}=-\frac{H_k^2}{8\pi}\,\delta\,\operatorname{th}\frac{d}{\delta}<0 \]

(in the case of the boundary of a massive superconductor with the normal phase, it follows from this, or directly from the expression for \(\sigma_{ns}^{H}\), that

\[ \sigma_{ns}^{H} = \int_{0}^{\infty} \left( \frac{H^2(z)}{8\pi} + \frac{\Lambda j_s^2}{2} - \frac{H_k\cdot H(z)}{4\pi} \right)\,dz = -\delta\cdot\frac{H_{km}^{2}}{8\pi}, \]

since for \(H(z)\) and \(j_s(z)\) here one must use expression (2,26)). But when \(\sigma_{ns}^{H}<0\), the formation of separation surfaces is only advantageous and, as is not difficult to see\(^{87,86}\), the destruction of superconductivity of a massive specimen at \(H=H_{km}\) would not take place (instead the specimen would break up into alternating superconducting and normal layers). In order to avoid these obvious contradictions, it is necessary to assume that at the boundary between the superconducting and normal phases there must also be, so to speak, a “localized” surface energy \(\sigma'_{ns}\), which, to ensure the positivity of the total surface energy \(\sigma_{ns}\), must satisfy the inequality

\[ \sigma'_{ns}=\sigma_{ns}-\sigma_{ns}^{H}>\delta\cdot\frac{H_{km}^{2}}{8\pi}. \tag{2,44} \]

It follows from experiment (see, for example,\(^{37}\)) that \(\sigma_{ns}\sim \delta\,\dfrac{H_{km}^{2}}{8\pi}\). At the same time, at the boundary between the normal and superconducting phases of one and the same metal, the appearance of such a comparatively enormous “localized” surface energy \(\sigma'_{ns}\), unrelated to the distribution of the magnetic field, is already quite unnatural. On the contrary, one must think that any rational theory of superconductivity should automatically lead to the possibility of expressing the surface energy \(\sigma_{ns}\) in terms of constants characterizing the superconductor. The same applies to the energy \(\sigma'_n-\sigma'_s\) in (2,42). Let us also note that the scarcity of experimental data does not allow one to assert that even formula (2,42) and the expression \(\sigma_{ns}=\sigma'_{ns}-\dfrac{H_k^2}{8\pi}\,\delta\,\operatorname{th}\dfrac{d}{\delta}\) with constant-

MODERN STATE OF THE THEORY OF SUPERCONDUCTIVITY

them at a given temperature $\delta$ and $\delta'_{ns}$ make it possible to give a quantitatively correct description of the destruction of superconductivity and all the features of the intermediate state.

The theory based on equation (2.6), even with the introduction of an additional surface energy, likewise does not make it possible to consider the destruction of superconductivity in films by a current$^{88}$, since this problem is not of a thermodynamic character*). And, finally, in the developed scheme the depth of penetration of the field into the superconductor $\delta$ does not depend on the field strength and on the dimensions of the specimen. Meanwhile, theoretical considerations (see $^{70,4}$ and below) lead to the conclusion that the depth of penetration of the field into a superconductor, especially for $H \sim H_k$, must depend on $H$, and this effect should in the first place be noticeable for thin specimens. Some indications of the existence of such an effect apparently already exist$^{17}$.

To summarize, we may say that the phenomenological theory of superconductivity, based on equation (2.6), which we call the “old theory,” is unsatisfactory. More precisely, the old theory is certainly inapplicable to the consideration of the destruction of superconductivity in thin objects, of the interface between the normal and superconducting phases and, in general, in strong fields when $H \sim H_k$. In weak fields $H \ll H_k$ and for currents $I$ much smaller than the critical current $I_k$, the use of the old theory contradicts no known facts, although it cannot be regarded as sufficiently well founded experimentally. As follows from the more general theory set out below—

*) The derivation of formulas for the critical current and the critical magnetic field of a film on the basis of considering the equilibrium of the boundary between the normal and superconducting phases of a metal$^{3}$ is incorrect for thin films, since this boundary itself, by virtue of what has been said above, cannot be considered on the basis of equation (2.6). Therefore we believe that the criticism of formula (2.42) undertaken by Laue$^{89}$ is unfounded. As to the unsuitability of the formula proposed by Laue$^{89}$,

$$ H_k = H_{km}\operatorname{ctg}\frac{d}{\delta} $$

one can easily be convinced of it at once from the fact that it cannot explain the phenomenon of the destruction of superconductivity at all. Indeed, according to this formula superconductivity should begin to be destroyed at the boundary of the film in the field

$$ H_k = H_{km}\operatorname{ctg}\frac{d}{\delta}. $$

Upon the destruction of superconductivity a normal phase will be formed, and the thickness of the superconducting “core” will become already less than $2d$. But when $d$ is decreased the critical field increases and the superconducting “core,” and hence also the superconductivity of the film in general, according to Laue, will disappear only when $H_k \to \infty$ (!). Therefore, within the framework of the theory based on equation (2.6), the destruction of superconductivity can be treated only thermodynamically, by considering the transition of the entire film as a whole, which leads to formula (2.42). Our criticism of this formula proceeds in an entirely different direction—it points to the unsuitability of the initial equation (2.6) itself.

...theoretical scheme, called for brevity simply the “new phenomenological theory,” the old theory in weak fields must be applicable and simply be the limiting case of a more complicated system of equations determining the field \(\mathbf H\) and the current \(\mathbf j_s\) in superconductors. The new theory\(^4\) makes it possible, at least in principle, to eliminate all the indicated shortcomings of the old theory, but, in its turn, has not yet been sufficiently tested experimentally. The question of testing the new theory, which, if successful, will make it possible to speak of a certain completion of the construction of the phenomenological theory of superconductivity, is now, in our opinion, of the greatest importance. We therefore turn to a rather detailed exposition of this new theory.

§ 3. NEW PHENOMENOLOGICAL THEORY: BASIC RELATIONS*)

The inadequacy of the old theory is manifested especially vividly in the fact that it leads to a negative surface tension \(\sigma^{H}_{ns}\). Therefore, first of all, it is necessary to find the reason which could change this situation. In this respect one may convince oneself that changing equation (2.6) in the direction of taking account of a possible dependence of \(\Lambda\) on \(H\) or \(j_s\) does not lead to positive results. On the contrary, they can, at least in principle, be produced by taking into account the quantum effect.\(^{10}\) Indeed, localization of the electrons participating in the transport of the superconducting current (i.e. localization of the “superconducting electrons”) in a surface layer of thickness of order \(\delta\) makes their momentum along the \(z\)-axis uncertain by an amount \(\sim \hbar/\delta\); the “zero energy” of the superconducting electrons in the surface layer corresponding to this spread of momentum values

\[ \sim \frac{\hbar^2 n_s}{m\delta^2}\sim 10^4 \frac{\mathrm{erg}}{\mathrm{cm}^3}, \]

for \(m \sim 10^{-27}\), \(\delta \sim 10^{-5}\) and \(n_s \sim 10^{21}\), where \(n_s\) is the concentration of superconducting electrons and \(m\) is their effective mass. At the same time, the magnetic energy

\[ \frac{H^2}{8\pi}\sim 10^4 \frac{\mathrm{erg}}{\mathrm{cm}^3} \]

only for \(H\sim 5\cdot 10^2\) gauss and, thus, the zero energy under ordinary conditions is in any case no smaller, and rather larger, than the magnetic energy. This positive zero energy can, in principle, ensure the positivity of \(\sigma_{ns}\). The meaning of the remark made is obviously that quantum effects in a superconductor, in view of the relative smallness of the penetration depth \(\delta\), may be very substantial. Hence the limited domain of applicability of the old phenomenological theory, which has as its basis classical hydrodynamics supplemented by the “rigidity” requirement (2.17), which selects only part of the solutions possible in classical theory, becomes understandable.

*) The contents of §§ 3, 4 are based on work \(^{4}\).

For a transition to a quantitative theory with quantum effects taken into account, it is necessary to express the energy of a superconductor in terms of quantities characterizing the superconducting state. The principal such quantity is the already mentioned number or, more precisely, the concentration of superconducting electrons \(n_s\), which plays a role analogous to the number of free electrons in the elementary theory of the electron gas in metals. The quantity \(n_s\), as is clear from (2.14), and also from what follows, is expressed in terms of an experimentally measured parameter of the superconductor. In the absence of a magnetic field it is natural to assume that the free energy of the superconductor \(F_{s0}\) depends only on the number \(n_s\), which in turn changes with temperature in such a way that for \(T \geq T_k\), where \(T_k\) is the critical temperature, \(n_s=0\), and for \(T<T_k\), \(n_s>0\).

