Site symmetry approach to lattice dynamics of semiconductor superlattices

Superlattices

and Microstructures,

SITE

SYMMETRY

A.

F.

APPROACH

0. Ioffe

Zentralinstitut

M.

TO

LATTICE

H. Bairamov, R.,A. Physical-Technical fur

211

Vol. 9, No. 2, 1997

E. Elektronenphysik, Berlin,

DYNAMICS

OF

Evarestov, Institute,

SEMICONDUCTOR

SUPERLATTICES

Yu. E. Kitaev Leningrad 194021,

Jahne Akademie DDR - 1086

der

Wissenschaften

USSR der

DDR,

Delaney, T. A. Gant*, M. V. Klein, D. Levi, J. Klem**+, H. Morkoc** Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign, (UIUC), 1110 W. Green Street, Urbana, IL 61801, USA (Received

13

August

1990)

We present here the site symmetry approach to lattice dynamics of superlattices which permits us to connect by symmetry the local atomic displacements and normal vibrational modes over the entire Brillouin The atomic zone. arrangements over the Wyckoff positions for (GaAs),(AlAs), and (Si),(Ge), superlattices oriented along COO11 for different sets of m and n are determined. We obtained the phonon symmetry for k # 0 using the developed theory of the band representations of space groups and derived the selection rules for the first and second-order infrared and Raman spectra. Raman spectra obtained for the (GaAsJ7(A1AsI18 and (Si)S(Ge) superlattices are interpreted in terms of the presente d theory.

1.

Introduction

Success in growing (GaAs) (AlAs), and superlattices (!3L's) with (Si) (Gel varyPng p?imitive cell, consisting of m and n monolayers of GaAs and AlAs (Si by molecular beam and Ge), respectively,. epitaxy have been recently demonstrated by several groups cl-191. Raman scattering techniques have proved to be an excellent too1 for studing electronic properties and lattice dynamics of these SL's and for obtaining a detailed information on crystalline quality and strain field distribution. To analyze the experimental data on first and second-order Raman scattering we have to study the SL's phonon symmetry and the corresponding selection rules for Raman scattering.

?? Present address: Research National KlaOR6, Canada

* Coordinated

??

+ Present Laboratory, USA

Division of Council,

Science address: Albuquerque

Laboratory, Sandia NM

Physics, Ottawa UIUO

National 87185-5800,

In SL'a the new periodicity has consequences both in a reduction of point symmetry with respect to constigroup tuent bulk materials and in an increase of the number of atoms primitive per cell resulting in a new space group symmetry. The (GaAs)m(AlAs) SL's grown along COO13 constitute two s P ngle cry Ital families with space groups D2d (nt+n=2k) and Dg (m+n=2k+l) depending on even or odd to2%!al number of monolayers

The atomic arrangement over the Wyckoff positions in the primitive cell for the SL's is governed by the specific values of m and n for each SL-family being a series of single crystals with the same space group. Thus, in terms of symmetry SL's belonging to the same family are distinct crystals differing by the arrangement of atoms in the primitive cell. We have derived general formulae for the atomic arrangements for the

SL's

in question

[17-191.

To

obtain

0 1991 Academic Press Limited

212

Superlattices

the SL'.s phonon symmetry ponding selection rules theory of the band (l3H's) of space groups. Theory of 2. groups space lattice

of to

complex crystals with the For large of atoms in the primitive cell number method of BR of space groups proved the trato be much more effective than the ditional factor group approaches c211. This method is especially efficient for families belonging to the same crystal space group but having different atomic over the Wyckoff positions arrangements the generation of l3R's does since not involve an information about the distriatoms in the primitive cell bution of over the symmetry positions. In terms of the group theory, the HR's repreof a space group F are the of F induced by the sentations irredu-

Table

1:

Dg

The simple BR's of the space group Dzd (I?m2)

i

2d

I&Q

Vol. 9, No. 2, 1997

cible representations (irreps) of its site symmetry subgroups MqC F. In terms dynamics it corresponds to of lattice normal vibrational modes induced by the local atomic displacements. the simple BR's may be All induced from the irreps of the site symmetry Mq.of relatively small number of groups rn the Wigner-Seitz unit cell q-points of a direct lattice belonging to the set The set Cl includes all the inequivau. lent points of the Wigner-Seitz unit cell and one representative point from all those inequivalent symmetry axes and planes which have no symmetry symmetry points. As an example we consider the generat&on of simple t3R's for the space group The set U includes 4 points D2d* The set K includes 5 a,b,c.d. points The set of simple BR's l-',M,X,P, and N. could be obtained by an induction from irreps of the site symmetry ClrOuDS the points of the biMb’DZdj, a(M,=D2d ) , C(MC=Dzd), d(Md=Dzd).

and the correswe developed the representations

band representations and its application dynamics of SL's

and Microstructures,

P

(000)

BT (O&u

(;1- $ $)

T2m

T2m

222

a,b,c,d

a,b

a(0001

a,

c,d

1

12

b(O+

b2

2

2

12

0(4+)

a2

d(+T2mt)

9 8

:

f

:

a,b

c,d

13

4

I.4

m

T

a

i

c

d

a,b,c,d

7234

1

2143

1

1:,,3~43~41:,.-1

,Ij

Superlattices

and Microstructures,

213

Vol. 9, No. 2, 1991

The complete results are presented in Table 1. The first column contains the of Wyckoff positions together notations with their coordinates and corresponding Columns 3-7 consite symmetry groups. EIR's simple tain indices of the the induced by the corresponding irreps of

3.

Similarity

of

analysis

of

superlattices

some The atomic arrangements for and (Si) (Gel SL's be(GaAs),(AlAs) longing to !he familie! wit{ the same space group reveal many similarities. As the atomic arrangements for an example, ~~;~;)z~A;~;~~S2and (Si)3(Ge)? ~~:~,,~;~ . The crysta and the BZ for the latter are shown in Fig. 1 and Fig. 2 respectively.

the site symmetry groups given in column The points 2. corresponding symmetry together with their coordinates and little groups are given in the headings of columns 3-7.

Table 2:

Symmetry

phoaon eymmetry in (oaA~)~(:Uis),~

and (Si),(Ge)3 superlatticee

tub/Au0

ga

woe

I&Q

P

<00&l ()$,,

B

cog,

T

P

2

2

2

1

5

5

3.4

3.4

192

b2kP)

2

1

4

4

1

e(x,j)

5

5

1,2

1,2

1.2

(00s)

al(b)

1,2

1,2

1,2

1,2

l,l

(00;)

bl (~1

5

5

3.4

3,4

192

b2W

5

5

3,4

3.4

1*2

c$!jd

a.+

1,2

1,2

3,4

3,4

191

(+O:,

b, (11

5

5

192

1,2

112

b2W

5

5

1,2

1.2

1‘2

bZ(d

em

em

2

1Si 72m 10

IA6

(000)(~~~)

x

222

$0) lh

M

IQ.

(#I

d4J)

?2m

2e

6Ga

2Ga

18Al

BIP

2f

24As

251

mm

214

Superlattices

2

x

The results of the analysis are presented in Table 2. Columns 1 and 2 contain information about the arrangement of atoms over the corresponding Wyckof f positions labelled in column 3. Columns BR's 5-9 contain the indices of the (i.e. the symmetry of normal vibrational modes at the corresponding symmetry points of the BE) induced by those irreps (given in column 4) of site symmetry groups according to which the local atomic displacements x, y, and z are transformed. The notations in Table 2 correspond to those in Table 1. From Table 2 we can easily write down the normal modes at any arbitrary point of the BZ. E.g. for the (Si13(Ge)3 SL we obtain 2rl

+ 4r2

3Ml

+3M2

4x1

+5x2

+ 4x3

+5p2

+4P3

.& 5x4; + 5P,;

+6N2.

The analysis performed shows that even for the SL's belonging to the same families the nwber of phonon crystal branches with a given symmetry, as well as the contributions of the displaceof the specific atoms to the phoments with a given symmetry depend on m nons and n since varying the number of mono-

a,

Fig.

1 Centered tetragonal unit cell of (Si) (Gel3 superlattice. The atoms labele a according to Wyckoff notation have their counterparts (not labeled) in the ;;;;ered unit cell shifted by (l/2 1/2 The axes x, y, and z are directed along CllOl, CllO], and COO11 of the bulk Si (or Gel.

D BE

of

in

the

posi-

the

as

(y'y') parallel to the Cl001 and CO101 directions are 45O-rotated around the z axis with respect to the basic translation vectors the SL-layer plane; the r (El )-phoizns are allowed in (xx) and(y;)/?x'y')scattering geometries while the r5(E)-

’

the

Wyckoff

Once the symmetry of phonons at k=O known, the selection rules for the ;Trst and second-order Raman scattering could be derived. Since the phonons with a given symmetry are associated with the vibrations of a specific group of atoms,

geometries

X

l/B of lattice

the atoms the

tions.

tries

N +y ----y P

/’

layers we rearrange cell among primitive

they convey information about the sublattice formed by this group of atoms. the first-order Raman scattering theIn rl(Al)-phonons are allowed in and (zz) scattering (xx), (YY), geome-

Z

tetragonal

+ 6r5; +6M5;

4Pl 12Nl

Fig. 2 The

Vol. 9, No. 2. 799 I

LL Y

/

and Microstructures,

centered

the simple BR's for the space given in Table 1 and detersymmetry of phonons at the corresponding symmetry points of the BZ.

well

as

in

(x'x')

and

where x' and y' axes

phonons in (x2) and (yz)/(x'z) and (y'z)-scattering geometries. We found that the following phonon combinations are allowed in the tvophonon infrared absorption with polarization of the radiation being shown in parentheses: (2) r,xr,, MlxM2,

RlxR2.

Superlattices

and Microstructures,

215

Vol. 9, No. 2, 1991

In the second-order Raman spectra the allowed phonon combinations in the (xx), (yy) and (zz) scattering geometries are: CKjl*,

KjXKj

(j=1,2);

with the following also allowed phonons scattering geometries: K1xK2;

K=r,M,P combinations in (xx) and

of (YY)

K==f;M,P.

In (xy) scattering phonon combinations

geometry are:

the

allowed

PlXP2. In

(x'x') and

CKjI*, In

(x'y'l

KlxK2; 4.

KjxKj

(y'y') (j=1,21;

geometries: K=f,M,P;

frequency

geometry:

K=r,M,P.

Experimental

results

and

shift

o (cm-’ ) -

PlxP2.

disussions

Raman scattering by longitudinal LO1phonons confined in GaAs layers of (GaAsl (AlAs) SL's has been studied in 17-19, most d%ail [P-4, 27-321. It is more difficult to observe experimentally LO -phonons confined in AlAs layers un 3 er the condition of resonance excitation with excitons due to manifestation interof spatially extended (surface) face modes c331. We present results of Raman scattein AlAs ring by LO -phonons confined layers high quality in (E aAsl7CAlAs) SL with an interface wid i"h of 3.64 8 and layer thicknesses dl'20.75 R for with d =51.45 R for AlAs under an GaAs and far from resoexcitation 0 z the spectra nances with the exciton transitions. The SL's were grown by molecular beam epitaxy on n-GaAs(001) substrates. The Raman SL-period was repeated 100 times. measurements were performed at 10 K 4880 R and 5145 R lines of a cv using in a Brewster's angle reflecAr+ -laser tion scattering geometry 2(x9x9)2 and where x' [lOOI, y' CO101 and Z(X'Y')Z, 2 COOll. 3 shows a typical Raman spectra Fig. of the (GaAs)7(AlAs)l8 SL in t;;oni;=quency region of Alhs optical . at the same experimental For comparison, also measured the freconditions, we zf the LO,-phonon line @JCL0 1 = quency 1 in the bulk MBE-grown ATAs. 405.8 cm it is shown by an arrow. A In Fig. 3, sharp peaks at the low freseries of line quency side of the bulk LO,-phonon can be attributed to the Guhntixed LO wi-1 h in AlAs layers phonons confined rl(A1)-symmetry (odd 1) for (x'x'l scattering geometry and with f2(B21-symmetry (even 11 for (x'y') scattering geometry. The measured frequencies of the confined and 4 are AlAs LOl-modes for 1 = 1,2,3,

First-order Ranan spectra of Fig. 3 Albs phonons confined for superlattice taken for (GaAs!j~AlA~~l; and z(x'x')z scattering confiZCX'Y gurations = 10 K andh&li = 2.409 eV.

405.0, 402.0, 397.5, arld 396.0 cm-l, respectively. These values are in a qood agreement with recent calculations for (GaAs),(A1As), Sl's on the basis of a rigid model with short-range ion and long-range interaction C34, 351. From Table 2 we can determine which groups of atoms in the primitive cell contribute to the phonons of a given symmetry. E.g., the full mechanical representation at the f-point (with rl and r2-ph onons observed experimentally in the first-order scattering processes) for the (GaAs)7(AlAs)l8 SL is 24rl(GaE; As:

f)

Al:;

As:)

+ 50f'S(Ga~~,;

+ 26r2(Gag,,; A1zY;

Al:;

AsErf).

We see that only z-components of displacements of Ga and Al atoms at epositions as well as of As atoms at fpositions contribute to the rl-mode. Forr2-mode there are also additional contributions from Ga atom at a-position and from As atom at c-position. For the mode there should be contributions fs from xy-displacements of all the atoms in the primitive cell. For (Si13(Ge)3 SL, the full mechanical representation at r-point is *rl(Si$; 6r5(Sizjf;

Gez)

+ 4r2(Sit

Gezyel.

f; Ge: '

We see that for the r, should be contributions fr;m ments of all the atoms in the cell whereas only Si and Ge and e positions contribute Since these atoms form mode. face, the f-l-mode probes the quality.

e)

+

‘ mode there x-displaceprimitive atoms &t. f to the f-lthe interinterface

216

Superlattxes

In recent first-order Raman measureultrathin ments of s strain-symmetrized subSL grown on (100) Ge (Si) (Gel3 stra 5 e, sharp peaks due to the phonons confined in Si and Ge layers were obserin r(xx)z scattering geometry ved (x!CllOl, y/l[1101, zl~cOOl1) C361. The observed spectral assignment of the lines could be performed in terms of the above theory. that we should note In conclusion, scattethe experimental study of Ranan spectra in SL's and their analysis ring spats the theory of the BR's of within up a new possibility to opens groups about SL useful information obtain a aonosingle vithin a microstructure layer.

c71

Cal

c91

Cl03

Cl11 Cl21

Cl31 - We should like to Acknowledgements much G. Abstreiter and K. very thank Walter Schottky Institut, TechniEberl, sche Universitat Munchen for giving us the experimental results on (Si)3(Gel3 SL's Raman scattering prior to publication. These data and helpful discussions stimulated this work. We are also grateKessler, Dittmer and F. R. ful to K. Institut fur Halbleiterphysik und Optik, for Technische Universitat Braunschweig The work at Illinois many discussions. Founwas supported by National Science dation under DMR 85-06674, 88-03108, and 86-12860, by JSEP and by AFOSR.

Cl43

Cl51

Cl61 Cl71

cl81

REFERENCES Cl1 c21

c31

c41 c51

[61

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Vol. 9, No. 2, 1991

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