Current-induced Nonequilibrium Phenomena in Quasi One-dimensional Superconductors (Springer Tracts in Modern Physics)

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 <[1>: 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
3

o

E

Vc_g_____. sc I &(

O

VA-.~----->& <

I,PZ

O

I

=/ i

,,

2

o

b)

f


Xps C

Fig. 38: Slope o f electrochemical quasiparticle and pair potentials in sample Sn 4 in the experimental situation o f Fig. 3S. (a) Before the appearance o f phase-slip centers in part 1 o f the sample. (b) Change with the occurrance o f a phase-slip center in part 1.

potential probe.) Then, following Fig. 37b, we get the slopes of 0 and (lz>~ as drawn in Fig. 38a. Since <~)l.p2>(~tp> 0, a small voltage Vc~l<2,p2 b u t (~t)l,p2>(~tp>0, the voltages VAc and VcE have different signs. In the case of sample Sn 4 also ,BT could be used to explain the experimental observations. However, similar experiments have been performed with sample S n 2 [367]. In these experiments the t r a n s p o r t c u r r e n t flows through part l only, while the SC/NC p o t e n t i a l - p r o b e pair P 2 / P 3 indicates a difference between ([l > and <~t> at the site of probe P3. In this sample the NC probe is situated in a much larger distance compared to sample Sn 4. As expected the r e s u l t s can only be understood assuming that probe P3

162

measures the quasiparticle p o t e n ti a l a f te r its swinging over (or pair potential. It is remarked that the t r a n s p o r t cu r r en t e n t e r s t hr o u g h the normal conducting probe P1 and t h a t the slope of N C / S C boundary has been a s s u m e d to be similar to t h a t near

below) the the sample (11) at this a phase-slip

c e n t e r (see Fig. 1r of ref. 367). Now we come to a discussion concerning a theoretical explanation. No s y s t e m a t i c a l l y derived theory is known to the author that r e s u l t s in a slope of (11) as proposed in the p r e s e n t work. The TDGL theory predicts a continuously decaying quasiparticle potential. Also the r e s u l t s of the KSS model do not show any evidence for the 'swinging-over' (or ' s w i n g i n g - b e l o w ' ) e f f e c t. Thus, we briefly sketch the main ideas of o u r own model c a l c u l a t i o n E474]. As already mentioned above, the idea is that a p h a s e - s l i p c e n t e r ex ci t es a c o l l e c t i v e e x c i t a t i o n ( C a r l s s o n - G o l d m a n mode) in addition to a diffusive behaviour o f the quasiparticles. The f e a t u r e s of the C a r l s s o n - G o t d m a n mode are described in detail in sectionS.6, where we also pointed o u t that the mode leads to d if f e r e n c e s of the e l e c t r o c h e m i c a l quasiparticle p o t en t i al and pair p o t en t i al at a given locus. The reason is that the mode g e n e r a t e s regions of enhanced and lowered C o o p e r - p a i r density. To maintain overall charge neutrality more e l e c t r o n - l i k e excitations are removed from regions of enhanced C o o p e r - p a i r density into regions of lowered C o o p e r - p a i r density and more h o l e - l i k e e x c i ta ti o n s are moving vice versa. Thus, in regions of enhanced C o o p e r - p a i r density there is an excess of more h o l e - l i k e over more e l e c t r o n - l i k e e x c i t a t i o n s ( Q * > O , v ~ ) . It is as s u m ed t h a t the core region of a p h a s e - s l i p c e n t e r is the source of the c o l l e c t i v e excitation, because this region periodically em i t s more e l e c t r o n - l i k e (or more h o l e - l i k e ) quasiparticle ex ci t at i o n s into the bordering s u p e r c o n d u c t o r r e p r e s e n t i n g a space charge causing C o o p e r pairs to leave the injection region (or pulling Cooper pairs into the injection region). Thus, on the side of the p h a s e - s l i p c e n t e r with an e x c e s s of more e l e c t r o n - l i k e quasiparticles a kind of C o o p e r - p a i r c o m p r e s s io n wave is g e n e r a t e d , while on the side with an excess of more h o l e - l i k e quasiparticles a kind of C o o p e r - p a i r dilution wave builds up. To simplify the description we a s s u m e that b o t h waves are g e n e r a te d at the core of the p h a s e - s l i p c e n t e r and we thus n e g l e c t the e x t e n s i o n of the injection region. To f o r m u l a t e this idea in a quantitative form, the p h a s e - s l i p c e n t e r may be l o cat ed at x=O and the polarity of the b a t t e r y may be chosen as in Fig. 12c (the negative pole to the l e f t where x < O and the positive pole to the r i g h t where x > O ) . Since the problem is s y m m et r i c in space, only the one half of the slope of the e l e c t r o c h e m i c a l potential is calculated which x ~ 0 for. The t o t a l t i m e - a v e r a g e d e l e c t r o c h e m i c a l quasiparticle potential, ( ~ > , is a ssu m ed to be the sum of t w o terms. The first t e r m is the quasiparticle potential p r o p o s e d by SBT, (~>sBr- The second term, (~t> K, is g e n e r a t e d by the C a r l s s o n - G o l d m a n mode. Thus, 163

<1~> = SBT+K

(203)

The 'SBT potential' is given by eq. (135) to be

(V>s.T = ([~p>r +

eV.~ sinh [ ( x . - x ) / A ] 2 sinh(xR/A)

Since VRc can be obtained from experiment, this equation describes of s0T(x) for fixed b u t arbitrary value of VRL if the point <[Z)sBT=<~p)r is fixed in a reasonable way. The calculation of the ' c o l l e c t i v e - o s c i l l a t i o n potential', (V)K, more involved: Assuming that < i x ) r remains c o n s t a n t in space, we Ansatz

(204)

the slope x R where is much make the

( < V > . - < V . > r ) ( x ) = -CK f, (x) Re < f2 (x) Us(X, t ) >

(205)

(SBT--<[I >r)• o = C K R e < u s ( O , t ) >

(206)

and

Here C k is a c o n s t a n t and us (x, t) is the space and time d e p e n d e n t displacement amplitude of the superfluid part (in one d i m e n s i o n ) as introduced in eq. (94) of sectionS.6, to describe the C a r l s s o n - G o l d m a n mode as a propagating longitudinal wave in which the density of the superfluid part (Cooper pairs) oscillates. This Ansatz can be justified by the following argument: Equation(206) expresses that the collective excitation is excited at x = 0 by an excess of quasiparticles which ((Ix>s~T-fiZp>r)x=0 is a measure for. This excess of quasiparticles generates a proportional e n h a n c e m e n t of the C o o p e r - p a i r density compared to the equilibrium situation. With regard to eq. (205), the change of the Cooper-pair density caused by the excess of quasiparticles at x=O leads to a propagating density f l u c t u a t i o n of the Cooper pairs, resulting in a difference between quasiparticle and pair potentials. This density f l u c t u a t i o n is influenced by the diffusion of the quasiparticles. Thus, in the vicinity of the phase-slip center the density f l u c t u a t i o n does n o t completely build up, because there is an excess of more e l e c t r o n - l i k e quasiparticles and the Cooper pairs escape from this region. We consider the fact by introducing the function f2(x)- Furthermore, the e s c a p e - m o t i o n of more e l e c t r o n - l i k e quasiparticles from regions of enhanced Cooper-pair density is hindered by the diffusing more electron-like excess quasiparticles. This fact is considered by inserting the function f~(x). Following SBT, the diffusing excess quasiparticle population is spatially reduced in proportion to s i n h [ ( x , - x ) / A ] / s i n h ( x R/A). Thus, we assume f,(x) = f 2 ( x ) = 1-

164

sinh [( x , - x) / A ] sinh(xa/A)

(207)

Finally, the negative sign in eq. (205) has been inserted because for the case of enhanced Cooper-pair density (that means ( U s ( X , t ) ) > O ) it yields an excess of more hole-like quasiparticles over more e l e c t r o n - l i k e quasiparticles, which should lead to ( ~)K < ( g~ )r" To calculate < U s ( X , t ) ) we consider Us(X,t) at an arbitrary b u t fixed site x as a f u n c t i o n of time. We assume that together with each p h a s e - s l i p process a new excitation of the collective mode takes place and only take into a c c o u n t c o n t r i b u t i o n s from the just excited wave. In other words, we take the following time average: ( us(x, t) ) = - -1

~ t+%sc Us(X,t) d t

(208)

"%sc At

Here %,c is the period of the phase-slip process and At=x/c~G, where ccG is the velocity of propagation of the mode into the s u p e r c o n d u c t o r (see sectionS.6 for a discussion of Cc~). Thus, we average over one period of the phase-slip process, s t a r t i n g at the m o m e n t at which the density f l u c t u a t i o n begins to build up at the site x. The f u n c t i o n u s ( x , t ) is obtained from eq. (94) by a linear combination of two terms, one term with the positive sign and the other one with the negative sign in the expression for ~0c6 (see eq.(95)). It is assumed that the amplitude of both terms is equal. We f u r t h e r m o r e assume that the collective mode is forced to have the same frequency, 2X/%sc=h/2eV~L, as the phase-slip process, by s e t t i n g 2 n / % s c =Re ~0c~=(G- F=)'/2. Due to eq.(97) it is 2 2 G=ccG qcG leading to qcG = (1/CcG) [(2rc/%sc )2 + F2] '~2. The further evaluation is s t r a i g h t f o r w a r d and r e s u l t s in the expression [474],

Qz>K =
sinh[(x"-x)/A]}~. [cos(qc.x)] [-Fcos(2--E~at) + zp,-"~2x sin("~ -~.--at)12n sinh ( X R / A )

%sc

exp [-F(/~t+Zpsr ]-exp (-FAt)

-F[exp(-F Zpsc)-I ]

(209)

To verify that this r e s u l t can at least qualitatively describe the experiments, we calculated (~)K using eq.(209), (~t)sBT using eq.(204), and then ( p > 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 <'gp)r, (tZ)ssr, (t~>K, and ( t J ) as a function o f the site on the right side o f a phase-slip center situated in part 3 o f sample Sn2. The phase-slip center is located at x=O. We set ([lpfr=O. The potentials are measured in (ze V=IO -6 electron-volt. The voltage Voc (leA) is taken from the experiment at the given temperature. The polarity o f I~A is indicated by the polarity ( ( ~ ) o f the battery[4742 potential ( F > shows the p o s t u l a t e d space dependence: Near the p h a s e - s l i p center its slope is similar to ([l)s. T and at a certain distance it 'swings' below <11.>r. At larger distance from the phase-slip c e n t e r it approaches
The f a c t t h a t there may be a f a c t o r t w o b et w een c a l c u l a t e d and measured values is n o t worrisome, since one should not e x p e c t t o o much from our 'rough' model. More i m p o r t a n t than the equality of numerical values is the f a c t that, given the possibilities of the sample g e o m e t r y (the p h a s e - s l i p c e n t e r has to be localized in p a r t 3 ) , a distance b e t w e e n the normal conducting p o t e n t i a l probe and the p h a s e - s l i p c e n t e r can be found that leads to an a g r e e m e n t of calculated and measured v o l t a g e s in o r d er of magnitude. Naturally, our model calculation may be criticized. The main p r o b l em is t h a t we a s s u m e the collective mode to be damped by the quasiparticle motion only (which is dissipative due to s c a t t e r i n g f r o m impurities), a l t h o u g h the inelastic e l e c t r o n - p h o n o n s c a t t e r i n g time z E is n o t much larger than the s u p e r c u r r e n t r e s p o n s e time %,. Therefore, one would e x p e c t t h a t a damping of the mode due to inelastic e l e c t r o n - p h o n o n p r o c e s s e s should be of i m p o r t a n c e (compare sections 5.0, 5.7, and 5.8.4) and an unknown mechanism has to be a s s u m e d which switches o f f these processes. A n o t h e r p r o b l e m is t h a t we simply combine the s o lu ti o n s of tw o limiting cases, namely the limit of exponential diffusive charge imbalance decay and the limit of the q u a s i p a r t i c l e - m o t i o n damped C a r l s s o n - G o l d m a n mode. It is not clear w h e t h e r this is al l o wed to come to a description of the general case. Thus, the calculation can only be the f ir s t step toward an understanding of the phe n o m en a observed. The m o d el needs an active p h a s e - s l i p c e n t e r and, therefore, does n o t describe the e x p e r i m e n t a l situation where a c u r r e n t leaves or e n t e r s the sample t h ro u g h a normal conducting probe ('dc injection'). However, also in this case C o o p e r pairs should leave a region with an overpopulation of more e l e c t r o n - l i k e e x c i t a t i o n s and more e l e c t r o n - l i k e excitations should leave regions of enhanced C o o p e r - p a i r density. Moreover, it is possible that more h o l e - l i k e quasiparticles from regions lying deeper in the s u p e r c o n d u c t o r migrate toward a region of more e l e c t r o n - l i k e quasiparticle excess. Our r e s u l t s bring to mind the screening problem of an excess charge in a solid. It also turns o u t in this case t h a t the s m o o t h l y decaying T h o m a s - F e r m i s o l u t i o n does n o t seem to be the c o m p l e t e description o f the problem, b u t t h a t l o n g - r a n g e order c h a r g e - d e n s i t y o s c i l l a t i o n s appear, s o - c a l l e d 'Eriedel o s c il la ti o n s ' E4751. The final question is, why measurements performed for long microbridges do not show any evidence for the ' s w i n g i n g - o v e r ' e f f e c t we have observed. The reason may be that the e l e c t r o n mean free path is much s m a l l e r in microbridges and, therefore, the C a r l s s o n - G o l d m a n mode is damped much s t r o n g e r than in our 'clean' samples.

167

8. The Limit of Long Quasiparticle Relaxation Times

8.1.

Introductory Remarks

The characteristic relaxation times of nonequilibrium quasiparticles in a su p er co n d u ct o r , such as the charge imbalance relaxation time, ZQ~, the order p a r a m e t e r relaxation time, zA, the recombination time, z r, and the thermalization time, Zth, all scale with the inelastic e l e c t r o n - p h o n o n collision time, ~E" These times become large if ~E grows (see sections 5.4 and 5.5 and ref. lO3, where z0=8.4z~). Furthermore, the dynamics of charge imbalance depends on z~, because the damping of the collective excitations in a s u p e r c o n d u c t o r decreases with increasing z E (see sections 5.6, 5.7, and 5.8.4). We decided to make an experimental study with whiskers made of t w o d i f f e r e n t materials with extremely d if f e r e n t inelastic electron-phonon collision times. Suitable materials are Zn with large z E and Pb with small zE (see Tab. A1). If we use Tinkham's e s t i m a t e for z E of the pure bulk materials, the z E values of Zn and Pb should differ by a f a c t o r of about 3.3" 10 4. If we take the r e s u l t of Kaplan et al. the values are predicted to differ by a f a c t o r of ab o u t 4 . 1 0 a. Thus, the inelastic collision times z E of these materials differ by several orders of magnitude. At the same time Zn and Pb r e p r e s e n t the limit of long and s h o r t zE in our experiments. While zE is of order lO-"s for Pb, it is of oder 10-~~ for Sn and In and of order tOTs for Zn. It is remarked t h a t ~E for Al is of order lO'8s. (A1 whiskers have not been investigated, b u t e x p e r i m e n t s with long microbridges are r e p o r t e d in the literature.) Ex p eri m en t s on Zn and Pb require enhanced cryogenic e f f o r t s , because the critical t e m p e r a t u r e To0 of Zn is below 1K, while To0 of Pb is above 4.2K. Therefore, these materials cannot be investigated in a bath c r y o s t a t and special c r y o s t a t s were developed which are described in chap. 3 of the p r e s e n t work. While e x p e r i m e n t s on Pb whiskers are discussed in chap. 9, r e s u l t s f o r Zn whiskers are given in the p r e s e n t chapter. Whiskers of Zn are e x p e c t e d to have much larger charge imbalance decay lengths (quasiparticle diffusion lengths) than all samples investigated before, because not only z E and thus ~Q, are large but, due to the ideal crystalline structure, t h ese specimens also have a large e l e c t r o n mean free path s

168

8.2.

Experiments on Zn and Z.&n-Ag Whiskers

Qualitatively, whiskers of Zn and Z n - A g alloys behave similar to tin whiskers [20,41]*': For small measuring currents of ab o u t l[tA or less, the V-T transition curves of pure Zn whiskers have a very small transition width, o f t e n only a fraction of one millikelvin. For Z n - A g alloys this t r a n s i t i o n width is usually some millikelvins. This indicates t h a t the sam p l es c o n s i s t of very h o m o g e n e o u s material. In both cases the transition curves show a shape typical f o r a f l u c t u a t i o n - g o v e r n e d phase transition. For larger c u r r e n t s the transition width of the curves rises and v o l ta g e steps build up. These v o l t a g e s t e p s are also observed in the V-I c h a r a c t e r i s t i c s which we have studied in more detail. The number of v o l t ag e st ep s and their p r o p e r t i e s s t r o n g l y depend on the length, L, of the sample. S h o r t sam p l es only show a single v o l t a g e step (Figs. 40 and 41). The s h o r t e r a sample is, the l o w e r the t e m p e r a t u r e has to be chosen until a sharp v o l t ag e jump can be observed. For very s h o r t samples, down to several millikelvins b el o w To0, the v o l t a g e step is rounded, although f l u c t u a t i o n s should n o t dominate the transition (Fig. 40). As long as no sharp v o lt a g e jump occurs, no h y s t e r e t i c behaviour is observed and for increasing and decreasing cu r r en t the same c h a r a c t e r i s t i c is obtained.

VlpV

ATImK -o.,'.o 1.21 1.61 2.Ol

3-

~ VIpV ~:/~,,./1- o.5o ..~"~:~'// I ~'-'~/ I ~/S~~// / I

2.,1

f

to.25

Zn15 Tco=0.843 K

T=Tco-AT

_

05

10

1Z5 ~ / . A ~

,,/

_

0

I

i

i

t

7 8 I/pA 2 3 4 5 6 Rg. 40: The V-I characteristics o f a very short zinc whisker (Zn IS, L=SOym) f o r several fixed temperatures, T, below the critical temperature, T~o , f o r increasing current[2OJ. The inset s h o w s the l o w current region at higher magnifica t/on. *' There is an err o r in the right hand scale of Fig. 4 in ref. 41. The numbers given have to be multiplied by a f a c t o r of 1.144. Moreover, it is remarked t h a t sample Z n 2 of ref. 41 is called Z n S in ref. 20.

169

VIpV

ATImK

5-

-2.03 0.29 0.57 0.86 1.03 1.15 1.43 1.72 2.00 2.28 2.57

4-

3

2

i

0

i

S S

Znl Tco=O.802K

T=Tco-/1T

,,

i

I

5

i

I

10

i

I

15//pA

Fig. ~l: The V-I characteristics o f a short zinc whisker (Znl, L=226pm) f o r several fixed temperatures, T, below the critical temperature, To, f o r increasing current[412. (The value o f L given here is obtained f r o m S E M pictures [20] and s l i g h t l y differs from that one o f ref. 41 which was measured with a light microscope.)

With increasing length also the number of voltage st ep s increases. A regular v o l t ag e step s t r u c t u r e with a series of steps, similar to whiskers o f Sn or In, is in the case of Zn only observed f o r samples with l en g t h s above ~1200~m (Fig. 42). Whiskers of Zn with Ag impurities qualitatively behave like samples of pure zinc. Generally, the number of v o l ta g e steps in the c h a r a c t e r i s t i c s of Zn and Z n - A g whiskers is much l o w e r than for whiskers of Sn or In with comparable lengths. Naturally, long samples also show hysteresis as soon as the v o l t a g e s t e p s b e c o m e sharp. As we discussed in chap. 6, for Zn and Z n - A g whiskers the critical c u r r e n t obeys the GL theory and the behaviour of the critical t e m p e r a t u r e , To0, is u n d e r s t o o d within the theory of Markowitz and Kadanoff. Now we turn to the properties of an isolated p h a s e - s l i p c e n t e r r e p r e s e n t e d by the f i r s t v o lt a g e step in the V-I characteristics. As discussed in detail in section 7.2, the first voltage step is characterized by its differential resistance (yielding a n o r m a l - l i k e length, LAn 1, which may be re l a t ed to the quasiparticle diffusion length, or charge imbalance relaxation length, within the SBT model) and the zero v o l t ag e intercept, I 0 (which may be i n t e r p r e t e d within the SBT model as a time averaged supercurrent). For all Zn and Z n - A g whiskers both quantities are independent of t e m p e r a t u r e . As an ex am p l e we show the r e s u l t s for a very s h or t and a long pure Zn whisker in Fig. 43.

170

VIpV- ATImK

-5.05 o.t,6 1.10 175

--

Zn21

/

/

//

/

(

~ /~

~.~ r:~o-,~r/ / / / / ~:~

303 lO-

6:17

/ o

,

0

,

100

,

200

300 IIpA

~g. 42: The V-I characteristics o f a long zinc whisker (Zn21, L=137Stzm) f o r several fixed temperatures T (decreasing from left to right) and increasing current [20].

0.5-

0.2 0.03

r:r~o-~r

a)

Zn15

o o oo ~162176

0.01

I

q

~

~

~

I

~

i

b)

0./, 0.1

~

7n15

++ ++ +++++{+{++I+l+++ t

0"7I 0.4 ,

I

10

'

'

i

i

I

5

i

i

i

i

ATImK

0

Fig. 4.3: Differential resistance ( d V / d l ) , and ratio I o / I ~ o f the first voltage step in the V-I characteristics o f a very short zinc whisker (Zn15, T~o=O.843K) and a long zinc whisker (Zn21, T~o=O.B14K) as a function o f the temperature difference A T = T o - T , with To the critical temperature o f the mentioned sample f203.

171

0 1.0-

0.5-

'

0

'

9

'

'

500

'

'

q

I

'

1000

i

L

,

I

'

1500

i

,

,

]

,

2000 L/~m

Fig. 44: Normalized normal-like length LAn I S "2 and ratio I o / I c as a function o f the sample length L. The sofid fines indicate values f o r a phase-slip center appearing unaffected by the contacts. Full squares in (a) indicate

results manifested by the investigation of higher voltage steps.

Since ( d V / d I ) , and I0/Io are t e m p e r a t u r e independent, averaged values can be c a l c u l a t e d f o r each sample. From the r esu l t i n g average of the differential resistance the n o r m a l - l i k e length of the sample, LAnl, can be calculated. In Fig. 44 we plot LAnl/21/2 and the averaged I 0 / L as a function of the length of the sample, L. Both quantities increase with increasing L. As we will discuss in the f o l lo w i n g section, this length dependence is caused by an influence of the s t r o n g l y superconducting Wood metal co n t act s. While LAnl/~pl/2 rapidly increases toward a l e n g t h - i n d e p e n d e n t value, the increase to a l e n g t h - i n d e p e n d e n t behaviour is more slow for I 0 / I 0. For L>1200[~m both quantities are length independent. We conclude t h a t above this l e n g t h p h a s e - s l i p c e n t e r s occur in our whiskers w i t h o u t being a f f e c t e d by the c ontact s . For t h e s e unaffected phase-slip cen t er s mean values are LAnl /2 */2=196 + - 23~tm ~/2 and I0/Io=0.56+ 0.06. It is remarked t h a t samples marked in Fig. 4 4 a by a full square have well developed higher v o l ta g e steps with a differential r e s i s t a n c e which is a multiple of ( d V/ d I ) ~ . This behaviour is expected for weakly interacting p h a s e - s l i p centers. However, in section 7.5 we also found that active p h a s e - s l i p cen t ers may not change the differential r e s i s t a n c e of an influenced c e n t e r although its critical c u r r e n t is enhanced.

172

In Fig. 4 4 a we normalized LAn 1 by 21/2, because we a s s u m e t h a t LAn 1 changes p r o p o r t i o n a l to 2 ~/2. This a s s u m p t i o n has been verified by a l o g - l o g p l o t of LAnI(2). In this p l o t (Fig. I0 of ref. 20) LAnI f o l l o w s a s t r a i g h t line with slope 1 / 2 f o r p h a s e - s l i p c e n te r s appearing .unaffected from the c o n t a c t s in long whiskers. Moreover, there is a group of r e s u l t s f r o m very s h o r t whiskers ( L < l O O p m ) joining a d i f f e r e n t l o w e r lying s t r a i g h t line, also with slope 1 / 2 . For I 0 / I o no dependence on the e l e c t r o n mean free path, ~, has been observed. For all clean whiskers 2 as calculated from the residual r e s i s t a n c e is much larger than the sample diameter d w. From the behaviour of To0 we know t h a t s u r f ace s c a t t e r i n g a f f e c t s the s u p er co n d u ct i n g p r o p e r t i e s of the sample. To include the influence of the surface on the relaxation behaviour of q u as i p ar t i cl e excitations, we assumed t h a t the e l e c t r o n mean free path is limited by the diameter, thus replacing 2 by d w for all sam p l es with 2>>d w in Fig. 4 4 a and in the f u r t h e r discussion of LAn 1. We did n o t undertake a detailed investigation of the interaction o f p h a s e - s l i p c e n t e r s as in the case of tin whiskers ( section 7.5). However, the manner in which the V-I characteristics of our sam p l es develop with decreasing t e m p e r a t u r e indicates an interaction of p h a s e - s l i p c e n t e r s also in the case of Zn whiskers: Once the f i r s t voltage step has developed, the distance in c u r r e n t to the second step increases and then decreases until finally b o t h s t e p s o c c u r at the same c u r r e n t (Fig. 42). For a q u a n t i t a t i v e investigation we p l o t t e d the o n s e t currents, I~/3=I~ ~3 and I~/3, f o r the f i r s t and the second step, respectively, as a function of t e m p e r a t u r e (Fig. 45). Very c l o s e to To and f o r low t e m p e r a t u r e s both st ep s appear t o g e t h e r at I . In the i n t e r m e d i a t e region I s is larger than Io, indicating an e n h a n c e m e n t of I s due to the influence of the f ir s t p h a s e - s l i p center.

8.3.

Interpretation by Comparison with Theoretical Work

First we will discuss the dependence of LAn I and I 0 / I ~ on the sample length L within the f r a m e w o r k o f the SBT model: The relation L A n l ~ 2 A is only valid f o r a p h a s e - s l i p c e n t e r with an undisturbed quasiparticle re l a x at i o n , where the distances b e t w e e n the site of the p h a s e - s l i p center, Xpsr and the points x, and x L, where [z=~,, are large c o m p a r e d to A. In the more general case it f o l l o w s from eq. (137) (dV/dI), =

(2PnA/A)tanh(L/2A)

(210)

and, thus, LAn 1 = 2 A t a n h ( L / 2 A )

(211)

These r e s u l t s are valid for a p h a s e - s l i p c e n t e r situated in the middle of the whisker and the a s s u m p t i o n that the t i m e - a v e r a g e d e l e c t r o c h e m i c a l p o t e n t i a l s 173

I2/3/ pA 213 40-

4. A 40 I 304

I I

3O

~ 7o 1

20

T___n QI/ tz /CO=U'U

\

14 I%

~

*

T:Tco-AT 10

I .... I

o f i-r s t

0

10

20

~.~

voltage step

~

30

40

50

r 213. --213 ~C / I J A

,

9 second voltage step 0

.

.

.

.

.I 5

.

