plasma waves in anisotropic superconducting films below and above the plasma frequency

PHYSICAL REVIEW B 1 AUGUST 1997-IVOLUME 56, NUMBER 5

Plasma waves in anisotropic superconducting films below and above the plasma frequency

Mauro M. DoriaInstituto de Fı´sica, Universidade Federal Fluminense, Avenida Litoranea s/n., Niteroí 24210-340 Rio de Janeiro, Brazil

Gilberto HollauerDepartamento de Fı´sica, Pontifıćia Universidade Cato´lica do Rio de Janeiro, Rio de Janeiro 22452-970 Rio de Janeiro, Brazil

F. Parage and O. BuissonCentre de Recherches sur les Tre`s Basses Tempe´ratures, Laboratoire Associe´ a l’Universite Joseph Fourier, C.N.R.S., Boîte Postale 166,

38042 Grenoble-Ce´dex 9, France~Received 20 November 1996!

We consider wave propagation inside an anisotropic superconducting film sandwiched between two semi-infinite nonconducting bounding dieletric media such that along thec axis, perpendicular to the surfaces, thereis a plasma frequencyvp below the superconducting gap. Propagation is assumed to be parallel to the surfacesin the dielectric media, where amplitudes decay exponentially. Belowvp , the amplitude also evanesces insidethe film, and we retrieve the experimentally measured lower dispersion relation branch,v}Ab, and therecently proposed higher frequency branch,v}1/Ab. Abovevp , propagation is of the guided wave type, i.e.;a dispersive plane wave confined inside the film that reflects into the dielectric interfaces, and the modes areapproximately described byv'vpA11(b/b0)2, whereb0 is discussed here.@S0163-1829~97!03426-7#

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I. INTRODUCTION

The experimental measurement of a plasma edge1 in theinfrared reflectivity of high-Tc bulk superconductors,2–4

La22xSrxCuO4, YBa2Cu3O82x ,5 and Bi2Sr2CaCu2O8,6–9 hasbrought a renewed interest in collective oscillations withCooper pair density. Previously, such studies were hampby the well-known argument10 that the Coulomb interactionshifts the frequency of such oscillations to above the gfrequency. Recent theoretical studies11–15 support the viewof a plasma oscillations along thec axis, the direction or-thogonal to the CuO2 planes, caused by a large supercoducting gap and a high anisotropy in these materials.

In superconducting films, plasma oscillations have dtinct properties. They exist in films regardless of their anisropy or layered structure, and regardless of the type of ping or even of the critical temperature value. This mode wpredicted16,18,17some time ago and was observed both in tgranular aluminium films, in the hundreds of MHz range19

and in thin YBa2Cu3O82x films20 in the higher frequencyrange of hundreds of GHz.

The study of propagating modes in films is not a nresearch field, they have been measured since long agseveral materials ranging from metals21–23 tosemiconductors,24 and also discussed in the theoreticliterature.26,25,27The novelty for the superconducting filmthe very low frequency where these modes are observedthe strong temperature dependence, explained by a colleoscillation with the Cooper pair density. Recently28 it hasbeen proposed that, similarly to metals and semiconductwo branches of propagating modes should be observehighly anisotropic superconducting films, e.g., the high-Tccompounds. The experimental observation of thesemodes can provide a method to independently measure

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London penetration lengths, parallel and perpendicular tofilm surface.

In this paper we include in the study of mode propagatin films the existence of ac-axis plasma frequency,vp ,inside the superconducting state. We only consider thec axisperpendicular to the surfaces since this configuration ismost easily grown nowadays. The propagation of plasmodes above and belowvp , their dispersion relations, arour goals in the present paper. We restrict our study to wathat propagate along the surfaces of the superconductingand evanesce in the dielectric media perpendicularly tosurfaces. Hence the present work studies a situation diffefrom that of Tachikiet al.,29 who have also considered aelectromagnetic wave with polarization along thec axis.Here the dielectric-superconductor interface has its noralong thec axis, whereas in their case, it is orthogonal to tc axis. Besides here the wave travels along the interfwhile in their case, wave propagation is perpendicular tointerface. Hence, the wave behavior outside the supercductor is quite distinct: here the wave is totally evanescethat is, it decays exponentially with the distance to the intface~surface plasmon!, whereas in their case it is just a planwave.

In this paper we show that there are two very distinbehaviors according to the field amplitudeinside the film:either it is evanescent away from the interfaces (v,vp) ornot. We call this last case confined propagation since itcurs inside the superconductor along an oblique direccausing reflection at the dielectric-superconductor interfa(v.vp).32 In many aspects this last case resembles progation in optic fibers. In conclusion this paper considerplasma frequencyvp inside the superconducting state astudies its effects into wave propagation in films.

The study of plasma modes in superconducting films wa plasma frequencyvp has been previously considered b

2722 © 1997 The American Physical Society

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56 2723PLASMA WAVES IN ANISOTROPIC SUPERCONDUCTING . . .

Artemenko and Kobel’kov30 in the context of kinetic equations for Green functions generalized to the case of layesuperconductors with weak interlayer coupling. The prespaper provides a more complete study of such modes, shthat there are symmetric and antisymmetric modesv,vp and discusses their several distinct regimes. Thetisymmetric mode is intimately connected to the transvecurrent component, thus being highly sensitive to the traverse London penetration length, in the same fashion thasymmetric mode depends on the longitudinal London petration length. We claim that these properties can be usegain a better understanding of the transverse and longitudsupercurrent components in anisotropic and layered filBesides, the present study is done in a framework distfrom those authors, we take the simplest possible theoryLondon-Maxwell theory for anisotropic materials, suppoedly valid when all lengths are larger than the interlayseparationa.

