˜

G> (1, 1 ) =i † † †

’e’ie(Λ(1)+Λ(1 )) ψ“ (1)ψ‘ (1 ) ’e’ie(Λ(1)’Λ(1 )) ψ“ (1)ψ“ (1 )

= eieΛ(1)„3 G> (1, 1 ) e’ieΛ(1 )„3 . (8.92)

The other Green™s functions in Nambu space, GR,A,K , are linear combinations of

>

G< , and therefore transform similarly. The gauge transformation then transforms

the matrix Green™s function in Keldysh space according to

G(1, 1 ) = eieΛ(1) „ G(1, 1 ) e’ieΛ(1 ) „ .

(3) (3)

˜ (8.93)

The ¬‚exibility of gauge transformations allows one to choose potentials that min-

imize the temporal variation of the order parameter, such as facilitating transfor-

mation to the gauge where the order parameter is real, the real ” gauge, where the

13 The technique has also been used to derive kinetic equations for quasi-one-dimensional conduc-

tors with a charge-density wave resulting from the Peierls instability [38].

232 8. Non-equilibrium superconductivity

phase of the order parameter vanishes, χ = 0. This is achieved by choosing the gauge

transformation

1

’e• ’ ¦ = χ ’ e•

™ (8.94)

2

and

e 1

’ A ’ vs = ’ (∇ χ + 2eA) (8.95)

m 2m

introducing the gauge-invariant quantities, the super¬‚uid velocity, vs , and the electro-

chemical potential, ¦, of the condensate or Cooper pairs.

8.2 Quasi-classical Green™s function theory

The superconducting state introduces the additional energy scale of the order pa-

rameter, which in the BCS-case equals the energy gap in the excitation spectrum.

In a conventional superconductor, as well as in super¬‚uid He-3, this scale is small

compared with the Fermi energy. The peaked structure at the Fermi momentum of

the Green™s functions thus remains as in the normal state, and the arguments for the

super¬‚uid case that brings us from the left and right Dyson equations, Eq. (8.80) and

Eq. (8.81), to the subtracted Dyson equation for the quasi-classical Green™s function

are thus identical to those of Section 7.5 for the normal state, and we obtain the

matrix equation, the Eilenberger equations,

’1

[g0 + iσ —¦ g]’ = 0 ,

, (8.96)

which gives the three coupled equations for g R(A) and g K where

’1 ’1

g0 (ˆ , R, t1 , t1 ) = g0 (ˆ , R, t1 ) δ(t1 ’ t1 ) (8.97)

p p

and14

’1

g0 (ˆ , R, t1 ) = „ (3) ‚t1 + vF · (∇R ’ ie„ (3) A(R, t1 )) + ieφ(R, t1 ) (8.98)

p

and the ξ-integrated or quasi-classical four by four (4 — 4) matrix Green™s function

i

ˆ

g(R, p, t1 , t1 ) = dξ G(R, p, t1 , t1 ) (8.99)

π

is de¬ned in the same way as in Section 7.5, capturing the low-energy behavior of

the Green™s functions.15 The equations for g R(A) determines the spectral densities

and the equation for g K is the quantum kinetic equation.16

14 The A2 -term is smaller in the quasi-classical expansion parameter »F /ξ0 ∼ ”/EF , the ratio of

the Fermi wavelength and the superconducting coherence length, e2 A2 /m ∼ evF A»F /ξ0 .

15 In Section 8.3, the quasi-classical Green™s functions will be introduced not by ξ-integration but

by considering the spatial behavior of the Green™s functions on the scale much larger than the

inter-atomic distance.

16 We follow the exposition given in reference [3] and reference [9].

8.2. Quasi-classical Green™s function theory 233

Writing out for the components, we have for the spectral components

’1

[g0 + iσ R(A) —¦ g R(A) ]’ = 0

, (8.100)

and for the kinetic component the quantum kinetic equation

’1

[g0 + i eσ —¦ g K ]’ = 2iσ K ’ i [(σ R ’ σ A ) —¦ g K ]+ .

