J. Phys. Chem. 1991, 95, 10772-10776
10772
3:l composition, Bi6Mo2OI5predominates and has a MOO, tetrahedron with bond lengths of 3 X 1.747 (16) and 1 X 1.793 (16) A. At the 4:l and 38:7 compositions, the major phase is Bi3*Mo7078and its Mool tetrahedron is essentially identical to that in Bi6Mo2OI5with bond lengths of 3 X 1.752 (16) and 1 X 1.793 (16) A. At the 14:l composition, the molybdenum-stabilized sillenite structure is the major phase and its MOO, tetrahedron has bond lengths of 2 X 1.748 (16), 1 X 1.765 (32), and 1 X 1.788 (20) A. The systematic method used in the present study for interpreting the Raman spectra of molybdate species by using empirical Raman stretching frequency/bond length/bond strength relations is a very effective way of determining the coordinations and bond lengths of molybdate species from their Raman spectra. In cases where diffraction techniques fall short of providing the oxygen positions
around cations in a metal oxide system, Raman spectroscopy may be used to generate this vital structural information. Used in this way, Raman spectroscopy becomes an important complementary technique to diffraction methods in structural investigations of transition-metal oxide systems. Acknowledgment. We are indebted to D. A. Jefferson (University of Cambridge), D. A. Buttrey (University of Delaware), and J. M. Thomas (The Royal Institute) for supplying the bismuth molybdate samples. Financial support from the Texaco Philanthropic Foundation and the Sherman Fairchild Foundation is acknowledged by F.D.H. Registry No. BizO,, 1304-76-3; Moo3, 1313-27-5; (Y-B~~Mo~O!~, 13595-85-2; &Bi2Mo2O9,16229-40-6; yBi2Mo06,13565-96-3; BI,Mo2O,(,51682-19-0 Bi38M07078, 1037 15-01-1.
Site Group Analysis of Normal Modes in Semiconductor Superlattices B. H. Bairamov, R. A. Evarestov, Yu. E. Kitaev, A . F. Ioffe Physical- Technical Institute, Si.Petersburg 194021, USSR
E. Jahne,* Zentralinstitut fuer Elektronenphysik, Berlin 1086, Germany
M. Delaney, T. A. Cant: M. V. Klein, D. Levi, J. Klem,t,i and H. Morkod Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign (UIUC), 110 W. Green Street, Urbana, Illinois 61801 (Received: November 12, 1990; In Final Form: June 25, 1991)
We present a site group analysis of normal modes in semiconductor superlattices which permits us to connect by symmetry the local atomic displacements and normal vibrational modes over the entire Brillouin zone. The arrangements of atoms over the Wyckoff positions for (GaAs),(AlAs), and (Si),(Ge), superlattices oriented along [Ool] are determined for different sets of m and n. We obtain the symmetry for k # 0 phonons using the theory of the band respresentations of space groups and derive selection rules for one- and two-phonon infrared absorption and for first- and second-order Raman scattering. Raman spectra are presented for ( G ~ A S ) , ( A ~ Aand S ) ~(Si),(Ge), ~ superlattices and interpreted in terms of the elaborated theory.
1. Introduction New techniques allow the fabrication of rather complex crystals with a large number of atoms in the primitive cell. Success in growing (GaAs),(AlAs), and (Si),(Ge), superlattices (SL's) with varying primitive cell, consisting of m and n monolayers of GaAs and AlAs (Si and Ge), respectively, by molecular beam epitaxy as well as characterization by Raman scattering of these SL's has been demonstrated recently by several groups.1-20 Improvements in growth techniques and growth control give the ability to produce ultrathin high-quality SL's with period ( m n)do (do being an interatomic spacing) comparable or even less than the de Broglie wavelength of electrons. In such short-period SL's, quantum size effects with tailored band gaps can be observed. In the case of (Si),(Ge), SL's, there is a built-in asymmetrically distributed strain due to the large lattice mismatch (-4.2%), although the Si and Ge surfaces are chemically similar and bonded perfectly. Symmetrization of strain of short-period (Si),(Ge), SL's was achieved very recently by incorporation of a thin homogeneous Sil..,Ge, buffer layer of an appropriate thickness and comp~sition.'~Due to the new periodicity and folding of the
+
'Present address: Division of Physics, National Research Council, Ottawa KlaOR6, Canada. *Coordinated Science Laboratory, UIUC. Present address: Sandia National Laboratory, Albuquerque, NM 87185-5800.