This conclusion about the temperature dependence of \(n_s\) is an obvious consequence of the fact that at \(T=T_k\) there is a second-order phase transition from the superconducting phase to the normal one*). The quantity \(n_s\) thus plays the role of that positive parameter which enters the general theory of phase transitions of the second order\(^{90}\) and which, for example, in the case of ferroelectrics is equal to the square of the spontaneous polarization \(P_s^2\), and in the case of ferromagnets to the square of the spontaneous magnetization \(M_s^2\)\(^{91}\). The free energy \(F_{0s}(n_s)\), which we refer to unit volume, must for \(T<T_k\) have a minimum at \(n_s\ne 0\), and for \(T\geq T_k\)—at \(n_s=0\). However, before writing \(F_{s0}\) as a function of \(n_s\), let us note that, in the spirit of the theory of second-order transitions, it is convenient, and for what follows practically necessary, to use instead of the quantity \(n_s\) a quantity \(\Psi\), defined by the relation

\[ |\Psi|^2=n_s \tag{3.1} \]

and playing the role of a certain “effective” wave function of the superconducting electrons. Of course, \(\Psi\) can be expressed in terms of the exact wave function of the electrons in the metal only on the basis of a microscopic theory (see also \(^{4}\)). We shall choose the normalization of our \(\Psi\)-function somewhat below.

Taking into account that at \(T=T_k\), \(|\Psi|^2=0\), and assuming that the function \(F_{s0}(|\Psi|^2)\) can be expanded near this point in a series, we may, in the neighborhood of \(T_k\), write \(F_{s0}\) in the form**)

\[ F_{s0}=F_{n0}+\alpha|\Psi|^2+\frac{\beta}{2}|\Psi|^4. \tag{3.2} \]

*) The indicated temperature dependence of \(n_s\) is also in accordance with the temperature dependence of \(\delta=\sqrt{\dfrac{mc^2}{4\pi e^2 n_s}}\); see (2.14) and (2.23).

**) Let us note that, without explicitly taking into account any electrostatic energy, we thereby consider the superconducting charge \(ne=n_s e\Psi\) [?] to be completely compensated by all the other charges of the metal, which is quite natural in the stationary state.

In the equilibrium state

\[ \frac{\partial F_{s0}}{\partial |\Psi|^2}=0,\qquad \frac{\partial^2 F_{s0}}{\partial^2 |\Psi|^2}>0, \]

and it must be that \(|\Psi|^2=0\) for \(T>T_k\) and \(|\Psi|^2>0\) for \(T<T_k\). Hence it follows that \(\alpha(T_k)=0\), \(\beta(T_k)>0\), and for \(T<T_k\), \(\alpha<0\). Therefore in the equilibrium state for \(T<T_k\)

\[ \left. \begin{aligned} |\Psi|^2&=|\Psi_\infty|^2=-\frac{\alpha}{\beta} =\frac{\left(\dfrac{d\alpha}{dT}\right)_k (T_k-T)}{\beta_k},\\[6pt] F_{s0}&=F_{n0}-\frac{\alpha^2}{2\beta} =F_{n0}-\frac{\left(\dfrac{d\alpha}{dT}\right)_k^{2}(T_k-T)^2}{2\beta_k}, \end{aligned} \right\} \tag{3,3} \]

where it has been taken into account that, within the limits of validity of the expansion (3,2), \(\alpha(T)=\left(\dfrac{d\alpha}{dT}\right)_k (T-T_k)\) and \(\beta(T)=\beta(T_k)\equiv\beta_k\). The choice of the sign \(\infty\) for \(\Psi\) in (3,3) is dictated by considerations that will be clear from what follows. The quantity \(F_{n0}\) in (3,2)—(3,3) is, obviously, the free energy of the normal phase, which in the range of applicability of formula (3,3) may be regarded as independent of temperature. Therefore, as is clear from (2,36) and (3,3),

\[ H_{km}^2=\frac{4\pi\alpha^2}{\beta} = \frac{4\pi\left(\dfrac{d\alpha}{dT}\right)_k^{2}(T_k-T)^2}{\beta_k}. \tag{3,4} \]

This formula, as is known, is fully confirmed by experimental data (see (1,2a)), which also serves as a justification of the assumptions made above.

Let us now consider a superconductor situated in a magnetic field constant in time.

In this case, in order to obtain the density of the free energy \(F_{sH}\), one must add to \(F_{s0}\) the field energy \(\dfrac{H^2}{8\pi}\) and the energy associated with the possible appearance, in the presence of the field, of a gradient of the function \(\Psi\). This latter energy, at least so long as the quantity \(|\nabla\Psi|^2\) is small, can, as a result of an expansion in powers of \(|\nabla\Psi|^2\), be represented in the form \(\mathrm{const}\cdot|\nabla\Psi|^2\), i.e., it has the same form as the density of kinetic energy in quantum mechanics. Therefore we shall write the corresponding expression in the form \(\dfrac{\hbar^2}{2m}|\nabla\Psi|^2\), where \(m\) is some coefficient which may be called the effective mass. In doing so, however, the interaction of the current associated with the presence of \(\nabla\Psi\) with the magnetic field has not yet been taken into account. In view of the foregoing, and proceeding from the requirement that the whole scheme be gauge invariant, to take account of the influence of the field it is necessary to make the usual replacement \(-i\hbar\nabla\) by \(-i\hbar\nabla-\dfrac{e}{c}\mathbf A\), where \(\mathbf A\) is the vector potential.

field and \(e\)—a charge, which there is no reason to regard as different from the electron charge. Thus, the energy density associated with the presence of \(\nabla\Psi\) and of the field \(\mathbf H=\operatorname{rot}\mathbf A\) has the form:

\[ \frac{H^2}{8\pi}+\frac{1}{2m}\left|-i\hbar\nabla\Psi-\frac{e}{c}\mathbf A\Psi\right|^2 . \]

As a result

\[ F_{sH}=F_{s0}+\frac{H^2}{8\pi}+\frac{1}{2m}\left|-i\hbar\nabla\Psi-\frac{e}{c}\mathbf A\Psi\right|^2 . \tag{3.5} \]

The equation for \(\Psi\) can now be found from the requirement that the total free energy of the body \(\int F_{sH}\,dV\) be minimal.

Thus, varying the total free energy with respect to \(\Psi^*\), we obtain:

\[ \frac{1}{2m}\left(-i\hbar\nabla-\frac{e}{c}\mathbf A\right)^2\Psi+ \frac{\partial F_{s0}(\Psi^*\cdot\Psi)}{\partial \Psi^*}=0, \tag{3.6} \]

and at the boundary of the superconductor, in view of the arbitrariness of the variation \(\delta\Psi^*\), the condition must be satisfied

\[ \mathbf n\cdot\left(-i\hbar\nabla\Psi-\frac{e}{c}\mathbf A\cdot\Psi\right)=0, \tag{3.7} \]

where \(\mathbf n\) is the vector normal to the boundary (the grounds for choosing such a boundary condition will be discussed further below).

As for the equation for \(\mathbf A\), putting \(\operatorname{div}\mathbf A=0\) and varying the free energy with respect to \(\mathbf A\), we obtain the usual expression:

\[ \Delta\mathbf A=-\frac{4\pi}{c}\mathbf j = \frac{2\pi i e\hbar}{mc}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right) + \frac{4\pi e^2}{mc^2}\mathbf A\cdot\Psi^*\cdot\Psi, \tag{3.8} \]

where

\[ \mathbf j=-\frac{i e\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right) -\frac{e^2}{mc}\mathbf A\cdot\Psi^*\cdot\Psi \]

is the quantum-mechanical expression, usual in form, for the current density. Taking into account that an expression analogous to (3.7) is also obtained for the conjugate quantity, it is easy to see that at the boundary \(\mathbf n\cdot\mathbf j=0\), as it should be.

The solution of the problem of the distribution of the field and current in a superconductor reduces to the simultaneous integration of equations (3.6) and (3.8).

Below we shall consider only the one-dimensional problem; the \(z\)-axis will be directed normal to the boundary separating the superconducting phase from the normal phase or from vacuum; the field \(\mathbf H\) will be assumed directed along the \(y\)-axis, and the current \(\mathbf j\) and vector potential \(\mathbf A\) along the \(x\)-axis (in this case \(H_y=dA_x/dz\), or simply \(H=dA/dz\)). In the one-dimensional problem it is natural to take \(|\Psi|^2\) to depend only on \(z\), and hence \(\Psi=e^{i\varphi(x,y)}\cdot\Psi(z)\). But then, taking into account the gradient invariance of the equations used, by choosing the corresponding potential \(\mathbf A\) one can achieve \(\Psi=\Psi(z)\), and thus

\[ \mathbf j=-\frac{e^2}{mc}\mathbf A\cdot\Psi^*\cdot\Psi \]

