Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
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7 Nanostructured Superconductors 277
neighboring Rutgers group, particularly with Pe-
ter Lindenfeld and the late William McLean, and
their students and post docs, amongst them Tom
Worthington and Tom Filler. At Tel Aviv Univer-
sity, Moshe Dayan, David Bergman, Henri Fenichel,
Alexander Gerber, Meir Gershenson, Ilan Grave, En-
rique Grunbaum, Y. Imry, Zvi Ovadyahu, Michael
Rappaport, Yoash Shapira, O. Entin Wohlman all
made important contributions.Inspiring discussions
with the late Shlomo Alexander took place on a per-
manent basis. I take the opportunity to express my
sincere gratitude to all of them. It is known that our
work on granular Aluminium was one of the triggers
that prompted Alex Muller to look for high tempera-
ture superconductivity in oxides.This may well have
been the most important outcome of this work.
This work was supported by the Oren Family
Chair for Experimental Solid State Physics.
References
1. B.Abeles,R.W.CohenandG.W.Cullen,Phys.Rev.Lett.17, 632 (1966)
2. Y. Shapira and G. Deutscher, Thin Solid Films 87, 29 (1982)
3. G. Deutscher, A. Kapitulnik and M. Rappaport, In: Percolation, Structures and Processes,Eds.G.Deutscher,R.Zallen
andJ.Adler,Ann.IsraelPhys.Soc.5, 207 (1983)
4. B. Abeles, Adv. Phys. 24, 407 (1975);Applied Solid State Science 6, 1 (1976)
5. A.Kapitulnik,M.Rappaport andG.Deutscher,J.Phys.Lett.42, L-541 (1981)
6. G. Deutscher, H. Fenichel, M. Gershenson, E. Grunbaum and Z. Ovadyahu, J. Low Temp. Phys. 10, 231 (1973)
7. U.Dai et al.,Phys. Rev. B 36, 790 (1987)
8. D.J. Bergman et al., Phys.Rev. Lett.50, 1512 (1983)
9. G. Deutscher, M. Rappaport and Z. Ovadyahu, Solid State Commun. 28, 247 (1978)
10. B. Bandyopadhyay et al., Phys. Rev. B 26, 3476 (1982)
11. A. Fulton and G.J. Dolan, Phys. Rev. Lett. 59, 109 (1987)
12. H. Devoret et al., Phys. Rev. Lett. 61, 1824 (1990)
13. G. Deutscher and M. Cyrot, unpublished
14. N.F. Mott and Davis,In: Electronic processes in non cristalline materials (Clarendon, Oxford, 1971)
15. B. Abeles, Phys. Rev. B 15, 2828 (1977)
16. N.F. Mott, Phil.Mag. 26, 1015 (1972)
17. J. Ioffe and A. Regel, Prog. Semiconduct. 4, 237 (1960)
18. G. Falci,A. Fubini and A. Mastellone, to be published
19. Y. Shapira, PhD Thesis,Tel Aviv University (1986)
20. D.C. Ralph, C.T. Black and M. Tinkham, Phys. Rev. Lett. 74, 3241 (1995)
21. G. Deutscher, O. Entin-Wohlman,S. Fishman and Y.Shapira,Phys. Rev. B 21, 5041 (1980)
22. R.L. Filler et al.,Phys.Rev. B 21, 5031 (1980)
23. Y. Shapira and G. Deutscher, Phys. Rev. B 27, 4463 (1983)
24. P.NozieresandF.Pistolesi,Eur.J.Phys.B10, 649 (1999)
25. G. Deutscher, Nature 397, 410 (1999)
26. L. Nechustan, MSc Thesis, Tel Aviv University (2002)
27. G. Deutscher, M. Gershenson,E. Grunbaum and Y. Imry,J.Vac.Sci. Technol. 10, 674 (1973)
28. G. Deutscher, Phys. Lett. 35A, 28 (1971)
29. G. Deutscher,Y. Imry and L. Gunther, Phys. Rev. B 10, 4042 (1974)
30. G. Deutscher, O. Entin-Wohlman and Y. Shapira, Phys. Rev. B 22, 4264 (1980)
31. A. Gerber and G. Deutscher, Physica B 126, 477 (1984)
32. M. Dayan and G. Deutscher, Proceedings of Low Temperature Physics LT14, Vol. 2, 241 (North-Holland,Amsterdam,
1975)
33. A. Gerber et al., Phys. Rev. Lett. 78, 4277 (1997)
8 Proximity–Coupled Systems: Quasiclassical Theory of Superconductivity
Venkat Chandrasekhar Dept. of Physics and Astronomy, Northwestern University, Evanston, USA
8.1 Introduction .............................................................................279
8.2 Transport Equations in the Diffusive Approximation .........................................280
8.3 The Keldysh Green’s Functions .............................................................283
8.4 The QuasiclassicalApproximation ..........................................................288
8.5 Non-Equilibrium Green’s Functions for Superconducting Systems .............................291
8.6 QuasiclassicalSuperconducting Green’s Functions ...........................................294
