Kopnin N.B. Theory of Nonequilibrium Superconductivity

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The regular functions are determined by the Usadel equations (5.98), (5.99).

Assume that the order parameter gradients are small Dk

. We also assume

that the electron–phonon relaxation is slow such that

which is

usually the case for temperatures not extremely close to T

. The Usadel

equations given the expressions for the Green functions as in a homogeneous

case

(10.104)

end p.210

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and

(10.105)

in the leading approximation in small gradients. As a result, we have from eqn

(10.75) for

> | |

We can neglect the second term in the r.h.s. since the vector part of the Green

function is small compared to the averaged one. Equation (10.101) gives

(10.106)

Using eqn (10.100) we find

(10.107)

The vector part of the Green function is small compared to the averaged

component and can be neglected.

Equation (10.102) should be used together with eqn (10.103) which determines

the vector part f

. The impurity collision integral vanishes because f

f = 0

for slow gradients. The part

gives

(10.108)

The function f

is obtained from eqn (10.102) where one inserts f

from eqn

(10.108). Note that the Green functions g± and f± in eqn (10.102) should be

taken with the proper account of the phonon scattering rate since the last term

in the l.h.s. is proportional to

. The so-obtained function f

should then be

used in the charge neutrality condition to find the equation which determines the

scalar potential

Kopnin, Nikolai, Senior Scientist, Low Temperature Laboratory, Helsinki University of

Technology, and L.D. Landau Institute for Theoretical Physics, Moscow

Theory of Nonequilibrium Superconductivity

Print ISBN 9780198507888, 2001

pp. [211]-[215]

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10.5.2 Heat conduction

Equation (10.106) allows one to calculate the thermal conductivity. Using the

definition of the internal energy current, eqn (10.66), we have

end p.211

We find from eqn (10.106)

Let us assume that the distribution function is f

(1)

= f

(0)

= tanh( /2T) where T is

a slow function of coordinates. The gradient becomes

The heat current takes the form j = k T where the thermal conductivity is

(10.109)

This expression for the thermal conductivity was first derived by Bardeen et al.

(1959).

end p.212

11 THE TIME-DEPENDENT GINZBURG-LANDAU THEORY

Nikolai B. Kopnin

Abstract: This chapter specifies the conditions when the time-dependent

Ginzburg–Landau (TDGL) model can be justified microscopically. The TDGL model

is shown to be exact for gapless superconductors. It is not exact, however, for

systems with a finite energy gap. The role of nonequilibrium excitations is

elucidated in the dynamics of superconductors. The generalized version of

TDGL-like model is derived for superconductors with relatively strong

pair-breaking effects due to inelastic relaxation. The (different) characteristic

relaxation times for the order parameter and for the superconducting phase are

identified. The TDGL-like theory is developed for d-wave superconductors. The

charge imbalance, the decay of a d.c. electric field in a superconductor, and the

surface resistance are discussed.

Keywords: time-dependent Ginzburg–Landau model, gapless

superconductor, inelastic relaxation, pair-breaking, relaxation time,

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charge imbalance, surface resistance

We find out the conditions when the time-dependent Ginzburg–Landau

model can be justified microscopically. We demonstrate that the TDGL

model is exact for gapless superconductors. It is not justified, however, for

systems with a finite energy gap. The role of nonequilibrium excitations is

elucidated in the dynamics of superconductors. The charge imbalance and

decay of a d. c. electric field in a superconductor is discussed.

11.1 Gapless superconductors with magnetic impurities

We already discussed the time-dependent Ginzburg–Landau (TDCL) model which

generalizes the usual Ginzburg–Landau (GL) theory and provides a simple

description of nonstationary processes in superconductors (see Section 1.2.1).

Consider now how and under which conditions, the microscopic theory can justify

the phenomenological TDGL model.

We start with the simplest example that requires a minimum of calculations if we

use the preparatory work done in the previous chapters. This is the case of dirty

superconductors which have a large concentration of magnetic impurities such

that

(Gor’kov and Eliashberg 1968). Note that this is exactly the

condition of gapless superconductivity. In this case also

. The regular

functions are found from eqn (6.45):

(11.1)

within the first-order approximation in and the zero-order approximation in

spatial gradients. The characteristic energy scale in eqn (11.1) is

Therefore, the external frequency

~ is small compared to the energy scale of

the regular Green functions. One can thus safely use the kinetic equations

obtained by expansion in small

. The collision integrals for magnetic scattering

in eqns (10.100), (10.102) become

All other terms are small in .

Kinetic equations (10.100), (10.102) become

(11.2)

end p.213

and

(11.3)

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The phonon collision integrals are small as compared to the magnetic scattering

and can be neglected. We see that the corrections f

and f

to the equilibrium f

(0)

are proportional to and are thus small as compared to the terms with f

(0)

the Keldysh function eqn (10.62). Therefore

(11.4)

As a result, the equation for the order parameter takes the form

(11.5)

It is instructive to compare this equation with eqn (1.85). These two equations

coincide if one replaces the imaginary part

in eqn (11.1) for the functions

R(A)

with an infinitely small i . However, the expression (11.4) for the Keldysh

function which has resulted in eqn (11.5) is not exact; it requires that the

deviations from equilibrium f

1, 2

are small. It is the gapless condition

that makes f

1, 2

small. We thus confirm the conclusion of Section 1.2.2 that the

TDGL scheme works only for gapless superconductors when deviations from

equilibrium are negligible.

