Quantum fluctuations and the universal criterion for the onset of superconductivity in Josephson arrays

Physica C 152 ( 1988) 361-377 North-Holland, Amsterdam

QUANTUM FLUCTUATIONS AND THE UNIVERSAL CRITERION FOR THE ONSET OF S U P E R C O N D U C T I V I T Y I N J O S E P H S O N ARRAYS Behzad M I R H A S H E M and Richard A. F E R R E L L Centerfor Theoretical Physics of the Department of Physics and Astronomy, University of Maryland, Collge Park, MD 20742, USA Received 14 March 1988 Revised manuscript received 29 April 1988

A detailed treatment of the zero-temperature quantum fluctuations of a granular film, modeled by a regular lattice of Josephsoncoupled grains, is presented. For small grains, whose junction reactance is determined by virtual quasi-particle tunneling, mean field theory predicts a universal threshold normal state resistance, R ~F, above which zero temperature superconductivity is suppressed by phase fluctuations. Cluster calculations, which allow for the correlated fluctuations of a fraction, f<< 1, of the z neighbors of each grain, reduce R MF to R ~ -----R~F ( 1--3f/5Z). By exploiting analogies with the treatment of thermal fluctuations in the Ising model, we conclude that this two-term expansion remains valid for the case of physical interest, f= 1.

I. Introduction Recent experiments on granular films have revealed the existence o f an apparently universal criterion for the onset of superconductivity. Twodimensional films of globular tin [ 1 ], gallium [2], or aluminum [3], deposited on an insulating substrate, were observed to become superconducting, at temperatures much lower than their bulk transition temperature, but only if R o, the normal state sheet resistance o f a square, did not exceed a threshold value, R ~ = 6.5 kfL The universality o f critical exponents is a familiar phenomenon in the theory of phase transitions. A universal resistance threshold, on the other hand, would correspond in classical statistical mechanics of spin systems, for example, to a critical temperature that is independent o f the coupling constant. In other words, in granular films, we are dealing with a different kind of universality. Because the relevant critierion is the normal state resistance, some authors have inferred that dissipation must play an important role and have constructed theories which focus on the interplay between q u a n t u m fluctuations and an undetermined source of dissipation, at temperatures far below the gap, whose strength is determined by the normal state

conductance [ 4]. We have taken a different point o f view [5,6]. In a granular system, charge transfer among the grains occurs by the tunneling of quasiparticles and Cooper pairs. At absolute zero, only Cooper pairs are present. I f the grains are large, pair transfer requires no energy and the ground state is globally phase-coherent. If the grains are small, however, the capacitive energy due to the charge imbalance across a junction may not be neglected [7]. Indeed, since pair number and phase are complementary variables, it was pointed out by Abeles that the resulting q u a n t u m fluctuations of the pair phase can destroy superconductivity [ 8 ]. A mean field theory of this effect was then developed by a number of investigators who obtained a criterion for superconductivity expressible in terms of the ratio of the Josephson coupling energy to the charging energy, the latter being determined by the geometrical capacitance [ 9-12 ]. The reactive response of a Josephson junction is not, however, determined entirely by its geometrical capacitance. The virtual transfer of quasiparticles across the junction can lead to an additional reactance, which is equivalent to an effective capacitance parallel to the geometrical mutual capacitance [ 13-15 ]. In the granular films that are o f current interest, the metallic globules are sufficiently

0921-4534/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

362

B. Mirhashern and R.,4. ki'rrell / L.'niver;al criterion.lbr the onset ql'superconduclivlO'

small that their geometrical capacitance can be neglected, as demonstrated in Appendix A. We have shown how, in this limit, mean field theory' leads [ 5,6 ] to a threshold which is in good agreement with the experimental results. The criterion's universality and its expressibility in terms of the normal state resistance follow from the fact that both the Josephson coupling energy and the effective capacitance are functions of the BCS gap and the inter-grain tunneling matrix element• In the final result, the gap cancels out, leaving only the tunneling matrix element, which can be expressed entirely in terms of the normal state resistance, The microstructure of the experimental samples has, thus far, not been established. Therefore, the extent to which the arrangement of the grains is disordered is unknown. While the relevance of such disorder to the onset of superconductivity is an important and unresolved issue, a model in which identical grains are considered to lie at regular lattice sites deserves continued attention. This is because, first of all, it may be feasible to realize experimentally an approximately regular structure and, second, because the semi-quantitative agreement between theory and experiment suggests that such a model may, in any case, capture the essential features of the experimental system, at least in so far as the threshold criterion is considered*. It is, therefore, important to determine how the mean field results for the perfect lattice are altered if the many-body problem is solved more accurately. After a review of our mean field treatment in section 2, we devote section 3 to certain cluster approximations which may be used, to some extent, to incorporate fluctuation effects into the calculation of the transition temperature of the lsing model. We then develop, in section 4, a parallel treatment of fluctuation corrections to the mean field threshold resistance.

2. T h e m e a n f i e l d t h r e s h o l d r e s i s t a n c e

The electrostatic energy of a regular array of identical metallic grains is * There exist very preliminary indications that the grains may indeed form a regular array [ 16 ]. If the grains are charged during deposition, for example, their mutual electrostatic repulsion may lead to the formation of an ordered lattice.

W=½ E CoV;z+ ½ ~ C,,( I.%- [', )'- . t

(2.1)

ti

where ;', is the electrostatic potential of the ith grain and C() and C,~ are the self- and mutual capacitances, respectively. At temperatures below T,, the bulk superconducting transition temperature, V, determines the time dependence of the c o m m o n Cooper pair phase of the electrons on the ith grain, 0,, according to the Josephson relation 2e

•

where - e

and 2xh are the electron charge and

Planck's constant, respectively• The Josephson coupling energy of the grains is U=-EI

~ cos(0,-0j),

(2.3)

/J

where the sum is restricted to nearest neighbors, with each pair of grains counted once. The Josephson current flowing into the ith grain is - 2e 0U

Q'-

h

00,

(2.4)

From the definition of the capacitance as a linear response function, Q,=col~+

Z c,j( f ' , - l",)

(2.5a)

]VI

2e d OW

-

h dt O0i "

(2.5b)

so that d 0 I4r dt 80i -

0U 00i "

(2.6)

This is just the Euler-Lagrange equation for minimizing the action S = j L dt, where the Lagrangian is L ( O, O) = W - U. Q u a n t u m fluctuation effects may now be examined by using e ~s/t' as the amplitude for a particular path {0,(t)}. However, as usual, the Hamiltonian formalism is more convenient for computational purposes. From eq. (2.5b), the momentum conjugate to the coordinate 0~ is essentially the charge on the ith grain so that the quantization prescription for the dimensionless momentum,

B. Mirhashem and R.A. Ferrell / Universal criterion for the onset of superconductivity

IOL 1 0 P/-- h0-~/--" i 00,'

