A quantitative analysis of the enhancement of superconductivity

ENHANCEMENT OF SUPERCONDUCTIVITY

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P. VAN DEN HAMER

TR diss

1539

ENHANCEMENT OF SUPERCONDUCTIVITY

proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft

op gezag van de rector magnificus,

prof.dr. J.M. Dirken,

in het openbaar te verdedigen

ten overstaan van een commissie

aangewezen door het College van Dekanen

op donderdag 7 mei 1987 te 16.00 uur

door

PETER VAN DEN HAMER

Natuurkundig ingenieur

geboren te Utrecht

TR diss ^

1539

prof.dr.ir. J.E. Mooij en prof.dr.ir. T.M. Klapwijk

Financiële steun voor het in dit

proefschrift beschreven onderzoek is verleend door de

Stichting voor Fundamenteel Onderzoek der Materie (FOM).

Table of Contents Preface 1 Introduction 2 I Enhancement of Superconductivity 4 1. Introduction 4 2. Gap Equations 5 3. Photons, Phonons and Quasiparticles 6

4. Evaluating the Distribution Function 10

5. The G-Function 12 6. Heating Effects 14 7. Diamagnetic Effect 16 8. Frequency Dependence of Nonequilibrium Effects 17

9. Gap Enhancement - A Quantitative Analysis 19 10. Critical Current of a Superconductor 24

References 31

II Microwave-Enhanced Critical Current in Superconducting Aluminum Strips 33

1 . Introduction ' 34 2. Sample Fabrication and Experimental Setup 34

3. Theory of I -Enhancement 36 4. Coupling of the Microwaves 39 5. Results and Discussion 41

6. Conclusions 44 Acknowledgements 45 References 45

III Self-Aligning Resist Techniques for Shadow Evaporation of a Supercon­

ducting Three-Terminal Device 47

1. Introduction 48 2. Shadow Evaporation 49 3. Three-Terminal Device Requirements 51

4. Fabrication 51 5. Conclusions 56

IV Enhancement of Superconductivity by Quasiparticle Injection.

Part I. Model Analysis 59

1. Introduction 60 2. Model 61 3. Gap Enhancement 66

4. Critical Current for (a/7) =0 71 5. Critical Current for enhanced A 77

6. Discussion 79 7. Conclusions 82

Acknowledgements 82 References 83

V Enhancement of Superconductivity by Quasiparticle Injection.

Part 2 . Critical Current Experiments 85

1. Introduction 86 2. Sample Design 90 3. Measuring I 94 c 4. Sample C h a r a c t e r i z a t i o n 96 5. I Enhancement Data 100 c 6. B i a s i n g a t I n t e r m e d i a t e V o l t a g e s 105 7. D i s c u s s i o n 106 8. C o n c l u s i o n s 110 Acknowledgements 111 References 111 Summary 114 Samenvatting 116 Curriculum Vitae 118

Preface

The research project described in this thesis started in January of 1982. The work was part of the research program of the 'Vaste Stof - Supergeleiding' (Solid State Physics) group at the Applied Physics Department of the Delft University of Technology. The group's senior scientists during this period were J.E. Mooij and T.M. Klapwijk.

The primary goal of work described here was to perform quantitative experi­ ments on the "enhancement of superconductivity" and to compare the data to the available theoretical models. The reason for undertaking this project was that on the one hand there was a notable lack of quantitive studies on the phenome­ non, while on the other hand, the results of one of the more recent and more extensive studies had led the researchers to conclude that there was a major discrepancy between theory and experiment.

A significant part of the work was performed in collaboration with the following students of the VS-SG group: P.B.L. Meyer (sample design and resist technology), E.A. Montie (enhancement measurements and modeling), H.M. Appelboom (bias electronics) and A.I. van Heek (artificial barriers).

We hope that this thesis will contribute to the insight that, although the experimental study of the enhancement of superconductivity is, at best, a intricate field of study, the phenomenon's basic principles appear to be well understood.

The work was financed by the 'Stichting voor Fundamenteel Onderzoek der Materie' (FOM) which in turn is supported by the 'Nederlandse Organisatie voor Zuiver Onderzoek der Materie' (ZWO).

INTRODUCTION

The experimental study of superconductivity is a somewhat paradoxical field. On the one hand, a number of the basic effects are totally absolute and precise. The central phenomenon of zero resistivity itself, for instance, is for all practical purposes absolute. Another familiar example, the so called AC Josephson effect, provides such a tight link between the quantities voltage and time that it is used for calibration purposes by many of the world's national bureaux of standards.

The murky side of superconductivity, however, is that it is such a frail and low energetic phenomenon that it can be influenced by almost anything. Thus, if one is doing an experiment on a superconducting sample in an environment with a stray magnetic field, some light and interference from a local radio station, these conditions can have a significant effect on the experiment. Fortunately such external influences can generally be readily eliminated. From an experimental point of view, the really serious experimental problems are the numerous - often unexpected - effects that can arise within the sample itself. The reader will encounter a few prime examples in this thesis.

This thesis is about a positive aspect of this murky side of superconducti­ vity: it examines a class of external perturbations which actually increases the degree of superconductivity in a sample as long as the perturbation is applied. These nonequilibrium phenomena are known as the "enhancement" or "stimulation" of superconductivity. There are three ways to evoke enhancement: a superconductor can be subjected to microwave irradiation, to high frequency accoustic waves or the superconductor can be injected with electrons with a slightly elevated energy.

Although the theoretical basis for enhancement by microwave absorption, phonon attenuation and quasiparticle injection has now become more or less estab­ lished and many experiments have been reported in which the various enhance­ ment phenomena were observed, few experimentalists have actually been able to perform experiments with which the theory could be quantitatively verified. The main reason for this is that it is very difficult to set up such experi­ ments in a sufficiently well defined way that one knows exactly what is going on. Typical problems might be that one can not directly determine the magni­ tude of the applied perturbation or that the measurements are stongly affected

by details of the sample geometry. Which just goes to prove my point that experimental work on superconductivity certainly has its murky side.

A few brief comments on the various chapters may help to guide the reader (the words in bold type can be found at the top of the pages of that chapter):

I. "Enhancement of Superconductivity" is intended as an introduction. It presents the basic theory and models used throughout this thesis. The three differenct sources of enhancement (photons, phonons and quasiparticle injection) are introduced and quantitatively compared.

II. "Microwave-Enhanced Critical Current in Superconducting Aluminum Strips" An experimental paper published in the Journal of Low Temperature Physics (54 607, 1984) describing an exploratory attempt to quantita­ tively study enhancement using microwave irradiation.

III. "Self-Aligning Resist Techniques for Shadow Evaporation of a

Superconducting Three-Terminal Device" describes the techniques devel­ oped in order to fabricate the samples used in the final chapter. The paper was presented orally at the Microcircult Engineering Conference (Rotterdam, 1985) and was published in Microelectronic Engineering 3_ 427 (1985). Due to the nature of the target audience, the employed shadow evaporation techniques are highlighted.

IV. "Enhancement of Superconductivity by Quasiparticle Injection.

I. Model Analysis.". Part one of a pair of papers submitted to the J. Low Temp. Phys. This pair of papers constitutes the central part of this thesis. In addition to presenting a model with which the experi­ ments in Chapter V can be compared, it contains a detailed analysis of the quasiparticle injection-induced enhancement of the gap and of the critical current of superconducting strips.

