Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors

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Time-Dependent Ginzburg–Landau

Equations

7.1. ORDER PARAMETER, ELECTRON EXCITATIONS, AND

PHONONS

The external fields acting on a superconductor may lead to nonstationary phenom-

ena that have to be described by dynamic equations. However, as was shown in the

previous chapters, the set of nonstationary equations in the general case is very

complicated and in addition to the equations for the main parameters characterizing

superconductivity (such as it includes generalized kinetic equations for

distribution functions (see Sect. 3.3). In the vicinity of the critical temperature (in

analogy with the stationary case, Sect. 1.3), one can simplify the general time-de-

pendent Ginzburg-Landau (TDGL) equations by considering the gapless case (Sect.

2.2). For finite-gap superconductors, the attempt to simplify the general scheme

encounters serious difficulties connected with the nonlocal kernels of the integral

equations governing the order parameter. To derive the equations for such super-

conductors, one needs to account simultaneously for the condensate, the excita-

tions, and the interaction between them. The success achieved in this direction

1–6

is due to progress in the kinetic description of single-particle excitations in non-

equilibrium superconductors (see the review articles in Refs. 7–10). The dynamic

equations for the order parameter were obtained in their most complete form by

Watts-Tobin et al.

But in some respects the theory still had some deficiencies,

which we have tried to correct.

In many situations, the possible deviation of the phonon system from equilib-

rium should be taken into account. The role of phonons in the problem considered

is twofold. First, the nonequilibrium in the phonon system may be essential for the

dynamics of the order parameter. Second, the time variations of the order parameter

modulus might lead to excess phonon generation and to phonon exchange between

a superconductor and its environment.

151

152 CHAPTER 7. TIME-DEPENDENT GINZBURG–LANDAU EQUATIONS

7.1.1. Basic Kinetic Equations

We will use here the generalized kinetic equations

11,12

for energy-integrated

Green–Gor’kov functions. As was shown in Chap. 4, these equations are still valid

in the case where the phonon system is not at equilibrium. In a real-time approxi-

mation,* the equations may be written in a very compact form:

Here

where is the Fermi momentum and r is the quasi-classical coordi-

nate, the Fourier transform of which is denoted by k. In Eq. (7.1) (A, ) are the

electromagnetic field potentials

7.1.2. Normalization Condition

Equation (7.1) must be supplied with the normalization condition, which

allows us to select the necessary solution of homogeneous (relative to the

equations (7.1):

In Eq. (7.1) the integration over the intrinsic spatial coordinate (or, depending on representation, over

the momentum variable) is assumed in general.

SECTION 7.1. ORDER PARAMETER, ELECTRON EXCITATIONS, AND PHONONS 153

We wrote this condition in Sect. 3.3 (see Eq. 3.68). It may be proven in the following

way (see, e.g., Ref. 5). Equation (7.1) may be presented in the form

where the operator as follows from Eqs. (7.6) and (7.1), is

Since the convolution * is commutative (which follows directly from its definition

in 3.70), it is easy to see that the condition

is compatible with the equation

which follows from Eq. (7.6). The value of a constant in (7.8) can be obtained by

considering Eq. (7.8) either in a superconducting region that is in an equilibrium

state, or in a normal area, where . The latter option is simpler, and it is possible

to calculate the constant immediately. Larkin and Ovchinnikov

introduced the

normalization in which Because a particular value of this constant is of

no importance, we will retain the normalization const , used earlier.

7.1.3. Definition of Order Parameter

We will now use the results obtained in Chaps. 3 and 4. The self-energy

function is an additive quantity that contains certain terms corresponding to the

interaction of electrons with impurities, with phonons, with each other, and so on.

The nonequilibrium order parameter in a weak-coupling limit is the

dimensionless electron–phonon coupling parameter) is defined by the formula

where the self-energy function representing the interaction of electrons with pho-

nons is (see Sect. 4.5):

154 CHAPTER 7. TIME-DEPENDENT GINZBURG–LANDAU EQUATIONS

The phonon propagator is expressed in terms of the nonequilibrium phonon

distribution function by the relations:

When phonons are in equilibrium, the contribution of the second term in Eq. (7.11)

is small by the parameter so to find one can use the simplified equation

that follows from Eqs. (7.10) to (7.13)

If the phonon distribution function is localized at energies and has

no singularities as a function of a real argument (this will be assumed further),

Eq. (7.14) may be applied to the situations with nonequilibrium phonons.

7.1.4. Nondiagonal Collision Channel

To obtain the propagator , one can use the equation that follows from

Eq. (7.1) for the nondiagonal “Keldysh” component. Separating in Eq. (7.1) the

virtual processes (see Sect. 4.1), which leads to Eq. (7.14), and ignoring the

renormalization terms, one finds the expression for the (4.2):

where the coefficients are connected with the self-energy functions [the defini-

tion of these quantities follows from a comparison of Eqs. (7.15) with (4.2)]. Taking

into account the nondiagonal channel in the kinetic equation for the electron-hole

distribution function we get the canonical form of the collision integral. We

recall here that the general expression for the which satisfies the

normalization condition (7.8), was discussed in Sect. 3.3, where the functions ,

and were introduced [see Eq. (3.72)]. These functions connect Green’s functions

with the distribution functions of electronlike and holelike excitations.

SECTION 7.1. ORDER PARAMETER, ELECTRON EXCITATIONS, AND PHONONS 155

7.1.5. Spectral Functions

According to Eqs. (3.77), (3.79), and (3.80), the functions

and

(as well as

are of a general type, i.e., they have definite only in

absence of an

external electromagnetic field. In the latter case they are equal to

where is the energy damping of electrons. Introducing also the functions

we can express the as

7.1.6. Charge Density

In expression (7.22), only the lowest convolution corrections are kept (the

contribution from the is negligible).

