Current State of the Theory of Superconductivity*

V. L. Ginzburg

§ 5. Normal Current in Superconductors

As was indicated in § 2, a normal current \(\mathbf{j}_n\) in a superconductor can arise only in a nonstationary field or in the presence of a temperature gradient.

Let us first consider the second of these cases\({}^{2,67}\). If, in a nonsuperconductor, in addition to the electric field there is a temperature gradient \(\nabla T\), then the current density is proportional not only to the electric field \(\mathbf{E}\), but also to \(\nabla T\), i.e. \(\mathbf{j}=\sigma \mathbf{E}+b\cdot\nabla T\). The thermoelectromotive force obtained in this way appears in any model of a metal that, in the absence of \(\nabla T\), leads to Ohm’s law \(\mathbf{j}=\sigma \mathbf{E}\). Therefore there seems to be no reason to doubt that, in a superconductor as well, in the presence of \(\nabla T\), equation (2.5) must be replaced by the following:

\[ \mathbf{j}_n=\sigma \mathbf{E}+b\nabla T, \tag{5.1} \]

where \(b(T)\) is a certain coefficient. As for the superconducting current, it follows from all experimental data that it is not connected with heat transport, i.e. is not connected with an entropy flow and, like the flow of the superfluid part of helium II, is reversible. Therefore the presence of a temperature gradient does not make it necessary to change equations (2.6) and (2.8) for the superconducting current, and we must still assume that

\[ \frac{d\Lambda\mathbf{j}_s}{dt}=\mathbf{E}. \]

It follows from this that in the stationary case \(\mathbf{E}=0\) and, therefore, according to (5.1),

\[ \mathbf{j}_n=b\cdot\nabla T. \tag{5.2} \]

Taking account of the possible presence of a gradient of the chemical potential does not change this conclusion and leads only to the universal replacement

* Continuation. See UFN, vol. XLII, issue 2, p. 169.

\(\mathbf E\) by \(\mathbf E-\nabla \dfrac{\mu_0}{e}\), where \(\mu_0\) is the chemical potential in the absence of a field and \(e\) is the elementary charge.

The normal current (5.2), flowing in a nonuniformly heated superconductor, usually cannot be observed from the magnetic field that it creates, since it is compensated by the superconducting current

\[ \mathbf j_s=-\mathbf j_n=-b\nabla T . \tag{5.3} \]

Indeed, under condition (5.3), in full agreement with experiment, the total current \(\mathbf j=\mathbf j_s+\mathbf j_n=0\), the magnetic field is absent, and all equations (5.1), (2.6), and (2.8) are satisfied. Thus, in a nonuniformly heated superconductor there is a circulation of currents (Fig. 9), analogous to the circulation of the superfluid and normal parts of a liquid in a nonuniformly heated helium II \([^{13,14}]\). At first glance it seems that this effect is impossible to detect even in principle, since the total current and the magnetic field are zero. This conclusion, however, is incorrect, since in an inhomogeneous or anisotropic superconductor the total current, although small, is, generally speaking, no longer equal to zero \(^{2,67}\). Moreover, if the rod shown in Fig. 9 is in the normal state, then \(\mathbf j=\mathbf j_n=0\) and heat transfer occurs only by the usual mechanism of thermal conductivity. In the superconducting state, however, heat is also transported by the normal current. Therefore the presence of the described circulation of currents must affect the thermal conductivity in the superconducting state \(^{3}\), which is composed of the lattice thermal conductivity, the ordinary electronic thermal conductivity, and the thermal conductivity associated with the current \(\mathbf j_n\).

Fig. 9.

The lattice thermal conductivity in the normal and superconducting states may be regarded as the same [this equality, of course, holds only at the same temperature, i.e. if the normal state is realized by placing the superconductor in a magnetic field \(H>H_k(T)\)]. The electronic thermal conductivity not associated with the current \(\mathbf j_n\) must be smaller in the superconducting state than in the normal state and, moreover, must fall sharply as the temperature is lowered. This conclusion follows from the experimental fact that the “normal conductivity” \(\sigma\), appearing in (2.5) and (5.1), decreases steeply with temperature and tends to zero as \(T\to 0\). The decrease of \(\sigma\) in the present case can be due only to the decrease in the number of “normal electrons” in the superconductor and to the tendency of this number to zero as \(T\to 0\) (see § 6). Hence it follows that the coefficient of thermal conductivity, likewise proportional to the number of normal

electrons, must fall sharply as the temperature is lowered. The connection between the conductivity \(\sigma\) and the electron part (not associated with the current \(j_n\)) of the thermal-conductivity coefficient \(\lambda_{e0}\), in all probability, can be expressed quantitatively by means of the well-known relation called the Wiedemann–Franz law:

\[ \frac{\lambda_{e0}}{\sigma T}=\frac{\pi^3 k^2}{3e^2}=2.71\cdot 10^{-13}\ \mathrm{CGSE}. \tag{5,4} \]

Indeed, relation (5,4) is obtained (see, for example, \({}^{96}\), p. 151) under very general assumptions, reducible to the possibility of introducing a mean free path of the electrons and to the assumption that the electrons form a degenerate gas (this latter assumption, moreover, is essential only from the point of view of obtaining in (5,4) a definite numerical coefficient). It may therefore be thought that relation (5,4) holds also in superconductors, at least when it is valid in the same metal at \(T>T_k\)*).

The quantity \(\lambda_{e0}\) in (5,4) is the thermal-conductivity coefficient for \(j_n=0\). If, however, one takes into account that in a superconductor \(j_n=b\nabla T\), then the total electronic thermal conductivity, as can be shown, is equal to

\[ \lambda_e=\lambda_{e0}+\lambda_{e j_n} =\lambda_{e0}+\frac{b\varepsilon T}{\sigma} =\lambda_{e0}+\left(\frac{d\varepsilon}{dT}\right)^2\sigma T= \]

\[ =\left[\frac{\pi^2 k^2}{3e^2}+\left(\frac{d\varepsilon}{dT}\right)^2\right]\sigma T, \tag{5,5} \]

where

\[ \frac{d\varepsilon}{dT}=\frac{b}{\sigma} \]

is the differential thermoelectromotive force, and the transition to the last expression has been made by using formula (5,4).

For \(T\sim T_k\) in tin,

\[ \frac{d\varepsilon}{dT}\sim 10^{-8}\frac{\text{volt}}{\text{degree}} =3\cdot 10^{-11}\ \mathrm{CGSE}. \]

In this case the term

\[ \frac{\pi^2 k^2}{3e^2} \]

is larger than

\[ \left(\frac{d\varepsilon}{dT}\right)^2 \]

by approximately \(10^8\) times and, consequently, the convective mechanism of heat transfer is completely insignificant, i.e. \(\lambda_e=\lambda_{e0}\). This conclusion is in complete agreement with the fact that in tin the coefficient of thermal conductivity in the superconducting state is smaller than in the normal state \({}^{24}\). The same holds also in other pure metals. However, in alloys, in some cases the thermal conductivity in the superconducting state is greater than in the normal state \({}^{25}\).

* ) Relation (5,4) is valid in a normal metal in the region of low temperatures provided that the resistance is determined by impurities (i.e. is the so-called residual resistance) and that, thereby, there exists a definite mean free path independent of temperature. Under such conditions it is natural to suppose that in the superconducting state as well the concept of a mean free path is applicable, and that it is equal to the mean free path in the normal state.

This occurs, for example, in the alloy \(90\%\,\mathrm{Pb}+10\%\,\mathrm{Bi}\)*). We see no way in which such an effect could be explained without invoking a convective mechanism of heat transfer.

Unfortunately, the absence of values of \(\dfrac{d\mathcal{E}}{dT}\) for alloys for which the thermal conductivity was measured in \(^{25}\) does not permit any direct conclusions to be drawn here. Nevertheless, it is unlikely that the value of \(\dfrac{d\mathcal{E}}{dT}\) increases so much that in (5.5) the term \(\lambda_{e j_n}\) becomes comparable with \(\lambda_{e0}\). It is more probable that the disagreement of formula (5.5) with experiment is connected with the fact that relation (5.5) is obtained only for an ideal superconductor. In alloys, however, owing to their inhomogeneity, in addition to the current \(\mathbf{j}_n=b\nabla T\), local currents \(^{2,67}\) also appear, and in general the whole picture becomes greatly complicated. Therefore the role of the convective mechanism of heat transfer in alloys remains unclear and evidently requires special investigation. Let us note in this connection that if, in alloys, conditions are actually realized under which local normal currents appreciably affect heat transfer, then the magnetic fields \(^{67}\) associated with these currents may prove sufficiently strong that they could be observed experimentally.