36 3

~2

(2.7)

reflects the complementarity of the two canonically conjugate variables, the number of pairs on each grain [ which can be O ( 10 6 ) ] and their common phase, 0/. Thus, quantum-mechanical fluctuations reflect the effect of the uncertainty principle on a "mesoscopic" level. (The choice of sign in eq. (2.7) parallels that for the mechanical analogue of the granular system, a collection of coupled quantum pendula. In the derivation of the Josephson effect using the tight-binding analogy [17] on the other hand, the phase is identified with the crystal momentum, so that the opposite convention is used.) Because the n-grain Hamiltonian, H= Y N,=, h p ~ , - L is non-linear, it is not feasible to obtain an exact solution. The long wavelength aspects of the problem are particularly difficult because of the presence of Goldstone bosons and critical fluctuations. As in refs. [5] and [6], therefore, we continue to focus our attention on the short-wavelength features of the problem. We consider a sequence of cluster approximations in which some such localized fluctuations are taken into account and the corresponding resistance threshold is determined. The simplest approximation is of the Hartree type, in which the two-grain interaction is replaced by an effective single-grain potential, i.e., for the ith grain, we replace

LL = ~e2( Co + zA )O~ + zp.Es cos 0/,

(2.9)

where z is the lattice coordination number. From eq. (2.7), h

p,= ~---e2(Co + zA)0/,

(2.101

so that the Hartree, or mean field, Hamiltonian is

H,

~---

4e 2

1

2

C o + z A C ( ~ p , - g ~ cos 0i) 4e2

(2.11)

~

(2.12)

=- Co +zAC HI .

In the reduced Hamiltonian/7,, the self-consistent field acting on the ith grain is

g~=z

z+--~

go,u=2/1,

(2.13)

where the parameter go is the ratio of the Josephson coupling energy to the charging energy: z~CEj

go-

4e 2 •

(2.14)

COS( 0 i - Oj ) = ½( eioie -iO:+ e -io, eiOJ)

The Hamiltonian H~ is just that of a pendulum of unit length and mass in a gravitational field of strength g~. The transformation g ~ - g ~ and 0/---'0/+ ~ leaves unchanged EG, the ground state eigenvalue of H~, which is therefore an even function of gj. From perturbation theory, one obtains

by

EG=__g2+7

½[eiO, +e-i~' ] = / / c o s ( 0 i - ] ~ ) ,

(2.8)

where
z g 4~ - Y58g , 6 + ' "

(2.15)

for the ground state energy. (For arbitrary g,, the eigenvalues of Mathieu's equation have been tabulated [18]. Elsewhere, we provide a convenient variational calculation of the ground state eigenvalue [ 19 ].) Thus, /t-

-

-

116_5 dE° dgl _ 2g I _ 7g3 + - f - g l

~ .

... (2.16)

Comparison with eq. (2.13) shows, that for 2 > ½, a nonzero value o f / t = ( 4 / x / 7 ) (2-½)~/2 is possible. According to eq. (2.15), this phase-ordered state of the granular film, which can sustain current in the absence of any applied voltage, has lower energy than

B. ~ltirhashem and R. 4. Ferrell / Universal criterion jor the onset e~fsuperconductivitv

364

the phase-incoherent one. The threshold criterion, ,~= ½, corresponds to 1 1 g~i= 2 z [ z + ( C o / A C ) ] ~- ~ z 2,

(2.17)

where the latter approximate equality holds for a film of very small grains, so that Co << AC. Combining the familiar expression for the maximum Josephson current,

~

E j = 7r2A

(2.18)

where d is the BCS energy gap and rYN is the normal state conductance, with the zero-frequency result for /xC [13-15],

quency dependence of AC. The relevance of disorder remains an open problem. Finite-frequency effects have, on the other hand, been investigated by one of us [20] and do not seem to alter substantially the results. It is important, therefore, to examine the accuracy of the self-consistent single-grain treatment to determine whether the approximate agreement with the experimental results persists as fluctuations are treated more accurately. Equation (2.17) suggests a scheme for studying such fluctuation corrections. As z, the number of neighbors, becomes very large, fluctuations average out and the self-consistent theory becomes, as a solution of the model we are considering, exact. For finite z, it is natural to develop an expansion in inv e r s e p o w e r s o f z:

h

2.19) g~,=~

yields go = i@~r-2 .

2.20)

The small effect of the phase dependent term in A C which has been omitted in eq. (2.19), is evaluated in Appendix B. In terms of the dimensionless resistance, 0"~ t

r= h/4e2 -

RN

Ro "

(2.21)

the phase coherence criterion, eq. (2.17), becomes r
(2.22)

where we have introduced the convenient reference resistance, R0 = h / 4 g 2= 6.5 k~. For a square lattice, z = 4 , we obtain, for the threshold sheet resistance, R~ = R ~ =5.7 k ~ ,

(2.23a)

while, for a hexagonal lattice, z = 6 , RE - R~ , ~ =4.9 kfl

(2.23b)

Thus, the simple self-consistent treatment not only explains the universality of the superconducting threshold, i.e., the fact that it is independent of the energy gap, A, but is also gives values for R~ which are in fair agreement with the experimental result, R ~ -~ 6.5 kfL Our treatment has neglected disorder and the fie-

1+--+--#+....

(2.24)

In a previous publication [ 6 ], we obtained c~L= 0 and c~2= 6/5 by allowing for the correlated fluctuations of the phase of a given grain with that of one of its neighbors. In this paper, we improve upon the above calculation by considering larger grain clusters. We will show that c~2 is apparently 6/5 times the number of neighbors that are correctly treated so that, when this number is taken to be equal to z, the actual expansion acquires the coefficient c ~ = 6 / 5 . We will then make an independent calculation, along the lines of Bethe's treatment of the Ising model [21] which confirms this conjecture. It is clear that the cluster expansion that we seek to develop has a close analogue in the classical statistical mechanics of Ising spins. Indeed, the Hartree approximation is equivalent to mean field theory, while improvements on the mean field result, by considering spin clusters, have a long history [21 ]. Therefore, it is conceptually helpful to revisit this familiar terrain. Of course, our approximations are primitive in the lsing context, where the task is the easier one of counting configurations rather than that of computing the quantum mechanical ground state energy. We pay particular attention to the large-z expansion of the cluster calculations of the Ising transition, although such an expansion is in no way needed for solving the two-dimensional Ising model. We study this expansion because, in our analogous perturbative treatment of the quantum mechanical

B. Mirhashemand R.A. Ferrell/ Universalcriterionfor the onsetof superconductivity problem in section 4, the expansion in powers of z - l is the only method at hand for attacking the problem.

365

For b << 1, the eigenvalues of the 2 × 2 transfer matrix F, where Z=tr(FU), are 2, = 2 (cosh K) [1 + ½b2 e2K+ ... ]

(3.9a)

and

3. Ising clusters

22 =2 (sinh K) [ 1 - ½b2 eZK+ ... ] .

