Superconducting microbridges and radiation-stimulated superconductivity

i M I I i i l '

*" o

o o

SUPERCONDUCTING MICROBRIDGES AND

RADIATION - STIMULATED SUPERCONDUCTIVITY

SUPERCONDUCTING MICROBRIDGES AND

RADIATION - STIMULATED SUPERCONDUCTIVITY

PROEFSCHRIFT

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Delft, op

gezag van de rector magnificus prof ir L Huisman, voor een commissie aangewezen door het college van dekanen, te verdedigen op

woensdag 16 maart 1977 te 16 00 uur

door

TEUNIS MARTIEN KLAPWIJK Natuurkundig ingenieur geboren te Berkel & Rodenrijs

Krips Repro B V. Meppel

Dit proefschrift is goedgekeurd door de promoter Prof, dipl.-ing. J.B. Westerdijk

Het onderzoek, waarover dit proefschrift verslag doet, is uitgevoerd in hechte samenwerking met dr. ir. J.E. Mooij

""

CONTENTS

CHAPTER I

GENERAL INTRODUCTION 1

References 6

CHAPTER II

REGIMES IN THE BEHAVIOUR OF SUPERCONDUCTING MICROBRIGDES . . . . 7

II. 1 Introduction 7 11.2 Theoretical models 8

II.2-1 Zero-voltage state 8 II.2-2 Voltage-carrying state 11

II. 2-2.1 Flux flow 12 II.2-2.2 Relaxation oscillation of the order

parameter 14 II.2-2.3 Voltage across phase-slip centers in

one-dimensional strips 15 II.2-2.A Resistively shunted junction model . . . . 17

11.3 Sample preparation and experimental set-up 17

11.4 Long aluminium microstrips 19 11.5 Short aluminium microbridges 22

II.3-1 Preliminary considerations 22

II.5-2 Static properties 25 II.5-3 Voltage-carrying state 28 II.5-4 Comparison of long and short microbridges 32

II.5-5 Hot spot formation 33 II. 6 Short tin microbridges 37 II.7 High-frequency limit of the relaxation oscillation . . . . 41

II. 8 Conclusion and summary 45

References 46

CHAPTER III

RADIATION-STIMULATED SUPERCONDUCTIVITY 51

III. I Introduction 51 1X1.2 The Dayem-Wyatt effect 52

III. 3 Eliashberg's theory of gap enhancement 54 III.4 Critical current of one-dimensional strips 59

III. 5 Experimental details 61 111.6 Results and discussion 63

III.6-1 General picture at 3 GHz 63 III.6-2 Frequency dependence 69 III.6-3 Stability of superconductivity above Tj, 73

III.6-4 Concluding remarks 74 111.7 Confrontation with microbridge results 75

III. 8 Conclusion and summary 76

References 77

Summary 79

CHAPTER I

GENERAL INTRODUCTION

In 1962 Josephson^ predicted that a supercurrent can flow through an oxyde barrier separating two superconductors. In addition he show-ed that this system carries an oscillating supercurrent in the pre-sence of a dc voltage. Finally, he showed that the maximum supercur-rent depends strongly on the applied magnetic field. Later on these predictions were included in a general picture of weakly coupled su-perconductors^. The experimental investigations followed the theore-tical predictions closely and confirmed them extensively. Since that time, besides the oxyde junctions, a variety of weakly coupled super-conductors has been introduced. In particular we would like to mention the superconductor-normal metal-superconductor junction (S-N-S), the superconductor-semimetal-superconductor junction (S-SM-S), the point contact, a constriction in a thin film (microbridge), a thin-film strip locally weakened by a normal-metal overlay (proximity-effect bridge),and finally a thin-film strip locally weakened by ion implantation. Re-cently a new type has been added to these: a weak link with variable coupling strength realized by injection of phonons, photons or quasi-particles. All these structures show aspects which are in line with the predictions of Josephson. On the other hand many aspects are pre-sent which are not described by theory and many deviations are also observed. Therefore, in particular cases an additional analysis is required. Moreover, a more or less satisfactory theoretical analysis up to now exists for oxyde junctions only. This treatment cannot be used for the description of other systems without further argument-ation and extensions.

In this thesis we concentrate the analysis on microbridges. Our initial aim was to use Josephson junctions, as they are called now, for emission and detection of high-frequency electromagnetic radiation. To that end junctions are required which have a small intrinsic capa-citance. It was shown that point contacts satisfy this requirement to a sufficient extent. They showed the Josephson effects in an excellent

way and have proven to be useful for detection. An important disad-vantage of these pressure contacts was their extreme sensitivity to all kinds of external disturbances such as mechanical and thermal shock. An obvious step to remove this disadvantage seemed to be the realization of a point contact in a thin-film structure. A microbridge therefore seemed to be very promising. Unfortunately the behaviour of these microbridges turned out to be much more complicated than in the case of point contacts and sometimes the Josephson effects were completely absent. Although at first discouraging such a situation forms a challenge to further research. The main theme of the follow-ing chapters is therefore an attempt to realize well-defined experi-ments which should be able to reveal the origin of the behaviour of microbridges.

To give an impression of the kind of phenomena we are looking for we will introduce the Josephson effect in very general terms. For a more thorough and complete introduction we refer the reader to Waldram The superconductor can be described as a macroscopic quantum system with one Schrodinger wave function i|j = | >|j | e . The absolute value \ii\

is a measure of the number of superconducting electrons in the metal. The importance of the phase becomes clear (not mentioning other cases) when two superconductors are weakly joined together. Superconductor 1

is described with ijj = | i(j | e '• and superconductor 2 with I(J2 = l^iple 2 If these superconductors are completely joined (ji, will be equal to (j),. If they are fully separated ()> will in general be unequal to ()> . In the case of a weak coupling between the superconductors, for instance by an oxyde barrier, the Josephson effects are present in the follow-ing way. Through the oxyde barrier a supercurrent I can flow which depends exclusively on the phase difference 4 = (fi, ~ 'l>2 according to the equation:

Ig = I^ sin <t, (1,1)

I is called the critical current, because it is the maximum value which the system is able to carry resistanceless. This current depends

on the absolute value of the wave functions, or order parameters, i|) and i() . According to equation (I.l) a supercurrent between +1 and -I can

exist by adaptation of the phase difference. If this maximum value is exceeded the system carries a voltage. Now Josephson found that the time derivative of the phase difference is proportional to the voltage applied:

i i = 2 ^ v (1.2) dt n

If a dc voltage is applied the phase difference grows linearly with time. Substitution in equation (I.l) gives an oscillating supercurrent

Ig = I^ sin ((2eV/h)t + (t>g} (1.3)

<{i(, is an arbritrary constant of integration. The time-averaged super-current is equal to zero. In principle the weak link will be able to emit radiation at a frequency 2eV/h, or about 0.5 GHz/yV. Having found an oscillating supercurrent it seems to be of interest to investigate the response of the weak link to external radiation of frequencies in the GHz-domain. If we assume application of a sum of dc and ac volta-ges: V = Vg + V, cos ait and substitute this voltage in equation (1.2), we find an expression for the supercurrent, which can be reduced to:

f2eV,1 , ,2eVo , ,

^s = Ic I -In k r ^i" {*0 * (-T- ^ ""^J (^-^^

In this equation J^^ stands for the n-th order Bessel function, with n an integer. The sum is taken from n = -"> to +<». From equation (1.4) it follows that in this case the time-averaged supercurrent is zero, except for dc voltages V equal to a rational number times the fre-quency of the applied radiation multiplied by — . Then a dc component of the current can be observed. The amplitude of the n-th component depends on the power of the applied radiation through the n-th order Bessel function. For n = 0 equation (1.4) gives the way in which the critical current depends on the power of the applied radiation; it is determined by J , which behaves in a periodic way.

