RAMAN INTENSITY OF FOLDED ACOUSTIC VIBRATIONS IN SUPERLATTICES WITH COMPLEX UNIT CELL

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RAMAN INTENSITY OF FOLDED ACOUSTIC VIBRATIONS IN SUPERLATTICES WITH

COMPLEX UNIT CELL

Bernard Jusserand, F. Mollot, M. Joncour, B. Etienne

To cite this version:

Bernard Jusserand, F. Mollot, M. Joncour, B. Etienne. RAMAN INTENSITY OF FOLDED ACOUS-

TIC VIBRATIONS IN SUPERLATTICES WITH COMPLEX UNIT CELL. Journal de Physique

Colloques, 1987, 48 (C5), pp.C5-577-C5-580. �10.1051/jphyscol:19875125�. �jpa-00226709�

Colloque C5, supplement au noll, Tome 48, novembre 1987

RAMAN INTENSITY OF FOLDED ACOUSTIC VIBRATIONS IN SUPERLATTICES WITH COMPLEX UNIT CELL

B. JUSSERAND, I?. MOLLOT'

,

CNET, Laboratoire de Bagneux, 196, Avenue Henri Ravera, F-92220 Bagneux, France

" c . N . R . s . , Laboratoire de Micro6lectronique et Microstructures,

196,Avenue Henri Ravera, F-92220 Bagneux, France

R6sum6: Nous prbsentons e t a n a l y s o n s q u a n t i t a t i v e m e n t d e s r k s u l t a t s d e d i f f u s i o n Raman obtenus s u r d e s superr6seaux GaAs-A1As d o n t l a s u p e r m a i l l e e s t c o n s t r u i t e B p a r t i r d e d i f f 6 r e n t e s g 6 n e r a t i o n s de l a s u i t e de Fibonacci. L ' i n t e n s i t 6 d e s r a i e s a c o u s t i q u e s r e p l i e e s r e f l & t e l a q u a s i p k r i o d i c i t 6 de c e t t e s u i t e jusqu'8 d e s g k n k r a t i o n s Blevkes, c e q u i d6montre l a grande e x t e n s i o n s p a t i a l e d e s v i b r a t i o n s a c o u s t i q u e s r e p l i C e s . A b s t r a c t : We p r e s e n t and q u a n t i t a t i v e l y a n a l y s e Raman s c a t t e r i n g r e s u l t s on GaAs-A1As p e r i o d i c s t r u c t u r e s with s u p e r c e l l s c o n s t r u c t e d a c c o r d i n g t o s u c c e s s i v e g e n e r a t i o n s o f t h e F i b o n a c c i sequence. The f o l d e d a c o u s t i c l i n e i n t e n s i t i e s r e f l e c t t h e sequence q u a s i p e r i o d i c i t y and change from sample t o sample up t o t h e h i g h e s t c o n s i d e r e d gene- r a t i o n which demonstrates t h e long range e x t e n s i o n o f t h e f o l d e d a c o u s t i c v i b r a t i o n s .

The v i b r a t i o n n a l p r o p e r t i e s of s u p e r l a t t i c e s have been e x t e n s i v e l y s t u d i e d i n t h e p a s t few y e a r s and a r e now w e l l u n d e r s t o o d ( 1 ) . The low frequency v i b r a t i o n s i n t h e s e s t r u c t u r e s , which a r e b u i l d from t h e a c o u s t i c modes o f both bulk c o n s t i t u a n t s , propagate a l o n g t h e s u p e r l a t t i c e a x i s . They t h u s a r e very s e n s i t i v e t o t h e long range o r d e r i n p e r i o d i c s t r u c t u r e s and then appear a s b e t t e r c a n d i d a t e s t h a n e l e c t r o n i c s t a t e s t o probe t h e s t a c k i n g sequence i n s u p e r l a t t i c e s with complex u n i t c e l l . The t h r e e dimensionnal c h a r a c t e r o f t h e e l e c t r o n i c s t a t e s indeed is l i m i t e d by t h e i r v a n i s h i n g n a t u r e i n t h e b a r r i e r s and h a s been only evidenced on s t r u c t u r e s with s m a l l b a r r i e r t h i c k n e s s and h e i g h t . For f o l d e d a c o u s t i c modes, t h e l o c a l wavevector is r e a l i n a l l t h e l a y e r s , i t s amplitude being modulated a c c o r d i n g t o t h e l o c a l a c o u s t i c p r o p e r t i e s . The coherence o f t h e s e modes along t h e s u p e r l a t t i c e a x i s is t h e n expected t o be very l a r g e .

