MEASUREMENT OF POTENTIAL DIFFERENCES IN NONEQUILIBRIUM SUPERCONDUCTORS

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MEASUREMENT OF POTENTIAL DIFFERENCES IN NONEQUILIBRIUM SUPERCONDUCTORS

W. Skocpol, A. Kadin, M. Tinkham

To cite this version:

W. Skocpol, A. Kadin, M. Tinkham. MEASUREMENT OF POTENTIAL DIFFERENCES IN

NONEQUILIBRIUM SUPERCONDUCTORS. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-

1421-C6-1425. �10.1051/jphyscol:19786582�. �jpa-00218074�

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JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8, Tome 39, a062 1978, page (-6-1421

MEASUREMENT O F POTENTIAL DIFFERENCES IN NONEQUILIBRIUM SUPERCONDUCTORS

W.J. Skocpol, A.M. Kadin and M. Tinkhanr

Department of Physics, Harvard University, Cambridge, Mass. 02138, U.S.A.

Rdsum6.- Nous rdsumons briPvement les mesures antdrieures des diffdrences de potentiel dans les su- praconducteurs hors d'dquilibre et nous d6crivons comment la rdsistance diff6rentielle des centres de phase glissants peut Ztre utilisde pour en ddduire le temps de relaxation en mode transverse

(d6sdquilibre des branches). Ensuite nous reportons nos mesures rdcentes de la d6pendance en champ magngtique de ce temps de relaxation, qui est en bon accord avec la prddiction de Schmid et Schgn.

Abstract.- We briefly review previous measurements of potential differences in nonequilibrium superconductors and describe how the differential resistance of current-induced phase-slip centers in superconducting filaments can be used to infer the transversemode (branch-imbalance) relaxation time. We then report our recent measurements of the magnetic field dependence of this relaxation time, which is found to be in good agreement with the prediction by Schmid and Schgn.

1. INTRODUCTION.- A superconductor may be descri- bed as consisting of three interacting systems : the superconducting condensate, the quasiparticle excitations, and the phonons. The quasiparticles can carry only the usual dissipative current dri- ven by a spatial gradient of the electrochemical potential, but the condensate can carry a dissipa- tionless supercurrent even without such a gradient.

The strength of the supercurrent is affected by the number and energy distribution of the quasiparticle excitations. These excitations have the usual electronlike or holelike character well outside or inside the Fermi surface, but have a mixed character near the Fermi surface where their energy is altered by the superconducting energy gap. In thermal equilibrium, brought about by inelastic collisions between the quasiparticles and the phonons, the energy distribution of quasiparticles is given equally on both branches of the excitation spectrum by the usual Fermi function, so that there is no net charge in the quasiparticles. An external perturbation which drives the system away from equilibrium may alter the distributions of both the quasiparticles and phonons.

As far as the quasiparticles are concerned, there are two general types of perturbation, as discussed below.

An "uncharged" perturbation such as the ab- sorption of light may alter the number and energy distribution of the quasiparticles, producing similar effects on the distributions of both electronlike and holelike excitations. Like a change of temperature, this will alter the magnitude of

A, the energy gap in the excitation spectrum, which is proportional to the order parameter in the

system, and relaxation will occur by electron-phonon processes involving scattering and recombina- tion of the quasiparticles. The study of such situations, referred to as having a "longitudinal"

character, is an area of active research outside the scope of this paper.

In contrast, a "charged" perturbation, such as the tunnelling injection of quasiparticles or the interconversion of normal currents and super- currents, may in addition involve the preferential buildup of quasiparticles on either the electronlike or holelike branch of the excitation spectrum.

Like a change of electrochemical potential this will alter the time dependence of the phase of the order parameter, so that these situations are des- cribed as having a "transverse" character

.

^Rela-

xation of this "branch imbalance" can occur by a subset of the electron-phonon processes, dominant- ly those which involve the scattering of quasiparticles of mixed electron-hole character near the gap edge. In order to maintain electrical neutrali- ty, it is necessary that a region with a nonzero branch imbalance have a difference of electrochemical potential between the condensate and the quasiparticles. In recent years it has been demonstrated that this potential difference can be observed using normal and superconducting probes in such nonequilibrium region. This phenomenon leads to the appearence of resistance on the superconducting side of a current-carrying normal/superconducting (N/S) boundary and to the current-induced

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786582

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C6- 1422 JOURNAL DE PHYSIQUE

appearance of resistance in superconducting filaments.

In this paper.we briefly review prior work jn this area, and then report new measurements of resistance in current-carrying superconducting filaments which explore for the first time the effect of a magnetic field on the branch-imbalance relaxation time.

