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† † i
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


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

(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 .

= 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)


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

”=’ 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

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