A normal current always arises even in the absence of a temperature gradient if a superconductor is placed in an alternating electromagnetic field. Indeed, if the magnetic field penetrating the superconductor varies with time, then the electric field \(\mathbf{E}\), which produces the normal current, is necessarily different from zero,

\[ \mathbf{j}_n=\frac{\mathbf{E}}{\sigma}. \]

If the alternating field in the superconductor is weak (i.e. \(H\ll H_k\)), then, also regarding the metal as homogeneous, isotropic, and uniformly heated, we may write all the necessary equations in the form

\[ \left. \begin{aligned} \operatorname{rot}\mathbf{H} &= \frac{4\pi}{c}\left(\mathbf{j}_s+\mathbf{j}_n\right) +\frac{\varepsilon_0}{c}\cdot\frac{\partial \mathbf{E}}{\partial t},\\ \operatorname{rot}\Lambda\mathbf{j}_s &= -\frac{1}{c}\mathbf{H},\\ \frac{\partial \Lambda\mathbf{j}_s}{\partial t} &= \mathbf{E},\\ \mathbf{j}_n &= \sigma \mathbf{E}, \end{aligned} \right\} \tag{5.6} \]

where \(\varepsilon_0\) is the dielectric constant in the superconducting state, not connected with the current \(\mathbf{j}_s\). The considerations given in § 6 show that the quantity \(\varepsilon_0\) may be very large, owing to

*) In \(^{25}\) the same result was obtained at some temperatures also for niobium. One may, however, think that in this case too the matter is not a pure superconductor, but an alloy, although with a small amount of impurities.

there is, generally speaking, no reason to neglect the term \(\dfrac{\varepsilon_0}{c}\dfrac{\partial \mathbf E}{\partial t}\) in (5.6), as is often done. The solution of system (5.6) is of interest primarily for a field sinusoidal in time. By virtue of the linearity of the equations, the more general problem is also reduced to this case. If all fields and currents are proportional to \(e^{i\omega t}\), then, as is easy to see, for \(\omega \ne 0\) system (5.6) is equivalent to the following:

\[ \begin{aligned} \operatorname{rot}\mathbf H &= \frac{4\pi}{c} \left( \sigma+\frac{1}{i\omega\Lambda}+\frac{i\omega}{c}\varepsilon_0 \right)\mathbf E = \frac{i\omega}{c}\,\varepsilon'\mathbf E, \\ \operatorname{rot}\mathbf E &= -\frac{i\omega}{c}\mathbf H . \end{aligned} \tag{5.7} \]

As is well known, equations of the same form are valid in the case of any medium characterized by a complex dielectric constant \(\varepsilon'\) (and permeability \(\mu=1\)). The specific properties of the medium are reflected only in the value of \(\varepsilon'\). In the case of superconductors, according to (5.7),

\[ \varepsilon'=\varepsilon-i\frac{4\pi\sigma}{\omega} = \varepsilon_0-\frac{4\pi}{\omega^2\Lambda} -i\frac{4\pi\sigma}{\omega}. \tag{5.8} \]

We note that equations (5.7) are somewhat more general than (5.6), since they are also suitable in the presence of dispersion, when \(\varepsilon_0\), \(\Lambda\), and \(\sigma\) depend on the frequency \(\omega\). The heat released in the metal, as always, is determined by the expression \(\dfrac{E^2}{\sigma}\). Taking (2.14) into account, we may write the dielectric constant of a superconductor \(\varepsilon\) in the following form:

\[ \varepsilon=\varepsilon_0-\frac{c^2}{\omega^2\delta_0^2} = \varepsilon_0-\frac{4\pi e^2 n_s}{m\omega^2}, \tag{5.9} \]

where \(\delta_0\) is the penetration depth into the superconductor of a weak static magnetic field, and \(n_s\) is the number of superconducting electrons. As was already mentioned in § 3, formula (5.9), where \(e\) and \(m\) are the charge and mass of a free electron, is essentially a definition of the number \(n_s\). The appropriateness and meaning of such a definition are clear if one recalls the expressions for \(\varepsilon\) and \(\sigma\) in the case of a classical electron gas, i.e., specifically, for an electron-ion plasma. For such a gas, as is known (see, for example, \(^{23}\)),

\[ \varepsilon=1-\frac{4\pi e^2 n}{m(\omega^2+\nu^2)}, \qquad \sigma=\frac{e^2 n\nu}{m(\omega^2+\nu^2)}, \tag{5.10} \]

where \(e\) and \(m\) are the charge and mass of a free electron, \(n\) is the electron concentration, \(\omega\) is the cyclic frequency, and \(\nu\) is the effective number of electron collisions per second. In (5.7), only the contribution to \(\varepsilon\) and \(\sigma\) made by the electrons is taken into account.

At sufficiently low frequencies, when the presence of the term \(\varepsilon_0\) in (5.9) and of unity in (5.10) is inessential, expression (5.9) for the dielectric constant of a superconductor is analogous to the expression for \(\varepsilon\) in the case of an electron gas with \(\nu=0\). In this case the number \(n_s\),

introduced according to (5.9), evidently has the meaning of the concentration of free electrons that would provide the value of \(\varepsilon\) observed for a superconductor. A completely equivalent definition of the “effective number of free electrons” can be given on the basis of the general expression for \(\varepsilon'\), obtained in dispersion theory for any medium:

\[ \varepsilon' = 1+\frac{4\pi e^2}{m}\sum_k \frac{n_{0k}}{\omega_{0k}^2-\omega^2+i\nu_{0k}\omega}, \tag{5.11} \]

where \(\omega_{0k}\) are the eigenfrequencies corresponding to transitions from the state under consideration \(0\) (for which \(\varepsilon'\) is calculated) to all other states \(k\), \(\nu_{0k}\) are the corresponding constants characterizing absorption, and \(n_{0k}\) is the effective number of electrons for the transition \(0 \to k\). Instead of \(n_{0k}\) one usually introduces the oscillator strength \(f=\dfrac{n_{0k}}{n}\), where \(n\) is the total concentration of all electrons in the medium.

According to the sum rule \(\sum_k f_{0k}=1\) and \(\sum_k n_{0k}=n\). In those cases where in (5.8) there is a term with \(\omega_{0k}=0\), we may speak of the presence of free electrons. The effective number of these free electrons is the number \(n_{0k}=n_0\), corresponding to the frequency \(\omega_{0k}=0\); moreover, in metals in the normal state \(\nu_{0k}\ne 0\), whereas in the superconducting state in (5.11) there is a term for which simultaneously \(\omega_{0k}=0\) and \(\nu_{0k}=0\). The number \(n_{0k}\) corresponding to this term is precisely the number of superconducting electrons \(n_s\) appearing in (5.9).

From what has been said and from (5.11) it is clear that the term \(\varepsilon_0\) in (5.9), as \(\omega\to 0\), tends to a finite limit. Therefore, according to (5.8), where the conductivity \(\sigma\) is also finite, for sufficiently small frequencies

\[ \varepsilon' \simeq \varepsilon \simeq -\frac{4\pi}{\omega^2\Lambda} = -\frac{c^2}{\omega^2\delta_0^2} = -\frac{4\pi e^2 n_s}{m\omega^2} = -3.18\cdot 10^9\,\frac{n_s}{\omega^2}. \tag{5.12} \]

As is known, and as will also be shown in § 6, an electromagnetic field in a medium with dielectric constant \(\varepsilon<0\) varies according to the law

\[ e^{-\frac{\omega}{c}\sqrt{|\varepsilon|}\,z}, \]

i.e., in the case (5.12), according to the law \(e^{-z/\delta_0}\), which was obtained in § 2 directly for the static case. Thus, in a field of any frequency and, in particular, in a stationary field, the electromagnetic properties of a superconductor are completely determined by relation (5.8), which for weak fields may be regarded as the basic equation of the phenomenological theory of superconductivity\(^*\).

\(^*\) In the stationary case, when \(\omega=0\) and \(\varepsilon'=\infty\), the correct solution of the problem is obtained if, in its solution valid for \(\omega\ne 0\), one lets \(\omega\) tend to zero. A similar condition, imposed as \(\omega\to 0\) on the solution of equations (5.7), is equivalent to condition (2.6).

The determination of \(\xi_0\) and \(\sigma\) in (5.8) is in principle possible in an alternating field, when \(\omega \ne 0\). For \(T \sim (1 \div 10)^\circ\), for pure metals in the nonsuperconducting state, usually \(\sigma \sim 10^{20}\) CGSE. Since near \(T_k\) in the superconducting state one may expect a value of \(\sigma\) of the same order of magnitude, the influence of the conductivity, as is clear from (5.8), is large for frequencies of the order of, or greater than,

\[ \frac{c^2}{4\pi\sigma_0^2} \sim 10^{10} \]

(since \(\xi_0 \sim 10^{-5}\)), i.e., for wavelengths shorter than \(20 \div 25\) cm. Thus, to measure \(\sigma\), it is necessary to work first of all in the centimeter range. Recently, in connection with the development of radiophysics, such measurements have been carried out intensively \(^{26-29б,\ 93}\).

To determine \(\sigma\) from measurements at high frequency it is necessary to know how the field penetrates into the metal, i.e., to use the theory of the skin effect. In this connection, as has become clear, at low temperatures and high frequencies, both in superconducting and in nonsuperconducting metals, the usual theory of the skin effect is inapplicable and the so-called anomalous skin effect occurs. The question of the anomalous skin effect, consideration of which is necessary for the interpretation of experimental data and for finding the quantities \(\varepsilon(T,\omega)\) and \(\sigma(T,\omega)\) in the superconducting and normal states, is of rather great interest. We shall therefore dwell on it in somewhat more detail.

