TH E U N I V ER S I T Y F M I C H I G A N COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report GROUP THEORETICAL ANALYSIS OF LATTICE VIBRATIONS OF ZINC-BLENDE TYPE CRYSTALS G. Venkataraman L. A. Feldkamp J. S. King ORA Project 01358 supported by: NATIONAL SCIENC E FOUNDTATION GRANT NO. GK-1945 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1968

ACKNOWLEDGMENTS This work was supported in part by the United States National Science Foundation. One of us (G.V.*) would like to thank The University of Michigan for the award of a fellowship made possible through funds provided by the Institute of Science and Technology and the General Electric Foundation. *Visiting scientist from Bhaba Atomic Research Centre, Trombay, Bombay, India, now returned. iii

TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES ABSTRACT Chapter I. INTRODUCTION!TL. A- I'TLICATlIO(N OF GHROUl! TH'EORY TO TI-iE STUDY OF LATTICE DYNATICI IL1I. SYMMIETRY OF THE ZINC-BLENDE LATTITCT; IV. RESULTS FOR ZINC-BLENDE STRUCTURE V. SELECTION RULES IN OPTICAL SPECTRA A. Infrared Absorption B. Raman Scattering v V vi vii 1 10 45 4 5 60 66 REFERENCES iv

LIST OF TABLES Table Page I. Phase Factors Used in Constructing the Multiplier Representations for Various Symmetry Points 18 II, Irreducible Multiplier Representations of the Group G (A) 20 0 III. Irreducible Multiplier Representations of the Group G (X) 27 IV. Irreducible Multiplier Representations of the Group G) V. Irreducible Multiplier Representations of the Group G (A) 35 V. Irreducible Multiplier Representations of the Group G (Z) 32 VI. Irreducible Multiplier Representations of the Group G (W) 34 VII. Irreducible Multiplier Representations of the Group G (W) 57 o VIII. Irreducible Multiplier Representations of the Group G (S) 39 IX. Character Table for the Group Go(r) 41 X. Compatibility Tables Connecting Representations at Different Symmetry Points 43 XI. Compatibility Tables for the Representations of the Single Groups Connecting the Zinc-Blende (T2) with the Face-Centered-Cubic (O0) and the Diamond (0) Structures 44 XII. The Form of the Matrices M (qsaxs'a'X') for Points L and W. 58 XIII. Summary of Two-Phonon Infrared Absorption Selection Rules Deduced by Birman(3) 59 V

LIST OF FIGURES Figure Page 1. Cubic unit cell of the zinc-blende lattice. 11 2. Primitive cell of the zinc-blende lattice and its relation to the cubic unit cell. 12 3. Portion of the reciprocal lattice of the zinc-blende structure. 13 4. Central Brillouin zone and the symmetry points of the zone. 15 5. Irreducible prism of the Brillouin zone. 17 6. Schematic plot of the dispersion relations for the zinc-blende lattice. 42 7. Transmission of zinc sulphide. 47 vi

ABSTRACT In this report is presented a group theoretical analysis of the lattice dynamics of crystals possessing the zinc-blende structure. The form of the dynamical matrix demanded by symmetry is derived for several symmetry points in the Brillouin zone. The construction of symmetry vectors and the block diagonalization of the dynamical matrix is shown in detail. Selection rules for two-phonon infrared absorption are presented for several critical points. vii

CHAPTER I INTRODUCTION We wish to present in this report a group theoretical analysis of the lattice vibrations of the zinc-blende lattice. The results described here were obtained in connection with the experimental study of phonon dispersion relations in cubic ZnS now in progress in this laboratory.(l) They are, however, equally applicable to all diatomic crystals having the zinc-blende (or sphalerite) structure; this includes several I-VII, II-VI, and III-V compounds. The symmetry properties of the zinc-blende structure were first considered by Parmenter(2) in relation to electron band structure. Subsequently, Birman(5) classified the phonons in terms of the symmetry of the lattice for purposes of working out selection rules in optical spectra. No detailed discussion of the eigenvectors and the diagonalization of the dynamical matrix based on symmetry considerations has been presented so far for this lattice, and the purpose of this report is to fill in the vacuum.(4) We begin in the following chapter with a brief resumi of the general principles involved in application of group theory to lattice dynamics. For a more detailed discussion of this topic we refer the reader to two recently published reviews on this subject.(5,65 Chapter III is devoted to a description of the geometry of the zinc-blende lattice and its reciprocal lattice. Chapter IV forms the core of this report and contains a discussion of the eigenvectors and the dynamical matrix at several symmetry points in the Brillouin zone. The concluding chapter is devoted to selection rules for two phonon processes in optical spectra. 1

