Current State of the Theory of Superconductivity

Yu. B. Rumer, Moscow

§ 1. Introduction

The theory of superconductivity faces two tasks. First, to give a phenomenological picture of the phenomenon, i.e. to elucidate the quantitative laws characterizing superconductivity; second, to establish the mechanism responsible for the appearance of superconductivity.

Whereas all attempts to find a way toward solving the second task have so far proved unsuccessful, on the path toward solving the first task we have recently had considerable successes.

Since in 1911 Kamerlingh-Onnes1 first discovered that at a temperature of \(4.22^\circ\mathrm{K}\) mercury becomes superconducting, the phenomenon of superconductivity has been found in the following elements (Table 1).

Table 1

It is still unclear whether superconductivity is a property only of a definite group of metals, or whether it appears in any conductor upon sufficiently strong cooling. In addition, superconductivity has been found in a whole series of alloys; however, the phenomena in them prove to be considerably more complicated, and we shall not touch on them in our review, restricting ourselves only to pure superconductors.

In what follows we shall briefly dwell only on those experimental facts that are of importance for the modern theory of superconductivity. The reader will find a more detailed exposition in the articles by W. Keesom and W. Meissner[^2].

§ 2. Magnetic Properties of Superconductors

Experience shows that superconductors possess very peculiar magnetic properties.

In 1933 Meissner and Ochsenfeld[^3], and independently of them L. V. Shubnikov[^4], discovered that the magnetic induction \(B\) inside a superconductor changes upon the transition to the superconducting state. They placed a cylinder of tin in a perpendicular magnetic field and cooled it below the transition temperature into the superconducting state. As long as the cylinder was in the normal state, the magnetic lines had the form shown in Fig. 1a. As soon as it passed into the superconducting state, a change occurred in the magnetic lines, and a distribution was obtained (Fig. 1b) approximately coinciding with that which is obtained if the field is calculated on the assumption that at all points inside the superconductor the magnetic induction \(B\) is equal to zero.

Fig. 1.

a — In the absence of superconductivity
b — In the superconducting state

Fig. 2.

The transition from the normal to the superconducting state and back can be carried out not only by changing the temperature, but also, at constant temperature, by changing the external magnetic field. If a given rod is in the superconducting state, then its resistance is restored when a sufficiently strong external longitudinal field \(H_{\mathrm{cr}}\) is applied. The lower the temperature, the greater the field strength required to restore the normal state. The curve of the dependence of \(H_{\mathrm{cr}}\) on temperature for tin is given in Fig. 2.

The transition of a superconductor to the normal state also occurs when a sufficiently strong current is passed through it. As Silsbee[^5] first noted, the transition takes place at the moment when the magnetic field of the current reaches its critical value at the surface of the conductor. Denoting by \(r\) the radius of the conductor and by \(j_{\mathrm{cr}}\) the critical current strength, we obtain the relation \(\frac{2j_{\mathrm{cr}}}{r}\), well confirmed by experiment.

With the reverse reduction of the external magnetic field below the critical value, hysteresis phenomena are observed. Fig. 3 shows a diagram obtained by L. V. Shubnikov for tin. We see that before the critical value of the external field is reached, the magnetic induction is equal to zero, taking on, at the transition point, its normal value \(B = H\) by a jump. In the reverse transition we observe a certain residual magnetic induction, which decreases only gradually to zero.

§ 3. Thermodynamics of Superconductors

As early as 1911, at the Solvay Congress, P. Langevin expressed the idea that the transition from the normal state to the superconducting state is a phase transition. This idea has now received full confirmation.

Fig. 3.

Fig. 4.

The experiments of Keesom and Ende\(^{6}\) and of Keesom and Kok\(^{7}\) revealed a jump in the heat capacity upon transition from the normal state to the superconducting state (Fig. 4). Rutgers gave a qualitative and quantitative

explanation of this jump, proceeding from the conception of a reversible transition from one phase to another.