V. "Enhancement of Superconductivity by Quasiparticle Injection.

II. Critical Current Experiments". Part two of a pair of papers submitted to the J. Low Temp. Phys. A new sample design is introduced. Experimental enhancement data are presented which are compared to the models presented in Chapter IV.

ENHANCEMENT OF SUPERCONDUCTIVITY

1. Introduction

In 1911 Kamerligh Onnes discovered that mercury can display an infinite electrical conductivity at temperatures below 4.2 Kelvin. It lasted 46 years until a detailed theory for the phenomenon superconductivity was available. In a landmark paper in 1957 [l], Bardeen, Cooper and Schrieffer (BCS) proposed an expression for a new ground state for the conduction electrons which, for suitable materials, would have a lower free energy than the traditional Fermi sea model. In this BCS ground state, all the conduction electrons condense into bound Cooper pairs consisting of two electrons with opposite momentum and spin. Thus the occupancy of an electron state is correlated to the occupancy of its corresponding state with opposite momentum and spin. Even though the BCS model contains certain major simplifications regarding the interaction between the conduction electrons and the crystal lattice, the BCS theory has been found to provide an adequate quantitative description for most supercon­ ducting materials [2].

At finite temperatures, thermal excitations with respect to the BCS ground state are generated. In their 1957 paper, BCS calculated that the energy E of an elementary excitation, in which an electron state at energy £ (relative to the Fermi energy) is occupied and the corresponding state with opposite momentum and spin is empty, can be expressed as

E = [ C2 + A2 j (1)

in which the parameter A is known as the energy gap because there are no states with excitation energies between 0 and A. These elementary excitations are commonly called quasiparticles and have Fermion properties. Thus, in thermal equilibrium, the occupation probability of a state with excitation energy E is given by the Fermi-Dirac distribution

f

t h

( E )

= I

1 + e

*P<iTT>] ' •

(2)

Note that the minimal energy required to generate extra excitations is 2A rather than A because Cooper pairs dissociate into two quasiparticles each.

2. Gap Equations

BCS showed that if the temperature is swept from T=0 to the superconductor's critical temperature T , A(T) will decrease and ultimately vanish due to the increased number of quasiparticles which limit the number of states to which Cooper pairs can scatter. This coupling between the energy gap and the quasi-particle population is described by the BCS gap equation if one assumes that N V is smaller than unity:

O BCS N V 0 BCS dE 1-2f(E) (E' A2) *

(3)

in which N is the normal state electron density of states' near the Fermi o

energy, VD/^C is a measure for the phonon-mediated attraction between elec-BCS

trons in the BCS theory and u is the BCS cutoff frequency. In equilibrium, the temperature dependence of A can be computed (see Fig. 1) by replacing f(E) by ft h( E ) . <3

1.5

0.5

Gap equations:

■ BCS

-Ginzburg-Landau

0.2

0.4

0.6

0.8

T/T.

Figure 1. Temperature dependence of t h e energy gap of a s u p e r c o n d u c t o r . At t e m p e r a t u r e s c l o s e t o T , t h e BCS theory ( s o l i d l i n e ) can be approximated using a power law (dashed l i n e ) d e r i v e d from t h e Ginzburg-Landau t h e o r y .

At temperatures close to T , the BCS gap equation reduces to the more manage­ able Ginzburg-Landau gap equation, which will be used throughout the rest of this work. For a distribution function defined as f(E)=f (E)+6f(E), the nonequilibrium Ginzburg-Landau gap equation becomes

CD 2 0 = A { T -T c 7C(3) kT 2 6f(E) dE (Ec A2)1 (4)

By substituting the BCS expression for the quaslpartlcle density of states N(E)

E N(E) =

(E" - ZT) equation (4) can be rewritten as

2,| o (5) 0.1066 kT T -T c 2 f N(E) öf(E) dE c' (6)

One should keep in mind that, by dividing out A in Eq. (4), we have lost a stable solution at A=0. Equation 6 shows that a reduction in the quaslpartlcle density, N(E)f(E), leads to an enhancement of A. This effect is known as the Parmenter [3] mechanism for the enhancement of superconductivity. The same equation also shows that if quasiparticles are moved to higher energy levels, this will also produce a net enhancement effect due to the 1/E weighing factor. This effect is called the Eliashberg mechanism after the Russian theoretician [4] who first pointed out that a redistribution of the quaslpar­ tlcle population can lead to gap enhancement. We will often use the term enhancement to denote all phenomena attributable to the nonequilibrium term in Eq. (6) - regardless of the net sign of its contribution.

3- Photons, Phonons and Quasiparticles

The Eliashberg mechanism for gap enhancement is observable when the quasipar­ ticles are pumped up to higher energy levels by the absorption of photons or phonons with energies nio. For frequencies for which nu)«kT, the enhancement effects are negligible because the distribution function barely changes. On the other hand, frequencies with niu»kT not only pump existing quasiparticles up to higher energy levels, but also produce large amounts of additional quasiparticles due to pair breaking, thereby reducing the enhancement effects.

Thus one is interested in frequencies for which hu~2b(7) because the enhance­

ment effects are largest when hco is just below 2A and because the onset of pair breaking phenomena at h<u=2A can be studied there. For materials like aluminum this corresponds to frequencies of a few gigaherz (microwaves).

As pointed out by Gray [ 5 ] , the quasiparticles tunneling between the two electrodes in a voltage biased symmetrical tunnel junction can also produce the Eliashberg effect. Such applications of tunnel junctions, in which a junction is used to drive at least one of its electrodes out of equilibrium, are commonly called quasiparticle injection (QPI). In the next section we will describe QPI for the case of a symmetric tunnel junction, being a tunnel junction with an identical pair of electrodes. The main conclusions will be that, at low voltages, QPI is highly analogous to photon or phonon absorption and that, although the symmetry appears to be broken by the polarity of the applied voltage, the observable enhancement in both electrodes is indistin­ guishable.

In order to satisfy the energy conservation principle, an electron with an energy E on one side of the barrier can only tunnel to a level with energy E'=E+eV on the opposite side of the barrier (E and E' are both measured from their electrode's respective Fermi levels). The rate at which electrons tunnel from left to right is proportional to

lJE)

- f(E)x O-f(E'))

N ( E ) N ( E t ) (7) in which 1/R is the junction's conductance per unit area. The rate at which

the inverse process occurs is given by

I J E ) - f ( E'} X e^ "f ( E )-1 N(E) N(E') (8)

Thus the net electron flow out of level E or into level E' is found by subtracting Eq. (8) from Eq. (7):

I J E ) ' - I J E ) <* f ( E ) e Rf ( E , ) N ( E ) N ( E , ) ( 9 )

Up t i l l now, we have discussed the tunnel probabilities in terms of electrons

rather that in terms of q u a s i p a r t i c l e s . A conversion from electrons to quasi­

p a r t i c l e s i s necessary in order to be able to use the standard tools of the

trade like the BCS density of s t a t e s . This transformation [6] complicates the

picture somewhat by replacing the simple electron tunneling process by four

Figure 2. Two quasiparticle tunneling processes (large arrows) superimposed on

the quasiparticle excitation spectra of both electrodes. „Single and double

dots indicate quasiparticles and Cooper pairs respectively. The dotted hor­

izontal lines represent the Fermi energy.

d i s t i n c t quasiparticle processes. Two of these processes only occur when eV>2A

or eV<-2A (pair breaking). Figures 2a and 2b show the two other processes,

both of which occur a t a r b i t r a r y voltages. In Fig. 2a, a q u a s i p a r t i c l e with

energy E simply tunnels through the barrier to a state with energy E'=E+eV (and vice versa) with a total rate given by

1 N(E) N(E')

1(E) = — ^ [f(E) - f(E')J (10) eR N

o

In Fig. 2b a quasiparticle is effectively transferred from the right hand side to the left hand side by removing a Cooper pair on the left and creating an new Cooper pair on the right. Note that if one only monitors the flow of

Figure 3. "Semiconductor" representation of the two tunneling processes shown in Figure 2. The large horizontal arrows at the top and bottom respectively correspond to Fig. 2a and 2b.

quasiparticles in Fig. 2b, one sees a quasiparticle with energy E' on the right tunnel to a state with energy E=E'+eV on the left. Because it can be shown that the processes shown in Figs. 2a and 2b take place at equal rates (for a symmetric junction and for equal initial energies), the quasiparticles in both electrodes are effectively pumped up to an energy eV above their original level. Thus the quantity eV in QPI-induced enhancement plays the same role as ncu does in photon or phonon induced enhancement. Similarly the conductivity per unit area of the tunnel junction corresponds to the incident photon/phonon power density.