CHAPTER II APPLICATION OF GROUP THEORY TO THE STUDY OF LATTICE DYNAMICS In this chapter, we present a brief discussion cohcerning the application of group theory to the study of lattice dynamics. This is done more for the sake of completeness than as a detailed exposition of the subject, and follows closely the work of Maradudin and Vosko( 5) referred to earlier. The problem of finding the phonon frequencies and eigenvectors is essentially one of solving the eigenvalue problem D.z) (1) where %j(q) is the normal mode frequency corresponding to the phonon wave vector q and branch j, e(q,j) is the associated eigenvector, and D(q) is the dynamical matrix whose elements are given by < p (A.X) ( fM M if E(S e X (ZZ q> (2) where x(Q') denotes the position of the P'th primitive cell, MK and MK, denote the masses of the Kth and K'th atoms, respectively, in the primitive cell, and 0g ( ~ r,) is the oath element of the three-dimensional force constant matrix connecting the Kth atom in the cell at the origin and the K'th atom in the cell 1'. If n is the number of atoms in the primitive cell, then D(q) is a 3n x 3n matrix, and correspondingly there are 5n eigenvalues @?(q) (j = 1,2,...3n), and 3n associated eigenvectors each having 3n components. Arranging the eigenvectors into a 3n x 3n matrix e(q), ~i_ —- [ e et),, _~,z,,.. _ ( (,3,.j, where e(q,j) is a 3n element column vector, we see that e,) ^ P ^ e ( a>) -_ -Q- (3a) ~ = -^ ^ dt o (% ), (3 ) ). (3b) 2

3 Thus to bring the dynamical matrix into the diagonal form Q what we need to find is the transformation matrix e(q). It turns out that if the crystal has some point group symmetry (as it generally does), then for q corresponding to symmetry points or along directions of high symmetry it is possible using the techniques of group theory to find a transformation matrix S(q) which serves to brin:g D(q) ito a blok diagonal form. The matrix S(q) depends on crystalline symmetry aloner, uw..ike e(q) wh.ich depends orn the force field between the atoms. Letm = S v(S) x(m) be.one of thie elements of the space group of the crystal where S denr.iotes tt'e rotatiornai par t, x(m) a lattice translation, and v(s) a frai:ti.or.al trat:slaticcn (which is zero for symmorphic groups). Then under t'hi:s syrnmetry operation, th-e eigenvec.tor efqj) transforms as where the elements of the'5i x 3-n transformation matrix ^ are given by (x) tc)s L o -L( x ( -<^ x J ( )* (5) Here SczX is the axth eleme-it of the three-dimensional rotation matrix S, xx() and x(K?) denote the positions of the atoms K and. in the primitive cell, and, X' (vii' S X ( i.') V(S) - yX (mn)'~ (6) Further, m( i'") deroites the sablatti,-e inwto which the sublattice < is transformed by the cperatiorL$m. The matrix' is unitary.*'Under Sm D(q) is transformed into - (_5 tsr)s> )r;'t{ s.., —.(S. (7) *Njote that for symmorphic:,, grjoups, the atoms always stay i:n the same sublattice anrd:cj:onsequen-tli y I' ihas o'r; iy diag onall boxes.

Consider now those particular space group elements.m = IV(R (R) + x(m)) for which + G - Tl, T ) (8) where G is a vector of the reciprocal lattice. We find that r(q;Rm) commutes with D(q) since D(q) = D(q + G). The elements of the type R/ Torm a group G(q) called the group of the wave vector, and we note that corresponding to every element of this group there is a unitary matrix operator r which commutes with D(q). It can be shown that the set of matrices r furnish a 3n-dimensional unitary representation of G(q) which is reducible. We observe in passing that because of the commutation of D(q) with r(q;Rm) there result interrelationships between the elements of D(q). Specifically, i) ( K) ) t, r )!SSP In addition one has In addition one has D )= T) k - - r~~~c JI (9b) which in some cases leads to additional interrelations. Define now a matrix T(q;R) by T(6; 2 )= XP _. -... pA-) + x(W),-..7;.(en (10) e Like the r matrices, the T matrices also commute with D(q). Further they obey the multiplication rule _C7 (i; KB; ) T(c U) Zj k,) T h )L (1)( (where Ri and Rj are rotation matrices corresponding to the ith and jth elements of G (q7) o

5 and provide a unitary multiplier representation of the point group Go(q) made up of only the rotational parts of Pm. The multiplier O(q;RiRj) is given by _ a - p-t R(I)- * 2;)] (12) -1 G(qR. ) being defined through the equation g7 = -L IS.) (13) Clearly = = 1 for symmorphic groups always. For nonsymmorphic groups, 4 = 1 for q within the Brillouin zone, while on the surface of the zone it can be different from unity having the value given by Eq. (13). The representation (multiplier) provided by the T's is a reducible representation, and the number n(s) of times the irreducible multiplier representation (IMR) occurs in the reducible representation can be found from the formula ns> = - i X (s)(. t> Trt f $)? w(14) where h is the order of Go(q) and X( l s) ^ S - Ct (S;(15) is the character corresponding to the element R of Go(q) in the sth IMR, T(S)(q;R) being the matrix representative in the IMR s. Except in the case of accidental degeneracy (which is rare), the eigenvectors belonging to the same eigenfrequency transform according to a unitary IMR of Go(q), and consequently we can replace the branch index j by a triplet of symbols s, a, X, where s denotes the IMR, a its occurrence, and \ the partner in that representation. X takes the values l,2,...fs, where fs is the dimensionality of the IMR. We thus have ()'(^ %j; -) e ~ ~ej.s~,~) T i -(16)'6= R; (5 j6)e(; A'-A 1 The eigenvector e(q;sa\) is a unit vector in 3n-dimensional space in which the basis vectors are