Experiments on oxyde junctions confirm these briefly mentioned con-cepts. For instance with external radiation the critical current be-haves according to the given predictions and current steps can be in-duced at the specific values of the voltage. Also for other types of weak links, microbridges for instance, these predictions remain quali-tatively valid. One problem arises from the fact that most experiments

have been carried out with current sources. In that case equation (1.4) needs to be modified. The dependence on power deviates from the Bessel function although it remains oscillatory. Another important complication with systems such as microbridges is that in the presen-ce of a voltage a relatively large ohmic quasiparticle current is also present. For such a complicated system a satisfactory theoreti-cal description is still lacking and the experimental side of this problem is the subject of this thesis.

Up to now experiments by others have shown that microbridges can behave as Josephson junctions. To that end the temperature-dependence of the critical current is primarily investigated as well as the res-ponse to microwave radiation. All information about their behaviour is extracted from the current-voltage characteristic. These character-istics are highly complicated and it turns out to be important to in-vestigate them at various temperatures. In order to have systems which can be treated theoretically it is necessary to fabricate bridges with sizes smaller than the characteristic lengths of the superconductor. To that purpose aluminium is excellently suited, which material plays an important part in this thesis.

Aluminium is an interesting material for another reason too. From investigations of Josephson effects in long, narrow thin-film strips the following picture has emerged. At a certain place along the strip over a short distance an oscillating process occurs. Outside this region the superconductor turns out to be out of equilibrium. Out of equilibrium in this context means that the quasiparticle and pair distribution are no longer thermal. Recovery of equilibrium occurs over a-characteristic distance A, the so-called quasiparticle diffu-sion length. In aluminium this length is very large. Therefore the supposed influence should be clearly observable with this material.

As already indicated it is important to investigate systems show-ing the Josephson effects in the presence of microwave radiation. Precisely in this case the most remarkable deviation from the given equations can be observed. The critical current appears to increase

with increasing microwave power instead of decreasing according to a zeroth order Bessel function. The remarkability is underlined when

one realizes that the critical current depends on the absolute value of the wave functions on either side of the bridge. The increase in critical current seems to indicate that microwave radiation is able to enhance superconductivity.' One of our aims became to clarify the origin of this phenomenon. To that end it was necessary to test the recently developed general theory of superconductors in the presence of radiation of Eliashberg and co-workers. We were able to find ex-perimental evidence for the correctness of their description and the results are primarily important for the rapidly developing field of non-equilibrium superconductivity. The explanation of the just mention-ed behaviour of microbridges follows as a consequence of this funda-mental effect. More importantly it was found that the critical tempe-rature of a superconductor can be increased by application of micro-wave radiation.

As a consequence of the before mentioned change in the experimental programme this thesis is divided into two parts. In Chapter II the microbridges as such are discussed. The experimental results on tin

and aluminium microbridges are compared with currently available theo-retical models. Due to the relatively small dimensions of aluminium microbridges we are able to separate some complicating mechnanisms from

the intrinsic behaviour. The nature of the subject requires frequent citing of previously published results and detailed comparison with our results. This turned out to be unavoidable and as a consequence this chapter is somewhat technical. For a more general discussion of time-dependent processes in superconductors and relevant quantities we would like to refer to Langenberg"*'^. Naturally this chapter is not the final word, if such words exist in physics, about microbrid-ges. Our experimental conditions are, however, defined well enough to prepare the way for theoreticians.

In Chapter III our experimental test of the theoretical model of Eliashberg and co-workers is described. After a brief description of the theoretical model our method of testing is given. The results in-dicate that the elements of the model used in the description of the phenomena are valid. We show that the available results on the micro-wave-enhanced supercurrent of microbridges can be explained with this

mechanism. In this particular field much remains to be done, which falls outside the scope of microbridge behaviour. It is however to be expected that from a continuation of research in this direction much can be learned about superconductors out of equilibrium.

Chapter III will also be published in the Journal of Low Tempera-ture Physics 2_5, 385 (1977). Chapter II is submitted to the same journal.

References

1. B.D. Josephson, Phys. Letters _1_, 251 (1962) 2. B.D. Josephson, Adv. Phys. \A_, 419 (1965)

3. J.R. Waldram, Rep. Prog. Phys. 39, 751-821 (1976)

4. D.N. Langenberg, in Festkorperprobleme - Advances in Solid State Physics, Vol. XIV, p. 67, H.J. Queisser, Ed. (F. Vieweg & Sohn, Braunschweig, 1974)

5. D.N. Langenberg, Low Temperature Physios - LT14, Vol. V, M. Krusius and M. Vuorio, Eds. (North-Holland Publ. Co., Amsterdam/Oxford

CHAPTER II

REGIMES IN THE BEHAVIOUR OF SUPERCONDUCTING MICROBRIDGES

JJ. 1 Introduction

Superconducting microbridges have frequently been investigated since the original work of Anderson and Dayem . A wealth of facts is available demonstrating that these weak links have many propert-ies in common with Josephson oxyde tunnel junctions. Since in con-trast to oxyde junctions they have a small intrinsic capacitance they are attractive for applications with electromagnetic radiation. At the same time they are interesting from a theoretical point of view. The voltage carrying superconducting state occurring in these bridges is one of the intriguing problems of non-equilibrium super-conductivity. The purpose of this chapter is to show which regimes can be distinguished in the current-voltage characteristics of mi-crobridges and to clarify which physical process is occurring in which regime. In most interpretations the coherence length plays a

crucial part. Most experiments have been conducted on tin and indium microbridges. With these materials it is difficult to obtain an appreciable temperature range in which the bridge dimensions are small in comparison with the coherence length. In aluminium this is much easier to achieve. We have investigated aluminium microbridges in detail and compare the results with results on tin microbridges. This allows us to weigh the flux flow concept, introduced by Ander-son and Dayem, against its one-dimensional limit: phase slip, as introduced by Notarys and Mercereau^ for proximity-effect bridges. The results lead to the conclusion that it is extremely important in the interpretation to pay attention to complicating mechanisms such as self-heating, variations of critical temperature, inhomogeneous current distributions. The resistively shunted junction model turns out to be poorly applicable to the basic current-voltage character-istic of microbridges.

zero-voltage case and for the voltage-carrying state. Sample pre-paration and experimental set-up are described in Section II.3. In Section II.4 we deal with results on long aluminium strips, which enable us to investigate further the concept of phase-slip centers recently developed by Skocpol et al''. In Section II.5 the results on short aluminium microbridges are presented. Short tin microbridges are treated briefly in Section II.6. The high-frequency limit to phase slip has been discussed recently in various papers. In Section II.7 we deal with this problem and compare the results with possibly relevant relaxation times. In Section II.8 we try to arrive at some definite conclusions.

II. 2 Theoretical models

In our terminology, a microbridge consists of a narrow constrict-ion in a superconducting film. All material constants such as criti-cal temperature, electronic mean free path, penetration depth and coherence length are supposed to be the same in the film and in the constriction. If a current flows through the system the current dens-ity will be maximum in the bridge and there at a certain critical value, the purely superconducting state will collapse leading to a voltage-carrying state. There are differences in behaviour between a long microbridge, hereafter called microstrip and a short micro-bridge, hereafter called microbridge. These differences concern both the zero-voltage state, limited by a critical current, and the vol-tage-carrying state. In both cases the behaviour depends primarily on the dimensions in relation to the coherence length £ and penetrat-ion depth X.

