Macroscopic Interpretation of Superconductivity

F. London (Oxford)

The theory of which I shall speak deals exclusively with the macroscopic interpretation of superconductivity. Apparently, the chief obstacle that stands in the way of understanding this phenomenon should be sought in the generally accepted macroscopic interpretation of it as a certain limiting case of ordinary conductivity.

The present state of the theory may be characterized in such a way that the impossibility of constructing a theory of superconductivity on the basis of the generally recognized ideas of the electron theory of metals and with the usual interpretation of the observed phenomenon becomes quite obvious.

Bloch and Landau formulated a theoretical program which, apparently, is prescribed by the facts. They believe that one should seek such a model of a metal which, in its most stable ...

state would include a permanent current in the absence of any external field whatsoever. In this connection ferromagnets are often cited as an analogy, since in the most stable state they possess permanent magnetization. The thermodynamic stability of the superconducting state and, in particular, the stability of non-decaying persistent currents, apparently admits of no other conclusions. However, as an objection to such a representation there was adduced (including by Bloch himself) a general theorem of electron theory, according to which in the most stable state of any system formed from electrons the presence of a current appears very unlikely. Hence Bloch concluded that the only theory that can be proved for superconductivity consists in the fact that any theory of superconductivity can be refuted. Up to now this theorem has always been confirmed by experiment.

We shall see that the “insoluble” problem of Landau and Bloch in fact was never posed by nature, and that the conclusions made on the basis of experimental facts were somewhat premature. We shall show that this phenomenon need not at all be interpreted as a limiting case of high conductivity, despite the fact that an electric current exists in the absence of an electric field \((J \ne 0, E = 0)\), and that the observed phenomenon can be given a formulation which eliminates the contradiction with generally accepted physical notions.

The experiment of Meissner and Ochsenfeld revealed to us new and quite unexpected properties of the superconducting state. From the fact of infinite conductivity it follows only that the magnetic flux in a superconductor must be constant and, consequently, must depend on how the superconductor passes through the critical curve. However, the experiments of Meissner showed, in addition, that the magnetic flux in a superconductor is in all probability equal to zero, provided only that the experiment is carried out under “ideal” conditions. What is meant by “ideal” conditions is not so easy to determine. The fact that under “non-ideal” conditions the magnetic field proved to be frozen into the superconductor is connected, apparently, with the presence of non-superconducting inclusions, or with the fact that the substance at the very beginning was insufficiently homogeneous, or else with the fact that, on condensation, the magnetic lines in some parts of the superconductor give a field exceeding the critical value.

Thus the preservation of the initial magnetic flux should be regarded not as a certain elementary phenomenon, but as a complex effect caused by the presence of several components or phases, which are taken into account in a microscopic treatment. In contrast to this, the elementary phenomenon in a pure superconductor should be regarded as a much simpler effect. According to Meissner’s experiments, the transition from the non-superconducting phase to the superconducting phase, carried out in a magnetic field, is, apparently, microscopically reversible, since the magnetic flux may be considered equal to zero in any element of volume of the superconducting phase, irrespective of how the transition through the critical curve was made. The magnetic behavior of a superconductor recalls the behavior of a metal with very strong diamagnetic properties, possessing a susceptibility \(\chi = -\dfrac{1}{4\pi}\) or a permeability \(\mu = 0\).

In a diamagnetic atom we find precisely an example of a permanent current flowing in a system that is in its most stable state. There is no contradiction here with Bloch’s theorem, since this theorem relates to systems without an imposed external electric or magnetic field. We see that in a magnetic field this theorem would obviously not be satisfied.

It seems very tempting to consider the current in a superconductor as a kind of diamagnetic current. This idea, timidly

which was expressed earlier, acquires a special appeal now after Meissner’s experiment, which apparently gives us a more elementary effect into which one may hope to reduce the still so enigmatic phenomenon of superconductivity.