6'l 0 0 0 1 0 I - o1 0 * * * I * f * - 0 0 0*- 3n basis 3n elements in each basis vector veco r vectors —-^ Using the results of group theory we can construct vectors, the so-called symmetry adapted vectors, which have the same transformation properties with respect to Go(q) as do the eigenvectors. The prescription for doing this is as follows: Form first the projection operator <U) > ( -) ~s -V c C-c,~(S Re r,\g \ (c;, jz7 (17) When this operator is applied successively to the 3n basis vectors mentioned above, it will project out only n(s) linearly independent vectors. Label them after normalization as 4(q;sl\), 4(q;s2x),...*(q;sn(s)X). All these symmetry adapted vectors transform according to the Xth row of the IMR's. The partners associated with each of these *'s may be found using the result .!7 ). (P / (V 1 (s, 5 LI), A _ s;.-ag, A,') (18) ) where( (s )(q) is obtained from (17) by replacing ((s) by T(St. This gives the required symmetry vectors. Next as with the e(q,j)'s, form a 3n x 3n matrix S(q) in which the columns are made up of the symmetry adapted vectors 4. Arrange the symmetry vectors so that all those which transform according to the same row X of the same IMR are grouped together. Then St (i ) ( 11 )`-S % — X ca) (19) will be block diagonal. In particular, (1) if an IMR occurs only once in the reducible representation T, the block of D' associated with this species will be completely diagonal no matter what the dimension of the IMR. The symmetry vectors are the required eigenvectors in this case;

7 (2) if the IMR occurs more than once, n(s) times say, then there will be fs boxes each of dimension n(s) associated with this species. The n(s) different frequencies must be obtained by diagonalizing one of the n( s)-dimensional blocks of D'(q). The eigenvectors in the case n(s) > 1 are obtained as follows: The n(s) eigenvectors e(q;sa%)(\ = l,...n(s)) all transform according to the Xth row of the IMR s. They may therefore be expressed as linear combinations of the n(s) symmetry vectors which also transform according to the same row of the same IMR, i.e., S e t e a, i i e t t c - Since the eigenvectors are orthonormal, it is evident that the complex coefficients ci must satisfy the condition E Cl a(B; 5-X) CZ (So) (21) The coefficients are obtained by feeding (20) and the eigenvalue appropriate to the mode (determined previously by diagonalizing the n(s)-dimensional block of D'(q)) into Eq. (1). This yields n(s) complex homogeneous equations in the ci's which are then solved for subject to the restraint in (21). It should be noted that the block diagonalization of D(q) and the determination of the eigenvectors could be achieved equally well through the use of the matrices r(q;Rm) and the irreducible or small representations yS(q;pm) of G(q) rather than T(q;R) and ( S)(q;R) of Go(q). In the former case, we have instead of (14), rib) = — r 7 _)^ ^'^,(^ TrLrs;^,) (22) k E t'-(r,) where' is the order of the translational group, and (zTr-C)pi-' Y(}C }b )' (25) - 7 Ccxp(-;~ p~x,', _'(-v)d (C?~, $ 23

8 Equation (22) can be easily seen to reduce to r) (s) -LE' ~ 1?rC~x iR~Y j TT- F( -b,, C) ) e p tn o ts a f te f The projection operators are formed using the formula (s) *,\ A -F *51-K;: ^R oL fI i~~~~~~~~~ 3 r. = ) F? "G, 0 Techniques for obtaining both the irreducible multiplier(7) and the small representations(8) are available. For symmorphic groups the representations are identical and in fact are nothing but the representations of the crystallographic point group to which Go(q) belongs. There exists besides the space group symmetry one other symmetry, viz., time reversal (i.e., the invariance of the equations of matrices under the transformation t - -t) which also sometimes leads to degeneracies. By this we mean a degeneracy of two frequencies not required by space group symmetry considerations, i.e., Cs0 ( ) = (I 0.1t* ol ) (' ( - i ) - (/ s/ I S Without going into details, we shall note that the criterion for additional degeneracy may be stated as follows.(5) (i) q inside the Brillouin zone: ^_____________ Compute Q = 1/h ~ x(S)(q;A2) where Aq = -q, and theAsummatlon is over there is no additional degeneracy. degeneracy of the form, A is an element such that all such elements. If Q = 1, If Q = -1 there is an additional JSC( ) C=C. K ) K') C 4 Q (24)

9 If Q = O, then also there is an additional degeneracy with (a(q) = wsga,(q), st' s. The extra degeneracy introduced in the last two cases is referred to as time reversal degeneracy. (ii) q on surface of the Brillouin zone: Compute Q = 1/h E(exp-i[q + A q].v(A)) x(S)(q;A2) where once again A is a rotational element which sends q into -q. The occurrence or nonoccurrence of extra degeneracies due to time reversal symmetry follows the same pattern as in case (i) depending on whether Q = 1, -1, or 0. (iii) q on the surface of the zone and equal to Q/2: If q = (G/2), then Q must be computed according to the formula i. p+ -R t ( Q = L jer -ll e X P and the test applied as before.