II.2-1 Zero-voltage state

Several calculations have been performed on microbridge-like structures in the zero-voltage state. These calculations aim at determining the critical current, its temperature dependence and the relation between the current and the phase difference across the

bridge. Most calculations are based on the Ginzburg-Landau equations, sometimes simplified, and are therefore valid close to the critical temperature only. The influence of the magnetic field associated with the currents is neglected in all cases.

As is well known, for a long, thin and narrow microstrip the supercurrent is proportional to the gradient of the phase. For such a one-dimensional system the temperature dependence of the critical current** is I = 1 (1 - T/T )3/2 j^^ which I is a proportionality

C C O c C O t- r J

constant, T is the temperature and T the critical temperature. With simplified Ginzburg-Landau equations Aslamazow and Larkin^ calculated the critical current of a narrow and short constriction. They found that the supercurrent depends sinusoidally on the phase difference. For a dirty superconductor the maximum value is equal to I = TTA^/(4eR kT ) , with R the normal state resistance of the

con-c n con-c n

striction and A the energy gap of the superconductor. This result is exactly equal to the result of the microscopic theory for a tunnel junction" close to T . The same result was also found for the com-•^ c

plete GL-equations by Christiansen et al^ and Gregers-Hansen et al^. They considered the case of a microbridge clamped between two bulk superconductors, assuming that the order parameter at the ends of the bridge is equal to the equilibrium value. As will be shown in the following, conventional microbridges have equal thickness of the bridge and the adjacent film. These are called uniform thickness bridges. Apart from these bridges other types have been introduced^ > ^'^ with a more three-dimensional structure: thin bridges between thick

slabs, called variable thickness bridges. One would like to know what consequences this difference in geometry has on the supercon-ducting properties. A disadvantage of the just mentioned calculat-ions is their inability to account for these differences in geome-try.

For these reasons we prefer a comparison of our experimental re-sults with calculations by Mooij et al^^. They calculated the sta-tic properties of two- and three-dimensional constrictions by solv-ing the GL-equations in elliptic cylindrical and oblate spheroidal coordinates respectively. For small three-dimensional constrictions

a sinusoidal current-phase relation is found, where the phase dif-ference between two points far away on either side of the bridge is well-defined. The critical current in the dirty limit turns out to be exactly equal to the expression given by Ambegaokar and Baratoff^, and depends linearly on temperature. Recently Kulik and Omel'yanchuk^' obtained the same result.

Considerable differences are found in the two-dimensional case. The phase difference across the constriction can only be defined by taking two specific points at a finite distance from the bridge or by subtracting certain contributions. Similar difficulties were experienced by Baratoff et al'^ in their calculations on one-dimens-ional weak link structures. The temperature dependence of the crit-ical current in the two-dimensional case is not linear, even for very small constrictions. In a certain range I is roughly proport-ional to (T - T ) ^ ' ^ , a result lying between that for one-dimensproport-ional and three-dimensional systems.

In the neck of the constriction the current density can exceed the critical value of a one-dimensional system. The order parameter is sustained by the adjacent regions where it is less disturbed because the current density is smaller. This enhancement due to the proximity effect is illustrated in Fig. II. 1. The parameter a is a measure of the constriction and is equal to w/(25 sin 8 ) , with w the width of the neck and 6 the angle which limits the hyperboloid. (See inset Fig. II. I). The dashed line of Fig. II.1 gives the value of the cri-tical current density in a one-dimensional strip (in the reduced units used equal to 2/3/3). The full curve gives the critical current density of a three-dimensional constriction at various values of a. From this figure we are able to estimate for a certain microbridge, at a given temperature to what extent the current density of the bridge exceeds that of a strip.

Summarizing the main features, one can state that a careful theo-retical analysis shows that three-dimensional constrictions have a sinusoidal current-phase relation, a critical current exactly equal to the expression from the microscopic theory, and a critical current density largely exceeding the value for a one-dimensional strip. The

I l _ I J

^ a

Fig. II. 1 Normalized critical current as a function of normalized width. The dashed line designates the Value for a one-dimensional etrip. The full line gives the result of a Ginzburg-Landau calculation on the geometry shown in the inset. The width o - w/(2^ sin 6 ) decreases with in-oreaaing 6 .

current-phase relation has found experimental confirmation in the work of Jackel et al^^. The temperature dependence seems to be in agreement with experimental results of Gregers-Hansen et al^ and Song et al^^, although their bridges probably do not have a purely three-dimensional character. Up to now the absolute value of the current density has received little or no attention.

II.2-2 Voltage-carrying state

The voltage a sample carries when the critical current is exceeded may be due to a variety of causes. For instance, the material might

lose its superconducting properties and become completely normal. However, application of microwave radiation reveals that current

10^ c

10'

steps can be induced in the current-voltage characteristic similar to those observed in a Josephson oxyde tunnel junction. Apparently, a voltage-carrying superconducting state exists, in which phase coherence is maintained. In that case, according to the Josephson relation•'•^, d(t>/dt = 2eV/h, the phase difference increases with time and the system is forced into a dynamical state. To describe such a state various models have been introduced, some of a purely phenome-nological character, others using some microscopic aspects.

II.2-2. 1 Flux flow

In the original paper of Anderson and Dayem^ on the observation of ac Josephson steps in the current-voltage characteristics of mi-crobridges, the following explanation is introduced: At the critical current the magnetic field at the edges is large enough to create vortices, which move across the bridge driven by the Lorentz force. When microwaves are able to synchronize this vortex motion the just mentioned steps occur. This model has subsequently been developed in more detail by Likharev^'' and Aslamazow and Larkin^^. In a par-ticular experiment we need criteria to decide whether we are in a flux flow regime or in the regime of one-dimensional phase slip. Therefore a simple description of flux flow phenomena is chosen which for type I superconductors has been adopted among others by Chandrasekhar et al^° and Tholfsen and Meissner^".

The force exerted by the current on a single vortex is the Lorentz force, F = j (j) , in which j is the current density and (ji is the elementary flux quantum. A friction term is also present, which is assumed to be viscous: F = n v, , with V- the mean velocity of the

V L L ' vortex motion and n a viscosity parameter. Finally, a pinning force

F = j ((i is supposed to be present, independent of the current density. We end up with the balance of forces:

n vj^ = (j - jp) *^ n can be written as^^:

in which a is the conductivity in the normal state and 5 is the tem-perature dependent coherence length.

The time dependent voltages within such a model consist of voltage pulses, which by interference with external microwave radiation lead to current steps in the I,V characteristics. The time-averaged vol-tage is V = iji V, with V the number of vortices crossing the bridge in one second. On the other hand the relation v = v /w should hold where v~^ is the time needed to cross the bridge. This relation is based on the assumption that, due to asymmetries and small static magnetic fields, the vortex is formed on one side of the bridge and

leaves the bridge at the opposite side. By combining these relations the following expression for the current-voltage characteristic is found:

V = -iS_ (I - I ) aw2d P

in which I = j wd and I = jwd, with d the thickness of the bridge. A linear relation between current and voltage is found (Fig. II.2) with a temperature dependent differential resistance. This result is in agreement with Likharev's calculation^^, provided that the number of vortices in the bridge is limited to one. For the case of a dirty superconductor the differential resistance is equal to:

dV ^(0-85)^g, \ 1 '^^ w2d T - T '^

c

with 5 the BCS-coherence length and 1 the mean free path.

