F N (x, t+ ; x , t) = ’i N + 2|ψ“ (x, t) ψ‘ (x , t)|N = e2 μt

F μ (x, t; x , t)

the term ’2μ F N (x, t; x , t ) would appear on the left in Eq. (8.30).

8.1. BCS-theory 225

8.1.1 Nambu or particle“hole space

In order to write the BCS-Hamiltonian, Eq. (8.24), in standard quadratic form of a

¬eld, we introduce with Nambu the pseudo-spinor ¬eld

ψ‘ (1) Ψ1 (1)

Ψ(1) ≡ ≡ , (8.32)

†

Ψ2 (1)

ψ“ (1)

where, introducing condensed notation, 1 ≡ (t1 , x1 ) comprises the spatial variable

and in the Heisenberg picture also the time variable. The adjoint Nambu ¬eld is

Ψ† (1) ≡ (ψ‘ (1), ψ“ (1)) ≡ (Ψ† (1), Ψ† (1)) ,

†

(8.33)

1 2

where in the last de¬nition we have introduced the matrix notation for the Nambu

or particle“hole space.

The BCS-Hamiltonian is

⎛ ⎞

HBCS = dx1 ⎝ ψσ (1)h(1)ψσ (1) + ”(1)ψ‘ (1)ψ“ (1) + ”— (1)ψ“ (1)ψ‘ (1))⎠ ,

† †

†

σ=‘,“

(8.34)

where

1

(’i∇x1 ’ eA(x1 , t1 ))2 + eφ(x1 , t1 ) ’ μ

h(1) = (8.35)

2m

and the presence of coupling of the electrons to a classical electromagnetic ¬eld has

been included.

Consider the quadratic form in terms of the Nambu ¬eld

h(1) ”(1)

H = dx1 Ψ† (1) Ψ(1) , (8.36)

” (1) ’h— (1)

—

where h— (1) denotes complex conjugate of the single-particle Hamiltonian

1

h— (1) =

2

(i∇x1 ’ eA(x1 , t1 )) + eφ(x1 , t1 ) ’ μ . (8.37)

2m

The o¬-diagonal terms are identical to the ones in the BCS-Hamiltonian, but only

the ¬rst of the diagonal terms

† †

dx1 (ψ‘ (1)h(1)ψ‘ (1) ’ ψ“ (1)h— (1)ψ“ (1)) (8.38)

gives the corresponding kinetic energy term. In the second term partial integrations

are performed, giving

† †

dx1 ψ“ (x1 , t1 )h— (1)ψ“ (1) = dx1 h(x1 , t1 )ψ“ (1 )ψ“ (1) . (8.39)

x1 =x1

Using the equal-time anti-commutation relations for the electron ¬elds produces the

wanted order of the operators but an additional delta function

†

dx1 h(x1 , t1 ) δ(x1 ’ x1 ) ’ ψ“ (1)ψ“ (1 ) , (8.40)

x1 =x1

226 8. Non-equilibrium superconductivity

which, however, is just a state independent (in¬nite) constant that has no in¬‚uence

on the dynamics and can be dropped. We have thus shown that the BCS-Hamiltonian

can equivalently be written in terms of the Nambu ¬eld as

h(1) ”(1)

HBCS = dx1 Ψ† (1) Ψ(1). (8.41)

”— (1) ’h— (1)

Two by two (2 — 2) matrices are introduced in Nambu space according to

ψ‘ (1) †

Ψ(1) Ψ† (1 ) ≡ (ψ‘ (1 ), ψ“ (1 ))

†

ψ“ (1)

†

ψ‘ (1)ψ‘ (1 ) ψ‘ (1)ψ“ (1 )

= (8.42)

† † †

ψ“ (1)ψ‘ (1 ) ψ“ (1)ψ“ (1 )

or, in Nambu index notation,

(Ψ(1) Ψ† (1 ))ij ≡ Ψi (1) Ψ† (1 ) . (8.43)

j

For the opposite sequence we de¬ne the 2—2-matrix

Ψ† (1) Ψi (1 )

(Ψ† (1) Ψ(1 ))ij ≡ j

†

ψ‘ (1)ψ‘ (1 ) ψ“ (1)ψ‘ (1 )

= . (8.44)

† † †

ψ‘ (1)ψ“ (1 ) ψ“ (1)ψ“ (1 )

The Nambu ¬eld is seen to satisfy the canonical anti-commutation rules

[Ψ(x) , Ψ† (x ))]+ = δ(x ’ x ) 1 (8.45)

and

[Ψ† (x) , Ψ† (x ))]+ = 0 = [Ψ(x) , Ψ(x ))]+ (8.46)

where 1 and 0 are the unit and zero matrices in Nambu space, respectively.

