- Email: [email protected]
Phase-slip centers in superconducting aluminium
Volume 57A, number 1
PHYSICS LETTERS
17 May 1976
PHASE-SLIP CENTERS IN SUPERCONDUCTING ALUMINIUM T.M. KLAPWIJK and J.E, MOOIJ Department of Applied Physics, Deift University of Technology, Deift, The Netherlands Received 29 March 1976 Measurements on aluminium strips confirm that the length of phase-slip centers is determined by inelastic scattering of quasiparticles. The relaxation time is found to be about iO~ s independent of temperature.
Recently Skocpol et a!. [11 demonstrated that a narrow superconducting tin strip biased above the critical current (1~)enters a voltage-carrying state with spatially localized voltage units called phase-slip centers. Each center carries a time-averaged supercurrent of about 0.5 I~and exhibits a temperature-independent differential resistance due to normal current flow governed by the quasiparticle diffusion length. Their data on tin indicate that the relaxation time associated with the nonequilibrium is typical of inelastic scattering times for electrons near the Fermi surface of norma! tin. In the course of our experiments on microwave-enhanced superconductivity of narrow aluminium strips [2] we observed similar regular steplike structures in the current-voltage characteristics. The voltage across the strip increased discontinuously with increasing current. Specifically, the differential resistance increased in steps of equal size, according to a relation dV/dJ = n Ru with n integer and R~a value characteristic of each strip. This is in agreement with the concept of spatially localized voltage units. As
some uncertainty has remained concerning the concepts of Skocpol et al. and as in aluminium the judastic scattering time is three orders of magnitude larger, our observations seem of interest. The time-averaged supercurrent carried by the voltage units was approximately 0.35 I~.In the case of tin the values observed ranged from 0.5 1~to 0.7 1~• This difference might be important in view of the fact that present theoretical models predict 0.5 f~or higher. The basic unit R~of the differential resistance should be determined by the quasiparticle diffusion length. As in [11 no temperature dependence was observed, even close to T~.Table 1 gives data of the vanous strips. R~is the differential resistance associated with one voltage unit, I the mean free path at helium temperatures calculated from the experimentally determined resistance ratio RRR, assuming 1300 1 56 A [3]. L~,the effective length of a voltage unit [1] follows from R~by taking L~= L (R~/Rr). Fig. I gives a plot of L~against l. Reasoning from the con-
Table 1 Experimental data of aluminium microstrips. (Length L, width W, thickness d, RT normal state resistance at Tc, RRR
300/R l’c). Sample no.
L(mm)
W(~m)
d(~m)
1 2 3 4 5 6 7 8 9 10
3.15 3.31 2.03 2.02 2.90 2.02 2.00 2.Q2 2.90 2.92
6.8 7.2 3.7 4.9 3.8 4.1 4.2 4.4 3.8 4.8
1.7 1.7 0.23 0.23 0.4 0.23 0.23 0.80 0.40 0.20
RTc(t~) 0.28 0.28 7.35 6.90 7.40 6.10 7.94 2.31 7.84 31.89
Ru(t2)
RRR
l(~sm)
L0CUrn)
0.035 0.027 0.88 0.48 0.61 0.64 0.64 0.15 0.50 1.66
26.0 22.0 9.7 7.7 7.6 9.7 7.3 11.6 7.9 4.2
0.39 0.34 0.15 0.12 0.12 0.15 0.11 0.18 0.12 0.07
394 316 238 140 239 211 164 190 185 152
97
Volume 57A, number 1 1000
PHYSICS LF1TFRS
p
I
10
I
•
I
•
~.