A few final words about the application of this vortex concept to microbridges should be added. Firstly, the experiments concern type I superconductors, which in general have no vortex state, but an intermediate state with normal domains containing many flux quanta. However, theoretically and experimentally it has been shown^^ that a thin film of type I material has type II behaviour below a certain thickness. For Sn the order of magnitude is about 1.000 A. Naturally, another condition for the existence of flux flow pheno-mena is that the width of the microbridge should be sufficiently

/ /

/ /

' /

/

/

/

c

Fig. II.2 Current-voltage characteristics of various theoretical models; resistively shunted junction model (RSJ model), phase slip model of Rieger, Scalapino and Meraereau^^

(RSM phase slip) and flux flaw model.

section. Finally, it is not obvious that formulas derived for flux flow in infinite films should apply in a situation where the width approximates the size of the vortex itself.

II.2-2.2 Relaxation oscillation of the order parameter

Josephson behaviour is also observed when the microbridge is nar-rower than the coherence length and penetration depth. To explain this apparently dynamic superconducting state Notarys and Mercereau^ introduced the concept of relaxation oscillation of the order para-meter, which is essentially the one-dimensional limit of flux flow. The (naive) physical picture is as follows.

The supercurrent is proportional to the gradient of the phase of the order parameter of the superconductor. If a voltage difference exists between two points, the phase difference will increase in time, leading to an acceleration of the supercurrent. As soon as

the critical current is reached, the superconducting state becomes unstable and the absolute value of the order parameter falls to zero. At that moment the total current is carried as a normal current, which fact allows the order parameter to restore itself. It is assum-ed that this whole process, collapse and restoration is occurring in a small period of time. During this period the phase difference changes with a value of 2ii. After this "phase slip" the electric field brings the Cooper pairs in a new accelerative motion leading to a new collapse etc. The repetition frequency of these phase slip events is given by the Josephson relation v = 2 eV/h in which V, the voltage across the weak section, is fully determined by the nor-mal state properties of the weak link. On the basis of time depend-ent Ginzburg-Landau equations, the picture of what happens in a bridge of finite length has been expanded by Rieger et al^^ and by Likharev and Yakobson^ . The latter authors have shown that it is not necessary to assume a sudden jump at the moment of "phase slip". Nevertheless we will continue using the term phase slip for the dy-namic, periodic process occurring in one-dimensional systems at fin-ite voltage.

The phase slip model as developed by Rieger et al is consistent with attributing to the system a dynamic current phase relation I = 0.5 I (1 + cos 4) instead of the usual Josephson relation

s c

I = Ij, sin ()i. A consequence is the prediction that even for high currents, the time-averaged supercurrent is not zero (Fig. II. 2 ) . Experimentally, in many weak links an asymptote is approached in the I-V characteristic at high currents which is shifted from an ohmic line through the origin. This shift is called "excess current". At present, the reality of the "excess current" is disputed both as regards the experimental evidence and as regards the theoretical predictions. We will come back to this question in the discussion of our experimental results on small aluminium microbridges.

II.2-2.3 Voltage across phase-slip centers in one-dimensional strips In the case of a one-dimensional strip for currents larger than the critical current a voltage is measured smaller than the normal

state voltage. In fact, the voltage increases stepwise in a very re-gular way. Skocpol et al^ have demonstrated that this behaviour cor-responds with the development of spatially localized, essentially identical, voltage units. In their interpretation at the center of each unit the order parameter executes a relaxation oscillation as in the model of Notarys and Mercereau. However, they show that the voltage across each unit, different from the model of Section II.2-2.2 is equal to about (2pA/S) (I - I ) , with p the normal resistivity, A a quasiparticle diffusion length and I = |l the time-averaged supercurrent. This equation states that the current can only be con-verted to a full supercurrent over a distance A, which is determined by inelastic scattering processes. In their model this length is put equal to /1/3 1 1 , in analogy to the one-dimensional diffusion at a superconductor-normal interface treated by Pippard et al^^. In the latter paper 1 stands for the Fermi velocity v„ times T „ , with T „

^ 2 -^ F Q Q

the time for relaxation of branch-imbalance in the superconductor. This quantity is temperature dependent and diverges near T . However, the measurements of Skocpol et al revealed a temperature independent length. Therefore they identified A with a length associated with in-elastic scattering processes already present in the normal state. The numerical values found for tin are in agreement with this hypothesis. The relaxation of the order parameter in the voltage unit and the associated acceleration and deceleration of the pairs was demonstrat-ed by the observation of ac Josephson steps in the current-voltage characteristics with application of microwave radiation.

Although the whole picture is open to further exploration it is clear that this discussion is relevant to the case of short micro-bridges. The same dynamic process leading to quasi-particle

non-equi-librium in the voltage units is expected to occur in short microbrid-ges. It is therefore reasonable to expect the same diffusion pheno-mena, except for the effectively smaller range due to the two- or

three-dimensional geometry. We conducted measurements in long alumi-nium strips, on the one hand to test the just mentioned ideas in a significantly different material, and on the other hand to be able to compare the characteristics of long and short microbridges.

II.2-2.4 Resistively shunted junction model.

In order to describe the behaviour of microbridges use is very of-ten made of a model which was originally developed for point contacts, the resistively shunted junction model. In this model a current from a current source is distributed between a normal resistance R and a so-called Josephson element defined by the equations (f = 2 eV/h and I = Ij, sin (|>. In fact, the quasiparticle current of an oxyde tunnel junction is replaced by an ohmic conductance, which is in agreement with a physical model of point contacts. They can be considered as consisting of an oxyde barrier with metallic shorts. Aslamazov and Larkin^ have shown that the time-averaged voltage in this model is equal to R/f^ - I^, a hyperbolic shape. For high voltages the asymptote is the voltage of the normal state (Fig. II.2). Roughly, the differ-ence between the current-voltage characteristics of an oxyde junction and a point contact can be understood on the basis of this model.

Gregers-Hansen et al^ have used this model in order to describe the properties of microbridges. They arrive at the conclusion that the response to microwave radiation can be described with this resistively shunted junction model in which R is the resistance of the bridge in the normal state. Continuing this line many attempts have been under-taken >2 to explain other aspects of the behaviour of microbridges. Self-inductances and capacitances were added to the basic RSJ circuit, without impressive succes. Recently, some aspects of the superfluid dynamics have been included^^»^^. It is assumed that the Josephson-oscillation is too fast to rebuild the superconducting state within one period leading to a deterioration of the amplitude of the super-current (RSJT model). The merits of these models will be discussed later on in this chapter, in the analysis of our experimental results.

II. 3 Sample preparation and experimental set-up.

The investigations reported on in this chapter concern microbridges of aluminium and tin and microstrips of aluminium, all prepared from

evaporated films. The pressure during evaporation was about I0~° Terr and the evaporation rates ranged from 50 to 100 A/s. For Sn films substrate cooling with liquid nitrogen was employed. To im-prove the adhesion to the substrate a thin layer of tin oxyde was laid down before the actual evaporation started^''. The thickness of the films was determined with a crystal quartz oscillator and usual-ly glass-coated alumina substrates were used. A diamond needle with well-defined angles in a scratching apparatus designed especially

for this purpose provided a convenient way of fabricating the mi-crobridges with the double scratch method^. If necessary it was pos-sible to make additional scratches in the film, for instance to measure the voltage at different places near the bridge. As discuss-ed later on it was necessary to improve the heat transfer from the bridges, which was done by fabricating thin bridges clamped between thicker films. The resulting geometry is shown in Fig. II.3. The scratched strips had a cress section in the shape of a trapezoid.