CHAPTER III SYMMETRY OF THE ZINC-BLENDE LATTICE The zinc-blende lattice can be looked upon as two interpenetrating face centered cubiclattices displaced relative to each other along the cube diagonal by an amount (a/4, a/4, a/4) where a is the cube edge, each lattice containing the same species of atoms. Figure 1 shows the arrangement of atoms in the cubic unit cell. Here the positive ions are indicated by filled circles and the negative ions by open circles. The face centered lattice formed by the positive ions is clearly evident. With a little effort, it can be seen that the negative ions also form a F-C-C lattice and that the latter is displaced along the cube diagonal with respect to the lattice of positive ions, Note that our choice of locating a positive ion at the origin is arbitrary, We might just as well have constructed the unit cell with the negative ion at the origin. Figure 2 shows the basis vectors al, a2, a3 of the direct lattice, where 9 = = ( 4 ) / 2L -L A (,) (25) i, j, k being unit vectors along the cartesian axes. Also shown in the figure is the trigonal primitive cell, which we note contains two atoms in contrast to the cubic unit cell which contains eight atoms. The coordinates of the two atoms in the primitive cell are: t ( ) = (0oV0,0) 4 (z)' (c /4. ) /4 ) 4 C )1 The reciprocal lattice, as is well known, has the body centered cubic structure; a portion of this is shown in Figure 3 along with the basis vectors bl, b2, b3, where bi = -) ( be~~~,~~~~ 2x (_S~.b(26, 2"t(L-j ). (26) eL 10

11 -y Figure 1. Cubic unit cell of the zinc-blende lattice.

12 - I:::::::::: M:.::......: ~::i: Figure 2. Primitive cell of the zinc-blende lattice and its relation to the cubic unit cell. Also shown are the basis vectors a1, a2, and, 3' I -~-~~.~ ~....'~s~':',* i::";';-;.:::~~:';:::..........~:::~.~~.....1 J ~~~~~~~~~~~:'!!/.!!to the cubic unit cell. Also shown are the basis vectors al, a2, and a3 of the direct lattice.

13 lattice of the zinc-blen.e strc lattice of the zinc-blende struc Figure 3. Portion of the reciprocal ture. The basis vectors bl, b2, and b3 are also shown

14 The first Brillouin zone is shown in Figure 4 with the symmetry points labelled essentially as in Figure 1 of Parmenter's paper. The space group G of the zinc-blende lattice is Td(F 43m), and the underlying point group Go is Td(43m). The 24 elements of the group Td are listed below and grouped into classes. The notation employed for the group elements follows the standard pattern, C denoting pure rotation, a denoting reflections, and S standing for roto-reflections. The arguments and subscripts serve to specify the operation with respect to the cubic axes. E] C3( ))C () C3' C )TC3(TI ) C3(TTi), C3TTI ), E c II), a2 (I),C 3, 3 T ~ s tTTI),1 C M-0 c I(I T O c (x) ( Ly) ( -?- C3 3 r), L ).,I I — ~~~~-I -1:~, ( x, -14 ()''), ( ) )<t (,.' ) S4 (t), ) ) s:~~~~4 (K,144-( [u~7i 7 x )T 3 -7 — zx ) x1 7;'- ZT-X I The three-dimensional matrices corresponding to the various group are given below. operations /100 E = O 1 0 \0 0 1/ C3(111) = \1 1 0\ 01 0/ _1 /0 0 (_ill) /0 0 (111) = (-1 0 0 1 \0 -1/ __3 (hz): 0 0o 1 0 00 -1 __ ) / o C3(111) = o 1 1 0 0 -1 0 -1 (ll) = C3 11 (0 \0 -1 _ /0 -1 A C3 (111) =100 -1 \1 0 0 C(x) = 0 0 0 -1 0 o -1 /-1 9(y) = o -1 S4 (x) = ( -' {'i 0 0 1 0 0 -Iy 0 -1 01 /-1 0 0\ C2(z) = O -1 0 -\ 0 1/ /0 0 -1 s4(y) = 0-1 0 0 / /-1 0 0 S4(x) = 0 o 1 \ 0 -10/

* 9uoz Tsq jo s3uTod JXacujumzs 9aqp pue Guoz uTlnoTTTJa TeJueaD' ansT. ( 0 T-q xz= (L ) I 0 = -D \- 0 0/ o I1- 0 0 0 z= ~ o- -'. 0 T0 0 T 0 I V 0/ 100 xz=. 0 0 10 z= 100 = =D \0 - T/ I- 0 0 00T \\ 10/ ~) 0I O ~ - /x= |00 T- = -D 0 1- 0/ O o xO I O (0 T 1- 0 - ) = () S T 0 0 - 0 0 T = (z)_s O/