8.7 The Dirty Limit: The Usadel Equation .......................................................297
8.8 Parametrization of the Quasiclassical Green’s Function .......................................302
8.9 Applications of the Quasiclassical Equations to Proximity-Coupled Systems ....................304
8.9.1Proximity-CoupledWire..............................................................305
8.9.2Superconductor-MetalBilayer.........................................................309
8.9.3TheSNSJunctionandAndreevInterferometers..........................................310
8.10 Summary ................................................................................311
References...............................................................................312
8.1 Introduction
Although the physics of normal metals (N) in close
proximity to superconductors (S) has been studied
extensively for more than thirty years, it is only in
the past decade that experiments have been able to
probe directly the region close to the NS interface at
temperatures far below the transition temperature T
c
of the superconductor. These experiments have been
made possible by the availability of microlithogra-
phy techniques that enable the fabrication of het-
erostructure devices with submicron scale features.
This size scale is comparable to the relevant physi-
cal length scales of the problem, and consequently, a
number of new effects have been observed. A vari-
ety of systems have been studied, the variation pri-
marily being in the type of normal metal coupled to
the superconductor. Canonical normal metals such
as copper and gold, semiconductors, insulators, and
ferromagnets have been employed. Although a vari-
ety of theoretical techniques have been used to de-
scribe proximity-coupled systems, the quasiclassical
theory of superconductivity [1–12] has proved to be
a remarkably powerful tool in understanding the mi-
croscopic basis for the remarkable effects observed
in these systems.
A number of excellent recent articles [13–19] have
explored the application of the quasiclassical theory
of superconductivity to proximity coupled systems.
In this review, a self-contained development of the
quasiclassical theory is presented,starting from non-
equilibrium Keldysh Green’s functions for normal
metal systems. If the normal metal is clean, quasi-
particles in the normal metal travel ballistically over
long length scales. In the samples studied in the ma-
jority of recent experiments, however, the quasipar-
ticles are scattered elastically within a short distance,
so that the quasiparticle motion is diffusive. Here,
280 V. Chandrasekhar
we shall concentrate on this diffusive case, where
the elastic scattering length is the shortest rele-
vant length scale in the problem. We shall also re-
strict ourselves to the case where the superconduc-
tor is of the canonical s-wave type, avoiding compli-
cations with non-spatially symmetric order param-
eters within the superconductor.
8.2 Transport Equations
in the Diffusive Approximation
As an illustration of some of the issues that arise
in dealing with non-equilibrium transport in meso-
scopic diffusive systems, we consider first the classi-
cal Boltzmann equation in the diffusion approxima-
tion. Consider a one-dimensional diffusive normal
metal wire of length L. We assume that all dimen-
sions of the wire are larger than , the elastic mean
free path of the conduction electrons.f (E, x),the dis-
tribution function of electrons at energy E and at a
point x along the wire, obeys the diffusion equation
D
d
2
f (E, x)
dx
2
+ C(f )=0. (8.1)
Here, D =(1/3)v
F
is the three-dimensional electron
diffusion constant, with v
F
being the Fermi velocity
of the electrons. C(f ) is the collision integral, which
takes into account inelastic scattering of the elec-
trons,and itself depends on the distributionfunction
f . If we consider a mesoscopic wire whose length L
is much shorter than any inelastic scattering length
L
in
, this term can be set to zero.