§ 6. NORMAL AND ANOMALOUS SKIN EFFECT IN METALS

Let us consider, first of all, the propagation in a medium with complex dielectric constant \(\varepsilon'\) of plane waves. Substituting the solution in the form of a plane wave into (5.7), we have:

\[ \mathbf{H}=\mathbf{H}_0 e^{i(\omega t-\mathbf{q}\mathbf{r})},\quad \mathbf{E}=\mathbf{E}_0 e^{i(\omega t-\mathbf{q}\mathbf{r})} =-\frac{c}{\omega\varepsilon'}[\mathbf{q}\mathbf{H}],\quad q^2=\frac{\varepsilon'\omega^2}{c^2}, \tag{6.1} \]

or, more explicitly,

\[ \left. \begin{aligned} \mathbf{H} &=\mathbf{H}_0 e^{\,i\left(\omega t-\frac{\omega\sqrt{\varepsilon'}}{c}z\right)} =\mathbf{H}_0 e^{-\frac{\omega}{c}kz+i\left(\omega t-\frac{\omega}{c}nz\right)},\\ (n-ik)^2&=\varepsilon',\quad n=\sqrt{\frac{\varepsilon}{2}+\sqrt{\left(\frac{\varepsilon}{2}\right)^2+\left(\frac{2\pi\sigma}{\omega}\right)^2}},\\ k&=\sqrt{-\frac{\varepsilon}{2}+\sqrt{\left(\frac{\varepsilon}{2}\right)^2+\left(\frac{2\pi\sigma}{\omega}\right)^2}},\quad \delta=\frac{c}{\omega k}, \end{aligned} \right\} \tag{6.2} \]

where the \(z\)-axis has been chosen as the direction of propagation, and the plus sign must always be taken before the roots.

If an electromagnetic wave is incident on a plane surface of a metal perpendicular to the \(z\)-axis, then in the metal, where the modulus \(\varepsilon'\)

is very large, and the derivatives of the field components with respect to \(z\) are much greater than the derivatives with respect to \(x\) or \(y\). Hence, as is seen from (5.7), it follows that in the metal only the tangential components of the fields \(\mathbf E\) and \(\mathbf H\) are substantial; they are related to one another in the same way as in the case of a plane wave (6.1)\(^*\). In this case, on the surface,

\[ E_x=\frac{H_y}{\sqrt{\varepsilon'}},\qquad E_y=-\frac{H_x}{\sqrt{\varepsilon'}}, \tag{6.3} \]

where the \(z\)-axis is directed into the body. The ratio of the tangential components of the fields \(\mathbf E\) and \(\mathbf H\) at the surface is directly measured (from the attenuation and the velocity of propagation of waves in a waveguide, from the reflection coefficient, etc.) and, according to (6.3), makes it possible to determine \(\varepsilon'\). Instead of this latter quantity, in radiophysics one often uses the “surface impedance”:

\[ Z=R+iX=\frac{4\pi}{c}\left[\frac{E_x}{H_y}\right]_{z=0} =\frac{4\pi}{c\sqrt{\varepsilon'}} =\frac{4\pi(n+ik)}{c(n^2+k^2)}. \tag{6.4} \]

In metals in the normal state, at radio frequencies usually

\[ \frac{2\pi\sigma}{\omega}\gg |\varepsilon|,\qquad n\simeq k\simeq \sqrt{\frac{2\pi\sigma}{\omega}} \tag{6.5} \]

and

\[ \left. \begin{aligned} Z=Z_{\mathrm{cl}}&=\frac{2\pi}{ck}\,(1+i) =(1+i)\left(\frac{2\pi\omega}{c^2\sigma}\right)^{1/2} =(1+i)\frac{2\pi\omega\delta_{\mathrm{cl}}}{c^2},\\[4pt] \delta_{\mathrm{cl}}&=\frac{c}{\omega k} =\left(\frac{c^2}{2\pi\omega\sigma}\right)^{1/2}, \end{aligned} \right\} \tag{6.6} \]

where \(\delta_{\mathrm{cl}}\) is the skin-layer thickness in the metal, introduced in the usual way under condition (6.5), sometimes called the classical skin-layer thickness.

In a superconductor at sufficiently low frequency

\[ \varepsilon'=-\frac{c^2}{\omega^2\delta_0^2}\quad \text{(see 5.12)} \]

and

\[ n=0,\qquad k=\sqrt{-\varepsilon}=\sqrt{\frac{4\pi}{\omega^2\Lambda}}=\frac{c}{\omega\delta_0}; \qquad Z=iX=i\frac{4\pi}{ck} =i\frac{4\pi\omega\delta_0}{c^2}. \tag{6.7} \]

In the region where formula (6.6) is valid, by measuring \(Z\) one can at once determine the conductivity \(\sigma\); moreover the “surface resistance” \(R=\operatorname{Re} Z\sim \sigma^{-1/2}\). Further, in (6.6), at not too high frequencies (more precisely, see below), \(\sigma\) is the static conductivity, which, of course, can be measured without using the theory of the skin effect. Thus relation (6.6) can be checked as a result of independent measurements of \(Z\) and of the static conductivity \(\sigma\). Such a check has shown that at high

\(^*\) The question of the accuracy of this assertion is considered in \(^{84}\).

at temperatures \(>50^\circ\) K the classical (ordinary) theory of the skin effect proves to be correct. However, at low temperatures a sharp disagreement is observed between formula (6.6) and experiment, clearly illustrated by Fig. 10, which refers to the frequency \(\omega=2\pi\cdot 2.4\cdot 10^{10}\). At a sufficiently low temperature (in the region of hydrogen and helium temperatures) the surface resistance, despite the continuing increase of \(\sigma\), ceases to decrease and tends to a finite limit. The reason for such an anomalous skin effect is as follows \(^{27,95}\). In deriving expressions (6.4) and (6.6), phenomenological electrodynamics was used and, in particular, Ohm’s law

\[ \mathbf{j}=\sigma \mathbf{E}. \]

But the use of Ohm’s law is possible only in the case when the electric field changes little over the mean free path of the current carriers, in the present case electrons. If, on the contrary, the field \(\mathbf{E}\) changes appreciably over distances of the order of the mean free path \(l\), then the mean current at a given point is no longer determined by the field taken only at that same point.

Fig. 10.

In the case of the skin effect the field changes appreciably over distances of the order of the depth of the skin layer \(\delta_{\mathrm{sk}}\). Therefore the ordinary theory of the skin effect, when \(\delta_{\mathrm{sk}}=\delta_{\mathrm{cl}}\), is valid only if (more precisely, see below)

\[ l \ll \delta_{\mathrm{cl}}. \tag{6.8} \]

Meanwhile, at low temperatures and high frequencies, as it turns out, not only does inequality (6.8) fail to hold, but even the reverse inequality is satisfied. Thus, at helium temperatures typical values are \(\sigma\sim 10^{20}\), \(l\sim 10^{-3}\ \mathrm{cm}\), for which at \(\omega\sim 10^{10}\) \((\lambda\sim 20\ \mathrm{cm})\) \(\delta_{\mathrm{cl}}\sim 10^{-5}\ll l\). Under conditions where inequality (6.8) is not satisfied, in order to find the impedance \(Z\), one must, in addition to Maxwell’s equations, use the expression for the current density obtained in the electron theory of metals.

The electron theory of metals in the normal state is based, as is well known, on the electron-gas model, which we shall also use, without dwelling here on the question of the limits of its applicability. It is essential only to emphasize that, entering upon the path of a model theory, we cannot avoid introducing into

the results obtained are to some extent arbitrary, being connected with particular properties of the chosen model. In the theory of an electron gas\(^{96—98}\) its state is determined by the distribution function of the electrons over momenta \(\mathbf p\) and coordinates \(\mathbf r=x\mathbf i+y\mathbf j+z\mathbf k\):

\[ f(\mathbf p,\mathbf r)=f_0(\mathbf p)+f_1(\mathbf p,\mathbf r),\qquad |f_1|\ll f_0, \tag{6,9} \]

where \(f_0\) is the equilibrium distribution function that obtains in the absence of a field, and \(f_1\) is an addition reflecting the influence of the field, which is regarded as weak; in (6,9) the spin variable is not taken into account, and no distinction is made between momentum and quasimomentum, since this is immaterial for what follows.