Fig. II. Z Variable thickness bridge fabricated with the double scratch method.

Electrical measurements were carried out with a standard four terminal method. With a current source the current-voltage character-istic was determined, and displayed either on an oscilloscope icreen, or on an XY recorder. Modulation of the current source with a small

ac signal provided the opportunity to measure the differential re-sistance with the aid of a lock-in amplifier. Microwave radiation of 10 and 35 GHz could be applied to the sample through waveguides.

The cryostat was surrounded by two concentric mumetal cylinders. The sample in direct contact with the helium bath was confined in a superconducting lead can. The temperature was determined with a ca-librated Ge thermometer and was kept constant to within 0.1 mK with an electronic control unit.

II. 4 Long aluminium microstrips.

To test the recently developed model of Skocpol et al (Section II.2-2.3) experiments were conducted on one-dimensional aluminium strips. In the experiments with tin strips it was found that the in-elastic scattering time is the relevant relaxation time determining the effective length. Since this time for aluminium has a value three orders of magnitude larger than for tin, a significantly different diffusion length is to be expected. Moreover, the value for the dif-fusion length in one-dimensional strips can be used to analyse the results of short microbridges. In that way it is possible to test whether non-equilibrium mechanisms influence these results as well.

In Fig. II.4 the current-voltage characteristic of such a strip is shown. Above the critical current, the value of which is governed by the usual (T - T)^'^-temperature dependence, the voltage increas-es in a curiously steplike manner. A careful analysis shows that the stable voltage-carrying parts have slopes which are related. If the differential resistance of the first part with finite slope is call-ed Rjj, then the slopes of the next stable parts are 2Ry, "iK^ etc. The dashed lines in the figure are drawn according to this relation. The behaviour is consistent with the picture of Skocpol et al of essentially identical, localized voltage units, called phase-slip centers. The intersection of the dotted lines with the V = 0-axis gives the time-averaged supercurrent, I , which is equal to about 0.35 I , somewhat lower than in tin strips. The value of R is in

K t i A )

200

100

/ /

OL

10

20

30

V(HV)

Fig. II. 4 Current-voltage characteristic of long aluminium strip.

agreement with the theoretical model as will be shown in the follow-ing.

In Table II.1 the dimensions of the investigated strips are given together with the appropriate values for the resistance ratio RRR, the resistance at the critical temperature R.^ and the differential

c

resistance of our voltage unit R^. The effective length of the vol-tage units Ljj follows from R^ by taking Lj^ = L (R^/R.j> ) . The mean free path for elastic scattering 1 is determined with the resistance ratio and the mean free path at room temperature. If we take

1 = 156 A, as given by Fawcett^^ the values given in the table are found. According to Skocpol et al the effective length L should be equal to 2/1/3 Ivp T2. with Vp the Fermi velocity and T the re-laxation time of the non-equilibrium distribution. In Fig. II.5 L is plotted against 1 on a double logarithmic scale. The dashed line gives the relation Lj^ <>: /l and the data are in reasonable agreement

Table II. 1 Experimental data thickness Sample 1 2 3 4 5 6 7 8 9 10 no of aluminium d, Rr|, normal c L(mm) 3.15 3.31 2.03 2.02 2.90 2.02 2.00 2.92 2.90 2.92 W(um) 6.8 7.2 3.7 4.9 3.8 4.1 4.2 4.4 3.8 4.8 microstri state resistance d(um) 1.7 1.7 0.23 0.23 0.4 0.23 0.23 0.80 0.40 0.20 R^ {Q.) 0.28 0.28 7.35 6.90 7.40 6.10 7.94 2.31 7.84 31.89 ps. (Length L, at T^, \{U) 0.035 0.027 0.88 0.48 0.61 0.64 0.64 0.15 0.50 1.66 RRR = RRR 26.0 22.0 9.7 7.7 7.6 9.7 7.3 11.6 7.9 4.2 width ^ 3 0 0 / ^ l(pm) 0.39 0.34 0.15 0.12 0.12 0.15 0.11 0.18 0.12 0.07

w,

with this relation. The appropriate relaxation time is equal to 1.9 x 10"^ s.

Since R^ was found to be independent of temperature the relaxat-ion time is also independent of temperature. If we adopt the inter-pretation of Skocpol et al the relaxation time should be identical to the inelastic scattering time of quasiparticles in the normal

Fig. II. 5 Mean free path dependence of L . The dashed line repre-sents L^ = 2(1/S IVp Tj/ with T = 1.9 X 10~'' s and V„ = 1.3 X 10^ am/s, assuming I - 156 A. 1000 - 1 — I — I — I I I I 1 1 1 — I — I — I — r — r l(im) 100 _ i — I — I — 1 I I

state. As can be shown^'> the value of this time is of the order of 10~^ s, according to experimental and theoretical results. So these data on aluminium confirm the picture, drawn by Skocpol et al, that the voltage across phase-slip centers is determined by inelas-tic scattering of quasiparinelas-ticles. The effective length is about 200 ym, whereas with tin it is about 10 ym. So this length indeed scales with the square root of the inelastic scattering time.

Finally, we point out some possibly important details of the cha-racteristic of Fig. II.4. The time-averaged supercurrent is about 0.35 I^, lower than the values observed for tin and also lower than the values found for short aluminium microbridges. In addition, it is important to take notice of the discontinous transition from the zero-voltage state to the state of the first phase-slip center. This transition is hysteretic (not shown in the figure) and is also ob-served in strips with lengths of 50-200 ym in which only one phase-slip center can exist. This hysteretic transition differs from the results on short microbridges.

II. 5 Short aluminium microbridges

II.5-1 Preliminary considerations.

The measurements were performed on short aluminium microbridges with lengths, thicknesses and widths below 0.5 ym. Since the BCS co-herence length 5 for this material is equal to 1.6 ym vortex

format-o

ion and consequently flux flow as the origin of the voltage-carrying state can be excluded. For the static case the calculation of Mooij et al can be applied, provided that the dimensions are small enough with respect to the penetration depth to rule out effects of the

self field. With a mean free path of 0.2 ym and a penetration depth at T = 0 of X(0) = 5 X 10~° m this requirement can be satisfied down to about T/T^, = 0.9. So microbridges of aluminium are attractive candidates to test both the static and dynamic theories of micro-bridge behaviour.

the expected Josephson ac steps at finite voltages were absent. At all temperatures it was impossible to induce current steps in the current-voltage characteristics with 35 GHz radiation. This situat-ion changed drastically when the geometry of the microbridges was modified. The original bridges were made in a film of thickness 0.1 ym. With the double scratch method and a scratching depth of about 0.3 ym this leads to bridges with an almost uniform thickness. If the fabrication process was started with a film thickness of 1 ym the same method led to variable thickness bridges with bridges thin-ner than the adjacent films. In these variable thickness bridges the Josephson behaviour was clearly present.

To illustrate the differences Fig. II.6 is given for the case of uniform thickness and Fig. II.7 for the variable thickness case. In

-mA /

' ^ /

Ici

/

w

•

0 0.1 0.2 0.3 0.4 0.5 m V

Fig. II. 6 Current-voltage characteristic of uniform thickness alu-minium microbridge. Bath temperature 1.05 K, T ~ 1.22 K.

