Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors
Подождите немного. Документ загружается.
SECTION 10.5. WEAKLY COUPLED BRIDGES 291
4. A. F. Volkov, Nonequilibrium states in superconducting tunnel structures, Sov. Phys. JETP 41(2),
376–380 (1975) [Zh. Eksp. i Tear. Fiz, 68(2), 756–765 (1975)].
5. A. M. Gulyan and G. F. Zharkov, Electron and phonon kinetics in a nonequilibrium Josephson
junction, Sov. Phys. JETP 62(1), 89–97 (1985) [Zh. Eksp. i Teor. Fiz. 89(7), 156–171 (1985)].
6. B. I. Ivlev, Resistive states in superconductor junctions, Sov. Phys. JETP 48(5) 893–896 (1978)
[Zh. Eksp. i Teor. Fiz. 75 [5(11)], 1771–1777 (1978)].
7. A. I. Larkin and Yu. N. Ovchinnikov, Tunnel effect between superconductors in an alternating
field, Sov. Phys. JETP 24
(11),
1035–1040 (1967) [Zh. Eksp. i Tear. Fiz. 51[5(11)], 1535–1543
(1966)].
8. J. Clarke, Experimental observation of pair-quasiparticle difference in nonequilibrium supercon-
ductors, Phys. Rev. Lett. 28(21), 1363–1366 (1972).
9. M. Tinkham, Tunneling generation, relaxation and tunneling detection of hole-electron imbalance
in superconductors, Phys. Rev. B. 7(5), 1747–1756 (1972).
10. I. K. Kirichenko and V. P. Seminozhenko, Distribution of nonequilibrium excitations and phonon
generation in superconducting tunnel junctions, Sov. J. Low Temp. Phys.
3
(8), 479–486 (1977)
[Fiz. Nizk. Temp. 3(8), 986–1000 (1977)].
11. V. F. Elesin and Yu. V. Kopaev, Superconductors with excess quasiparticles, Sov. Phys. Uspekhi
24(2), 116–141 (1981) [Usp. Fiz. Nauk 133(2), 259–307 (1981)].
12. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, pp. 313–343, Pergamon,
Oxford (1960).
13. J.-J. Chang and D. J. Scalapino, Nonequilibrium superconductivity, J. Low Temp. Phys. 31(1/2),
1–32(1978).
14. P. Otschik, H. Eschrig, and F. Lange, Sound wave amplification in superconductors, J. Low Temp.
Phys. 43(3/4), 397–408 (1981).
15. A. G. Aronov and B. Z. Spivak, Properties of superconductors with nonequilibrium excitations,
Sov. Phys. Solid State 18(2), 312–320 (1976) [Fiz. Tverd. Tela 18(2), 541–553 (1976)].
16. K. E. Gray and H. W. Willemsen, Inhomogeneous state of superconductors induced by intense
tunnel injection in superconducting lead films, J. Low Temp. Phys. 31(5/6), 911–925 (1978).
17. J. A. Pals, K. Weiss, P. M. T. van Attekum, R. E. Horstman, and J. Wolter, Nonequilibrium
superconductivity in homogeneous thin films, Phys. Rep. 80(4), 323–390 (1982).
18. R. H. Parmenter, Enhancement of superconductivity by extraction of normal carriers, Phys. Rev.
Lett. 7(7), 274–277 (1961).
19. R. G. Melton, J. L. Paterson, and S. B. Kaplan, Superconducting tunnel junction refrigereator,
Phys. Rev. B. 21(5), 1858–1867 (1980).
20. M. Nahum, T. M. Eiles, and J. M. Martinis, Electronic microrefrigeration based on a normal
metal-insulator-superconductor tunnel junction, Appl. Phys. Lett. 65(24), 3123–3125 (1994).
21. P. A. Fisher, J. N. Ullom, and M. Nahum, Design of a novel on-chip electronic refrigerator based
on a normal-insulator-superconductor tunnel junction, J. Low Temp. Phys. 101(3/4), 561–565
(1995).