The contour-ordered Green™s function is de¬ned in particle“hole or Nambu space

according to

G(1, 1 ) = ’i Tct (ΨH (1) Ψ† (1 )) , (8.47)

H

>

where the subscript indicates the ¬eld is in the Heisenberg picture. For t1 c t1 the

contour-ordered Green™s function becomes

’i (ΨH (1) Ψ† (1 ))

G> (1, 1 ) = H

†

ψ‘ (1)ψ‘ (1 ) ψ‘ (1)ψ“ (1 )

’i

= . (8.48)

† † †

ψ“ (1)ψ‘ (1 ) ψ“ (1)ψ“ (1 )

In Nambu index notation the greater Green™s function simply becomes

G> (1, 1 ) = ’i Ψi (1) Ψ† (1 ) . (8.49)

j

ij

8.1. BCS-theory 227

The lesser Green™s function is then

i (Ψ† (1 ) ΨH (1))

G< (1, 1 ) = H

†

ψ‘ (1 )ψ‘ (1) ψ“ (1 )ψ‘ (1)

= i (8.50)

† † †

ψ‘ (1 )ψ“ (1) ψ“ (1 )ψ“ (1)

and in matrix notation

G< (1, 1 ) = ’i Ψ† (1) Ψj (1 ) . (8.51)

i

ij

To acquaint ourselves with Nambu space we consider the dynamics of the Nambu

¬eld governed by the BCS-Hamiltonian. In the presence of classical electromagnetic

¬elds, the free one-particle Hamiltonian is

H0 (t) = dx Ψ† 0 (x, t) h(x, t) ΨH0 (x, t) , (8.52)

H

where

h(1) 0

≡ hij (1) .

h(1) = (8.53)

’h— (1)

0

From the equations of motion for the free Nambu ¬eld

i‚t1 Ψi (1) = hij (1)Ψj (1) (8.54)

and

i‚t1 Ψ† (1) = ’h— (1)Ψ† (1) , (8.55)

ij

i j

where the ¬elds are in the Heisenberg picture with respect to H0 (t), the equations of

motion for the free Nambu Green™s functions become

(i‚t1 ’ h(1))G> (1, 1 ) = 0 (8.56)

0

and

←—

i‚t1 G> (1, 1 ) = ’G> (1, 1 ) h (1 ) , (8.57)

0 0

where the arrow indicates that the spatial di¬erential operator operates to the left.

<

Identical equations of motion are obtained for G0 (1, 1 ).

The presence of the pairing interaction then leads to the appearance of a self-

energy which is purely o¬-diagonal in Nambu space

0 ”(1)

ΣBCS = . (8.58)

—

” (1) 0

In order to get more symmetric equations we perform the transformation

G(1, 1 ) ’ „3 G(1, 1 ) ≡ G , (8.59)

228 8. Non-equilibrium superconductivity

where „3 denotes the third Pauli matrix in Nambu space. The equations of motion

for the free Nambu Green™s functions then become

> >

i„3 ‚t1 G< (1, 1 hN (1)G< (1, 1

)= ) (8.60)

0 0

and

←—

> >

’G< (1, 1

G< (1, 1

i„3 ‚t1 )= ) (1 ) , (8.61)

hN

0 0

where

1 ’2

hN (1) = h(1)„3 = ’ ‚ (1) + eφ(1) ’ μ (8.62)

2m

and ’

‚ (1) = ∇x1 ’ ie„3 A(x1 , t1 ) . (8.63)

The BCS-self-energy, describing the pairing interaction, then becomes

’”(1)

0

ΣBCS = . (8.64)

—

” (1) 0

Introducing the Nambu ¬eld facilitates the description of the particle“hole coher-

ence in a superconductor. Next we introduce the real-time formalism for describing

non-equilibrium states as discussed in Chapter 5, here for the purpose of describing

non-equilibrium superconductivity. For a superconductor this means adding to the

Nambu-indices of Green™s functions the additional Schwinger“Keldysh or dynamical

indices.

Exercise 8.4. Show that, in equilibrium, the retarded Nambu Green™s function has

the form (unit matrices in Nambu space are suppressed)

GR (E, p) = (E „3 ’ ξ(p) ’ ΣR (E, p))’1 . (8.65)

In a strong coupling superconductor the self-energy has, according to the electron“

phonon model, the form

ΣR (E) = (1 ’ Z R (E))E „3 ’ i¦R (E) „1 , (8.66)

and show as a consequence that the retarded Nambu Green™s function becomes

’E Z R (E) „3 ’ ξ(p) ’ i¦R (E) „1

R

G (E, p) = . (8.67)

(ξ(p))2 ’ E 2 (Z R (E))2 + (¦R (E))2

8.1.2 Equations of motion in Nambu“Keldysh space

The contour-ordered Green™s function in Nambu space is de¬ned according to

GC (1, 1 ) = ’i„3 Tct (ΨH (1) Ψ† (1 )) (8.68)

H

and is mapped into real-time dynamical or Schwinger“Keldysh space according to

the usual rule, Eq. (5.1),

G11 (1, 1 ) G12 (1, 1 )

GC (1, 1 ) ’ G(1, 1 ) ≡ , (8.69)

’G21 (1, 1 ) ’G22 (1, 1 )

8.1. BCS-theory 229

where the Schwinger“Keldysh components now are Nambu matrices

Ψ† (1 ))

G11 (1, 1 ) = ’i „3 T (ΨH (1) (8.70)

H

and G-lesser

†

G12 (1, 1 ) = G< (x1 , t1 , x1 , t1 ) = i „3 ψH (x1 , t1 ) ψH (x1 , t1 ) (8.71)

and G-greater

†

G21 (1, 1 ) = G> (x1 , t1 , x1 , t1 ) = ’i „3 ψH (x1 , t1 ) ψH (x1 , t1 ) (8.72)

and

Ψ† (1 ))

˜

G22 (1, 1 ) = ’i „3 T (ΨH (1) (8.73)

H

and the Pauli matrix appears because of the convention, Eq. (8.59).