. 1

t 0

' -1

z1TImK

Fig. 4S: C u r r e n t at t h e o n s e t o f the f i r s t and s e c o n d voltage s t e p s o f a z i n c w h i s k e r ( s a m p l e Z n 21) as a f u n c t i o n o f d T= T o - T. Data taken f r o m Fig. 42. The s o l i d line r e p r e s e n t s I Z / J ~ d T . The i n s e t s h o w s the d i f f e r e n c e b e t w e e n the o n s e t c u r r e n t o f b o t h s t e p s as a f u n c t i o n o f IJ/3. For the i n s e t data w e r e taken f r o m Fig. 42 and f r o m f u r t h e r m e a s u r e m e n t s .

of pairs and quasiparticles are equal at the c o n t a c t edges. This equality o f the e l e c t r o c h e m i c a l p o t e n t i a l s of pairs and quasiparticles is the c o n s e q u e n c e of an Andreev r e f l e c t i o n p r o b l e m at the w h i s k e r / c o n t a c t interface as will be discussed within the i n t e r p r e t a t i o n of the behaviour of I 0 / I o. Thus, the relation LAnI=2A, where A = ( ( 1 / 3 ) s '/2, only can be applied in the region where LAn t is independent of the sample length and the p h a s e - s l i p c e n t e r s are not a f f e c t e d by the contacts. With v~=9.1, lOs m / s one has % = ( 3 . 2 + 0 . 7 5 ) ' 1 0 - 8 s which may be called ZOZnW according to the s y s t e m a t i c n o t a t i o n introduced in section 7.2. With this value for % we calculated A for each sample and then invoked eq. (211) for a prediction of LAn I for the sample. The ratio o f measured and thus calculated values of LAn I is close to unity. This indicates t h a t the length dependence o f LAn I is described by the SBT model and s u p p o r t s our evaluation o f %. Let us now discuss the behaviour of I 0 / I ~ which f o r an u n a f f e c t e d p h a s e - s l i p c e n t e r has a magnitude similar to that one measured for whiskers of Sn and In (not t o o close to To0) and to the SBT prediction, b u t which decreases fo r decreasing sample length. In our opinion this reduction of I 0 / I c is the c o n s e q u e n c e of an enhanced overpopulation of the excitation s p e c t r u m with nonequilibrium quasiparticles: For s h o r t samples a large number o f nonequilibrium quasiparticles generated by the phase-slip process reach the super co n d u ct i n g contacts. At the interface between the whisker and the c o n t a c t the energy gap rises rapidly (typical length ~(T)) from the small gap

174

of the s a m p l e t o w a r d the large gap of the Wood metal. (The t e m p e r a t u r e is c los e to the critical t e m p e r a t u r e of the Zn whisker, b u t very low c o m p a r e d to the critical t e m p e r a t u r e of W o o d ' s metal of about 8.2K [19].) In the c o n t a c t region we, thus, have an Andreev r e f l e c t i o n p r o b l em [275]. A quasiparticle can only p e n e t r a t e into the c o n t a c t region until its energy is equal to the l o w e s t energy of the spatially varying excitation spectrum. At this point t h ere has to be a direct conversion from the quasiparticle c u r r e n t into a s u p er cu r r en t. Essentially all of the quasiparticle ex ci t at i o n s have to undergo this conversion process. The conversion process is called Andreev reflection, because in this p r o c e s s the quasiparticle which tries to p e n e t r a t e into the s t r o n g s u p e r c o n d u c t o r vanishes and a quasiparticle with o p p o s i t e e f f e c t i v e charge is e m i t t e d out of the s t r o n g s u p e r c o n d u c t o r into the o p p o s i t e direction. The quasiparticles seem to be r e f l e c t e d at the i n t e r f a c e with their e f f e c t i v e charge changed to the o p p o s i t e [94,275]. Thus, if a c u r r e n t o f e l e c t r o n - l i k e quasiparticles p e n e t r a t e s into the interface, a c u r r e n t of h o l e - l i k e e x c i t a t i o n s leaves the interface in the o p p o s i t e direction. The c o n s e q u e n c e is a reduction of the charge imbalance to zero in the c o n t a c t region, leading to an equality b e t w e e n the e l e c t r o c h e m i c a l p o t e n t i a l s of quasiparticles and C o o p e r pairs. In the whisker the charge imbalance is reduced by the r e f l e c t e d excitations, but the density of nonequilibrium q u a s i p a r t i c l e s is enhanced compared to an undisturbed relaxation in a long sample. Therefore, an overpopulation of the excitation s p e c t r u m is created at the core of the p h a s e - s l i p c e n t e r which we assume to be characterized by an e f fe ct i v e t e m p e r a t u r e T*> 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<~0(T)). The f i r s t condition is not the m o s t serious restriction. Although s "~o is only s t r i c t l y f ulf i l l ed f o r our Z n - A g alloy whiskers, also for our pure samples it is ~ ~ ~0. The local equilibrium condition, however, r e s t r i c t s the applicability of the theory to a very small t e m p e r a t u r e interval of only some o n e - h u n d r e d t h parts of one millikelvin below the critical t e m p e r a t u r e [201. Thus, we c a n n o t e x p e c t a q u a n t i t a t i v e description of our m e a s u rem en t s. KR give r e s u l t s as a function of the pair-breaking p a r a m e t e r T. The p r o b l e m is t h a t f o r o u r samples it is ~'> 1000. For such high ~, values KR do not give a prediction for the n o r m a l - l i k e length and the z e r o - v o l t a g e intercept. For 7 ~ 1000, KR obtain that LAn 1 should be close to 2 A Q , i n ~ A T -'/4. Furthermore, I 0 should be c l o s e to zero. Both predictions do not agree with o u r experiments. B a r a t o f f calculated the behaviour o f a phase-slip c e n t e r (in a g ap l ess q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r ) in a situation, where the local equilibrium approximation does not hold. This approach has been widely discussed in sections 5.9 and 7.2. It predicts 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 (of a given sample) as soon as the t e m p e r a t u r e is n o t t o o c los e to To0, so that ~GL~ tE (that means AE < ~0 (T) ). In this t e m p e r a t u r e - i n d e p e n d e n t regime z 2 may be identified with ze which does n o t depend on the t e m p e r a t u r e (for a given sample). From the condition ~GLgs it f o l l o w s that the t e m p e r a t u r e - i n d e p e n d e n t regime f o r zinc should already be p r e s e n t if the t e m p e r a t u r e is lowered to only some o n e - h u n d r e d t h parts of one millikelvin below Tc0[20]. Moreover, for pure p o l y cry s t a l li n e zinc we g e t s 9 l O T s using Tinkham's f o r m u l a and zE=9.3"lO-Ss a f t e r Kaplan et al. (compare s e c t i o n S . 4 and Tab. A1). Our e x p e r i m e n t a l r e s u l t for % is only a f a c t o r of 3 sm al l er than Kaplan's value for zE" B a r a t o f f ' s theory, therefore, gives a possible explanation for o u r measurements. Since Zn is a material with a large inelastic e l e c t r o n - p h o n o n s c a t t e r i n g time, rE, we also calculated the 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 time, t e e , f r o m t he o r et i cal results given in section 5.4. For pure three-dimensional polycrystalline zinc at its critical temperature, t e e is of order 10 .6 s. In the 176

t h r e e - d i m e n s i o n a l case with finite 2. at T = O K it is Zee of order lO-Ss f o r an e l e c t r o n with energy ~K= A ( T 0 0 = O - 8 K , A T = I 0 m K ) , if it is assu m ed t h a t 2.=ll.tm. Here, &(To0=0.8K, A T = I 0 m K ) is the BCS r e s u l t f o r the energy gap of 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 with T 0 = 0 . S K at a t e m p e r a t u r e which is 10 mK b el o w To0, as may be reasonable for a Zn whisker. Furthermore, one may evaluate the expression for t h i ck er t w o - d i m e n s i o n a l disordered systems. (For polycrystalline zinc with 2 ~ l ~ m and a film thickness of d ~ l p m at 0.875K it is d 2 < . h v r 2 / 3 k T and k v d , > 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


0

I l9

0

0

~

0

000

00000 0

0

9

o 9o

5'0

Pbl

o

Pb3

9

160

i213, / p A^213

FIg. SS-- C u r r e n t

range A ] which the f i r s t linear p o r t i o n ( o r f o r l o w e r t e m p e r a t u r e s the f i r s t voltage s t e p ) exists in, as a f u n c t i o n o f I~2~ f o r t w o lead whiskers, samples Pb l and Pb 3 E3SPZ

c u r r e n t . W h i l e h i t h e n d e c r e a s e s for P b 3 ( s i m i l a r to t h e i n v e s t i g a t e d Zn w h i s k e r in s e c t i o n 8.2), it s e e m s to s a t u r a t e f o r s a m p l e Pb 1. Thus, a l s o in t h e c a s e o f Pb w h i s k e r s t h e d i s t a n c e in c u r r e n t b e t w e e n t h e a p p e a r a n c e o f t h e f i r s t p h a s e - s l i p c e n t e r and the n e x t p h a s e - s l i p c e n t e r (or p h a s e - s l i p c e n t e r s , if t h e y a p p e a r t o g e t h e r at t h e s a m e c u r r e n t ) is s m a l l f o r s m a l l c r i t i c a l c u r r e n t s b u t g r o w s if t h e critical c u r r e n t b e c o m e s l a r g e r a t

187

lower t e m p e r a t u r e s . From this behaviour we conclude that the p r e s e n c e of a p h a s e - s l i p c e n t e r enhances the critical c u r r e n t of the f o l l o w i n g centers. Thus, also in the case of Pb whiskers a stabilizing e f f e c t g en er at es the large c ur r en t region b e t w e e n the f i r s t onset of vo l t ag e and the normal state. However, also in this case the mechanism is still unknown.

9.2.4.

Pb with Small Impurity Concentrations of In (or Bi)

The V-I c h a r a c t e r i s ti c s of Pb whiskers with small impurity c o n c e n t r a t i o n s of In or Bi are very similar to those of pure Pb whiskers. In Fig. 59 the c h a r a c t e r i s t i c s f o r a Pb whisker with 0.4at% In (sample P_b_blnS) are s h o wn [359]. Results for a Pb whisker with 1.4Sat% Bi (sample P b B i 3 ) are given in ref. 359. The main difference to the case of pure Pb is t h a t In and Bi impurities act to shift the t e m p e r a t u r e at which the transition be c o m es unstable at l toward the critical t e m p e r a t u r e and t h a t the c h a r a c t e r i s t i c is much more s t r a i g h t a f t e r the v o l t ag e jump. For l o w er t e m p e r a t u r e s a distinct v o l ta g e step s t r u c t u r e similar to that one of Sn or In is observed. Evaluating the differential resistance of the first linear p o r t i o n (or at low er t e m p e r a t u r e the first voltage step) one finds t h a t the d i f f e r e n t i a l r e s i s t a n c e is t e m p e r a t u r e independent. There may be a slight increase o f the differential r e s i s t a n c e close to To0 and at low t e m p e r a t u r e s . Close to To0 where the characteristics show linear portions I 0 / I c is cl o se to unity. According to the d e v e l o p m e n t of voltage st ep s in the characteristics, the ratio I 0 / I ` then decreases with decreasing t e m p e r a t u r e . For lower t e m p e r a t u r e s , 10/I ` b e c o m e s independent of t e m p e r a t u r e . As well for sample P__bbln5 as for P__bbBi3, I 0 / I ~ roughly equals 0.85 in the low t e m p e r a t u r e range [359].

15-

P._bbIn5 AT/mK Tco=71931K

5.6 8.5 11.7

13.6

15.9 I

T = Tco-A T >, -1 \

10-

5-

0 3O

80

140

200

300

350

I/pA Fig. Sg: V - I c h a r a c t e r i s t i c s o f a Pb w h i s k e r w i t h 0 . 4 a t X In i m p u r i t i e s P_bbin 5) f o r increasing and decreasing c u r r e n t [359].

188

(sample

E 30 EL

20-

_.~1lO-

Pb

o, 9

Pbln

[]

PbBi

A

5

t.01

0:1 I /~m

Fig. 60: N o r m a l - l i k e length, LAn 1 , as a function path, Z, using l o g - l o g coordinates.

o f the electron

mean free

The p l o t s h o w s r e s u l t s f o r w h i s k e r s o f P__bb-In, denoted "(LAnl)OPblt, W; P__b_b-Bi, d e n o t e d "(LAn 1 )OP___bBJW~ and pure Pb, denoted "(LAn I )OPbW" The r e s u l t s f o r the alloys are f r o m ref. 359 as well as the light s y m b o l m a r k e d Pb values, while the f u l l s y m b o l marked Pb r e s u l t s are f r o m ref. 40.

In Fig. 60 we p l o t t e d the n o r m a l - l i k e lengths for several P__bb-ln and P_bb-Bi whiskers as a function of the e l e c t r o n mean free path, adding r e s u l t s of pure Pb whiskers. For the alloys, we calculated LAn 1 by averaging the d i f f e r e n t i a l r e s i s t a n c e in the t e m p e r a t u r e region where it is t e m p e r a t u r e independent. The r e s u l t s show the familiar 2~/2dependence. We also i n v e s t i g a t e d the mean free path dependence of the averaged low t e m p e r a t u r e values ( I 0 / I o ) 0 f o r Pb, P__bb-In, and Pb-Bi whiskers [359]. With decreasing 2 the r e s u l t s decrease from about unity to about 0.85 for s h o r t e l e c t r o n mean free paths.

9.3.

Interpretation within Phase-Slip Models and the TDGL Theory

First, we will i n t e r p r e t our e x p e r i m e n t a l r e s u l t s in the f r a m e w o r k of the SBT model. For In-rich alloys the n o r m a l - l i k e length LAn 1 exhibits a variety of t e m p e r a t u r e dependences. For several samples we found a certain t e m p e r a t u r e range where L A n l ~ A T -1/4. As we pointed o u t in sectionT.2, LAn 1 may be identified with 2AQ~in in this case (where AQ~m is the charge imbalance r e l a x a t i o n length due to inelastic electron--phonon p r o c e s s e s ) and an e x p e r i m e n t a l value f o r zQ~in(O)~z ~ may then be evaluated. It is found t h a t the e x p e r i m e n t a l values for %.in(O) as evaluated from the e x p e r i m e n t and as c a l c u l a t e d t h e o r e t i c a l l y (using Tinkham's e s t i m a t e of eq.(72) for z E) have the

189

same order of magnitude (see Tab. I of ref. 40). Moreover, the e x p e r i m e n t a l r e s u l t s obtained for ZE' using zQ,in(O) = 0.42 z E, systematically fit into Fig. A 2 concerning the dependence of z E on the e l e c t r o n mean free path. Therefore, we conclude that in the region where the n o r m a l - l i k e length is p r o p o r t i o n a l to AT -1/4 the healing of quasiparticles is governed by the relaxation of charge imbalance due to inelastic phonon processes. In this c o n t e x t it is remarked that in the case of In-rich alloys we did not s t r i c t l y f o l l o w Tinkham's formula to calculate z E in ref. 40. N ev er t h el ess, the r e s u l t s obtained seem to be 'closer to the truth' and, thus, will also be used in the discussion of the experimental r e s u l t s given in the p r e s e n t work. The p r o b l e m is explained in detail in appendix 5 where we also give the values of zE which would f o l l o w f r o m a s t r i c t application of Tinkham's formula. For sufficiently low t e m p e r a t u r e s the n o r m a l - l i k e length of In-rich alloys b e c o m e s independent of t e m p e r a t u r e and shows the 21/2dependence predicted by the SBT model. As described in section 7.2 a t e m p e r a t u r e independent quasiparticle relaxation time z 2 can be evaluated f r o m the s t r a i g h t line of Fig. 48, r e p r e s e n t i n g a graphically averaged value f o r all In-Pb whiskers considering the r e s u l t s of pure indium whiskers. Assuming t h a t the value of vF f o r indium as given in Tab. A 2 also holds for In-rich alloys, it f o l l o w s z 2=1.90. |0 -~ s [ 4 0 ] , s u b s e q u e n t l y referred to as ZOinPbW. For whiskers of Pb and Pb with small c o n c e n t r a t i o n s of In or Bi impurities, the n o r m a l - l i k e length is independent of t e m p e r a t u r e and shows the s predicted by SBT. Also in this case a t e m p e r a t u r e independent quasiparticle relaxation time can be evaluated. For Pb whiskers alone it f o l l o w s f r o m Fig. 15 of ref. 40 z2=5.42-10-~~ (called ZOPbW in the following). For P___bb-In and P__bb-Bi whiskers, considering also samples of p u r e Pb and assuming t h a t the value for vF of Pb (Tab. A2) also holds for P b - r i c h alloys, it f o l l o w s from Fig. 60, t h a t z2=4.93-10-~~ in the following).

[359] (called ZOP_~bln,Bi w

In s ect i o n 7.2 we discussed that a t e m p e r a t u r e independent quasiparticle relaxation time which may be identified with z E is e x p e c t e d from theory f o r sufficiently low t e m p e r a t u r e s where zG~< ZE' This i n t e r p r e t a t i o n is n o t s u c c e s s f u l f o r the samples discussed in the p r e s e n t chapter: In the case of In-rich alloys samples 2, 5, and 8 show a d i s t i n c t region where L A n l ~ A T -1/4 besides the t e m p e r a t u r e independent behaviour (Fig. 46). As m e n t i o n e d above, an experimental value of z E can be evaluated in this case which may be used to calculate the t e m p e r a t u r e b o r d e r below which a 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 l e n g t h should occur. Using ~Q.in(O)=O.42~E, we g e t from Tab. 1 of ref. 40 z E = l . 0 9 - l O l ~ 0.95. l O l ~ and 2.57-10-~~ f o r samples 2, S, and 8, leading to t e m p e r a t u r e b o r d e r s o f hT=27.6 mK, 31.5 mK, and 11.7 mK, respectively. While the r e s u l t is r e a s o n a b l e for sample 8, the change to a t e m p e r a t u r e independent behaviour occurs much c l o s e r to T0 for the o t h e r specimens (Fig. 46). Moreover, in all cases the values o f z E are an o r d e r of magnitude s m a l l e r than ZOI_nPbW" 190

For whiskers of Pb and Pb with small c o n c e n t r a t i o n s of In and Bi, only a t e m p e r a t u r e independent behaviour was observed. Therefore, values for ZE of Pb cannot be obtained from our e x p e r i m e n t s but r e s u l t s from Tab. A1 may be used to e s t i m a t e the t e m p e r a t u r e border. Llsing ~E=0-0555" 10-~~s, as calculated by applying Tinkham's e s ti m a t e , we g et AT=540mK. For zE=O.38.10-~~ as measured, we get AT=79mK. In both cases the change to a t e m p e r a t u r e independent behaviour is predicted for t e m p e r a t u r e s far below the range of our m e a s u r e m e n t s . Furthermore, ZOPbln,BiW~l:OPbW is at l east an order of magnitude larger than all known values f o r z E of lead (Tab. A1). Next, also identifying LAn 1 with two times the t e m p e r a t u r e independent l e n g t h A~,(O), as discussed in section 7.2, does n o t lead to an explanation of the t e m p e r a t u r e independent regime. Already in section 7.2 we discussed t h a t AKo(O) is much t o o large in the case of pure In whiskers and t h a t its p r o p o r t i o n a l i t y to s c o n t r a d i c t s the observed # 1/2dependence of the n o r m a l - l i k e length. Since pure In whiskers and I n - P b whiskers join the same line LAn I ~21/2 (Fig. 48), the length AKo(O) is also not appropriate to explain the e x p e r i m e n t s p e r f o r m e d with whiskers of In-rich alloys. The same conclusion is drawn in the case of Pb and Pb with small c o n c e n t r a t i o n s o f In and Bi. Hsing the material p a r a m e t e r s of pure bulk Pb and s ~ l g m as typical f o r a pure whisker, we g e t A,u(O)=312tlm which is much larger than LAn 1 / 2 in this case (Fig. 60). Moreover, the predicted 2 dependence c o n t r a d i c t s the observed 2 '/2 law. Thus, also in this case e l e c t r o n - b o g o l o n s c a t t e r i n g is no appropriate explanation for the t e m p e r a t u r e independence of the n o r m a l - l i k e length. It is remarkable that the temperature independent quasiparticle r e l a x a t i o n times f o r In-rich and Pb-rich alloys join the empirical rule which c o n n e c t s the quasiparticle relaxation times with the scaling times t 0 introduced by Kaplan et al. [103]. For a detailed discussion see section 7.2. T o g e t h e r with the r e s u l t of t h a t section it f o l l o w s ~OSnW/Z0(Sn) = (ZOinW'~,'EOInPbW)/'Eo(ln) = ('COPbW~'EOPloln,BiW)/'Co(Pb)

(212)

where ~ 0 ( P b ) = 0 . 1 9 6 ' l O - g s [103]. It is s o m e w h a t astonishing that r e s u l t s obtained from s t r o n g - c o u p l i n g and w e a k - c o u p l i n g materials join a c o m m o n rule. As we have d i scu ssed in detail in ref. 40, this r e s u l t indicates that the charge imbalance relaxation in lead is caused by s c a t t e r i n g p r o c e s s e s in which low energy phonons are involved f o r which the e l e c t r o n - p h o n o n coupling function is ap p r o x i m at el y p r o p o r t i o n a l to the quadrat of the phonon frequency as in the case o f weak-coupling superconductors. Thus, only the AT -1/4 law observed for LAn 1 of several In-Pb alloy whiskers in a certain t e m p e r a t u r e range can be u n d e r s t o o d within the SBT model. In this t e m p e r a t u r e region inelastic e l e c t r o n - p h o n o n s c a t t e r i n g governs 191

the charge imbalance relaxation. The t e m p e r a t u r e independent r e s u l t s of the n o r m a l - l i k e length lead to experimental values of a t e m p e r a t u r e i n d ep en d en t quasiparticle relaxation time. However, these times cannot be interpreted. Also the s t r o n g e r t e m p e r a t u r e laws observed for several In-rich alloys c a n n o t be u n d e r s t o o d within the SBT model. One runs into the same p r o b l e m s as discussed in detail in section 7.2 for the case of pure In whiskers. Also the t e m p e r a t u r e dependence observed f o r I 0 / l r for In-rich alloys cannot be u n d e r s t o o d within the SBT model. In this model it is I o / I ` ~ 0.65. This value is in a g r e e m e n t with the t e m p e r a t u r e independent value of I 0 / I r for pure In whiskers (Fig. S0). However, the e l e c t r o n mean free path dependence of t h e s e values (Fig. 50) also cannot be u n d e r s t o o d in the SBT model. Moreover, the r e s u l t that I 0 / I " is close to unity as observed f o r whiskers of pure Pb and Pb_b-In or P b_b-Bi whiskers close to To0 is n o t predicted by SBT. The shape of the V-I c h a r a c te r is t ic s of Pb, P b-ln, and P__.bb-Bi whiskers can, however, qualitatively be u n d e r s t o o d within the KSS model widely discussed in s ect i o n 5.8.4: Ix Fig. 16 we p l o t t e d r e s u l t s of the analytical approximation of the KSS model. The shape of the I - V characteristics depends s t r o n g l y on the ratio ~E/~0R. For ~E/'Co,,~l (T close to To0), the KSS model predicts t h a t the characteristic is a s t r a i g h t line, developing continuously at I , so that I0=I o. For increasing % / % , (decreasing T) the characteristic b en d s up until for ~E/'r it rises vertically at the critical c u r r en t and then bends over to a q u a s i - o h m i c behaviour for I~Io. By a f u r t h e r decrease of the t e m p e r a t u r e (zE/zo,>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 T
the s a m p l e s w o u l d be dirty. The l a s t r e q u i r e m e n t is n o t f u l f i l l e d for all s a m p l e s , b u t it may be t h e c a s e f o r t h e Ln-Pb w h i s k e r s 6.1, 7, 8, 11, 13, 14, and 18. The In w h i s k e r s and t h e I n - P b w h i s k e r s s h o w a variety o f t e m p e r a t u r e laws f o r t h e n o r m a l - l i k e length. For a fixed s a m p l e t h e t e m p e r a t u r e d e p e n d e n c e o f LAn 1 b e c o m e s w e a k e r f o r l o w e r m e a s u r i n g t e m p e r a t u r e . If t h e t e m p e r a t u r e is s u f f i c i e n t l y far away f r o m T~ t h e n o r m a l - l i k e l e n g t h b e c o m e s t e m p e r a t u r e i n d e p e n d e n t . All t e m p e r a t u r e laws ( e x c e p t t h e t e m p e r a t u r e i n d e p e n d e n c e ) e x p e r i m e n t a l l y o b s e r v e d are a l s o p r e d i c t e d by KR. Thus, t h e r e is a q u a l i t a t i v e a g r e e m e n t b e t w e e n the TDGL t h e o r y and o u r e x p e r i m e n t s , e x c e p t t h a t t h e t e m p e r a t u r e i n d e p e n d e n c e of LAn ! c a n n o t be u n d e r s t o o d within t h e TDGL r e s u l t s of KR. U s u a l l y a s a m p l e d o e s n o t s h o w all t e m p e r a t u r e laws ( L A n l ~ A T "~, AT - ~ , 2 , AT-"*, and t e m p e r a t u r e i n d e p e n d e n t ) , if the m e a s u r i n g t e m p e r a t u r e is l o w e r e d ( s t a r t i n g f r o m T,0). T h e r e are w h i s k e r s with L A n I ~ A T -~j4 and LAn 1 temperature independent only. O t h e r s directly change from exhibiting LAn I ~ AT -~ to a t e m p e r a t u r e i n d e p e n d e n t n o r m a l - l i k e length. F o r a m o r e q u a n t i t a t i v e c o m p a r i s o n o f the TDGL t h e o r y w i t h o u r m e a s u r e m e n t s we t a b u l a t e d the t e m p e r a t u r e b o r d e r s and 3' v a l u e b o r d e r s w i t h i n which a given t e m p e r a t u r e law f o r Lnnl is e x p e r i m e n t a l l y o b s e r v e d f o r the d i f f e r e n t s a m p l e s [40]. It t u r n s o u t t h a t L A n i ~ A T -~ is u s u a l l y o b s e r v e d if 3' is b e t w e e n 5 and ll and LAn 1 ~ AT -ty2 for 3' b e t w e e n 9 and 22. This may be r e g a r d e d as an a g r e e m e n t with t h e TDGL t h e o r y - * ' which p r e d i c t s a c h a n g e to a AT -~ b e h a v i o u r if 3' f a l l s b e l o w 10 and a AT -~j2 law f o r 3' b e t w e e n 10 and 20. (AKR/~D(T) only s l i g h t l y c h a n g e s if 3' is' e n h a n c e d f r o m 10 to 20.) A n o r m a l - l i k e l e n g t h LAnl ~ AT -tz" is o b s e r v e d for 3' v a l u e s b e t w e e n 7 and 39. This o b s e r v a t i o n is n o t in a g r e e m e n t w i t h t h e TDGL t h e o r y which e x p e c t s a AT -'~4 law for T > I O 0 . In the m e n t i o n e d 3' r a n g e a AT -~/2 law o r s o m e t h i n g b e t w e e n a AT -1/2 and AT "~4 law is p r e d i c t e d by t h e TDGL theory. In t h e r e g i o n of t h e AT -~j2 law one may c o m p a r e the a b s o l u t e v a l u e s o f t h e m e a s u r e d n o r m a l - l i k e l e n g t h s w i t h the p r e d i c t i o n o f t h e TDGL theory. The c a l c u l a t e d v a l u e s are l a r g e r t h a n t h e m e a s u r e d o n e s {by up to a f a c t o r of 2.6). It may be i n t e r e s t i n g to s u m m a r i z e t h e r e s u l t s f o r dirty w h i s k e r s w h i c h t h e TDGL t h e o r y can be s t r i c t l y applied for: T h e s e s a m p l e s u s u a l l y s h o w L A n t ~ A T -~ f o r T b e t w e e n S.4 and 11.2 and L A n I ~ A T -lz= f o r T b e t w e e n 8.9 and 17.9. F u r t h e r m o r e , t h e r e is one s a m p l e with LAn I ~ AT -*~" h o w e v e r , a t v a l u e s of 3' b e t w e e n 6.5 and 18.1. The c a l c u l a t e d v a l u e s f o r t h e n o r m a l - l i k e l e n g t h in t h e AT -*/2 t e m p e r a t u r e law r e g i o n are l a r g e r than t h e m e a s u r e d o n e s (by up t o a f a c t o r o f 2.6).

