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Superconductivity in nearly one-dimensional tin wires
SUPERCONDUCTIVITY
IN NEARLY TIN WIRE3
ONE-DIMENSIONAL
G. E. POSSIN Department of Physics, Stanford University, Stanford, California 94305
Synopsis We have prepared tin wires with diameters less than in mica. The resistance ratios indicate low temperature the geometry. We have measured the resistance in the find good agreement with both the predicted temperature
1000 A by electroplating into small holes electron mean free paths limited only by Aslamazov-Larkin region above T, and dependence and magnitude.
We have prepared tin wires with diameters between 0.04 and 0.1~ by electroplating into holes in mica. The holes are prepared by etching high quality natural muscovite’) mica in 20% HF at 24’C2). The acid selectively etches out damage tracks produced by the spontaneous fission of natural U235impurities. The etch process as described by Price and co-workers proceeds in three steps. Within the first few seconds a hole about 25 8, in diameter is formed. No more etching occurs for about 100 seconds during which the reaction products diffuse out of the hole. During the last stage the hole widens at a uniform rate. The best available evidence indicates that the holes have uniform cross section. This is consistent with the immediate formation of a 25 A hole, which indicates very rapid diffusion of the acid. The length of each hole which is formed from the back to back track of two fission fragments is about 15 cc. We have obtained hole size calibrations from the formula d = (T - 100 s) (1 A/ s)+25 A where d is the hole diameter in A and T is the etching time in seconds. This is consistent with the data of Price et al. and our own measurements for long etching times. We estimate the accuracy of our diameters to be & 10% in the 0.1 p range. The holes are filled with tin by electroplating from a concentrated SnSO, acid plating solution. One side of a 12 or.thick piece of mica is covered with a tin film in a vacuum evaporator. A few drops of plating solution is added to the other side and a voltage applied to the cell through a tin electrode. The holes are filled with tin in about 15 seconds with an estimated plating efficiency of 1%. The first hole to fill begins to form a small cap, and because of the large series resistance in the plating control circuit, the formation of additional wires is stopped. We believe that plating occurs for the same reason that the holes t Research supported by the Office of Naval Research.
340
G. E. POSSIN
are of uniform diameter: there is rapid diffusion along the hole. This is further supported by the observation that strong bases will attack mica but will not form holes and that we have never observed wire formation with a basic gold cyanide plating solution. We have, however, successfully grown zinc and indium wires from acid plating solutions. Electrical contact was made to a single wire by subsequently electroplating gold onto the small (= 5 p) cap and immediately evaporating a contact film of indium. The time fromgold electroplating to cooling of the sample to 77 K was always less than 20 minutes. The maximum diffusion of gold in this time into single crystals is less than 4 p along the a axis and less than 0.3 p along the c axis3). Since the size of the cap was 4 CL,it is unlikely that the wires were contaminated by the gold. The samples were immersed in liquid helium in a standard low temperature apparatus. The resistance of the samples was measured with a 140 Hz kelvin double bridge, using a phase sensitive lock-in amplifier. The resolution and reproducibility of the bridge, which was limited by the Johnson noise of the 2 kohm resistors in the bridge, was 0.002 Sz at the typical measuring current of 0.4 rms PA. The entire apparatus was operated in a copper screened room to eliminate possible noise problems. The bridge current was always adjusted so that the sample obeyed Ohm’s law within the accuracy of the bridge. For R/RN = 1 this was typically for currents less than 1 rms PA or about 0.1% of the zero-temperature critical current. The resolution and reproducibility of the bridge was 0.002 fi at 0.4 A runs. Wires with maximum diameters between 0.04 and 0.1 w have been successfully cooled to 4 K. The resistance ratio r = R(300 K)lR(4.2 K) was always less than or equal to that predicted by Mathiessens rule, assuming an electron mean free path at room temperature of 100 A. For most samples r was within 30% of the ideal. The room temperature mean free path calculated from anomalous skin effect data”) is between 90 and 140 A. The close to ideal resistance ratios indicated that the electron mean free path is limited by the diameter. Because the wires are embeded in the mica, a direct measurement of the actual wire diameter is not possible. However, both the resistance ratios and room temperature resistances are consistent with the assumption that the wire and hole diameters are the same. The differential thermal contraction from 300 to 4 K is 44.8 X 10e4 for tin4) while for mica it is nearly zero. The resultant stress should enhance the transition temperature from 3.72 to about 3.85 K”*7). T,as defined by the midpoint of the transition was between 3.8 and 3.9 K in all cases, and for the better samples between 3.82 and 3.86 K. For some wires the effect of non-uniform strains were clearly detectable as steps in the transition curve. The transition curve for a particularly good 1000 A wire is shown in fig. 1. Because most of the samples exhibited a large low temperature tail, the
SUPERCONDUCTIVITY
IN ONE-DIMENSIONAL
TIN
341
.:
13.7
. . . 13.6
‘L
3.78
3.00
3.02
TEMPERATURE
Fig. 1. Resistance
uer~us temperature
3.84
K
3.8
4.0
4.:
TEMPERATURE
for a 1000 A diameter tin wire approximately Sample 024.
