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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)


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

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