Folded, confined, interface, surface, and slab vibrational modes in semiconductor superlattices

Superlattices and Microstructures, Vol 5, N o 1, 1989

FOLDED,

CONFINED,

27

INTERFACE, SURFACE, AND SLAB SEMICONDUCTOR SUPERLATTICES

N. Cardona M a x - P l a n c k - l n s t i t u t fur F e s t k ~ r p e r f o r s c h u n g , D - 7 0 0 0 S t u t t g a r t 80, Federal R e p u b l i c (Received

August

8,

VIBRATIONAL

Heisenbergstr. of G e r m a n y

MODES

IN

I,

1988)

Vibrational modes o f s e m i c o n d u c t o r superlattices are related to those of their bulk constituents. Reduction in the translational symmetry enables observation by o p t i c a l spectroscopy of symmetry forbidden modes o f t h e b u l k m a t e r i a l s . Interesting but subtle concepts, such as B r i l l o u i n zone folding and mode c o n f i n e m e n t , arise. Moreover, electrostatic interface phonons exist in ideal infinite superlattices. In practical s u p e r l a t t i c e s with a finite number of periods, slab and surface v i b r a t i o n a l modes also occur. Most of the past e x p e r i m e n t s have been p e r f o r m e d by means of R e N a n s p e c t r o s c o p y on G a A s / A l x G a l _ x A s superlattices; recent work on Ge/Si systems reveals new features. The p r o p e r t i e s of these modes, their s y m m e t r i e s and relationship to the modes of the bulk c o m p o n e n t s are discussed.

I. I n t r o d u c t i o n

Semiconductor superlattices have become one o f t h e most f r u i t f u l objects of physical investigation since they were proposed in the pioneering work of Esaki and Tsu. 1 In this work, vibrational spectroscopy has prominent role since 1977. 2~a R fs. 2 and 3 G a A s / A I A s s m a l l period s u p e r i a t tices were prepared and their vibrational modes i n v e s t i g a t e d by means of Raman and ir s p e c t r o s c o p y . The a u t h o r s found modes r e l a t e d to the ( = 0 R e N a n p h o n o n s of the bulk and also modes at lower f r e q u e n c i e s w h i c h they a t t r i b u t e d to the 'folded' d i s p e r s i o n relations of a c o u s t i c phonons. The latter, however, were p a r t l y spurious, due to s c ( t t e r i n g f r o m air at the sample surface. ~ Nevertheless the c o n c e p t of B r i l l o u i n zone folding has remained very useful to interpret v i b r a t i o n a l spectra of superlattices.

~e~ :

In t h e p r e s e n t p a p e r we e m p h a s i z e the difference between the copcepts of zone-folded modes, applicable to acoustic-like phonons, and c o n f i n e d modes a p p l i c a b l e to p h o n o n s r e l a t e d to optical b r a n c h e s of the bulk m a t e r i a l s . Much o f been p e r f o r m e d

the subsequent work has on G a A s / G a l _ x A l x A S s u p e r -

0 7 4 9 - 6 0 3 6 / 8 9 / 0 1 0 0 2 7 + 16 $02.00/0

lattices (in c o n t r a s t to the original GaAs/AIAs work of Refs. 2-3)). These superlattices are easier to grow and are more stable for x
© 1989 Academic Press Limited

28 the p o l a r i z a t i o n o f the s c a t t e r e d beam, plus the possibility of varying the s c a t t e r i n g wave v e c t o r , have given Raman spectroscopy a lead over o t h e r t e c h n i ques in investigating superlattices. Very r e c e n t l y , h i g h - r e s o l u t i o n e l e c t r o n energy l o s s s p e c t r o s c o p y data (HREELS) have become a v a i l a b l e , u They are p a r t i cularly s u i t a b l e to e x c i t a t i o n s local i z e d in the immediate v i c i n i t y o f the s u r f a c e (440 A). The GaAs/AIAs s u p e r l a t t i c e s are easy t o p r e p a r e by m o l e c u l a r bea T e p i t a x y (MBE) on GaAs s u b s t r a t e s . ~ They have the p r o p e r t y o f a v e r y small l a t tice c o n s t a n t mismatch b e t w e e n GaAs (a o = 5.653 A) and AIAs (a 0 = 5.660 A). This f a v o r s growth and absence o f misfit dislocations. Recently, considerable i n t e r e s t has been devoted t o o t h e r types o f s u p e r l a t t i c e s in which the a o mismatch is larger, such as GaSb-AISb (0.65% misma~Rh),9 o r S i / G e l - x x S i (V~ t o 4% mismatch) ~ and GaAs/InxGal_xAS. ~ I f the s u p e r l a t t i c e p e r i o d s are t h i n , the in-plane lattice constants equalize themselves t h r o u g h the e s t a b l i s h m e n t o f t e n s i l e and compressive s t r a i n s in both components. As the period grows, it becomes e n e r g e t i c a l l y m o r e f a v o r a b l e t o r e l e a s e the s t r a i n t h r o u ~ the b u i l d - u p of mi s fi t dislocations. ~ Raman s c a t t e r i n g is an i d e a l t o o l to i n v e s t i g a t e these phenomena: s t r a i n s s h i f t and s p l i t the Raman phonons o f the b u l k m a t e r i a l . These s h i f t s and s p l i t t i n g s can be used t o e v a l u a t e the s t r e s s once t h e i r dependence on s t r e s s for the bulk is known. 13 We m e n t i o n h e r e an i n t e r e s t i n g peculiarity o f the GemSin s u p e r l a t t i c e s whose consequences have not y e t been i n v e s t i g a t e d . For the [ 0 0 1 ] growth d i rections three different space (and p o i n t ) groups are p o s s i b l e : i f both m and n a r e even t h e space group i s D)h , if o~ index even and the o t h e r odd ~t is D~, and i f both i n d i c e s are odd D~d (M.I. Alonso, unpublished). Superlatt i c e s made o f two amorphous components (e.g., S i L ~ i 1 _ x N x) have a l s o r e c e i v e d a t t e n t i o n . 14 T h e i r a c o u s t i c photons a l s o exhibit '.folding' effects, I" a fact which confirms the quasicontinuous, e l a s t i c n a t u r e o f these v i b r a t i o n s . The elastic continuum theory of acoustic phonons in s u p e r l a t t i c e s was worked out by Rytov in 1 9 5 5 ( p r o p a g a t i o n o f ~eismic waves t h r o u g h s t r a t i f i e d media), l Recently, quasiperiodlc superlatt i c e s , r e p r e s e n t i n g the s i m p l e s t p r o t o type of quasiperiodic crystals, have been c o n s i d e r e d . P a r t i c u l a r l y ~nter~ting are the F i b o n a c c i s u p e r l a t t i c e s .

Superlattices and Microstructures, Vol. 5, No. 1, 1989 This a r t i c l e d i s c u s s e s e x p e r i m e n t a l and t h e o r e t i c a l d e v e l o p m e n t s in t h e field of vibrations in superlattices, w i t h emphasis on the GaAs/AIAs system. A number o f m o r e e x t e n s i v e r e v i e w a r t i c l e s has a l r e a d y appeared. 5,18-21

2.

2.1

INTRODUCTION TO LIGHT SCATTERING IN BULK CRYSTALS S e l e c t i o n Rules and Coupling Constants

L i g h t s c a t t e r i n g in s o l i d s i s t h o r o u g h l y d i s c u s s e d in Ref. 21. The mechanism o f s c a t t e r i n g by phonons is easy t o u n d e r s t a n d . Semiconductors have a v e r y high electronic susceptibility. The phonons modulate t h i s s u s c e p t i b i l i t y X at t h e i r f r e q u e n c y Wp by ' t i c k l i n g ' the atoms. Thus we have components o f X modulated like Xp - e x p ( ± i ~ p t ) . When m u l t i p l y i n g t h i s by the i n c i d e n t l a s e r field, o f f r e q u e n c y ~$, a p o l a r i z a t i o n ~ e x p [ - i ( w L ± W p ) t ] r e s u l t s . This a c t s as a r a d i a t i n g d i p o l e and thus s c a t t e r e d radiation a t t h e f r e q u e n c y ~L±mp a p p e a r s . T h e - ( + ) s i g n c o r r e s p o n d s t o the Stokes (anti-Stokes) c o m p o n e n t . The scatterin 9 efficiency contains a factor 1+nB(mp) in the Stokes and nB(mp) in the a n t i - S t o k e s case, where nB is the BoseE i n s t e i n phonon o c c u p a t i o n number. It a l s o c o n t a i n s as the c o u p l i n g c o n s t a n t the square o f the s o - c a l l e d Raman susceptibility~ i.e., the derivative of X(mL) w i t h r e s p e c t t o t h e phonon c o o r d i n a t e d i s p l a c e m e n t . X(~L) has s i n g u l a rities at the s o - c a l l e d i n t e r b a n d gap e n e r g i e s or c r i t i c a l points. Its derivat i v e w i l l be even m o r e s i n g u l a r at these p o i n t s . T h u s resonance e f f e c t s in Raman s c a t t e r i n g by phonons a r i s e . In a c r y s t a l l i n e sample t h e r e i s an i m p o r t a n t s e l e c t i o n r u l e which o b t a i n s from w a v e v e c t o r or c r y s t a l momentum conservation: ~S = ~L ± ~'