Proceeding from the conception of two phases, superconducting and normal, let us denote the thermodynamic potentials of both phases by \(\Phi_s\) and \(\Phi_n\). At a temperature below the critical one, \(T_0\), the first phase is stable, \(\Phi_s < \Phi_n\) (Fig. 5).

Consider a long rod in an external longitudinal magnetic field \(H_a\). The field strength \(H\) inside the superconductor will be \(H = H_a\), since the tangential component of the vector \(\mathbf H\) is continuous. When the magnetic field \(H_a\) is switched on, the magnetic moment per unit volume of the superconductor will be (since \(B = H + 4\pi M = 0\)) \(M = -\dfrac{1}{4\pi}H\). Consequently, when the magnetic field is switched on, the energy of the superconductor increases. The additional energy per unit mass of the superconductor is equal to

\[ -\frac{1}{\rho}\int_0^H M\,dH=\frac{H^2}{8\pi\rho}, \]

where \(\rho\) is the density. The thermodynamic potential per unit mass of the superconductor placed in a magnetic field will therefore be

\[ \Phi_s+\frac{1}{8\pi\rho}H^2 . \]

At the transition point the equality must hold

\[ \Phi_s+\frac{1}{8\pi\rho}H_{\mathrm{cr}}^2=\Phi_n . \]

Fig. 5.

According to the known formulas of thermodynamics, we obtain for the difference of the heat capacities in the superconducting and normal phases

\[ C_s-C_n=-T\frac{d^2}{dT^2}(\Phi_s-\Phi_n), \]

whence, substituting \(\Phi_n-\Phi_s=\dfrac{1}{8\pi\rho}H_{\mathrm{cr}}^2\), we obtain Rutger’s formula

\[ C_s-C_n = T\frac{d^2}{dT^2}\left(\frac{H_{\mathrm{cr}}^2}{8\pi\rho}\right) = \frac{T}{4\pi\rho} \left\{ \left(\frac{dH_{\mathrm{cr}}}{dT}\right)^2 + H_{\mathrm{cr}}\frac{d^2H_{\mathrm{cr}}}{dT^2} \right\}. \]

Since at the transition point \(H_{\mathrm{cr}}=0\), from Rutger’s formula we finally obtain

\[ C_s-C_n = \frac{T}{4\pi\rho} \left(\frac{dH_{\mathrm{cr}}}{dT}\right)^2_{T=T'} . \]

In Table 2 the results of experiments and calculations are compared; they agree excellently with one another. Thus the conception of two

Table 2

in phases and of a reversible transition from one phase to another is fully confirmed.

The heat of transition from the superconducting state to the normal state in the presence of a magnetic field is determined by the formula

\[ Q=T(S_n-S_s), \]

where \(S_s\) and \(S_n\) denote the entropy of the superconducting and normal states. By the formulas of thermodynamics we have

\[ Q=T\frac{d}{dT}(\Phi_s-\Phi_n)=-\frac{T}{4\pi\rho}H_{\mathrm{cr}}\frac{dH_{\mathrm{cr}}}{dT} \]

and obtain a numerical dependence between the heat of transition and the course of the dependence of the critical magnetic field on temperature. Keesom’s measurements revealed, in this case as well, agreement between theory and experiment.

§ 4. Intermediate State

The phenomenon becomes somewhat more complicated if we pass from long rods to a body of arbitrary shape. As experiment shows, in this case we have not two but three states:

1) superconducting, for which \(\mathbf{B}=0\), 2) an intermediate state, for which \(\mathbf{B}\) is different from zero but has not yet reached its normal value, and, finally, 3) the normal state, for which \(B=H^{1)}\). In order to explain the dependence of the magnetic properties of a superconductor on its geometrical shape, one must take into account the demagnetizing factor.