•
._-,—_ -
100-
—
[6] A I
=
590 A, r
dominated by electronic conduction and of which the electron-phonon part is equal to AT2. According to
I
10
~lcm2) we would have found 1300
2 0.5 X 10 ~ sand TF (Ta) 1.5 X 10 7 s. In the litera ture other values for p1 can be found, of the same order of magnitude but differing as much as a lacto TE of three. We conclude therefore that both r2 and are of the order of 10 ~s, but that a moie accurate determination is not possible. An estimate of the inelastic scattering time can be made independently from the experimentally determined low temperature thermal resistivity, which is
L (i~m)
11
17 May 1976
ml
.~
is about
4.4 X 10
5
cm/WK foi samples with a
resistance ratio smaller than 100. The thermal resist-
i
ivity is related to the relaxation time by the equation Fig. 1. Mean tree path dependence of Ln. The dashed line represents 2(’lva’T2)”2 with r 1.3 X ~ cm~s,assuming l3~Ø 21561.9 A. X 10 ~ sand CF
2 =(~Tv~rth)I With ~
cepts of Skocpol et a!. I~is expected to be equal to 2 (~lv 12 in which 0F is the Fermi velocity (1.3 X 1. T,)h 108 cm/s [3]) and r 2 the relaxation time of the nonequilibrium process. Although the scattering quite large the genera] trend is in accordance with isthis relation. From these experimental data r 2 can be estimated. The dashed line in fig. 1 corresponds to ~2 = 1.9 x 10 ~ s. The temperature independent relaxation of the nonequilibnium indicates a connection with the inelastic scattering time for electrons, rather than with the relaxation times characteristic of the superconducting state. According to Tinkham [41the inelastic scattering time for electrons near thewhich Fermir surface is 3 in given by r1 (7~= (r0/8.4) (6/fl 0 = (1300/vt.) (300/6) stands for the elastic scattering time at the Debye temperature 6. For aluminium with 0 400 K and at T = T~we obtain r1 0.4 x 10 ~ about a factor of five less than our experimental value
1.47 X 10 ~
—
for r7. However, the determination of ~2 from the experimental data and the theoretical estimate of r~depends strongly on the value chosen for the mean free path 1 at 300K. In the preceding calculations we used 1300 = 156 A. Skocpol et al. determined I and r~from the >( 1 = 1.6 product p1. Choosing the value of ref. [5] (p
—
0.17 X 10 ~ s. Because it is to be expected that in phase-slip centers relaxation of quasiparticles at the Fermi surface dominates and the thermal resistivity also involves processes with the Tth should be electrons scaled up far by from a factor Fermi surface 93/8.4 [1]. This gives TF 1.8 X 10 ~ s, again of the —
same order of magnitude. In conclusion, the relaxation time associated with the non-equilibrium region in aluminium strips is independent of temperature and of the order of magnitude 10 ~ s. This gives strong support to the hypothesis of Skoepol et a!. that the relevant time is the inelastic scattering time for electrons at the Fermi surface.
References Ill
—
98
—
joule/cm3 Pth —AT K2 [5] and T — T~— 1.18K we find Tth
W.J Skocpol, MR Bea~leyand M Tinkham. J. I ow Temp. Phys. 16 (1974) 145.
(21
T.M. Klapwijk and i.E. Mooij, Physica 81(1976)132, and to be published. [3] E. Fawcett, in The Fermi surface, Proc. Intern. Conf.,
141 151 161
Cooperstown, New York 1960, eds. W.A. Harrison and M.E. Webb (Wiley, New York 1960) p. 201. M. Tinkham, Phys. Rev. B6 (1972) 1747. J.L. Electron transport in metals (Interscience, New Olsen, York 1962). T.Q7~) Ainundsen, A.references Myhre andcited J.A.M. Salter, Phil. Mag. 25 (1 513, and therein.
PHYSICS LETTERS
17 May 1976
PHASE-SLIP CENTERS IN SUPERCONDUCTING ALUMINIUM T.M. KLAPWIJK and J.E, MOOIJ Department of Applied Physics, Deift University of Technology, Deift, The Netherlands Received 29 March 1976 Measurements on aluminium strips confirm that the length of phase-slip centers is determined by inelastic scattering of quasiparticles. The relaxation time is found to be about iO~ s independent of temperature.
Recently Skocpol et a!. [11 demonstrated that a narrow superconducting tin strip biased above the critical current (1~)enters a voltage-carrying state with spatially localized voltage units called phase-slip centers. Each center carries a time-averaged supercurrent of about 0.5 I~and exhibits a temperature-independent differential resistance due to normal current flow governed by the quasiparticle diffusion length. Their data on tin indicate that the relaxation time associated with the nonequilibrium is typical of inelastic scattering times for electrons near the Fermi surface of norma! tin. In the course of our experiments on microwave-enhanced superconductivity of narrow aluminium strips [2] we observed similar regular steplike structures in the current-voltage characteristics. The voltage across the strip increased discontinuously with increasing current. Specifically, the differential resistance increased in steps of equal size, according to a relation dV/dJ = n Ru with n integer and R~a value characteristic of each strip. This is in agreement with the concept of spatially localized voltage units. As
some uncertainty has remained concerning the concepts of Skocpol et al. and as in aluminium the judastic scattering time is three orders of magnitude larger, our observations seem of interest. The time-averaged supercurrent carried by the voltage units was approximately 0.35 I~.In the case of tin the values observed ranged from 0.5 1~to 0.7 1~• This difference might be important in view of the fact that present theoretical models predict 0.5 f~or higher. The basic unit R~of the differential resistance should be determined by the quasiparticle diffusion length. As in [11 no temperature dependence was observed, even close to T~.Table 1 gives data of the vanous strips. R~is the differential resistance associated with one voltage unit, I the mean free path at helium temperatures calculated from the experimentally determined resistance ratio RRR, assuming 1300 1 56 A [3]. L~,the effective length of a voltage unit [1] follows from R~by taking L~= L (R~/Rr). Fig. I gives a plot of L~against l. Reasoning from the con-
Table 1 Experimental data of aluminium microstrips. (Length L, width W, thickness d, RT normal state resistance at Tc, RRR
300/R l’c). Sample no.