141.~long.
transition width was defined by AT, = (dR/dT),,,[R(4.2 K)- R(3.6 K)]. The maximum slope always occurred between R/RN = 0.8 and 0.4. Many of the wires were discarded because they had a very broad transition width and a large amount of structure. We do not claim that any of our samples have had transition widths limited only by fluctuations, but only that the observed A Tcw’s represent an upper limit to the fluctuation limited widths. A convenient semi-quantitative comparison of theory can be made by calculating AT..., from the relation Ag( = a k,T, where: AR(T), = H ( T)2/8r is the superconducting condensation energy, R ( T) = 5 (T) (7r/4) d2 is the characteristic fluctuation volume, t(T) = 0.85 (&d )112( Tc/AT)1’2 (we assume l,rr = d) is the Landau-Ginzburg coherence length, T,- T = AT,, and (Y is a parameter of the order of unity. This gives AT, a dV5j3which is characteristic of classical fluctuation theories. The calculation of Langer and AmbegaokaI-8) for the onset of resistance at zero current can be expressed in this form. If we take the L & A parameter y = 45 as determined by Webb and Warburtons) we find (Y= 11.6. Taking Ag(T) = (( 1.85)2/87r) H,(O) (AT/T,)* (ref. lo), H,(O) = 306 Oe, and &, = 2300 A, gives AT, = 1.03d-5’3 K,
(1)
where d is 100 A units. Some wires with diameters as small as 450 A had transition widths, as defined above, within 20% of that given by eq. (1). The transition widths reported by Webb and Warburtong) for their tin whiskers were approximately of this magnitude. At the present time we regard this as only suggestive evidence that fluctuation broadening is responsible for our observed transition widths. The conductivity for T > T, has been treated by Aslamazov and LarkinlO).
342
G. E. POSSIN
They find
(j&)‘”=(g-J
(2)
where RN is the normal resistance, n = (4 - 0)/2, and D is the dimension of the sample, i.e. n = 4 for a wire with diameter d + t(T). For a dirty wire they give ATo = TJ5.4 fi/(r kBT,)“2p,2~]2’3 DC&5’3,
(3)
where r is the electron mean free time, p0 the Fermi momentum and s the cross-sectional area. The theory is expected to hold only for RN/RN-k %=1. To evaluate eq. (3) we take V, from v,=- rr2kB2 -tre - 1 e2 ( 1, > 7~’ where y = 1040 ergs/cm3K is the electronic specific-heat coefficient and p. = m* V,/fi where m*/m = 1.2”). This gives AT, = 1.3 X 10m4cV3 K where d is in microns. We experimentally determine RN from the resistance in a high magnetic field. From O-6 kOe the total magnetoresistance p is of the order of 2 x 10M3 and roughly parabolic in the applied field. The critical fields for these wires are of the order of 1 kOe so a sufficiently accurate determination of RN is
TEMPERATURE
Fig. 2. RN = 13.931.003~ determined from magnetic field measurements;
Sample024.
SUPERCONDUCTIVITY
IN ONE-DIMENSIONAL
TIN
343
possible. The magnetoresistance is also distinguished by its lack of temperature dependence. Figure 2 shows the measured dependence of (RN/AR)‘13 on temperature. The agreement with the predicted temperature dependence is quite good to 3.88 K. If the same data is plotted as RN/AR us. temperature, the fit is clearly not as good over as wide a temperature range. The data for several samples is summarized in table I. The value of AT, is corrected for the lowis taken directly from the plots of (RN/AR) 2’3.AT”“” 0 TABLE I
Sample 22 24 7 12 11 13
Diam.