The experimental data on the plasma edge clearly shostrong temperature behavior of the plasma frequency. This no plasma edge in the normal state while it occurs immdiately below the critical temperature. Because of strong clisions the electronic scattering hides any signal ofplasma edge in the normal state.31 In the superconductingstate the situation changes significantly because part ofelectrons turn into Cooper pairs forming a collective nondsipative state. Since we do not take into account this fracof normal electrons, expected to strongly contribute inneighborhood of the critical temperature, our results ohold at very low temperatures, where the normal electrcan be safely ignored. For instance, our model only appfor La22xSrxCuO4, x50.16 (Tc534 K! at temperatures below 20 K, according to the Fig. 3~b! of the work ofTamasakuet al.2: only below this temperature can the scatering rate of the carriers along thec axis be safely ignored

In the context of this theory we determineexactly themode frequency,v, as a function of its wavelength along thsurfaces,b, and of the remaining parameters: the two Lodon penetration lengths, transverse (l') and longitudinal(l i) to the surfaces; the dielectric constant of the noncducting media exterior to the film« , the film thicknessd,and the isotropy and frequency independent«s , the simplestphenomenological choice for the superconductor’s staticeletric constant.

This paper is organized as follows. In Sec. II, we derthe dispersion relation equations, above and belowplasma frequency, using the London-Maxwell theory, asolve themexactly. Our study is restricted to identical toand bottom dielectric media. Similar conclusions should aapply to the most general asymmetric case. Section III dwith the v,vp case, where we study the two possibbranches, the so-called symmetric~lower! and antisymmetric~upper! branches, and their three possible regimes: opticoupled, and asymptotic. For each branch we derive, frour exact solution, useful approximated expressions for eof the above regimes. In this fashion many of the previofilm studies are shown to be suitable approximations ofexact solutions. In Sec. IV we study wave propagation abvp and, obtain, from our exact dispersion relation, the seral dispersion relation branches found by ArtemenkoKobel’kov.30 We show that symmetry still plays an impo

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tant role and the two branches belowvp split into severalones, the symmetry of the state being determined bynumber of half-wavelengths that fit perpendicularly to tfilm. In Sec. V we apply the present theory to the high-Tcsuperconductors, suitably choosing the parameter valuesnally in Sec. VI we summarize our major results. The proof some of the results from our exact solution, such asapproximated expressions, are the subject of the Append

II. THE LONDON-MAXWELL THEORY APPLIED TOANISOTROPIC SUPERCONDUCTING FILMS

In this section we apply the London-Maxwell theorydescribe wave propagation in a superconducting film sawiched between two identical nonconducting dielectmedia.33 An external electromagnetic wave of frequencyv,and wave numberk[v/c, is absorbed by the superconducing film, producing a mode whose dispersion relationv(b). We choose a coordinate system where the two plparallel surfaces separating the superconducting film todielectric media are atx5d/2 and x52d/2 . Propagationtakes place along thez axis such that all fields can be expressed asFi(x)exp@2i(bz2vt)#. At this point we introducethe time dependence exp(ivt) to all fields into the basic equations governing the system. Current transport inside the fiis described by the first London equation,

ivm0l i2Ji5Ei , ivm0l'

2 J'5E' , ~1!

whereEi andE' are the field components parallel and pependicular to the film surfaces, respectively. The electromnetic coupling, given by Maxwell’s equations,

“•D50, ~2!

“•H50, ~3!

“3E52 ivm0H, ~4!

“3H5 ivD ,

where

D5esE2 iJ/v, ~5!

shows that the superconductor dielectric constant is tensoD5e0«•E.

«5S «' 0 0

0 « i 0

0 0 « i

D ,

«'5«s21

~kl'!2, « i5«s21

~kl i!2 . ~6!

Phenomenological theories, such as the present, desthe superconductor only at energies much lower thanpair-breaking threshold. The frequencyv is much smallerthan the frequency defined by the superconducting gap. C

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2724 56DORIA, HOLLAUER, PARAGE, AND BUISSON

sequently the frequency is much smaller than the plasfrequency along the CuO2 planeskl i!1, thus rendering anegative, and large in modulus, dielectric tensor componparallel to the surfaces:

u« iu@1« i'21

~kl i!2 . ~7!

In the present model thec-axis plasma frequency is reachewhen the dielectric tensor component perpendicular tosurfaces becomes null:

«'~v5vp!50, vp5c

A«sl'

. ~8!

Hence«' changes sign asv crossesvp .Solving Maxwell’s equations for this particular geomet

gives two independent sets of field componen(Hx ,Ey ,Hz), the transverse electric~TE!, and (Ex ,Hy ,Ez),the transverse magnetic~TM! propagating modes. For thTM mode the superconductor-dielectric interfaces acqusuperficial charge densities. Curiously anisotropy alsoplies in a volumetric charge density inside the filmr52 i“•J/v5e0(12«' /« i)]E' /]x, although it is not re-sponsible for these low-frequency propagating modes. Insuch modes were observed in nearly isotropic thin granaluminum films.19 In conclusion the TM mode supports lowfrequency traveling waves because of the oscillating supcial charge densities that couple the nonconducting dielecmedium to the superconducting film, their ratio given by

TABLE I. The four possible dispersion relations andEz .

Ez Dispersion relation

Eozcosh(tx)t«

t«i52tanhSt d

2DEozsinh(tx)

t«

t«i52

1

tanhSt d

2DEozcos(t8x)

t8«

t«i52tanS t8

d

2DEozsin(t8x)

t8«

t«i5

1

tanS t8d

2D

TABLE II. Dimensionless variables.

