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

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SECTION 5.1. SOURCE OF EXCITATIONS 121

dependence of g on the angle between the vectors p and A may be approximated

by the first spherical harmonic:

Below we will consider the monochromatic field: As follows from

(5.3), the elements

containing even powers of the field are nonzero in the equation for and the

elements

containing odd powers of the field amplitude A are nonzero in the equation for

5.1.3. Excitation Source in Normal Metals

The collisions with impurities disappear from the equation for In contrast,

impurities play a major role in the equation for For one gets from (5.3):

Because the value of the

functions (the equations for them contain on the left-hand side) are small

compared with in the frequency range (5.2):

Hence only may be retained in the equation for g, and it follows finally from

(5.3) and (5.6)*:

where is the elastic scattering time of electrons entering (5.11) in view of (5.9),

(4.5), (4.2), (2.37), and (2.8). Substitution of (5.11) into (5.9) gives the equation

* As follows from Eqs. (5.9) and (5.10), at only single-quantum transitions must be taken into

account in the absorption acts of photon by electrons. The n-quanta absorption probability is small in

parameter

122 CHAPTER 5. BASIC EQUILIBRIUM PROPERTIES

where we have taken into account that is the (even

over ) excitation distribution function.

5.1.4. "Dirty" Superconductors

We turn now to the case of a superconductor.* Retaining in the isotropic part

only the diagonal term we will consider equations for separate components

of From (3.63) it follows immediately that:

In this chapter we will consider states that are symmetric in the electron-hole

branches. Then the collision integral takes a simplified form [cf. Eqs. (4.2) and

(4.7)]:

and, as earlier, the impurity contributions disappear from (5.14).

In the equations for we can neglect the phonon and electron self-energies,

compared with the one related to the impurities. By analogy with the normal state,

we will use a zero-field approximation for isotropic spectral functions and a linear

approximation for anisotropic spectral functions:

where

Solving now (3.63) subject to (5.15) to (5.18), we find

Introducing, in accordance with Sect. 3.2, the distribution function :

*The specifics of the problem considered in this chapter permit the use of a gauge with a real order

parameter.

SECTION 5.2. STIMULATION EFFECT 123

we find from (3.63) and (5.14) the kinetic equation

in which

The collision integral in (5.21) corresponds to electron–phonon and electron–elec-

tron collisions:

and in accordance with (5.14) we should put in (4.119) and

(4.36). In the general case, without any restriction on the function these integrals

were found in Sect. 4.2.

5.2. STIMULATION EFFECT

5.2.1. Nonequilibrium Self-Consistency Equation

The kinetic equations derived in the previous section contain the nonequili-

brium gap connected with by the relation (4.6). Using the expressions for

one can obtain a self-consistency equation

which differs from the equilibrium equation by replacement of tanh

sign by sign where is the nonequilibrium distribution

function [cf. (1.154)].

We will consider in more detail how the electromagnetic field of frequency

influences the nonequilibrium gap A formal condition allows

one to go far enough analytically. At the field term in (5.21) [we will denote

it ] is the difference of some function at the points shifted along the -axis by

the value . Substituting in i the equilibrium function

124 CHAPTER 5. BASIC EQUILIBRIUM PROPERTIES

instead of , expanding and keeping only the first, nonvanishing two terms in the

series, we obtain:

The first term is defined mainly in the region and the second in the range

5.2.2.

Relaxation-Time Approximation

To find a solution for the linearized equation, we will subdivide the variation

of distribution function into parts:

For the function the collision integral in (5.21) allows us to use the

relaxation time approximation. Indeed, substituting one can

verify that the terms containing as integrands are smaller than other terms in

which is simply a factor. Omitting the smaller terms, we find an approximation

(the approximation; for more details see Sect. 11.3)

The damping may be found using explicit expressions for collision integrals

(4.36) and (4.119). It is determined by the range of integration over , which is

comparable with

and, up to the corrections has the same value as in a

normal state:

5.2.3.

Solution for Distribution Function at

Introducing the notation we find at

At the value of does not depend on Substituting (5.28) into the

integral part of (omitted in the approximation), one can verify that the

correction has the relative smallness but is nonzero in the region

Subsequent iterations do not change this result.

The correction which corresponds to the second term in (5.26), may be

investigated in the same manner. This correction has a structure analogous to the

above-mentioned correction to (5.28). Thus the full variation of the distribution

SECTION 5.2. STIMULATION EFFECT 125

function consists of the part (5.28), which dominates at and of a small

“tail,” diminishing at To calculate this tail, one must go beyond the

approximation. The function enters the expression (5.25) for with the weight

Consequently, the main contribution comes from the region

This allows one to take into account only the part (5.28) when determining

and confirms the applicability of the approximation in the vicinity of

5.2.4. Enhancement of the Gap

We substitute now the expression (5.28) into the gap equation (5.25), trans-

forming it preliminarily to the form

where now Accounting for

we find

where

Comparing (5.31) with the dynamic contribution, we can estimate the relative value

of the kinetic effect:

As (the latter inequality is the condition of a single-quantum

absorption of the field’s energy), one may conclude that high-frequency electro-

magnetic radiation should stimulate superconducting ordering in the film. The

enhancement of a superconductor’s critical parameters (critical temperature, cur-

rent, etc.) was indeed observed experimentally.