The function \(f_1\) is determined by the kinetic equation

\[ \frac{\partial f_1}{\partial t}+e\mathbf E\nabla_{\mathbf p}f_0+\mathbf v\nabla_{\mathbf r}f_1+\frac{f_1}{\tau}=0, \tag{6,10} \]

where \(\mathbf E\) is the electric-field strength; the magnetic field \(\mathbf H\) is assumed absent; \(e\) is the electron charge; \(\mathbf v=\nabla_{\mathbf p}W\) is the electron velocity; \(W(\mathbf p)\) is its energy; \(\tau=\frac{l}{v}\) is the free time, and \(l(\mathbf v)\) is the mean free path (as for the notation, it is the usual one, i.e.

\[ \nabla_{\mathbf p}=\frac{\partial}{\partial p_x}\mathbf i+ \frac{\partial}{\partial p_y}\mathbf j+ \frac{\partial}{\partial p_z}\mathbf k \quad\text{and}\quad \nabla_{\mathbf r}=\frac{\partial}{\partial x}\mathbf i+ \frac{\partial}{\partial y}\mathbf j+ \frac{\partial}{\partial z}\mathbf k). \]

We shall assume that isotropy obtains, i.e. \(f_0\) depends only on \(W\), and consequently

\[ \frac{\partial f_0}{\partial p_i} = \frac{\partial f_0}{\partial W}\cdot \frac{\partial W}{\partial p_i} = v_i\frac{\partial f_0}{\partial W}. \]

Assuming further, in accordance with the problem with which we shall have to deal, that the field and all other quantities are proportional to \(e^{i\omega t}\) and depend only on the coordinate \(z\), we obtain (the field is, moreover, assumed to have a component only along the \(x\)-axis):

\[ \frac{\partial f_1}{\partial z}+\frac{1+i\omega\tau}{\tau v_z}\,f_1 = -eE\frac{\partial f_0}{\partial W}\cdot\frac{v_x}{v_z}. \tag{6,11} \]

If the field decreases into the depth of the metal over a length of order \(\delta_{\mathrm{sk}}\), then

\[ \frac{\partial f_1}{\partial z}\sim\frac{f_1}{\delta_{\mathrm{sk}}}, \]

since the distribution function under the action of the field will also change appreciably over the same distance \(\delta_{\mathrm{sk}}\). Hence it follows that the first term in (6,11) is much smaller than the second, if

\[ l\ll \delta_{\mathrm{sk}}\left(1+(\omega\tau)^2\right)^{1/2}, \tag{6,8a} \]

where \(l=\tau v\) is the mean free path.

At sufficiently low frequencies in the normal state \(\delta_{\mathrm{sk}}=\delta_{\mathrm{kl}}\) [see (6.5)—(6.6)], \(\omega\tau \ll 1\), and (6.8a) passes into (6.8). Condition (6.8a), which is always satisfied at sufficiently low frequency (since, as \(\omega \to 0\), \(\delta_{\mathrm{sk}} \to \delta_{\mathrm{kl}} \sim \omega^{-1/2}\)), is also satisfied in the other limiting case, when \(\omega \to \infty\). However, in the intermediate frequency region condition (6.8a) may also fail to be satisfied (see the example given above).

If condition (6.8a) is fulfilled, as is usually assumed, then the solution of equation (6.11) is as follows (we simply neglect the first term):

\[ f_1=-\,\frac{eE\tau v_x\,\dfrac{\partial f_0}{\partial W}}{1+i\omega\tau} \tag{6.12} \]

and the current \(j\), directed, like the field \(E\), along the \(x\)-axis, is equal to:

\[ j=e\int f_1 v_x\cdot 2\,\frac{dN(W)}{dW}\,dW =\left(\sigma+\frac{i\omega(\varepsilon-1)}{4\pi}\right)E = \frac{2e^2v_0^2\tau_0\left(\dfrac{dN}{dW}\right)_0E} {3(1+i\omega\tau_0)}\,, \tag{6.13} \]

where \(2\,\dfrac{dN(W)}{dW}\,dW\) is the number of states in the energy interval \(dW\) (the factor 2 takes account of the two possible spin orientations), and in the integration it has been taken into account that, for the Fermi distribution, \(\dfrac{\partial f_0}{\partial W}\) is a function close to the \(\delta\)-function (therefore \(\int \varphi(W)\dfrac{\partial f_0}{\partial W}=-\varphi(W_0)\), where \(W_0\) is the energy at the Fermi boundary, and it is taken into account that \(\int \dfrac{\partial f_0}{\partial W}\,dW=f_0(\infty)-f_0(0)=-1\)); the subscript zero in (6.13) indicates that the corresponding quantities are taken at the Fermi boundary, i.e. for the energy \(W_0\). According to (6.13):

\[ \left. \begin{aligned} \varepsilon &=1- \frac{4\pi\cdot \dfrac{2}{3}e^2v_0^2\left(-\dfrac{dN}{dW}\right)_0} {\omega^2+\nu_0^2} =1-\frac{4\pi\sigma}{\nu_0}, \\[6pt] \sigma &= \frac{\dfrac{2}{3}e^2v_0^2\left(\dfrac{dN}{dW}\right)_0\nu_0} {\omega^2+\nu_0^2} = \frac{\sigma(0)\nu_0^2}{\omega^2+\nu_0^2}, \end{aligned} \right\} \tag{6.14} \]

where \(\nu_0=\dfrac{1}{\tau_0}=\dfrac{v_0}{l}\) is the number of collisions, and \(\sigma(0)=\dfrac{2}{3}e^2v_0^2\left(\dfrac{dN}{dW}\right)_0\cdot\dfrac{1}{\nu_0}\) is the static value of the conductivity. These expressions have the same form as in the case of the classical electron gas [see (5.10)], and turn out to be completely identical with the latter,

if the number of conduction electrons is defined by the relation

\[ n_0=\frac{2}{3}mv_0^2\left(\frac{dN}{dW}\right)_0, \tag{6,15} \]

where \(m\) is the mass of a free electron.

The definition of the number of free electrons (6,15) is in complete agreement with the general definition of this quantity given in § 5. Experimentally \(n_0\) can be found by measuring \(\varepsilon\) and \(\sigma\) and using formulas (6,14) and (6,15). In this case an especially clear picture obtains if

\[ \omega^2 \gg \nu^2 \tag{6,16} \]

and, consequently,

\[ \varepsilon \simeq \varepsilon_0-\frac{4\pi e^2 n_0}{m\omega^2}, \tag{6,17} \]

where, instead of unity in (6,14), we write \(\varepsilon_0\) in order to take into account the part of \(\varepsilon\) not connected with the conduction electrons. At room temperature \(l \sim 10^{-5}\div 10^{-6}\), \(\nu_0=\dfrac{v_0}{l}\sim 10^{13}\div 10^{14}\) \((v_0\sim 10^8)\), and condition (6,16) is already fulfilled in the far infrared part of the spectrum. If condition (6,16) is fulfilled and quantum absorption has not yet come into play (since \(\omega\) is less than the threshold frequency \(\omega_{0k}\) for the internal photoelectric effect), then with the aid of formula (6,17), applicable in this case, \(n_0\) is immediately determined. Corresponding, sufficiently complete and reliable measurements are available only for Ag, Au, and Cu. The results of these measurements are given in Table IV,

where \(n_a\) is the concentration of atoms and \(f=\dfrac{n_0}{n_a}\).

Table IV

	\(n_0\)	\(n_a\)	\(f=\dfrac{n_0}{n_a}\)	Remark	\(n_s\)	\(\dfrac{n_s}{n_a}\)
Ag	\(5.7\cdot 10^{22}\)	\(5.9\cdot 10^{22}\)	0.97	see \(^{97}\)	—	—
Au	\(4.66\cdot 10^{22}\)	\(5.9\cdot 10^{22}\)	0.79	see \(^{97}\)	—	—
Cu	\(3.1\cdot 10^{22}\)	\(8.5\cdot 10^{22}\)	0.37	see \(^{97}\)	—	—
Hg	\(13.5\cdot 10^{22}\)	\(4.3\cdot 10^{22}\)	3.14	see \(^{99}\)	\(6.5\cdot 10^{21}\)	0.05
Sn	\(5.8\cdot 10^{22}\)	\(3.7\cdot 10^{22}\)	1.53	calculated according to data \(^{100}\)	\(5\div 10\cdot 10^{21}\)	\(0.08\div 0.17\)

It is interesting to compare the number of free conduction electrons \(n_0\) with the number of “superconducting electrons” \(n_s\) [see (5,12)]:

\[ n_s=\frac{mc^2}{4\pi e^2\delta_0^2} =\frac{2.83\cdot 10^{11}}{\delta_0^2}\ \mathrm{cm}^{-3}. \tag{6,18} \]

Using the values \(\delta_0=6.6\cdot 10^{-6}\) cm for mercury (see § 4) and \(\delta_0=5.2 \div 7.5\cdot 10^{-6}\) for tin (see \(^{17,19}\)), we obtain the values of \(n_s\) indicated in Table IV. There are also given the values of \(n_0\) for Hg and Sn, calculated from very approximate and incomplete data \(^{99,100}\). However, it is difficult to doubt that for Hg and Sn, just as for Ag, Au, and Cu, \(n_0\sim n_a\). At the same time

\[ \frac{n_s}{n_0}\ll 1 . \]

This result is of fundamental significance; it shows that in superconductivity (even at \(T\to 0\)) only a small fraction (about 0.1) of those “free” electrons which participate in ordinary conductivity takes part. The question arises as to what happens in the superconducting state to the remaining free electrons, which do not make their contribution to \(n_s\).

