Localized and resonant acoustic modes in semi-infinite superlattices

Vacuum/volume 45/numbers 2/3/pages 341 to 344/1994 Printed m Great Britain

0042-207X/94S6 00+ 00 © 1993 PergamonPress Ltd

Localized and resonant acoustic modes in s e m i - i n f i n i t e superlattices B D j a f a r i - R o u h a n i , E H El B o u d o u t i a n d E M K h o u r d i f i , Equ~pe de Dynam~que des Interfaces, Laboratoffe de Dynamtque et Structure des Mat~rmux Mol~culatres, UFR de Physique, UmversR~ des Sctences et Technologtes de Ldle, 5 9 6 5 5 Vdleneuve d'Ascq, France

We mvestlgate the exmtence of Iocahzed and resonant acousttc modes assoctated with the surface of a semlmfmtte superlatttce or its interface with a substrate, consMermg the case of shear horizontal wbrattons These modes appear as well-defmed peaks of the density of states, either ms/de the mmtgaps or mstde the bulk bands of the superlatttce The denstty of states, whmh are calculated as functtons of the frequency ~o and the wave vector kll (parallel to the mterfaces), are obtained from an analyttc determmatton of the Green funcOon for a semHnhntte superlattwe w~th or wtthout a cap layer, as we//as for a superlatOce m contact w~th a substrate Bestdes, we show that the creatton from the/nftmte superlatttce of a free surface or of the substrate-superlatt~ce interface gtves rise to ~ peaks of wetght 1/4 m the density of states, at the edges of the superlatttce bulk bands Finally tt wtll be emphasized that the same calculations can be transposed to the study of the electromc structure of superlatttces m the Kromg-Penney m o d e l or to the propagation ofpolarttons

The acousuc vibrations m superlattices (SL) have been investigated during the last decade using b o t h the transfer matrix a n d the G r e e n ' s function m e t h o d ~ The extended states p r o p a g a t i n g In the whole SL form bulk b a n d s which are separated by small gaps Localized modes associated with a p e r t u r b a t i o n of the perfect SL m a y exist inside these gaps In particular, the existence of surface acousuc waves in SL have been d e m o n s t r a t e d ', as well as the possibility of localized m o d e s associated with the interface between a SL a n d a substrate 2, or with a p l a n a r defect m an otherwise perfect SL The knowledge o f the G r e e n function in these heterostructures 4 enables us to calculate b o t h the local a n d total density of states (DOS) Then, m a d d m o n to the dispersmn o f extended a n d localized states, one can also o b t a i n the spatial distribution of the m o d e s and, in particular, the possibility o f r e s o n a n t m o d e s which m a y a p p e a r as well-defined peaks o f the DOS inside the bulk b a n d s In this p a p e r we are anterested in a calculaUon of DOS and dispersion curves in a semi-infinite SL with or without a cap layer at the surface, or m contact with a substrate W e hm~t ourselves to the s~mplest case o f shear horizontal w b r a U o n s where the atomic displacements are along the axis .~_~, when v~ is the axis o f the SL a n d the wave vector kll (parallel to the interfaces) is &rected parallel to -, i After a brief i n t r o d u c t i o n to the theory, we present a discussion o f the general b e h a v i o u r o f DOS as well a~ a few illustrations of the dispersion curves, m particular in the case o f r e s o n a n t modes induced by the cap layer In this p a p e r we calculate the D O S by using the theory of interface response in composite materials 5 In t h ~ theory the Green function q o f a composite system can be written as

g(D, D) = G(D, D) - G(D, M ) G +G(D,M)G

I ( M , M ) G ( M , D)

'(M,M)g(M,M)G-'(M,M)G(M,D)

(1)

