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P3 Rg. 36: S k e t c h o f sample S n 2 f o r the discussion o f the slope o f the time-averaged electrochemical p o t e n t i a l f o r quasiparticles and (gp > f o r Cooper pairs. The p h a s e - s l i p c e n t e r is s i t u a t e d at Xpsc. Here ([zp > changes in a s t e p - l i k e way f r o m the value ([zp>1 on the l e f t side o f the p h a s e - s l i p c e n t e r to (l~p)r on the right side. The dashed line s h o w s ([~>ssr as p r o p o s e d by S B T (see sectionS.8.3). The solid line s h o w s our proposal ( g ) , which we can explain our experimental r e s u l t s with. The numerals 1- 4 mark the p a r t s o f the s a m p l e and P 1 - P 3 the p o t e n t i a l probes; NC, normal conducting; SC, superconducting. The polarity o f the battery c o n n e c t e d with the Wood m e t a l c o n t a c t s (shaded) is indicated by @-), C-~D, leading to a current leA. The circles at the c r o s s o v e r p o i n t s o f the electrochemical p o t e n t i a l s with the p o t e n t i a l p r o b e s indicate which potential is measured by the probe: Probe P I m e a s u r e s ( p ) (xp, )=: ([z>p~ ; probe P 3 measures 1. discuss how the slope of has to be modified to explain our experimental results. For this purpose we use qualitative sketches and only discuss the sign of the measured voltages. We s t a r t with the discussion of the potential m e a s u r e m e n t s in the presence of a t r a n s p o r t c u r r e n t (Fig. 34). In Fig. 36 we sketch the experimental situation. The phase-slip center is situated in p a r t 3 and the voltage foot is detected with the N C / S C potential probe pair P 1 / P 2 applied to p a r t 2 . One has to explain that the voltage step generated by the phase-slip center in the characteristics of p a r t 3 has the same sign as the voltage foot. This can be achieved with our proposal for : The voltage foot has been interpreted as the difference between the electrochemical p o t e n t i a l s of quasipartictes and Cooper pairs at the site of the normal conducting probe and is, thus, given by VcBf = ( ( g p > l - < l ~ > p , ) / ( - e ) and the voltage step by
160
(201)
V0c = ( ( F ) . 3 - ( M , ) l ) / ( - e )
(202)
Since (p.>p~>(#p>l>(p)p3, we get the same sign for VcBf and VDc. This has been observed in the experiment. Hsing (#)ssT instead of (I~), we have to replace the quasiparticle p o t e n t i a l s at the probes P1 and P3 in eqs.(201) and(202) by (#)sBv.Pl and (~>SBT,P3" AS (Mp)I>(V)SBT, pI>(IZ)SBT,~3 we would get VcBf and VDc with different signs which would be contrary to our experimental observation. From Fig. 36 we also see that Vc~f=O if probe P1 has become superconducting, because then P1 and P2 measure the spatially c o n s t a n t (11~>i. With the proposed slope of ( # ) we can also u n d e r s t a n d the voltage foot appearing in p a r t 3 of sample Sn2 if a phase-slip center generates a voltage step in part l. Also in this case the signs of the voltage of the step and the foot are equal. To explain this, one may imagine that the phase-slip center is situated in part l. Then probe P2 measures (Mp)r and probe P3 measures the lower lying ( # ) , : r Now we show that also the m e a s u r e m e n t s in the absence of a t r a n s p o r t c u r r e n t can be u n d e r s t o o d with the proposed slope of ([z). In the experiments performed with sample S n r (Fig. 35) the t r a n s p o r t c u r r e n t leaves the sample through the superconducting probe P I and does not flow through p a r t 3 of the whisker where the differences between ( l z ) and ( ~ p ) are detected. We assume that a phase-slip center appears in the disturbed border region b e t w e e n the potential probe and the sample (Fig. 37a), before p h a s e - s l i p centers in the sample develop, leading to a slope of the electrochemical p o t e n t i a l s as drawn in Fig. 37b. (A similar slope would also be' p r e s e n t with the phase-slip center localized at a point Xpsr situated in the
OBE
SAMPLE
_•PR |
SC i
SAMPLE
(P'p)r
< |
(o}
PROBE
SC
Xpsc
|
(b)
Fig. 37: Slope o f the time-averaged electrochemical quasiparticle potential ( ~ ) and pair potential (lip} f o r a current f l o w i n g through one part o f the sample only and then leaving the sample through a potential probe. (a) S k e t c h o f the sample geometry. The arrows indicate conventional curren ts. (b) Slope o f the p o t e n t i a l s i f the potential probe and the sample are superconducting (SC) and a phase-slip center develops at Xpsc in the contact region.
161
Sn&
|
t
IAB 1
m l 2lp 2
IGI using eq.(203) for sample S n 2 in the experimental situation of Fig. 34 where a t r a n s p o r t c u r r e n t passes the whole sample. For this calculation we insert xR=65Opm , A=6S.S~m and, furthermore, the clean limit expressions for F, coG, and G with s and plot the spatial dependence of the electrochemical p o t e n t i a l s for different values of the t e m p e r a t u r e and V,L. To compare the calculated r e s u l t s with the m e a s u r e m e n t s , t e m p e r a t u r e s are considered for which experiments were carried out and values for VRL=VDc~ VOB are taken from the m e a s u r e m e n t [4743. An example for such a calculation is shown in Fig. 39. Only the right side of the phase-slip center has been p l o t t e d and one may think of a symmetric branch to get the sketched slope of ( ~ ) in Fig. 36. The calculated
165
CPIpeVI
Sn 2
0"5 1
Tco=3.693 K+0.SmK T = 3,693K-4.8mK
Voc= O.59pV -~
/EA= 2201JA
"I ~
sBT
IJp
-0.2
I
0
'
= sBT+ '
'
I
.
I
500 xlpm
Rg, 39: Slope o f the electrochemical potentials (CP), namely T in the sense of Parker's model (see sectionS.S). In this case the whisker behaves as if it had the t e m p e r a t u r e T*, so t h a t its critical c u r r e n t is reduced to I (T ~) < Io (T). This e f f e c t has to be considered in the SBT model (Fig. 13c), because the a m p l i t u d e of the oscillating s u p e r c u r r e n t in the core now is the critical c u r r e n t Io(T *) and n o t Ir The c o n s e q u e n ce is a reduction of the time averaged s u p e r c u r r e n t I 0, leading to a s m a l le r ratio I0/I c. The T * e f f e c t also c o n t r i b u t e s to the hysteresis and leads to the obvious broadening o f the hysteresis width for sh o r t zinc whiskers (chap. 11). Next, we discuss the properties of an isolated p h a s e - s l i p c e n t e r in a zinc whisker which is n o t a f f e c t e d by the contacts. Such a p h a s e - s l i p c e n t e r is related to the f i r st v o l t ag e s t e p in a long whisker. As found for all samples, this step is characterized by a temperature independent ratio I 0/ I and a temperature i n d ep en d en t differential resistance. This differential resistance leads to a t e m p e r a t u r e independent n o r m a l - l i k e l e n g t h and, thus, within the SBT model to a t e m p e r a t u r e independent quasiparticle relaxation time z2. The time has to be identified with the charge imbalance relaxation time zQ.. As already pointed out in s e c t i o n 7 . 2 no t e m p e r a t u r e independent steady s t a t e charge imbalance relaxation time is known. For Sn and In (not too c l o s e to To0) we also obtained a t e m p e r a t u r e independent quasiparticle relaxation time (section7.2). In that case we found 175
an empirical rule which c o n n e c t s the value of z2 with the scaling time z 0 of Kaplan et al. [103]. This rule is n o t valid for Zn whiskers, because it would require a value of % for Zn whiskers which is about t w o orders o f magnitude larger than the measured r e s u l t E20, 41]. Also Kulik's e l e c t r o n - b o g o l o n approach, widely discussed in section T.2, does not explain the observed phenomena. The r e s u l t i n g n o r m a l - l i k e l e n g t h 2AKu(O) is t wo orders o f magnitude too large. Next, we are going to discuss the predictions of the TDGL theory (sectionS.9) as given by Kramer and Rangel (KR): The calculations of KR are based on the generalized TDGL equations (eqs.(172) and (173)) and are valid for dirty f i l a m e n t s (2,,~0) and in local equilibrium (AE l . ) In this case ~:ee is of the order of 10 -8 s. Thus, in the t h r e e - d i m e n s i o n a l case and also in the t w o - d i m e n s i o n a l case (assuming values f o r 2 and d which are similar to the properties in our whiskers) it f o l l o w s a value for Zee which is larger than ~E and much l a r g e r than our measured z2. We conclude t h a t inelastic e l e c t r o n - e l e c t r o n s c a t t e r i n g should not play an i m p o r t a n t role in our experiments. Finally, also in the case of Zn whiskers t h er e is evidence f o r the e n h a n c e m e n t of the critical c u r r e n t of a p h a s e - s l i p c e n t e r by an active p h a s e - s l i p center. Considering the large quasiparticle relaxation length in our zinc whiskers one may s p e c u l a t e E22], t h a t the e n h a n c e m e n t e f f e c t is caused by q u as i p ar t i cl e c u r r e n t s generated by the active p h a s e - s l i p center. This mechanism has been widely discussed in sections S.8.3 and 7.S. The r ed u ct i o n of the e n h a n c e m e n t may then be caused by a weakening p r o cess due to an o v e r p o p u l a t i o n of the excitation s p e c t r u m with nonequilibrium quasiparticles (sectionS.S). A p r o o f of this explanation is n o t possible, because the distance b e t w e e n the involved p h a s e - s l i p c e n te r s is unknown. Nevertheless, also in the case o f zinc whiskers a stabilizing e f f e c t explains the large c u r r e n t region b e t w e e n the f i r s t o n s e t of v o lt a g e and the fully normal conducting state.
177
9. The Limit of Short Quasiparticle Relaxation Times
9.1.
Introductory Remarks
In section 8.1 we discussed that the quasiparticle r e l a x a t i o n times scale with the inelastic e l e c t r o n - p h o n o n scattering time, z E, and t h a t also the dynamics of charge imbalance d e p e n d s on this characteristic time. Moreover, we remarked t h a t Pb whiskers were chosen for the experimental study of the limit of s h o r t quasiparticle relaxation times, because the value of % is very s h o r t in this material. For the m e a s u r e m e n t s of Pb whiskers a special 4He o v e r p r e s s u r e c r y o s t a t had been developed because highly stabilized t e m p e r a t u r e s around 7.2K, t h a t means above the normal boiling point of "He, were needed (chap. 3). The astonishing r e s u l t of th e s e e x p e r i m e n t s is t h a t t h er e are no v o l t a g e st e p s visible in the transition characteristics. At a f i r st view we t h o u g h t t h a t there was n o t any s t r u c t u r e in the characteristics. However, m o r e detailed investigations with a higher resolution showed, that, a l t h o u g h there are no v o l t a g e jumps in the V-I characteristics, one can clearly recognize p o r t i o n s where the v o l t a g e depends linearly on the current. In our opinion also t h e s e linear portions are generated by p h a s e - s l i p centers: There is a reasonable behaviour of the n o r m a l - l i k e length assigned to their differential resistance. Moreover, the characteristics exhibit ' c u r r e n t steps' (section7.3) in a HF-radiation field. Nevertheless, the absence of v o lt a g e jumps and a ratio I 0 / I o ~ 1 seemed so unusual to us t h a t we s t a r t e d a systematic investigation of whiskers made of materials with d i f f e r e n t e l e c t r o n - p h o n o n coupling s t r e n g t h s . For this purpose we investigated numerous whiskers made of In-Pb alloys with lead c o n c e n t r a t i o n s spread over the whole alloy system. As In is a w e a k - c o u p l i n g s u p e r c o n d u c t o r and Pb is a s t r o n g - c o u p l i n g superconductor, it is, thus, possible to investigate the continuous change of properties toward the s t r o n g coupling behaviour of Pb. Special a t t e n t i o n has been paid to the investigation of In-rich alloys and Pb with small In concentrations. To prove the universality of the r e s u l t s of Pb with small In impurity content, we also investigated Pb whiskers with Bi impurities. The r e s u l t s of In-rich alloys may be partly i n t e r p r e t e d within the SBT model and contain interesting information for a comparison with the TDGL
178
r e s u l t s of KR. The r e s u l t s obtained for whiskers of pure Pb and P_b_b-ln or P___b_b-Bi whiskers may be interpreted within the SBT and KSS model. The KSS model leads to a qualitative understanding of the results. It t u r n s out t h a t it is indeed the s t r o n g - c o u p l i n g nature of the material with its small inelastic e l e c t r o n - p h o n o n scattering time, ~E, which makes visible the observed, somewhat u n u s u a l phenomena over a large range of measuring temperatures. The small value of ~E acts to spread the t e m p e r a t u r e range where there are no voltage jumps but, nevertheless, linear portions. In principle this behaviour should also be p r e s e n t in the t r a n s i t i o n behaviour of filaments of other materials. However, usually it c a n n o t be investigated, because the related t e m p e r a t u r e range is only small and very close to To0 and the p h e n o m e n a seem to be disturbed by f l u c t u a t i o n effects. In the following the basic r e s u l t s of our investigations will be summarized. For f u r t h e r information see refs. 40, 359, and 372. All experiments were performed in the mentioned overpressure cryostat.
9.2.
9.2.1.
Change from Weak to Strong Coupling Behaviour: Experiments Whiskers of the In-Pb Alloy System
on
In-Rich Alloys
All whiskers from In-rich alloys (that means from the In crystalline solid s o l u t i o n which extends up to a lead c o n c e n t r a t i o n cpb=12.7at%) show characteristics very similar to those of samples from pure indium [16]. Thus, their shape is similar to the pure tin case (Figs. 5 and 6) except that the differential resistance (dV/dl)~ and the ratio I 0 / I ~ of the first voltage step depend on t e m p e r a t u r e (section 7.2). The differential resistance and, thus, also the n o r m a l - l i k e length, LAn1, show the whole variety of temperature dependences known from the investigations of In whiskers (section7.2): The n o r m a l - l i k e length may be i n d e p e n d e n t of t e m p e r a t u r e or change proportional to AT -~/4, AT -~/2, and AT "~, respectively. Some examples are shown in Figs. 46 and 47. For larger values of AT the normal-like length does not show any t e m p e r a t u r e dependence. These temperature independent r e s u l t s may be averaged for each sample (the average is called (LAnl)OinebW) and p l o t t e d as a f u n c t i o n of the electron mean free path 2 of the sample together with the r e s u l t s for pure In whiskers (called (LAnt)OinW). These 'low temperature values' show the familiar 2~J2 dependence (Fig. 48). The ratio I 0 / I c exhibits the behaviour expected from the fact that V1(I,) is a s t r a i g h t line, requiring that 1- I 0 / I : compensates the t e m p e r a t u r e dependence of ( d V / d I ) , (section7.2). Thus, I 0 / I ~ is temperature i n d e p e n d e n t for large values of /iT and shows an increasing tendency if the t e m p e r a t u r e approaches To (Fig. 49). Also in the case of I 0/I~ the temperature
179
50-
Rg. 46.
Sample 1 o
fn-Pb
E
21..
'M
8
o
9
~
50:~
-Pb
Sample
-4 10-
10
~A 5-
5
~AT -~f2
,.,AT-~/~" 2
i
50_
10-=
r,
I J~,tl~llll
5
I
I
--
10 20 30 In-Pb Sample 2 o 5 o
10-
~
5
10 20 30 rn-Pb Sample
_ AtI 1[1~~ ~~ i ~ T T
181/ 1361 -., ,~, o,,,u
5-
5--
2
50-
,v AT -1/~ I
I"1
I I
5
10
I-t 20 30 a T/mK
2
I i Iil 5
~,llllq I' I 10 20 30
,~ TI mK
F i g . 4 6 : Normal-fike length, L A n l ,
a s a function o f the temperature difference d T=T,o-T, using l o g - l o g coordinates, f o r several In-Pb whiskers with t,*'4 I,A n l ~ d T , represented by a straight line with slope - 1 / 4 [40Z Horizontal lines indicate that LAn 1 is independent o f temperature. S a m p l e 9 s h o w s this beha viour only. For Tea o f each sample see refs. 14 and 40. -
P i g . 4 7 : Normal-like length, L A n l ,
as a function o f the temperature difference dT=T~o-T, using log-log coordinates, for several l nn-Pb whiskers with other temperature behaviour besides the d T - ~ law f40Z Horizontal lines indicate that LAn I is independent o f temperature. For T O o f each sample see refs. 14 and 40.
independent values may be averaged for each sample. In Fig. 50 we p l o t t e d these ' l o w t e m p e r a t u r e values' (called ( I 0 / 1 ) O i n P b W ) as a f u n c t i o n of s t o g e t h e r with the r e s u l t s of ( I 0 / I c ) o l n w of pure indium whiskers. This p l o t shows t h a t indeed there is a slight decrease of the pure indium r e s u l t s with decreasing 2, as mentioned in section 7.2. In the l n - P b whiskers 2 is f u r t h e r decreased, leading to a f u r t h e r decrease and then to an increase o f the ratio. A similar behaviour was observed f o r S nn-ln whiskers (Fig. 27).
180
E
100"
~ so-
in,ZD.pb n_
I~
10-"
5-
1
i
I
100
I lltl~
1
L
J I lllll
500 1000
I
5000 10000 .~/10 -10m
Flg. 48:
Length (LAnl)OInPbW as a function o f the electron mean free path, ~, f o r several l n - P b whiskers, using l o g - l o g coordinates [402 R e s u l t s f o r (LAn! ) O l n W o f pure In whiskers are added. ( A ) : Pure In whiskers from ref. 17 (0), (0), (,): l n - P b whiskers from ref. 40. In that work we introduced different s y m b o l s to distinguish between samples where the first voltage step is well developed (light circle) or occurs together with the second step (full circle) and the second and third step (full diamond), respectively. The f u l l line represents an ~,~2 dependence.
1.0
{{tttt
Zo/Zc
In- Pb
Sample 11 u
0.90.8-
t
0.7I
I
I
10 5 0'6 t
0.5
0.~, 1:11g.49:
0 ,~T/mK I
I
25 20
I
15
I
10 ATI mK
I
5
0
Ratio I o / I ~ as a f u n c t i o n o f the temperature difference A T = T o f o r t w o I n - P b alloy whiskers [40]. For T~o o f each sample see ref. 14.
T 181
1.0 I n , I n - Pb ~o~
% ~
~o
2
0.5-
0.2 100
J , l],~,, [ I ' 'I'"'I 1000 10 000 / / 1 0 -lo m
Fig. SO: R a t i o ( l o / I c ) o i n P b W as a f u n c t i o n o f t h e e l e c t r o n m e a n f r e e p a t h , Z, f o r s e v e r a l I n - P b w h i s k e r s [402 R e s u l t s f o r ( l o / l , ) o l n w o f p u r e In w h i s k e r s are added. T h e s y m b o l s are t h e s a m e a s t h o s e o f Fig. 48.