The macroscopic description proposed by me jointly with H. London^26 shows that one can work out a program which is to some extent free from Bloch’s dilemma. Here the superconducting current manifests itself as a diamagnetic current maintained by a magnetic field. In Meissner’s experiment this field is the external magnetic field. In a permanent current circulating in a ring, the magnetic field is created by the current itself. In the most stable state of the ring there are no currents until a magnetic field is applied. The state in which there is a permanent current in the ring is not a state of lowest energy, but under macroscopic conditions it is metastable.

We propose the relation between the magnetic field \(\mathbf{H}\) and the current density in the superconducting state \(\mathbf{J}\) in the form of the equation:

\[ c\,\operatorname{rot}\Lambda \mathbf{J}=-\mathbf{H}, \tag{1} \]

where \(\Lambda\) is a certain positive constant characteristic of the superconductor.

Thus we characterize a superconductor not by some value of the permeability \(\mu\), but by a differential equation. As we shall see, this means that the entire superconductor is regarded as one large diamagnetic atom, but the screening of the applied magnetic field is effected by volume currents and not by atomic magnetization.

Using Maxwell’s equation

\[ \operatorname{rot}\mathbf{H}=\frac{1}{c}\mathbf{J} \tag{2} \]

(we neglect here the displacement current and, eliminating \(\mathbf{J}\) from (1) and (2), obtain

\[ \Lambda c^2 \operatorname{rot}\operatorname{rot}\mathbf{H}=-\mathbf{H} \]

or, since \(\operatorname{div}\mathbf{H}=0\),

\[ \Lambda c^2 \Delta \mathbf{H}=\mathbf{H}. \tag{3} \]

Solutions of this equation that are regular inside the superconductor decrease exponentially as one moves inward from the surface, while at the surface they pass into the values of the external field. At a distance \(c\sqrt{\Lambda}\) from the surface the field is practically equal to zero. As we shall see, the microscopic interpretation gives for this distance a value of about \(10^{-5}\) cm. Thus Meissner’s experiments fit equation (1) with one quite natural distinction: the magnetic flux disappears not at the surface, but within a very thin surface layer. For simplicity we take the magnetic induction \(\mathbf{B}\) everywhere equal to the field strength

\[ \mathbf{B}=\mathbf{H}. \tag{4} \]

Taking \(\operatorname{rot}\) of (3), we obtain the analogous equation for \(\mathbf{J}\)

\[ \Lambda c^2 \operatorname{rot}\operatorname{rot}\mathbf{J}+\mathbf{J}=0. \tag{5} \]

The screening diamagnetic currents always flow very close to the surface.

Differentiating (1) with respect to time and taking into account that \(\dot{\mathbf{H}}=-c\,\operatorname{rot}\mathbf{E}\), we obtain

\[ \operatorname{rot}(\Lambda \dot{\mathbf{J}}-\mathbf{E})=0, \]

i.e. \(\Lambda \mathbf J-\mathbf E\) must be the gradient of a certain scalar \(\varphi\)

\[ \Lambda \mathbf J-\mathbf E=\operatorname{grad}\varphi . \tag{6} \]

For what follows it is only necessary to know that \(\varphi\) is a single-valued scalar. Initially we assumed also that \(\varphi\) is equal to the product of the electric charge by \(-\Lambda c\). Then equations (1) and (6) could be given the usual symmetric form

\[ \Lambda c\cdot\left(\frac{\partial J_i}{\partial x_k}-\frac{\partial J_k}{\partial x_i}\right)=f_{ik}, \]

according to which the electric field, like the magnetic field, must, starting from the surface, decrease to zero. The a priori fact of such slight penetration of the electric field apparently cannot meet with objections. However, since in some processes such electric fields lead to a negative value of Joule heat, we must proceed to consequences with which it is in no way permissible to agree\({}^{27}\). True, by choosing appropriate boundary conditions for the electric field, we can satisfy the requirements of thermodynamics.

On the other hand, the theory can be given a somewhat different formulation, in which electric fields will be excluded in principle. For everything that follows, these details concerning the behavior of electric fields are of no interest, and without bringing in new experimental facts about them no final conclusions can be drawn.