Figure 3 shows an alternative representation of the quasiparticle tunneling processes called the semiconductor model [2]. The inverted copies of the quasiparticle excitation spectra are used in the same way as the "normal" branches. Thus, for example, the expression for the total tunneling current in the semiconductor representation is

"r N(|E|) N(|E+eV|)

I(V) = -!_ dE - (f(|E|) - f(|E+eV|)) . (11)

i

(N

r

Although the semiconductor model makes the processes shown in Fig. 2 or the two pair breaking processes that occur when |eV|>2A seem quite natural, it should be kept in mind that the extra branches below the Fermi levels are simply duplicates of the actual spectra that have been inverted to reflect minus signs in the energy conservation conditions of processes involving Cooper pairs.

t. Evaluating the Distribution Function

In the previous section we have sketched how an expression can be derived for the net rate at which quasiparticles are injected into or extracted out of a particular energy level by the tunneling current. For photon or phonon absorp­ tion, the corresponding pumping rates depend on the absoption probability of a quantum with energy nu> by a quasiparticle with initial energy E in a supercon­ ductor with energy gap A. These "coherence factors" are known from BCS theory [2] and are somewhat different for photons than they are for phonons.

To calculate 6f(E) in Eq. (4), one uses a steady-state Boltzmann equation in which the sum of the pumping and pair breaking rates are balanced by

inelas-tic relaxation processes. For most application one can employ a so-called relaxation time approximation for the latter rate:

2 N(E) 6f(E) I .(E)

rel (12)

in which x is the effective inelastic quasiparticle scattering rate. A second E

major approximation is that one substitutes f (E) for f(E) in Eq. (10). The latter implies that our model will be to first order in the intensity of the applied perturbation only [7].

Figure 4 shows the nonequilibrium distribution for microwave absorption computed in this way. The quantity 6f(E)N(E) represents the change in the quasiparticle density per unit energy and is significant here because it occurs in Eq. ( 6 ) . Note that at the higher of the two microwave frequencies, pair breaking causes new quaslparticles to be injected into the energy range between A and nu>-A. At the lower frequency the total number of quaslparticles is conserved. 3

*

0 . 1 •

UJ

0

- 0 . 1 ■

10

20

30

40

50

E foieV)

Figure 4 . Nonequilibrium q u a s i p a r t i c l e d i s t r i b u t i o n i n a s u p e r c o n d u c t o r i r r a ­ d i a t e d w i t h microwaves (A=10 yeV; kT=100 ueV). The 15 ueV photons i n c r e a s e t h e o c c u p a t i o n a t high e n e r g i e s by d e p l e t i n g t h e lower energy l e v e l s . The 25 ueV photons a r e a l s o c a p a b l e of g e n e r a t i n g new e x c i t a t i o n s because nu>>2A.

5. The G-Function

We will now present the nonequilibrium terms in the gap equation in a form meant to illustrate the similarities as well as to highlight the differences

between the three forms of enhancement. In order to arrive at a common notation, we will use the quantity hui to represent either the photon energy, the phonon energy or the quantity eV in QPI experiments. The intensity of the applied perturbation will be characterized by the dissipated power per unit volume P . For a symmetric tunnel junction with electrode thickness d/2, P is

v v

2 - 1

defined as V /Rd, where R is the junction's normal state conductance per unit area. For photon or phonon absorption in films with thickness d, P represents the power that the metal would absorb per unit volume in the normal state. The gap dependence of the dissipation will be accounted for in the various nonequilibrium expressions for the integral term in Eq. (6).

Especially when fitting model calculations to experimental data, it is convenient to use an alternative quantity to describe the strength of the applied perturbation [8]:

b " (13)

y 2 N (hu))2 V

Note that, unlike P , a/^ is independent of eV for tunnel junctions. It can be regarded as the ratio between the quasipartlcle pumping and the relaxation rates.

All nonequilibrium terms discussed in the next four sections of this paper will be presented in a form that can be added to the right hand side of the Ginzburg-Landau gap equation:

0.1066 kT cy T -T + terml + term2 + (14) T c

Analytical expressions for the integral in Eq. (6) were first given by Schön and Tremblay [9] and Eckern, Schmid, Schmutz and Schön [10] in papers in which they discussed the stability of enhanced superconductivity. To first order in Fiio/kT they found

1 P T „ A

V E r •>

terml = G[ J (15) 1 2 N kT Fin h u)

i n which the e x p r e s s i o n for the G-function (in terms of complete e l l i p t i c i n t e g r a l s ) depends on t h e enhancement t e c h n i q u e being u s e d . As shown i n F i g . 5 , t h e t h r e e types of G-functions a r e q u i t e s i m i l a r when h<D<2A. At high f r e q u e n c i e s , however, p a i r breaking c a u s e s t h e Parmenter e f f e c t t o e i t h e r reduce t h e net enhancement ( p h o t o n s ) , t o e x a c t l y c a n c e l out t h e E l i a s h b e r g e f f e c t (QPI) or t o dominate the E l i a s h b e r g e f f e c t (phonons).

CD

- 2

- 4

0.5

-Photons

—Quasiparticles

—Phonons

1.5

2.5

A/fiu

Figure 5. G(A/nu>) for photons, phonons and quaslparticle injection. A positive value of the G-function corresponds to a potential enhancement of A and all related parameters.

6. Heating effects

It can be shown [10,11] that for all three types of enhancement the second order contribution to the integral in gap equation (6) is

term2 - 0.17

V E

(16)

2 N (kT ) o c

The second order contribution thus lowers the value of the energy gap. This is basically because the three mechanisms that evoke enhancement increase the average quasiparticle energy, thereby raising the electron temperature. Term2 can be interpreted as the heating of the electrons relative to the phonons in the sample film due to the finite thermal coupling between the two popula­ tions. The result shown in Eq. (16) was obtained by a variational method. A simpler derivation based on the BCS expression for the electronic heat capa­ city at T [2] produces an identical expression with a 25% lower prefactor.

As the temperature T in the gap equation will generally refer to the tempera­ ture of the sample's environment (e.g. the helium bath or the substrate), a second heating correction is necessary to account for the thermal boundary resistance 1/Y between the superconducting film and its surroundings:

termH

P d v Y T

(17)

It is convenient to combine term2 and termH into an expression describing the total heating of the electrons relative to the bath temperature as both terms have an identical frequency and temperature dependence and because it is generally impossible to experimentally distinguish between the two:

term2H = term2 + termH = - (0.17 + F,_) h

V E

2 N (kT ) o c

(18)

This defines F . Under typical conditions F is comparable to- or larger than 0.17.