Brillouin zone (BZ) the most interesting phenomenon in symmetrically strained (Si),(Ge), SL's is the band-gap conversion ~
~~~~~
(1) Klein, M. V. IEEE J . Quantum Electron. 1986, QE-22, 1760. (2) Jusserand, B.; Cardona, M. Light Scattering in Solids; Cardona, M., Giintherodt, G., Eds.; Springer Verlag: Heidelberg, 1989; Vol. V, p 49. (3) Bairamov, B. H.; Evarestov, R. A.; Ipatova, I. P.; Kitaev, Yu. E.; Maslov, A. Yu.; Delaney, M.; Gant, T. A.; Klein, M. V.; Levi, D.; Klem, J.; Morkoc, H. Proceedings of the Fourth Interational Conference on Superlattices, Microstructures and Microdevices, Trieste, 1988. Superlattices Microstruct. 1989, 6 , 227. (4) Wang, Z. P.; Han, H. X.;Li, G. H.; Jiang, D. S.;Ploog, K. Phys. Rev. 1988, 838, 8483. (5) Fasol, G.; Tanaka, M.; Sakaki, H.; Hirokoshi, Y. Phys. Rev. 1988,838, 6056. (6) Brigger, H.; Abstreiter, G.;Jorke, H.; Herzog, H. J.; Kasper, E. Phys. Rev. 1986,833, 5928. (7) Dharma-wardana, M. W. C.; Lockwood, D. J.; Baribeau, J. M.; Houghton, D. C. Phys. Rev. 1986, 834, 3034. (8) Ospelt, M.; Bacsa, W.; Henz, J.; Mader, K. A,; Kanel, H. von. Superlattices Microsrruct. 1988, 4 , 717. (9) Menhdez, J.; Pinczuk, A.; Bevk, J.; Mannaerts, J. J . Vac. Sri. Technol. 1988, 86, 1306. (IO) Alonso, M. I.; Cardona, M.; Kanellis, G Solid State Commun. 1989, 69, 479; Corrigendum 1989, 69, 70(7). (11) Fasolino, A.; Molinari, E. J. Phys. (Paris) 1987, C5, 569. (12) Alonso, M. A.; Cerdeira, F.; Niles, D.; Cardona, M. ADPI. .. Phys. . Lett. 1989, 55, 41 1. (1 3) Kasper, E.; Kibbel, H.; Jorke, H.; Briigger, H.; Fries, E.; Abstreiter, G . Phys. Rev. 1988, 838, 3599.
0022-3654/91/2095-10772%02.50/0 0 1991 American Chemical Society
Semiconductor Superlattices
The Journal of Physical Chemistry, Vol. 95, No. 26, 1991 10773
from an indirect-energy-gap material (such as Si and Ge) to a TABLE I quasi-direct material with a rather small gap. This makes the (Si),(Ge), SL’s interesting for new optoelectronic devices. 1 1 1 1 During the past years, Raman scattering techniques have proved 1 -1 -1 2 to be an excellent tool for studying electronic and lattice dynamical -1 1 -1 3 properties of semiconducting SL’s and for obtaining detailed -1 -1 1 4 information on crystalline quality and strain field distribution. (000) (no) (iio) Recently we performed an analysis of experimental data of first 1 1 1 and second-order Raman scattering in (GaAs),(AlAs), [OOl] 1 1 1 1 SL’s.I8 We studied the symmetry properties of the SL‘s and found 1 -1 -1 2 that the arrangement of atoms over the Wyckoff positions in the 1 -1 -1 2 primitive cell is determined by the specific values of m and n. To 1 1 1 1 obtain the symmetry of the SL phonons involved in the scattering -2 0 0 3.4 process and the corresponding selection rules we used the theory of the band representations (BR’s) of space g r o ~ p s . ~ ~ ’ ~ - ~ ~the set K. The set K includes all nonequivalent symmetry points The idea that the symmetry of SL’s depends on m and n was of the BZ and a representative point from each those nonequivalent applied to the (Si),(Ge), [OOl] SL’s in ref 10 in which the exsymmetry lines and symmetry planes which have no symmetry istence of five different types of Si-Ge SL’s with symmetry Did, points. At other arbitrary points of the BZ, a BR is defined D;dr Dih, D;:, and D S was established. The atomic arrangement unambiguously by compatibility relations. over the Wyckoff positions for several SL types was determined To simplify the analysis of all the possible types of BR’s for and the symmetry of phonons a t k = 0 was obtained with wella given space group, the concept of simple BR may be introduced. known factor group analysis.21 All the simple BR’s may be induced from the IR’s of the site In this paper we apply the theory of the B R s of space groups symmetry groups M , of relatively small number of 4 points in the to (Si),(Ge), SL’s in order to determine the phonon symmetry Wigner-Seitz unit cell of a direct lattice belonging to the set Q. at k # 0. We find the arrangements of atoms for these SL‘s and The set Q includes