Let us now consider the problem of a persistent current, or, more precisely, the problem of a persistent magnetic flux. For this we need a superconductor in the form of a ring, since a superconductor of ordinary form does not give an appreciable magnetic flux. Let \(C\) be a closed curve passing inside the ring which cannot be contracted to a point without leaving the superconductor. The surface \(S\), bounded by the curve \(C\), lies partly inside the superconductor and partly in the nonsuperconducting region. But Maxwell’s equations are valid throughout all space, and therefore, for the change of the magnetic flux penetrating the surface \(S\), we can write

\[ \iint_S \dot H_n\,d\sigma=-c\oint_C E_s\,ds . \]

The integral on the right-hand side extends only over the superconducting region. Therefore, using (6), we can replace \(E_s\) by

\[ \Lambda J_s-\frac{\partial\varphi}{\partial s}. \]

Then we obtain

\[ \iint \dot H\,d\sigma = -c\oint\left(\Lambda J_s-\frac{\partial\varphi}{\partial s}\right)\,ds = -c\Lambda\oint J_s\,ds, \]

or, integrating with respect to time,

\[ \iint H_n(t)\,d\sigma+c\Lambda\oint J_s(t)\,ds= \]

\[ = \iint H_n(t_0)\,d\sigma+c\Lambda\oint J_s(t_0)\,ds=\text{const}. \tag{7} \]

Thus the quantity \(\iint H_n\,d\sigma + c\Delta \oint J_s\,ds\) is invariant in time. In consequence of the relation \(\Delta c\,\operatorname{rot}\mathbf J=-\mathbf H\), applying Stokes’ theorem, we may find that the value of this quantity does not depend on the position of the curve \(C\) inside the ring. But the ring can always be made so thick, and the curve \(C\) placed so far from its surface, that the value of the current density, which decreases exponentially inward into the ring, may be neglected. Then we obtain

\[ \iint H_n(t)\,d\sigma=\iint H_n(t_0)\,d\sigma . \tag{7'} \]

The magnetic flux contained in the cavity of the ring remains constant at all times, irrespective of any disturbances of the external field. The ring behaves like a permanent magnet. However, since it possesses a certain amount of magnetic energy of this flux, it should be regarded as being in a metastable state. Only by a finite change of the parameters of the system (for example, by passing through the critical value) can it be transferred into an absolutely stable state containing no flux.

It is necessary to emphasize that our purely magnetic description differs fundamentally from the conception sometimes advanced, according to which superconductivity should be characterized by the particular value of the magnetic permeability \(\mu=0\). It is true that for superconductors of ordinary form, in the case of purely magnetic phenomena, both formulations give macroscopically identical results. However, for multiply connected superconductors these formulations differ sharply from one another. A ring characterized by the given value of the permeability has in its cavity a field proportional to the intensity of the applied external field, and in the particular case it is necessarily equal to zero if the external field is absent. From the single fact that \(\mu=0\), one cannot obtain the stability of an electric current.

open). However, of course, in reality the eigenfunctions of the electrons in a metal are strongly disturbed by a magnetic field, and consequently the expressions in brackets do not vanish. On the contrary, they become, in order of magnitude, equal to the terms containing the vector potential. Moreover, the energy values of the electronic states are perturbed, and their thermal distribution changes as a result. As a result we obtain only the very weak so-called Landau–Peierls diamagnetism.

But suppose that the electrons interact with one another in such a way that the very lowest state is separated from the excited states by a finite interval. Then the perturbing action of the field on the eigenfunctions may be considerable only if this perturbation is of the same order of magnitude as the forces of interaction of the electrons. As long as the magnetic field remains small, one may assume that the perturbation of the eigenfunction is proportional to the square or a higher power of the magnetic-field intensity. We shall write this in the form:

\[ \psi=\psi_0+\mathbf H^2\psi_1. \]

Here \(\psi_0\) is the eigenfunction in the absence of a field. With such an eigenfunction the brackets in (10) give us an expression, quadratic

In multiply connected superconductors (forming a completely superconducting contour, for example, in rings) this is, generally speaking, impossible. If, in the presence of a magnetic flux \(F\) through the space enclosed by them, one integrates along a curve enclosing this cavity, we obtain

\[ \oint A_s\,ds=\iint H_n\,d\sigma=F\ne 0. \]