At low temperature (\(T\to 0\)) and at not too high frequencies, the normal conductivity \(\sigma\) in a superconductor tends to zero (see below). Thus, the free electrons which do not pass into the superconducting state cannot remain free in the sense of this word that we discussed in § 5. Consequently, in the transition to the superconducting state a part of the free electrons of the metal, about 10% at \(T\to 0\), begins to participate in superconductivity, while the remaining \(\sim 90\%\) of the electrons “freeze,” i.e., pass into a bound state. This means that in expression (5.11) for \(\varepsilon'\), these “frozen” electrons correspond not to a natural frequency \(\omega_{0k}=0\), as in the free state, but to certain frequencies \(\omega_{0k}\ne 0\). In order of magnitude these frequencies must be equal to

\[ \frac{kT_k}{\hbar}\sim 10^{11}\div 10^{12}, \]

since the characteristic energy in the theory of superconductivity is, evidently, the energy \(kT_k\). The frequencies \(\omega_{0k}\) in question are considerably smaller than ordinary atomic frequencies. Therefore the “frozen” electrons must make a noticeable contribution to the value of \(\varepsilon\), namely, at \(\omega=\nu=0\) and \(T\to 0\), a contribution of the order

\[ \varepsilon_0 \sim \frac{4\pi e^2 n_0}{m\omega_{0k}^2}\sim 10^8 \div 10^{10}. \tag{6.19} \]

The quantity \(\varepsilon_0\), dependent on \(T\), of which we are speaking is, evidently, in meaning the constant \(\varepsilon_0\) in (5.9). The remark made, due to L. D. Landau (1945), may be quite essential also for the microscopic theory of superconductivity, which will be discussed in the second part of this article. At present it is important to emphasize that the value of \(\varepsilon_0\) in the superconducting state may be very large and, in any case, quite different from that in the normal state. Therefore, in expressions (5.8) and (5.9) for \(\varepsilon\) and \(\varepsilon'\) of a superconductor, the quantity \(\varepsilon_0\) cannot be discarded without further consideration.

Let us return, however, to the question of the theory of the skin effect in metals in the normal state. If condition (6,8a) is satisfied, then, as we have seen, one immediately obtains Ohm’s law (6,13) for the current density and formulas (6,14) for \(\varepsilon\) and \(\sigma\). In this case all the results obtained at the beginning of this section are valid (they were obtained on the basis of using Ohm’s law and Maxwell’s equations). If, however, condition (6,8a) is not satisfied, then the first term in the kinetic equation (6,11) cannot be neglected, Ohm’s law does not hold locally (since the current at a given point is determined by the field not only at this point, but also throughout the entire skin layer), and in order to determine the character of the penetration of the field into the metal one must solve jointly equation (6,11) and the field equations

\[ \operatorname{rot}\mathbf{E}=-\frac{i\omega}{c}\mathbf{H} \quad\text{and}\quad \operatorname{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{i\omega\varepsilon_0}{c}\mathbf{E}, \]

where

\[ \mathbf{j}=e\int f_1\mathbf{v}\cdot 2\left(\frac{dN}{dW}\right)_0\,dW \]

(see 6,13). In this case, when condition (6,8a) is not satisfied and the anomalous skin effect takes place, finding the field \(\bar E(z)\), or even only the impedance \(Z\), turns out to be a very complicated problem, a detailed discussion of which within the scope of this article is not possible (see \(^{95}\)). Therefore we shall restrict ourselves to considering only the limiting case, opposite to the classical case (6,8), namely the case when

\[ l\gg \delta_{\mathrm{ck}}\left(1+(\omega\tau)^2\right)^{\frac12}. \tag{6,20} \]

As has already been indicated, at helium temperatures in a pure metal the condition (6,20) is well satisfied for radio frequencies. In the limiting case (6,20)

\[ Z=Z_n=R_n+iX_n= \left(\sqrt{3}\,\pi\cdot\frac{\omega^2 l}{c^4\sigma(0)}\right)^{\frac13} (1+\sqrt{3}\,i), \tag{6,21} \]

where \(\sigma(0)\) is the static value of the conductivity.

Expression (6,21) was obtained in \(^{95}\) on the basis of the kinetic equation (6,11) for the electron gas in the metal and under the assumption that the reflection of electrons incident from within the metal on its surface occurs diffusely. If one assumes that the reflection is specular, then a common factor \(8/9\) appears in (6,21). Thus, the result depends only weakly on the character of the reflection at the surface. More important is the fact that the metal is assumed to be perfectly homogeneous, and its surface plane. These conditions in practice are often not satisfied, as will be discussed further below.

Since \(\sigma\) depends on the temperature \(T\) only through the mean free path \(l\), the impedance \(Z_n\) should not depend on temperature. For pure metals in the region of helium and sometimes even hydrogen temperatures precisely such behavior of \(Z(T)\) is observed experimentally (in this case \(Z\) no longer depends on \(T\) already where \(\sigma\), with decreasing temperature, still continues to increase). For example, for tin, investigated in work \(^{29}\), for \(\omega = 2\pi \cdot 2.4 \cdot 10^{10}\) at \(T=300^\circ\), \(R=\operatorname{Re} Z = 0.1217\) cm, at \(T=20^\circ\), \(R=0.0180\), and at \(T<15^\circ\) (down to \(T_k=3.71^\circ\)), \(R=0.0175\) cm; see also Fig. 10.

Thus, at helium temperatures in the centimeter range, consideration of the limiting case (6.20)—(6.21) is quite sufficient.

For what follows it is essential that a formula of the type (6.21) can be obtained from rather general considerations \(^{27}\). The skin-layer thickness and the impedance depend on the conductivity \(\sigma(0)\), proportional to the mean free path \(l\)*). It is clear, however, that from the standpoint of losses and of the magnitude of the impedance, only those collisions of electrons that occur in the skin layer are essential. Collisions occurring outside this layer, where the field is already zero (or, more precisely, sufficiently small), cannot play any role. In other words, if the condition (6.20) is satisfied, not the entire free path of an electron \(l\) is effective, but only the part of it lying in the skin layer, this part being of the order of the skin-layer thickness \(\delta_{\mathrm{sk}}\). This means that not the entire conductivity \(\sigma(0)\) will be effective, but a part of it of the order \(\sigma(0)\dfrac{\delta_{\mathrm{sk}}}{l}\). From the point of view set forth, one may expect that for \(l \gg \delta_{\mathrm{sk}}\), in order to obtain expressions for \(\delta\) and \(Z_n\), in the ordinary theory of the skin effect one must substitute, instead of \(\sigma(0)\), the quantity

\[ \sigma_{\mathrm{eff}}=\sigma(0)\frac{\delta_{\mathrm{sk}}\beta}{l}\,e^{\,i\left(\frac{\pi}{2}-\varphi\right)}, \tag{6.22} \]

where \(\beta\) and \(\varphi\) are certain real constants (the form of writing the complex constant \(\beta e^{\,i\left(\frac{\pi}{2}-\varphi\right)}\) introduced in this way is determined by convenience considerations). In the normal state, in the region of interest to us, the influence of \(\varepsilon\) may be neglected (for details see \(^{95}\)), i.e., one may put

\[ \varepsilon_{\mathrm{eff}}=\frac{4\pi\sigma(0)\delta_{\mathrm{sk}}}{\omega l}\,\beta e^{-i\varphi}. \]

Then, co-

*) It is essential here that throughout the entire radio range one must use the static value \(\sigma=\sigma(0)\). This follows \(^{95}\) from the fact that in the case (6.20) the condition for the absence of relaxation is not the usual condition \(\omega^3 \ll \nu_0^2\), evident from (6.14), but the condition

\[ \omega^3 \ll \nu_0^2\frac{l^2}{\delta_{\mathrm{sk}}^2}. \]

according to (6.4), and taking into account that \(\delta_{\mathrm{sk}}=\dfrac{c}{\omega k_{\mathrm{eff}}}\) [see (6.2)],

\[ \delta_{\mathrm{sk}}=\left(\frac{c^{3}l}{4\pi\sigma(0)\beta\omega\sin^{2}\frac{\varphi}{2}}\right)^{\frac13}, \]

\[ \begin{aligned} Z&=\left(\frac{16\pi^{2}l\omega^{3}\sin\frac{\varphi}{2}}{c^{4}\sigma(0)\beta}\right)^{\frac13} \left(\cos\frac{\varphi}{2}+i\sin\frac{\varphi}{2}\right) \\ &=\frac{2\pi\delta_{\mathrm{sk}}\omega}{c^{2}} \left(\sin\varphi+2i\sin^{2}\frac{\varphi}{2}\right). \end{aligned} \tag{6.23} \]

This expression for \(Z\) coincides with (6.21) if one puts

\[ \varphi=\frac{2}{3}\pi,\qquad \beta=\pi. \tag{6.24} \]

In this case [see (6.23) and (6.6)]:

\[ \delta_{\mathrm{sk}}=\delta_{n}= \left(\frac{c^{3}l}{3\pi^{2}\omega\sigma(0)}\right)^{\frac13} = \left(\frac{2l}{3\pi\delta_{\mathrm{cl}}}\right)^{\frac13}\delta_{\mathrm{cl}}, \tag{6.25} \]

where in \(\delta_{\mathrm{cl}}\) one sets \(\sigma=\sigma(0)\).

It is evident that, by virtue of (6.20), where in meaning \(\delta_{\mathrm{sk}}=\delta_{n}\), \(\delta_{n}\gg\delta_{\mathrm{cl}}\). From (6.21) three conclusions follow which admit experimental verification: the impedance \(Z_{n}\) must not depend on the temperature \(T\), \(Z_{n}\) is proportional to \(\omega^{2/3}\), and

\[ \frac{\operatorname{Im}Z_{n}}{\operatorname{Re}Z_{n}}=\frac{X_{n}}{R_{n}}=\sqrt{3}, \]

whereas in the usual theory [see (6.6)] \(X_{\mathrm{cl}}=R_{\mathrm{cl}}\). All these conclusions are in rather good agreement with experiment\(^{57}\), although there are still not very many data here. It is especially important that from equation (6.21) one can determine the quantity \(\sigma(0)/l\):

\[ \frac{\sigma(0)}{l}=\frac{\sqrt{3}\pi\omega^{3}}{c^{4}R_{n}^{3}}. \tag{6.26} \]

Since the conductivity \(\sigma_{0}\) can be measured by a static method (we are speaking of nonsuperconductors), measurements of \(R_{n}\) make it possible to determine the mean free path \(l^{*}\). Knowledge of \(l\), in turn, makes it possible to test the very foundations of the electron theory of metals which we used above.