, , (8.101)

The self-energy comprises the e¬ective electron“electron interaction, impurity

scattering and electron spin-¬‚ip scattering due to magnetic impurities. The impurity

scattering is in the weak disorder limit described by the self-energy17

ˆ

dp

σimp (ˆ , R, t1 , t1 ) = ’iπni N0 |Vimp (ˆ · p )|2 g(R, p , t1 , t1 )

pˆ ˆ (8.102)

p

4π

quite analogous to that of the normal state, Eq. (7.51), except that the Green™s

function in addition to the real-time dynamical index structure is a matrix in Nambu

space.

Even a small amount of magnetic impurities can, owing to their breaking of time

reversal symmetry and consequent disruption of the coherence of the superconduct-

ing state, have a drastic e¬ect on the properties of a superconductor, leading to

the phenomena of gap-less superconductivity, and an amount of a few percent can

destroy superconductivity completely [39]. We therefore include spin-¬‚ip scattering

of electrons, which in contrast to normal impurities leads to pair-breaking and the

quite di¬erent physics just mentioned. We assume that the positions and spin-states

of the magnetic impurities are random, and owing to the latter assumption we can

limit the analysis to the last term in Eq. (2.25).18 In terms of the Nambu ¬eld the

scattering o¬ the magnetic impurities then becomes

dx u(x ’ xa ) Sa Ψ† (x)Ψ(x) ,

Vsf ’ z

(8.103)

a

where as usual in the Nambu formalism, an in¬nite constant has been dropped. The

spin-¬‚ip self-energy has the additional feature, compared to the impurity scattering,

zz

of averaging over the random spin orientations of Sa Sa . Assuming that all impurities

have the same spin, Sa = S, the averaging gives the factor S(S+1)/3 and the spin-¬‚ip

self-energy then becomes

ˆ

dp

σsf (ˆ , R, t1 , t1 ) = ’iπnmagn.imp.N0 S(S +1) |u(ˆ · p )|2 „ (3) g(R, p , t1 , t1 ) „ (3)

pˆ ˆ

p

4π

(8.104)

where nmagn.imp is the concentration of magnetic impurities. Since the exchange

interaction is weak, only s-wave scattering needs to be taken into consideration, and

17 The weak disorder limit refers to / F „ 1, and the neglect of localization e¬ects, but we could

of course trivially include multiple scattering by introducing the t-matrix instead of the impurity

potential. For a discussion see for example chapter 3 of reference [1].

18 Magnetic impurity scattering was discussed in Exercise 2.5 on page 37, and for example in

chapter 11 of reference [1].

234 8. Non-equilibrium superconductivity

the spin-¬‚ip self-energy becomes

i dˆ (3)

p

σsf (ˆ , R, t1 , t1 ) = ’ „ g(R, p , t1 , t1 ) „ (3) ,

ˆ (8.105)

p

2„s 4π

where the spin-¬‚ip scattering time is

ˆ

1 dp

|u(ˆ · p )|2 .

pˆ

= 2πnmag.impN0 S(S + 1) (8.106)

„s 4π

When inelastic e¬ects are of interest, they are for example described by the

electron“phonon interaction through the self-energy whose matrix components of

the matrix self-energy are

»

R(A)

dˆ (g K (R, p , t1 , t1 ) DR(A) (R, pF (ˆ ’ p ), t1 , t1 )

ˆ ˆ pˆ

σe’ph (R, p, t1 , t1 ) = p

8

+ g R(A) (R, p , t1 , t1 ) DK (R, pF (ˆ ’ p ), t1 , t1 ))

ˆ pˆ (8.107)

and

»

dˆ (g R (R, p , t1 , t1 )DR (R, pF (ˆ ’ p ), t1 , t1 )

K

ˆ ˆ pˆ

σe’ph (R, p, t1 , t1 ) = p

8

g A (R, p , t1 , t1 )DA (R, pF (ˆ ’ p ), t1 , t1 )

ˆ pˆ

+

g K (R, p , t1 , t1 )DK (R, pF (ˆ ’ p ), t1 , t1 ))

ˆ pˆ

+ (8.108)

or

»

dˆ ((g R ’ g A )(DR ’ DA ) + g K DK )

K

σe’ph = (8.109)

p

8

since

g R (t1 , t1 ) DA (t1 , t1 ) = 0 = g A (t1 , t1 ) DR (t1 , t1 ) . (8.110)

The di¬erence of the self-energies in comparison with the normal state is that the

electron quasi-classical propagators are now matrices in Nambu space.