If, however, we choose the path of integration sufficiently deep inside the superconductor, then we shall have

\[ \oint J_s\,ds=0. \]

Thus \(\Lambda cJ\) and \(-A\), although having equal curls, must differ from one another by the gradient of a multivalued scalar \(\nu\), whose increment on going around is equal to the magnetic flux through the cavity,

\[ \Lambda cJ + A = \operatorname{grad}\nu . \tag{9} \]

However, in this case too, by generalizing our formula for simply connected conductors, \(A+\Lambda cJ=0\), we can choose a definite expression for \(A\). In addition to the equations \(\operatorname{rot}(A+\Lambda cJ)=0\) and \(\operatorname{div}(A+\Lambda cJ)=0\) inside the superconductor, we postulate that on the surface

\[ A_n+\Lambda cJ_n=0. \]

This gives the following conditions for \(\nu\):

\[ \Delta \nu=0 \]

inside the superconductor,

\[ \frac{\partial \nu}{\partial n}=0 \]

on its surface.

According to the condition that the increment of \(\nu\) after going around the \(i\)-th cavity must be equal to the magnetic flux \(F_i\) through this cavity, or, more precisely, equal to the value of the quantity conserved in time,

\[ \iint H_n\,d\sigma+\oint \Lambda cJ_s\,d_s \]

for this cavity, \(\nu\) is determined in a completely definite way and proves to be independent of time.

Equations (8) and (9) are equivalent to the basic equation (1).

No justification of our macroscopic equations with the aid of the theory of electrons in a metal has yet been undertaken. However, we would like to outline the research program following from them in a microscopic approach.

As is known, quantum mechanics gives for the density of the electric current the expression

\[ J=\frac{he}{4\pi im}\left(\psi\,\operatorname{grad}\psi^*-\psi^*\,\operatorname{grad}\psi\right)-\frac{e^2}{mc}\psi\psi^*A. \tag{10} \]

If \(\psi\) is the wave function of an individual electron in the self-consistent field of the remaining electrons, then \(\psi\psi^*\) gives the statistical probability of finding this electron at each point of space. Summing over all electrons, we obtain the number \(n\) of electrons in \(1\ \mathrm{cm}^3\), \(\sum \psi\psi^*\).

If ordinary eigenfunctions of the free electron in a metal are taken, then the expression in parentheses, when summed over all electrons, vanishes for reasons of symmetry, and equation (10) becomes identical with equation (8), the constant \(\Lambda\) being equal to

\[ \frac{m}{ne^2} \]

(we leave aside the question of the exact value of \(n\)

open). However, of course, in reality the eigenfunctions of the electrons in a metal are strongly perturbed by the magnetic field, and, consequently, the expressions in brackets do not vanish. On the contrary, they become, in order of magnitude, equal to the terms with the vector potential. Moreover, the energy values of the electronic states experience perturbation, and their thermal distribution changes as a result. Consequently we obtain only the very weak so-called Landau–Peierls diamagnetism.

But let us suppose that the electrons interact with one another in such a way that the lowest state is separated from the excited states by a finite interval. Then the perturbing action of the field on the eigenfunctions may be significant only if this perturbation is of the same order of magnitude as the forces of interaction of the electrons. So long as the magnetic field remains small, it may be assumed that the perturbation of the eigenfunction is proportional to the square or a higher power of the magnetic-field strength. We shall write this in the following form:

\[ \psi=\psi_0+\mathbf{H}^2\psi_1. \]

Here \(\psi_0\) is the eigenfunction in the absence of the field. With such an eigenfunction the brackets in (10) give an expression quadratic in \(\mathbf{H}\), which may be neglected in comparison with the term containing \(\mathbf{A}\). Thus the relation (8) between the current and the magnetic field proposed by us could be explained by this “rigid” behavior of the eigenfunctions.