9 1 C o n c e r n i n g the AT ~ d e p e n d e n c e this c o n c l u s i o n c o u l d n o t be d r a w n in ref. 40, b e c a u s e t h e r e s u l t s o f t h e TDGL t h e o r y f o r T <10 w e r e n o t k n o w n at t h a t time. 196

Now we will discuss the r e s u l t s of Pb, P10-1n, and Pb-Bi whiskers. The value of ~E f o r t h e s e materials is so small that all e x p e r i m e n t s are p e r f o r m e d for ~" values below ~'=3 [40, 359]. For sample 7 * and 12", the t e m p e r a t u r e is, moreover, so c l o s e to T~ t h a t 0.48 < ~ < 0.74 and 0.21< q, < 0.62, respectively. In this -( range no prediction for the n o r m a l - l i k e length of a h o m o g e n e o u s f i l a m e n t is given by KR. Only r e s u l t s of the static approximation are available, but also only down to -/=1. Therefore, no comparison b e t w e e n e x p e r i m e n t and theory can be done if the specimens are regarded as h o m o g e n e o u s filaments. However, also the static approximation would n o t yield the t e m p e r a t u r e independence of the n o r m a l - l i k e length observed experimentally. Finally, the ratio [~K,=10/Ic is predicted to grow with decreasing ~" ( T a b . ! of ref. 316). For a fixed sample I0/I ~ thus should increase if approaching T00. This is in qualitative a g r e e m e n t with our experimental o b s e r v a t i o n s for In and I n - P b whiskers. Moreover, Pb, P__bb-In, and P..b_b-Bi whiskers show this tendency. One may also argue that the increasing tendency of the t e m p e r a t u r e independent values of I0/I.- of whiskers from the ln-Pb alloy s y s t e m with increasing lead c o n t e n t (Fig. S0) is a consequence of the decrease of T with decreasing ~E" The p r o b le m is t h a t the t e m p e r a t u r e independence o f I0/I ~ cannot be u n d e r s t o o d within the TDGL theory.

197

10. Universal Behaviour of an Isolated Phase-Slip Center: Experiment and TDGL Theory

One of the basic problems in the scope of the cu r r en t induced dissipative s t a t e in a q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r is to come to an understanding of the behaviour of an isolated p h a s e - s l i p center. Within the TDGL theory an isolated phase-slip c e n t e r is a p h a s e - s l i p s t a t e with infinite periodic length. In o t h e r words, an isolated p h a s e - s l i p c e n t e r is a single p h a s e - s l i p c e n t e r in a f i l a m e n t which develops w i t h o u t being influenced by o t h e r cen t ers or a f f e c t e d by the contacts. The f i rs t v o l ta g e step in the characteristics of a whisker or a long microbridge may be regarded as generated by an isolated p h a s e - s l i p center. However, if the quasiparticle relaxation length is very large, as in Zn whiskers, the sample has to be sufficiently long to exclude influences o f the s u p e r c o n d u c t i n g co n t a c ts . There are t wo basic characteristic properties of the first v o l t a g e step in a V-[ c h a r a c t e r i s t i c which have been widely discussed in the preceding chapters. These are the n o r m a l - l i k e length, LAnl, and the ratio of the e x t r a p o l a t e d zero voltage i n t e r c e p t and the critical current, I0/1 ~. The e x p e r i m e n t s show that the behaviour o f both quantities may be very d i f f e r e n t for samples of d i f f e r e n t materials. Both quantities may be t e m p e r a t u r e independent over the whole t e m p e r a t u r e range investigated. In o t h e r cases a t e m p e r a t u r e independent behaviour is only observed at s o m e w h a t lower t e m p e r a t u r e s , T, and there is a s t r o n g t e m p e r a t u r e dependence if one approaches the critical temperature, To0. In the case of the n o r m a l - l i k e length, a variety of t e m p e r a t u r e laws is observed. Besides being t e m p e r a t u r e independent, LAn ! may be proportional to AT -1, AT -1;2 or AT -1;4, where AT=To0-T. For a given sample the n o r m a l- l ik e length usually runs t h r o u g h d i f f e r e n t t e m p e r a t u r e dependences with increasing s t r e n g t h if T b eco m es c l o s e r to To0. The situation s e e m s to be rather envolved. Therefore, we looked for a suitable p a r a m e t e r t h a t governs the behaviour of both c h a r a c t e r i s t i c properties. Theoretical r e s u l t s for L A n l / ~ o and I 0 / l c elaborated by Kramer and Rangel (KR) f r o m the TDGL equations for dirty s u p e r c o n d u c t o r s are p l o t t e d or t ab u l at e d as a function o f the pair-breaking parameter, 3" (see sectionS.9). Here ~0 is the GL coherence length in the dirty case. In the preceding ch ap t er we found a g r e e m e n t with several predictions of KR. Although n o t all phenomena observed in our e x p er i m en t s could be explained, we a s s u m e that ~f is a suitable parameter. Therefore, we p l o t t e d e x p e r i m e n t a l 198

values for L A n l / ~ and I 0 / I : as a function of 3, for whiskers of d i f f e r e n t materials [147,477]. Here, ~ is the GL coherence length for arbitrary e l e c t r o n mean free path. We s u m m a r i z e d e x p e r i m e n t a l r e s u l t s from whiskers made of Pb, In, Sn, and Zn and of ln-Pb, Sn-In, and Z n - A g alloys. Due to the very d i f f e r e n t values o f the inelastic e l e c t r o n - p h o n o n collision time, zE, in t h ese materials the i n v e s t i g a t e d 3, range e x t e n d s from 0.1 in the case of Pb whiskers up to 20000 for Zn whiskers. We found that tAnl/~ and I 0 / I o s y s t e m a t i c a l l y develop with growing 3". Our p l o t s may be regarded as an experimentally obtained universal prediction f o r the behaviour of an isolated p h a s e - s l i p center, only depending on the p a r a m e t e r T. Moreover, it turns out that our e x p e r i m e n t s in a certain 3" range are in qualitative and reasonable quantitative a g r e e m e n t with the r e s u l t s of the TDGL theory. First we are going to discuss the range of small and medium pair-breaking parameters, 3", [147]. The corresponding universal p l o t s are shown in Figs. 61, 62, and 63. In Figs. 61 and 63 experimental r e s u l t s for 24 d i f f e r e n t samples are shown t o g e t h e r with the predictions of the TDGL theory ~1. Some r e p r e s e n t a t i v e r e s u l t s of the single sample behaviour of L A n l / ~ are redrawn in Fig. 62. The theoretical prediction of LAnl/~0 is just t h a t one given in Fig. 20 of the p r e s e n t work for the case of an isolated p h a s e - s l i p c e n t e r (dp=oo). The quantity I 0 / I was c a l c u l a t e d by KR f o r a periodic array o f p h a s e - s l i p c e n te r s only. As already pointed o u t in section 5.9, I0/1 c is nearly independent of the periodic length, dp, f o r not too small periodic lengths. Therefore, the r e s u l t s for dp=12~0 as given in Tab. 1 of ref. 316 may be applied f o r an isolated p h a s e - s l i p c e n t e r and, thus, are p l o t t e d in Fig. 62. In Figs. 61 and 63 we do not use d i f f e r e n t symbols for the d i f f e r e n t samples. Therefore, it is remarked that due to the small value of zE, whiskers of Pb and P b - I n alloys have values for 7 b e t w een 0.22 and 2.7. The I nn-Pb alloys have 3" values b e t w e e n 5 and 43. For pure In whiskers 3" is b et w een

9 1 All values f o r 3" belonging to the experimental r e s u l t s in these figures are c a l c u l a t e d using Tinkham's e s t i m a t e f o r ~E given in section 5.4. As m en t i o n ed in section 9.3 and discussed in detail in appendix S, we did n o t s t r i c t l y f o l l o w Tinkham's f o r m u l a in the case of I n - P b whiskers, b u t i n s e r t e d the t o t a l resistivity at room t e m p e r a t u r e instead of the p h o n o n - i n d u c e d part. The same was done for Sn and S_nn-In whiskers. In this case the t o t a l resistivity is calculated from the p h o n o n - i n d u c e d part (which we identify with the bulk material resistivity of pure tin) and the measured residual resistance ratio, applying Mathiessen's rule (section 5.4). In t h e case of Sn whiskers the r e s u l t s nearly do n o t deviate from those s t r i c t l y evaluated, because the residual r e s i s t a n c e ratios of t h e s e samples are very small. 199

100" o : Experiment

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a function o f the pair-breaking parameter. Y, f o r sm a l l and medium values o f this parameter [1472. Experimental values f o r an isolated phase-slip center in whiskers o f different materials (Pb, In, Sn, Pb-In, I_n-Pb, S n - l n ) and at different temperatures, T, close to To, together with the prediction o f the TDGL theory. The straight solid lines (dashed i f estimated) indicate the borders o f experimental values and at the same time indicate the dependence experimentally f o u n d f o r L A n t / ( on T. We observed f o r the regions ' a ' - " d , respectively, that LAnI / ( ~ y, ~ y1/2, independent o f 7, and ~ 7-', corresponding to a temperature dependence o f LAn I for a sample in these regions given by LAn I independent o f temperature, ~ d T- t/4, ~ d T- ,/2, and ~ d T " ' To see the correspondence o f the dependence o f L A n l / ( on Y and o f LAn ! on AT, note that y ~ A T ,/2 while ( ~ A T ''/2, where AT=T~o-T. A s we did not introduce different s y m b o l s f o r the different samples, the behaviour o f each single sample is not directly visible.

10.5 and 39, while T ranges between 19 and 56 for pure Sn whiskers and be t ween 25 and 54 for Sn-In alloy whiskers. First let us discuss the behaviour of L A , 1 / ~ given in Fig. 61. The e x p e r i m e n t a l r e s u l t s of all samples are lying within a certain region of the ( L A n l / ~ ) - T plane. The s t r a i g h t lines mark the borders of this region and at the same time indicate which dependence of LAnl/~ on T (and, thus, which t e m p e r a t u r e dependence of LAn 1) has been observed in the experiments. For y between 9 and 30 the indicated dependences of L A n l / ~ on T overlap. This means the o b s er v ati o n of several d i f f e r e n t dependences in this y range. The behaviour of the single samples cannot be distinguished from each othe r in Fig. 61. However, the dependence of LAn 1 on the t e m p e r a t u r e was published b e f o r e fo r m o s t samples. A c o m p l e t e list of r e f e r e n c e s is given in 200

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~, 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

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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
t e m p e r a t u r e independent LAn I occurs much c l o s e r to To0 than

given by the

b o r d e r 8T~. Next, we are going to discuss the Sn and S_o-In specimens f o r which r e s u l t s are also p l o t t e d in Fig. 61. Using again z E as c a l c u l a t e d a f t e r Tinkham's e s t i m a t e (see appendix A1 of ref. 147 for the material p a r a m e t e r s needed) it is ATE=8.lmK ( o r T m a x = 5 | . 4 ) for samples Sn [191 and Sn 3 [223, while ATE=II.lmK ( o r Tmax=43.9) for sample Sn 2 [221 and tin w h i s k e r l of ref. 23. Furthermore, it is ATE=|O.8mK, l l. 4 m K, and l l . g m K ( o r Ymax=43.9, 42.9, and 42.0) for samples S_~n-In7 [153, S__nn-InlO [15], and S_nn-Inl7 [153, respectively. As already discussed in section 7.2, it is 8T E=8.4 mK ( o r Ymax=50.6) for polyerystalline pure tin. It turns out t h a t for Sn and S_n_n-In whiskers a t e m p e r a t u r e independent behaviour of LAn 1 (or the dependence LAnl/~T) is found up to t e m p e r a t u r e s which are much c l o s e r to To0 than the b o r d e r given by 8T E (or for T much s m a ll e r than Tmax)" For experimental data which d e m o n s t r a t e this s t a t e m e n t see Fig. lO of ref. 16 ( s a m p l e s Sn, S n - l n 7 (1.1at%In), S__~n-lnl7 (3.2at%In)), the intrinsic data in Fig. 9 of ref. 23 (tin w h i s k e r i ), and Fig. 62 of the p r e s e n t work (samples Sn 3, S n - l n 7). Thus, only the In whiskers mentioned and some of the I n - P b whiskers show a change of their n o r m a l - l i k e length from a t e m p e r a t u r e d ep en d en t to a t e m p e r a t u r e independent behaviour at the limits of the local equilibrium a ppro x i m at i o n given above. For the o t h e r samples LAn 1 already b e c o m e s independent of t e m p e r a t u r e (Ln-Pb whiskers) or is already t e m p e r a t u r e independent (Pb, Pl0-In, Sn, S__n_n-In whiskers) at t e m p e r a t u r e s c l o s e r to To. Before we shall try to draw a conclusion, we recall some r e s u l t s o f o t h e r materials: For Zn and Zn-Ag whiskers it is ATE only some hundredth parts o f a millikelvin (section 8.3), so that all e x p e r i m e n t s are p e r f o r m e d outs i d e the local equilibrium approximation. For all measuring t e m p e r a t u r e s the s e s p eci m en s deliver a t e m p e r a t u r e independent n o r m a l - l i k e length. The r e s u l t i n g q u as i p ar t ic le relaxation time z 2 is not t o o d i f f e r e n t f r o m z E. The identification of ~ with zE was proposed by Baratoff, who calculated the p r o p e r t i e s of a p h a s e - s l i p c e n t e r outside the local equilibrium ap p r o x i m at i o n (see s e c t i o n s 5.9, 7.2, and 8.3). While a c r o s s - o v e r f r o m a t e m p e r a t u r e dependent to a t e m p e r a t u r e independent behaviour of LAn I is n o t observed for Zn and Z_nq-Ag whiskers, this t r a n s i t i o n occurs in the Al microbridges of Liengme et al. [1363 (see the discussion of their e x p e r i m e n t s given in section 7.2). The quasiparticle relaxation time z 2 determined from the r e s u l t s of the temperature independent regime is in a g r e e m e n t with zE as calculated f o r A1 a f t e r Tinkham's e s t i m a t e (section 7.2). From the t e m p e r a t u r e d ep en d en t regime where L A n l ~ A T -'/4 they evaluated a time ~E which is much smaller. Set t i n g ~L equal to the latter r e s u l t for ~, they found t h a t the c r o s s - o v e r from the t e m p e r a t u r e d ep en d e n t to the t e m p e r a t u r e independent regime occurs at the b o r d e r of the local equilibrium approximation range given by this criterion. Naturally, it is n o t easy to draw a conclusion from these r e s u l t s . There are, however, several e x p e r i m e n t s which give evidence that the sample is out s i d e the local equilibrium approximation in the t e m p e r a t u r e range where 207

the n o r m a l - l i k e length is t e m p e r a t u r e independent. Therefore, with s o m e caution, we s t a t e that 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 indicates that the sample has left the t e m p e r a t u r e range (or ~" range) o f the local equilibrium approximation and, thus, cannot be described by the TDGL equations. There are several a r g u m e n t s which may explain t h a t there are a lot of sa mp l es for which a change to the t e m p e r a t u r e independent behaviour occurs at a t e m p e r a t u r e (or t" value) which differs from the upper border of the local equilibrium range given by the condition AE=~0(T) which means t h a t A T = A T E, l'=~(max, OF I:GL=~E. One problem is that this condition gives the border wh ere the b r e a k - d o w n of the local equilibrium approximation at l east occurs. The more rigorous condition for the validity of the local equilibrium approximation is AE<<~D(T). Therefore, it may be possible t h a t a sample leaves the local equilibrium range at a t e m p e r a t u r e which is c l o s e r to To0 than ATE (or at a T value which is s m a ll e r than Tmax)- Moreover, only a few of our s am p l es are in the dirty limit. Besides this there is a su b st an t i al uncertainty in the k n o w le d g e of the material p a r a m e t e r ~E {see section 5.4 t o g e t h e r with ap p e n d ix 3 of the present work and refs. 147 and 477 for a critical discussion of this quantity). In the remainder of this c h a p t e r we will discuss the universal behaviour of an isolated p h a s e - s l i p center in a material with large pair-breaking p a r a m e t e r s ~', such as Zn, Z_~n-Ag, and AI [4.77]. The universal p l o t s f o r L A n l / ~ and I 0 / I o as a function of T are shown in Figs. 64 and 65. (Again was c a l c u l a t e d using Tinkham's e s t i m a t e for zE' For details and material p a r a m e t e r s see ref. r In Fig. 64 we p l o t L A n l / ~ versus ~" for our Zn and Zn - A g whiskers (see chap. 8) t o g e t h e r with r e s u l t s for Al microbridges of Liengme et al. [136] and Stuivinga et al. [73]. The curve of the TDGL theory ( a f t e r KR [3161) for high Y values p l o t t e d in this figure is valid f o r an isolated p h a s e - s l i p c e n t e r in a h o m o g e n e o u s f i l a m e n t as well as in the static limit. The curve is also given in Fig. 20 of the p r e s e n t work. Calculated values are only available up to ~'~lO00. The prediction of the TDGL theory is close to L A n i / ~ "1/2 which is also e x p e c t e d for even higher 7 values [3161. For the AI microbridges of Liengme et al. [136] in a certain ~' range a A T " / ' t e m p e r a t u r e dependence for LAn 1 has been observed leading to L A n l / ~ "1/2, in a g r e e m e n t with the theory. The "{ range of the AT -~4 t e m p e r a t u r e law depends on the homogeneity of the sample. For very h o m o g e n e o u s samples the range is only small and an early c r o s s - o v e r to a t e m p e r a t u r e - i n d e p e n d e n t behaviour ( LAn I / ~ ~ T) is observed. The microbridges of Stuivinga et al. [73] only deliver a AT -~/4 law. These samples contain a notch to pin the p h a s e - s l i p center. It may be assumed that the c r o s s - o v e r to a t e m p e r a t u r e independent behaviour takes place at higher ~" values which no m e a s u r e m e n t s were carried o u t for. For whiskers of Zn and Z n - A g alloys the s m a l l e s t ~( values are beyond the c r o s s - o v e r region of the AI microbridges. The t e m p e r a t u r e independence 208

#

TDGL-THEORY

. . . . . . . .

. . . . . . . .

'

'

Fig. 64: Normal-like length, LAnI, normalized by the GL coherence length, ~, as a function o f the pair breaking-parameter, y, f o r large 7 #4 77Z Experimental values for an isolated phase-slip center in whiskers o f Zn and Z n - A g and in AI microbridges for different temperatures, T, close to To, together with the prediction o f the TDGL theory: ( A ) Zn whiskers, ( V ) Z n - A g whiskers, ( n ) AI microbridge ( s a m p l e 3 from ref. 73), ( 0 ) homogeneous AI microbridges from ref. 136 (the result with the largest 7 value is from sample 4, the f u r t h e r p o i n t s from sample II), ( 0 ) inhomogeneous AI microbridge ( s a m p l e 9 from ref. 136). S y m b o l s with the same error bars mark the range o f a measurement. The experimental values ly on the connecting lines between the symbols. For LAnl / ( ~ 7 and ~ 7 ~2 it is LAn 1 temperature independent and ~ d T -'~4, respectively, where d T = To - T. The dashed line indicates the estimated slope for the temperature independent normal-like length (normalized by ( ) o f AI microbridges.

o f LAn I f o r w h i s k e r s s u p p o r t s t h e m e a s u r e m e n t s o f m i c r o b r i d g e s a n d e x t e n d s the y range of our universal behaviour study to values of about20000. The a b s o l u t e v a l u e s o f L A n l / ~ a r e s m a l l e r t h a n t h o s e f o r AI m i c r o b r i d g e s . F u r t h e r m o r e , we f o u n d t h a t f o r a f i x e d ~ t h e v a l u e of L A n l / ~ d e c r e a s e s w i t h decreasing grade of homogeneity of a whisker. In Fig. 65 w e p l o t t h e r a t i o I 0 / I ~ as a f u n c t i o n o f y f o r o u r Z n a n d Z n - A g w h i s k e r s a n d f o r AI m i c r o b r i d g e s . S i n c e f o r h i g h ~f v a l u e s ( y > 100) t h e s t a t i c a p p r o x i m a t i o n c a n b e a p p l i e d , t h e TDGL t h e o r y e x p e c t s 10/I c c l o s e t o z e r o E316]. T h e m e a s u r e m e n t s d o n o t c o n f i r m t h i s p r e d i c t i o n . F o r a n i s o l a t e d p h a s e - s l i p c e n t e r , 10/I o s c a t t e r s a r o u n d 0.,5. W h i l e I 0 / I o d o e s n o t d e p e n d o n t e m p e r a t u r e f o r o u r w h i s k e r s , a t e m p e r a t u r e d e p e n d e n c e o f t h i s q u a n t i t y is o b s e r v e d f o r t h e m i c r o b r i d g e o f S t u i v i n g a e t al. [73]. F o r t h i s s a m p l e 1 o / l ~ i n c r e a s e s f r o m 0.3 c l o s e t o To0 t o 0.7 a t l o w e r t e m p e r a t u r e s . T w o o f o u r 209

015

0

2

. . . . . . .

'13

10

. . . . . . .

'1_4

10

'

i

I

!

~,

I

I

I I

I05

Rg. 6S: Ratio l o / I c as a function o f the pair-breaking parameter, Y, f o r large y. Experimental values f o r a single phase-slip center in whiskers o f Zn and Zn-Ag and in A1 microbridges f o r different temperatures close to To. Same notation as used in Fig. 64. The single light circle is from sample 4 o f ref. 136.

samples also show values of about 0.3. This r e s u l t is not typical f o r an isolated p h a s e - s l i p c e n t e r in a whisker. It can be explained by a nucleation of the p h a s e - s l i p c e n t e r c lo s e to one of the superconducting c o n t a c t s (see the discussion in section 8.3). With some caution an analogous explanation may be applied concerning the low values for the A1 microbridges of Stuivinga et al. [73] (see ref. 20 for a detailed discussion). It is remarked t h a t in the two whiskers with a depressed ratio I0/I o the distance b et w een the p h a s e - s l i p center and the c o n ta c t is large enough for an undisturbed quasiparticle relaxation. Thus, they show a value for LAn ! which is typical for an isolated p h a s e - s l i p center. The investigation of the universal behaviour of an i s o l a t e d p h a s e - s l i p center p r es en t ed in the p r e s e n t c h a p te r gives us an overview over a Y range of about six decades. There are three y ranges for which the n o r m a l - l i k e length LAn1 does not depend on temperature, separated by i n t e r m e d i a t e regions with t e m p e r a t u r e dependent LAn !. The ratio 10/I ~ d ecr eases with increasing y f r o m unity to a saturating value of about 0.55. The behaviour of the t e m p e r a t u r e dependent regime of LAn I may be u n d e r s t o o d within the TDGL theory. Also a decrease of I0/I c with increasing y is predicted by the TDGL theory, but not the saturating tendency. In the temperature independent regime of LAn 1 the samples seem to be outside of local equilibrium. In this case o t h e r theoretical met h o d s have to be applied. However, only some f i r s t approaches of this kind exist.

210

11. Hysteresis of the Critical Current

11.1.

Introductory Remarks

One of the characteristic f e a t u r e s of the c u r r e n t - i n d u c e d breakdown of super co n d u ct i v i t y in a q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r is its hysteretic behaviour. Our whiskers show a hysteresis as well in their V - T transition curves as also in their V - I characteristics. In the case o f the V-T characteristics the cu r r en t has a fixed value and the v o l t a g e is measured as a function of the t e m p e r a t u r e , T. For increasing t e m p e r a t u r e the sample enters the v o l t a g e - c a r r y i n g s t a t e at the transition t e m p e r a t u r e , To: ( N o t e t h a t T~ is the transition t e m p e r a t u r e in the c u r r e n t - c a r r y i n g s t a t e and, therefore, more or less lower than the critical t e m p e r a t u r e , Tr The difference b e tw e e n T c and T~ depends on the magnitude of the t r a n s p o r t current.) A f u r t h e r increase of the t e m p e r a t u r e leads to the normal c o n d u c t i n g state. If then the t e m p e r a t u r e is lowered, the sample recovers 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 which is s m a l l e r than T c. In the case o f a V-I characteristic the t e m p e r a t u r e is fixed and the volt ag e is measured as a function of the current. As sketched in Fig. 25, fo r increasing c u r r e n t the dissipative p h a s e - s l i p s t a t e is entered at the critical c u r r e n t Ic, while for decreasing c u r r e n t the superconducting s t a t e is recovered at the j u m p -b ack current, I,, which is s m a l le r than the critical current. The p r o b l e m is to find out which mechanisms g e n e r a t e the observed hysteresis. Since energy is dissipated in the f i l a m e n t a v e r y i m p o r t a n t q u e s t i o n is to e s t i m a t e how far J o u l e heating e f f e c t s can warm up the lattice of the sample and, thus, may lead to hysteretic behaviour. For this p u r p o se the t h erm al boundary resistance ('Kapitza resistance') between the whisker and the surrounding helium has to be determined. If the c o n t r i b u t i o n of the thermal hysteresis is known, the remaining intrinsic n o n t h e r m a l h y s t e r e t i c behaviour of a p h a s e - s l i p c e n t e r can be investigated. It t u r n s o u t t h a t this part c a n n o t be described by a single mechanism but seems to be g e n e r a t e d by 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 quasiparticle e x c i t a t i o n s and by the influence of charge imbalance waves on the p h a s e - s l i p process. Before we go into details, we should briefly sketch the history of the investigations of hysteresis e f f e c t s of p h a s e - s l i p c e n t e r s and then add some remarks on the organization of the p r e s e n t chapter. 211

Already J.D. Meyer [19] observed hysteresis effects in his investigations of pure Sn whiskers (see Fig. 6 of the present work). In the case of V-I characteristics he found that the hysteresis width, I - I , , increases if the temperature is lowered. Moreover, he reportet that the jump-back voltage, VR(IR), does not show the linear increase as observed for the height of the first voltage jump for which V,(Io) is a straight line. The same behaviour was found for whiskers of S n - I n alloys [15, 28] and pure In [478]. For these specimens I ~ R grows linearly with increasing critical c u r r e n t (after an initial nonlinear increase). Thus, we characterized the development of the hysteresis width in the V-I characteristics of a sample by d ( l o - I R ) / d l ~ and plotted the r e s u l t s as a function of the electron mean free path 2 of the samples. Moreover, the s a t u r a t i n g value, VRs, of the jump-back voltage, V~, was determined for different samples and p l o t t e d as a f u n c t i o n 2. In the case of S__nn-ln alloys, d ( I , - I ~ ) / d I = is roughly 0.5 for s between 0.045 and 0.2~tm and then smoothly decreases to about a value of 0.2 at 2=3~m. The r e s u l t s for V,s first increase with increasing s ( f r o m V~s ~ 6 #V at s ~ 0.045 #m to VRs~ 19#V at 2 ~ l[~m) and then decrease again (to VRs~0.7#V at 2 ~ 3 # m ) . In the case of In whiskers, d ( I c - I R ) / d l ` scatters between O.l and 0.3 for ~ between about O.1S and 1.6#m, while VRs c o n t i n u o u s l y decreases with decreasing ~ from about 4.6 to 1.2 #V in the same range of mean free paths. An interesting question is whether a r e - e n t r a n t part of the V-I characteristics with negative differential resistance which cannot be passed for impressed c u r r e n t leads to the sharp voltage jumps at I~ and I~, respectively. J.D. Meyer [479] found evidence for such a r e - e n t r a n t part by s h u n t i n g a tin whisker with a small parallel normal conducting resistor, thus, working with impressed voltage instead of impressed current. Also the Sn microbridges of SBT [32] showed hysteresis effects in their V-I characteristics. SBT argue that these effects may be partly u n d e r s t o o d as heating effects, but that in addition n o n t h e r m a l intrinsic hysteresis m u s t be present. Subsequently, Kadin, Skocpol, and Tinkham [84] also investigated the hysteresis of Sn microbridges and conclude that for their specimens thermal effects alone are sufficient to u n d e r s t a n d the whole hysteresis width. The TDGL theory [316] predicts an intrinsic n o n t h e r m a l hysteresis (see section 5.9). Also the KSS model [158] leads to an intrinsic hysteretic behaviour (as discussed in detail in section S.8.r It is remarkable that the KSS model predicts a s a t u r a t i n g behaviour of the j u m p - b a c k voltage [158]. These theoretical r e s u l t s will be discussed in detail in the p r e s e n t chapter. The characteristic saturating behaviour of the jump-back voltage was also observed for whiskers of I n - P b alloys [40]. For these samples we, furthermore, compared the measured onset temperature, T Hexp y OF AT~yP=T,0-THyP, of the hysteretic behaviour in the V-I characteristics with 212

theoretical predictions. In the KSS model the hysteretic regime is predicted for ~ E / % , > 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"~

~<~

%~

SnH3 Tco=3.6908 K

o..,%~,

-I#'3 ~THo~o~ ~

~,'2'a

T=Tco-AT

loo-

0

TR

"%,

Tc i

T;;PTco

[

i

i

i

i

[

ATImK

50

0

Pig. 68: Critical current, I~2/3, and j u m p - b a c k current, I~2/J, as a function o f A T for a pure tin whisker (sample S n H 3 o f ref. 332). Large magnification close to T~o The full lines indicate the estimated slope through the measured values. The horizontal line between the two branches represents the temperature hysteresis, ST,, at arbitrary but fixed current Ir. The quantities T~(I~) and (Ir) are the transition temperature and the j u m p - b a c k temperature (into the superconducting state) f o r fixed current Ix , respectively. The temperature THy'P indicates the onset o f hysteresis observed experimentally.

lo 200" j.