CHAPTER IV RESULTS FOR ZINC-BLENDE STRUCTURE The general considerations of Chapter II will now be applied to the various symmetry points in the irreducible portion of the Brillouin zone illustrated in Figure 5. This prism occupies 1/48 of the volume of the first zone, and the results for points in the rest of the zone may be obtained through the application of Eqs. (4) and (7). We begin by noting that since Td is a symmorphic group, the T matrices have nonzero diagonal boxes only, and have the typical form T (-; P, r 0 )> (28a) where 9. =exp * Cyc >-I X(t)J =/,z (28b) Clearly, G1 = 1 always since l 1) = O. The values for 02 for the elements of Go(q) for q corresponding to the various symmetry points under consideration are listed in Table I. We begin the symmetry analysis with the point A. A: q = 2j/a( To, O0) 1> I>O The point group Go(q) associated with A has the elements (E I0) (C2(x) 0o) _yzlO), (_iz l0), and is isomorphous to the molecular point group C2v. The T matrices are: TJa E ) e o ( o C,(x) Ci C(X) \ V CTk J (i, 27j — ) j e _ o.a. (29) 16

*a uoz UTno-mIT aq p jo uIsTaid -TqTonp@I 5.rL jT M I'7 T' *e -.:#-~i: ~.a n9 /'*'**" jo 1'A'/.'/: ( -1 z

18 TABLE I PHASE FACTORS USED IN CONSTRUCTING THE MULTIPLIER REPRESENTATIONS FOR VARIOUS SYMMETRY POINTS Point _q' Operation R 2(2).q [x(2) - Rx( 2) ] 2 r x A A (0,, 0) -(1,0,0) a ( 1, 0,0) a a All 24 E C2(x) c2(y) C( z) S4(x) (-1 S4 (X) yz Cy( C (x) a yz (.4, -$ - )a 11 1 (-y, - )a 111. 4 4' 4 1 1 1 ($-$,-)a (4' 4'a 111,iii'iT-4) a')a 411 4 1 1 1,_ _ _)a 1 1 1 (~,~,^)a 1 4'4' 4 1 1 1l 4 4, 4)a 1 1 1 ( 4'4 )a ^^^^ /i ^ Ic' 4' (^ ^ ^W8 0 0 0 TC O Tr 0 0 0 0 0 0 0 0 0 0 0 n~ 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 E C3(111) C3(111) xy a yz xz xz 0 1

19 TABLE I (Concluded) Point q Operation Rx(2) q.[x(2) - Rx(2)] G2 C2(y) (,,-)a -1 2~ 1 1 W 0) E a 1 2=(t 111, S - lnS4(y)) ) a S4 (Y)-) 1 (-4'- 4)a 2 S4(y) -1,- )a 2 41 4 2Tt (4I,,) E )a 0 a 4 4' 0) xy 4 4 4 S2(1, n, r) (44,)a yz 4 4'4 The irreducible representations for this group and the number of times they occur in the reducible representation (T(A;R)) are listed in Table II.

20 TABLE II IRREDUCIBLE MULTIPLIER REPRESENTATIONS OF THE GROUP G (A) E C2(x) r~ a n(A.) A1 1 1 1 1 2 2 1 1 -1 -1 0 A3 1 -1 -1 1 2 A 1 -1 1 -1 2 Tr T 6 -2 2 2 - -,__,_, The dynamical matrix can be written quite generally as ID (t) D ( I I IIL;e ) -- 1 (30) where K = 1, 2 denotes positive ion and negative ion, respectively. taking q = A and R -= yz and using Eq. (9a) we obtain, Next, DX. V 4l)'D 6 Y~ (A ) a (;,6x I LyX*e IQ ) 0~~KS D a)- -? Z;~ x x t (XX 1)'(&. I.) xx?x'Dzx, (6tl), I w 1) LIX ( 6 I ) ( kt L D4 Xt e \ ) DXX') a(C< ) De a(ai 5- yfI )D y(2 a)

21 With R = a, however, we obtain = Syz Ig D D Ie ) (O -b, d -D 6l1) D (6 I I aIKI) -) 1 -D a & IKX(.,tw),e.(a ng -b u a I) D, iHb6 ) b 1i: I(V) )D y (,) From these we find: PX? T (~4) =.( )= 0 and D) a I) - )tD (A C) D f (8K x) = ( )tI ) J The remaining elements in Go(A) D(q). We may thus write it as m. do not provide any further simplification in A 0 & G0 0 0 B C H.T 0 C B 0 T H 531a) 6 00 O o H* z o' H' 7D 0 0 0 o F F E avoiding the subscripts. Here A, B, C, D, E, and F are completely real. The matrix given in (31a) must be consistent with the general requirement that