Meanwhile, the mechanism considered is by no means absolutely new. This mechanism characterizes a superconductor taken as a whole as one large diamagnetic atom. In a diamagnetic atom placed in a magnetic field, too, the eigenfunction undergoes a perturbation proportional only to \(\mathbf{H}^2\), which in no way can explain diamagnetism. The diamagnetic current is completely expressed by the term contained in (10),

\[ \frac{e^2}{mc}\psi\psi^*\mathbf{A}, \]

where \(\psi\) is the unperturbed eigenfunction. The moment of this current directly gives the known formula for the diamagnetic susceptibility of the atom. But ordinary atoms are so small that the external magnetic field is only weakly screened by the internal one. Only if the atom had a diameter \(\gtrsim 10^{-5}\) cm (which corresponds to an electron density of \(10^{23}\ \mathrm{cm}^{-3}\)) would the external field almost completely fail to penetrate inside it.

It is still necessary to consider separately a superconductor having the form of a ring. In this case, instead of equation (8), we have equation (9)

\[ \Lambda c\mathbf{J}+\mathbf{A}=\operatorname{grad}v, \tag{9} \]

where the term \(\operatorname{grad}v\) is needed to describe the magnetic flux through the ring.

Let us consider an atom having the form of a ring, with magnetic flux \(F\) passing through its opening. We shall first suppose that this flux occupies a very small region of the opening, so that it nowhere touches the ring itself. Since for any closed curve enclosing the magnetic flux, in particular also for one which passes inside the ring, the relation

\[ \oint A_s\,ds=F, \]

must hold, the vector potential \(\mathbf{A}\) of this magnetic field will inevitably be different from zero everywhere, even in the ring, although in fact there is no field there. Therefore the wave equation of the electron ring will inevitably contain a vector potential, despite the absence of a magnetic field in it. The presence of such a physically unreal vector potential, which can always be pre-

take it to be the gradient of some many-valued scalar, we compensate by an “adaptation” of the proper function. In that case
\(\mathbf{A}=\operatorname{grad}\nu\), and the correctly “adapted” proper function has the form

\[ \psi=\psi_0 e^{-\frac{2\pi i}{h}\frac{e}{c}\nu}, \]

where \(\psi_0\) is the proper function of the electron ring in the absence of flux. One can verify directly that such a \(\psi\) is an exact solution of the wave equation including the physically real vector potential \(\mathbf{A}=\operatorname{grad}\nu\), and that the current obtained from equation (10) with such a \(\psi\) vanishes if the current is not given by \(\psi_0\) itself.

Let us now suppose that the magnetic flux penetrates slightly into the ring, but that it is so weak that nowhere does it exceed the critical value. Then the vector potential may be written in the following form:

\[ \mathbf{A}=\operatorname{grad}\nu+\mathbf{A}_1 . \]

But the term \(\mathbf{A}_1\) represents only a weak additional field, which perturbs the proper function quite insignificantly, as we also assumed in the general case of weak magnetic fields. Therefore it alone gives the current in (10)

\[ \mathbf{J}=-\frac{e^2}{mc}\sum \psi\psi^{*}\mathbf{A}_1 = \frac{1}{\Lambda c}(\operatorname{grad}\nu-\mathbf{A}). \]

This coincides exactly with our fundamental relation for the ring (9). Thus the nondamping currents in the ring should in fact be regarded as a diamagnetic effect stabilizing itself.

Thus, the future foundation of this theory, proceeding from consideration of the electrons in the metal, should apparently attempt to explain an entirely simple state of affairs, namely, that when a superconductivity current is produced or maintained, practically nothing occurs; in any case the wave functions of the superconducting state undergo no changes. More precisely, it would be sufficient to show that the wave function in this state, under the action of a magnetic field, undergoes a perturbation proportional only to the higher powers of \(\mathbf{H}\). For the wave description of matter it is very characteristic that waves do not directly express the motion of a particle in the presence of magnetic fields. The de Broglie wave number determines the total momentum \(p\), which in the case of a magnetic field is composed of the “kinetic” momentum \(m\mathbf{v}\) and the “potential” momentum \(\frac{e}{c}\mathbf{A}\). If in a degenerate electron gas the first of these is equal to zero and, by virtue of the presence of an interaction which still remains to be explained, remains equal to zero also in the presence of a magnetic field, then the current is proportional, with the opposite sign, to the potential momentum, and our equations are valid.