R(300 K)
RN
R(3.6 K) AT, X 103
7OoA 1000 1000 1000 700 700
242 R 131 140 159 280 362
42.1.5-c0.005Q” 13.93 -c0.003= 12.30*0.1b 15.26-c0.1b 67.3 ?0.15b 57.8 f 1.2b
20R 4 3 4 40 15
9+2K 5.5kO.5 5*2 4.3* 1.5 4.5* 1.5 S&3
a from magnetic field measurements b picked for best fit c the indicated errors reflect the errors in the measured resistances values of RN.
ATyx 14a3K 7-1-l 6&2 5.222 8.2k2.5 1023.5
AL Theory lose AT, x 103 10.5 K 5.9 5.9 5.9 10.5 10.5
and the uncertainty in the
temperature residual resistance which clearly does not contribute to the transition. In spite of the wide variation in transition widths the values of APO”= are consistent. This is reasonable since the fit to the AL theory is made several AT,‘s above T,. We hope to eliminate the experimental problem of the low-temperature residual resistance which we believe is due to the gold involved in the contact procedure. This should enable a more sensitive quantitative test of the theory and also improve the measured transition widths.
REFERENCES
1) Spruce Pine Mica Company. 2) Price, P. B. and Walker, R. M., J. appl. Phys. 33 (1962) 3407; Fleischer, R. L., Price, P. B. and Walker, R. M., Rev. Sci. Inst. 34 (1963) 5 10. 3) Dyson, B. F., J. appl. Phys. 37 (1966) 2375. 4) Lyall, K. R. and Cochran, J. F., Phys. Rev. 159 (1967) 5 17. 5) Conuccini, R. J. and Friewek, J. J., Thermal Expansion of Technical Solids at Low Temperatures, National Bureau of Standards Monograph 29 (1961). 6) Lock, J. M., Proc. Roy. Sot. (London) A208 (195 1) 39 1. 7) Hall, P. M., J. appl. Phys. 36 (1965) 247 1. 8) Langer, J. S. and Ambegaokar, V., Phys. Rev. 164 (1967) 498. 9) See original manuscript: Webb, W. W. and Warburton, R. J., Phys. Rev. Letters 20 (1967) 461. 10) Aslamazov, T. G. and Larkin, A. I., Soviet Physics-Solid State 10 (1968) 875. 11) Daunt, J. G., Progress in Low Temperature Physics, C. J. Garter, ed. North-Holland (Amsterdam, 1965), Chapter XI.
IN NEARLY TIN WIRE3
ONE-DIMENSIONAL
G. E. POSSIN Department of Physics, Stanford University, Stanford, California 94305
Synopsis We have prepared tin wires with diameters less than in mica. The resistance ratios indicate low temperature the geometry. We have measured the resistance in the find good agreement with both the predicted temperature
1000 A by electroplating into small holes electron mean free paths limited only by Aslamazov-Larkin region above T, and dependence and magnitude.
We have prepared tin wires with diameters between 0.04 and 0.1~ by electroplating into holes in mica. The holes are prepared by etching high quality natural muscovite’) mica in 20% HF at 24’C2). The acid selectively etches out damage tracks produced by the spontaneous fission of natural U235impurities. The etch process as described by Price and co-workers proceeds in three steps. Within the first few seconds a hole about 25 8, in diameter is formed. No more etching occurs for about 100 seconds during which the reaction products diffuse out of the hole. During the last stage the hole widens at a uniform rate. The best available evidence indicates that the holes have uniform cross section. This is consistent with the immediate formation of a 25 A hole, which indicates very rapid diffusion of the acid. The length of each hole which is formed from the back to back track of two fission fragments is about 15 cc. We have obtained hole size calibrations from the formula d = (T - 100 s) (1 A/ s)+25 A where d is the hole diameter in A and T is the etching time in seconds. This is consistent with the data of Price et al. and our own measurements for long etching times. We estimate the accuracy of our diameters to be & 10% in the 0.1 p range. The holes are filled with tin by electroplating from a concentrated SnSO, acid plating solution. One side of a 12 or.thick piece of mica is covered with a tin film in a vacuum evaporator. A few drops of plating solution is added to the other side and a voltage applied to the cell through a tin electrode. The holes are filled with tin in about 15 seconds with an estimated plating efficiency of 1%. The first hole to fill begins to form a small cap, and because of the large series resistance in the plating control circuit, the formation of additional wires is stopped. We believe that plating occurs for the same reason that the holes t Research supported by the Office of Naval Research.
