Electric field dependence of paraconductivity

ELECTRIC

FIELD

DEPENDENCE M. A. KLENlN

OF PARACONDUCTIVITYt

and M. A. JFNSENS

Department

of Physics,

and Laboratory for Research

on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania, USA

Synopsis We have studied the suppression caused by an applied electric field of the fluctuations (which are responsible for the paraconductivity seen above the superconducting transition) in aluminum films. Of particular interest is a comparison between data taken in the temperature range where the zero-bias conductivity is approximately that predicted by Azlamazov and Larkin and data taken in the temperature range (further from T,) where significant deviations from their expression are observed.

The large number of publications which have dealt with fluctuation phenomena in superconductors just above the transition temperature have led to a somewhat bewildering array of datalm6), much of which agree remarkably well with theory and some of which 5*6)do not. A systematic discussion of deviations from the, by now, well-known results of Azlamazov and Larkin7) (A-L) has been carried out by Masker and Parks?). These authors observe, in aluminum films, transition widths which vary with the effective mean free path of conduction electrons and which, in the cleanest films studied, are typically an order of magnitude larger than the A-L value. Most of the experiments to date have concentrated on studying the zero-current behavior and have neglected non-linear effects of an applied electric field. We have measured current-voltage characteristics of thin, moderately clean aluminum films in the vicinity of T, and have found measurable deviations from linearity at several tenths of a degree from T,, even for small electric fields. The films measured in this investigation were typically of the order of 100 A thick and were prepared in several different ways. In some cases the films were deposited on a substrate previously coated with SiO, in others, they were deposited directly on glass. Most films were prepared simply by masking the slides. Several later specimens consisted of areas whose edges had been t Research supported by the United States Air Force Office of Scientific Research under Grant AFOSR-1149-66 and by the Advanced Research Projects Agency. $ Alfred P. Sloan Foundation Postdoctoral Fellow 1968-l 970. 279

280

M. A. KLENIN

AND M. A. JENSEN

removed with a sharpened tungsten needle. In terms of E = (T- TJT, temperature ranged from E = 0.0 to E = 0.5 (or from 2.0 K to 3.0 K). Electron microscope studies indicated that grain size was between 50 and 100 8, and that averages of grain size, taken over selected areas whose linear dimension was of the order of the coherence length, were constant over the entire sample. (This was in most cases an area 0.16 cm x 0.01 cm.) Selected area electron diffraction studies corroborated grain size estimates. The rings obtained were sharp, indicating lattice parameters from the given area varied less than l/2% and variation from region to region was well within the accuracy of the technique (about 2%). A d.c. method was used to measure voltage across the film. Because of the method of nulling out the normal-state resistance, maximum accuracy was obtained for small (less than one percent) deviations from that value. Conductivities were taken from the slopes of the resulting Z-V plots, giving a sensitivity in the region of interest, of about one part in 105. Measurements were made with the sample inside a mu-metal shield, so that magnetic fields were reduced to less than two milligauss. Fig. 1 shows a plot of the inverse of the deviation from normal-state conductivity 6~ (normalized to normal-state conductivity crO)uersus temperature for several different films, all in zero applied electric field. The A-L prediction is that such a plot should yield a straight line whose slope is inversely proportional to &, the resistance per square area of the sample. Far from the transition our results are more or less in agreement with those of Masker and Parks? who find transition widths (inverse of the slope) are about five to ten times the I

I

I

I

I

I

I

I

I

1

500 400 -

300

0

Ro=16.6

-

200-

IOO-

2.00

2.20

2.40

2.60

2.90

T (K)-P

Fig. 1. Inverse fluctuation conductivity ha, normalized to the normal-state conductivity (T,,versus temperature for several samples with value of R as indicated. T~/R~ determined from the slope of the curve, is predicted by A-L to have the value 1.52 x 10-5W1. The lines are arbitrarily drawn through the data points.

ELECTRIC

FIELD

DEPENDENCE

OF

PARACONDUCTIVITY

281

A-L value. Near T,, however, the curve becomes considerably sharper so that for 0.0 1 ~5 ha/u,, L 0.1 the slope approaches A-L. We have drawn straight lines through what are apparently the two segments of each curve. Recent work by ThompsotP) has yielded expressions for thin-film superconductors in which one has terms in the conductivity in addition to the A-L term. In general, these anomalous terms increase the transition width by roughly an order of magnitude far from T,,but are dominated by the A-L term close to the transition and the curve approaches T, with a slope more nearly that of A-L. It is found that as T, is shifted by a pair-breaking field (intrinsic or external), the curve scales in such a way that the A-L expression becomes approximately correct over a larger portion of the temperature range. In fig. 2, we show the effect of applying a small electric field to one of the samples of fig. 1, the two curves are drawn for E = 0 and E -L1.4 mV/cm.

