Kopnin N.B. Theory of Nonequilibrium Superconductivity

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p.202

where N and N

are densities of electrons and ions, respectively. In equilibrium,

N = N

Introduction of an additional electronic charge results in a shift of the

chemical potential = e such that

The Poisson equation becomes

It demonstrates that the potential together with N decay at distances of the

order of the Debye screening length

which is of the order of interatomic distance in good metals. Therefore, variations

in the electronic charge density are practically zero at distances of the order of

the coherence length, and we have to put the constraint N = 0 or

(10.72)

The latter follows from the continuity equation

Since N

= N

where N

is the normal-state electron density, the charge

neutrality condition requires

(10.73)

10.4 Collision integrals

Collision integrals describe interaction of excitations with impurities, phonons

and, with each other. These interactions are responsible for establishing

equilibrium in an electronic subsystem in a superconductor assuming that the

crystal lattice (phonons) together with impurities are themselves in equilibrium

and form a heat bath. In practical superconducting compounds, impurity

concentrations are usually such that the most effective relaxation is brought

about through scattering of electrons by impurities; the electron–phonon

relaxation rate is usually much smaller, to say nothing about electron–electron

collisions. If not specified otherwise, we assume in what follows that the impurity

collision integral is the largest source of relaxation.

In this section we consider various collision integrals and derive expressions for

them in terms of the distribution functions f

(1)

and f

(2)

which will be used later in

our discussions. We note that all the collision integrals vanish for the equilibrium

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distribution f

(1)

(0)

and f

(2)

= 0. They are thus proportional to deviations from

equilibrium. Therefore, one can use the stationary Green functions

R(A)

in the

collision integrals.

end p.203

10.4.1 Impurities

10.4.1.1 Nonmagnetic scattering

We shall assume for simplicity that the impurity scattering is isotropic. Remind

that the self-energies for nonmagnetic impurities are

Here we use notation instead of

imp

for brevity. The collision integrals

are found from eqns (9.14) and (9.16) using the definitions eqns (10.43),

(10.45), and (10.48):

where

(10.74)

(10.75)

and

In addition

(10.76)

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Note that the collision integrals and J

differ in two respects. First, the

collision, integral J

vanishes after integration, over the momentum directions

while the integral

does not:

At the same time vanishes for any function f

(2)

independent of the

momentum directions while does not.

end p.204

10.4.1.2 Scattering by magnetic impurities

Keep in mind that the self-energy in eqn (5.76) for magnetic impurities is

(10.77)

The spin-flip collision integrals become

(10.78)

(10.79)

and

10.4.2 Electron–phonon collision integral

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For small deviations from the equilibrium distribution function and for , we

can consider only the terms linear in f

(2)

. The electron–phonon self-energies

have the form of eqns (9.12) and (9.13). One obtains for the collision integrals:

(10.80)

Here is the electron–phonon interaction constant. Terms linear

in f

(2)

do not appear in . The second collision integral contains both f

(1)

and f

(2)

. It has a form . The first part is

end p.205

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(10.81)

The function, f

(1)

here should be considered equilibrium: f

(1)

= tanh ( /2T). The

part

only depends on f

(1)

(10.82)

The functions (p, r) and (p, r) in eqns (10.80)–(10.82) have the same

momentum since the relative change in p during interaction with the acoustic

phonons is small.

We note that the collision integrals

and vanish for the

equilibrium f

(1)

= tanh ( /2T). In turn, the collision integral vanishes

for any function of the form of f

(2)

= const (df

(0)

/d ); the latter corresponds to

the invariance with respect to variations in the chemical potential.

10.4.2.1 Simplifications for T

Near the critical temperature when T, one usually needs the distribution

functions for energies

~ , i.e., for T, while the deviation from equilibrium

(1)

(0)

is small for ~ T. The phonon collision integral can be simplified

considerably in this case.

For

T, and for T, ~ T we have and . As a

result

(10.83)

where the electron–phonon mean free time is

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. [206]-[210]

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(10.84)

The second collision integral becomes

end p.206

(10.85)

Here

(10.86)

The electron–phonon relaxation rate, eqn (10.84), is .

Usually the Debye frequency is higher than the critical temperature, T

and

also

. The electron–phonon relaxation rate is thus small compared to the

order parameter for temperatures which are not very close to T

. High-temperature superconductors can be exceptions because the critical

temperature is roughly of the same order as

. Nevertheless, experimentally,

the inequality

is believed to hold. This should be especially

correct for d-wave superconductors where it is crucial for the very existence of

d-wave superconductivity as we have pointed out in Section 6.3.

10.4.3 Electron–electron collision integral

Self-energies for the electron–electron collisions can be obtained from the

expressions established (Eliashberg 1971) for the electron–electron interaction in

Section 8.3. The transformation to the quasiclassical functions is made according

to the general rules described earlier. The only difference is that the self-energy

contains three G functions and only two integrations over the momenta.

However, we can put

because we are only interested in energies close to the Fermi surface. We obtain

(10.87)

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(10.88)

and

(10.89)

(10.90)

Here A and B are the matrix elements of the interaction potential; different signs

appear because of different spin structure of the matrix elements between the

end p.207

states and as compared to those between the states and . The curly

brackets are defined as

The energies and momenta satisfy the conservation laws:

(10.91)

For a point interaction V(r) = V (r) one has

The electron–electron interaction constant is defined as

= V. We introduce

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the operator

(10.92)

Self-energies become

(10.93)

(10.94)

(10.95)

(10.96)

We write down only one electron–electron collision integral

To calculate we can put f

(2)

= 0. Using that Tr Î

R(A)

= 0 we obtain

(10.97)

where

end p.208

and

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Here the index 0 refers to , p, while the index 1 is for

, p

, etc.

The equilibrium distribution function f

(0)

= tanh ( /2T) turns the function F( ,

) to zero. If we put now

where is a small correction, we obtain

10.5 Kinetic equations for dirty s-wave superconductors

To make an example, we apply our kinetic equations to dirty s-wave

superconductors such that the mean, free time satisfies the condition

The starting equations have the form of eqns (10.55) and (10.44). We write

them again

(10.98)

(10.99)

The collision integrals are given byeqns (10.74)–(10.75) for scattering by

impurities, and byeqns (10.80)–(10.82) for electron–phonon relaxation.

We use the method employed earlier in Sections 5.6 and 9.3 for superconductors

with a short mean free path. Let us first average eqn (10.98) over the

end p.209

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directions of v

. The nonmagnetic impurity collision integral vanishes and we

obtain

(10.100)

This equation contains the vector component of f

in the left-hand side and the

average component of f

in the right-hand side. Now we multiply eqn (10.99)

with v

and average it:

(10.101)

Here we keep the largest collision term due to scattering by nonmagnetic

impurities. The vector components of regular functions in eqn (10.101) are

neglected as compared to the averaged components. This equation also contains

the averaged f

and the vector f

The second set of equations for vector f

and averaged f

components is obtained

as follows. First, we average eqn (10.99):

(10.102)

Next, we multiply eqn (10.98) with v

and average it:

(10.103)

The four equations (10.100), (10.101), (10.102), and (10.103) determine the

four functions f

, f

, and f

10.5.1 Small gradients without magnetic impurities

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