This diffusion equation has a simple solution in
some specific geometries.Consider the case of a wire
of length L sandwiched between two normal ‘reser-
voirs,’ which we shall call the left (at x =0)and
right (at x = L) reservoirs.The reservoirs are defined
as having an equilibriumdistributionfunction f
L
(E)
and f
R
(E) respectively.Then the solution of the diffu-
sion equation, Eq.(8.1), under the condition L L
in
(so that C(f ) = 0), and subject to the boundary con-
ditions that f (E, x)=f
L
(E)attheleftreservoir,and
f (E, x)=f
R
(E) at the right reservoir, is given by [20]
f (E, x)=
f
R
(E)−f
L
(E)
x
L
+ f
L
(E) . (8.2)
The electrical current I through the wire in the dif-
fusion approximation is given by
I =−eAD
N(E)
@f
@x
dE , (8.3)
where A is the cross-sectional area of the wire. With
f (E, x) given by Eq. (8.2), this can be written
I =−
eAD
L
N(E)[f
R
(E)−f
L
(E)]dE . (8.4)
In order to obtain a finite current, we must apply a
voltage across the wire.Let us apply a voltage V to the
left reservoir, keeping the right reservoir at ground
(V = 0). This has the effect of shifting the energies
of the electrons in the left reservoir by −eV,sothat
the electron distribution function there is given by
f
L
(E)=f
0
(E − eV), where f
0
(E) is the usual equilib-
rium Fermi distribution function
f
0
(E)=
1
e
E/k
B
T
+1
(8.5)
and the energy E is measured from the Fermi en-
ergy E
F
. Figure 8.1 shows the distribution function
f (E, x) in the wire.It has a step-functionform,which
varies linearly along the length of the wire. Evidence
for such a non-equilibrium distribution has recently
been seen experimentally in a series of beautiful ex-
periments by the Saclay group [21].
Fig. 8.1. Nonequilibrium electron distribution function f
in a normal wire with a potential V applied across it as
a function of energy E and position x. The position x is
normalized to the length L of the wire
8 Quasiclassical Theory of Superconductivity 281
In the limit of small voltages V, the difference of
distribution functions in Eq. (8.4) can be expanded
as
f
R
(E)−f
L
(E)=f
0
(E)−f
0
(E − eV) ≈ f
0
(E)
−
f
0
(E)+eV
−
@f
0
@E
=−eV
−
@f
0
@E
(8.6)
so that
I =
e
2
AD
L
V
N(E)
−
@f
0
@E
dE
≈
N(0)e
2
AD
L
V , (8.7)
where N(0) is the density of states at the Fermi en-
ergy. In performing the integral, we have assumed
that we are at low enough temperatures so that
the derivative of the Fermi function (−@f
0
/@E)can
be approximated by ı(E). The electrical conduc-
tance is then given in the Nernst-Einstein form
G = N(0)e
2
D(A/L)=
0
(A/L).
The thermal current through the wire is given by
an expression similar to Eq. (8.4):
I
T
=
AD
L
EN(E)[f
R
(E)−f
L
(E)]dE . (8.8)
The critical difference between the expression for
the thermal current and the expression for the elec-
trical current is the presence of an additional factor
of E in the integrand in Eq. (8.8), which has impor-
tant consequences in the calculation of the thermal
properties.
To obtain expressions for the thermal coefficients,
let the temperature of the left reservoir be T and its
voltage V, and let the temperature of the right reser-
voir be T + T and its corresponding voltage V =0.
The distributionfunction in the left electrode is then
f
L
(E)=f
0
(E − eV, T), and the distribution function
in the right electrode is f
R
(E)=f
0
(E, T + T). In the
limit of small T, V, the difference in the distribu-
tion functions in Eq. (8.4) can be expanded as
f
R
(E)−f
L
(E)=
f
0
(E, T)+
E
T
T
−
@f
0
@E
−
f
0
(E, T)+eV
−
@f
0
@E
=
E
T
T − eV
−
@f
0
@E
. (8.9)
Putting this expression into the equations for the
electrical and thermal currents, Eqs. (8.4) and (8.8),
one obtains the two transport equations
I =
e
2
AD
L
N(E)
−
@f
0
@E
dE
V
−
eAD
TL
EN(E)
−
@f
0
@E
dE
T , (8.10a)
I
T
=−
eAD
L
EN(E)
−
@f
0
@E
dE
V
+
AD
TL
E
2
N(E)
−
@f
0
@E
dE
T . (8.10b)
These equations are equivalent to the usual form
for the transport equations for a metal:
I = GV + T , (8.11a)
I
T
= V + T , (8.11b)
with the linear-response thermoelectric coefficients
defined by
G =
e
2
AD
L
N(E)
−
@f
0
@E
dE
, (8.12a)
=−
eAD
TL
EN(E)
−
@f
0
@E
dE
, (8.12b)
=−
eAD
L
EN(E)
−
@f
0
@E
dE
, (8.12c)
=
AD
TL
E
2
N(E)
−
@f
0
@E
dE
. (8.12d)
If we approximate the derivative of the Fermi func-
tion by a ı function at low temperatures,as we did for
the electricalconductance G, we see that all other co-
efficients vanish. In order to obtain a finite value, we
use the Sommerfeld expansion for the integrals [22]
282 V. Chandrasekhar
¥ (E)
−
@f
0
@E
dE = (8.13)
¥ (0) +
2
(k
B
T)
2
6
@
2
¥ (E)
@E
2
E=0
+ ...
