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

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SECTION 4.2. INELASTIC ELECTRON–ELECTRON COLLISIONS 101

Coefficients

entering (4.37) to (4.39), are given by the following relations:

102 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES

The quantities and are defined by the expressions

Factors a and b, entering Eqs. (4.40) to (4.43), are numbers (of an order of

unity) and are connected with

and

by relations of the type

4.2.5.

Essence of Elementary Acts

The meaning of the elementary acts described by the collision operator (4.36)

is quite transparent. Consider, for example, the term in that is proportional to

With a positive sign of the first component in this term describes the merger

of three electron excitations into a single electron-type excitation. With a negative

sign of three merging electron excitations create an excitation on the hole branch.

Thus in the first case the difference in the number of electrons and holes changes

by 2, while in the second case it changes by 4. Analogous processes are described

by other items in this term. It vanishes in the case of a normal metal when

hence the channel of homogeneous relaxation of electron-hole imbalance

is closed in a normal metal.

Note that the collision integral (4.36) has obtained

such a transparent meaning, owing to the specific selection of the form of the

functions in the expressions for in (3.81) to (3.88).

4.3. KINETIC EQUATION FOR PHONONS

4.3.1. Application of Keldysh Technique

The phonon Green–Keldysh function

is introduced in the usual manner*:

*We omit further below the index of phonon polarization. It may be restored in the final expressions.

As mentioned in Sect. 2.3, in the isotropic model of metals, the electrons interact only with longitudinal

acoustic phonons. This interaction is implied in this chapter. In case of transverse phonons, the

coherence factors would change their signs (see Chap. 12).

SECTION 4.3. KINETIC EQUATION FOR PHONONS 103

The Keldysh indices i, k are the signs minus or plus, according to the location of

the time coordinate on each of the two time axes Recall that

the time on the second axis (with the sign plus) is greater than any time on the first

axis (with the sign minus), and the

-ordering on the second axis proceeds in the

reverse order. The free phonon field operators are real see Sect. 3.1):

Here is the volume of the system; is the dispersion of phonons in normal

metal; and and are the phonon creation and annihilation operators.

The “bare” Green-Keldysh functions, defined by Eqs. (4.47) and (4.48), may

be easily found in the homogeneous and stationary cases. For instance, the expres-

sion for is (cf. Ref. 6):

where is the nonequilibrium phonon distribution function.

In addition, we introduce an operator which acts on the first (second)

argument of the phonon propagator (u is the phonon velocity):

where

and is the third of the Pauli matrices

In general the phonon function obeys the Dyson equation

where all the functions are matrices in Keldysh indices. Note that owing to their

definition (4.47), the Green–Keldysh functions are linear dependent

and consequently the polarization operators are also

linear dependent:

104 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES

The electron Green–Keldysh–Nambu function is defined analogously:

as mentioned in Sect. 3.2. Here and are the Nambu indices of the

field operators

The Green’s function thus introduced (in the absence of interactions that depend

explicitly on spin variables) has the symmetry property

where the bar above the index indicates its reversion [i.e., From

Eqs. (4.54) and (4.56) it follows that:

The functions are defined according to the relation (3.55):

from which [taking into account Eqs. (4.57) and (4.58)] equalities follow:

SECTION 4.3. KINETIC EQUATION FOR PHONONS 105

4.3.2. Quasi-classical Approximation

In homogeneous and stationary cases, the Green–Keldysh functions depend

on the difference of space-time coordinates. If the evolution of the phonon system

is taking place sufficiently slowly, one can assume that all the quantities depend

only weakly on the summary variable and are the functions mainly of the

difference variable , Separating these variables [we use the notations of type

one can perform the Fourier transformation over the difference variables

where obviously, Acting by the operator on Eq.

(4.52) and by on Eq. (4.53), and subtracting, we obtain the result for the

component:

Consider first the right side of Eq. (4.68) and transform it with the help of

quasi-classical conditions. For the phonon system, these conditions mean that the

quantities characterizing its evolution in time and space must be large in

comparison with the characteristic phonon reciprocal frequencies and wave

numbers

This is a good approximation when the perturbation of the phonon system is caused

by the superconducting electron system. Condition (4.69) permits us to simplify in

the usual manner (cf., e.g., Ref. 6) the left side of (4.68). Taking into account that

the operator on the left side of (4.68) may be presented in the form

and carrying out the Fourier transformation of (4.68), we obtain the expression

106 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES

(the arguments of all the functions in parentheses are we have used here

the linear interdependence of and mentioned earlier).

4.3.3. Phonon Distribution Function

To obtain the kinetic equation in terms of the distribution function N(q, r, t), it

is necessary to find a relation between functions N and In the quasi-classical

case, such a relation may be found rather simply. As noted in Sect. 3.1, the

superconducting transition negligibly influences [because

the bare phonon spectrum: Implying the quasi-classical condition for

the phonons, we assume that and obey the relation

For acoustic phonons (only these are important in the isotropic case) and for

momenta that are small compared with the extreme value in the crystal, the

following relation is valid:

Using also the property

[which follows from (4.47)], substituting Eqs. (4.72) to (4.74) into (4.71) and

integrating over within the limits one obtains the following kinetic

equation:

where the quantity

is the inequilibrium source.

