Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors
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SECTION 4.3. KINETIC EQUATION FOR PHONONS 111
Thus, for polarization operator (4.83), the complete set of functions
is found, which describes the nonequilibrium case. One can show that
they are identical to those obtained earlier on the basis of the Keldysh formula.
4.3.6. Equivalence of Keldysh and Eliashberg Approaches
Consider for this purpose any one of the components in which follow
from (4.83), for example,
and make a Fourier transformation to the spatial and temporal variables. As a result,
one obtains
Now we write down all the terms obtained from the analytical continu-
ation. Taking into account Eqs. (4.83) and (4.97) one finds
To compare this with result (4.82) obtained by the Keldysh technique, we must
make the substitution in the second multiplier of each of the
components in braces (or in brackets) either in (4.98) or in (4.82). In the latter case,
this should be done using the relations (4.63) to (4.65). We use the first possibility,
noting that the Eliashberg functions have the properties
[in the absence of the spin-dependent interactions
After removing the parentheses certain components vanish [for example,
, so Eq. (4.98) can be reduced to the form
112 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES
Comparing this expression with Eq. (4.82), we see that they coincide if the functions
are replaced by Because
these functions coincide up to the sign, the polarization operators (which are
quadratic in Green’s functions) coincide identically.
4.3.7.
Transition to Energy-Integrated Propagators
Consider now an arbitrary component [e.g., the first one in the expression for
, which follows from (4.83), taking into account (4.94). In this
expression we can move from the integration over to angle and energy
integrations based on the relation
Using the auxiliary
we may integrate with respect
to the variable This makes it possible to express the quantity
in terms of energy-integrated functions determined by a relation of the type
since the function in (4.102) restricts mainly the angular integration and hence it
may be factored out of the -integral. Thus we have
To shorten the notations, we introduce the operator
and the convention
SECTION 4.4. INELASTIC ELECTRON–PHONON COLLISIONS 113
obtaining thus the final expressions for the Fourier components of the quantities,
which define the inequilibrium source in the phonon kinetic equation:
Before bringing the equation for phonons to the canonical form, we will obtain the
expression for the collision integral of electrons with phonons.
4.4. INELASTIC ELECTRON–PHONON COLLISIONS
The self-energy functions for an electron–phonon interaction were derived in
Sect. 3.2 assuming an equilibrium phonon distribution. We will consider now the
general case when the phonon system is not in equilibrium.
4.4.1. Electron–Phonon Self-Energy Parts
In the representation of discrete imaginary frequencies
m and n are integers], we have:
which corresponds to the diagram of Fig. 4.3. The functions (as well as ) in
(4.109) are assumed to be complete, including both external field and the self-energy
parts and polarization operators. Assuming that initially in the absence of an external
field the system is in equilibrium, we expand and in a power series over the
field and consider the analytical structure of the diagram as the function
of the complex variable at fixed imaginary frequencies
114 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES
. The manifold of cuts of the object under consideration
consists of the cuts of the internal and the -function. We denote this
manifold by These cuts may be considered as situated on the lines
in the complex plane between the uppermost line
and the abscissa (as was assumed earlier in accordance with the causality principle).
The combinations of which constitute are defined by the distribution of the
field vertices over the internal lines of the diagram Let us assume that
manifolds of cuts correspond to and
functions. Replacing in (4.109) the summation over by contour integration and
shifting as usual the integration contour to the banks of the cuts, we find the resulting
expression
where and are the jumps in Green’s functions on the corre-
sponding cuts (hereafter for brevity we omit the second indices of Green’s func-
tions). Continuing now analytically in Eq. (4.110) over from the upper bank of
the uppermost cut (the lower bank of the lowermost cut), we obtain (after returning
to real shifting the integration variable, summing over all the orders of
perturbation theory, and integrating over the energy the expression
in which the is defined as
Introducing as
we obtain, starting from Eq. (4.110), the expressions for the matrix elements of
which may be presented in the form (omitting for simplicity the second arguments)
SECTION 4.4. INELASTIC ELECTRON–PHONON COLLISIONS 115
In the diagonal over frequencies (quasi-classical) approximation, the phonon
propagators may be expressed through the function
4.4.2. Canonical Form for Electron–Phonon Collisions
Using the relations (4.114) to (4.118), (3.83) to (3.88), (4.7), (4.9), and (4.13),
one arrives at the following form of the electron–phonon collision integral:
where the factors are
116
CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES
Having ascertained that the function introduced by (4.117) plays the role
of a phonon nonequilibrium distribution function, we will now obtain the kinetic
equation for this quantity. We start from expression (4.112) for and (4.92)
for a polarization operator Separating the anomalous parts
one obtains
Regular functions in (4.123) and (4.124) (for example, the advanced function
can be determined by the diagram expansion, in which all the functions (propaga-
tors and polarization operators) are advanced ones. Separating in the diagram
expansion for the left free line we have the equation
where the following notation is used
(an integration over internal momentum or a coordinate variable is also assumed).
Separating the right free line in the same manner as subtracting the result
from (4.125), and using formulas (4.123) and (4.124) together with the expression
for regular functions, one obtains the relation
which is the desired general form of the kinetic equation for phonons. Expressions
(4.75) and (4.77) follow from (4.127) after integration over the positive half-axis
in a quasi-classical limit.
Note that at this stage the “bath” temperature T, which enters into imaginary
frequency variables of initial equations, is eliminated both from (4.127) and from
(4.114) to (4.116). The situation here is fully equivalent to that obtained by the
Keldysh technique. In the technique of analytical continuation, the bath temperature
plays the role of the equilibrium density matrix in Keldysh’s method—this matrix
is also eliminated from the final expressions.