Using the fact that N
(0) = N(0)/2E
F
,weobtainthe
following expressions for the last three coefficients
=−
2
k
2
B
T
6
eAD
L
N(0)
E
F
, (8.14a)
=−
2
(k
B
T)
2
6
eAD
L
N(0)
E
F
, (8.14b)
=
2
k
2
B
T
3
AD
L
N(0) . (8.14c)
Experimentally, the quantities often measured are
the thermopower S and the thermal conductance G
T
.
The thermopower is defined as the thermal voltage
generated by a temperature differential T,under
the condition that no electrical current flow through
the wire (I = 0). Putting this condition into the first
transport equation, Eq. (8.10a), we obtain
S =
V
T
=−
G
=
2
6
k
B
e
k
B
T
E
F
. (8.15)
Note that since the expression for S contains a factor
k
B
T/E
F
, the thermopower of a typical normal metal
is very small.
The thermal conductance G
T
is defined as the ra-
tio of the thermal current I
T
to the temperature dif-
ferential T, under the same condition of no electri-
cal current flow (I = 0). From the equations above
G
T
=
I
T
T
= + S . (8.16)
For typical metals, the second term is much smaller
than the first, and is usually ignored, so that G
T
≈ .
If we take the ratio of the electrical to the thermal
conductance, we obtain
G
T
G
=
2
3
k
2
B
e
2
T. (8.17)
Consequently, one finds that the Wiedemann–Franz
law holds,even though the scattering lengths for mo-
mentum relaxation () and energy relaxation (L
in
)
are quite different. This is because the reservoirs are
assumed to be perfect, so that any electron entering
a reservoir immediately equilibrates with the other
electrons in the reservoir.
Before we go on to discuss normal metals in con-
tact with superconductors, it is worthwhile to re-
view some importantassumptions in the calculations
above.First,inourcalculations,wehaveassumedthat
the diffusion coefficient D is a constant independent
of the energy E and position x. When coherence ef-
fects are important, as in the case of the proxim-
ity effect, the diffusion coefficient becomes a func-
tion of both these parameters, D(E, x). The differen-
tial equation for the distribution function, Eq. (8.1)
is then modified. D(E, x) itself in general is deter-
mined by the distribution function f (E, x), so one
must solve a set of coupled differential equations to
obtain a solution.This is difficult todo analytically in
all but the simplest of cases, and more often must be
done numerically. The remainder of this chapter will
be devoted in great part to deriving the appropriate
differential equations for the distribution functions
and diffusioncoefficientin the caseof normal metals
in contact with superconductors,using the quasiclas-
sical equations for superconductivity.
Second, apart from the electrical conductance G,
the thermal coefficients derived above would all van-
ish if the density of states at the Fermi energy N(E),
were assumed to be constant,so that it could be taken
out of the integrals. The small variation in the den-
sity of states at the Fermi energy is responsible for
finite(butsmall) values of the off-diagonaltransport
coefficients and . For example, the thermopower
S, which is non-zero only if there is an asymmetry
between the properties of particles and holes near
the Fermi energy, vanishes if N(E) is assumed con-
stant. The small difference in N(E)aboveandbelow
E
F
gives rise to the small but finite thermopowers of
typical normal metals. The conventional quasiclas-
sical theory of superconductivity assumes particle-
hole symmetry aprioriin the definition of the qua-
siclassical Green’s functions, in that N(E) is assumed
constant at E
F
. Consequently, thermoelectric effects
cannot be calculated in the conventional quasiclas-
sical approximation; an extension of the theory is
required.
8 Quasiclassical Theory of Superconductivity 283
Finally,we have been discussing here cur rents and
conductances, rather than current densities and con-
ductivities. These are the more relevant quantities,
since we will be discussing mesoscopic samples in
which the measured properties are properties of the
sample as a whole. This will be particularly impor-
tant for the proximity effect, where long range phase
coherence means that non-local effects are impor-
tant.