The information contained in the various Schwinger“Keldysh components of the

matrix Green™s function is rather condensed and it can be useful to have explicit

expressions for two independent components, say G-lesser and G-greater, from which

all other relevant Green™s functions can be constructed.

Exercise 8.5. Show that, in terms of the electron ¬eld, we have

†

ψ‘ (1) ψ‘ (1 ) ψ‘ (1) ψ“ (1 )

G (1, 1 ) = ’i„3

>

(8.74)

† † †

ψ“ (1) ψ‘ (1 ) ψ“ (1) ψ“ (1 )

and

†

ψ‘ (1 ) ψ‘ (1) ψ“ (1 ) ψ‘ (1)

<

G (1, 1 ) = i„3 . (8.75)

† † †

ψ‘ (1 ) ψ“ (1) ψ“ (1 ) ψ“ (1)

The matrix Green™s function on triagonal form

GR GK

G= (8.76)

GA

0

has, according to the construction in Section 5.3, the retarded and advanced compo-

nents

†

’iθ(t ’ t ) „3 [ψH (x, t) , ψH (x , t )]+

GR (x, t, x , t ) =

θ(t ’ t ) G> (x, t, x , t ) ’ G< (x, t, x , t )

= (8.77)

and

†

iθ(t ’ t) „3 [ψH (x, t) , ψH (x , t )]+

GA (x, t, x , t ) =

’θ(t ’ t) G> (x, t, x , t ) ’ G< (x, t, x , t )

= (8.78)

230 8. Non-equilibrium superconductivity

and the Keldysh or kinetic Green™s function

†

’i„3 [ψH (x, t) , ψH (x , t )]’

GK (x, t, x , t ) =

G> (x, t, x , t ) + G< (x, t, x , t ) .

= (8.79)

Exercise 8.6. Write down the retarded, advanced and kinetic components of the

Nambu Green™s function in terms of the electron ¬eld.

The equation of motion, the non-equilibrium Dyson equation, for the matrix

Green™s function becomes

(G’1 — G)(1, 1 ) = δ(1 ’ 1 ) + (Σ — G)(1, 1 ) (8.80)

0

and for the conjugate equation

(G — G’1 )(1, 1 ) = δ(1 ’ 1 ) + (G — Σ)(1, 1 ) , (8.81)

0

where the inverse free matrix Green™s function in Nambu“Keldysh space

G’1 (1, 1 ) = G’1 (1) δ(1 ’ 1 ) (8.82)

0 0

is speci¬ed in the triagonal representation by

G’1 (1) = (i„ (3) ‚t1 ’ h(1)) , (8.83)

0

and

„3 0

„ (3) = (8.84)

0 „3

is the 4—4-matrix, diagonal in Keldysh indices and „3 the third Pauli matrix in

Nambu space. Here

12

h(1) = ’ ‚ (1) + eφ(1) ’ μ , (8.85)

2m

where

‚(1) = ∇x1 ’ ie„ (3) A(x1 , t1 ) . (8.86)

Written out in components the matrix equation, Eq. (8.80), this gives

G’1 (1) GR(A) (1, 1 ) = δ(1 ’ 1 ) + (ΣR(A) — GR(A) )(1, 1 ) (8.87)

0

and

G’1 (1) GK (1, 1 ) = (ΣR — GK )(1, 1 ) + (ΣK — GA )(1, 1 ) . (8.88)

0

Subtracting the left and right Dyson equations, Eq. (8.80) and Eq. (8.81), we

obtain an equation identical in form to that of the normal state, Eq. (7.1), an equation

for the spectral weight function and a quantum kinetic equation of the form Eq. (7.3).

However, they are additionally matrix equations in Nambu space. Generally they are

8.1. BCS-theory 231

too complicated to be analytically tractable. It is therefore of importance that the

quasi-classical approximation works for the superconducting state, at least excellently

in low temperature superconductors where the superconducting coherence length,

ξ0 = vF /π”, is much longer than the Fermi wavelength. In other words, the small

distance information in the above equations is irrelevant and should be removed.

After discussing the gauge transformation properties of the Nambu Green™s functions,

we turn to describe the quasi-classical theory of non-equilibrium superconductors

which precisely does that.13

8.1.3 Green™s functions and gauge transformations

The ¬eld operator representing a charged particle transforms according to

˜

ψ(x, t) ’ ψ(x, t) eieΛ(x,t) ≡ ψ(x, t) (8.89)

under the gauge transformation

‚Λ(x, t)

•(x, t) ’ •(x, t) + A(x, t) ’ A(x, t) ’ ∇x Λ(x, t) .

, (8.90)

‚t

The probability and current density of the particles will be invariant to this shift;

quantum mechanics is gauge invariant.

