Inelastic light scattering in GaAsAlAs superlattices

Superlattices and Microstructures, VoL 6, No. 2, 1989

227

INELASTIC LIGHT SCATTERING IN GaAs-A1As SUPERLATYICES B. H. Bairamov, R. A. Evarestov, I. P. Ipatova, Yu. E. Kitaev, A. Yu. Maslov A. F. Ioffe Physico-Technical Institute, Leningrad 194021, USSR M. Delaney, T. A. Gant*, M. V. Klein, D. Levi, J. Klem**t, H. Morkoq** Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign, (UIUC), 1110 W. Green Street, Urbana, IL 61801, USA (Received 9 August 1988)

The first- and second-order Raman scattering by confined LO phonons in (GaAs)m(A1As)nsuperlattices are studied theoretically and experimentally. It is pointed out that the frequencies of these modes of (GaAs)21(AlAs)6supcrlattice only roughly correspond to bulk GaAs phonons. From linewidth measurements we fred the life time of confined LOI phonons ~ LOI = (3.60 + 0.40) ps; "~LOfr') = (6.30 + 0.40) ps for bulk (MBE) GaAs. To analyze the symmetries of confined phon0ns and to obtain the selection rules, band representations of space groups D9 and D~d have been developed. We find that there are eight non-equivalent cases for the arrangement of atoms in the primitive cell over Wyckoff positions corresponding to different sets of m and n. We also establish that the contribution of specific atoms to the phonons with definite symmetry as well as vibrational representation, i.e. the number of phonon branches with given symmetry, depend on m and n. It is shown that the analysis of phonons from the Brillouin zone considerably increases the number of independent atomic groups and thus potentially increases the analytical applications of Raman scattering and thca'ebyopens up new possibilities to obtain broad information about superlattice microstructure.

There isconsiderablecurrenttheoreticaland experimental interestin the study of inelasticlightscatteringin semiconductor superlattices(SL's)in order to understand the rmlan~of various vibrati~al modes and their interaction with the confined electrons forming a quasi-two dimensional system. These properties are important not only from the fundan~ntalpoint of view but also because of the influence they have on transport phenomena. The electronphonc~ interactions can be considerably modified relative to those in ti~ bulk, not only due to reduced ditmnsionalityof the electron system, but also because the patterns of the nC~L1 vibrational modes are themselves modified by the presence of spatial periodicity. To date most of the studies have concentrated on fu-storder Raman scanering.l-9 To our knowledge only one work 2 has been published on second-order Raman scattering by confined optical phonons in GaAs-AIAs SL's. For laser energies in resonance with the first electron-heavyhole exciton the experiments show a series of sharp peaks * ** I"

Present address: Division of Physics, National Research Council, Ottawa Kla OR6, Canada. Coordin~_mdScience Laboratory, UIUC. Present add~ss: Sandia National Laboratory, Albuquerque NM 87185-5800, USA.

0749-6036/89/060227 + 05 $02.00/0

that can be attributed to combinations of confined GaAs LOI phonons with FI(AI) symmetry. In this paper we ~ extensive experimental and theoretical study of first- and second order Raman scattering by confined modes in G a A s - AlAs SL's. To analyze the spectra and to obtain the selection rules for the combination of phonons f,cm~the entire Brillouln zone (BZ) we have developed band representations (BR's) theory of space groups. 12 2. EXPERIMENTAL DETAILS The parameters of our SL's are NI: dl = 58.87 (m = 21), d2 = 18.05 (n = 6), do = 2.22; N2: dl ---20.75 (m = 7), d2 = 51.45 (n= 20), do = 3.64, where dl, d2~ do are the thicknesses in A of GaAs, AlAs slabs and interface widths, respectively. The SL period was repeated 100 times. We present here results for one sample N1. The resuits for sample N2 are qualitatively similar to the ones given here. The Raman spectra were excited at 10K f{trfrom resonance with the excitons using 5145 and 4880 Alines of a ew Ar laser in a Brewster's angle reflection scattering geometry from ((301) surface. The axes are: x(100), y(010) and z(001). The spectral resolution varied between ~1 and © 1989 Academic Press Limited

228

Superlattices and Microstructures, Vol. 6, No. 2, 1989

- 4 cm -1 for first- and second-order Raman scattering, respectively.

