SURFACE AND BULK PLASMON-POLARITONS IN PERIODIC METALLIC HETEROSTRUCTURES

(1)

HAL Id: jpa-00224160

https://hal.archives-ouvertes.fr/jpa-00224160

Submitted on 1 Jan 1984

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

SURFACE AND BULK PLASMON-POLARITONS IN PERIODIC METALLIC HETEROSTRUCTURES

G. Giuliani, J. Quinn, R. Wallis

To cite this version:

G. Giuliani, J. Quinn, R. Wallis. SURFACE AND BULK PLASMON-POLARITONS IN PERIODIC METALLIC HETEROSTRUCTURES. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-285-C5- 289. �10.1051/jphyscol:1984542�. �jpa-00224160�

(2)

JOURNAL DE PHYSIQUE

Colloque C5, supplément au n04, Tome 45, avril 1984 page C5-285

SURFACE AND BULK PLASMON-POLARITONS I N P E R I O D I C METALLIC HETEROSTRUCTURES

G.F. G i u l i a n i , J . J . Quinn and R.F. all lis'

Physics Department, Brown U n i v e r s i t y , Providence, R I 02922, U.S.A.

' p h y s i c s Department, U n i v e r s i t y of CaZifornia, Iruine, CA 92727, U.S.A.

Résumé : Nous avons étudié l a dispersion des excitations de charge d'une hétérostructure métallique périodique dans l e cadre d'une théorie locale. On trouve un spectre riche qui se c m p s e d'excitations de volume e t de surface. La fréquence de ces excitations dépend des propriétés diélectriques des milieux qui composent l a structure. Nous discutons l a correspondance de notre analyse avec l e s résultats d'une théorie non-locale a i n s i que l e s e f f e t s r e l a t i v i s t e s .

Abstract: We have investigated the theory of the plasmon-plariton niodes of a p r i o d i c metallic heterostructure d i n g use of a local theory. A rich spectrum is found of b t h buU: and surface excitations whose freguency depends upon the di- e l e c t r i c properties of each constituent. We discuss the connections of our analysis with the results obtained within a non-local theory and the effects of retardation.

In t h i s paper we report on the r e s u l t s of a theoretical investigation of the collective plamn-polariton spectrum i n a periodic metallic heterostructure con- s i s t i n g of an array of alternate metallic slabs characterized by dielectric constants and ^{c g .}^Aschematic i l l u s t r a t i o n of t h i s system i s provided in Pig. 1.

The unit ce11 of t h i s structure is r-epresented by t w slabs of different material.

Such a system can be realized i n pmctice by an a r t i f i c i a l l y structured sanicon- ductor. Highly do@ semiconductor heterostructures l i k e C,aAs/G+All-&s and InAs/GaSb, o r modulation do@ structures 1il;e n-n+-n-n+. . . layers i n S i or GaAs, seem a t the m n t t o offer the mst flexible and viable solution. An extreme case i s representeà by type-I and type-II semiconducting superlattices i n which a pericdic array of quasi two-dimentional electron and/or hole layers i s akedded i n a dielectric background. l

The plasna modes of an i n f i n i t e pericdic ~ t a l l i c l?eterostructure have been recently studied within a non-local theory based upon a generalized hydrodynanic a p p r ~ a c h . ~ We present here a simpler approach which M e s use of the usual local t h e ~ r ~ . ~ We w i l l assume the e l e c t r i c f i e l d in the A-type region of the nth ce11 of the system t o have the forn

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984542

(3)

JOURNAL DE PHYSIQUE

Fig. 1 - S e h a t i c of a pericdic metallic heterostructure consisting of twa different t p s of di- e l e c t r i c slabs .

Here q i s the wavenLnnber along the layers, "A= (q2 - ^C-22 ^w ^Ea) , anü E* i S the amplitude of the forward or bachlard going waves. In (1) d = dA + dB is the width of the unit c e l l , and O<z<dA. The z axis is chosen o r - q o n a l to the layers. An analogous formula can be written for the e l e c t r i c f i e l d i n the B-type region of the same ce11 i n te- of the corresponding quantities argand3 + in. The spectrum of the bulk plaçma d e s of the system is then obtained imposing s a -

4 .

dard boundary conditions on Eyand D Z a t the interfaces with the aid of the mn- s t i t u t i v e equation

where I k / < i l /d. This equation i s the analogue of the Bloch theoren for the e l e c t r i c f i e l d associated to the plasmons of a periodic m t a l l i c structure. The dispersion relation f o r the bulk p l a m modes i s found t o ke

where TI =CA% / @ B %. I f we choose for the dielectric constants the usual local

2 2

f o m ~ = 1

-gIB

¹

.

^where ^W is the local value of the pla- frequency AIB

in the A(B)-type regions, in the electrostatic l i m i t Eq. (3) leads t o

As i l l u s t r a t e d i n Piq. 2 and 3, the spectrum consists in tm separate bands of plasma modes (shaded regions), which are spanned by Eq. ( 4 ) as k i s varied kt- ween O (correspnding t o w+(q,O), - the two interna1 edges) , t o IT /d (correspoding

(4)

Fiq. 2 - A p l o t of f r e q e n c y w vs. q.