As terml and term2H are proportional to nw and (RID) respectively, term2H will dominate the enhancement term at high frequencies. The conditions under which

(positive) gap enhancement is possible are given by

> 4 ft Ü) F +0.17 f T h c Vo.1066

I

F.

(19)

This inequality can be demonstrated graphically by plotting both terms along a niu/A axis (Fig. 6 ) . The magnitude of the right-hand side of Eq. (19) is represented by the dashed line. The slope of the line depends on the reduced temperature and the the sample's thermal coupling to its surroundings (F )

h For the slope shown in Fig. 6, the net effect of terml and term2H is positive (gap enhancement) when nw<2A. For photon induced enhancement only, the net effect stays positive up to I W A « 2 . 6 .

<3 CD

-2

-4

-Photons

—Quasiparticles

—Phonons

0.5

1.5

2.5

3.5

tlü/A

Figure 6. The frequency range in which positive enhancement can occur is de­ termined by the relative magnitude of the G-function (same data as Fig. 5) and the heating of the electrons (dotted line). The slope of the latter de­ pends on F and T/T (shown for F, =0.2 and T/T =0.95).

n c n c

Figure 7 summarizes the conditions under which the net enhancement is posi­ tive. The quantity along the horizontal axis corresponds to the slope of the straight line shown in Fig. 6. All points in Fig. 7 below (inside) the appropriate curve will exhibit a positive net enhancement effect. The vertical line corresponds to the numerical example shown in Fig. 6. Figure 7 shows that phonon absorption will only produce a net enhancement effect at a given temperature for sufficiently low values of F,. For the other two forms of

enhancement, a large heating effect will lower the maximum frequency at which enhancement can be observed.

0 0.1 0.2 0.3 0.4 0.5

t0.17+F

h

)KV(l-T/T

c

)

Figure 7. Boundaries between gap enhancement (below curves) and gap depression for the three forms of enhancement. The dotted arc and vertical line are explained in the text.

The highest frequency at which enhancement can occur in a given sample is found at the temperature at which the traces exhibit the discontinuity. The dotted arc in Fig. 7 illustrates how the graph is traversed for a given sample as a function of temperature. The intersection with the photon curve indicates the temperature at which the highest possible frequency with a net enhancement effect is obtained.

7. Diamagnetic effect

For microwave enhancement, an additional correction term is required to describe the pair breaking caused by the root mean square value of the electromagnetic field [ 8 ] . The expression for this term,

nh P

termD = — — (20) 4 N kT (h<u)

o c

differs considerably from previous terms because the phenomenon is unrelated to the nonequilibrium quasiparticle distribution. Because termD is independent of hu), it determines the lowest frequency at which (positive) enhancement is possible. The minimum frequency for microwave gap enhancement has been employed by van Son et al. [12] to measure T in aluminum films with varying

E resistivities.

8. Frequency Dependence of Nonequilibrium Effects

In section 7 we compared the relative magnitude of termi and term2H as a function of frequency and found that above a certain frequency the latter is dominant. In this section we will discuss how the absolute magnitude of the terms vary as a function of frequency - or for QPI, as a function of voltage. The basic relationships can be readily summarized as follows:

Termi Term2H TermD

«

cc oc P /hü) p V P / ( h w )2 V oc oc

-Flu) * a/-y 2 (fiu)) *a/-y 01/7 (21)

The important point here, is to decide which quantity to keep constant during a frequency sweep. In a QPI experiment, for example, one presumably wants to know how the magnitude of a nonequilibrium term in the gap equation changes when the applied voltage V is varied. The junction's resistance is obviously considered fixed. Because the dissipated power per unit volume P is

propor-2 v

tional to V /R, we find that Termi is proportional to eV/R. Thus, in QPI experiments, the quantity that one keeps fixed is 01/7 rather than P .

For microwave or phonon experiments, the quantities that one would want to keep constant in an ideal experiment are respectively the magnitude of the applied electromagnetic field and the phonon generation rate. Because the rate with which photons or phonons are absorbed is proportional to nw and because the energy of the absorbed quantum also equals nw, we again find that cx/y is

the quantity to keep fixed. Thus the rightmost column in Eq. (21) is represen­ tative for the actual energy dependence of the nonequilibrium terms in the gap equation.

From an experimental point of view, the above premises are somewhat overopti-mistic for the case of photon- and phonon-induced enhancement. In practice it is very difficult to keep the field intensity within the sample constant across a significant frequency range. For photons, for example, one connects a microwave source to a waveguide which is either terminated in the vicinity of the sample or is directly soldered onto the sample film. Due to reflections, attenuation and outward radiation it is very difficult to achieve a well-defined microwave coupling in such systems. Even if one is able to either control or measure the microwave power close to the sample, it will still be very difficult to determine the coupling between field and sample because this depends on the details of the geometry of the sample and its related wiring. The main reason why a significant part of this thesis is devoted to work on QPI is precisely because such experiments provide accurate control over the absolute magnitude of the applied perturbation.

9- Gap Enhancement - A Quantitative Analysis

We will now quantitatively compare the three forms of enhancement and examine the relative importance of the different nonequilibrium terms in the gap equation. In all numeric examples we will use the following set of values based on typical values for aluminum:

frequency = 1 GHz (Fm>=4 ueV) or 10 GHz (nü)=4l ueV) a/y = 0.3 (for 1 GHz) or 0.03 (for 10 GHz) Fh = 0.17

t = 5 ns

1 1)7 _1 -7,

N = 1.1*10 ' J m 3

o

Table 1. Parameters used in all numerical examples.

The two values of 01/7 at 1 and 10 GHz correspond to values of P of 6x10 and

7 3 V

6x10 W/m respectively. Scaled to a volume of 10x10x0.01 um one finds values of 6 and 60 pW. If this same volume represents one of the electrodes of a symmetric tunnel junction, the barrier impedances required to obtain these values of 0/7- in QPI experiments would be 1.5 fi for 4 nV and 15 fl for 40 yV. Although it appears to be in contradiction to the discussion in the previous section, the value of a/f will not be kept constant as a function of frequency for practical reasons: Because the main enhancement term, terml, has a prefactor of na>*oi/Y, enhancement at 10 GHz is of the order of ten times as effective as enhancement at 1 GHz. In QPI-induced enhancement this corresponds to the statement that the quasiparticle current flowing through the junction is, to first order, proportional to the applied voltage (Ohm's law). The specific dependence of the tunnel current on A and V in a superconductor is incorporated in the G-function part of terml. For photon or phonon enhance­ ment , the absorption rates in a non-superconducting metal are also proportion­ al to no). We will therefore compensate for the basic proportionality between the enhancement and hui by keeping n<D*a/-y constant. Thus, one should keep in mind when interpreting frequency dependent results in the following sections that a factor of nu> in the frequency dependence has been artificially sup­ pressed. Put differently, the degree of enhancement shown for 1 GHz has been exaggerated by a factor of 10 relative to the enhancement data shown for 10 GHz.

Figure 8 shows the calculated temperature dependence of the energy gap of a superconductor in equilibrium as well as in the presence of microwave radia­ tion. The choice of temperature range is based on the validity of. the Ginzburg-Landau gap equation (see Fig. 1 ) . Although higher levels of enhance­ ment are experimentally possible (shifts in T of up to 10% have been observed), such large perturbations can not be accurately described with models which are only to first order in a/y.

