Kopnin N.B. Theory of Nonequilibrium Superconductivity

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energy becomes

(7.37)

where

(7.38)

The first term in eqn (7.37) is independent of the vortex position. However, the

energy U

is a function, of the vortex coordinate along the z axis.

For a weakly layered case, to the first approximation in s

, vortex solution

has the same form as for an anisotropic superconductor. Indeed, let us take the y

axis along the magnetic field and put

where

and

Assume that =

is large, 1. We can neglect the magnetic field in the

equation for f and obtain from eqn (7.21), (7.29), and (7.30)

end p.138

(7.39)

This equation coincides with eqn (1.48) in Section 1.1.2 which we derived for a

single vortex in an isotropic superconductor. We see that the anisotropy of the

vortex core scales with the coherence length in the corresponding directions.

To calculate the pinning energy, we insert the exact vortex solution of eqn (7.20)

into the free energy expression. In the small position-dependent part of the

energy in eqn (7.38) one can use the approximate solution eqn (7.39) for an

anisotropic case. To calculate the integrals in eqn (7.38), we shift the integration

contours C

and C

to the upper and lower complex half planes of the integration

variable z, respectively. The integral is then determined by a singularity of the

function f(z) which is the closest to the real axis. The function f(z) has poles on

the imaginary

axis:

(7.40)

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This fact can easily be understood from a model expression for the vortex-core

function

. It has poles at i

= i( /2+ k). The numerical

solution (Barone et al. 1990) shows that the closest poles to the real axis have

= 2.51 in eqn (7.40).

Let the vortex be located at the point x = 0, z = z

. The vortex amplitude f is a

function of the shifted variable

(7.41)

and its phase is now

To the leading approximation in , the energy density is determined by

the highest singularity which comes from the gradient term:

To calculate the integrals in eqn (7.38) we write

Consider first the integral over the contour C

. At , the imaginary part of

z = z

+ iz is large: z /s 1, therefore,

The integral over the contour C

gives

end p.139

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Here L is the vortex length along the y axis. During the integration we use that

and calculate the derivative only of the most rapidly

varying exponential part of the function under the integral. Together with the

integral over the contour C

, this equation gives the energy per unit vortex

length

(7.42)

where

(7.43)

We see that the vortex prefers to sit in-between the layers z

= s/2 where its

energy is minimal. If the vortex is not exactly in the middle, z

= s/2 + z, a

restoring (pinning) force appears. The pinning force per unit vortex length is

(7.44)

The pinning energy is exponentially small because the layered structure is

weakly pronounced: the distance between the layers is small compared to the

coherence length

end p.140

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Part III Nonequilibrium superconductivity

end p.141

end p.142

8 NONSTATIONARY THEORY

Nikolai B. Kopnin

Abstract: This chapter considers two methods designed for calculating the

real-time (retarded, advanced, and Keldysh) Green functions of nonstationary

superconductors: (i) the method of analytical continuation worked out by

Eliashberg; and (ii) the Keldysh diagram technique. The Eliashberg phonon model

of superconductivity is discussed and the equation for the order parameter is

derived. Expressions for self-energies of electron-phonon and electron-electron

interactions are obtained. Transport-like equations for the retarded (advanced)

and Keldysh Green functions of particles interacting with impurities, phonons,

and with each other are derived. Expressions for the electron density, electric

current, internal energy, and energy current are derived in terms of the Keldysh

Green functions.

Keywords: analytical continuation, Keldysh function, phonon model,

electron-phonon interaction, electron-electron interaction, self-energy,

impurity, electric current, energy current

We consider both the method of analytical continuation worked out by

Eliashberg and the Keldysh diagram technique, which are designed for

calculating the realtime Green functions of nonstationary superconductors.

The Eliashberg phonon model of superconductivity is discussed. We derive

equations for the Keldysh Green function for particles interacting with

impurities, phonons and with each other.

8.1 The method of analytical continuation

This part of the book is devoted to introduction of general principles and tools of

the microscopic nonstationary theory of superconductivity and is based on the

general concepts we learned in the previous chapters. A physical system which

evolves in time may be out of equilibrium with respect to the heat bath;

moreover, various parts of the system may be not in equilibrium with each other.