In view of the importance of this question, and because it is usually presented in a distorted form, we shall dwell on it in somewhat greater detail\(^{92a}\).

The electron theory of metals in its usual form, based on equation (6.11) and taking for \(f_{0}\) the Fermi distribution, leads to expressions (6.14) for \(\varepsilon\) and \(\sigma\). For

\[ \text{*) Another method for determining }l,\text{ based on measurements in thin films}^{101\text{–}101b},\text{ is also of undoubted interest.} \]

in this, apart from the basic assumption that the electrons in the metal form a gas, in passing from (6.10) to (6.11) the assumption of isotropy was made, i.e. it was assumed that \(f_0\) depends only on the energy of the electrons \(W\). In addition to the expressions for \(\varepsilon\) and \(\sigma\), the theory, under the same assumptions, leads to the following expression for the electronic part of the heat capacity\({}^{98}\):

\[ c_n^e=\frac{2\pi^2 k^2}{3}\,T\left(\frac{dN}{dW}\right)_0=\gamma T, \tag{6.27} \]

where \(T\) is the absolute temperature, \(k=1.38\cdot 10^{-16}\) is Boltzmann’s constant, and the heat capacity is referred to unit volume. The quantity \(c_n^e\), as is known, is measured experimentally (see § 1).

The expressions (6.14) and (6.27) for \(\varepsilon\), \(\sigma\), and \(c_n^e\) contain three unknown quantities \(v_0\), \(\nu_0=\dfrac{1}{\tau_0}\), or \(l=\dfrac{v_0}{\nu_0}\), and \(\left(\dfrac{dN}{dW}\right)_0\). Thus we cannot test the theory on the basis of measurements of only \(\varepsilon\), \(\sigma\), and \(c_n^e\) (we are now disregarding the possibility of testing the frequency dependence of \(\varepsilon\) and \(\sigma\) and their relation to one another). The role of the necessary fourth independent measurement may be played by the determination of \(R\), i.e., according to (6.26), by the determination of \(\dfrac{\sigma(0)}{l}\).

For convenience we write here once more all the necessary expressions [see (6.14), (6.15), and (6.27)]:

\[ \varepsilon=1-\frac{4\pi e^2 n_0}{m(\omega^2+\nu_0^2)},\qquad \sigma=\frac{e^3 n_0 v_0}{m(\omega^2+\nu_0^2)} =\sigma(0)\frac{\nu_0^2}{\omega^2+\nu_0^2}, \]

\[ \sigma(0)=\frac{e^3 n_0}{m\nu_0}=\frac{e^2 n_0 l}{m v_0},\qquad n_0=\frac{2}{3}mv_0^2\left(\frac{dN}{dW}\right)_0, \]

\[ \gamma=\frac{2\pi^2}{3}k^2\left(\frac{dN}{dW}\right)_0 =\frac{\pi^2 k^2 n_0}{m v_0^2},\qquad \nu_0=\frac{v_0}{l}. \tag{6.28} \]

From this, as is readily seen, there follows the relation\({}^{92a}\)

\[ \frac{\left(\dfrac{\sigma(0)}{l}\right)^2}{\gamma n_0} =\frac{e^4}{\pi^2 k^2 m}=3.08\cdot 10^{20}\ \mathrm{CGSE}. \tag{6.29} \]

On the left-hand side of this relation there appear quantities measured independently of one another: \(\dfrac{\sigma(0)}{l}\) from (6.26), \(n_0\) by optical methods [see (6.17)], and \(\gamma\) from calorimetric measurements (or, for superconductors, using the value \(\Delta c=c_s-c_n\), obtained from experiments on the destruction of superconductivity by a magnetic field; see §§ 1 and 2). On the right-hand side of equality (6.29) stand universal constants, and thus a test of the theory indeed becomes possible. In this respect relation (6.29) is analogous to the Wiedemann–

Franz:

\[ \frac{\lambda_e}{\sigma(0)T}=\frac{\pi^2 k^2}{3e^2}=2.71\cdot 10^{-13}\ \mathrm{CGSE}, \tag{6,30} \]

where \(\lambda_e\) is the coefficient of electronic thermal conductivity\(*\).

Instead of using formula (6.29) to test the theory, in papers \(^{26-29,95}\) it is assumed that the electrons in a metal form a completely free electron gas. In such a model the concentration of free electrons \(n_0\) has a direct meaning and is equal to

\[ n_0=\frac{4\pi}{3}\cdot 2\left(\frac{mv_0}{2\pi\hbar}\right)^3;\quad \text{further,}\quad \frac{dN}{dW}=\frac{2^{5/2}m^{3/2}W^{1/2}}{8\pi^2\hbar^3} \]

and

\[ \frac{\sigma(0)}{l} = \frac{e^2 n_0^{2/3}}{3^{1/2}\pi^{2/3}\hbar} = 7.1\cdot 10^7 n_0^{2/3}. \tag{6,31} \]

Thus, in the model of a free electron gas, for which, of course, there are no grounds, by measuring \(\frac{\sigma(0)}{l}\) one can immediately find the concentration \(n_0\) (in the model under consideration the same value \(n_0\) also appears in formula (6.17)). The values of \(n_0\) obtained with the aid of (6.31) and (6.26) turned out to be considerably smaller than could have been expected. Thus, for example, for Sn \(n_0=0.2 n_a=0.72\cdot 10^{22}\), whereas from optical data \(n_0\simeq n_a\) (see Table IV). The same discrepancy also occurs for the noble metals. It is especially important that this contradiction is in no way removed either by abandoning the model of completely free electrons or by treating the material on the basis of formula (6.29). Thus, for Sn, taking

\[ \gamma=3.5\cdot 10^{-4}\ \frac{\mathrm{cal}}{\mathrm{mol}\cdot \mathrm{deg}^2} \]

(see \(^{32}\), Table 10), \(n_0=5.8\cdot 10^{23}\ \mathrm{cm}^{-3}\) (see above, Table IV) and \(\frac{\sigma}{l}=2.7\cdot 10^{23}\) CGSE (see \(^{29}\)), we have

\[ \frac{\left(\frac{\sigma}{l}\right)^2}{n_0\gamma}=0.14\cdot 10^{20}, \]

instead of \(3.08\cdot 10^{20}\), according to (6.29). The inaccuracy of formula (6.26), in the derivation of which it is assumed that the electrons are reflected diffusely from the surface of the metal, as well as possible errors in all the accepted experimental values, evidently cannot explain the discrepancy obtained.

\(*\) It should be noted that relation (6.30) is valid independently of the assumption of isotropy, i.e. the assumption that the distribution function \(f_0\) depends only on the energy \(W\) (the assertion made in \(^{98}\) that the Wiedemann–Franz law is valid only under the assumption of isotropy is erroneous; see \(^{96}\), ch. 5, § 4). Meanwhile, formula (6.29) is obtained only under the assumption of isotropy; otherwise an additional numerical factor appears in (6.29).

On the contrary, as is now becoming clear\(^{26a}\), the dependence of the measured value of \(R\) on the state of the metal surface (method of treatment and cleaning, etc.) is so significant that it may well be responsible for the failure of formula \((6,26)\)\(^*\). The great sensitivity of the value \(R_n\) to the properties of the surface may substantially devalue the possibility of determining the mean free path \(l\) in this way. In any case, further discussion of this question seems to us at present, pending the appearance of new experimental data, premature.