Currents and densities are in the quasi-classical description, just as in the normal

state, split into low- and high-energy contributions. The charge density becomes, in

terms of the quasi-classical Green™s function,

∞

1 dˆ

p

ρ(R, T ) = ’2eN0 dE Tr(g K (E, , p, R, T ))

ˆ

e •(R, T ) + , (8.111)

8 4π ’∞

where N0 is the density of states at the Fermi energy, and Tr denotes the trace with

respect to Nambu or particle“hole space. The current density is given by

∞

eN0 vF dˆ

p

j(R, T ) = ’ dE p Tr(„3 g K (E, , p, R, T )) .

ˆ ˆ (8.112)

4 4π ’∞

8.2. Quasi-classical Green™s function theory 235

The order parameter is speci¬ed in terms of the o¬-diagonal component of the

quasi-classical kinetic propagator according to (absorbing the coupling constant)

i» dˆ

p

”(R, T ) = ’ dE Tr((„1 ’ i„2 ) g K (E, p, R, T )) .

ˆ (8.113)

8 4π

Exercise 8.7. Show that the quasi-classical retarded Nambu Green™s function in the

thermal equilibrium state is

E Z R (E) „3 + i¦R (E) „1

g R (E) = . (8.114)

E 2 (Z R (E))2 ’ (¦R (E))2

In the strong coupling case the order parameter is

¦R (E)

”= . (8.115)

Z R (E)

8.2.1 Normalization condition

In the superconducting state the retarded and advanced quasi-classical propagators

do not reduce to scalars (times the unit matrix in Nambu space) as in the normal

state, and the quantum kinetic matrix equation, Eq. (8.96), constitutes a compli-

cated coupled set of equations describing the states (as speci¬ed by g R(A) ) and their

occupation (as described by g K ). Since the quantum kinetic equation, Eq. (8.96),

is homogeneous and the time convolution associative, the whole hierarchy g —¦ g,

g —¦ g —¦ g,..., are solutions if g itself is a solution. A normalization condition

to cut o¬ the hierarchy is therefore needed. For a translationally invariant state

in thermal equilibrium it follows from Exercise 8.7 (or see the explicit expressions

obtained in Section 8.2.3) that (g R (E))2 and (g A (E))2 equal the unit matrix in

Nambu space, (g R (E))2 = 1 = (g A (E))2 , and the ¬‚uctuation“dissipation relation,

g K (E) = (g R (E)) ’ g A (E)) tanh(E/2T ), then guarantees that the 21-component in

Schwinger“Keldysh indices of g —¦ g vanishes, g R (E) g K (E) + g K (E) g A (E) = 0.

Since the quantum kinetic equation, Eq. (8.101), is ¬rst order in the spatial variable,

the solution is uniquely speci¬ed by boundary conditions. Since a non-equilibrium

state can spatially join up smoothly with the thermal equilibrium state we therefore

anticipate the general validity of the normalization condition

(g —¦ g)(t1 ’ t1 ) = δ(t1 ’ t1 ) . (8.116)

The function g —¦ g is thus a trivial solution to the kinetic equation, but contains the

important information of normalization. Section 8.3 provides a detailed proof of the

normalization condition.

The three coupled equations for the quasi-classical propagators g R,A,K in equa-

tions Eq. (8.101) and Eq. (8.100) constitute, together with the normalization con-

dition, the powerful quasi-classical theory of conventional superconductors. Writing

out the components in the normalization condition we have

g R(A) —¦ g R(A) = δ(t1 ’ t1 ) (8.117)

and

gR —¦ gK + gK —¦ gA = 0 . (8.118)

236 8. Non-equilibrium superconductivity

8.2.2 Kinetic equation

The normalization condition, Eq. (8.118), is solved by representing the kinetic Green™s

function in the form

gK = gR —¦ h ’ h —¦ gA , (8.119)

where h so far is an arbitrary matrix distribution function in particle“hole space.

