The critical current of superconducting contacts in a microwave field

The critical current of superconducting contacts in a microwave field


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Volume 67A, number 3


7 August 1978

THE CRITICAL CURRENT OF SUPERCONDUCTING CONTACTS IN A MICROWAVE FIELD L.G. ASLAMAZOV and A.1. LARKIN Landau Institute for Theoretical Physics, Moscow, USSR Received 23 February 1978 Revised manuscript received 25 May 1978

Effective cooling of electrons enclosed in the contact region with a depressed value of the energy gap is shown to produce significant variations of the critical current in an external microwave field.

Experiments [1—7]on superconducting contacts (microbridges, point contacts, etc.) reveal that microwave radiation increases the critical current. This phenomenon is explained by the creation of a nonequilibrium electron energy distribution: n(e) = [1—f(e)]/2, instead of the equilibrium Fermi distribution: f= th(/2T) [8]. lt is shown below that, unlike the case of a spatial uniform superconductor [8], the effective cooling of electrons enclosed in the contact region with a depressed gap value is essential. Electrons with energy ~< where is the modulus of the order parameter in the banks reflect near the contact edges, namely at e = z~.If the contact is placed in an alternating field, electron energy diffusion appears. The energy diffusion is described by the following kinetic equation [9]:

The superconducting current 1~flowing through the contact is determined by the Ginzburg— Landau equation, which for the nonequilibrium state under the above mentioned limitation on the contact size is [11,141: D ra2~ T 1 ~ — 2 f(LI)J~j+ ~ BL\3 + ~(z~)= O, C r d (2) ~ = f(e) th = C Vp, —



i r —




/ («2


~2y1/2>~— ~

2/~r2)—~(aO/at)), 5





— —

(.~i~j2) ‚

where the brackets (>denote averaging over the contact region in which0Fe > ~‚ is thethe overbar averaging 1tr/3 spatialdenotes diffusion coeffiover time, D= cient; the relaxation time is assumed large as compared to the field period and to the characteristic time of electron diffusion through the contact. Besides this, the validity of formula (1) is limited by condition: w «Dia2 ~ ~‚ which determines the possible size a of contacts.


C îrepSDi~f(~), 8ir2T2 where p is the density of states, S the contact cross section, p the phase of the order parameter. =




When the radiation power is large, the energy diffuequation sion is also(1)large can and be considered the left hand to be sideequal of the tokinetic zero. Then the particle flow is D 5 0f/8 = const. On the other hand, the boundary condition for the kinetic equation implies that the particle flow is zero when the energy equals the minimum value of the energy gap, which isofreached at the centre of the contact the bottom the potential well). Therefore ~f/8 (at = O and f() = const. A large radiation power leads to a large energy diffusion, which equalizes the population of the levels. That is why the distribution function does not depend on energy. The second boundary condition: f(z~ 0)= th(~0/2T) ~/2T, means that for energies e > ~ the distribution is the equilibrium one,


e e e

e e

a a a a a a a Volume 67A, number 3


because of the rapid spatial diffusion. Therefore, for the distribution function at high levels of radiation, we havef(e < i~) = ~ 0/2T. The nonequilibrium term is calculated by means of this distribution function and formula (2): _______

l+~/1_~2/~2 O


1 a2t~ ~° 0.66 _O.62e2/L~ Pf .~42_/2 de,



r /a 0\ P~—~) [1_O.O4~,—~---)



2L~ ~/~2 2 / T~ ~





(3) We can find the superconducting current by substituting this expression in the Ginzburg—Landau equation; if a at high radiation power, only the nonequilibrium term is significant. The critical current is determined from the condition of the maximum of the superconducting current. At high radiation power the critical current does not depend on power and equals

e where the dimensionless parameter P = r 2/Tis proportional to the radiation power. The suppression of 5~ superconductivity in the contact can be explained in the following way. The energy diffusion coefficient is a monotonically decreasing function of energy. As a result, the particle flow also decreases with energy. The occupation of states n (e) increases: f — th (/2 T) <0, and the nonequiibrium term becomes negative. The critical current of the contact decreases. The sign of f— th (e/2T) becomes positive only when the energies


are very these energies close to significant the bottom deviations of the potential of the distribution well. For




/ ~ ln

7 August 1978






043e P SDh/2I~SI2/T O

(4\ „ ‚

The modulus of the order parameter in the contact is equal to O.7z~ 0and weakly depends on the coordinate. 5!4. We can see from formula that 1‘ —~(T~ — T) The critical current of the(4) contact without radiation is I~ (T~— T)3!2 if a > ~‚ and ~ (Te— T) if a < Therefore the ratio I‘/I~reaches its maximum value at the temperature when a ~and this corresponds to experimental results [2—4].Near T~,when ~ 2, 0~Da and at low temperatures, when T~<~ T~,the changes in the distribution function do not significantly influence the critical current. At low radiation power, significant deviations of electron distribution occur only at energies close to the magnitude of the gap in the center of the contact. For all other energies perturbation theory is valid, and the kinetic equation can be solved by substituting f th e/2T in the right hand side of the equation. To calculate the energy diffusion coefficient, it is necessary to know the dependence of~on the coordinate. l‘his dependence can be obtained at low radiation power from the equilibrium Ginzburg— Landau equation (the nonequilibrium term is assumed to be small). The dependence of ~ on time is..determined by the rate of change of the phase difference x of the order parameter in the banks of the contact, due to the alternating voltage Von the contact: Ç = 2eV/1~.As a corollary for short contacts, for example: ~ > a > ~ where ~ is the coherence length, we have: -~


function occur and it becomes impossible to use perturbation theory. But this region does notradiation give a cornparably large contribution to Ø(I~) at low power. The energy region where the essential deviations of the distribution function occur widens with the increase of the radiation power. For short contacts: a < ~‚ the gap ~ depends strongly on the coordinate. So, the electrons with energies close to the magnitude of the gap in the contact center are localized near the centre. Only these electrons are cooled and the order parameter increases only near the centre of the contact. For all other regions of the contact the electrons are hot and the order parameter decreases. Therefore for short contacts the decrease of the critical current continues up to large radiation powers, when noticeable deviations of the distribution function occur for all important energies e The situation differs for long contacts: a> The order parameter in such contacts depends weakly on the coordinate and is close to the value ~ in the centre of the contact. Electrons can diffuse through the entire contact if their energy is not too close to Then, even at low radiation power, contributions to the nonequiibriurn term from cool electrons with energies close to ~ and from hot electrons with other energies nearly cancel and the rest gives the positive sign of ~ The critical current then increases. To find the nonequilibriurn term from eq. (8), it is necessary to solve the kinetic equation; before this, however, ~.



Volume 67A, number 3


the energy diffusion coefficient has to be determined, Assuming that the bottom of the alternate potential well is flat, we have the following expression for D 5 from formula (1):






4D ~2



Xi/~~2\ ~


dx] 2) (6)


0.23 ~2T~/(e2






Here ~ is close to the magnitude of the order parameter in the centre of an infinitely long contact: \/~7~ ~. Substituting this expression in the kinetic equation and its solution in formula (2), we have: =

0.lPTln ——0.08 ~/~~O.2TO.3p0.4ln ~a2

(7) The value p TP215(1 — T/T~)‘/‘° determines the width of the energy region near the bottom of the potential well where essential deviations from the equilibrium distribution occur and perturbation theory is not valid. The second term in expression (7) becomes significant only at low powers:P< (~/a)5/3((T~T)/T~) and this gives the suppression of superconductivity. For sufficiently long contacts, a> ~(TcI(Tc T))3~0, this suppression is small and the second term in expression (7) is negligible. The critical current can be found from formula (2). Using expression (7), we have jo Te 3/2 ~q-~ ~ ~=0.O6 (Te— T) Pin—. (8) —

Therefore in long contacts the critical current increases already at low radiation power and is proportional to it.

In accordance with theory, experiments [3,4] reveal the transition from the increase to the suppression of superconductivity in the contact, when the temperature becomes very close to T~and the coherence length ~ becomes larger than the contact size a. The increase is linear. But the critical current saturation cannot always be approached in experiment, because of the direct heating of the contact by radiation [12]. Our results are valid if the alternating part of the order parameter z~is small compared to the part depending on the coordinate (in long contacts the


7 August 1978

last part is of the order of ~0~2/a2). In this case, it follows from the Ginzburg—Landau equation that ~i strongly depends on the coordinate in the entire contact region. The smaller the frequency, the bigger the alternating of theoforder parameter. When ~i > 2/a2 thepart bottom the potential well oscillates as and ~~2/~t ~0~a whole and in formula (6) (a~2/at) cancel. The energy diffusion coefficient becomes small and Eliashberg‘s term can become significant [8]. This term gives a linear increase even near T~this is possibly why the increase taking place even near T~occurs at low frequencies and in very long contacts [2—6]. It can be also noted that the changes of the critical current in the microwave field and the appearance of tics current‘s the of the contact plateau[4,13,141 on the volt—ampère have to be correlated: characterisboth effects become noticeable only when the contact sizeisa>v~7~[4]. The authors are very grateful to Dr. Yu. Ovchinnikov for valuable discussions. References [1] A. Wyatt, V.M. Dmitriev, W. Moore and F. Sheard, Phys. Rev. Lett. 16 (1966) 1166. [2] A. Dayem and J.J. Wiegand, Phys. Rev. 155 (1967) 419.


Yu. Latishev and F. Nadj, Zh. Eksp. Teor. Fiz. 71(1976) 2158. [41V. Gubankov, V. Koshelets and G. Ovsyannikov, Zh. Eksp. Teor. Fiz. 73 (1977) 1435. [5] K. Shepard, Physica 55 (1971) 786. [6] T. Klapwijk and J.E. Mooij, Physica 81B (1976) 132; T. Klapwijk, Temp. Phys. J. 26van (1977) der Bergh 385. and J. Mooij, J. Low. [7] R. Fjordboge, T. Clark and P. Lindelof, Phys. Rev. Lett. 37 Eliashberg, (1976) 1302. [8] G. Soy. Phys. JETP Lett. 11(1970)114; Soy. Phys. JETP 34 (1972) 668. [9] A. Larkin and Yu. Ovchinnikov, Zh. Eksp. Teor. Fiz. 68 (1975) 1915.

[10] B. Ivlev and G. Eliashberg, Soy. Phys. JETP Lett. 13 (1971) 333. [11] A. Larkin and Yu. Ovchinnikov, Zh. Eksp. Teor. Fir. 73 (1977) 299. [12] M. Tinkham, M. Octavio and W. Skopol, J. Appi. Phys. 48 (1977) 1311. [13] M. Octavio, W. Skocpol and M. Tinkham, Nonequilibriumenhanced critical currents in a short superconducting weak link, preprint. [14] L. Aslamazov and A. Larkin, Zh. Eksp. Teor. Fiz. 70 (1976) 1340.