5

E

g 0

o

i

I

i

lo

20

30

t

i

7'0

tllg. 69: Width o f the temperature hysteresis, 3 ~ , as a function o f 17 for a tin whisker (sample S n H 3 o f ref. 332). The inset s h o w s the l o w e r part in larger magnification in order to make conspicuous the f o o t l i k e structure. The f u l l line is the estimated slope through the measured values. The dashed line is the case o f only Joule heating.

223

heating should clearly be visible in such a plot. For sample SnH3 t h er e se e m s to be the e x p e c t e d linear region f o r IF2> 50 mA 2. For s m a l l e r c u r r e n t s ~T. s h o ws a more c o m p l ic a t e d behaviour. At very small c u r r e n t s the me n t i o n ed f o o t l i k e s t r u c t u r e can be seen. It is this f o o t l i k e s t r u c t u r e which indicates intrinsic hysteresis. In the foUowing section we will p r e s e n t a phenomenological model which is able to explain the hysteretic properties of a p h a s e - s l i p c e n t e r in the ' f o o t - s t r u c t u r e region'.

11.4. 11.4.1.

Phenomenological Hysteresis Model A Model with Two Fitting Parameters

In this subsection we will discuss the basic ideas which lead to a p h e n o m e n o l o g i c a l hysteresis model with two fitting parameters. The discussion is based on the ideas given in ref. 332 and the i m p r o v e m e n t s made on this subject in ref. 483. In subsection 11.4.3 we will p r e s e n t a model w i t h o u t fitting parameters. Naturally, the second case is of advantage. The p a r a m e t e r free model is, however, r e s t r i c t e d to a s m a l l e r c u r r e n t range. The reason is t h a t detailed i n f o r m a t io n about the v o l t ag e developed by the p h a s e - s l i p c e n t e r is needed in this case. Thus, the "two fitting p a r a m e t e r s version' of the model is still an i m p o r t a n t t o o l to get i n f o r m a t i o n ab o u t the contribution of the d i f f e r e n t mechanisms which generate the hysteresis of a p h a s e - s l i p center. As already pointed o u t in section 11.1, our model considers three d i f f e r e n t mechanisms which lead to a hysteretic behaviour: the s e l f heating o f the p h a s e - s l i p center, charge imbalance waves ('KSS mechanism') and an overpopulation of the excitation s p e c t r u m with nonequilibrium quasiparticles ( ' T * mechanism'). We s t a r t with the e s t i m a t i o n of the s e l f - h e a t i n g of a phase-slip center: The p h a s e - s l i p s t a t e is a dissipative state. The energy dissipation in detail is e x p e c t e d to be a r a t h e r involved problem. We only consider a very rough model which gives an upper boundary for the effect. Much si m p l er than the t r e a t m e n t o f Stuivinga et al. [515], we assume that a region of the length o f the quasiparticle diffusion length A at both sides o f the p h a s e - s l i p c e n t e r is equally warmed up. The entire s u p e r c o n d u c t o r remains at the bath t e m p e r a t u r e T. Due to eq. (214) it is (~ = ~ K S ( T L - T ) where ctK is the heat t r a n s f e r coefficient, T L the lattice t e m p e r a t u r e 224

(221) of the

whisker, and S the s u r f a c e of the warm region through which the dissipated energy is t r a n s m i t t e d into the surrounding helium bath. Assuming again a c i r c u l a r - c r o s s sectional area of the whisker with radius r w, which is the s m a l l e s t surface possible, it is S = 4r:rwA

(222)

To c a l c u l a t e the energy dissipation 0 in the p h a s e - s l i p center, we r e m e m b e r t h a t in t i m e - a v e r a g e a part I 0 o f the t o t al c u r r e n t I is carried as a supercurrent. T h e r e f o r e t~ = V ( I ) ( I - I 0), where V(I) is the time averaged volt ag e g en erat ed by the p h a s e - s l i p center. As discussed in s e c t i o n 7 . 2 (see eqs.(187) and (188)) it is V ( I ) = ( 2 P n A / A ) ( I - I 0 ) . With R n = P n L / A it f o l l o w s (~ = ( 2 R n A / L ) ( I - l o ) 2

(223)

With 8TL:=TL-T we, thus, g e t gT L= ( I - l o ) 2 R n / 2 ~ Z r w ~ K L

(224)

This e x p r e s s i o n for the e n h a n c e m e n t ~TL o f the lattice t e m p e r a t u r e above the bath t e m p e r a t u r e may o v e r e s t i m a t e the w a r m i n g - u p of the lattice, because charge imbalance relaxation does not necessarily include a c o m p l e t e energy relaxation of the excitations. The length A, however, is the relaxation length f o r the quasiparticle charge imbalance. It is remarked t h a t f o r I0=0 eq. (224) yields eq. (219), describing the t e m p e r a t u r e e n h a n c e m e n t in a s e l f - h e a t i n g h o t spot. To calculate ~TL in the p h a s e - s l i p regime, where I 0 is nonzero, measured values of I 0 are used. Naturally, the w a r m i n g - u p of the lattice leads to a hysteresis. For fixed c u r r e n t I F the jump back into the superconducting s t a t e takes place at a bath t e m p e r a t u r e T = T R for which T,+~TL(I~)=To(I~). Thus, ~TL(I ~) is equal to the t e m p e r a t u r e h y s t e r e s i s ~TH(I F) g e n e r a t e d by a s e l f - h e a t i n g o f the p h a s e - s l i p center. Considering this mechanism only, there usually is no a g r e e m e n t with the measured ~TH(IF2). The t e m p e r a t u r e rise in the p h a s e - s l i p regime, where the c ur ren t is typically only some hundreds of ~A, is usually much too small, especially in the case of whiskers o f pure material. Next we discuss the charge imbalance wave induced hysteresis ('KSS m e c h a n i s m ' ) : The KSS model predicts an intrinsic nonthermal hysteresis. The origin of this hysteresis is a reactive e f f e c t of the charge imbalance waves excited by the phase-slip c e n t e r on the p h a s e - s l i p process. We d i s cu s s ed the KSS model and its hysteretic behaviour in detail in section 5.8.4. The o n s e t of hysteresis is given by the criterion z0,=~E, where z E is the inelastic e l e c t r o n - p h o n o n collision time and %.=(2/2vF)~)(1-T/Too )-~ the 225

s u p e r c u r r e n t response time. (This expression for zo. is valid for the w e a k - c o u p l i n g case, see eq.(ll2), but will be used for all materials to discuss their hysteretic behaviour.) The condition %.=% yields the o n s e t KS$ t e m p e r a t u r e THy below which the KSS model predicts hysteretic behaviour. Using the KSS model it is also possible to calculate the t e m p e r a t u r e hysteresis 8TH as a f u n c t i o n of arbitrary fixed c u r r e n t I F. For this purpose we first calculated the j u m p - b a c k current, I,, from the theoretical r e s u l t for the V-I characteristics at given fixed bath temperature T : T , . For this purpose we use the analytical approximation of the KSS model given in eq.(160) and insert the c u r r e n t from eq.(163), yielding I = I 0 e X / ( e X - l ) + ( V / R e f f ) [ 1 - ( z E / ~ 0 . ) 1/2]

(225)

where x ( V ) = I Ref f ( % . / z E)I/=V-I, with Ref f=2AQ.in/OA. This is an equation for the I-V characteristics at fixed t e m p e r a t u r e yielding reversal V-I characteristics for the hysteretic case where z0, < z E (see Fig. 16). From the condition d I / d V = O we get the voltage V. where the sample jumps back into the superconducting state. Calculating d l / d V from eq. (225) and s e t t i n g the r e s u l t equal to zero, yields the condition x~ e X a / ( e •

1) 2 = 1- (l:oR/~c) l'z

(226)

where x.:=x(V.). This problem has to be solved by iteration. Reinserting V. into the I-V characteristic, eq. (225), we get the j u m p - b a c k c u r r e n t I.. Using the r e s u l t s of the GL theory for a q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r (see eqs. (1S) and (17)) we then get the t r a n s i t i o n t e m p e r a t u r e corresponding to I. as To(I.) = T c o - l . = / 3 1 d l y 3 / d T o 1 - 1

(227)

The KSS prediction for the temperature hysteresis 3TH(I F) then follows from eq.(220) as ST,.ss(IF=I.) = Too- I . = / 3 1 d I ~ t 3 / d T o l " - T .

(228)

For an i l l u s t r a t i o n of the different quantities used in this equation see Fig. 68. Usually the KSS mechanism alone does not lead to a quantitative description of the hysteresis, because the KSS model predicts the o n s e t of hysteresis for too low temperatures and the width of the hysteresis is too small. Moreover, the qualitative shape of ~T~s~(IF2) does n o t show the characteristic footlike structure. Finally, we explain our attempt to consider a quasiparticle overpopulation induced hysteresis ( ' T ' m e c h a n i s m ' ) : During the p h a s e - s l i p 226

cycle t h er e is a periodic production of nonequilibrium quasiparticles. We a s s u m e t h a t t h es e quasiparticles create a charge balanced 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 in the c o r e of the p h a s e - s l i p c e n t e r so t h a t they are c ha r act eri zed by an e f f e c t i v e t e m p e r a t u r e T*> T in the sense of Parker's model, widely d i s c u s s e d in section S.S. In this case the s u p e r c o n d u c t o r behaves as if it had the t e m p e r a t u r e T*, so that its critical c u r r e n t is reduced to Ic(T *) < I t ( T ) . Hysteresis in the V-I characteristics at fixed bath t e m p e r a t u r e should be observed, because the c u r r e n t has to be lowered to lc(T *) b e f o r e the sample jumps back into the superconducting state. As we have shown in section 5.5, close to the critical t e m p e r a t u r e T~0 the e n h a n c e m e n t of T* above the bath temperature, T, is approximately given by ( see eq. (90)) ST* = ( 2 / 3 ) T o 0 ( N / N T - 1 ) (229) where S T * : = T * - T , N the actual quasiparticle number per volume, and N T the quasiparticle number per volume at the t e m p e r a t u r e T in thermal equilibrium. The quasiparticle density in t h e r m a l equilibrium is given by N T = Nma x - n s e

(230)

where n s e is the ' s u p e r e l e c t r o n density', that means the number of e l e c t r o n s per volume bound to C o o p e r pairs, and Nma x is the maximum quasiparticle density which is simply the total e l e c t r o n density, n, of e l e c t r o n s in the normal state. The ratio n s e / n is given in eq. (99). If we f u r t h e r m o r e use the e x p r e s s i o n for the energy gap of 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 in the vicinity of the critical t e m p e r a t u r e ( s e e eq. (69)) it f o l l o w s f r o m eq. (230) NT = N m a x ( 1 - 2 X ( l - T / T o o ) )

(231)

where 3( is given by eq.(8) of the p r e s e n t work. Now we i n t ro d u c e the two f it ti n g p a r a m e t e r s mentioned in the heading of this section by assuming that in the hysteretic case it is N/Nr-1

= [(Nmax/NT)sH-1][3 H

(232)

Before we s o m e w h a t discuss the meaning of the p a r a m e t e r s sH and ~H and their determination, we give our final r e s u l t f o r ST* which is ST*

=

(2/3)T~o[eH/(1-2~((1-T/Too))-I][B ,

(233)

For fixed c u r r e n t I~, the j u m p - b a c k into the su p er co n d u ct i n g s t a t e occurs as soon as the bath t e m p e r a t u r e is lowered to a value T R for which TR+ST*(TR)=Tc(IF). Thus, ST* is equal to the width of the t e m p e r a t u r e hysteresis ST. (I F) g e n e r a t e d by the T * mechanism. The p a r a m e t e r s. describes the o n s e t of the T* hysteresis. It takes into consideration t h a t a quasiparticle overpopulation induced hysteresis may only 227

be observed below a certain temperature TI~IPy. (Here 'op' s t a n d s 'overpopulation'.) This temperature has to be calculated from experimental data. At THPy one should have N=NT, requiring sHNma• yielding ~. = 1 - 2 X ( 1 - T H y P / T o o ) Naturally, this r e s u l t also follows directly from eq.(233),

for the

(234) because

~T*=0

at

The existence of a temperature T~_lPy
r e - e n t e r s the s u p e r c o n d u c t in g s t a t e ST.(I~2=I 2) is given by the sum of the In the o t h e r case where the KSS the j u m p - b a c k cu r r e n t in the KSS t e m p e r a t u r e the d i f f e r e n t quantities of

and the width of the hysteresis actual values STL and ST*. model predicts hysteresis we cal cu l at e model. - [A p r o b l em is, at which the KSS model h a v e - t o be calculated.

Our actual opinion is, due to the discussion given in section ll.4.3, that the t e m p e r a t u r e T * is not g e n e r a t e d t h r o u g h o u t the p h a s e - s l i p center, b u t is r e s t r i c t e d to its core. Thus, T* should only e n t e r the e x p r e s s i o n f o r the critical c u r r e n t of the core. The o t h e r quantities of the model should be taken at the lattice t e m p e r a t u r e T , = T + S T , . These quantities are %, and Aq.in, The reason is t h a t in the t r a n s m i s s i o n line picture of the KSS model the s e quantities determine the impedance of outside the core.] - Naturally, the j u m p - b a c k c u r r e n t is s m a l l e r than I. Therefore, a f u r t h e r s t e p - b y - s t e p reduction of the c u r r e n t is necessary. This reduction changes T~ and T* and the whole procedure has to be repeated from the beginning until the j u m p - b a c k c u r r e n t of the KSS model is equal to the actual c u r r e n t I. The t otal hysteresis ST.(IF==I =) is then given by the sum of the actual values STy, ST*, and ST~ss. For the d e t e r m i n a t i o n of ST ~ we need values for the p a r a m e t e r s e. and ~. from the experiment. In the general case the fitting procedure has to be p e r f o r m e d with r e s p e c t to the hysteresis caused by the KSS mechanism and the s e l f - h e a t i n g of the p h a s e - s l i p center. The procedure is s o m e w h a t involved Er In the case o f EH, f i r s t the c u r r e n t IF, where STY=0 has to be e s t i m a t e d from the shape of the measured STH(IF2) curve. Knowing this current, the r e l a t e d j u m p - b a c k t e m p e r a t u r e T,(IF,) can be determined. It can be obtained by a h o r i z o n t a l cut at 12~3F, t h r o u g h an [c2 / 3 , I.2 / 3 ( A T ) plot. - [ A n o t h e r p o s s i b i l i t y is to calculate T.(I~,), using S T . , : = ~ T . ( I F , ) = T ( I . , ) - T . ( I r ~ ) and I~(3=ldI~/dTol(To0-To(I.,)), where dI~'3/dT, is the measured slope of the 1.2~3(To) s t r a i g h t line.] Then, STL(IF~) can be calculated using I 0 as measured f o r T.. The onset temperature of the T ~ hysteresis then is T ~ p = Ta(IF1) +ST~(IF~) and then ~. is calculated from eq. (234). In the case o f 13H, a point (I~22, ST.=) of the measured ST.(I~) curve is chosen, with IF2 s o m e w h a t larger than I2,. In general, at this point it is ~TH2=~TL(I~2)+ST*+~TKss. The aim is to separate ST ~ so t h a t 13H can be calculated f r o m eq.(233) with the known value o f e.. The f i r s t step is to c a lcu l at e TR(IF2) and 8TL(I~2) as described in the case of the ~. fit. The remaining problem is to separate the known width 8T~2=ST.2-STL(IF2) into the c o n t r i b u t i o n s ST * and 8TKss, where STKss depends on ST *. This p r o b l e m can be solved by an iteration procedure. As a s t a r t i n g t e m p e r a t u r e we take the lattice t e m p e r a t u r e TR(IF2)+STL(IF2) and calculate the c o n t r i b u t i o n S TKss for this t e m p e r a t u r e . Subtracting this r e s u l t f r o m STy2 gives a f i r s t a ppr o x i m at i o n for S T*-. Now the actual temperature is set to T,(IF2)+~TL(IF2)+~T ~ and ST~ss is calculated again leading to a b e t t e r e s t i m a t e for ST*. The procedure is repeated until the accuracy of ~T ~ is

229

b e t t e r than 0.05mK. The r e s u l t for ~T* is inserted into eq. (233) t o g e t h e r with the lattice temperature T,(Ir2)+~TL(IF2) and the value for ~., to obtain the parameter ~.In the following section the predictions of our phenomenological hysteresis model with two fitting parameters will be compared with experimental data of different materials. It t u r n s out that the model is able to give a quantitative description of the measured hysteresis including the limits in which one of the mechanisms dominates.

11.4.2.

Comparison with Experiment

In this s u b s e c t i o n we compare the predictions of the phenomenological hysteresis model with two fitting parameters (which will be briefly called ' f i t t e d version') with m e a s u r e m e n t s of the temperature hysteresis of whiskers from Sn, In, I n - P b alloys, Pb, and Zn. The calculations are performed with the m o s t recent version of the model as described in the preceding subsection. It contains all improvements introduced in ref. 483. Thus, the r e s u l t s given in the p r e s e n t work may be somewhat different from those published priviously [332,4823. The characteristic properties (including the I0/l o profiles, as a function of the temperature, and the r e s u l t s for the parameters e N and [3H) of all samples mentioned are summarized in ref. 483. In Fig. 70 r e s u l t s for the pure tin whisker of section 11.3 are s h o w n * ' Now, circles mark the experimental values, while the full and dashed lines are the r e s u l t s of the phenomenological model and its single contributions, respectively. The footlike s t r u c t u r e as well as the s u b s e q u e n t increase are predicted by the model. Over the whole range the quantitative a g r e e m e n t is satisfactory. Close to T00 (low c u r r e n t s ) the hysteresis is governed by quasiparticle overpopulation, while Joule heating becomes i m p o r t a n t for large currents. The hysteresis caused by the KSS mechanism is of minor importance. In Figs. 7 1 - 7 4 we present experiments and calculations for pure In, I n - P b alloys, and pure Pb. For pure In (Fig. 71), ~T,(IF2) behaves very similar to the case of pure tin. Again the experimental data show a pronounced footlike s t r u c t u r e which is well described by the fitted version of the phenomenoiogical model. In the low c u r r e n t region the largest c o n t r i b u t i o n is due to the T ' m e c h a n i s m , while for larger Currents heating effects become more and more important. Again the KSS contribution is only small t h r o u g h o u t the investigated current range.

*' The sample is called ' S n H 3 ' in ref. 332 and in the p r e s e n t work, while it is called 'Snl" in ref. 483. 230

20

o

v

~ 9 o ^ o 7

Sn H3

o;y

,-3:

Phen.Model o o

10-

CF,.Vo~., o-"...y.~

/I /

0

i 2

0

4

6

IF21mA2

Rig. 70: Width o f the temperature hysteresis, 3TH , as a function of If for a tin whisker (sample SnH3). The circles represent measured values while the full line is calculated from the fitted version ('Fit. Vers.' ) o f the phenomenological hysteresis model, using s~ and ~H as fitting parameters. The dashed lines are the single contributions from the mechanisms of

quasiparticle overpopulation (3T~), self-heating o f the phase-slip center (STL), and charge imbalance waves from the KSS model (3TKss). The solid line is given by the sum 3TL+ST~+3T~ss. For this sample it is ~=0.9956 and #H=0.II99,

10

,nl

~176 /

0

/

Modei--~ _o (Fit.Vers.} - / 00 0

~o

Phen.

f / \ g T x

o

....... 0

_~

"

//

XT

: ; :; . . . . . . 0.5

/

.... 1

-1.5

I//mA2 Fig,/l.

Measured width o f the temperature hysteresis, 3T~, as a function o f l / for an indium whisker (sample Inl, T~o=3.3938K) in comparison with the fitted version o f the phenomenological hysteresis model. Same notation as used in Fig. 70. For this sample it is zH= 0.9970 and flH =0.1645.

231

0

4-

InPbl

o

k~:

Phen.Model-~.0 (Fit, Vers.J

2-

oj

0

"

O~

/ / /

.''---~TL

0.2

0.4 ZZ/mA2

RIB', 72:

Measured width o f the temperature hysteresis, ~T,, as a function o f 2 _~ a l~ f o r an indium-lead alloy whisker (sample l nnPbl with cp~-,.8at~, T~o=3.S311K) in comparison with the f i t t e d version o f the phenomenological model. Same notation as used in Fig. 70. For this sample it is ~, = 0.9997 and ~H = 0.2218.

~d

15o

~ 10"

5-

TnPb3

y/

~M0del

...,~oo .../o ~

0

/ o ~

1

(FiLVers.I

,

2

,

3 ZF2/10-3mA2

4

Rg. 73: Measured width (circles) o f the temperature hysteresis, 3T., as a function o f 17 f o r an indium-lead ahoy whisker (sample In Pb 3 with ce~ = 78.5at%, T o=6.8693K) in comparison with the f i t t e d version o f the phenomenological m od el (full line). In this case Joule heating is the only contribution to the hysteresis.

By alloying Pb to In, the footlike structure becomes smaller with increasing Pb content (that means decreasing electron mean free path 2) and the influence of the s e l f - h e a t i n g grows. Sample InPb ! only s h o w s a small gT* and heating is the largest contribution over the whole measured range (Fig. 72). The limiting case is represented by sample I n P b 3 , where only Joule heating o c c u r s (Fig. 73). In this sample no voltage steps have been

232

xE '.-o

/

30 Pbl 20 Phen.Model

0

10-

0

-

o

i

1

2

2

i

RE. 74: Measured width (circles) o f the temperature hysteresis, 3 TH, as a function o f I7 f o r a lead whisker (sample Pb l) in comparison with the f i t t e d version o f the phenomenological model (full line). In this case Joule heating is the only contribution to the hysteresis.

observed and the t~TH(IF2) plot does not show any footlike s t r u c t u r e . The prediction of the phenomenological model is a straight line through the origin because we set I0=O. In Fig. 74 we show r e s u l t s obtained for a pure Pb whisker (sample Pb 1). Since the V-I characteristics of sample Pb 1 show linear portions close to To0 w i t h o u t any voltage jumps and without any hysteresis, TH(I ~) is equal to zero in the low c u r r e n t range. For lower t e m p e r a t u r e s a voltage step and hysteresis occur. However, no intrinsic hysteresis (absent footlike s t r u c t u r e ) is seen and the whole width gTH is well described by heating. Finally, r e s u l t s for pure zinc whiskers are shown in Figs. 75 and 76. The examples given are representative for the hysteretic behaviour of long (sample Zn2S, Fig. 75) and s h o r t (sample Zn18, Fig. 76) samples. In both cases a footlike s t r u c t u r e indicates an intrinsic hysteresis. For the long sample, Zn 25, the KSS mechanism explains the hysteresis for small c u r r e n t s completely. For higher c u r r e n t s the quasiparticle overpopulation induced hysteresis becomes important. The warming up of the lattice is very small for all currents. For the short sample, Znl8, the hysteresis width is dominated by the T ' m e c h a n i s m . The absolute value of ST.(It 2) is more than an order of magnitude larger compared to sample Zn25. The reason for such large ST* is probably the Andreev reflection of the quasiparticle excitations at the s t r o n g l y s u p e r c o n d u c t i n g Wood metal c o n t a c t s (compare the discussion given in section 8.3). Thus, the quasiparticle excitations are trapped within the whisker. The various materials investigated enable us to t e s t the validity of our hysteresis model in the limits where only one mechanism governs the 233

2O 9,..,. ~

o

Zn25 Phen.M o d e l ~ (Fit.V e r ~ ~

10-

I

0.02

o

o:o4

0:06

o.oa

I/ IrnA2 Fig. 7$: Measured width o f the temperature hysteresis, ST., as a f u n c t i o n o f I / f o r a ( l o n g ) zinc whisker (sample Zn2,5, L=ISOOItm, T~o=O.BIO2K) in c o m p a r i s o n w i t h t h e f i t t e d version o f t h e p h e n o m e n o l o g i c a l m o d e l . S a m e n o t a t i o n a s u s e d in Fig. 70. F o r t h i s s a m p l e i t is s. = 0.9764 a n d #. = 0.3512.

40 ~c

~

Zn18

E

Phen'M~

o

j~#~

I ~ ~ ~ ~~ ~

20-

_-: :/:.:_ : : :

............. 0

.....

0

I

5

I

10 I2110-Z.rnA 2

15

Fig. 76: Measured width o f the temperature hysteresis, 3T. , as a f u n c t i o n o f I / f o r a (short) zinc whisker (sample Zn 18, L -- 225[zm, To = 0.8161K) in comparison with the f i t t e d version o f the phenomenological model. Same notation as used in Fig. 70. For this sample it is s. = 1 and 8. = 0.8038.

hysteresis: The limit o f quasiparticle overpopulation ( T ' m e c h a n i s m ) induced hysteresis occurs for whiskers from pure Sn or In in the low current range. Charge imbalance wave (KSS mechanism) induced hysteresis generates the hysteresis in the l o w current range of long whiskers from pure Zn. The only heating case is observed in dirty ln-Pb whiskers and s p e c i m e n s o f pure Pb. In all cases the m o d e l calculation leads to a satisfying description of the experimental results.