Fig. II.6 strong hysteresis is observable, which starts already close to T . No Josephson behaviour is observed. In Fig. II.7 the hystere-sis has disappeared and Josephson steps are easily induced. We

Fig. II. 7 Current-voltage characteristics of variable thickness aluminium microbridge at various levels of microwave power (35 GHz). Bath temperature 1.05 K, T^ = 1.22 K.

attribute these differences in behaviour to the influence of self-heating. In the variable thickness bridge the heat generated in the bridge in the voltage-carrying state is more efficiently transferred to the helium bath than in the uniform thickness bridge. Skocpol et al^^ have shown that large parts of the current-voltage characteris-tics of long and short tin microbridges are dominated by self-heat-ing. In our case the heating effects are even more pronounced. The operating temperature of aluminium is much lower than of tin and consequently the thermal boundary resistance between the metal and helium substrate is higher. In the following it will be shown that even in the variable thickness case at high voltages self-heating remains important. In the experiments described below variable thickness bridges were used as a rule on account of the better heat transfer.

II.5-2 Static properties.

In Fig. II.8 the critical current of a microbridge is plotted as a function of temperature. This curve is linear over an appreciably large range of temperatures, except very close to the critical

temperature. In Fig. 11.9 thi-s part is plotted again,with the

Ic (mA)

t '

2

1

105 1.10 115

— T I K )

Fig. II. 8 Critical current of variable thickness aluminium micro-bridge (AIM 214:2) as a function of temperature.

critical current to the power 2/3. In this temperature region the bridge seems to behave in a striplike manner, with some deviations even closer to the critical temperature. This behaviour strongly vio lates the predictions of theory (Section II.2-1) and is due to the fact that the critical temperature of the bridge T ° is higher than that of the adjacent film, T^, . Experiments with tin microbridges revealed that 1^,° can be importantly lower than T^'. A variety of causes is currently available, as intrinsic fluctuations, thickness effects, strain due to differences in expansion coefficients between metal and substrate, dislocations introduced by the fabrication pro-cedure etc. In our case the reverse behaviour is observed. The cri-tical temperature of the bridge is higher than that of the film,

Fig. II.9 Critical

cur-rent of Fig. II, 8

at temperatures close to

the critical temperature

plotted to the power 2/3.

115 116 117 118

- ^ T ( K )

about 15 mK. Probably, this phenomenon is associated with the well-known thickness dependence of the critical temperature of aluminium films. (See for instance Strongin et al^^ and Naugle et al^''). In our bridges the films are about a factor of 5 thicker than the bridges themselves. Therefore, such effects are not unlikely,

al-though it is surprising that it is still active in a small volume clamped between two bulk normal regions.

By simultaneous measurement of the critical temperature of the films, and of the bridge system as a whole we were able to establish that the linear temperature dependence of Fig. II.8 starts as soon as the films become superconducting. However, this would imply at the same time that for temperatures between 1.162 K and 1.170 K the measured critical current is not a true supercurrent. A voltage would be developed across the normal films. With the aid of an ac-method it was established that the resistance was less than 10~ ft, which is rather low in comparison with the resistance to be

expect-ed for the geometry usexpect-ed. Therefore we are lexpect-ed to suppose that expect- edge-effects ° play an additional part. At the edges and along the scratch the evaporated film thickness decreases gradually. These thin edges, will also have a higher critical temperature and will act as

super-conducting channels carrying the current up to the bridge with zero resistance. A temperature dependence of the critical current accord-ing to a 3/2-power law seems therefore not unreasonable.

The theory of Section II.2-1 concerns three-dimensional constrict-ions, which are superconducting as a whole with uniform T^. Experi-mentally, is is not possible to satisfy this requirement. Neverthe-less in the region in which both bridge and films are superconduct-ing a linear temperature dependence is observed. Apparently the re-latively small difference in critical temperatures is not very im-portant. One should note that the temperature regime in which a li-near dependence is found is large in comparison with the difference in Tj,. From Table II.2 it is clear that for all bridges, in spite

Table II. 2

Experimental data of short aluminium microbridges Film thickness d, bridge length 1, bridge

the bridge triangular

is approximately , which leads to

iw. The shape

width w of the a surface of the cross is the resistance of the bridge at 1.2 K.

al values for the constants appearing in critical current I^ = (B/R)(T^ - T) and j pectively. Sample no AIM 210:3 213:1 213:2 214:2 222: 1

T^ is the critical temperature

d(ym) l(ym) 1.3 0.5 1.5 0.4 1.4 0.3 1.4 0.4 1.7 0.6 w(ym) R(n) 0.4 0.23 0.6 0.05 0.5 0.11 0.5 0.05 0.5 0.26 B and j : . The thickness of cross section is section of Iw2. R are experiment-the expressions for

c ~ Jcio of the BxlO^ 1.09 1.15 1.12 1. 14 1.16 (1 - T/T^) bridge.

JclO->°"^^

of a difference in resistance of a factor 5, the coefficient

B = Ij,R/(T - T) is almost constant, as predicted by theory. The ab-solute value is equal to about 1.1 x 10~ eV/K whereas theory pre-dicts 0.6 x 10"3 eV/K.

critic-al current density of the bridge may be much larger than that of a one-dimensional strip. The bridge of Fig. II.8 behaves according to the relation j (, = j^.^^ (1 - T/T^,) with j ^ ^ ^ = 8.7 x 10^° A/m^, where-as a strip is governed by the relation j (, = j ^.^^ (1 - T/T^)^'^ with Jco = l-S X lO^l A/m2. At t = T/T^, = 0.95 we find j (, (bridge)/j j, (strip) = 2.4. With other bridges, dependent on the dimensions, even a fac-tor of ten can be found. Generally this enhancement effect is larger in the shorter bridges. The a values (Fig. II. 1) corresponding with the observed enhancement are not unreasonable, considering the co-herence length and the bridge sizes.

Summarizing it can be stated that microbridges of aluminium smal-ler than the coherence length and penetration depth have critical currents linearly dependent on temperature. Moreover the theoretical-ly predicted relation between critical current and normal state re-sistance appears to be present. These results confirm the results found in tin and indium microbridges by Gregers-Hansen et al^ and Song and Rochlin^^. In addition it is found that the critical cur-rent density of the microbridges may exceed that of a one-dimens-ional strip by an order of magnitude. An important complication in the interpretation of the results is the difference in critical temperature between bridge and films.

II.5-3 Voltage-carrying state.