9.2.2.
The Range of Substantial Alloy Contents
The In-Pb alloy s y s t e m shows a nearly c o m p l e t e solubility over the whole range of concentrations. There is a small range b et w een the In crystalline solid solution and the Pb crystalline solid solution which a c c 1 mixed crystal e x i s t s in. The t w o - p h a s e regions are located b e t w e e n lead c o n c e n t r a t i o n s o f 12.7 and 13.7 at% and between 28.5 and 35.5at%, respectively [14, 476]. We investigated several samples with a lead c o n t e n t b et w een 13.5 and 97.S at%. It appears to be impossible to evaluate t h e characteristic p r o p e r t i e s of v o l t ag e s t ep s generated by phase-slip centers. The reason is t h a t ' p h a s e - s l i p s t r u c t u r e s ' disappear within the rise of big v o l t ag e st ep s (which we call ' r e s i s t i v e s t e p s ' ) . In the u, mixed crystal region evidence f o r the residue of a p h a s e - s l i p s t r u c t u r e may be d e t e c t e d E40]. This was n o t possible in the ch ar act er i s t i c s of whiskers from the Pb crystalline solid solution. The behaviour of the resistive steps is very d i f f e r e n t from the behaviour of a p h a s e - s l i p center. We assume that the observed s t r u c t u r e s are g e n e r a t e d by so called ' s e l f - h e a t i n g hot s p o t s ' E40]. Thus, a l t h o u g h cpb was enhanced up to 97.Sat% in the investigations of ref. 40, the In c o n t e n t was still too large to observe v o l t ag e st ep s g e n e r a t e d by p h a s e - s l i p centers. We recently p e r f o r m e d ex p er i m en t s with small In impurity c o n c e n t r a t i o n s which allow the study of p h a s e - s l i p s t r u c t u r e s [359]. The r e s u l t s will be r e p o r t e d in section 9.2.4. It is more c o m f o r t a b l e f i r st to discuss the results of pure Pb whiskers, because small impurity c o n c e n t r a t i o n s of In (or Bi) act to modify the behaviour of pure samples.
182
9.2.3.
Pure Pb
Also in the case of pure Pb whiskers, the V-T t r a n s i t i o n curves for small measuring c u r r e n t s show a sharp, f l u c t u a t i o n governed t r a n s i t i o n from the s u p e r c o n d u c t i n g to the normal state. If the measuring c u r r e n t is adjusted to higher fixed currents, the transition width (in temperature) is broadened. However, no voltage steps are observed [40]. Also the V-I characteristics do not show any voltage steps. Nevertheless, in the low voltage region of the characteristics, portions of c o n s t a n t differential resistance are observed, where the V-I characteristic is straight. In Figs. 51-S3 the V-l characteristics of sample P b 7 * are shown in three different solutions [40]. There is no hysteresis so that the characteristics follow the same line for increasing and decreasing current. For larger c u r r e n t s there is a separation of the c h a r a c t e r i s t i c s for increasing and decreasing c u r r e n t in the medium range, while close to the critical c u r r e n t both curves still join together {Fig. S1). This behaviour is generated by Joule heating effects. It is clear from Fig. 53 that the slope of the first linear portion is well described by V ( I ) = ( d V / d I ) ~ ( l - 1 0 ) with I0~I" for l ~ I c, leading to V(Io)=0. The r e s u l t I0/Ir ~ 1 is typical for pure Pb whiskers. The figures also show that the first linear portion is followed by others and that there is no separation of the linear portions by voltage steps. In sample P b 7 * the first linear portion is generated by a single phase slip center while the second linear portion is generated by three phase-slip centers. In Fig. 54 we p l o t t e d the corresponding normal-like lengths LAn 1 and LAn3 as a function of the temperature. We added r e s u l t s obtained from another lead whisker (sample Pb 12 ~) in which already the first linear portion is generated by three phase-slip centers.
4.00 Pb7*
VlpV300
/
Tco=7.2057 K T = Tco -,~ T
-
200-
100-
26.4 K
0
' 0
I 100
'
I 200
'
I 300
'
I 400
I
I 500
~
I i _.T/IJA
F i g . $1-" The V-1 characteristics
(increasing and decreasing current) o f the lead w h i s k e r 7 ~ f o r s e v e r a l f i x e d t e m p e r a t u r e s TE40J 183
J/
3O
Pb 7 *
VlpV
/ / "
///
T = Tco-,~T
,o
///"
10'
o
0 Fig. S2: L o w e r p a r t
10
20
30
50 I I pA
40
o f the V - I c h a r a c t e r i s t i c s (increasing and decreasing c u r r e n t ) o f lead w h i s k e r 7 ~. The dashed lines r e p r e s e n t the linear p o r t i o n s w i t h c o n s t a n t differential r e s i s t a n c e [401.
1.0
~'~g.,~
VIpV-
~t 100
Pb 7*
I/ Tco=7.2057 K / I T= TCO- 2.33 inky
..,,.J
//
0.5-
Pb
Fig. s4
50-
, ,,
0
, i ,,
10
C/,/t
, , ,,
20
0
i,,,,
30 I I ~A
4TImK 5
I
I
4
I
3
2
i
1
-
0
Fig. $3" O n s e t o f a V - I characteristic (increasing c u r r e n t ) o f lead w h i s k e r 7~ Dashed
lines
are parallel
to
the
lineal" p o r t i o n s
with
constant
differential
r e s i s t a n c e [40].
Fig. S4: (0): N o r m a l - l i k e
l e n g h t s LAn! and LAn 3 f o r s a m p l e P b 7 ~" as a f u n c t i o n o f t e m p e r a t u r e [40]. For this s a m p l e LAn ~ has been c a l c u l a t e d w i t h the differential r e s i s t a n c e o f the f i r s t linear portion, while LAn 3 is the r e s u l t f o r the s e c o n d p o r t i o n . The i n d e x "3' indicates that three p h a s e - s l i p c e n t e r s g e n e r a t e this p a r t o f the characteristics. (I) : Results
for
sample
P b l 2 ~* as
calculated
from
p o r t i o n [40]. Here, zlT=Tco-T., w i t h ~o the clqtical t e m p e r a t u r e given in refs. 14 and 40 (or also in Fig. 51 for Pb 7 *).
184
the
first
linear
o f each s a m p l e
as
It is c h r a c t e r i s t i c for lead whiskers that the n o r m a l - l i k e l e n g t h LAn 1 does n o t depend on t e m p e r a t u r e . Thus, again an averaged n o r m a l - l i k e length LAn I may be calculated f o r each sample and p l o t t e d as a function of the e l e c t r o n mean free path 2, showing the familiar 2'~2dependence [40]. The r e s u l t s obtained f o r pure Pb whiskers are summarized in a c o m m o n p l o t t o g e t h e r with n o r m a l - l i k e l e n g t h s of P b - l n and Pb-Bi whiskers, shown in the f oll o wi n g section. There are several samples where the f i r st linear p o r t i o n is not g e n e r a t e d by a single p h a s e - s l i p center. We r ef er to refs. 40 and 359 f o r a discussion how to handle this problem and how to evaluate a suitable value f o r LAn I in this case. F u r t h e r detailed investigations of the linear portion p h e n o m e n o n in the V-1 c h a r a c t e r i s t i c s of pure Pb whiskers show t h a t a curvature may appear in t h e beginning of the portion. To d e m o n s t r a t e this we r ef er to Fig. S5: This figure shows r e s u l t s of sample Pb 1 from ref. 359. The first linear portion is ge n er at ed by a single p h a s e - s l i p c e n t e r in the whisker. As the differential resistance of the second linear portion is twice that of the f i r st one, we conclude t h a t the second linear portion is g en er at ed by t w o p h a s e - s l i p centers. C l o s e to To0 the linear portions are only very slightly bended. (This is probably the reason why no curvature can be recognized in Fig. S4.) However, the curvature b e c o m e s s t r o n g e r for decreasing t e m p e r a t u r e until the slope b e c o m e s infinite at the critical current. Below this t e m p e r a t u r e the transition b e c o m e s hysteretic. Due to the bending of the linear portion, the ratio I 0 / I ~ slightly falls be l o w unity. At the t e m p e r a t u r e where the slope of the c h a r a c t e r i s t i c s at the critical c u r r e n t has become infinite, I 0 / I c is still ab o u t 0.94 and seem s to saturate.
L-
C
Pbl
d
e
f
Tco= 7.2157 K 3-
T =Tco-AT
:>
s
AT~InK a b c d e f
1-
0 , 2O
75
,// 210
L90
730 f/pA
1120
3.2 7.1 1L.1 25.2 32.9 L3.9 1150
Fig. ,5,5: V-I charactelqstics o f the pure lead whisker Pb I (for increasing and decreasing current) traced down to lower measuring temperatures to demonstrate the bending o f linear portions [3S9Z
185
3"
/
Pb3 Tco=7-1793 K
T-T~
~
b: AT=15.SmI~
/
AT
/
.
AT=iS.
o/ 200
/,
,,, 250
I
150
200
I/pA FIg. S6: V - I c h a r a c t e r i s t i c s o f t h e p u r e l e a d w h i s k e r P b 3 for increasing c u r r e n t [372]. (a) W i t h o u t H F r a d i a t i o n (b) With applied HF radiation of v = 300MHz. Current steps appear at Vj~ = ~ ( h / 2 e ) v , w i t h 1~ =1 a n d 2.
We a l s o a p p l i e d a H F - r a d i a t i o n field to a p u r e lead w h i s k e r ( s a m p l e P b 3 o f ref. 372), using a loop t u r n e d a r o u n d the w h i s k e r to c o u p l e the r a d i a t i o n t o t h e s a m p l e (see s e c t i o n 7 . 3 f o r d e t a i l s a b o u t this t e c h n i q u e ) . The r e s u l t o f such a m e a s u r e m e n t is s h o w n in Fig. 56. W i t h o u t HF r a d i a t i o n t h e V-I c h a r a c t e r i s t i c s e x h i b i t a linear p o r t i o n g e n e r a t e d by a s i n g l e p h a s e - s l i p c e n t e r in the w h i s k e r . In the HF field ' c u r r e n t s t e p s ' are o b s e r v e d . As w e l l the main s t e p at VjI = ( h / 2 e ) v as also the f i r s t harmonic step at V j 2 = 2 ( h / 2 e ) v occur, w h e r e v is t h e f r e q u e n c y of the HF field. In Fig. 57 a s e q u e n c e o f V-I c h a r a c t e r i s t i c s o f t h e s a m e s a m p l e in a HF field is s h o w n , w h e r e t h e f r e q u e n c y o f t h e r a d i a t i o n is a d j u s t e d to d i f f e r e n t v a l u e s f r o m 220 to 6 0 0 M H z . The c h a r a c t e r i s t i c s e x h i b i t the main c u r r e n t s t e p at V~I. We have widely d i s c u s s e d in s e c t i o n 7.3 t h a t t h e o b s e r v a t i o n of t h e s e ' i n v e r s e ac J o s e p h s o n - l i k e e f f e c t s ' e s t a b l i s h the e x i s t e n c e of p h a s e - s l i p p r o c e s s e s at J o s e p h s o n f r e q u e n c y in o u r s a m p l e s . Finally, a l s o in p u r e Pb w h i s k e r s t h e r e is e v i d e n c e for an i n t e r a c t i o n o f t h e p h a s e - s l i p c e n t e r s . Similar to the c a s e o f Zn w h i s k e r s ( s e c t i o n 8 . 2 ) we did n o t u n d e r t a k e a d e t a i l e d i n v e s t i g a t i o n o f t h e i n t e r a c t i o n o f p h a s e - s l i p c e n t e r s as in t h e c a s e of tin w h i s k e r s ( s e c t i o n 7 . S ) , b u t we d r a w this c o n c l u s i o n f r o m t h e b e h a v i o u r of t h e c u r r e n t r a n g e AI o v e r which t h e f i r s t linear p o r t i o n e x i s t s until the n e x t p o r t i o n a p p e a r s [3591. In Fig. 58 we p l o t t e d t h e r a n g e AI as a f u n c t i o n o f I~ ~3 f o r s a m p l e s P b l and P b 3 , w h e r e t h e f i r s t linear p o r t i o n is g e n e r a t e d by a s i n g l e p h a s e - s l i p c e n t e r in the s a m p l e . A l s o f o r Pb w h i s k e r s I~/3 is p r o p o r t i o n a l to AT=To0- T. For b o t h s a m p l e s ~I f i r s t i n c r e a s e s with i n c r e a s i n g c r i t i c a l
186
Pb3 Tco= 7"1793 K
/
r-- T o- A r
/
3-
A T=7.7+0.2 mK >1 2-
v/MHz :
/
220.
28(
2~
1-
0 ,
,
20
40
50
/,
,
30 80 Z/HA
ld0
189
FIg. S7:
V - I c h a r a c t e r i s t i c s o f the pure lead w h i s k e r P b 3 f o r increasing c u r r e n t in a H F radiation f i e l d at several d i f f e r e n t frequencies v between 220 and 500 M H z [372Z
[],r .,'
609
l) the KSS characteristic bends back so t h a t a v o l t a g e e x i s t s f o r c u r r e n t s less than I . For m e a s u r e m e n t s with an impressed c u r r e n t a v o l t ag e jump would now appear at Io and the transition would be hysteretic. T h e d e v e l o p m e n t of the characteristics with decreasing t e m p e r a t u r e is in qualitative a g r e e m e n t with the experimental r e s u l t s given in this chapter. Especially the s t r a i g h t - l i n e behaviour of our V-I ch ar act er i st i cs close to To0 and, thus, I0=I , would be predicted by the KSS model. The transition to a hysteretic behaviour, however, occurs already for t e m p e r a t u r e s much c l o s e r to T00 than predicted by KSS. In ref. 359 we showed that this observation may be caused by heating e f f e c t s and, therefore, does not necessarily c o n t r a d i c t the KSS model. In whiskers made of Pb the linear-portion phenomenon is observed over a wide t e m p e r a t u r e range. Small impurity c o n c e n t r a t i o n s of In or Bi act to shift the t e m p e r a t u r e below which voltage jumps appear toward To0. For whiskers o f o t h e r materials (Sn, In, and Zn) the d e v e l o p m e n t of the V-I c h a r a c t e r i s t i c with decreasing t e m p e r a t u r e seems to be qualitatively similar. However, the t e m p e r a t u r e range which linear portions or rounded v o l t ag e steps are observed in is very small and very close to To0, where f l u c t u a t i o n e f f e c t s may influence the characteristics, and the e f f e c t is only observed in a few experiments. There may be evidence for the phenomenon in the V-I characteristics c l o s e s t to To0 given in refs. t6,36,37, and 20 Fig. 2 (equal to 192
Fig. 42 of the p r e s e n t work) and in the c har act er i st i cs of s o m e of our unpublished e x p e r i m e n t s . A l s o t h es e f e a t u r e s can qualitatively be u n d e r s t o o d within the analytical a pp ro x i m at i o n of the KSS model: The t e m p e r a t u r e range of the l i n e a r - p o r t i o n phenomenon is wide in the s t r o n g - c o u p l i n g materials because zc is so small. From Tab. A1 one can see that z E of Pb is at l e a s t one order of magnitude s m a l l e r than the values for Sn or In. Thus, there is a much wider t e m p e r a t u r e region in which ZE/ZO, is much s m a l l e r than unity. If the e l e c t r o n mean free path, 2, of a Pb whisker is reduced by impurities, the ratio zE/z0, is larger than in the clean case, for a given t e m p e r a t u r e difference h T = T c 0 - T , and the t e m p e r a t u r e range shrinks which linear portion phenomena are predicted in by KSS. The reason for the increase of zE/Zo, is its proportionality to :~/.e which increases with decreasing ~? as long as the sample is not in the dirty limit. The ratio zE/Zo,, f u r t h e r m o r e , depends on To and z E. The critical t e m p e r a t u r e of our Pb whiskers with small c o n c e n t r a t i o n s of In or Bi is similar to t h a t one of pure Pb whiskers so t h a t its influence on the change of z E/z0. is of no importance. A decrease of z~ of Pb with decreasing 2 (Fig. A l l , if there really is any (compare Tab. A i) would weaken the increase of zE/Zoa. The influence of impurities on the shape of the V-I c h a r a c t e r i s t i c s o f Pb whiskers, thus, is in qualitative a g r e e m e n t with the KSS model. We should, however, remark that the o n s e t of a sharp hysteretic transition into the v o l t a g e - c a r r y i n g s t a t e and its dependence on the e l e c t r o n mean free path is quantitatively b e t t e r described by J o u l e heating e f f e c t s [359]. Thus, the general shape of the V-I characteristic of Pb whiskers (and Pb whiskers with In and Bi impurities) is qualitatively predicted by the analytical approximation of the KSS model. The t e m p e r a t u r e independence of the differential resistance, however, cannot be u n d e r s t o o d within t h e KSS model either. We do n o t only s t a t e the mentioned qualitative agreement, b u t also s p e c u l a t e s o m e w h a t a b o u t a physical i n t e r p r e t a t i o n of the r e s u l t I 0 / I r l: In the KSS model the expression for the dependence of the time averaged v o l t a g e on the t o t a l c u r r e n t is formally the same as the SBT result. Therefore, the measured i n t e r c e p t I 0 also in the KSS model may be i n t e r p r e t e d as time averaged s u p e r c u r r e n t in the core of the p h a s e - s l i p center. However, as discussed in detail in sectionS.8.4, the SBT r e s u l t is actually only recovered in the high v o lt a g e dc limit. The reason is t h a t in the SBT model it is a s s u m e d t h a t the phase angle difference b e t w e e n t w o points of the s u p e r c o n d u c t o r develops linearly in time. In this case the time average of the s u p e r c u r r e n t in the core can be calculated by averaging its dependence on the phase angle within one period. In general {especially for low voltages) it is n o t al l o wed to replace the time averaged s u p e r c u r r e n t in the core, (Is(Xpsc, t ) ) , by its phase angle average. The reason is t h a t the v o l t a g e across the core region and thus (via the J o s e p h s o n relation) the phase angle difference is a c o m p l i c a t e d function of time due to the excitation of charge 193
imbalance waves by the p h a s e - s l i p cycle of the core. Therefore, in general ( I s (X~sc, t ) ) is not independent of the t i m e - a v e r a g e d v o l t ag e (as in the SBT model) b u t changes along the v o l t a g e - c u r r e n t characteristics. This leads to a significant deviation from the high voltage dc s t r a i g h t line behaviour f o r low voltages. The experimental r e s u l t 10~l o can for instance be u n d e r s t o o d if the s u p e r c u r r e n t in the core increases very rapidly in time to a value c l o s e to Io, then creeping t o w a r d Io which is reached at the end of the p h a s e - s l i p cycle. This behaviour m u s t be a c o n s e q u e n c e of the time d e v e l o p m e n t of the phase difference across the core region (related to the time d e v e l o p m e n t of the vol t ag e across the core region by the j o s e p h s o n relation) in connection with the c u r r e n t - p h a s e relation connecting the phase angle difference with the s u p e r c u r r e n t in the core. In o t h e r words, if the c u r r e n t - p h a s e relation has a simple form (for instance sinusoidal), the voltage across the core region m u s t be s t r o n g l y time dependent. (One may c o n s u l t the r e s u l t s of the RSM paper [265] in this c o n te x t . ) On the o th e r hand, if the v o l t ag e is nearly time independent, the c u r r e n t - p h a s e relation m u s t be complicated. From the e x p e r i m e n t s p e r f o r m e d with Pb whiskers in a HF radiation field one would expect a sinusoidal c u r r e n t - p h a s e relation, because no subharmonic c u r r e n t steps are observed in the V-I characteristics (see Figs. 56 and 57, and compare the discussion in section 7.3). This c o n c l u s i o n was drawn considering our calculation of the t i m e - a v e r a g e d s u p e r c u r r e n t in the core which a s s u m e s a time independent voltage across the core region (see section T.3). If the p r e s e n t a r g u m e n t a t i o n concerning I 0 / I ~1 should hold, a c om p l i cat ed s t r o n g l y time dependent behaviour of the v o l t ag e across the core would be required. Let us assume that our a r g u m e n t concerning the I0/Io prob l em given above is correct. Then this discrepancy seems to indicate t h a t the c u r r e n t steps occurring in the V-I c h a r a ct er i st i cs in a HF field can be described by the calculation of section 7.3 a l t h o u g h the properties in the core are much more involved in detail than considered in that calculation, Finally, the experimental r e s u l t s of the p r e s e n t ch ap t er may be compared with the predictions of the TDGL theory as elaborated by KR and widely discussed in section 5.9. The conclusions of this comparison will be summarized briefly and refs. 40 and 359 may be c o n s u l t e d for f u r t h e r details *'. Moreover, the f o ll o w in g chapter deals with a c o m p a r i s o n of measured p r o p e r t i e s of an isolated phase-slip c e n t e r with the predictions of the TDGL theory. The central r e s u l t elaborated by KR from the TDGL theory is a prediction AK, f o r the behaviour of the n o r m a l - l i k e length LAn 1 and a prediction ~k, Io for the zero voltage in te r c e p t I0. (One may co m p ar e eqs. (174) and (t87) to see t h a t LAnl=AKR and I0=[3~,1o.) The r e s u l t s for AK~/~0(T) as a function of the pair-breaking parameter, y, are p l o t t e d in Fig. 20, while the
9 1 Note t h a t the quantity LAnI,,, introduced in ref. 40 is equal to AK.. 194
ratio I0/Io=[3K, is t a b u l a t e d in the work of KR [316]. The quantities are discussed in detail in section 5.9 and at the end of section 7.2. For an i s o l a t e d phase slip center in a h o m o g e n e o u s f i l a m e n t it is L A n I = 2 A Q ~ i n ~ A T -~t" for very large p a i r - b r e a k i n g p a r a m e t e r s ( ~ > 1 0 0 0 ) only. However, also f o r s o m e w h a t s m a l l e r `( values the AT -~4 t e m p e r a t u r e law is a ppr o x i m at el y obtained, because LAn l only changes to about 2.6AQ~in if ~" is lowered to ab o u t lO0. If ~ is f u r t h e r decreased a continuous change to a s t r o n g e r t e m p e r a t u r e law is predicted. For T ~ 10 it is LAn 1 ~ 7~D(T) ~ AT -~2. For T1< b ---.I I
1 '
I
~'~ 100
l:)lg. 62: N o r m a l - l i k e length, LAn I , normafized by ~he GL coherence length,
~, as a function o f the p a i r - b r e a k i n g parameter, y. Experimental values f o r an isolated phase-slip center in whiskers o f different materials and at different t e m p e r a t u r e s close to T~o, t o g e t h e r with the prediction o f the TDGL theory. This figure shows some selected samples appearing in the medium y range o f Fig. 61. To demonstrate the single-sample behaviour we used different symbols f o r the different whiskers: ( 0 ) ln17, (r'l) In20, ( O ) l_p_n-Pb2, ( 9 Ln - Pb S, ( ~ ) In-Pb 6. I, ( 0 ) l__q-Pb 13, ( V ) ln-Pb 14, ( ~7) Sn 3 part 3, ( 9 ) S__~n-ln7. The meaning o f the f u l l lines and the regions ' a ' - ' d " is the same as explained in the caption o f Fig. 61.
