Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors
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SECTION 9.4. EMISSION FROM PHASE-SUP CENTERS 251
252 CHAPTER 9. PHASE-SLIP CENTERS
PSC structure as a whole along the filament, the velocity of the motion was
cm/s. According to this picture, the electric field pulses
arise periodically at the ends of the filament, defining the generation frequency.
Another explanation of the low-frequency part of the spectrum is that such pulsa-
tions are due to small differences in oscillation frequencies of separate phase-slip
This difference may be caused by variations in geometrical or physical
parameters along the filament. A complete explanation of the mechanism of
low-frequency oscillations is yet to be found.
9.4.2.
Phonon Radiation
Another interesting consequence of the PSC regime is the possibility of phonon
radiation from the PSC. Expressions (7.68) and (7.58) can be used to calculate the
spectral dependence of the phonon emission from the phase-slip center
for arbitrary moments of time. Since is a periodic function of time, the phonon
emission from the active region oscillates and the phonon flux periodically reverses
its direction, as is evident from the sign alternations of entering into (7.68).
[Note that the time-averaged value of the phonon flux is nonzero because of the
interdependence of
The intensity of the phonon flux emitted from the unit volume may be
estimated as For the voltages (e.g.,
which is of the order of Joule’s
heat where is the specific resistivity of the active region with the length
Thus this kinetic effect,
39
though not observed experimentally as yet, has a
noticeable scale.
It is necessary to acknowledge that the sign-alternating pulsating phonon fluxes
are not easy to detect. One approach would be to use a high-resolution technique
[the detection time of phonons must be shorter than the period of oscillations given
by Eq. (9.54)]. Another approach would be to detect the phonon flux by a method
that is sensitive to only one direction of the phonon flux (for example, a method
based on using the fountain effect in superfluid
4
He.)
40
The detection of the phonon emission, which is spatially modulated and
periodically reversed in time, might provide additional confirmation of the validity
of current theoretical understanding of the physics of resistive states in “one-dimen-
sional” superconductors. The discovery of phase-slip centers in high-temperature
raises such expectations.
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SECTION 9.4. EMISSION FROM PHASE-SLIP CENTERS 253
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254 CHAPTER 9. PHASE-SLIP CENTERS
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10
Josephson Junctions
10.1. TUNNEL SOURCE OF EXCITATIONS
10.1.1. Josephson Effect
Consider a superconductor that occupies all the space, with the current j flowing
along the direction x. In Chap. 1 expression (1.53) was obtained for a superfluid
current in an equilibrium superconductor. Because the problem considered here is
formally one-dimensional, one can choose a gauge where
is the phase of the condensate wave function (1.52). From this expression it
follows that the current is defined by the phase difference between the points along
the direction of motion. Let now two superconductors occupy two isolated half-
spaces and each of them be described by its own wave function. If these supercon-
ductors are brought into contact, then a current will flow perpendicular to the
boundary because the phases of wave functions in each of superconductors, in
general, may differ from each other. Since the states in (1.52) with where
n is an integer, are physically identical the current between superconductors must
be a periodic function of the phase difference with the period In addition,
the current must vanish when the phase difference is zero. Hence the functional
dependence of the current between superconductors, at least in a first approxima-
tion, may be expressed by the relation this is the first, or the
stationary Josephson effect.
1,2
If the superconductors occupying the half-spaces are connected by the tunnel-
ing barrier, the potential difference may appear on the barrier in the
current-carrying state. Bearing in mind the relation between the phase of the wave
function and the scalar potential (see 8.3), according to which in equilibrium
we arrive at the conclusion that the current between superconductors oscillates
in time: The frequency of oscillations
where we have restored the fundamental constant previously set at unity
(the second, or nonstationary Josephson effect).
255
256
CHAPTER 10. JOSEPHSON JUNCTIONS
The phenomena that take place in the Josephson junctions have been studied
in numerous papers (see bibliographical review, Ref. 3). Before moving to the
description of a nonequilibrium Josephson junction, note that the equilibrium
theory of superconductivity (see Sect. 1.2) cannot describe the situation when the
potential difference exists at the superconductor’s boundary (the potential and
the normal current do not enter the equilibrium equations at all). In the case
considered the boundary between superconductors appears as a
nonequilibrium source. In the presence of massive banks, the nonequilibrium
carriers may dissipate by diffusion and the nonequilibrium phenomena cannot
noticeably change physical properties of the junction, which thus could be consid-
ered as equilibrium one. The picture differs for thin-film junctions whose thickness
is comparable with the diffusion length of the electron excitations. In this case the
junction banks are in a spatially homogeneous nonequilibrium state and, as we will
see in this chapter, the nonequilibrium effects become essential.
10.1.2. Tunneling Hamiltonian and Self-Energies
Let the voltage be applied to the tunnel junction and this voltage
be constant in time. We will describe the tunnel process by the Hamiltonian
where the indices 1 and 2 correspond to different banks of the junction and
integration is assumed over their volumes. The self-energy parts corresponding to
the tunneling in the superconductor–insulator–superconductor junction
may be obtained with the help of Hamiltonian and in our notations they
have the form*
where is the tunneling frequency related to the tunneling operator and
the function corresponds to the superconductor injector.