where D a n d M are, respectively, the whole space a n d the space

of the interfaces In the composite m a t e r i a l , G is a block-diagonal matrix in which each block G~ c o r r e s p o n d s to the bulk G r e e n function of the subsystem ~ In o u r case, the SL IS c o m p o s e d o f slabs of materials t (~ = 1,2) with thickness d, In e q u a t i o n (1), the c a l c u l a n o n o f g ( D , D) requires, besides G,, the knowledge of q(M,M) In practice, the latter is o b t a i n e d 5 by inverting the m a m x 9 ~(M, M ) which can be simply built from a j u x t a p o s m o n of the matrices q,, ~(M, M), where 9~,(M, M ) is the interface G r e e n ' s function o f the slab z alone 5 In the geometry o f the SL, the elements of the G r e e n function take the form 9(~, k,lntv3, n't' ~'~), where u9 is the frequency o f the acoustic wave, kbl the wave vector parallel to the interfaces, a n d n, t, ",~ indicates a position along the axis of the SL in m e d m m t belonging to the unit cell n W~th these n o t a t i o n s a n d for a given value of the wave vector k> q~,(M, M) are 2 x 2 matrices and, therefore, q - t(M, M ) becomes a m d l a g o n a l m a m x which is the analogue of the H a m l l t o n l a n for a h n e a r chain problem with nearest n e i g h b o u r interactions The 'atomic positions' are represented by the positions of the interfaces in the SL This similarity enables us to derive in closed form the G r e e n funcUon of the infinite a n d semi-infinite SL Let us iiotme that a semi-infinite SL can be reahzed m the same way as for a linear chain, l e by cutting the interaction between two n e l g h b o u r i n g a t o m s , the addition of a cap layer to the SL is equivalent to the a d s o r p t i o n of a n a t o m at the surface of the linear c h a i n , the interlace between the SL a n d a substrate is obtained by changing the & a g o n a l element of the surface a t o m m the h n e a r chain The details of these calculauons a n d the analytic expressions of the G r e e n functions wdl be g~ven elsewhere The knowledge o f # in these systems enables us to calculate in each case the local DOS n((o, kll,~0 = - [ p ( v 0 / ~ ] 2~o Im q(u),k:llmv~, n l v 0 , as well as the total DOS for a given value of the wave vector kll n(u), k 0 = .[ d-~ ~n(oJ, kl,, x 0 p(.x ~) Is the mass dens=ty at the point -~, In the following we shall focus on a few applications o f these results Figure I(a) gives an illustration of the bulk b a n d s a n d surface

341

B D/afar/Rouham et al Acoustic modes

11 lO

9 60

8

L1

L2

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7

40 6 30 H 20 4 3

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2

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1

-20

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I

I

I

I

2

3

I

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I

I

I

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(b)

-30 70

T1 I 75

) 80

I I I 85 9 0 95 toD/Ct(Ga As)

100 105 110

Figure l. (a) Bulk bands and surface vva~e dispersion for a GaAs AlAs SL ~lth d~ - d: The shaded areas are the bulk bands The surface modes are presented rather for a GaAs la)er (sohd line) or a AlAs layer (dashed lines) at the surthce Dimensionless quantmes are used on both axes q(GaAs) is a transverse velocity of sound m GaAs given b) . 'C44:p (b) Density of states (DOS) of the semHnhnlte SL with an AlAs layer at the surface, for kllD = 7 The contribution of the lnfimte SL has been subtracted The DOS is gl~ en m units of D,'t~(GaAs) B, and T, refer to 0 peaks of weight - 1,,4 at the edges of the bulk bands, L, indicate the modes locahzed at the surface

m o d e s for a G a A s AlAs SL, with d~ = dz and period D = d~ 4- d2 The slab at the surface o f the SL is either G a A s or AlAs with the same thickness as in the bulk One can observe that the surface modes are very d e p e n d e n t on the type o f material which is at the surface Assuming that the latter is AlAs and choosing k ID = 7, Figure 1(b) shows the variation o f the total DOS between the semi-infinite and infinite SL The delta functions in this figure are b r o a d e n e d by adding a small imaginary part to the frequency .) The delta functions o f weight 1 associated to the localized surface modes are labelled L, Besides. we have shown analytically and checked numerically that 5 peaks o f weight - 1 / 4 appear In An((o. k 0 both at the b o t t o m and the top o f ever~ bulk b a n d s o f the SL (these peaks are labelled B, and T,. respectively) On the other hand, like in any one dimensional system. An(o). k 0 presents a 1/v/Au) divergent behaviour near the edges o f the bulk bands In Figure 1(b) one can observe a c o m b i n a t i o n o f the latter divergences with the artificially broadened 3 peaks o f weight - 1/4 this is why the shapes of these peaks are slightly different from one to a n o t h e r A p a r t from the above 6 peaks and the particular behavlour near the band edges, the variation kn(~o, k H) of the DOS does not show any other slgmficant effect inside the bulk bands o f the SL The material AlAs being still at the surface o f the SL, one can imagine that its thickness d . is now different from that dt o f the layers in the bulk W h e n d~) increases from d~ for example, the frequencies o f the localized modes in Figure 1(a) decreases until the corresponding branches merge into the bulk bands and become resonant states (see also Figure 2(c) to be discussed below), at the same time new localized branches are extracted from the bulk bands However, the resonant modes remain welldefined features o f the DOS only as far as their frequencies 342

