Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors
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SECTION 12.3. PHONON INSTABILITY 341
quadratures, one may see that changes its sign at , while phonon instability
is expected at B > 0.54. Thus, for the appropriate choice of parameters, the
instability related to the sign reversal of might not be destructive. For this reason
we will not study the problem of the superfluid velocity fluctuations in detail. Note
only that the sign reversal of if any, does not necessarily indicate the develop-
ment of instability. The interface between the film and the bulk superconductor can
electrodynamically stabilize the situation in a thin film if (cf. Sect. 11.5).
Here and are the magnetic field penetration depths connected to the London
penetration depth by the relation
In the presence of competing processes it is highly important that for a certain
choice of parameters the self-consistent kinetic equations allow only one solution
with a small [see Eq. (12.46) and Fig. 12.10]. Otherwise, in the case of multiple
solutions, a spatially inhomogeneous state of the superconductor may arise (see
Chap. 11 and Sect. 12.2), which would complicate the situation.
Note that the longitudinal electric field in superconductors (which is related to
the branch imbalance in the electron excitation spectrum) might also fluctuate (cf.
Sect. 8.4). However, in our case (T = 0 and “narrow” distributions) the imbalance
is expected to be small and no instability is likely to be developed as a result of the
fluctuations of the longitudinal field. The problem of the stability of the pattern
arising relative to the high-frequency fluctuations of a transverse electromagnetic
field (photons) will be considered in Sect. 12.4.
12.3.6. Estimated Gain
We now provide some numerical values for parameters relevant to the acoustic
quantum generation regime. From the viewpoint of possible experiments, the
nonequilibrium state can be efficiently realized in aluminum films as a result of the
relatively long lifetime of the electron excitations in this material. Characteristic
values of for aluminum are estimated to be (see Table 10.1 in Sect.
10.2), and if then for aluminum at cm we have from (10.12) RS
Such low-resistance junctions may be implemented in practice (see,
e.g., Refs. 32 and 33). The necessity to satisfy the inequality imposes
additional restraints on both metals comprising the SIS´ junction.
Note that the velocity of sound in a superconductor is significantly less than
the speed of light, while the dimensionless electron–phonon interaction constant
exceeds the corresponding constant of electromagnetic interaction. Because of that,
the gain of the acoustic quantum generation could be quite high. Using (12.53) we
may obtain the following expression for the amplification factor (i.e., the gain):
342 CHAPTER 12. INVERSE POPULATION INSTABILITIES
where is the quantity in braces in (12.57), and for characteristic metals (see
Table 10.1 in Sect. 10.2) we have from (12.64) during pumping,
which corresponds to This is higher than the working gain of gaseous,
solid-state, and semiconductor lasers.
*
12.4.
PHOTON INSTABILITY
Photons represent a Bose field, which has a lot of analogy with the phonon
field. At the same time there are major distinctions, which makes the case of photon
instability worth a separate discussion.
12.4.1. Two Channels of Electron–Photon Interaction
Consider a photon of frequency propagating through a superconductor
with a nonequilibrium electron population. We assume that the distribution function
is isotropic in the momentum space. As mentioned earlier, a photon can
participate (in a first approximation) in two types of processes, which are shown in
Fig. 12.13. They can be described by the collision operators (cf. Sect. 5.3):
(these operators are written in the limit of large photon numbers assuming
the absence of branch imbalance: Here is a (positive) constant propor-
tional to the interaction cross section of the photon with an electron quasi-particle.
*
Hence the acoustic quantum generation, in principle, might be achieved without using a resonator
(superluminescence), with the coherent phonon flux propagating along the thin film. Moreover, because
the phonons suffer total internal reflection at the metal–liquid helium interface, it seems expedient to
try to implement a ring generator scheme.
SECTION 12.4. PHOTON INSTABILITY 343
(these operators are written in the limit of large photon numbers , assuming
the absence of branch imbalance: ). Here is a (positive) constant propor-
tional to the interaction cross section of the photon with an electron quasi-particle.
12.4.2. Photons Versus Phonons
Taking into account that
and integrating in (12.65) and (12.66) with the help of one obtains for
the photon absorption coefficients and in recombination and
relaxation channels, respectively, the following relations:
Interchanging the variables and – epsilon in the integrand of (12.68), one can
rewrite it in the form (the same procedure was applied in Sect. 12.3 to get 12.55):
344 CHAPTER 12. INVERSE POPULATION INSTABILITIES
and thus
According to Eq. (12.73) one cannot expect to obtain any amplification of electro-
magnetic waves at frequencies because the gain in the recombination
channel is completely suppressed by the relaxation losses. This is a major difference
in the behavior of photon and phonon fields in nonequilibrium superconductors.
The physical reason* is in the transverse nature of the photon field as opposed to
the interaction of longitudinal phonons within an isotropic metal.
12.4.3.
"Clean" Limit Kinematic Restrictions
The result (12.73) is based on Eqs. (12.65) and (12.66), which could be violated
in the limit of perfect crystals (i.e., in specimens with a small concentration of elastic
scatterers; cf. Chap. 5). In this limit the relaxation process of the type
which is described by the collision operator (12.65), is forbidden owing to the
energy-momentum conservation law subject to the condition where is the
Fermi velocity and c is the speed of light in the medium. In fact, the last inequality
(the Cherenkov condition) suppresses the relaxational channel (Fig. 5.1) while
leaving the recombinational channel active. This suppression effect was treated in
detail in Sect. 5.4. Hence in this particular case of perfect crystalline samples one
can expect a negative absorption coefficient for photons at
12
.4.4. Experimental Feasibility
The detection of stimulated emission of electromagnetic radiation from a
nonequilibrium point microjunction NIS was reported in Refs. 35–37 (the scheme
of the experiment is depicted in Fig. 12.14). The nonequilibrium population was
created by excess quasi-particles injected by the current flowing from a normal
metal to the superconductor. The recombination of two quasi-particles across the
gap (see Fig. 12.13a) creates a new pair, which is added to the Cooper condensate.