The matrix Green™s function therefore transforms according to

†

eie(Λ(1)’Λ(1 )) ψ‘ (1 )ψ‘ (1) eie(Λ(1)+Λ(1 )) ψ“ (1 )ψ‘ (1)

˜

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 (8.91)

and similarly for

†

eie(Λ(1)’Λ(1 )) ψ‘ (1)ψ‘ (1 ) eie(Λ(1)+Λ(1 )) ψ‘ (1)ψ“ (1 )

˜

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

where ζ is Riemann™s zeta function.20

We note that, in the electron“phonon model, the superconductor is always gap-

less as the interaction leads to pair breaking and smearing of the spectral densities.

The inelastic collision rate is ¬nite, the pair-breaking parameter, and N1 is nonzero

for all energies.

8.3 Trajectory Green™s functions

A physically transparent approach to the quasi-classical Green™s function theory of

superconductivity revealing the physical content of ξ-integration and providing a

general proof of the important normalization condition was given by Shelankov, and

we follow in this section the presentation of reference [40]. The quasi-classical theory

for a superconductor is based on the existence of a small parameter, viz. that all

relevant length scales of the system: the superconducting coherence length, ξ0 =

vF /π”, and the impurity mean free path, l = vF „ , are large compared with the

microscopic length scale of a degenerate Fermi system, the inverse of the Fermi

momentum, p’1 , the inter-atomic distance, kF /ξ0

’1

1 (throughout we set = 1).

F

In addition, the length scale for the variation of the external ¬elds, »external , as well

as the order parameter are smoothly varying functions on this atomic length scale.

The 4 — 4 matrix Green™s function (matrix with respect to both Nambu and

Schwinger“Keldysh index) can be expressed through its Fourier transform

dp ip·r

G(x1 , x2 , t1 , t2 ) = e G(p, R, t1 , t2 ) , (8.138)

(2π)3

where on the right-hand side the spatial Wigner coordinates, the relative, r = x1 ’x2 ,

and center of mass coordinates, R = (x1 + x2 )/2, have been introduced. For a

degenerate Fermi system, we recall from Chapter 7 that the Green™s functions are

p’1 the exponential is in general

peaked at the Fermi surface, and for distances r F

rapidly oscillating and we can make use of the identity

e’ip r

eip·r eip r

δ(ˆ + ˆ) ’ δ(ˆ ’ ˆ) ,

= (8.139)

pr pr

2πi pr pr

20 The electron“phonon collision rate can be modi¬ed owing to the presence of disorder, as we will

discuss in Section 11.3.1.

8.3. Trajectory Green™s functions 239

where a hat on a vector denotes as usual the unit vector in the direction of the

p’1 the matrix Green™s function can be expressed in the form

vector. Thus for r F

(suppressing here the time coordinates since they are immaterial for the following)

m eipF |x1 ’x2 | m e’ipF |x1 ’x2 |

G(x1 , x2 ) = ’ g+ (x1 , x2 ) + g’ (x1 , x2 ) , (8.140)

2π |x1 ’ x2 | 2π |x1 ’ x2 |

p’1 ,

where, assuming |x1 ’ x2 | F

∞

i

vF d(p ’ pF ) e±i(p’pF )|x1 ’x2 | G(±pˆ, R)

g± (x1 , x2 ) = (8.141)

r

2π ’∞

and the rapid convergence of the integrand limits the integration over the length of

the momentum to the region near the Fermi surface.

The equations of motion for the slowly varying functions, g± , are obtained by

substituting into the (left) Dyson equation, which gives

±ivFˆ · ∇x1 g± (x1 , x2 ) + H(±ˆ, x1 ) —¦ g± (x1 , x2 ) = 0 , (8.142)

r r

where (re-introducing brie¬‚y the time variables)

‚

’ eφ(x, t1 ) + evF „3 n · A(x, t1 ) δ(t1 ’ t2 )

H(n, x, t1 , t2 ) = i„3

‚t1

’ Σ(n, x, t1 , t2 ) (8.143)

and we have used the fact that the components of the matrix self-energy are peaked

p’1 , i.e. slowly varying functions of the

for small spatial separations, |x1 ’ x2 | F

momentum as discussed in Section 7.5, and

dr eipF n·r Σ(x + r/2, x ’ r/2, t1 , t2 ) .

Σ(n, x, t1 , t2 ) = (8.144)

The circle in Eq. (8.142) denotes, besides integration with respect to the internal

time, an additional matrix multiplication with respect to Nambu and dynamical

p’1 , the second spatial derivative is negligible because

indices. Since |x1 ’ x2 | F

the envelope functions, g± , are slowly varying, and consequently the di¬erentiation

acts only along the straight line connecting the space points in question, the classical

trajectory connecting the points. Only the in¬‚uence of the external ¬elds on the phase

of the propagator is thus included and the e¬ects of the Lorentz force are absent,

as expected in the quasi-classical Green™s function technique. Thermo-electric and

other particle“hole symmetry broken e¬ects are also absent just as in the normal

state as discussed in Chapter 7.