O.B

" L

O.B

-

0.4

0.2

-a. 10 GHz photons

(O/T-0.03)

■b. 1 GHz photons (a/7-0.3)

Equllibrlua

0.9

0.92

0.94

0.96

0.9B

T/T.

Figure 8. Effect of 1 GHz and 10 GHz microwave radiation on the energy gap A of a superconductor. The dotted line indicates the equilibrium temperature dependence of A. The values of other parameters are shown in Table 1. In the remainder of this chapter we will express enhancement phenomena using a Quantity 6 [8] defined as M J exp 0.1066 kT c T -T c exp (22)

Thus, while 6 represents the sum of the nonequilibrium terms in the gap exp

equation, it can also be regarded as the net drop in the normalized quasipar-ticle temperature caused by the various nonequilibrium effects. Positive

values of 6 correspond to an increased energy gap. From an experimental point of view, 6 can be calculated by comparing the enhanced value of A to its equilibrium value A at the same ambient temperature:

exp

T -T c

(23)

Figure 9 shows the same d a t a a s in F i g . 8 when r e p r e s e n t e d as 6 . Thus, the o r - e x p

parameters shown in Table 1 result in an apparent temperature shift of the order of 0.5%. In addition, Fig. 9 provides a breakdown of the net enhancement into the various nonequilibrium terms at both frequencies. Because we selected F =0.17, termH always equals term2 (see Eq. 18) so there is no need to show these terms separately.

O O O

10

8

■ a. 10 6Hz Teritii ■b. Terml+Term2H . c . Terml+Term2H+TermD ■d. 1 GHz Terml ■e. Terml+Term2H • f. Terml+Term2H+TernD ^

---v

0.9

0.92

0.94

0.96

0.98

T/T.

Figure 9. Microwave enhancement of the energy gap expressed as 6 (see Eq. 2 2 ) . The results of the full- as well as partial expressfoRs are shown to indicate the relative importance of the heating and diamagne-tic terms in the gap equation. Curves c and f correspond to the data shown in Fig. 8. Curves b and c almost coincide.

As shown in Fig. 9, at 10 GHz the primary correction to the basic enhancement phenomenon, termi, is the heating term, term2H. At 1 GHz the heating term is

2

less important due to its proportionally to (Rio) . The diamagnetlc term, termD, is negligible at 10 GHz, but becomes the largest correction at 1 GHz.

O O O

10

B

6

2 ■

0.9

_

_.—-:

E

-T--a:

-b--c:

-d:

10

10

1

1

GHz

GHz

GHz

GHz

Termi

Terml+Term2H

Terra 1

Terml+Term2H

■

/ a

^ b

=

^

:

0.92

0.94

0.96

0.98

T/T.

Figure 10. Phonon-induced enhancement of the energy gap. The curves shown are directly comparable to those in Figures 9 and 11.

A more interesting conclusion that one can draw from Fig. 9, however, is that the curves only exhibit a limited gap- and temperature dependence, so that, by rough approximation, the nonequilibrium phenomena can be regarded as an increase in the effective critical temperature of the superconductor. It is also interesting to compare the magnitude of the enhancement at 1 GHz with that at 10 GHz. Even though the value of 01/7 was adjusted at both frequencies in such a way that niu*a/7 was kept constant (see Table 1 ) , the nonequilibrium effects are still significantly stronger at higher frequencies. This is simply because the nonequilibrium effects per absorbed photon are larger for higher values of ncu. This is obviously the case for the heating effects in term2H. In termi, this dependence is incorporated in the G-function and arises because

high values of FKU lead to larger changes in the quasiparticle distribution function.

Figures 10 and 11 shows the equivalent traces for phonon- and QPI-lnduced gap enhancement. Only two curves per frequency are shown because termD is only applicable for microwave-induced enhancement. The bias voltages for the QPI curves are related to the frequencies for the photon and phonon absorption by eV=hio. O O O

s

10

B

6

■ i i ■a: 10 6Hz Termi -b: 10 GHz Ter«l+Terffl2H ■c: 1 GHz Termi d: 1 GHz TerBl+Term2H

0.9

0.92

0.94

_0.96

_0.98

y\

Figure 11. Enhancement of the energy gap by quasiparticle injection. The calculated gap enhancement is observable in both electrodes. Frequencies of 1 and 10 GHz correspond to bias voltages of 4.1 and 41 yV respectively. Comparing the three forms of enhancement shown in Figs. 9 to 11, one finds that the main difference is the magnitude of termi. This term is roughly twice as large for microwave absorption as it is for phonon absorption. QPI lies in between these two. These differences in magnitude in termi are due to differ­ ences in the values of the respective G-functions (see Fig. 5 ) , which in turn are due to differences in the quantum transition rates of the processes involved. The important point here is that the ratio between the different G-functions is amplified in the final values of 6 because of the rather large

correction terms which cancel out a substantial portion of terml (the impor­ tance of this amplification effect strongly depends on the value of F ) . Thus the values that one measures in phonon experiments, and to a lesser degree in QPI experiments, are actually the leftovers of a larger effect that has been largely counteracted by heating effects.

10. Critical Current of a Superconductor

The infinite conductivity of a superconductor is due to the special, highly coherent, nature of the Cooper pairs. If a current is applied to a supercon­ ductor, it is therefore carried entirely by the Cooper pairs. The current density of the Cooper pairs is proportional to the density of Cooper pairs times their momentum. Because both quantities vary with increasing current, computation of the maximum obtainable current density in a superconductor is nontrivial. In the following we will assume a superconducting strip whose width and thickness are small enough to ensure a homogeneous current density across the strip's cross section. The critical current density in larger samples is largely determined by magnetic phenomena and by the geometry of the conductor's cross section.

Applying a current to a superconductor leads to a non-zero momentum of the Cooper pairs and a decrease in A relative to its equilibrium value A at that temperature. At a critical value of the current, a sudden transition to the normal, dissipative, state takes place: the energy gap vanishes and the applied current is carried as a normal current. In practice, this transition is hysteretic (see Fig. 12) due to the temperature rise caused by Ohmic dissipation in the normal state. The term "critical current" is therefore defined as the maximum current that can be passed through a superconducting sample without dissipation. The current I at which the sample switches back from the normal to the superconducting state when the applied current is reduced, largely depends on the thermal properties of the sample and will not be discussed here.

Within the framework of the Ginzburg-Landau description of superconductivity, the dependence of the gap on the supercurrent density can be calculated by taking the kinetic energy of the Cooper pairs into account when minimizing the free energy of the system. An expression for the supercurrent density which is

0) O) (0

o

>

Current

Figure 12. C u r r e n t - v o l t a g e c h a r a c t e r i s t i c of a superconducting s t r i p . The c u r r e n t induced t r a n s i t i o n between the s u p e r c o n d u c t i n g and normal s t a t e s i s h y s t e r e t i c . The s u p e r c o n d u c t i n g - t o - n o r m a l t r a n s i t i o n i s c a l l e d t h e c r i t i c a l c u r r e n t I .

c

more c l o s e l y l i n k e d t o t h e microscopic p i c t u r e of a superconductor was given by Schmld, Schon and Tinkham [ 1 3 ] . I t r e l a t e s t h e c u r r e n t d e n s i t y to both the normalized momentum Q of t h e Cooper p a i r s and t h e gap A:

Y~ = 0.1066 V27 ( l - 2 f ( A ) ) £- Q

GL c (24)

In this equation the Ginzburg-Landau critical current density J„, is a material parameter and f(A) represents the distribution function evaluated at the gap edge. Under equilibrium conditions, and near T , this reduces to

_J_ 0.1066 , A 2 n

J^ = ~ V 2 7 (k T_ ) Q "

UL C

(25) The i n f l u e n c e of a s u p e r c u r r e n t on the gap can be added t o gap equation (14) with an a d d i t i o n a l term:

,2

1 "

0.8

S 0.6 ■

0.4

0.2

/

^<_

0.2 0.4 0.6 0.8

A/A.