Up to now the discussion has concerned exclusively the anomalous skin effect in the nonsuperconducting state. Let us now turn to the case of superconductors\(^{26, 27, 29, 92a}\). The mechanism of normal conductivity in the superconducting state is still unclear. In particular, it is not clear to what extent in this case, for the “normal electrons,” one may use the electron-gas model which is used in the case of nonsuperconductors. It is therefore desirable to consider the question of the skin effect in superconductors while making the minimum number of additional assumptions. This requirement is best met by the method used above for the normal state, namely the introduction of an effective conductivity \((6,22)\). This method, it is true, is valid only in the limiting case \((6,20)\), when the mean free path is much greater than the thickness of the skin layer. But if this condition is already fulfilled above \(T_k\), as is usually the case, then it will also be fulfilled for \(T<T_k\). We shall assume condition \((6,20)\) to be satisfied. Then it is natural to think that the behavior of a superconductor in a high-frequency field is determined by the effective complex dielectric constant

\[ \left. \begin{aligned} \varepsilon'_{\mathrm{eff}} &= \varepsilon - i \frac{4\pi\sigma\delta_{\mathrm{sk}}}{\omega l}\, \beta e^{\,i\left(\frac{\pi}{2}-\varphi\right)} = (n_{\mathrm{eff}}-ik_{\mathrm{eff}})^2,\\ \varepsilon &= \varepsilon_0-\frac{4\pi}{\omega^2\Lambda},\qquad \delta_{\mathrm{sk}}=\frac{c}{\omega k_{\mathrm{eff}}},\qquad Z=\frac{4\pi}{c\sqrt{\varepsilon'_{\mathrm{eff}}}}, \end{aligned} \right\} \tag{6,32} \]

where \(\sigma=\sigma(0)\); unlike the normal state, the quantity \(\varepsilon\) can no longer be neglected. The coefficients \(\beta\) and \(\varphi\) in the superconducting and normal states may, generally speaking, be different, but for definiteness we shall now assume that they remain unchanged, and shall immediately take

\[ \varphi=\frac{2}{3}\pi \]

and \(\beta=\pi\) (see \((6,24)\)). The value of \(\varepsilon\) in \((6,32)\) is in no way changed in comparison with \((5,5)\), where

\[ \varepsilon'=\varepsilon-i\frac{4\pi\sigma}{\omega}. \]

This is explained by the fact that \(\varepsilon\) in a superconductor may be regarded as independent of the mean free path

\[ {}^*)\ \text{Here it is essential that, according to }(6,26),\ \left(\frac{\delta}{l}\right)^2\sim R_n^{-6}\ \text{and, thus, a twofold change in }R_n\text{ changes the ratio }\frac{\left(\frac{\delta}{l}\right)^2}{n_0 l}\text{ by a factor of 16.} \]

of the mean free path \(l\). Therefore the relation between \(\delta_{\mathrm{sk}}\) and \(l\) is quite immaterial from the point of view of the effective value of \(\varepsilon\).*)

In view of the rather cumbersome nature of the general formulas for \(Z\) and \(\delta_{\mathrm{sk}}\), we shall consider separate special cases of §2a.

At not too high frequencies and as \(T \to 0\), losses in a superconductor vanish, i.e. \(\sigma \to 0\). At high frequencies there must appear absorption of a quantum character, not connected with conductivity, but due to transitions of electrons in the metal to excited levels. The characteristic frequency at which this absorption should arise is

\[ \omega_{\mathrm{k}} \sim \frac{kT_{\mathrm{k}}}{\hbar} \sim 10^{11} \div 10^{12} \left( \lambda=\frac{2\pi c}{\omega}\sim 0.1 \div 1\ \mathrm{cm} \right), \]

since \(k=1.38\cdot 10^{-16}\) and \(T_{\mathrm{k}}\sim 1\div 10^\circ\). In experiment, at the highest frequency achieved \(\left(\omega=1.5\cdot 10^{11},\ \lambda=1.25\ \mathrm{cm};\right.\) tin was investigated, for which \(T_{\mathrm{k}}=3.7\)\()\), some absorption at low temperature is already noticeable. However, it is still not clear whether this absorption is not parasitic, connected with impurities, etc. In the infrared part of the spectrum, as is known, the losses in the superconducting state are, within the accuracy of the experiments, the same as in the normal state, i.e., in agreement with the estimate made, \(\omega_{\mathrm{infrar}}>\omega_{\mathrm{k}}\). At frequencies \(\omega<10^{11}\) \((\lambda>3\ \mathrm{cm})\), absorption as \(T\to 0\) is imperceptible, and in this region we may indeed assume that \(\sigma\to 0\).

In this limiting case \(\sigma=0\):

\[ \left. \begin{aligned} \delta_{\mathrm{sk}}=\delta_{s0} &=\frac{c}{\omega\sqrt{|\varepsilon|}} =\frac{c}{\omega\sqrt{\dfrac{4\pi}{\omega^{2}\Lambda}-\varepsilon_{0}}} =\frac{c}{\omega\sqrt{\dfrac{4\pi e^{2}n_s}{m\omega^{2}}-\varepsilon_{0}}} \\[6pt] &=\frac{\delta_{0}}{\sqrt{1-\dfrac{m\varepsilon_{0}\omega^{2}}{4\pi e^{2}n_s}}}, \\[10pt] Z=Z_{s0}=iX_{s0} &=\frac{4\pi i}{c\sqrt{|\varepsilon|}} =\frac{4\pi i\delta_{s0}\omega}{c^{2}}, \qquad \delta_{0}^{2}=\frac{mc^{2}}{4\pi e^{2}n_s}, \end{aligned} \right\} \tag{6,33} \]

where it is assumed that \(\varepsilon<0\) (otherwise the field does not attenuate into the depth of the metal).

At low frequencies, when \(\varepsilon_{0}\ll \dfrac{4\pi e^{2}n_s}{m\omega^{2}}\), formulas (6,33) pass into (6.7), i.e. \(\delta_{s0}=\delta_{0}\). If \(\varepsilon_{0}\sim 10^{9}\) [cf. (6.19)], then \(\delta_{s0}\) will begin to differ from the depth \(\delta_{0}\), which does not depend on \(\omega\)

*) In the normal state, as is clear from (6.14), \(\varepsilon\) depends on \(l\). A term of the same type should also appear in the superconducting state, but here, at the frequencies of interest to us, the contribution to \(\varepsilon_{\mathrm{ef}}\) due to this term is considerably smaller than the contribution associated with the term

\[ -\frac{4\pi}{\omega^{2}\Lambda}. \]

at \(\omega\sim 10^{11}\). If quantum absorption did not set in, then, by increasing the frequency, one could reach the region where \(\varepsilon_0=-\dfrac{4\pi e^2 n_s}{m\omega^2}\) and the superconductor would become transparent. But, in all probability, quantum absorption already sets in at lower frequencies. In any case, the study of the impedance \(Z\) at low temperatures and at various frequencies should lead to the determination, or at least to an estimate, of the quantity \(\varepsilon_0\), which is very important for the theory of superconductivity.

At low temperatures, in the following approximation, taking into account the conductivity \(\sigma\), which is regarded as small, we have:

\[ \left. \begin{gathered} \delta_{\mathrm{ck}}=\delta_{s0}+\delta_{s1},\quad |\delta_{s1}|\ll \delta_{s0},\\[4pt] \delta_{s1}=-\frac{\delta_{s0}^{4}}{3\delta_n^3},\quad \delta_n=\left(-\frac{c^2 l}{3\pi^2\omega\sigma}\right)^{\frac13},\quad \delta_{s0}=\frac{c}{\omega\sqrt{|\varepsilon|}},\\[6pt] Z=Z_{s1}=\frac{4\pi i}{c\sqrt{|\varepsilon|}} \left(1-\frac{\delta_{s0}^3}{3\delta_n^3} -i\frac{\delta_{s0}^3}{\sqrt{3}\delta_n^3}\right) =R_{s1}+iX_{s1},\\[6pt] R_{s1}=\frac{X_{s0}}{\sqrt{3}}\frac{\delta_{s0}^3}{\delta_n^3},\quad X_{s1}=X_{s0}\left(1-\frac{\delta_{s0}^3}{3\delta_n^3}\right),\\[6pt] X_{s0}=\frac{4\pi}{c\sqrt{|\varepsilon|}} =\frac{4\pi\omega\delta_{s0}}{c^2}. \end{gathered} \right\} \tag{6.34} \]

In another limiting case, when \(\dfrac{4\pi\sigma\delta_{\mathrm{ck}}}{\omega l}\gg |\varepsilon|\), the formulas (6.21) and (6.25) derived above apply, i.e. \(\delta_{\mathrm{ck}}=\delta_n\) and \(Z=Z_n\).

In the next approximation, when the value of \(\varepsilon\) is small, but still not negligible,

\[ \left. \begin{gathered} \delta_{\mathrm{ck}}=\delta_n+\delta_{s2},\quad |\delta_{s2}|\ll \delta_n,\quad \delta_{s2}=-\frac{\delta_n^3}{4\delta_{s0}^2},\\[6pt] Z=Z_{s2}=Z_n\left\{1+\frac{\delta_n^2}{16\delta_{s0}^2}(3\sqrt{3}\,i-1)\right\} =R_{s2}+iX_{s2},\\[6pt] R_{s2}=R_n\left\{1-\frac{5\delta_n^2}{8\delta_{s0}^2}\right\},\quad X_{s2}=X_n\left\{1+\frac{\delta_n^2}{8\delta_{s0}^2}\right\},\\[6pt] Z_n=\left(\sqrt{3}\pi\,\frac{\omega^2 l}{c^4\sigma}\right)^{\frac13}(1+\sqrt{3}\,i) =R_n+iX_n\\ =\frac{\sqrt{3}\pi\delta_n\omega(1+\sqrt{3}\,i)}{c^2}. \end{gathered} \right\} \tag{6.35} \]

Expressions (6.35) are evidently valid near the critical temperature \(T_k\) as long as \(\delta_n\ll\delta_{s0}\), i.e. as long as the thickness of the skin layer is significantly smaller than the depth of penetration into the superconductor of a static magnetic field (recall that, for...