L(mm)
W(~m)
d(~m)
1 2 3 4 5 6 7 8 9 10
3.15 3.31 2.03 2.02 2.90 2.02 2.00 2.Q2 2.90 2.92
6.8 7.2 3.7 4.9 3.8 4.1 4.2 4.4 3.8 4.8
1.7 1.7 0.23 0.23 0.4 0.23 0.23 0.80 0.40 0.20
RTc(t~) 0.28 0.28 7.35 6.90 7.40 6.10 7.94 2.31 7.84 31.89
Ru(t2)
RRR
l(~sm)
L0CUrn)
0.035 0.027 0.88 0.48 0.61 0.64 0.64 0.15 0.50 1.66
26.0 22.0 9.7 7.7 7.6 9.7 7.3 11.6 7.9 4.2
0.39 0.34 0.15 0.12 0.12 0.15 0.11 0.18 0.12 0.07
394 316 238 140 239 211 164 190 185 152
97
Volume 57A, number 1 1000
PHYSICS LF1TFRS
p
I
10
I
•
I
•
~.
•
._-,—_ -
100-
—
[6] A I
=
590 A, r
dominated by electronic conduction and of which the electron-phonon part is equal to AT2. According to
I
10
~lcm2) we would have found 1300
2 0.5 X 10 ~ sand TF (Ta) 1.5 X 10 7 s. In the litera ture other values for p1 can be found, of the same order of magnitude but differing as much as a lacto TE of three. We conclude therefore that both r2 and are of the order of 10 ~s, but that a moie accurate determination is not possible. An estimate of the inelastic scattering time can be made independently from the experimentally determined low temperature thermal resistivity, which is
L (i~m)
11
17 May 1976
ml
.~
is about
4.4 X 10
5
cm/WK foi samples with a
resistance ratio smaller than 100. The thermal resist-
i
ivity is related to the relaxation time by the equation Fig. 1. Mean tree path dependence of Ln. The dashed line represents 2(’lva’T2)”2 with r 1.3 X ~ cm~s,assuming l3~Ø 21561.9 A. X 10 ~ sand CF
2 =(~Tv~rth)I With ~
cepts of Skocpol et a!. I~is expected to be equal to 2 (~lv 12 in which 0F is the Fermi velocity (1.3 X 1. T,)h 108 cm/s [3]) and r 2 the relaxation time of the nonequilibrium process. Although the scattering quite large the genera] trend is in accordance with isthis relation. From these experimental data r 2 can be estimated. The dashed line in fig. 1 corresponds to ~2 = 1.9 x 10 ~ s. The temperature independent relaxation of the nonequilibnium indicates a connection with the inelastic scattering time for electrons, rather than with the relaxation times characteristic of the superconducting state. According to Tinkham [41the inelastic scattering time for electrons near thewhich Fermir surface is 3 in given by r1 (7~= (r0/8.4) (6/fl 0 = (1300/vt.) (300/6) stands for the elastic scattering time at the Debye temperature 6. For aluminium with 0 400 K and at T = T~we obtain r1 0.4 x 10 ~ about a factor of five less than our experimental value
1.47 X 10 ~
—
for r7. However, the determination of ~2 from the experimental data and the theoretical estimate of r~depends strongly on the value chosen for the mean free path 1 at 300K. In the preceding calculations we used 1300 = 156 A. Skocpol et al. determined I and r~from the >( 1 = 1.6 product p1. Choosing the value of ref. [5] (p
—
0.17 X 10 ~ s. Because it is to be expected that in phase-slip centers relaxation of quasiparticles at the Fermi surface dominates and the thermal resistivity also involves processes with the Tth should be electrons scaled up far by from a factor Fermi surface 93/8.4 [1]. This gives TF 1.8 X 10 ~ s, again of the —
same order of magnitude. In conclusion, the relaxation time associated with the non-equilibrium region in aluminium strips is independent of temperature and of the order of magnitude 10 ~ s. This gives strong support to the hypothesis of Skoepol et a!. that the relevant time is the inelastic scattering time for electrons at the Fermi surface.
References Ill
—
98
—
joule/cm3 Pth —AT K2 [5] and T — T~— 1.18K we find Tth
W.J Skocpol, MR Bea~leyand M Tinkham. J. I ow Temp. Phys. 16 (1974) 145.
(21
T.M. Klapwijk and i.E. Mooij, Physica 81(1976)132, and to be published. [3] E. Fawcett, in The Fermi surface, Proc. Intern. Conf.,
141 151 161
Cooperstown, New York 1960, eds. W.A. Harrison and M.E. Webb (Wiley, New York 1960) p. 201. M. Tinkham, Phys. Rev. B6 (1972) 1747. J.L. Electron transport in metals (Interscience, New Olsen, York 1962). T.Q7~) Ainundsen, A.references Myhre andcited J.A.M. Salter, Phil. Mag. 25 (1 513, and therein.