Figure 13. Ginzburg-Landau relationship between the gap A and the supercur-rent I. The critical cursupercur-rent I is reached when A reaches V(2/3) of its equilibrium value. The dashed part of the curve is thermodynamically un­ stable. Both A and I are temperature dependent.

The relationship between J and A obtained after eliminating Q using Eqs. (14), (25) and (26), leads to an equilibrium critical current J occurring at A=A

co ° c As shown in Fig. 13, Ac is a factor of 0.816 smaller than the equilibrium value of the gap A at that temperature. The temperature dependence of J

° co is given by r J 11 J- i2/3 "GL T -T c (27)

For nonequillbrium conditions, the value of f (A) in Eq. (24) can be computed as the sum of the Fermi function at the gap edge and its perturbation öf(A) as used in Eq. (6). For arbitrary value of ftu), photons, phonons as well as QPI pump quasiparticles away from E=A towards E=A+nu>. At frequencies below 2A, no

quasiparticles are pumped into energy level E=A because the density of states at E=A-nu> is zero. Thus, a strong Eliashberg effect is always accompanied by a a negative value of 6f(A). This gives rise to a significant additional enhancement mechanism in critical current experiments that is not available in gap enhancement. This extra enhancement effect is particularly important in QPI experiments with asymmetric tunnel junction biased at |eV|~|A —A j [14].

0.1

0.0B

~k 0-OB h

t — I

\

~ 0.04

0.02

0

0

• a. 10 GHz Photons (a/fO.03)

-b. 1 GHz Photons (a/r-0.3)

Equilibrium

0.92

0.94

0.96

0.9B

T/T.

Figure 14. Effect of 1 GHz and 10 GHz microwave irradiation on the critical current I of a superconducting strip. Dotted line indicates the equili­ brium Ginzburg-Landau temperature dependence of I .

c

Figure 14 shows the photon-induced critical current enhancement at 1 and 10 GHz for the parameters shown in Table 1 . In order to perform a quantitative comparison between I -enhancement (Fig. 14) and A-enhancement (Fig. 8 ) , we will express both phenomena using the previously introduced quantity 6 . If we maintain the convention that 6 represents the normalized shift in

exp

temperature corresponding to the observed enhancement, the definition of 6 for I -enhancement becomes (see Eqs. 22 and 2 7 ) :

^JGL 2/3 _{T -T} c + 6 exp exp (28)

A more operational expression relating 6 to the ratio of the nonequilibrium and the equilibrium value of I at the same temperature is:

c 6 = { exp l 2/3 T -T - i } - 2 _ (29) C OJ C

Figure 15 shows the data of Fig. 14 expressed as 6

exp In all traces the full gap equations (with heating and diamagnetic terms) have been used. The lowest and highest trace at each frequency show the amount of gap enhancement and I enhancement respectively. The third trace shows the computed amount of I -enhancement if the Fermi function rather than the nonequilibrium distribution function is used for f(A).

O O O

10

B

6

4

2

0

0.9

• -»^ ~~~*"*^—«. :___ . ■ —" — — a. b. c. d. e. f. ^ = ~ ~

10

10

10

1 1 1 * * • ■ * - * ^^ GHz Ie GHz I' GHz Ae GHz I GHz I" GHz Ae a f (A) If (A)

a

b

c

""'--"--=0

-0

— Z

-"

'

^-^^^

— "

•

^"' /

e ^-^>' •

f

0.92

0.94

0.96

0.98

T/T.

Figure 15. Enhancement of A and of I expressed as 6 (see text) for two c exp microwave frequencies. I -enhancement can thus be significantly larger than A-enhancement. Curves b and e represent simplified expressions for I -enhancement.

c

As illustrated in Fig. 15, I -enhancement tends to be larger than A-enhance­ ment under identical conditions. This is partly due to the contribution of the nonequilibrium occupation probability 6f(A) and partly because the value of

the gap is lowered by the transport current, thereby increasing the main enhancement term (see Fig. 5 ) . At low frequencies (e.g. 1 GHz in Fig. 15) the 6f(A)-based enhancement can actually exceed the enhancement derived from the Eliashberg effect because the latter is proportional to hw*a/y while 6f(A) is

proportional to a/f.

It is also worth noting that - unlike the Eliashberg effect - the contribution of the óf(A) term to I -enhancement can diminish with increasing temperature.

c

This is partly because the rate at which quasiparticles are pumped from energy level E=A to E=A+nu> is proportional to the quasiparticle density of states N(E) for E=A+hw. Because the density of states falls off with increasing energy on a scale of A(T), N(A+nu)) decreases when A is lowered while fico is held fixed. Because, for microwave enhancement, the Eliashberg and 6f(A) mechanisms respectively increase and decrease with rising temperature, the temperature dependence of the net I -enhancement can exhibit either type of behaviour. O O O

10

8

6

4

a. 10 GHz I

b. 10 GHz A

e

c. 1 GHz I

d. 1 GHz A

<y,

•

0.9

0.92

0.94

0.96

0.9B

T/T.

Figure 16. Comparison of A- and I -enhancement by phonon atentuatlon at two phonon frequencies. Traces b and d correspond to traces b and d in Fig. 10.

As shown in Fig. 16, phonon-induced enhancement of I is also larger than c

phonon-induced gap enhancement. The difference between the two seems to be rather constant as a function of temperature. The difference between I

-c enhancement and gap enhancement by quasiparticle injection (see Fig. 17) is much larger: critical current enhancement can be up to an order of magnitude

larger than gap enhancement.

O O O

u

8

6

4

2

n

a . b. d . 10 GHz I 10 GHz Ae 1 GHz A

" a

/ b '

■

_ - - - "

d

'

0.9

0.92

0.94

0.96

0.98

T/T.

Figure 17. Comparison of QPI-induced enhancement of A and I at two bias voltages corresponding to 1 and 10 GHz. The I -enhancement trace for

1 GHz (not shown) lies at 6 «0.05 for T=0.9T °.

References

1. J . Bardeen, L.N. Cooper and J . R . S c h r i e f f e r , Phys. Rev. 108, 1175 (1957) 2 . For a more d e t a i l e d d e s c r i p t i o n of BCS theory see Chapter 2 of I n t r o d u c ­

tion t o S u p e r c o n d u c t i v i t y , M. Tinkham (McGraw-Hill, New York, 1975) 3- R.H. Parmenter, Phys. Rev. L e t t . ]_, 274 (1961)

4 . G.M. E l i a s h b e r g , Zh. Eksp. Teor. F i x . Pis'ma V1_, 186 (1970); JETP L e t t e r s ri_, 114 (1970); B . I . I v l e v and G.M. E l i a s h b e r g , Zh. Eksp. Teor. F i x . Pis'ma 2 1 - 464 (1971 ); JETP L e t t e r s 2 1 , 333 (1971 ) B . I . I v l e v , S.G. L i s i t s z y n and G.M. E l i a s h b e r g , J . Low Temp. Phys. 2£. ^ 9 (1973) 5 . K.E. Gray, Solid S t a t e Commun. 26_, 633 (1978)

6 . K.E. Gray, in Nonequilibrium S u p e r c o n d u c t i v i t y , Phonons and Kapitza Boundaries, K.E. Gray, e d . (Plenum P r e s s , New York, 1981), p . 131

7 . This approximation i s t h e reason why the s i n g u l a r i t i e s i n N(E) r e s u l t in s i n g u l a r i t i e s i n 6f(E) - r e g a r d l e s s of t h e s i z e of the a p p l i e d p e r t u r b a ­ t i o n .