(i)

where ~S L a r e t h e wave v e c t o r s o f the s c a t t e r e d ' and i n c i d e n t l i g h t , respectiv e l y , and ~ t h a t o f the phonon. T h e - ( + ) sign applies to Stokes (anti-Stokes) s c a t t e r i n g . The o r d e r of magnitude o f q at a g e n e r a l p o i n t o f the B r i l l o u i n zone (BZ) i s 2 ~ / a 0 . The magnitudes o f ~ and E L a r e 2 ~ n / ~ S ~ In i s t h e r e f r a c t i v e index o f the medlum, ~ the wavelength in vacuum - 5 0 0 0 A), Hence k S and k L a r e nearly negligible compared w i t h the dimension o f the B r i l l o u i n zone and o n l y phonons w i t h ~ v e r y c l o s e t o the c e n t e r o f the BZ (q=O, F - p o i n t ) can be seen. In

Superlattices and Microstructures, Vol. 5, No. 1, 1989 the case o f a c o u s t i c phonons w i t h a s m a l l v a l u e o f q, Up = qvp (vp i s t h e speed o f sound) the phenomenon Is c a l l e d Brillouin scattering. Translational invariance requires t h a t f o r a c o u s t i c phonons w i t h q=O ( a l l atoms move by the same amount u~ pure t r a n s l a t i o n ) X not be m o d i f i e d by u: the 'Raman s u s c e p t i b i l i t y ' should vanish. A non-vanishing differential susceptibility i s o b t a i n e d f o r q@O, however. In this case du/dz = ¢ = iqu (z i s the p r o p a g a t i o n d i r e c t i o n and • a component of the strain tensor). The s t r a i n e multiplied by the a p p r o p r i a t e s t r a i n optical c o n s t a n t ( P i j ) gives a time-dependent component o f the s u s c e p t i b i l i t y which is r e s p o n s i b l e f o r B r i l l o u i n s c a t tering. This Brillouin susceptibility e x h i b i t s resonance phenomena near e l e c tronic transitions, somewhat s i m i l a r t o those found in Raman s c a t t e r i n g . These resonances and t h e i r t h e o r y are d i s cussed in Ref. 22. The s c a t t e r i n g e f f i c i e n c y i s u s u a l l y w r i t t e n as: @2S

= les'~'eLl 2

,

(21

where ~ i s the s o l i d a n g l e , & the path in t h e s a m p l e , e S ! the u n i t v e c t o r s o f the s c a t t e r e d (S) 'and the i n c i d e n t (L) electric fields and ~ t h e Raman ( o r Brillouin) t e n s o r f o r the phonons under consideration. The t e n s o r ~ must have the same symmetry as the c o r r e s p o n d i n g phonon. I t i s a symmetric t e n s o r e x c e p ~ possibly, very close to resonance. Phonons which b e l o n g t o r e p r e s e n t a t i o n s o t h e r than those o f a symmetric second rank t e n s o r are s a i d t o be Raman-(dip o l e ) f o r b i d d e n . For phonons which are simultaneously ir and Raman a c t i v e (i.e., in the absence o f i n v e r s i o n symmetry) t h e i r l o n g i t u d i n a l component may be seen in d i p o l e f o r b i d d e n c o n f i g u r a t i o n s c l o s e t o resonance. I t is a quad r u p o l e e f f e c t which can be r e p r e s e n t e d by a c o m p l e t e l y s y m m e t r i c t e n s o r ~ in Eq. (2). The c o r r e s p o n d i n g s c a t t e r i n g efficiency is proportional t o q2 f o r small q. 2.2

Circumventing S e l e c t i o n Rule

the

T-Conservation

Equation (1) i s a consequence o f the s t r i n g e n t t r a n s l a t i o n a l symmetry o f the p e r f e c t c r y s t a l . S e v e r a l t e c h n i q u e s are used t o c i r c u m v e n t i t . Tetrahedral semiconductors can be p r e p a r e d in e i t h e r crystalline or amorphous form. The l a t t e r possesses the same t e t r a h e d r a l s h o r t range o r d e r as the c r y s t a l l i n e form but

29 no l o n g - r a n g e o r d e r a t a l l . As a r e s u l t , i n s t e a d o f the sharp o p t i c a l phonons a t q=O the Raman (and a l s o i r ) s p e c t r a o f amorphous t e t r a h e d r a l semiconductors c o n t a i n f o u r broad bands which c o r r e spond t o the TA, LA, LO, and TO phonon b r a n c h e s . These s p e c t r a r e p r e s e n t ( a f t e r removing the dispersion of coupling c o n s t a n t s and B o s e - E i n s t e i n f a c t R { s ) the d e n s i t y o f phonon s t a t e s vs. Up. L4 Another way o f o b s e r v i n g phonons away from q=O w i t h o p t i c a l t e c h n i q u e s is by measuring s e c o n d - o r d e r s p e c t r a in which two phonons are g e n e r a t e d : one phonon d e s t r o y s the t r a n s l a t i o n a l invar i a n c e w h i l e the o t h e r samples the thus distorted crystal. All q's can be reached s i n c e o n l y the sum o f the twophonon w a v e v e c t o r s must be n e a r l y z e r o (q1+q~ ~ 0 ) . I f both phonons are o f the same ?requency ( s c a t t e r i n g by o v e r t o n e s ) one o b t a i n s a Raman spectrum which r e presents the density of one-phonon s t a t e s ~ j t h the energy s c a l e m u l t i p l i e d by two. The t r a n s l a t i o n a l symmetry can a l s o be d e s t r o y e d t h r o u g h the random a d d i t i o n of i m p u r i t i e s . P a r t i c u l a r l y interesting in c o n n e c t i o n w i t h s u p e r l a t t i c e s is the case of mi x e ~ c r y s t a l C s u c h as Al 1 _ x G a x A s . 2 6 , 7 F i g u r e shows t h e appearance o f d i s o r d e r a c t i v a t e d TA, LA, LO, and TO bands (DATA, DALA, . . . ) in Ga0 8 A l o 2As. These bands r e p r e s e n t the c o r r e s p o n d i n g broadened d e n s i t y o f phonon s t a t e s o f GaAs. A better c o n t r o l a b l e way o f des t r o y i n g or r e d u c i n g the t r a n s l a t i o n a l symmetry i s by growing a o n e - d i m e n s i o n a l superlattice. This lifts most o f the original translational symmetry o p e r a t i o n s and l e a v e s o n l y a few which c o r r e spond t o the p e r i o d o f the s u p e r l a t t i c e . A (001) GaAs/AIAs s u p e r l a t t i c e with a p e r i o d o f n I l a y e r s o f GaAs (monolayer thickness al) and n 2 l a y e r s o f AIAs ( t h i c k n e s s a 2) i s shown s c h e m a t i c a l l y in F i g . 2. The new p e r i o d becomes ( n l a I + n2a 2) = d i n s t e a d o f a t ( a 2 ) o f the bulk m a t e r i a l s . The reduced BZ o f the s u p e r l a t t i c e now extends from q=O t o q = ±~/d i n s t e a d o f the much l a r g e r b o u n d a r i e s ~/al 3 a)nd ~ / a 2 o f the b u l k m a t e r i a l s (see Fig. . The ~ i s p e r s i o n r e l a t i o n s o f the b u l k c r y s t a l s must now be mapped o n t o the BZ o f the s u p e r l a t t i c e (Fig. 3). While the b u l k c r y s t a l has o n l y one set o f o p t i c a l phonons (one LO and two TO phonons each s e t ) at q=O the s u p e r l a t t i c e has 2 ( n 1 + n 2 ) - 1 s e t s . A l l o f these q=O phonons may, in p r i n c i p l e , be Raman active. In t h i s manner one can see by Raman s p e c t r o s c o p y m o r e phonons than in the c o r r e s p o n d i n g bulk c r y s t a l s .