all the nonequivalent points of the Wigner-Seitz derive the selection rules for k # 0 phonons. It turns out that unit cell and one representative point from all those nonequivalent these SL‘s reveal many similarities with the (GaAs),(AlAs), ones. symmetry axes and symmetry planes which have no symmetry points. The B R s induced by the IR’s of site symmetry groups of 4 points which do not belong to the set Q are composite and 2. Theory of Band Representations of Space Groups and Its may be obtained as a direct sum of the simple ones. Application to Lattice Dynamics of SL’s The procedure of generation of simple BR’s is the following. To analyze the symmetry of phonons in SL‘s the method of BRs The characters of the representation of the space group F induced of space groups was used.22.23For complex crystals with a large by the IR 8 of the site symmetry group Mq for a symmetry element number of atoms in the primitive cell this method proved to be (gla) of the small group Fkarezz much more efficient than the well-known factor group ap“I proaches.zl It is especially efficient for crystal families belonging xpl(dn)= exp(ikiiJJ,)ii‘”(g,-’ggJ) to the same space group but having different atomic arrangements J‘ 1 over the Wyckoff positions, since the generation of B R s does not involve information about the distribution of atoms in the primitive where cell over the symmetry positions. SL‘s are an ideal model system %[”(g,-’ggJ) = O? gJ-’ggJ# (1) to demonstrate the advantages of the method of the B R s since this method enables the generation of BR’s for the whole SL = xB(m), g;lggJ = m family. In terms of the group theory, the BR’s of a space group F a r e and iiJJ= ( d a ) l - qJ. the representations of F induced by the irreducible representations Here, 4, are t e points of the set Q, m are the symmetry ele(IRs) of its site symmetry subgroups M , C F. In terms of lattice ments of the site symmetry group M,, xo(m) are the characters dynamics they correspond to normal vibrational modes induced of the IR for m, gJ are the coset representatives in the decomby the local atomic displaccments. position of the space group F with respect to its subgroup M,, n, In the reciprocal space (k basis), the BR can be completely is the number of the coset representatives in this decomposition specified by the I R s of the space group F (labeled as the small and @a) are the elements of the small group Fk.-Decomposition IR’s of the little group Fi) only in points of the BZ belonging to of the induced representation into the IR’s for k C K gives the index of the BR in k basis. As an example we consider the generation of simple BR’s for (14) Cerdeira, F.; Alonso, M. I.; Niles, D.; Garriga, M.; Cardona, M. Phys. the space group D;& The set Q includes four points a, b, c, and Reu. 1989, 840, 1361. d. The set K includes five points r, M , X,P,and N . The set of (15) Friess, E.; Eberl, K.; Menczigar, U.; Abstreiter, G. Solid State simple BR’s could be obtained by induction from the IR’s of the Commun. 1990, 73, 203. (16) Dettmer, K.; Kessler, F. R. Private communication. site symmetry groups of points a ( M , = DZd),b ( M b = DZd),c (17) Kitaev, Yu. E.; Evarestov, R. A. Fiz. Tuerd. Tela. 1988, 30, 2970; (Mc = D2d)9 d (Md = D2d). Sou. Phys. Solid State 1989, 30, 17 12. The characters of the induced representations can be calculated (18) Bairamov, B. H.; Gant, T. A.; Delaney, M.; Kitaev, Yu.E.; Klein, according to (1). M. V.; Levi, D.; Morkoc, H.; Evarestov, R. A. Zh. Eksp. Teor. Fiz. 1989,68, 2200; Sou. Phys. JETP. 1989, 68, 1271. Let us consider the BR’s induced by the IR’s of the site sym(19) Bairamov, B. H.; Gant, T. A.; Delaney, M.; Kitaev, Yu.E.; Klein, metry group Mb = DU of lattice point b(OOI/,), with coordinates M. V.; Levi, D.; Morkoc, H.; Evarestov, R. A. Pis’ma Zh. Eksp. Teor, Fir. given in units of the translation vectors of the centered tetragonal 1989, 50, 32; JETP Lett. 1989, 50, 37. unit cell. We define the labels of the BR’s in the X point of the (20) Sapriel, J.; Michel, J. C.; Toledano, J. C.; Vacher, R.; Kervarec, J.; Regreny, A. Phys. Rev. 1983, 828, 2007. BZ, X(OO’/z) (Fx = D2) with coordinates given in units of the (21) Rousseau, D. L.; Bauman, R. P.; Porto, S. P. J . Raman Spectrosc. basic translation vectors of the reciprocal lattice. By doing that, 1981, I O , 253. we obtain Table I as intermediate result. In the upper part of (22) Evarestov, R. A.; Smirnov, V. P. Metody teorii grupp u kuantouoi Table I, the characters of small IR’s XI 2,3,4 are givenz4for the khimii tuerdogo tela (Group Theory Methods in Solid State Quantum