3.1. Two-spin cluster

(3.9b)

NOW,

The simplest approximation to the nearest-neighbor Ising model partition function

exp(K~ a,aj~

Z=

\

{a,=_+ 1}

ij

(3.1)

/

J

(3.2)

In this approximation, the critical value of the coupling constant K, K~ v, is (3.3)

For a two-spin cluster, the effective partition function is Z=

~

2(tanh K)N 1 1 + ~ - n h - - ~ ~3"

( 3.11 )

so that, for b~O,

-N

J

K~V=l/z.

(3.10)

10lnZ

is the mean field replacement a, Z aj--,a, Z = z < a ) a , .

k,~(] _1

exp {K[ ( z - 1 ) < o') (o', + o'z) + 6,o'z ] } .

Oh

= b e2X~ 1

Therefore, from eq. (3.8), the critical coupling constant, Ki N) , is given by

2(tanhK~N))N 1 K~N)(z--2) exp[ZK~ u)] 1 - 1T(-~anh--~u)~-ul=l (3.12) Thus, since N_> 3,

{a~,2 = -+ 1 }

(3.4) A simple calculation gives for the critical coupling constant, K~ 2) , 2K~ 2) ( z - 1 ) = 1 +exp[ -2K~ 2) ] ,

(3.5)

so that

K~2)=

1+ ~'2 + ' "

1 =KcMF(1 + ~--5+ ' ' ' ) .

(3.6)

~

exp(K~iaiai+,+b~i ai) ,

(3.7,

where

b=K(z-2)
(3.13)

independent of N. Note that the chain approxima. tion, which allows for the fluctuations of two neighbors of a given grain, results in a coefficient of the z -2 term in K~/K~ v which is twice that of the twospin treatment. This suggests that if the fluctuations of a fraction f of the neighbors of a given spin are taken into account, the result might be

K~=K~V(l + f + ' " )

Kc = K ~ F ( 1 + 1 ) .

For a closed chain of N-spins,

{a,=_+ J}

,

(3.14a)

Taking all z neighbors into account (f= 1 ) would then give

3.2. Closed N-chain

Z=

Ki N)=K~ v 1+-~ +...

(3.8)

(3.14b)

3.3. Multiple neighbors To check the above idea, we must allow for each grain to interact with more than two fluctuating

B. Mlrhashem and R.A. Ferrell/ Universal criterionjar the onset qf .~uperconducttvtty

366

neighbors. Thus, we consider two neighboring closed chains o f interacting grains so that each grain has z' =fz " a c t i v e " b o n d s and z - z ' m e a n field bonds. Some realizations of this m o d e l are as follows: on a t w o - d i m e n s i o n a l hexagonal lattice, z = 6, z' = 4; on a cubic lattice in D dimensions, z = 2 D , z' = 3 . We will derive the desired result in a fashion which parallels our subsequent t r e a t m e n t o f the granular film. F o r the expectation value of a given spin, say O'1, w e have Z e x p ( K 7 2 ' a, ai) e x p ( b 2 a~.)al (O. 1 ) =

:a,:

_u_

k

exp ( K ~ ' o',a~) exp (b ~ o'k) '

ia, l

(a) =b[1 +z'tanh K+z' (5-

(3.15)

where

The expansion in brackets is similar to the familiar high-temperature expansion of the cubic lattice zerofield susceptibility [22], (3.21b)

Using eq. ( 3.16 ), self-consistency gives, for the critical coupling constant, /(~,

l=(z-z')K,[l+z'K,.+z'(z'-l)Kc+'"]

b=K(z-z')(a)

,

(3.16)

and, in the p r i m e d sum, every pair of spins connected by a b o n d is counted once. F o r d e t e r m i n i n g the threshold value o f K, we need ( a l ) to first order in b:

:-,I

~

K~.-~-

,;

In,[

1+

( X

(3.17)

Z exp ( K Z' a,a,)

, (3.22a)

or

v ~ exp (KV'~ aiai)/aka l

{d'L ) =

1) tanh2K+...] . (3.21a)

Zo = 1 + 6 tanh K + 3 0 t a n h 2 K + ... .

1,

I/

is some neighbor of site 1 and z' - 1 terms of the form a~.a~.., where 1" is some neighbor of 1', other than 1. In the sum over m, only the term where a,,,=a~ survives, so that

i

+

z' l-

+'"

z'2 + -¢

z'(2'-l) _2

+...)

U

In the d e n o m i n a t o r of eq. (3.17), we have Z=

3~ ( c o s h K ) X = ' , " 2 [ l + t a n h K ~ ' a, aj + t a n h 2 K ~ '' a, a t a k a l + " ' ] ,

(3.18)

This result, restricted to f<< 1, strengthens the case for the validity o f the conjecture o f eq. (3.14a) for the full interval f < 1.

u/,/

where N is the total n u m b e r of spins on both chains. In the d o u b l e - p r i m e d sum, sites i and k are distinct, with j and l their respective " a c t i v e " neighbors, and, if sites k a n d j coincide, i and l are distinct. This implies that

3.4. Bethe-type duster The generalization o f Bethe's expression [21 ] for K~, for an arbitrary c o o r d i n a t i o n number, is KL'=-~In(I-2)_

Z = 2 ~(cosh K) x--/211 + 0 ( K 3 ) The n u m e r a t o r o f eq. (3.17) is

Z(a)=

~ X a,,,a,(coshK)'~:/2[l+tanhK~' : a l '~ nl

a,a~

/I

+ t a n h 2 K ~ " a, ajakal + ' " ]

(3.23)

(3.19)

] .

.

(3.20)

l/l
I n the O ( t a n h K ) term, i and j must coincide with 1 or m for a nonzero contribution. There are z' such

terms, in the O(tanh2K) term, in the double-primed sum, there are z' terms o f the form a Ial. , where 1'

which, e x p a n d e d in powers o f z ~, does indeed verify eq. (3.14b). In the traditional Bethe approxim a t i o n [21 ], one allows for lhe interactions o f a central spin, ao, with its nearest neighbors, which we now denote by a,, with u = 1, 2 .... z, where each of the latter spins is acted on by a mean field b due to all o f the other spins in the lattice, so that Z=

~"

exp

a,,+

.

(3.24)

B. Mirhashemand R.A. Ferrell/ Universalcriterionfor the onsetof superconductivity One then computes ( a o ) and (a~) and eliminates b by requiring ( a o ) = (a~). The above equation is then satisfied, with ( a o ) = ( a ~ ) > 0 , for K>KBc, where K~ is given by eq. (3.23). Because the analogous calculation for the determination of the superconductivity threshold resistance runs into certain complications which will be discussed below, we seek an alternative derivation of Bethe's result, which can be applied in a straightforward fashion to the quantum mechanical problem at hand. As we will shortly verify, the key ingredient of the Bethe calculation appears to be the choice of a cluster for which ( a ) is not the same at all lattice sites for an arbitrary value of the effective field b. This necessitates the adjustment of b in order to restore the inherent symmetry of the complete lattice. Thus, b is no longer tied to K, as it is in self-consistent expression of the type b = K ( z - z ' ) ( a ) , and, in fact, becomes adjusted to a value smaller than the self-consistent expression. This partially compensates for the shortcomings of effective-field theories in allowing for the disruptive fluctuations, and has the effect of raising Kc. To investigate this idea, we consider an open threespin Ising chain so that

Z=

~

{a,,a2,o~)

exp[K(ala2+a2aa)+b(z-1)(al+a3)

+b' (z-2)42],

(3.25)

where we have temporarily distinguished between b and b' in order subseqently to be able to obtain ( at ) by differentiation. Performing the sum over spin configurations gives Z = 8 [ (cosh2KcoshZb(z - 1 )

tanhKc-

1 z-l'

4. Cluster calculations of the threshold resistance

4.1. Two-grain cluster The mean field treatment of section 2 shows that, for small grains, Co may safely be set equal to zero in eq. (2.1). The Lagrangian is then given by

Ac2

L=--~- o (Vi-Vj)2+Ej ~ c o s ( ~ , - 0 s )

h2AC _ ~ e2 ~ ( 0 i - ~ j ) 2 + E j ~ c o s ( 0 i - ~ j ) (4.1b) 6 6 with both double sums restricted to nearest neighbors. We proceed in analogy with the spin cluster calculations of section 3. For future reference, we first perform the Legendre transformation to H for the general case of an open chain of N grains and later specialize to N = 2. As before, we set ( 0 j ) = 0 for the off-chain grains so that the chain kinetic energy is N--I

~. ~),~,+,.

To simplify the equations, we have temporarily set 4e2/AC= h = 1. The "velocities" 0g may be collected into a colums vector ~. Rewriting eq. (4.2) as

W= li)XM~

(o"1)=(0"3) =b[ ( z - 1 ) + ( z - 2 )

defines the symmetric matrices (3.27)

(4.2)

i=1

so that, setting b=b' << l,

× t a n h K + ( z - 1 ) tanh2K],

(4.1a)

-

i=1

(3.26)

(3.28)

which is equivalent to eq. (3.23). From eqs. (3.27a, b), b = (1/z) (a)
N

+ 2 cosh Ksinh Kcosh b ( z - 1 ) ×sinh b ( z - 1 ) sinh b' ( z - 2 ) ]

Eliminating b by setting ( a t ) = ( a2 ) = ( a 3 ) , gives

W=lz E ~-

+sinhEKsinh2b(z - 1 ) ) cosh b ' z - 2 )

367

M = z ( I - I K)

(4.3)

(4.4)

while

(a2)=b[(z-2)+2(z-1)

tanh K ] .

(3.27)

and K. In the above equation, I is the identity matrix while the entries of K are all zero except for those

368

B. Mirhashem and R.A. Ferrell I Universal criterion jor the onset olsuperconducttvtty

that border the diagonal. The latter are all equal to one. For example, for N = 4 ,

/t2 = [ l ~Pl2 + ½ p 2. + 71P l P2 L

K=

1 0 0

--g2 H'I'2 --g, (H] + H ~ ) ] ,

"

1

with

The dimensionless m o m e n t a , p,= ( 1/h) (OL/O0;), may also be collected into a column vector (4.5)

p=MO,

W : ½pTM - ~p.

H',:'j= cos (0; - 0j )

( 4.6

(4.14)

The two-grain coupling constant is g2=(z-z-l)go

From eq. (4.4), ~ ( / + I K +z

1~ z- K 2 + ' " )

'

(4.7

,

4e 2 [

w=x-G

1 N-- 1 L½, = I ~ P 2 + z , 2 ~ P ; P ' + ' =

1 "~ 1

+ ~ L ,=2

1 x-2

+75

Z

1

p,~+ - - (p~, + p ~ . ) 2z 2 P;P,+2

1

(4.8)

.

/=1

Specializing now to the case of N = 2 , we have K 2 = I and iI

M-'=

( 1 1 ) I 1-t- 75 + )q + . . . 1

+

1-F

z K

z

- z2--1

'..

(4.9)

(I+~K)

(4.15)

while gl = ( z - 1 )/tg2 ,

To order 1/Z 3,

(4.13)

and H', = c o s 0, •

so that the kinetic energy becomes

M-'=

(4.12)

(4.16)

proportional to the order parameter kt = (cos 0 ) , is the effective field due to z - 1 non-fluctuating neighbors of each grain. Unfortunately, even for this relatively simple case, the Hamiltonian cannot be diagonalized exactly. We resort, therefore, to the perturbative calculation of g~ in inverse powers of z. From eqs. (4.15) and (2.17), the threshold value for g2 is O ( 1 / z ) , while g~ is infinitesimal near threshold. Threrefore, the calculation of e~ j) and e~2) in A E G = - 2 g 2 [1 +e~ ~)g2 +e~2~g2 + ' " ] + O ( g l 4) , (4.17) where AEG = E~ ( g l , g2 ) -- Ec; (0, g2 ), determines c~j and c~: ofeq. (2.24). To calculate these coefficients, we employ the Wigner-Brillouin perturbation theory. In the free-rotors basis, the energy eigenfunctions are

1

q/(01,02 ) = ~ exp(inlgh ) exp(in202)

so that W=_~z2_

1

½p2+,~p2+ 2

PIP2

(4.10)

Thus, the two-grain Hamiltonian is 4e 2 z /~2 , H2 -- AC g 2 -- 1 where

(4.11)

-u~, ( 0 , ) u,2(02) ,

(4.18)

with, from eq. (4.12), unperturbed eigenvalues equal to ½ ( n Z + n 2 ) + ( 1 / z ) n ~ n 2 . The non-zero matrix elements of H; and H;~ are just - ½gl and - ½g2, respectively. The angular m o m e n t u m q u a n t u m number n; is the n u m b e r of excess Cooper pairs on the ith grain.

B. Mirhashem and R.A. Ferrell / Universal criterion for the onset of superconductivity X

(a)

X

×

369

0AEG -=2/L=-- - =4g~ [1 + 2 g ~ + ' - ' ] .

(b)

X

X

(c)

0g,

X

(4.22a)

Substituting g~ from eqs. (4.15 ) and (2.17 ) into eq. (4.22a) yields

X

X X

(d)

X X

(e)

/t=2

1 IX

f)

Fig. 1. Graphs a through e are for the third and fourth order twograin contributions to the ground state energy. The two solid lines represent a pair of neighboring grains and the dashed lines correspond to their Josephson coupling. The crosses indicate the effective field due to non-fluctuating neighbors. Graph f is for a fourth order three-grain contribution.

The first term in brackets in eq. (4.17), the 0 (g2gO) contribution, is just twice the leading term

in eq. (2.15 ). Diagrams a and b in fig. 1 give the corrections to AE6 which are first order in g2, which is indicated by the dashed lines connecting the solid lines representing the two fluctuating grains. The effective field gl is represented by a cross. From diagram b, we obtain - 2 g 2 g 2 e ~ ~b) = ( _ ½g~ )2( _ ½g2 ) ( 2 ) 2 = _ 2 g 2 g 2 ,

(-½)(-½)

(4.19) so that ~ b ) = 1. In eq. (4.19), the factors of ( - ½) in the denominator are the energy differences for intermediate states with n~= + 1. One factor of 2 comes from the two directions, in which H" can transfer a pair, and another accounts for the contribution of the time-reversed diagram. Similarly, the four diagram of type a contribute

1+

g,-2

z-I

gl,

(4.22b)

effectively replacing the factor z - 1 in eq. (4.16 ) by z and thereby restoring the single-grain Hamiltonian to its full mean-field strength. It follows, by substitution ofeq. (4.16) into eq. (4.22b) that

,4 2c, An alternative, simplified method for arriving at eq. (4.21), which will be useful to have for later application to the larger clusters, is based on grouping graphs la and lb together. Let us first evaluate the sum of these two graphs only insofar as the upper line is concerned, which we take to correspond to grain 2. We denote by AUo the first-order change induced in the wave function of grain 2 by - g l H~. For the corresponding partial expectation value of the inter-grain interaction we find, integrating only over the coordinates of grain 2, ( H ' ( 2 ) 2 = (Uo +Auo, HI'2 (Uo +AUo ) )2

= (Uo (2), H'(2AUo (2))2 + (AUo (2), H72uo (2))2 ,

(4.23)

with the first and second terms corresponding to graphs la and lb, respectively. On the other hand, with the interaction written out as

1

~l~) _ -1 -~z '

(4.20)

where the z-dependent factor in the denominator arises from the P~P2 term in /12. To calculate the O ( z -~) correction to the threshold, z-~ may be set equal to zero in the denominator ofeq. (4.20) since it contributes an order z-2 term to the expansion in eq. (4.17). Thus, ~ G = - - 2 g ~ [1 +2g2 + ' " ] + O ( g ~ ) , SO that (COS 01 +COS 02 )

(4.21)

- g 2 H 7 2 = - g 2 COS(¢l - - 0 2 )

= - g 2 cos 01 cos 02 - g 2 sin 0~ sin 02 (4.24) and with the local gauge (sin 02 ) 2 ~ 0, w e obtain --g2

chosen

such

that

(H72)2 = --g2 cos 01 (COS 02 )2 = - - / t g 2 COS 01 •

(4.25)

Added to -g~H'~ this effectively replaces z - 1 , the number of neighbors in eq. (4.16 ), by z, thereby restoring the strength of the single-grain Hamiltonian to

370

B. Mirhashem and R.A. Ferrell / Universal criterion for the onset O['superconductivity

its full mean-field value, as already noted following eq. (4.22b). By substituting ( c o s 02) = / ~ = 2 g j from eq. (2.16), we bring eq. (4.25) into the form - g 2
cos 0~ .

(4.26)

The integration over the coordinates o f grain 1 can now be carried out, treating 2 as an effective single-grain perturbation, o f the same form as g~H~, but with coupling constant gT =2g~g,. Because - g e a can act either before or after g~ H't, the complete t h i r d - o r d e r term ( p r o p o r t i o n a l to g ~ & ) in the ground state energy, from the sum o f all graphs o f the type l a and l b, is - 2gl g'~ --- - 4g~g2,

( 4.27 )

in agreement with eq. (4.21). Again, it is evident that the net effect o f the interactions in diagrams l a and l b is to restore the mean field acting on each grain to its full value, zing2. Calculating the average interaction with an " a c t i v e " neighbor serves to group the latter with the z - 1 mean field neighbors so as, in effect, to increase the total n u m b e r of such neighbors to z. D i a g r a m s c, d, and e in fig. 1 are of orderg~g_~. To obtain the 1 / z e correction to the m e a n field threshold, we can ignore the m o m e n t u m cross term. We then obtain, for the contributions to e~2~ of eq. (4.17), e~2c~ =.~, 6 E(2d}__ 3 and eC"'~ 6 There are e --Ta, further contributions to e~2~ from the expansion o f the p e r t u r b e d energy d e n o m i n a t o r s of the terms of 0 order g~gO and g~g;. These c o r r e s p o n d to the insertion of the O ( g l g ° ) graph in the O ( g ° g 2 ) one, which contributes - ½ to e~:~, and the insertion of the O ( g ~ g 2 ) graph into the O ( g 2 g °) one, which contributes - 1 to e~-~. It is natural to c o m b i n e these insertions with d i a g r a m s d and c, respectively, to obtain g~2~ +½ and g~_ed~=-- 3, SO that the entire c o n t r i b u t i o n to e~2~ comes from d i a g r a m e in which H ' and H" alternate. Inserting e~~'~ =e~ 2~ = 6 into eq. ( 4.17 ), we obtain AEc=-2gl

[

l+g2+

+~gs+O

+54~].

(4.28b)

The self-consistency condition on ,u gives 1=2(z-1)g~

+ 725~e+ O

,

(4.29)

so that g~=2zz

1+

z ~ +O

,

(4.30)

or, using eq. (4.15),

,[

- 2z 2

l+

Comparison

z- + with

o(')1 z3

eq.

"

(2.24)

(4.31) gives ~Yt=0

and

OQ~g.

4,2. Closed N-chain We were led [6] by the above calculation to conclude that the corrections to the mean field result. g~) = l / 2 z 2, were O ( z - 4 ) , and, therefore, very small for z = 4 or 6. But, by analogy to the Ising calculations o f section 3, we must proceed more cautiously and examine larger grain clusters. As we now proceed to demonstrate, this study will lead us the conclusion that the correction is O ( z - 3 ) _ still small, but not as small as we earlier supposed. In the Ising case, the closed N-chain calculation in section 3 above gave a K~!H/z 2 correction to the mean-field result which was twice that o f the two-spin case. Let us, therefore, consider the analogous p r o b l e m for the superconducting grains. To obtain the kinetic energy expression for this case, we replace N by N + 1 in the second sum in eq. ( 4 ) a n d identify the first and ( N + l ) t h velocities, to obtain, from eq. (4.8) the closed N-chain expression

w=~

~ p2+1-

P, Pi+l + 25

PT+"" (4.32)

l_z_ 1 + O ( g 4) ,

~-2g~[+-I

(4.28a)

Inside the brackets in eq. (4.32), we keep diagonal m o m e n t u m terms to order 1 / z z and off-diagonal terms to o r d e r 1/z, since the latter do not contribute

-

B. Mirhashemand R.A. Ferrell/ Universalcriterionfor the onsetof superconductivity to zeroth order in g2. Working to the stated accuracy, the 1/z 2 modification of the diagonal term factors out and we obtain 4e2( 2)[~ W=~Tz 1+75 :

i=l

1~ 1 p2 + _ PiPi+l • Z

bution from the three-grain diagrams to the ground state energy is, therefore, - 2N(2g~g2 )2 = _ 8Ng~g 2. Thus, in place of eqs. (4.28a, b), we now have