8.3 The Keldysh Green’s Functions
The starting point for developing the quantum ana-
log to the classical Boltzmann transport equation is
the Keldysh diagrammatic technique. We shall begin
our discussion of the Keldysh technique in the no-
tation of Lifshitz and Pitaevskii [23]. As in the equi-
librium case, we define a non-equilibrium Green’s
function
G
1
2
(X
1
, X
2
)=−i
9
n
T ˆ
1
(X
1
) ˆ
+
2
(X
2
)
n
:
. (8.18)
Here X
1
and X
2
take into account the three spatial
coordinates (denoted by r
1
and r
2
respectively), and
the time coordinate (t
1
and t
2
). The difference be-
tween the non-equilibrium Keldysh Green’s function
above and the usual equilibrium Green’s function is
that the average is taken over anyquantum state |n >
of the system, rather than just the ground state |0 >.
ThetimeorderingoperatorT has the effect
G
1
2
(X
1
, X
2
) = (8.19)
1
−i
9
n
ˆ
1
(X
1
) ˆ
+
2
(X
2
)
n
:
, if t
1
> t
2
,
+i
9
n
ˆ
+
2
(X
2
) ˆ
1
(X
1
)
n
:
, if t
2
> t
1
,
for fermion operators (the case of interest here). For
simplicity, we label the spin indices and the coor-
dinate indices by numbers. We shall define a set of
Green’s functions that will be useful later:
G
˛˛
12
=−i
9
T ˆ
1
ˆ
+
2
:
, (8.20a)
G
ˇˇ
12
= i
˜
T ˆ
+
1
ˆ
2
, (8.20b)
G
˛ˇ
12
=−i
9
ˆ
1
ˆ
+
2
:
, (8.20c)
G
ˇ˛
12
= i
9
ˆ
+
2
ˆ
1
:
. (8.20d)
The first Green’s function is just the one defined
above,in this new notation.The second Green’s func-
tion is similar in definition to G
˛˛
12
,exceptthatitis
defined with operator
˜
T instead of T, which orders
the operators in reverse chronological order.The last
two Green’s functions are defined without any time-
ordering operators.The four Green’s functionsso de-
fined are not linearly independent, but are related by
linear equations of the form
G
˛˛
+ G
ˇˇ
= G
˛ˇ
+ G
ˇ˛
. (8.21)
The retarded and advanced Green’s functions G
R
and
G
A
can be defined as in the equilibrium case
G
R
12
=
1
−i
9
ˆ
1
ˆ
+
2
+ ˆ
+
2
ˆ
1
:
, if t
1
> t
2
,
0 , if t
2
> t
1
(8.22)
and
G
A
12
=
1
0 , if t
1
> t
2
,
i
9
ˆ
1
ˆ
+
2
+ ˆ
+
2
ˆ
1
:
, if t
2
> t
1
.
(8.23)
and are related by
G
A
12
=
G
R
21
∗
. (8.24)
G
R
and G
A
canbewrittenintermsoftheKeldysh
Green’s functions defined earlier as (we set =1)
G
R
= G
˛˛
− G
˛ˇ
= G
ˇ˛
− G
ˇˇ
, (8.25a)
G
A
= G
˛˛
− G
ˇ˛
= G
˛ˇ
− G
ˇˇ
. (8.25b)
G
˛˛
satisfies the equation of motion
G
−1
01
G
(0)˛˛
12
= ı(X
1
− X
2
) , (8.26)
where G
−1
0
is the differential operator (in the free
electron approximation)
G
−1
0
= i
@
@t
+
∇
2
2m
+ (8.27)
and the second subscript to G
−1
0
denotes that the dif-
ferentials in Eq. (8.27) are with respect to coordi-
nates corresponding to this subscript. The argument
of the delta function Eq. (8.26) includes space, time
and spin coordinates, and the notation G
(0)˛˛
12
(with
284 V. Chandrasekhar
a superscript 0) signifies a Green’s function for an
ideal gas.
The ı function in Eq. (8.26) arises from the dis-
continuities in G
˛˛
at t
1
= t
2
. G
R
and G
A
have similar
discontinuities, and obey a similar equation. G
(0)ˇˇ
has a discontinuity of the opposite sign, and hence
obeys the equation
G
−1
01
G
(0)ˇˇ
12
=−ı(X
1
− X
2
) . (8.28)
G
˛ˇ
and G
ˇ˛
have no discontinuities at t
1
= t
2
,and
hence obey the equations
G
−1
01
G
(0)ˇ˛
12
=0, (8.29a)
G
−1
01
G
(0)˛ˇ
12
=0. (8.29b)
The diagram technique for Keldysh Green’s func-
tions is similar to that for equilibrium Green’s func-
tions, except that one needs to sum over the internal
Fig. 8.2. Diagrams corresponding to the first order correc-
tions to the Keldysh Green’s function G
˛˛
12
in the presence
of an external potential U ,which is represented by a dashed
line
indices ˛ and ˇ, with a corresponding increase in
the number of diagrams. For example, Fig. 8.2 shows
two diagrams corresponding to the first ordercorrec-
tions to G
˛˛
12
in the presence of an external potential,
which is represented by a dashed line.
