Kopnin N.B. Theory of Nonequilibrium Superconductivity

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(3.85)

Equation (3.85) is nothing but eqn (1.57). To see this we use eqn (3.42) and

express

in terms of e through eqn (3.76). Equation (3.85) gives the same

value of T

as eqn (3.81), of course. It can also be used to get the order

parameter value at zero temperature:

(3.86)

Therefore, at T = 0,

(3.87)

3.5 Perturbation theory

3.5.1 Diagram technique

Let us assume that the operator consists of a main part plus a small

correction. For example, we can consider the case when the magnetic field and

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the order parameter are small. We can separate the operator into the part

corresponding to a free particle

(3.88)

and a perturbation defined by eqn (3.70) so that . There

may be other possibilities; for example,

is not small but the magnetic field is

small, etc. Let us look for the Green function in the form of an expansion in

powers of perturbation. We write

where Green function of a free particle:

and the correction satisfies the equation

The full Green function can be written as a series

or in the form of the Dyson equation

(3.89)

The Dyson equation (3.89) is equivalent to the Gor’kov equation (3.18). The

same expansion holds also for retarded and advanced functions, as can be seen

from eqns (3.40).

The series expansion and the Dyson equation can also be represented as graphs.

Let us ascribe a single line to the Green function

, a double line to the full

Green function

. Two ends represent two coordinates x

and x

We denote the

interaction

by a wavy line with one coordinate. As a result, the full Green

function can be presented either as an infinite series of diagrams, Fig. 3.4(a), or

as an graphical Dyson equation, Fig. 3.4(b).

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. [61]-[65]

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3.5.2 Electric current

The electric current can be calculated using eqn (3.38) and the Green function

which can be found, for example, by means of the perturbation theory. We have

end p.61

FIG. 3.4. Diagrammatic representations for the Green function.

(a) The perturbation series. (b) The Dyson equation.

In the momentum representation we have

(3.90)

The last term is called the diamagnetic term. It should disappear in the normal

state if one neglects the Landau quantization in a magnetic field. We prove this

using the Green function of the normal state. In the presence of magnetic field

in the leading approximation in A. Since

p k

k, we get

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The first term does not give a contribution to the current. Since p k, we have

We use the identity

Therefore,

end p.62

The corresponding contribution to the current is

This cancels the diamagnetic term.

end p.63

4 SUPERCONDUCTING ALLOYS

Nikolai B. Kopnin

Abstract: This chapter explains how to incorporate scattering by random

impurity atoms into the general Green function formalism of the theory of

superconductivity. The cross-diagram technique based on the averaging over

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random impurity positions is derived using the Born approximation for the

scattering amplitude. Impurity self-energy is derived. Homogeneous state of an

s-wave superconductor is considered.

Keywords: alloys, impurity, Born approximation, cross-diagram

technique, self-energy

We learn how to incorporate scattering by random impurity atoms into the

general Green function formalism of the theory of superconductivity. The

cross-diagram technique is derived using the Born approximation for the

scattering amplitude.

4.1 Averaging over impurity positions

We describe a simple and powerful method how one can incorporate the

interaction between electrons and random impurity atoms into the general BCS

scheme. The method has been developed by Abrikosov and Gor’kov (1958). It is

assumed that physical properties of a superconductor containing a large amount

of random impurities can be obtained by averaging over realizations of the

disordered impurity potentials. Another assumption is the Born approximation

which implies that the scattering potential is small compared to the characteristic

atomic potential the latter being of the order of the Fermi energy of the host

material.

Each impurity interacts with electrons via a potential u(r

) where r

is the

position of an impurity atom. Let us consider this interaction as perturbation and

calculate the Green function. We assume for simplicity a homogeneous order

parameter. Generalization for a spatially dependent order parameter is

straightforward. The Green function has the form

Here

(0)

refers to a superconductor without impurities. Note that the Green

functions on the both sides of each impurity potential have the same frequency

because the impurity scattering is elastic. We introduce the Fourier

transformed Green functions and find

end p.64

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FIG. 4.1. Graphic representation of interaction with impurities.

(4.1)

We omit the index

for brevity. This series can be presented graphically as

lines separated by crosses, where each cross designates the scattering potential

u, Fig. 4.1

Since the impurity atoms are distributed randomly, summation over positions of

impurity atoms can be substituted with integration over their coordinates, i.e.,

with the averaging over positions of impurities and multiplication by the number

of impurity atoms

where n

imp

is the concentration of impurity atoms. The integration results in

-functions of the corresponding differences of momenta in eqn (4.1). The

first-order correction in eqn (4.1) gives

(4.2)

In the second-order correction we separate first the terms with r

, i.e., the

terms where the impurity scattering occurs on different atoms. The average gives

the -functions (p p

) and (p p

). The second-order term becomes

(4.3)

If we now collect the terms of all orders in u where scattering; occurs on different

atoms we obtain the effective correction to the Hamiltonian in the form

This term is not important: it can be incorporated in the chemical potential. We

shall omit this contribution in what follows.

Consider now the second-order term where scattering occurs at the same atom r

= r

. The sum gives

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end p.65

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(4.4)

Here we introduce the “self-energy”

(4.5)

Since both p and p

are close to the Fermi surface, the scattering amplitude

depends only on the angle between p and p

: |u(p p

= |u( )|

The scattering amplitude determines the scattering mean free time. In the

normal state it is

(4.6)

One can introduce the scattering cross section

p, p1

. In the Born approximation

The scattering mean free time becomes

For an isotropic scattering by impurities |u( )|

= const, the self-energy is

independent of the momentum direction:

(4.7)

Here the scattering mean free time is

(4.8)

The corresponding contribution to the Green function can be shown on, a

diagram as a dashed line connecting two crosses belonging to the same atom

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. [66]-[70]

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(see Fig. 4.2).

Consider now higher-order terms in u. The terms where all the atoms are

different have already been taken into account. The rest can be separated into

two groups. The first contains such scattering processes that each atom only

participates twice. On a diagram this corresponds to various combinations of

dashed lines connecting two crosses. The second group involves processes where

each atom participates more than twice. One can show that the contribution from

more than two crosses per atom is smaller than that from two crosses per

end p.66

FIG. 4.2. (a) Averaging over positions of one atom. (b)

First-order self-energy.

FIG. 4.3. Averaging over positions of two atoms. (a) and (b)

Dashed lines do not cross. (c) Dashed lines cross, which

restricts the interval of the momentum directions.

atom by a factor u/E

for each extra cross. Indeed, the diagram with three

crosses gives

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We put r

= r

= r and integrate over d

r. We get

where

end p.67

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