Kopnin N.B. Theory of Nonequilibrium Superconductivity

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Sometimes, one uses the equations for anomalous functions, as well. We obtain

in exactly the same way

(9.40)

Here

end p.180

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The vector part of the equation is

(9.41)

9.4 Stimulated superconductivity

We consider here an example that demonstrates how the quasiparticle

distribution affects the superconducting order parameter. This example is closely

related with experiment and is instructive in that it treats a nonlinear response

of a superconductor to an applied time-dependent perturbation. Consider a

superconducting film placed in a microwave electromagnetic field. A part of

electromagnetic energy is absorbed in the film. The intuitive picture is that, the

superconducting being heated, its temperature grows and the order parameter

decreases. However, this is not always the case. The real behavior is more tricky.

The temperature, indeed, can increase. However, the overall rise in temperature

depends on the cooling rate provided by the thermal contact of the film with the

heat bath. If the heat contact is good, the temperature does not increase

considerably. What happens is as follows. The microwave irradiation with a

frequency of the order of the energy gap produces redistribution of energy

between excitations and shifts the quasiparticle distribution to higher energies.

The states with

~ become depleted. This causes an increase in the energy

gap as compared to its value without irradiation. This effect is called stimulated

superconductivity. It was first theoretically predicted by Eliashberg (1970) and

then observed in many experiments. The detailed discussion of stimulated

superconductivity can be found in the review by Eliashberg and Ivlev (1986).

We consider the simplest case of temperatures close to T

, and restrict ourselves

to dirty superconductors with the order parameter homogeneous in space. We

first find the (averaged over time) distribution function taking into account

absorption of energy from an external electromagnetic field with a frequency

. For such frequencies, the order parameter performs only small oscillations

near its average value (see Section 11.2), therefore we can take

= const.

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. [181]-[185]

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To find the tone-averaged distribution function we use the kinetic equation for

the anomalous function taken at zero frequency. Within the linear approximation

in A

we have from eqn (9.40)

(9.42)

end p.181

We assume that magnetic impurities are absent. The vector parts of both the

regular functions

and

and of the anomalous function

(a)

are proportional

to the vector potential A and have Fourier components at the frequency of the

external field. Therefore, we obtain from eqn (9.41)

where

The vector components of regular Green functions are found from eqn (9.39).

Since the averaged Green functions do not depend on time

we find

Making the corresponding shifts in the frequencies, we have

The vector part of the anomalous function is to be found from eqn (9.41) which

gives

within a linear approximation in A

. Consider the relation

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similar to eqn (9.32). We can neglect the last two terms with the vector

components of regular functions which are small in A

. We find from eqn (9.4)

Making the corresponding shifts in frequencies, we have

end p.182

Finally, inserting

(a)

together with

R(A)

into eqn (9.42) we obtain the equation

for the distribution, function

For temperatures close to T

, the phonon collision integral at zero frequency can

be taken in a

-approxination (sec Section 10.4 for more detail)

where f

= f

(1)

– f

(0)

is the correction to the distribution function f

(1)

defined in

eqn (9.28). As a result,

We omit the averaging brackets for brevity.

The regular functions are given by eqns (5.48), (5.47), and (5.50), (5.51).

Therefore, we find for

> 0

(9.43)

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The term

in eqn (9.43) is proportional to the absorbed power. Indeed, since

the dissipated energy per volume is

end p.183

multiplied by f ho inelastic relaxation time

. This energy is distributed among

N ~ (0) particles. Therefore, the quantity ( / ) is the absorbed energy

per particle. The last term in eqn (9.43) only exists if

> 2 . It corresponds to

creation of new excitations over the energy gap

The nonequilibrium correction to the distribution function is negative. This means

that the number of excitations with energies

~ decreases. It is equivalent to

a decrease in an “effective temperature” of excitations. Let us study how this

affects the order parameter. We define the function

We have for T

where the function P( / ) is

with w = / . The last term is nonzero for > /2. As we mentioned, it

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corresponds to creation of quasiparticle excitations by the external irradiation.

The function

It has a maximum at w = 2 where its first derivative is discontinuous.

Let us consider the equation for the order parameter. We obtain the Keldysh

function from eqns (9.25) and (9.28) in the form

Let us insert this into the order-parameter equation (9.17)

The equilibrium part of the distribution function gives the usual stationary

Ginzburg–Landau equation (6.23). The resulting equation has the form

(9.44)

where

One can generally neglect the first term in eqn (9.44) when .

end p.184

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FIG. 9.1. The order parameter as a function of temperature. The

long-dashed line

(T) shows the order parameter in the absence of

irradiation. The increasing part of W

( ) shown by short-dashed line

is unstable. The arrows indicate the order parameter jumps with

varying the temperature.