the appendix o f ref. 147. Several r e s u l t s are also given in the p r e s e n t work, such as for t wo of the Pb whiskers (Fig. Sr all I nn-Pb whiskers (Figs. 46 and 47), and all In whiskers (Fig. 30). The r e s u l t s for a single sample do not s c a t t e r very much, so that the t e m p e r a t u r e dependence of LAn I can clearly be observed. For some r e p r e s e n t a t i v e samples we p l o t t e d L A n l / ~ as a function of y in Fig. 62, making the behaviour of the individual sample visible by using d i f f e r e n t s y m b o l s for each specimen. The s y s t e m a t i c d e v e l o p m e n t o f LAn 1 on the t e m p e r a t u r e s t a t e d above leads to the s y s t e m a t i c change of L A n l / ~ o n 7. It appears, t h a t for d i f f e r e n t specimens of the same type of material the magnitude o f the quantity L A n l / ~ at the same value o f y differs by a f a c t o r of up to four. We usually found t h a t whiskers of very h o m o g e n e o u s pure material show values of LAnl/~ close to the upper border marked in Fig. 61. Alloy whiskers have l o w e r values, down to the lower border. Samples which are less h o m o g e n e o u s generally have low values of LAnl/~.
201
A very i m p o r t a n t information which may be obtained from Fig. 60 concerns the t e m p e r a t u r e dependence of LAn 1 for a given sample. This t e m p e r a t u r e dependence can be concluded from a known dependence of LAnl/~ on y, considering that for a fixed sample it is y ~ A T 1/~, while ~ ~ A T "1~2. In the range where it is L A n l / ~ y , it is LAn 1 independent of the temperature. If L A n l / ~ ~ Y"2 it follows LAn 1 ~ AT .... , while LAnl/~ independent of y leads to LAn 1 ~ AT 1'2 and LAnl/~ ~ y - ' y i e l d s LAn 1 ~ AT -1. Thus, Fig. 61 is also a map for the t e m p e r a t u r e laws of the n o r m a l - l i k e length, giving information about the observed t e m p e r a t u r e law of a fixed sample in a certain y range. While LAn I is found to be t e m p e r a t u r e independent for small and large values of y (regions 'a'), t h er e is an intermediate range where LAn 1 shows a AT "~/4 law ( r e g i o n ' b ' ) or a AT -~2 law ( r e g i o n ' c ' ) or even a AT ~ law ( r e g i o n ' d ' ) , depending on the value of y. If there is a certain y range where the mentioned regions overlap, d i f f e r e n t t e m p e r a t u r e laws were observed f o r d i f f e r e n t specimens. Whiskers made of Pb and P_.bb-ln alloys have very small values of y whereas samples of Sn and S..nn-In alloys have large values of y. Therefore, t he s e specimens show a length LAn I that is independent of t e m p e r a t u r e . On the o t h e r hand, samples made of pure In and I n - P b alloys have y values in the i n t e r m e d i a t e range and, thus, their n o r m a l - li k e length may show d i f f e r e n t t e m p e r a t u r e laws. The prediction of the TDGL theory is within the range of our experimental data, except the r e s u l t for the static limit for y s The static limit is the lower b o r d e r down to which inhomogeneities can d ep r ess the value of LAnl/~ below the r e s u l t for the homogeneous filament. The main difference b et ween e x p e r i m e n t and theory is that the theory does n o t predict LAnl/~y that means LAn ~ independent of temperature. Furthermore, the increase of LAnl/~ with growing y is not as s t r o n g as observed experimentally. Therefore, the change from LAn~/~ independent o f T to L A ~ I / ~ T '/2 (related to a change of L A n l ~ A T -''2 to L A n l ~ A T -~/') is predicted for much larger values of y than observed in our experiments. At the same time the intermediate region b e tw e e n both dependences is predicted to be much b ro ad e r than experimentally observed. Now let us discuss the behaviour of I 0 / I 0 p l o t t e d in Fig. 63: The experimental r e s u l t s decrease from I0/Io=1 at very small values of y to I0/I ~ ~ 0 . 5 at y ~ 5 0 . The prediction of the TDGL theory is b el o w the experimental results. Nevertheless, f o r ~" between 2 and 20 the ratio I0/Io decreases with the same slope as the theoretical prediction for ~" b e t w e e n 10 and 80. Then the dependence of our experimental r e s u l t s b e c o m e s much weaker. Indeed, the e x p e r i m e n ts p e r f o r m e d with Zn whiskers discussed b e l o w indicate t h a t there seems to be no continuous decrease of I0/I c with decreasing y. Next we will discuss how far long microbridges of Sn and In join our universal plots. For the m o s t part we only will consider the behaviour o f L A n l / ~ on y because data f o r 10/I c are too rare. The history of e x p e r i m e n t a l 202
1,0--
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I ''"
100
Fig. 63: Ratio Io / I as a function o f the pair-breaking parameter, % f o r small and m e d i u m values o f 7 EI47Z Experimental values f o r an isolated phase-slip center in whiskers o f different materials (Pb, In, Sn, P_bb-In, ln-Pb, S n - l n ) and f o r different temperatures close to T~o , together with the prediction o f the TDGL theory. We did not use different s y m b o l s For the different samples in this figure. investigations of p h a s e - s l i p c e n te r s in long microbridges was r e p o r t e d ill section7.2, containing i n f o r m a t i o n s concerning the d i f f e r e n t experiments. If not explicitly given, detailed information about the evaluation procedure of the values fo r L A n l / ~ and T of microbridges given below may be obtained from ref. 147. First, we are going to discuss a group of m e a s u r e m e n t s where the n o r m a l - l i k e length has been evaluated from the differential resistance of the V- I characteristics. The Sn microbridges of SBT showed a t e m p e r a t u r e i n d ep en d en t n o r m a l - l i k e length [321. A typical r e s u l t f o r their bridge would be T ~ S1 and LAnl/~S. The T value lies in region 'a' of Fig. 61. The ratio L A n l / ~ is s o m e w h a t s m a l l e r than we observed in our experiments. In this range of T values our s am p l e s also show a t e m p e r a t u r e independent LAn 1. Since the bridge m en t i o n ed does not c o n s is t of h o m o g e n e o u s material (see Fig. 2 of ref. 32) we should not worry a b o u t the small value of LAnl/~. On the o t h e r hand, Kadin, Skocpol, and Tinkham measured an Sn microbridge showing a differential resistance changing proportional to AT -1/4 b e t w e e n AT=28.6 and 72.4mK (that means for T ranging f r o m 77.8 to 123.8) *~ and becoming t e m p e r a t u r e independent for even l o w er t e m p e r a t u r e s *-1 For the d e t e r m i n a t i o n of y we calculated r E a f t e r Tinkham's e s t i m a t e given in sectionS.4, with To0=3.998K as obtained from Fig. g b of ref. 84. The o t h e r material p a r a m e t e r s are the same as t h o se used f o r the o t h e r Sn microbridges discussed (see appendix A2 of ref. 147).
203
(see Fig. q b of ref. 84). Since the t e m p e r a t u r e dependence of the r el at ed n o r m a l - l i k e length is the same as that one of the differential resistance, it f o l l o w s LAn i ~ AT -1/4 or L A n l / ~ 2 ' ~/2 in the mentioned 2" range. (The a b s o l u t e magnitude of LAnl/~ cannot be calculated, because the sample p a r a m e t e r s needed are not given in ref. 84.) The m e a s u r e m e n t s of Kadin, Smith, and Tinkham are p er f o r m e d in a y range which lies above the region of experimental values of Sn and S__n-In whiskers in Fig. 61 or 62. In this y range no experimental r e s u l t s from whiskers are available. A t e m p e r a t u r e d e p e n d e n t n o r m a l- l ik e length L A n l ~ A T -'~" ( t h at means L A n l / ~ Yl/:) also was obtained from m e a s u r e m e n t s of the differential resistance o f a long indium microbridge by Jillie [33], with LAn~/~ increasing from 3.3 to 5.7 for 2' growing from 11.4 to 33.8. Since t h ese y values range within region ' b ' of Fig. 61 we would e x p e c t the observed t e m p e r a t u r e law. The ratios L A n l / ~ are within the borders of our experimental results, cl o se to the l o wer border. Also Weissbrod et al. [369~ investigated phase-slip cen t er s in super co n d u ct i n g indium microbridges. In Fig. 7 ~1 o f their paper they give r e s u l t s fo r L A n l / ~ as a function of ~ for two d i f f e r e n t samples, called WInIV (lOOvm long) and W l n X b (13Otlm long). To characterize the behaviour of the samples one may fit s t r a i g h t lines through the p l o t t e d r e s u l t s . In the case o f sample W l n l V it is L A n l / ~ y ~/= (that means LA n I ~ A T ) f o r 2" between 36.6 and 78.1 and L A n l / ~ y ( t h a t means LAn 1 independent of t e m p e r a t u r e ) f o r ~" b e t w e e n 78.1 and 128.1. The a b s o l u t e values f o r L n n l / ~ are 15.5, 23.1, and 37.8 f o r 2"=36.6, 78.1, and 128.1, respectively. In the case of sample W l n X b it is L A n l / ~ T ~/2 over the w ho l e 2" range f r o m 25.6 to 70.3, with L A n l / ~ = S . 6 and 9.1 for 2'=23.6 and 70.3, respectively. In b o t h cases t e m p e r a t u r e dependent n o r m a l - l i k e le n g th s are observed in regions where our whiskers already show a t e m p e r a t u r e independent behaviour o f this quantity. The change to a t e m p e r a t u r e independent behaviour observed in one of the samples occurs at a T value which is outside the range of our m e a s u r e m e n t s . The a b s o l u t e values for both samples are lying within the b o r d er s (or e x t r a p o l a t e d borders) of our experimental r e s u l t s (close to the upper b o r d e r in the case of sample W l n l V and close to the lower border for W l n X b ) . Weissbrod et al. also evaluated the ratio I 0 / I ~ (for two In microbridges of 30Fm length) and p l o t t e d the r e s u l t s in Fig. 5 of ref. 369 as a f u n c t i o n of -
1/4
9 1 The r e s u l t s for L A n l / ~ given in that figure have to be multiplied by a f a c t o r of 2~0/~. If one uses ~0=440nm as taken in t h a t work, this f a c t o r yields 4.94 and 3.74 for sample WInIV and W I n X b , respectively. For our discussion we prefer to take ~0=0.2944ilm [40] so that the f a c t o r is S.91 f o r WInIV and 4.33 for W I n X b . The authors of ref. 369 use ~E=8Ops. For our discussion we p r e f e r to cal cu l at e zE a f t e r Tinkham's e s t i m a t e given in sectionS.4., leading to z~=250ps f o r both samples, if we use their measured critical t e m p e r a t u r e and the material parameters for pure In as given in ref. 40. Thus, we multiplied the Z values obtained from Fig. 7 of ref. 369 by 250 p s / 8 0 ps =3.125. 204
the t e m p e r a t u r e . For decreasing t e m p e r a t u r e decrease and then increase again. We will
the not
experimental discuss the
values f i r s t temperature
d e p e n d e n t behaviour in detail, but only s t a t e that I 0 / I o of their sam p l es shows values b e t w e e n 0.7 and 0.5 in the t e m p e r a t u r e range b e t w e e n 3.3954 K and 3.3354K, t h a t m e a n s for 7 b e t w e e n 30.5 and 96.7 (if we again c a l c u l a t e ~E as described in the last f o o t n o t e ) . This r e s u l t is in a g r e e m e n t with our m e a s u r e m e n t s given in Fig. 63, as far as there is an overlap of the ~f ranges, and e s t a b l i s h e s the saturating tendency of 1 0 / I . Now, we are going to discuss a group of e x p e r i m e n t s where the quasiparticle diffusion length, A, was obtained from spatially r e s o l v e d m e a s u r e m e n t s o f the e l e c t r o c h e m i c a l quasiparticle potential near a p h a s e - s l i p c e n t e r and where the n o r m a l - l i k e length may be calculated as L A n l ~ 2 A according to SBT: In all cases th e s e m e a s u r e m e n t s yield a quasiparticle diffusion length A (and, thus, also a n o r m a l - l i k e length LAn 1) changing proportional to A T - ' / ' ( th a t means proportional to "("2). From the m e a s u r e m e n t s of Dolan and Jackel [70] for a tin bridge we g et L A n l / ~ increasing f r o m 5.7 to 7.9 for y growing from 63.8 to 121.3. M e a s u r e m e n t s o f Aponte and Tinkham [723 lead to LAnl/~ increasing from 3.4 to 6.9 for T growing from 30.1 to 123.5. These r e s u l t s are below the b o r d e r of our experimental values in Fig. 61. As far as there is an overlap of the y ranges with o u r experiments, the y values are so large already, t h a t our m e a s u r e m e n t s show a t e m p e r a t u r e independent n o r m a l - l i k e length. A l t h o u g h in some cases the behaviour of Sn and In microbridges are in a g r e e m e n t with our m e a s u r e m e n t s (SBT and Jillie), the c h a r a c t e r i s t i c d i f f e r e n c e b e t w e e n whiskers and Sn and In microbridges seems to be t h a t microbridges show a t e m p e r a t u r e dependent n o r m a l - l i k e length in a ~" range where LAn 1 f o r whiskers has already become t e m p e r a t u r e independent (Weissbrod et al., Aponte and Tinkham). In microbridges this t e m p e r a t u r e dependence was found up to a ~" value of about l20 (Kadin et al., Dolan and j a c k e l , Aponte and Tinkham). The reason fo r the d i f f e r e n t behaviour of microbridges and whiskers is finally not clear. For whiskers there should be no masking of a AT -1/4 law by heating e f f e c t s which are usually invoked to explain the observation of a t e m p e r a t u r e independent n o r m a l - l i k e length in the case of microbridges. We have already discussed this point s o m e w h a t in detail in section 7.2. May be t h a t the d i f f e r e n t grade of homogeneity plays a role. Already in section 7.2 we discussed that compared to whiskers, microbridges are usually much more i n h o m o g e n e o u s samples with s t r o n g variations of the critical t e m p e r a t u r e along the bridge and that pinning centers, such as notches, are widely used to fix the p h a s e - s l i p c e n t e r in a bridge. The influence of inhomogeneities may be indeed a probable explanation. As already m en t i o n e d in section 7.2, the e x p er i m en t s of Liengme et al. discussed b el o w seem to indicate this. Moreover, very r e c e n t e x p e r i m e n t s on tunable weak links seem to establish this s t a t e m e n t (chap. 12). In t h ese e x p e r i m e n t s the properties of a phase-slip c e n t e r were studied which 205