(from the conditions \(\operatorname{div}\mathbf j=\dfrac{dj_z}{dz}=0\) and \(\mathbf j\cdot\mathbf n=0\) it follows—

gives that \(j_z=0\). In addition, in the present case the equations do not contain the imaginary unit \(i\) (since \(\mathbf A\cdot\nabla\Psi=0\)), and the function \(\Psi\) may be regarded as real, as we shall assume. As a result, equations (3.6) and (3.8) take the form:

\[ \left. \begin{aligned} &\frac{d^2\Psi}{dz^2}+\frac{2m}{\hbar^2}|a|\left(1-\frac{e^2}{2mc^2|a|}A^2\right)\Psi-\frac{2m}{\hbar^2}\beta\Psi^3=0,\\ &\frac{d^2A}{dz^2}-\frac{4\pi e^2}{mc^2}\Psi^2\cdot A=0, \end{aligned} \right\} \tag{3.9} \]

where the expression (3.2) has been used and it has been taken into account that \(a<0\). Let us also write down the expression for the surface energy; it differs from (2.40) by replacing the energy \(\dfrac{H^2}{8\pi}+\dfrac{\lambda j_s^2}{2}\), present in the old theory, by the expression \(F_{sH}-F_{s0}\), clear from (3.2), (3.3), and (3.5):

\[ \sigma=\int_0^d\left( a\Psi^2+\frac{\beta}{2}\Psi^4+\frac{a^2}{2\beta} +\frac{\hbar^2}{2m}\left(\frac{d\Psi}{dz}\right)^2 +\frac{e^2}{2mc^2}A^2\Psi^2+ \frac{H^2}{8\pi}-\frac{H_{\kappa}H}{4\pi} \right)\,dz. \tag{3.10} \]

From the condition of minimality of \(\sigma\), which is the free energy per unit surface area, we, of course, also obtain both the first equation (3.9), by varying (3.10) with respect to \(\Psi\), and the second equation (3.9), by varying (3.10) with respect to \(A\). At the same time, since the function \(\Psi\) is not specified at \(z=0\), it is necessary that the condition

\[ z=0,\qquad \frac{d\Psi}{dz}=0, \tag{3.11} \]

be satisfied, which also follows directly from (3.7). The presence of the last term in (3.10) leads to the fact that no additional conditions are imposed on the field at \(z=0\), apart from the usual electrodynamic requirements of continuity of \(A\) and \(H\) (the values of \(A\) and \(H\) at \(z=0\) in the general case, when \(H\ne H_{\kappa}\), will be denoted by \(A_0\) and \(H_0=\left(\dfrac{dA}{dz}\right)_0\)).

Let us now introduce the following parameters \(H_{km}\), \(\delta_0\), and \(\chi\), and also the new variables \(z'\), \(\Psi'\), \(A'\), and \(H'\).

\[ \left. \begin{aligned} &\Psi'^2=\frac{\Psi^2}{\Psi_\infty^2}=\frac{\Psi^2}{|a|/\beta},\qquad z'=\frac{z}{\delta_0},\qquad A'=\sqrt{\frac{e^2}{2mc^2|a|}}\,A=\frac{A}{\sqrt{2}\,H_{km}\cdot\delta_0},\\[4pt] &H'=\frac{dA'}{dz'}=\frac{1}{\sqrt{2}}\frac{H}{H_{km}},\\[4pt] &\delta_0^2=\frac{mc^2\beta}{4\pi e^2|a|},\qquad H_{km}^2=\frac{4\pi a^2}{\beta},\\[4pt] &\chi^2=\frac{1}{2\pi}\left(\frac{mc}{e\hbar}\right)^2\beta =\frac{2e^2}{\hbar^2c^2}H_{km}^2\delta_0^4. \end{aligned} \right\} \tag{3.12} \]

Then equations (3.9) take the form

\[ \begin{split} \frac{d^{2}\Psi'}{dz^{2}}&=\varkappa^{2}\left[-(1-A'^{2})\Psi'+\Psi'^{3}\right],\\ \frac{d^{2}A'}{dz^{2}}&=\Psi'^{2}A', \end{split} \tag{3.13} \]

Using these variables, expression (3.10) must be written in the form:

\[ \sigma=\frac{H_{k m}^{2}}{4\pi}\cdot \delta_{0}\int_{0}^{d'} \left\{\frac{1}{2}-(1-A'^{2})\Psi'^{2}+\frac{\Psi'^{4}}{2}+ \right. \]

\[ \left. +\frac{1}{\varkappa^{2}}\left(\frac{d\Psi'}{dz}\right)^{2}+H'^{2}-2H'\cdot H'_{k}\right\}\,dz'. \tag{3.10a} \]

If \(\varkappa=0\), then, by virtue of (3.13) and (3.11), \(\Psi^{2}=n_s=\mathrm{const}\), and our equations pass into the equations of the old theory with \(\delta^{2}=\delta_{0}^{2}=\dfrac{mc^{2}}{4\pi e^{2}n_s}\) (cf. (2.25) with the second equation (3.9)). The same holds in the general case: if in (3.8) one sets \(\nabla\Psi=0\), then this equation, if the operation \(\operatorname{rot}\) is applied to it, is equivalent to equation (2.26), or, as is immediately clear, for \(\nabla\Psi=0\)

\[ \mathbf{j}=-\frac{e^{2}}{mc}|\Psi|^{2}\mathbf{A}, \]

which leads to equation (2.6). The condition \(\varkappa\to 0\) corresponds to the case \(\hbar\to\infty\) (see (3.12)). But also for \(\hbar\to 0\) one again obtains equation (2.6), but with \(|\Psi|^{2}\), depending on \(\mathbf{A}\); the latter is clear from (3.6). Hence it is also seen that equation (2.6), which does not contain \(\hbar\), nevertheless cannot be regarded as classical, just like the relation

\[ \mathbf{j}=-\frac{e^{2}}{mc}|\Psi|^{2}\mathbf{A}. \]

As was indicated in § 2, this conclusion is also clear from general considerations. Although for \(\varkappa=0\) our scheme passes, from the point of view of the form of the equations, into the usual theory, it differs essentially from the latter even in this limiting case. The point is that in the old theory the quantity

\[ \Lambda=\frac{4\pi\delta^{2}}{c^{2}} \]

at a given temperature is fixed and does not depend on the field. In our case, however, even for \(\varkappa=0\), the quantity \(\Psi^{2}\), equivalent to \(\delta\) (see (3.12)—(3.12a)), is determined from the condition of minimum free energy, which, in the presence of a field, leads for superconductors of finite dimensions to a dependence of the penetration depth \(\delta\) on \(H\) (see § 4).

From the limiting case \(\varkappa=0\), and also from what follows, it is clear that the experimentally measured quantity is the parameter

\[ \delta_{0}^{2}=\frac{mc^{2}\beta}{4\pi e^{2}|\alpha|} =\frac{mc^{2}}{4\pi e^{2}|\Psi_{\infty}|^{2}} =\frac{mc^{2}}{4\pi e^{2}n_s} \]

(\(\delta_{0}\) is the penetration depth of a weak magnetic field). It is precisely this quantity that also enters into the expression for the dielectric constant

\[ \varepsilon=\varepsilon_{0}-\frac{4\pi e^{2}n_s}{m\omega^{2}} \]

of a superconductor in a time-dependent field not

too high a frequency \((\varepsilon_0\) is a certain constant corresponding to the contribution to \(\varepsilon\) of all particles except the “superconducting electrons”; see § 5). The parameter \(\Psi_\infty^2=n_s\) itself is not a measurable quantity, just as the number of “free electrons” in the usual quantum theory of metals is not. Therefore in both these cases one can speak only of the “effective number of electrons,” which can be determined from the value of \(\varepsilon\) or \(\delta_0^2\), assuming in the corresponding expressions the quantities \(e\) and \(m\) to have the same values as for a free electron (for more detail see § 6). Proceeding in this way, we shall relate \(n_s=|\Psi_\infty|^2\) to the observed quantity \(\delta_0^2\) \((e=4.8\cdot10^{-10},\ m=9.1\cdot10^{-28})\):

\[ \delta_0^2=\frac{mc^2\beta}{4\pi e^2|\alpha|} =2.84\cdot10^{11}\frac{\beta}{|\alpha|} =\frac{2.84\cdot10^{11}}{|\Psi_\infty|^2}\ \mathrm{cm}^2. \tag{3,12a} \]

From (3,12a) and from measurements of the critical field

\[ H_{\mathrm{km}}=\frac{\sqrt{4\pi|\alpha|}}{\sqrt{\beta}} \]

one can find \(\alpha\) and \(\beta\). In addition to \(\delta_0\) and \(H_{\mathrm{km}}\) (or \(\alpha\) and \(\beta\)), one more dimensionless parameter enters the theory,

\[ \chi^2=\frac{2e^2}{\hbar^2c^2}\,H_{\mathrm{km}}^2\cdot \delta_0^4. \tag{3,14} \]

The charge \(e\) entering the theory was introduced into the original equation (3,5) by the usual substitution \(-i\hbar\nabla\) by \(-i\hbar\nabla-\frac{e}{c}\mathbf A\). One may think that the charge \(e\) coincides with the charge of the electron, since with respect to the charge, in contrast to the mass, there is no reason to distinguish it from the true charge of the current carriers. In this case

\[ \chi^2=4.64\cdot10^{14}H_{\mathrm{km}}^2\cdot \delta_0^4, \tag{3,14a} \]

where \(\delta_0\) is measured in cm and \(H_{\mathrm{km}}\) in oersteds.