Demagnetizing factor. If a body having the shape of an ellipsoid is placed in an external magnetic field \(H_a\), directed along one of its axes, then inside the body the magnetic-field strength \(\mathbf{H}\) and the magnetic induction \(\mathbf{B}\) will be expressed by

\[ \left. \begin{aligned} \mathbf{H} &= \mathbf{H}_a-4\pi n\mathbf{M},\\ \mathbf{B} &= \mathbf{H}_a-4\pi n\mathbf{M}+4\pi\mathbf{M} = \mathbf{H}_a+4\pi(1-n)\mathbf{M} \end{aligned} \right\} \tag{*} \]

\(^{1)}\) In what follows we shall always assume that in the intermediate state the magnetic permeability \(\mu=1\).

where \(M\) is the magnetization, and \(n\) is the so-called demagnetizing factor1

\[ n=\frac{abc}{2}\int_{0}^{\infty} \frac{d\tau}{(a^{2}+\tau)\sqrt{(a^{2}+\tau)(b^{2}+\tau)(c^{2}+\tau)}} . \]

\(a, b, c\) are the semiaxes of the ellipsoid; the magnetic field is directed along the axis \(a\).

We are interested in special cases:

1. A circular long cylinder in a longitudinal field

\[ a\gg b;\qquad b=c;\qquad n=0. \]

1. A circular long cylinder in a transverse field

\[ c\gg b;\qquad b=a;\qquad n=\frac{1}{2}. \]

1. A thin plate in a transverse field

\[ b\gg a;\qquad b=c;\qquad n=1. \]

1. A sphere

\[ a=b=c;\qquad n=\frac{1}{3}. \]

Eliminating \(M\) from equations (*), we obtain the condition which \(H\) and \(B\) inside the body must satisfy if the external field is equal to \(H_a\)

Fig. 6.              Fig. 7.

\[ B_n+H(1-n)=H_a. \]

Let us now see how \(H\) and \(B\) are related to each other in a superconductor. From experiments with a long rod in a longitudinal field we have found that, since for it

\[ B=0 \quad \text{for } H<H_{\mathrm{cr}}, \]

\[ B=H \quad \text{for } H>H_{\mathrm{cr}}, \]

and, consequently, the dependence of H on B is represented by the discontinuous function \(H=f(B)\), shown in Fig. 6.

In order to clarify the course of the curve \(H=f(B)\) in the region \(0<B<H_{\mathrm{cr}}\), let us consider a body with a nonzero demagnetizing factor. Experiment shows the following (Fig. 7). Up to some value of the external field \(H_a\), less than the critical one, we have \(B=0\), i.e., the superconducting state. Then, as the external field is increased to the critical value, the induction B is different from zero and grows linearly, reaching its normal value at \(H_a=H_{\mathrm{cr}}\). We have before us the so-called intermediate state. Finally, for \(H_a>H_{\mathrm{cr}}\) the specimen passes into the normal state. Substituting into the equation

\[ B_n+H(1-n)=H_a \]

the function \(H=f(B)\), we have:

\[ B_n+f(B)(1-n)=H_0 \]

and from the linear increase of \(B\) observed experimentally in the intermediate state we conclude that in it as well \(f(B)\) will be a linear function. Consequently, the course of the dependence \(H=f(B)\) is represented by the broken curve \(OABC\) (Fig. 8). Intersecting it with the straight line

\[ B_n+H(1-n)=H_a, \]

which cuts off on the axes H and B, respectively, the segments \(\dfrac{H_a}{1-n}\) and \(\dfrac{H_a}{n}\), we obtain the values of \(H\) and \(B\) corresponding to the external field \(H_a\). For example, in the case of a long rod in a transverse field,

\[ n=\frac{1}{2}, \]

and we have

\[ \frac{H_a}{\dfrac{1}{2}}<H_{\mathrm{cr}} \quad \text{superconductor} \]

\[ \frac{1}{2}H_{\mathrm{cr}}<H_a<H_{\mathrm{cr}} \quad \text{intermediate state} \]

\[ H_a>H_{\mathrm{cr}} \quad \text{normal state} \]

In the case of a sphere,

\[ n=\frac{1}{3}, \]

and we have

\[ \frac{H_a}{\dfrac{2}{3}}<H_{\mathrm{cr}} \quad \text{superconductor} \]

\[ \frac{2}{3}H_{\mathrm{cr}}<H_a<H_{\mathrm{cr}} \quad \text{intermediate state} \]

\[ H_a>H_{\mathrm{cr}} \quad \text{normal state}. \]

Fig. 8.

