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

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SECT/ON 2.1. SCATTERING ON IMPURITIES 41

where

is the (electron) elastic scattering time. Thus the main contribution arises from the

diagrams containing crosses, which correspond to the same atoms. It is convenient

to link these crosses by broken lines. The diagrams with three crosses provide

nothing new. The fourth order of the perturbation theory generally is represented

by the diagram

A comparison of contributions from the diagrams

shows that the diagrams with intersected broken lines contain a small parameter

, where l is the electron’s mean-free-path. Indeed, for the first of

the diagrams in (2.10) we have

After averaging over the impurity positions, this transforms to

42 CHAPTER 2. BASIC EQUILIBRIUM PROPERTIES

At the same time, the third of the diagrams in (2.10) yields

or, after averaging over the impurities,

Restrictions that follow from the angle integration in Eq. (2.14) require that

one of the G-functions be of the order Meanwhile in expression (2.13)

the same function is of the order in the region important for integration. This

circumstance confirms the statement on diagrams with intersections. The situation

is analogous to the case of the second and third diagrams (2.10).

2.1.3. Born's Approximation

Apart from the diagrams considered (2.10), there is another one of the fourth

order:

SECTION 2.1. SCATTERING ON IMPURITIES 43

The contribution of such diagrams is essential in the case of non-Born scattering.

We consider the opposite situation

where one can omit contributions such as Exp. (2.15).

Let us now sum the selected diagrams. We have

It is not difficult to see that the result of the graphical summation of series (2.17)

may be depicted as

or in analytic form

This assumption is satisfactory for our immediate purposes. However, when we discuss electron

scattering by magnetic impurities later, continuing to make this assumption results in the failure to

predict the Kondo effect

and its very interesting consequences for transport phenomena in metals,

especially in the case of thermoelectricity.

5–10

The solution of Dyson’s equation (2.19), as usual, may be presented in the form,

where, since it follows from Eqs. (2.19) and (2.20), the self-energy part is

defined by the equation

Assuming is purely imaginary, we find in analogy with Eq. (2.7):

(2.22)

Comparing Eq. (2.22) with the limiting case , one finds = sign and,

consequently,

Moving now from (2.23) to the coordinate representation:

and taking into account formula (1.170), we rewrite (2.24) in the form

Closing the integration contour over in the upper and lower half-planes for the

first and second integrals in (2.25), respectively, and noting that the first integral is

nonzero at and the second at only, we find

where is the electron mean-free-path.

44 CHAPTER 2. BASIC EQUILIBRIUM PROPERTIES

SECTION 2.1. SCATTERING ON IMPURITIES 45

2.1.4. Equations for a Superconducting State

For superconductors we have the system of equations

in analogy with (2.18) (here we set Let us make the series expansion in

(2.27) and (2.28) in powers of the Bose field for example, in an equation for the

G-function:

Separating the free line in this diagram, we obtain a remaining series

plus a class of diagrams with even numbers of

which enter the sum after the sign of the cross (we do not distinguish here

between and

Two options are possible when the external broken line is separated from such

a diagram: there is either an even or an odd number of vertices in the inner and

in the outer regions of this broken line. Summation of these two classes of diagrams

yields

Thus, we get the equation for the G-function

46 CHAPTER 2. BASIC EQUILIBRIUM PROPERTIES

(we have used here the rule concerning the diagram signs mentioned in Sec. 1.4).

In the same manner one can obtain for the expansion

Again, the left free line may be separated, which is followed by the vertex or by

a cross. Summing the first class of diagrams, one obtains If a vertex

one can separate the first external broken line:

(in the inner part of the diagram series there are broken lines and vertices If there

is an even number of in the inner part, then summation of the diagrams gives the

function , and the function obviously emerges in the outer part. If the broken

line embraces the odd numbers of then this class of diagrams yields the function

and the function G emerges in the outer part.

Thus

where Taking into account the definition of functions and

and analogously for other two elements of the -matrix (1.114), we obtain the

system of equations for superconductors with impurities:

SECTION 2.2. MAGNETIC IMPURITIES 47

The self-energy matrix here is:

as follows from Eqs. (2.34), (2.35), and (2.8).