We r e c e n t l y a n a l y s e d ( 1 ) t h e e f f e c t of t h e t h i c k n e s s e s r a t i o on f o l d e d a c o u s t i c Raman l i n e s f r e q u e n c i e s and i n t e n s i t i e s i n simple GaAs/AlAs s u p e r l a t t i c e s . I n t h i s communication we f i r s t r e c a l l t h e s a l i e n t r e s u l t s o f t h i s s t u d y , which demonstrates t h e g r e a t s e n s i t i v i t y of t h e Raman i n t e n s i t i e s , r a t h e r t h a n f r e q u e n c i e s , on t h e s t r u c t u r e of t h e s u p e r c e l l . We t h e n apply our quantitative model o f t h e Raman i n t e n s i t i e s t o s u p e r l a t t i c e s with complex u n i t c e l l . We p a r t i c u l a r l y c o n s i d e r a s e r i e s o f samples which a r e p e r i o d i c s t a k i n g s o f s u c c e s s i v e g e n e r a t i o n s o f t h e Fibonacci sequence. T h i s sequence h a s been i n t r o d u c e d and a p p l i e d t o t h e e l a b o r a t i o n o f GaAs-A1As complex s t r u c t u r e s by Merlin e t a 1 . ( 2 ) and e x h i b i t q u a s i p e r i o d i c o r d e r . We t a k e advantage o f t h e p e c u l i a r p r o p e r t i e s o f t h i s sequence t o g e n e r a t e a s e r i e s of complex samples with s u f f i c i e n t l y simple and w e l l c o r r e l a t e d Raman s p e c t r a . We o b t a i n evidence of t h e f r a c t a l c h a r a c t e r o f t h e F o u r i e r s p e c t r a o f t h i s sequence, each Raman spectrum being dominated by a few l i n e s whose f r e q u e n c i e s and i n t e n s i t i e s a r e f i x e d beyond a given g e n e r a t i o n o r d e r . Furthermore t h e Raman s p e c t r a change from sample t o sample up t o t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19875125

C5-578 JOURNAL DE PHYSIQUE

l a r g e s t p e r i o d we s t u d i e d , which demonstates t h e long range coherence a l o n g t h e s u p e r l a t t i c e a x i s o f t h e f o l d e d a c o u s t i c modes.

The d i s p e r s i o n c u r v e s of t h e folded a c o u s t i c modes a r e o b t a i n e d i ) by f o l d i n g t h e a c o u s t i c d i s p e r s i o n curve of an average bulk compound and i i ) by opening s m a l l gaps a t zone c e n t e r and zone edge due t o t h e c o u p l i n g between t h e d i f f e r e n t f o l d e d branches.

The d i s p e r s i o n c u r v e s of t h e f o l d e d modes a r e then h i g h l y s e n s i t i v e t o t h e p e r i o d o f t h e s t r u c t u r e through t h e s i z e of t h e f o l d i n g wavevector and t o t h e d e t a i l s o f t h e modulation through t h e average a c o u s t i c a l v e l o c i t y and t h e gaps magnitude. As we demonstrated p r e v i o u s l y ( l ) , t h i s modulation is t i n y i n t h e GaAs-AlAs system and consequently t h e zone edge gaps a r e extremely s m a l l and d i f f i c u l t t o observe by l i g h t s c a t t e r i n g . Moreover t h e i r o b s e r v a t i o n i s impossible i n t h e u s u a l b a c k s c a t t e r i n g c o n f i g u r a t i o n a s , i n s u p e r l a t t i c e s , t h e involved wavevector i s never n e g l i g i b l e r e l a t i v e t o t h e B r i l l o u i n zone e x t e n s i o n and can be even much l a r g e r t h a n it f o r l a r g e p e r i o d samples.

On t h e o t h e r hand, t h e Raman i n t e n s i t y o f t h e f o l d e d a c o u s t i c modes is due t o a p h o t o e l a s t i c p r o c e s s which a l l o w s t h e o b s e r v a t i o n i n p a r a l l e l c o n f i g u r a t i o n of a l l t h e f o l d e d v i b r a t i o n s and t h e l a r g e d i f f e r e n c e between t h e p h o t o e l a s t i c c o e f f i c i e n t s o f GaAs and A l A s f o r t u n a t e l y makes t h e Raman i n t e n s i t i e s very s e n s i t i v e t o t h e s u p e r c e l l s t r u c t u r e . The Raman i n t e n s i t y per u n i t l e n g t h a t temperature T of a mode o f wavevector q i n branch j r e a d s ( 3 ) :

2

I oc

W

where u ( z ) is t h e displacement o f t h e mode and P ( z ) t h e l o c a l r e l e v a n t p h o t o e l a s t i c c o e f f i c i e n t a t p o s i t i o n z along t h e sample p e r i o d d. n(w) is t h e t h e r m a l Bose p o p u l a t i o n f a c t o r a t t h e frequency w o f t h e considered v i b r a t i o n . This c a l c u l a t i o n can be performed e x a c t l y f o r s u p e r l a t t i c e s with piecewise c o n s t a n t composition p r o f i l e . We however p r e f e r t o perform a F o u r i e r a n a l y s i s o f t h e problem which is v a l i d f o r any given p e r i o d i c p r o f i l e and a l l o w s t o i n v e s t i g a t e t h e o r i g i n o f each l i n e i n t e n s i t y . We then o b t a i n :

2

W

where Pn and un a r e t h e n-th F o u r i e r components o f P ( z ) and u ( z ) and where G=2n/d.

From t h i s formula we can understand t h e r e s p e c t i v e e f f e c t o f t h e a c o u s t i c and p h o t o e l a s t i c modulations. I f we n e g l e c t t h e p h o t o e l a s t i c one, t h e o b s e r v a t i o n o f t h e f o l d e d modes is o n l y due t o t h e i r c o u p l i n g t o t h e t r u e a c o u s t i c branch and is r e l a t e d t o t h e POUO term. The c a l c u l a t e d i n t e n s i t i e s a r e t h e n extremely s m a l l , i n complete disagreement with experiment. Neglecting t h e a c o u s t i c a l modulation a s done i n Ref.3 g i v e s a b e t t e r d e s c r i p t i o n o f t h e Raman s p e c t r a , t h e i n t e n s i t y o f f o l d e d l i n e n b e i n g

Fig.1: Raman s p e c t r a on two d i f f e r e n t "simple" s u p e r l a t t i c e s compared with t h e corresponding c a l c u l a t i o n s o f t h e l i n e i n t e n s i t i e s .

45 ' 35 25 15 5 FREQUENCY SHIFT [ern+)

however falls in quantitatively reproducing some experimental features such as the intensity difference between modes +n and -n. Taking in account a large set o f Fourier components allowed us to quantitatively fit the Raman intensities. These quantities are then the result of the interference between both modulations: the acoustical one which induces a contribution proportionnal to the average photoelastic coefficient PO and the photoelastic one which induces contributions proportionnal to the difference between these coefficients in both bulk constituants. This result is illustrated on Fig.1 which displays low frequency Raman spectra on two different samples compared with the relevant predictions of our model. In such "simple" samples with two layers by unit cell, the structure can be described by its period d and the relative thickness of the layers a. For both samples the calculated curves have been obtained for fixed period and wavevector corresponding to the presented spectrum and illustrate the intensity variation as a function of a. The vertical line in each case show the actual value of a in the sample. As clearly appear on the Figure, the model well describes both the relative intensity of the doublets, mainly reflecting the photoelastic modulation, and the doublet "asymmetry" arising from the acoustic modulation.

To further illustrate the previous analysis and test the limits in the sensitivity of the folded acoustic modes on complex periodic structures, we performed Raman backscattering experiments on a series of GaAs-A1As periodic structures constructed according to successive generations of the Fibonacci sequence. This mathematical sequence is based on two building elements A and B stacked according to the following procedure: the generation n+l is obtained by replacing in the generation n each block A by the sequence AB and each block B by the A one. One then obtains for instance: G2=ABA, G4=ABAABABA,

...

~=(1+45)/2. The Fourier spectrum o f such a sequence is not a broad band as could be assumed for a non periodic system but consist in a discrete set of Fourier components corresponding to the rationnal approximations of T. As real samples are finite, they must be constructed according to a given Fibonacci generation. Due to the fractal character of the spectrum, the successive generations have very similar Fourier spectra: the most intense components remain almost constant in frequency and intensity from a given generation order, new smaller components appearing in between.

Opposite to samples of Ref.2, we used single layer building blocks A and B defined as follows: A is made of 8 monolayers o f GaAs and B of 13 monolayers of AlAs.

The most frequent sequence A is here the thinnest one and the average A1 content goes to 0.5 as the order is increased. As our aim is to analyse the folded lines intensities, the coalescence of some neighbouring A building blocks due to their single layer nature is indifferent. We can then minimize the thickness of the sequences and thus to expand the Raman spectra to less low frequencies. We present experimental results on samples G2,G4,G6,G9 based on generations 2,4,6,9, the corresponding nominal sequence length and average A1 contents being respectively 82.0, 223.4, 588.2, 2494.2A and 0.45, 0.49, 0.5, 0.5. The samples, grown by Molecular Beam Epitaxy, consist of the stacking o f several times the same generation in order to keep a constant total thickness (-0.5pm) from sample to sample. They have been first studied by X-Ray diffraction and the spectra around the (002) order of GaAs exhibit a large number of satellites as a consequence of the superperiodicity. From these spectra we can easily deduce the real A and B block lengths which appear to be very slightly smaller than the nominal ones, Moreover, the diffraction patterns exhibit characteristic features related to the Fibonacci construction as will be described elsewhere.

The low frequency Raman spectra on the four samples are shown on Fig.2 and all

C5-580 JOURNAL DE PHYSIQUE

exhibit a large number o f folded lines. On G2, which only contains two layers by unit cell (the periodic ABA is equivalent to a periodic AAB), we recover the usual result that the first doublet is the more intense one (see Fig.1). This property disappears

on G 4 which contains 6 layers by sequence and the Raman spectra is dominated by the +3 doublet, at slightly lower energy than the dominant one in 62. In this sample the backscattering wavevector is larger than half the Brillouin zone extension and the folded lines, though easily indexed, do not clearly appear a s a succession o f doublets. Moreover,as evident from a comparison with G6 and G9 spectra, the dominant features o f the quasiperiodic stacking are already defined at this stage: lines at frequencies -13, 18, 22 and 27 cm-1 have about the same frequency and intensity in the three spectra, even if they

correspond to increasing folding indexes. To

21

I

analyse the Raman spectrum on G6, for which the backscattering wavevector is close to twice the Brillouin zone extension, a comparison with a

2

calculated spectra is very useful, also shown on Fig.2. As a matter o f fact, there is a good

2

correspondance between the frequencies o f the 2 G6

dominant lines o f the measured and calculated spectra and the trends of the intensities are well reproduced. Some discrepancies in the relative intensities o f the lines however appear, in particular the large intensity o f Line -3, which are not yet understood. The sensitivity of the spectra on defects in the structure, such as interface broadening and layer thickness irreproducibility, should be in particular investigated in this frame.

Nethertheless, G6 sample already exhibits a Raman spectrum which rather reflects the quasiperiodic order than the sequence

periodicity

.

o

experimental spectrum, though not yet indexed, RAMAN SHIFT (cm-1)

gives further evidence o f the discrete nature o f Fiq.2: Measured spectra on the Fourier transform of large order Fibonacci samples G2 to G9 (bottom) sequences. It also gives evidence o f the large compared with the calculated one spatial coherence o f the folded acoustic modes. for G6 (top). The higher energy The spectra on G6 and G9 clearly differ, which part of the spectra has been proves the coherence o f the modes on at least enhanced using a larger slit the period of G9 i.e. 2500A. This result is aperture.

consistant with other evidences obtained by phonon transmission measurenents(4) or deduced from the intrinsic width o f the folded lines(5).

References:

(1) See 8.Jusserand et al., Phys.Rev.35,2808 (1987) and references therein.

(2) R.Merlin et al., Phys.Rev.Lett. 55,1768 (1985).

(3) C.Colvard et al., Phys.Rev.831, 2080 (1980).

(4) V. Narayanamurti et al., Phys.Rev.Lett.43, 2012 (1979).

(5) 3.Sapriel et al., in Proceedings of the 18th Int. Conf. on the Physics of Semiconductors, edited by 0. Engstrom (World Scientific, Singapore, 1987) p. 723.