From the experimental data discussed in § 4 it follows that for mercury

\[ \chi^2=0.027,\quad \chi=0.165,\quad \sqrt{\chi}=0.406. \tag{3,14b} \]

§ 4. NEW PHENOMENOLOGICAL THEORY. CONCRETE RESULTS

Passing to the application of the equations obtained to the solution of concrete problems, let us first of all consider the case of a superconducting half-space bordering on vacuum (the superconductor for \(z>0\), the boundary \(z=0\)). The corresponding solution will, of course, also apply to a sufficiently thick plate whose half-thickness \(d\gg\delta_0\).

For \(z' = 0\), \(H' = H'_0\); for \(z' = \infty\), \(H' = A' = 0\) (the choice in the present case of the value \(A'(\infty)=0\) is quite natural and, most importantly, possible). Further, for \(z'=\infty\) we are dealing with a superconductor without a field and far from any boundaries and, consequently, the solution (3.3) must hold, i.e. \(\Psi'_\infty = 1\), \(d\Psi'/dz' = 0\). Thus, for \(z'=\infty\):

\[ \Psi_{\infty}^{\prime 2}=1,\qquad \frac{d\Psi'}{dz'}=0,\qquad H'=A'=0. \tag{4,1} \]

This solution, of course, satisfies equations (3.13). As for the boundary with the vacuum at \(z'=0\), condition (3.11), obtained directly from the variational principle, must be fulfilled on it. Here, however, several remarks must be made. Condition (3.11) is obtained if no additional requirements are imposed on \(\Psi'\) at the boundary (natural boundary conditions); but if from the very beginning one requires that at the boundary with the vacuum \(\Psi'=0\), then condition (3.11) is no longer obtained. However, the condition \(\Psi'=0\) or \(\Psi'=\mathrm{const}\) is inadmissible within the framework of the scheme being developed, since in that case the problem of a superconducting plate has no solution except for certain special values of the plate thickness \(2d\). Therefore we impose no additional conditions on \(\Psi'\) at the boundary with the vacuum and thus arrive at (3.11). In the absence of a field, when \(H=A=0\), it follows from (3.11), (4.1), and equations (3.13) that

\[ H'=A'=0;\qquad \Psi^{\prime 2}=\Psi_{\infty}^{\prime 2}=1, \tag{4,2} \]

i.e. the presence of the boundary with the vacuum does not influence \(\Psi'\). At first glance this result may seem unacceptable, since it is natural to require that the \(\Psi\)-function vanish at the boundary of the metal. The point, however, is that our \(\Psi\)-function is a certain averaged quantity, the averaging being carried out over regions large compared with the lattice constant but small compared with the penetration depth \(\delta_0\). Therefore our \(\Psi\)-function must reflect the distribution of the mean density of “superconducting electrons,” which in the absence of a field may be regarded as constant—if not necessarily, then at any rate quite naturally.

In the presence of a magnetic field the solution (4.2), of course, no longer holds, and it is necessary to integrate equations (3.13) with the boundary condition (4.1) for \(z'=\infty\) and the condition for \(z'=0\):

\[ H'=H'_0;\qquad \left(\frac{d\Psi'}{dz'}\right)_0=0. \tag{4,3} \]

The values \(A'_0\) and \(\Psi'_0\) are not known in advance. Equations (3.13), to со-

unfortunately, are not integrable in quadratures, and we can indicate only one of their integrals:

\[ (1-A'^2)\Psi'^2-\frac{\Psi'^4}{2} +\left(\frac{dA'}{dz'}\right)^2 +\frac{1}{\varkappa^2}\left(\frac{d\Psi'}{dz'}\right)^2 =\operatorname{const}. \tag{4,4} \]

For the problem that interests us at present, by virtue of (4,1) \(\operatorname{const}=\dfrac12\), and, thus,

\[ H'^2=\left(\frac{dA'}{dz'}\right)^2 = \frac{1}{2} -\frac{1}{\varkappa^2}\left(\frac{d\Psi'}{dz'}\right)^2 -(1-A'^2)\Psi'^2+\frac{\Psi'^4}{2}. \tag{4,5} \]

As has been said, it does not seem possible to solve equations (3,13) in general form, and we shall give here an approximate solution, valid for small \(z\) (more precisely, the solution will be valid if the product \(zH_0'^2\) is small). To find this solution we put

\[ \Psi'= \Psi'_0+\varphi'=1+\varphi',\qquad |\varphi'|\ll 1. \tag{4,6} \]

Then in the first approximation, accurate to terms of order \(\varphi'A'\) and \(\varphi'^2\), the system (3,13) assumes the form

\[ \frac{d^2\varphi'}{dz'^2}=\varkappa^2(2\varphi'+A'^2),\qquad \frac{d^3A'}{dz'^3}=A'. \tag{4,7} \]

This system is immediately integrated, and its solution may be used to find the next approximation, etc. Let us give the solution of the problem with the conditions (4,1) and (4,3), accurate through terms of order \(H_0'^3\) inclusive:

\[ \left. \begin{aligned} \Psi'&=1+\frac{\varkappa H_0'^2}{\sqrt{2}\,(-\varkappa^3)} \left(\frac{\varkappa}{\sqrt{2}}e^{-2z}-e^{-\sqrt{2}\varkappa z}\right),\\[4pt] A'&=-H_0'e^{-z} -\frac{\varkappa H_0'^3}{\sqrt{2}\,(2-\varkappa^3)} \left\{ \frac{\varkappa}{4\sqrt{2}}e^{-3z} -\frac{e^{-(\sqrt{2}\varkappa+1)z}}{\varkappa(\varkappa+\sqrt{2})} \right.\\ &\hspace{4.2cm}\left. -\frac{3\varkappa^3+3\sqrt{2}\varkappa^2+8\varkappa-4\sqrt{2}} {4\sqrt{2}\varkappa(\varkappa+\sqrt{2})}\,e^{-z} \right\},\\[4pt] H'&=\frac{dA'}{dz'}. \end{aligned} \right\} \tag{4,8} \]

For \(z'=0\), of course,

\[ \frac{d\Psi'}{dz}=0,\qquad H'=H'_0 \]

and

\[ \Psi'_0=1-\frac{\varkappa H_0'^2}{2(\varkappa+\sqrt{2})}, \]

\[ A'_0=-H'_0-\frac{\varkappa(\varkappa+2\sqrt{2})}{4(\varkappa+\sqrt{2})}\,H_0'^3. \tag{4,9} \]

The largest terms not written in (4,8) in the expression for \(\Psi\) have order \(\varkappa^2 H_0'^4\), and in the expression for \(A'\)—order \(\varkappa^3 H_0'^5\); for \(\varkappa=\sqrt{2}\) there is no essential singularity in the solution, but it has a somewhat different form, on ...

CURRENT STATE OF THE THEORY OF SUPERCONDUCTIVITY

we shall not dwell on. The field \(H'_0\) in the equilibrium state is less than or equal to the critical field for the massive conductor, which in the new variables is equal to \(H'_{\mathrm{cm}}=\dfrac{1}{\sqrt{2}}\) (see (3,12)). Therefore, according to (4,9), for \(\chi=0.165\) (see (3,146)) \(\Psi'_0 \geqslant 0.974\) (equality holds for \(H'_0=H'_{\mathrm{cm}}=\dfrac{1}{\sqrt{2}}\)), and thus the use of formulas (4,8) in this case is quite permissible if an accuracy of the order of one or several percent is sufficient. At present such accuracy in the measurement of \(\delta_0\) is still far from being attained. As an investigation of the original equations (3,13) shows, for \(\chi>\dfrac{1}{\sqrt{2}}\) there arises a peculiar instability of the normal phase of the metal, which for \(\chi<\dfrac{1}{\sqrt{2}}\) is stable in the field \(H'_0>\dfrac{1}{\sqrt{2}}\) (for more detail see \(^{4}\)). In view of this circumstance, all our results, generally speaking, are applicable only for \(\chi<\dfrac{1}{\sqrt{2}}\). This restriction is apparently not essential, since from the available experimental data it follows that \(\chi \ll 1\).

Using the solution (4,8), it is easy to determine the dependence of the depth of penetration of the magnetic field into a massive superconductor on the field strength. In accordance with the experimental method of measurement \(^{18,19}\), let us define the depth of field penetration into a massive superconductor by the expression

\[ \delta= \frac{\displaystyle\int_0^\infty H\,dz}{H_0} = \delta_0 \frac{\displaystyle\int_0^\infty H'\,dz'}{H'_0} = \delta_0\frac{|A'_0|}{H'_0}, \tag{4,10} \]

where \(H_0\) is the external field (the field at \(z=0\)), and in the first expression the usual units are used, while in the second and third the new (reduced) units are used for \(H\), \(H_0\), \(A_0\), and \(z\). Substituting the field (4,8) into (4,10), we have (in ordinary units)

\[ \delta=\delta_0\left( 1+\frac{\chi(\chi+2\sqrt{2})}{8(\chi+\sqrt{2})^2} \left(\frac{H_0}{H_{\mathrm{cm}}}\right)^2 \right) = \delta_0\left( 1+f(\chi)\left(\frac{H_0}{H_{\mathrm{cm}}}\right)^2 \right), \]

\[ \frac{d\delta}{dT} = \frac{d\delta_0}{dT} + f(\chi)\left(\frac{H_0}{H_{\mathrm{cm}}}\right)^2 \left( \frac{d\delta_0}{dT} - \frac{2\,\dfrac{dH_{\mathrm{cm}}}{dT}}{H_{\mathrm{cm}}}\,\delta_0 \right). \tag{4,11} \]

Hence it is clear that the quantity \(\delta_0\), as already mentioned, is the penetration depth in a weak field. The function \(f(\chi)\) can—

monotonically grows with \(\varkappa\) in such a way that \(f(0)=0\), \(f(\infty)=\dfrac{1}{8}\) and, for \(\varkappa \ll 1\),

\[ f(\varkappa) \simeq \frac{\varkappa}{4\sqrt{2}} . \]

Thus, for \(H_0=H_{\mathrm{cm}}\), even for \(\varkappa=\dfrac{1}{\sqrt{2}}\), \(\delta=1.07\delta_0\), while for \(\varkappa=0.165\), \(\delta=1.028\delta_0\). We see that the expected change of \(\delta\) with \(H_0\) for mercury, for which according to our estimate \(\varkappa=0.165\), is very small and lies within the limits of accuracy of the measurements carried out in \({}^{19}\) (the data of \({}^{18}\) concerning the dependence of \(\delta\) on \(H_0\), probably, for the reasons indicated in \({}^{19}\), are incorrect; this is also seen from the fact that in \({}^{18}\) in a number of cases \(\delta\) is proportional to \(H_0\), and not to \(H_0^2\), whereas in a weak field in any case \(\delta\sim H_0^2\), since \(\delta\) is an even function of \(H_0\)). As we shall see below, for thin superconductors the dependence of \(\delta\) on \(H_0\) is considerably greater than for massive ones, and can be observed in experiments of the type described in \({}^{17}\) (therefore it is possible that the dependence of \(\delta\) on \(H_0\) noted in \({}^{17}\) is real, which does not contradict the absence of a noticeable effect in \({}^{19}\)).

Let us turn to the determination of the surface energy \(\sigma_{ns}\) at the boundary separating the superconducting phase from the normal one.

To calculate \(\sigma_{ns}\) it is necessary to find the solution of equations (3.13) for a superconducting half-space bordering on a half-space filled with the normal phase of the same metal. There is no reason to suppose that at the boundary of the superconducting and normal phases the condition (4.3) for the boundary of a superconductor with vacuum will be satisfied. Moreover, since the entire difference between the two phases formally reduces to the fact that in one of them \(\Psi\ne 0\), while in the other \(\Psi=0\), one must think that the transition between the phases occurs continuously, so that at the boundary \(\Psi=0\)*).

It can be shown that the equations of the problem do not admit a solution in which \(\Psi\) vanishes at some point located at a finite distance. On the contrary, the equations have a solution corresponding to the case when the function \(\Psi\) smoothly and continuously changes from the value \(\Psi=1\) at \(z=\infty\) to the value \(\Psi=0\) at \(z=-\infty\). Thus, the transition from the superconducting phase

*) A jump of \(\Psi\) would be associated with infinitely large kinetic energy, since in this case

\[ \frac{\hbar^2}{2m}\left(\frac{d\Psi}{dx}\right)^2 \to \infty ; \]

actually, of course, a jump of our function \(\Psi\) means its change over a distance of the order of atomic dimensions \(a\sim 10^{-8}\) and

\[ \frac{\hbar^2}{2m}\left(\frac{d\Psi}{dx}\right)^2 \sim \frac{\hbar^2\Psi^2}{2ma^2} = \frac{\hbar^2 n_s}{2ma^2}. \]

At the boundary of the superconducting phase with vacuum this energy is part of the surface energy of the metal, which we do not consider (it is many orders of magnitude greater than \(\sigma_{ns}\) or \(\sigma_n' - \sigma_s\)). In the case of the boundary of the superconducting and normal phases, however, there is no surface energy except \(\sigma_{ns}\), and the appearance of a jump of \(\Psi\), and hence also of an energy

\[ \sim \frac{\hbar^2 n_s}{2ma^2}\,a \sim \sim 10^2\ \frac{\mathrm{erg}}{\mathrm{cm}^2} \quad (\text{for } n_s\sim 10^{21}) \]

is completely inadmissible.

to the normal state occurs in the transition layer and corresponds to the solution of equations (3.13) with the boundary conditions

\[ \begin{gathered} z \to \infty:\quad \Psi'=\Psi'_{\infty}=1,\qquad H'=A'=\frac{d\Psi'}{dz'}=0,\\ z \to -\infty:\quad \Psi'=0,\qquad \frac{d\Psi'}{dz'}=0,\qquad H'=H'_0=\frac{1}{\sqrt{2}},\\ A'=H'_0 z'+\mathrm{const}. \end{gathered} \tag{4.12} \]

In fact, of course, the transition layer is not infinite, but has a depth of order \(\delta_0\) (more precisely, of order \(\delta_0/\varkappa\); see \(^{4}\)), just as the magnetic field penetrates into the superconductor practically to a depth of order \(\delta_0\), although it is strictly equal to zero only at \(z=\infty\). The surface energy \(\sigma_{ns}\) is equal to (see (3.10a))

\[ \begin{aligned} \sigma_{ns} &= \frac{H_{\mathrm{km}}^2}{2\pi}\,\delta_0 \int_{-\infty}^{+\infty} \left\{ \frac{1}{2}-(1-A'^2)\Psi'^2+\frac{\Psi'^4}{2}-H'_0H' \right\}dz' \\ &= \frac{H_{\mathrm{km}}^2}{2\pi}\,\delta_0 \int_{-\infty}^{+\infty} \left\{ \frac{1}{\varkappa^2}\left(\frac{d\Psi'}{dz'}\right)^2+H'^2-H'_0H' \right\}dz', \end{aligned} \tag{4.13} \]

where the relation (4.5), \(H'_0=H'_{\mathrm{km}}=1/\sqrt{2}\), has been used, and all quantities under the integrals are expressed in reduced units.

In view of the fact that the system (3.13) is, in general, not integrable, we can give an analytic expression for \(\sigma_{ns}\) only for sufficiently small \(\varkappa\). Namely, it can be shown (see \(^{4}\)) that, to within terms of order \(1/\sqrt{\varkappa}\),

\[ \sigma_{ns}=\frac{\delta_0 H_{\mathrm{km}}^2}{\sqrt{2}\cdot 3\pi\varkappa}; \qquad \Delta=\frac{\sigma_{ns}}{H_{\mathrm{km}}^2/8\pi} = \frac{1.89}{\varkappa}\,\delta_0 \tag{4.14} \]

\[ \left(\text{for } \sqrt{\varkappa}\ll 1\right). \]

It is especially important to emphasize that for small \(\varkappa\), \(\sigma_{ns}>0\), which is absolutely necessary and whose attainment was our principal aim. For sufficiently large \(\varkappa\), on the contrary, \(\sigma_{ns}<0\) (this is immediately clear from (4.13), since \(H'^2<H'H'_0\)), which also indicates the inadmissibility of such large \(\varkappa\). The value of \(\varkappa\) adopted by us for mercury (3.14b), very small from all other points of view, may still prove insufficiently small for the applicability of formula (4.14), since in this case \(\sqrt{\varkappa}=0.407\). If, nevertheless, one uses formula (4.14) in this case, then \(\Delta=11.4\delta_0\). More exact calculations have not yet been carried out. According to a very rough experi-

mental estimate\(^3\) for mercury \(\Delta \sim \delta_0\). The thickness of the transition layer, as it turns out\(^4\), is of the order

\[ \frac{\delta_0}{\chi}\sim 10\delta_0 . \]

In addition to the half-space, the problem is one-dimensional for plane plates and films. Here it is of interest to calculate the critical field \(H_k\) for the destruction of superconductivity of a film and the magnetic moment of the film \(\mu\) in any field; moreover, in the case where a resultant current \(I\) flows through the film, it is necessary to find the critical value of the current \(I_k\) which destroys superconductivity, as well as the dependence of \(I_k\) on the applied additional field \(H_0\).