After exceeding the critical current the voltage across the weak link develops in the manner shown in Fig. 11.10 for different tempe-ratures. A large variety of phenomena is observed, which will first be mentioned briefly and will later be discussed in more detail. At not too low temperatures one has a continuous transition from the zero-voltage state to a region where the I-V characteristic is given by a straight line. This straight line does not coincide with the normal ohmic characteristic. The extrapolation of this line inter-sects with the I-axis at a point between 0.4 I^ and 0.8 Ij,, depen-dent on temperature and on the microbridge involved. At lower tem-peratures structure becomes visible on this line. Part of the struc-ture shifts with temperastruc-ture, part remains at a fixed voltage. The

Fig. 11.10 Current-voltage characteristics of an aluminium micro-bridge at various temperatures. At fixed voltages self-induced steps are visible. Also a temperature dependent structure can be observed, which turns out to be the sub-harmonic energy gap structure. The noisy structure visi-ble at T = 1.140 K at higher voltages is due to the

form-ation of a normal hot spot in the films. Similar struct-ure can be observed at higher temperatstruct-ures.

former turns out to be the well-known subharmonic energy gap struct-ure. The fixed structure is due to the so-called self-induced steps, which are caused by the electromagnetic interaction of the bridge with the environment. The linear part of the characteristic ends in a hysteretic, discontinuous jump to a higher voltage. This jump re-flects the formation of a normal domain. Due to the dissipation the critical temperature is exceeded. In the following these results are analyzed in connection with the theories mentioned in Section II.2. The hot spot aspects will be discussed separately. In our

experi-mental situation flux flow can evidently be excluded, since the di-mensions of our bridges are too small. This is confirmed by the agree ment between the theoretical and experimental values of I^. in the static case. The I-V curve up to the discontinuous jump shows close resemblance with the characteristic of RSM phase slip (Fig. II.2). Over an extended range of currents we observe a straight line which is shifted with respect to the origin, unlike the predictions of the RSJ model. This can be interpreted as evidence for the existence of a true excess current in small variable thickness microbridges. Jackel et al^° have argued that all observations of excess current can be explained on the basis of the RSJ model in combination with a non-sinusoidal (static) current-phase relation. With a purely si-nusoidal current-phase relation no excess current should be expect-ed. Both theory (Section II. 2-1) and experiment (Jackel et al^**) indicate that the static current-phase relation of small variable thickness microbridges is sinusoidal. Our experiments are therefore in disagreement with the interpretation suggested by Jackel et al^^.

As indicated earlier, structure appears on the straight line at lower temperatures. The temperature independent structure has the appearance of current steps at constant voltage and arises because the oscillations of the supercurrent generate an electromagnetic field, which interacts with the metallic surroundings. In Table II.3 for the successive steps the voltages, frequencies and corresponding free space wavelengths are given. In addition the ratio is given of the diameter of the accidental cylindrical cavity (29.9 mm) to the wave length. The sample was mounted half-way the cylinder, perpendi-cular to the axis and in fact divided the cavity into two almost equal parts. As a consequence of the ill-defined geometry attempts to identify the modes had no succes. The given ratio's of D/X are in the expected range. Moreover these steps were absent when the same sample was used without the surrounding can while the other parts of the characteristic remained unchanged. On account of these data the temperature independent structure can be identified as self-induced steps, which were observed earlier for microbridges by Levinsen^^. At the same time they clearly demonstrate the existence

Table II. 3

Data of self-induced steps, corresponding length. D/X is frequency and the ratio of to the wavelength. Step no 1 2 3 4 5 6 7 8 V(yV) 17.0 20.6 25.5 29.0 30.4 32.9 34.9 37.8 observed X is the at the voltage V. v corresponding the diameter v(GHz) 8.2 10.0 12.3 14.0 14.7 15.9 16.9 18.3 free is the space waver of the cylindrical cavity

X (mm) 36.5 30.1 24.3 21.4 20.4 18.8 17.8 16.4

1

of an oscillating electromagnetic field generated by the microbrid-ge. The voltages where the temperature-dependent structure is observ-ed are plottobserv-ed to the second power against temperature in Fig. 11.11.

Fig. 11.11

Voltages of the tem- 5

V llfl" u perature dependent ' * , structure of Fig. | 4 II. 10 as a function of temperature. 3 105 i-s _J i s _ 110 1 15 • T ( K ) 1 2 0

They appear to follow a straight line, which implies a dependence of the voltage on the square root of temperature. When we call the larger voltage Vg„ and the smaller one Vg , their ratio Vg./Vg is equal to 0.67 for the whole range of temperatures. This type of be-haviour could very well be associated with the well-known " subhar-monic energy gap structure. In that case one expects structure at voltages equal to 2A/ne with n integer. In view of the given ratio it is to be expected that Vg is equal to 2A(T)/3e and Vg is equal to 2A(T)/2e. The corresponding value of A(0) following from

A(T) = 1.74 A(0) /I - T/T^ is equal to 0.169 x lO"^ eV. This value agrees quite well with the values known from literature. With an effective critical temperature of 1. 174 K the ratio A(0)/kT^ is equal to 1.67 close to the BCS value. On account of the temperature depen-dence and the absolute value this structure can be identified as sub-harmonic energy gap structure. The gap itself at 2A(T)/e is not ob-served in these curves. Structure for higher values of n is present but is less clearly discernible.

II.5-4 Comparison of long and short microbridges.

The differential resistance in the characteristic of long micro-strips, appeared to be determined by a quasiparticle diffusion length in agreement with the model of Skocpol et al. A temperature independ-ent length of about 100 ym was found. As is evidindepend-ent from Fig. 11.10 the differential resistance of the microbridges is, apart from the already mentioned structure, also independent of temperature and voltage. In Table II.4 the ratio of the differential resistance to the total resistance in the normal state is given. In analogy to the procedure followed in Section II.4 an effective length can be defined for the intrinsic microbridge behaviour. To that end we need an expression for the resistance in a variable thickness bridge. If we suppose that the microbridge can be approximated by a constrict-ion described in oblate spheroidal coordinates, the resistance is given by

„/ v o 2y sin 9 R(y) 2 • , -^ e,

m'- ^ ''"'^ ^—^-^)

Table II. 4 D a t a concerning t h e ratio of the t a n c e . Lj^ is the r e g i o n . Sample n o . AIM 210:3 213:1 213:2 222:1 the di

voltage across aluminium Eferential resistance associated effective 1 dV R dl 0.79 0.76 0.81 0.78 to length mi the of u -A 1 dV . =robridges. ^ ^ ^s n o r m a l state r e s i s -the dissipative L (ym) n 1.4 1.7 1.9 1.6

with w the width of the bridge, y the distance from the center in the direction of the current and 6 the angle which limits the hy-perboloid (See inset Fig. II. 1). R(<») gives the resistance of the total system (y ->•'»). The effective length L = 2y defined in this way gives the length necessary to produce the observed voltage by

flow of quasiparticles. As shown in Table II.4 this length is about 2 ym which is considerably longer than the bridge length. (We used 0 = Tr/3. Other reasonable values of 6 do not change the picture

g g qualitatively). A translation of the one-dimensional diffusion

pro-blem to three-dimensional diffusion shows that this effective length is in reasonable agreement with A = 100 ym.

This analysis shows that the effective length of the non-equili-brium region in the microbridge is of the same order of magnitude as the coherence length. Therefore a separate treatment of the time dependent phase slip and the time independent quasiparticle diffus-ion as carried out by Skocpol et al seems to be questdiffus-ionable in this case.

II.5-5 Hot spot formation.

The end of the linear portion of the characteristics of Fig. 11.10 is marked by a discontinuous, hysteretic jump. The sample enters, as we suppose, a new state due to the formation of a normal hot spot in the films. In order to achieve such a situation the dissipation

0.05mA

25(iV

F%g. 11.12 Current-voltage characteristics of an aluminium microbridge at various levels of microwave power (35 GHz). Note the microwave-induced steps at voltages higher than the discontinuous jump.

should be high enough to reach a temperature increase of about 10 mK, which is not unreasonable in view of the thermal conductivity and heat transfer coefficients of aluminium. However, a problem in this interpretation arises from the observation of microwave-induced steps for voltages higher than the discontinuity, already observable in Fig. II.7 and also given in Fig. 11.12. This apparent contradict-ion disappears when one assumes that a hot spot is formed only in the films in the neighborhood of the bridge, although the bridge it-self remains superconducting. The following experiment, together with some general considerations, led to this point of view.