AEo=-Ng21

i=1

371

I

292 1+2~2+ l - z - '

A_ 12.7,2 T~-~2

(4.33) +8g~ + O ( ~ 3 ) ] + O ( g 4 )

The chain Hamiltonian is, therefore, 4e2(~) HN = X-(~z 1+

(4.34a)

/~U,

~__Ng2[l+492+ 292z +!~g21 "

(4.37)

The self-consistency condition, - d E J d g l =N× (cos 0) =N/z, gives

where

HN=l ~ P 2 + -1 ~ PiP,+l--& ~ H~5--gl Zi=l

i=1

ij

H,', (4.34b)

with H;] and H', as defined in eqs. (4.13 ) and (4.14 ), and z

g2-- l + ( 2 / z 2 ) g o ~- Z-gl = ( Z-- 2 )fig2 •

go,

l = 2 ( z - 2 ) g ~ [ l + 4 g ~ + 2 g ~ z +~g~2]._-

(4.38)

i=1

(4.35) (4.36)

Compared to the two-grain case, there are now N / 2 as many O(gEg °) contributions and, since each grain on the closed chain can now interact with either of two neighboring grains, N times as many contributions from diagrams a through e. (The energy denominator corrections go through as before with improper contributions to the perturbed energy denominators canceling against disconnected diagrams.) The new feature is the appearance of threegrain Feynman diagrams, (such as shown in fig. l f ) , in which one grain interacts once with each of the two neighbors on its sides. Each of the neighbors is acted upon once by the mean field. There is no mean field acting on the central grain. Since these diagrams are of order g~, we need only their leading conbtibution. The momentum cross-terms can consequently be ignored in calculating the three-grain diagrams. The argument leading to eq. (4.27) can be used to sum all such diagrams. We simply have two external fields of strength g* = 2g~g2 acting on the central grain and an additional factor of 2 to account for the relative order of the fields. The total contri-

Calculating g~, and using eq. (4.35), we find

1

1(1+ 75]~zz\ 2~1(1 + ~2) z2

g8 - z[ 1 - ( 2 / z 2) ] g~ = z 1(1+12) - 2z 2 ~

,

(4.39)

so that the correction to the mean field result, as already reported [6], is twice as big as that given by the two-grain calculation. This parallels the result that we found above, in section 3, for the Ising model.

4.3. Multiple neighbors The next approximation we are led to consider is one in which the interactions of a given grain with some z' << z neighbors are correctly treated. The case z' = 1 is the two-grain case, which gave for the coefficient a2 eq. (2.24), a2 = 6, while z' = 2 is the closed chain, with a 2 = ~ . As mentioned in section 3.3 above, the case z' -- 3 can be realized on a square lattice by considering two adjacent rows. On a hexagonal lattice, this arrangement would be a realization of the z' = 4 case. Equation (4.2), is now replaced by (4.40)

W = 1 E Mij~i()J, /J

where

Mo=z

(1)

I o - zKo

.

(4.41)

372

B. Mirhashem and R.4. Ferrell / Universal criterion.for the onset of superconductivity

As for the single chain, l,s=O unless sites i and j coincide and K0 = 0 unless sites i and j are connected by an "active" bond, in which case, it equals 1. To have (cos ~;) be identical on all the sites, we imagine that the two chains are wrapped on a cylinder. M, is an element of a symmetric operator M, so that (4.42)

W=½ Z ( M - I ) , J P , PJ

.']+O(~).

+½(z'(z'-l))'8g~+

(4.48)

To O ( 1/z 2), self-consistency gives 1 = 2 ( z - z ' ) ~ [1 +2z'~,~ ]. Therefore,

U

with M -~ given by eq. (4.7). We need the off-diagonal elements of M-~ to order 1/z 2 and the diagonal elements to order 1/z 3. Now, (4.43)

( K 2 ) , = • K,sKj, = z ' , J

so that the Hamiltonian is

4e ( H=~-~z 1+)5

- 2z

1+O

(4.49)

as before, and for the same reason. In the next order, 1 = 2 ( z - z ' )g~[ l + ~ ( 2 ) ( 1 + ; )

,

+ ~z'(~z2 )

(4.44) + Z' (Z' -- 1 )'4' 4@~2+ O ( ; )

]

with ~ 2 Z ~ { 1--3 : +O ( ~ ) ]

.KI= ½ ~. p 2 --t- 1 ~ii p, p j _ ~ l ~. H,< - R 2 ~ H,'j ;

Z

t

U

(4.45) The double sums are over "active" pairs, each counted once and H~ and H~} are given in eqs. (4.14 ) and (4.13), respectively. Also, g~ = ( z - z' )/2~2 ,

z

l+(z,/z2)go

SO that gD~-

(z)

~- z -

(4.47)

go.

We denote the total number of grains on both chains by N. Since the structure of/~ is very similar to the single closed chain Hamiltonian, (4.34b), we can easily write down the expression for the ground state energy by comparing the contributions for the two cases. The O (g2gO) contribution is unchanged while that of the two-grain interactions (diagrams a-e) must be multiplied by ½z' since each grain now has z' active neighbors instead of just 2. The three-grain contribution must be multiplied by the number of pairs of active neighbors, ½(z' (z' - 1 ) ). The appropriate modification of eq. (4.37) is, therefore, z'g2

g2~C .

(4.50)

6 r We thus obtain c~2= 3z , which is consistent with the single-chain calculation, z'--2, considered earlier. Thus, the Ising analogy continues to hold (cf. eqs. 3.5 and 3.13).

4.4. Bethe-type cluster

Finally, we consider the three-grain Bethe-type approximation. The off-diagonal velocity matrix for this case is K=

0

,

1

which has the property that K then sums to

1~ -2

AEG = -NR12 l + z ' g 2 + l _ z _ 1 + ½z' ( - r )g2

1+ -

- 2z 2 1+ 3

(4.46)

with g2-

,

![ M

1=

_

3 ---

2K. Expansion (4.7)

1 (K K2)] I + 1 - (2/z 2) + ~5_ -

(4.51,

B. MirhashemandR.A. Ferrell/ Universalcriterionfor the onsetofsuperconductivity Denoting the phase of the central grain by 0c and that of its neighbors by 0L.R, the Hamiltonian becomes

4ze 2 H = AC(z2_2)/-7,

(4.52)

1-(2/z) ,l'~, = ~n 2E°2+ 1 - - ( l / z ~ En jg2.

1

H=[lp2 + I ( I - - -~)(p2 +pE )+ ~ Pc(PL +PR)

(4.60)

To O ( 1 / z ) , this condition is 4(1 - 1)E~° +2El'g~ =2E °2-al'"cwEl ~;2-

where

373

(4.61)

The O(g ° ) contributions are Eo2° = - 1

(4.62)

1

+ -~SPLPR--g2 (COS(0L --G )

and

+ C O S ( 0 R - - 0 c ) ) - - g c COS 0c

%02=

- g l (cos 0L +COS OR )1 ' with

(4.53)

gc = ( z - 2 ) h B

(4.55)

and gl = ( z - 1 ) h l .

nrnl

(4.57)

.Cn m l ~~c m ~ d;1~52 l,vn ,

we have m--I

x" ..... ~.mtfZ--2~ --
..t+m-lg~ (4.58) 61

\ Z-- 1 /

4El+1 'zl

x" t. ml(Z--2)m.l+m--lon 81 62.