The field \(H_k\) is determined by formulas (2,40) and (3,10a):

\[ \frac{H_{\mathrm{km}}^2}{8\pi}=\frac{H_k^2}{8\pi}+\frac{\varepsilon}{d}, \]

\[ \sigma=\frac{H_{\mathrm{km}}^2}{4\pi}\,\delta_0 \int_0^{d'}\left\{\frac{1}{2}-(1-A'^2)\Psi'^2+ \frac{\Psi'^4}{2}+\frac{1}{\chi^2}\left(\frac{d\Psi'}{dz'}\right)^2+H'^2-2H'H_k'\right\}\,dz', \tag{4,15} \]

where the thickness of the plate is equal to \(2d\), and under the integral sign all quantities are expressed in reduced units (the \(z\)-axis is perpendicular to the film and \(z=0\) at its middle). The magnetic moment of the film, referred to a unit of its surface, in an external field \(H_0\) parallel to the film, is equal to (see (2,30))

\[ \mu=\int_{-d}^{d}\frac{H(z)-H_0}{4\pi}\,dz =\frac{1}{2\pi}\left[A(d)-H_0d\right], \tag{4,16} \]

where, in passing to the second expression, it has been taken into account that for a film (with no resultant current) in an external field \(H(z)=H(-z)\), and

\[ A(d)=\int_0^d H(z)\,dz, \]

since below the potential \(A\) is chosen so that \(A(0)=0\).

To find \(H_k\), \(\mu\), and \(I_k\), one must find the solution of equations (3,13) with boundary conditions at \(z'=\pm d'\):

\[ \frac{d\Psi'}{dz'}=0,\quad H'=H_0'\pm H_I',\quad H_I'=\frac{2\pi}{c}I', \tag{4,17} \]

where \(H_0'\) is the external magnetic field directed along the \(y\)-axis, \(I'\) is the total current \(\left(I'=\int j'\,dz',\ j' \text{ is the current density}\right)\), flowing through the film in the direction of the negative \(x\)-axis, and \(2H_y'=\dfrac{4\pi}{c}I'\)—the difference between the values of the total field on the two sides of the film—

which is associated with the presence of the current \(I'\) (since \(H'_I=\dfrac{H_I}{\sqrt{2}H_{\mathrm{cm}}}\), it is clear that \(I'=\dfrac{I}{\sqrt{2}H_{\mathrm{cm}}}\)). If the current \(I\) and the field \(H_0\) are not mutually perpendicular, then two components of the potential \(\mathbf A\) (the components \(A_x\) and \(A_y\)) are nonzero, instead of the single component \(A_x\) in the case considered above. In this case, instead of (3.13), we have the system of initial equations in the form

\[ \left. \begin{aligned} \frac{d^2\Psi'}{dz'^2} &=\chi^2\left[-\left(1-A_x'^2-A_y'^2\right)\Psi' + \Psi'^3\right],\\ \frac{d^2 A'_x}{dz'^2} &=\Psi'^2 A'_x,\qquad \frac{d^2 A'_y}{dz'^2}=\Psi'^2 A'_y \end{aligned} \right\} \tag{4.18} \]

These equations must be solved under the conditions, for \(z'=\pm d'\):

\[ \left. \begin{aligned} \frac{d\Psi'}{dz'}&=0,\qquad H'_x=H'_{x0},\\ H'_y&=H'_{y0}\pm H'_I,\qquad H'_I=\frac{2\pi}{c}I', \end{aligned} \right\} \tag{4.19} \]

where the axes are chosen in such a way that the total current has a component only along the \(x\)-axis and, consequently, the field \(H_I\) is directed along the \(y\)-axis; \(H_{x0}\) and \(H_{y0}\) are the components of the external field along the \(x\)- and \(y\)-axes.

For sufficiently thick plates, i.e. if \(d\gg \delta_0\), the value of \(H_k\) can be obtained immediately by substituting the solution (4.8) into (4.15) and putting \(d\to\infty\) in the integral. As a result, for \(d\gg\delta_0\) we have

\[ \frac{H_k}{H_{\mathrm{cm}}} = 1+\frac{\delta_0}{2d}\left(1+\frac{f(\chi)}{2}\right), \tag{4.20} \]

where \(f(\chi)=\dfrac{\chi(\chi+2\sqrt{2})}{8(\chi+\sqrt{2})^2}\) is the same function as in (4.11), and formula (4.20) is valid to within terms of order \(\left(\dfrac{\delta_0}{d}\right)^3\). In the same approximation, in the old phenomenological theory one obtains expression (4.20) with \(\chi=0\) (see (2.41)).

For films of arbitrary thickness the solution of equations (3.13) must be carried out anew*). In this case, the solution (4.8) suggests that for thin films as well, at small \(\chi\), the function \(\Psi'\) changes only weakly with change of \(z'\). Starting from this assumption, which is justified post factum, we put

\[ \Psi'=\Psi'_0+\varphi',\qquad |\varphi'|\ll \Psi'_0,\qquad \varphi'(z'=0)=0. \tag{4.21} \]

*) In the absence of a field for films, by virtue of condition (4.17), the solution (4.2), i.e. \(\Psi^2=1\), still holds. But, of course, the introduction of our function \(\Psi\) is possible only if \(d\gg a\), where \(a\) is an atomic dimension. This condition is also necessary in order that one may speak of a superconducting film with the same parameters \((H_{\mathrm{cm}}, T_k, \delta_0, \chi)\) as for a bulk metal. We consider only films satisfying this condition.

Then equations (3.13) in the first approximation take the form

\[ \begin{gathered} \frac{d^{2}\varphi'}{dz'^{2}}=\varkappa^{2}\left\{\Psi_{0}'^{3}-\Psi_{0}'+(3\Psi_{0}'^{2}-1)\varphi'+A'^{2}\Psi_{0}'\right\},\\ \frac{d^{2}A'}{dz'^{2}}=\Psi_{0}'^{2}A'. \end{gathered} \tag{4.22} \]

From the second equation (4.22), taking into account the boundary conditions (4.17), we find the values of \(A'\) and \(H'\):

\[ \begin{gathered} A'=\frac{H_{0}'\,\operatorname{sh}\Psi_{0}'z'}{\Psi_{0}'\operatorname{ch}\Psi_{0}'d'}+ \frac{H_{1}'\,\operatorname{ch}\Psi_{0}'z'}{\Psi_{0}'\operatorname{sh}\Psi_{0}'d'},\\ H'=\frac{dA'}{dz'}= \frac{H_{0}'\,\operatorname{ch}\Psi_{0}'z'}{\operatorname{ch}\Psi_{0}'d'}+ \frac{H_{1}'\,\operatorname{sh}\Psi_{0}'z'}{\operatorname{sh}\Psi_{0}'d'}. \end{gathered} \tag{4.23} \]

Substituting (4.23) into the first equation (4.22), we find \(\varphi\), and from the requirement that for \(z'=\pm d'\), \(d\varphi'/dz'=0\), a transcendental equation determining \(\Psi_{0}'\) is obtained.

As we shall see, practically only the limiting case \(\varkappa=0\) is of interest. Therefore we shall not write out the function \(\varphi\), which for \(\varkappa=0\) is equal to zero. In this case, if \(\varkappa=0\)*):

\[ \Psi_{0}'^{2}(\Psi_{0}'^{2}-1)= \frac{H_{0}'^{2}\left(1-\dfrac{\operatorname{sh}2\Psi_{0}'d'}{2\Psi_{0}'d'}\right)} {2\operatorname{ch}^{2}\Psi_{0}'d'} - \frac{H_{1}'^{2}\left(1+\dfrac{\operatorname{sh}2\Psi_{0}'d'}{2\Psi_{0}'d'}\right)} {2\operatorname{sh}^{2}\Psi_{0}'d'}. \tag{4.24} \]

If \(\varkappa\ne0\), the equation for \(\Psi_{0}'\) is more complicated; for illustration we give it for the case when \(H_{1}'=0\):

\[ \Psi_{0}'^{2}-1= \frac{ 2H_{0}'^{2}\left\{ 1-\dfrac{\operatorname{sh}2\Psi_{0}'d'}{2\Psi_{0}'d'}\cdot \dfrac{\varkappa\sqrt{3\Psi_{0}'^{2}-1}\,d} {\operatorname{sh}(\varkappa\sqrt{3\Psi_{0}'^{2}-1}\,d)} \right\} } { \operatorname{ch}^{2}\Psi_{0}'d'\left\{4\Psi_{0}'^{2}-\varkappa^{2}(3\Psi_{0}'^{2}-1)\right\} }. \tag{4.25} \]

*) For \(\varkappa=0\) the equation for \(\Psi'=\Psi_{0}'=\mathrm{const}\) can be obtained at once from the condition of the minimum of the free energy, i.e. from the condition \(d\sigma/d\Psi=0\).