At a distance of 10 and 100 ym of the bridge to be investigated scratches are made as drawn in the inset of Fig. 11.13. As a funct-ion of current through contacts 1 and 8 the voltage is measured

Fig. 11.13 Current-voltage characteristic of the system shown in the inset. The voltages are measured at various spots on the film. When a normal hot spot reaches the scratch-es the measured voltagscratch-es start to differ.

between contacts 3 and 6, 4 and 5 and 2 and 7 respectively (see in-set Fig. 11.13). If at high voltages a hot spot is formed starting at the bridge, V , V and V. .. will have the same values as long as the radius of the hot spot is smaller than 10 ym. As soon as the size becomes larger V will outgrow V while V, and V remain the same. The latter two will also start to differ when

the radius exceeds 100 ym. This expected behaviour is precisely what is observed in the experiment (Fig. 11.13), with the

understand-ing that V, . and V, ^ diverge immediately at the discontinuity. Ap-parently the hot spot, formed at the discontinuity, has a minimum radius of 10 ym.

As already mentioned phase coherence remains, expressing itself in the microwave-induced current steps. Even periodicity can be ob-served (Fig. II. 12). Wyatt et al**' suggested, that the Josephson behaviour in a microbridge is caused by the existance of a thermally grown superconductor-normal-superconductor junction. However, this explanation is highly improbable for our observations, because in SNS junctions the bariers have a thickness of at most 1 ym . If these thicknesses are exceeded^, only at very low temperatures a supercurrent can be observed. Attempts to demonstrate phase coherence across such a normal region in a direct way were unsuccesful. There-fore the explanation has to be found in another direction.

In Section II.5-2 it is mentioned that the critical temperature of an aluminium microbridge is higher than of the films themselves. This difference may be of the order of 15 mK. For aluminium the thermal healing length in the film n = /Xd/o is about 300 ym

(X = 102 W/mK, a = 10^ W/m^K and d = 1 y m ) . This means^S that in the dissipative state (also if dissipation is localized to within a ra-dius of for instance 2 ym) the temperature in the bridge can hardly differ from the temperature in the adjacent film up to a radius of about 30 ym. The curves of Fig. 11.10 have been measured a few tens of mK's below T . Therefore it is imaginable that due to the dissip-ation normal domains are formed in the films, although the bridge itself remains superconducting as drawn roughly in Fig. 11.14. In view of the large thermal healing length, the radius of these domains

Fig. 11.14 Hypothetical form of the normal domains around a microbridge in the dissi-pative state at higher voltages. The dotted area is superconductive. This form is due to the lower cri-tical temperature of the films com-pared with that of the bridge.

Al-though normal domains are present phase coherence can be maintained.

should be about several tens of microns, in agreement with the ex-perimental results of Fig. 11.13. In drawing Fig. 11.14 we took account of the fact that the critical temperature of the edges is also higher than in the films, because they are thinner. The picture is consistent with the size of the hot spot observed and with the fact that phase coherence is maintained. It follows logically from the differences in critical temperature and the large thermal heal-ing length. In a situation with these peculiar hot spots the Joseph-son behaviour is of course not easily described. Both the supercur-rent and the possible quasiparticle cursupercur-rents are governed by com-plicated boundary conditions which should be taken into account in an accurate interpretation of the post-hot-spot behaviour.

JJ.6 Short tin microbridges

In all investigated cases the results on tin microbridges differ-ed drastically from those on aluminium bridges, measurdiffer-ed in the same range of reduced temperatures. In our opinion, these differences find their origin in the different values of coherence length and penetration depth in comparison with the sizes. For instance in tin, with a mean free path of 0.2 ym, 5 = 0.23 ym, X(o) = 0.05 ym and a reduced temperature t = T/T^, = 0.96 we find C(T) = 1 ym and

X(T) = 0.16 ym. With microbridge sizes of the order of 0.5 ym these figures indicate that already close to T^ an inhomogeneous current density might develop due to self-field effects. This situation be-comes even worse if instead of the calculated BCS coherence length £ the measured value of Hauser'*'* is taken: 5 = 0 . 9 ym. This leads

^o o to a penetration depth and coherence length of 0.18 ym and 0.3 ym

respectively. Even a variation of the order parameter perpendicular to the direction of the current, in other words the formation of a vortex, cannot be completely excluded.

In Fig. 11.15 a set of I-V curves is given at different bath temperatures. The sizes of the bridge are rather large, 0.6 ym long and 0.7 ym wide. The critical current follows mainly a (T - T ) V 2

Fig. 11.15 Current-voltage characteristics of a tin variable

thick-ness microbridge at various temperatures. Note the

init-ial straight line at V - 0 at the lower temperatures.

with results of Song and Rochlin^^ and Jahn and Kao"*^ for relative-ly large bridges. Very close to the critical temperature (T^, = 3.975 K, T = 3.850 K) the characteristics have many aspects in common with those found for aluminium. At lower temperatures the slope of the characteristics near V = 0 increases, accompanied by a subse-quent bending at higher voltages. Ultimately a characteristic re-sults, which above the critical current starts with a steep line ending in a discontinuous, hysteretic jump to another stable state. In Fig. 11.15 the subharmonic energy gap structure can also be ob-served.

These characteristics are influenced by self-heating as can be illustrated by monitoring the changes of the characteristic by pass-ing through the X-point of liquid helium (2.17 K ) . The heat trans-fer from a solid body to liquid helium is higher just below T, than just above. In Fig. II. 16 it is shown that the characteristics at 2.2 and 2.3 K coincide at higher voltages, while at 2.1 K the vol-tage at the same current is considerably smaller, corresponding with a smaller hot spot. Due to the strong hysteresis we were unable to settle a lower limit to the voltage where this difference could still be observed.

Apparently microbridges of aluminium and tin have many aspects in common, but there are also important differences. Note that the characteristics of Fig. 11.10 and 11.15 are measured at comparable reduced temperatures. One of the intriguing differences is the over-all shape of the current-voltage characteristic. Several authors^^' 29>'*S have tried to explain the pattern of tin and indium micro-bridges (as shown in Fig. II. 15) by incorporating the dynamics of the superfluid into the resistively shunted junction model. A total-ly different approach has been followed by Guthman et al ', who try to explain the observations with a flux flow concept. We agree with the latter view, although our interpretation of the observed curves differs from their treatment.

If flux flow occurs, which in view of the sizes might be possible, flux entrance determines the critical current and the voltage just above I^ will be determined by moving flux tubes. In Fig. 11.17 the

Fig. II. 16 Set of current-voltage characteristics of a tin microbridge showing the influence of self-heating. At 2.2 K and 2.3 K the curves coincide, whereas at 2.1 K the voltage at the same current is lower. This is due to the fact that below the \-paint of liquid helium (2.17 K) a better heat trans-fer from bridge to bath exists.

differential conductance of the initial steep part of Fig. 11.15 is shown as a function of temperature. This conductance seems to vary li-nearly with reduced temperature, which is also predicted by the flux flow theory (Section II.2-2.1). The proportionality constant is about 530 n~^, which according to theory should be equal to

— • w^d

^ TT(0.85)2^0

The value of a/1 for tin has been found to be 10^^ Q~^ m~2 from the

anomalous skin effect. With w = 0.7 ym and wd = 2.5 x 10"^^ m2 the

value of 530 Cl~'^ corresponds with S Q = 0.15 ym, in fair agreement

with theoretical values. In this respect the flux flow concept leads to reasonable results. Additional evidence follows from a comparison with the results on aluminium microbridges.