The existence of such a representation is provided by the normalization condition,

Eq. (8.117) and Eq. (8.118), as the choice

1R

(g —¦ g K ’ g K —¦ g A )

h= (8.120)

4

solves Eq. (8.119). This choice is by no means unique, in fact the substitution

h ’ h + gR —¦ k + k —¦ gA (8.121)

leads to the same g K for arbitrary k.19

Using the equation of motion for g R(A) , Eq. (8.201), and the fact that the time

convolution composition —¦ is associative, the kinetic equation, Eq. (8.101), is brought

to the form for the distribution matrix

g R —¦ B[h] ’ B[h] —¦ g A = 0 , (8.122)

where

’1

B[h] = σ K + h —¦ σ A ’ σ R —¦ h + [g0 —¦ h]’ .

, (8.123)

The quasi-classical equations are integral equations with respect to the energy

variable, and only in special cases, such as at temperatures close to the critical tem-

perature, are they amenable to analytical treatment. However, they can be solved

numerically and provide a remarkably accurate description of non-equilibrium phe-

nomena in conventional superconductors. The quantum kinetic equation is thus a

powerful tool to obtain a quantitative description of non-equilibrium properties of

superconductors.

Before we unfold the information contained in the quantum kinetic equation we

consider the equation for the spectral densities or generalized densities of states,

Eq. (8.100), as they are input for solving the kinetic equation.

8.2.3 Spectral densities

The equation of motion for the retarded and advanced propagators in Eq. (8.96)

becomes

’1

[g0 + iσ R(A) —¦ g R(A) ]’ = 0 .

, (8.124)

In the static case, we note in general that it follows from Eq. (8.124) that g R(A)

is traceless, so that

g R(A) = ±R(A) „3 + β R(A) „1 + γ R(A) „2 . (8.125)

19 A choice making the resemblance between the Boltzmann equation and Eq. (8.122) immediate

in the quasi-particle approximation has been introduced in reference [40].

8.2. Quasi-classical Green™s function theory 237

The quantities ±R(A) , β R(A) and γ R(A) denote generalized densities of states.

We need to consider only one set of generalized densities of states since from the

equality

GR(A) (1, 1 ) = „3 (GR(A) (1, 1 ))† „3 (8.126)

it follows in general that

±A = ’(±R )— β A = (β R )— γ A = (γ R )— .

, , (8.127)

In a translationally invariant state of a superconductor in thermal equilibrium, the

spectral densities depend only on the energy variable, E, and the real and imaginary

parts of the spectral densities are even and odd functions, respectively. In general,

the equations for the spectral functions have to be solved numerically, for which they

are quite amenable, and they then serve as input information in the quantum kinetic

equation.

To elucidate the information contained in Eq. (8.124), we solve it in equilibrium

and take the BCS-limit, obtaining

g R(A) = ±R(A) „3 + β R(A) „1 (8.128)

as

E „3 + i” „1

√

g R (E) = . (8.129)

E 2 ’ ”2

Splitting in real and imaginary parts

±R(A) = +

β R(A) = N2 (E) +

N1 (E) + i R1 (E) , i R2 (E) , (8.130)

(’) (’)

where

|E|

N1 (E) = √ ˜(E 2 ’ ”2 ) (8.131)

2 ’ ”2

E

is the density of states of BCS-quasi-particles, and

”

N2 (E) = √ ˜(”2 ’ E 2 ) (8.132)

”2 ’ E 2

and

E ”

R1 (E) = ’ N2 (E) , R2 = N1 (E) (8.133)

” E

with ” being the BCS-energy gap.

Exercise 8.8. Show that in the weak coupling limit, the equilibrium electron“phonon

self-energy is speci¬ed by (recall the notation of Exercise 8.4 on page 228)

eZ R (E) = 1 + » (8.134)

and

1+» 1

≡

m(E Z R (E)) = , (8.135)

2„ (E) 2„in

238 8. Non-equilibrium superconductivity

where » = g 2 N0 is the dimensionless electron“phonon coupling constant and the

inelastic electron“phonon collision rate is given by

∞

(E ’ E)|E ’ E| cosh 2T

E

1 »π

= dE N1 (E ) . (8.136)

’E)

4(cpF )2 sinh (E2T

„ (E) E

cosh 2T

’∞

For temperatures close to the transition temperature, ” T , the rate becomes equal

to that of the normal state and we obtain for the collision rate for an electron on the

Fermi surface

∞

E2 »T 3

1 »π 7π

= dE = ζ(3) (8.137)

sinh E

(cpF )2 (cpF )2

„ (E = 0) 2

0 T