30

Superlattices and Microstructures, Vol. 5, No. 1, 1989 AI As AI ASGaASGoASGoASAI ASAI As

xI[10o]

o2

1 Fig. tice

[001]

°i

d2

d1

2: S k e t c h o f (nl, n 2) w i t h

a GaAs/A1As superlatn i = 3 and n 2 = 2.

I

20

q:4qz ' , i DATO "OAO 100 200 300 400 WAVE NUMBER (cm -1)

i

~LO

TO

F i g . I : Raman s p e c t r a o f Gan RAIn )As separated into irreducible ~'~m~pon ~ s, showing disorder-induced first order spectra (from Ref. 26).

LA

j !LA i

0

q

~_

O

Oi

3.

D i s p e r s i o n R e l a t i o n : F o l d i n g and Phonon Confinement

Let us c o n s i d e r the s u p e r l a t t i c e o f F i g . 2. The d i f f e r e n c e in the l a t t i c e dynamical parameters of both constituents is not large, otherwise they would not grow e p i t a x i a l l y on each o t h e r . In f a c t , the main d i f f e r e n c e l i e s in the atomic mass o f Al (27 atomic mass u n i t s ) vs. Ga (70 amu). As f a r as the acoustic phonons a r e c o n c e r n e d , t h e atomic mass e n t e r s in t h e i r f r e q u e n c y as the d e n s i t y and thus i s t o be averaged w i t h the mass o f As (75 amu): the d e n s i t i e s o f GaAs and AIAs o n l y d i f f e r by 30% ( t h e c o r r e s p o n d i n g f r e q u e n c i e s by 15%). These d i f f e r e n c e s can be t r e a t e d by p e r turbation theory. Let us consider a fictitious b u l k c r y s t a l w i t h the average p a r a m e t e r s o f GaAs and AIAs, in p a r t i c u l a r the average d e n s i t y . The s u p e r l a t r i c e o f F i g . 2 can be o b t a i n e d by applying a square _Have p e r t u r b a t i o n t o these p a r a m e t e r s . TM There i s a s t r o n g f o r m a l a n a l o g y between the l a t t i c e dynamical problem o f a c o u s t i c phonons in a s u p e r l a t t i c e and t h a t o f e l e c t r o n s in a c r y s t a l l a t t i c e .

~ 2(QI+Q?) 2(01+o2)

O ___ 2(el+o))

Fig. 3: S c h e m a t i c r e p r e s e n t a t i o n of the folding of the Brillouin zone for a (2,2) superlattice. Note that for the optical modes to be of the mfoldedm type, as shown here, the d i f f e r e n c e in the cation messes w o u l d have to be much smaller than that of the masses of Al and Ga.

Let us first c o n s i d e r free e l e c t r o n s in a crystal with a given structure but vanishingly small potential. The elect r o n s . . h a v e a ) a r a b o l i c energy d i s p e r s i o n E = (12/2m)q w h i c h must be folded into the reduced B r i l l o u i n zone of the crystal: E

~

(±3 + ~)2

,

(3)

w h e r e ~ are reciprocal lattice vectors. Switching on a weak potential (nearly free e l e c t r o n or weak b i n d i n g a p p r o x i m a tion, WBA) leaves the menergy band m of Eq. (3) nearly u n c h a n g e d , except for the a p p e a r a n c e of s p l i t t i n g s at the center and the edge of the reduced BZ of the superlattice (the s o - c a l l e d minizone).

Superlattices and Microstructures, VoL 5, No. 1, 1989 Similarly, for a c o u s t i c p h o n o n s in the bulk crystal one has a linear d i s p e r s i o n relation (u = vpq w h i c h must be 'folded' into the m i n i z o n e of the s u p e r l a t t i c e in order to treat the c o r r e s p o n d i n g p e r t u r bation (square wave m o d u l a t i o n of parameters). For small perturbations one obtains splittings at the center and edges of the o n e - d i m e n s i o n a l minizone (Fig. 3), i.e., at

q m of

the

31

),Low=6471 ,0

sTl~,,,z' --

62A

z

73A

-'-,

;

~

/

F

/ ,"

= nla|~n~azm = ~; O, bulk

.,

=,

...

(4)

crystal.

The 'folding' of the BZ o n t o a minizone will substantially alter the optical (Roman) s e l e c t i o n rule based on conservation (Eq. (1)I: w h i l e in the bulk crystal only a c o u s t i c p h o n o n s of vary small frequency (up to u = Vp ( 4 w n / X L ) f o r b a c k s c a t t e r i n g ) are allowed (Brillouin scattering), many acoustic phonons of the bulk become 'optic' p h o n o n s of the s u p e r l a t t i c e and for q=O (reduced m i n i z o n e ) they can be optically allowed. The new selection rules b e c o m e in the b a c k s c a t t e r i n g configuration q = )(41n/X)

21 + n l a l + n 2 a 2 m,

m = O,

...

I,

2,

+ ~)

O- Si : H I a-SiNx : H IduU.iloyers

6O

J 40

.

All

~

~ 20

0

RAMAN SHIFT (crn -1) F i g . 4: F i r s t doublet i n t h e Raman s p e c trum of the folded LA p h o n o n s o f a m o r phous Si/SiN1. 3 superlattices. From R e f .

(5)

The a r g u m e n t j u s t g i v e n i s b a s e d on the continuum~ elastic limit of acoustic vibrations, valid for q (bulk) << 2 1 / a o (a o = bulk lattice constant ; a I + a2). Hence i t should not matter whether the material is a m o r p h o u s or c r y s t a l l i n e . The d i s p e r s i o n relation of the a c o u s t i c modes in the s u p e r l a t t i c e should d e p e n d only on m a c r o s c o p i c parameters well d e f i n e d in both cases, such as the density and the speed of sound. As an example we show in Fig. 4 the first folded LA doublet, at f r e q u e n c i e s :

u = vp(±4wn/~

~91

(6)

obtained for longitudinal acoustic phonons in a s e r i e s of amorphous sup~zlattices with different periods. Ib The doublets o b s e r v e d a g r e e w i t h Eq. ( 6 ) . The perturbative treatment just described is valid if the amplitude of the p e r t u r b a t i o n is s m a l l c o m p a r e d with the w i d t h of the band under c o n s i d e r a tion. This is the case for the a c o u s t i c phonon b r a n c h e s of the bulk m a t e r i a l s constituents of typical superlattices, whereas it is no longer true for the c o r r e s p o n d i n g optical branches. In the

15.

case of GaAs/A1As, for instance, t h e TO (q=O) optical frequency o f b u l k GaAs i s 267 cm -I (~t 296 K) w h i l e that of AIAs is 361 c m - ' ( The w i d t h s of the TO bands are <30 c m - ' , i.e., m u c h less than the d i f f e r e n c e in the FO (q=O) f r e q u e n c i e s . Hence averaging both frequencies and t r e a t i n g the f l u c t u a t i o n s as a p e r t u r b a tion is m e a n i n g l e s s . Instead one treats the v i b r a t i o n s as l o c a l i z e d on either the GaAs slabs, with no v i b r a t i o n of the AIAs, or on AIAs slabs, with no p a r t i c i pation of GaAs. The p r o c e d u r e of localizing or c o n f i n i n g the p h o n o n s is similar to the tight b i n d i n g m e t h o d of calc u l a t i n g e n e r g y bands of m a t e r i a l s with two atoms per unit cell. When the difference between the e n e r g i e s of corresponding electronic states in the two atoms is larger than the o v e r l a p integrals, this method, and not the WBA, gives the better d e s c r i p t i o n since the electrons are n e a r l y l o c a l i z e d in one type of atom or the other. Thus we have i n t r o d u c e d the somewhat o r t h o g o n a l concepts of zone folding, appropriate to

Superlattices and Microstructures, Vol. 5, No. I, 1989

32 a c o u s t i c phonons, and confinement, app r o p r i a t e to f l a t o p t i c a l branches. In the former case, and f o r XL >> d, acoust i c phonons with: q = ~ - m, m = O, ± I ,

±2,

~L--iB~ ,v

T01

z(x,x)i

//~'; /

(7)

are a c t i v a t e d by the s u p e r l a t t i c e p e r t u r b a t i o n . In the case of c o n f i n e d TO p h o n o n s we may assume t h a t , approximat e l y , the v i b r a t i o n a l amplitude vanishes at the i n t e r f a c e s (see, however, Ref. 29). Thus we observe v i b r a t i o n a l f r e quencies n e a r l y equal t o those of each bulk m a t e r i a l w i t h q l = ~ i m, q2 = ~2 m; m = ± I ,

z

±2.