Retardation Anisotropy Dielectric Thickness Wave numb

g[v/b

vR[Sl'

liD2

r[«

«sA[Sli

d D2

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ctar

fi-ic

s~x5d/2!

s~x52d/2!52

]Ez /]xux5d/2

]Ez /]xux52d/2. ~9!

The TM field equations are given below, for the dielectmedium, (x>d/2 andx<2d/2),

Ex5 ib

t 2

]Ez

]x, Hy5 i e0

v «

t 2

]Ez

]x,

]2Ez

]x2 2 t 2Ez50, t 25b22k2 « , ~10!

and for the superconducting strip (2d/2<x<d/2),

Ex5 ib« i

t2«'

]Ez

]x, Hy5 i e0

v« i

t2

]Ez

]x,

]2Ez

]x2 2t2Ez50,

t25« i

«'

b22k2« i . ~11!

In this paper we seek propagating modes that evanescthe dielectric medium, namely display exponential decaythe dielectric. This condition ist 25b2@12(v/bv)2#.0,and can be interpreted as demanding a phase velov/b, smaller than the speed of light in the dielectr

v[c/A« . Hence above the film (x>d/2) one gets that

Ez5E0exp~2 t x!. ~12!

From the other side, exponential decay inside the film issure to occur because the sign oft2 is not uniquely defined,and this is expected, since this sign determines differphysical regimes. For this reason we introduce the belowabove plasma condition and study their respectivet2 re-gions. Hereafter, in order to simplify further discussions,will choose the nonconducting dielectric media to havelargest static constant,« .«s .

v,vp. In this case both« i and «' have the same signIntroducing the Eq. ~7! approximation, one gets that2'u« i /«'ub211/l i

2.0. The planes of constant phase az5const and field amplitudes also evanesce inside the suconductor:

Ez5E0exp~2tx!1F0exp~tx!, t2.0. ~13!

v.vp. In this case« i,0 and «'.0. Even within theEq. ~7! approximation,t2'2u« i /«'ub211/l i

2 has no defi-nite sign. The mode may be evanescent or propagative inthe superconductor. However the evanescent modes dpear for « .«s and then one gets thatt2,0. In this case the

r

TABLE III. The dispersion relations forv,vp .


Eozcosh(tx/d) AAXg2

A12g25tanh~t/2!

t

Eozsinh(tx/d)AAX

g2

A12g25

1

ttanh~t/2!

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planes of constant phase arebz6t8x5const.(t8252t2.0), and we fall into the case of confined propgation, with an oblique incidence well defined at the intfaces:

Ez5E0exp~2 i t8x!1F0exp~ i t8x!,

t8252« i

«'

b21k2« i.0. ~14!

In summary for the present purposes the sign oft2 isuniquely determined below and abovevp by the sign of theratio « i /«' .

The dispersion relations follow from the continuity of thratio Hy /Ez at a single interface, sayx5d/2, once assumedthat the longitudinal fieldEz is either an even or an odfunction with respect to thex50 plane. This is possible because the two dielectric media, above and below the film,equal. At this point we introduce some dimensionless vables useful in the study of the above dispersion relatioWe cast some of our previous results in the dimension

TABLE IV. The dispersion relationsv.vp .


Eozcos(t8x/d) AAXg2

A12g25

tan~ t8/2!

t8

Eozsin(t8x/d)AAX

g2

A12g252

1

t8tan~ t8/2!

-

rei-s.ss

variables formalism. The condition of evanescence alongdirection orthogonal to the surfaces is 0<g<1,since according to Eq. ~12!, one has thatEz5E0exp(2AX/AA12g2x/d). The dominance of the inci-dent wavelength over London’s penetration along the sface, which gave Eq.~7!, becomesg2!r /X. The dielectriccomponent ratio, within the Eq.~7! approximation, is«' /« i'(1/R)(12Rg2X/r ). Thus thev,vp andv.vp re-gimes correspond toRg2X/r ,1 and Rg2X/r .1, respec-tively. The c axis ~high-Tc compounds! is perpendicular tothe surfaces and soR>1.

In the following, we obtain theexactsolution for each ofthe dispersion relations in Table I considering the dimensiless variablesdt (v,vp) anddt8 (v.vp) as a parametes, whose range is still to be determined below and abovp . Therefore we seek the dimensionless curve,g(X),in parametric form, @X(s),g(s)#, and immediatelyobtain the dispersion relation, @b(s),v(s)#5@AX(s)/l i ,(v/l i)g(s)AX(s)#.

v,vp. We introduce the dimensionless parameter,

t[dt51

AAA XR

12g2XR/r11. ~15!

Using the dimensionless variables of Table II, the dispsion relations of Table I become that seen in Table III.find their solution, first obtainX(t,g) from Eq. ~15!, andthen introduce it into Table III. We obtain a second degrequation forg(t)2 with two roots: one negative, the othepositive. The negative root does not correspond to propaing modes and only the positive is left:

the

ess

g I~ t !25~At221!2r 1A@~At221!1r #214~Ar2/R!~At221!gI~ t !

2~At221!@11~Ar/R!gI~ t !#, ~16!

X~ t !5r

R

At221

g I~ t !2~At221!1r. ~17!

I 5$S,A% labels theEz symmetry:gA(t)5t2tanh2(t/2) andgS(t)5t2/tanh2(t/2) are associated with the antisymmetric andsymmetric modes, respectively. According to the above equations the parametert range is@1/AA,`#.

v.vp. Through Eq.~14! we define a new dimensionless parameter

t8[dt851

AAA XR

g2XR/r 2121. ~18!

Introducing the Table II dimensionless variables into the~Table I! dispersion relations gives Table IV. To solve them, exprEq. ~18! asX(t8,g) and introduce it back into Table IV obtaining a second degree equation forg(t8)2, whose solution is


g I~ t8!25~At8211!1r 6A@~At8211!2r #224~Ar2/R!~At8211!hI~ t8!

2~At8211!@11~Ar/R!hI~ t8!#, ~19!

XI~ t8!5r

R

At8211

g I~ t8!2~At8211!2r. ~20!

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Like in the previous case,I 5$S,A% gives theEz parity:hA(t8)5t82tan2(t8/2) andhS(t8)5t82/tan2(t8/2) for the an-tisymmetric and symmetric modes, respectively.

The discussion of the suitable parameter range is minvolving in the present case. According to Table IV tfunction t8tan(t8/2) must remain negative for the antisymmetric problem whereas tan(t8/2)/t8 must be positive for thesymmetric case. This restricts the parameter ranget8 to theintervals, @2Np,(2N11)p# for symmetric and@(2N11)p,2(N11)p# for antisymmetric, whereN is aninteger larger or equal to zero.

Our exactparametrized solutionsv(b) are ready for ap-plications, just requiring numerical values for the parametThis is only done in Sec. V, and before we find conveniendescribe the physical properties of these solutions in thetwo sections.

III. DISPERSION RELATION BELOW THE PLASMAFREQUENCY

In this section we study the properties of propagatmodes belowvp . We find it convenient to summarize themajor physical properties and introduce some approximaexpressions, each describing a different regime of the ev,vp dispersion curve. We leave to the Appendix the prothat the previous section’s exact results do justify our pictand yield the approximated expressions. Below the plasfrequency the field amplitudes inside the film evanesce frthe surfaces, and this exponential decay is characterizet, according to Eq.~13!. The choice of film thickness,d, isvery important in order to assure sufficient coupling betwethe two surfaces. For extremely thick films the surfacescouple and the symmetric and the antisymmetric modesthe same, only independent surface plasma modes exithis situation. Indeed, in the case of strong coupling betwthe two surfaces three distinct regimes are possible for bsymmetric and antisymmetric modes. In sequence of incring b we call themoptical, coupled, andasymptotic~see Fig.2!.

Close to the originb'0, the modes are optical, that ithey are essentially plane waves traveling in a dielecmedium. There is almost no exponential decay in the srounding dielectric semispaces up to appreciable distant '0, according to Eq.~12!. Propagation occurs with thspeed of light in the dielectric, (g'1). The kinetic energy ofthe superconducting carriers is negligible compared tomagnetic energy of the mode. The superconducting film ctributes very weakly to this regime. In the optical regime ohas approximately the linear behaviorv'vb.

Slowly the condensate’s kinetic energy increases withbuntil finally both energies become comparable and a fi

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crossover takes place. This is the onset of thecoupledre-gime, a slow mode (g,1), where the condensate’s kinetenergy dominates over the magnetic energy, the film anddielectric are strongly coupled. This is the true plasma moregime. Such evanescent propagating modesknown,16–18,28here we discuss the effects of ac-axis plasmafrequencyvp below the gap.30 In the coupled regime insidethe film, the fields evanesce from the surfaces very smooresulting into two types of coupling between the superficplasmons, which are the antisymmetric and the symmemodes. For the symmetric mode its dispersion relationbeen theoretically studied by many authors16–18 and experi-mentally observed19,20 in the past:

vs'vl iAdb

2. ~21!

Recently28 it has been proposed that the symmetric modejust the lowest frequency branch. In fact, there is an up~antisymmetric! branch mode, that can also be experimetally observed for highly anisotropic high-Tc superconduct-ing films:

va'v

l'A 1

db/21«s / «. ~22!

Because the antisymmetric branch is the highest inquency, it is more sensitive to the effects of the plasmaquencyvp . For db/2 comparable~or smaller! to «s / « , thisbranch becomes difficult to observe sinceva'vp . Fordb/2 dominant over«s / « , the symmetric and antisymmetrirelations are proportional toAb and 1/Ab, respectively.

The crossover frequency between the optical and the smetric regime is obtained at the intersection of the tbranches:

vcross,s5vd

2l i2 . ~23!

The crossover between optical and antisymmetric modeimplicitly given by the unique real solution:

S d

2v Dvcross,a3 1S «s

«Dvcross,a

2 2S vl'

D 2

50. ~24!

For the antisymmetric mode, the optical and the coupledquencies are, respectively, an increasing and a decreafunction of b. Thus the crossovervcross,a also gives an esti-mate of the maximum frequency, associated with the pseen in the antisymmetric curve~Fig. 2!.

By increasingb, evanescence inside the superconducbecomes stronger (t is large!, up to the point where the

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surfaces are nearly decoupled. We have just reachedasymptotic regime. Thus for sufficiently highb, symmetric,and antisymmetric modes converge to the same asympfrequency,

vas5vpAS 11A114~ « l i /«sl'!2

112~ « l i /«sl'!21A114~ « l i /«sl'!2D .

~25!

Notice that it is always true thatvas,vp . It is easy to checkin the following extreme cases of a very high dieletric costant (« /«s@l' /l i), and of the opposite case, namewhen anisotropy plays a more important role than the dietric constant, (« /«s!l' /l i). In the former case we obtain28

vas5v/Al'l i, and in the lattervas5vp . In summary wehave just reviewed the major features of the three possregimes for both the symmetric and the antisymmetric dpersion relations.