8 . J . E . Mooij, i n Nonequilibrium S u p e r c o n d u c t i v i t y , Phonons and Kapitza Boundaries, K.E. Gray, e d . (Plenum P r e s s , New York, 1981), p . 191 9- G. Schön and A.-M. Tremblay, Phys. Rev. L e t t 42_, 1086 (1979)

10. U. Eckern, A. Schmid, M. Schmutz and G. Schön, J . Low Temp. Phys. 36_, 643 (1979)

11 . W. Dupont, Diplom T h e s i s , Karlsruhe (1977)

12. P.C. van Son, J . Romijn, T.M. Klapwijk and J . E . Mooij, Phys. Rev. B 29 1503 (1984)

13- A. Schmid, G. Schön and M. Tinkham, Phys. Rev. B 2J_, 5076 (1980)

14. I t i s i n t e r e s t i n g to n o t e t h a t Eq. (24) d i v e r g e s when t h e nonequilibrium d i s t r i b u t i o n f u n c t i o n a t the gap edge goes through 0 . 5 . This c o n d i t i o n occurs when nfi~2A, but should not be i d e n t i f i e d with t h e s i n g u l a r i t y a t hO.=2A i t s e l f . Although t h e d i v e r g e n c e i s presumably an a r t i f a c t a r i s i n g from a p p r o x i m a t i o n s in t h e d e r i v a t i o n of Eq. ( 2 4 ) , i t w i l l not vanish i f the d i s t r i b u t i o n f u n c t i o n a t t h e gap edge i s r e p l a c e d by a weighted i n t e g r a l . Although i t i s of l i m i t e d consequence for the enhancement c a l c u l a t i o n s p r e s e n t e d i n t h i s t h e s i s because the s o l u t i o n of the gap e q u a t i o n are e f f e c t i v e l y " r e p e l l e d " by t h e d i v e r g e n c e , i t i s r e l e v a n t for free energy c a l c u l a t i o n s of c e r t a i n n o n e q u i l i b r i u m s t a t e s .

Microwave-Enhanced C r i t i c a l Current i n Superconducting Aluminum S t r i p s

P . van den Hamer, T.M. Klapwijk, J . E . Mooij Department of Applied Physics

D e l f t U n i v e r s i t y of Technology D e l f t , The Netherlands

A study has been made of t h e enhancement of the c r i t i c a l c u r r e n t i n s u p e r c o n d u c t i n g aluminum m i c r o s t r i p s by microwave r a d i a t i o n . The observed enhancement i s compared t o c a l c u l a ­ t i o n s based on t h e E l i a s h b e r g theory modified for a p p l i c a t i o n t o c r i t i c a l c u r r e n t enhancement. Theory and experiment a r e i n r e a s o n a b l e agreement i f t h e s t r o n g l y t e m p e r a t u r e - d e p e n d e n t microwave impedance i s taken i n t o a c c o u n t .

1 . Introduction

After the initial theoretical predictions by Eliashberg et al. [1], the enhancement by microwaves of the energy gap of a superconductor has been studied extensively [2]. Experimental support was provided by measurements of the enhancement of the energy gap, the critical temperature and critical pair-breaking current. In the theory, expressions were given for the energy gap. More recently Schmid, Schön and Tinkham [3] derived an expression for the supercurrent in a nonequilibrium superconductor. Based on these methods, one can calculate the microwave-enhanced critical current.

In the work presented here, measured values of the enhanced critical current are compared to theory. A simple heating correction is applied which assumes a single effective temperature increase above the bath temperature. The micro­ wave intensity in the theory is expressed in terms of the electric field. In actual experiments this electric field depends strongly on the value of the energy gap [4].

2. Sample Fabrication and Experimental Setup

Measurements have been performed on 200 \im long Al strips between broad Al

banks. The width of 1 .5 um and thickness of 150 nm were chosen sufficiently small to ensure a homogeneous current density. Optical contact printing was used to transfer the pattern onto thermally oxidized silicon substrates. The

150 nm Al film was evaporated from tungsten boats at a rate of 1 nm/s and at a pressure of 4x10 Pa. All measurements reported on here pertain to one particular strip with a critical temperature T of 1.224 K and a 10% to 9 0 %

c

resistive transition of 0.8 mK. Its room temperature resistance of 29.6 ft reduced to 4.4 ft at 4.2 K. Measurements using other strips with somewhat different dimensions gave similar results.

Measurements were performed in a standard He-4 cryostat. The substrate was mounted on a copper bar within a lead-lined cylinder and was in direct contact with the liquid helium. The temperature was regulated by a feedback loop consisting of a germanium thermometer, a resistance bridge and an electric heater in the helium bath. The germanium sensor was enclosed in a separate

-4

metal casing. The temperature fluctuations were well below 10 K as estimated from the short-term variations in the strip's critical current.

The frequency range from 1 to 10 GHz was covered with 3 separate sources. The microwaves passed through an attenuator, a ferrite isolator and a coaxial cable which was terminated above the sample by a wire loop. Effects due to impedance mismatches caused the microwave power to vary strongly with frequen­ cy (typically 20 dB for a 100 MHz shift in frequency). In order to minimize the power variations due to frequency drift, the frequency was adjusted for maximal power, as indicated by the enhancement. Measurements performed at an adjacent minimum were not significantly different. The microwave power typically drifted 0.2 dB within one session. It is important to note that the absolute microwave power within the sample could only be determined by fitting it at each frequency against the theoretical model.

The critical current I was measured by passing a sawtooth current through the strip and monitoring the resulting voltage. This time-dependent voltage was fed to a signal averager. The halfway point in the voltage transition was used as a criterion for I This choice is correct if the width of the transition is mainly due to temperature fluctuations. External noise would cause a hysteretic transition to occur at a lower value than predicted by theory. However, no significant deviations from the equilibrium temperature dependence were detected within the temperature range of interest.

0.93 0.94 0 9 5 096 097 0.98 0.99 1.00 v0.98 0.99 1.00 v0.97 0.98 0.99 1.00

Temperature ( T / TC)

Figure 1 . Measurements of the microwave-enhanced critical current J of an Al strip at 1, 3 and 10 GHz. Leftmost line in each set is with no radiation. Subsequent curves are at 3 dB increments in the relative applied power.

Near T and at high power levels the microwaves caused thermal latching of the c

strip: after exceeding the critical current, the strip is unable to return to the superconducting state when the DC current is lowered. Pals et al. [5] have attributed this to a rise in temperature due to the RF dissipation. To extend our measurements into this range we reduced the microwave power after the current exceeded I . The full power was reapplied during the next ascending slope of the current sawtooth while the strip was in the superconducting state. Use of this pulsing technique or the effective duty cycle of the applied microwaves had no measurable effect on the value of I .

c

Fig. 1 shows the critical current measured at 3 dB (factor 2) steps in the relative applied microwave power for 1, 3 and 10 GHz.