22 ^A) As noted earlier, this condition cations (see remarks after 9b). operation C_(y) takes q to -q if _- J — ->V^~ 0l) (9c) in some instances provides further simplifiLet us explore this possibility. Now the q is along A.* Therefore, A _?r-h = I ~ I i ( % Ozone 4d (see Eq. (9a)). This together with (31) above gives us D^ (r= C ACC I I, Since C is real, this means C = In a similar fashion, we can show F = 0 The element J, however, does not vanish. Noting the additional simplifications deduced above, we may write (31a) as _ (_ 0 0 B o o 0 G* o o 14~~ T 0 a I* U0 J 1* 6- -O O O H T O HT 0 3- H 1D 0) 0 0 CE (3lb) *In general, the requirement is that Sq be equal to -q + G. = AI

Our task now is to block diagonalize the matrix above and for this purpose we need S(q). Let us first construct the projection operators(^l, PA3, and IA4. Using Eq. (17), the matrices in (27) and (29) and Table II, we get 4 0 O 00 0 C O 0 0 I 4 o 40 o O (.) o ('30 (32) 0 0 0 2.z-Z.0-2 Z 0 _4 4.00 0 0 2L-Z,0-2- z (33) 0 00 0 2-2 0 A4 I 4 0 0 0 C) Z Z C) 27 2 (34)

24 Using these projection matrices, we obtain the following symmetry vectors. 0 0 0 (') I 0 0 0 0 O 0 0O 0 0 0 3;a -1/ O C) 0 0 A4: O 0 0 0 0 0 0'o L (55) From these we find the transformation matrix S(~) to be, / 000 00 o o I/oz O 0 o'/fI 0 I/~L 0 o o -/ C 1/f 0 S () 0 / 00 0 o 0 00 o'/2 0 o rz o - //r 0'/ L 3 6 4 (36) and the block diagonal form D'(q) of the dynamical matrix to be A GTh(6) 0 *H-T 0 H R~, f: O (37)

25 By diagonalizing successively the three 2 x 2 blocks in (37) above, we may obtain w2(A;Al1), a2(A;A12), etc. Thus (^i^_ = aI + __ _ -4(I (a - -4(3) K- a) co) (dAL ou) ~ aI -j - 4 r2.-h I ) (58a) where for A1, (38b) O(,= A C>o2-2 - 17) ) Xcz - G ) for A3, (38c) 0( E r'<I and for A4, % = B, z - E.,, 2.-= - (38d) The index a takes on the values 1 and 2 corresponding to the two occurrences of each of the representations Al, A3, and A4. The eigenvectors in each case can be written as ~A,~~~ (&%)t + -B3L,(a)2(/\, ), (39) A;(a,'f'(' z^l) + B^ ^zf;. (;5 az Substituting this and the corresponding frequency into Eq. (1), we have IA / () 4 12 i (ti ) (= lt, a ) A)} (&) 2 - f t & (2 =; (I * t Lc) g ) FL. (40a)

26 Also, fFIA alZ ( )| B. (X)| = I (40b) + B/)( = I Solving these for Ai(a) and Bi(a), {it IL (2, - l; )'- 4,(I (41) {K~ r+ -i~'-'5<' 2. t Finally we test for possible extra degeneracies due to time reversal symmetry. Since A is within the zone, we apply the criteria of (i), p. 8. The elements A which send q to -q are C2(z), C2(y), S'l(X), and S4(X). Therefore Q = Xd ( A ) - { k \ 1 + )-Z (EC) +: (ci,(x)) (x )) = 1 for A1 = 0 for A3 = 0 for A4. Thus A3 and A4 are related by time reversal symmetry,(2) and as is to be expected, both occur in (T) the same number of times. By examining the symmetry vectors, we note that thie Al modes are longitudinal while A3 and A4 are transverse. X: q = 2i/a(1,0,0) The point group Go(q) is isomorphous to the molecular point group Vd and has the elements (Eo0], a(K) o)], [C2(y)10] (a(z) lo, {S4(x) O) (S4 (x)J0), ( 0yzl {(a-zO). The T matrices may be written down readily using Table I and the three-dimensional matrices in (27). The irreducible representations and the number of times each occurs in the reducible representation are given in Table III. The dynamical matrix after simplifications has the form

TABLE III IRREDUCIBLE MULTIPLIER REPRESENTATIONS OF THE GROUP G (X) E C2(z) C2(y) C2(x) S4(X) S (x) a n(X..... y yz 1 X 1 1 1 1 1 1 1 1 X2 1 1 1 1 -1 -1 -1 -10 X3 1 -1 -1 1 -1 -1 11 1 x4 1 -1 -1 1 1 1 -1 -1 X jrll ro- oi rl,3 r-1 r0 -1 ll ~ll r~ 2 X5 [lo1 oi] LoJ] L-] [;O-1JJ oJ ~J] IIJ 2 Tr T 6 0 0 -2 0 0 2 2 T-) -1

28 q 00 o B oo 0 0.0 O 0E o e'o X)(~ C 0 0 oo C o E* OD E o I o 0 0 7L> (42) while the matrix S(X) constructed from the symmetry vectors has the form. X< X3 XS S(x) 0 0 0 O> 0 C) 0 0 O 0000 O OC 0 0 00 / oo C) 00 0 I oo (43) From these two matrices, we obtain the block diagonal form as.c O5.0 0 ~5~ 0.... D(X) — Z I (D 0 O 0 O O BE O " 4 X,,. 0 O I B E 0 I. t) I -j (44)