In the same manner, Dyson’s equation for the
Green’s function can be represented as shown in
Fig. 8.3, where the ellipse represents the self-energy
function £
˛ˇ
. In order to make the notation more
compact and tractable, it is useful to introduce a ma-
trix Green’s function and corresponding self-energy
matrix:
ˇ
G =
G
ˇˇ
G
˛ˇ
G
ˇ˛
G
˛˛
,
ˇ
£ =
£
ˇˇ
£
˛ˇ
£
ˇ˛
£
˛˛
. (8.30)
Dyson’s equation can then be written in matrix form
as
ˇ
G
12
=
ˇ
G
0
12
+
ˇ
G
0
14
ˇ
£
43
ˇ
G
32
d
4
X
3
d
4
X
4
, (8.31)
where the usual rules of matrix multiplication are
used.
As we saw earlier, the components of the Green’s
function matrix are not independent,but are linearly
related. One can therefore perform a transformation
to set one of the matrix components to zero. There
are many ways to do this; the one we shall use here
is the one employed in most recent literature on the
quasiclassical theory of superconductivity [11]. The
Fig. 8.3. Dyson’s equation in diagrammatic form for the Keldysh Green’s function G
˛˛
12
.Thethick line represents the exact
Green’s function G
˛˛
12
,thethin line represents the Green’s function for an ideal gas G
(0)˛˛
12
, and the ellipse represents the
self-energy £
8 Quasiclassical Theory of Superconductivity 285
matrices above are first rotated in Keldysh space us-
ing the transformation
ˇ
G →
3
ˇ
G,wherethe matri-
ces are identical in form to the Pauli spin matrices
0
=
10
01
,
1
=
01
10
,
2
=
0−i
i 0
,
3
=
10
0−1
, (8.32)
and then transformed
ˇ
G → Q
ˇ
GQ
†
,wherethematrix
Q is given by
Q =
1
√
2
(
0
− i
2
) . (8.33)
The resulting matrices have the form
ˇ
G =
G
R
G
K
0 G
A
,
ˇ
£ =
£
R
£
K
0 £
A
, (8.34)
where the retarded and advanced Green’s functions
G
R
and G
A
have been defined above, and the new
Keldysh Green’s function G
K
is given by
G
K
= G
˛˛
+ G
ˇˇ
= G
˛ˇ
+ G
ˇ˛
. (8.35)
£
R
, £
A
and £
K
are defined in a similar manner. In
what follows, we shall use the “check” to denote the
Keldysh matrices. In order to avoid writing integrals
over the space and time coordinates, we introduce
the binary operator ⊗, which has the effect of inte-
grating over the free space and time coordinatesand
performing matrix multiplication when applied be-
tween two Keldysh matrices. Thus, the Dyson equa-
tion for the Keldysh Green’s function written above
can be represented as
ˇ
G =
ˇ
G
0
+
ˇ
G
0
⊗
ˇ
£ ⊗
ˇ
G . (8.36)
ˇ
G
0
obeys the equation of motion
G
−1
01
ˇ
G
0
12
= ı(X
1
− X
2
) . (8.37)
Operating G
−1
01
on the Dyson equation (8.36) from the
left, we obtain
G
−1
01
ˇ
G = ı(X
1
− X
2
)+
ˇ
£ ⊗
ˇ
G . (8.38)
The conjugate equation is
1
ˇ
GG
−1
02
= ı(X
1
− X
2
)+
ˇ
G ⊗
ˇ
£ . (8.39)
Subtracting the two equations, we obtain
(G
−1
0
⊗
ˇ
G −
ˇ
G ⊗G
−1
0
)−(
ˇ
£ ⊗
ˇ
G −
ˇ
G ⊗
ˇ
£)=0, (8.40)
where we have suppressed the subscript “1” of G
−1
01
.
Thiscanbewrittenasacommutator
[G
−1
0
#
ˇ
G]−[
ˇ
£ #
ˇ
G]=0,
or
[G
−1
0
#
ˇ
G]=[
ˇ
£ #
ˇ
G] , (8.41)
where the # operator defines the commutator of two
operators [A # B]=A ⊗ B − B ⊗ A.