2. REVIEW OF PRIOR WORK.- The conceptually most direct experiment in this area /1,2/, involves a dou- ble-tunnel-junction sandwich in which a quasiparticle current is injected into the middle (superconducting) film of the sandwich through one junction, and the electrochemical potential of the quasiparticles in the middle film is probed by the normal- metal electrode of the other junction. A potential difference can be observed between the superconducting lead to the middle film and the normal-metal electrode. The injection is spatially uniform within the nonequilibrium region, and the steady-state potential difference is proportional to the injection current and the branch-imbalance relaxation time 131. The measured relaxation time is consistent with the expression

T = (4k,T/aA)~

q ⁽¹⁾

where -rE is the equilibrium-averaged electron-phonon (inelastic) scattering time of quasiparticles near the Fermi surface. The factor of kgT/A arises because branch-imbalance relaxation occurs primari- ly via stages between

A

and -2A. This conceptually straightforward situation has been analyzed from a number of theoretical perspectives / 4 , 6 / , with generally consistent results.

A related set of experiments have investiga- ted the quasiparticle/condensate potential difference observed in the vicinity of a current-carrying N/S boundary. Observations of the resistance of the intermediate state / 7 / and of S/N/S sandwiches 181, demonstrated that a measurable excess resistance could be associated with the superconducting side of a current-carrying

SIN

boundary. Observations /9/ with superconducting and normal probes placed within a few micrometers on the superconducting side of such a boundary in a thin-film strip confir- med the existence of a potential difference between the quasiparticles and the condensate. If the superconducting probe was placed closer to the boundary than the normal probe was, the observed potential drop changed sign depending on whether the su-

perconducting side was above or below its T

.

^As

shown in figure 1, this result is consistent with a spatially uniform electrochemical potential of the condensate (pc, measured by the superconducting probe) and an exponential relaxation of the quasi-

Fig. 1 : Schematic representation of measurements by Yu and Mercereau /9,10/ of electrochemical potentials near a superconducting/nod boundary.

particle electrochemical potential ( p measured q'

by the normal probe) to the condensate value with a characteristic length scale much longer than the superconducting coherencelength. Subsequent theoretical analyses /5,10-131 have concluded that this exponential spatial variation of the quasiparticle/

condensate potential difference should be governed by the quasiparticle diffusion length A =(DT ) "

Q Q

where D = 113 v R is the diffusion constant, al- F

though the situation is complicated somewhat by the spatial dependence of A on the scale of the coherence length near the SIN interface. The additio- nal resistance contribution from the superconducting side of the SIN boundary is thus essentially the normal resistance of a length A of the mate-

Q rial on that side.

A similar situation prevails when the critical current of a one-dimensional superconducting filament (long and thin compared to the coherence length) is exceeded. A sufficiently large current through such a sample will drive it totally normal, but the transition between the fully superconducting and the fully normal states takes place over a substantial range of current. In zero magnetic field near Tc, this transition is characterized by the appearance of a series of voltage steps, each followed by a plateau of approximately cons-

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tant differential resistance 114,151. An example is shown in figure 2 .

Fig. 2 : Current-voltage characteristics for tin bridge # 25 A near Tc(H). A) H=O, T=3.839 K ; B) H=480 Oe, T=3.315 K. In each case, the differen-

tialresistance in the plateau region of the se- cond step is twice that of the first, since each increment corresponds to a similar phase-slip center (PSC)

.

Skocpol, Beasley and Tinkham /15/ have esta- blished that this voltage profile is due to the successive onset of a series of discrete, spatially localized dissipative units, called phase-slip centers (PSC). Each PSC first appears at its own local value of the critical current, as determined by the local critical temperature and cross-sectio- nal area, and by the presence of other PSC's near- by. Although these critical currents may vary con- siderably, the increment of differential resistance associated with each PSC is quite similar to that of each of the others. This increment is approximately equal to the normal resistance R of

n a length 211 of the sample where A is the appropri- ate quasiparticle diffusion length. This descrip- tion has been supported by experiments on tin 115,

161, indium /16/, and aluminum /17/ microbridges, and tin-alloy whisker crystals /18/.

In the specific model proposed by Skocpol

g

al. to explain these features, the electrochemical

-

potentials

u

of the quasiparticles and of the 9

condensate, which must become equal far from the center of the dissipative region, show fundamental- ly different behavior in the nonequilibrium region near a phase-slip center. A voltage drop across the PSC must appear as an equal change in both of these potentials,but the change occurs over two different length scales (figure 3). A gradient in

u

produces

S S S S S S S S

Fig. 3 : Schematic representation of measurements by Dolan and Jackel /16/ of electrochemical potentials near a phase-slip center.

an acceleration of the supercurrent J and leads to a relaxation oscillation (or phase-slip cycle) of the order parameter within a region of the size of the coherence length <(T). The time-averaged condensate potential

Tc

changes abruptly within this region and is constant outside. The phase-slip process causes the time-averaged supercurrent

7

at the center to be depressed below the value far away, so that if the total current is to remain constant the dissipative normal current

3

=- o-

n

ev$

must make up the difference (o is the normal me- tallic conductivity). The interconversion between this normal current and the supercurrent must occur over a scale of the diffusion length A, since inelastic processes are necessary. The change in - p therefore takes place over a range 2A, so that

4

the differential resistance dV/dI contributed by the PSC should be equal to the normal resistance Rn of the length 2A as stated above.