We find that the dielectric constant ratior must be suffi-ciently high in order to prevent that theb range be extremelylimited for the coupled regime. In Sec. V we study succrossovers in the case of the high-Tc parameter values. Thcriterion for the disappearance of the coupled antisymmeand symmetric regimes is just given by an asymptotic fquencyvassmaller thanvcross,s andvcross,a , respectively. Asan example consider the case of an extremely large anropy (l'→`), where we get thatvas'vp→0, eventuallysmaller than the crossover frequencies of Eqs.~23! and~24!.In this limit the dispersion relation goes almost directly frothe optical to the asymptotic regime.

IV. DISPERSION RELATION ABOVE THE PLASMAFREQUENCY

In this section propagating modes inside the supercducting film with the frequency larger thanvp are consid-ered. We summarize their major physical properties,leave to the Appendix the derivation of such propertiesterms of the exact parametrized dispersion relation. Weagain interested in modes that evanesce in the dieletricdia. As discussed before,t8 is always real with the experimental parameter choice of« .«s . The propagation of lightinside the film can be understood by a superposition ofdispersive plane waves, exp$2i@6t8(b)x1bz2vt#%, regardedas incident and reflected waves, thus with a well definangle of incidence at the superconductor-dielectric infaces. Thus while outside the film light is still evanescentits interior the propagation can be pictured through ray op~see Fig. 1!, similarly to an optical fiber.

The exact parametrized plasma relation, given by E~19! and ~20!, leads to the simple approximated dispersirelation, in the vicinity ofv/b'v,

vM5vpA11~bd!2

~pM !21~d/l i!2S l'

l iD 2

, ~26!

whereM is a positive integer, odd for the symmetric mod(M52N11) and even for the antisymmetric mod@M52(N11)#. This approximated expression was obtain

he

tic

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dr-

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d

by Artemenko and Kobel’kov.30 The M th curve satisfies theevanescence condition,g<1, for b.b init(M ):

b init~M !51

l'A«s / « 21/~pM !2~d/l i!

2. ~27!

At the particular valueb5b init(M ), the wave vector alongthex direction is given byt8d5Mp whereM is exactly thenumber of half-wavelength that fit perpendicularly to tfilm. For b.b init(M ), the relation betweent8 andM is notso simple because the wave is also in the dielectric mediathis caseM just determines the number ofEz extrema alongthex direction~see Fig. 1!. For instance, when just one halwavelength fits into the film we are facing theM51 sym-metric mode, which has just a singleEz extremum~maxi-mum!. We notice that the frequency of the plasma moddecreases for increasingM . For b,b init(M ) we have thatg.1 and the modes are not evanescent, they are pwaves traveling in the dielectric media. They have an olique incidence at the surfaces and to see this just takeplane wave exp$2i@ tx1bz2vt#% with the speed of light in

the dielectric, v5v/A t 21b2. One obtains thatt 5bAg221, showing that we are in ag regime not studiedhere.

FIG. 1. A pictorial view of wave propagation in a superconduing film surrounded by two equivalent nonconducting media. Fv,vp the instantaneous electric field and the superficial charare shown for symmetric~a! and antisymmetric~b! modes. Forv.vp the optical ray associated with the plane wave travelinside the film is shown here~c! for M51, 2, 3, and 4. The sym-metry of each state is also shown here, for both cases~a! and ~b!,and ~c!, through the diagramEz versusx.

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The present theory eventually breaks down because lt8 means an infinitesimally small wavelength inside the fialong thex axis. For this reason a physical cutoff must be pinto this theory, the interplane separationa. The upper limittmax8 is determined when ax direction half-wavelength fitsinto the interplane distancea:

tmax8 5p/a5Mmaxp/d. ~28!

Introducing Mmax in the approximated dispersion relatioEq. ~26!, gives the lowest frequency plasma modvmin5vpA11(blJ /p)2, wherelJ[al' /l i is the Joseph-son penetration depth.

Beyondtmax8 more elaborate models should provide a dscription of the condensate in such scale. This can rendetheory’s usage quite limited, in case the ratiod/a is small,because for very few half-wavelengths inside the filmreach the cutoff limit. However the applicability of thimodel abovevp is not restricted to a small ratiod/a becausethe propagating mode has constant amplitude inside theand so, no matter how far apart the surfaces are, therealways be waves reflected at the surfaces. For this reasomajor conclusions of this section, namely, the existenceslow modes above the plasma frequency, correspondindispersive plane waves inside the film, must remain vaeven near the cutoff limit. In the next section we take tstandard parameter values for the high-Tc material and dis-cuss the properties of such propagating modes belowabove the plasma frequency.

V. APPLICATIONS

For our applications we choose the following set of prameters for the high-Tc ceramic superconductors. The stadielectric constant2 is «s'30. The zero-temperature Londopenetration length along the CuO2 planes is lCO50.15mm.34 While the anisotropy, l' /l i , is 5 forYBa2Cu3O82x ,34 it has been changing in the past fBi2Sr2CaCu2Ox , ranging from 102, mainly from torque mea-surements studies,34 to much higher values.36 We show herethat choosing the anisotropy between these two valueshave important effects on the properties of the dispersrelations for v,vp . There is also the compounTl2Ba2CaCuOx compound35 with l' /l i'90.