234

Joule heating effects occur in all samples. While these effects are negligibly small in clean samples (for n o t too high currents}, they become the largest c o n t r i b u t i o n to the hysteresis in dirty whiskers. From our phenomenological model it can be seen that the thermal hysteresis does n o t only mask the intrinsic hysteresis b u t also suppresses its magnitude. The reason is that the hysteresis caused by intrinsic mechanisms increases with decreasing temperature. Joule heating, however, enhances the lattice temperature. For Pb whiskers we measured I 0 / I c = l close to To0, leading to ~TL=0 at I=Io, so that no heating induced hysteresis can occur, in a g r e e m e n t with the experiment. Since the measured hysteresis is well described by our phenomenological hysteresis model, we believe that the model contains the essential mechanisms which generate the hysteretic behaviour of a phase-slip c e n t e r in a q u a s i - o n e - d i m e n s i o n a l superconductor. Naturally, one would desire to have a model w i t h o u t fitting parameters. An a t t e m p t to come to a f i t t i n g - p a r a m e t e r - f r e e description is given in the following subsection.

11.4.3.

A Model without Fitting Parameters

For the development of a hysteresis model w i t h o u t fitting parameters a more detailed description of the generation and healing processes of quasiparticles during the phase-slip cycle is needed. The description m u s t be explicit enough to derive an expression for the T ' c o n t r i b u t i o n to the hysteresis which does not contain any fitting parameter. The following a t t e m p t (elaborated in ref. 483) will yield such a result. A comparison with the former expression for ~T* leads to analytical r e s u l t s for ~. and ~.. A p h a s e - s l i p center has a 'core region' of some GL coherence l e n g t h s , ~(T), where the relaxation oscillations of the order parameter lead to s t r o n g variations of the C o o p e r - p a i r density. In this region the nonequilibrium quasiparticles are produced during the phase-slip cycle. Then these quasiparticles diffuse away from the core region into the s u r r o u n d i n g superconductor, while they relax to reach an equilibrium occupation of the excitation spectrum. Depending on the polarity of the t r a n s p o r t c u r r e n t ( t h a t means of the applied battery) more hole-like quasiparticles and more e l e c t r o n - l i k e quasiparticles diffuse into opposite directions. The consequence is t h a t a charge imbalance is created at both sides of the c e n t e r ('core'} of the core region. In the core itself, the paths of both kinds of quasiparticles cross, c o m p e n s a t i n g their charge so that there is no charge imbalance b u t a charge balanced quasiparticle overpopulation which will be described by an effective t e m p e r a t u r e T* in the sense of Parker's model (discussed in detail in section 5.5). It is assumed that the temperature T* in the core determines whether the s u p e r c o n d u c t o r recovers the superconducting state or remains in the dissipative p h a s e - s l i p state. The jump back into the s u p e r c o n d u c t i n g state c a n n o t occur as long as I > I o ( T * ) , where I is the total current. 235

The overpopulation of the excitation spectrum in the core (and t h e r e f o r e in principle also T*) is time dependent, because it depends on the generation, the migration and the relaxation of nonequilibrium quasiparticles. For a rough e s t i m a t e we take the spatial quasiparticle distribution in the m o m e n t of the p h a s e - s l i p event and trace its time evolution w i t h o u t considering the f u r t h e r d e v e l o p m e n t of the order parameter. Thus, we assume that the quasiparticles are g en er at ed i n s t a n ta n e o u s ly at the beginning of the p h a s e - s l i p cycle. F u r t h e r m o r e we a s s u m e that the charge is balanced in this m o m e n t at each point. In the m o m e n t o f the p h a s e - s l i p event the a b s o l u t e value IqJI of the s u p e r c o n d u c t i n g order p a r a m e t e r is zero in the core ( l o c a t e d at x=O) and increases exponentially with the characteristic length ~(T) on both sides o f the core to its equilibrium value Id2rl far away from the p h a s e - s l i p center. Thus, Ikb(x)l = I ~ r l ( l - e x p ( - I x l / ~ ( T ) ) (235) where ~(T) is given by eq.(6). This shape of I~b(x)l is s u p p o r t e d by r e s u l t s of the TDGL theory (see Fig. 5 of ref. 305). Identifying the square of the absolute value of the order p a r a m e t e r with the density of C o o p e r pairs leads to an expression f o r the local value o f the quasiparticle overpopulation in the m o m e n t of the p h a s e - s l i p ev en t (t=O), namely Nop(X,t=O) = (Nmax-Nr)(l-(l~(x)l/l~rl) 2) (236) yielding with I r

from eq. (235) Nop(X,t=O) = (Nmax-Nr) (2e -Ixl/5 _ e-21xl/~)

For t=O so t h a t there everywhere in diffusing into quasiparticles.

(237)

both branches of the excitation s p e c t r u m are equally p o p u l at ed is neither a branch imbalance nor a charge imbalance. For t > 0 the core region the quasiparticles are divided into t w o halves, opposite directions depending on the ef f ect i v e charge of the At the core both halves cross and we g et

Nop(X= 0, t) = ( N m a x - N T) ( 2 e - A ( t ) / ~ - e - 2 A ( t ) / ~ )

(238)

where A ( t ) = ( ( 1 / 3 ) s is the distance over which a quasiparticle excitation travels along the x-axis in the time, t, by random walk. Until now we have n e g l e c te d that Nop also relaxes by a recombination of the quasiparticles. However, actually on their way to the core the number of excess quasiparticles relaxes by recombination. In the sense of a relaxation time approximation we assume that there is an exponential decay of the number of travelling quasipartictes so that the quasiparticle o v e r p o p u l a t i o n in the core is given by 236

N o p ( X = 0 , t) = ( N m a x - N r ) ( 2 e - A ( t ) / ~ - e-2A(t)/~)e-t/Zeff

(239)

The e f f e c t i v e quasiparticle recombination time, 1:eff, is the p r o d u c t of the intrinsic recombination time, z'r, and the phonon trapping f a c t o r which takes into a c c o u n t t h a t phonons created by recombination p r o c e s s e s can break a pair again and, t h e r e f o r e , delay the relaxation of Nop (see f o r instance refs. 103, 104, 110, 131, and 132). Kaplan et ai. [103] have c a l c u l a t e d the time z r b e t w e e n t w o recombination events in the dirty limit. It is Zr=Zr/2, because z r ch arac te r iz e s the decay of a quasiparticle population and t w o quasiparticles of this population vanish in each recombination event. The time z r depends on the t e m p e r a t u r e and on the quasiparticle energy. For T approaching Tr the limiting value for z r of a quasiparticle at the gap edge is given by Zr=Zo/4.2. The scaling time z 0 is r e l a t e d to ~E by z0=8.4z E [80] leading to Zr= 2z E or ~r = zE" The use of this limiting value should be a good approximation in our experimental situation s o m e millikelvin b e l o w To0. Close to To0 the t e m p e r a t u r e dependence of z r is very weak. Moreover, it is reasonable to assume t h a t the quasiparticles created by the p h a s e - s l i p p r o c e s s have energies n o t very far away from the gap edge, much s m a l l e r than the zero energy gap P*(O). Even if a quasiparticle would have an energy A(0), zr would only be ab o u t a f a c t o r of two s m a ll e r [103]. The p ro b l em of phonon trapping has been investigated in detail for thin films [104, 110, 131, 132, 516, 517]. In our model p h o n o n - t r a p p i n g e f f e c t s are n e g l e c t e d so that Z e f f = l : r leading to Z e f f = Z E. To with an zpsc, of vol t ag e

r e l a t e the time dependent quasiparticle o v e r p o p u l a t i o n in the core e f f e c t i v e t e m p e r a t u r e T ~, we average N o p ( x = O , t ) over one period, the p h a s e - s l i p cycle. The time zpsr is r el at ed to the t i m e - a v e r a g e d V at the c o n t a c t s of the sample by the J o s e p h s o n relation so t h a t

Zpsc=h/2eV. F u r t h e r m o r e , using ~ T * = ( 2 / 3 ) T c 0 ( N / N v - I ) from eq.(229), the actual t o t al quasiparticle density, N = Nop+NT, leads to

where N is

~PSC ~T* = 2 Tr ~ N m aNxT- N T ZPsclf ( 2 e - A ( t ) / r

e-2A(t)/~tT))e-t/zE dt

(240)

o Due to eq. (231) it is ( N m a x - N + ) / N . r = 1 / [ 1 - 2 X ( 1 - T / T o 0 ) ] - I . Thus, comparing the r e s u l t for $T* given in eq.(240) with t h a t one of eq. (233) yields ~H = 1

and

~H = ~ I I: PSC

(241)

"~PSC F ( 2 e -A(t)/~(T) - e -2A(t)/~(T) ) e-t/~E d t

J

(242)

o

237

Thus, eH is unity and [3. becomes a function of the voltage and the temperature. In the presence of Joule heating, T has to be replaced by the lattice temperature, TL, which is enhanced over the bath temperature, T, by an a m o u n t STc. Since E.=I, the onset of the T ' h y s t e r e s i s (that means the exceeding of a certain threshold so that the effect can be detected in our experiments) and the f u r t h e r development of its magnitude are determined by [3M which relates the effective temperature T* with the migration and recombination behaviour of the quasiparticles in a phase-slip center. For a comparison with the experiment, the same computing method as before is used (see sectionll.4.1), but ~. is set to unity and [5H from eq. (242) is inserted into eq.(233). To calculate the c o n t r i b u t i o n of Joule heating we again insert I 0 from the experiment. Moreover, the voltage V( I ) = (d V/d I )1 ( I - I 0) is also taken from the measurement. Here ( d V / d I ) 1 is the differential resistance of the V-I characteristics after the first voltage jump. With this voltage the time %sc is known, so that the integral in [3. can be evaluated by numerical methods. Although the experimental shape of V(I) is used, the decision for the jump back into the superconducting state is given by the model calculation. In the calculations made for our samples the jump-back occurs between Io and I 0. Therefore, there is no problem with the point I = I 0, where ~Tc becomes zero due to the influence of an experimental parameter and ~T * also vanishes, because V(I)=0, leading to %sr In the following subsection we will compare the predictions of our phenomenological model in the 'analytical version' (i.e. with E.=l and the 'analytical' expression for 13. from eq.(242)) with experimental data. A satisfactory agreement will be obtained which is remarkable, because in the analytical version the model does n o t contain any fitting parameter.

fl.4.4.

Comparison with Experiment

Using the hysteresis model without fitting parameters, we have performed calculations for all materials investigated which show a hysteresis not only caused by Joule heating [4833. For all samples (except I0 2, the characteristic properties of this sample are given in ref. 483), ' f i t t e d ' model calculations, using the fitting parameters z. and 13. are also shown in the present work (see section 11.2). The 'analytical' computations, using ~.=l and the 'analytical' expression for [3. from eq. (242) are restricted to a smaller range of IF2 than the ' f i t t e d ' computations, because reliable r e s u l t s for the differential resistance are only available in a restricted range of temperatures. Throughout there is a satisfactory agreement between the e x p e r i m e n t and the analytical c o m p u t a t i o n s (Figs. 7 7 - 8 0 ) . This is remarkable, because no fitting parameters are used. 238

o 1-

SnH3

9,e" E

o o

/-

~o

0.5-

o

o Phen. Nodel (An.Vers.} ~ ~

o

-

arL\ 0

o

. . . . . . . I. . . . . . . . .

1

0

0.1

0.2

0.3

Z71mA= r i g . 77: Measured width o f the temperature hysteresis,

3T,, as a function o f 17 for a tin whisker (sample S n H 3 ) in comparison with the 'analytical' version ('An. Vers.') o f the phenomenological hysteresis model, using s, = 1 and the analytical expression f o r ft,. The notation used is the same as introduced in Fig. 70.

"" E

In2

o

o

o

o

o

o~//~ (3" 0 0

Phen. M o d e l (An.Vers.)

i 1

....

T ..... 2

~rL\ ,. . . . . . 3

,..... 4 I7/mA 2

5

Fig. 78: Measured width o f the temperature hysteresis, 3T., as a function o f I7 for an indium whisker (sample In2, T,o=3.3938K) in comparison with the "analytical' version ('An. Vers.') o f the phenomenological model. The notation used is the same as introduced in Fig. 70.

For t h e pure w h i s k e r s S n H 3 , In2, and Z n l 8 the c a l c u l a t e d h y s t e r e s i s is s o m e w h a t s m a l l e r t h a n the m e a s u r e d one, w h i l e t h e r e is an e x a c t a g r e e m e n t in t h e c a s e o f t h e a l l o y w h i s k e r I n P b 1. T h e 'analytical' c o m p u t a t i o n f o r t h e l o n g Zn w h i s k e r ( s a m p l e Z n 2 S ) is n o t s h o w n . T h e r e is an a g r e e m e n t b e t w e e n the c a l c u l a t e d and t h e m e a s u r e d 239

InPbl ~ o Phen. Model~ (An.Vers.J ^~/,.v

E ""

o

--

0.5

..J.o'J . . . . .

0 0

,

,

0.05

0.1

0

0

~,

~TKss~__ 0.15 Z~/mA 2

Fig. 19- Measured width o f the temperature hysteresis, 3T,, as a function o f 17 f o r an indium-lead alloy whisker (sample l_.nPbl with %~=2.8at%) in comparison with the "analytical" version ('An. Vers.') o f the phenomenological model. The notation used is the same as introduced in Fig. 70.

I Zn18

o o o

o

o~ ~176 10-

0 0

o~~~

~

1 ~ ~-

7

~-

Phen. Modet

(An.Ve~. ~)j / t 6

..... " 6 / "'

~TL -, . . . . . . . 1

,. . . . . . . . 2

-; 3

zT110-%A2

4

[zig. 80" Measured width o f the temperature hysteresis, 3T,, as a f u n c t i o n o f 17 f o r a ( s h o r t ) zinc whisker (sample Zn l8) in comparison with the "analytical" version ('An. Vers.') o f the phenomenological model. The notation used is the same as introduced in Fig. 70.

hysteresis width. In the calculated range of small c u r r e n t s I F the hysteresis is given by the KSS mechanism as in the ' f i t t e d ' model calculation. There is a vanishing small contribution of ST*. The reason is t h a t I b eco m es c l o s e to I0, leading to a very small V(I), so that ~T* b e c o m e s small due to the large value of ~,sc" This r e s u l t shows that the relation between the voltage and gT* is reasonable. However, in this case the limit of the e x p e r i m e n t a l V-1 c h a r a c t e r i s t i c s s t r o n g l y influences the result. 240

For sample Zn t8 we can explain why the c a l c u l a t e d hysteresis width is s m a l l e r than the measured one. The reason is t h a t this sample is very s h o r t so t h a t nonequilibrium quasiparticles are A n d r e e v - r e f l e c t e d at the s t r o n g l y s u p e r c o n d u c t i n g Wood m e ta l c o n t a c t s and return to the core (compare the discussion given in section 11.4.2). As the r e f l e c t i o n occurs at both co n t act s, r e f l e c t e d quasiparticles of both charges c r o s s at the core and c r e a t e an additional c o n t r i b u t i o n to ~T* which has not been included in the model b u t is p r e s e n t in the experiment.

11.5.

The Hysteresis in the TDGL Theory

The TDGL theory predicts an intrinsic hysteresis for the p h a s e - s l i p center, b ecau s e f o r fixed t e m p e r a t u r e o s c i ll a t o r y p h a s e - s l i p s o l u t i o n s e x i s t b e t w e e n a minimal and a maximal c u r r e n t density, called Jmin and Jmax (see Fig. 19 in s ect i o n 5.9)*~. The dissipative s t a t e is en t er ed at the critical c u r r e n t density jo of the time independent GL theory. However, fo r an isolated phase-slip c e n t e r in a h o m o g e n e o u s f i l a m e n t it is Jmax<Jo as long as T < y o ~ 5 . 5 , while for ,f>~(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 <'( < 80 an ap p ro x i m a t e l y s t r a i g h t line, leading to (in physical units*2) Jmin = (j0/0.385) 90.48 ~,-0.265 Since J, " 2/a ~ AT be p r o p o r t i o n a l to s o f t l y decreasing t e m p e r a t u r e T (i.e.

(243)

and y ~ ATe/2, we thus find out, t h a t Jrnin" 2/3 is predicted to AT ~ that means, it is a nearly s t r a i g h t line with a a b s o l u t e value of its slope with decreasing bath increasing AT= To0-T).

*' In section 5.9 we usually used normalized units, while physical units are used t h r o u g h o u t the p r e s e n t section. In the TDGL theory the c u r r e n t densities are normalized with J0 as given in eq. (171). It is Ij01=(1/0.385)j~ where jo is the GL critical current density. *2 In Tab. 1 o f ref. 316 values for Jmin are given in normalized units and, therefore, they were multiplied with jc/0.385 in o r d er to g e t c u r r e n t densities in physical units. 241

To co m p ar e the predictions of the TDGL theory with our experiments, we d e t e r m i n e d the values ~/'HyeXp. . .~[t . LXlHyeXpT c 0 - T ~ x Py ) , - at which the o n s e t o f hysteresis has been observed experimentally. Already in sectionll.1, we exp mentioned t h a t for whiskers of In-Pb a l l o y s [ 4 0 ] ~'Hy is found to be larger than yo, indicating t h a t the measured o n s e t of hysteresis occurs f u r t h e r below To0 than predicted by KR. The same observation is made f o r the I_En-Pb whiskers of ref. 483 as well as for whiskers of Sn [332~, In [4833, and Zn[4833. Also in these samples the o n s e t of hysteresis is predicted f o r t e m p e r a t u r e s which are t o o close to To0. No co r r el at i o n b e t w e e n this discrepancy and the ratio 2 / ~ o ( t h a t means the 'dirtyness' of a sample) has been observed. For the Pb whiskers, P b l and Pb2, and the dirty sample In P b 3 of ref. 483 the TDGL theory predicts an o n s e t of the intrinsic hysteresis at very low t e m p e r a t u r e s , out of the measured range. The measured hysteresis f o r these samples is indeed due to J o u l e heating only [483] ( s e e Figs. 73 and 74 in section 11.r For a quantitative comparison o f the hysteresis width a prediction f o r STH has to be evaluated from the TDGL theory. For this purpose we p l o t the 12z3(hT) s t r a i g h t line observed in our experiment t o g e t h e r with the KR prediction 12~3 using eq.(243). Here, A is the c r o s s - s e c t i o n a l R,TDGL = ( A j m i n ) 2/~ area of the sample. From this p l o t the width ~TH at arbitrary fixed c u r r e n t IF can be d e t e r m i n e d so t h a t a prediction of the TDGL theory for ~TH(IF2) is I

obtained. In Fig. 81 r e s u l t s for a pure tin whisker (sample SnH3) are shown. In the whole range p l o t t e d we used eq. (243), although f o r y below lO and above 80 (corresponding to AT=0.3 and 20inK, respectively) we do n o t know w h e t h e r this approximation is still valid. The width of the t e m p e r a t u r e hysteresis, ~TH(I~2), is shown in the inset. Although there is no quantitative a g r e e m e n t b e t w e e n the KR prediction and our m easu r em en t s, the shape of the I,2~3 curve and the f o o t s t r u c t u r e in the ST.(IF 2) plot are similar to the experimental results. One reason for the d i s a g r e e m e n t may be t h a t sample S n H 3 is a pure s u p e r c o n d u c t o r while the KR theory is only valid in the dirty limit. The observation t h a t the hysteresis predicted by the TDGL theory is much t o o large was not only observed for pure tin whiskers [332] b u t also for samples of pure In [483] and l n - P b alloys [483] which are more dirty so t h a t 2 < ~ 0. With decreasing 2 / ~ o the a g r e e m e n t b eco m es b et t er . In Fig. 82 r e s u l t s f o r sample l n P b 2 are shown, for which 2 / ~ o = 0 . 2 8 . This r e s u l t is the b e s t a g r e e m e n t we found for the samples investigated in ref. [483]. Again we used eq. (243) in the whole range plotted. ]n this case the condition t h a t y should be b e t w e e n l0 and 80 is fulfilled f o r all values plotted. Very recently KR published a s h o r t note dealing with the d i s a g r e e m e n t b e t w e e n the hysteresis width as predicted by the TDGL theory and as observed in the experiments[S18]. KR argue that the d i s a g r e e m e n t is n o t really alarming because intrinsic f l u c t u a t i o n s and ex t r i n si c f l u c t u a t i o n s ( t h at 242

\ \ 2"2/3 TDGL

~o-

,r213..

200

-~C

v E

.213 ~ \

~

~<=_

~~.~ ~

~ ~'W

11i ///

TDGL

/

/

Experiment

o -1/ \~

, 1

2.~ T D G L ' ' - .

,

I7/mA~ SnH3

0

,

50

ATImK

0

]Fig. 81:

Critical current I j "~ and jump-back current IZ/3 f o r a pure tin whisker (sample S n H3 ) as a function o f AT [332]. The f u l l fines indicate the estimated slope through the measured values. The dashed fines mark results o f the TDGL theory as calculated from the results o f KR (for details see the text).

In the i n s e t the width o f the temperature hysteresis, &T., as a f u n c t i o n o f 17 is compared with the prediction o f the TDGL theory.

10-

I_~nPb2

5-

.. T

" ~

D

G

L

~Experiment

0 0

0.05

0.1 ZF2/mA2

Fig. I]2: Measured width o f the temperature hysteresis, ~T., as a function

of I 7 f o r an indium-lead alloy whisker (sample l n P b 2 with cpb=3.Sat% and T~o=3.S881K) in comparison with the prediction o f the TDGA theory [483].

243

means external noise) may reduce the hysteresis. The idea is t h a t the ideal low er limit Jmin is not reached, because these disturbances co n v er t the s t a b l e p h a s e - s l i p solution below jr into a m e t a s t a b l e st at e with finite lifetime. KR pointed o u t that an inclusion of f l u c t u a t i o n s in the theory would be important. Moreover, KR p r o p o s e an idea for an experiment to t e s t their claim. This e x p e r i m e n t needs, however, a fast registration method f o r the volt ag e at the f i l a m e n t and it seems to be impossible to realize this in the vV range. Naturally, f l u c t u a t i o n s may influence the hysteresis width. However, it se e m s to us as if the d i s a g r e e m e n t between the predicted and measured hysteresis width had a more fundamental origin: A l o w e r border f o r the j u m p - b a c k current I, of the f i r st v o l t ag e step o f a V-I characteristic is given by I0 (see Fig. 25). This s t a t e m e n t is in a g r e e m e n t with the TDGL theory (see Fig. 19). Thus, I , / I ~ should be larger than 10/I :. If we cal cu l a t e I,,TD~JIc--jmm/j: from eq. (243) for a given y, we find t h a t I,,TD~L/Io is larger than the prediction of the TDGL theory for 1 . / I (as e x p e c t e d ) , but is s m a l l e r than the measured ratio I 0 / l . To be more explicit, it is IR,TDGL/Io=0.68, 0.56, 0.51, 0.47, 0.44, and 0.42 f o r y=lO, 20, 30, 40, 50, and 60, respectively. These values are smaller than the e x p e r i m e n t a l r e s u l t s for I 0 / I c, lying close to the lo w e r border of the m e a s u r e m e n t s p l o t t e d in Fig. 63. Since the experimental value of I0 is larger than I.,TD~L, the measured hysteresis width can never have the magnitude predicted by the TDGL theory. The experimental width would be s m a l l e r than the theoretical width even if the hysteresis in the e x p e r i m e n t would e x t e n d down to I0. Therefore, even if f l u c t u a t i o n s would force the sample to r e - e n t e r the superconducting s t a t e at a larger c u r r e n t than in the undisturbed situation, this e f f e c t would n o t be s u f f i c i e n t to explain the d i s a g r e e m e n t b e tw e en the TDGL theory and the experiments. Since not only the prediction of the TDGL theory f o r IR,T0OL/Io but also that one for I 0 / I 0 is l o w e r than observed experimentally, it seem s to us as if there was a more rudimentary difference between the behaviour of an array o f p h a s e - s l i p centers in the TDGL theory o f KR (only f o r an array of p h a s e - s l i p cente r s r e s u l t s for I,,TD~L and I0 are available) and a single p h a s e - s l i p c e n t e r in our samples.

244

12. Tunable Weak Links

The regular voltage step structure which characterizes the c u r r e n t - i n d u c e d breakdown of superconductivity in a q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r is generated by spatially localized phase-slip centers. The first p h a s e - s l i p center appears at the point of weakest superconductivity in the sample. The next phase-slip center occurs at a point, where the local critical c u r r e n t has its s m a l l e s t value in the presence of the first phase-slip center and so on. In an ideal homogeneous filament with strongly s u p e r c o n d u c t i n g c o n t a c t s in the absence of fluctuations, the first phase-slip center is expected to develop in the middle of the filament and the appearance of f u r t h e r p h a s e - s l i p centers is only governed by stabilizing and destabilizing effects of the phase-slip centers already present. In a real specimen there may be material inhomogeneities leading to places of (more or less) weakened superconductivity. The first p h a s e - s l i p center then appears at the weak link with the lowest critical current. The development of s u b s e q u e n t phase-slip centers is then governed by stabilizing and destabilizing effects and the s t r e n g t h of superconductivity in the other weak links. In (long) microbridges there are inhomogeneities which influence the appearance of phase-slip centers in these samples. In whiskers, inhomogeneities are of minor importance. These samples are very homogeneous. If we, nevertheless, assume that also in whiskers the p h a s e - s l i p c e n t e r s develop at localized weak links, one should keep in mind that the s u p e r c o n d u c t i n g properties of these weak links differ only very slightly from those of the whole whisker. A weak link in whiskers only seems to be a kind of 'pinning center' for the phase-slip center (see sections 7.2, 7.4, 7.5, and chap. 10 for a detailed discussion). The weak links pinning the phase-slip centers in a q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r usually have a fixed and in most cases f o r t u i t o u s strength. Therefore, the question how the behaviour of the phase-slip center depends on the properties of the weak link usually c a n n o t be answered. However, already in section 7.2 and chap. 10 we discussed that the grade of homogeneity of a specimen may influence the behaviour of a p h a s e - s l i p center and that such an influence is also expected by the TDGL theory widely discussed in section 5.9.

245

Therefore, we p e r f o r m e d m e a s u r e m e n t s of the V-I c h a r a c t e r i s t i c s of a p h a s e - s l i p c e n t e r developing in a whisker at a weak link with a continuously adjustable s t r e n g t h of superconductivity ('tunable weak link', abbreviation 'TWL'). Naturally, such an experimental a r r a ng em en t is a su p er co n d u ct i n g device. It may be not only a tool for basic research but also f o r application p u r p o s e s in superconducting electronic circuits. In the p r e s e n t chapter we only report ab o u t the r e s u l t s of a tunable weak link in a tin whisker [23]. We refer to chap. 13 for remarks on ongoing experimental investigations dealing with tunable weak links in whiskers of tin, indium, and zinc.

12.1.

Experiments

The sample, r e s u l t s o f which are given in the p r esen t chapter, was made using t wo tin whiskers. Each whisker is held by t w o isolated Wood metal blocks which are s u p e r c o n d u c t in g during m e a s u r e m e n t s (see Fig. l d or the inset of Fig. 83). At the c r o s s - o v e r the whiskers were co n n ect ed by a kind o f s p o t welding procedure: The filaments are b r o u g h t into touch with each o t h e r and then a discharge c u r r e n t is allowed to flow across the touching region, leading to a m e t a l l i c c o n t a c t between the two whiskers. Now one whisker is used as 'measuring whisker' while the o t h e r one is the ' c o n t r o l l i n g whisker'. In the m e a s u r e m e n t s shown in the p r e s e n t chapter, w h i s k e r l is the measuring whisker (voltage and c u r r e n t d en o t ed by V and I, respectively) and w h i s k e r 2 is the c o n t r o l l i n g whisker (voltage and c u r r e n t denoted by Vt and 1t , respectively). Through the c o n t r o l l i n g whisker a 'control current' (or ' tu n i n g current'), It, flows, influencing the s u p e r c o n d u c t i n g properties of the measuring whisker near the c o n t a c t region. As the c o n t r o l c u r r e n t can be continuously adjusted, a w e a k - l i n k s t r u c t u r e with tunable s t r e n g t h ('TWL') can be induced in the measuring whisker. W i t h o u t c o n t r o l c u r r e n t ( I t = O ) , the intrinsic properties can be studied. In order to investigate the behaviour of the TWL one then may for instance measure V-I c h a r a c t e r i s t i c s of w h i s k e r l with T fixed and I t as a parameter. An ex am p l e for such a m e a s u r e m e n t is shown in Fig. 83: For I t = O, the V-I c h a r a c t e r i s t i c s show the typical intrinsic v o l t a g e - s t e p structure. For It#O, the basic o b s er v at io n is now, that for a certain magnitude of the c o n t r o l current, I t, an additional voltage step appears b e f o r e the o n s e t of the intrinsic characteristic. This additional v o l ta g e step is caused by the TWL. It vanishes when the c o n t r o l current is switched off. With increasing value of the c o n t r o l c u r r e n t the critical c u r r e n t of the TWL decreases and the slope of its V-I c h a r a t e r i s t i c s changes. For very high values o f I t the c ha r act eri s t i cs s t a r t with zero critical c u r r e n t from the origin. The o n s e t of the intrinsic characteristics, however, remains nearly unaltered. 246

Zt,/mA

TcoI = 3.688 K T = Tcol - A T AT = 1/,.4 mK

>

:t~ Vt

2~

L.6

3.8__8__ 3./, 3.2 3.0 2.8

TWL.A

2.0

J 9

1.0

O.~JO" O intrinsic .z~ O

500

_.T/ IJA

Fig, 8 3 : V - 1

characteristics o f a tunable weak link (sample T W L S n ) at fixed temperature for different values o f the control current, It [232 The inset s h o w s the sample geometry.. Whisker l is the 'measuring whisker" while w h i s k e r 2 acts as "controlling whisker'. The s y m b o l s "~.-~ " and " - ~ ' indicarte the polarity o f the current. The left part o f w h i s k e r l (from the l e f t W o o d m e t a l contact to the crossing point o f both whiskers) is 250[zm long, while the length o f the right part is 5631zm. The length o f the l o w e r part o f w h i s k e r 2 is 400gin and that one o f the upper part is 225~m. Here, T,o , is the critical temperature o f whisker l. The critical temperature o f w h i s k e r 2 is 3 . d m K higher. Further characteristic properties are given in ref. 23. For a certain a m o u n t o f the current It an tunable critical current ( ' T W L ~ ) appears before voltage step. The zero voltage point varies fo r It is indicated by the fine horizontal line which The current I is increasing.

additional voltage step with the first natural ( ' i n t r i n s i c ") the different characteristics. a characteristic s t a r t s f ro m .

These measurements were carried out for several fixed temperatures. The critical current of the TWL and the critical current belonging to the o n s e t o f t h e i n t r i n s i c c h a r a c t e r i s t i c a r e p l o t t e d in Fig. 84. W h i l e t h e c r i t i c a l c u r r e n t o f t h e i n t r i n s i c c h a r a c t e r i s t i c is n e a r l y n o t c h a n g e d b y t h e c o n t r o l current (except of a slight decrease for large values of It) , the critical

247

TWLSn Tcol = 3 . 6 8 8 K

100-

N~ :x 5"

T

1 ~--~ 2 IF- ~

~:intrinsic char. ,:TWL ATImK

=TcoI-AT

J~

.

.

.

.

.

~'-'~=--~-~=,,~~176176176176 ,~--,~ __--~-r~-tt~,-0~-7-o ~~

3 "-~. . . . ~ - ' ~ - , ~ ' ~ 50-

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14.4 12.1 9.4

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6 ..... 0

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.

.

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84,: 12/3 as a function o f the control current, It, f o r the intrinsic characteristics o f w h i s k e r l and f o r the tunable weak rink ( T W L ) ot r sample TWLSn f o r several different fixed temperatures [23]. The f u l l lines indicate the estimated slope through the measured critical currents. The horizontal line holds fo r cases in which the phase-slip center at the T W L does n o t appear before the onset o f the intrinsic characteristics. The d o w n w a r d bending full lines indicate the critical current o f the TWL. Fig,

c u r r e n t of the TWL depends s t r o n g l y on It. For all t e m p e r a t u r e s investigated Iff 3 initially depends linearly on I t. For t e m p e r a t u r e s not too cl o se to To0, the linear dependence changes into a more complicated behaviour b e f o r e lo goes to zero. With decreasing t e m p e r a t u r e b o t h the co n t r o l c u r r e n t at which the voltage step caused by the TWL appears in the V-I c h a r a c t e r i s t i c s and the value belonging to zero critical c u r r e n t of the TWL are shifted to higher values. Not only the critical current, but also the o t h er characteristic properties of the v o l t ag e step g e n e r a te d by the phase-slip c e n t e r at the TWL depend on the co n t ro l current, I t, t h a t means on the s t r e n g t h of superconductivity in the tunable weak link. As can be seen from Figs. 83 and 84, this s t r e n g t h decreases with increasing c o n t r o l current. In Fig. S5 we p l o t t e d the differential resistance (dV/dD~ o f the f i r s t volt ag e step of the V-I characteristics as well as r e s u l t s for the ratio I 0 / I c. For l o wer values of I t the v z I characteristics s t a r t with the intrinsic v o l t ag e step, whereas for higher I t, the phase-slip c e n t e r at the TWL g e n e r a t e s the f i rst v o l t ag e step. Both quantities are clearly sm al l er for the p h a s e - s l i p c e n t e r at the TWL, with a downward tendency for increasing c o n t r o l current. It is remarkable t h a t for the intrinsic voltage step as f i r st step of the c ha ract er i s t i cs t h e s e properties are not changed by the c o n t r o l c u r r e n t which flow s t h r o u g h the TWL. 248

{

E

o :intrinsic char.

{I{{+++,,+v,.,.,.

4

I

~

I

'

I

'

I

~

I

1.0-

t'tt+,i+"

0.5TcoI = 3.688 K =Too~-z~T TWLSn 0.0

I

0

1

~

1

2

L

I

3

J

I

l

4

ItlmA

I

5

Fig. 85: Differential resistance ( d V / d I ) ~ and ratio I o / I , f o r the f i rst voltage s t e p o f the V-I characteristics o f whisker 1 o f sample T W L Sn [23J. For l o w e r control currents, I t, the V-I characteristic s t a r t s with an intrinsic voltage step, while f o r larger values o f I t the f i rst voltage s t e p o f the characteristic is generated by the phase-slip center at the TWL.

We also investigated the hysteretic properties of the p h a s e - s l i p c e n t e r at the TWL. Our e x p e r i m e n t a l observation is that the hysteresis width, I o - I , , of the f i rs t v o l t a g e step of the V-I characteristics seems to be independent of I t as long as the step is g e n e r a te d by an intrinsic p h a s e - s l i p center. However, I . - I . decreases rapidly with increasing I t for the p h a s e - s l i p c e n t e r at the TWL [23]. An i n t e r e s t i n g question is w h e t h e r the t e m p e r a t u r e dependence of ( d V / d I ) I and I 0 / I ~ of the TWL differs from t h a t one of the intrinsic v o l t a g e step. To answer this question, the intrinsic V-I characteristic for It= 0 has to be measured at several fixed t e m p e r a t u r e s as well as the V-I c h a r a c t e r i s t i c for a fixed c o n t r o l c u r r e n t It=~0 which is sufficiently large so t h a t the p h a s e - s l i p c e n t e r at the TWL g e n e r a t e s the f i r st v o l t ag e step of the characteristic. Naturally, the desired information can be obtained by p e r f o r m i n g m e a s u r e m e n t s o f the kind p l o t t e d in Fig. 83 f o r several fixed t e m p e r a t u r e s . A no t h er possibility (which may be more c o m f o r t a b l e ) is to m easu r e V-I characteristics with I t fixed and T as a p a r a m e t e r (Fig. 86).

249

0

100

200

300

400 I I pA

Fig. 86: V-I characteristics o f whisker l o f TWL Sn for several fixed temperatures, without a control current (upper part o f the figure) and with a control current l~=3.0mA flowing (lower part o f the figure). The current I is increasing f23].

The V - I characteristics w i t h an applied c o n t r o l c u r r e n t l o o k s i m i l a r to t h o s e w i t h o u t c o n t r o l c u r r e n t , b u t they s h o w m o r e n o i s e and c h a n g e d characteristic properties: W i t h o u t a c o n t r o l c u r r e n t t h e critical c u r r e n t s h o w s t h e typical dependence I~3(To)~AT. With a control current, the first intrinsic voltage s t e p s h o w s t h e s a m e t e m p e r a t u r e d e p e n d e n c e with an a b s o l u t e value which is only very s l i g h t l y r e d u c e d . The b e h a v i o u r o f t h e TWL, h o w e v e r , is c h a n g e d by I t. Its critical c u r r e n t d o e s n o t f o l l o w an I.=/3(To) s t r a i g h t line and s h o w s a r e d u c e d value d e p e n d i n g on the m a g n i t u d e o f t h e c o n t r o l c u r r e n t [23]. The 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 l ) , and t h e r a t i o I 0 / I o o f t h e TWL b o t h are s m a l l e r t h a n f o r the i n t r i n s i c p h a s e - s l i p c e n t e r at [ t = 0 . As w e l l f o r the intrinsic p h a s e - s l i p c e n t e r at I t = O as a l s o f o r t h e TWL, b o t h q u a n t i t i e s do n o t d e p e n d on t e m p e r a t u r e [23]. This s t a t e m e n t is valid f o r t e m p e r a t u r e s which are n o t t o o c l o s e t o To0. In t h e c a s e o f ( d V / d I ) , we c o n s i d e r e d only c h a r a c t e r i s t i c s o f t h e T W L w i t h a n o n z e r o c r i t i c a l c u r r e n t . C l o s e r to To0, 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 s o m e w h a t i n c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e (see t h e l o w e r p a r t o f Fig. 86 f o r

250

8 T < 4 . 4 m K ) . In the case of I 0 / l : we only considered c h a r a c t e r i s t i c s of the TWL f o r which a v o l t a g e jump is visible at the critical current. C l o s e r to To, where the c h a r a c t e r i s t i c s more or less continuously s t a r t at I~ the ratio I 0 / I o is c l o s e to unity or not defined for vanishing critical c u r r e n t (see the low er part o f Fig. 86 f o r 8 T < 7 . g m K ) . We should, however, remark, t h a t the re are very r e c e n t e x p e r i m e n t s p e r f o r m e d with tin whiskers showing a t e m p e r a t u r e d e p e n d e n t d i f f e r e n t i a l resistance. This t e m p e r a t u r e dependence is p r e s e n t in the intrinsic case and for the TWL. The s t r e n g t h of the t e m p e r a t u r e dependence of the differential resistance of the TWL is a l t e r e d by the c o n t r o l current. We refer to c h a p t e r 13 for a detailed discussion. The height of the f i r s t voltage step, V~=V(I~ depends linearly on the critical c u r r e n t Io, f o l l o w i n g d i f f e r e n t s t r a i g h t lines for the intrinsic voltage step at I t = 0 and for the TWL E23]. It is remarkable t h a t all values f o r the TWL join the same s t r a i g h t line, independent of the magnitude of I t , t h a t means of the weakness o f the TWL. Since V(Io) can be w r i tt e n as V(I~ o, this obs er v at i o n indicates t h a t the p r o d u c t of (dV/dl)~ and ( l - I 0 / I c) does not, or only weakly, depend on the s t r e n g t h o f the TWL, a l t h o u g h both quantities change with increasing weakness of the TWL. This constancy of the p r o d u c t s e e m s to be c h a r a c t e r i s t i c for a p h a s e - s l i p c e n t e r under conditions where ( d V / d I ) 1 and I 0 / I o change. It has also been observed for indium whiskers, where both q u a n t i t i e s grow if the t e m p e r a t u r e approaches the critical t e m p e r a t u r e (see section 7.2).

12.2.

Interpretation

The experimental r e s u l t s obtained for tunable weak links are not easy to interpret. In the f o ll o w in g we will try to give an explanation f o r the weakening mechanism and s o m e w h a t s p e c u l a t e ab o u t an explanation f o r the behaviour of the c h a r a c t e r i s t i c properties of a p h a s e - s l i p c e n t e r occurring at a TWL. However, m o s t of our r e s u l t s m u s t remain empirical s t a t e m e n t s at the m o m en t . One of the basic q u e s t i o n s concerns the weakening mechanism which is r e s p o n s a b l e for the weakening of superconductivity in the TWL. There are several e f f e c t s of the c o n t r o l c u r r e n t which may lead to a weakening of sup erco n d u ct i v i t y and, therefore, have to be discussed, namely c u r r e n t - i n d u c e d depairing, pair breaking by the magnetic field, simple heating, and so called 'nonequilibrium heating' which is nothing else but the e x c i t a t i o n of the energy mode (discussed in sectionS.S) by an o v e r p o p u l a t i o n of q u as i p ar t i cl e excitations. In the l i t e r a t u r e several m e th o d s of continuously weakening the p r o p e r t i e s of a s u p e r c o n d u c t o r are reported: illumination [104, 231-236, 519], quasiparticle injection t h r o u g h a tunnel c o n t a c t [198, 2 0 0 - 2 0 6 , 5 2 0 - 5 2 4 ] , 251

phonon injection from a normal conducting heater strip crossing the s a m p l e [224-230], a locally applied magnetic field [525], and the weakening of super co n d u ct i v i t y by a t r a n s p o r t c u r r e n t [108, 526]. In several cases c o n t r o l l a b l e weak links were generated [201, 202, 204, 206, 2 2 4 - 2 3 0 , 233, 235, 519-526]. Also their application to a dc squid is r e p o r t e d [522, 524]. The cited experimental works show results which We also observe in our experiments: With increasing weakness of the s u p e r c o n d u c t o r its critical c u r r e n t is reduced and the current-hysteresis of the transition b e c o m e s smaller. However, there is no m e a s u r e m e n t of the reduction of the critical c ur ren t with increasing weakness p a r a m e te r which shows the same sl o p e as we have obtained in Fig. 84. Therefore, an explanation f o r the weakening mechanism c a n n o t be found by a comparison of our r e s u l t s with an e x p e r i m e n t described in the literature. The reason may be t h a t in our e x p e r i m e n t s we g e n e r a t e a tunable weak link in a q u a s i - o n e - d i m e n s i o n a l superconductor. The samples in the cited literature are not q u a s i - o n e - d i m e n s i o n a l ( e x c e p t perhaps those used in ref. 526). Therefore, we tried to find an explanation f o r the weakening mechanism in our e x p e r i m e n t by a discussion of all relevant weakening p r o c e s s e s , however, r e s t r i c t i n g our analysis to the region of c o n t r o l c u r r e n t s where I~~3 depends linearly on I t . It turns o u t [23], that our o b s e r v a t i o n s cannot be explained by simple heating, where the t e m p e r a t u r e of the sample is locally enhanced over the bath t e m p e r a t u r e and all phonons are characterized by this higher t e m p e r a t u r e . Moreover, the magnetic field of the c o n t r o l c u r r e n t can be excluded as well as c u r r e n t - i n d u c e d depairing due to a part of the c o n t r o l c ur ren t flowing across the measuring whisker. In our opinion 'nonequilibrium heating' caused by quasiparticle injection is a probable explanation f o r the weakening mechanism in o u r experiment. The appearance of the p h a s e - s l i p center at the TWL b e f o r e the f i r st intrinsic p h a s e - s l i p c e n t e r in the V-I c h a r a c te r is t ic s occurs at control c u r r e n t s f o r which the c o n t r o l l i n g whisker has nearly reached the normal conducting state. Therefore, we assume that nonequilibrium quasiparticles or normal e l e c t r o n s which are p r e s e n t in the c o n t r o l l i n g whisker are injected into the measuring whisker, leading to an overpopulation of the excitation spectrum. The injection can be caused by a diffusion through the metallic c o n t a c t b e t w e e n both whiskers or by the e f f e c t t h a t a part of a normal c u r r e n t tries to take its path t h r o u g h the s u p e r c o n d u c t in g measuring whisker. The weakening of superconductivity by a quasiparticle overpopulation has been widely dealt with in section 5.5. In t h a t section t w o q u a s i - t h e r m a l models are discussed, the so called ~ model of Owen and Scalapino and the T* model of Parker. Both models predict a decrease of the gap p a r a m e t e r with increasing excess quasiparticle density. While in the (1*model the gap shows a s t e p - l i k e transition to zero at high quasiparticle densities, the T * model predicts a continuously decreasing gap. Since in our e x p e r i m e n t the critical c u r r e n t continuously goes to zero with increasing co n t r o l current, I t , we prefer to apply the T ~ model to our problem. In this model the p r o p e r t i e s 252

of the nonequilibrium s u p e r c o n d u c t o r are obtained as the thermal equilibrium p r o p e r t i e s of t h a t s u p e r c o n d u c t o r at the t e m p e r a t u r e T*. What we, therefore, need is a relation b e t w e e n the c o n t r o l current, I t , and the t e m p e r a t u r e T*. Then the critical c u r r e n t of the TWL can be c a l c u l a t e d f o r a given I t as GL critical c u r r e n t of the measuring whisker at the t e m p e r a t u r e T*. As far as the critical c u r r e n t of the TWL is nonzero, T* can only have values b e t w e e n the bath t e m p e r a t u r e , T, and the critical t e m p e r a t u r e , To0. Since T is only s o m e millikelvins below To0, ST*= T * - T cannot exceed a few millikelvin. As T is several kelvins, this implies ST*((T. As we pointed o u t in se c ti o n 5.5, P a r k e r ' s model then yields (244)

ST*~ (2/3) T Nop/N r

where N o p = N - N t is the quasiparticle overpopulation of the e x c i t a t i o n spe ct r u m . This equation gives us the relation b e t w e e n the actual quasiparticle number per volume, N, and the quasiparticle t e m p e r a t u r e , T*. Here, N r is the thermal equilibrium quasiparticle number per volume at the bath t e m p e r a t u r e , T. If we f u r t h e r m o r e would use T mTo0, we would g et eq .( 9 0 ) o f sect i o n 5.5 which we a p p l i e d for the description of the T* c o n t r i b u t i o n to the hysteresis. To r e l a t e the quasiparticle overpopulation Nop o f the e x c i t a t i o n s p e c t r u m in the TWL with the c o n tr o l c u r r e n t I t , we a s s u m e t h a t Nop = {3rw L ( I t where

13twc

characterizes

the

-

ltA)

quasiparticle

production

(245) due

to

the

co n t r o l

current.

This equation e x p r e s s e s the experimental observation, t h a t a certain c o n t r o l c u r r e n t ItA is needed until the p h a s e - s l i p c e n t e r at the TWL appears b e f o r e the o n s e t of the intrinsic V-I characteristic. We, thus, a s s u m e t h a t only the a m o u n t of the c o n t r o l current, It, flowing above the "appearance current', ItA, c r e a t e s a quasiparticle o v e r p o p u l a t i o n which is large enough to lead to a remarkable e n h a n c e m e n t of the e f f e c t i v e temperature, T*, above the bath t e m p e r a t u r e , T. According to the GL theory the critical c u r r e n t of a q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r is given by (see section 5.1) I2~'(L) = -(dI2~VdL)(To-T)

(246)

where T, is the transition t e m p e r a t u r e at c u r r e n t I c. Usually To is equal to the bath t e m p e r a t u r e T at which the V-I c h a r a c t e r i s t i c s is measured. In the TWL, however, it is T = T + $ T * so t h a t T c 0 - T , = h T - S T * , w h e r e 5 T = T 0 - T . With eqs. (244) and (245) it f o l l o w s f o r the critical c u r r e n t of the tunable weak link 253

1 .... .....

d ]2/3 ( A T ' I t ) = - ( - ~ - ) oc

tAT-2

~T ~TwL ( I t - l t A ) )

(247)

Here, the index ' 0 ' expresses, that the value has to be taken for the intrinsic first v o l t a g e step at It= 0. 2/3 For fixed bath t e m p e r a t u r e , I,,rw L depends linearly on I t as observed in our e x p e r i m e n t for the initial change of the critical c u r r e n t of the TWL. From eq. (247) we g e t the s l o p e d I:./T:~L( AT, It) dI~ '3 dlt = (..~../.)o 32_ T

(248)

Since T ~ T , 0 f o r all m e a s u r e m e n ts , we should g et the same initial slope for all m e a s u r e m e n t s independent o f the special AT. This was indeed observed experimentally (Fig. 84). The quantity 13rwL is a fitting p a r a m e t e r which has to be d e t e r m i n e d from the experiment. For this purpose one may use N r from eq.(231) and apply eq. (247) or (248). It may be i n te r e s t in g to know s o m e t h i n g about the magnitude of ~T*. This i n fo rm at i o n can be directly obtained from Fig. 84: Let us, for instance, discuss the critical c u r r e n t of the TWL p l o t t e d in m e a s u r e m e n t no. 2 p e r f o r m e d at a bath t e m p e r a t u r e T2=Tc0,-12.1mK. For a certain c o n t r o l c u r r e n t the critical c u r r e n t o f the TWL b e c o m es equal to the critical c u r r e n t of the f i r s t intrinsic voltage step at It= 0 of m e a s u r e m e n t no. 3, occurring at the bath t e m p e r a t u r e T3=To0,-9.4mK. This fact indicates t h a t the e f f e c t i v e t e m p e r a t u r e T * in the TWL is enhanced to the bath t e m p e r a t u r e of m e a s u r e m e n t no. 3. Thus, T2+~T *=T 3 or $T x=T 3 - T 2=2.7mK. It is also possible to calculate ~T* for arbitrary value of I t in the linear region. Hsing eqs. (244), {245), and (248) we g e t

ST*=

di2'3 ( A T ' I t ) dla/Z -' I : .... ( r )0 ( t - I t A ) dI t

(249)

Naturally, this f o r m u l a also yields a value f o r ST* in the situation described above. From Fig. 84 we get d I2,~T~vL/dIt= - 20.5 gA2~3/mA. Furthermore, [23], (d I~3/dTr )0 = 6121 gA2/3/K. At the point where the m e a s u r e m e n t no. 2 yields a critical c u r r e n t of the TWL equal to the critical c u r r e n t of the f ir s t intrinsic v o l ta g e step of m e a s u r e m e n t no. 3, it is I t - I t A = 0 . 8 mA. As e x p e c t e d we g e t ~T*= 2.7 mK f r o m eq. (249). We think to have found a probable explanation f o r the weakening mechanism, whereas the o t h e r properties of the p h a s e - s l i p c e n t e r at the TWL remain unexplained: The reduced differential resistance of the p h a s e - s l i p c e n t e r at the TWL seems to indicate that the inelastic s c a t t e r i n g time % for the quasiparticles

254

has a s m a l l e r value than in the intrinsic case (see section 7.2 for the relation of ~2 and (dV/dI)~). This may be caused by an enhanced quasiparticle r e c o m b i n a t i o n rate due to the quasiparticle overpopulation E206], but can also be g e n e r a t e d by the m a g n e t i c field of the c o n t r o l c u r r e n t (see the discussion of the m e c h a n i s m s c o n t r i b u t i n g to zQ~ given in section 5.4). The l a s t e x p l a n a t i o n s m u s t remain speculative, because t h er e do n o t e xis t any t h e o r e t i c a l calculations concerning the p r o p e r t i e s of a p h a s e - s l i p c e n t e r with a weak link of tunable s t r e n g t h as its origin which can be directly compared with the p r e s e n t experiment. The only t h e o r e t i c a l approach to such a problem known to the a u t h o r has been r e c e n t l y done by Kramer and Rangel (KR) in the f r a m e w o r k of the TDGL theory (see ref. 316 and sectionS.9) for localized weak links with reduced mean free path for the e l e c t r o n s ( ' M F P - t y p e weak r e g i o n ' ) or reduced critical t e m p e r a t u r e ('To0-type weak r e g i o n ' ) . The c a l c u l a t i o n s lead to a s m a l l e r critical cu r r en t and a s m a l l e r n o r m a l - l i k e l e n g t h (and, thus, to a s m a l l e r ( d V / d I ) 1) f o r the weaker link. These r e s u l t s are obtained for both kinds of weak regions ( ex cep t f o r large reductions o f the e l e c t r o n mean free path in MFP-type weak regions, where the n o r m a l - l i k e l e n g t h is predicted to increase). They are in qualitative a g r e e m e n t with our m e a s u r e m e n t s . The ratio I0/I o is decreased by M F P - t y p e weak regions and appears to be increased by To0-type weak regions. Thus, in the case of the ratio I J I o a qualitative a g r e e m e n t with the predictions of KR for M F P - t y p e weak regions may be remarked. For To0-type weak regions KR found t h a t "the situation is sh i f t ed toward the s t a t i c limit". In addition to the mentioned reduction of the a b s o l u t e value of the n o r m a l - l i k e length this prediction implies a weaker t e m p e r a t u r e dependence of this quantity. Also for M F P - t y p e weak regions KR pre d i ct a r ed u ct i o n of the t e m p e r a t u r e dependence of the n o r m a l - l i k e length. In the e x p e r i m e n t s discussed in the p r e s e n t chapter we found t h a t ( d V / d I ) l (and, thus, the n o r m a l - l i k e length) and I0/I ~ do not depend on t e m p e r a t u r e , but we r e f e r to the f o ll o w in g section for f u r t h e r e x p e r i m e n t s carried o u t with whiskers of Sn and o t h e r materials. Thus, there is some qualitative a g r e e m e n t between the calculations of KR and our m e a s u r e m e n t s . One should, however, keep in mind t h a t the theory does not consider the weakening mechanism which we e x p e c t to govern the p r o p e r t i e s of the TWL in our experiment. Furthermore, there remains the p ro b l e m t h a t the calculations are carried o u t in the dirty limit, while our s a m p l e s are s u p e r c o n d u c t o r s being in the clean limit. Finally, we should remark t h a t the tunable weak links are the subject of c u r r e n t investigations and we r e f e r to chapter l3 for additional r e s u l t s on this scope.

255

13. Remarks on Ongoing Work

In this ch ap t e r we will briefly sketch the p r o b l e m s we are dealing with at the m o m en t . These are the investigation of 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 center, e x p e r i m e n t s on tunable weak links in whiskers of Sn, In, and Zn, studies of m i c r o c o n t a c t s b e tw e e n two whiskers and m e a s u r e m e n t s of the interaction of p h a s e - s l i p c e n te r s in Zn whiskers. In m o s t cases we can only p r e s e n t our experimental results, because m o m e n t a r y there is no i n t e r p r e t a t i o n available. The research work r e p o r t e d in the f o ll o w i n g has been done by X. Yang (and the author) in our laboratory and has n o t yet been published elsewhere.

13.1.