To treat a nonequilibrium situation, one needs to know the distribution of

nonequilibrium excitations in addition to their spectrum. Moreover, the spectrum

itself can be distorted as the system gets far from equilibrium. The specifics of

superconductors are that the excitations which affect the response of a

Kopnin, Nikolai, Senior Scientist, Low Temperature Laboratory, Helsinki University of

Technology, and L.D. Landau Institute for Theoretical Physics, Moscow

Theory of Nonequilibrium Superconductivity

Print ISBN 9780198507888, 2001

pp. [141]-[145]

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superconductor to an external field can accumulate near the energy gap when

their relaxation is relatively slow. The major problem of the nonstationary theory

is thus to find the distribution function of these excitations. In a nonlinear case,

one needs this distribution also to calculate the modified spectrum in a

self-consistent way.

Within the Green function formalism of the microscopic theory of nonstationary

superconductivity, all these different tasks are combined into a single basic

problem of finding the time-dependent Green function of the system. However,

we encounter a major complication in this way: there is no simple recipe for

calculating the real-time Green functions because the Dyson equations for them

cannot be constructed by simply repeating the previous argumentation. There

are two general methods developed so far for calculating the time-dependent

Green functions. The first is due to Keldysh (1964). It deals directly with the set

of real-time Green functions defined on the time axis with special rules of time

ordering of the particle-field operators. The second method has been developed

by Gor’kov and Eliashberg (1968) and by Eliashberg (1971). It operates with the

Matsubara Green functions which, as functions of frequencies, are analytically

continued from imaginary onto the real-frequency axis. We shall start with the

analytical continuation technique. We demonstrate in Section 8.5 that

end p.143

both those methods are completely equivalent. In the present chapter, we derive

the equations for those real-time Green functions which are appropriate for the

superconducting response. The following chapters describe how to define and

calculate the distribution function. We derive the kinetic equations for the

distribution function; we solve them for various cases and show how to use the

obtained distribution for further calculations.

As already mentioned, there is no simple way to calculate the real-time Green

function. Instead, comparatively simple equations and the corresponding diagram

technique exist for the Green functions which are defined for an imaginary time

(Matsubara functions). This is because, in contrast to its imaginary

counterpart, a real time cannot be directly time-ordered with the inverse

temperature –i/T and thus the Wick theorem cannot be directly applied for the

real-time field operators. However, solving the imaginary-time equations or,

equivalently, calculating all the diagrams, one can obtain the Matsubara

functions. Since the field operators in both real and imaginary time

representations obey the same equations with the substitution

, one

expects that the real-time Green function can be obtained from the Matsubara

function by an analytical continuation from imaginary time onto the real-time

axis provided the necessary initial conditions are satisfied. We assume that all

the interactions between particles are switched on at the time

Therefore, the Green functions at

coincide with those for free

particles which are in equilibrium with each other and with the heat bath with a

temperature T.

For practical purposes, the Matsubara Green functions are usually calculated as

the Fourier transforms for discrete imaginary (Matsubara) frequencies

for Fermi particles, and for Bose

particles and external fields. The analytical continuation in time is equivalent to

the analytical continuation in frequencies

and from imaginary onto the real

frequency axis. To satisfy the initial conditions, one has to make the continuation

in such a way that all the external field frequencies have an infinitesimal positive

imaginary part. In this case, all diagrams containing the field operators vanish for

according to the causality principle. In other words, one has to

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continue all the field frequencies from the upper half-plane of complex .

In principle, one can first solve the particular physical problem and find the

Matsubara Green function for it. After that, one can continue the Matsubara

function analytically and obtain the required Green function in the real-time

representation. This method, however, is not convenient. It operates with

quantities which do not have a simple physical meaning and thus can cause

confusion, which sometimes leads to incorrect results. The modern microscopic

theory, instead, works with the real-time Green functions from the very

beginning. In this representation, we can always keep track of the real physical

phenomena involved in the particular problem. The price for that is more

complicated equations needed to find the real-time functions. This chapter

describes how to derive and solve equations for the real-time Green functions of

superconductors. We shall follow the approach suggested by Eliashberg (1971).