22. A. Bardas and D. Averin, Peltier effect in normal-metal-superconductor microcontacts, Phys. Rev.
B, 52(17), 12873–12877(1995).
23. M. M. Leivo, J. P. Pekola, and D. V. Averin, Efficient Peltier refrigeration by a pair of normal
metal/insulator/superconductor junctions, Appl. Phys. Lett. 68(14), 1996–1998(1996).
24. A. J. Manninen, M. M. Leivo, and J. P. Pekola, Refrigereation of a dielectric membrane by
superconductor/insulator/normal-metal/insulator/superconductor tunneling, Appl. Phys. Lett.
60(14), 1885–1887(1997).
25. A. M. Gulyan, G. F. Zharkov, and G. M. Sergoyan, Thermal balance in the tunnel contact of a
superconductor, Sov. Phys. Lebedev Inst. Reports, 10, 38–42 (1984) [Kratk. Soobsch. poFiz, FIAN,
10, 33–36(1985)].
292
CHAPTER 10. JOSEPHSON JUNCTIONS
26. L. G. Aslamasov and A.I. Larkin, Josephson effect in superconducting point contacts, JETP Lett.
9, 87–91 (1969) [Pis ’ma v Zh. Eksp. i Teor. Fiz. 9, 150–154 (1968)].
27. L. G. Aslamasov and A. I. Larkin, Superconducting contacts with a nonequilibrium electron
distribution function, Sov. Phys. JETP 43(4), 698–703 (1976) [Zh. Eksp. i Tear. Fiz 70(4),
1340–1347 (1976)].
28. A. Schmid, G. Schön, and M. Tinkham, Dynamic properties of superconducting weak links, Phys.
Rev. B 21(11), 5076–5086 (1980).
29. A. Barone and G. Paternò, Physics and Applications of the Josephson Effect, pp. 47–49, Wiley,
New York (1982).
30. C. M. Falco, W. H. Parker, and S. E. Trullinger, Observation of a phase modulated quasiparticle
current in superconducting weak links, Phys. Rev. Lett. 31(15), 933–936 (1973).
31. V. Ambegaokar and B. I. Halperin, Voltage due to thermal noise in the Josephson effect, Phys. Rev.
Lett. 22(25), 1364–1366(1969).
32. H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions,
Physica 7(4), 284–304 (1940).
33. A. V. Zorin, I. O Kulik, K. K. Likharev and J. R. Schrieffer, The sign of the interference current
component in superconducting junctions, Sov. J. Low Temp. Phys.
5
(10), 537–546 (1979)
[
Fiz.
Nizk. Temp. 5(10), 1138–1157 (1979)].
34. J. 1. Pankove, New effect at superconducting contacts, Phys. Lett. 21(4), 406–407 (1966).
35. A. I. Akimenko, V. S. Solov’ev, and I. K. Yanson, ac Josephson current as a function of voltage,
Sov. J. Low Temp. Phys.
2
(4), 238–241 (1976) [Fiz. Mat. Temp. 2(4), 480–485 (1976)].
36. M. Octavio, W. J. Skochpol, and M. Tinkham, Nonequilibrium-enhanced supercurrents in short
superconducting weak links, Phys. Rev. B 17(1), 159–169 (1978).
37. V. N. Gubankov, V. P. Koshelets, and G. A. Ovsyannikov, Properties of Josephson thin film
variable-thickness microbridges, Sov. Phys. JETP 46(4), 755–760 (1977) [Zh. Eksp. i Teor. Fiz.
73[4(10)], 1435–1444(1977)].
38. Yu. Ya. Divin and F Ya. Nad’, A model of real superconducting point contacts, Sov. J. Low Temp.
Phys. 4(9), 520–525 (1978)
[
Fiz. Nizk. Temp. 4(9), 1105–1114 (1978)].
