Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors
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SECTION 10.2. OSCILLATORY PROPERTIES OF A TUNNEL SOURCE 271
The transformation coefficient P is independent of the frequency of the test
radiation (Fig. 10.7) at sufficiently high frequencies but at lower frequen-
cies it increases sharply, which ultimately is due to the singularity in the supercon-
ductor’s electron density of states (in the calculations this singularity was cut off at
energies The inspection shows that the intensity of the satellites increases
with diminishing (Fig. 10.8).
The dependence of P on the injector’s gap is a peculiar one (Fig. 10.9). If
(this numeric value corresponds to adopted in calcula-
tions), P drops by an order of magnitude. However, if I by 1%, P increases
noticeably, but diminishes with further increases in This circumstance may be
of use in the detection of satellites. Indeed, as noted earlier, the estimate (10.37)
was obtained for the response of a single film, and, generally speaking, is not valid
for a real Josephson junction, shown schematically in Fig. 10.8. However, in
asymmetric junctions, the oscillations of the electron excitation’s density manifest
themselves only in a film with a smaller gap. (The difference between and
must be quite small, about 1%; such a difference may exist for films of
identical metals owing to some difference in deposition conditions, thickness, etc.)
In these cases the estimate of (10.37) may be used directly.
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CHAPTER 10. JOSEPHSON JUNCTIONS
SECTION 10.2. OSCILLATORY PROPERTIES OF A TUNNEL SOURCE 273
For a lead film with and we have
Values of the same order were obtained for niobium
and tantalum. Such values can be easily measured spectroscopically. For metals
with low (and large , such as aluminum) , is relatively large while
is small and hence In such junctions, the level of nonequili-
brium excursion is quite high and other interesting effects may also be observed
(satellites with multiple frequencies, etc.).
Note the following. In principle, the appearance of satellites in an electromag-
netic wave scattered by the Josephson junction may be explained within the usual
electrodynamics by accounting for the interference between the ac current induced
by the external field and the Josephson current. The proper functional dependencies
(similar to those shown in Figs. 10.5–10.9) may obviously be different. It is also
important to emphasize the fundamental difference between these effects (the
kinetic and electrodynamic one). The “kinetic effect” holds if we consider scattering
of the acoustic wave at the junction, instead of electromagnetic radiation
The high-frequency acoustic wave would also generate satellites with
frequencies However, the “electrodynamic effect” cannot occur during the
scattering of an acoustic wave. Thus the satellites caused by the oscillations in the
excitation’s density, in principle, can be singled out and unambiguously interpreted.
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CHAPTER 10. JOSEPHSON JUNCTIONS
10.3. SELF-CONSISTENT SOLUTION OF KINETIC EQUATIONS
10.3.1. Analytic Solution with a Branch Imbalance
Now we
will
consider
the
voltages
when
In the entire superconducting temperature range, except for a very small vicinity of
the transition point, the scale of the inverse frequencies corresponding to (10.38) is
small compared to the time intervals that characterize the kinetics of single-particle
excitations. Hence at the voltages of Eq. (10.38), the coherent effects in the
excitation system are negligible (the results presented in the preceding sections
implicitly confirm this).
In spite of the reduced role of quantum oscillations, the range of voltages in
Eq. (10.38) is interesting, because the applied electric field is capable of pair
breaking during pair tunneling. This breaking process is a resonant one and the
resulting degree of nonequilibrium may be quite high.
In general one must consider (10.13) and (10.14) simultaneously with the
analogous equations for the injector, so the number of coupled equations doubles.
Indeed, the function (10.7) entering into (10.4) (it is enough to consider only its
contribution) contains the distribution functions of both superconductors. In two
limiting cases—symmetric and highly asymmetric junctions—the situation sim-
plifies. In this chapter we consider the case of a symmetric SIS junction. Then in
the force of the symmetry the following identities hold:
where all distribution functions correspond to the test superconductor. We will not
provide explicit expressions for the quantities save space. These
expressions are obvious from a comparison of (10.40) and (10.41) with (10.13).
In representations (10.40) and (10.41), one may explicitly identify terms
related to resonant pair breaking. This representation is convenient, because it
allows exact accounting for these processes. For this purpose we will move to new
arguments in (10.40) and (10.41), making the transposition We get thus
two additional equations defined in the same energy region
SECTION 10.3. SELF-CONSISTENT SOLUTION OF KINETIC EQUATIONS 275
with the determinant
which is nonzero in the region
where subject to (10.39)
Composing now the determinants and for a system of four equations, one
finds in the usual manner in the region (10.45):
At low temperatures the “tail” of the distribution function at
is small. Neglecting this tail, as well as the tunneling redistribution of excitations
and the relaxation processes, one obtains the solution that generalizes (accounting
for the population imbalance) the expression derived by Aronov and Spivak
15
for
the case of ultra high frequency (UHF) pumping. The iterations of the solution prove
that accounting for the “tail” produces small changes in the resulting picture.