Figure 1 provides a pictorial intuitive view of wavpropagation in the film below and above the plasma fquency. Belowvp the instantaneous electric field, as wellthe componentEz , are shown for both symmetric and ansymmetric fields. The surface charges are the sole sourcpropagating electric fields. The superficial charge arranment has strong consequences for the electric-field distrtion inside the film, leading to symmetric and antisymmetmodes, found at distinct frequency ranges, the latter beinupper mode. To understand the effects of anisotropy, olayered structure, into these modes, where the CuO2 planesare parallel to the film surface, consider the transverselongitudinal field and supercurrent components.Ex and Jxare very intense for the antisymmetric mode and nearly zfor the symmetric one, andEz and Jz , are the dominantcomponents for the symmetric mode but not for the antisymetric one. Thus the relevant penetration depths for the s

ge

t

:

-he

,illtheftode

nd

-

ann

-

ofe-u-

ana

nd

ro

--

metric and the antisymmetric modes must bel i and l' ,respectively. The anisotropyl'.l i hardens the systemalong thec axis thus making the antisymmetric mode lowin frequency. Abovevp a dispersive plane waves traveinside the film that undergoes total reflection at the intfaces. This figure also depicts the number of extrema forelectric componentEz , which determines the symmetry othe mode.

In Fig. 2, the symmetric and antisymmetric dispersionlations are shown for a very thin filmd510 nm and the threeanisotropies,l' /l i55, 102, and 103. In order to slow downas much as possible light in the dielectric, and so, lowercoupled regime frequency range, we choose SrTiO3 as theexterior nonconducting dielectric medium. At low tempertures its dielectric constant is known to be high up to tGHz frequency,19 « '20 000. The symmetric state crossovfrequency,vcross,s , between the optical and the square roregimes is found to be 75 GHz for the three anisotropdisplayed here. The antisymmetric state crossover frequevcross,a , between the optical and the coupled regime,pends on the anisotropy, being 2.3 THz, 290 GHz, andGHz for l' /l i55, 102, and 103, respectively. Because othis anisotropy dependence they are not indicated in thisure. Notice that the symmetric mode~dashed lines! is alwaysfound at a frequency range lower than its correspondingtisymmetric mode~continuous lines!. In the low wave-vectorlimit, all the dispersions collapse into the same linear~opti-cal! regime. In the opposite limit,b very large, both sym-metric and antisymmetric curves converge to the saasymptotic frequency. This asymptotic frequency is justsurface plasma frequency, strongly affected by anisotropTHz for Y-Ba-Cu-O, 200 GHz for Bi-Sr-Ca-Cu-O and 5GHz for l' /l i5103. This saturation frequency is independent of film thickness, as expected, according to Eq.~25!.Notice that the wave vector signaling the onset of tasymptotic regime decreases for increasing anisotropy, raing from 10 mm21 for Y-Ba-Cu-O to 100 mm21 for themaximum anisotropy considered here. Between the optand the asymptotic is the coupled regime, the true plasmode, found to extend over a large range of frequencywave number for a Y-Ba-Cu-O thin film, according to th

FIG. 2. The symmetric and antisymmetricv,vp dispersionrelations are shown for three anisotropies. A very thin supercducting film,d510 nm thick, surrounded by SrTiO3, is considered.

sththe

ftha

w

nc-

mnyCthAlu

nanio

a

p.of

e-heinthet

has

desdi-

itssf aapthehepa-toffe-ncy.avetheon-a

-lmingoutce inre-gerdeshusudyur

s inetricex-

tedre-In

e

tiv


figure. In this regime the symmetric plasma mode followsquare root and the antisymmetric mode clearly showsinverse square-root dependence. For Bi-Sr-Ca-Cu-O,coupled regime is shorter than in Y-Ba-Cu-O, ranging btween 0.1 and 5mm21. For the maximum anisotropy o103, the coupled regime disappears thus only survivingoptical and the asymptotic regimes. We conclude thatextremely large anisotropy inhibits coupling between the tsurfaces and enhances the surface plasma modes.

Figure 3 shows the asymptotic frequency versus« , forseveral anisotropies. Notice that the asymptotic frequefor small « is just vp . For Y-Ba-Cu-O, the asymptotic frequency drops over an order of magnitude when« changes bythree decades. The choice of a nonconducting media of sdielectric constant can render impossible the observatiothese plasma modes because the asymptotic frequenccomes larger than the superconducting gap. In Bi-Sr-Cu-O, and other extreme anisotropy compounds,asymptotic regime is always much smaller than the gap.« increases, we have determined the crossover va«sl' /l i , where the asymptotic frequency acquires a« de-pendence, thus being strongly affected by anisotropy. Hein order that the inverse square-root dependence of thesymmetric coupled regime be observable, the condit« .«sl' /l i must be satisfied.

Figure 4 shows the dispersion relations abovevp for a100-nm-thick film. The dielectric constant« is taken equalto «s. We have chosen a Bi-Sr-Ca-Cu-O compound withanisotropyl' /l i 5 102. Therefore thec-axis plasma fre-quency is much smaller than the superconducting ga37

Above vp , plasma modes branch into a large numbermodes when film thickness increases. For the choice o100-nm-thick film, the present London-Maxwell theory rmains valid for the first 66 modes. We plot in this figure tM51,2,3,4,7,12, and 66 dispersion relations. For increasM the plasma mode becomes slower, indicating thatnumber of reflections that the confined plane wave undgoes at the interfaces has also increased. Notice thatM566 plasma mode is very nearvp . The M51 branch is

FIG. 3. This figure shows the asymptotic frequency versus« ,

for three anisotropies. For small« the asymptotic frequency is jusvp . The asymptotic frequency for Y-Ba-Cu-O is the most sensitto changes in the substrate dielectric constant.

aee-

eno

y

allofbe-a-ese,

ceti-n

n

fa

ger-he

very near to the optical branch showing that it essentiallythe speed of light in the dielectric media.

