Bias current dependence of the electromagnetic response of thin granular films and Josephson arrays

Physica C 161 (1989) 354-366 North-Holland

BIAS CURRENT D E P E N D E N C E OF T H E E L E C T R O M A G N E T I C RESPONSE OF T H I N GRANULAR F I L M S AND J O S E P H S O N ARRAYS Behzad MIRHASHEM and Richard A. FERRELL Centerfor Superconductivity Research, Department of Physics and Astronomy, University of Maryland, College Park, Maryland, 20742-4111, USA Received 24 August 1989

The mean field theory of global superconductivity in Josephson junction arrays is used to calculate the electromagnetic response and surface plasmon spectrum in the presence of a bias DC current. The results are expressed in terms of a current-dependent kinetic inductance which is found to exhibit a simple scaling behavior for all normal state resistances that satisfy the criterion for global Cooper pair phase coherence.

I. Introduction

Superconductivity in inhomogeneous systems is a subject of current experimental and theoretical interest [ 1 ]. In particular, the observation of a normal state sheet resistance criterion for the occurrence of global superconductivity [ 2-4 ] in ultra-thin films has led to a number of theoretical investigations of the interplay [5] between the Josephson effect, which energetically favors Cooper pair phase coherence, and zero-point charge and phase fluctuations, which disrupt such coherence, in accordance with Heisenberg's Uncertainty Principle, as applied to the relevant complementary mesoscopic variables, pair phase and pair number. In granular structures, this interplay has been shown [ 6-8 ] to lead to a state of global superconductivity provided the separation between the grains is sufficiently small. In the limit of very small grain size, a universal normal state resistance criterion [ 9,10 ], of about 4 kf~, independent of the BCS energy gap, has been derived, in agreement with measured values ranging from [3] 4 to [2] 6.5k~. This universality arises from the cancellation of the energy gap in the ratio of the Josephson and charging strengths [ 11 ], in the small-grain limit in which the capacitance determining the charging energy is entirely non-geometrical in origin, arising from virt.ual quasi-particle fluctuations [ 12,13 ] required by causality [ 14 ]. For the grain radii of interest, of the order 0921-4534/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

of 100 A [3], the capacitance is indeed dominated by this quantum mechanical effect [ 15 ]. Various theoretical approaches [ 10,11,16 ] have led to resistance criteria that are, more or less, in numerical agreement with the various experimental determinations of the normal state threshold resistance. In order to discriminate among the different theories it is useful to extend them to predictions of other experimentally observable properties, so that the correct theoretical picture can objectively emerge. Our investigations along this line have focused on the broken symmetry state's response to DC and AC perturbations. Specifically, within the self-consistent mean field approximation, the critical current [ 17 ] and the kinetic inductance [ 18 ] of a granular film, modeled by a lattice of grains, have been computed. These zero-temperature calculations are experimentally relevant in the limit in which the thermal energy, kRT, is sufficiently smaller than all of the relevant energy scales, set by the zero-temperature BCS gap, do = 1.76kn Tc,

(1.1)

the Josephson coherence energy, Ej, and the charging energy, Ec= ( - 2 e ) 2 / ( A C ) , with - e the electron charge and Tc the bulk transition temperature. The Josephson coupling strength is Ej = 3 o R o / 2 R N ,

(1.2)

B. Mirhashem, R.A. Ferrell / Thin granular films and Josephson arrays.

with RN the normal state resistance of a square and Ro = h/4e 2 ~- 6.5 kf~.

(1.3)

The zero-frequency quasi-particle capacitance is [ 1214] A C = 3h / 64AoRN .

(1.4)

Planck's and Boltzmann's constants have been denoted by h and kB, respectively. As thermal quasiparticle excitations which reduce the gap and Ej ( ___Ao for RN values of interest) are exponentially, suppressed, according to the BCS and AmbegaokarBaratoff theories, respectively, the assumption of complete quasi-particle freeze-out is, in practice, rather accurate for temperatures T< Td2. On the other hand, the basic coupling constant of the global transition

go =EjIEc =ACE/4e 2

( 1.5 ) (1.6)

355

the current-biased kinetic inductance. This leads naturally to the calculation of the spectrum of the self-sustained electromagnetic excitations of the array. Section 3 concludes with a proposed experimental arrangement using a film deposited on a corrugated substrate. Surface plasmons that are excited by electron bombardment in such films can be expected to emit detectable radiation. Section 4 is a brief summary, while appendices A, B, and C are devoted to the effect of retardation on the surface plasmon dispersion relation, the photon yield for electron bombardment, and the demonstration of the applicability of the kinetic inductance scaling function to fabricated arrays, respectively.

2. The kinetic inductance

The Lagrangian for a granular film, modeled as a regular two-dimensional array of grains, each described by BCS theory, is

has a critical mean field value [ 9,10 ] of (2.1)

L=T-V,

g ~ = 1/2z 2,

(1.7) with charging energy

with the lattice coordination number z = 4 for a square lattice of grains. Thus, in the neighborhood of the transition, the mean field requirement that Ec/ z be large compared to kBT is even less restrictive so that, in practice, the zero-temperature theory is of direct experimental relevance. In section 2 of this paper, after a brief review of the theory of refs. [ 10,17,18], we calculate the kinetic inductance in the presence of a bias DC current. For RN near the normal resistance threshold R~, i.e., r~< 1, with the reduced sheet resistance defined as r=RN/R~,

(1.8)

a scaling function for the kinetic inductance is obtained in which the dependence on r is absorbed in the scale for the DC current. We demonstrate that this scaling function is quite accurate for the entire range 0 < r < 1. Experimentally, if shorting connections can be avoided in the fabrication of the films, it should prove possible to probe the entire course of the scaling function. In section 3, we study the response of the array to an external electromagnetic field, as determined by

T = --C• ( V,-- ~j) 2 2,j

(2.2)

and V= - E j )-" cos(0,-q~j).