All physical quantities like the current, the particle number, or the order

end p.144

FIG. 8.1. Diagrammatic presentation of the Green function

expansion, in powers of the external field H (shown by wavy

lines) up to the third-order term.

parameter are expressed through sums of certain matrix elements of the Green

function

over the Matsubara frequencies:

(8.1)

In our case,

= 2 iT(n + 1/2) are Matsubara frequencies for Fermionic

operators. The frequencies

refer to the “external” Bose fields and

to the time-dependent order parameter. Our goal is to find an analytical

continuation of eqn (8.1) from the upper half plane of the external frequency

To make the analytical continuation in a general case, we consider the full

function

as a perturbation series expansion in terms of a generalized

“field”. We start with the Dyson equation for the Matsubara functions

(8.2)

Here and

(8.3)

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and, is an, “external field” matrix as defined by eqn. (5.57). At the beginning,

we do not consider the interaction with impurities and phonons.

8.1.1 Clean superconductors

Let us consider the N-th order term in the expansion of eqn (8.2) which we write

(8.4)

where =

+ ... +

and is the non-perturbed Green function of the

normal state,

The diagrammatic representation of eqn (8.4) is shown in Fig. 8.1. The

summations of the Matsubara frequencies

and integration over the

corresponding

end p.145

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on the plane of complex z. Contours around the points z = 2 iT (k + 1/2)

(circles) are unrolled into contours going above and below the cuts."

title="Fig. 8.2. The cuts (shown by thick solid lines) along the singular

lines of the function

on the plane of complex z. Contours around

the points z = 2

iT (k + 1/2) (circles) are unrolled into contours going

above and below the cuts." class="figure">

IG. 8.2. The cuts (shown by thick solid lines) along the

singular lines of the function

on the plane of complex

z. Contours around the points z = 2

iT (k + 1/2) (circles) are

unrolled into contours going above and below the cuts.

momenta k

are implicitly assumed. The sum of the type of eqn (8.1) can be

written as the contour integral

(8.5)

taken around each of the poles of the function tanh (z/2T), i.e., around all z =

Kopnin, Nikolai, Senior Scientist, Low Temperature Laboratory, Helsinki University of

Technology, and L.D. Landau Institute for Theoretical Physics, Moscow

Theory of Nonequilibrium Superconductivity

Print ISBN 9780198507888, 2001

pp. [146]-[150]

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2 iT (n + 1/2). Consider the poles of as functions of z. The poles are

. Since

is real, they are

determined by equations

, where l = 0, ...,N and

= 0.

Let us make N + 1 horizontal cuts on the complex plane z passing through all

shown in Fig. 8.2. Since

=2 iTk

, these cuts do

not go through the poles of tanh (z/2T). We shall continue from positive

imaginary values of all

, therefore we can consider all k

to be positive, and

numerate the cuts from Imz = 0 to Imz = Im

upwards in order of increasing k

Now we can unroll the contours going around en eh z = 2 iT (n + 1/2) into the

contours going along the lower and upper coasts of each of N + 1 cuts. At each

cut labeled by l. the variable ; the lower contour

goes from right to left while the upper goes from left to right (see Fig. 8.2). As a

result, we have

end p.146

Here

is the jump of the Green function at the l-th cut where

. Because of periodicity of tanh (z/2T) along the

imaginary axis with the period 2

iT, the argument of the hyperbolic tangent, is

just

at each cut. Let us write one of the jumps explicitly. For example,

Here all

are still discrete imaginary. When we unroll the contours, we make

the analytical continuation in the variable z. At the cut l = 1 for example, we

have, for a real

, Im( +

) > Im( + i0) > 0 while Im( + w

–w) < ... < Im(

–

) < Im( – i0) < 0. Therefore, we obtain

(8.6)

Now we can consider all

as real continuous values. The sums over discrete

Matsubara frequencies

= 2 iTk

in equations of the type of eqn (8.6)

transform into the integrals according to the rule

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We make the substitution of integration variable at the

l-th cut to restore the previous notation of incoming and outgoing frequencies

and obtain

Here are constructed according to the scheme shown in eqn.

(8.6) but with shifted frequencies:

end p.147

(8.7)

Therefore, the sum in eqn (8.5) transforms into

where

(8.8)

Writing ail the jumps explicitly, we obtain

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