appeared in a weakly superconducting section induced into a whisker. The temperature dependence of the differential resistance of the V-I c ha ract er i s t i cs g e n e r a t e d by this phase-slip center shows a dependence on the s t r e n g t h of superconductivity in the weakly superconducting section. Also from the TDGL theory one would e x p e c t a s t r o n g influence of inhomogeneities on the properties of a phase-slip c e n t e r developing in an i n h o m o g e n e o u s section o f a specimen (see the discussion in the p r e s e n t section and section 5.9). The problem is that the theory predicts a weaker t e m p e r a t u r e dependence of the n o r m a l- l ik e length if i n h o m o g en ei t i es are considered. Naturally we should also discuss why microbridges only yield n o r m a l - l i k e lengths with LAn 1 ~ AT "1/4, but why there is no e x p e r i m e n t r e p o r t e d where LAn 1 changes proportional to AT -''~ or AT -~ for T approaching To0. The reason is probably that the values of y for microbridges were not small enough to observe a n o r m a l - l i k e length with one of the s t r o n g e r t e m p e r a t u r e laws. Concerning the TDGL theory a very i m p o r t a n t question is, how far the change to a t e m p e r a t u r e independent n o r m a l - l i k e length may have s o m e t h i n g to do with going beyond the validity range of the local equilibrium approximation. The local equilibrium range is at l east l e f t if ~o(T) b eco m es sma l l er than A E which means that AT exceeds ATe, or y exceeds ~'max, or ZGL be c o m es s m a l l e r than z e (see sectionS.9). For all Pb and P___b_b-In whiskers mentioned in Fig. 61 it is Ymax = 8.8. The y values of these samples are much smaller than this limit. For all In and I n - P b whiskers of this figure (and for sample In) we have already discussed in section 9.3, that AT l , while the criterion is y > y c ~ 5 . S in the TDGL theory. Thus, the criterion for the o n s e t of hysteresis is given by ~E/%R=1 and "(=To, respectively. We calculated x J % , ( A T H y P ) a n d x a : " = ~ ( A T a : . ~) for several I n - P b whiskers [40] and found that the KSS criterion is fulfilled within a factor of 2. Here, the experimental r e s u l t s xr p) show values b e t w e e n O.S and 1,9. In the case of the TDGL criterion we observed that exp ~(Hy is always larger than ~'o (except for one sample). This means that the measured o n s e t of hysteresis occurs at lower t e m p e r a t u r e s than predicted by the TDGL theory. For Pb, P__bb-In, and Pb-Bi whiskers no intrinsic hysteresis was observed. If there is any hysteretic behaviour present, it seems to be caused by thermal effects [60, 359]. This observation is c o n s i s t e n t with the KSS criterion KSS "%/%,=1 which leads to a prediction ATHy for the o n s e t of hysteresis which is larger than 8 0 0 m K for pure Pb whiskers. In the case of P b - I n and P_bb-Bi whiskers ATHy Kss ranges between 160 and 370mK, at least an order of Texp ~ The observation is, however, also magnitude larger than the measured h_Hy consistent with the TDGL criterion, because for Pb, Pb-ln, and P__bb-Bi whiskers the measuring temperatures are so close to Tr that the related values are much smaller than ~" [40,147,359]. It is remarked that Tinkham's estimate for ~E (see section S.4) is used throughout. We also investigated the hysteretic behaviour of V-I characteristics of our whiskers under the influence of HF radiation for whiskers made of Sn E371], In [16,372], and l n - P b alloys [40] ~L In all cases HF-radiation field induced ' c u r r e n t steps' are also observed in the hysteretic part of the characteristics so that the occurrance of phase-slip processes at Josephson frequency is established also in the hysteretic regime. Recently, Weissbrod et al. [369] investigated the hysteretic behaviour of the V-I characteristics of long indium microbridges. They found the development of IR/I o with decreasing temperature to be c o n s i s t e n t with the
*' See the long version of that work (denoted by ref. 9 in that paper). 213
KSS model which predicts a saturating behaviour of this quantity. The o n s e t of hysteresis in their samples occurs much closer to To0 than predicted by KSS *~. For a s y s t e m a t i c investigation of hysteresis e f f e c t s in whiskers we f i r s t studied the heat t r a n s f e r properties of whiskers from the In-Pb alloy s y s t e m to the surrounding helium E48OL For this purpose we evaluated the width of the t e m p e r a t u r e hysteresis, ~TH, at fixed c u r r en t I F. If the c u r r e n t s are sufficiently large, s e l f - h e a t i n g hot s p o t s appear in the sample and the hysteresis is g e n e r a t e d by thermal e f f e c t s . Since in t h ese e x p e r i m e n t s the heat removal occurs mainly through the surface of the sample, the heat t r a n s f e r c o e f f i c i e n t and thus the Kapitza resistance can be easily evaluated from these experiments. Similar investigations have been p e r f o r m e d with Zn and Z n - A g whiskers to obtain the Kapitza resistance of zinc below 1K [481]. Then we s t a r t e d detailed investigations of the hysteretic behaviour for lower measuring currents, where the appearance of p h a s e - s l i p c e n t e r s governs the dissipative state. Also in this case the t e m p e r a t u r e hysteresis, ~T., was chosen to characterize the hysteretic behaviour of the dissipative p h a s e - s l i p state. The reason is that the behaviour of this quantity in the case of a thermally dominated hysteresis is well known from our investigations of the heat t r a n s f e r properties of our filaments. Therefore, deviations f r o m t h a t behaviour are e x p e c t e d to be caused by intrinsic, nonthermal e f f e c t s . Already in the e x p e r i m e n t s concerning the heat t r a n s f e r p r o p e r t i e s of l n - P b whiskers we observed a 'footlike' s t r u c t u r e in the ~T.(IF 2) p l o t at small c u r r e n t s and we s u g g e s t e d that this s t r u c t u r e is caused by c o n t r i b u t i o n s of a n o n t h e r m a l hysteresis [4803. Therefore, we p e rf o r m ed detailed m e a s u r e m e n t s of 8T H and p l o t t e d the quantity as a function of Ir2. This can be done by a direct observation of ~ T . ( I F ) = T ~ ( I ~ ) - T , ( I ~) in a V-T curve at fixed c u r r e n t I F. An o t h er possibility is to measure V-I c h a r a c t e r i s t i c s at several fixed bath t e m p e r a t u r e s and to plot Ij/~ and IR"2/3 as a function of the bath t e m p e r a t u r e (see for instance Fig. 21). Then the quantity ~TH(I F) is obtained as the distance in t e m p e r a t u r e b e t w e e n the I02/3 and IR2/~ branch of such a p l o t at an arbitrary but c o n s t a n t value I~2/3. Naturally, also the directly measured T ( I F) and TR(IF) values join this plot. Thus, our usual procedure is to summarize b o t h kinds of m e a s u r e m e n t s in a common plot, then to draw
*~ We do n o t agree with the calculation of "~o. as p e r f o r m e d in ref. 369. Since the microbridges are not 'dirty', the expression for arbitrary s as given in eq.(ll2) of the p r e s e n t work should be applied and n o t the dirty limit result. Using the material p a r a m e t e r s given in ref. 369 and VF for pure In E40], AT Kss fo r the microbridges investigated ranges b et w een 28 and 104 mK for Hy the choice zE=8Ops and b e tw e e n 20 and 75mK for the choice z~=llOps. We also e s t i m a t e d z~ a f t e r Tinkham using the measured critical t e m p e r a t u r e s of ref. 369 and the o t h e r p a r a m e t e r s for pure In as given in ref. 40, r es u l t i n g in z E~ 2.5- 101~ s f o r all microbridges. With ~0 =0.2944 ~tm Kss between 9 and 33 mK. and the measured ~? and 'F,o it is f i T Hy 214
estimated slopes through the experimental points of the lc2~3 and I, 2/3 branch, and then to determine ~TH(I ~) and, thus, gTH(IF2) from such a plot. We will i l l u s t r a t e the method somewhat in detail in section 11.3. Using this method we first studied the hysteretic behaviour of pure Sn whiskers [332]. Indeed a footlike s t r u c t u r e appeared in the gT,(I~=) p l o t for small values of I~2. Since this s t r u c t u r e is assumed to be generated by intrinsic hysteretic effects of the phase-slip process, we compared the experiments with the theoretical predictions of KSS [158] and the TDGL r e s u l t s of KR [316]. It t u r n s out that the m e a s u r e m e n t s are not described by one of the existing theories [332]. Therefore, we developed a phenomenological hysteresis model [3321, considering s e l f - h e a t i n g effects of the phase-slip center, hysteresis effects generated by charge imbalance waves ('KSS mechanism'), and an overpopulation of the excitation spectrum by nonequilibrium quasiparticles ('T* mechanism'). It t u r n s out that none of the mechanisms alone is able to describe the observed hysteresis. A combination of the three mechanisms is needed for a successful explanation of the measured hysteresis effects in tin whiskers. We s u b s e q u e n t l y investigated the intrinsic hysteresis of phase-slip centers in Zn whiskers [482]. Also in this case our phenomenological hysteresis model is able to describe the observed effects and to give us i n f o r m a t i o n how s t r o n g the different mechanisms c o n t r i b u t e to the t o t a l hysteresis width. As a next step we somewhat improved the phenomenological model and then again successfully applied the model to the materials listed above [483]. Moreover, we measured the hysteresis of whiskers from In, In-Pb, and Pb and again s u c c e s s f u l l y applied the phenomenological model [483]. In this version, our phenomenological model still contained two fitting parameters. These parameters are needed for the calculation of the T * c o n t r i b u t i o n to the hysteresis. To get a phenomenological model without any fitting parameter, we developed a more detailed description of the generation and healing processes of nonequilibrium quasiparticles during the phase-slip cycle. The calculations r e s u l t in an expression for the T* c o n t r i b u t i o n to the hysteresis in which one of the former fitting parameters is equal to unity, while the other parameter is not simply a constant, b u t a function of the voltage developed by the phase-slip center [483]. A comparison of the fitting parameter free version of the phenomenological hysteresis model with experimental hysteresis widths of Sn, In, In-Pb, and Zn whiskers r e s u l t s in a satisfactory agreement b e t w e e n m e a s u r e m e n t and model calculation [483]. This agreement indicates that our model seems to contain the basic mechanisms which generate the hysteresis of a p h a s e - s l i p center in our q u a s i - o n e - d i m e n s i o n a l filaments. I n the following sections the ideas of our systematic investigation of hysteresis effects will be discussed somewhat in detail. We shall s t a r t with a 215
description of the heat t r a n s f e r or Kapitza resistance problem in section 11.2. Then m e a s u r e m e n t s of the hysteretic behaviour of a tin whisker will be shown and the evaluation procedure of the t e m p e r a t u r e hysteresis will be described (section ll.3). In section ll.4 our phenomenological hysteresis m o d el is explained. We s t a r t with the version containing two fitting p a r a m e t e r s which we compare with experiments, before the d e v e l o p m e n t of the model w i t h o u t f i t t i n g p a r a m e t e r s will be described. Then, also this model will be compared with experiments. Finally, there is a separate section which we discuss the predictions of the TDGL theory in and compare the c a l c u l a t e d hysteresis with our experiments.
11.2. H e a t T r a n s f e r f r o m a M e t a l l i c F i l a m e n t Into H e l i u m - T h e K a p i t z a R e s i s t a n c e
While the c u r r e n t - i n d u c e d breakdown of superconductivity in thin f i l a m e n t s very cl o s e to the critical temperature, To0, is characterized by a region of nonequilibrium superconductivity between the su p er co n d u ct i n g and the normal conducting state, the transition is governed by the f o r m a t i o n of s e l f - h e a t i n g hot s p o ts f o r t e m p e r a t u r e s far below To. There are detailed t h e o r e t i c a l and experimental investigations of the h o t - s p o t p h e n o m e n o n r e p o r t e d in the l i t e r a t u r e [260, 4 8 4 - 4 9 8 ] . In the h o t - s p o t regime there is a large t h er m al hysteresis, 8T H, which may be measured directly by the observation of V-T transition curves or evaluated from the behaviour of the V-I c h a r a c t e r i s t i c s as described in the preceding section. The width of the hysteresis is determined by the heat t r a n s f e r properties to the environment of the sample. In the case of whiskers, in principle, there are two mechanisms f o r heat removal, heat t r a n s f e r t h r o u g h the surface, and thermal conductivity along the sample. However, the heat t r a n s f e r t h r o u g h the surface dominates and the heat flow into the c o n t a c t s can be neglected. There are t w o reasons why the heat flow into the c o n t a c t s is only small. One reason is that the hot s p o t may not extend over the w h o l e length of the whisker. Since the thermal conductivity in the whisker is much s m a l l e r than in pure bulk material (due to the smaller value of the e l e c t r o n mean free path in a whisker) .1, already a small distance b et w een the hot s p o t and the c o n t a c t s (which can be e s t i m a t e d from experiment) is s u f f i c i e n t to reduce the heat flow to the c o n t a c t s to a negligible value compared to the surface heat transfer. See ref. 480 for a detailed discussion. A n o t h e r reason is that for sufficiently low t e m p e r a t u r e s there is nearly no heat flow due to thermal conductivity via e l e c t r o n s into the s t r o n g l y supe rco n d u ct i n g contacts, even if the s e l f - h e a t i n g normal conducting region ~ Note that the main contribution to thermal conductivity at low t e m p e r a t u r e s comes from the e l e c t r o n s . The thermal conductivity of the phonons is very small. 216
e x t e n d s to the co n ta c t s . There is a large boundary resistance f o r the heat t r a n s f e r t h r o u g h a normal/superconducting boundary. The heat flux t h r o u g h such a boundary has been calculated by Andreev E499, 500]. The p r o b l em is discussed in detail in ref. 481. In the case of Zn and Zn-Ag whiskers usually the whole whisker becomes normal conducting so that the self-heating region extends to the strongly superconducting Wood metal contacts. We evaluated Andreev's r e s u l t f o r a Zn whisker, finding that the heat flux through both c o n t a c t s is ten o r d e r s of magnitude s m a l l e r than the total energy dissipation ~4813. For whiskers of pure In, the In-Pb alloy s y s t e m and pure Pb (of refs. 480, 483, and section 3.3.1 of ref. 40) the heat flux into the s u p e r c o n d u c t i n g c o n t a c t s in nearly all cases would not be sufficiently d e p r e s s e d if the hot spot w o u ld extend to the contacts. The reason is that the measuring t e m p e r a t u r e is n o t low enough compared to the critical t e m p e r a t u r e of the c o n t a c t material *~. However, in these samples the h o t s p o t actually does not e x t e n d to the contacts. As explained in ref. 480, we e s t i m a t e d the distance b e tw e e n the borders of the hot s p o t and each c o n t a c t bl oc k and c a l c u l a t e d the ratio of the heat flow t o w a r d the c o n t a c t s and the t o t a l heat flow. For all samples we find the ratio to be s m a l l e r than O.01. Thus, less than 1% of the t o t a l heat flow occurs into the co n t act s. For the pure Sn whiskers investigated in ref. 332 ( c o n t a c t e d with W o o d ' s metal) it is finally not clear which of the two mechanisms s u p r e s s e s the heat f l o w into the c o n t a c t s more effectively. If the hot s p o t would e x t e n d to the c o n t a c t s f o r all samples less than 8 7o of the t o t al heat dissipated in the sample would f l o w into the c o n ta c t s . Actually the hot sp o t s do not e x t e n d to the c o n t a c t s . Due to this e f f e c t for all samples less than 8% of the t o t al heat flow should occur into the contacts. Since in our samples the heat t r a n s f e r through the surface dominates, the heat t r a n s f e r c o e f f ic ie n t and, thus, the Kapitza resistance [SO1-505] can be ev al u at ed easily f r o m m e a s u r e m e n t s of the t h er m al hysteresis. Imagine the m e a s u r e m e n t of a V - T characteristic at fixed c u r r e n t I F. As already p o i n t ed o u t in the beginning of the preceding section, a t e m p e r a t u r e hysteresis, ~T.( I r ) = T ( I F) - 1, ( I F), is observed, because for increasing temperature the sample e n t e r s a dissipative s t a t e at the transition t e m p e r a t u r e T,(I~), while the specimen for decreasing t e m p e r a t u r e r e c o v e r s the s u p e r c o n d u c t i n g s t a t e at a bath t e m p e r a t u r e TR(I ~) which is lower than To(It). If the hysteresis is simply caused by a w a r m i n g - u p of the whisker due to J o u l e h e a t i n g , the r e - e n t e r i n g of the superconducting s t a t e at T, indicates t h a t at the bath t e m p e r a t u r e T, the sample temperature, Tw, is j u s t equal to the transition t e m p e r a t u r e T c. Thus, in this m o m e n t the difference *" We used squeeze c o n t a c t s made of In and W o o d ' s metal f o r pure In whiskers and pure Pb whiskers, respectively. For whiskers o f In-Pb alloys usually Pb contacts were used, e x c e p t for samples l n - P b 2 6 of ref. 480 and I n - P b 3 o f ref. 483 which were c o n t a c t e d with W o o d ' s metal.