2.1.5. Anderson's Theorem

The solution of Eqs. (2.36) and (2.37) may be found in the same manner as

was done for the normal state. It gives the same formal result: the appearance

of exponential factors in the Green’s functions. However, the gap in the

energy spectrum of the superconductor is subject to the self-consistency equa-

tion (1.144), which includes the superconducting propagator at So

evidently nonmagnetic impurities do not influence the thermodynamics of a

superconductor.

2.1.6. "Londonization" by Elastic Scattering

Another important consequence follows from the comparison of Eqs. (1.169)

and (2.26). At / the electron correlation radius in superconductors becomes

less than . We have mentioned this circumstance in Chap. 1 as the “Londoniza-

tion” of superconductors by the scattering of elastic impurities. This aspect of the

influence of impurities is important for superconductors, making their electrody-

namics local.

2.2.

MAGNETIC IMPURITIES

When the paramagnetic part of the potential (r) (2.2) is “switched on,” the

interaction becomes explicitly dependent on the electrons’ spins. Consequently, the

spin variables should be preserved in the intermediate calculations of Sect. 2.1.

Using the Hamiltonian

*In the theory of superconductivity, this result is sometimes called the Anderson theorem.

48 CHAPTER 2. BASIC EQUILIBRIUM PROPERTIES

the following equations of motion for the Heisenberg operators and can

be obtained:

we have included the summation over impurities and the

exchange potential in 1. Starting from the definition of Green’s functions, taking

the derivative of (1.110) and account of (2.38) and using “Gor’kov splitting”

(1.112), one obtains the equation

where the function obeys the equation

Now, making the series expansion for the functions G and F

on the basis of (2.39)

and (2.40), one can see that enters into the diagrams

whereas enters into the diagrams

2.2.1. Averaging over Spin Directions

Let us return now to the definition of functions

(see Sect. 2.1). Besides averaging over the spatial distribution of impurities, one

must also average over the spin directions, assuming their random orientation. In

the absence of impurities it follows that ~ If the dashed line is spin

dependent, then

Thus, the averaging of diagrams for G-functions adds the term to the

potential The situation with functions and

is different. The corresponding

diagrams contain an additional factor and a line ; for example,

SECTION 2.2. MAGNETIC IMPURITIES 49

As a result, the part of the diagram represented by a dashed line and containing

- — - will have a sign opposite to Indeed, (see 1.119 and 1.120). But

and this causes a chain of relations:

2.2.2 Spin-Flip Time

The “magnetic” part in the averaged diagrams for F and has a sign opposite

to the part produced by the usual impurities. Accounting for this, the value in

the diagonal components of in Eq. (2.37) is replaced by

and in nondiagonal components by

The difference in values (2.46) and (2.47) is due exclusively to the magnetic part

of the interaction and defines the reciprocal spin-flip time

2.2.3. Depression of Transition Temperature

Let us now consider the influence of paramagnetic impurities on the thermo-

dynamic properties of superconductors. The initial equations (prior to the impurity

averaging) in the representation of the imaginary discrete frequencies

have the form

CHAPTER 2. BASIC EQUILIBRIUM PROPERTIES

As before, we use the potential (2.2). It will be shown now that the critical

temperature remains unchanged if and diminishes if

Since the temperatures close to the critical one are of interest, in the expansion

of in powers of the field it is sufficient to retain only the lowest-order diagram:

Substituting the corresponding analytic expression into the self-consistency equa-

tion (2.51), we find

The equation for must have a nonzero solution at the critical temperature.

Averaging (2.53) over the impurity positions and taking into account that is a

smooth function and ) is a rapidly oscillating one, one may write

The averaging procedure in (2.54) produces the broken lines connecting the

crosses not only on the same propagation line, but also on different lines (recall that

thepotential corresponds to crosses on the G-function's line). In the first case,

we have and a factor arises for the diagram.

This leads to the substitution in expression (2.26):

Correspondingly, the Fourier component of Eq. (2.55) has the form [compare Eq.

(2.23)]:

In the second case, one must calculate in (2.54) a “ladder” diagram of “dressed”

functions: