Kopnin N.B. Theory of Nonequilibrium Superconductivity
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if it has an energy | | = and moves within the range of angles >
0
counted from one of the nodes. The angle
0
is defined such that | | =
|sin(2
0
)|. For classically localized quasiparticles, the derivative
n
/ =
n
( ) is the quasiclassical interlevel spacing. We shall consider low
temperatures such that the level n = 0 is only important. Such temperatures
correspond to
. Keep in mind that according to ecpi (6.100)
(15.36)
where , and L denotes the logarithmic function of .
The kinetic equation for the distribution function of localized excitations in a
moving vortex has the form of eqn (15.24). We use a
-approximation eqn
(14.8) for the collision integral. We look for a solution in the form f
1
= f
1
[
0
( ,
), ] which does not contain an explicit dependence on . Equation (15.24)
gives
(15.37)
where
(15.38)
We take the nonequilibrium distribution function again in the form of eqn (14.9).
The factors
O
and
H
depend now on . We assume a Fermi surface with not
less than the tetragonal symmetry in the plane perpendicular to the vortex axis.
For simplicity, we consider only the particle-like Fermi surface. Taking into
account that, for the tetragonal symmetry, the responses do not depend on the
direction of the vortex motion with respect to the crystal lattice, one finds two
coupled first-order differential equations for
O
( ) and
H
( ):
(15.39)
Equations (15.39) can easily be solved. We have
H
( ) = ReW;
O
( ) = Im W
where
Kopnin, Nikolai, Senior Scientist, Low Temperature Laboratory, Helsinki University of
Technology, and L.D. Landau Institute for Theoretical Physics, Moscow
Theory of Nonequilibrium Superconductivity
Print ISBN 9780198507888, 2001
pp. [316]-[319]
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with
In the moderately clean limit, , the potential U( ) is always large,
and we obtain the local solution as in an s-wave superconductor
(15.40)
Consider temperatures which correspond to the energy
range
. For these temperatures, excitations are mostly
localized
end p.316
in the vortex core as we discussed in Sect ion 6.4.2. Let us find the distribution
function in the region of angles
not specifically close to the gap nodes. In the
superclean limit
, the potential U( ) is small almost everywhere
except for vicinities of the gap nodes where it becomes large. For angles larger
than
from the nodes one can neglect the potential U. As a
result, one has for
< < /2 where
1
1,
(15.41)
The constants A and B and the overall behavior of the distribution function are to
the highest extent determined by what happens in a close vicinity of the gap
nodes. It is this region which is responsible for the whole build-up of
nonequilibrium distribution. The constants can be found by matching with the
solution in the vicinity of a node,
1, where . This
provides the boundary condition
across the node. Here 2
0
= F(+ ) F( ). For energies
, the region of angles <
0
with delocalized trajectories
is not important. The range of angles
~
1
sets the width of the transition
region near a gap node where the distribution function jumps from its value at
= to its value at = + . We obtain using eqn (6.100)
(15.42)
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The factor
0
contains exactly the integral which determines the true bound
states in eqn (6.99). It converges and is determined by angles
. The interlevel spacing
0
is as in eqn (6.102)
The solution for
O,H
should be periodically continued to the rest of angles with
the period
/2 since the response function has the same tetragonal symmetry as
the underlying system:
( /2 ) = ( ), The behavior of the factors
O,H
as functions of the angle is illustrated schematically in Fig. 15.1.
Together with the above boundary condition, this gives
(15.43)
We have after averaging over the azimuthal angle
(15.44)
end p.317
FIG. 15.1. The factors
H,O
functions of . The limit
0
1 is shown
when
O
cos while
H
1 sin .
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15.5.2 Conductivity
The force on a vortex from the environment where it moves is the momentum
transferred from the excitations, eqn (14.4). Only the low energy levels with n =
0 are excited. The force takes the form
(15.45)
Since
O
and
H
do not depend on , the integration over d in eqn (15.45) can
be reduced to the integration over d
. This results in the components of the flux
flow conductivity in the form
(15.46)
where the average over the Fermi surface
H,O
is defined by eqn (14.13).
In the moderately clean case, the factors
O
and
H
are given by eqn (15.40).
The conductivities are
(15.47)
This is similar to the results for an s-wave superconductor (see Section 14.6).
In the superclean limit
the factors
O
and
H
for low temperatures T
T
c
are determined by eqn (15.44). The universal regime is reached when
0
1, i.e., when . One has from eqn (15.44)
H
= 1 2/ ,
O
= 2/ . The Ohmic and Hall conductivities become
end p.318
(15.48)
These conductivities are universal: they do not depend on the scattering time.
The only material characteristics Is the number of carriers.
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If the mean free path is further increased, the parameter
0
decreases. In the
limit
0
1, i.e., for , we have from eqns (15.44,
15.46)
(15.49)
The longitudinal conductivity decreases and finally vanishes while the Hall
conductivity reaches its nondissipative value.
The true interlevel distance
0
introduces a new energy scale which does not
exist for superconductors without gap nodes. For conventional superconductors,
there is a single characteristic value, the minigap
0
, which marks a crossover in
the relaxation rate
between two regimes: a dissipative viscous flux flow for
, and a dissipation-less vortex motion for . For
superconductors with gap nodes, there are two critical values: the average
quasiclassical interlevel distance
and the true interlevel distance
0
which is less than
0
. Therefore one has three regimes: (i) moderately clean
regime
with a high dissipation, (ii) intermediate universal regime
while with a finite dissipation, where the quasiclassical
levels are well resolved on average but the true minigap is smaller than the
scattering rate; and (iii) the extreme super clean case
when the true
levels are well separated and thus dissipation is small.
end p.319
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Kopnin, Nikolai, Senior Scientist, Low Temperature Laboratory, Helsinki University of
Technology, and L.D. Landau Institute for Theoretical Physics, Moscow
Theory of Nonequilibrium Superconductivity
Print ISBN 9780198507888, 2001
pp. [322]
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