The t m shaded reqions a r e t h e l o c i of bulk plasna ex- c i t a t i o n s , Eq. (4) . ^Here

bth WA and WB a r e f i n i t e and w~zwg. wW ( w i +w$ +/

/ ^2%;^WAB is defined in the t e x t . The thick lines a r e surface p l a m n s . The one higher i n frequency irierges into t h e continuum a t qd

-

3.4. Not shown i s t h e low l y i n g single p a r t i c l e continuum which is r e s p n s i b l e f o r t h e Landau damping of t h e surface p l a m n s .

to - ( q , ~ / d ) , t h e two external ones). The 'cm plasmon bands tend asymptotically to wAB = ( wz+ /2+ a s qd becams large. For d~ =dg , t h e gap between t h e t c i ~

bands closes and t h e plasmon continuum i s sinply connected. In t h i s respect this continuum is similar to t h e one of type-II smiconductinq superlattices. These modes a r e collective excitations of t h e e n t i r e array and can be viewed a s super- p s i t i o n s of the surface plasrnons which characterize an isolated A-B q t e r f a c e . This can e a s i l y l x seen from Eq. ( 3 ) by taking t h e appropriate l i m i t s . I f one of t h e tm frequencies i s z e r o , t h e lower band w i l l consist of acoustic plasmons and the contin.~.~.m w i l l extend between w=O ;mrl wc.ti)-(q,O) .

A s i n the familiar case of a hmgenous system this picture does not change significantly when retardation e f f e c t s a r e allowed for.

The l o c a l theory discussed above leads t o two plasmon branches f o r any given value of q. In a more appropriate non-local theory i n which t h e quantpn nature of t h e conduction electron gas is accounted for, propagating plasma waves a r e allowd,toqether~with interface plasmons i n each meta1li.c s l a b . I n t h i s case plas- mn-wave-guide phenomena occw which lead t o a m c h richer spectrum a s f o r each value oE q several extra branches e x i s t whose number de-wnds u p n t h e geometrical situation.

Consider now the case in which t h e periodic array i s terminated a t t h e interface between an A-type s l a b and a s e n i i n f i n i t e rnedim characterized by a dielec-

(5)

JOURNAL DE PHYSIQUE

Fig. 3 - Same a s Fig. 2. Here w ~ < w g . ("AB The t m surface p l a m n

branches are in t h i s case well detached £rom the 'cm continua.

W~

("AO

t r i c constant s o . Following the procedure discussed i n reference 6 leads i n a straiqhtforward way t o the dispersion relation f o r the surface modes of t h i s system. We find

2 -2 2

where no= ~ ~ awith a0 ~ /= (q ~ - ~c w a ~ Eq. (5) ~ i s cqlemented by the expression for a which i s a coinplex quantity whose real part i s positive definite ancl represents the inverse penetration lenqth. m r e precisely ~&=ex~(-ad)~&,+, +

From Eq. (3) we have

2 2

I f we take tz0= l-uO/u , Eq. (5) leads i n the non-retarded limit t o a simple quadratic equation for w 2 . The spectrum which results is extranely rich anù depends i n an essential way u p the s p c i f i c values of uA, W O , dA and dB .

ûnce w i s known, a is determincd by Eq. ( 6 ) . It m s t be stressed here t h a t care m s t be taken in disregavding s p i o u s solutions which are characterized by n q a - tive values of Re(a ) . Figs. 2 and 3 show C m surface plasmon branches correspnding t o tw~ different situations i n the electrostatic lisnit. For large values of qd the two branches tend t o the frequency of the s u f a c e plasnons a t isolated A--B and A-O interfaces. A d i r e c t inspection of -[. (6) leads tothe conclusion

(6)

that I m ( n ) = x/d, for %<w<o-(q,a/d) and w+(a,T/d)<w<wA, for UA>%. Im(n) = 0 elsewhere. Unlike the case of semiconducting superlattices7 the surface excitations discussed here do suffer Landau damping due to their three-dimensional cha- racter .

Surface plasrîionscannot exist within the light cone and their dispersion relation is severely mdified for m l 1 values ofcq/w when retardation effects are accounted for. As d is much larger than the typical atomic distance,relativistic effects can be hprtanteven for qdQl. The lower branch in Figs. 2 and 3 is par- ticularly interesting as the interaction with the radiation field could allow the detection of these modes by means of the usual attenuated total reflection method.

Further suitable experimental procedures include resonant Raman scattering and electron enerqy loss.

A more detailed analysis of the retardation limit and of the effects of non- locality on the dispersion relation of the surface plasmons of periodic metallic heterostructures will be reported in a forthcming publication.

Acknowiedcpnents: We are qrateful to the National Science Foundation for supprt of this work under Grants D W 81-21069, and INT 81-15141.

References :

* On leave fram the Scuola N o m l e Superiore, Pisa, Italy.

1. L. Esaki, "Novel Materials and Techniques in Condensed Vatter", ed. by G. W. Crabtree and P. Vashista, (North Iiolland, NY, 1982) , page 1.

2. G. F. Giuliani, J. J. Quinn, and R. F. Wallis, Bull. Am. Phys.

Soc. E, 448 (1983) ; and preprint.

3. R. E. C d e y and D. L. Mills, Bull. Am. Phys. Soc. 28, 408 (1983);

these authors have apparently carried out an analysis simil= to ours.

4. K. W. Chiu and J. J. Quinn, Phys. Rev. El 4727 (1974).

5. G. Qin, G. F. Giuliani, and 5. J. Quinn, to be published in Physical Review .

6. G. F. Giuliani and J. J. Qiunn, Phys. Rev. Lett.5&919(1983); and to be published in Surface Science.

7. R. Fuchs and K. L. Kliewer, Phys. Rev. - 3, 2270 (1971); D. E.

Beck, Phys. Rev. - 4, 1555 (1971).