Specifying a linear trajectory by a position, R, and its direction, n, the positions

on the linear trajectory, r, can be speci¬ed by the distance, y, from the position R

r = R + yn . (8.145)

For the propagator on the trajectory we then have

g± (n, R, y1 , y2 ) = g± (R + y1 n, R + y2 ) (8.146)

240 8. Non-equilibrium superconductivity

and we introduce the matrix Green™s function on the trajectory

§

⎨ g+ (R + y1 n, R + y2 n) y1 > y2

g(n, R, y1 , y2 ) ≡ (8.147)

©

g’ (R + y1 n, R + y2 n) y1 < y2 .

p’1 ,

Then, according to Eq. (8.141), and again with |y1 ’ y2 | F

∞

i

vF d(p ’ pF ) e±i(p’pF )(y1 ’y2 ) G(p n, R + (y1 + y2 )n/2)

g(n, R, y1 , y2 ) =

2π ’∞

(8.148)

and we observe that the trajectory Green™s function describes the propagation of

particles with momentum value pF along the direction n, and satis¬es according to

p’1 , the equation

Eq. (8.142), for |y1 ’ y2 | F

‚

g(y1 , y2 ) + H(n, y1 ) —¦ g(y1 , y2 ) = 0 ,

ivF (8.149)

‚y1

where the notation

g(y1 , y2 ) ≡ g(n, R, y1 , y2 ) (8.150)

has been introduced. Equation (8.149) is incomplete as we have no information at

the singular point, y1 = y2 . Forming the quantity

∞

’vF

g(y + δ, y) ’ g(y ’ δ, y) = d(p ’ pF ) G(p n, R + n(y + δ/2)) sin((p ’ pF )δ)

π ’∞

(8.151)

p’1 ,

and assuming ξ0 δ we can neglect the dependence in the center of mass

F

coordinate on δ, and as the contribution from the momentum integration comes from

the regions far from the Fermi surface in the limit of vanishing δ, we can insert the

normal state Green™s functions to obtain (recall Eq. (7.125))

g(y + δ, y) ’ g(y ’ δ, y) = δ(t1 ’ t2 ) , (8.152)

where the unit matrix in Nambu“Keldysh space has been suppressed on the right-

hand side, and δ ξ0 , »external . This result can be included in the equation of

motion, Eq. (8.149), as a source term, and we obtain the quasi-classical equation of

motion

‚

g(y1 , y2 ) + H(n, y1 ) —¦ g(y1 , y2 ) = ivF δ(y1 ’ y2 ) .

ivF (8.153)

‚y1

Together with the similarly obtained conjugate equation

‚

’ivF g(y1 , y2 ) + g(y1 , y2 ) —¦ H(n, y2 ) = ivF δ(y1 ’ y2 ) (8.154)

‚y2

we have the equations determining the non-equilibrium properties of a low-temperature

superconductor.

8.3. Trajectory Green™s functions 241

Exercise 8.9. Show that the retarded, advanced and kinetic components of the

trajectory Green™s function satisfy the relations

g R (n, R, y1 , t1 , y2 , t2 ) = ’„3 (g A (n, R, y2 , t2 , y1 , t1 ))† „3 (8.155)

and

g K (n, R, y1 , t1 , y2 , t2 ) = „3 (g K (n, R, y2 , t2 , y1 , t1 ))† „3 (8.156)

and for spin-independent dynamics

g R (n, R, y1 , t1 , y2 , t2 ) = „1 (g A (’n, R, y1 , t2 , y2 , t1 ))T „1 (8.157)

and

g K (n, R, y1 , t1 , y2 , t2 ) = „1 (g K (’n, R, y1 , t2 , y2 , t1 ))T „1 . (8.158)

From the quasi-classical equations of motion, Eq. (8.153) and Eq. (8.154), it

follows that for y1 = y2

‚

g(y1 , y) —¦ g(y, y2 ) =0 (8.159)

‚y

and the function g(y1 , y) —¦ g(y, y2 ) jumps to constant values at the ¬xed positions

y1 and y2 . Since we know the jumps of g we get

§

⎪ g(y1 , y2 ) y1 > y > y2

⎪

⎪

⎪

⎨

y ∈ [y1 , y2 ]

g(y1 , y) —¦ g(y, y2 ) = 0 / (8.160)

⎪

⎪

⎪

⎪

©

’g(y1 , y2 ) y1 < y < y2

where the value zero follows from the decay of the Green™s function as a function of

the spatial variable as the positions in the quasi-classical Green™s function satisfy the

constraint |y1 ’ y| l, and in a disordered conductor the Green™s function decays

according to g(y1 , y) ∝ exp{|y1 ’ y|/2l}, where l is the impurity mean free path

(recall Exercise 7.4 on page 192).

Introducing the coinciding argument trajectory Green™s functions (suppressing

the time variables)

g± (n, r) ≡ lim g(±) (n, R, y ± δ, y) (8.161)

δ’0

we observe that their left“right subtracted Dyson equations of motion according to

Eq. (8.153) and Eq. (8.154) are

±ivF · ∇r g± + H —¦ g± ’ g± —¦ H = 0 (8.162)

and according to Eq. (8.152) and Eq. (8.160) they satisfy the relations

g± —¦ g± = ± g± (8.163)

242 8. Non-equilibrium superconductivity

and

g± —¦ g“ = 0 = g“ —¦ g± (8.164)

and

g+ ’ g’ = 1 , (8.165)

where 1 is the unit matrix in Nambu“Keldysh space.