(2.3)

0

The sums are such as to count each pair of neighbors only once. In eq. (2.2), the mutual capacitance is C = C~ + A C where C~ is the geometrical capacitance and AC is the virtual quasi-particle tunneling contribution, eq. (1.4). The voltage on the ith grain, V,, is related to the pair phase q~,, by Josephson's equation, (2.4)

Vi

so that, in the Hartree, or mean field, approximation, eqs. (2.1-4) lead [ 10 ] to an effective one-body Hamiltonian for, say, grain "1", HMF = with

4e 2 ~ Hi

,

(2.5)

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B. Mirhashem, R.A. Ferrell/ Thin granularfilms and Josephson arrays.

H1 = ½p2 - g c o s 0~,

(2.6)

where (2.7)

g = zZgolt

and g o - C E j / 4 e 2 is given by eq. (1.6) in the limit C1 << AC. The coordination number for the square lattice is z = 4. The ground state expectation value of the order parameter is /t= (cos ¢t )

(2.8)

and the m o m e n t u m conjugate to qh is 0 Pl = i - -

(2.9)

001 "

The reduced mean field Hamiltonian, H~ ofeq. (2.6), is that of a pendulum of unit mass and length and angular m o m e n t u m - P l [5 ]. In the weak "gravitational field" limit, g<< l, the ground state energy is Eo= _g2+ 7g4

~ g 6 . b ....

(2.10)

of eq. (2.11 ), determines the self-consistent order parameter/t for all r < 1. The resulting [ 18 ] function /~2(r), reproduced here as fig. 1, is a monotonically decreasing function between the "strong coupling" and "weak coupling" limits of/z(0) = 1 and/t( 1 ) = 0 , respectively. Now, the flow of DC current parallel to one o f the lattice vectors corresponds to a mean phase difference, 6, in the phases qh and ~2 of neighbors so situated relative to one another. As there is no mean phase difference in the perpendicular direction, the coupling constant ofeq. (2.15 ) is weakened by a factor o f (2 + 2 cos 8 ) / 4 = cos 2 5/2, leading to a currentnormalized coupling constant [ 17 ] ]A

25

(2.16)

]A

g= F r~c°s 5 = ~2

where ? - r sec 5/2. Let ~ ~°2) be the fluctuating phases in the zero-current state of the array, so that ~1 - ~2 = 0 ~o) _ ~ o ) + & The current flowing across a strip o f unit length is given by I= -KG

From the Feynman-Hellman theorem, we obtain, for the mean value of the cosine, the "pendulum function",

-OEG -f(g)=2g-7g3+ 0g

< COS ~1 > -- - -

(2.17)

where, in mean field approximation, the function

... < 2 g .

(2.11) This leads, using eq. (2.7), to a mean field threshold i " g~ = 1/222 _ 32

(2.12)

From eqs. (1.2,4,6), for a film consisting of grains sufficiently small that C1 is negligible compared to AC, go = I-~(Ro/RN) 2 ,

~2 0.5

05 r

1.0

(2.14)

(Various corrections reduce R ~ from its mean field value, as discussed in ref. [ 15 ].) Introducing the reduced resistance o f e q . (1.8), eq. (2.7) becomes g=it/2r 2

IQUANTUM

(2.13)

so that the critical normal state sheet resistance is [9,10] g ~ = ½x/~go ~ 5.7 k ~ .

t.O

(2.15)

which, combined with the pendulum function, f ( g )

Fig. 1. Square of the self-consistent order parameter,/~2, versus the reduced normal state sheet resistance of a granular film, r=Rr~/R~. Neglecting quantum fluctuations yields the horizontal dashed line, corresponding to the "classical" value,/fl= 1. In the presence of a bias DC current, f12=f12(~), where ~=r sec 5]2, with 8 the bias phase. In the case of an artificially structured array with the mutual capacitance dominated by the geometrical contribution, the abscissa is v/~ and ~= r s e c 2 5 / 2 .

B. Mirhashem, R.A. Ferrell / Thin granular films and Josephson arrays.

357

4e 2

G=

_ h2 rEjYiiEll ,

1 r

where the reduced inverse kinetic inductance is defined as the derivative function

sin O

= < c o s ( e l °) - ¢ d °)) > - -

r

cos 6 + < sin (01 °) _¢~o)) > _ _

0G = G ' (r, 6) Yu = -fig

r

sin d

= - - ?2

= /z" sin 6 r

(2•18)

is of universal form. The material constant, specific for a particular gap do, is nAo

r 4~e

_

K = -ZaeR~, - a -h l~J= --rlj a '

(2.19)

with Ij being the maximum Josephson current for an individual junction in the absence of quantum fluctuation and a being the separation between the grain centers. Maximizing G(r, (5) has led to the Critical current [ 17 ], G¢(r). By considering the case of current flow in the direction of a lattice diagonal, we found that quantum fluctuations wash out the directional dependence of Gc for r~> 0.7. In the presence of a bias phase 6, the square of the order parameter is, according to eqs. (2.8,11,16), #z(?), which is the same function that is displayed in fig. 1 except for r being replaced by ?. An externally applied AC voltage difference, A V, along the lattice direction that is parallel to the DC bias current produces, according to eq. (2.4), a variation in the mean phase difference between two neighboring grains, 2e

2ea _

~= ~ - A V = - --~-E,

(2.20)

where E, = - - A V / a is the AC electric field, which we take throughout to be infinitesimally small. According to eqs. (2.17,18,19) the current oscillates at the rate •

•

0G

Ill = -- K G = - K - - ~ ~ 2ea K OG

(2•22)