1,2,24

As follows from (5.31), in weak

fields the enhancement effect grows with the intensity of radiation as One can

expect that the enhancement of by an order of its value may be reached

in fields with the intensity

126 CHAPTER 5. BASIC EQUILIBRIUM PROPERTIES

(assuming that the experiment is carried out at optional frequencies At such

intensities, as can be seen from (5.1), the dynamic influence of the field is

insignificant. The influence of the field on the superconductor’s single-particle

electron excitations is also small [i.e., the change of spectral functions (3.66) is

small and becomes essential only at significantly higher intensities:

Ac--

cordingly, the model equation (5.25) remains valid in fields with intensities (5.34).

Nevertheless, the experimentally observed values of are only on the order of

several percent,

24,25

even in optimal cases. The reason probably is that the “heating”

processes in the superconductor’s electron subsystem become essential at relatively

moderate intensities of the electromagnetic field. Such processes may occur, for

example, when the electron with the energy (an energy the electron can

obtain by sequential stages of photon absorption) collides with the Cooper pair and

decays into three electronic excitations. This mechanism is analogous to the

mechanism of “shock ionization” and is described by the electron–electron colli-

sion integral (4.36) (by terms that are proportional to the factor ). Such processes

increase the total number of excitations and lead to the effective damping of the gap

(see, e.g., Ref. 26).

5.3.

PHOTON–ELECTRON INTERACTIONS

As was established in the preceding section, high-frequency electromagnetic

radiation influences a superconductor by the Eliashberg mechanism, effectively

redistributing electrons and holes in the momentum space, so that the “center of

gravity” of the Fermi distribution function is shifted to higher energies, while the

total number of excitations remains constant. As a result, the states near the Fermi

surface at the gap edge become unoccupied and this leads to an increase in the

superconducting order parameter, according to the self-consistency condition. Even

though the enhancement of the order parameter is rather small, the detection of the

effect allows us to estimate the characteristic time scales that characterize the

microscopic processes in superconductors. For example, the frequency range of

electromagnetic radiation, which can produce the enhancement effect, is restricted

by the value from above and by the value from below. Further theoretical

analysis of the enhancement mechanism also reveals the existence of other condi-

tions that are important for its realization. One of these conditions is the smallness

of an electron’s mean-free-path:

where is the electron’s elastic scattering time on impurities. In real superconduc-

tors, the condition (5.35) need not be fulfilled. Moreover, the reversed inequality

may take place. We will consider now this situation, which would arise in the latter

case.

SECTION 5.3. PHOTON–ELECTRON INTERACTIONS 127

To study the enhancement problem in perfect specimens, it is necessary first

to derive the appropriate expression for the nonequilibrium source because the

kinetic equation (5.21) describes the case of dirty superconductors. To solve that

problem, it is expedient to use the quantum description for the interaction between

electrons and photons.

5.3.1. Quantum Description

The Hamiltonian of this interaction in superconductors has the form (the gauge

is used):

Introducing the operator

where and are the photon creation and annihilation operators (k and

correspond to the momentum and polarization of the photon), and

Moving to the Nambu representation the Hamiltonian ac-

quires the form:

For nonequilibrium superconductors, we will use the Keldysh diagram tech-

nique (mentioned in Sect. 3.3), in which the electron self-energies are represented

by the matrices

where the quantities are matrices in the Nambu space. The

electron and photon Green’s functions are defined by the equations

128 CHAPTER 5. BASIC EQUILIBRIUM PROPERTIES

For the Fock states of an electromagnetic field, the photon propagators

have the forms

where is the photon occupation number, The

vertices where the energy-momentum transfer takes place are presented in the

Keldysh technique

by objects of the kind where the two upper and all the

lower indices correspond to electrons, while l corresponds to Bosons. In the case

of electron–photon interaction, one can find using (5.36):

Using standard graphic techniques and assuming that the propagator links two

vertices with electron momenta and those due to the transverse character of

electromagnetic field we find, taking into account (5.37) and

(5.40),

Summing over the photon’s polarizations and integrating over using also the

relation between the -function and the distribution function of the electron-hole

excitations (see Sect. 3.3), the nonequilibrium source ) may be found in the

form of the collision integral of electrons with photons:

SECTION 5.3. PHOTON–ELECTRON INTERACTIONS 129

where

and In the “dirty” limit, for the case of a monochromatic field in the

absence of branch imbalance expression (5.48) must be converted into

the field term of the Eliashberg kinetic equations, obtained by the classical descrip-

tion of an electromagnetic field in Sect. 5.1. We will confirm this now.

5.3.2.

Collision Integral as a Nonequilibrium Single-Electron

Source

In the presence of impurity scattering, the energy-momentum conservation

laws [for the relaxation channel in (5.48)] may be presented in the form

130 CHAPTER 5.

BASIC EQUILIBRIUM PROPERTIES

where

is a momentum that is transferred to the scattering center (the scattering is

assumed to be elastic). In the limiting case of a classical electromagnetic field, one

has . Retaining in (5.48) the terms proportional to and averaging over

the directions of

, we find

Because we can write

where is the unity vector in the direction p. Denoting the angle between and

as we obtain

where cos Dealing in the same manner with other

we find

This form would be convenient for further angle averaging in the next section.

5.3.3. Classical Field Action in a "Dirty" Limit

Let the external electromagnetic field be a quasi-monochromatic wave having

a spectral width around the “carrying” frequency and an angle distribution

around the direction

. Then