217
b e t w e e n the sample and the bath temperature, T w - T ., is equal to T~-TR=~TM, so t h a t the t e m p e r a t u r e o f the whisker is related with the t e m p e r a t u r e hysteresis by Tw= T, + STs. We a s s u m e that the w a r m i n g - u p of the sample is caused by the f o r m a t i o n of a s e l f - h e a t i n g h o t spot. If the heat t r a n s f e r through the surface of the sample into the 4He dominates, the hot s p o t can be ap p r o x i m at el y regarded as a normal conducting region with sharp edges and a uniform t e m p e r a t u r e Tw over its whole length, Lspot. In this case [505, 506] t ~ = A s [ C ( T w ) T ~ - C ( T ) T * ] , where (~ is the heat t r a n s m i t t e d t h r o u g h the boundary surface area, A s, C characterizes the heat t r a n s f e r properties, and T is the helium bath temperature. This implies the usual simplification t h a t the phonon t r a n s m i s s i o n probability is independent of phonon frequency. If C is a slowly varying function of the temperature, this equation leads to
(~ = C A s ( T ~ v - T')
(213)
If the t e m p e r a t u r e difference, ~Tw, b e tw e e n sample and bath is very small, then T w - T 4 ~ 4T3~Tw. This leads to = aKAs ( T w - T)
(214)
where a ~ = 4 C T 3 is the heat t r a n s f e r c o e f f i c i en t which is r el at ed to the Kapitza resistance, R K, by ct~=l/R~. Thus, equation(213) can be r e w r i t t e n as
(~ = (4RKT~)-'As(T~v-T ')
(215)
If Rn is the residual resistance of the sample and L its t o t a l length, the dissipated energy is given by (~ = (R n L s p o t / L ) I~
(216)
Assuming that the whisker is a cylindrical filament with radius r w, it is As = 2 ~ r w L s p o t
(217)
We thus g et 4
(T w - T ' ) ~ r w L / I ~ 2 R n = R KT a
(218)
as our final result, valid for arbitrary t e m p e r a t u r e differences b e t w e e n whisker and helium bath. In situations where the difference b e t w e e n sample and bath t e m p e r a t u r e is very s m al l it is ( T ~ v - T 4 ) ~ 4 T 3(T w - T ) . We f u r t h e r m o r e introduce R n = P n L / A and ~K:I/RK, where A=Trrw2 is the c r o s s - s e c t i o n a l area of the whisker and Pn the residual resistivity. Then we g et from eq. (218)
(2aK/r w)(Tw-T) 218
~ O.(I~/A) 2
(219)
This e q u a t i o n is only valid w his k er and the helium bath.
for
small
temperature
d i f f er en ces
between
a
Our m o s t s y s t e m a t i c investigations of the heat t r a n s f e r p r o p e r t i e s have p e r f o r m e d with Zn whiskers, including a Z n - A g specimen, in a superfluid 4He bath [481]. For this p u rp o s e we d e t e r m i n e d the l e f t - h a n d side of eq. (218) at T = T R from the m e a s u r e m e n t of the hysteresis width of a specimen (using Tw=TR+~T.) and p l o t t e d the values as a function of the t e m p e r a t u r e . The r e s u l t s f o r sample Zn 18 are shown in Fig. 66. For t e m p e r a t u r e s b e t w e e n 0.52 and 0.70K the q u o t i e n t has the c o n s t a n t value 1 4 . 0 c m 2 K ' W 1. The l e f t - h a n d side of eq. (218) is equal to RKT ~. Thus, we have found t h a t for s a m p l e Z n l 8 in the m en t i o n ed t e m p e r a t u r e interval it is RKT3=14.0cm~K4W-I, indicating that R K~ T -3 in this t e m p e r a t u r e range. C l o s e to T~0 the hysteresis is g e n e r at ed by intrinsic n o n - t h e r m a l m e c h a n i s m s of the nonequilibrium p h a s e - s l i p state. In this t e m p e r a t u r e region the evaluation p ro c e d u r e cannot be applied. Moreover, we do not e x p e c t eq. (218) to be valid in the intermediate region in which no c o n s t a n t value of its l e f t - h a n d side has been observed. From our e x p e r i m e n t s the mean value RKT3= (15.2 +-3.1) cm 2 K 4 W -~ may be calculated, valid f o r pure zinc whiskers b e t w e e n ab o u t 0.5 and 0.7 K. In ref. 481 we compared our r e s u l t s with t h eo r et i cal e x p r e s s i o n s f o r RK and m e a s u r e m e n t s on copper at 1K (in the literature we did n o t find a value o f R K for zinc). First, some remarks on the theoretical r e s u l t s : The
been
v
F I cm2K4IW
o
500
o o
F=fT~-T~)?TrL o
Zn18 o
rc0 =0.816 K
o o o
100
o
/F=R KT3:14.0cm 2K~/W
O.
oo~~~
~--o---o--o--o--o--o--o--oo-o-o--o.c~oooo
'
0.5
I
0.6
'
o o
1
OY
o~
o
'
TIK
I
0.8
Fig. 66: Plot o f the l e f t - h a n d side o f eq.(218) versus the t e m p e r a t u r e T f o r a
zinc w h i s k e r (sample Z n l 8 ) . The sample parameters are given by R n = 0 . 7 3 0 f o r the residual resistance, L=22Sllm f o r the length, and rw=O.29gm f o r the radius. The solid line r e p r e s e n t s F = R K T 3 = I 4 . 0 c m 2 K" W -1. 219
Kapitza r e s i s t a n c e is a thermal impedance that occurs at the boundary be t ween t wo d i f f e r e n t materials, leading to a t e m p e r a t u r e discontinuity across the interface. The microscopic reason for the t e m p e r a t u r e discontinuity is that p h o n o n s on b o t h sides of the interface are only partially t r a n s m i t t e d through the boundary ES01-SOS]. A t h e o r e t i c a l e x p r e s s i o n for the Kapitza resistance b et w een liquid helium and a solid has been calculated by Khalatnikov within the f r am ew o r k of an acoustic mismatch theory [S01,50S]. The theory predicts t h a t R K ~ T "3 as observed f o r our zinc whiskers. However, the a b s o l u t e value measured f o r RgT 3 in o u r e x p e r i m e n t s is a f a c t o r of 40 sm al l er than the t h e o r e t i c a l prediction [481]. This is no accidental result. Except for very low t e m p e r a t u r e s ( T < 0 . 1 K ) the e x p e r i m e n t a l values are one or t w o orders o f magnitude s m a l l e r than the r e s u l t as calculated by the acoustic m i s m a t c h theory [501, 505]. The explanation is that the acoustic mismatch mechanism is shunted or bypassed by a more efficient heat t r a n s f e r mechanism. Recent theories on this scope have been proposed which c o n c e n t r a t e on ex ci t at i o n s a s s o c i a t e d with the helium at the interface [505,507,508]. This additional mechanism b e c o m e s more and more i m p o r t a n t between 0.1 and 1 K, leading to a s t r o n g t e m p e r a t u r e dependence of RKT3 in this t e m p e r a t u r e range. M e a s u r e m e n t s p e r f o r m e d with a c o p p e r / * H e boundary show the predicted behaviour [507, 508]. Our r e s u l t s for zinc b e t w een O.S and 0.7K do n o t show any t e m p e r a t u r e dependence for R~T ~. The reason may be a s h i f t of the t e m p e r a t u r e d ep en d e n t range to lower t e m p e r a tu r es. A lower limit for RKT:' can be calculated by assuming t h a t there is no r e f l e c t i o n o f phonons at the interface. This a s s u m p t i o n is called 'phonon radiation limit' [509,510]. The resulting value of RKT 3 for zinc is much ( a b o u t a f a c t o r o f 6) s m a l l e r than our experimental r e s u l t [481]. Experimentally it is found that the dependence o f RK on the Debye t e m p e r a t u r e , O, shows the tendency R ~ O (see refs. S02 and St|). Since the Debye t e m p e r a t u r e o f copper and zinc are nearly equal [512, 513], RKT 3 o f both materia[s should also be very similar. As no experimental value f o r R K of zinc is known from the l i t e r a t u r e we compared our r e s u l t s with t h o s e of copper near I K, where RKT ~ f o r copper is independent of t e m p e r a t u r e ES07, SO8,514]. For copper it is RKT3 ~ 1 0 - 2 0 c m 2K 4W -1, in a g r e e m e n t with the experimental r e s u l t for zinc. For whiskers of pure In, In-Pb alloys, pure Pb and pure Sn we do n o t have such detailed experimental r e s u l t s about the t e m p e r a t u r e dependence of the Kapitza r e s i s t a n c e as for our zinc whiskers. Nevertheless, for a l o t of samples a heat t r a n s f e r c o e f f i c i e n t ~K= I / R K has been evaluated f r o m m e a s u r e m e n t s of the thermal hysteresis. Results for c~K are given in refs. 480, 483, s ect io n 3.3.1 of ref. 40, and in ref. 332. For t h e s e specimens we simply assumed that c% is a fixed temperature independent quantity for a given sample at all measuring t e m p e r a t u r e s . This may be allowed, because the f a c t o r by which the t e m p e r a t u r e TR is varied in t he s e m e a s u r e m e n t s is much s m a ll e r than in the case o f zinc. For all 220
s a m p l e s ( e x c e p t the Sn whiskers) the ratio TR/T0 of the m e a s u r e m e n t s involved in the evaluation o f a N does not fall b el o w 0.98. For the Sn whiskers the minimum value of TR/T,0 is 0.81, 0.93, and 0.91 for sam p l es S h i l l , S n H 2 , and S n H 3 , respectively. The usual procedure is then to plot the hysteresis width ~T.(I~ 2) and to look for a region where the p l o t is a s t r a i g h t line which e x t r a p o l a t e s back to the origin. Using eq.(219) a fixed value for ctK f o r this range is then evaluated which is a s s u m e d to hold also f o r all t e m p e r a t u r e s c l o s e r to To0. The use of eq.(219) should be allowed, because ~ T w ( T , ) = ~ T . is much s m a l l e r than T R f o r t h e s e specimens. In this c o n t e x t we should remark that in our early d e t e r m i n a t i o n of ~K for In-Pb and a pure Pb whisker (ref. 480 and section 3.3.1 of ref. 40) we did not t race the behaviour of ~TH(IF2) to such high values of I~2 for which 8TH(IF2) is a s t r a i g h t line e x t r a p o l a t i n g back to the origin. We found, however, t h a t already for lo w e r values of IF2 all samples investigated s h o w e d a range where ~T.~IF2-1H ~, where IH2 is the e x t r a p o l a t e d zero hysteresis width i n t e r c e p t [480]. Then we calculated c~K using eq. (219) but replacing I,2 by 1 , 2 - I , 2. This implies the idea t h a t in this region t h er e is a certain c o n t r i b u t i o n of the dissipated energy which does n o t lead to a t h e r m a l hysteresis. This procedure was the f i r s t empirical a t t e m p t to separate J o u l e heating induced h y s t e r e s i s e f f e c t s from intrinsic ones. We c o m p a r e d o u r r e s u l t s f o r a K with values of 1 / R E obtained by e x t r a p o l a t i n g l i t e r a t u r e data to 4 K of the Kapitza r e s i s t a n c e f o r In, Pb, Pb with oxidized surface, and Cu as measured in superfluid helium [480]. This c o m p a r i s o n shows t h a t our r e s u l t s have a reasonable magnitude. In Fig. 67 the r e s u l t s f o r the heat t r a n s f e r c o e f f i c i e n t ctK are summarized for whiskers of In, ln-Pb, Pb, and Sn. For specimens o f In, In-Pb, and Pb
I0 Z
o:Sn
A'In *: In-Pb u : Pb
? E~)
5
9
o
O []
o I
9
l I
AoO
9 ,,, i
3
4
;
&
'
T/K8
R g , 67: Heat transfer coefficients, ~
=I/R~,, as a function o f the temperature f o r whiskers o f Sn [3323, In [4833, In-Pb [40, 480, 483J, and Pb F480, 4833. 221
the heat t r a n s f e r c o e f f ic ie n t has been determined for t e m p e r a t u r e s very cl o se to Tr (in all cases evaluations have been carried out up to T J T c 0 > 0 . 9 9 ) . Therefore, we p l o t t e d a K as a function of the critical t e m p e r a t u r e of the sample. In the case of Sn the heat t r a n s f e r co ef f i ci en t was c a l c u l a t e d f o r T , / T o 0 = 0 . 8 0 9 - 0 . 8 6 9 , 0.930, 0.914-0.926 f o r samples SnH1, S n H 2 , and S n H 3 of ref. 332, respectively. For these samples we p l o t t e d cQ at the t e m p e r a t u r e border c l o s e s t to To0. The r e s u l t s seem to s c a t t e r between I and S W/cm2K. However, within the values f o r specimens of In, In-Pb, and Pb t h er e seem s to be an increasing tendency from roughly 1.25W/cm 2 K at T ~ 3.4 K to ab o u t 4 . S W / c m 2 K at T ~ 7 . 2 2 K . This increase is much weaker than a T 3 law. A possible reason f o r these o b s e r v a t i o n s may be oxide at the surface of the whisker. A d i f f e r e n t oxidation of the specimens may lead to the s c a t t e r i n g of the r e s u l t s . There are m e a s u r e m e n t s r e p o r t e t in the l i t e r a t u r e f o r Pb which indicate a reduction of cQ if the surface of a sample is oxidized [480]. Assuming that the disturbance of heat t r a n s f e r t h r o u g h the surface o f a whisker from the In-Pb s y s t e m (including the pure materials) b eco m es s t r o n g e r f o r increasing lead c o n t e n t would be a possible explanation f o r the slow increase o f uK with increasing temperature. The investigation of the heat t r a n s f e r properties of a whisker to the surrounding helium bath enables us to e s t i m a t e the c o n t r i b u t i o n of J o u l e heating to the hysteresis in the p h a s e - s l i p regime. W i t h o u t this i n f o r m a t i o n it would n o t be possible to separate thermal and intrinsic c o n t r i b u t i o n s to the hysteresis o f a p h a s e - s l i p center.
11.3.
Hysteretlc Behavtour o f a Tin Whisker
In this section we give an example f o r the e x p e r i m e n t a l investigations of the h y s t e r e t i c behaviour o f a whisker. For this purpose a pure tin w h i sk er (sample SnH3 of ref. 332) has been chosen. An overview o f the d e v e l o p m e n t of I~/3 and IRa/a with decreasing t e m p e r a t u r e has already been given in section 6.1, where we discussed the behaviour o f the critical c u r r e n t o f our filaments (Fig. 21). All values r e s u l t from V-I characteristics. In Fig. 68 we p l o t the vicinity of the critical t e m p e r a t u r e in more detail. Moreover, in this figure we indicate how the t e m p e r a t u r e hysteresis, 8TH, at arbitrary fixed cu r r en t I r is evaluated. As can be seen f r o m the figure, 3T H depends on the magnitude o f I~ and is given by 8T~(I~) = To(I~)-TR(I r)
(220)
In Fig. 69 we p l o t ST. as a function o f IF2. As discussed in the preceding section, a simple heating model with a c o n s t a n t value of the heat t r a n s f e r coefficient, aK, would predict ~TH~ I / and, therefore, deviations from simple 222
200-
~ "~2"~n"~
~~(o it is }max=J: (where T is the p a i r - b r e a k i n g p a r a m e t e r of the TDGL theory). For the f i r st voltage step of our V-I characteristics, which may be regarded as being g e n e r a t e d by an isolated p h a s e - s l i p center, we, therefore, should observe no hysteresis b el o w h'c. Above % the hysteresis should be given by ]c-Jmin" Kramer and Rangel (KR) in Tab. l of ref. 316 give values of Jrnin for a periodic array of p h a s e - s l i p centers. As KR point o u t t h a t Jnain is r a t h e r insensitive to the spatial period of the p h a s e - s l i p centers, we may use t h e s e values for a comparison with our experiments. If we make a l o g - l o g p l o t of Jmin against T, we obtain f o r the region 10 ,0
].5-
I
0
It=0.7mA
AT: Tco-T I
I
o
i
s
Flg. 92: D i f f e r e n t i a l r e s i s t a n c e ( d V / d I ) , and ratio 1 o / I ~ f o r t h e f i r s t
voltage s t e p o f s a m p l e T W L Z n 3 in the i n t r i n s i c case (It=O) a n d w i t h a p p l i e d c o n t r o l c u r r e n t I t = 0 . 7 m A . Here, To is t h e critical t e m p e r a t u r e o f the m e a s u r i n g whisker,
265
This e x p e r i m e n t also shows that the shift of the whole characteristic observed for zinc is not caused by J o u le heating e f f e c t s due to a heat t r a n s f e r f r o m the c o n t r o l l i n g whisker th r o u g h the superfluid helium to the measuring whisker. In this experiment the c o n t r o l l i n g whisker is normal conducting. Therefore, J o u l e heating already occurs for small c o n t r o l currents. The heat generation is still p r e s e n t af t er the d e s t r u c t i o n of the metallic connection b e t w e e n the whiskers. The dissipated heat does, however, not influence the measuring whisker.
13.3.
Microcontacts
In specimens used for tunable weak link measurements, t h er e is a metallic m i c r o c o n t a c t b e t w e e n the measuring and the c o n t r o l l i n g whisker (see chaps. 2.2, 12, and 13.2). This m i c r o c o n t a c t is located at the crossing point of the two filaments. It is created by a discharge cu r r en t which is allowed to flow across the touching region of the whiskers. This discharge c u r r e n t d e s t r o y s the oxide layer on the surface of the whiskers and g e n e r a t e s a s p o t welding o f the filaments. Already in our early studies of tunable weak links [23] we mentioned that the V-I c h a r a c t e r i s ti c s of such a m i c r o c o n t a c t seem to indicate its metallic character. Recently we s t a r t e d a more detailed investigation of the properties of these microcontacts. For this purpose we measured the V-I characteristics at several fixed t e m p e r a t u r e s for m i c r o c o n t a c t s between t w o whiskers of Sn, In, and Zn, respectively. In t h e s e e x p e r i m e n ts the measuring current I enters the sample (for instance) through the measuring whisker, flows through the m i c r o c o n t a c t into the c o n t r o l l i n g whisker, and then leaves the sample through this whisker. The v o lt a g e V is then d e t e c t e d b et w een the measuring and the c o n t r o l l i n g whisker at the t w o free squeeze c o n t a c t s which no c u r r e n t f l o ws through. As an example, the V-I characteristics of a m i c r o c o n t a c t b e t w e e n two Sn whiskers (sample T W L S n 8 ) are shown in Fig. 93. The g e o m e t r y of the sample is sketched in the inset. The lengths of the l e f t and r i g h t side o f whisker l and t h o s e of the lower and upper part of w h i s k e r 2 are 400, 413, 288, and 425[1m, respectively. Furthermore, it is for w h i s k e r l (whisker2): R298K= 8.9 f/ (34.5fD, Rn=O.O31f/ (O.077fD, A = 9 . 0 S v m 2 (3.1S[tm2), 2 = 2 . 9 0 ~ m (2.94~tm), dI~/3/dT.=-1202OtIA2/3/K (-7180t~A2/3/K), and the o r i e n t a t i o n s are [100] ([001]). For all ch ar a c te r is t ic s shown, both whiskers are in the s u p e r c o n d u c t i n g state. This is clear f o r the n o n - c u r r e n t - c a r r y i n g parts, because the measuring t e m p e r a t u r e s are lower than the critical t e m p e r a t u r e s of the whiskers. Also the c u r r e n t - c a r r y i n g parts remain fully superconducting, because the t r a n s p o r t c u r r e n t s are not large enough to cause their transition into the dissipative state. 266
VIIaV
TWLSn8
100-
Tcol = TC02
ATImK 111.7132.0 60 4 F ) . 8 ~ o y ~ , / / . 12.3 50 604758 908 1518
=3.6980K
A T= Tcol - T
31.o
//G J'/
1 F/-~~
Oi
0
I
100
I
I
200
I
I
300
I
1
I IIJA
I
500
Fig. 93: V-I characteristics (increasing current) o f a microcontact between two tin whiskers (sample TWLSn8) for several fixed temperatures. The inset shows the geometry o f the sample. Here To, and T~o2 are the critical temperatures o f w h i s k e r l and 2, respectively. In this sample the critical temperatures are equal f o r both whiskers. Both whiskers are superconducting throughout the measurement. The dashed fines are guides to the eyes to make the portions o f c o n s t a n t differential resistance visible.