T (K) Fig. 2. Effect

of a small electric

ness = 150 A). The sample

field applied

edges were removed

-

to one of the samples by scribing.

of fig. 1 ( Rn = 7.3 R, thick-

The solid lines are obtained

from eq.

(I), with the value of T,, indicated.

(This corresponds to a current density of roughly 150 A/cm2.) We have attempted a quantitative comparison between our data points and.the expression derived by Thompson for the case of a parallel magnetic field.

(1) r0 is the A-L transition width and 6 is (T,,,T,), where Te,, is the transition temperature in the absence of any pairbreaking mechanism. We have used this equation, rather than the one derived for perpendicular magnetic field case, for two reasons: (i) it apparently fits reasonably well;

282

M. A. KLENIN AND

M. A.JENSEN

near T, it tends to the A-L straight line whereas the equation for a film in a perpendicular field tends to a line with a slope about one third that of the A-L value; (ii) the theoretical difference between the parallel and perpendicular field cases arises because of the discrete excitation spectrum of the system in a perpendicular field. Thus one expects the zero E field curve to be the limit of the parallel magnetic field calculation with its use of a continuous spectrum. Obviously, there is one free parameter for fitting the given function. The E = 0 curve is drawn setting S = 0.13; i.e., the shift in T,due to internal pairbreaking interactions is about 0.26 K. For E = 1.4mV/cm we observe an additional shift of about 80 millidegrees and in fact, the curve drawn for 6 = 0.17 fits quite well. The data for E = 0 show deviations from the theoretical curve at z = 0.01 which we do not believe to be due to experimental error. This can be seen as a sharp change of slope in fig. 1. We do not understand the reason for this anomaly; one possibility is the limitation on the temperature range over which the expression used is in fact valid. We emphasize that we have no a priori reason to believe that the effect of an electric field on the conductivity ought to be identical to the predictions for the magnetic field case. It does, however, seem relevant that the theory, which deals with the suppression of the anomalous terms, predicts an increasingly steep curve as the pair-breaker is applied, with the most obvious effects occurring relatively far from T,.In any case, it would be reasonable to suppose that given a pair-breaking parameter 6, the form given should yield the correct temperature dependence, and electric fields do cause pair-breaking. We have found it convenient to consider separately 1-v characteristics in the two temperature ranges defined by the two segments of the E = 0 curve shown in fig. 1. If it is true that in the region near T,,the fluctuation conductivity is predominantly the A-L conductivity, then one would expect the nonlinearities to be nearly of the form predicted by considering only the A-L term, or equivalently, that predicted by a time-dependent Ginzburg-Landau formalism, as is the result of a calculation by Huraultg). In essence, this theory is a description of the damping of fluctuation effects as the lifetime of a fluctuation becomes limited by the electric field, rather than by the temperature-dependent free energy terms. Physically, this corresponds to the effect of accelerating the superfluid, within the lifetime of a fluctuation, to a critical velocity at which the electron pairs become unstable; the situation is analogous to critical-current effects below T,.Unlike the conductivity in the linear region of the I-V curve, the conductivity in the non-linear region does depend upon the sample parameters to, the bulk zero-temperature correlation length 1 the mean free path, and T,.These in turn are used to determine y, the coefficient which defines the relationship between the time scale and the .free energy terms which appear in the time-dependent Ginzburg-Landau equation. Given these parameters one may define a critical field E,(T)above

ELECTRIC

FIELD

DEPENDENCE

OF

283

PARACONDUCTIVITY

which one has non-linear behavior due to the depairing effects of the electric field. From Hurault we have

(2)

esti-

For to = 16,000 A, I= 150 A, T, = 2 K .one has the order-of-magnitude mate for our samples -C(T)

= 19e3j2(V/cm).

(3)

In order to avoid heating effects we were restricted to current densities well under 103- 1O4A/cm2 and so were limited to relatively small fields, of the order of millivolts per centimeter. Thus, we expected to see linear f-I/ characteristics for E > 10m2. For smaller E and for E > E,(t) the expression for the conductivity takes the (temperature-independent) form:

(4) -

(E in mV/cm)

0.035E -2i3 0.1.