VI. CONCLUSION

In this paper we have studied propagating plasma moin a superconducting film surrounded by two identicalelectric media. The superconductor is anisotropic havinguniaxial direction (c axis! perpendicular to the interfacewith the dielectric media. We consider the existence oplasma frequency along the uniaxial direction below the gand study its effects into the propagating modes usingLondon-Maxwell theory, which gives a good account of tphysical situation for scales larger than the interplane seration. In fact we do use the interplane separation as a cufor the present theory. We only consider low incident frquency compared to the superconducting gap frequeMoreover the wavelength associated with the incident wis much larger than London’s penetration length alongsurfaces. Therefore the dielectric constant of the supercducting film along the plane is always negative and withlarge modulus. Along thec axis, the situation is quite distinct. The dielectric constant of the superconducting fialong this direction vanishes at the plasma frequency, bepositive above. In this paper we were only concerned abmodes that propagate parallel to the surfaces and evanesthe dielectric media. Under this condition, and the requiment of the nonconducting media dielectric constant larthan the superconductor’s static constant, we find that moare either totally above the plasma frequency, or below, tnever crossing the plasma frequency line. Thus we stthese two regions of frequency separately. Within oLondon-Maxwell theory framework we find theexact ex-pressions for the dispersion relations of the plasma modethe two cases. The exact expressions are found in paramform, the parameter being the one that characterizes theponential behavior inside the superconductor. Approximaexpressions for some especial regimes of the dispersionlations are obtained from our exact parametric solution.this way we retrieve well-known results in th

e

FIG. 4. The dispersion relations abovevp is shown for a 100-nm-thick film and anisotropyl' /l i 5 100. The M51 modepropagates with speedv and at theM566 mode is the upper limitfor the validity of the present theory.

afifr

st ithtiptymartrthtor

ioonntthtionvheis

vaesce-thdiice

laa

th

onayfra

.

herehe

nIn

--

ly;on

ery

ym-me

dd ae

ricter-r-

-

ht

rs

n

et-s

tion

a-the-tq.thebuton

m


literature16–18,28,30to the context of an anisotropic film withplasma frequency inside, and also, derive new ones. Wethat propagating modes below and above the plasmaquency have quite distinct physical properties.

Below the plasma frequency, the amplitude also evaneinside the superconductor, the film thickness is importanorder to assure a sufficiently strong coupling betweensurfaces. There are two branches of the dispersion relacorresponding to the two possible arrangements of the suficial charge densities. The symmetric branch is the lowesfrequency and has the two superficial charge densities smetrically disposed. The highest branch has opposite chfacing each other at the interfaces thus being antisymmeWe have studied here in detail the three possible regimescan exist for these two branches, their crossover, andconditions for the existence of the coupled regime, the minteresting of the three regimes. In the so-called coupledgime both symmetric and antisymmetric dispersion relatbranches provides independent information on the two Ldon penetration lengths. The antisymmetric mode is imately connected to the transverse current component,being highly sensitive to the transverse London penetralength, in the same fashion that the symmetric mode depeon the longitudinal London penetration length. We beliethat this remarkable property can be used to gain furtunderstanding of the transverse current component in antropic and layered films.

Above the plasma frequency the amplitudes do not enesce inside the film, propagation is geometrically undstood, and an oblique ray can represent the plane wave inthe superconductor which is totally reflected at the interfa~see Fig. 1!. However outside the film, in the dielectric media, the wave remains evanescent, and we concludeabove the plasma frequency the superconducting filmplays confined propagation, similar to an optic fiber. Notthat the film thickness is not a crucial parameter becausematter how apart the surfaces are, there will always be pwaves traveling in its interior. Contrary to below the plasmfrequency, above there are many modes correspondingsentially to the number of half-wavelengths that fit insidefilm along thec axis.

In summary we have shown in this paper that supercducting films surrounded by a dielectric medium displvery interesting plasma mode propagation properties atquencies below the gap frequency and such propertiesquite distinct below and above thec-axis plasma frequency

ACKNOWLEDGMENT

This work was done under a CNRS~France!-CNPq~Brazil! collaboration program.

APPENDIX

In this appendix we provide further details on how texact parametrized solutions of Sec. IV lead to the pictudeveloped in Secs. III and IV for below and above tplasma frequency, respectively.

v,vp. The three possible regimes, optical coupled aasymptotic follow from the exact parametrized solution.particular we show that Eqs.~21! and ~22!, for the coupled

nde-

ceneoner-in

-geic.at

heste-n-

i-usndsero-

-r-ides

ats-

none

es-e

-

e-re

s

d

regime, and Eq.~25!, for the asymptotic regime, follow fromEqs.~16! and~17!. We derive both Eqs.~21! and~22! withinthe following approximations:~i! retardation effects are neglected (g!1) (A12g2'1); ~ii ! fields evanesce slowly inside the film,t!1. Therefore the~Table III! dispersion rela-tions become AAXg2'1/2 and AAXg2'2/t2 for thesymmetric and for the antisymmetric modes, respectiveand~iii ! the film thickness is much smaller than the Londpenetration length along the surfacesA@1 so that eventhough t is small we can approximate Eq.~15! byAt2'AXR/(12g2XR/r ). Direct elimination of the param-eter t leads to both Eqs.~21! and ~22!. In the asymptoticregime fields inside the superconductor fade away vquickly from the surfaces, thus corresponding tot@1. Thesurface decoupling is seen from the antisymmetric and smetric dispersion relations which converge to the saasymptotic frequency vas. According to Eq. ~16!gA(t→`)5gS(t→`)→t2 and Eq. ~25! follows in astraightforward way from this argument.