The quantity g(n, r) = g+ (n, r) + g’ (n, r) therefore satis¬es the equation of mo-

tion

ivF · ∇r g + H —¦ g ’ g —¦ H = 0 (8.166)

and, according to Eq. (8.163), Eq. (8.164) and Eq. (8.165), the normalization condi-

tion

g —¦ g = 1. (8.167)

The equation of motion is the same as that for the ξ-integrated Green™s function

and the above analysis provides an explicit procedure for the ξ-integration as

∞

i

vF d(p ’ pF ) G(pn, R) cos((p ’ pF )δ) .

g(n, r) = lim (8.168)

π δ’0 ’∞

The integral is convergent when δ is ¬nite and independent of δ for δ ξ0 . The

dropping of the high-energy contributions in the ξ-integration procedure is in this

procedure made explicit by the small distance cut-o¬.

The quantum e¬ects included in the quantum kinetic equation for g K is thus the

particle“hole coherence due to the pairing interaction whereas the kinetics is classical.

8.4 Kinetics in a dirty superconductor

A characteristic feature of a solid is that it contains imperfections, generally referred

to as impurities. Typically superconductors thus contain impurities, and of relevance

is a dirty superconductor. The kinetics in a disordered superconductor will be dif-

fusive. In the dirty limit where the mean free path is smaller than the coherence

length, or kTc < /„ , the integral equation with respect to the ordinary impurity

scattering, i.e. the non-spin-¬‚ip impurity scattering, can then be reduced to a much

simpler di¬erential equation of the di¬usive type.21 We therefore return to the cou-

pled equations for the quasi-classical propagators g R,A,K , Eq. (8.96), supplemented

by the normalization condition, Eq. (8.116).

In the dirty limit, the Green™s function will be almost isotropic, and an expansion

in spherical harmonics needs only keep the s- and p-wave parts

g(ˆ , R, t1 , t1 ) = gs (R, t1 , t1 ) + p · gp (R, t1 , t1 )

ˆ (8.169)

p

and

|ˆ · gp (R, t1 , t1 )| |gs (R, t1 , t1 )| . (8.170)

p

21 Quite analogous to deriving the di¬usion equation from the Boltzmann equation as discussed

in Sections 7.4.2 and 7.5.5.

8.4. Kinetics in a dirty superconductor 243

The self-energy is then

σ(ˆ , R, t1 , t1 ) = σs (R, t1 , t1 ) + p · σ p (R, t1 , t1 ) ,

ˆ (8.171)

p

where

ˆ

dp

p · σ p (R, t1 , t1 ) = ’iπni N0 |Vimp (ˆ · p )|2 p · gp (R, t1 , t1 )

ˆ pˆ ˆ (8.172)

4π

and

i

σs = ’ gs + σs , (8.173)

2„

where

i

σs = ’ e’ph

„3 gs „3 + σs (8.174)

2„s

contains the e¬ects of spin-¬‚ip and electron“phonon scattering.

Performing the angular integration gives

’i 1 1

σp = ’ gp , (8.175)

2 „ „tr

where „tr is the impurity transport life time determining the normal state conduc-

tivity

ˆ

1 dp

|Vimp (ˆ · p )|2 (1 ’ p · p ) .

pˆ ˆˆ

= 2πni N0 (8.176)

„tr 4π

The inverse propagator has exactly the form

’1 ’1

ˆ ’1

g0 = g0s + p · g0p , (8.177)

where

’1

g0s = („3 ‚t1 + ieφ(R, t1 )) δ(t1 ’ t1 ) (8.178)

and

’1

g0p = vF ‚ ‚ = (∇R ’ ie„3 A(R, t1 )) δ(t1 ’ t1 ) .

, (8.179)

The kinetic equation in the dirty limit can be split into even and odd parts with

ˆ

respect to p

1

’1

[g0s + iσs —¦ gs ]’ + vF [‚ —¦ gp ]’ = 0

, , (8.180)

3

and

1

[gs —¦ gp ]’ + vF [‚ —¦ gs ]’ = 0 .

, , (8.181)

2„tr

Using the s- and p-wave parts of the normalization condition gives

gs —¦ gs = δ(t1 ’ t1 ) (8.182)

and

[gs —¦ gp ]+ = 0

, (8.183)

244 8. Non-equilibrium superconductivity

and we get

gp = ’l gs —¦ [‚ —¦ gs ]’ ,

, (8.184)

where l = vF „tr is the impurity mean free path.

Upon inserting into Eq. (8.180), an equation for the isotropic part of the quasi-

classical Green™s function is obtained, the Usadel equation [41],

’1

[g0s + iσs ’ D0 ‚ —¦ gs —¦ ‚ —¦ gs ]’ = 0 .

, (8.185)

We have obtained a kinetic equation which is local in space, an equation for the

quasi-classical Green™s function for coinciding spatial arguments. This equation is

the starting point for considering general non-equilibrium phenomena in a dirty su-

perconductor.