Recent experiments by Dolan and Jackel /16/

have provided direct and striking evidence for the above model. Samples were fabricated which included arrays of voltage probes closely spaced along

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JOURNAL D E _PHYSIQUE

the length of the thin-film microbridge (figure 3).

Both superconducting and normal probes were used, so that it was possible to make independent measurements of

11

and

c

as a function of position.

q

As expected, the value of

y

jumped abruptly at the center of the dissipative region, while

y

^{was found}

4

to relax exponentially back to the equilibrium value on each side of the center with a decay length which fit the form

A

==with T = lxl~-~~s.x(l

-

Q Q Q

T/T~)-' for their tin sample. Furthermore, when one of the normal probes was used to inject quasiparticles into the bridge, measurements of the re- sulting spatial variation of in the vicinity of

4

the injection yielded exactly the same decay length.

In our minds, this settles the question whether T~ or T~ is the relevant relaxation time.

Within the limited temperature range over which changes in Rn accurately reflected changes in A Dolan and Jackel found that the differential

Q'

resistance R of the PSC did indeed correspond (within about 20 %) to the normal resistance expected from a section of the bridge of length 2A Thus

Q

-

observation of the differential resistance of phase-slip centers can be used to measure the nonequilibrium relaxation time.

Previous measurements on phase-slip centers were made in zero magnetic field. The research discussed in the following section was undertaken to determine whether PSC's also appear when a magnetic field H parallel to the plane of the thin-film microbridge depresses Tc, and what effect the magnetic field, as an example of a pair-breaking perturbation, has on the nonequilibrium processes.

3. MAGNETIC FIELD DEPENDENCE OF THE TRANSVERSE RE- LAXATION TIME.- We find 1191 that the I-V characteristics of tin microstrips in a parallel magnetic field, for T near Tc(H), are qualitatively the same as those for H=O near T (figure 2). Quantitati- vely, the major effect of increasing the magnetic field is the decrease of R together with the cor-

n'

responding increase in the number of PSC's which can "Fit" along the length of the bridge. Typical- ly, the length decreases from about 5

urn

for H=O to about 0.5 um for large fields, and the corres- ponding time decreases from about 10-'s. almost to

10-'Is. The temperature and field dependence of R n is shown in figure 4 for a typical sample with parameters as follows : length L=50 um, width

W=2 um, thickness d=500

1,

mean free path R=200

i,

Fig. 4 : Differential resistance Rn of the first PSC in tin bridge # 15 A, as a function of temperature for several values of the parallel magnetic field : 0 , H=O ; A, H=1.2 kOe ;

a ,

^{H=2.0 kOe}^;

V , H=3.2 kOe. The solid curves represent the theoretical predictions of equations (2)-(6) with T~(T,'

= 3x10-'~s. The solid symbols are values corrected for heating-induced distortion of the I-V curves.

parallel critical field at zero temperature H (0) = 4350 Oe, and Tc = 3.905 K. The measured

C I I

values of R are plotted as a function of bath tem- n

perature T for various constant values of parallel magnetic field H. We have also included values for R (solid points) corrected /I91 for distortions of

n

the shape of the experimental I-V curves due to the progressive reduction of the critical current I (T) with increasing dissipation VI.

An expression for the relaxation time in the presence of a pair-breaking perturbation such as a magnetic field has been derived by Schmid and Schgn /5/ for spatially homogeneous transverse mode (branch imbalance) disequilibrium. If T is the charac-

E

teristic relaxation time for electron-phonon relaxation processes near T (H) and near the Fermi surface, and T is the magnetic pairbreaking time

(H/

1.76 k B ~ c ~ E / ~

(oU-',

then the expression gi- c

ll

ven by Schmid and Schon, neglecting a factor which is close to unity in our domain of investigation,

-.

where A = A(T,H) and

r

= E 2 T )-' + T

-3.

For E

H=O this reduces to equation (I), while for large values of the field, it reduces to

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Similar expressions have been derived by Artemen- ko Snd Volkov 1131 and by Shelankov 1201. Taking

the approximate temperature dependences

T~ (T) = T~ (Tc) (Tc/T) (4) A(T,H) = A ( T ) ~

-

H ~ / H ~

C I I

( T ~ J ~ ~ ⁽⁵⁾

and treating T (T ) as a single fitting parameter E c

for the data in figure 4, we obtain the solid curves in figure 4. The good agreement shown is obtained for T~ = 3 x 10-I0s. This is in reasonable accord with other experimental and theoretical es- timates of -cE for tin /2,4,21/. Similar agreement has been found in a variety of samples,

with Hcll(0) ranging from 700 Oe to 4000 Oe. Thus we find substantial agreement with the formula first derived by Schmid and Schsn, although we note that the agreement has been obtained outside the range for which .cR > T ~ , which was the domain in which the formula was initially derived.

ACKNOWLEDGMENTS.- This work was supported in part by the National Science Foundation and the Office of Naval Research.

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