(8}

i

RAMAN SHIFl'(cm -1)

Fig. 5: TO-phonons confined to GaAs observed in a (7,7) GaAs/AIAs superlattice. The o b s e r v a t i o n is only p o s s i b l e very close to an e x c i t o n i c resonance (see F i g . 16). From Ref. 7.

N o t e t h a t f o r d I ~ : n ~ Eqs. (8) and (7) coincide, and thus usion between the c l e a r l y d i s t i n c t c o n c e p t s of f o l d i n g and c o n f i n e m e n t may arise. As an e x a m p l e of the Raman s p e c t r u m of TO p h o n o n s w i t h q (bulk) # 0 in sup e r l a t t i c e s we show in Fig. 5 the GaAslike TO p h o n o n s for a G a A ~ - A I A s superl a t t i c e ~ i 2 h n~ = n 2 = 7. They correspond to___. (8/ w i t h m = I , 2, . . . , 5. In F i g . 6 we d i s p l a y the f r e q u e n c i e s of the LO phonons vs. q o b t a i n e d in t h i s manner and compare them w i t h t h e d i s p e r sion relation o f b u l k GaAs and A l A s . The agreement is excellent. We s h o u l d p o i n t o u t ~ h o w e v e r , t h a t n 1 and n 2 may h a v e t o be ~ p l a c e d by n 1 + I and n2 + 1 i n Eqs. (4)" as was done i n F i g . 6 - Most o f t h e superlattices so i n v e s t i g a t e d are of the [001] type. Recentj~, data ha~ been obtained for [ 1 1 0 ] V~ a n d [ 1 2 0 ] ~ " GaAsAlAs systems.

400.-

380

Phonons

4.1 Zone f o l d i n g

can

The zone f o l d e d a c o u s t i c phonons be t r e a ~ d w i t h the e l a s t i c c o n t i -

nuum m o d e l or t h e l i n e a r c h a i n mod e l . 2 8 ' 3 ~ The former i s exact f o r small q's. The l a t t e r a p p l i e s to a r b i t r a r y q's. The l i n e a r chain a p p l i e s t o 3d c r y s t a l s p r o v i d e d one r e p l a c e s i n t e ~ atomic by i n t e r p l a n a r f o r c e c o n s t a n t s . ~ Folded a c o u s t i c modes in s u p e r l a ~ t i c e s are u s u a l l y observed in the backs c a t t e r i n g c o n f i g u r a t i o n w i t h q along the s u p e r l a t t i c e a x i s . We l i m i t ourselves t o t h i s case. The atomic d i s placement ~ for e i t h e r l o n g i t u d i n a l or t r a n s ~ ) r s e modes fulfill L a p l a c e ' s equationZ°, hence t h e y can be w r i t t e n as

t

""'.

""'" %

c~

"" -.

360

"'"~

~ "'.

".. o

"

o

'E 340

240 4. A c o u s t i c

oe~

o

o.

,.o

F i g 6: C o n f i n e d G a A s - t y p e and A l A s - t y p e LO f r e q u e n c i e s measured for different GaAs/AIAs s u p e r l a t t i c e s (various symb o l s ) p l o t t e d vs. q c o m p a r e d with the GaAs d i s p e r s i o n curve m e a s u r e d by neutron s c a t t e r i n g (solid line) and with two d i f f e r e n t c a l c u l a t i o n s of the dispersion relation of AIAs (dashed and dashed-dotted lines). From Z.P. Wang, D.S. Jiang, and K. Ploog, Solid State Commun. 65, 661 (1988).

SUmS o f sines and cosines of zQ 1 2 where the O's are related to the v i 6 @ a t i o n a l f r e q u e n c y Up t h r o u g h Up = Q i , 2 V l , 2 (v I B 2 is the a p p r o p r i a t e speed o f sound in m e d i u m I or 2). After imposing b o u n d a r y

Superlattices and Microstructures, VoL 5, No. 1, 1989

33

conditions a t the interface (continuity of ~ and of the components of the stress tensor which correspond to forces along x, y) and using Bloch's theorem to relate u(q) in one period to u(q) .i~ the next (through t h e phase f a c t o r e l q ° ) we fi~..v the secular equation for q vs. Up cos(qd)

upd 1

i

!

upd 2 cos v2

= cos~

I

FORWARD ERING

Xi~567nm

T~250K BAC!SCATTERINO

mpd I upd 2 a sin--~TI s i n ~ v 2 , (9)

where =

1 + 6 = ½

PLY1|

IP2V2

,p-T~T + p--~T~,.

(1o)

Equations (9, 10) are v a l i d f o r both l o n g i t u d i n a l and t r a n s v e r s e phonons p r o v i d e d one uses the a p p r o p r i a t e v a l u e s o f v 1 . 2. E q u a t i o n ( 9 ) i s s i m i l a r to that fodnd f o r o t h e r p r o p a g a t i o n phenomena in

stratified

media,

e.g.,

interface

modes.

As a l r e a d y m e n t i o n e d , P2V2 i s u s u a l l y not v e r y d i f f e r e n t from P l V l , hence 6 << 1. Equation (9) can thus be r e w r i t t e n as: cos(qd)

• upd 1

7:

First

LA

Raman d o u b l e t

of

a

GaAs-A1As s u p e r l a t t i c e w i t h d = 37 J and dl/d = 0 . 2 3 as o b s e r v e d i n f o r w a r d and backscattering. The f o r w a r d scattering spectrum displays the even-odd selection rule discussed in the text: Only the high frequency component of the doublet i s s e e n . The weak l o w f r e q u e n c y signal i s due t o b a c k s c a t t e r i n g l e a k s . From B. ausserand et el., Phys. Rev. B 33, 2897 (1986).

upd 2 sin-~-- . ( 1 1 )

For q not t o o c l o s e t o the c e n t e r o r edge o f the minizone (q # ~m/d) and 6 small we may s e t 6 = 0 in Eq. ( 1 1 ) . We thus f i n d :

= i~" m ~ q l

w h e r e <> r e p r e s e n t s

the

-1,

(12)

average over

the

two superlattice components. Equation (12) r e p r o d u c e s the r e s u l t s o b t a i n e d in Sect. 3, i.e, the folded dispersion relation of a bulk material with an a v e r a R e speed o f s o u n d . The e r r o r comm i t t e d in Eq. (12) i s q u a d r a t i c in 6 p r o v i d e d q @ O, a / d . For q = 0 or ~/d we find a correction linear in 6 which produces the s p l i t t i n g o f the d e g e n e r a t e frequencies:

~Up = ~

Fig.

44 STOKES SHIFT

dI d2 c o s [ u p i ~ 1 + ~-~1] -

:

a s~n~

Up(q)

55

6

sin

m~dl

dVl

.

(13)

This splitting i s n o t easy t o see i n backscattering since q is usually sufficiently l a r g e t o make Eq. ( 1 2 ) an e x c e l lent approximation. In forward scattering, however, q becomes negligible compared with 2~/d and we can see a

splitting o f the Raman-active modes o f Eq. ( 1 2 ) . A measurement f o r a GaAs-AIAs superlattice is shown in 2 ~ig. 7. The

observed splitting Amp = well with the predictions for this s y s t e m . We n o t e

. of that

cm-tagrees Eq. ( 1 3 ) only the

upper component of the f o r w a r d s c a t t e r i n g d o u b l e t i s seen. This i s due t o the f a c t t h a t f o r q = 0 the upper comp o n e n t u z i s odd ( ~ s i n q z ) . The s c a t t e r e d i n t e n s i t y i s o b t a i n e d by a v e r a g i n g o v e r a p e r i o d the s t r a i n (aUz/aZ) times the a p p r o p r i a t e p h o t o e l e s t i c c o n s t a n t . The l o w e r c o m p o n e n t , f o r w h i c h u z i s even, y i e l d s an odd s t r a i n which a v e r ages t o z e r o . The upper component is even in the s t r a i n and g i v e s a s t r o n g s i g n a l . Away from q = 0 the odd and even modes mix and both s i g n a l s become n e a r l y equal. The e x i s t e n c e o f minigaps a t the BZ boundaries of superlattices has been demonstrated by measuring the t r a n s m i s s i o n o f monochromatic phonons produced

by a suoerconducting tunnel junction. 15,35 Recent work has revealed minigaps inside the Brillouin zone due t o LO-TO ~ t i c r o s s i n g for off-axis propagation.