Exercise 8.10. Show that the current density in the dirty limit takes the form

∞

eN0 D0

dE Tr(„3 (gs —¦ ‚ —¦ gs + gs —¦ ‚ —¦ gs )) ,

R K K A

j(R, T ) = (8.186)

4 ’∞

which by using the Einstein relation, σ0 = 2e2 N0 D0 , can be expressed in terms of

the conductivity of the normal state.

8.4.1 Kinetic equation

In the dirty limit, the kinetic equation

g R —¦ B[h] ’ B[h] —¦ g A = 0 (8.187)

is speci¬ed by

(g0 )’1 —¦ h ’ h —¦ (g0 )’1 ’ iσe’ph

R A K

B[h] =

D0 ‚ —¦ g R —¦ [‚ —¦ h]’ ’ D0 [‚ —¦ h]’ —¦ g A —¦ ‚ ,

’ , , (8.188)

where

’1 1

R(A) R(A)

ˆ

= ’iE„3 + ie•(r, t) + ” + iσe’ph + „3 g R(A) „3 .

g0 (8.189)

2„s

Inelastic e¬ects are included through the electron“phonon interaction.

In the low frequency limit, the problem simpli¬es, and we discuss this case in

order to show how the matrix distribution function enters the collision integral. For

superconducting states close to the transition temperature, the Ginzburg“Landau

regime, the component γ is negligible, as discussed in the next section, and the

distribution matrix h can be chosen diagonal in Nambu space

h = h1 1 + h2 „3 . (8.190)

We then perform a Taylor expansion in Eq. (8.187), and linearize the equation with

respect to h1 ’h0 and h2 . To expose the kinetic equations satis¬ed by the distribution

8.4. Kinetics in a dirty superconductor 245

functions we multiply the kinetic equation with Pauli matrices and take the trace in

particle“hole space, in fact for the present case we take the trace of the equation and

the trace of the equation multiplied by „3 , and obtain the two coupled equations for

the distribution functions

™ ™

N1 h1 + R2 e”‚E h1 + 2R2 m” h2 ’ D0 ∇R · M1 (E, E) ∇R h1

D0 ∇R · (∇R h2 + ps ‚E h0 ) ’ 4N2 R2 ps · (∇R h2 + ps ‚E h0 )

™ ™

+

= K1 [h1 ] (8.191)

and

™ ™ ™

N1 (h2 + ¦ ‚E h0 ) + 2N2 e” h2 ’ N2 m” ‚E h0 ’ 4D0 N2 R2 ps · ∇R h1

’ D0 ∇R · M2 (E, E) (∇R h2 + ps ‚E h0 ) = K2 [h2 ] ,

™ (8.192)

where the collision integrals are given by, i = 1, 2,

∞

2T ’ hi (E ) cosh 2T

hi (E) cosh2 2E

E

’π dE μ(E ’ E ) Mi (E, E )

Ki [hi ] = ,

E’E E E

sinh 2T cosh 2T cosh 2T

’∞

(8.193)

where

N1 (E) N1 (E ) + R2 (E) R2 (E ) i=1

Mi (E, E ) = (8.194)

N1 (E) N1 (E ) + N2 (E) N2 (E ) i=2

and μ is the Fermi surface average of the function in Eq. (7.134), the Eliashberg

function, ±2 F (E ’ E ) = μ(E ’ E ),

i» dˆ

p

μ(E ’ E ) = (DR (ˆ · p, E ’ E) ’ DA (ˆ · p, E ’ E)) ,

pˆ pˆ (8.195)

2π 4π

or in general the Fermi surface weighted average of the phonon spectral weight func-

tion and the momentum-dependent coupling function (recall Eq. (7.134)).

Together with the expressions for charge and current density and Maxwell™s equa-

tions, the kinetic equations for the distribution functions supplemented with the

equations for the generalized densities of states and the order parameter equation,

constitute a complete description of a dirty conventional superconductor in the low-

frequency limit.

Exercise 8.11. Show that in the Debye model of lattice vibrations, the Eliashberg

function becomes

»

E|E| θ(ωD ’ |E|) .

μ(E) = (8.196)

4(cpF )2

246 8. Non-equilibrium superconductivity

8.4.2 Ginzburg“Landau regime

In this section we shall derive the time-dependent Ginzburg“Landau equation for the

order parameter.22 First we further reduce the equation determining the components

of the spectral part of the Usadel equation

’1

’ D0 ‚ —¦ g R(A) —¦ ‚ —¦ g R(A) ]’ = 0

R(A)

[g0 + iσ , (8.197)

by considering the case where temporal non-equilibrium is slow.

R(A)

We shall treat the pairing e¬ect (contained in e σe’ph ) in the BCS-approximation

and approximate the electronic damping by the equilibrium expression Eq. (8.136).

Then the retarded (advanced) electron“phonon self-energy reduces to

i

’

R(A) ˆ

’»E „3 ’ i” ,

σe’ph = (8.198)

(+)

2„in

where „in is the inelastic electron“phonon scattering time, and the gap matrix is

0 ”

ˆ

”= , (8.199)

”— 0

where (from now on we drop the s-wave index)

ωD

i»

”=’ dE Tr („1 ’ i„2 )g K (8.200)

8

’ωD

is the order parameter.