One can see that a nonzero order parameter exists even if T > T

because W > 0

for not very high

. For the irradiation intensity such that the

temperature dependence of the order parameter is determined mainly by the

first term in W

. The maximum temperature T

max

in this limit is determined by

At this temperature, = /2. If the temperature is increased further the order

parameter jumps to zero. The increasing branch of W (

) is unstable. The order

parameter approaches the equilibrium value with decreasing temperature. The

temperature dependence

(T) is shown schematically in Fig. 9.1.

end p.185

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10 KINETIC EQUATIONS

Nikolai B. Kopnin

Abstract: This chapter derives the kinetic equations for the two-component

distribution function in a gauge-invariant form. The collision integrals for

interaction of excitations with impurities, phonons, and with each other are

written down. The gauge-invariant expressions for electron density, electric

current, heat current, and order parameter are obtained. Kinetic equations for

dirty superconductors are derived. Heat conduction in superconducting state is

considered.

Keywords: kinetic equation, gauge invariance, distribution function,

collision integral, impurity, phonons, electron-electron interaction, heat

current

We derive the kinetic equations for the two-component distribution

function in a gauge-invariant form. We write down the collision integrals

for interaction of excitations with impurities, phonons, and with each other.

10.1 Gauge-invariant Green functions

In this section we introduce a more convenient description of a time-dependent

state of a superconductor in terms of the gauge-invariant Green functions. We

know that the gauge invariance is the basic property of the superconducting

state. The Eilenberger equations are gauge-invariant and so are the regular

(retarded and advanced) quasiclassical Green functions for a time-independent

state of a superconductor which are determined by these equations. Of course,

the gauge-invariance holds for a nonstationary state, as well. However, the

generalized distribution function introduced in Chapter 9 and the regular

functions for a nonstationary state, treated separately, can become non-gauge-

invariant if special precautions are not taken.

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. [186]-[190]

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Remind that according to the general gauge-invariance (see also page 19), the

electromagnetic potentials can be changed according to

(10.1)

where f is an arbitrary function, without changing the electromagnetic fields

(10.2)

In superconductors, the transformation eqn (10.1) is accompanied by the

transformation of the order parameter phase

(10.3)

Therefore the combinations

(10.4)

are also gauge-invariant; they are called the gauge-invariant potentials. Of

course, the order parameter magnitude is gauge-invariant, too. Therefore, the

equations

end p.186

for the Green functions and for the distribution function should contain, in

addition to the electric field E and magnetic field H, the vector and scalar

potentials in proper combinations with the spatial and time derivatives of the

order parameter phase such that they appear as the gauge-invariant potentials

eqn (10.4). Note that the time-dependent Ginzbnrg–Landau model discussed

earlier in Section 1.2.1 is an example of such gauge invariant theory because it

only contain the potentials (10.4) and the order parameter magnitude.

In this section we define the Green functions and the generalized distribution

functions in a gauge-invariant way. We start with the transport-like equations

obtained in Section 8.4 and derive the kinetic equations for the distribution

function in a state which slowly varies in time. More particularly, we assume that

the time variations of the superconducting parameters arc such that the

characteristic frequency is

. These equations will make the basis for

further discussions of nonequilibrium properties of superconductors.

Let us turn back to the definitions of non-quasiclassical Green functions in

Chapter 8. We recall the operator eqn (8.3) acting on the matrix Green function

in the Nambu space. It can be written as

(10.5)

where

(10.6)

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and p = i is the particle momentum operator. The Hamiltonian is

(10.7)

where

(10.8)

see eqn (5.57). The velocity is v = (p)/ p, and m* is the effective mass of an,

electron. As distinct from eqn (3.70), the effective Hamiltonian contains the

scalar potential

which should be separated from a spatially independent part of

the chemical potential, E

The equation for the regular Green function can be written as

(10.9)

The full convolution in the coordinate space is

(10.10)

It is equivalent to the shortcut [AB] used in the momentum frequency

representation. The equation of motion for the total Green function is obtained

from eqn (8.61):

end p.187

(10.11)

The collision-Integral matrix is

(10.12)

The regular Green functions satisfy eqn (10.11) with the collision integral

(10.13)

As before, we use the mixed coordinate-momentum representation

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