Superlattices and Microstructures, VoL 5, No. 1, 1989

34 4.2 Slab Modes We show in Fig. 8 Raman data obt a i n e d f o r a [001] s u p e r l a t t i c e composed o f 10 p e r i o d s o f Ge2Si 2 on a Si subs t r a t e , both uncapped and capped w i t h 100 monolayers of S i . 37 In t h i s case low frequency slab modes, corresponding t o the l o n g i t u d i n a l v i b r a t i o n of s u p e r l a t tice (+capping) are observed t o g e t h e r w i t h the B r i l l o u i n s i g n a l of the subs t r a t e . Chain model c a l c u l a t i o n s of the slab modes are presented in Ref. 37.

0.0

200.0

400.0

5. O p t i c a l Phonons: Confinement 5.1 Boundary C o n d i t i o n s As mentioned in Sect. 3 the o p t i c a l GaAs-AIAs s u p e r l a t t i c e s are c o n f i n e d e i t h e r t o the GaAs or the AIAs slabs. For TO phonons i t is usually assumed t h a t the a p p r o p r i a t e boundary c o n d i t i o n i s u = 0 a t the i n t e r f a c e . However, some p e n e t r a t i o n i n t o the mforb i d d e n ' l a y e r s occurs. This p e n e t r a t i o n can be e a s i l y e s t i m a t e d by c o n s i d e r i n g a p a r a b o l i c expansion of the d i s p e r s i o n r e l a t i o n of the bulk m a t e r i a l :

phonons of

2 ~TO(q) = mTo(O) - BTOq

(14)

For TO phonons in GaAs BTO = 88 cm- I A2. For Al~s ~TO i s e s t i m a t e d to be around 60 cmA"_ For q r e a l Eq. (14) y i e l d s the standard d o w n - d i s p e r s i n g TO bands. For q pure imaginary (q = i q , q r e a l ) i t represents up-dispersing bands which decay in space l i k e e x p ( - q z ) . The penet r a t i o n depth of the A I A s - l i k e mode i n t o GaAs can be e s t i m a t e d w i t h Eq. (14) from the d i f f e r e n c e between the mTO~S of GaAs ( 2 6 7 cm - ~ ) and AIAs (361 c m - 1 ) . We write: 361 - 267 = 94 cm- I

= BTO~2.

(15)

For the A I A s - l i k e mode Eq. (15) y i e l d s a p e n e t r a t i o n d e p t h i n t o GaAs q - " = I A, not q u i t e n e g l i g i b l e when compared w i t h a I = 2.8 A. A s i m i l a r e s t i m a t e can be made f o r the LO phonons. For the GaAsl i k e mode the e s t i m a t e is somewhat more difficult. The TO bands of AIAs now ben~ d o w n w a r d s , t o w a r d s t h e ~TO o f GaAs. Since i t gets c l o s e t o the l a t t e r , however, the corresponding penetration depth would be expected to be l a r g e r than t h a t f o r the AIAs-mode i n t o GaAs. An e s t i m a t e g i v e s 2.8 A f o r the GaAsl i k e mode i n t o A I A s . 38 Since d l = d~ -2.8 A, a s l i g h t c o r r e c t i o n to E~q_ ( 0 ) . i n t h e sense o f i n c r e a s i n g n I from the nominal value t o about n I + I , may be in o r d e r (see Eq. (16) below).

0.0

5.0

10.0

15.0

20.0

25.0

~ (cm") Fig. 8: Experimental Raman spectrum of a sample h a v i n g 10 l a y e r s of Ge2Si2 on a (O01)-Si s u b s t r a t e . Top panel-, spectrum from the uncapped p a r t of the sample. Bottom p a n e l : spectrum from the capped p a r t of the sample. The arrow i n d i c a t e s the exp r j ~ e n t a l B r i l l o u i n frequency of 5.3 cm-~. j / The above d i s c u s s i o n has a macroscopic c h a r a c t e r . When we t a l k about p e n e t r a t i o n depths of the order of a f r a c t i o n o f d I 2 ' h o w e v e r , one s h o u l d look at the m i c r o s c o p i c d e t a i l s of u, i.e., at the vibrational eigenmodes. This has b e e n done in Ref. 29. In the comment b y J u s s e r a n d and P a q u e t a simple, l i n e a r chain c a l c u l a t i o n of the K r o n i g - P e n n e y - t y p e1 9 , 2 9 , 3 9 y i e l d s f o r the allowed wave v e c t o r q1: ql = ~

~

m, m=±l , ±2 . . .

(16)

instead o f Eqe.qu(ZtIi.0 For l a r g e d~ = n l a >>a b o t h ns g i v e simi ar results but considerable deviations are o b t a i n e d f o r small d I . Equation (8) corresponds to s e t t i n g the v i b r a t i o n a l amplitude equal t o zero at the beginning and the end of a p e r i o d (i.e., from As t o As). Equation (16), however, i m p l i e s t h a t u f i r s t vanishes at the f i r s t Al a t o m s o u t s i d e of the GaAs l a y e r . This is p h y s i c a l l y reasona b l e s i n c e Al is much l i g h t e r than

Superlattices and Microstructures, VoL 5, No. 1, 1989 e i t h e r Ga o r As and thus w i l l not be a b l e t o f o l l o w the GaAs v i b r a t i o n . The As atoms t i e d t o Ga, however, s h o u l d follow it. A microscopic i n v e s t i g a t i o n of t h i s boundary c o n d i t i o n e f f e c t was p e r f o r m e d by M o l i n a r i e t an. 29 They c a l c u l a t e d the dispersion relations of a GaAs-AIAs superlattice with n l = n 2 = 7. T h i s c a l c u l a t i o n D which can be r e g a r d e d as 'ab i n i t i o ' , was based on p l a n a r f o r c e c o n s t a n t s o b t a i n e d from f i r s t principles p s a u d o p o t e n t i a l c a l c u l a t i o n s . The a g r e e ment w i t h t h e b u l k d i s p e r s i o n r e l a t i o n c a l c u l a t e d w i t h the same f o r c e c o n s t a n t s using Eq. (16) i s e x c e l l e n t , b e t t e r than that f o u n d with £q. (8). Figure 9 illustrates the cancellation of u for An which is o b t a i n e d in the mab initio' calculation provided that m is small. However, for larger values of m (e.g., m = 7 in Fig. 9) more leakage into AIAs results. In the zincblende structure the q = 0 optical p h o n o n s are both Raman and infrared active. B e c a u s e of the infrared activity the Raman phonons are split into a LO s i n g l e t and a TO doublet. The electrostatic fields associated with these p h o n o n s are also s u b j e c t to boundary c o n d i t i o n s . For the LO phonons, and neglecting retardation, the electrostatic field ~ is related to a potential 0: ~ = - g r ; d ¢. Sometimes these modes are treated as c o n s i s t i n g only of electrostatic fields and t h u s the boundary condition._¢.~ 0 at the interface is imposed 4U,nl Since the polarization P, and t h u s t h e d e p o l a r i z i n g field E, a r e proportional t o u, t h e p o t e n t i a l 0 and u are out of phase (Fig. 1 0 } . I t i s , however, unjustifiable to impose boundary conditions on 0 and n o t on u. We s h a l l find that the boundary condition u = 0 at the interface, neglecting the boundary condition on O, i s j u s t i f i a b l e . The use of incorrect boundary c o n d i t i o n s l e a d s t o i n c o r r e c t t r e a t m e n t o f phenomena in which the d e t a i l s o f the wavef u n c t i o n are i m p o r t a n t , e . g . , e l e c t r o n phonon i n t e r a c t i o n - " (and t h u s Raman s c a t t e r i n g , see b e l o w ) . In the absence o f f r e e charges and neglecting retardation the electric field and t h e e l e c t r i c displacement and ~ f u l f i l l :

cu)l

~ = O;div

0 = 0;~

= ¢~,

(17)

where e, t h e dielectric constant, includes electron and phonon contributions. We may derive ~ from a scalar p o S t n t i a l O or ~ from a vector p o t e n t i a l I~ and Eqs. (17) become:

35 UJ a

~" <

I'

II

CI

J

I--

z .~O

q = 2 ~/(7a) ' '

'

q=

"

"

•

2-r/a

i ........................... I" (GaAs) 7 (AlAs) 7 ~ [ 1 0 0 ]

Fig. 9: LO-phonon amplitude in a (GaAs)7(AIAs)7 superlattice for m = 1 and m = 7. From Ref. 29, M o l i n a r i e t e l .