340
G. E. POSSIN
are of uniform diameter: there is rapid diffusion along the hole. This is further supported by the observation that strong bases will attack mica but will not form holes and that we have never observed wire formation with a basic gold cyanide plating solution. We have, however, successfully grown zinc and indium wires from acid plating solutions. Electrical contact was made to a single wire by subsequently electroplating gold onto the small (= 5 p) cap and immediately evaporating a contact film of indium. The time fromgold electroplating to cooling of the sample to 77 K was always less than 20 minutes. The maximum diffusion of gold in this time into single crystals is less than 4 p along the a axis and less than 0.3 p along the c axis3). Since the size of the cap was 4 CL,it is unlikely that the wires were contaminated by the gold. The samples were immersed in liquid helium in a standard low temperature apparatus. The resistance of the samples was measured with a 140 Hz kelvin double bridge, using a phase sensitive lock-in amplifier. The resolution and reproducibility of the bridge, which was limited by the Johnson noise of the 2 kohm resistors in the bridge, was 0.002 Sz at the typical measuring current of 0.4 rms PA. The entire apparatus was operated in a copper screened room to eliminate possible noise problems. The bridge current was always adjusted so that the sample obeyed Ohm’s law within the accuracy of the bridge. For R/RN = 1 this was typically for currents less than 1 rms PA or about 0.1% of the zero-temperature critical current. The resolution and reproducibility of the bridge was 0.002 fi at 0.4 A runs. Wires with maximum diameters between 0.04 and 0.1 w have been successfully cooled to 4 K. The resistance ratio r = R(300 K)lR(4.2 K) was always less than or equal to that predicted by Mathiessens rule, assuming an electron mean free path at room temperature of 100 A. For most samples r was within 30% of the ideal. The room temperature mean free path calculated from anomalous skin effect data”) is between 90 and 140 A. The close to ideal resistance ratios indicated that the electron mean free path is limited by the diameter. Because the wires are embeded in the mica, a direct measurement of the actual wire diameter is not possible. However, both the resistance ratios and room temperature resistances are consistent with the assumption that the wire and hole diameters are the same. The differential thermal contraction from 300 to 4 K is 44.8 X 10e4 for tin4) while for mica it is nearly zero. The resultant stress should enhance the transition temperature from 3.72 to about 3.85 K”*7). T,as defined by the midpoint of the transition was between 3.8 and 3.9 K in all cases, and for the better samples between 3.82 and 3.86 K. For some wires the effect of non-uniform strains were clearly detectable as steps in the transition curve. The transition curve for a particularly good 1000 A wire is shown in fig. 1. Because most of the samples exhibited a large low temperature tail, the
SUPERCONDUCTIVITY
IN ONE-DIMENSIONAL
TIN
341
.:
13.7
. . . 13.6
‘L
3.78
3.00
3.02
TEMPERATURE
Fig. 1. Resistance
uer~us temperature
3.84
K
3.8
4.0
4.:
TEMPERATURE
for a 1000 A diameter tin wire approximately Sample 024.
141.~long.
transition width was defined by AT, = (dR/dT),,,[R(4.2 K)- R(3.6 K)]. The maximum slope always occurred between R/RN = 0.8 and 0.4. Many of the wires were discarded because they had a very broad transition width and a large amount of structure. We do not claim that any of our samples have had transition widths limited only by fluctuations, but only that the observed A Tcw’s represent an upper limit to the fluctuation limited widths. A convenient semi-quantitative comparison of theory can be made by calculating AT..., from the relation Ag( = a k,T, where: AR(T), = H ( T)2/8r is the superconducting condensation energy, R ( T) = 5 (T) (7r/4) d2 is the characteristic fluctuation volume, t(T) = 0.85 (&d )112( Tc/AT)1’2 (we assume l,rr = d) is the Landau-Ginzburg coherence length, T,- T = AT,, and (Y is a parameter of the order of unity. This gives AT, a dV5j3which is characteristic of classical fluctuation theories. The calculation of Langer and AmbegaokaI-8) for the onset of resistance at zero current can be expressed in this form. If we take the L & A parameter y = 45 as determined by Webb and Warburtons) we find (Y= 11.6. Taking Ag(T) = (( 1.85)2/87r) H,(O) (AT/T,)* (ref. lo), H,(O) = 306 Oe, and &, = 2300 A, gives AT, = 1.03d-5’3 K,
(1)
where d is 100 A units. Some wires with diameters as small as 450 A had transition widths, as defined above, within 20% of that given by eq. (1). The transition widths reported by Webb and Warburtong) for their tin whiskers were approximately of this magnitude. At the present time we regard this as only suggestive evidence that fluctuation broadening is responsible for our observed transition widths. The conductivity for T > T, has been treated by Aslamazov and LarkinlO).