As is clear from (4.15), this condition gives

\[ \Psi'^{2}-1= \frac{-\displaystyle\int_{0}^{d'} A'^{2}\,dz'}{d'}. \]

Let us now dwell in somewhat greater detail on the destruction of superconductivity of films by an external field in the absence of a total current. If \(z=0\), then \(\Psi'=\Psi'_0=\mathrm{const}\), and for the field the solution (4.23) with \(H'_I=0\) is valid. Substituting this solution into (4.15), we readily find

\[ \left(\frac{H_k}{H_{km}}\right)^2 = \frac{\Psi_0^{\prime 2}\left(2-\Psi_0^{\prime 2}\right)} {1-\dfrac{\operatorname{th}\Psi'_0 d'}{\Psi'_0 d'}} . \tag{4.26} \]

In the same case \((z=0,\ H'_I=0)\), equation (4.24) or (4.25) for

\[ H_0=H_k \quad \text{or} \quad H'_0=\frac{H_k}{\sqrt{2}H_{km}} \]

gives

\[ \left(\frac{H_k}{H_{km}}\right)^2 = \frac{4\Psi_0^{\prime 2}\left(\Psi_0^{\prime 2}-1\right)\operatorname{ch}^2\Psi'_0 d'} {1-\dfrac{\operatorname{sh}2\Psi'_0 d'}{2\Psi'_0 d'}} . \tag{4.27} \]

From (4.26)—(4.27), using the measured values of \(\dfrac{H_k}{H_{km}}\) and knowing \(d\), one can determine \(\Psi'_0\) and \(\delta_0=\dfrac{d}{d'}\). It is easy to see that, for small values of

\[ \Psi'_0 d'=\Psi'_0\frac{d}{\delta_0}, \quad \text{for } H=H_k\ \Psi'_0=0 \]

and

\[ \frac{H_k}{H_{km}}=\frac{\sqrt{6}\,\delta_0}{d}. \tag{4.28} \]

Thus, in this case a second-order phase transition takes place: with increasing field \(\Psi'_0\) decreases and at the transition point \(\Psi'_0=0\). As is clear from (4.24), for \(H_I=0\), to accuracy up to terms of order \(d'^2\) (taking into account that \(H_0^{\prime 2}d'^2\) may be of order unity), we have

\[ \Psi_0^{\prime 2} = \frac{ 1-\left(\dfrac{H_0}{H_{km}}\right)^2\dfrac{d^2}{6\delta_0^2} }{ 1-\dfrac{2}{15}\left(\dfrac{H_0}{H_{km}}\right)^2\dfrac{d^4}{\delta_0^4} }. \tag{4.29} \]

The transition to the normal state is a second-order transition for \(d\ll d_k\), where, as is readily shown from (4.26)—(4.27),

\[ d_k=\frac{\sqrt{5}}{2}\,\delta_0 . \tag{4.30} \]

The point \(d=d_k\) is a kind of Curie critical point\(^{90}\), and for \(d>d_k\) a first-order transition takes place, i.e. for \(H_0>H_k\), \(\Psi'_0>0\), and the latent heat of transition is released (for \(d<d_k\), at \(H=H_k\) there is a jump in the heat capacity; the heat capacity of thin films evidently depends on \(H_0\)).

The penetration depth of the field, as is clear from (4.23), is equal to

\[ \delta=\frac{\delta_0}{\Psi'_0}, \tag{4.31} \]

and we see that for sufficiently thin specimens at \(H_0\sim H_k\) the penetration depth may be considerably greater than for a bulk metal. In this case (see (4.16) and (4.23) with \(H_i=0\)):

\[ \mu=-\frac{H_0 d}{2\pi}\left(1-\frac{\delta}{d}\operatorname{th}\frac{d}{\delta}\right)= \]

\[ =-\frac{H_0 d}{2\pi}\left\{\frac{1}{3}\left(\frac{d}{\delta}\right)^2-\frac{2}{15}\left(\frac{d}{\delta}\right)^4+\ldots\right\}. \tag{4.32} \]

By measuring \(\mu\), one can find the depth \(\delta\), which, according to (4.31) and (4.24), depends on \(H_0\). Thus, as was already indicated, even for \(\chi=0\) the new phenomenological theory in the case of thin superconductors differs substantially from the old theory.

For \(\chi\ne 0\) all expressions become, in general, very cumbersome. However, for small \(\chi\), which alone are of interest to us, and not too large \(d\), all expressions can be expanded in a series in \(\chi d\). As a result, in the range of thicknesses where a second-order transition takes place, instead of (4.28) we have

\[ \left(\frac{H_k}{H_{kM}}\right)^2 =6\left(\frac{\delta_0}{d}\right)^2-\frac{7}{10}\chi^2+ \frac{11}{1400}\chi^4\left(\frac{d}{\delta_0}\right)^2-\ldots . \tag{4.33} \]

The value of \(d_k\) is now equal to:

\[ d_k^2=\frac{5}{4}\left(1-\frac{7}{24}\chi^2+\ldots\right)\delta_0^2 . \tag{4.34} \]

If for \(\chi^2\) one adopts the value (3.146), then it is practically unnecessary to take into account the terms containing \(\chi^2\) in (4.33)—(4.34), as well as in other analogous expressions.

Experimental data suitable for quantitative treatment on the destruction of superconductivity of films in an external field are available only in \(^{21}\) and refer to mercury. However, even in this case the scatter of points in the measurements was rather large and, moreover, owing to the absence in \(^{21}\) of tables, the values of \(\frac{H_k}{H_{kM}}\) had to be taken from a graph; the principal source of inaccuracy consists in the fact that the film thickness indicated in \(^{21}\) represents a certain mean value and may, especially for thin films, differ substantially from the thickness \(d\) entering into our formulas, in which it is assumed, of course, that the film is ideally homogeneous.

Table II gives directly the values of \(\delta_0\), obtained by means of formula (4.28) on the basis of the data for \(\dfrac{H_k}{H_{km}}\) and \(d\),

Table II

Values of \(\delta_0\) for mercury (the values of \(\delta_0\) and \(d\) are in \(10^{-5}\) cm)

\(T\)	\(2d\): 0,596	\(2d\): 0,840	\(2d\): 1,178	\(2d\): 1,423	\(2d\): 1,690	\(2d\): 2,400	\(2d\): 4,390	\(2d\): 10,880	\(2d_k = \sqrt[?]{5}\delta_{0\mathrm{пл}}\)
4,13	5,13	4,61	4,07	4,17	3,80	3,37	3,08	(3,72) 3,56	6,9
4,12	4,12	4,06	3,47	3,36	3,27	3,11	2,72	(3,52) 3,30	6,1
4,10	3,47	3,38	2,87	3,02	2,79	2,53	2,28	(3,21) 2,50	5,1
4,05	2,66	2,62	2,32	2,27	2,08	1,86	(1,80) 1,80	(2,86) 1,95	4,0
4,00	2,28	2,31	1,92	1,82	1,76	1,56	(1,57) 1,57	(2,72) 1,70	3,5
3,80	1,69	1,62	1,40	1,28	1,24	1,10	(1,31) 1,15	(2,63) 1,39	2,5
3,50	1,27	1,24	1,08	0,99	0,98	(0,87) 0,87	(1,23) 0,99	(2,50) 1,16	1,95
3,00	1,10	1,10	0,92	0,84	(0,83) 0,83	(0,77) 0,72	(1,16) 0,84	—	1,61
2,50	0,92	0,94	0,86	0,80	(0,75) 0,75	(0,73) 0,66	(1,13) 0,78	(2,45) 1,0	1,48

given in \(^{21}\); here the values for whose calculation formula (4.28) is no longer applicable are placed in parentheses, since

\(d>d_k\). Under these quantities, the values obtained directly from formulas (4.26)—(4.27) are given below in parentheses.

In the last column are given the values \(2d_k\), obtained by formula (4.30), using the minimum value \(\delta_0\) indicated in the corresponding row. From Table II, as well as directly from the graph of the dependence of \(\ln \dfrac{H_k}{H_{km}}\) on \(\ln 2d\) in \({}^{21}\), a sharp break is evident in the course of this dependence, which occurs precisely when \(d\) passes beyond \(d_k\) (in Table II the single values of \(\delta_0\) and the values in parentheses, according to (4.28), are simply quantities proportional to \(\dfrac{H_k}{H_{km}}\,d\); this product decreases up to \(d_k\), and for \(d>d_k\) begins to increase sharply). In this circumstance we are inclined to see confirmation of the conclusion concerning the different character of the transition for \(d<d_k\) and \(d>d_k\). The decrease of the values of \(\delta_0\) with increasing \(d\), clearly expressed in Table II, for \(d<d_k\) can quite well be explained by the already mentioned difference between the values of \(d\) indicated in \({}^{21}\) and the effective values \(d_{\mathrm{eff}}\). Qualitatively it is clear that the thinner the film, the more strongly \(d_{\mathrm{eff}}\) differs from \(d\), with \(d_{\mathrm{eff}}<d\). The observed dependence of \(\delta_0\) on \(d\) for \(d<d_k\) is in agreement with this picture. As for the increase of the values of \(\delta_0\), calculated from formulas (4.26)—(4.27), with increasing \(d\) for \(d>d_k\), we, on the contrary, see no grounds for it. It is necessary, however, to bear in mind two circumstances: first, our whole scheme, based on the expansion of \(F_{s0}\) in (3.6) in a series in powers of \(\Psi^2\) with accuracy up to terms \(\Psi^4\), is, generally speaking, applicable only near \(T_k\), while relation (3.4) holds, and

\[ \delta_0^2=\frac{\mathrm{const}}{T_k-T} =\frac{\delta_{00}^2}{1-\dfrac{T}{T_k}}, \tag{4.35} \]

where \(\delta_{00}\) is a certain constant (see (3.3) and (3.12)).