(o)

(b)

m=2

m=1

u

u

rn=/~

m=3

F i g . 10: Diagrams o f the e l e c t r o s t a t i c p o t e n t i a l $ and the mechanical d i s p l a c e ment u f o r c o n f i n e d LO phonons. (a) c o r r e s p o n d s t o Eq. (20a) and (b) t o Eq. (20b).

¢V20 = O; Equation (IBa) making e i t h e r :

e-lv×vx~ can

be

= 0

fulfilled

8(~) = 0 or V20 = 0 Equation (19a) leads longitudinal modes of

(18a,b) by

{19a,b)

t o the c o n f i n e d either component

modes.3 42s o l u t i o n o f Eq. (18b) obtained for e-I = 0 (i.e., ¢ = ®) y i e l d s the c o n f i n e d TO modes. We s h a l l now examine the d e t a i l s o f TO and LO c o n f i n e d modes and the boundary c o n d i t i o n s a t the i n t e r f a c e . Since the t r e a t ment i s macroscopic we s h a l l not make any a l l o w a n c e f o r the d i f f e r e n c e between Eqs. (8) and ( 1 6 ) . For LO modes o f s l a b 1 (s ( I ) = O) the p o t e n t i a l in s l a b I can be expanded as a sum o f the f o l l o w i n g F o u r i e r comp o n e n t s , t a k i n g the c e n t e r o f s l a b I as origin: ¢l(X,Z)

= ¢oeikxcos qz;

(20a)

Superlattices and Microstructures, Vol. 5, No. 1, 1989

36 ¢l(X,Z)

= ¢ o e i k x s i n qz.

(20b)

F o r simplicity, we have only included in Eqs. (20) the t r a n s v e r s e dependence on x. The b o u n d a r y c o n d i t i o n for O z leads to ¢ = constant in slab Z. The boundary condition for E. yields; zu qa #2(±dl/2) = ¢oelkXcos ~ = 0 (21el ¢2(±dl/2)

= ±¢oeikxsin

-~

= 0

(21b)

Equations (21a,b) correspond to Eq. (20a,b), respectively. We now i m p o s e t h e condition u x z = 0 at the interfaces. Equation (21a) yields: ux = ik¢o e i k x cos ~-~ = O;

(22a)

uz -

(22b)

ikO0 e i k x sin ~

= O.

It is o b v i o u s l y impossible to fulfill Eqs. (22) s i m u l t a n e o u s l y except in the case k = O. In this case we find q = ~ m, m = 2,

4,

6.

(23)

We n o t e t h a t f o r o p t i c a l l y excited phonons e v e n i f k # O, k w i l l be much smaller than w/d ini typical superl~ttices (k ~ x/lO00 A "; q m 2w/100 A-'). Thus t h e b o u n d a r y c o n d i t i o n o f Eq. ( 2 2 a ) is more stringent than that of Eq. ( 2 2 b ) . We may n e g l e c t the latter, which will h a v e t o be f u l f i l l e d through small admixture ~ other modes around the interface. The same a r g u m e n t a p p l i e s t o Eq. ( 2 0 b ) . We f i n d i n t h i s c a s e : q = ~ m,

m = 1,

3,

5

...

(24)

The p o t e n t i a l s ¢(z) and t h e d i s placements u(z) corresponding to Eqs. (20a,b) a r e shown s c h e m a t i c a l l y in Fig. 10. The b o u n d a r y c o n d i t i o n appropriate t o Raman b a c k s c a t t e r i n g i s u = 0 and t h u s # must be a maximum ( n o t a n o d e ) a t the boundary. 5.2

Raman

Selection Rules

Conventional GaAs/AIAs superlattices are grown with the z-axis along [001] (see however Refs. 30,31). Backscattering on the [001] face of the bulk crystals is only allowed for phonons which vibrate p e r p e n d i c u l a r to the surface. A c c o r d i n g to the discussion in 5.1 superlattice phonons vibrating along [001] are LO-like provided k << 21/d (k in the layer plane). TO-like phonons must be polarized along x or y for k << 2w/d. The point group of this infinite superlattice is D2d (symmetry operations: improper f o u r - f o l d rotation along z, two-fold rotations perpendi-

cular to z, two p e r p e n d i c u l a r reflection planes containing z): the vibrations along x, y are two-fold degenerate and belong to t h e E r e p r e s e n t a t i o n D2d (Table 2.1 of Ref. 22). They have Raman tensors of the form:

0

'

I

(25)

It is not possible to couple to these TO-like phonons in b a c k s c a t t e r i n g , i.e., w i t h (L and ~S a l o n g x a n d / o r y. However, they are weakly seen in Fig. 5 This is possible only under e x t r e m e l y resonant conditions, i.e., with the laser frequency very close to that of the first excitons of the superlattice. The m e c h a n i s m for this breakdown of the dipole selection rule is not known. The TO phonons of a G a A s / A I G a A s superlattice were observed in an allowed c o n f i g u ~ $ t i o n (90 o scattering) by Zucker et el. n b ( l a s e r i n c i d e n t a l o n g [001], scattered light collected sidewise along [010]). The s u p e r l a t t i c e had been c l a d with AIGaAs so as t o ' g u i d e ' the scattared light along the y direction. Under these conditions, one c o u p l e s to the t e n s o r s o f Eq. ( 2 5 ) . One can a l s o c o u p l e to the LO-like mode whose Raman t e n s o r has t h e f o r m :

by taking

(LN[OIO],

BSII[IO0].

The LO phonons of the [001] superl a t t i c e s can have, within the D2d point group, two types of symmetries: B 2 which c o r r e s p o n d s to Eq. (26) and leads to the same dipole selection rules as in bulk materials, and AI, which would be dipole forbidden in the bulk. The (a) modes in Fig. 10 h a v e A 1 s y m m e t r y w h i l e t h e ( b ) modes h a v e B 2 s y m m e t r y . The A 1 modes a r e forbidden in the bulk. T h e i r Raman t e n sor for the superlattice is diagonal with Rxx = R... Hence they should appear for p a r a l l e l ~ ( L and (S polarizations. W h i l e the A I modes are dipole forbidden in the bulk, for parallel polarization quadrupole allowed, dipole forbidden modes can be seen near resonance f o r ~Llles ( S e c t . 2 ) . T h e i r s t r e n g t h should be p r o p o r t i o n a l to the square of q. For the LO modes in the s u p e r l a t t i c e the s c a t t e r i n g wave v e c t o r must be r e placed by Eqs. (24) and the modes become d i p o l e a l l o w e d . By analogy to the bulk case one may e x p e c t , however, t h a t these A I modes s h o u l d r e s o n a t e s t r o n g e r than the B2 modes.

37

Superlattices and Microstructures, Vol. 5, No. 1, 1989 l

3

I

WL: 2./.leV

- ---z(x,y)z

)

~l~

,, "

/,'~v

Interface

and S u r f a c e

Modes

I n t e r f a c e modes, p r o p a g a t i n g along x , y, and a l s o z, are found f o r an i n f i n i t e s u p e r l a t t i c e from the s o l u t i o n of Eq. (19b). We t r e a t f i r s t a single int e r f a c e between GaAs and AIAs. Let us use the f o l l o w i n g s o l u t i o n of Eq. (19b) which decays on both sides of the i n t e r face:

'

I .~/

6.

LOI (o)

i',

¢ = l i e i k x eqz

(z < O)

0 = 12 e i k x e-qz

(z > O) .

(27)

i

~'

wL--i.92eV

A p p l y i n g the c o n d i t i o n of c o n t i n u i t y of E. andi Dz we f i n d the f r e q u e n c y Ul o f t~ese n t e r f a c e modes by s o l v i n g :

L02(

z_ 60 --- z(x,y)i z(x,x)E &O

20

e(1)(~ I)

LO6

__ ~

/

q'._ FLO/. I,m ,,

"

'*'L~

Using f o r uLO(q) the standard d i s persion relations of the b u l k , the h i g h e s t LO phonon s h o u l d be a B2 mode ( m = 1, Eq. ( 2 4 ) , followed by A1 (m = 2 ) , f o l l o w e d by B2 (m = 3 ) , e t c . , in a l t e r n a t i n g sequence. 7 We i l l u s t r a t e t h i s in F i g . 11. In the upper p a r t of t h i s f i g u r e one can see in the BZ s c a t tering configuration [z(x,y)z] the h i g h e s t LO mode f o r m = I (L01). In the AI c o n f i g u r a t i o n [ z ( x , x ) z ] we see the m = 2 (L02) mode which peaks below L01, as e x p e c t e d . We a l s o see m = 3, 5 modes. These o b s e r v a t i o n s were o n l y p o s s i b l e away from the s t r o n g resonance of the lowest e x c i t o n . Close t o t h i s resonance t h e A 1 modes become v e r y s t r o n g and a p p e a r even i n the B2 s c a t t e r i n g c o n f i guration: s c a t t e r i n g by d e f e c t s seems t o induce t h i s breakdown in the s e l e c t i o n rules. We should keep in mind t h a t the AI p h o n o n s m u s t c o u p l e t o the i n c i d e n t and scattered fields mainly through

their electrostatic potential ¢ (Frithlich interaction) w h i l e B( c o u p l e s through u (deformation potentlal interaction) (see Fig. IQ): the former resonates more strongly than the latter.