342
G. E. POSSIN
They find
(j&)‘”=(g-J
(2)
where RN is the normal resistance, n = (4 - 0)/2, and D is the dimension of the sample, i.e. n = 4 for a wire with diameter d + t(T). For a dirty wire they give ATo = TJ5.4 fi/(r kBT,)“2p,2~]2’3 DC&5’3,
(3)
where r is the electron mean free time, p0 the Fermi momentum and s the cross-sectional area. The theory is expected to hold only for RN/RN-k %=1. To evaluate eq. (3) we take V, from v,=- rr2kB2 -tre - 1 e2 ( 1, > 7~’ where y = 1040 ergs/cm3K is the electronic specific-heat coefficient and p. = m* V,/fi where m*/m = 1.2”). This gives AT, = 1.3 X 10m4cV3 K where d is in microns. We experimentally determine RN from the resistance in a high magnetic field. From O-6 kOe the total magnetoresistance p is of the order of 2 x 10M3 and roughly parabolic in the applied field. The critical fields for these wires are of the order of 1 kOe so a sufficiently accurate determination of RN is
TEMPERATURE
Fig. 2. RN = 13.931.003~ determined from magnetic field measurements;
Sample024.
SUPERCONDUCTIVITY
IN ONE-DIMENSIONAL
TIN
343
possible. The magnetoresistance is also distinguished by its lack of temperature dependence. Figure 2 shows the measured dependence of (RN/AR)‘13 on temperature. The agreement with the predicted temperature dependence is quite good to 3.88 K. If the same data is plotted as RN/AR us. temperature, the fit is clearly not as good over as wide a temperature range. The data for several samples is summarized in table I. The value of AT, is corrected for the lowis taken directly from the plots of (RN/AR) 2’3.AT”“” 0 TABLE I
Sample 22 24 7 12 11 13
Diam.
R(300 K)
RN
R(3.6 K) AT, X 103
7OoA 1000 1000 1000 700 700
242 R 131 140 159 280 362
42.1.5-c0.005Q” 13.93 -c0.003= 12.30*0.1b 15.26-c0.1b 67.3 ?0.15b 57.8 f 1.2b
20R 4 3 4 40 15
9+2K 5.5kO.5 5*2 4.3* 1.5 4.5* 1.5 S&3
a from magnetic field measurements b picked for best fit c the indicated errors reflect the errors in the measured resistances values of RN.
ATyx 14a3K 7-1-l 6&2 5.222 8.2k2.5 1023.5
AL Theory lose AT, x 103 10.5 K 5.9 5.9 5.9 10.5 10.5
and the uncertainty in the
temperature residual resistance which clearly does not contribute to the transition. In spite of the wide variation in transition widths the values of APO”= are consistent. This is reasonable since the fit to the AL theory is made several AT,‘s above T,. We hope to eliminate the experimental problem of the low-temperature residual resistance which we believe is due to the gold involved in the contact procedure. This should enable a more sensitive quantitative test of the theory and also improve the measured transition widths.
REFERENCES
1) Spruce Pine Mica Company. 2) Price, P. B. and Walker, R. M., J. appl. Phys. 33 (1962) 3407; Fleischer, R. L., Price, P. B. and Walker, R. M., Rev. Sci. Inst. 34 (1963) 5 10. 3) Dyson, B. F., J. appl. Phys. 37 (1966) 2375. 4) Lyall, K. R. and Cochran, J. F., Phys. Rev. 159 (1967) 5 17. 5) Conuccini, R. J. and Friewek, J. J., Thermal Expansion of Technical Solids at Low Temperatures, National Bureau of Standards Monograph 29 (1961). 6) Lock, J. M., Proc. Roy. Sot. (London) A208 (195 1) 39 1. 7) Hall, P. M., J. appl. Phys. 36 (1965) 247 1. 8) Langer, J. S. and Ambegaokar, V., Phys. Rev. 164 (1967) 498. 9) See original manuscript: Webb, W. W. and Warburton, R. J., Phys. Rev. Letters 20 (1967) 461. 10) Aslamazov, T. G. and Larkin, A. I., Soviet Physics-Solid State 10 (1968) 875. 11) Daunt, J. G., Progress in Low Temperature Physics, C. J. Garter, ed. North-Holland (Amsterdam, 1965), Chapter XI.