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