3. Theory of I_-Enhancement

Near the critical temperature a DC current can be included in the Eliashberg model of gap enhancement:

(1a) * , A - ) 2 T - T 1ocn<D

°-

107

i ? r j = - T - -

Q +

? 7 k ^

G [

^

] "■ c J c ' c *

where T is related to the actual bath temperature T by *

T - T T -T no o a r fi 00 -,2 a f n m ,2

^ T - = ^ T - " 2 k T "

F

h o - 7 kï- - ° -

1 7

7 [ k T -

( 1 b )

c c c o ^ c J ^ c J

where Q, A and 10 are the normalized super f luid momentum, the energy gap and the microwave frequency, respectively. The last term in Eq. 1a represents the main nonequilibrium contribution. The microwave power a in the strip can be expressed in terms of the amplitude E of the electric field as

2 2 e D E

a = - j (2) 2 h <o

where D=v„je/3. The excitation of quasiparticles by the microwaves is balanced _r by relaxation with an effective rate y=f\/-z , where i„ is the inelastic

E E scattering time. The function G(A/Fuu) is given by

2nA X2 ,„2 A2 \ i 2A < hoi (n2u)2-A2 )' G(A/nu>) = (3) — 7 — { ln(r—) - 1 + 0.26 — } 2A > nu) A nci) A

original expression containing elliptic integrals [ 6 ] . A more complete discussion of various terms in Eq. 1 can be found in Ref. 2. The heating of the sample is described by the second term from the right in Eq. 1b and assumes a single effective temperature for phonons and electrons. The heating parameter F. is defined as

h

d k T n

F. = 1 . 5 v T C T (1)

h YK TE TF

where d, n and T„ are the film thickness, the density of conduction electrons

r

and the Fermi temperature, respectively. Note that the heating depends on the real component of the conductivity o . As will be described below, o can differ considerably from the normal state value a .

o

The s u p e r f l u i d momentum Q i s r e l a t e d t o t h e s u p e r c u r r e n t d e n s i t y J [ 3 ] as no 8kT ,

JS = - 2 ! ( Ï D Ï T J Q A C 1 - 2 f0( A ) - 2 6 f ( A ) ) ( 5 )

where f (E) is the Fermi occupation probability. Note that the nonequili-brium distribution 6f(E) not only effects A (Eq. la) but also changes the relationship between j and Q. In equilibrium and for A « kT Eq. 5 reduces to

no 8kT i A2

J

o= if W ^

Q

2kT

(6)

From Eqs. 7 and 8 in Ref. 2, one finds for the nonequillbrium distribution at E=A to first order in nto/kT

c

6

V * > = - 7 ï ï k V U

1 +

! S

è

- 0-i)

è

<Kn

U

-2A)]

where 8 is the Heaviside step function. Consequently the nonequillbrium contribution to Eq. 5 is

no ( 8kT ,. 01 fi u) r „. , , ,

C a l c u l a t i o n s of t h e d i s t r i b u t i o n function to second o r d e r i n n<i)/kT yield a c

second order contribution to 1 £

8kT -.J A^ 1 a ha) 2 s

. 2

no r OKI -ij a i a no) c

J

2 = -

0

-

i )

IT [Tmrj

Q

k T 7 i T 7

C

k T ^

The t o t a l s u p e r c u r r e n t j =j + j + j _ i s now given by:

JS(A,Q) A — : = 6V3 Q 0.107 Jco kT c v c e o . J (10)

where j is the zero-temperature extrapolation of the Ginzburg-Landau critical current. The current j as a function of A is obtained by elimi­ nating Q from Eq. 10 using Eq. 1. In Fig. 2 the inset shows some typical j (A) curves at various microwave intensities. The barely visible bend at A/kT =0.2

c is due to pair-breaking when 2A < nu>. The critical current j is found by maximizing j with respect to A. The corresponding value of A will be referred to as A . In equilibrium this gives the familiar results j (T)=j x(1-T/T )1"5

c 2 2 a c co c occurring at A = -A where A is the zero-current value of the gap. Fig. 2 shows computed values of j at fixed levels of a/7 for 10 GHz. Note that

0.93 0.94 0.95 0.96 0.97 0.98 0.99

Temperature (T*/T

c

)

1.00

Figure 2. Inset shows j (A) for several values of the power parameter 01/7 (10 GHz, T/Tc= 0.97).SThe critical current j is found at the maxima of

although the temperature scale in this figure doesn't include the corrections of Eq. 1b, their inclusion would give a similar set of roughly parallel lines.

H. Coupling of the Microwaves

As pointed out recently [4], when interpreting microwave enhancement experi­ ments it is important to analyse the nature of the microwave coupling to the sample. For this case we represent the microwave source, as seen by the sample, as a voltage source V with an unknown internal impedance Z. (see

Rr 1 inset Fig. 3 ) . The strip's geometrical inductance L is about 1 pH/pm and is

S

insensitive to changes of the cross-section. The response of a superconductor to a microwave field can be described with a complex- conductivity o=o -io . In a strip with cross-section S and length d the equivalent circuit consists of a resistance R=do S and an inductance uiL =do S~ . Mattis and Bardeen [7] determined a general expression for o"(w,A). For Txi),A « kT these integral equations can be reduced to

a n u> A 5 " = 1 +kT~L^-> ( 1 1 a ) , o c and a it A 2 kT

-

- 2

(

kT-

) (

KT>

(11b) o c

where L(A/nu)) consists of complete elliptic integrals of the first (K) and second (E) kind:

LCA/RuO = ± - AK( § £ ^ ) - 1 ( 1 ± |Aj E(!£*ÜJ ( 1 2 )

hui ^2A+n(j/ 4 *■ PKI)-" '~2A+nu)J

The minus signs should be used when fico > 2A and the plus signs when hu <_ 2A. It is crucial to note that because both a and o depend strongly on the gap, the microwave intensity a in the strip can vary with temperature. If the strip's impedance Z is much larger than Z., the electric field E in the strip is constant for varying A. In this constant voltage approximation 01 is proportional to the applied power P__. In the opposite limit however, where

nr

Z « Z , the coupling is of the constant current type and results in

PRF , ,

" = - 5 — r (13)

V°2

Note that for our purposes the geometric inductance L should be considered S

external to Z . As 10L is of the same order of magnitude as u>L, and R and as s g ° k all three are proportional to the length of the strip, the constant voltage

l i m i t cannot be reached by simply l e n g t h e n i n g t h e s t r i p . F i g . 3 i l l u s t r a t e s t h e t e m p e r a t u r e dependence of a for given P assuming a v a l u e of t h e gap of

nr

A . Thus, particularly at low frequencies, lowering of the temperature leads to a dramatic reduction in absorbed microwave power. This explains qualita­ tively the difference between Figs. 1 and 2. Formal calculations of the critical current for given P_„ and bath temperature TL consist of maximizing

nr b Eq. 10, where a/7 and Q are determined by Eq. 13 and Eq. 1 respectively.

2

1

0

2

1

0

2

1

0

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

Temperature (T/T

c

)

Figure 3- Temperature-dependence of the real (o ) and imaginary (o ) compo­ nents of the conductivity as well as the resulting (Eq. 12) relative micro­ wave power a . Inset shows equivalent circuit for a superconductor in

a microwave field. The strip itself is represented by a resistance, the "kinetic" inductance of the superfluid and a classic "geometrical" induc­ tance. I l l »

3

1 J - i — r y 1 T "

\ °

2

GHz \ ^ >

i ■ i i i i

10 GHz

" ^ \ ^ ~ ~ — — _oo___

i i i i i

_ .