For the case of no DC bias, the function G' (r, 0) was calculated in ref. [ 18 ]. We now calculate it for the general case• Because fi, the phase difference between neighboring grains in the direction of the bias current, is not directly observable, it is necessary to use eq. (2.18) in order to eliminate d in terms of G, which represents the observable DC bias current. In other words, we need to express the derivative G' as a function of G itself. In carrying this out, it is convenient to write the desired function, Y(r, G ) , in terms of the scaled variable y, = Y, (r, G) / Yll(r, 0 ), where Y~(r, 0), the inverse kinetic inductance at zero bias, is already known [ 18 ] as a function of r. We similarly introduce the reduced phase, 0 / 6 ¢ ( r ) , and the reduced bias current, G / G ¢ ( r ) , scaled by the critical phase and current, 6¢(r) and G¢(r), respectively. For a given value of r and of 6, and thus of ?= r sec d/2, it is a straightforward procedure to determine the order parameter/z self-consistently from eq. (2.16) and the "pendulum f u n c t i o n " , / z = f ( ~ ) . The differentiation required by eq. (2.23) is facilitated by the analytic expression f o r f ( ~ ) that we have obtained from our variational treatment [19] of Mathieu's equation. It is, however, worthwhile to recall that in refs. [17] and [ 18] we found that the strong coupling and weak coupling approximations sufficed, together, to cover most of the parameter range 0 < r < 1. In the extreme strong coupling limit, as r ~ 0 , the pendulum function saturates at f(oo) = 1. The quantum fluctuations become then totally suppressed and eq. (2.18) reduces trivially to G=r -l sind,

(2.24)

with 6¢ = x / 2 and Gc = r - '. The scaled current-phase relation is therefore G = s i n ( 2 . ~d ) . G--~

(2.21)

(2.23)

(2.25)

B. Mirhashem, R.A. Ferrell/ Thin granularfilms and Josephson arrays.

358

Differentiation and elimination of d/6¢ gives then

~ = [~( 1 - r ) ]w2

G 2 ]/2 y,, = [ 1 - ( ~ - ) ] .

and

(2.26,

Gc = ~32( 1 - r ) S c These last two equations are shown by the dot-dashed curve in fig. 2 and the lower dot-dashed curve in fig. 3, respectively. As already noted [ 17 ], the weak coupling approximation o f eq. (2.11 ), f ( ~ ) _~2 ~ - 7~ [ 3 ], that is appropriate for ~<< 1 and 0 < l - r < < 1, yields Ginzburg-Landau type behavior. The phase-dependence of the order parameter is given by /~2=218(1-r)-52 ]

(2.27)

and G = ~ [ 8 ( 1- r ) 5 - 5 3 ] ,

(2.28)

from which we obtain the m a x i m u m values

= g64 i(]

(2.29)

,1/2(

1 - r ) 3/2 •

The scaled version of eq. (2.28) is therefore

GIGc = 35/5c - ½( 5 / ~ c )

3 ,

G/Gc 0.5

G/Gc = 3 ( 1 - y , )1/2_ ½( 1 - y , )3/2,

I

0.5

1.0 8/8c

Fig. 2. ScaledDC current, G (r, 5 )/G¢ (r), versus scaled bias phase, O/Sc(r). The solid and dot-dashed curves show the weak and extreme strong coupling approximations, respectively. t •

i

i

"

"

/

Y(r,G)

0.5

i

0

(2.32)

Eliminating 5/5c between eqs. (2.32) and (2.31) yields the desired functional relationship between Y~l and the current,

1.0

1.0

(2.31)

as shown by the solid curve in fig. 2. The two curves in fig. 2 that represent the two different approximations, which in principle correspond to quite disparate extreme cases, are in fact very close to one another. At no point are they separated by more than a few percent. This is illustrated by the initial slopes, r~/2 and 1.5, ofeqs. (2.25) and (2.31), respectively. The top curvatures are n2/4 and 3, respectively. By differentiating eq. (2.31) we obtain the weak coupling scaled inverse kinetic inductance function y, = 1 - (5/~c)2.

i

(2.30)

i

t.0

015 G/G c

Fig. 3. Scaled inverse kinetic inductance y=Y(r, G)/Y(r, 0), versus scaled DC bias current, G(r, 5)~Go(r). The upper two curves show y±, for the DC current perpendicular to the AC field while the lower two curves show y±, for the parallel case. In both cases, the solid and dot-dashed curves correspond to the weak and extreme strong coupling limits, respectively.

(2.33)

as shown by the lower solid curve in fig. 3. As is the case in fig. 2, the two lower curves in fig. 3 are also very close together. By detailed numerical computations for the specific parameter choices of r = 0.4, 0.6 and 0.8, we have verified that the resulting curves, when scaled as in fig. 3, fall between the strong coupling and weak coupling curves (lower dot-dashed and solid, respectively). It therefore follows that, for practical purposes (i.e., to an accuracy of a few percent), either eq. (2.26) or eq. (2.33) can be used over the entire interval 0 < r < 1. There is, of course, a substantial difference between the strong and weak coupling approximations for the scale factors G~(r) and Y, (r, 0). We conclude this section by calculating Y±, the inverse kinetic inductance for an AC field E . that is perpendicular to the DC bias current. We can continue to use eq. (2.20), which now relates the phase difference in the perpendicular direction to E . . The perpendicular AC current is therefore given by 4e 2 I . = - K ~ - ~ 5= ~ rEj Y . E . ,

(2.34)

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B. Mirhashem, R.A. Ferrell / Thin granular films and Josephson arrays.

with (2.35)

Y.L = ~ 2 / r ,

which, for comparison with the parallel case of eq. (2.22), can be written as Y± = G/sin J .

1012 electrons/cm 2, several orders of magnitude smaller than the conduction electron density in the film.

3. Electromagnetic properties

(2.36)

3.1. Transmission coefficient In the strong coupling limit the scaling function Y± (r, G) y± - - -1, Y. (r, 0)

(2.37)

independent of G/Gc, as shown by the dot-dashed straight line in fig. 3. For weak coupling, eq. (2.27) yields y± = 1 - 1 (J/So) 2 .

(2.38)

By eliminating J/J¢ from eqs. (2.31 ) and (2.38), we obtain the desired connection G 3x/~ _ y ± )3/2 Gc2 [ ( 1 - - y ± ) 1 / 2 - - (1 ],

(2.39)

which yields the plot of y± versus G/Gc that is shown by the upper solid curve in fig. 3. As in the parallel case, this scaling relation is a monotonically decreasing function of G/Gc. The maximum drop in y± at G = G¢ is, however, only one-third, in contrast to the full disappearance of Yll at G = G~. These results for the kinetic inductance can be interpreted in terms of an effective planar superconducting electron density, ns, defined by rewriting eq. (2.21) as I=

nse 2 Ex, m

(2.40)

Historically, measurements [20] of the far infrared transmissivity of thin films have played an important role by establishing the existence of the superconducting energy gap and confirming the frequency dependence of the electrical conductivity predicted [21 ] on the basis of the BCS theory. The fractional transmissivity of a thin film, d~, is given by [22]

4rt6_, 12 F

F ¢--1=L1+ (n+l)cJ

4m

ns(r, G ) = -7-s-rEjY=ns(r, O)y ,

(2.41)

?l-

where 4m

ns(r, 0 ) = -;-7rEjY(r, O)

(2.42)

?l"

was

estimated

[18]

to

be

of the

order

of

12'

(3.1)

with 0 = 01 + i02 being the two-dimensional electrical conductivity (bulk conductivity times film thickness) and c/n the speed of light in the substrate. The above expression suggests the possibility of investigating the bias-current dependence of the kinetic inductance of a granular film that has been studied above in section 2 by a measurement of the transmissivity of the film. Indeed, eqs. (2.21,22) imply 0~ = 0 and 02= (2eKa/ho))Y(r, G), with the orientation of the AC field relative to the DC current yet unspecified. From eqs. (2.19) and (1.3)

02 = a c

( ~ ¢ ~ -Y(r,- O)y \RN] So2 '

(3.2)

SO (T_I

so that

4r~02 + L(~I)cA

1)1/2 =

4 h a |/~__~_ / R - x, Y(r,O)._______jy (n+l) \RN,/ .(2 '

(3.3)

where a = e 2 / h c ~ _ 1/137 is the fine structure constant and f2= h~o/2Ao-e)/o)g. We see that the transmissivity drops below unity for R N < R ~ and that, for fixed RN, it should have a monotonically increasing dependence on the bias current. The current dependence of the transmissivity is determined by the functions y plotted in fig. 3. The adiabatic current response assumed in the calculation of the kinetic inductance in section 2 restricts the applicability of

360

B. Mirhashem, R.A. Ferrell / Thin granularfilms and Josephson arrays.

eq. (3.2) to values of Q significantly less than one. The observation of a substantial drop in the transmissivity requires, in any case, rather small values of because of the factor of a in eq. (3.3).

the wave equation, from which it follows, for a sinusoidal variation parallel to the plane of the film at wave number q, that the z-component, normal to the plane of the film at z = 0 , is given, above and below the plane, by

3.2. Surface plasmons

E(+) z,ln _ -- + -- __1 V(2).Ei, ' As an alternative approach to the measurement of the current-dependent kinetic inductance, we turn to a study of the response of the array to a spatiallyvarying longitudinal field. We will demonstrate that a non-zero inverse kinetic inductance leads to the existence of collective modes in the array. These "surface plasmons" are the analogues of the Josephson plasmons in a single junction. In fact, it will be seen that the resonant frequency of these excitations is a function of the bias current so that probing their spectrum will provide a means for studying the current dependence of the kinetic inductance. This behavior of an array is similar to that of a single junction, which also has a bias-current-dependent resonance, as was demonstrated by Dahm et al. [23 ]. A non-uniform current density I in the plane of the film will generate a two-dimensional charge density s according to the continuity equation 0s -0t

= -- V ( 2 ) . I .

(3.4)

Differentiating with respect to time permits the introduction of the inductive response of the film according to eq. (2.21). For the subsequent analysis it is convenient to exhibit explicitly a factor of C . T h e resulting equation is 02"7

0t 2 --

-- V ( 2 ) . I =

-ff92CV(2)'E,

(3.5)

where the frequency unit, by virtue of eqs. (1.2,4), is

2e(~_

Co= - ~

rY

)1/2

~-2

~_~

~ , [ r Y ( r , 0)y] 1/2,

(3.6)

for Cl << AC. We can consider that the field E that drives I is the superposition of an external field Ee, and the induced field Ein that is generated by s, i.e., E=Eex +Ein .

(3.7)

Outside the film, Ein is divergence-free and satisfies

(3.8)

K+

where x+=

(

q2_

(3.9)

and E+ are, respectively, the extinction coefficients and the dielectric constants in the half-spaces z = + Izl. Gauss' law requires, in consequence of E(-+ z,ln ) , the perpendicular flux densities, that there be in the plane the following two-dimensional charge density: 1

~7(.+)

~--~6+ ~ z , ln I z = o + - ~ =

(~+-+

E-)

E+ + ~-_

1

E-- E z,(:-) ln [z=o-

V(2)'Ein

4n

(3.10)

Here, we have employed eq. (3.8) in order to eliminate the perpendicular components in terms of the parallel component. The problem is considerably simplified by neglecting retardation and using Laplace's equation instead of the wave equation to determine the extinction coefficients. This approximation, which is shown in appendix A to be quite accurate in the present context, replaces eq. (3.9) by x+ = x_ = q, so that the ratio of the charge density to V(2)'Ein in eq. (3.10) becomes the frequency independent quantity ( / ( 2 n q ) , with g= (e+ + e_ )/2. If this relationship is used to eliminate V(2).Ein from eq. (3.5) in terms of s, the resulting equation of motion yields the surface plasmon dispersion relation Ogqoc x/~ ,

(3.11)

a familiar result [24 ] for a thin metal film, in the absence of damping. The above analysis is, however, not complete for a Josephson array because the potential variation within the plane produces an additional charge density in consequence of C. The full charge density is therefore

B. Mirhashem, R.A. Ferrell / Thin granularfilms and Josephson arrays.

(3.12)

= ( 1 "4- q s ) Ci7(2).Ein ' q/ where

q;-~ = 2 n C / g .

(3.13)

Substitution of eq. (3.7) and the elimination of the induced field from eq. (3.5) by means ofeq. (3.12) yields the inhomogeneous wave equation

02S +to2S=-(~.)2CV2"Eex 0t 2

t o q - ( a - ~ a ) 1/2 q q~ •

(3.15)

For an external forcing field oscillating at angular velocity to, the solution of eq. (3.14) is ~2 2 2 CV(2)'Eex , to --toq

(3.16)

with the surface plasmon resonance manifesting itself by the diverging response at co=toq. This resonance is a two-dimensional analogue of the one-dimensional propagating plasma mode discussed by Mooij and SchiSn [ 25 ]. Substituting eq. (3.16) into eq. (3.12 ) expresses the induced field in terms of the external field by V(Z).Ein

-

q Cs q+q~

2 O)q 2 2 to --toq

F(2).Eex

"

(3.17)

The total electric field is therefore given in the z = 0 plane by E(2) -P~2) --t-b"(2) = ( 1 - to2/(D2) - 1]q'(2) I

~

in

--

2ao~-13.6eV the Rydberg constant. As Ao/ry is of order Tc (K) / 105, this wavelength is of order 10 ~tm. For q >> qs, eq. (3.15 ) predicts a fiat spectrum. For q << qs, eq. (3.15 ) reduces to the square root dispersion relation of eq. (3.13 ). As discussed in appendix A, the group velocity singularity at q = 0 is removed once proper account is taken of retardation effect. From eq. (3.6) we observe that the entire spectrum drops to zero as ~ 0)oc ( 1 - r ) l/z at fixed bias current and as y~/2 for fixed r. 3.3. Optical coupling o f surface plasmons

(3.14)

with the surface plasmon dispersion relation

s=

361

~ e x

~ e x

(3.18)

where the superscript denotes the components parallel to the plane. This result is similar to the familiar relationship between the electric and displacement fields in a bulk three-dimensional plasma, E=E -1 ( t o ) D = ( 1 -- t o 2 / t o 2 ) - 1D .

(3.19)

From eq. (3.13 ) the characteristic wave number qs corresponds, for Ct <
From the preceding section, and the discussion of appendix A, it is evident that the phase velocity of the surface plasmons is very much less than that of light. There is therefore no direct coupling, for a uniform array, of the surface plasmons to the radiation field. They could, however, be excited by the inelastic scattering of electrons, which would serve to provide the external field Eex of the right hand members of eq. (3.18). Because of the very low quantum energy of these plasmons, their observation by the usual method of measuring characteristic energy losses is, however, not feasible. We are therefore led to consider the possibility of observing the radiative decay of surface plasmons that have been excited by inelastic electron scattering, in analogy to the observation of plasma oscillations [26] in metal films. In quantum language, in the present context, it is necessary to remove the excess momentum of the surface plasmons in order for them to transform into photons. This can be accomplished by preparing the array on a spatially periodically corrugated substrate of wave number qo. Such a corrugation can be expected to impose on d, the two-dimensional conductivity of the array, the fractional modulation flcos qo'r, where r is the two-dimensional space coordinate in the z = 0 plane and fl is some dimensionless parameter of order unity that corresponds to the depth of the corrugations. The local current-field relation in the plane of the array becomes thereby, assuming a lossless response,

l(r, t ) = d 2 ( t o ) ( l + f l c o s q o . r ) E ( 2 ) ( r ,

t) .

(3.20)

If now, by inelastic electron scattering, a surface plasmon is excited with wave number also equal to qo and of amplitude Eqo, the second term in paren-

362

B. Mirhashem, R.A. Ferrell / Thin granular films and Josephson arrays. (~2

theses in eq. (3.20) will generate the current

s = flC

fl0g~) cos 2 qo" re-i~o, -

-

fl OEqo ~ ( 2 ) -e- i t o t

2

(3.21) The r-independent term describes a spatially uniform current sheet that radiates perpendicular to the sheet. As discussed in appendix B, the excitation and subsequent radiative de-excitation of plasmons with wave numbers near to qo can be expected to yield an angular distribution of photons. A further alternative for the observation of surface plasmons in a spatially modulated array, or thin granular film, of the type envisioned above is to excite the surface plasmons by normally incident radiation of frequency to. For the analysis of this process it is convenient to return to eq. (2.21), which, in modulated form, becomes (3.22)

(The subscript on the wave number now serves no purpose and can be dropped.) The modulated version of eq. (3.5) is therefore 02s Ot--5 = - f f 9 2 C V ( 2 ) . [ ( l + f l c o s q . r ) E

(2) ] .

(3.23)

Carrying out the differentiation and substituting 17(E).E from eq. (3.12) yields 02s Ot--5 + ( 1 + flcos q . r ) t o 2 s = f l ~ Z C sin q . r q . E (2) . (3.24) At this point we adopt a perturbation approach and expand E (z) (taken without loss of generality, to be parallel to qo) in the Taylor's series E ( Z ) = E o + flE~ + ....

(3.25)

The first term is the r-independent optical component, while the second term has wave number q, etc. To O(fl), we can ignore the modulation in the lefthand member of eq. (3.24) to obtain 02s + t o E s = f l q ~ 2 C s i n q x E o 0t 2

OJq --0)

2 q sin q" r Eo .

(3.27)

This requires, according to eq. (3.16), an in-plane modulated field

+ 2 ~E(¢2°) c°s2q°'re-i°~t

l ( r , t) = ( 1 +flcosq'r)~2CE(2)(r, t) .

2

(3.26)

The solution of this driven oscillator equation is the oscillating charge density, sinusoidal in space,

Eo (1_092/0)2) cosq-r.

E l =-

(3.28)

Substituting eqs. (3.28) and (3.25) into eq. (3.20) and taking the spatial average yields, to O(f12), the current sheet response to the optical component of the field as Io=(I(r,t))

= 02 (o~) ( ( 1 + f l c o s q . r ) (Eo +fiE1 ) ) =02(09) (Eo + f12(Elcos q . r ) ) =0eff(to)Eo,

(3.29)

where the effective optical conductivity of the film is

~

[

aeff= 1

B2/2

1

(l_w2/to~)_l 02(to) ,

(3.30)

the form of which may be expected to have a validity beyond that of the Taylor's expansion used in deriving it. Using 0err for 02 in eq. (3.1) gives 100% transmission at o)2= (1-flE/2)to2q, followed by a dip to zero transmission at the resonance to = toq. All of the above work is predicated on the working hypothesis that 0~ ( t o ) = 0 for all to
4. Summary In summary, we have reported on the calculations of the kinetic inductance of an ultra-thin granular film, modeled as a lattice of superconducting grains, as a function of the normal state resistance and bias DC current. Near the threshold for global superconductivity, the kinetic inductance exhibits, in the mean field approximation, Landau-Ginzburg type scaling. We have shown further that for the case of parallel AC field and bias current, the scaling function re-

B. Mirhashem, R.A. Ferrell/ Thin granularfilms and Josephson arrays. mains accurate arbitrarily far into the state of broken symmetry. Measurement of the kinetic inductance of Josephson arrays consisting of grains large enough for the charging effects central to our considerations to be negligible is an established technique [27 ]. Experiments should also prove to be feasible in the smallgrain limit considered. In the usual set-up, it is the mutual inductance of a pair of coils coupled via the sample which is measured. To study the DC bias dependence, the AC current flowing in the primary coil should modulate a variable DC current which would establish diamagnetic DC Josephson currents in the film. As demonstrated in section 3, it may also prove possible to study the kinetic inductance by electromagnetic transmissivity experiments of the type originally performed by Glover and Tinkham [20 ]. Also in that section, we calculated the surface plasmon spectrum of the array and noted the feasibility of studying this spectrum by detecting the radiation emitted as a result of electron bombardment of films deposited on a corrugated substrate.

363

As a consequence, eq. (3.15) becomes

O)q(t¢~

1/2

--:-

,

(A.2)

which determines coq implicitly as a function of q. Eliminating x from these two equations relates the wave number q to the associated resonant frequency COqby '

q

~Oq

O.)q

(A.3)

where e~b2

3n2 ea2rY(r, O ) y ,

(A.4)

~= c2q~ - ~ -

for C1 <
(A.5)

q~O

However, because of the appearance of a, the fine structure constant, in eq. (A.4), eq. (A.5) is relevant only at extremely long wavelengths. For coq>> x/%t3 and q >> rlqs, eq. (3.15 ) is recovered.

Acknowledgement This work has been supported by the National Science Foundation under Grant for Basic Research DMR 85-06009.

Appendix A. Effect of retardation on the surface plasmon dispersion relation To incorporate the effect of retardation into the calculation of the surface plasmon spectrum, we must allow for the magnetic field generated by the oscillating surface currents. This is accomplished above, in section 3, by using the wave equation to determine the extinction coefficients above and below the plane of the film. Here, in order to obtain an explicit expression for the effect of retardation, we limit ourselves to the special case of equal dielectric constant ~+ = ~_ = ~ on the two sides, so that eq. (3.9) reduces to x+ = x_ = to, where

l¢: (q2--~(.02/¢2)1/2.

(A. 1 )

Appendix B. Surface plasmon excitation by inelastic electron scattering As noted in section 3.3, the radiative decay of surface plasmons is kincmatically forbidden for a film that is uniform on the scale of the plasmon wavelength. If the film is deposited on a corrugated surface, however, a plasmon can lose its momentum and photon emission becomes possible. To calculate the photon yield, we follow the approach of ref. [28 ] to compute the probability that a single normally incident electron will excite one quantum of plasma oscillation in the film. Considering a film of unit area, the electron-plasmon interaction can be written as

H ' = --e ~ 0fe-iq'" e-qlzl ,

(B. 1 )

q

where the electrostatic potential Fourier amplitude Oq is a linear combination of plasmon creation and annihilation operators. The electron's position operator is (r, z), with r and q being two-dimensional vectors. Quantizing the electron's momentum p in a box of length L in the z direction, the matrix element

364

B. Mirhashem, R.A. Ferrell / Thin granular films and Josephson arrays.

for the electron to make a transition from a state ~i to a state ~f while the plasmon oscillator at wave number q makes a transition to its first excited level is

so that the excitation amplitude is

-i i

n'~q) ,nt (oo) = ~

(B.6)

(H~-nt(t))lod/

(H')~q) = --e(0q) lO(~//f,e-iq"e -qLzl gi) or

_ - e ( Loq ) to f d2rdz eiO, i _ t n ) . V / n e _ i q . r e _ q l z l '~q)(OO)mt = ~ie (0q)10 --

-- e( ~)q ) l ° sg(2)'q[ q

+ q--~21 ' (B.2)

where K = ( p ~ - p f ) / h . By energy conservation, K~=ogq/v, where v is the electron's velocity. Closure requires I (0q)lol2= (I~bql2)oo --- < t~bq[2>o. Thus, for

K(2)=q,

IH~I2__ e2< 10¢ L2 12)o .4qE/(q2+O92q/V2)2.

(B.3)

dP L 2n __ _ IH~I2p(E) . v h

dE2~t

The density of electron states is p(E)=tp2dff2el/vh 3 while, by momentum conservation, the solid angle subtended by the scattered electron is dO~t= ( h i p)2d2q so that 4e2q2v2 < I(bq 12>0

dP=

h2(q2v2+(.o2)2 d2q.

(B.4)

In the above calculation it was assumed that the electron passes at constant velocity through the film and substrate. To study how eq. (B.4) is modified if the electron becomes trapped in the substrate, we consider the more general problem of the electron, treated as a classical source, interacting with the plasmon oscillator, assuming that the electron's speed, which is v before reaching the film ( t < 0), smoothly decreases subsequent to its passage through the film at t = 0. However, first, we demonstrate that such an approach reproduces eq. (B.4) in the appropriate limit. The motion of the electron along the classical trajectory r H=0, z = v t corresponds, in the interaction picture, to a perturbation of the plasmon Hamiltonian at each wave number q,

H'(q)["~ lnl I,t] -~" -eOqeit°qte -qvltl

(B.5)

1)

qv-(- iogq + qv--iWq

.

(B.7)

Thus, the total excitation probability is

4e2v2q 2 P= ~ h

< 10q 12>o

(B.8)

~q (q2/)2+O)q2)2"

With the replacement ~¢-~(2~) -2 f d2q, eq. (B.4) is recovered. More generally, the factor in parentheses in eq. (B.7) is dq= - -l

qv + io9o

By Fermi's Golden Rule, the number of transitions per incident electron is

(1

+

i

dt ei'°¢e -qlz<°t

(B.9)

0

and, to evaluate it, a convenient integrable approximation to e -qlz(t) I, for t> 0, that simulates the slowing down and stopping of the electron is ye -qvt/~ + ( 1 - y ) . The case y = 1 corresponds to no change in speed, leading to eq. (B.8), while y = 0 is the limiting case of abrupt stoppage at the film, with negligible penetration of the electron into the substrate. For the general case, 0 < y< l, the rate of penetration is represented by ( _ q ) - t I n ( l - y ) . Performing the integration in eq. (B.9) yields

1 72 f(Y'Y)=-°')qaq=-~711W y - i 7 + i ( l - y ) ,

(B.10)

where y = vqo/COqo = V/%o is the ratio of the electron's incident speed to the plasmon phase velocity. Because the photon momentum is very small, only plasmons that have wave number almost exactly equal to the corrugation wave number qo can lose the fight amount of momentum to emit radiation. As the solid angle subtended by the emitted photon is dg2vh = (c/oj)2d2q, the photon yield on the side of the film in vacuum is (d--~ph) = ~e2 < [~b~ol2> Ijq 2.

(B.11)

The ground state expectation value, < 10qo12>o can be evaluated by identifying the energy stored in the

B. Mirhashem, R.A. Ferrell/ Thin granularfilms and Josephson arrays. electric field with the oscillator potential energy, htoqo/4. The field energy is equal to the work done in assembling the surface charge distribution, ( 1 +qo/ q~)qolOqol2/4n, so that IOqol2=nhogqo/qo( l +qo/ q~) =hvq/[2( 1 +qo/q~) 1. Thus, (B.16)

= 4n \ c J ( l + q o / q ~ ) " For qo>> q,, Vqo= ~ / q o and, using eq. (A.4),

Ol2(qs) 2 / 3 r Y ( r , O ) y

(dN)

(B.17)

which is of order 10-20~2202, where 20 is the corrugation wavelength, 2nq~ ~, in p.m and we have used the estimate 2nq; -~ _ 10 lxm of section 3. (It is clear from eq. (B. 10) that ]j']2 is generally of order unity for y~ 1 and y>~ 1.) The photon yield equation, eq. (B.11 ), is predicated on the assumption that there are no other channels available for plasmon decay. Otherwise, it is reduced by the branching ratio F r / ( F r-l- F D ) where Fi- ' is the radiative lifetime and FD is the rate of decay by non-radiative processes. From eq. (3.21 ), the energy radiated per second per unit area of the film is

--

,[,2~2[~2

8C

[ ~"(2) ]2

t--qo J

2-2 2 :'( 2fl 2 qoa2Oqo/8¢

(B.18)

where z is the film thickness. Dividing by the plasmon energy per unit area, ( l +qo/qs ) qo~o /4n, gives the radiative decay rate

Fr= nr2fl2q°O2-a#2(q°z)(qsZ)2c

Appendix C. Kinetic inductance of fabricated Josephson junction arrays Quite recently, quantitative results on the transition to global superconductivity in artificially structured Josephson junction arrays have been reported [29]. To account for these measurements, the geometrical mutual capacitance, C1, of the junctions has to be taken into account [30]. Indeed, the appropriate limit for a theoretical treatment of these systems, at the current level of miniaturization, is C1 >> AC. In this appendix, we study how the results of section 2 are modified in this limit. Using eq. (1.2), the mean field critical value of the coupling constant, eq. (2.12), gives, as a non-universal threshold for the onset of global superconductivity,

R ~ =hAoCl/e 4 .

-~N Yy~og, (B.19)

a rate much smaller than the gap frequency.

(C.I)

The self-consistency condition of eq. (2.15) now becomes

g= l~/ 2r

(C.2)

so that f12 is still given by fig. 1, but with the abscissa now being x/~. In the presence of a bias current, the renormalized coupling constant of eq. (2.16 ) is now ~=/z/27

1 S = ~c [Tfla2Eq(2)/2]2

365

(C.3)

where g=rsec2t~/2. Following through the subsequent calculations in section (2), it is seen that, in the ultra-strong coupling case, eqs. (2.25,26) continue to hold with the same values of Gc and tic. In the weak coupling limit, eq. (2.33) still applies as it is a general consequence of mean-field theory. Thus, both the ultra-strong and weak coupling limiting forms of the scaling function Yll, eqs. (2.26,32), remain valid and our basic result, fig. 3, is unchanged. (A similar analysis shows that the perpendicular scaling function is also unmodified.) The scale factors, Y(r, 0) and Gc(r), are, however, changed. The zero bias inverse kinetic inductance Y(r, 0 ) = ll2/r is reduced by a factor of two in the limit r ~ 1 and can be obtained for arbitrary r from fig. l, reading the abscissa as x/~. For r~-O, Y(r, 0) = (1 - x / ~ ) / r. Similarly, eqs. (16) and (21 ) of ref. [ 17 ], for the r - 1 and r-~0 limiting behaviors of the critical current, now become

B. Mirhashem, R.A. Ferrell/ Thin granularfilms and Josephsonarrays.

366

Gc(r)~- ~

32

(l-r)

3/2

(C.4)

and

Gc(r)~- 1 ( l _ x / ~ ) r

,

(c.5)

respectively.

References [ 1 ] Percolation, Localization, and Superconductivity, eds. A.M. Goldman and S.A. Wolf (Plenum Press, New York, 1984). [ 2 ] B.G. Orr, H.M. Jaeger and A.M. Goldman, Phys. Rev. B 32 (1985) 7586; H.M. Jaegar, D.B. Haviland, A.M. Goldman and B.G. Orr, Phys. Rev. B 34 (1986) 4920; H.M. Jaegar, D.B. Haviland and A.M. Goldman, Physica B 152 (1988) 218. [3] S. Kobayashi and F. Komori, J. Phys. Soc. Jpn. 57 (1988) 1884. [4] R. Barber Jr. and R.E. Glover, III, Bull. Am. Phys. Soc. 34 (1989) 1021 and to be published. [ 5 ] 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. [6] B. Abeles, Phys. Rev. B 15 (1977) 2828. [7] E. Simanek, Phys. Rev. B 22 (1980) 459. [8] K.B. Efetov, Sov. Phys. JETP 51 (1980) 1015. [9] R.A. Ferrell and B. Mirhashem, in: Proceedings of the Ninth International Conference on Noise in Physical Systems, Montreal, 1987, ed. C.M. Van Vliet (World Scientific, Singapore, 1987) p. 179.

[ 10] R.A. Ferrell and B. Mirhashem, Phys. Rev. B 37 (1988) 648. [ 11 ] S. Chakravarty, S. Kivelson, G.T. Zimanyi and B. Halperin, Phys. Rev. B 35 (1987) 7256. [ 12 ] A.I. Larkin and Yu.N. Ovchinnikov, Phys. Rev. B 28 (1983) 6281. [ 13 ] U. Ekern, G. Sch~n and V. Ambegaokar, Phys. Rev. B 30 (1984) 6419. [ 14] R.A. Ferrell, Physica C 152 (1988) 10; ibid., 152 (1988) 231. [ 15 ] B. Mirhashem and R.A. Ferrell, Physica C 152 ( 1988 ) 349. [ 16 ] M.P.A. Fisher, Phys. Rev. B 36 (1987) 1917. [17] R.A. Ferrell and B. Mirhashem, Phys. Rev. B 38 (1988) 8703. [18] R.A. Ferrell and B. Mirhashem, Phys. Lett. A 130 (1988) 43. [ 19] R.A. Ferrell and B. Mirhashem, to be published. [20] R.E. Glover and M. Tinkham, Phys. Rev. 104 (1956) 844; D.M. Ginsberg and M. Tinkham, Phys. Rev. 118 (1960) 990; L.H. Palmer and M. Tinkham, Phys. Rev. 165 (1968) 588. [21 ] D.C. Mattis and J. Bardeen, Phys. Rev. 111 (1958) 412. [ 22 ] M. Tinkham, Introduction to Superconductivity, (McGraw Hill, 1975) p. 70. [ 23 ] A.J. Dahm, A. Denestein, T.F. Finnegan, D.N. Langenberg and D.J. Scalapino, Phys. Rev. Lett. 20 ( 1968 ) 859; ibid., 20 (1968) 1020. [24] R.H. Ritchie, Phys. Rev. 106 (1957) 874. [25] J.E. Mooij and G. Sch6n, Phys. Rev. Lett. 55 (1985) 114. [26] R.A. Ferrell, Phys. Rev. 111 (1958) 1214. [27] C.J. Lobb, Physica B 152 (1988) 1; Ch. Leeman, Ph. Lerch, B.-A. Racine and P. Martinoli, Phys. Rev. Lett. 56 (1986) 1291. [28] R.A. Ferrell, Phys. Rev. 101 (1956) 554. [29] L.J. Geerligs, M. Peters, L.E.M. de Groot, A. Verbruggen and J.E. Mooij, Phys. Rev. Lett. 63 (1989) 326. [30] R.A. Ferrell and B. Mirhashem, Phys. Rev. Lett. 63 (1989) 1753.