39. H. A. Notarys and J. E. Mercereau, Proximity effect bridge and superconducting microcircuitry,
J. Appl. Phys. 44(4), 1821–1830 (1973).
40. T. M. Klapwijk, M. Sepers, and J. E. Mooij, Regimes in the superconducting microbridges, J. Low.
Temp. Phys. 27(5/6), 801–835 (1977).
41. K. K. Likharev and L. A. Yakobson, Dynamical properties of superconducting filaments of finite
length,
Sov.
Phys.
JETP
41(3),
570–576
(1975)
[Zh.
Eksp.
i
Tear.
Fiz.
68(3),
1150–1160
(1975)].
42. S. N. Artemenko, A. F. Volkov, and A. V. Zaitzev, On the excess current in microbridges S-C-S
and S-C-N, Solid State Commun. 30(12), 771–773 (1979).
11
Influence of Laser Radiation
It is not surprising that a nonequilibrium state is created when photons of energy
much larger than the pairing energy are absorbed by the superconductor. What is
surprising is how strongly the behavior of this still superconducting state depends
on the “tiny features” of the superconducting state.
11.1
ELESIN APPROACH TO QUASI-PARTICLE DISTRIBUTIONS
Consider a thin superconducting film irradiated homogeneously by an electro-
magnetic field. The frequency of the radiation is supposed to be substantially
higher than the value of the gap
*
: so the excess quasi-particles in the
electron system are created by the field in the energy range As a
consequence, the source “density” in the kinetic equation for electrons [Eq.
(5.21)] is small and the shape of the electron excitation distribution function is
formed mainly by the inelastic collision operators. For simplicity we will consider
only inelastic electron-phonon collisions (4.119), assuming that the phonons are
in equilibrium as a result of coupling with the heat-bath.
11.1.1.
Spectral Function of Electron–Phonon Interaction
We will start with the case of temperature Denoting
one can write the kinetic equation (5.21) for the quasi-stationary state in
the form:
*
In this chapter a gauge with the real order parameter will only be used.
293
294 CHAPTER 11. INFLUENCE OF LASER RADIATION
In Eq. (11.1) the electron-phonon interaction function is introduced.
The approximation
corresponds to the cases considered in the previous chapters. In (11.2), the depend-
ence of the electron-phonon interaction matrix element on the wave vector q is
taken into account ( for ionic crystals). Note that in the adopted notations, the
nonequilibrium source in (11.1) differs by the factor from the canonical
form (5.59).
11.1.2. Excess Quasi-particles: Normalization Condition
The concentration of excess quasi-particles is determined by the integral form
of Eq. (11.1). The normalization condition
1
is obtained by integrating Eq. (11.1) over the relaxation terms disappear on doing
so. The explicit expression for may be significantly simplified using the
condition Omitting the U-factors in (5.59) [i.e., ignoring in Eq. (11.1) the
redistribution of excess quasi-particles by electromagnetic radiation—these factors
cancel each other in Eq. (11.3)] and simplifying the term that corresponds to the
creation of the excess quasi-particles from the condensate by the photon, one can
write [cf. Eq. (5.54)]:
11.1.3. Separation of "Coherent" Contributions
For [i.e., for k
=
–1 in Eq. (11.2)], the kinetic equation
(11.1) can be solved analytically.
1
We will analyze only this model case, when Eqs.
(11.1) and (11.3) acquire the forms:
SECTION 11.1. ELESIN APPROACH TO QUASI-PARTICLE DISTRIBUTIONS 295
In this case, as can be seen from (11.5) and (11.6), the solution depends
explicitly on the order parameter through the coherence factors
11.1.4.
Analytic Solution for
Assuming in these equations, we will neglect for the moment the
“coherent” contribution to the function We will denote the solution of (11.5)
with To define this function, one has the equation
It is useful to introduce the function
which obeys the nonlinear differential equation, following from (11.7):
Introducing further one finds from (11.9) the equation:
from which
or, returning to the original function y
,
296
CHAPTER 11. INFLUENCE OF LASER RADIATION
Only the solution with a positive sign before the square root satisfies the boundary
condition [see (11.8)], and thus the integration constant is defined as
So we have obtained for the function
The characteristic parameter a determines the behavior of the solution To
obtain this quantity, we rewrite the normalization condition in the form:
from which it follows* [neglecting corrections of the order of
Thus the maximum value of the function increases, the
function decreases and becomes small at a. As mentioned earlier, the function
does not depend on A. Thus one can conclude that at optical
pumping.