Low-Temperature Behaviour of a Phase-Slip Center

Usually, detailed investigations of the behaviour of a p h a s e - s l i p c e n t e r in our whiskers are r e s t r i c t e d to a t e m p e r a t u r e range in the direct vicinity o f the critical t e m p e r a t u r e . In the case of pure tin whiskers the t e m p e r a t u r e range does n o t exceed about 15 mK below T,0. An i n t e r e s t i n g problem should be to trace the behaviour of a p h a s e - s l i p c e n t e r down to much lower t e m p e r a tu r e s . Then, the sample leaves the q u a s i - o n e - d i m e n s i o n a l case more and more. In section 6.1 we discussed m e a s u r e m e n t s of the critical c u r r e n t of a tin whisker (sample SnH3) which were carried out down to more than 3 0 0 m K below T,0 (Fig. 21). In section 11.3 we showed the hysteretic behaviour of t h a t tin whisker for the w h o le t e m p e r a t u r e range (Fig. 69). In the p r e s e n t sect i o n we will discuss the behaviour of the characteristic properties of the f i r st voltage step in the V-t characteristics g e n e r a t e d by a single p h a s e - s l i p c e n t e r in t h a t sample. We e x t e n d e d our m e a s u r e m e n t s down to about 40 mK below To0. The c h a r a c t e r i s t i c properties of the first voltage step are (besides the critical current, I c, and the j u m p - b a c k current, I,) the differential r e s i s t a n c e (dV/dI)~, the e x t r a p o l a t e d zero voltage intercept, I 0 (and, thus, the ratio 10/Io), the height of the v o lt a g e jump at the critical current, V~=V(Io), and the j u m p -b ack voltage, VR=V(I,). A sketch of the f i r st v o l t ag e step is given in section 7.2 (see Fig. 2S).

256

Sn H3 Tco=3.6908K

1.0-

~#~#~%'~ ~

h T=Tco- T

0.5.

I

I

I

I

I

I

I

1

2010L

0

i

1()

L

2~0

J

3~0

[

A Tim K

50

Fig. 87: Differential resistance ( d V / d l ) , and ratio I o / I ~ f o r the first voltage s t e p in the V - I charateristic o f a tin whisker (sample SnH3), traced f r o m temperatures very close to the critical temperature To down to ' l o w

temperatures ".

In Fig. 87 w e p l o t t e d ( d V / d l ) ~ a n d 10/I o as a f u n c t i o n o f t e m p e r a t u r e . 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 is c o n s t a n t u n t i l AT e x c e e d s a b o u t 1 4 m K . T h e n , (dV/dI) 1 increases, reaches a maximum value at AT~21mK, and then decreases toward a constant value. T h e m a g n i t u d e o f 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 in t h e t e m p e r a t u r e i n d e p e n d e n t r e g i m e a t l o w t e m p e r a t u r e s ( l a r g e AT) is s l i g h t l y h i g h e r t h a n c l o s e t o To0 ( s m a l l AT). T h e b e h a v i o u r o f 10/I ` is very s i m i l a r . W h i l e t h e m a x i m u m is o b s e r v e d a t t h e s a m e t e m p e r a t u r e , t h e i n c r e a s e is o b s e r v e d s o m e w h a t c l o s e r t o T,0 a n d the decrease extends to somewhat lower temperatures. Moreover, the c o n s t a n t b e h a v i o u r is n o t s o c l e a r l y e x p r e s s e d a s in t h e c a s e o f t h e differential resistance. To g e t a d d i t i o n a l i n f o r m a t i o n we m a d e a l o g - l o g p l o t o f ( d V / d l ) l a g a i n s t AT. T h i s p l o t s h o w s t h a t t h e i n c r e a s e o f ( d V / d I ) ~ is p r o p o r t i o n a l t o AT s/4, w h i l e t h e d e c r e a s e o c c u r s p r o p o r t i o n a l t o AT -2. F u r t h e r m o r e , w e e v a l u a t e d t h e h e i g h t o f t h e f i r s t v o l t a g e j u m p , V~, a n d p l o t t e d i t a s a f u n c t i o n o f t h e c r i t i c a l c u r r e n t , I c. We o b s e r v e d t h a t t h e r e is a l i n e a r i n c r e a s e o f V~ ( I , ) w i t h i n c r e a s i n g c r i t i c a l c u r r e n t up t o a m a g n i t u d e of I C which corresponds to that one at the temperature of the maximum of 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 . A t t h i s c r i t i c a l c u r r e n t t h e r e is a s u d d e n c h a n g e in

257

the slope of the s t r a i g h t inclination as before.

line which now increases

with several times the

Finally, cl o s e to the t e m p e r a t u r e of the maximum of the d i f f e r e n t i a l r e s i s t a n c e also the behaviour of the j u m p - b a c k voltage, VR, changes. As long as the j u m p - b a c k current, I,, is s m a l l e r than the value co r r esp o n d i n g to t h a t t e m p e r a t u r e , Va(I R) shows the typical saturating behaviour (there may be a slight increase with increasing jump-back current). As soon as I R s o m e w h a t exceeds the mentioned value, V~(I R) increases linearly. The behaviour r e p o r t e d is not a f o r t u i t o u s event of a single specimen, but s eem s to be typical for the low t e m p e r a t u r e behaviour o f a p h a s e - s l i p c e n t e r in a tin whisker. We p e r f o r m e d the same m e a s u r e m e n t s with a d i f f e r e n t tin whisker and obtained the same results. The only difference is that the maximum of the differential resistance is sh i f t ed some millikelvin toward l o w e r t e m p e r a t u r e s and t h a t a slight increase of ( d V / d I ) 1 is observed f or very small values of AT ( t h a t means very close to T~0). At the m o m e n t we do n o t know which mechanism g e n e r a t e s the observed behaviour.

13.2.

Tunable Weak Links

Also tunable weak links (introduced in chap. 12) are the subject of c u r r e n t research. On the one hand we continued to investigate the p r o p e r t i e s of p h a s e - s l i p cen te r s occurring at a tunable weak link (TWL) in pure Sn whiskers, on the o t h e r hand we extended our m e a s u r e m e n t s to o t h e r materials, such as In and Zn. First let us d i s c u s s recent e x p e r i m e n ts p e r f o r m ed with tin whiskers: To ge t additional i n f o r m a ti o n about the weakening mechanism in the TWL we built a sample where the measuring whisker and the c o n t r o l l i n g whisker ( b o t h pure tin whiskers) touch each o t h e r but where there is no m e t a l l i c c o n t a c t b e t w e e n both whiskers, because no discharge c u r r e n t is al l o w ed to flow across the touching region. In this case the application of a c u r r e n t through the c o n t r o l l i n g whisker does n o t generate a tunable weak link in the measuring whisker. The only e f f e c t is that in the case of very large c o n t r o l c u r r e n t s the ch ar a c t e r i s ti c s of the measuring whisker b e c o m e s o m e w h a t noisy. The reason may be t h a t the heat dissipation of the c o n t r o l whisker leads to disturbances of the t e m p e r a t u r e constancy in the en v i r o n m en t of the measuring whisker. This e x p e r i m e n t e s ta b l is h e s our i n t e r p r e t a t i o n t h a t nonequilibrium quasiparticles generated by injection p r o c e s s e s through the 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 are responsible for the weakening of superconductivity in the tunable weak link. We, therefore, continued to investigate samples with a m e t a l l i c m i c r o c o n t a c t b e t w e e n both tin whiskers. A l o t of e x p e r i m e n t a l data were 258

obtained for t h e s e specimens. They behave similar to sample T W L S n of c h a p t e r 12. There are, however, some remarkable new results. To give an example, we shall discuss the m e a s u r e m e n t s p e r f o r m e d with sample T W L S n l 0 : The sample c o n s i s t s of tw o pure Sn whiskers and was prepared as described in chapter 12. The g e o m e t r y (that means the size of the d i f f e r e n t lengths) is comparable with sample TWLSn. The critical t e m p e r a t u r e s are T 0 = 3 . 6 9 9 5 K and T00t=3.6998K for the measuring and the c o n t r o l l i n g whisker, respectively. Moreover, it may be i n t e r e s t i n g to know s o m e p r o p e r t i e s of the measuring whisker. One part is 263Fm long, while the l e n g t h of the o t h e r part is 488Fm. The c r o s s - s e c t i o n a l area is A = 3 . 9 4 F m 2, the o ri en t at i o n is [0013, the residual resistance is R n = 0 . 0 4 S S ~ , and the e l e c t r o n mean free path is 2 = 4 . 2 ~ m . The slope of the 1,2~3(To) s t r a i g h t line of the measuring whisker without a control current is d I~/3/dTc = - 6800 gA2/3/K. An example for the V-I characteristics of sample T W L S n l 0 at fixed t e m p e r a t u r e but f o r d i f f e r e n t c o n t r o l c u r r e n t s is given in Fig. 88. In this specimen (and in some of the o th e r s ) we succeeded to trace the behaviour of the v o l t a g e step g e n e r a t e d by the p h a s e - s l i p c e n t e r at the TWL while this step still appears within the V-I characteristics above the critical c u r r e n t of the fi rs t few intrinsic v o l t a g e steps. Again the c o n t r o l l i n g whisker is in the dissipative s t a t e (close to being fully normal conducting) as soon as the r e du ct i o n of the critical c u r r e n t of the TWL with increasing c o n t r o l current, I t, can be traced in Fig. 88. We should remark t h a t t h er e are also m e a s u r e m e n t s of sample T W L S n l 0 (at l ow er t e m p e r a t u r e ) where the m o v e m e n t of the p h a s e - s l i p c e n t e r at the TWL within the intrinsic c h a r a c t e r i s t i c can be already observed while the c o n t r o l l i n g whisker is still in the s u p e r c o n d u c t i n g state. In all m e a s u r e m e n t s, however, the c o n t r o l l i n g whisker is in the dissipative s t a t e when the critical c u r r e n t of the TWL is s m a l l e r than the intrinsic critical c u r r e n t so t h a t the p h a s e - s l i p c e n t e r at the TWL appears prior to the first intrinsic p h a s e - s l i p center. The r e s u l t s f o r l ~ ( I t ) are similar to t h o se given in Fig. 84 f o r TWLSn. The d i f f e r e n c e is, naturally, that in the case of T W L S n l 0 the critical c u r r e n t of the TWL can already be traced while it is s m a l l e r than t h a t one of the o n s e t of the intrinsic characteristic. The influence of the c o n t r o l c u r r e n t on the critical c u r r e n t of the f i r s t intrinsic v o l tag e step is s o m e w h a t s t r o n g e r than in sample TWLSn. The reason may be that the f i r st intrinsic p h a s e - s l i p c e n t e r is localized at a place which is c l o s e r to the TWL than in sample TWLSn. As soon as a c o n t r o l current is applied the intrinsic critical c u r r e n t is s l i g h t l y reduced. This reduction becomes s t r o n g e r with i n c r e a s i n g magnitude of I t . There is a stabilizing interaction b e t w e e n the p h a s e - s l i p c e n t e r at the TWL and the first intrinsic p h a s e - s l i p center. As soon as the p h a s e - s l i p c e n t e r at the TWL appears prior to the first intrinsic v o l t a g e step a j u m p - l i k e increase of the critical c u r r e n t of the f i r s t intrinsic p h a s e - s l i p c e n t e r and a reduction of its c o n t r o l c u r r e n t induced downward tendency is observed.

259

"3

10

.

O

~

0

100

200

300

I /tJA

500

Fig, 88"

V-I characteristics o f a tunable weak link (sample T W L S n l O ) at fixed temperature f o r different values o f the control current, I t. For a s k e t c h o f the sample g e o m e t r y see Fig. l d or the inset o f Fig. 83. The measuring whisker and the controlling whisker both are pure tin whiskers. In this specimen a voltage step with tunable critical current ( ' T W L ' ) can already be traced while it is still within the intrinsic characteristic. The zero voltage point varies f o r the different characteristics. It is indicated by the fine horizontal line f r o m which a characteristic starts. The current I is increasing. Here, T~o is the critical temperature o f the measuring whisker.

As in the case of sample TWLSn of chapter 12, also for TW LS n 10, the differential resistance (dV/dI)~ and the ratio I 0/Io of the v o l t a g e step ge ne r at ed by the p h a s e - s l i p c e n t e r at the TWL (for c o n s t a n t t e m p e r a t u r e ) are s m a l l e r than fo r the first intrinsic v o lt a g e step and a downward tendency is observed fo r increasing value of I t . There is, however, a very remarkable difference in the t e m p e r a t u r e dependence of the differential resistance (Fig. 89). Already the differential resistance of the f i r s t intrinsic voltage step (for It=O) is only t e m p e r a t u r e independent for t e m p e r a t u r e s which are not t o o close to To0. In the direct vicinity of To0 a s t r o n g increase of ( d V / d l ) 1 is observed. Also the differential resistance of the v o lt a g e step generated by the p h a s e - s l i p c e n t e r at the TWL is only t e m p e r a t u r e independent s o m e w h a t f u r t h e r away f r o m To0, b u t 260

l

TWLSnl0 Tco= 3.6995 K

10

5 It = O mA intrinsic

E i

~

I

L

[

I

i

I

i

I

o: I t = 3.1mA. TWL ~,: I t =3.5mA. TWL

5

AT=7-co- T

0

I

0

I

~

I

I

i

,

,

,

5

I

10

,4T ImK Fig. 89: Differential resistance ( d V / d l ) , o f the first voltage step o f the V-I characteristics o f the measuring whisker o f sample T W L S n 10. For It =0, the first voltage step is generated by an intrinsic p h a s e - s l i p center. This case is shown in the upper part o f the figure. In the lower part the first voltage step is generated by the TWL. Results for two different control currents, It=3.1 and 3.5mA are shown. Here, T~o is the critical temperature o f the measuring whisker.

increases when the critical temperature is approached. Compared to the intrinsic voltage step, the o n s e t of the increase of the differential resistance of the TWL occurs at lower temperatures (larger AT) and is shifted to even lower t e m p e r a t u r e s for larger I t . Moreover, the increase of its differential resistance with increasing temperature (decreasing AT) becomes s t r o n g e r for larger control current. The ratio I0/I c is t e m p e r a t u r e independent for the intrinsic first voltage step (It=O). In cases where a c o n s t a n t control current, It, is flowing and the first voltage step is generated by the phase-slip center at the TWL, its ratio 10/I r remains t e m p e r a t u r e independent somewhat f u r t h e r away from T00, b u t decreases if the temperature becomes very close to the critical temperature. Our m e a s u r e m e n t s show that the kind and s t r e n g t h of the t e m p e r a t u r e dependence of the differential resistance (and, thus, of the n o r m a l - l i k e length, LAn1) depend on the s t r e n g t h of superconductivity in the weak link at which a phase-slip center develops. The role of inhomogeneities on the behaviour of phase-slip centers has been discussed in several sections of the p r e s e n t work (see sections 5.9, 7.2, 7.4, 7.5, and chapters 10 and 12). With some caution, we now draw the following conclusion: The differential 261

resistance of a p h a s e - s l i p c e n t e r in an ideal h o m o g en eo u s or nearly h o m o g e n e o u s filament o f pure tin is t e m p e r a t u r e independent. If the p h a s e - s l i p c e n t e r s develop at weak links in the sample with su p er co n d u ct i n g p r o p e r t i e s which do not only differ very slightly f r o m t h o se of the whole whisker, they g e n e r a t e voltage steps with a differential resistance which is more or less t e m p e r a t u r e dependent, depending on the weakness of superconductivity in the weak link. This conclusion implies, that in sample TWLSntO also the f i r s t intrinsic p h a s e - s l i p c e n t e r develops at a weak link with su p er co n d u ct i n g p r o p e r t i e s which deviate s o m e w h a t s t r o n g e r from the properties of the sample than this is the case in o th e r specimens, in which (dV/dl)~ is not t e m p e r a t u r e dependent. This indeed may be the reason: The tin whiskers used for the e x p e r i m e n t s r e p o r t e d in the p r e s e n t section are 'new' whiskers while in o t h e r e x p e r i m e n t s 'old" whiskers are used. New whiskers are freshly grown specimens while old whiskers were grown several years before they are picked up from the screw (see chap. 2) and used for a m e a s u r e m e n t . In the new whiskers the p h a s e - s l i p c e n t e r s seem to be pinned at weak links with p r o p e r t i e s which deviate s o m e w h a t more from t h o se of the whole whisker as in the case of old whiskers. May be that d e fect s in a whisker are healing while the f i l am en t s are kept at room temperature. This i n t e r p r e t a t i o n is s u p p o r t e d by the observation t h a t new whiskers used f o r the investigation of the low t e m p e r a t u r e behaviour of a phase-slip c e n t e r (section l3.1) show an increase of the differential resistance (dV/dl)~ very close to To0. As already mentioned in the beginning of this section we not only investigated tunable weak links in tin whiskers but also in whiskers o f indium and zinc. In both cases measuring and c o n t r o l l i n g whisker are of the same material, t h a t means in the case of indium the specimen is made f r o m two pure In whiskers, while in the case of zinc two pure Zn whiskers are used. The fabrication procedure of the TWL is the same as before, e x c e p t that in the case of indium the c o n t a c t blocks which hold the whiskers are made o f indium. Let us f i r s t discuss the indium specimen, TWLIn4: The behaviour of the critical c u r r e n t Ir at fixed t e m p e r a t u r e looks similar to the case of TWLSn. The difference is that the c o n t r o l currents needed to sh i f t the critical cu r r en t of the TWL to zero are much smaller. For 5 T = 9 . 4 m K the critical c u r r e n t of the TWL b e c o m e s s m a l ler than that one of the f i r st intrinsic v o l t ag e step already for I t ~ O . 2 m A and a c o n t r o l cu r r en t o f only I t = l m A is needed to suppress the critical c u r r en t to zero. The c o n t r o l l i n g whisker is in the dissipative s t a t e in situations where the properties of the measuring whisker are influenced by the c o n tr o l current. For c o n s t a n t t e m p e r a t u r e the differential resistance of the v o l t a g e step ge nerat ed by the p h a s e - s l i p c e n t e r at the TWL is s o m e w h a t s m a l l e r than that one of the intrinsic first voltage step for It= O. The same o b s e r v a t i o n is made f o r the ratio I0/I r 262

As usually observed for indium whiskers, the differential resistance of the f i r s t intrinsic voltage step is t e m p e r a t u r e d ep en d en t and s t r o n g l y increases if the t e m p e r a t u r e approaches To0. Contrary to the observation f o r tin specimens r e p o r t e d above, the differential resistance of the v o l t a g e s t e p g e n e r a t e d by the p h a s e - s l i p center at a TWL in an indium whisker for c o n s t a n t I t s h o ws a much weaker t e m p e r a t u r e dependence than t h a t one of the f i r s t intrinsic v o l t a g e step. The behaviour of the ratio I0/I o f o r c o n s t a n t I t is similar to t h a t one of the differential resistance: In the intrinsic case ( I t = 0 ) it increases if T approaches To0. The increase is weaker for the p h a s e - s l i p c e n t e r at the TWL than for the f i r s t intrinsic p h a s e - s l i p center. Finally,

we

shall

discuss

the

properties

of

a

TWL

in

a

pure

zinc

whisker: We may f i r s t give some information about the specimen called T W L Z n 3 . The l en g th s of the two parts of the measuring whisker are S00~m and 337.5~m, respectively. Its c r o s s - s e c t i o n a l area is A = S . 9 3 ~ m 2, the orien t at i o n is [ll~0], the resistance at room t e m p e r a t u r e is R298K= 8.26 f~, the residual r e s i s t a n c e is Rn= 0.03 fl, and its e l e c t r o n mean free path as calculated f r o m the residual resistance is s =lO.35gm. Furthermore, for the measuring wh i s k er it is dI~/dTc=-81OOpA2~3/K. The critical t e m p e r a t u r e o f the measuring w h is k e r is T0=0.8182 K. The critical t e m p e r a t u r e of the c o n t r o l l i n g whisker, To0t, is 0.8182 K, too. Also in this case the c o n t r o l l i n g whisker is in the dissipative s t a t e in situations where the c o n t r o l c u r r e n t influences the properties of the measuring whisker. Experimental r e s u l t s f o r the V-I c h a r a c t e r i st i cs of sample T W L Z n 3 at fixed t e m p e r a t u r e but d i f f e r e n t c o n t r o l c u r r e n t s , It, are shown in Fig. 90. The behaviour is very d i f f e r e n t from that one observed for specimens of tin and indium. No real TWL seems to establish in a zinc whisker. Instead of this the wh o l e c h a r a c t e r i s t i c is shifted toward s m a l l e r critical currents. In Fig. 90 we traced the behaviour until the critical c u r r en t of the f i r s t v o l t a g e step is c o m p l e t e l y suppressed. In our opinion the long l i f e - t i m e of nonequilibrium quasiparticles in zinc is re s p o n s i b l e for the observed effect. The whole measuring whisker seems to be f l o o d e d by nonequilibrium quasiparticles entering the whisker t h r o u g h the m i c r o c o n t a c t to the c o n t r o l l i n g whisker, because their relaxation is so slow. Nev ert h el es s , we evaluated the properties of the f i r st v o l t a g e step with applied c o n t r o l c u r r e n t and compared the r e s u l t s with the intrinsic case where It=O. We found t h a t very small c o n tr o l c u r r e n t s do not change the critical c u r r e n t at which the f ir s t o n s e t of voltage occurs. For s o m e w h a t larger c o n t r o l c u r r e n t s I~ is then more and more reduced (Fig. 91). The behaviour o f I~/3(It) then is similar to t h a t one obtained for the critical c u r r e n t of the TWL in sample TWLSn. It may be i n t e r e s t i n g to know t h a t the c o n t r o l l i n g whisker f o r AT= 3.8, 3.2, 3.S, and 2.0 inK, respectively, e n t e r s the dissipative s t a t e at I t , = ll0, 80, 55, and 40~A and the c o m p l e t e l y normal c o n d u c t i n g s t a t e at It,.= 265, 2S2, 23S, 222~A.

263

/ 5.

/

/

..Js "_Y //

/, / /

.

,0.8..

0.7 0.6 0.5

.

0.4 0.3 .

0.2

AT:3.8 mK

0.0

0

J

I

I

~

200

100

0

IIpA

I

300

R g . 90: V-I characteristics o f sample T W L Z n 3 , a tunable weak link arrangement with t w o pure zinc whiskers used as measuring whisker and controlling whisker, respectively. The m e a s u r e m e n t s were carried out at f i xed temperature and increasing current, for several control currents, I t . Here, To is the critical temperature o f the measuring whisker, The zero voltage point varies for the different characteristics. It is indicated by the horizontal line which a characteristic starts from.

AO

TWLZn3 Tc°=0'8182K

ATImK

¢0

3.8 1

o o

3.2

/1T=Tco-T

%O%o

0 0 O0

20i 10I 0 Fig. 91:/j/3

00

2.5

Oo

°o

2o,00000 °Ooo °o ° ooop, °o 0 o ~°Ooo ° o o ° o 00 O0 0 0 O00 0 O0 0 000 o 0 ° 0 000000000 00 0 0 00 I

0

'

J

'

t

I

'

O.5

~

~

'01

9

1.0

g

g

i

/~/mA

as a function o f the control current, It, f o r the onset o f the characteristics o f the measuring whisker o f sample T W L Z n 3 f o r several f i xed temperatures. Here, T~o is the critical temperature o f the measuring whisker.

264

For c o n s t a n t temperature, the differential resistance (dV/dI)~ and the ratio I0/I o are nearly i n d e p e n d e n t of the control current. There may be a slight increase of both quantities with increasing 1t. Furthermore, we investigated the t e m p e r a t u r e dependence of ( d V / d I ) , and I0/I ,. In the intrinsic case (It=O) these quantities do n o t depend on temperature. With applied control current, both quantities become t e m p e r a t u r e dependent. Their temperature dependence becomes s t r o n g e r for larger c o n t r o l current. An example for the intrinsic (It=O) and the quasiparticle overflooded case (It# O) is plotted in Fig. 92. The experiments performed with zinc establish our a s s u m p t i o n that the weakening mechanism in our experiments is due to quasiparticle overpopulation. Somewhat fortuitously an experiment has been performed which shows that the metallic connection between c o n t r o l l i n g and measuring whisker is needed to c o n t r o l the properties of the measuring whisker. In this experiment the measuring whisker was a pure zinc whisker and two c o n t r o l l i n g whiskers were applied at two different places. As c o n t r o l l i n g whiskers we used Z n - A g whiskers which are n o r m a l conducting during the m e a s u r e m e n t s . By a f o r t u i t o u s event the metallic connection between one of the c o n t r o l l i n g whiskers and the measuring whisker was destroyed during the m e a s u r e m e n t s . Before the destruction of the microcontact, the properties of the measuring whisker could be c o n t r o l l e d very effectively by the mentioned c o n t r o l l i n g whisker. Only small control c u r r e n t s already lead to s u b s t a n t i a l shifts of the critical c u r r e n t of the measuring whisker. After the disconnection of the microcontact no shift of I t by the c o n t r o l c u r r e n t was observed.

20-

TWLZn3 Tco= 0.8182 K

10-

~r162162 Ir--Orn,~

E

0

1.0"

I

I

I

I

TWLZn3

].5i..J

It=0mA

"

0

It =0.TmA

10-

4***;,>,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

/

<~

/

0

ATImK

5.4

50

/

100

ZlpA

big, 94" I7-I characteristics (increasing c u r r e n t ) o f a m i c r o c o n t a c t between t w o indium whiskers (sample T W L I n 4 ) f o r several fixed temperatures. The inset s h o w s the g e o m e t r y o f the sample. Here, To , and To2 are the critical temperatures o f whisker I and 2, respectively. The dashed lines are guides to the eyes to make the p o r t i o n s o f c o n s t a n t differential resistance visible.

s k e t c h e d in the inset. The l e n g t h s of the l e f t and right side of whisker l and t hos e o f the l o w e r and upper part of w h i s k e r 2 are 438, 288, 412, and 338 Fro, respectively. Furthermore, it is for whisker 1 (whisker 2): R~sK= 34.1f~ (61.4 f/), Rn= 0.448 C/ (0.980 f)), A= 1.95~m 2 ( 1.13 ~tm2), s = 1.2311m (1.01 t~m), and d I ~ / 3 / d T , = - 4 S 8 0 g A 2 ' 3 / K ( - 3360 ~A2/3/K). It is remarked that w h i s k e r l is used as measuring whisker in the TWL e x p e r i m e n t s mentioned in the preceding section. For all ch ar a c t e r i s ti c s shown the n o n - c u r r e n t - c a r r y i n g parts of the whiskers are in the superconducting state, so that the v o l t ag e across the microcontact is measured with superconducting probes. Also the c u r r e n t - c a r r y i n g part of whisker l remains fully superconducting during the m e a s u r e m e n t s , while the c u r r e n t - c a r r y i n g part of w h i s k e r 2 probably changes into the dissipative state. The characteristics in Fig. 94 were recorded with high r e s o l u t i o n and, therefore, show the linear portion phenomenon m o s t clearly. The dashed lines indicate our e s t i m a t e d slope through the sections of c o n s t a n t differential resistance. For the characteristics at AT=5.4mK our e s t i m a t e f o r the differential resistance of the indicated linear portions is (1.87 + -0.30), (7.43+0.7), and (30.O+-l.3)mfL For the characteristics at A T = 8 . 3 m K the values are (3.5-+0.7), (10.6+-1.6), and (28.2-+1.5), while at A T = 9 . 8 m K we g et (7.8-+0.7) and (30.9+l.O) mfl. Within our limits of accuracy, we, thus, see that the obtained differential resistances are m u l t i p l es of the s m a l l e s t differential r e s i s t a n c e observed in a characteristic: For &T=S.4 mK the l a r g e r 268

differential r e s i s t a n c e s are r and 16 times the l o w e s t differential resistance. For AT= 8 . 3 m K we observe 3 and 8 t im e s the l o w e s t d i f f e r e n t i a l resistance, whereas for A T = 9 . 8 m K the larger differential r e s i s t a n c e is equal to 4 times the s m a l l e r one. These r e s u l t s are not u n d e r s t o o d at present.

13.4.