156 CHAPTER 7. TIME-DEPENDENT GINZBURG–LANDAU EQUATIONS

Using Eqs. (3.99), (3.77), (3.79), and (3.80), one can establish that in super-

conductors the charge density in the above approximation must have the form

which is strictly gauge-invariant. [Note that in Refs. 2–6 a slightly deficient

expression for is given; it may be obtained from Eq. (7.23) if which is

not fulfilled at

Separating in (7.14) the equilibrium part and making standard calculations (see

Sect. 1.4), we get an equation for the order parameter near

where is the diffusion constant and is the Riemann

7.1.7. Gap-Control Term

Equation (7.24) for (

,t) contains a “gap-control” term:

where . The nonequilibrium functions and should be found

from the kinetic Eq. (7.1), where one can assume the phonon system to be initially

in equilibrium. Note that the terms generated by make insignificant contribu-

tions to Eq. (7.24) because the function is nonzero at Only the values

of play a major role in the integrand of (7.25). For this reason, one can neglect

the terms proportional to in expression (3.77), which then takes the form

From the kinetic equations for and in the absence of the potential in the local

equilibrium approximation, it follows that

We will briefly follow the derivation procedure of these relations to clarify the

essence of local equilibrium approximation.

SECTION 7.1. ORDER PARAMETER, ELECTRON EXCITATIONS, AND PHONONS 157

7.1.8. Local-Equilibrium Approximation

If the characteristic frequencies and gradients of the electron system perturba-

tion obey the relations*

where is the damping caused by inelastic processes in the electron system, then

in the kinetic equations [(7.1) and (3.63)] that define the functions and , one can

neglect the left-hand sides and the terms connected with the Hamiltonian This

means that the functions and do not depend explicitly on the space

coordinate

and time

. Only implicit dependence on

and t remains through

the parameter

,t), which enters into (7.1). This means that owing to effective

inelastic collisions, the behavior of single-particle electron excitations in an

external field is fully determined by the evolution of the order parameter that

governs the formation of the distribution function (and does not depend, e.g., on

the diffusion mechanism). In other words, local equilibrium between the system of

single-particle excitations and the pair-condensate is taking place.

7.1.9. Determination of the

In this approximation from the diagonal components of (3.63), the equation

for the follows:

[the series expansion of functions and in (7.29) may be restricted to the

first terms owing to the quasi-classical conditions]. Inserting expression (7.22) into

(7.29) and omitting convolution corrections, one finds

where the transformation rule is used and the inequalities

have been taken into account. The functions are expressed through the

collision operators obtained in Chap. 4:

We also assume the quasi-classical character of the external fields

(

) and

158 CHAPTER 7. TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS

Furthermore, we will assume that the electron-phonon collisions provide the most

effective channel of inelastic relaxation and represent in the

form

where

In Eqs. (7.34) to (7.36), the function is the distribution function of phonons,

which as yet is assumed to be an equilibrium one:

In the vicinity of critical temperature, where for the collision integral

the relaxation-time approximation may be used*

*This opportunity occurs because the perturbation of the distribution function is localized in an energy

range smaller than the temperature diffusion scale. Because of this, the term containing the nonequili-

brium distribution function in the integral form is smaller than the “free” term. We stress this

circumstance because it remains valid also in derivation of the function (see later discussion).

However, sometimes the approximation is criticized and, moreover, negated (see, e.g., Ref. 6) as

violating a condition related to the particle number conservation. In our calculation scheme, the missing

term automatically appears from the gauge-transformation rules for the functions and which were

established in Sect. 3.3. (An approach equivalent to that of Ref. 6 is used in Sect. 13.3.4.)

SECTION 7.1. ORDER PARAMETER, ELECTRON EXCITATIONS, AND

PHONONS

159

where

Using Eqs. (7.32), (7.33), and (7.38), we find from (7.30) the first expression in

(7.27).

7.1.10. Determination of the

Let us now determine the function The nondiagonal elements of Eqs. (3.63)

and (7.15) are essential, because the first term (proportional to in Green’s

functions and (7.22) does not contribute to the equation

Accounting for this, one finds

The same approximations are used here as in deriving Eq. (7.29). Using the

relation

one obtains from Eqs. (7.38), (7.41), and (7.42) the second part of expression (7.27).

The potential may be restored now in (7.27) with the help of Eq. (7.26),

where one should make Omitting the term proportional to [its

contribution to Eq. (7.27) is small], one finds

As for the function the quadratic in the term may be omitted—it is

proportional to The linear term, which takes into account the transformation

rule for gives the equation

160 CHAPTER 7. TIME-DEPENDENT GINZBURG–LANDAU EQUATIONS

7.1.11. Order Parameter Equation

Utilizing Eqs. (7.43) and (7.44), the equation for an order parameter [Eqs.

(7.24) and (7.25)] takes the final form

The function in (7.45) is due to the contribution of the nonequilibrium

phonon subsystem.

7.1.12. Contribution of Nonequilibrium Phonons

We will now trace the origin of If the phonons are shifted from

equilibrium, the collision integral (7.38) acquires the contribution

which follows from Eq. (7.33). The factor is the functional and is linked with

the deviation of the phonon distribution function from the equilibrium

This leads to the redefinition of the function (7.43), which now has the form

The function (7.44) remains unchanged. Substituting (7.48) into (7.25), one finds

for ) the form