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Temperature dependence of the order-parameter relaxation time of a superconducting indium film
Solid State Communications, Printed in Great Britain.
TEMPERATURE
Vol. 60, No. 2, pp. 147-149,
DEPENDENCE OF THE ORDER-PARAMETER RELAXATION SUPERCONDUCTING INDIUM FILM* H. Weimrod,
Physikalisches
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
Institut
II, Universitlt
TIME OF A
R. Gro/3 and R.P. Huebener
Tiibingen D-7400 Tiibingen
1, Federal Republic of Germany
(Received 20 February 1986, in revised form 14 May 1986 by B. Miihlschlegel)
We have measured the critical d.c. current of a superconducting indium bridge in the presence of a superimposed a.c. current for frequencies up to 400 MHz. The temperature dependent order parameter relaxation time has been determined from these experiments, yielding the value rE = 80~s for the inelastic electron phonon scattering time. RECENTLY Pals et al. [l] reported on measurements of the critical d.c. current of a superconducting aluminum film in the presence of a superimposed a.c. current with different amplitude and frequency. They demonstrated that the order parameter relaxation time rA can directly be obtained from these measurements. Starting from the time-dependent Ginzburg-Landau equation, as proposed by Tinkham [2], the authors calculated the maximum value je of the d.c. current yielding a periodic nonzero solution for the order parameter when an a.c. current jr cos wt is present (total current j = je -jr cos at). Here je and ji are normalized to the critical current for each temperature without superimposed a.c. component. In the limit or,, < 1, and WrA % 1 Pals et al. [l] obtained the following approximations
.Z j0
=
l-;((wr&
a?-‘$ > 1.
(lb)
In the limit c#rA < 1, the relaxation time rL\ can directly be obtained from the slope of the straight line expected at low frequencies. In the present paper we report on similar measurements performed using an iridium microbridge. Since in In the relaxation time rA is about one hundred times shorter than in Al, the measurements require correspondingly higher frequencies. The In microbridge was 30~ long, 0.8~ thick, and 3.2j.n-n wide and was placed between much wider In-film sections for attaching the current and voltage leads. The samples geometry is shown schematically in the inset of Fig. 1. The sample was prepared on a sapphire substrate using standard
* Supported by a grant of the Deutsche meinschaft.
Forschungsge-
147
photolithographic techniques and a single-step evaporation procedure (evaportion rate * 8-l 0 rims-’ ). During the evaporation the substrate was in thermal contact with a cooling trap filled with liquid nitrogen. The sample resistance at 4.2K was 14.2ma (residual resistance ratio = 108). During the measurements the sample was in direct contact with the liquid helium bath. The temperature was obtained from the vapour pressure measured with a MKS Baratron pressure gauge. The a.c. current was generated using a Marconi TF 2015 fm/am signal generator and an ENI 503 L power amplifier. The a.c. current amplitude was measured using a Tektronix 7904 oszilloscope. Keeping the superimposed a.c. current constant, the d.c. current was raised monotonically until a nonzero voltage could be detected with a Keithley 140 Nanovolt amplifier. In this way the maximum value j. was obtained. During our experiments the normalized amplitude ji of the a.c. current was taken in the range ji = 0.18-0.25. Our measurements were extended up to frequencies of 400MHz. Here the impedance mismatch represented some problem. The a.c. current was supplied via 5Ofi lines connected to 5Oa striplines on the sapphire substrate. Since the microbridge was much smaller than the stripline, some reflection occurred due to impedance mismatch. Standing waves were generated on the a.c. lines (depending upon the length of the high-frequency line), resulting in a slight frequency dependence of the a.c. amplitude in the microbridge. We have minimized the influence of this frequency dependence of the a.c. amplitude by performing the measurements with three different lengths of the a.c. lines for each temperature and by evaluation the average of these measurements. This method almost entirely eliminated the influence of the impedance mismatch in the frequency range between 10 and lOOMHz, which was taken into account for a quantitative analysis. In Fig. 1 we show this average of
148
RELAXATION
0.7 -
TIME OF A SUPERCONDUCTING
Vol. 60. No. 2
INDIUM FILM
Sample W-In-IX Temperature U9L K I&O 0.33mA
0.6 -
OSL
50
100
1M
200
250
Fig. 1. Normalized
critical d.c. current vs the frequency a.c. current. The data points represent average values obtained from three measurements with different lengths of the highfrequency line. T = 3.394K. The inset shows the sample geometry.