not too high frequencies: \(\delta_{s0}^{2}=\delta_0^{2}\). At the point \(T_k\) itself the depth \(\delta_{s0}\to\infty\), and, as it should be, \(\delta_{sk}=\delta_n(T_k)\) and \(Z=Z_n(T_k)\). Below \(T_k\), at a small distance from this point, according to (6.35),

\[ \frac{R(T)}{R(T_k)} = \frac{R_s(T)}{R_n(T_k)} = -\frac{\delta_n(T)}{\delta_n(T_k)} \cdot \left(1-\frac{5}{8}\cdot\frac{\delta_n^{2}(T)}{\delta_{s0}^{2}(T)}\right) = \]

\[ = \left[ \frac{\sigma(T_k)\,l(T)}{\sigma(T)\,l(T_k)} \right]^{\frac{1}{3}} \left\{ 1-\frac{5}{8} \frac{ \left(1-\dfrac{m\omega^{2}\varepsilon_0}{4\pi e^{2}n_s}\right) c^{\frac{4}{3}} l^{\frac{2}{3}} }{ \delta_0^{2}(3\pi^{2}\omega\sigma)^{\frac{2}{3}} } \right\}. \tag{6.36} \]

As \(T\to T_k\), \(\dfrac{d\delta_0}{dT}\to\infty\), and thus the derivative \(\dfrac{d}{dT}\left(\dfrac{R(T)}{R(T_k)}\right)\) is determined mainly by the derivative of the second term in (6.36), whence, if the term with \(\varepsilon_0\) is not taken into account,

\[ \frac{d}{dT}\left(\frac{R(t)}{R(T_k)}\right)\sim \omega^{-\frac{2}{3}}. \]

At a sufficiently low temperature, according to (6.34), \(R(T)\sim\omega^{3}\) and

\[ \frac{R(T)}{R(T_k)}\sim\omega^{\frac{4}{3}} \]

(the influence of the term with \(\varepsilon_0\) is again not taken into account). Experimentally\(^{27--29b}\) the frequency dependence of

\[ \frac{d}{dT}\left(\frac{R(T)}{R(T_k)}\right) \]

is found to be somewhat less sharp than follows from the theory (rather a law \(\omega^{-1/2}\) instead of \(\omega^{-2/3}\)).

Fig. 11.

This discrepancy, as is easy to see, cannot be explained by the influence of the term with \(\varepsilon_0\). It must be borne in mind, however, that up to now the comparison has been made chiefly of measurements by different authors, each of whom worked at a single frequency; moreover, the data of different authors, in those cases where they refer to the same frequency, sometimes do not coincide (thus in \(^{26}\) the quantity

\[ \left.\frac{d}{dT}\left(\frac{R(T)}{R(T_k)}\right)\right|_{T_k} \]

changes by a factor of 2.95 in passing from the frequency 1200 to the frequency 9200 hertz, while in \(^{29b}\) it changes by a factor of 3.4; the ratio

\[ \left(\frac{9200}{1200}\right)^{\frac{2}{3}}=3.9 \]

). Taking into account the strong dependence of the impedance on the state of the surface,\(^{29a}\) it is clear that comparison of experimental data at different frequencies must be carried out for surfaces prepared in the same way. Since

CURRENT STATE OF THE THEORY OF SUPERCONDUCTIVITY

since this has not yet been done, the question of testing the theory from this side remains open.

Assuming the formulas (6.32)—(6.36) given to be valid, we can, knowing \(R\) and \(X\), find \(\varepsilon\) and \(\dfrac{\sigma(0)}{l}\). In practice, \(R(T,\omega)\) is measured (the values of \(R\) for tin\(^{26}\) at frequencies of 1200 and 9200 hertz are shown in Fig. 11) and the difference \(X(T,\omega)-X(T_k,\omega)\), since measurements of the quantity \(X\) itself are very difficult. Therefore it is also necessary to know from independent measurements, at some temperature, the value of \(\delta_{s0}(T,\omega)\). In the frequency range where \(\delta_{s0}=\delta_0\), for this purpose one may use the value \(\delta_0\) determined from measurements in a constant magnetic field.

To find the conductivity \(\sigma(0)\), and not only the ratio \(\dfrac{\sigma(0)}{l}\), additional assumptions are needed.

It is natural, for example, to assume that in those cases when for \(T>T_k\) \(\sigma(0)\) no longer depends on temperature, i.e. the mean free path \(l\) is determined by impurities, the value of \(l\) in the superconducting state is the same as in the normal state. But in the normal state we can, in principle, determine both \(\sigma(0)\) and \(l\) (\(\sigma(0)\) from static measurements and \(\dfrac{\sigma(0)}{l}\) from the value of \(R_n\)).

Thus the mean free path in the superconducting state becomes known, and consequently, if the ratio \(\dfrac{\sigma(0)}{l}\) is known, so is the quantity \(\sigma(0)\) itself.

At the same time it must be emphasized that the formulas obtained are based on the initial relation (6.32) and, moreover, on quite definite values of the quantities \(\varphi\) and \(\beta\) \(\left(\varphi=\dfrac{2}{3}\pi\right.\) and \(\left.\beta=\pi\right)\). Meanwhile in (6.32), instead of \(\delta=\dfrac{c}{\omega k_{\mathrm{eff}}}\), one may apparently, with no less justification, choose the value

\[ \delta=-\,\frac{ic}{\omega\sqrt{\varepsilon_{\mathrm{eff}}}} =\frac{c}{\omega\left(k_{\mathrm{eff}}-in_{\mathrm{eff}}\right)} . \]

To obtain in this case formula (6.21), it is necessary to assume that \(\varphi=\dfrac{1}{2}\pi\) and \(\rho=\dfrac{2}{\sqrt{3}}\pi\), while the corresponding expressions for \(Z\) in the superconducting state differ from (6.34) and (6.35) by numerical coefficients of order unity. Thus, by specifying a definite frequency dependence of all quantities, we cannot determine the numerical coefficients quite exactly. Essentially the same situation occurs also in the rigorous kinetic approach\(^{95,29}\), since the results of calculations in this case depend on assumptions about the character of the reflection of electrons from the metal surface and on the initial distribution function of the electrons over momenta. However, such uncertainty in the values of the numerical coefficients,

in general, does not prevent one from finding the temperature dependence of \(\sigma\) and from making a fairly accurate (with an accuracy of \(\lesssim 50\%\)) estimate of the value of \(\sigma\) itself, not to mention the dielectric constant \(\varepsilon_0\), which can be determined from the exact formula (6.33). In addition, some refinement of the formulas for \(R\) and \(X\) can be achieved as a result of comparing the experimental and theoretical values of the ratio \(\dfrac{X}{R}\).

Finally, a known check on the values of \(\sigma\) obtained as a result of impedance measurements can be obtained by measurements of the thermal conductivity, followed by determination of \(\sigma\) with the aid of the Wiedemann–Franz relation (see § 5).

In the available works \(^{29,296}\), the processing of the experimental data was carried out on the basis of the additional assumption that the number of “normal electrons” in a superconductor decreases with temperature to the same extent as the number of “superconducting electrons” increases. In other words, the validity of the relation

\[ \frac{\sigma(T)}{\sigma(T_k)} = \frac{n_0(T)}{n_0(T_k)} = 1-\frac{\delta_0^2(0)}{\delta_0^2(T)} = 1-\frac{n_s(0)}{n_s(T)}, \tag{6.37} \]

is assumed, where \(n_0\) is the effective number of “normal electrons.”

This relation is, of course, valid in both limiting cases \(\left(\dfrac{\sigma(T)}{\sigma(T_k)} \to 1\right.\) as \(T \to T_k\), since \(\delta_0(T \to T_k)\to\infty\); in addition, \(\sigma(0)=0\)). But at an arbitrary temperature there are no special grounds for relation (6.37). Consequently, this relation should not be made the basis of calculations, but should be checked by an independent determination of \(\sigma\) and \(\delta_0\). In \(^{29,296}\) the possible difference between \(\delta_{s0}\) and \(\delta_0\), associated with the presence of the dielectric constant \(\varepsilon_0\), is also not taken into account. And finally, the experimental data themselves obtained so far are quite unreliable in the sense that they depend strongly on the properties of the surface and therefore cannot, without further investigation, be used to determine quantities (such as \(\varepsilon\) and \(\sigma\)) characterizing the bulk metal. For all these reasons we shall not discuss here the results obtained in the corresponding works \(^{29-296}\). We shall only indicate that these results, within the limits of the accuracy to which they can lay claim, do not contradict the known static measurements of \(\delta_0(T)\) and show that \(\sigma(T)\) for \(T<T_k\) falls rather rapidly with decreasing temperature.

In summary, it must be said that impedance measurements \(Z\) in the superconducting state are very interesting and may lead to the determination of the magnitude and temperature dependence of the conductivity \(\sigma\), and also to clarification of the role of the static dielectric constant in the superconducting state \(\varepsilon_0\). Measurements

at sufficiently high frequencies will make it possible, moreover, to determine the magnitude of the natural frequencies in the superconducting state and the possible frequency dependence of $\varepsilon_0$. At the same time it is clear that solving all these problems requires a great deal of experimental work, which has only just begun.

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PROOF-CORRECTION NOTE

In a new paper by Pippard (Proc. Roy. Soc. 203, 98, 195 (1950)) the impedance was measured for tin single crystals, and a noticeable anisotropy was observed. The resulting nonmonotonic dependence of \(X_{s0}\) on \(\Theta\) for \(0 \leq \Theta \leq \pi/2\) is completely incomprehensible and must be carefully investigated.