10.3.2. Inclusion of Self-Consistency Equation
The procedure for finding the solutions is not entirely reduced to simple
iterations. Note that the nonequilibrium gap enters the kinetic equation for
this gap itself depends on the form of It may be shown that when
Eq. (10.14) is equivalent to the system
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CHAPTER 10. JOSEPHSON JUNCTIONS
This system was included in the general iteration scheme that also utilizes (10.47)
at
The behavior of the excitation distribution function is illustrated in Figs.
10.10–10.14. Note the imbalance between the electronlike and the holelike
excitations, which occurs in both the subthreshold and superthre-
shold regimes.
10.3.3. Analysis of Numerical Solutions: Subthreshold Voltages
In the subthreshold case, in addition to the appearance of “spike,” the distribu-
tion function shows a tendency for a global displacement toward higher energies
(Fig. 10.10), which ultimately stimulates superconductivity (see Sect. 5.2) and
creates a phonon deficit (see the following discussion). As the temperature drops,
the nonequilibrium features become more pronounced: a larger number of spikes
appear, which are separated by a distance V in the energy space (cf. Figs. 10.11 and
10.12). At very low temperatures, the distribution function acquires a “sawtooth”
shape and is nonzero even in the range where the equilibrium Fermi “tail” is
SECTION 10.3. SELF-CONSISTENT SOLUTION OF KINETIC EQUATIONS 277
negligible. Note that the number of observed spikes increases with a drop in voltage
(Fig. 10.13).
10.3.4. Analysis of Numerical Solutions: Superthreshold Voltages
In the superthreshold range, virtually all the excess excitations are concentrated
in the region above the gap and, as Fig. 10.14 shows, the “tail” adjacent to this
region is negligible. Hence we may speak of a “quasi-local” distribution of
nonequilibrium quasi-particles. The behavior of the nonequilibrium gauge-invariant
potential is illustrated in Fig. 10.15.
Note two characteristic features of the behavior of the nonequilibrium gap
First, there is a range of V where subthreshold and superthreshold values coexist
(compare curves 1 and 2 in Fig. 10.16). This produces the hysteresis in the
current-voltage characteristics (see Fig. 10.17). Second, at with an increase
in voltage V the curve increases insignificantly, rapidly saturating. Here we
have the familiar phenomenon of superconductivity enhancement due to the tun-
neling absorption of
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CHAPTER 10. JOSEPHSON JUNCTIONS
10.4. PHONON EMISSION FROM A TUNNEL JUNCTION
Using the electron-hole distribution function found in the preceding section,
one can calculate the spectrum of phonons emitted from the tunnel junction
(analogously to the electromagnetic field, considered in Chap. 6).
10.4.1. Phonon Deficit in a Subthreshold Regime
The spectrum of phonons emitted from a nonequilibrium film is shown in Fig.
10.18. The dip in the spectrum at reflects the presence of the phonon
deficit effect. The same effect, though caused by the action of the UHF field, was
discussed in Chap. 6. Since the essence of the effect remains the same in the present
case, we will not provide a detailed commentary.
One small difference in the present case is that there are now two (rather than
one) relaxation peaks* (see Fig. 10.18, curve 1). Their origin is related to the
*Such peaks would most likely occur under UHF pumping also (cf. Chap. 5), if one could perform the
calculations with the appropriate accuracy.
SECTION 10.4. PHONON EMISSION FROM A TUNNEL JUNCTION 279
sawtooth shape of the excitation distribution function (see the inset to Fig. 10.18).
In some cases the peaks are indistinguishable at the scale used (Fig. 10.18, curve
2).
10.4.2. Superthreshold Regime: Preconditions of Deficit
The behavior of the phonon emission spectra in the superthreshold regime
(Figs. 10.19 and 10.20) is peculiar. Note that curves similar to those in Fig. 10.20
(curve 1) were first obtained by Chang and (in a simplified model
ignoring imbalance). However, they did not mention that at small values of the
phonon fluxes became negative: this is illustrated in Fig. 10.20 (curve 3).
The phonon deficit effect when the excess quasi-particles are generated by the
field from the condensate is not a trivial one. Evidently it is due to the strong
localization of excess excitations near the gap edge in the energy space (see Fig.
10.14). As a result, in the scattering of phonons by electrons, the emission of
phonons with frequencies higher than the boundary value is forbidden. At the same
time, scattering with the absorption of phonons is possible: at high energies there
is no energy gap. Hence the scattering mechanism causes a phonon deficit in a
certain spectral range. A question arises in connection with this: Doesn’t the deficit
280 CHAPTER 10. JOSEPHSON JUNCTIONS
reflect the phonon instability at the relaxation frequencies (i.e., the sign reversal of
the phonon absorption coefficient)? As the calculations show (Fig. 10.20, curve 3),
the sign of the absorption coefficient does not change. An inspection of the collision
integral (4.128) shows that the small (equilibrium) “tail” of the excitation distribu-
tion function does not contribute to the nonequilibrium emission of phonons, while
it does contribute to the absorption coefficient, compensating for the small dip that
results from nonequilibrium.
Note that effects of the type presented in Figs. 10.19 and 10.20 hold for a broad
range of junction parameters. We will not dwell here on further quantitative
analysis.