v.vp. To derive Eq.~26! from the exact parametrizesolution is in fact very simple, because it does not demandetailed study of theg(t8) curve. To do so we reparametrizthe exact solution of Eq.~19!, replacingt8 by a new param-eter t1, t85t I2t1 where tS5(2N11)p, andtA52(N11)p. The advantages are twofold: both symmetand antisymmetric modes are now defined in the same inval 0<t1<p; and the approximation is under control, coresponding to the limitt1 /t I!1. Notice that att150 allcurves satisfy the conditiong51, thus justifying our claimthat Eq.~26! is a good approximation for the exact parametrized dispersion relation of Eqs.~19! and ~20! when thephase velocity is approximately given by the speed of ligin the dielectric. Indeed to obtain Eq.~26! from Eqs.~19! and~20!, notice thatg(t I2t1) can be Taylor expanded in poweof t1 because the functionshI(t8), introduced in Sec. IV, arewell behaved in this neighborhood: hI(t8)5(t I2t1)2tan2(t1/2). We introduce the approximatiot1 /t I!1 into Eq. ~20! obtaining that XI'(r /R)/$g I

2

2r /@A(t I)211#%. The key issue here is that all the param

ric dependence ofX is now ong, because the last term waapproximated by a constant,r /A(t I2t1)2'r /AtI

2 . Thereforewe obtain the approximated parametrized dispersion rela

v~ t1!'vpA g I~ t1!2

g I~ t1!22r /AtI2, ~A1!

b~ t1!'bcA 1

g I~ t1!22r /AtI2, ~A2!

from where the family of curvesv(b) of Eq. ~26! follow, bysuitably removing the functiong(t1).

Finally we would like to get some more detailed informtion about the parametrized exact solution, in particulartwo following issues:~i! v(b) is always an increasing function of b; ~ii ! nearvp it suffices to consider the positive rooof Eq. ~19! because the positive and negative roots of E~19! are just parts of the same curve and meet wheresquare root vanishes. The claims are of general validitywe derive them under some further working assumptionsEq. ~19!, which in leading order, acquires a very simple for

i-

oua

tc-rt

cal

end

the


~ t I2t1!2@r @1,

g I~ t1!2516A124~r 2/R!tan2~ t1 /2!

2@11Ar/R~ t I2t1!2tan2~ t1 /2!#. ~A3!

To have the conditionr @1 satisfied one must choose a delectric media of sufficiently high constant:« @«s . To haveat least one mode described by the above inequality,must require that the minimal frequency satisfies the ineq

ity: tmax8 @Ar /A, which isA« /«s!2pl i /a. To show that the

d.

.

.

.

hi

B

e

.

.

nel-

curve never saturates, taket150, where one finds thag51 andg50 for the positive and negative roots, respetively. Obviously the positive root is the low-frequency paof the curve. Notice that the negative root is not physibecause forg50, b(X) is negative according to Eq.~20!.This just means that the negative square root curve mustat a finite nonzerog, where the denominator of Eq.~20!vanishes and soX diverges.g is limited between zero andone, so it must also be finite at this point. Consequentlyvmust also diverge, and so, we conclude from this thatcurve never saturates inv or b.

ys.

A

ps

a C

1The experimental observation of the plasma edge was founY-Ba-Cu-O by B. Koch, M. Dueler, H.P. Geserich, Th. Wolf, GRoth, and G. Zachman, inElectronic Properties of High-Tc Su-perconductors, edited by by H. Kuzmany, M. Mehring, and JFink ~Springer, Berlin, 1990!.

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4J. H. Kim, H. S. Somal, M. T. Czyzyk, D. van der Marel, AWittlin, A. M. Gerrits, V. H. M. Duijn, N. T. Hien, and A. A.Menovsky, Physica C247, 297 ~1995!.

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10P. W. Anderson, Phys. Rev.112, 1900~1958!.11T. Mishonov, Phys. Rev. B44, 12 033~1991!; 50, 4004~1994!.12S. N. Artemenko and A. G. Kobelkov, JETP Lett.58, 445~1993!.13H. A. Fertig and S. Das Sarma, Phys. Rev. Lett.65, 1482~1990!;

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~1994!; 74, 1493~E! ~1995!.

in

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v.

20F. J. Dunmore, D. Z. Liu, H. D. Drew, and S. Das Sarma, PhRev. B52, R731~1995!.

21H. Boersch, J. Geiger, A. Imbush, and N. Niedrig, Phys. Lett.22,146 ~1966!.

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23D. Sarid, Phys. Rev. Lett.47, 1927~1981!.24S. Ushioda and R. Loudon, inSurface Polaritons, edited by V.M.

Agranovich and A.A. Maradudin~North-Holland, New York,1982!, p. 573.

25E. N. Economou, Phys. Rev.182, 539 ~1969!.26K. L. Kliewer and R. Fuchs, Phys. Rev.153, 498 ~1966!.27P. K. Tien and R. Ulrich, J. Opt. Soc. Am.60, 1325~1970!.28M. M. Doria, F. Parage, and O. Buisson, Europhys. Lett.35, 445

~1996!.29M. Tachiki, T. Koyama, and S. Takahashi, Phys. Rev. B50, 7065

~1994!.30S. N. Artemenko and A. G. Kobelkov, Physica C253, 373

~1995!.31S. V. Pokrovsky and V. L. Pokrovsky, J. Supercond.8, 183

~1995!.32P. Yeh,Optical Waves in Layered Media~Wiley & Sons, New

York, 1988!.33O. Buisson, F. Parage, and J. Richard, inMacroscopic Quantum

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819 ~1990!.36E. H. Brandt, Rep. Prog. Phys.58, 1465~1995!.37Tunneling and proximity effect studies give the frequency ga

5.031012, and 7.531012 Hz, for Y-Ba-Cu-O and Bi-Sr-Ca-Cu-O, respectively. See, for instance, M. R. Beasley, Physic185, 227 ~1991!.

plasma waves in anisotropic superconducting films below and above the plasma frequency

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