The Mixed or Wigner Representation
The Green’s function
ˇ
G
12
oscillates rapidly with the
difference r
2
- r
1
, on the scale of the inverse Fermi
wave vector k
−1
F
. In the physical systems to be dis-
cussed, we are interested in variations on much
longer length scales. We therefore perform a trans-
formation to center-of-mass coordinates, R, T,and
difference coordinates, r, t, defined by the equa-
tions [24]
r
1
= R − r/2 , r
2
= R + r/2 , (8.42a)
t
1
= T − t/2 , t
2
= T + t/2 , (8.42b)
and then define a Fourier transform of
ˇ
G
12
with re-
spect to the variables r and t:
ˇ
G(R, T; r, t)=
e
−iEt
e
ip·r
ˇ
G(R, T; p, E) dp dE .
(8.43)
In Eq. (8.41) the part of the commutator involving
G
−1
0
canbewrittenas
G
−1
01
⊗
ˇ
G −
ˇ
G ⊗ G
−1
02
=
i
@
@t
1
+
@
@t
2
(8.44)
−
1
2m
(∇
2
1
− ∇
2
2
)
ˇ
G ,
1
From the right, we operate on the coordinate 2, hence we use the operator G
−1
02
. G
−1
02
operated on the right is equivalent
to G
−1∗
02
operated on the left.
286 V. Chandrasekhar
using Eq. (8.27) in the free electron approximation.
Transforming to the coordinates, R, T, r, t,thiscan
be written
i
@
@T
+
1
m
∇
r
·∇
R
ˇ
G . (8.45)
From Eq. (8.35), the Keldysh component G
K
of the
Green’s function can be written as the sum of the
Green’s functions G
˛ˇ
and G
ˇ˛
. Keeping in mind the
definitions of these functions given in Eq. (8.20), we
can define a non-equilibrium distribution function
f (R, T, r), which is related to the function G
ˇ˛
by
f (R, T, r)=
9
ˆ
+
(R + r/2, T + t/2)
׈ (R − r/2, T − t/2
:
t=0
=−iG
ˇ˛
(R, T, r, t)
t=0
. (8.46)
Similarly, we have
9
ˆ (R + r/2, T + t/2) ˆ
+
(R − r/2, T − t/2
:
t=0
=1−f (R, T, r)=iG
˛ˇ
(R, T, r, t)
t=0
. (8.47)
Subtracting the first equation from the second, we
obtain
1−2f (R, T, r)=h(R, T, r)
= i
G
˛ˇ
(R, T, r, t)
t=0
+ G
ˇ˛
(R, T, r, t)
t=0
= iG
K
(R, T, r, t)
t=0
, (8.48)
where we have defined a new distribution function
h(R, T, r). In terms of the mixed Fourier transform,
Eq. (8.43), this can be written as
h(R, T, r)=iG
K
(R, T, r, t)
t=0
=
i
2
e
ip·r
G
K
(R, T; p, E)dp dE . (8.49)
Taking the Keldysh component of Eq. (8.45) at t =0,
we obtain
i
@
@T
+
1
m
∇
r
·∇
R
dp
dE
2
e
ip·r
G
K
(R, T; p, E),
(8.50)
or, in terms of the Fourier components with respect
to r, corresponding to momentum p,
@h(R, T, p)
@T
+
p
m
·∇
R
h(R, T, p) , (8.51)
where the Fourier transform of the Wigner distribu-
tion function h(R, T, p)isgivenby
h(R, T, p)=
i
2
G
K
(R, T; p, E)dE . (8.52)
In equilibrium, h is given by
h
0
(
p
)=1−2f
0
(
p
)=tanh(
p
/2k
B
T) . (8.53)
Equation (8.51), which is the Keldysh component of
thelefthandsideofEq.(8.41),hastheformoftheone
side of the classical Boltzmann equation for the dis-
tribution function.(Usingthe definitionof the func-
tion h(R, T, p), this can also be written in a more
conventional form in terms of f (R, T, p).) The right
hand side of the Keldysh component of Eq. (8.41)
must therefore correspond to the collision terms.
TherighthandsideoftheKeldyshcomponentcan
be written as 2(£
ˇ˛
G
˛ˇ
− £
˛ˇ
G
ˇ˛
). Taking the limit
at t = 0, and writing in terms of the distribution
function f (R, T, p) using Eq. (8.45) and Eq. (8.46),
this Keldysh component of the right hand side of
Eq. (8.41) can be written as [23]
−2i
£
ˇ˛
(R, T, p)(1 − f (R, T, p)) (8.54)
+ £
˛ˇ
(R, T, p)f (R, T, p)
.