--300 TE o

:329C

3. RESULTS AND DISCUSSION a) FIRST ORDER SCATTERING Fig. 1 shows the first-order Raman scattering spectra of (GaAs)21 - (AlAs)6 SL in the region of confined GaAs LO t modes. The sharpness of the confined phonon lines illustrates the quality of our SL and means a high degree of crystalline perfection. We measured the linewidth (FLH1) of LO1 phonons and obtain FLOI = (1.50 + 0.05) cm -1 corrected for instrumental resolution. For comparison, we also give our result FLO1 = (0.85 + 0.05) cm -1 and frequency shift O~Ofl-3 = 296 cm -1 of the LO (F) phonon of bulk MBE GaAs. Taking into account that the linewidths are caused by cubic anharmonicity, we found for the confined phonon lifetime '~LO1= (3.60 + 0.40) ps and '~LOff) = (6.30 + 0.40) ps for bulk GaAs. The observed confined GaAs LOI phonon frequencies versus the quantized wave vector are presented in Fig. 2. The solid line is the result of linear chain model calculation based on a phonon effective mass approximationl,7,9 with error function concentration profiles and interface width derived from the fit to X-ray data. It is worth pointing out that the frequencies of confined GaAs phonons for (GaAs)21 (AlAs)6 SL only roughly correspond to bulk GaAs phonons, deviating towards higher energy for large q. Strictly speaking, the SL phonon spectra cannot be deduced from that of bulk GaAs or AlAs simply as the spectra of perturbed system with lowered

1.0

I LO 1

-"-r

:

0.5'

xl/&5

LOsil ) LO z

TOI L011L09L07/'::.-!! ! I

I

I

. ;

!::".L04

....................... •..... I i "...;........ 0

......

; . . . .

'

'

xx

0

'

'

~:

:

"

LOt, I i

:

TOz IF; :' I LOa I l i :;-, L0101 .

!

,

,

270

290

Fig. 1. First-order Raman (GaAs)2t phonons depolarized z(xy)z configurations. T

O

270 ,

q (units of

2if/a)

i

10 X

Fig. 2. Observed frequencies of confined LO1 phonons of (GaAs)21 (AlAs)6 SL versus the quantized wave vector q = ~..~ (n+l)ao (filled circles) and q = Q.~ nao (shown) only for ~ = 7). The solid line is the result of our calculation (see the text). For comparison we have also plotted the low temperature (T = 10K) neutron scattering data 10 for the bulk GaAs (squares) and Rarnan data for (GaAs)7 (AlAs)7 SL-f'dled squares. 3

symmetry since both the number and type of atoms in the primitive cell and therefore the interatomic interactions are distinct in these systems. The observed values of AlAs confined LO1 phonon frequencies are in agreement with the dispersion curve of bulk AIAs.ll, lla b) APPLICATION OF BR OF SPACE GROUPS IN THE ANALYSIS OF SL PHONON SPECTRA

-II-

=

289

~ , c m "~

scattering spectra of confined of (GaAs)21 (AlAs)6 SL for and polarized z(xx)z scattering = 10K, Zi - 5145]L

The symmetry of SL's (GaAs)m (AlAs)n is DS2d (m+n=2k) and D 9 (m+n=2k+l) 18 depending on the pfimitve ceil. Up to now only F-point phonons (k--0) in SL's were discussed in terms of point group (D2d) symmetry. 1 However, the general analysis of phonon symmetry over the Brillouin zone (k~O) and its dependence on m and n have not been published. We develop here an application of BR's of space groups12 in the analysis of SL phonon symmetry and derive the second-order Raman scattering selection rules based on such a symmetry. BR of space groups establish the correlation of system local properties (here, the local atom displacements) with its band properties (here, normal vibrational modes). In terms of group theory, the BR is the reducible representation of infinite dimension constructed in the k-basis by combination of irreducible representations (i.r.'s) of the space group over the Brillouin Zone (BZ). However, it can be completely specified by the i.r.'s in all the inequivalent symmetry points of the BZ and one representation point from all those inequivalent symmetry lines and symmetry planes which have no symmetry points. In other arbitrary

229

Superlattices and Microstructures, VoL 6, No. 2, 1989 points of the BZ, the BR may be derived from compatibility relations. The SL's are the ideal model crystals to demonstrate the advantages of band representation theory of space groups in the application to lattice dynamic problems. In terms of symmetry, the SL's with different m and n, even those belonging to the same space group, are distinct crystals differing by atomic arrangement in the primitive cell. The phonon symmetry analysis with well-known method of constructing the vibrational representation of the crystal and its subsequent decomposition into i.r.'s of crystal factor-group as well as the analysis with the so called "site symmetry" method 19 do not allow solution of this problem in a general way for arbitrary m and n but require one to perform the calculations for each given SL all over again. Moreover, the above methods are very cumbersome especially for the crystals with a large number of atoms in the primitive cell. In the approach used in the present work one needs to construct the BR's only for two space groups D 5 and D 9 and then to derive the formula for arrangements of atoms in the primitive cell over the Wyckoff positions for arbitrary m and n. Combining them one can easily perform the phonon symmetry analysis for any given SL's. We have established that there are eight nonequivalent cases of atomic arrangement corresponding to different sets of m and n and constructed the BR's 20 of space groups D 5 and D9 d. Then we performed the phonon symmetry analysis and established that the types of phonon symmetry for all the SL's belonging to the same space group do not depend on the specific values of m and n. However, the contribution of specific atoms to the phonons with definite symmetry depends on m and n due to the change of atomic arrangement. The vibrational representation, i.e. the number of phonon branches with given symmetry, changes as well. The results of the analysis will be published elsewhere. Below, we apply them for the analysis of experiments on the SL with specific values of m and n. The symmetry of SL (GaAs)21 (AlAs)6 (m+n=27) is D 9 . The primitive cell contains 54 atoms. The number of vibrational modes is 162. The arrangement of atoms over the Wyckoff position in the primitive cell is the following: one Ga atom is at the point a (000), the site symmetry is 42m; an As atom is at the point lc(0,1/2,1/4)-42m, 20 Ga atoms and 6 A1 atoms are in pairs at the position 2e(00z) (00z-)-mm and 26 As atoms are in pairs at the position 2f(0,1/2,z) (1/2,0,g)-mm (Fig. 3). The phonon symmetries of (GaAs)21 (AlAs)6 SL are given in Table 1. The atoms in the primitive cell with corresponding Wyckoff position (in parentheses), the Wyckoff position components along the basis translational vectors tl, t2, t3 of the double primitive cell given in terms of tl, t2, t3, units, and the Wyckoff position site symmetry are given in the first column. The i.r.'s of the site symmetry groups according to notation20 with corresponding atomic displacement components are given in the second column. The symbols on BR's in k-basis indexed from the corresponding i.r.'s of the site symmetry groups, i.e. the indices of the i.r.'s of the group of wave vector k according to notation22 defining the phonon symmetry at corresponding BZ points, are given in the 3rd and 4 th etc.

~

AI (2el

t3 ~ .

.

.

.

.

Go

(2el

.

....

As(2f) ~As (2f} ~ ~

~

z

Go(2e) At (2el

(o) Fig. 3. The structure of (GaAs L)21 (AlAs)6 SL and the positions of the atoms in Wyckoff notation showing cubic (a) and centered tetragonal (b) unit cells. columns. The symbols of BZ points, their components in terms of reciprocal lattice vector units and the corresponding point groups of wavevector k are given in the rifles of each column. It is seen, that definite groups of atoms contribute to the phonons of given symmetry, e.g., the vibrational representation at F-point is 26F1 (Ga ze ; A1ze ; As~) z + 28F2 (Gaa, e) Alez ASc,f ) + 54F5 (GaXY ;AlXY'e,AsXY,f )'

(1)

where superscripts give the components of atomic displacements, and subscripts are the Wyckoff positions occupied by the atoms contributing to the mode of a given symmetry. We see from (1) that only z-components of Ga and A1 atoms at e positions as well as of As at f-positions contribute to Fl-mode. The analysis of phonons from the other points of the BZ whose combinations could be seen in second-order Raman scattering considerably increase the number of independent atomic groups. From these data, broad information about SL atomic structure can be obtained. To analyze the second order Raman scattering spectra we have obtained the selection rules for Raman-active phonons with symmetry given in Table 2.

Superlattices and Microstructures, Vol. 6, No. 2, 1989

230

Table 1. Symmetry o f phonons in (GaAs)21 (AlAs)6 superlattice

F

-M 1 1 1

X

~2m

42m

222

4

m

b2(z)

2

2

2

2

1

42m

e~x,y)

5

5

3,4

3,4

1,2

1As(c)

b2(z)

2

1

4

4

1

e(x,y)

5

5

1,2

1,2

1,2

10 x 2Ga(e) 3 x 2AI(e)

al(z)

1,2

1,2

1,2

1,2

1,1

(OOz)

bl(x)

5

5

3,4

3,4

1,2

(OOz-')

b2(y)

5

5

3,4

3,4

1,2

13 x 2As (0

al(z)

1,2

1,2

3,4

3,4

1,1

(o ~.1 z)

bl(x)

5

5

1,2

1,2

1,2

b2(y)

5

5

1,2

1,2

1,2

14m2 1Ga(a)

P 1 1

N

(01 0)

(OO0)

I 1 (o ~~-)

42m

mm

(1 o-~) mm

Table 2. Assignments and frequencies of the confined GaAs phonons in the second-order Raman spectrum of (GaAs)21 (AlAs)6 SL

Scattering geometry xx

Phonon combination

r~equency (cm-1)

Phonon combination

Frequency (cm"1)

LO1 + LO1 (I'2 x r2)

591.4

LO1 + LO2 (F1 x r2)

586.3

LO2 + LO2 (rl x r l )

590.2

LO1 + LO4 (Fl x r2)

582.4

LO3 + LO3 (F2 x F2)

588.8

LO2 + LO3 ( r l x r2)

580.5

Superlattices and Microstructures, VoL 6, No. 2, 1989 1.0

560 ! LO • LO i~ '1 2 3 ii ~'LO I-LO

5t.8i~, i~ 1

6

REFERENCES

l

5t, t, lr ! j~l LO 1 + LO z

I--q

O.S ¸ r

" ....

"7i

HI--

1.

xy

2. 3.

:

,.....',..' /I,:, , / \ ........... LO3*L03~,5';C ,_ "........................ ,~,LU + L U

It, 2 2 ~i LO, * LO,

0 O,S

•

ilJ

4. 5. 6.

5601) It

5151 I .,i

.._~..t',.

231

1660

"

xx'

7.

k

/ "..

i

i

i

:

765.__

I tlOO

8.

9. !

co (cm -1)

10.

Fig. 4. Second-order Rarnan scattering spectra of GaAs)21 (AlAs)6 SL for depolarized z(xy)z and polarized z(xx~ scattering configurations. T = 10K, 2ti 5145A.

11.

500

700

First, we determined the critical points of the phonon density of states and established that the Z-point is critical for F1- and F2-mode, M point for M1- and M2-mocle and P point for P1- and P2-mode. Second, we determined that in (xx) scattering geometry the Raman-active combinations are F1 x F1, F2 x 1"2; M1 × M1, M2 x M2; P1 x P1, P2 x P2, P1 x P2 and in (xy) F1 x F2; M I x M2 and PI x P2.

lla. 12. 13. 14. 15. 16. 17. 18.

c) SECOND-ORDER RAMAN SCATI'ERING A Typical Ram,an Spectrum in the frequency region corresponding to the second order scattering by confined optical phonons of (GaAs)21 (AlAs)6 SL is shown in Fig. 4. Their assignments and frequencies are given in Table 2. In conclusion, it is shown that the experimental study of fn'st- and second order Raman scattering in SL's and their analysis with BR's theory of space groups potentially increases the analytical applications of Raman scattering and opens up new possibilities to obtain broad information on atomic structure of SL's. Moreover, monocrystalline nearly lattice matched, SL's - being a series of distinct man-made crystals - are good model systems to enlarge the field of applications of BR's theory itself. Acknowledgement - The work at Illinois was supported by the National Science Foundation under DMR 85-06674, 88-03108, and 86-12860, by JSEP and by AFOSR.

19. 20. 21. 22.

M.V. Klein, IEEE J. Quant. Electr., 1986, QE-22, N9, p. 1760-1770. A.K. Sood, J. Menendez, M. Cardona, K. Ploog, Physical Review B, 32, 2, 1985, p. 1412-1414. A.K. Sood, J. Menendez, M. Cardona, K. Ploog, Physical Review Letmrs, ~ 19, 1985 p. 2111-2114. K. Kubota, M. Nakayama, H. Katoh, N. Sano, Solid State Communications, 49, 2 (1984) p. 157-159: C. Colvard, T. Gant, M. V. Klein, R. Merlin, R. Fisher, H. Morko~, A. C. Oossltrd, Physical Review B31.4, (1985), p. 2080-2091. B. Jusserand, D. Paquet, A. Regreny, Physical Review B30. 10, (1984). p. 6245-6247. B. Jusserand, D. Paquet, A. Regreny, Superlattices and Microstructures, Jo 1 (1985) p. 61-66. A. Ishibashi, M, Itabashi, Y. Moil, Y. Kaneko, S.

Kawado, N. Watanabe, Physical Review B33, 4, (1986) p. 2887-2889. A.C. Maciel, L. C. C. Cruz, J. F. Ryan, Journal of Physics C20, 7, (1987) p. 3041-3046. E. Richter, D. Strauch, Solid State Communications 64, 6 (1987) p. 867-870. S.K. Yip, Y. C. Chang, Physical Review B30, 12 (1984) 7037-7059. H. Chu, S. F. Ren, Y. C. Chang, Physical Review B37, (1988) p. 10746. R.A. Evarestov, V. P. Smirnov, Metody teorii grupp v kvantovy khimii tverdogo tela. - L.: Izd. LGU, 1987. - 375 s. E. Molinari, A. Fasolino, K. Kunc, Physical Review Letters 56, 16 (1986) p. 1751. B. Jusserand, D. Paquet, Physical Review Letters 56, 16, (1986) p. 1752. A.K. Sood, J. Menendez, M. Cardona, K. Ploog, Physical Review Letters 56, 16, (1986) p. 1753. G. Dolling, J. I. T. Waugh, In: Lattice Dynamics, ed. R. F. Wallis, (Pergamon, London, 1965) p. 19. B.H. Bairamov, Preprint FTI N 1192, 1987. - 31 s. J. Sapriel, J. C. Michel, J. C. Toledano, R. Vacher, J. Kervarec, A. Regreny, Physical Review B28, 4 (1983) p. 2007-2016. G.N. Zhizin, B. N. Marvin, B. F. Shabanov, Opticheskie kolebatelnye spektry kristallov.-M.: Nauka, 1984.-232 s. O.V. Kovalev, Neprivodimye i indutsyrovannye predstavleniya i kopredstavleniya fedorovskikh grupp.-M.:Nauka, 1986.-368 s. International Tables for Crystallography/F_xLTheo Hahn. vol. A Space-group symmetry.-DordrechtBoston, Reidel, 1983.-854 p. S.C. Miller, W. F. Love, Tables of irreducible representations of space groups and corepresentation of magnetic space groups.-Colorado, Boulder, 1967.1095 p.