f = w/2n of the superimposed
the normalized critical current plotted vs the frequency of the a.c. current. The data agree well with the behavior predicted in equation (1). From the data such as shown in Fig. 1 we have calculated the relaxation time rA using the slope of the linear branch at low frequencies and the approximation of equation (la). Evaluating the temperature dependence of 7~ requires the accurate measurement of the critical temperature T,. The value of T, was obtained by measuring the critical current of the sample (for zero a.c. current) at several temperatures near T,. The results followed quite well the (1 - T/Tc)3’2 behavior expected from the Ginzburg-Landau theory. By extrapolating the Z,(T) curve to zero current, the value T, = 3.405 K was found for the sample. In Fig. 2 the relaxation time rA is plotted vs (1 TIT,)- “’ . We see that the relation TA
=
1.2*~~.(1
-T/Tc)-1’2,
0
300 f IMHzl
(2)
expected theoretically [3], is well satisfied by our data. Here rE is the inelastic electron-phonon scattering time. We note that the data in Fig. 2 cover the range down to 90mK below the critical temperature T,. Assuming a reasonable value of the heat transfer coefficient between the In film and the sapphire substrate, the Joule heating effects of the applied high-frequency current were found to be negligible. From the straight line fitted to the data points in Fig. 2 we deduce the value rn = 80~s. We note that the results indicated in equation (1) are obtained under the assumption jr /jO Q 1. In our experiments disscussed so far the ratio jr/i0 was always kept smaller than about 0.3. However, we have found that the value of the time rA determined using equation
0
I
10
20
30 ^_ (l-T/ki-w
Fig. 2. Order parameter relaxation time vs (1 - T/Tc)-“2. (1) remained practically unchanged if jr /ie was taken as large as about 1.O. Our experimental value of TV for indium indicated above is slightly smaller than the results reported previously. From resistance measurements on SNS junctions Hsiang and Clarke [4] obtained r3 = 1 lops. Frank et al. [S] studied .the transient response of indium microbridges to supercritical current pulses and reported rE = 140~s. Klein et al. [6] obtained the value rE = loops from the nonlinearity in the flux-flow behavior of indium films. Theoretical calculations by Kaplan et al. [7] yielded the value rE = 100 ps. We note that in our calculation of the inelastic scattering time rE based on equation (2) we have neglected any gap enhancement effect due to the highfrequency current [2,8]. Such a gap enhancement represents an effective cooling of the quasiparticles and thereby causes a reduction of the value of T in equation (2) which has to be modified accordingly. Hence, the value of the inelastic scattering time rn obtained from the modified expression for rA will tend to be larger than our value calculated above. Whether this effect can explain the fact that our value of the time r3 is slightly smaller than the previous results, can only be determined by further systematic experiments. REFERENCES 1. 2.
3. 4. 5.
J.A. Pals, J.A. Geurst & J.J. Ramekers, P&s. Rev. B23,6184 (1981). M. Tinkham, in Nonequilibrium Superconductivity, Phonons and Kapitza Boundaries, p. 23 1, (Edited by K.E. Gray), NATO Advanced Study Series B, Vol. 65 Plenum, New York, (1981). A. Schmid 8c G. Schon, J. Low Temp. Phys. 20, 207 (1975). T.Y. Hsiang & J. Clarke, Phys. Rev. B21, 945 (1980). D.J. Frank, M. Tit&ham, A. Davidson & S.M. Faris, Phvs. Rev. Lett. 50. 1611 (1983).
Vol. 60, No. 2 6. 7.
RELAXATION
TIME OF A SUPERCONDUCTING
W. Klein, R.P. Huebener, S. Gauss & J. Parisi, J. Low Temp Phys. 61,413 (1985). S.B. Kaplan, C.C. Chi, D.N. Imgenberg, J.H. Chang, S. Jafarey & D.J. Scalapino, Phys. Rev.
8.
INDIUM FILM
149
B14,4854 (1976). R.E. Horstman, J. Wolter & M.C.H.M. Wouters, Solid State Commun. 42,133 (1982).