(28)

We n o t e t h a t t h e r e a r e two ~ rs one GaAs-like and the o t h e r A I A s - l i k ~ . These m l ' S are i n d e p e n d e n t o f k and l i e b e t ween mTO and wLO. I f t h e e l e c t r o n i c c o n t r i b u t i o n s to the dielectric conas r ~2,equal st1~s for both materials (e® = ® )) t h e i n t e r f a c e f r e q u e n c y becomes:

RAIdAN SHIFT(ca"I) F i g . 11: ( a t non-resonant s c a t t e r i n g in a (7,7) GaAs/A1As s u p e r l a t t i c e showing the difference between A 1 and B phonons. ( b ) r e s o n a ~ t ¢ © s c a t t ~ r i n g : o2nly A 1 p h o n o n s a r e seen. ",~=

= -e(2)(Ul)

=I For air:

an

=I

(=f0 +Z

]

interface

of

=

112

(29)

semiconductor

s=u~+u~ 1/2 = I ¢=-TZ+'T---I

to

(30)

S i n c e s® ~ I 0 , Eq. ( 3 0 ) y i e l d s for u I a v a l u e r a t h e r c l o s e t o u L. F o r t h e i n t e r f a c e modes o f t h e s u p e r l a t t i c e we must apply t h e E x , Dz c o n t i n u i t y conditions at two interfaces and t h e B l o c h c o n d i tion for the propagation alone ~ We

obtain the secular equation: 34,4~, cos qd = co~h kd

Z~

zn

cosh

kd 2 +

s 1

inh kxd 2

1311

with =

n(u)

e(1)(Ul ) 61Z)(~l)

(32)

This equation is similar to Eqs. (g,171. In the limit q + 0 Eq. (31) splits into: tanh

kd I kd 2 Z cosh z = -n;

tanh kdl Z c o s h kd2 Z = -n -I

(33a) (33b)

tends r~pidly to near u(1) and wL~) w h i l e n -i has the ®same b e h a v i o r near

Superlattices a n d Microstructures, Vol. 5, No. 1, 1989

38

t l~ u o )n s a o n df

; ! ~ ) ' .(33~1 are t h e s found, e r e g i oThusS°lU-four ns

~ j modes a p p e a r f o r each k and q, near t h e uLO and UTO modes o f b o t h c o n s t i tuents, respectively. In the l i m i t the simple form:

k *

0 Eqs.

(33)

take

(2)-like o n e s f r o m u(2JLO • t o u~021. For k + ® all ui's converge at the v a l u e g i v e n by:

¢ i d 2 + ¢2di = O; s l d I + e2d 2 = 0 (34a,b) Or,

using t h e average symbol

Another i n t e r e s t i n g limit of Eq. (31) is t h a t f o r q = w/d (edge o f the minizone): Eqs. (33a,b) are o b t a i n e d w i t h the minus signs r e p l a c e d by p l u s ses. In the l i m i t k + 0 the s o l u t i o n s _ I) (1) ~ 2) ~ 2). the are e I - u~0 , UL0 , ~0 ' 40 " spectra of i I ext~ fo~.~he (1)-like vibrations f r o m u + 6 / , u~6 ! and for the

_,

<>:

,

-

<¢-1> = O; = O.

,

.e.,

=

oz0;- 0

"{36)

(35a,b)

Equation (35a) corresponds t o TO modes o f a m a t e r i a l w i t h t h e average d i e l e c tric c o n s t a n t < ¢ - I > - I w h i l e Eq. ( 3 5 b ) represents L0 modes f o r an a v e r a g e <¢>. In the case d I = d 2 Eqs. (35a,b) yield the same s o l u t i o n , namely t h a t o f Eq. ( 2 9 ) ! It may seem p u z z l i n g that modes which decay around the i n t e r f a c e become l i k e bulk modes o f a n a v e r a g e m a t e r i a l . This is so because we are in the l i m i t o f q ~ 0 and thus the modes a c t u a l l y do n o t d e c a ~ . We n o t e t h a t f o r d 1 = d~ E q s . (33a,b) l e a d t o t h e same r e s u l t . ~orre spondingly, no g a p a p p e a r s i n t h e c a l c u lated dispersion relations v s . k: a g a p a p p e a r s when d 1 # d 2 ( s e e F i g . 1 2 ) .

The i n t e r f a c e modes f o r k = 0 h a v e definite parity with respect to the 2 - f o l d r o t a t i o n f o r q = O, ad. The parity is i n d i c a t e d in F i g . 12: I t r e verses f o r a g i v e n band in going from m a t e r i a l I t o 2. A l s o , both modes q = 0 and q = w/d o f a g i v e n branch (LO or TO) have the same p a r i t y f o r the m a t e r i a l w i t h the t h i c k e r l a y e r s , o p p o s i t e p a r i ties f o r the t h i n n e r l a y e r s . L i k e in F i g . 7, the p a r i t y d e t e r m i n e s the coup-

I

'

I

'

1

r

'

'

SAMPLE A

410

~J" --- k z d , O

2 ' ~ " I "

I-.

380

SAMPLE

~Z

B

.~

;',

j

-'-u~L=2'4OgeV (c) ~

--

(d) SAMPLEB

A '

I

, '

~,11

tel

t

//

A

..

/ /

I!

I

t

I

\I

t

q

I

,

• %1t"

• i

270

330

26O 0 kx dl Fig. 12: modes o f

LU

l\

~Au~r ~ ', / ,,,t', !! ........ ~ ~ J.~" Io I I ---WL= 1-832 eV,l# jr/ ~w !' ") I/'\l,!! t

2eOi~

.

r,

Dispersion GaAs/A1As

d(rlight- )dj4. ( l e f t )

I

I

1

i

1

2

3

4

5

of the interfacesuperlattices with

and

with

dI < d2

I

350

I

I

370

i

I

390

I

I

~0

,

430

RAMAN SHIFT (cm"11 Fig. 13: Rmman s c a t t e r i n g by interface modes o f G a A s / A 1 A s s u p e r l a t t i c e s reported in Ref. 34. Sample A (B) is that whose dispersion relation is given in the left (right) o f F i g . 12.

Superlattices and Microstructures, Vol. 5, No. I, 1989

39 Standing and s u r f a c e waves have been r e c e n t l y observed f o r plasmons in GaAsAIAs s~Rerlattices by light scattering.~° S i m i l a r phenomena s h o u l d be o b s e r v a b l e f o r the i n t e r f a c e phonons. A r e c e n t p u b l i c a t i o n r e p o r t s the o b s e r v a tion of s u r f a c e phonons a t the a i r superlattice interface of GaAs-AIGaAs s u p e r l a t t i c e s by means o f high r e s o l Ht i o n e l e c t r o n energy l o s s s p e c t r o c o p y , v

l i n g s t r e n g t h : modes in which ¢ i s odd should c o u p l e little if the c o u p l i n g occurs v i a 0. F i g u r e 13 shows t h a t t h i s i s indeed the case. The continuum t r e a t m e n t o f i n t e r f a c e modes j u s t g i v e n s h o u l d be a c c u r a t e at best for very large wavelengths. These m o d e s appear a u t o m a t i c a l l y and exactly in any 3-d treatment of the lattice dynamics w i t h long range e l e c trostatic f o r c e s and the u n i t c e l l of t h e s u p e r l a t t i c e 4 6 , 4 7 as modes p r o p a gating p e r p e n d i c u l a r (or o b l i q u e l y ) to the s u p e r l a t t i c e a x e s . Examples o f such c a l c u l a t i o n s are shown in F i g . 14. The signature of i n t e r f a c e modes i s t h e i r strong, nonanalytic dispersion as a function o f t h e a n g l e o f ~ w i t h the superlattice a x i s f o r ~ + 0 (F - r l i n e of Fig. 14}. The i n t e r f a c e modes are l a b e l e d by ' I ' in F i g . 14. Note t h a t each c o n f i n e d mode o f odd index m f o r k = 0 g e n e r a t e s an i n t e r f a c e mode f o r k @ O: the Coulomb e f f e c t s are p a r t i c u l a r l y s t r o n g f o r m = 1. For even m the nonanalytic effects of Coulomb i n t e r action vanish.