I

I

0.06 0

I

I

E N 2x IO+

0 0

l

t

0.03 -

%

E

Fig. 3. Fluctuation

.0236

conductivity

vs. electric

T, - E s 0.02 where the E = 0 fluctuation through

(mV/cm)

these points in fig.

4

-

field at several conductivity

E- “’

temperatures

is nearly

A-L.

in the range (The straight

1 yields ro/Rn = 1.y X. l,0-5fl-1).

(‘I-

T,)/

line drawn

284

M. A. KLENIN

AND M. A. JENSEN

In fig. 3 we have a log-log plot of the fluctuation conductivity versus E field at several temperatures in the range where Hurault’s results predict observable suppression of the fluctuation conductivity with electric field. The values of E are as indicated on the graph. Considering just the curve taken at the temperature closest to T,(E = 0.002, T - T, = 0.004 K) we find that for large E the conductivity does vary as E-2’3 as expected. The straight line drawn through the points is the function 0.0236 E- 2’3,from which we obtain E, (E = 0.002) = 0.94 mV/cm. On the basis of our sample parameters we expected the value of the critical field to be about 1.7 mV/cm. The effect of increasing temperature is that EC should become larger and deviations from constant (+ within our field range should become smaller. At E = 4 x 10P3, EC = 2.63 mV/cm, beyond our range of measurement, and here as in the curves taken at progressively higher temperatures, the conductivity falls off more slowly as a function of field. Farther from T,, however the Z-k’ characteristics no longer conform to the pattern expected solely on the basis of the Ginzburg-Landau equation. The expected pattern, from the expressions given above, is simply that, as the zerobias fluctuation conductivity decreases (at higher T), the effects of an E field become less noticeable and ought to be virtually unobservable for values of E of the order of 0.1. In fig. 4, we see that this, in fact, is not the case. The coordinates are the same as those in fig. 3. At E - 4 X 1O-2EC should be at 144 mV/cm, but apparently the fluctuation effects are suppressed strongly by the E field, even at one millivolt per centimeter. All of the temperatures at which these curves were taken correspond to points in fig. 2 for which the E = 0

‘o-zI

6~16~

t 3x10-3

b

,.-.I .

E-&3x10-2

XE"

0.0678

9.3x lo-2

E

Fig. 4. Fluctuation conductivity

0.678

0.408

0.204

(mV/cm)

2.04

+

for E 2 0.02, where the E = 0 fluctuation conductivity is significantly larger than the A-L value.

ELECTRIC FIELD DEPENDENCE

OF PARACONDUCTIVITY

285

conductivity is significantly larger than that predicted by A-L, and one is led to suspect that only the anomalous part of the fluctuation conductivity is being suppressed. Several alternative (and very distasteful) explanations of the data present themselves. First, there arises the possibility of sample heating with increased current. This seems an unlikely explanation for several reasons. Since the behavior near T, conforms quite well to theoretical predictions, it seems reasonable to believe that, in this region at least, we are observing true current (or E field) effects. It is doubtful that the thermal characteristics of the material undergo an abrupt change twenty millidegrees above the resistive transition, suddenly making heating effects important. We have observed what we believe to be heating effects in similar films above the transition at much higher current. The current-voltage plots in these high-current cases were characterized by very noticeable, and not very reproducible hysteresis. Power inputs in these cases were about two orders of ‘magnitude larger than the power inputs here, and the data obtained from the samples shown here were independent of whether the current was increasing or decreasing. Another disagreeable possibility is that we are really observing the superconducting transition of the filamentary edges of the films (of width perhaps equal to the correlation length) despite the fact that we have removed the edges of the evaporated area (for the sample of figs. 2, 3 and 4). It is true that one might expect T, at such edges to be somewhat higher than in the bulk of the sample, inasmuch as the edges ought to be thinner and more disordered; both of these conditions tend to raise the T, of aluminumlO). One might therefore try to explain the data by claiming that current densities through these filaments are greatly enhanced compared to the rest of the film. Evidence that the-data are not the result of edge effects comes from some measurements made of the effects of applied magnetic fields. Fig. 5 shows some data taken in collaboration with Crow, et al. at Brookhaven National Laboratorys,ll). The sample studied in this case was somewhat cleaner (I?, per square = 4 a) and thicker (about 250 A) and its transition temperature was lower. The two curves were taken in zero magnetic field and in a parallel field of 500 gauss, respectively. In this case, as in the case of the electric field, the slope of the plot is increased considerably as the fluctuation conductivity is suppressed. Where the zerofield slope is smallest (i.e., where the additional fluctuations are predominant) the effect of the parallel field is greatest. Since the edges, if they have a higher T,, ought also to have a higher critical field, one would expect their contribution to the change in conductivity to be less strongly affected by the field, than is the contribution of the bulk sample. We thus find it difficult to attribute the excess width of the transition to edge effects (or to other inhomogeneities in the sample). It is also worth noting that the behavior in parallel and perpendicular magnetic fields is qualitatively similar, but perpendicular fields of several gauss cause significant changes in the shape of the transition compared with