TEMPERATURE
Vol. 60, No. 2, pp. 147-149,
DEPENDENCE OF THE ORDER-PARAMETER RELAXATION SUPERCONDUCTING INDIUM FILM* H. Weimrod,
Physikalisches
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
Institut
II, Universitlt
TIME OF A
R. Gro/3 and R.P. Huebener
Tiibingen D-7400 Tiibingen
1, Federal Republic of Germany
(Received 20 February 1986, in revised form 14 May 1986 by B. Miihlschlegel)
We have measured the critical d.c. current of a superconducting indium bridge in the presence of a superimposed a.c. current for frequencies up to 400 MHz. The temperature dependent order parameter relaxation time has been determined from these experiments, yielding the value rE = 80~s for the inelastic electron phonon scattering time. RECENTLY Pals et al. [l] reported on measurements of the critical d.c. current of a superconducting aluminum film in the presence of a superimposed a.c. current with different amplitude and frequency. They demonstrated that the order parameter relaxation time rA can directly be obtained from these measurements. Starting from the time-dependent Ginzburg-Landau equation, as proposed by Tinkham [2], the authors calculated the maximum value je of the d.c. current yielding a periodic nonzero solution for the order parameter when an a.c. current jr cos wt is present (total current j = je -jr cos at). Here je and ji are normalized to the critical current for each temperature without superimposed a.c. component. In the limit or,, < 1, and WrA % 1 Pals et al. [l] obtained the following approximations
.Z j0
=
l-;((wr&
a?-‘$ > 1.
(lb)
In the limit c#rA < 1, the relaxation time rL\ can directly be obtained from the slope of the straight line expected at low frequencies. In the present paper we report on similar measurements performed using an iridium microbridge. Since in In the relaxation time rA is about one hundred times shorter than in Al, the measurements require correspondingly higher frequencies. The In microbridge was 30~ long, 0.8~ thick, and 3.2j.n-n wide and was placed between much wider In-film sections for attaching the current and voltage leads. The samples geometry is shown schematically in the inset of Fig. 1. The sample was prepared on a sapphire substrate using standard
* Supported by a grant of the Deutsche meinschaft.
Forschungsge-
147
photolithographic techniques and a single-step evaporation procedure (evaportion rate * 8-l 0 rims-’ ). During the evaporation the substrate was in thermal contact with a cooling trap filled with liquid nitrogen. The sample resistance at 4.2K was 14.2ma (residual resistance ratio = 108). During the measurements the sample was in direct contact with the liquid helium bath. The temperature was obtained from the vapour pressure measured with a MKS Baratron pressure gauge. The a.c. current was generated using a Marconi TF 2015 fm/am signal generator and an ENI 503 L power amplifier. The a.c. current amplitude was measured using a Tektronix 7904 oszilloscope. Keeping the superimposed a.c. current constant, the d.c. current was raised monotonically until a nonzero voltage could be detected with a Keithley 140 Nanovolt amplifier. In this way the maximum value j. was obtained. During our experiments the normalized amplitude ji of the a.c. current was taken in the range ji = 0.18-0.25. Our measurements were extended up to frequencies of 400MHz. Here the impedance mismatch represented some problem. The a.c. current was supplied via 5Ofi lines connected to 5Oa striplines on the sapphire substrate. Since the microbridge was much smaller than the stripline, some reflection occurred due to impedance mismatch. Standing waves were generated on the a.c. lines (depending upon the length of the high-frequency line), resulting in a slight frequency dependence of the a.c. amplitude in the microbridge. We have minimized the influence of this frequency dependence of the a.c. amplitude by performing the measurements with three different lengths of the a.c. lines for each temperature and by evaluation the average of these measurements. This method almost entirely eliminated the influence of the impedance mismatch in the frequency range between 10 and lOOMHz, which was taken into account for a quantitative analysis. In Fig. 1 we show this average of
148
RELAXATION
0.7 -
TIME OF A SUPERCONDUCTING
Vol. 60. No. 2
INDIUM FILM
Sample W-In-IX Temperature U9L K I&O 0.33mA
0.6 -
OSL
50
100
1M
200
250
Fig. 1. Normalized
critical d.c. current vs the frequency a.c. current. The data points represent average values obtained from three measurements with different lengths of the highfrequency line. T = 3.394K. The inset shows the sample geometry.