The first term in Eq. (8.55) with the factor (1 −
f (R, T, p)) has the usual form of a scattering-in term,
corresponding to the gain of particles,while the sec-
ond term has the form for a scattering-out term,
corresponding to a loss of particles. Consequently,
we see that the Keldysh component of the right-left
subtracted Dyson equation gives the transport equa-
tion for the distributionfunction.Fromthe diagonal
components of the same equation, one can obtain
solutions for the other components of
ˇ
G and
ˇ
£.More
typically, the scattering terms on the right hand side
of the Boltzmann equation make it difficult to solve,
and some approximations must be employed. If the
variation of the system with the center-of-mass co-
ordinates T and R is small, then one can expand the
Green’s functions and self-energies, which are func-
tionsof R, r, T, t about R and T in a Taylor’sseries to
first order in r and t. This is the gradient expansion
8 Quasiclassical Theory of Superconductivity 287
discussed by Kadanoff and Baym [24] and Larkin
and Ovchinnikov [4], and we shall return to it at the
end of this section.
If instead of taking the differenceof Eq.(8.38) and
its conjugate equation, Eq. (8.39), we take the sum,
we obtain the equation
[G
−1
0
⊕
ˇ
G]=2ı(X
1
− X
2
)+[
ˇ
£ ⊕
ˇ
G] , (8.55)
where the operator ⊕ defines the Keldysh anti-
commutator, in the same way as the operator # de-
fines the Keldysh commutator. The left hand side of
Eq. (8.55) can be written as
G
−1
01
⊗
ˇ
G +
ˇ
G ⊗ G
−1
02
=
i
@
@t
1
−
@
@t
2
(8.56)
−
1
2m
∇
2
1
+ ∇
2
2
ˇ
G .
Transforming to the mixed representation,we obtain
2i
@
@t
+
1
m
1
4
∇
2
R
+ ∇
2
r
ˇ
G(R, T, r, t) . (8.57)
Now, the assumption we are making is that the vari-
ations of G on the scale of R are much slower than
the variations on the scale of r. Hence, the terms
in equation above involving derivatives with R con-
tribute much less than the those with r,andcanbe
neglected in this approximation. If we consider the
equation for G
0
, for which the terms involving £ on
the righthand side of Eq.(8.55) are 0,we then obtain,
after transforming to Fourier components [25]
(E −
p
)G
0
(R, T, p, E) = 1 (8.58)
or
G
0
(R, T, p, E)=
1
(E −
p
)
, (8.59)
where
p
=
p
2
2m
− ∼ v
F
(p − p
F
) . (8.60)
So far, we have assumed a free-electron model. If
there is a slowly varying potential U(R, T), the equa-
tion above can be modified to
G
0
(R, T, p, E)=
1
(E −
p
− U (R, T))
. (8.61)
Thisequation has form of a Green’s function for a free
particle, but in a slowly varying potential U (R, T).
The operator G
−1
0
in the mixed representation can
therefore be written as
G
−1
0
=(E −
p
− U (R, T)) . (8.62)
More accurately, one transforms Eq. (8.27) using the
following equations
@
@t
1,2
=
1
2
@
@T
±
@
@t
, (8.63a)
∇
1,2
=
1
2
∇
R
±∇
r
, (8.63b)
∇
2
1,2
=
1
4
∇
2
R
±∇
r
·∇
R
+ ∇
2
r
. (8.63c)
Again, assuming the functions in the mixed repre-
sentation are slowly varying functions of R,wecan
ignore the second derivatives with respect to R,to
obtain
G
−1
0
=
1
2
@
@T
+
@
@t
+
∇
2
r
2m
+
∇
r
·∇
R
2m
+ . (8.64)
In most applications of interest here, we also ignore
theslowT dependence.Adding the potential U(R, T),
and writing interms of the partial Fourier transform
with respect to p and E,weobtain
G
−1
0
= E −
p
+
i
2
v
F
·∇
R
+ − U(R, T) , (8.65)
wherewehavereplacedp/m by v
F
, since the impor-
tant region of interest is at the Fermi surface. Note
that we still keep account of the direction of the mo-
mentum.
To conclude this section, we derive some expres-
sions for the convolution of two operators in the
mixed representation, the gradient expansion dis-
cussed above. Consider the convolution of two op-
erators defined as an integral over the internal space
and time coordinates
A ⊗ B =
dr
3
dt
3
A(r
1
, r
3
, t
1
, t
3
)B(r
3
, r
2
, t
3
, t
2
) .
(8.66)