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

Подождите немного. Документ загружается.

SECTION 3.3. QUASI-CLASSICAL APPROXIMATION 81

Here If the external momentum in (3.56) is close to the

Fermi surface: then the main contribution to integral (3.56)

is provided by the region (the integration over in the regions remote

from the Fermi surface renormalizes the chemical potential, which is insensitive to

details of the electron distribution in the vicinity of The D-function now depends

only on the angle between Using the chain

of equalities

it is easy to establish that is expressed by Green’s functions, integrated over the

energy variable:

Similar conclusions follow for other self-energy functions, so that one has:

3.3.2. Eliashberg Kinetic Equations

Now let us transform Eq. (3.51). Ignoring the quadratic terms in A, and moving

to the quantities , one finds by multiplying Eq. (3.52) by from the

left and by from the right:

82 CHAPTER 3. NONEQUILIBR/UM GENERAL EQUATIONS

where Subtracting from Eq. (3.62) Eq. (3.61) and integrating the result

over we find the equation for the function (3.58):

The set of arguments of the functions entering this equation is analogous to

that of electron distribution function. For this reason Eq. (3.63) may be called the

generalized kinetic equation. The quantity

is the collision integral (at present between electrons and phonons). The equation

for the function g

(

)

[which may be obtained from Eq.(3.38) by the procedure used

above] is similar to Eq. (3.63), although the quantity must be replaced by

In the equilibrium case, when the field

is absent:

i.e., , is proportional to the density of single-particle excitation states (for

this reason are called spectral functions

As for the functions from Eq. (3.58) in the equilibrium case, the relation

follows

where is the distribution function of the electronlike and holelike

Fermi excitations. In the nonequilibrium case, as we will see, in general

does not necessarily coincide with However, the correct generalization of Eq.

(3.67) cannot be achieved by the trivial replacement as

might be thought. Such a replacement would retain as an odd function in

whereas in the general case can have an even in the part also. The necessary

generalization can be obtained with the help of the normalization condition for

g-functions, as was shown for the equilibrium case by Eilenberger

and for the

nonequilibrium case by Larkin and Ovchinnikov.

SECTION 3.3. QUASI-CLASSICAL APPROXIMATION 83

3.3.3. Normalization Condition

In a real-time representation (see Eq. 3.53), this condition has the form

where

and the symbol * is the convolution in time according to

The normalization condition (3.68) is satisfied identically by the following

substitution

where is an arbitrary matrix function of which may be represented as the

sum of the Pauli matrices, the diagonal matrices only participating in this decom-

position:

The functions and are linked with the distribution function of electron-hole

excitations Before showing this relation, we consider some general properties

of and establish their gauge transformation laws.

3.3.4. Gauge Transformation

The basic gauge transformation law for a field operator under the

transformation of scalar potential

(where is an arbitrary function). From (3.73) and (3.53) it follows that the

propagator (as well as the spectral functions and is transformed according

*To avoid interrupting the presentation, we will prove this statement in Chap. 7. It is worth noting that

in the theory of superconductivity, the normalization condition is proved only at the “physical level”

of rigor.

84 CHAPTER 3. NONEQUILIBRIUM GENERAL EQUATIONS

In the quasi-classical limit, when the propagators are fast-varying functions of

a difference variable and slow-varying functions of a sum

the expression (3.70) may be presented in the form

In Eq. (3.75) the following notations are used: and the

frequency corresponds to the Fermi transform over the difference argument

. From Eqs. (3.74) and (3.75) a transformation law follows for diagonal

components of propagators*:

Hereafter the terms proportional to are omitted owing to the assumed quasi-clas-

sical character of

At the same time, one can make a gauge transformation of the function defined

by expression (3.71). Taking into account that the functions and transform in

analogy to (3.74), one can obtain the coincidence of the corresponding result with

(3.76). This provides the transformation laws for the functions and

The functions and in (3.77) are defined by the relations

If we have in accordance with (3.74), (3.71), and (3.78),

*At this stage it becomes clear that expression (3.74) is an equivalent form of the usual relation for

Green’s function: in the gauge transformation. Expanding the expo-

nent over the “fast” time and finding the Fourier transforms over the difference variable, one

can obtain expression (3.76) for the appropriate matrix component.

SECTION 3.3. QUASI-CLASSICAL APPROXIMATION 85

Note that, owing to Eqs. (3.77) and (3.79), the functions and (in analogy with

and are functions of a general type. They have definite parity only in the

absence of external fields.

3.3.5. Electron and Hole Distribution Functions

In the absence of external fields, as may be seen from the definition of these

functions and Eq. (3.63), and are odd functions of and

are even functions of Introducing an arbitrary function (here

we can write (making

Because the function should be determined further from the kinetic equations,

there is still an arbitrariness in the choice of coefficients and It is convenient

to choose them in the form

where

Then the expressions for take the form

where

86 CHAPTER 3. NONEQUILIBRIUM GENERAL EQUATIONS

The constant in expression (3.68) may be chosen to be Without a loss of

generality, this ensures a limiting transition to expressions (3.66) and (3.67) in the

absence of imbalance, and simultaneously assigns to the function a transparent

meaning for the energy distribution function in “pure” superconductors (this will

be shown later). At the same time, it might be noted that the description of a

superconductor in terms of integrated over the is also valid

in cases when the concept of the energy spectrum turns deficient and becomes a

bad quantum number (e.g., in the case of superconductors containing very many

impurities).

3.3.6. Kinetic Equations: Keldysh Option

As shown by Keldysh,

in a nonequilibrium system, the Green’s function

technique allows us to develop kinetic formulas without integrating over energies.