f = w/2n of the superimposed
the normalized critical current plotted vs the frequency of the a.c. current. The data agree well with the behavior predicted in equation (1). From the data such as shown in Fig. 1 we have calculated the relaxation time rA using the slope of the linear branch at low frequencies and the approximation of equation (la). Evaluating the temperature dependence of 7~ requires the accurate measurement of the critical temperature T,. The value of T, was obtained by measuring the critical current of the sample (for zero a.c. current) at several temperatures near T,. The results followed quite well the (1 - T/Tc)3’2 behavior expected from the Ginzburg-Landau theory. By extrapolating the Z,(T) curve to zero current, the value T, = 3.405 K was found for the sample. In Fig. 2 the relaxation time rA is plotted vs (1 TIT,)- “’ . We see that the relation TA
=
1.2*~~.(1
-T/Tc)-1’2,
0
300 f IMHzl
(2)
expected theoretically [3], is well satisfied by our data. Here rE is the inelastic electron-phonon scattering time. We note that the data in Fig. 2 cover the range down to 90mK below the critical temperature T,. Assuming a reasonable value of the heat transfer coefficient between the In film and the sapphire substrate, the Joule heating effects of the applied high-frequency current were found to be negligible. From the straight line fitted to the data points in Fig. 2 we deduce the value rn = 80~s. We note that the results indicated in equation (1) are obtained under the assumption jr /jO Q 1. In our experiments disscussed so far the ratio jr/i0 was always kept smaller than about 0.3. However, we have found that the value of the time rA determined using equation
0
I
10
20
30 ^_ (l-T/ki-w
Fig. 2. Order parameter relaxation time vs (1 - T/Tc)-“2. (1) remained practically unchanged if jr /ie was taken as large as about 1.O. Our experimental value of TV for indium indicated above is slightly smaller than the results reported previously. From resistance measurements on SNS junctions Hsiang and Clarke [4] obtained r3 = 1 lops. Frank et al. [S] studied .the transient response of indium microbridges to supercritical current pulses and reported rE = 140~s. Klein et al. [6] obtained the value rE = loops from the nonlinearity in the flux-flow behavior of indium films. Theoretical calculations by Kaplan et al. [7] yielded the value rE = 100 ps. We note that in our calculation of the inelastic scattering time rE based on equation (2) we have neglected any gap enhancement effect due to the highfrequency current [2,8]. Such a gap enhancement represents an effective cooling of the quasiparticles and thereby causes a reduction of the value of T in equation (2) which has to be modified accordingly. Hence, the value of the inelastic scattering time rn obtained from the modified expression for rA will tend to be larger than our value calculated above. Whether this effect can explain the fact that our value of the time r3 is slightly smaller than the previous results, can only be determined by further systematic experiments. REFERENCES 1. 2.
3. 4. 5.
J.A. Pals, J.A. Geurst & J.J. Ramekers, P&s. Rev. B23,6184 (1981). M. Tinkham, in Nonequilibrium Superconductivity, Phonons and Kapitza Boundaries, p. 23 1, (Edited by K.E. Gray), NATO Advanced Study Series B, Vol. 65 Plenum, New York, (1981). A. Schmid 8c G. Schon, J. Low Temp. Phys. 20, 207 (1975). T.Y. Hsiang & J. Clarke, Phys. Rev. B21, 945 (1980). D.J. Frank, M. Tit&ham, A. Davidson & S.M. Faris, Phvs. Rev. Lett. 50. 1611 (1983).
Vol. 60, No. 2 6. 7.
RELAXATION
TIME OF A SUPERCONDUCTING
W. Klein, R.P. Huebener, S. Gauss & J. Parisi, J. Low Temp Phys. 61,413 (1985). S.B. Kaplan, C.C. Chi, D.N. Imgenberg, J.H. Chang, S. Jafarey & D.J. Scalapino, Phys. Rev.
8.
INDIUM FILM
149
B14,4854 (1976). R.E. Horstman, J. Wolter & M.C.H.M. Wouters, Solid State Commun. 42,133 (1982).