The Interaction o f Phase-Slip Centers in Zn Whiskers

The influence of an active phase-slip center in a whisker on the s u p e r c o n d u c t i n g p r o p e r t i e s of the filament is one of the unresolved p r o b l e m s in the s t u d i es of nonequilibrium superconductivity in q u a s i - o n e - d i m e n s i o n a l s u p e r c o n d u c t o r s . Before we discuss our very r e c e n t e x p e r i m e n t a l r e s u l t s obtained for Zn whiskers, we should briefly summarize our k n o w l e d g e a b o u t this phenomenon. E x p e r i m e n t s on tin -whiskers p e r f o r med with m u l t i - p o t e n t i a l - p r o b e c o n t a c t s clearly show the e x i s te n c e of such an influence: There is a mutual activation of p h a s e - s l i p c e n te r s leading to v o l t ag e steps appearing synchronously at the same c u r r e n t (section7.4). There is a stabilization of superconductivity by active p h a s e - s l i p c e n t e r s leading to an e n h a n c e m e n t of the critical c u r r e n t of o t h e r p h a s e - s l i p centers. S o m e t i m e s also d e p r e s s i o n e f f e c t s are observed. It seems as if p r e s e n t p h a s e - s l i p cen t er s m i g h t change the differential r e s i s t a n c e of o t h e r p h a s e - s l i p cen t er s (section7.5). For o t h e r materials we did not p e r f o r m such detailed investigation of the influence of active p h a s e - s l i p c e n t e r s on the s u p e r c o n d u c t i n g p r o p e r t i e s of the filament. N e v e r t h e le s s , also for whiskers of Zn there is evidence f o r a s t i m u l a t i o n of superconductivity by an active p h a s e - s l i p center. To show this, we simply evaluated the distance in c u r r e n t b e t w e e n the f i r st and second volt ag e jump in the V-I characteristic of a Zn whisker f o r several fixed t e m p e r a t u r e s . We found t h a t this distance is very small f o r t e m p e r a t u r e s very c l o s e to T,0. It f ir s t increases and then decreases again if the t e m p e r a t u r e is lowered (section 8.2). A similar behaviour is found f o r the width of the f i r s t linear p o r t i o n o f a Pb whisker. In a d i f f e r e n t Pb whisker the width also increases with decreasing t e m p e r a t u r e but then shows a s a t u r a t i n g behaviour. In this sample the d e c r e a s e is not observed in the measured temperature range (section 9.2.3). Also In whiskers show an increase of the distance b e t w e e n the f i r s t two v o l t a g e jumps of their V-I characteristics, if the t e m p e r a t u r e is l o w er ed (Fig. 3 of ref. 16). Similar e f f e c t s are observed for S__n_n-ln and I n - P b whiskers. For sample S__~n-ln7 the distance grows with decreasing t e m p e r a t u r e (Fig. 4 of ref. 15) while an increase with a s u b s e q u e n t decrease is observed for sample l_nn-Pb 7 (Fig. 1 of ref. 40). Finally, there may be evidence for an interaction of p h a s e - s l i p c e n t e r s from the behaviour of Sn and In whiskers in a HF radiation field. These 269

specimens show i n v e r s e - a c - j o s e p h s o n - l i k e e f f e c t s also in cases where there is more than one p h a s e - s l i p c e n t e r present. One idea to explain this observation might be based on the assumption of a periodic s t i m u l a t i o n (or weakening) of the superconducting properties of the filament by p r e s e n t p h a s e - s l i p centers (section 7.3). Lip to now the mechanism responsible fo r the influence of an active p h a s e - s l i p c e n t e r on the properties of the f i l a m e n t is not understood. The main p ro b l em is that our e x p e r im e n t s p e r f o r m e d with Sn whiskers show t h a t this influence acts over so large distances that it cannot be explained by e f f e c t s due to quasiparticle diffusion currents. In this situation we recently decided to st ar t detailed investigations o f the interaction of phase-slip centers in Zn whiskers, using m u l t i - p o t e n t i a l - p r o b e contacts. In this material the quasiparticle relaxation times are very long. T h e r e f o r e quasiparticle diffusion induced e f f e c t s are e x p e c t e d to be large in this material. In the following we will briefly sketch the main r e s u l t s obtained up to now.

WWZn7 P1 P2

1 ...L

P3 P4

3

5

A B C

DE

Rg. 9$: S k e t c h o f sample W W Z n T, a zinc whisker with f o u r additional poten t i a l probes made o f tin whiskers. The zinc whisker is held by" the Wood m e t a l c o n t a c t s A and E while the potential probes P I - P 4 are held by the Wood m et a l c o n t a c t s B - E . The num b er s I - 5 denote the whisker parts. The crosses indicate all phase-slip centers which may appear, t w o in part ! and one in each o f the other parts. The phase-slip centers will be referred to as P S C l a , P S C l b , PSC2, PSC3, PSC4, and PSCS. Here, the n u m b e r indicates the whisker part which the phase-slip center is located in. In part 1 there may be t w o phase-slip centers. A s it is not k n o w n which one is the l e f t and which one is the right one, they are characterized by their appearance current which is s m a ll e r f o r PSC l a than f o r PSC I b. The l e n g t h s o f the whisker p a r t s as measured with the optical microscope are: L,=(413+~])gm, L2=(194 +-6)gm, L3=(344 +-6)~zm, L~ = (12S +-6)[zm, and L 5 - - (1SO+~e)Izm. 270

For these experiments we used m u l t i - p o t e n t i a l - p r o b e contacts, where the Zn whisker is held by two Wood metal squeeze c o n t a c t s and where there are four additional potential probes made from tin whiskers. At the measuring t e m p e r a t u r e s both the Wood metal contacts and the tin whisker potential probes are strongly superconducting. The potential probes are s o m e w h a t pressed against the Zn whisker and then spot welded with the whisker by a discharge c u r r e n t across the touching point. Several specimens of this kind were investigated. All r e s u l t s given below are from sample W W Z n 7. The geometry of the sample is sketched in Fig. 95. The l e n g t h s of the different whisker parts are given in the figure caption. The resistance of the whole zinc whisker at room temperature is R208K=32.3f), its residual resistance is Rn= O.18f). Its c r o s s - s e c t i o n a l area is A = 2.23 #m 2, its mean free path is 2 =6.74[~m, and its crystallographic orientation is [11~0]. Furthermore, it is dI~/3/dT, = - r The critical t e m p e r a t u r e as extrapolated from the I~/3(T) straight line is Tc0=O.8187K. The t r a n s i t i o n width of a V-T characteristic at small c u r r e n t s is a b o u t 4.8 mK for lIIA. First we studied the transition behaviour of the whole Zn whisker by measuring some of its V-T transition curves and several of its V-I characteristics (Fig. 96). During these m e a s u r e m e n t s a c u r r e n t IAF is flowing

~

1,

29

3

O-

,o- o

2.3

P ~ 1b2

1.5

1o0

I

Vf,J/S,.zy '---->

),j /

WWZn7 4.9

I

0

50

100

I

150

i

i

IAFIpA

Fig. 96:

V~F-14, characteristics o f sample W WZn 7 (a Zn whisker with 4 additional S n whiskers as superconducting potential probes) at several fixed temperatures f o r increasing current. The main figure only shows the lower part o f the characteristics. The whole characteristic f o r one o f the temperatures is p l o t t e d in the inset. The numbers at this characteristic indicate which one o f the phase-slip centers sketched in Fig. 95 generates a certain voltage jump. 271

through the Zn whisker entering the specimen through contact A and leaving it through c o n t a c t F, and a voltage VAF iS detected between c o n t a c t s A and F (conventional c u r r e n t and voltage as introduced in section 7.6). The VAr-IA~ characteristics of the whole Zn whisker exhibit six voltage steps until the completely normal conducting state is reached. To find out in which part of the specimen the phase-slip centers are situated which generate these voltage steps, the V-IA~ characteristics of all parts of the whisker (that means the YAs, Vsc, Vr VoE, VE~-IA~ characteristics) have been measured. The r e s u l t is that for increasing c u r r e n t the first two voltage steps are generated by the two phase-slip centers in part 1, called PSC l a and P S C l b , while the third, f o u r t h , fifth, and sixth voltage step is generated by PSC2, PSC3, PSC4, and PSCS, respectively. There are also cases in which PSCS appears before PSC 4, so that the fifth voltage step is generated by PSCS and the sixth voltage step by PSC 4. Thus, for increasing c u r r e n t IAF, the p h a s e - s l i p centers in W W Z n 7 (except perhaps those in p a r t l ) are usually switched on from left to right. With this knowledge about the transition behaviour of our sample, we performed measurements concerning the interaction of neighbouring phase-slip centers. At first we investigated how the c u r r e n t at which a phase-slip center appears (the 'appearance current' or 'critical c u r r e n t of the phase-slip center') is influenced by an active phase-slip c e n t e r in its neighbourhood. To see how the first phase-slip center, PSC la, influences the appearance c u r r e n t of the second phase-slip center, PSC l b, we evaluated the distance in current, AI, between the voltage jumps generated by these phase-slip centers in the VA~--IA~ characteristic (Fig. 97). For decreasing measuring t e m p e r a t u r e (increasing AT) the distance hi first increases, then

WWZn7 Tco= 0.8187 K <

0

50"

~o@

~0 l

o~ o

o

300

20-

o

0 0

10-

o 0

i

I

10

IO

I

/1T=Tco-T i

I

I

5

'

~

I

i

I

0

/1TImK Fig. 97: Difference z~l between the currents at which the f i rst and second voltage s t e p appear in the V-I characteristics o f sample W W Z n 7, p l o t t e d as a function o f the temperature. 272

d e c r e a s e s , f a l l i n g b e l o w its ( n o n z e r o ) value o b s e r v e d e x t r e m e l y c l o s e to To0, and f i n a l l y b e c o m e s zero. Thus, f o r d e c r e a s i n g t e m p e r a t u r e t h e c r i t i c a l current of the second phase-slip c e n t e r is f i r s t e n h a n c e d , t h e n the e n h a n c e m e n t d i s a p p e a r s until finally a d e p r e s s i o n of this c u r r e n t is o b s e r v e d . The n o n z e r o v a l u e o f AI e x t r e m e l y c l o s e to To0 i n d i c a t e s t h a t a c e r t a i n d i s t a n c e in c u r r e n t b e t w e e n the f i r s t t w o v o l t a g e s t e p s has its origin in a s l i g h t l y i n c o m p l e t e h o m o g e n e i t y o f t h e whisker. To see, h o w t h e t w o p h a s e - s l i p c e n t e r s in p a r t l, P S C I a and P S C l b , i n f l u e n c e t h e a p p e a r a n c e c u r r e n t o f t h e p h a s e - s l i p c e n t e r in p a r t 2 , PSC 2, a d i f f e r e n t m e t h o d was used: For fixed t e m p e r a t u r e , we f i r s t a p p l i e d a c u r r e n t IA~ and m e a s u r e d t h e VAF--IA~ c h a r a c t e r i s t i c . In this c a s e t h e c u r r e n t f l o w s t h r o u g h t h e w h o l e s a m p l e and PSC 2 a p p e a r s under t h e i n f l u e n c e o f t h e t w o p h a s e - s l i p c e n t e r s in p a r t l. Then we applied a c u r r e n t IB~ and m e a s u r e d t h e VA,-Isr c h a r a c t e r i s t i c . In this c a s e the c u r r e n t e n t e r s t h e Zn w h i s k e r t h r o u g h t h e f i r s t p o t e n t i a l p r o b e so t h a t P S C l a and PSC l b are n o t a c t i v a t e d and PSC 2 a p p e a r s w i t h o u t any i n f l u e n c e o f o t h e r p h a s e - s l i p c e n t e r s . The s a m e m e t h o d was used to s t u d y how P S C 2 i n f l u e n c e s t h e a p p e a r a n c e c u r r e n t o f PSC 3: In a VAF--IB~ c h a r a c t e r i s t i c t h e c u r r e n t e n t e r s t h e Zn w h i s k e r t h r o u g h the f i r s t p o t e n t i a l p r o b e and P S C 3 a p p e a r s u n d e r t h e i n f l u e n c e o f P S C 2 . In a VA~--IcF c h a r a c t e r i s t i c t h e c u r r e n t e n t e r s t h e Zn w h i s k e r t h r o u g h t h e s e c o n d p o t e n t i a l probe, so t h a t P S C 2 is n o t a c t i v a t e d and PSC 3 a p p e a r s w i t h o u t any i n f l u e n c e o f o t h e r p h a s e - s l i p c e n t e r s . W i t h this m e t h o d a l s o t h e i n f l u e n c e o f PSC 3 on PSC 4 was s t u d i e d (by c o m p a r i n g VAr-IcF and VAF--ID~ c h a r a c t e r i s t i c s ) and t h a t one o f PSC 4 on P S C S (by c o m p a r i n g VAF-10~ and VAF--1EFc h a r a c t e r i s t i c s ) . In t h e f o l l o w i n g we will give s o m e q u a n t i t a t i v e r e s u l t s a b o u t the i n f l u e n c e o f PSC l a and PSC l b on t h e a p p e a r a n c e or c r i t i c a l c u r r e n t I~ of P S C 2 and a b o u t t h e i n f l u e n c e of P S C 2 on t h e c r i t i c a l c u r r e n t I psr o f P S C 3 . In t h e s e c a s e s t h e e f f e c t s are c l e a r l y e x p r e s s e d . B e f o r e p r o c e e d i n g , we w o u l d like to r e m a r k t h a t we p l o t t e d n o t o n l y I~/S(T~)=I~,~(To) to e x t r a p o l a t e t h e critical t e m p e r a t u r e T~ o f t h e w h i s k e r (which we a l s o may c a l l Tc0,p a r t l , b e c a u s e t h e f i r s t p h a s e - s l i p c e n t e r s a p p e a r 2/3 These critical in p a r t l o f t h e w h i s k e r ) , b u t a l s o IsF,c(T o) and 12/3(Tr cF,o c u r r e n t s c h a r a c t e r i z e t h e u n i n f l u e n c e d o n s e t o f v o l t a g e in p a r t s 2 and 3, r e s p e c t i v e l y . A l s o in t h e s e c a s e s a s t r a i g h t line b e h a v i o u r is o b t a i n e d and o n e may e x t r a p o l a t e a l s o t h e s e s t r a i g h t lines to vanishing c r i t i c a l c u r r e n t , to g e t To0,p a r t 2 and To, part3, r e s p e c t i v e l y . The r e s u l t is t h a t To0 = To0, p a r t l < To0, p a r t 2 < Tr p a r t 3 with T~0, p a r t 2 - W:0, p a r t l = 1.S m K and T~0, p a r t 3 - Tr p a r t l = 2 . 4 mK. The m e a s u r e m e n t s d e s c r i b e d above s h o w t h a t t h e p r e s e n c e of P S C I a and PSC l b l e a d s t o a c h a n g e of Io.,sc2 and t h a t t h e p r e s e n c e of P S C 2 l e a d s to a c h a n g e of Io.p~cs. C l o s e t o To0 t h e c r i t i c a l c u r r e n t s are e n h a n c e d w h i l e s o m e w h a t f u r t h e r away f r o m To they are d e p r e s s e d . In Fig. 98 s o m e enhancement o f lr

characteristics and Ir Psc3

are s h o w n which d e m o n s t r a t e the w h i l e in Fig. 99 we plotted some

273

K

20

10AIPSC2 r.1 c psc3

__~ IB~ 2

I

o

---~IcF 0

0

I

I

50

100

I

150

I

200

I

Z/HA

Fig. 98: VAr-I characteristics ( w i t h 1 = IAF, IsF, and IcF ) o f s a m p l e W W Z n 7 (increasing current). The z e r o voltage p o i n t is d i f f e r e n t f o r the characteristics. It is given by the h o r i z o n t a l line which a characteristic s t a r t s f r o m . This p l o t d e m o n s t r a t e s that, very c l o s e to the critical t e m p e r a t u r e , T~o, o f the Z n whisker, the p h a s e - s l i p c e n t e r s P S C Ia and I b enhance the critical

c u r r e n t o f P S C 2 , and that P S C 2 e n h a n c e s t h e critical current o f P S C 3 . P$C2 The c h a n g e s in the critical c u r r e n t s are called d l c pscla'lb and dlcpsc 3 , P$C2 ,

respectively. Here, the subscript indicates which critical current is changed while the exponent indicates which phase-slip center (or centers) generates this change.

characteristics demonstrating t h e d e p r e s s i o n o f b o t h c r i t i c a l c u r r e n t s . In Fig. 100 w e s u m m a r i z e t h e r e s u l t s o b t a i n e d f o r d i f f e r e n t t e m p e r a t u r e s . T h e idea o f t h e n o t a t i o n u s e d t o c h a r a c t e r i z e t h e c h a n g e in t h e c r i t i c a l c u r r e n t s in t h e f i g u r e s is t h a t t h e s u b s c r i p t gives i n f o r m a t i o n a b o u t t h e p h a s e - s l i p c e n t e r , t h e c r i t i c a l c u r r e n t o f w h i c h is s h i f t e d , w h i l e t h e e x p o n e n t i n d i c a t e s t h e p h a s e - s l i p c e n t e r ( o r c e n t e r s ) w h i c h g e n e r a t e s t h e c h a n g e in

_AIPscla'lb_c,PSC2

Ir

t h e c r i t i c a l c u r r e n t . M o r e e x p l i c i t l y , it is ---c.Psc2][Pscla'lband AIPSC2 = IoPsc2 c r i t i c a l c u r r e n t s w i t h an e x p o n e n t a r e t h o s e o f t h e c,PSC3 , P~c3-Io Psc3- T h e p h a s e - s l i p c e n t e r in t h e s u b s c r i p t m e a s u r e d in t h e p r e s e n c e o f t h e p h a s e - s l i p c e n t e r s in t h e e x p o n e n t . T h e r e is no e x p o n e n t if t h e p h a s e - s l i p c e n t e r m e n t i o n e d in t h e s u b s c r i p t a p p e a r s as t h e f i r s t o n e in a c h a r a c t e r i s t i c in t h e a b s e n c e o f any o t h e r p h a s e - s l i p c e n t e r in t h e w h i s k e r . In t h e e x p e r i m e n t s r e p o r t e d a b o v e , we deal w i t h p h a s e - s l i p c e n t e r s in their 'natural sequence' T h e o n l y t h i n g we do, is t o f e e d t h e c u r r e n t through different contacts, so that some of the phase-slip centers do not a p p e a r . B e l o w , w e will d i s c u s s e x p e r i m e n t s w h i c h a l s o c o n c e r n c r i t i c a l 274

WWZn7 ~o=0.8187 K AT = Tco- T AT=3.1 mK

::(: 1.1

~

I

9 IcE 0

0

I

I

I

I

50

100

150

200

b"lg. 99: VA~-I characteristics (increasing current).

I

ilkl A

(with I = IAF, lsF, and ICF) o f sample

WWZn 7

The zero voltage point is different f o r the characteristics. It is given by the horizontal line which a characteristic s t a r t s from. This p l o t d e m o n s t r a t e s that, s o m e w h a t f u r t h e r f r o m T~o, the phase-slip centers P S C I a and I b reduce the critical current o f P S C 2 and that P S C 2 reduces the critical current o f PSC 3. The notation used to characterize the critical current changes has been explained in Fig. 98 and in more detail in the text.

c u r r e n t s h i f t s o f p h a s e - s l i p c e n t e r s by n e i g h b o u r i n g p h a s e - s l i p c e n t e r s , b u t in w h i c h t h e c u r r e n t in t h e i n f l u e n c i n g p h a s e - s l i p c e n t e r is h e l d f i x e d a n d in w h i c h w e s t u d y t h e c r i t i c a l c u r r e n t o f t h e i n f l u e n c e d p h a s e - s l i p c e n t e r as a function of the magnitude and the polarity of the fixed current. These experiments were performed with the phase-slip centers PSC2 a n d P S C 3 , s i t u a t e d in p a r t 2 a n d 3 o f s a m p l e W W Z n 7 , r e s p e c t i v e l y . F i r s t we s t u d i e d t h e i n f l u e n c e o f P S C 2 o n t h e c r i t i c a l c u r r e n t o f PSC 3. In Fig. 101 we s h o w VAF-IcD c h a r a c t e r i s t i c s w i t h I~c as a p a r a m e t e r . F o r IBc= O t h e v o l t a g e VAF is z e r o u n t i l PSC 3 a p p e a r s a t a c e r t a i n c r i t i c a l c u r r e n t IcDc,psc3. In t h i s c a s e P S C 2 is i n a c t i v e . F o r I,c= 12OgA, PSC 2 is a c t i v e a l r e a d y b e f o r e t h e a p p e a r a n c e o f P S C 3 . T h e r e f o r e , a c o n s t a n t v o l t a g e VAF>O iS o b s e r v e d u n t i l a v o l t a g e j u m p i n d i c a t e s t h e a p p e a r a n c e o f PSC 3 a t a c r i t i c a l c u r r e n t IcDo,psc3 s h i f t e d t o a s o m e w h a t l a r g e r value. F o r I , c = - 1 2 0 [ i A , a g a i n P S C 2 is a c t i v e , b u t n o w a constant v o l t a g e VAF
275

20-

< l

WWZn7 Tco=0.8187K AT= Tco-T

~V 10

I

,,,~ ~=~ ~,IL=. II

// -10-

9

0__0"

PSCIO,lb

o. ~z~.Psc2

,,I

,,i/ ,~

-20

.-0"0-...

-- -PSC2 9 .Zl..lC, PSC3 .

1

6-1

4 ZlTImK

FIE.100:Change

o f the critical current o f phase-slip centers P $ C 2 and P S C 3 ( o f sample W W Z n T ) by P S C l a , l b and PSC2, respectively, as a function o f the temperature. The notation used has been introduced in Fig. 98 and in more detail in the text. The dashed and d o t t e d lines are est i m a t ed slopes through the measured values.

:>

WWZn7

LL

10-

~A ~'II~ 8C

DE

_ PSC2

PSC3

IBc/HA PSCF~/~~

PSC3~ - ~-1200

V

PSC2 -5

,dT=7-co- T

Tco= 0.8187 K -10

I

0

I

I

I

AT=I.2mK l

tO0

J

I

IcolHA

Rg. lOl: ~r-lco characteristics o f sample W W Z n 7 (increasing current Ico ) with 1Be as a parameter, to demonstrate h o w the activation o f a phase-slip center in p a r t 2 o f the whisker ( P S C 2 ) influences the critical current at which a phase-slip center in p a r t 3 ( P S C 3 ) appears.

276

ATImK

f Tco WWZn7 = 0.8187 K Q AT=Tco-T 100 w"

,,, ,,,, i i

1.2

~II 9

Ot

9

I

-200

0.2

p

I

I~

-I00 '

o, [] : PSC2 inactive e, 9 : PSC2 active o

' 2&o

'

IsclpA Fig. 102: Critical current lcoc,pscj o f phase-slip center P S C 3 in p a r t 3 o f sample W W Z n 7 as a function o f the current IBc through p a r t 2 o f the sample, f o r t w o different f i xed temperatures. We performed these experiments at two different fixed t e m p e r a t u r e s for several values of the fixed c u r r e n t IBc. The r e s u l t s are summarized in Fig. 102. This figure clearly shows that the critical c u r r e n t of PSC3 is enhanced by PSC2 if IBc>O (that means in the 'series case', where IBc and Ic0 are flowing in the same direction), while the critical c u r r e n t of PSC3 is depressed if IBc

3LL

WWZn7

10-

A ~

BC

5-

PSC2 .r- 1 ~

PSC3

ICoI~A

~ F

DE

/

..140

p~.~/

0

0

-1 o s

- 5 PSC3

AT= Tco_T

Tco=O.8187K AT=I.2mK

-10

. . . .

0

j

IO0

,

,

.[ec l HA

~F-Igr characteristics o f sample W W Z n 7 (increasing c u r r e n t Igc ) with Ico as a p a r a m e t e r to d e m o n s t r a t e h o w the activation o f a p h a s e - s l i p center in p a r t 3 o f the w h i s k e r (PSC3) influences the c r i t i c a l c u r r e n t at which a p h a s e - s l i p center in p a r t 2 (PSC2) appears. F i g . 103;

277

series case and depressed in the opposed case. Results for three d i f f e r e n t fixed t e m p e r a t u r e s are summarized in Fig. 104. In this figure a measuring point is shown, fo r which Ico is nonzero, b u t not large enough to activate PSC3 (see the light circle at AT=O.32mK). In this case no sh i f t of the critical c u r r e n t of PSC2 is observed, indicating that the cu r r en t Ico i t s e l f does not influence the critical current of PSC2 as long as it is carried as a s u p e r c u r r e n t and PSC 3 is n o t activated. tip to now we have not done a detailed quantitative i n t e r p r e t a t i o n of the critical c u r r e n t m e a s u r e m e n t s discussed above. Therefore, at the m o m e n t we can only make some s t a t e m e n t s concluded from the qualitative behaviour. The discussion given in section 7.5 in mind, we expect that for the interaction e f f e c t s observed for neighbouring p h a s e - s l i p centers in our Zn whisker diffusing nonequilibrium quasiparticles are of importance. This a s s u m p t i o n would explain the d i f f e r e n t shift of the critical c u r r e n t in the series and opposed case observed in Figs. 101-i04. Besides the asymmetric quasiparticle diffusion effect, there seems to be a sy m m et r i c e f f e c t (perhaps a T ' e f f e c t ) p r e s e n t which leads to a depression o f the critical current, independent of the relative polarity of the c u r r e n t s flowing t h r o u g h the p h a s e - s l i p centers. The presence of this e f f e c t would explain why the e n h a n c e m e n t e f f e c t s of the critical c u r r e n t are smaller than the depression effects. The same explanation may hold for the e x p e r i m e n t s shown in Figs. 97-100, if one a s s u m e s t h a t the symmetric e f f e c t b eco m es s t r o n g e r at low er t e m p e r a t u r e s and finally o v e r c o m p e n s a t e s the a s y m m e t r i c effect. However, it may also be that the explanation of these ex p er i m en t s with its e n h a n c e m e n t of the critical c u r r e n t very close to To0 and the depression s o m e w h a t f u r t h e r away from the critical t e m p e r a t u r e is much more complicated due to the excitation of charge imbalance waves. Then there are not simply spatially decaying quasiparticle c u r r e n t s around the p h a s e - s l i p

WWZn7 <::L TCO--0.8187K AT= Tco-T ~"50

ATImK ~.Ai

1.2

,,mb

0.8

eeeO

0.3

0

o, u, a :PSC3 inactiv~

0t t -200 -100 I

i

~

I

o, -., A:PSC3 active '

0

'

10" 0

'

i 200

IcolpA

Fig. 104: Critical c u r r e n t IBc,,psr o f p h a s e - s l i p c e n t e r P S C 2 in p a r t 2 o f s a m p l e W W Z n 7 as a f u n c t i o n o f the current leo through p a r t 3 o f the sample, f o r three d i f f e r e n t f i x e d t e m p e r a t u r e s . 278

c e n t e r f l o wi n g into the same direction at any time. In this case one may s p e c u l a t e about an a lt e r n a t iv e explanation of the e f f e c t s shown in Fig. 100 by assuming t h a t the influencing p h a s e - s l i p c e n t e r causes quasiparticle c u r r e n t s at the locus of the influenced p h a s e - s l i p c e n t e r which are flowing parallel to the s u p e r c u r r e n t very c lo s e to To0 but antiparallel at s o m e w h a t l o w e r temperature. The c a l c u l a t i o n s o f the time averaged e l e c t r o c h e m i c a l quasiparticle pot en t i al <~> in the presence of collective e x ci t at i o n s for a tin whisker given in Fig. 39 of section 7.6 seem to indicate that s o m e t h i n g like that should be possible. This figure shows t h a t the gradient of <~> changes its sign s o m e w h a t away f r o m the p h a s e - s l i p center, where < g ) has a minimum. Following SBT ( section 5.8.3 ) the quasiparticle current is <j....~n)=(1/ePn)(d