Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors
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SECTION 9.3. ANALYSIS OF RESULTS 241
242
CHAPTER 9. PHASE-SLIP CENTERS
I I
SECTION 9.3. ANALYSIS OF RESULTS 243
9.3.9. More Features of Numeric Solution
s
The solutions for are shown in
Fig. 9.19 (one PSC) and Fig. 9.20(two PSCs). The phase difference at the filament’s
ends in the state with two PSCs increases by over the period of stable oscillations
(Fig. 9.21). The dynamics of the phase variation along the filament with two PSCs
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CHAPTER 9. PHASE-SLIP CENTERS
is demonstrated in Fig. 9.22. At the moment 5, the phase has a vertical tangent
at the point of PSC (the dotted line). At the next moment (6) the phase at this points
breaks and two branches of on both sides of PSC slip relative one another.
The left branch at the moment (see Fig. 9.21) remains in the position as shown
in Fig. 9.22, but the right branch at the moment occupies the position a, coinciding
with the right branch of the curve 1. Because the phase is a multivalued
function the right branch can be shifted from the
position a into position b (dashed line). This, however, produces the discontinuity
of in the regular point because the phase is assumed to be antisymmet-
ric relative to the middle of the interval: Figures 9.23–9.26
illustrate the solutions with the boundary condition (9.15) , and Fig.
9.27 corresponds to the cyclic boundary condition
9.3.10. Role of Interference Current Component
It is of interest to see how the solutions behave if the interference current
(9.8) is included (we have demonstrated some of the results in the preceding
drawings). Figures 9.28–9.31 illustrate the role of the interference current compo-
nent. One can see from the figures into account does not significantly change
the qualitative pattern of the resistive state, although some changes occur. In
particular, the oscillation periods and current-voltage characteristics change notice-
ably (Figs. 9.12, 9.13, and 9.17). The presence is revealed more markedly in
the behavior of the potential, in which an anomaly is formed in the vicinity of PSC
(Fig. 9.28), which in turn leads to peculiar behavior of the normal component of
the (Fjg. 9.30). Such behavior persists also in the case
of two PSCs (Figs. 9.29 and 9.31).
With increasing the magnitude of the anomaly also increases, but at
the interference contribution disappears. Although the normal current may
become rather large when vanishes, there is no anomaly Because the total
current is kept fixed, const, the anomaly is compensated for
by a corresponding anomaly Thus the total current j stays a finite and smooth
function.
A remark should be made concerning the anomalies The
anomalous behavior is related to the term in (9.8), which
grows when tends to zero and Q becomes infinite. The superconducting current
at these moments remains finite. Representing in the form
it is easy to conclude that enormously large values of
1, i.e., the characteristic frequencies in the vicinity of PSC
are greater than the energy damping of electrons. However, in deriving TDGL
equations, quasi-classical conditions were assumed, i.e., that the characteristic
frequencies should be less than the energy damping [see (7.28)]. In fact, this
condition becomes violated in the vicinity of the PSC, which demonstrates a certain
SECTION 9.3. ANALYSIS OF RESULTS 245
246 CHAPTER 9. PHASE-SUP CENTERS
SECTION 9.3. ANALYSIS OF RESULTS 247
248 CHAPTER 9. PHASE-SLIP CENTERS
inconsistency in the equations used. It should be noted that this difficulty is inherent
to TDGL equations, even Indeed, the large space and time gradients
always arise in the PSC structure, in contradiction to quasi-classical conditions.
One possible way to overcome this difficulty might be a consistent accounting for
time derivatives of higher order. In such an approach, the inclusion of an additional
term proportional to see footnote on page 161] in the
dynamic equation (7.45) may be reasonable. The inclusion of an interference
current component in the set of dynamic equations is in fact a step in this direction.
9.4. EMISSION FROM PHASE-SLIP CENTERS
As shown in the preceding section, the resistive state of a superconducting
filament is characterized by periodically placed phase-slip centers. In the vicinity
of these centers, having a length scale where is the
stationary value of the gap, all the quantities describing the electron system oscillate
in time with frequency, which is defined by the voltage drop on the PSC [see
expression (9.54)].
9.4.1. Generation of Electromagnetic Radiation
It is natural to suppose that electromagnetic radiation might be generated in
these conditions. Indeed, such radiation from narrow thin-film structures was
even prior to the creation of the microscopic theory of PSC (a typical
level of radiation was W). As was found further, the picture is complicated:
besides the high-frequency radiation (32.13) an emission at low frequencies was
also Assuming that this effect may be explained by the motion of the
SECTION 9.4. EMISSION FROM PHASE-SLIP CENTERS 249
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CHAPTER 9. PHASE-SUP CENTERS