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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
„R = . (8.218)
8 |T ’ Tc |

In the superconducting state the relaxation of the order parameter is, according
to Eq. (8.211), determined by the non-equilibrium distribution of the quasi-particles,
8.5. Charge imbalance 249


which in turn is in¬‚uenced by the time dependence of the order parameter. In the
spatially homogeneous situation where h2 vanishes, the kinetic equation, Eq. (8.191),
reduces to
N1
™ ™
N1 h1 + R2 ”‚E h0 = ’ h1 . (8.219)
„in
Calculating the control function gives
π ™
χ= „in ” (8.220)
4Tc
and according to the time-dependent Ginzburg“Landau equation the relaxation time
for the order parameter, ” „in 1, is

π 3 Tc
„R = „in . (8.221)
7ζ(3) ”
Experimental observation of the relaxation of the magnitude of the order parameter
can been achieved by driving the superconductor out of thermal equilibrium by a
laser pulse [44].


8.5 Charge imbalance
Under non-equilibrium conditions in a superconductor a di¬erence in the electro-
chemical potential between the condensate and the quasi-particles can exist, re¬‚ecting
the ¬nite rate of conversion between supercurrent and normal current. For example,
charge imbalance occurs when charge from a normal metal is injected into a super-
conductor in a tunnel junction. As an application of the theory of non-equilibrium
superconductivity, we shall consider the phenomenon of charge imbalance generated
by the combined presence of a supercurrent and a temperature gradient. We shall
limit ourselves to the case of temperatures close to the critical temperature where
analytical results can be obtained.23
The charge density is in the real ”-gauge, recall Section 8.1.3,
⎛ ⎞


ρ = 2eN0 ⎝¦ + dE N1 (E) f2 (E)⎠ , (8.222)
’∞

where the condensate electro-chemical potential in general is ¦ = χ/2 ’ e•, χ being

the phase of the order parameter. We have introduced distribution functions related
to the original ones according to h1 = 1 ’ 2f1 and h2 = ’2f2 .24 We could insert
instead the full distribution function, f = f1 + f2 , in Eq. (8.222) as f1 is an odd
function, and thereby observe that the charge of the quasi-particles described by the
distribution function is the elementary charge, the full electronic charge.
23 We essentially follow the presentation of reference [45]. For general references to charge imbal-
ance in superconductors, as well as other non-equilibrium phenomena, we refer to the articles in
reference [42].
24 In reference [46], they are referred to as the longitudinal and transverse distribution functions.
250 8. Non-equilibrium superconductivity


The strong Coulomb force suppresses charge ¬‚uctuations, but it is possible to
have a charge imbalance between the charge carried by the condensate of correlated
electrons and the charge carried by quasi-particles


Q— = 2eN0 dE N1 (E) f2 (E) . (8.223)
’∞

The presence of a temperature variation, T (r) = T + δT (r) creates a non-
equilibrium distribution in the thermal mode
E ‚f0
δf = f1 ’ f0 = ’ δT (8.224)
T ‚E
where f0 is the Fermi function. The presence of a supercurrent, ps = mvs = ’(∇χ +
2eA)/2, couples via the kinetic equation for the charge mode the thermal and the
charge mode. For a stationary situation with f2 homogeneous in space we have
according to Eq. (8.192)

2N2 ”f2 ’ 4D0 N2 R2 ps · ∇f1 = K2 [f2 ] . (8.225)

The ¬rst term on the left gives rise to conversion between the supercurrent and the
current carried by quasi-particles, while the second term is a driving term propor-
tional to vs · ∇T . Close to the transition temperature, Tc , the collision integral is
dominated by energies in the region E T . In this energy regime we have ± 1
˜˜
and β, γ · where we have introduced the notation · = ”(T )/T for the small
parameter of the problem. The collision integral then becomes proportional to the
inelastic collision rate
⎛ ⎞

N1 ⎝ ‚f0
dE N1 (E) f2 (E)⎠ .
K2 [f2 ] = ’ f2 + (8.226)
„ (E) ‚E
’∞

The last term is proportional to the charge imbalance, and we get the following
kinetic equation
N1 ‚f0 E ‚f0
(2”„ (E)N2 + N1 ) f2 + Q— = ’4mD0 „ (E)N2 R2 vs · ∇T. (8.227)
2N0 ‚E T ‚E
Integrating with respect to the energy variable gives
Q— 4mD0 A
=’ vs · ∇T , (8.228)
T 1’B
2N0
where

‚f0 N1 N2 R2 E„ (E)
A=’ dE (8.229)
‚E N1 + 2”„ (E)N2
’∞
8.6. Summary 251


and

2
‚f0 N1
B=’ dE . (8.230)
‚E N1 + 2”„ (E)N2
’∞

Assuming weak pair-breaking the quantities can be evaluated. To zeroth order in ·
we have B 1 as N1 1 and N2 0. From the structure of the densities of states
it is apparent that the main correction contribution comes from the energy range ”
up to a few ”. We can therefore use the high-energy expansion
”“
N2 = , (8.231)
E 2 + “2
where
1 1 1
+ D0 (p2 ’ ”’1 ∇2 ”)
“= + (8.232)
s
2„ (E) „s 2

is the pair-breaking parameter. In the limit of weak pair-breaking, “ ”, and ”
is small as we assume the temperature is close to the critical temperature, so that
”(„ (E)“)1/2 T , and we get

π”
B=1’ (2„ (E)“)1/2 . (8.233)
4T
In the BCS-limit, A is logarithmically divergent due to the singular behavior of
the density of states, but the pair-breaking smears out the singularity and gives a
logarithmic cut-o¬ at ln(4”/“), and we have

” 4”
+ 2(2„ (E)“ ’ 1)1/2 arctan((2„ (E)“ ’ 1)1/2 )
A= ln . (8.234)
8T “

In the limit where electron“phonon interaction provides the main pair breaking mech-
anism, “ ∼ 1/2„E , the charge imbalance thus becomes

2 pF l
Q— = 2N0 (vs · ∇T ) ln(8”„ (E)) . (8.235)
3π T
For a discussion of the experimental observation of charge imbalance we refer the
reader to reference [47].


8.6 Summary
In this chapter we have considered non-equilibrium superconductivity. By using the
quasi-classical Green™s technique, a theory with an accuracy in the 1% range was
constructed that were able to describe the non-equilibrium states of a conventional
superconductor. This is a rather impressive achievement bearing in mind that a su-
perconductor is a messy many-body system. In general one obtains coupled equations

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