--2(COSOL)= /.2, "~n / 7 , /

1

g~- 2z'

nml

\ Z-- I /

Near threshold, with gl and g¢ infinitesimal, (cos G ) = (cos 0L) implies

(1/z))E. +2E51

(4.66a)

or, from eq. (4.54), 1

g~_ 2z2,

(4.66b)

which is the usual mean field result. To O ( 1/z 2), eq. (4.60) gives 1

(4.59)

(4.65)

Equation (4.64) then gives

g~=~zz + ~ z 2(2+E2o - ~ 1,o2~ 2 J-

and

(4.64)

where E11 is calculated using diagrams of types a and b in fig. 1, except that, now, the field acting on the central grain is gc while it is gl for either of its 2 neighbors. Thus, El I is just twice the coefficient of the g2g2 term ofeq. (4.28a):

(4.56)

As in the Ising model, the effective Bethe field, ha=h'B, is adjusted to obtain (cos G ) = ( c o s 0L)=(COS 0R). We temporarily distinguish between ha and hh in order to obtain the individual expectation values of the grain order parameters using the Feynman-Hellman theorem. From the expansion of the ground state energy

(4.63)

Thus, to this order, -4 g ~ - zEll ,

2)

EG = ~,

-2 1_ (1/z 2) •

(4.67)

The two-grain contributions to E~° and e2°2, which are equal, come from diagrams c-e and, as explained before, sum to just the contribution of diagram e. From this diagram, we obtain exactly the two-grain result. (In the two-grain case, the solid line in diagram e on which the crosses lie may be either of the two grains. In the present case, it may be either of the two neighbors of the central grain. Diagrams a

B. Mirhashem and R.A. Ferrell / Universal criterion jbr the onset (?fsuperconductivtty

374

and b, on the other hand, have no such factor of 2 in the two-grain case.) Only co2 has three-grain contributions. In the case of e~°, both crosses must lie on the central line so that the outer lines do not return to the ground state. The three-grain contribution to co2 is one-third that of a closed chain with N = 3. Thus, to order 1/z 2, 1

1

l

1

g~ = 2z + ~ z 212+ ½((co2) 2-gr~in-- (e °2 ) 3-g~,i.)]

HR

P~

+ - Po

k=l

Pk + ~

Pt,

-g2k~, ~ cos(Oo-Ok )-g,

~:l ~ cos0kl'

where g~ = g o ( z - I )

2z + ~ 5 z 2 [ 2 + ½ ( -

12

+8)]

(4.68)

or

469,

Equation (4.54) then gives

,(6)

g~i = 2z----~ 1 +

.

(4.70)

Again, the Ising analogy is verified. Thus, in eq. (2.24), we have a ~ = 6, as conjectured. From eq. (2.20), the dimensionless critical resistance is given by

3 ~1/2 r= \ l~go J -~ 8 z l + ~ z 3

(4.71)

(4.72)

For a square the lattice, the threshold resistance from eq. (2.23a) is reduced by 15%. For the hexagonal lattice, the threshold from eq. (2.23b) is reduced by 10%. It is difficult, at this time, to make a quantitative estimate of the effect of higher order terms on the above results. For completeness, we conclude this section by giving the Hamiltonian for the traditional Bethe-type approximation consisting of a central grain surrounded by z neighbors. The mean field now acts only on the neighbors. The Hamiltonian for this model is

4.74)

and gl is eliminated by imposing (cos 0o ) = (cos Ok ) •

1 2z(l+ )

4.73)

4.75 )

The coupling of the central phase, 0o, to all of its : neighbors increases the multiplicity of otherwisenegligible high-order diagrams and complicates the calculation. The Ising analogy leads us to believe. however, that the final result for this model will agree with eq. (4.72).

5. S u m m a r y and c o n c l u s i o n s

We have attributed the apparent universality of the onset of superconductivity in granular films [ 1-3 ], Rx-~6.5 k~, to the disruption of phase coherence among Josephson-coupled grains. This "mesoscopic" quantum effect leads to a universal normal state resistance criterion for films consisting of very small grains whose reactive response to voltage fluctuations is determined by virtual quasi-particle tunneling. While mean field theory gives results that are in satisfactory agreement with experiment, fluctuation corrections, which we have calculated in analogy with cluster expansion approximations of the Ising model, reduce the predicted threshold by a factor of 1 - 3 / 5 z , to leading order in the inverse coordination number. This raises the likelihood that an exact treatment of the model under consideration will not quantitatively account for the threshold observed in refs. [ 1-3 ]. Model improvements, such as the inclusion of disorder, would therefore need to be considered. In the meantime, measurements of the critical current and the kinetic inductance will test the semi-quantitative mean field predictions [23,24] to establish whether the basic physical ingredients of

B. Mirhashem and R.A. Ferrell / Universal criterion for the onset of superconductivity

the model, Josephson phase fluctuations and virtual quasi-particle reactance, capture the most important features of small-grain two dimensional films. In summary, we have demonstrated that the fluctuations cause the onset resistance of a square array to be lowered by 15% relative to the mean field value of 5.7 kfl. The frequency dependence of AC produces a further decrease of approximately the same size [20]. Adding the phase effect calculated in Appendix B, we expect a total reduction by roughly onethird, or, say 1.9 kfl, yielding a net onset resistance of about R~z -- 3.8 kfl. It is gratifying that very recent measurements [25] on a set of granular tin films exhibit global superconductivity at a temperature of 1 K in those films whose normal state resistance is less than 4 kfl and do not show global superconductivity in the films with normal state resistance greater than this value.

2 ro

C--~ro

4

ro

375

3 ro

5

The above series is not useful for ro > b. Although an exact solution of Laplace's equation is available [26], the leading dependence of C on b/ro, for b << ro, can be easily calculated by observing that, in this case, the electric field lines are almost parallel to the line joining the centers of the spheres, which we take to be the z-axis. If the transverse distance, p, away from this axis is less than ro, the perpendicular distance from the z = 0 midplane to a point on the upper sphere is z ~- b+p2/2ro. The average field strength is = Vo/z, giving pc

Q". ~

Z~pdp= T

\2bro +1

)

.