Lip t o a c e r t a i n c r i t i c a l c u r r e n t t h e microcontact is s u p e r c o n d u c t i n g . T h e n t h e r e is a c o n t i n u o u s t r a n s i t i o n i n t o t h e v o l t a g e c a r r y i n g s t a t e . T h e r e is no step-like change of the voltage at the critical current. A remarkable o b s e r v a t i o n is t h a t s o m e o f t h e V - I characteristics o f t h e m i c r o c o n t a c t in t h e low voltage region exhibit straight parts of constant differential resistance ( s e e , f o r i n s t a n c e , Fig. 93, AT= 75.8 a n d 90.8 inK). T h e e f f e c t is q u a l i t a t i v e l y s i m i l a r t o t h e l i n e a r p o r t i o n p h e n o m e n o n o b s e r v e d in P b w h i s k e r s ( c h a p . 9). A l s o in t h e c a s e o f t h e m i c r o c o n t a c t s , t h e differential resistance of a linear portion at higher current seems to be an i n t e g e r m u l t i p l e o f t h o s e a t l o w e r c u r r e n t s . F o r t h e characteristic a t 5T=90.8mK in Pig. 93 t h r e e l i n e a r p o r t i o n s are v i s i b l e w i t h a d i f f e r e n t i a l r e s i s t a n c e d V / d I = 43, 89, a n d 178 mr), r e s p e c t i v e l y . T h e d i f f e r e n t i a l r e s i s t a n c e o f t h e s e c o n d l i n e a r p o r t i o n is r o u g h l y t w o t i m e s t h a t o n e o f t h e f i r s t l i n e a r portion (which can only be approximately determined}, and the differential r e s i s t a n c e o f t h e t h i r d l i n e a r p o r t i o n is e x a c t l y t w o t i m e s t h a t o n e o f t h e second linear portion. T h e e x p e r i m e n t s o n m i c r o c o n t a c t s b e t w e e n t w o In or t w o Z n w h i s k e r s yield qualitatively similar results (including the linear portion phenomenon). In Fig. 94 s o m e V - I c h a r a c t e r i s t i c s o f a m i c r o c o n t a c t between t w o In whiskers are shown (sample TWLInr T h e g e o m e t r y o f t h e s a m p l e is
267
VI!:t rco2=3.3790K/ TWLIn 4 Tcol= 3.3818 K ,,1T=Tcol- T
0.5.
8.3
/
22.4
12.7 11.6
12.7
9.1
14.7 21.9
12.0 14.9
20.9 14.9 14.4 9.9
8.7 12.2 10.3 8.4
0.877
0.621
Yf
28.2
11.3
19.8
11.9
29.0 32.8
23.5 14.7
21.8
7.8
Final Note
During the print of the p r e s e n t work we developed additional ideas f o r the i n t e r p r e t a t i o n of our ongoing work mentioned in chap. 13 (for details see ref. 531 ): 1. The low t e m p e r a t u r e behaviour of a p h a s e - s l i p c e n t e r (sec. 13.1) seem s to be governed by charge imbalance waves: We found t h a t the maximum o f the d i f f e r e n t i a l r e s i s t a n c e appears at a t e m p e r a t u r e for which the decay l e n g t h o f the charge imbalance wave, kd,Kss, in the high frequency limit (see p. 71 o f the p r e s e n t work) has its maximum value. 2. We roughly e s t i m a t e d the dimensions of our m i c r o c o n t a c t s (sec. 13.3). The length o f the m i c r o c o n t a c t is of the o r d e r 0.2pm, while the diameters of the c o n t a c t s range b e tw e e n 0.O1 and O.lO#m. As far as known to us there is not any phenomenon reported in the l i t e r a t u r e which would be able to explain the shape of the V-I ch ar act er i st i cs of our m i c r o c o n t a c t s . We a s s u m e t h a t the c o r e region of a p h a s e - s l i p c e n t e r e s t a b l i s h e s in the m i c r o c o n t a c t and t h a t this p h a s e - s l i p c e n t e r shows discrete excited st at es . For increasing c u r r e n t the phase-slip c e n t e r changes from one s t a t e to the other. Since the v o lt a g e is d e t e c t e d by superconducting parts o f the whiskers which e x t e n d to the borders of the microcontact, the voltage should be a measure f o r differences of the el ect r o ch em i cal pair p o t en t i al across the core region o f the phase-slip center. As far as we know, excited s t a t e s o f p h a s e - s l i p c e n te r s have not been observed up to now, and t h er e is not any t h eo r et i ca l prediction of t h e s e states. 3. We i n t e r p r e t e d the interaction of neighbouring p h a s e - s l i p c e n t e r s in Zn whiskers in more detail. For this purpose the shifts o f the critical c u r r e n t shown in Figs. 102 and 104 were split into an a s y m m e t r i c and a s y m m e t r i c part (as d i s cu s s e d on p. 147 of the p r e s e nt work). We found that the a s y m m e t r i c part is well described by time averaged quasiparticle c u r r e n t s g e n e r a t e d by the influencing p h a s e - s l i p center as given by the SBT model.
299
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317
List of Abbreviations
BCS GL KR KSS MKSA PSC RSM SBT SC/NC SEM TDGL TEM TWL
Bardeen, Cooper, Schrieffer Ginsburg, Landau Kramer, Rangel Kadin, Smith, Skocpoi Meter, kilogram, second, ampere Phase-slip center Rieger, Scalapino, Mercereau Skocpol, Beasley, Tinkham Superconductor / normalconductor Scanning electron microscope Time-dependent Ginsburg-Landau Transmission electron microscope Tunable weak link
319
List of Symbols
General Remarks 1. T h e
equations
are
usually
written
in a 'linear form',
that
means,
for
instance, 1 + ab/cd 2.
insteadof
I +
ab
The v e c t o r nature of a quantity is indicated by underlining (for instance,
A(r)).
Symbols A A(r) As a
_B Sc th BT,T~
IB~,o, IBK.o
bm
C
C(y) CK
C r o s s - s e c t i o n a l area of a sample Vector potential of the magnetic field at locus _r Boundary surface for heat t r a n s f e r (sec. ll.2) Factor (sec. 7.3) equal to I0/I o C o n s t a n t used in the SBT model Normalized m e a n - s q u a r e deviation of the gap, 0 value for clean bulk material Magnetic field Thermodynamicat critical magnetic field W e i g h t e d average of Bose distributions bT~ and bT (sec. 5.5) Creation and annihilation operator, respectively, for a quasiparticle excitation with wave number v e c t o r K and spin o (Bogolubov o p e r a t o r s ) C o e f f i c i e n t (sec. 7.3) 1. A c o n s t a n t used in the discussion of the e l e c t r o n mean free path dependence of z E 2. A c o n s t a n t used in sec. 6.2 3. C = C (T) Coefficient characterizing the heat transfer p r o p e r t i e s through the surface (sec. ll.2) A function defined in the TDGL theory Leakage capacitance {KSS model) A c o n s t a n t introduced in sec. 7.6 Leakage capacitance (Kulik, sec. 7.2)
321
CcG CKsS CKu CL~ C T
c~(~) Cpb Cp, C~ Y¢
CK_~, CE~
Bare wave phase velocity of the C a r l s s o n - G o l d m a n mode Bare wave phase velocity of the charge imbalance wave Bare wave phase velocity of the collective mode according to Kulik Longitudinal and transversal sound velocity, respectively Coefficients (KSS model) Lead c o n c e n t r a t i o n C o n s t a n t s used in the discussion of t~p (sec. 5.4) Creation and annihilation operator, respectively, for an electron in the state K4'
D D (T) d dp
Diffusion c o n s t a n t Deviation of the critical magnetic field from a parabolic law Film thickness Periodic length (distance of phase-slip centers in an array, TDGL theory)
E EK
Electrical field Quasiparticle excitation energy Absolute value of the electron charge (it is e>O, the electron charge is denoted by - e ) Twice the electron charge ( e' = - 2 e)
e
F
F*
FKs$ ~o F0 fK f, (x), f=(x) f(~÷_(t)) ~+_ G
GK$S
Damping factor of the C a r l s s o n - G o l d m a n mode Dimensionless factor introduced in the discussion of charge imbalance relaxation times (sec. 5.4) Damping factor of the charge imbalance wave Damping factor of the collective mode according to Kulik Free energy barrier according to Langer and Ambegaokar Fermi function Occupation probability of a K state by a quasiparticle excitation Functions introduced in sec. 7.6 C u r r e n t - p h a s e relation (KSS model) Phase angle average of the function f (KSS model) 1. Free enthalpy 2. Abbreviation used in the discussion of the C a r l s s o n Goldman mode (sec. 5.6) Leakage conductance (KSS model) Abbreviation used in the discussion of the charge imbalance waves
GK, GKu Gs
322
Abbreviation used in the discussion of the according to Kulik (sec. 5.7) Leakage conductance (Kulik, sec. 7.2) Ginsburg-Landau free enthalpy
collective mode
A c o n s t a n t used in the discussion of the e l e c t r o n mean free path dependence of zE
gL
It is h = h / 2 ~ , I AI
lAB
IAF IAF~¢
Io(T ~) Ie.PSC2
lc.vWL PSC2
AIo,
aI°~
I~,, I~
IH
Ik Im lrnax ltnin In(x)
where h is Planck's c o n s t a n t
Total c u r r e n t 1. Width of a current step (sec. 7.3) 2. Difference between the o n s e t c u r r e n t s of the first t w o v o l t a g e s t e p s in the V-I c h a r a c t e r i s t i c s o f a zinc whisker (sample Zn21, Fig. 45) 3. C u r r e n t range over which the first linear portion in the V-I c h a r a c t e r i s t i c s o f a Pb whisker e x i s t s (sec. 9.2.3) 4. Distance in c u r r e n t between the first two v o l t a g e steps in sample W W Z n 7 (sec. 13.4) A conventional current flowing from c o n t a c t A to c o n t a c t B (sec. 7.6, p. 156) C u r r e n t flowing from c o n t a c t A to c o n t a c t F (sample WWZnT, sec. 13.4) Critical current, where the c u r r e n t flows from c o n t a c t A to c o n t a c t F (sec. 13.4) Critical c u r r e n t Critical c u r r e n t at the t e m p e r a t u r e T* (sec. 8.3) Critical current at which the p h a s e - s l i p center PSC2 appears in the V-I c h a r a c t e r i s t i c s of sample W W Z n 7 (sec. 13.4) Critical c u r r e n t of the TWL (chap. 12) Shift o f the critical c u r r e n t of phase-slip c e n t e r PSC3 under the influence of p h a s e - s l i p c e n t e r PSC2 in sample W W Z n 7 (sec. 13.4) Shift of the critical c u r r e n t of bridge l in the closely spaced t w o - m i c r o b r i d g e system (sec. 7.5) A s y m m e t r i c part of AI,~ S y m m e t r i c part of A I , Fixed c u r r e n t (chap. 11) Fixed c u r r e n t s introduced in sec. 11.4.1 for the description of the fitting procedure in the case of the hysteresis model with two fitting p a r a m e t e r s C u r r e n t in a generalized resistivity shunted junction model (KSS model) E x t r a p o l a t e d zero hysteresis width i n t e r c e p t (sec. 11.2) A function characterizing the anisotropy e f f e c t (sec. 6.2) C o n t r o l c u r r e n t (sec. 7.5) Measuring c u r r e n t (sec. 7.5) Maximum c u r r e n t for the existence o f r~ p h a s e - s l i p c e n t e r s (SBT model) Limit of the hysteresis in the KSS model Normal c u r r e n t at locus x, (...) time average
323
J u m p - b a c k c u r r e n t (sec. 6.1, 11.1 and Fig. 25) J u m p - b a c k c u r r e n t as predicted by the TDGL theory (sec. 11.5) Supercurrent Is S u p e r c u r r e n t at locus x, time average Is(x) Spatial average of the s u p e r c u r r e n t in the weak link in the T,(t) RSM model S u p e r c u r r e n t in the core of a phase-slip center, time I s ( Xpsc, t) average Control (or tuning) c u r r e n t through the c o n t r o l l i n g whisker of It a t u n a b l e weak link a r r a n g e m e n t (chap. 12) Control c u r r e n t needed until the phase-slip center at the TWL ItA appears before the o n s e t of the intrinsic V-I characteristic (chap. 12) Zero voltage intercept obtained by back extrapolation of the l0 V-I characteristic after the first voltage step to V = 0 'Low temperature value' of the ratio 10/I ¢ for In whiskers (Io/I,)otnW (sec. 7.2) 'Low temperature value' of the ratio I0/I o for I n - P b whiskers ( Io/I=)oInPbW (sec. 9.2.1) 2/3 Slope of the Ic=':' ( T o ) straight line for the first intrinsic voltage ( d l , /dT¢) o step of the measuring whisker in a t u n a b l e weak link a r r a n g e m e n t (chap. 12) 1R
IR,TOGL
Jn('")
E L J¢ A. r, Jmax, Jtnin
J_. In
is
J0 J0
K_K_ KF
KGL KI 324
Bessel f u n c t i o n s of the first kind of integer order (sec. 7.3) Normalized c u r r e n t (TDGL theory) Critical c u r r e n t density (j¢=Io/A) Ginsburg-Landau critical c u r r e n t (normalized units, TDGL theory) IJpper and lower border, respectively, of the c u r r e n t range in which phase-slip solutions exist (normalized units, TDGL theory ) 'Normal' c u r r e n t density (i.e. quasiparticle c u r r e n t density), jn(X) space dependent absolute value, time average Absolute value of the normal c u r r e n t (normalized units, TDGL theory) Supercurrent density, ]six) space dependent a b s o l u t e value, time average Normalization c u r r e n t (TDGL theory) Extrapolated zero voltage intercept of a v o l t a g e - c u r r e n t characteristic (normalized units, TDGL theory) Wave number vector of an electron Fermi wave number C o n s t a n t appearing in the Ginsburg-Landau critical c u r r e n t density A parameter characterizing all effects besides the anisotropy effect (sec. 6.2)
25K
Range of nonzero interaction potential in the BCS theory
L
Sample length Series inductance per unit length (KSS model) N o r m a l - l i k e length related to the differential r e s i s t a n c e o f the first p h a s e - s l i p c e n t e r of a V-I characteristic. The s u b s c r i p t s 'OInW', 'OPbW', 'OI_nnPbW', and '0P_bbBiW' d e n o t e the "low t e m p e r a t u r e values' of the n o r m a l - l i k e length LAn 1 f o r whiskers o f In, Pb, In_-Pb, and P_bb-Bi, respectively (sec. 7.2, 9.2.4, 9.2.1, and 9.2.4, respectively) N o r m a l - l i k e length related to the increase of the differential resistance, M d V / d I ) 2, a f t e r the appearance of the second v o l t a g e step in a V-I characteristic. The s u b s c r i p t 'OInW' d e n o t e s the ' l o w t e m p e r a t u r e value' of the n o r m a l - l i k e l e n g t h LAn2 for whiskers of In (sec. 7.2) Normal-like length as calculated from the differential r e s i s t a n c e of the first linear portion of the V-I c h a r a c t e r i s t i c of a Pb whisker generated by 3 active p h a s e - s l i p centers (sec. 9.2.3) Series inductance (Kulik, sec. 7.2) Length of the hot s p o t (sec. 11.2) E l e c t r o n mean free path
LAni
LAn2
LAn3
L'Ku Lspot
£
m m
N
Nmax Nn(~K) Nop
Nq (E K) Ns NT No
n
nar
%(r)
E l e c t r o n mass (except in sec. 7.3, where m d e n o t e s an index) Twice the e l e c t r o n mass (m' = 2 m) Number of particles in a system (for instance e l e c t r o n s ) Maximum quasiparticle density (sec. 11.4.1) Normal density of s t a t e s for one spin direction Quasiparticle overpopulation, Nop(X, t= 0): spatial dependence at the m o m e n t of the phase-slip event, N o p ( x = O , t ) : time dependence at the core of the p h a s e - s l i p c e n t e r Quasiparticle density of s t a t e s in a s u p e r c o n d u c t o r (sec. A.2) Abbreviation used in the discussion of Kulik's work in sec. 5.7 Number of quasiparticles per volume at the bath t e m p e r a t u r e T Number of e l e c t r o n i c s t a t e s (in a free e l e c t r o n model) for one spin direction per volume and energy interval at the Fermi energy E l e c t r o n density (except in sec. 7.3, where n d e n o t e s an index) 1. Effective number of e l e c t r o n s added to the quasiparticle s y s t e m if a quasiparticle with wave number K is excited (sec. 5.2) 2. Positive integer or fractional number (sec. 7.3) Number of proximity e f f e c t bridges in an array (sec. 7.3) Local density o f the superconducting charge carriers in the Ginsburg-Landau theory
325
nse nso no
Superelectron density (i.e. n u m b e r of electrons b o u n d to Cooper pairs per volume) Density of particles described by ~ in the absence of c u r r e n t s or magnetic fields (Ginsburg-Landau theory) Maximum possible density of superconducting particles (Cooper pairs) as p r e s e n t at T=O K (Ginsburg-Landau theory)
P1, P2, P3, P4 Potential probes Fermi m o m e n t u m P~
Q Q
qKss
Branch imbalance Abbreviation (TDGL theory) Heat dissipation in a sample (chap. ll) Charge imbalance of the quasiparticle system per unit volume Quasiparticle charge Charge of the condensate Phonon wave n u m b e r vector Gradient of the phase in the SBT model Wave n u m b e r of the C a r l s s o n - G o l d m a n mode Wave n u m b e r vector of the charge imbalance wave (absolute
q~
value: qKss) Abbreviation used in the KSS model
5
Q* Q~
Qp q qcG
Rac Reff RK
Rn R298K
RD rw
S
Su Ss
T AT T* 326
Series resistance per unit length (KSS model) Impedance of the t r a n s m i s s i o n line in the high-frequency limit (KSS model) Normal resistance of length 2AQ,in of the t r a n s m i s s i o n line (KSS model) Kapitza resistance (sec. ll.2) Series resistance (Kulik, sec. 7.2) Residual resistance Resistance at room temperature Sheet resistance (sec. 5.4) Space vector Radius of a whisker assuming a circular c r o s s - s e c t i o n a l area Surface of the warm region around a phase-slip center through which the dissipated energy is transmitted into the s u r r o u n d i n g helium bath (sec. ll.4.t) Abbreviation (sec. A.2) Factor accounting for the nonexponential charge imbalance decay in the matching regions (sec. 5.9) Absolute temperature (bath temperature, sec. 11.2) Temperature difference of the actual helium bath t e m p e r a t u r e relative to the critical temperature ( A T = T , 0 - T ) Effective quasiparticle temperature