The i n t e r f a c e modes j u s t d e s c r i b e d are o f macroscopic n a t u r e . As seen in Fig. 15 t h e y e x t e n d c o n s i d e r a b l y i n t o the i n t e r i o r o f the l a y e r s . M i c r o s c o p i c s u r f a c e modes, c o n f i n e d t o one or two monolayers around the i n t e r f a c e , e x i s t in Ge-Si s u p e r l a t t i c e s . They c o r r e s p o n d to vibrations of the Ge/Si interface bonds which may f a l l in a gap o f a l l o t h e r p~des and thus be s t r o n g l y l o c a lized. 7.

Resonance E f f e c t s

Resonance e f f e c t s f o r phonon s c a t tering have been e x t e n s i v y ~ investigated experimentally. 7,34 ,51 T h e i r theoretical u n d e r p i n n i n g , however, is n o t y e t w e l l u n d e r s t o o d . 52 Resonances o c c u r w h e n e i t h e r wL o r ~S e q u a l s t h e f r e q u e n c y mex o f an i n t e r b a n d e x c i t o n ( i n g e i n g and o u t g o i n g r e s o n a n c e s ) , usua l l y a l s o l o c a l i z e d in the GaAs l a y e r s .

P r o p a g a t i n g i n t e r f a c e modes e x i s t in an i n f i n i t e s u p e r l a t t i c e . For p r a c t i cal, finite s u p e r l a t t i c e s , one must impose boundary c o n d i t i o n s a t the o u t s i d e surface, and s t a n d i n g waves a p p e a r . 400

4o0

350 -

Z,50

300

3OO

25O

250 200

Ik.

200 150 ~ / "

I-

i

150

IO0

100

,D

~ 3

C 0 m

so

o

o

40O

4OO

400

~ = ~

3OO

z.L._

500

-

25U

200 150 I O0 50 X

M

r

Z

o M

F i g . 14: Phonon d i s p e r s i o n r e l a t i o n s o f f o u r GaAs/AIAs s u p e r l a t ~ c e s w i t h i n t e r face modes l a b e l e d ' I ' . ~V

r

r

Superlattices and Microstructures, Vol. 5, No. 1, 1989

40

a) AlAs-like Interface Optical Modes

3x~o3F~-~rm-~- ~

~

rLO

T

k II [100] Uz i

~

n.1

HHIOUT)

0

~

280

290

300*

RAMAN SHIFTt crrf )

--kx~O - - kx= 218kxmax - - - kx= 3/8kxmax

-I

PHOTON ENERGY (eV)

Fig. 16: scattering Ga AI

Resonant Remen profile for by LO-phonons in a GaAsAs s u p e r l a t t i c e !n 1 = 37, • ncoming and outgoing resonances are observed. HH ( L H ) labels resonances with transitions originating in localized heavy (light) hole bands. The insert indicates the observed LOphonon lineshape and an i n t e r f a c e mode ( I F ) . 50

n20;7 s)O' S

l

l

)

Ux

0

I -I

-

I

-

---

i

kx: 2/8 kxmax I kx= 3/8 kxmax I

1041 GaAs/1251 Gao?sAIo25As T=2K z(x,x)2

--

Go 60 60 6o 60 A,

A, A't

60

Fig. 15: E i g e n v e c t o r s of interface ))des of an ( A I A s ) 5 / ( G a A s ) 5 s u p e r l e t t i c e . - "

,± This exciton must be composed of an electron and a h o l e w i t h t h e same l o n g i tudinal quantization n u m b e r (same n u m b e r o f n o d e s o f t h e wave f u n c t i o n ) , otherwise the resonance is very weak. The first r e s o n a n c e (m = I exciton) is much stronger than the subsequent ones (Fig. 16). Usually the 'outgoing resonance' for u S = Uex is strong;re}h~ the incoming one (u L = Uex). ",~"'~ T h i s has b e e n i n t e r p r e t e d as r e l a t e d to i m p u r i t y induce) (o)cth order s c a t t e r i n g prolasses -,3 , w h i c h a l s o o c c u r f o r mfor bidden' s~ttering by LO-phonons in bulk

s a m p l e s . ~ An a l t e r n a t i v e interpretat i o n , based on the incoming and o u t g o i n g photons r e s o n a t i n g w i t h d i f f e r e n t i n t e ~ mediate states has also been proposed. ~ S i n c e i n some s p e c i a l cases the incoming resonance dominates~ the latter mechan i s m can a l s o be o p e r a t i v e . We show i n Fig. 17 a t h e o r e t i c a l fit to the n = 2 resonance of Fig. 16 b a s e d on t h e impurity enhanced FrBhlich mechanism. 7.1

Double

and T r i p l e

Resonances

Resonance phenomena are also observed for interface m o d e s 34 as indi cated in Fig. 13. A l t h o u g h these modes are either leAs, or A l A s - l i k e , their e's

.~m3

lh ~ I-"~1

t

E(n:2.HH) i J I 1.65

L

hwL(eV)

J

I° I I70

F i g . 17: C o m p a r i s o n o f t h e n = 2 e x c i t o n resonance of Fig. 16 w i t h calculations based on the d e fcenct induced Fr@hl i c h interaction model. "v and u ' s e x t e n d into b o t h m e d i a and t h u s s h o u l d be r e s o n a n t a t b o t h GaAs a n d A l A s excitons. Since this is not the case for the confined modes (they resonate at electronic excitations of the Baterial where they are confined) one may use t h e fact that they resonate at electronic gaps of both materials as a s i g n a t u r e of the interface modes. Miller e t e l . 54 r e p o r t e d the observation in a GaAs-GaA1As superlattice, with appropriately chosen layer thicknesses, of a double resonance in which b o t h w L and w s e q u a l e x c i t o n i c energies. This becomes possible because of the splitting of the light hole and h e a v y hole excitons: w s can r e s o n a t e w i t h t h e

Superlattices and Microstructures, VoL 5, No. I, 1989 TRRS /

2.

E

_

e

C1

j

4

k

~s

3.

z

~

3

2h~L0 Lrll

zo

4.

L) o3 19O

1.92

194

196

1.98

5.

2.00

h w i (eV)

F i g . 18: Resonance p r o f i l e s of the 2LO phonon l i n e f o r z ( + , - ) z and z ( + , + ) ~ . Th~ l i n e s are drawn as a 9uide to the eye. The i n s e t shows the s c a t t e r i n g diagram for triply resonant Raman s c a t t e r i n g (TRRS).

6.

7. heavy h o l e and mL w i t h t h e l i g h t h o l e e x c i t o n . Very l a r g e s c a t t e r i n g e f f i c i e n c i e s are observed. A l s o , the p o l a r i z a t i o n selection rules are very i n t e r e s t ing for such s u p e r l a t t i c e s , the normal modes for the incident and scattered light being c i r c u l a r l y p o l a r i z e d . These phenomena are s i m i l a r to those of bulk GaAs under a u n i a x i a l s t r e s s which splits t h ~ valence bands by a phonon frequency.~ If the s p l i t t i n g between the two r e s o n a t i n g e x c i t o n s becomes equal t o two phonon f r e q u e n c i e s t r i p l e resonances are observed: i n i t i a l and f i n a l s t a t e s r e sonate w i t h the q = 0 e x c i t o n s w h i l e the i n t e r m e d i a t e one resonates w i t h a q @ 0 HH e x c i t o n (see Fig. 18). In t h i s case the incident and scattered 'normal modes' are circularly polarized and interesting selection rules are observed: for the strong, triple r e s o n a n c e the p o l a r i z a t i o n s of incident and scattared light are r e v e r s e d (with r e s p e c t to lab axes). This p h e n o m e n o n is Fr~hlich interaction induced and becomes p o s s i b l e t h r o u g h the m i x i n g of light and heavy hole off q = O. By analogy, one has been able to e x p l a i n the fact that only o n e - p h o n o n m = even modes are seen near the r e s o n a n c e in Fig. llb as due to fourth order processes in which one phonon and one i m p e r f e c t i o n scatter the e l e c t r o n s . 55 It is h o p e d that this phenomena will c o n t r i b u t e significantly to the understanding of the resonance mechanisms. 8. 1.

References L. Esaki and R. Tsu: Develop. 14, 61 (1970).

8.

9.

10.

11.

12.

13.

14. 15.

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