For Hg the region where relation (3.4) is valid and, in this connection, where the applicability of formula (4.35) should be expected, lies between \(T_k\) and \(T\simeq 3.80\div 4.0^\circ\). At smaller values of \(T\), generally speaking, one should take into account terms \(\sim\Psi^6\) in \(F_{s0}\), i.e. the terms \(\sim\Psi^6\) in (3.13), and the application of all the formulas obtained without replacing \(\dfrac{|a|}{\beta}\) by

\[ \left(\frac{d\alpha}{dT}\right)_k \frac{(T_k-T)}{\beta_k} \]

is possible only if the nonlinear dependence of \(\dfrac{|a|}{\beta}\) on \(T_k-T\) is more substantial than the influence of the terms with \(\Psi^6\). Such a situation is possible, but it will be justified only on the basis of an analysis of sufficiently extensive experimental data, which we cannot now carry out, owing to the absence of the latter. In view of what has been said, the data of Table II for \(T<3.80^\circ\) at \(d>d_k\) may be distorted.

The second circumstance that must be borne in mind is that \(T_k\) varies appreciably from film to film; in \(^{21}\) all the data were reduced to \(T_k=4.167^\circ\), and this operation, known to be inaccurate at \(T=4.12^\circ\) and \(T=4.13^\circ\), may affect the data of Table II at lower temperatures as well. The whole question clearly requires a detailed experimental investigation; for the present, however, we shall take as \(\delta_0\) the smallest values in Table II and compare them with the data obtained by other methods \(^{17,19}\). In doing so it must be borne in mind that in \(^{19}\) only the quantities \(\delta_0(T)-\delta(2.5^\circ)\) are measured directly, while the quantities \(\delta_0\) are calculated as a result of an extrapolation that is not a priori legitimate. The values of \(\delta_0\) obtained in \(^{17}\) are based on preceding measurements with colloids and are also inaccurate; in the experiment itself

Table III

\(T\)	\(\delta_0\) from Table 2	\(\delta_0-\delta_0(2.5^\circ)\) from Table 2	\(\delta_0\) from \(^{17}\)	\(\delta_0-\delta_0(2.5)\) from \(^{17}\)	\(\delta_0\) from \(^{19}\)	\(\delta_0-\delta_0(2.5^\circ)\) from \(^{19}\)
4.13	3.08	2.42	4.08	3.28	2.28	1.82
4.12	2.72	2.06	3.57	2.77	2.04	1.58
4.10	2.28	1.62	2.80	2.00	1.72	1.26
4.05	1.80	1.14	2.34	1.54	1.31	0.85
4.00	1.56	0.90	1.95	1.15	1.10	0.64
3.80	1.10	0.44	1.38	0.58	0.77	0.31
3.50	0.87	0.21	—	—	0.61	0.15
3.00	0.72	0.06	—	—	0.50	0.04
2.50	0.66	0.00	0.80	0.00	0.46	0.00

and in this case \(\delta_0(T)-\delta_0(2.5^\circ)\) is measured. As is seen from Table III, where all quantities must be multiplied by \(10^{-5}\,\mathrm{cm}\), within the accuracy presently attainable the data of Table II agree with those obtained by other methods (it should be specially emphasized that the data \(^{19}\) refer to massive samples). The values of \(\delta_0(T)-\delta_0(2.5^\circ)\) according to \(^{4,21}\) (curve II), \(^{17}\) (curve I), and \(^{19}\) (curve III) are also shown in Fig. 3.

Taking for \(\delta_0\) the values indicated in the second column of Table II, with the aid of formula (3.14a) and taking into account that near \(T_k\) for mercury \(H_{\mathrm{km}}=187(T_k-T)\), we find \(\chi^2\). In doing so, if we use, as the most reliable, the value of \(\delta_0\) at \(4^\circ\), we obtain the result (3.14b). Using the values of \(\delta_0\) indicated in \(^{19}\) for mercury and for tin, we obtain \(\chi^2 \simeq 0.015\).

Let us turn to the question of the destruction of the superconductivity of a film by a current. For \(\chi=0\), the function \(\Psi'_0\) in the presence of a current is determined by formula (4.24), which for \(d'\ll 1\) takes the form

\[ \Psi_0'^2=1-\frac{H_0'^2 d'^2}{3}-\frac{H_I'^2}{\Psi_0'^4 d'^2}. \tag{4.36} \]

The field \(H_I'\), as a function of \(\Psi'_0\), has a maximum, vanishing at \(\Psi'_0=0\) and at some \(\Psi'_0\ne 0\) (if \(H'_0=0\), then \(H'_I=0\) at \(\Psi'_0=1\)); in other words, for a given \(H'_I\) the function \(\Psi'_0\) may, according to (4.36), have two values. It is easy to see that the superconductivity of the film is stable only so long as the field \(H'_I\) increases as \(\Psi'_0\) decreases (in this case the free energy is smaller than for the same \(H'_I\), but with a smaller \(\Psi'_0\)). The critical field \(H_{I\kappa}\) is determined from the condition

\[ \frac{dH'_I}{d\Psi'_0}=0, \]

which leads to the relation

\[ \left(\frac{H_{I\kappa}}{H_{\kappa \mathrm{M}}}\right) = \frac{2\sqrt{2}}{3\sqrt{3}}\,\frac{d}{\delta_0} \left[ 1-\left(\frac{H_0}{H_\kappa}\right)^2 \right]^{\frac{3}{2}}, \tag{4.37} \]

where \(H_\kappa\) is the critical field for the given film in the absence of current, and \(H_0\) is the external field.

In the absence of the field \(H_0\),

\[ \frac{H_{I\kappa}}{H_{\kappa \mathrm{M}}} = \frac{2\sqrt{2}}{3\sqrt{3}}\,\frac{d}{\delta_0}, \qquad H_{I\kappa}=\frac{2\pi}{c}\,I_\kappa, \tag{4.38} \]

where \(I_\kappa\) is the critical current destroying superconductivity. In the case of an arbitrary orientation of \(H_0\) and \(H_I\), when one must use equations (4.18) with the boundary conditions (4.19), it is easy to see that the previous formulas (4.24), (4.36), and (4.37) are obtained, where \(H_0^2=H_{0x}^2+H_{0y}^2\) (the current \(I\) is directed along the negative \(x\)-axis, and the field \(H_I\) along the \(y\)-axis). In the case of a current flowing along a film on the surface of a circular cylinder, formula (4.38) remains valid, but one must remember that the magnetic field on the outer surface of the film is equal to \(2H_I=\frac{4\pi}{c}I\), where \(I=\frac{J}{2\pi R}\) (\(J\) is the total current through the film and \(R\) is its radius).

Let us note that, as follows from (4.28) and (4.38), for sufficiently thin films

\[ H_\kappa\cdot H_{I\kappa}=\frac{4}{3}\,H_{\kappa \mathrm{M}}^2. \tag{4.39} \]

Thus, while the values \(H_\kappa\) and \(H_{I\kappa}\) themselves for thin films may differ very greatly from \(H_{\kappa \mathrm{M}}\), the product-

The value \(H_k H_{Ik}\), equal for a thick specimen to \(H_{km}^2\), increases for the thinnest films only by a factor of \(4/3\).

Relations (4.37)—(4.38) are in qualitative agreement with experiments\({}^{83}\), from which it does not appear possible to extract quantitative data.

In summary, it may be pointed out that a whole range of possibilities is available for an experimental test of the theory: measurement of the critical field and critical current for films (formulas (4.26)—(4.27), (4.28), and (4.38)), measurement of the influence of a field on the critical current (see (4.37)), measurement of the magnetic moment (see (4.31)—(4.32)), measurement of \(\sigma_{ns}\), and, finally, measurement of \(\delta(H_0)\) for massive superconductors (see (4.11) and (4.20)). At the same time, in working with films, it is practically, apparently, impossible to determine \(\chi\) directly if the value of \(\chi\) is indeed small. Therefore, in order to determine \(\chi\), without using formula (3.14a), it is necessary either to determine \(\sigma_{ns}\)—a quantity especially sensitive to \(\chi\)—or to carry out precise measurements (with an accuracy up to \(\sim 1\%\)) of \(\delta\) as a function of \(H_0 \sim H_k\) for massive superconductors.

A comprehensive experimental verification of the new phenomenological theory is, of course, absolutely necessary. However, even now one may state that all the available data speak in its favor, and at the same time the new theory, unlike the old one, is free of internal contradictions (in particular, as we have seen, the new theory leads to a positive surface tension \(\sigma_{ns}\)). Therefore, until experimental data appear that compel one to doubt the validity of the equations obtained, which ultimately determine the field and current in a superconductor, one may consider that the phenomenological theory of superconductivity has essentially been constructed.

(To be concluded in the next issue)