ST*
E n h a n c e m e n t of the effective quasiparticle temperature, T *, above the bath temperature, T, i.e. ST *= T * - T T~ Transition temperature at c u r r e n t I0 Transition t e m p e r a t u r e at c u r r e n t I F (chap. 11) T,i!F) Critical temperature Too Critical t e m p e r a t u r e of p a t t i of sample W W Z n 7 (sec. 13.4) T¢o, p a r t 1 Critical temperature of the clean matrix (sec. 6.2) T¢0r Upper temperature border for the local equilibrium T~ approximation (TDGL theory) AT~ It is ATE=ITo0-TEI Temperature down to which the simple (i.e. gapless) version of TGL the TDGL theory may be used AT~L It is AT~L=ITo0-T~LI ST. (I~) Width of the temperature hysteresis at c u r r e n t I~ T exit:) Measured o n s e t temperature of hysteretic behaviour (sec. 11.1) Hy AT~yp Measured o n s e t of hysteretic behaviour with ATHyP= T¢0-T~yp (sec. 11.1) ATHK,s Prediction for the o n s e t of hysteresis after the KSS model y (see. 11.1) O n s e t temperature of the quasiparticle overpopulation induced hysteresis (sec. 11.4.1) ST.,, ,~TH2,ST."2 Quantities introduced in sec. 11.4.1 for the description of the fitting procedure in the case of the hysteresis model with two fitting parameters Charge imbalance wave induced hysteresis (KSS mechanism, TKs$ sec. 11.4.1) Lattice temperature (sec. 11.4.1) TL E n h a n c e m e n t of the lattice t e m p e r t u r e above the bath 8T L temperature T, i.e. ST L= TL- T T.(I~) J u m p - b a c k t e m p e r a t u r e of the helium bath at c u r r e n t I, ( chap. 11) The same as AT~ (see f o o t n o t e p. 195) ATv Sample temperature (sec. ll.2) Tw 8Tw E n h a n c e m e n t of the sample t e m p e r a t u r e above the bath temperature, i.e. 8Tw= T w - T A c o n s t a n t used in the calculation of l:ee (sec. 5.4) T1 t Time Normalized time (TDGL theory) Time defined in sec. 5.8.3 (SBT model) tsb Normalization time (TDGL theory) to U
luKI 2 u~ (L t)
Normalization c o n s t a n t (TDGL theory) Occupation probability of the s t a t e with wave n u m b e r vector K by a hole in a BCS s u p e r c o n d u c t o r (= u~) Displacement amplitude of the superfluid part in the C a r l s s o n - G o l d m a n mode, time average (sec. 7.6)
327
Ur
C o n s t a n t used in the SBT model
V
Voltage, time average Normalized voltage (TDGL theory), time average Radiation induced voltage (sec. 7.3) A conventional voltage measured between c o n t a c t s A and B (sec. 7.6, p. 156) Voltage measured between contacts A and F (sample W W Z n 7 , sec. 13.4) Scaling voltage (KSS model) Voltage of the voltage foot (sec. 7.6) Effective e l e c t r o n - e l e c t r o n interaction potential Time averaged voltage in a generalized resistivity s h u n t e d junction model (KSS model) Voltage of harmonic and subharmonic c u r r e n t steps (sec. 7.3) Voltage of the main c u r r e n t step (sec. 7.3) Voltage developed by ~a phase-slip centers (SBT model) J u m p - b a c k voltage Voltage between the points x R and x L (SBT model) Saturating value of the jump-back voltage, VR (sec. ll.l) Voltage across the controlling (or "tuning') whisker of a t u n a b l e weak link arrangement (chap. 12) Height of the first voltage step Voltage (i. e. electrostatic potential difference) b e t w e e n r , and
Va
v~8 V~ v Vc,f Veee (K, 9_)
Vj~
v. Vm
v. v., vt V,(I,)
% V_+(V+_) V~. co
(dV/dl),
VF
IvJ VKss
vs XG, XSG
XFsc x=x(V)
X L , XR
X 1, X r
328
_r~: V,2 = ¢~(L~) - ¢4 (r2) Voltage between the left (right) and right (left) side of the Josephson oscillator (KSS model) Voltage measured by probes far away from the phase-slip center (KSS model) Differential resistance of the characteristics after the first voltage jump (or of the first linear portion) in a V-I characteristic at c o n s t a n t temperature Fermi velocity Occupation probability of the state with wave n u m b e r vector K by an electron in a BCS superconductor (= v~) Phase velocity of the charge imbalance wave Superfluid velocity (absolute value, vs) Abbreviations used in eq. (89) of sec. 5.5 Locus of the core of the phase-slip center Abbreviation (sec. ll.4.1) with ×R =x(VR) Distance from the core of the phase-slip center (TDGL theory) Unit vector along the axis of the filament (SBT model) Point left and right from the phase-slip center where t~ = P~ (SBT model) Point left and right from the phase-slip center
Z(o~,,) Z(T)
Impedance of the t r a n s m i s s i o n line (KSS model) Abbreviation (sec. A.2) Impedance of the t r a n s m i s s i o n line (Kulik, sec. 7.2)
~t
C o n s t a n t of the Ginsburg-Landau theory Heat t r a n s f e r coefficient (sec. 11.2) E l e c t r o n - p h o n o n coupling function o f the Eliashberg t h e o r y
[3KR 13. [~$BT [~TWL
F
r0 T Yc exp "i'Hy ~'max
T, ~(_r)
A(T)
C o n s t a n t o f the Ginsburg-Landau theory Ratio of the 'excess current' I 0 and the critical c u r r e n t (TDGL theory) Fitting p a r a m e t e r of the phenomenological h y s t e r e s i s model (sec. 11.4.1) Ratio /I c of the SBT model A c o n s t a n t , relating the quasiparticle o v e r p o p u l a t i o n in a t u n a b l e weak link (TWL) with the c o n t r o l c u r r e n t (sec. 12.2) Pair-breaking p a r a m e t e r (sec. S.4), e x c e p t in sect7.3, where F d e n o t e s the Gamma function Abbreviation used in the discussion of the damping of the C a r l s s o n - G o l d m a n mode Pair-breaking p a r a m e t e r (TDGL theory) Euler's c o n s t a n t Value o f the p a i r - b r e a k i n g p a r a m e t e r for which the upper c u r r e n t b o r d e r of the p h a s e - s l i p s o l u t i o n is equal to the Ginsburg-Landau critical c u r r e n t Value of the p a i r - b r e a k i n g parameter, "f, at the o n s e t t e m p e r a t u r e of hysteretic behaviour (sec. 11.1) Hpper limit o f the p a i r - b r e a k i n g p a r a m e t e r ( c o n c e r n i n g the local equilibrium approximation, TDGL theory) Abbreviation (TDGL theory) Average value of the gap (angle average) Space d e p e n d e n t o r d e r p a r a m e t e r of the microscopic theory o f superconductivity (in the absence of c u r r e n t s and magnetic fields: A0 (r)) Energy gap of the superconductor at temperature T (A(0)
6,o
= A(T=0))
Normalized o r d e r p a r a m e t e r in the presence of magnetic impurities (TDGL theory) Equilibrium value of the order p a r a m e t e r (TDGL theory) Equilibrium order p a r a m e t e r in the presence of magnetic impurities (TDGL theory) M e a n - s q u a r e deviation of the gap from the average value ( a n g u l a r average)
329
~H EK E0 "QF
Fitting parameter of the phenomenological hysteresis
model
(sec. 11.4.1) Plane wave energy of a free electron state measured relative to the chemical potential, p~ Influence constant
Fermi energy Plane wave energy of a free electron state Debye temperature Phase of the order parameter (TDGL theory) Ginsburg-Landau parameter
A A(t) AE AKR
AKu a L
AQ= Ae~in X(T) k, ~'d, KS$
kL(O)
It* ~chem
~c,p [~c,q ~el chern
~tF
(tl)K
~Lp
330
Quasiparticle diffusion length (SBT model) Quasiparticle diffusion length, A(t) = ((1/3)2 vFt) '/2, sec. 11.4.3 Diffusion length related to ~ (TDGL theory) Normal-like length (TDGL theory) Decay length of the scalar potential or penetration depth of the electric field into a superconductor (according to Kulik) London's constant Charge imbalance relaxation length Charge imbalance relaxation length due to inelastic electron-phonon collisions Magnetic penetration depth Decay length of the Carlsson-Goldman mode Decay length of the charge imbalance wave London penetration depth at T = 0 K Electrochemical potential of quasiparticles, (...) time average Normalized electrochemical quasiparticle potential (TDGL theory), (...) time average Chemical potential in the quasi-thermal Owen-Scalapino model for quasiparticle overpopulation Chemical potential Chemical potential of Cooper pairs Chemical potential of quasiparticles Electrochemical potential Chemical potential of electrons in the normal state (often called 'Fermi energy' although rigorously 7]~=tIF(T=O)) Time averaged value of the contribution of collective excitations to the electrochemical potential of quasiparticles (sec. 7.6) Electrochemical potential of Cooper pairs, (...) time average Time averaged difference of the electrochemical pair potentials far away from the core of the phase-slip center (sec. 7.3)
[J*p, a
Ilp, d
('~ p, 12
p,
v vc
V¢,min vj VPSC
~(T)
~0
~0(T,0)
p
p Pn PnSnInZ
Slope of the electrochemical pair potential during the accelleration part of the phase-slip cycle {SBT model) Slope of the electrochemical pair potential during the decelleration part of the phase-slip cycle (SBT model) Time averaged electrochemical p o t e n t i a l s left and right of the phase-slip center (SBT model) Difference of the electrochemical pair potential between the locus r , and r 2 (sec. 7.3: between the core region of a phase-slip center), i.e. ~p,12=~p(r~)-vp(r2), time average Time averaged electrochemical quasiparticle potential at the locus of potential probe Pl (Fig. 36) quasiparticle potential as Time averaged electrochemical proposed by SBT (sec. 7.6) Induction constant, {l0 = 4 ~ • 10-~ Vs A-t m-' Frequency of the HF radiation field Low frequency border of HF s t i m u l a t e d (sec. 7.3) Lowest frequency that can lead to an superconductivity by HF radiation (sec. 7.3) Josephson frequency Phase-slip rate (TDGL theory)
superconductivity enhancement
of
Ginsburg-Landau coherence length Ginsburg-Landau coherence length in the dirty limit BCS coherence length (typical decay length of the probability density for a Cooper pair in a weak-coupling s u p e r c o n d u c t o r at T = O K ) Coherence length (typical decay length of the probability density for a Cooper pair in a weak-coupling s u p e r c o n d u c t o r at T = To) Resistivity (Density, in gL2, eq. (85), and Tab. A3) Residual resistance ratio Residual resistivity Residual resistivity of tin with indium impurities perpendicular to the tetrad axis of the tetragonal elementary cell of "white tin'
Po2o P2gag
~298K, tot 0
Product P ' £ at the Debye temperature, O Phonon induced temperature dependent part of the resistivity at room temperature (298 K) Total resistivity at room temperature (sometimes called O~T) Conductivity in the normal s t a t e (in sec. 5.2: spin index) Effective cross-sectional area for screened c o u l o m b interaction Impurity scattering time in the normal metal ( ~ = 2 / v F) 331
'[eb 1~ebO 'Ee l f
l:ee ~es ~GL PSC ZQ 1~qet Zqin
~q*ee 'tQ*el l:q~ 9
1;q~in Zq~in(O)
~Q*in, jc Zq~in, m ~R ~r
T$ ph
~s
332
Inelastic electron-phonon collision time, i.e. lifetime of electrons in the normal state at the Fermi level and at Too due to e l e c t r o n - p h o n o n collisions The value of zE for pure materials after Keck and Schmid (sec. 5.4) Electron-bogolon scattering rate Abbreviation used in the discussion of Kulik's work in sec. 5.7 Effective quasiparticle recombination time, including phonon trapping effects (sec. 11.4.3) Inelastic electron-electron scattering time Phonon escape time Ginsburg-Landauorder parameter relaxation time (gap relaxation time in the gapless limit) Period of the phase-slip process Branch imbalance relaxation time Branch imbalance relaxation time due to gap anisotropy Branch imbalance relaxation time due to inelastic electron-phonon scattering Charge imbalance relaxation time Charge imbalance relaxation t i m e due to inelastic electron-electron scattering Charge imbalance relaxation t i m e due to elastic scattering processes in the presence of gap anisotropy Charge imbalance relaxation time due to a spatial dependence of the gap Charge imbalance relaxation time due to inelastic e l e c t r o n - p h o n o n processes only Forefactor in the expression of the charge imbalance relaxation time due to inelastic e l e c t r o n - p h o n o n scattering in the case of a weak-coupling superconductor Charge imbalance relaxation time due to inelastic e l e c t r o n - p h o n o n collisions and a s u p e r c u r r e n t Charge imbalance relaxation time due to inelastic e l e c t r o n - p h o n o n collisions and magnetic impurities Relaxation time of the order parameter (TDGL theory) Intrinsic recombination time (between two recombination events) for the recombination of a given quasiparticle (sec. 11.4.3) Intrinsic quasiparticle recombination time characterizing the decay of a quasiparticle population (sec. 11.4.3) Elastic spin-flip scattering time Scattering time for a phonon with a quasiparticle Abbreviation used in the charge imbalance wave equation Order parameter relaxation time E l e c t r o n - p h o n o n scattering time extrapolated back to the Debye temperature from the high temperature limit where the resistivity is proportional to the temperature (sec. 5.4)
t o
%(...) ~OlnW
Z O I D_nP b W
ZOPb In, BiW
XOPbW ZoR ZOSnW
~OZnW 1:2
~o ¢ ¢ (r_, t)
eL, CT
q~a (P12
q~÷_(t)
Unit time of Kaplan et al. [103] Unit time of Kaplan et al. [103] for the material mentioned, (...) = (Sn), (In), sec. 7.2, (...) = (Pb), sec. 9.3 'Low t e m p e r a t u r e value' of the quasiparticle relaxation time zz for In whiskers (sec. 7.2) 'Low temperature value' of the quasiparticle relaxation time z~ for I_n-Pb whiskers (sec. 9.3) Temperature independent quasiparticle relaxation time ~2 evaluated for Pb, Pb-In, and Pb-Bi whiskers (sec. 9.3) Temperature independent quasiparticle relaxation time z2 evaluated for pure Pb whiskers only (sec. 9.3) S u p e r c u r r e n t response time Temperature independent relaxation time z~ evaluated for Sn and S n - I n whiskers (sec. 7.2) Temperature independent relaxation time z2 evaluated for Zn whiskers (sec. 8.3) Quasiparticle relaxation time (SBT model) Potential introduced in the TDGL theory Normalization potential (TDGL theory) Electrostatic potential Scalar potential of the electromagnetic field Charge imbalance induced part of the e l e c t r o s t a t i c potential Order parameter induced part of the electrostatic potential Functions used in the discussion of the electron mean free path dependence of z~ Space dependent phase of the order parameter of the Ginsburg-Landau theory Phase angle of the radiation of a HF field (sec. 7.3) Phase difference between r~ and 1-2, i.e. %2=q0(1-,)-q~(r~), in sec. 7.3: across the core region of a phase-slip center Abbreviation used in sec. 7.3 Difference between the phases of the order parameter at both sides of the Josephson oscillator (KSS model)
x
'Impurity function' of the Ginsburg-Landau theory (sec. 5.1) A quantity which is proportional to the density of impurities (sec. 6.2)
~'f BC$
Wave function of the superconducting ground state as proposed by BCS Complex order parameter of the Ginsburg-Landau theory at locus r Equilibrium value of the order parameter in the absence of c u r r e n t s or magnetic fields
~(_r) @0
333
(D L0a (De
(De6 ~c, r a i n
¢% % ~Kss ¢alKu IDOp [OPSC
Density of particles described by ~ in the absence of currents or magnetic fields (Ginsburg-Landau theory) Absolute value of the equilibrium value of the order parameter far away from the phase-slip center Normalized order parameter (TDGL theory) Time average of the absolute value of the normalized order parameter (TDGL theory) Volume of the superconductor Phonon frequency Angular frequency o f the radiation of a HF field (sec. 7.3) Scaling angular frequency (KSS model) Angular frequency of the Carlsson-Goldman mode Lowest angular frequency that can lead to an enhancement of superconductivity by HF radiation (sec. 7.3) Debye (angular) frequency Josephson (angular) frequency (KSS model) Angular frequency of the charge imbalance wave Angular frequency of the collective mode according to Kulik Frequency of order parameter oscillations Repetition (angular) frequency of the phase-slip process
Direction of electron spin (spin up, spin down)
334
Subject Index
Ac-driven p h a s e - s l i p s t a t e , 134, 135 Ac o r d e r p a r a m e t e r coupling, 133 Ac q u a s i p a r t i c l e coupling, 133 Ac q u a s i p a r t i c l e currents, 151 Acoustic m i s m a t c h theory (for the Kapitza r e s i s t a n c e ) , 220 Alloy whiskers, 7 Andreev reflection, 33, 174, 175,233, 241 A n i s o t r o p i c materials, 7, 103-106 A n i s o t r o p y effect, 104 A n o m a l o u s Green function, 79 A r b i t r a r y q u a s i p a r t i c l e disequilibrium, 46, 54