10.1.3. Derivation of Excitation Source
Taking into account relations (4.13) to (4.15) and (10.2), we find;
*The formal analogy between (10.1) and the Hamiltonian (2.1) describing the scattering of electrons on
impurities makes the detailed derivation of Eq. (10.2) excessive [cf. Eq. (2.37)].
SECTION 10.1. TUNNEL SOURCE OF EXCITATIONS 257
where Moving to the distribution function
according to (3.84) to (3.88), we find, based on (10.3),
In Eq. (10.4) the dimensionless factors are defined in the following way:
We emphasize that in these final expressions, which represent the nonequilibrium
tunnel source of the Josephson junction,
5
the quantity is defined positively. Here
258 CHAPTER 10. JOSEPHSON JUNCTIONS
appears as the electronlike excitation distribution function, denotes the
holelike one. [Expression (10.4) without the terms i.e., without taking
into account the macroscopic phase coherency in the junction, was derived in Ref.
6.]
The tunnel frequency in (10.4) should be expressed in terms of the detectable
parameters. To achieve this, it is necessary to have the expression for the current
flowing across the Josephson junction in a nonequilibrium state. Naturally, in
equilibrium it should transform into the expression obtained by Josephson. We will
now derive the expression for this nonequilibrium current.
10.1 .4. Expression for Tunnel Current
We start from the 11-component of Eq. (3.63). Integration over and sub-
sequent averaging over the angle variables transforms it into a continuity equation.
After such a procedure, the terms remaining in (3.63) contain the self-energy parts
representing (up to a numeric factor) the divergence of the current density, inte-
grated over the volume of the nonequilibrium film (the spatial homogeneity of
the pattern is assumed). Thus the integral tunnel current acquires the form
where For the current amplitude we find the following function-
als, (for simplicity the sign of the modulus in quantities in these expressions
is omitted):
j
SECTION 10.1. TUNNEL SOURCE OF EXCITATIONS 259
We stress that the expression derived for the current is not a trivial generaliza-
tion of the formula figuring in equilibrium theory. The last two terms in (10.11)
emerge from the branch imbalance and they characterize solely the nonequilibrium
situation. In the equilibrium approximation, [where
the formulas (10.8) to (10.11) transform to the familiar
expressions for the Josephson current.
1,7
10.1.5. "Tunnel Frequency" Parameter
We will now find the explicit form of v. Moving to the equilibrium case
and making the limiting transition in (10.8) to (10.11) to Ohm’s law we
get from which
Here S is the junction’s cross section, d is the thickness of the S-film, and R is the
resistivity of the junction in the normal state (caused by the dielectric interlayer of
an area S
)
. For clarity the electron charge e is restored in expression (10.12).
10.1.6. Complete Set of Equations
Nonequilibrium functions entering the expres-
sions (10.9) to (10.11) are yet to be determined. The following serves the purpose:
At strong deviation from equilibrium, the order parameter modulus (which may
be taken as real in spatially homogeneous states) must be determined from the
self-consistency equation. The self-consistent value of is determined by formula
(4.6), from which it follows (see Sect. 5.2) that
This relation will be used in conjunction with the kinetic equation (10.13).
Generally, the system of equations (10.13) and (10.14) should also be supple-
mented by Maxwell’s equations, which sometimes dictate the selection among
260
CHAPTER 10. JOSEPHSON JUNCTIONS
various solutions of the kinetic problem [since the current through the junction is
determined by the relations (10.8)–(10.11)].
10.2.
OSCILLATORY PROPERTIES OF A TUNNEL SOURCE
We will now study the kinetics of an electron system of nonequilibrium tunnel
Josephson junction. We assume that the Josephson junction consists of two suffi-
ciently thin films with a constant voltage V applied between them. Various applied
voltages will be considered: both (1) the supercritical values when the
applied electrical field generates excess quasi-particles from the condensate, and
(2) the case of subcritical voltages In the latter case, in addition to
the usual Josephson effect, another inherent effect appears: if the electron excita-
tions are out of equilibrium, quantum oscillations arise in the system. We will
discuss some manifestations of this effect in more detail.
10.2.1. Clark's Branch Imbalance
As first demonstrated by Clarke,
8
when electrons are injected from a normal
metal into a superconducting film (NIS), a potential difference between the Cooper
condensate and the excitations appears in the film. Tinkham
9
has shown that this
property of the NIS junction was caused by the population imbalance of electron–
hole excitation branches in the nonequilibrium superconducting film. We will
elucidate the physical meaning of this phenomenon.
Let us consider the tunnel junction consisting of films sufficiently thin
that the phonons are in equilibrium and the picture in each of the films is homoge-
neous across their thickness. Let the junction be coupled with a heat bath having a
temperature T. We will consider the solutions of Eq. (10.13), assuming arbitrary
deviation of the distribution function from the equilibrium Fermi function
First, however, we will examine some properties of the expression
1. The typical values of which determine the injection intensity, are
usually significantly smaller than the intensity of relaxation processes
(which may be roughly characterized by the damping where
is the transition temperature, and is the Debye frequency). In this
limiting case and we may assume the distribution
functions are equilibrium ones: Assum-
ing also we obtain the familiar expression
4,9
for
the tunnel source of the NIS junction. A remarkable property of this
expression is that this finally causes the branch imbalance
during the tunneling injection of excitations.
2. One can easily see that the property of the tunnel source
holds also in the case (even for a symmetrical SIS junction). However,