SECTION 4.3. KINETIC EQUATION FOR PHONONS 107

Note that expression (4.76) may also be presented in a somewhat different form

if one makes the standard transformation (see Ref. 6) from the matrix [in

analogy with (3.55)] to the linearly independent functions

Now we will specify the polarization operators in Eq. (4.77).

4.3.4. Polarization Operators in Keldysh's Technique

In the Keldysh–Nambu technique, the polarization operator is represented by

a diagram expansion:

Regular Feynman rules

are applied; the only difference is that all the quantities,

including the vertices, are matrices in Keldysh–Nambu indices. Since the transition

to the superconducting state (as well as the interaction with an external electromag-

netic field) affects only a minor smeared region [the bold lines in

(4.78)] are considered as exact functions (they contain electrons interacting among

themselves, with the external field, phonons, impurities, etc.). Because in this

technique the vertices have the matrix structure

we obtain

(

g is the electron-phonon interaction constant)

However, it is more convenient to deal with the linearly independent quantities

(3.55). Moving simultaneously to and omitting

components like (which are identically zero), we obtain

and correspondingly

108 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES

Expressions (4.81) and (4.82) allow one to obtain the collision integral for the

phonon kinetic equation in a superconducting system. All the influence of the

electromagnetic field is contained in Green’s functions for electrons, which are

exact and also account for impurities and other fields acting on the electron system.

As required by the kinetic equation, written in the form of (4.75), we should

move to the (x,p) representation. It is clear that the polarization operators can be

expressed in terms of the energy-integrated Green’s functions. Before making this

transformation, we will derive the expressions for the polarization operators in Eq.

(4.77), using the analytical continuation technique.

4.3.5. Polarization Operators: Analytical Continuation Technique

In a discrete imaginary frequency representation

we have the following expression for the polarization operator:

For brevity we omit the second arguments of Green’s functions which

may be reconstructed from the “decay” conservation law for internal variables:

Rule (4.84) is responsible for the appearance of in (4.83), which differ

from by the reversed directions of the arrows in the diagrams. In addition, the

pair in (4.83) is accompanied (cf. Sect. 1.4.1) by a change in the diagram’s

sign. Starting the analytic continuation of the polarization operator, we consider

each component in (4.83) as the infinite sum of the diagrams of various orders in

the external field. The entire procedure is analogous to that used earlier in deriving

the analytically continued self-energy parts of electron–electron collisions. The

only difference is that the external frequencies here are Bose frequencies (and

naturally there are two electron lines). Since the directions of arrows in the diagrams

do not influence the analytic continuation process, we will consider only the

expression

SECTION 4.3. KINETIC EQUATION FOR PHONONS 109

For simplicity of notation, we omit the signs of internal frequency integration

and summation at the vertices of interaction with the external field and phonons.

Remember that expression (4.85) corresponds to the diagram series of perturbation

theory and contains the sum of the various diagrams up to the infinite order; Green’s

functions for electrons in the polarization loops contain the phonon self-energy

insertions that in turn contain field vertices of an arbitrary order. The diagram of a

specific order in the external field (considered as a function of the complex variable

has a cut on the line Im for fixed imaginary frequencies of the field

vertices, which goes between the uppermost and lowermost banks:

As in the case of the electron–electron self-energy parts, the quantities

represent certain combinations of the field vertex frequencies; here the set of these

combinations and their total number depend on the distribution of vertices along

the electron lines. Assuming that the cuts with

correspond to these lines, we transform the frequency sum in (4.85) to the contour

integral

where C is the contour shown in Fig. 2.1. Deforming the contour which

goes along the banks of the cut, and noting that for the diagrams of any order the

integrals along the big arcs vanish when the corresponding radii tend to infinity,

after some straightforward calculations we obtain:

where is the jump in the Green’s function at the corresponding cut line. The

external variable and the field frequencies remain imaginary. Their combination

determines the set of cuts for the given diagram. Assuming in all diagrams

(the upper bank of the upper cut line), shifting the integration variable in all diagram

expressions (as was done in Sects. 2.3 and 4.2) and summing over all orders of the

perturbation series, we obtain

110 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES

where the G-function is determined as

while the functions are determined directly from the diagram expansion (or

the Dyson equation), where all the Green’s functions for electrons are retarded (or

advanced) and their entire set coincides with the functions figuring in

Sect. 2.3. The expression for also follows from (4.88) for (the lower

bank of the lower cut line):

Using for an expression analogous to (4.90), but with the external

frequencies representing now the Bose field, i.e.:

we obtain after the substitution of (4.88) into (4.92):

Shifting the integration variable in each of the terms in the first integral

in the second integral, and using the identity

(3.48), one gets

Summing in (4.94) all orders of perturbation theory and accounting for the defini-

tion of (4.89), we finally obtain