(A4)

o

Appendix A. Mutual capacitance of small spheres

Now, Pc -~ ro so that, to logarithmic accuracy, ro

To verify the asserion that the geometrical capacitance is negligible compared to the effective quasiparticle capacitance, we consider the electrostatic problem of 2 metallic spheres of radius ro whose centers are separated by a distance D = 2 (to + b). If the electrostatic potential on the surface of the spheres is _+ V0, the charge on each sphere, to lowest order in to~b, is _+roVo, so that the geometrical capacitance is just C=2V ° An expansion in powers ofro/b can be obtained [25 ] by placing successive images inside the spheres, along the line joining their centers. The first-order correction can be calculated by recognizing that, to this order, the sphere of potential + Bo "sees" a point charge of strength -roVo a distance D from its center. Therefore, an image whose charge is ( + ro Vo) (ro/D) must be placed a distance r g/D from its center. The capacitance is, therefore, 2

r°( C=-~

r° + 0 ( b ) ) 1+ ~-~

.

Continuing this fashion, one obtains

(A2)

ro

C=~ln

(A5)

b"

The equation

has the right limiting behaviors, eqs. (A2) and (A5), for r0 << b and ro >> b, and can be used to interpolate between them. It is clear, in any case, that for a wide range of values of the ratio ro/b, C ~ - ro. From eq. (2.19), on the other hand, the quasi-particle capacitance is 3h 3e 2 AC= ~-~ S aN -~ ]--~~- ,

(A7)

using ~ 1 =h/4e 2, which expresses accurately the experimentally observed threshold. Thus, the ratio of the geometical and quasi-particle capacitances is approximately C

16

AC-

A

3 (e2/ro) = "~

=8(1"76)

(eZ/2ao) e2/2ao '

376

B. Mirhashem and R.A. Ferrell / Universal criterion fl)r the onset of superconductivity

where ao-~ 0.5 A. is the Bohr radius, e2/2ao = 13.6 eV is the Rydberg constant and we have substituted the BCS formula for the energy gap. Thus, c

/z=2g'l =2gl - ~ /I z , giving 7 g~ = ~,u,

~ (3× AC -

lO-5)(r°~Tc (K). \a0 /

(A9)

If we take ro-~ 100 A, and T~_~3.7 K, the transtion temperature of tin, we find C/AC~_ 0.02, so that the geometrical capacitance is indeed negligible.

(B6)

an increase over its previous value by a factor of 7Self-consistency consequently requires an increase in g~, relative to eq. (2.17), by this same factor. This reduce the universal resistance criterion ofeq. (2.22) by the factor (-67)~/2-~0.93. If a fully symmetrized form is used for the perturbation, the redeuction factor becomes v 12/12/1/13-~0.96.

Appendix B. Effect of the phase dependence of the self-capacitance on the resistance threshold

As shown in refs. [ 14] and [ 15], the zero-frequency effective quasi-particle capacitance is A c ( ~ ) = z x c ( 1 - ~ cos 0 ) ,

(Bl)

with AC-AC(O) given in eq. (2.19). Making the mean-field replacement of eq.. (2.18 ) in A C ( 0 i - 0/), with fl set equal to zero, changes the effective Lagrangian of eq. (2.9) to h2 L; = g T [ C o + z ( 1 - ~/z cos ~, )~XCld~ +z/zEj cos 0, .

(B2)

(This form is sufficiently accurate to determine the effect of the phase-dependent capacitance on the threshold to leading in z ~. ) Thus, the reduced meanfield Hamiltonian of eq. (2.12) is now given by ( 1 ), 2 /1" = 1 - (/z/3) cos 0i ~p" -g~ cos 0~ =/q~ + l~ , (B3) where the symmetrized perturbation, Vs = ~/~(p2 cos 0, + c o s ~/p~ ) ,

(B4)

has no diagonal elements and the approximation is permitted near threshold, where /~<< 1. Equations (2.12 ) and (2.13 ) define/q~ and g~, respectively. The off-diagonal ground to excited state matrix elements of Vs can be taken into account by replacing gl by the effective coupling constant g'~ =g~ - - h / t . Therefore, at threshold,

(B5)

Acknowledgement

We are happy to acknowledge support by the National Science Foundation via Grant for Basic Research No. D M R 85-06009.

References [ l ] B.G. Orr, H.M. Jaeger, A.M. Goldman and C.G. Kuper, Phys. Rev. Left. 56 (1986) 378. [2] H.M. Jaeger, D.B. Haviland, A.M. Goldman and B.G. Orr, Phys. Rev. B 34 (1986) 4920. [3] A.M. Goldman, private communication. [4] See, e.g.M.P.A. Fisher, Phys. Rev. B 36 (1987) 1917. [5 ] R.A. Ferrell and B. Mirhashem, in: Proc. 9th Int. Conf. on Noise in Physical Systems, Montreal, 1987, ed. C.M. Van Vliet (World Scientific, Singapore, 1987) p. 179. [6] R.A. Ferrell and B. Mirhashem, Phys. Rev. B 37 (1988) 648. [7] P.W. Anderson, Special Effects in Superconductivity, in: Lectures on the Many-Body Problem, Ravello 1963. Vol. II, ed. E.R. Caianiello (Academic Press, New York, 1964) p. 113. [8] B. Abetes, Phys. Rev. B 15 (1977) 2828. [9] E. Sim~inek, Phys. Rev. B 22 (1980) 459. [ 10] S. Doniach, Phys. Rev. B 24 ( 1981 ) 5063. [ 11 ] K.B. Efetov, Sov. Phys. JETP 51 (1980) 1015. [ 12] P. Fazekas, B. Miilhschlegel and M. Schr6ter, Z. Phys. B 57 (1984) 193. [ 13 ] A.I. Larkin and Yu.N. Ovchinnikov, Phys. Rev. B 28 ( 1983 ) 6281. [14] U. Ekern, G. Sch6n and V. Ambegaokar, Phys. Rev. B 30 (1984) 6419. [15] R.A. Ferrell, Physicsa C 152 (1988) 10. [ 16] A.M. Goldman, private communication. [ 17 ] R.A. Ferrell and R.E. Prange, Phys. Rev. Lett. 10 ( 1963 ) 479.

B. Mirhashem and R.A. Ferrell / Universal criterion for the onset of superconductivity [ 18 ] Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors, Natl. Bureau of Standards, Applied Mathematics Series 59 (NBS, Washington, DC, 1967). [ 19] R.A. Ferrell and B. Mirhashem, to be published. [20] R.A. Ferrell, to be published. [21] H.A. Bethe, Proc. Roy. Soc. A 150 (1935) 552; J.M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979)p. 151. [22] M.E. Fisher, Scaling, Universality and Renormalization Group Theory, in: Critical Phenomena, ed. F.J.W. Hahne (Springer, Berlin, 1983) p. 1-137 (see also p. 54).

377

[23] R.A. Ferrell and B. Mirhashem, University of Maryland Department Physics and Astronomy Publication No. 88-135 (January, 1988), to be published. [24] R.A. Ferrell and B. Mirhashem, Phys. Lett. A, to be published. [25] S. Kobayashi and F. Komori, J. Phys. Soc. Jpn. 57 (1988), in press. [26] S.S. Attwood, Electric and Magnetic Fields (Dover, New York, 1967) p. 160. [27] G.R. Dean, Phys. Rev. 1 (1913) 316.