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

„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