SECTION 4.4. INELASTIC ELECTRON–PHONON COLLISIONS 117
4.4.3. Canonical Form for Phonon–Electron Collisions
The canonical form of the phonon–electron collision integral follows from
Eqs. (4.75), (4.77), (4.107), (4.108), and (3.84) to (3.88):
Expression (4.119) describes, besides the energy relaxation of electrons,
inelastic collision processes that produce the relaxation of electron-hole population
imbalance in superconductors. The situation here is fully analogous to that dis-
cussed in Sect. 4.2 and requires no further comments.
References
1. M. Yu. Reizer and A. V. Sergeev, Electron-phonon interaction in impurity-containing metals and
superconductors, Sov. Phys. JETP 63(3), 616–624(1986) [Zh. Eksp. i Teor. Fiz. 90(3), 1056–1070
(1986)].
2. G. M. Eliashberg, Inelastic electron collisions and nonequilibrium stationary states in supercon-
ductors, Sov. Phys. JETP 34(3), 668–676 (1972) [Zh. Eksp. i Teor. Fiz. 61[3(9)], 1254–1272
(1971)].
3. I. E. Bulyzhenkov and B. I. Ivlev, Nonequilibrium phenomena in junctions of superconductors,
Sov, Phys. JETP 47(1), 115–120 (1978) [Zh. Eksp. i Teor. Fiz. 74(1), 224–235 (1978)].
4. A. M. Gulian and G. F. Zharkov, Electron and phonon kinetics in a nonequilibrium Josephson
junction, Sov. Phys. JETP 62(1), 89–97 (1985) [Zh. Eksp. i Teor. Fiz. 89(3), 1056–1070 (1985)].
5. M. Tinkham, Tunneling generation, relaxation and tunneling detection of hole-electron imbalance
in superconductors, Phys. Rev. B 6(5), 1747–1756 (1972).
118 CHAPTER 4. BASIC EQUILIBRIUM PROPERTIES
6. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics
,
pp. 391–412, Pergamon, Oxford (1981).
7. A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskii, Quantum Field Theoretical Methods in
Statistical Physics
,
2nd ed., pp. 63–70, Pergamon, Oxford (1965).
8. L. V. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP
20
(4), 1018–
1026 (1965) [Zh. Eksp. i Teor. Phys. 47 [4(10)], 1515–1527 (1964)].
5
Microwave-Enhanced Order
Parameter
When a piece of normal metal is put into the high-frequency field, it is heated. This
is not surprising from the point of view of physics intuition. What was puzzling and
considered a “paradoxical quality” is that metals in superconducting state can
behave as if they are being cooled under these circumstances. Enhanced values of
critical currents were revealed by initial experimental measurements
1,2
performed
on microbridges. It was not until 1970 that Eliashberg
3
recognized that this
enhancement has nothing to do with spatial inhomegeneity. Instead, it is caused by
the effective cooling of electrons by high-frequency electromagnetic fields. This
specific mechanism, which will be described in this Chapter, was then further
elaborated for cases where the energy comes from electromagnetic
4–9
and acoustic
10
fields, as well as the tunneling process.
11–14
Experiments
10,15–21
well confirmed
these predictions. It was also recognized that this “gap enhancement” effect should
be accompanied by a “phonon deficit effect,” which will be discussed in the next
Chapter.
During many years the enhancement effects looked like “small” and “of no
practical significance,” though “fascinating from a physical point of view”.
22
However, rather recently experimentalists have been able to demonstrate the
“microrefrigeration” effect, which relies on physics that is closely related to that of
superconductivity enhancement (see Section 10.4).
5.1. SOURCE OF EXCITATIONS
5.1.1. Estimate for Magnetic Field Depairing Effect
An alternating electromagnetic field influences a superconducting order pa-
rameter in various ways. We start our discussion with the dynamic suppression of
119
120 CHAPTER 5. BASIC EQUILIBRIUM PROPERTIES
the order parameter modulus (denoted as in this chapter) by the effective mean
square of field amplitude
Near the action of is analogous to the action of a static magnetic field.
The time average variation of may be estimated by using the Ginzburg-Landau
theory (Sect. 1.2)
where is the diffusion coefficient of normal electrons. Second, the field
exerts a kinetic influence when the high-frequency quanta are absorbed by electron
excitations. That shifts their distribution over energies and changes the value of
according to the self-consistency equation.
As we will see in Sect. 5.2, at sufficiently high frequencies
where is the characteristic energy relaxation time (to be addressed later) in the
single-electron system, the kinetic effect dominates. Moreover, the variation of
has a sign that is opposite to that of Eq. (5.1).
5.1.2. Single-Quantum Transitions
We will consider this problem using the results of Eliashberg.
3,23
Because the
action of a high-frequency field (5.2) on the oscillating part of the order parameter
is negligibly small, the nonequilibrium order parameter may be taken as a
stationary one. We assume also that the superconducting film is sufficiently thin
(having a thickness d
),
so the picture does not depend on the z-coordinate, which
is perpendicular to the film surface. We assume also that the electron’s mean-free-
path is small: . For orientation, consider first the case of a normal
metal, setting ~ In this case the matrix functions and
u
are diagonal.
The system of equations determining reduces to a single equation
where
because
(in the case of a normal metal, all the field-containing terms in a diagram expansion
for vanish after the integration over . For simplicity we assume in
this case the high-frequency current density and also the vector potential A are
constant over the film’s cross section. Because the mean-free-path l is small, the