A pair-binding energy was assumed to be emitted in the form of millimeter-range
electromagnetic quanta, corresponding to the value of superconducting tanta-
*
To confirm this conclusion, one should bear in mind that the conductivity has a large imaginary part,
which stabilizes electromagnetic fluctuations.
11
SECTION 12.4. PHOTON INSTABILITY 345
lum, used in the experiments in Refs. 35–37. The power of the emitted coherent
radiation was reported to be about
Unfortunately, this result has remained unconfirmed by other authors. More-
over, in a subsequent work,
38
critical arguments were put forward to prove the
impossibility of “laser” action. As was pointed out, the relaxation loss channel (see
Fig. 12.13b) is very important in superconductors and it overdamps the recombi-
national gain channel. To have a gain, it is necessary to pay special attention to the
crystallinity of the sample. Also, it is necessary to use an injection scheme that
guarantees (at least theoretically) the possibility of an inverse population at the
operational regime.
As shown in this chapter, the most promising results might be obtained by
tunnel injection in nonsymmetric SIS´ junctions, which seems quite appropriate for
the creation of an inverse population in the “narrow” pumping case (Fig. 12.12).
The deviation from equilibrium is higher when the injector has a larger gap.
Recalling the results discussed, it seems evident that with a normal-metal injector
(used in Refs. 35–37), one cannot expect to fulfill the necessary conditions for
quantum generation.
From the experimental point of view, serious difficulties could arise even in
the case of an SIS´ junction. Indeed, as mentioned in Sect. 12.3, the favorable values
of the parameter B should be about 1. This means that the RS value of the junction
must be less than Such junctions, with an exceedingly high transpar-
ency of the tunneling barrier, are very sensitive to the processes of degradation and
346 CHAPTER 12. INVERSE POPULATION INSTABILITIES
breakdown. This is probably why the inverse population has not been observed
experimentally so far.
Possibilities for creating inversely populated states in superconductors exist
also in the wide pumping discussed in Sect. 12.1. Some factors could favor this
conclusion. Among them are sufficiently high quality of the crystalline structure of
the superconducting specimen, the non-Debye character of the phonon spectrum,
sufficiently large values of the Cooper gap compared with the characteristic phonon
frequencies, and significant participation of transverse phonon modes in the elec-
tron relaxation processes.
The fulfillment of the last condition is rather probable in superconductors with
an anisotropic crystalline structure, particularly in high-temperature superconduc-
tors. To make more strict predictions, one needs to analyze the problem from first
principles, using the partial functions for all the phonon modes The
polarizations of these modes must be known. For numerical calculations, it is
necessary also to know the energy band structure of electrons in the normal state
as well as the properties of the superconducting state. The positioning of phonon
peaks relative to the frequency plays a major role in the shape formation
of the function as the discussion in Sect. 12.2 revealed.
It must be acknowledged that only the main points of the superconductor-based
quantum generation have been outlined in this chapter. Among the omitted ques-
tions is the competition between phonon and photon instabilities. Both regimes of
instability have practical implications. To some extent, the result of competition is
determined by external factors, such as the design of the resonator, or, in a more
general sense, the nature of the feedback. This competition becomes of practical
importance when the instability threshold is crossed. Investigating beyond this
threshold is then a task for future theoretical work.
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13
Thermoelectric Phenomena
13.1. LINEAR RESPONSE TO THERMAL GRADIENT
Thermoelectric effects in charged Fermi liquids are small owing to the approxi-
mate symmetry between particle and hole branches of elementary excitations. In
thermal equilibrium, some weak asymmetry in the properties of these excitations
results only from a nonzero curvature of the Fermi surface.
13.1.1. Thermopower of Normal Fermi Liquids
We will use for this analysis the general expressions for current obtained in
Sect. 7.2, and consider first the case of a normal metal. For this case Eq. (7.83) may
be represented as*
Substituting the normal-metal limit we arrive at
In the case of a linear response we have
where so (13.2) takes the form:
*
Both a derivation and a discussion of this basic relation for thermoelectricity in normal metals may be
found elsewhere (see, e.g., Refs. 1 and 2).
349
350 CHAPTER 13. THERMOELECTRIC PHENOMENA
Everywhere in this book we treated the conductivity (7.84) as a constant. To
describe thermoelectricity, one must account for the dispersion of in the energy
space.
*
Keeping only the first two terms in the series expansion for we have
where
Thus we obtain
At thermodynamic equilibrium in normal metals, a symmetry between electron-
and holelike excitations is governed by the electroneutrality condition,
and the first integral in (13.7) vanishes. Evaluating the second integral in (13.7),
one obtains the differential thermopower of a normal metal (or the Seebeck
coefficient 5) in the following form:
(the electron charge is restored).
13.1.2. Response of a Superconductor's Normal Component
For superconductors, the calculations of S should be modified by using the
spectral functions (7.16); (7.19) to (7.21) in expression (13.7) for the current:
*
Ziman
3
describes as the “electrical conductivity which the metal would have, at absolute zero, if
the Fermi level, came at the energy in our notation).