286

M. A. KLENIN

600~

AND

M. A. JENSEN

.

I

I

I

.. .

. .

.

500-

. .

8. 400-

.

.

.

. . 300-

.

.

.

.

.

lm

. 200-

.

IOO-

0

I I.60

1.50

1.40

1.70

_T (K) Fig. 5. Effect shows

of a parallel

the inverse

magnetic

field applied

of the fluctuation

to an A film for which

conductivity

VS. temperature.

R. = 40.

The curve

Data are from ref. 1 I.

the hundreds of gauss required in the case of a parallel field. This too, we take to be an indication that we are dealing with a true two-dimensional system, rather than one-dimensional edges anda two-dimensional film with lower T,. In conclusion, we have observed the effects of an electric field on the fluctuation conductivity of a two-dimensional system, and have found that close to T, we are in reasonable agreement with a theory which bases its analysis on the simple time-dependent forms of the Ginzburg-Landau equation. Far from T,, however, our results seem more compatible with a theory which predicts an additional contribution to the conductivity, which is rapidly suppressed by electric field effects,, Additional work is in progress to extend our data to higher values of electric field.. Acknowledgements. We wish to’express our thanks to M.,Strongin, R. S. Thompson and J. E. Crow for many stimulating conversations on their work as’ well for providing us with their results prior to publication and to A. J. Heeger for some very helpful discussions.

REFERENCES 1) Glover,

R. E.; Phys.

28A(1968) 2) Strongin, Letters

Letters

25A (1967)

M., Kammerer, 20

5) Masker,

D. C. and Glover,

R. E., Phys.

0.

F., Crow,

J., Thompson,

R. S. and Fine,

Letters

H. L., Phys.

(1968)922.

3) Smith, R. O., Serin, B. and Abrahams, 4) Gittleman,

542; Naugle,

110.

J. J., Cohen, W. E. and Parks,

E., Phys. Letters

R. W. and Hanak, R. D., preprint.

28A

(1968)224.

J. I-t., Phys. Lettetx29A

ft969)

5t.

Rev.

ELECTRIC

FIELD

DEPENDENCE

OF PARACONDUCTIVITY

287

6) We presented preliminary data of the March, 1969 meeting of the American Physical Society [Klenin, M. A. and Jensen, M. A., A.P.S. Bulletin 14 (1969) 4371 and in a preprint noted that deviations from the predictions of ref. 7 occurred for ( T - T,)/T, 2 0.1. We attributed these deviations to thermal fluctuations and presented a calculation which we believed to support this point of view. Subsequently, a numerical error was pointed out by a referee in our estimate invalidating our argument for the case of aluminum. The situation has been confused somewhat further by the observation of similar deviations from theory in Pb films, as reported by Serin et al. [this conference]. Furthermore, the fit to our exponential function in their case yields a parameter which seems to be in agreement with estimates of the parameter obtained -from independent experiments. 7) Azlamazov, L. G. and Larkin, A. I., Phys. Letters 26A (1968) 238. Azlamazov, L. Cl. and Larkin, A. I., Soviet Physics-Solid State (Eng. transl.) 10 (1968) 875. 8) Thompson, R. S., this conference. 9) Hurault, J. P., Phys. Rev. 179 (1969) 494. 10) Strongin, M., Kammerer, 0. F. and Paskin, A., Phys. Rev. Letters 14 (1965) 949. 11) Crowe, J. E., Thomson, R. S., Klenin, M. A. and Bhatnager, A. K., Phys. Rev. Letters 24 (1970)371.