We assume that the characteristic non-equilibrium frequency, ω, satis¬es ω <

”, T, 1/„ . We can then make a temporal gradient expansion in Eq. (8.197) and

obtain to lowest order for the o¬-diagonal components

1

D0 (±D2 (β ’ iγ) ’ (β ’ iγ)∇2 ±)R(A)

R

2

R(A)

i i ™

’iE (’) (β ’ iγ) ’ ”± + ±(β ’ iγ) ’ ¦ ‚E (β ’ iγ)

+

= (8.201)

2„in „s

and

1

D0 (±D—2 (β + iγ) ’ (β + iγ)∇2 ±)R(A)

R

2

R(A)

i i

— ™

’iE (’) (β + iγ) ’ ” ± + ±(β + iγ) + ¦ ‚E (β + iγ)

+

= , (8.202)

2„in „s

where

D = ∇R ’ 2ieA (8.203)

22 We essentially follow reference [42] and reference [43].

8.4. Kinetics in a dirty superconductor 247

is the gauge co-variant derivative. We note that all time-dependent terms cancel

except the one involving the electro-chemical potential of the condensate. Together

with the normalization condition

(±R )2 + (β R )2 + (γ R )2 = 1 (8.204)

these equations determine the generalized densities of states. In view of Eq. (8.155),

say only the retarded components needs to be evaluated, and in the following we

therefore leave out the superscript.

Assuming the superconductor is in the Ginzburg“Landau regime where the tem-

perature is close to the critical temperature, ”(T ) T , we can iterate Eq. (8.202)

starting with the density of states for the normal state, i.e. ± ’ 1, and neglect

spatial variations. We then obtain to a ¬rst approximation

”—

β + iγ = (8.205)

’iE + 1/2„in + 1/„s

and similarly for β ’ iγ. Then using the normalization condition, Eq. (8.204), gives

the ¬rst order correction to ±. In the next iteration we then obtain

”—

β + iγ =

’iE + 1/2„in + 1/„s

D0 D— ”— (’iE + 1/2„in )|”|2 ”—

i

+ + (8.206)

(’iE + 1/2„in + 1/„s )2 (’iE + 1/2„in + 1/„s )4

2

and similarly for β ’ iγ. It follows from these equations that γ is smaller than the

other components by the amount ”/T , and can be neglected in the Ginzburg“Landau

regime.

The distribution matrix in Nambu space we assume to be of the form

h = h0 + h1 + h2 „3 . (8.207)

Making the slow frequency gradient expansion of the kinetic propagator, Eq. (8.119),

and keeping only linear terms in the distribution functions h1 and h2 we obtain

i

= h0 (g R ’ g A ) ’

gK [h0 , g R + g A ]p

2

+ h1 (g R ’ g A ) + h2 (g R „3 ’ „3 g A ) , (8.208)

where the Poisson bracket is with respect to time and energy variables. In the

expression for the order parameter, Eq. (8.113), we therefore obtain

∞

» i

dE (h0 (β ’ β — ) ’ [h0 , (β + β — )]p

”(R, T ) = ’

4 2

’∞

+ h1 (β ’ β — ) ’ h2 (β + β — )) . (8.209)

248 8. Non-equilibrium superconductivity

Using the known pole structure of h0 = tanh E/2T , terms involving this function can

be evaluated by the residue theorem, and we arrive at the time-dependent Ginzburg“

Landau equation for the order parameter

A ’ B|”|2 ’ C(‚T ’ D0 D2 ) + χ ” = 0 , (8.210)

where ∞

1

χ= dE R2 h1 (8.211)

”

’∞

is Schmid™s control function, controlling the magnitude of the order parameter, and

the coe¬cients can be expressed in terms of the poly-gamma-functions (ψ being the

di-gamma-function)

Tc

+ ψ(1/2 + ρT /Tc ) ’ ψ(1/2 + ρ)

A = ln (8.212)

T

and

1 1

B=’ ψ (2) (1/2 + ρ) + ρs ψ (3) (1/2 + ρ) (8.213)

(4πT )2 3

and

1

C=’ ψ (1) (1/2 + ρ) , (8.214)

4πT

where ρ = ρs + ρin

1 1

ρs = , ρin = (8.215)

2π„s T 4π„in T

and we have used the relation for the transition temperatures in the presence and

absence of pair-breaking mechanisms

Tc0

= ψ(1/2 + ρT /Tc) ’ ψ(1/2) .

ln (8.216)

Tc

Evaluating the coe¬cients gives the time-dependent Ginzburg“Landau equation

T ’ Tc 7ζ(3) ”2

π™

”(x, t) = ’ + ξ 2 (0) (4m2 vs ’ ∇2 ) + χ ”(x, t) ,

2

+ x

2 T2

8Tc Tc 8π c

(8.217)

2

where ξ (0) = πD0 /8Tc is the coherence length in the dirty limit.

In the normal state close to the transition temperature, there will be supercon-

ducting ¬‚uctuations in the order parameter. In that case, the ¬rst term in the

time-dependent Ginzburg“Landau equation, Eq. (8.217), dominates and the thermal

¬‚uctuations of the order parameter decays with the relaxation time

π 1

N