29 The frequencies for X1 and X3'are: Co) XX) =X - C A)( X; X.) A (45) The two frequencies for X5 are given by (58a) with (46) The eigenvectors are: O 0 C) 0 C) 0 O I 0 C) 0 C) 0 I (47) L Those for YX are given by Eq. (41) with the definitions of xll, etc., as in (46). Observe that in both X1 and X3, which occur only once, one species of atoms is stationary. This is consistent with the general observation of Elliott and Thorpe(9) that "if a representation occurs only once, then only one species is involved in that mode." Also note that both X1 and X3 refer to longitudinal modes. The "optic" or the "acoustic" natureof the mode depends upon the relative values of C and A. The modes belonging to Xs involve motions of both atoms, and are transverse. Time reversal does not introduce any additional degeneracies. A: q = T/a(r, rl,) 1 > r > 0 Go(q) is isomorphous to the point group C3v, and its irreducible representations are given in Table IV. Also presented in the table are the

occurrences of the irreducible representations in the reducible representation formed by the T matrices. TABLE IV IRREDUCIBLE MULTIPLIER REPRESENTATIONS OF THE GROUP G (A) E C3(111)ll) c() an(A.) 3 3 ____ ______ ___xy yz xz A 1 1 1 1 1 1 2 A2 1 1 1 -1 -1 -1 _ 0) ( t( 1 (- +0) ( /-( ) 2.___. 2 2 2 2 22 22 Tr T 6 0 0 2 2 The dynamical matrix for this point can be written as +D (/I A 3 B E F B 13 B F E SB A F F E+ F* F' C m IF* E* F"~ T' C:F* F * E+D * F E C (48)

31 while the S matrix has the form S(A) 1/3 0 oI4e /3 0 o'/r| 0l/S 0 1/43 /FG o I/lr o'/ k0 ofi 0 l/o,/f6 D 1/ C) //f6'/ O -S~ 0 c I — A~ )r - I (49) Upon transforming D(A) using S(A), we obtain the block diagonal form D'(A) as ~ _*+2F E +2F c - 2-D 0 0 I I(A) A-B E-F 0 0^ c 0 C-) *_'I c -:~ ( C 0 A-B (50) The eigenfrequencies and eigenvectors resulting from these 2 x 2 blocks may as before be obtained from Eqs. (38a) and (41), respectively, with

32 < 22 - (z12 E 4 2- P (51) for A1, and 2zz C -D (52) for A3. Timereversal does not introduce any additional degeneracies. L: q = T/a(l 1,,1) The results for the point L which is on the zone boundary in the (111) direction are identical with those deduced above for A, a general point in the same direction. This follows from the fact that Go() is the same for bothpoints Z: q = 2a/a(l,T,O) 0 < < 0.5 The group G0(q) for this point has just two elements, E and C2(y). The T matrices are readily written down using the rotation matrices in (27) and the phase factors in Table I.. The character table and the break-up of the reducible representation are shown in Table V. TABLE V IRREDUCIBLE MULTIPLIER REPRESENTATIONS OF THE GROUP Go(Z) E C.(y) Tn(zi) Z1 1 1 3 ZT 1 -1 3 Tr T 6 0 I I

The dynamical matrix assumes the form _ (Z)..... B O o C) CL 0C) O I\ I 0 0 0 0 C) C, 0 0 F ( 5~)~ where K* -'iK, = -e -e " L M = ei rl N* = e N. K, L* = L IAMC N* Oi~rN.For S(Z) we h-J.ave _(z) o 0C i (.) () o o / O _ C ~ C) I i o.... # / 0 0 C 0 / o 0 0 0 0 I ( C) 0 C ( C z i /0 o i (54) The resul t.i.ng block diagonal form is M' L M M 0: 0 D'(z) = _rs 0 A ok (55)

The two 5 x 3 matrices above must be diagonalized to obtain the three frequencies belonging to Z1 and the three frequencies belonging to Z2. Time reversal does not introduce any extra degeneracies. W: q = 2n/a(1,1/2,0) The point W has higher symmetry than Z, there being four elements in Go(q), which is isomorphous to S4. The character table and the reduction of the T matrices are given in Table VI. TABLE VI IRREDUCIBLE MULTIPLIER REPRESENTATIONS OF THE GROUP Go(W) ____ E S4(y) S'(y) Cp(y) n(wi) JE S4( SdlY(y) 41 (Y) C2 W1 1 1 1 1 1 W2 1 -1 - 1 2 W3 1 i -i -1 2 W4 1 -i i -1 1 Tr T 6 (-l+i) (-l-i) 0 Following the usual procedure, we find the dynamical matrix D(W) to be \A/) A Oo o oA o t.* o OLO D0 L 0 \S> L o 0 I< L 0 o0 D 0 O0 0 Li C) 0 0 0 (56) with K* = -iK, L* -iL.