B a r d e e n - C o o p e r - S c h r i e f f e r (BCS) coherence length, 7, 19, 195 gap equation, 46 g r o u n d s t a t e , 17, 21 theory, 17 Bessel functions, 128-129 Bogolubov o p e r a t o r s , 24 Boundary SC/NC, 33, 34, 42, 66 Branch crossing, 38 Branch imbalance, 39 Branch imbalance relaxation time, 39, 115, 116 - relation with the charge imbalance r e l a x a t i o n time, 39, 40
C a r l s s o n - G o l d m a n mode, S1-54, $6, 69, 153, 163, 167 decay length of the mode, 53, 54 Chaos, 135- 138 analog simulations, 136, 137 analytical solutions, 136
noise, 136 Charge imbalance, 17, 24, 28, 46, 69, 120 d i f f e r e n t definitions, 287 dynamics, 54 - 59 Charge imbalance relaxation, 34, 35, 192, 225 at lower t e m p e r a t u r e s , 39 due to a supercurrent, 36, 37 due to a spatial dependence of the gap, 3 6 - 38 due to e l a s t i c s c a t t e r i n g , 3 5 - 3 9 due to gap anisotropy, 35, 38, 39, 115, 116 due to inelastic e l e c t r o n - e l e c t r o n s c a t t e r i n g , 39 due to inelastic e l e c t r o n - p h o n o n scattering, 3 4 - 3 7 , 40, 113 due to s p i n - f l i p scattering, 36, 37, 39 r e l a x a t i o n length, 33, 66, 109, 113, 115, 119, 120, 151, 189 r e l a x a t i o n time, 36, 67, 111, 113, 115, 121, 175, 189-192 n o n - s t e a d y state, 45, 46 Charge imbalance wave, 56, 57, 6 9 - 7 4 , 133, 149, 193, 194, 211, 215, 224, 225, 278, 279, 299 Charge imbalance wave equation, 17, 54, 56, 69, 70, 117, 295 Charge imbalance wave induced hysteresis ('KSS mechanism'), 225 - 226, 229 - 241 Charge of the condensate, 28 Chemical p o t e n t i a l , 26 of pairs, 22, 2 6 - 3 0 local value, 29 relation with charge 335
imbalance, 2 8 - 30 of quasiparticles, 26, 29-31, 80, 288 physical interpretation, 30, 3l relation with charge imbalance, 29, 30 Coherence length, ~0 (T,0), 19 Collective excitations, 17, S l - 5 4 , 153, 155-167, 279 Collective excitation potential, 164 - 166 Conductivity, 55 Contacts, S C o n t r o l current, 246 C o n t r o l l a b l e weak links, 245 - 255, 258 - 266 Convention of c u r r e n t and voltage notation, 1S6 Cooper pairs, 21 Cooper-pair formation, 23 Coupled Josephson oscillators, 132 Critical current, 1, 14, 20, 90, 99-103, 125, 130, 135, 137, 144, 175, 211, 223, 2 4 6 - 2 4 8 , 2 5 0 - 2 5 4 , 259, 260, 262, 263, 264, 2 7 2 - 2 7 9 Critical c u r r e n t 'locking' or 'pulling', 148, 150, 153 Critical magnetic field, 100 Critical temperature, 1, 103-106 of different bulk materials, 293 of Wood's metal, 175 Cross sectional area, 5, 6 Cystallographic orientation, 7 C u r r e n t - p h a s e relation, 62, 63, 72, 7 6 - 7 8 , 125, 127-129, 194 C u r r e n t steps, 123-126, 129-138, 178, 186, 187, 194, 213
Debye temperature, 41, 220, 293 Density, 44, 293 Density of states a t the Fermi border, N O, 19 in the normal state, Nn(eK), 27, 288 of quasiparticle excitations, Nq(EK), 288 Diameter of a sample, 173 336
Differential resistance, 14, lS, 67, 108-114, 120-123, 170-173, 175, 178, 179, 188, 212, 248 - 251, 254 - 258, 260- 263, 2 6 5 , 2 6 7 - 269 Diffusion c o n s t a n t , 36, 43, 82 Dolan and Jackel experiment, 155 Double tunnel junctions, 41
Effective temperature, T ~, 48, 49 Electrical field in a superconductor, 33 Electrochemical potentials, 17, 22, 26, 107 c o n t r i b u t i o n of collective excitations, 163- 167 m e a s u r e m e n t with an N C / S C p o t e n t i a l - p r o b e pair, 33, 1S5-167 of pairs, 22, 25, 3 0 - 3 2 , 6 4 - 6 6 , 70, 71, 72-76, 118, 127, 128, lSS-167 of quasiparticles, 25, 42, 6 4 - 6 6 , 70, 71, 72-76, 80, 85, 8 8 - 9 0 , 118-120, 136, 1SS-167, 279 relation with charge imbalance, 30 spatially resolved m e a s u r e m e n t s of the quasiparticle potential near a phase-slip center, 15S, 205 Electromagnetic field in a superconductor, 58 - 59, 117 - 118 E l e c t r o n - b o g o l o n scattering (Kulik), 5 8 - 5 9 , 117-119, 176, 191 Electron density, 43 E l e c t r o n - e l e c t r o n scattering time, 39, 43, 44, 176, 177 E l e c t r o n - p h o n o n coupling function, 41, 44, 45, 191 Electrostatic potential, 33, 34, 80, 118 charge imbalance induced part, 33, 34, 70, 71, 7 3 - 7 5 , 80, 81, 94, 118 order parameter induced part, 33, 80, 81, 118 Elesin model, 49, 50 Energy gap, 22, 24 of a s t r o n g - c o u p l i n g superconductor, 195 of a weak-coupling s u p e r c o n d u c t o r
close to the critical t e m p e r a t u r e , 4O reduction by a t r a n s p o r t current, 40 Energy mode, 17, 46, 50, 25l Equilibrium p r o p e r t i e s , 9 9 - 1 0 6 Equivalent circuit of a p h a s e - s l i p center, 70 Euler's c o n s t a n t , 19 Even disequilibrium, 46, 83 Excess current, 91
H e a t - f l o w into the c o n t a c t s , 216, 217 Heating e f f e c t s , 121, 182, 183, t92, 193, 205, 211, 216-222, 251 Heat t r a n s f e r , 211, 214-222 t r a n s f e r coefficient, 214-222 t r a n s f e r t h r o u g h an N C / S C boundary, 217 High-frequency electromagnetic coupling, 133 H i g h - f r e q u e n c y radiation, 123, 126-128, 130, 137, 186, 187 High-frequency radiation s t i m u l a t e d superconductivity, 42, 46, 47, 130, 137, 152 low frequency border, 130, 152 High voltage dc behaviour, 72, 73, 75, 78, 118, 127, 193 Homogeneity of microbridges, 122, 142, 205, 245 of whiskers, 144-146, 205, 245 Hot s p o t s , 182, 214-222 Hysteresis, 16, 7 7 - 7 9 , 90, 108, 123, 135, 169, 170, 175, 183, 185, 192, 211-244, 249, 281
Fermi energy, 19, 43 function, 24, 27, 28 m o m e n t u m , 43 velocity, 19, 111, 174, 190, 293 wave number, 23, 24 F i r s t sound, 51 First voltage step, 15, 108, 109, 1t2, 114, 126, 175, 211, 212, 251, 256-258, 260 F l u c t u a t i o n s , 14, 60, 61, 87, 195, 242, 244 Flux flow, 42 Footlike structure, 214, 223, 224, 230, 232, 233, 242 F o o t s t r u c t u r e s , 42 Fourier Bessel series, 128 Illumination, 42, 47, 134, 251 Four t e r m i n a l weak links, 150 Impedance of the t r a n s m i s s i o n line F o u r t h sound, 51 according to the KSS model, 71, 74 Free charge density, 71 a f t e r Kulik, 118 Friedel oscillations, 167 Impurity function, ~(, 19 Frequency locking, 132 Impurity s c a t t e r i n g time, 38 Inelastic e l e c t r o n - p h o n o n collision time, ~E, 36, 37, 4 1 - 4 6 , 67, 71, 83, Gap enhancement, 46, 47 84, 113, 115, 116, 117, 119-121, 168, Gapless limit, 8 2 - 8 4 , 86 176, 178, 179, 189-191, 193, 195, 197, Ginsburg-Landau (GL) 199, 203, 204, 206, 207, 2 8 8 - 2 9 7 coherence length, 1, 7, 19, 82, 195 e l e c t r o n mean free path dependence, equations, 18 44 - 46, 293 - 295 free enthalpy, 80 summary of values, 2 8 8 - 2 9 2 o r d e r p a r a m e t e r r e l a x a t i o n time, Influence of the contacts, 37, 38 172, 173, 210 parameter, 8, 20 Inhomogeneities, 94, 122, 201, 202, regime, 84 205, 206, 245, 255, 261, 262 theory, 1, 17, 18 Interaction of p h a s e - s l i p centers, 130-132, 138, 139, 141, 142-154, 173,
337
177, 186-188, 259, 269-282, 299 ac quasiparticle effect, 149, 150 a s y m m e t r i c part, 147-149, 151, 278, 299 by quasiparticle currents, 147, 148, 177, 278, 299 change of the voltage, 146, 148, 279 - 282 depression of the critical current, 153, 272, 273, 275-279 enhancement of the critical current, 142-154, 173, 177, 186-188, 259, 269, 272 - 279 mechanism, 152-154, 270, 278-279, 281, 299 polarity dependence, 2 7 5 - 2 7 8 regenerative f e e d - b a c k effect, 149 s y m m e t r i c part, 147, 148, 151, 177, 278, 299 v o l t a g e and differential resistance, 146, 148, 150, 151, 2 7 9 - 2 8 2 Intrinsic hysteresis, 211-215, 219, 222, 223, 225 Inverse ac J o s e p h s o n effect, 124, 125, 134, 135, 137, 138, 186, 187 I s o l a t e d p h a s e - s l i p center, 15, 88, 90, 9 2 - 9 4 , 175, 194, 195, 198-210 Josephson frequency, 60, 69, 73, 79, 107, 123 junctions (chaos), 136, 137 analog simulations, 136, 137 e x p e r i m e n t s , 137 relation, 33, 60, 81, 97, 125, 128 o s c i l l a t o r , 69, 72, 127 tunnel junctions, 132, 136 arrays, 125, 132, 137 J u rap-back current, 99, 108, 2ll, 212, 223, 226, 258 t e m p e r a t u r e , 211, 223 voltage, 16, 108, 212, 226, 256, 258
K a d i n - S m i t h - S k o c p o l (KSS) model, 17, 69, 82, 109, 122, 123, 127, 133, 338
151, 179, 192, 193, 213-215, 225, 226, 228, 229 Kapitza resistance, 211, 214, 216-222
Lattice t e m p e r a t u r e , 224, 225 Length of a sample, 6, 172, 173 Linear p o r t i o n s , 178, 179, 183-187, 192, 193 Local critical current, 67, 245 Local equilibrium, 29 Local equilibrium approximation, 8 4 - 8 6 , 94, 122, 176, 195, 2 0 6 - 2 0 8 , 210 London c o n s t a n t , 31, 32 equations, 31 p e n e t r a t i o n depth, 7, 20 Longitudinal mode, 46 Low t e m p e r a t u r e techniques, 9
Macroscopic wave function, 18 Magnetic p e n e t r a t i o n depth, 1, 7, 19, 20 Material p a r a m e t e r s , 7 Mathiessen's rule, 41 Mean free path for the conduction electrons, 6, 173 Mean square deviation of the gap, 38, 104, 106, 115, 116 Microbridges arrays, 131- 133, 146 interaction of two (short) c l o s e l y spaced bridges, 146- 151 the 'other experiment', 148 long bridges, l, 14, 119-121, 125, 151, 2 0 2 - 2 0 6 , 208-210, 212, 213, 245 s h o r t bridges, 125, 131 special arrangement, 131 variable thickness bridges, 131 variation of the critical t e m p e r a t u r e along the sample, 142 wide bridges, 125 M u l t i p l e - g a p s t a t e s , 47 Multiple terminal weak links, 132, 150
Mu l t i - p o t e n t i a l - p r o b e contacts, S, 138, 269-271
Josephson current, 134 Phase locking, 132 Phase-slip center, 2, 5 9 - 95, t 0 7 - 167 Negative differential resistance, 77, 212 core, 62, 87, 107, 127, 175, 193, 194, Nonequilibrium heating, 251, 252 235, 236, 285 core region, 62, 69, ll8, 127, 193, Nonequilibrium potential, 71 Normal-like length, 91- 95, 108- 123, 194, 23S, 245, 285 170 - 181, 183 - 185, 189 - 210, 25,5 localization, 107, 138-142 low temperature behaviour, 2 5 6 - 2 5 8 quasiparticle distribution at the Occupation probabilities m o m e n t of the phase-slip event, in a BCS superconductor, 22 236 of a K state with a quasiparticle Phase-slip excitation, 27 cycle, 60, 69 Odd disequilibrium, 46, 83 model, 6 2 - 7 9 , 130 Opposed case, 146 phenomenon, 59-61 Order parameter period, 60 in the core region of a phase-slip process, 2, 60, 61, 81, 87 center, 236 repetition frequency, 60 in the TDGL theory, 88, 89, 136 solutions, 86, 87 of the GL theory, 18-20, 82 solutions of the TDGL theory, of the microscopic theory, 19 86 - 91 phase, 20 chaotic solutions, 134- 136 relaxation time, 80 s t a t e (ac-driven), 134, 135 relaxation time in the presence of state (ac and dr-driven), 135-137 a nonzero gap, 42, 50, 51 Phenomenological hysteresis model, 215, 224 - 241 Phonon escape time, 49 Pair breaking, 82 Phonon trapping effects, 237 Pair-breaking effects Phonon radiation limit, 220 of the magnetic field, 130, 137, 251 Plasma oscillations, 51 of the supercurrent, 69, 130, 251, 252 Point contacts, 125 Pair-breaking parameter Propagating collective oscillations, S1 parameter F, 36, 37, 83 Proximity effect bridges, 125, 131 parameter y, 84, 8S, 194-210 array, 125, 131 Pairing interaction, 21 BCS approximation, 22 Pairing time, 81 Q u a s i - o n e - d i m e n s i o n a l superPenetration depth of the electric field conductor, I, 99 into a superconductor, 58, 117 Quasiparticle density in thermal Period doubling cascade, 135 equilibrium, 227 Periodic array of phase-slip centers, Quasiparticle excitations, 8 7 - 9 2 , 136, 154, 199 2, 17, 21-25, 27, 82, 83, 107 Periodic irradiation, 134 charge, 24, 28, 29 Periodic modulation of the maximum density of states, 40, 287 339
excitation energy, 2 2 - 2 5 , 115 Quasiparticle diffusion length, 66, 108, 109, 152, 168, 205 Quasiparticle overpopulation, 47, 148, 174, 175, 211, 215, 224, 226 - 230, 234 - 238, 251 - 254 in the core of a p h a s e - s l i p c e n t e r at the m o m e n t of the p h a s e - s l i p event, 236 Quasiparticle overpopulation induced h y s t e r e s i s ('T* mechanism'), 226- 241 Quasiparticle recombination time, 42, 47, 119, 236, 237 r e l a x a t i o n time, 66, 67, 109, 111, 113, I t S - 117, 119- 122, 168, 174 177, 178, 190, 191, 254, 255 temperature, 24, 147, 226-241, 2 5 2 - 2 5 4 tunnel injection, 33 Quasiparticles e l e c t r o n - l i k e , 24 h o l e - l i k e , 24 more e l e c t r o n - l i k e , 24 more hole-like, 24 Q u a s i - p e r i o d i c p h a s e - s l i p solutions, 135, 136 Q u a s i - t h e r m a l m o d e l s for quasiparticle o v e r p o p u l a t i o n e f f e c t s ( ~ and T* model), 4 7 - 4 9 , 252 -
Radiation from a j o s e p h s o n junction, 132 c o h e r e n t radiation, t32 R e - e n t r a n t p a r t of the V-I characteristic, 77, 2t2 Residual resistance, 6 r e s i s t a n c e ratio, 41 resistivity, 6 Resistance at room t e m p e r a t u r e , 6 Resistive s t e p s , 182 Resistivity, 44 Resistivity at room t e m p e r a t u r e , 41, 293 340
Resistivity s h u n t e d junction model, 77, 129, 137, 149 capacitive, ac and dc drive, 136, 137 capacitive with s e l f - i n d u c t a n c e , 137 R i e g e r - S c a l a p i n o - Mercereau (RSM) model, 17, 62, 81, 86, 126, 128, 194 Screening of an excess charge, 167 Second sound, 51 Second v o l t a g e step, ll0, Ill S e l f - h e a t i n g o f a p h a s e - s l i p center, 215, 224, 225, 230-241 Series case, 146 Sheet resistance, 44 S k o c p o l - B e a s l e y - T i n k h a m (SBT) model, 17, 63, 72, 75, 82, 91, 109, 111, 113, 118, 127, 159, 160, 163, 164, 173-175, 178, 179, 189-193 Sound-like excitations in helium, 51 Sound velocity, 44, 293 Specular scattering, 104, 110 Spin-flip s c a t t e r i n g time, 36, 37 Squid, 252 Stabilization o f s u p e r c o n d u c t i v i t y (by a p h a s e - s l i p center), 142-154, 173, 174, 177, 186-188, 259, 269, 272-279, 299 Static approximation, 86, 9 2 - 9 4 , 197, 2 0 0 - 2 0 2 , 255 Stimulation of superconductivity, 47 S t r o n g - c o u p l i n g superconductors, 17, 9 6 - 9 8 , 178, 179, 183-197 Superconducting electronics, 246, 252 Superconducting ground state, 22 S u p e r c u r r e n t density, 60 Supercurrent r e s p o n s e time, 55, 56, 70, 71, 77, 78, 225, 226 Supereiectron density, 53, 227 Superfluid velocity, 20, 36, 37, 60 Synchronous voltage s t e p s , 139, 140, 142
Telegraph equation, 69, 117 T e m p e r a t u r e hysteresis, 214, 215, 217, 218, 222 - 226, 228 - 243
Temperature mode, 46 Thermal hysteresis, 211, 212, 2 t 6 - 2 2 2 Thermoelectric effects in a superconductor, 34 Thin-film bridges, 125 Third sound, 51 Thomas-Fermi screening, 167 T h r e e - t e r m i n a l weak links, 150 Time-averaged supercurrent, 62, 67, 68, 72, 73, 75, 9l, 107-109, 126-129, 170, 174, 175, 193, 194, 225 T i m e - d e p e n d e n t Ginsburg-Landau (TDGL) equations, 80, 82, 8 3 - 8 5 c o m p u t e r s i m u l a t i o n s of a combined dc and ac current drive, 134 s o l u t i o n s for a q u a s i - o n e dimensional superconductor, 86 - 95 s o l u t i o n for periodically driven c u r r e n t - c a r r y i n g filaments, 134, 135, 138 ac drive, 134, 135 ac and dc drive, 135, 136 two phase-slip centers with ac and dc drive, t35 T i m e - d e p e n d e n t Ginsburg-Landau (TDGL) theory, 17, 7 9 - 9 5 , 109, 122, 153, 154, 176, 178, 189, 194-210, 212, 213, 215, 236, 241-244, 255, 297 Transition characteristics, 14, t5 T r a n s i t i o n temperature, 1, 7, 211 T r a n s m i s s i o n line equivalent, 70 Transverse mode, 46 Tunable weak link (TWL), 122, 205, 245 - 255, 258 - 266 weakening mechanism, 251- 254, 258, 265 Tuning current, 246
Tunnel junctions, 125, 132, 251 Type I and Type I1 superconductors, 20 Unit (or 'scaling') time z 0 of Kaplan et al., 41, 191 empirical rule, 117, 176, 191 Valence effect, 104 Virtual phonons, 2i V o l t a g e - c u r r e n t (V-l) characteristics, 15, 62, 67, 68, 72, 75, 77, 78, 90, 9t, 118, 123-126, 140, t69-171, 182-188, 192, 211, 226, 227, 246, 247, 249-251, 256, 257, 259, 260, 263, 264, 266- 268, 271 Voltage foot, 157, 160, 161 Voltage locking, t32, 133, 138 Voltage standard, 125 V o l t a g e - t e m p e r a t u r e (V-T) curves, 14, 15, 139, 141, 169, 183, 211 Weak-coupling superconductors, 96, 97, 178, 179, 191, 195 Weak link, 62, 126, 131-134, 150, 245, 246 Whisker, 1 characteristic properties, 5 growth, 4 handling, 4 pictures, 5, 6 W h i s k e r / c o n t a c t interface, 174, 175 Zero voltage intercept, I 0, 15, 63, 68, 72, 90, 91, 108-113, 122, 126, 127, 170-176, 178-183, 185, 188, 189, 192 - 199, 202 - 210, 248 - 251, 255, 256, 257, 2 6 0 - 2 6 5 Zeta function, 19
341