c
Chemistry) (in Russian); Izd. LGU: Leningrad, 1987. (23) Kovalev, 0. V. Neprivodimiye i indutsirouanniye predstauleniya i kopredstauleniya fedorouskikh grupp (Irreducible and Induced Representations and Corepresentations of Fedorou Groups) (in Russian); Nauka: Moscow, 1986.
(24) Miller, S. C.; Love, W. F. Tables of Irreducible Representations of Space Groups and Co-Representations of Magnetic Space Groups; Pruett: Boulder, CO, 1968.
10774 The Journal of Physical Chemistry, Vol. 95, No. 26, 1991
Bairamov et al.
TABLE II: The Simple BR's of the Space Group Gd(14n12)~
r
(000) 42m
05 14m2 a(O 0 0) b(O0 1/2) ~ ( 1/2 0 1/4)
X
P
(00 1/2) 222
(1/4 114 4 1/41
a,b,c,d
a,b
a,
1
b2
2 3 4 5
1 2 3 4 5
a2 d(01/23/4) 42m
M ( i / 2 42m 1/2 1/21
b,
e
c,d 2 1 4 3 5
a,b
1 2 2 1 3,4
c,d 3 4 4 3 1,2
a
b
C
1 2 1 2 3,4
2 1 2 1 3,4
3 4 3 4 1,2
N (01/20) m d 4 3 4 3 1,2
a, b, c , d 1 1 2 2 1,2
"he labeling of the space group IRs corresponds to that by Miller and Love?4 the site symmetry group I R s are labeled according to Bradley and Cracknell;zsthe choice of origin and the setting of the space group as well as the Wyckoff position notations are that as given in ref 26. TABLE 111: Similarity of Phonoa Symmetry in (GaAs),(AIAs),8 and (Si),(&), Superlattices
GaAs/AIAs
D92d
Si/Ge
14m2
r
M
(000)
( i / 2 1/2 1/2) 42m
42m
X (001/21 222
P
N
(1/4 114 114) 4
(0 1/20) m
2
2
2
1
1, 2
3, 4
3, 4
1, 1
5
1, 2
1, 2
1, 2
5
1, 2
1, 2
1, 2
IC
1 As
1Ge
(0 1/2 1/4) 42m 2e
symmetry elements (E, C2, U,,, UXy) of the small group FP Since Mb = DZd,in the decomposition of the space group F = D;d with respect to its subgroup Mb the only coset representative is the element g = E . In this case, all the elements of the small group Fx are contained in the site symmetry group Mb. Hence, for any element of the small group, the character of the induced representation equals the character of the IR 0 of the site symmetry group Dzd for the small element multiplied by an exponential factor. To obtain the exponential factor, we calculate the vectors ajj
= (da)qb- 4 6
(g
c D2d)
and write them in the corresponding line in Table I. (Note that the coordinates of ajj are given in units of the primitive cell translation vectors.) Tie next line in Table I contains the exponential factors exp(ikajj) for a given wave vector. Using standard tables of IR's of point groups2Sand taking the characters of those elements of site symmetry group h f b = DZd which are contained in the small group Fx = D2, we calculate the characters of the representations induced by the corresponding IR's of Mb (last five lines in Table I). The first four representations are the irreducible ones whereas the last is the reducible one. Comparing them with the small I R s of the small group Fx,given in the upper part of Table I, we find their indexes and put them down in the last column. The same procedure should be implemented also for the other points of the set Q. The complete results are presented in Table 11. In Table 11, the first column contains the notations of the Wyckoff positions together with their coordinates and corresponding site symmetry groups. Columns 3-7 contain the indexes of the simple B R s induced by the corresponding I R s of the site symmetry groups given in column 2. The corresponding symmetry ( 2 5 ) Bradley, C. T.; Cracknell, A. P. The Mathematical Theory of Symmetry in Solids; Oxford University Press: London, 1972. (26) Hahn, T., Ed. International Tables for Crystallography, Vol. A , Space Group Symmetry; D. Reidel: Dordrecht, 1983.
points together with their coordinates and small groups are given in the heading of columns 3-7. 3. Symmetry Analysis of Superlattices In SL's the new periodicity has consequences both in a reduction of point group symmetry with respect to constituent bulk materials and in an increase of the number of atoms per primitive cell leading to a new space group symmetry. (GaAs),(Ah), SL's grown along the [Ool] direction constitute two single-crystal families with space groups D:d ( m + n = 2k) and D& ( m n = 2k + 1) depending upon even or odd total number of monolayers ( m + n) per primitive cell.20 The (Si),(Ge), [Ool] SL's constitute five singlecrystal families with symmetries D:d (m+ n = 4k; m, n odd); D9 (m + n = 4k + 2; m , n odd); D$h( m + n = 4k; m, n even); D2h 2g ( m + n = 4k + 2; m, n even), and Dii (m+ n = 2k +1).I0 In ref 17 we have pointed out that not only the space group symmetry but also the atomic arrangement over the Wyckoff positions in the primitive cell for the SL's are governed by the specific values of m and n for each SL family representing 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. The atomic arrangements for some (GaAs),(AlAs), and (Si),(Ge), SL's belonging to the families with the same space group reveal many similarities. The same is true for their phonon spectra. As an example, the atomic arrangements for (GaA S ) ~ ( A ~ A Sand ) , ~(Si)3(Ge)3SL's are given in Table 111. The crystal structure and the BZ for the latter are shown in Figure 1. The atomic arrangements for the other cases are given elsewhere. ' Having established the atomic arrangement over the Wyckoff positions we used the simple BR's for the space group Dzd given in Table I1 and determined the symmetry of phonons at corresponding symmetry points of the BZ. The use of the BR's of the space group Did allows us to obtain a general solution for any SL with symmetry D& without constructing the full mechanical
+
The Journal of Physical Chemistry, Vol. 95, No. 26, 1991
Semiconductor Superlattices
‘k X
Ge (IC)
I a3
Si ( 2 f ) Si (la)
~
Si ( 2 f l I
10775
second-order Raman scattering can be derived. Since phonons with a given symmetry are associated with vibrations of a specific group of atoms, they convey information about the sublattice formed by this group of atoms. It is obvious that the analysis of phonons from k # 0 points of the BZ whose combinations can appear in the two-phonon infrared and second-order Raman spectra increases significantly the number of possible variants of the atomic groups allowing us to obtain broad information about the SL microstructure. In the first-order Raman scattering the I’,(Al) phonons are allowed in ( x x ) , b y ) , and ( z z ) scattering geometries (in the parentheses the incident and scattered light polarizations are shown) as well as in (x’x’) and by’) geometries, where x’and y’axes parallel to the [ l o o ] and [OIO]directions are 45’ rotated around the z axis with respect to the basic translation vectors in the SL-layer plane; the r 2 ( B 2 )phonons are allowed in ( x x ) and by)/(x’y’) scattering geometries while the r5(E) phonons are allowed in ( x z ) and b z ) / ( x ’ z ) and b’z)scattering geometries. Using the well-known selection procedure we find that (with polarization of the radiation being shown in parentheses) the following phonon combinations are allowed in the two-phonon infrared absorption spectra: (2)
rl x r2, M , x M,,
P, x p2
In the second-order Raman scattering spectra in the ( x x ) , b y ) , and (zz) scattering geometries the following phonon combinations are allowed: [KjI2, K, X Kj (j = 1 , 2 ) ; K =
r, M , P
In the ( x x ) and b y ) scattering geometries the following combinations of phonons are allowed likewise:
K~ x K ~ ;K =
r, M , P
In (xy) scattering geometry the allowed phonon combinations are PI x p2
in (x’x’) and b’y’)geometries [Kj12, KjX Kj (j = 1 , 2); K = I?, M , P
X
Figure 1. Centered tetragonal unit cell of (Si)3(Ge)3superlattice and the 1/8 of the BZ of the centered tetragonal lattice. The atoms labeled according to Wyckoff notation have their counterparts (not labeled) in the centered unit cell shifted by ( I / * The axes x , y . and I are directed along [ l IO], [lie], and [OOI]of the bulk Si (or Ge).
representation for each S L all over again. The results of the analysis are presented in Table 111. Similar results were obtained for the SL’s from the other crystal families. In Table 111, columns 1 and 2 contain information about the arrangement of atoms over the corresponding Wyckoff positions given in column 3 . Columns 5-9 contain the indexes of the B R s (Le., the symmetry of normal vibrational modes at the corresponding symmetry points of the BZ) induced by those I R s (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 111 correspond to those in Table 11. From Table 111 we can easily write down the normal modes at any arbitrary point of the BZ. For example, for the (Si),(Ge), SL we obtain 2 r , 4 r 2 + 617,; 4Pl + 5P2 + 4p3 + 5p4 3 M , 3M2 6M5; 12N1 6N2 4 x 1 5x2 + 4x3 + 5x4 The analysis performed by us shows that even for the SL’s belonging to the same crystal families the structure of the vibrational representation; i.e., the number of phonon branches with a given symmetry as well as the contributions of the displacements of the specific atoms to the phonons with a given symmetry depend on m and n, since by varying the number of monolayers we rearrange the atoms in the primitive cell among the Wyckoff positions. When the symmetry of phonons at k # 0 is known, selection rules for one- and two-phonon absorption as well as first- and
+
+
+
+
+
P I X P2
and in (x,y’) geometry
K, X K2; K = r, M , P 4. Experimental Results and Discussion
Raman scattering by LOI phonons confirmed in GaAs layers of (GaAs),(Ak), SL’s has been studied exten~ively.’~J’-’~~~~-~~ The LO, notation refers to confined longitudinal optical (LO) modes with I half-wavelengths (I confinement number). Experimentally it is more difficult to observe LOI phonons confined in AlAs layers under the condition of resonance excitation with excitons due to manifestation of spatially extended (surface) interface modes.33 Furthermore, there are also disorder-induced first-order Raman lines which usually do not appear active in the Raman spectra of high-quality SL’s. To assign Raman lines clearly to confined modes in AlAs high structural quality SL’s are required. In this paper, we present new results of Raman scattering by LO, phonons confined in AlAs layers in a (G~AS),(AIAS),~ SL with layer thicknesses d , = 20.15 A for GaAs and d2 = 51.45 A for AlAs and with an interface width of 3.64 8, under excitation (27) Sood, A. K.; Menendez, J.; Cardona, M.; Ploog, K. Phys. Reo. Left. 1985, 54, 21 11; 1986, 56, 1753. (28) Jusserand, B.; Paquet, D.;Regreny, A. Superlaftices Microstruct. 19115. -. - -, -1., 62. -_ . (29) Ishibashi, A.; Itabashi, M.; Mor, Y.; Kaneko, Y.; Kawado, S.; Watanabe, N. Phys. Reu. 1986, 8 3 3 , 2887. (30) Maciel, A. C.; Cruz, L. C. C.; Ryan, J. F. J . Phys. 1987, C20, 3041. (31) Arora, A.; Ramadas, A. K. Phys. Reo. 1987, 836, 1021. (32) Chen, Y.; Jin, Y.; Zhu, X.; Zhang, S. L. Proceedings of the Third International Conference on Phonon Physics; Hunklinger, S . , Ludwig, W., Weiss, G., Eds.; World Scientific: Singapore, 1989, p 761. (33) Pokatilov, E. P.; Beril, S. I. Phys. Sfarus Solidi 8 1983, 118, 567.
10776 The Journal of Physical Chemistry, Vol. 95, No. 26, 1991
-3
scattering processes) for the (GaAs),(AlAs),, SL is
I.”
24r,(Ga:; Al:; As?) + 26r2(Ga&; Al:; As:,,) + 5Or,(Ga$; Aly; As:$) (2)
s
x
r
2c“ 0 5 c C
E“
0
E
Bairamov et al.
O
340
360
380
400
420
frequency shift w [cm-’)-
Figure 2. First-order Raman scattering spectra by LO, phonons confined in AlAs layer in (GaAs)7(AIAs),8superlattice for z(x’y’)Z (I?, phonons are active, odd I ) and z(x’x’)z scattering geometries (I’, phonons, even I). The x’, y’, and z axes are parallel to the [IOO], [OIO], and [OOl] directions of the bulk Si(Ge). T = 10 K;hw, = 2.409 eV.
of the spectra far from resonances with the exciton transitions. The single-crystal SL’s were grown by molecular beam epitaxy on the n-GaAs (001) substrates. The SL period was repeated 100 times. Raman measurements were performed at 10 K using 4880and 5145-A lines of a C W Ar+ laser in a Brewster’s angle reflection scattering geometry z(x’x’)z and z(xly’)r, where x’ll [loo], y ’ l ) [OlO], and z 11 [OOl]. Figure 2 shows Raman spectra of the (GaAs),(A1As),8 SL in the frequency region of AlAs optical phonons. For comparison, under the same experimental conditions, we also measured the frequency of the LOo phonon line (wLo0= 405.8 cm-I) in the bulk MBE-grown AlAs. It is shown in Figure 2 by an arrow. A series of sharp peaks at the low-frequency side of the bulk LOo phonon line can be attributed to the quantized LOIphonons confined in AlAs layers with r I ( A l )symmetry (odd r) for (x’x’) scattering geometry and with r2(B2)symmetry (even I ) for (xy’) scattering geometry. The measured frequencies of the confined AlAs LO, modes for I = 1, 2, 3, and 4 are 405.0, 402.0, 397.5, and 396.0 cm-l, respectively. These values are in good agreement with recent calculations for (GaAs),(AlAs), SL‘s on the basis of a rigid ion model with short-range interaction between nearest- and next-nearest-neighbors and with long-range interaction between ions in the nearest layer^.^^^^, From Table I11 we can determine which groups of atoms in the primitive cell contribute to the phonons of a given symmetry. For example, the full mechanical representation at the I’ point (with r, and r2phonons observed experimentally in the first-order (34) Yip, S. K.;Chang, Y. C. Phys. Rev. 1984, B30, 7037. (35) Chu, H.;Ren, S. F.; Chang, Y. C. Phys. Rev. 1988, 837, 10746.
From eq 2 we see that only z components of displacements of Ga and Al atoms at the e positions as well as of As atoms at f positions contribute to the r, mode. For the r2mode there are also additional contributions from Ga atom at the a position and from As atom a t the c position. For the rs mode there should be contributions from xy displacements of all the atoms in the primitive cell. For (Si)3(Ge)3 SL, the full mechanical representation a t the r point is 2rI(Sif;Ge:)
+ 4r2(Si;,,; Get,,) + 6I’,(Si;?;
Ge$)
(3)
From eq 3 we see that for the r2mode there should be contributions from z displacements of all the atoms in the primitive cell whereas only Si and Ge atoms at the f and e positions contribute to the rl mode. Since these atoms form the interface, the rl mode monitors the interface quality. In recent first-order Raman measurements of a strain-symmetrized ultrathin (Si),(Ge), SL grown on (100) Ge substrate, sharp peaks due to the phonons confined in Si and Ge layers were observed in z(xx)z scattering geometry ( x 11 [ 1101, y 11 [ 1IO], z 11 [OOl]).36 The assignment of the observed spectral lines could be performed in terms of the above theory. In conclusion, we should note that the experimental study of Raman scattering spectra in SL‘s and their analysis within the theory of the B R s of space groups open up a new possibility to obtain useful information about SL microstructure within a single monolayer. Acknowledgment. We thank very much G. Abstreiter and K. Eberl of Walter Schottky Institut, Technische Universitat Munchen, for giving us the experimental results on (Si),(Ge), SL‘s Raman scattering prior to publication. These data and helpful discussions encouraged our interest and stimulated this work. We are also grateful to K. Dettmer and F. R. Kessler of Institut fur Halbleiterphysik and Optik, Technische Universitiit Braunschweig, for many discussions. The work at Illinois was supported by the National Science Foundation under DMR 8506674, 88-03108, and 86-12860, by JSEP and by AFOSR. Registry No. AIAs, 22831-42-1; GaAs, 1303-00-0; germanium, 7440-56-4; silicon, 7440-21-3. (36) Abstreiter, G.;Eberl, K. Private communication.