In such cases, the energy distribution function ofexcitations is connected to Green’s

functions, integrated over the frequency variable. If the energy spectrum is well

defined, these two methods are usually adequate.*

For the causal Green’s function using the definition (3.53) and solving at

the Dyson equations by analogy to the normal metal case (cf. Ref. 12), we

have:

where the factors are defined by the relation (1.23), –by (1.132),

and corresponds to the distribution function of electronlike and holelike

excitations.

Following Aronov and Gurevich,

one can introduce a spectral representation

for the causal function

*We

have

noted

the

advantage

energy-integrated

functions

for

“dirty”

superconductors. At the same

particle cascading in superconductors. In such situations usual (Keldysh’s) formulation of the kinetic

scheme is preferable.

time, this technique fails when considering processes far from the Fermi surface, e.g., a high-energy

SECTION 3.3. QUASI-CLASSICAL APPROXIMATION 87

Comparing Eqs. (3.89) and (3.90), one can find

In the same manner

Using these relations one can obtain the canonical forms of the collision

integrals in superconductors

in a manner completely analogous to the case of a

normal metal.

We will obtain the same kind of collision integrals (in rather

than in representation) in the next chapter by employing the propagators

integrated over energies. When both representations are applicable, these collision

integrals are completely adequate.

3.3.7. Expressions for Charge and Current

In general, the technique of the energy-integrated Green’s functions is more

convenient for those problems where the kinetic processes occur in the vicinity of

the Fermi surface. In these cases it provides a powerful tool for the study of both

pure and dirty superconductors.

On the contrary, if the main processes occur in the regions remote from the

Fermi surface, a straightforward application of this technique may lead to erroneous

results. This difficulty may be overcome by properly accounting for the contribution

that results from the equilibrium in the Green’s function formula. Consider, for

example, an expression for the electron charge in a superconductor. In the repre-

sentation of the discrete imaginary frequencies, one can write an expression (2.86)

for the number of electrons in superconductors as

We will separate in this expression the contribution supplied by a zero-order Green’s

function includes the quasi-classi-

88 CHAPTER 3. NONEQUILIBRIUM GENERAL EQUATIONS

cal scalar potential in the system. The deviation in the number of particles induced

by the potential (in the first order in has the form

where is the equilibrium electron density. Thus the electron density in non-

equilibrium superconductors is

where [the prime in (3.98) may be omitted if the

integration over is assumed the principal value sense in symmetrical limits]. From

(3.98) the relation follows for a charge in nonequilibrium conditions:

The situation with expression (2.87) for the electric current is analogous. The

correct accounting of the contribution supplied by the regions remote from the

Fermi surface results in the disappearance of the contribution from the second term

in Eq. (2.87), which must be absent if the technique

is used.

References

1. A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskiy, Quantum Field Theoretical Methods in

Statistical Physics,

2nd

ed., pp. 75–76, Pergamon, Oxford (1965).

2. A. B. Migdal, Interaction between electrons and lattice vibrations in a normal metal, Sov. Phys.

JETP

7(6),

996–1001 (1958)

[

Zh. Eksp. i Teor. Phys.

(6), 1438–1446 (1958)].

3. A. S. Alexandrov and J.Ranninger, Bipolaronic superconductivity, Phys. Rev. B24(3), 1164–1169

(1981).

4. A. S. Alexandrov and N. F. Mott, Polarons and Bipolarons, pp. 179–187, World Scientific,

Singapore (1995).

5. E. G. Brovman and Yu. M. Kagan, Phonons in nontransition metals, Sov. Phys. Uspekhi 17 (2),

125–152 (1974) [Usp. Fiz. Nauk

112

(3), 369–426 (1974)].

6. G. M. Eliashberg, Temperature Green’s function for electrons in a superconductor, Sov. Phys. JETP

(5), 1000–1002 (1960) [Zh. Eksp. i Teor. Phys. 39 [5(11)], 1437–1441 (1960)].

7. E. G. Maximov, in: High-temperature Superconductivity, edited by V. L. Ginzburg and D. A.

Kirzhnits, pp. 1–364, Consultants Bureau, New York (1982).

8. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, pp. 245–310, Pergamon, Oxford (1982).

9. G. M. Eliashberg, Inelastic electron collisions and nonequilibrium stationary states in supercon-

ductors,

Sov.

Phys.

JETP

(3),

668–676

(1972)

[Zh.

Eksp.

Teor.

Fiz.

[3(9)],

1254–1272

(1971)].

10. L. V. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP

(4), 1018–

1026 (1965) [Zh. Eksp. i Teor. Phys.

[4(10)], 1515–1527 (1964)].

SECTION 3.3. QUASI-CLASSICAL APPROXIMATION 89

11. A. F. Volkov and Sh. M. Kogan, Collisionless relaxation of the energy gap in superconductors,

Sov. Phys. JETP 38(5), 1018–1021 (1974) [Zh. Eksp. i Teor. Phys.

[5(11)], 2038–2046 (1973)].

12. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, pp. 391–412, Pergamon, Oxford (1981).

13. G. Eilenberger, Transformation of Gor’kov equations for type II superconductors into transport-

like equations, Z. Phys.

214

(2), 195–213 (1968).

14. A.I. Larkin and Yu. N. Ovchinnikov, Nonlinear effects during the motion of vortices in supercon-

ductors, Sov. Phys. JETP

(1), 155–162 (1977) [Zh. Eksp. i Teor. Fiz.

[1(7)], 299–312 (1977)].

15. A. A. Aronov and V. L. Gurevich, Stability of nonequilibrium Fermi distributions with respect to

Cooper pairing, Sov. Phys. JETP

(3), 550–556 (1974) [Zh. Eksp. i Teor. Phys.

3(9),

1111–1124(1973)].