Equation (11.5) yields

(11.6)

Since , the derivative of the distribution function becomes f

(0)

= 2 ( ). The last term in eqn (11.6) gives

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The first term in the r.h.s. of eqn (11.6) gives the stationary part of the

Ginzburg–Landau equation (Abrikosov and Gor’kov 1960). We have from eqn

(11.6)

(11.7)

The expression for the current is obtained in exactly the same way as before. We

have for the vector part of the distribution function from eqn (10.103)

end p.214

Thus the normal current is j

E. The total current becomes

(11.8)

One can easily see that the system of equations (11.7) and (11.8) can be written

in the form of general equations of relaxation dynamics, eqns (1.65), (1.67) used

to derive the TDGL model in Section 1.2.1 with the superconducting free energy

The order parameter relaxation constant as defined in Section 1.2.1 is

The relaxation times are

Their ratio u = 12.

We conclude therefore that gapless superconductors with large concentration of

magnetic impurities can be exactly described by the TDGL model. The reason is

that the distribution of excitations is very close to equilibrium. It is the small

parameter

which makes the deviation from equilibrium negligible. We shall

see in the next example, that the TDGL model starts to fail as the deviation from

equilibrium increases.

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11.2 Generalized TDGL equations

The example which we discuss here is more interesting and more realistic in a

sense that it does not require strong limitations necessary for gapless

superconductivity. Of course, we need, some restrictions to simplify the general

kinetic equations. Consider a dirty superconducting alloy with the mean free path

or . We assume that the temperature is close to T

Moreover, we assume that variations in space and time are slow:

(11.9)

In our example, the inelastic scattering by phonons is of major importance. Since

the largest wave vector is k

(T) and D

(T)

, the condition of

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slow spatial variations is certainly satisfied if . The gradients

can satisfy eqn (11.9) also because the characteristic scale of the particular

problem may happen to be large compared to

(T) for one or another reason; the

condition

can be lifted in this case. As a reward to the

restriction of eqn (11.9) we obtain a possibility to derive a closed set of

time-dependent equations for the order parameter magnitude and phase which

resemble the TDGL model. On the other hand, they are more general and allow

us to follow the crossover from the simple TDGL scheme to a more realistic

dynamics for systems with strong deviations from equilibrium. The TDGL-like

equations in this case were derived by Watts-Tobin et al. (1981).

Consider first the regular Green functions. Within the leading approximation in

spatial gradients and in the external frequency, one has from Eilenberger

equation (5.84)

(11.10)

The phonon self-energies are

(11.11)

and

(11.12)

Energies ~ T dominate in the integrals in eqns (11.11) and (11.12). At these

energies, the functions g

R(A)

±1 and f

R(A)

~ /T 1, therefore,

and

~ ( /T

)

, where the electron–phonon mean free time

defined by eqn (10.84).

Equation (11.10) gives

Kopnin, Nikolai, Senior Scientist, Low Temperature Laboratory, Helsinki University of

Technology, and L.D. Landau Institute for Theoretical Physics, Moscow

Theory of Nonequilibrium Superconductivity

Print ISBN 9780198507888, 2001

pp. [216]-[220]

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Using the normalization [g

R(A)

]

R(A)

= f

R(A)

= 1 we obtain

(11.13)

We choose to be real for brevity at this point, and put f

R(A)

= f

R(A)

within the

leading approximation.

end p.216

Kinetic equations (10.100) takes the form

(11.14)

whence

(11.15)

Equation (10.102) becomes

(11.16)

We can neglect the spatial derivative in eqn (11.16) because it is small as

compared to the other terms when

. Fortunately, it is this range of

which gives the main contribution to the equation for the order parameter.

Indeed, for higher energies

~ T, the spatial gradient could no longer be ignored

since the other terms in eqn (11.16) also become small. However, as we shall

see in a moment, the function f

decreases rapidly as increases. It becomes

small for

~ T so that the energy range ~ T does not contribute much to the

integrals which contain f

. Equation (11.16) yields, after a little algebra,

(11.17)

where C is independent of . In contrast to f

, the function is not expected to

decay for

~ T. Equation (11.17) has been derived using the argumentation as

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follows. We note that the first term in f

satisfies eqn (11.16) exactly while

transforms into C f

(0)

/ for energies ~ T and larger. Since energies ~ T

dominate in the collision integral the expression (10.85) for J

takes the form

Remind that it is assumed T. Inserting this into eqn (11.16) we arrive at eqn

(11.17).

The constant C is to be found from the charge neutrality condition. The change in

the electron density (10.71) due to nonequilibrium processes is

(11.18)

The change in the electron density should be zero. The second term in the r.h.s.

of eqn (11.18) is determined by

~ T. For these energies, g 1. Using eqn

(11.17) we find

end p.217

Therefore

(11.19)

The function f

decreases with and becomes small for . This behavior is

exactly what we expect for f

Consider now the self-consistency equation for the order parameter. We have

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