1
We will now examine the “coherent” contributions to the function
11.2.
ORDER-PARAMETER AMBIGUITY
11.2.1. First-Order "Coherent" Correction
Assuming that one finds from (11.5) the equation
is small when which is of interest here.
SECTION 11.2. ORDER-PARAMETER AMBIGUITY 297
Taking into account that the term (11.17), containing in the integral form, is
small, one obtains to the first order in
Note that this function is negative at Taking into account zero-order solutions,
the self-consistency equation [of the type (10.14) that follows from (7.14)] can be
presented in the form
or, after simple transformations, as
11.2.2. Critical Pumping Intensity
The parameter a in (11.20) depends on the pumping intensity through the
relation (11.16). The critical value at which vanishes, may be determined
from (11.20):
or
11
.
2.3.
Multiple-Order-Parameter Solutions
As mentioned in Ref. 2, the solution at intensities is not a unique
one. Indeed, if the coherent contribution is neglected, the right side of Eq. (11.20)
vanishes. So at it follows from (11.20):
298 CHAPTER 11. INFLUENCE OF LASER RADIATION
Thus at the gap tends to zero monotonically and no nontrivial solutions o
f
Eq. (11.23) exist at If the right side of Eq. (11.20) is taken into account, Eq
.
(11.23) acquires additional terms:
where is the Katalan constant. In Fig. 11.1 the dependence of on
the order parameter is plotted according to Eq. (11.24). The above-mentioned trivia
l
solution normal state) also exists for any This state might be
realized if one approaches the superconducting state from the normal metal state
by decreasing the pumping intensity. Thus at there exist three solutions
of the self-consistency equation (11.19). Two nonzero branches merge a
t
as one can see from Eq. (11.24). In this case
11.2.4. Stability of Solutions
We will consider now the stability problem of various branches of the solution
of Eq. (11.24) relative to spatially homogeneous perturbations:
SECTION 11.2. ORDER-PARAMETER AMBIGUITY 299
As above, neglecting in Eq. (11.1), restoring on the left side the time derivative,
varying the right side over n and and omitting the small contribution due to the
term with in the integrand, one obtains the equation
Varying (11.19), one gets
Assuming, as usual,
we obtain from (11.19)
where . Substituting Eqs. (11.29) and (11.30) into (11.28), we get
Noting that the main contributions to the integrals come from small
(11.31) may be transformed to
Substituting into (11.27) and (11.32), one can obtain:
300 CHAPTER 11. INFLUENCE OF LASER RADIATION
11.2.5. Physics of Coherent Instability
As follows from (11.33), the parameter so that the correspond-
ing solution is unstable [here is the root of the equation which coincides
with (11.25)]. This means that the states of the system, indicated by the dotted line
in Fig. 11.1, are unstable against the transition to the normal state or to the state
with The region of instability coincides with the region of ambiguity of
This ambiguity arises, as mentioned earlier, from the coherence factors in the
kinetic equation, which reveal the physical cause of instability.
2
Indeed, let the gap
at be decreased slightly due to a fluctuation. Then the probability of excess
quasi-particle recombination decreases owing to the coherence factors. This in-
creases the number of excess quasi-particles and leads to a further decrease in
etc. The reverse conclusions are also valid. As a result, from the intermediate values
of the solution passes to the branch or to the branch This leads
to the “coherent” instability.
11.3
. FINITE TEMPERATURES
11.3.1. Inclusion of Thermal Phonons
As before, we will consider the kinetic equation (5.21), with a collision integral
of the form
The phonon distribution function in (11.34) remains in equilibrium: