Microwave-enhanced critical currents in superconducting microbridges explained by the electric-field induced quasiparticle-pair inequilibrium

Solid State Communications, Vol. 18, pp. 283—286, 1976.

Pergamon Press.

Printed in Great Britain

MICROWAVE-ENHANCED CRITICAL CURRENTS IN SUPERCONDUCTING MICROBRIDGES EXPLAINED BY THE ELECTRIC-FIELD INDUCED QUASIPARTICLE-PAIR INEQUILIBRIUM P.E. Lindelof Physics Laboratory I, H.C. Orsted Institute, University of Copenhagen, Denmark (Received 14 August 1975 by N.J. Meyer) A supercurrent through a superconducting microbridge gives rise to a minimum in the density of Cooper pairs in the middle of the bridge. Application of a microwave electric field creates a non-equilibrium situation and therefore smears the Cooper pair distribution. This allows for a higher critical current and explains the microwave-enhanced supercurrent observed in thin-film microbridges. ONE OF THE MANY fascinating phenomena in thinfilm microbridges is the possibility of increasing the maximum critical current of the bridges by applying a microwave field. This effect was first noticed by Wyatt, Dmitriev, More and Sheard1 and by Dayem and Wiegand.2 It has since been under investigation several times and a considerable amount of data exist. In order to explain this effect we propose a new type of nonequilibrium situation (and its subsequent decay) set up by a.c. currents and voltages in weak links, A thin-film microbridge with dimensions smaller than the temperature-dependent coherence length, exhibits the Josephson effect. The current—voltage characteristic is normally calculated on the basis of the resistively shunted model (RSJ). If the current is composed of a d.c. contribution ‘d.c. and a microwave current with amplitude I~and angular frequency w, we have for RSJ ‘d.c.

+ 1~sin wi’

=

h d~ 1o sin ~ + 2eR dt —

microbridge [Fig. 1(a)] the amplitude of the Ginzburg— (1) Landau order parameter a minimum in thethis middle the bridge [Fig. 1(b)] Inhas a two-fluid picture mini- of .

where I~is the maximum critical current, R the normal resistance of the bridge and 4 is the quantum mechanical phase difference across the bridge. The voltage is given by V = (h/2e)(d~/dt).If the parameter ç~= (hw/2eRI 0)> 1, the microwave current gives rise to a microwave voltage amplitude V1 = RI1 and the critical current I~depends on the microwave current (or voltage) as a zero order Bessel function ‘~= ‘~~0

I

1.

mum in the density of the Cooper pairs must be associated with a corresponding maximum in the density of the quasiparticles [Fig. 1(c)] to ensure the conservation of charge. This space distribution of the quasiparticles ,

drifts in an applied electric field and quasiparticles are injected into regions where inelastic relaxation processes are needed in order to gain equilibrium withup theinCooper 5 In fact the quasiparticles are moving energy pairs. along the energy dispersion relation to create a branch (2) imbalance which released inelastic scattering.5’6 An r.f. can fieldonly thusbe smears the by time-averaged dis-

hw J Experiments on thin-film microbridges deviate somewhat from the predictions based on equation (1). ~

time-dependent Ginzburg Landau theory. Recent experiment4 seems in fact to indicate that this effect is primarily connected with the normal electrons. If a voltage exists in a region of a superconductor where the energy gap changes with position, then the quasiparticles constantly drift into regions where they are not in equilibrium with the Cooper pairs and a relaxation takes place. Such a situation was considered by Pippard, Shepherd and Tindall5 to explain their excess resistances at superconducting normal interfaces and it was also the basis for the interpretation of experiments on long bridges by Skocpol, Beasley and Tinkham.6 The use of such a picture also seems worthwhile for short bridges. In the following a simple phenomenological theory is developed on this basis, with the aim of explaining the effect discovered in references 1 and 2. When a supercurrent is passed through a Josephson

A number of these deviations are explained3byhowever introducing relaxation effects for the current superfluid, the microwave-enhanced critital does not seem to be a straightforward consequence of a linearised

tribution (n~(x))of the quasiparticles, dependent on the inelastic relaxation time r 2 multiplied by the angular r.f. frequency w. Charge uniformity must always be prevalent (due to the very short screening length in metals) and it is preserved by motion of the superconducting ground state. Along with the decreased time-averaged 283

284

CRITICAL CURRENTS IN SUPERCONDUCTING MICROBRIDGES badge reglorl

(C)

I

Vol. 18. No. 3

/12

where n.~(x) the equilibrium density of superfluid atis xlnormalized > 1/2 [i.e.ton~(l/2) I at all teinperatures)1 n is determined by the current such that /~ (1 0n0)n0 and the critical current corresponds to

~x)

n0

.

2/2

(0

Kb

Cooper pars

enhanceme’~’~\\~~/

Quosip

i

~

tic es

~~AppI

~ The conclusions we shall reach are not crucially dependent on equation (3): in fact any variation with a minimum for n0(x) in the middle of the bridge will do The two-fluid model and charge uniformity then yield the space dependent part of the density of the quasi particles

—

~

n~(xj I n~(x). (4) Note that n~(x) 0 at H > 1/2. Note also that tile quasiparticle density must be considered an integrated value over all energies. 0

1

Fig. 1. Illustrations to the r.f.-enhancement of the critica] current in thin-film microbridges. (a) Superconducting microbridge with length / connecting two strong superconductors. (b) Variation through the bridge of the normalized density of Cooper pairs at zero current (1 0) and at the maximum critical current (I Jo). (c) Quasiparticle density variation through the bridge (only the x-dependent part). Application of a microwave (r.f.signal smears the distribution as shown. The Cooper pair density adjust to the quasiparticle density as illustrated in (b) and consequently the density of Cooper pairs in the supercurrent. cal bridge middle is enhanced, allowing for a higher criti—

density of quasiparticles in the middle of the bridge there is a corresponding increase in the time-averaged density (n 0(x)) of the Cooper pairs [Fig. 1(b)]. This increase in the density of the Cooper pairs in the middle of the bridge allows for a higher critical current, as observed. In order to make the above suggestion more quantitative, we consider a linear model of a bridge with a length /•7 If 1 < ~ where ~b is the coherence length in the bridge region the amplitude of the order parameter has only one minimum along the bridge for all values of applied current. The bridge region 1/2
In the electric field of an applied microwave uurrent T(t) (RI1 /1~sin wt, the quasiparticles will attain a drift velocity CT in EU)

v(t)

expressed as usual in terms of the electronic charge c. mass in and relaxation time T, The quasiparticles drifting through regions with a spatially dependent energy gap create a non-equilibrum situation, and relax back to the condensate over a characteristic time ~2 (‘~‘r) which 56 is related This non-equilito the relax ation time brum situation for inelastic is ruled by scattering. the following Boltzmann equatioll —[n~(x, t)]

-

n~(X,t)

v(t)

ax

n,~(x)

T2(x. t)

where n~(x,t) is the non-equilibrium density oJ the quasiparticles [n~(x , t) 0] There exists a quite general solution8of this equation, known as Chambers kinetic integral: -~

n~(x,t)

.

n~[x

~(t’)]

dt ~2

exp

[i

.

ds

r

2 k(s)]

J (7)

[x(t’)j

where

j

~x(t’) v(t”) dt’. section reduction for the bridge region (constriction) and y is the ratio of strengths of the superconductors 7 As a in For the present purpose we simply calculate the the bridge very approximate and background expressionregions for therespectively. variation of the den- time average value of n~(x,t) at x — 0 and with equations (3) (5) inserted. We put n 0 — ~ corresponding to sity of the Cooper pairs ~z5(x)at zero voltages, we use the maximum supercurrent before the microwaves are tJv//0

~

(‘~

)(~~

applied. We also assume we find

~2

to be independent ofx. Then

Vol. 18, No.3

CRITICAL CURRENTS IN SUPERCONDUCTING MICROBRIDGES (n~(0,t))

=

n~(0) ay (-~— 1) —

Jnn(0~i’) di’

~

=

s~_

(2)2 1 +W12

(erRIi)2

—

285 (

I’

Calculated

We see that the quasiparticle density in the middle of the bridge has decreased and consequently the density of Cooper pairs has increased by the same amount. The relative increase in the density of the superfluid corresponds to a possible higher critical current in the same proportion, namely from I~= 1o to:

1

2 Measured

Notarys. Yu and Mercereau) 0~

1

~

~

rf POWER (arbitrary

i.

units)

Fig. 2. Comparison of equation (11) in the text and the 2 2 2 enhancement of the critical current in a proximity = , + (2erEIi~ w 12 (9~ bridge measured by Notarys, Yu and Mercereau.9 The ° ~ay 2) ~ ml2w) 1 + w2r~ ‘ / only fitting involves scaling of the axes. r.f. frequency: 10 GHz. There are thus two competing mechanisms by which the critical current of a microbridge changes in a microin J 0(a) coincides with the second minimum in I~vs r.f. wave field: firstly there is the modulation of the cornpower. We have chosen A(T) 10(T) such that the sec2 corresponds to experiment. plex order parameter by the microwave current which in ond maximum in J~ vs a general decreases the critical current. If we take the limit An order of magnitud.e estimate of A based on the of very small bridges and assume (hw/2eRIo)> 1 this bridge geometry is difficult. Firstly y is not known and takes the form of equation (2); secondly the electric cannot simply be taken as the ratio of the link critical field associated with the microwave current causes a current to the critical current of the background film, as drift of the quasiparticles into regions where they are not the former is much enhanced by the proximity of the in equilibrium with the Cooper pairs, this leads to an background film. Secondly, and for the same reason, it increase in the critical current equation (9). The total is also uncertain how to estimate the length of the critical current can therefore be written bridge. If we nevertheless insert 1— 1 pm which is the

1-.~.--_I\

/2eRI’~2 =

J~1 + A

—~j

‘2 RI \ J 0

(

length of the normal metal overlayer, r

l0’3sec

corresponding to a residual resistance ratio of 10 and tally y. (a Based wr2 =we1 can we then get 7Aestimate = 10’° = 1, on y the 1).enhancement ExperimenA = ~/ 1 given in Fig. I of reference 9 at 2 GHz and using a = 10 I \ / ~ \2 we find that ‘y varies from 108 to l0~in the temperaThe type of bridges to which the above theory can ture region from 2 to 3 K. be most directly compared is the proximity bridges. For There are two more features in the experiment of these the order parameter varies only in one dimension Notarys, Yu and Mercereau9 which are noticeable. (a = 1) and is suppressed by proximity to a normal Firstly the enhancement is only seen for short bridges, metal in a very short region. The suppression of superin fact A in equation (10) is proportional to! Secconductivity in the bridge region is strongest close to T~, ondly the enhancement at fixed frequency close to T~ where equation (10) therefore can be written gives a maximum critical current which is essentially where

~)

!~

~

,/

(10)

-~-—--~

2r2 \ hw 1 + w w2r~

°~

‘~.

2eRI =

A(T) Jo(T)a2Jo(ci)

where

a

=

w

1

(11)

Notarys, Yu and Mercereau9 have measured the critical current close to T~for one of their tin proximity bridges as a function of microwave power at 10 GHz. Their experimental result is shown in Fig. 2. A quantitative estimate based on their bridge data shows that the assumptioncondition hw/2eRJo 1 holds. Although this is not necessary for>tile validity of our theory, it isa convenient because we have tabulated Besselfunctions. We have fitted equation (11) to their experimental curve as depicted in Fig. 2. The r.f. power is proportional to a2 and scaled in such a way that the second zero point

independent of T. This we think is connected with a similar of y and(Dayem ~ Fortemperature constrictiondependence type microbridges bridges) the mam problem of applying the above theory is to choose 1 and a(T). For very long bridges the maximum supercurrent is unstable for small voltages which immediately_create a “phase-slip center” of the length A = ~./‘ 3z3rr2 (VF = Fermi velocity) as discussed by 6 However for short Skocpol, Beasley Tinkham. bridges the order and parameter will have a stable minimum in the middle of the bridge, the depth of which (i.e. a) will depend on the geometry of the bridge and the length of coherence. There are two experimental observations in reference 2 which can be readily understood

286

CRITICAL CURRENTS IN SUPERCONDUCTING MICROBRIDGES

on the basis of equation (10) namely the linear power dependence of the enhancement for small microwave powers and the existence of a frequency below which the effect is not seen. Recently we have found that the low frequency cut-off for the enhancement close to T~is around 7 GHz for indium, in contrast 1.5 GHz found for tin,2 probably reflecting a different value of 12 in the two materials. We have also experimentally observed

Vol. 18, No. 3

that the higher the residual resistivity of our films (i.e. small r), the smaller the general enhancement, also consistent with equation (10).

Acknowledgements I have benefited from several helpful discussions with Dr. T.D. Clark, Professor J. Clarke and Professor H. HqSjgaard Jensen.

REFERENCES 1.

WYATT A.F.G., DMITRIEV V.M., MOORE W.S. & SHEARD F.W., Phy.s. Rev. Lett. 16, 1166(1966).

2. 3.

DAYEM A.H. & WIEGAND J.J.,Phys. Rev. 155, 419 (1967). JENSEN H.H. & LINDELOF P.E., Proceedings of LT 14, Finland 1975, in press.

4. 5.

TREDWELL T.J. & JACOBSEN E.H., Phys. Rev. Lett. 35, 244 (1975). PIPPARD A.B., SHEPHERD J.G. & TINDALL D.A.,P~’oc.R. Soc. London A324, 17(1971).

6.

SKOCPOLW.J., BEASLEY M.R. & TINKHAM M.,J. Low Temp. Phys. 16, 145 (1974).

7.

GREGERS-HANSEN P.E., LEVINSEN M.T. & PEDERSEN G.F., J. Low Temp. Phys. 7,99(1972).

8.

CHAMBERS R.,.Proc. Phys. Soc. (London) A65, 458 (1952).

9.

NOTARYS H.A., Yu M.L. & MERCEREAU J.E., Phys. Rev. Lett. 30, 743 (1973).