In the case of a plate \(n=1\), we have no superconducting state at all. Even an infinitesimal external field brings the superconducting state into an intermediate one, which passes into the normal state at \(H_a=H_{\mathrm{cr}}\).

§ 5. The nature of the intermediate state

What, then, is the nature of the intermediate state? Are we dealing with a third phase of matter, distinct from the superconducting and the normal? An answer to this question is given by one of L. D. Landau’s recent papers.\(^{10}\)

Let us consider a plate in a transverse magnetic field. Landau shows that if \(H_a\) lies in the interval \(0 < H_a < H_{\mathrm{cr}}\), then the thermodynamically stable state will be one in which the specimen is divided into alternating superconducting and normal layers, separated from one another by some distance \(a\). Let us examine these layers more closely. The lines of magnetic induction do not penetrate the superconducting layers, but pass only through the normal layers. At the boundary of two layers the normal component of the magnetic-field intensity is zero (as a consequence of the continuity of the normal components of the vector \(B\)). The tangential component of the vector field intensity \(H\) is, evidently, equal to the value of the critical field \(H_{\mathrm{cr}}\); otherwise part of the substance would pass from one phase into the other. These conditions make it possible to solve completely the problem of the shape of the layers for a given distance between them.

Fig. 9.

Fig. 10.

This problem is mathematically equivalent to the following problem of hydromechanics. A fluid stream, having at infinity the velocity \(H_a\), flows around the piers of a bridge situated at a distance \(a\) from one another. It is required to determine the shape of the piers such that the fluid flows around the piers with the given tangential velocity \(H_{\mathrm{cr}}\). The problem is solved, like other plane problems of hydromechanics, by the method of the theory of functions of a complex variable.

In order to determine the distance \(a\) between the separate superconducting layers, let us note that two factors influence this quantity:

1. Increasing the number of layers (i.e., decreasing the distance) leads to an increase in the capillary energy, which is energetically unfavorable.

2. Decreasing the number of layers (i.e., increasing the distance between layers) increases the curvature of the magnetic lines when they pass into the conductor, which, as calculation shows, causes an increase in the energy of the conductor.

Both these factors act in opposite directions, and the stable state will be the one in which the distance between layers corresponds to the minimum of the energy.

Calculation shows that the magnitude of the capillary energy is inversely proportional to the distance between layers, whereas the additional energy due to the curvature of the magnetic lines increases directly in proportion to the distance. Consequently, the dependence of the thermodynamic potential on \(a\) will have the form

\[ \Phi = \Phi_0 + \frac{A}{a} + B a . \]

The condition for a minimum of \(\Phi\) gives

\[ \Phi'(a) = 0; \qquad -\frac{B}{a^2} + C = 0, \]

whence we obtain for \(a\) the value

\[ a = \sqrt{\frac{B}{C}} . \]

References

1. Camerling-Onnes, Proc. Amster. Acad., 27, 75, 1911.
2. V. Keesom, Uspekhi fizich. nauk, 15, 181, 1935; V. Meissner, Uspekhi fizich. nauk, 15, 207, 1935.
3. Meissner u. Ochsenfeld, Naturwiss., 21, 787, 1933.
4. Schubnikow, Nature, 134, 286, 1934.
5. Silsbee, J. Wash. Acad. Sci., 6, 597, 1916.
6. Keesom a. v. Ende, Proc. Amst. Acad. Sci., 35, 143, 1932.
7. Keesom a. Kok, Physica, 1, 175, 1934.
8. Rutgers, Proc. Amster. Acad., 36, 153, 1933.
9. R. Peierls, Proc. Roy. Soc., London, 155, 613, 1936.
10. Landau. Phys. Z. d. Sowjetun., 11, 129, 1937.

1. See, for example, Ya. I. Frenkel, Electrodynamics, vol. II, p. 488. ↩↩