remain in the vicinity o f the band edges and they vanish by penetrating inside the bands Now we assume that a cap layer of SI. o f thickness d,), Is deposited on top o f the G a A s AlAs SL terminated by an AlAs layer F~gure 2(a) gives the dispersion o f localized and resonant modes induced by the cap layer in the case (do/D) = 4 Depending on their frequencies, these modes may propagate in both the SL and the cap layer, or propagate in one and decay in the other, or finally decay on both sides o f the SL-adla>er interface The latter branches, which correspond to interface localized modes, are labelled by the index The behavlour o f the localized and resonant modes in the DOS are illustrated in Figure 2(b) in the case k,D = 3 An({,>,k,I) is here the total DOS of our system from which the contribution o f the infinite SL have been subtracted B, and 7", again refer to 0 peaks o f weight - 1/'4 at the edges o f the SL bulk b a n d s . L, and R,. respectively, indicate the locahzed and resonant m o d e s induced by the cap layer The most intense resonance IS the lowest one situated just aboxe the sound hne of the S1 adlayer, the next resonances are less intense especially at higher frequencies where the separations between the successive branches increase Figure 2(c) shows the behaviour o f localized and resonant modes as a function o f the thickness do o f the cap layer, at k i d = 1 The first few branches become closer to each other when do increases, and as a consequence the intensities o f the c o n espondlng resonances increase Let us also notice that the curxes in this figure are almost horizontal when a localized branch is going to become resonant b5 merging into a bulk b a n d , the variation with d0 is faster when the resonant branch penetrates deep into the band, but then the intensity of the resonant state decreases, or may c~cn vanish m particular when d,, is small or

B Djafari

Rouhani

et al: Acoustic

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123456

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Figure 2. (a) Dispersion of localized and resonant modes induced by a cap layer of Si, of thickness d,, = 40, deposited on top of the GaAssAlAs SL terminated by an AlAs layer. The shaded areas are the bulk bands of the SL. The heavy line indicates the sound line of Si. The branches labelled (i) are localized at the interface between the SL and the cap layer. (b) Density of states, in units of D/c,(GaAs), at k,,D = 3 (the contribution of the infinite SL has been subtracted). B,, T, and f., have the same meaning as in Figure I(b) ; R, refer to resonant modes. (c) Frequencies of the modes induced by the cap layer as a function of the thickness do, for k,,D = I.

the frequency frcqucncy

is high.

w in Figure

Finally, 2(c),

let us mention there

is a periodic

that

for any

repetition

given of the

modes as a function of d,. When the thickness a’,,is going to infinity, we find the situation of a semi-infinite SL in contact with a substrate. The variation of the total DOS, between this system and the infinite SL and substrate alone, again contains delta peaks of weight - l/4 at the edges of the SL bulk bands as well as at the sound line of the substrate. We have also investigated the localized’ and resonant

modes associated with the SL-substrate interface. The detailed results for this system as well as for a tinite SL deposited on a substrate will be presented elsewhere. As a final remark, let us emphasize that the calculations presented here for elastic waves can be transposed straightforwardly to the electronic structure of SL in the Kronig-Penney model’, or to the propagation of polaritons in these heterostructures when each constituent is characterized by a local dielectric constant E(U)). This is because both the equations of motion and 343

B DlafaH Rouham et al Acous[~c modes

the b o u n d a r y conditions in the above problems involve similar mathemaUcal equaUons Therefore the general behaviour and conclusions obtained m this paper are also valid for the two other physical problems

Acknowledgement The authors are indebted to Dr L Dobrzynskl for helpful discussions

344

References t For a re~lew, ~ee for example J Saprlel and B Djafarl Rouham Sut/a~e Act Rep, 10, 189 (1989) -~E Khour&fi and B Djafarl Rouham, Sm/a~ e Set, 211/212, 361 (19891 ~S Tamura, Phlw Rev, B38, 1261 (1989), E M Khourdlfi and B Djafarl Rouham, J Phvs Condensed 44atter 1, 7543 (1989) 4 F Garcla Mohner, m this conference L Dobrzynskl, SutJa(e S~t, 175, 1 (1986), 180, 489 (19873 ~Hung-Slk Cho and P R Prucnal, Phys Rev, B36, 3237 (1987), M Steshcka, R Kucharczyk and M L Glasser Ph~ s Rel, B42, 1458 (199(11