35 The S(W) matrix has the form S(W) 0 0 0 0o C) -L-k C) /0 0 O0 C O C/.0 LA,/2 o / 000 -LI/ 0 L/Iy I o0 0 / 0 L 0 0 0 w, w\ \W W4 w (57) The resulting block diagonal matrix is 0 O O (\w) B L O 0 0 TaL -1 -.l I~...,, o o:,r.e.o............ - — 3 0 0 A (58) The eigenfrequencies for the modes W1 and W4 are given simply as (cz w,/1) = -f W4 ) - A (59)

36 Those for W2 and W3 are found by Eq. (38a), i.e., for W2 we take solving the appropriate 2 x 2 blocks as in B and - L (60a) while for W3 we take 22= E cr = -4 k and (60b) Regarding extra degeneracies due to W3 and W4 are degenerate. However, shows that for W3, =Q X C ( time reversal, application of Parmenter(2) remarks that criterion (ii) on p. 9 P-2 —) ) -+ X (C2(X)) + C 3..) +- 3 ( Q' \ ) w8 4~ r 4 W (E-) I.a

37 Thus W3 cannot be related by time reversal degeneracy to any other representation. The same can be shown to be true for W4. It is worth noting that if W3 and W4 were degenerate, we would expect them to occur the same number of times in the reducible representation which, as seen in Table VI, they do not. Z: q = 2T/a(T,T,0). 0 < r< < 0.75 The point group Go(q) is very simple having just two elements (EJO) and (a 10), and is isomorphous to the molecular point group C. The irreducible representations and their occurrences in the reducible representation furnished by T matrices are given in Table VII. TABLE VII IRREDUCIBLE MULTIPLIER REPRESENTATIONS OF THE GROUP Go(S) E ca n(i) E1 1 1 4 EZ 1 -1 2 Tr T 6 2 = I.....! The dynamical matrix after simplification has the form J> (s: ) C a t D — D - B oM' ML * w L J~ 1* m L T L E-tJ L M W k G- - E H AH F (61)

38 where J* = J +it L* Le +i2 M* = -Ne+i2 * =+i2 Nd K =Ke i2.T The matrix S which serves to bring D(Z) above into block diagonal form is found to be E cz: ) J/ o o 0 r// o /fj0 C) 0 o -O/t o o 0 o 000 o0 0 O - t o/o o -l 000 I00;-,1 ^^ (62) The block diagonal matrix is f 2- ( 4 +C) -F2. (T+ L ) D/) = -r (e+ L) — iL Ms~ Ir( I> - <l4t~b 4 N) -~ii F P 0 0 0 0 0 0 0 0 L _ - ~ ~ ~ ~ ~ ~ ~ ~ ~~ — 0 0 0 0 0 0 0 0 O O -z(4- c) - (Z L*) iz(U -L) r7- C- &2j 9 4F- z (63) The frequencies and eigenvectors Z2) may be determined using Eqs. for the 2 x 2 block (corresponding to species (38a) and (41) with,o<1/ = -. (A - C) ) 02^ -'" (e-6G) 7 o(z = (i (T-L). )2 (64)

59 The solutions for the 4 x 4 block are too lengthy to be given here. They may, however, be found easily for any given problem through the use of a computer. K: q = 2T/a(0,75,0,75,0) The results deduced above for E are applicable to the point K as well and indeed beyond K up to the point 27/a(1,l,0). S: q = 2Tr/a(l,,r) The results for S are very similar to those just deduced for E. The wave vector group has two elements E IO), ( |Io), with the following character table. - -yz TABLE VIII IRREDUCIBLE MULTIPLIER REPRESENTATIONS OF THE GROUP Go(S) E a vz n(S i) S1 1 1 4 2 1 -1 2 Tr T 6 2 The dynamical matrix is I I D(S) A C -C B -C D T'! L~ m C /\J M k E Cr GC -. F H - 6 H F T LL N M -I i (65) +i2L + ei2 e +i2=N with J* = ei J, L* = e L, M* = e M, and N* = -e 2 and after transformation by the matrix

/ 00 0 00 o o // O 0 //f / 0 o tl/L o o - /i o ) 0 0 (s) = C) C C Ilr (3 / C) 0C) C) /l O -/ s, & (66) assumes the form A T CCZ LZ T E N' i GZ Q -cz NfZ B+D K+M' L*Ni -GE Ik<'* F+ l 0 -' (S) 0 B-D k —M K-M F-H (67) which we note is very similar to that for E. r: q = (o,o,o) Finally we turn to the most symmetric point in the Brillouin zone, namely the center. At this point, the point group is the full tetrahedral group Td

having 24 elements. The character table and the decomposition of the reducible representation are given in Table IX. From this we see that corresponding to q = 0 there must be two triplets each belonging to the representation rP5. One of these corresponds to the acoustic modes of zero frequency. The remaining triply degenerate mode is the optic frequency. Experimentally, however, in zinc-blende type crystals two distinct frequencies are seen at q = 0 instead of a single frequency. It would seem from this that there is some contradiction. It turns out, however, that the splitting that is observed (which incidentally is due to coulombic effects*) is at values of which though small are yet not identically equal to zero. At q 0 there is in fact a triple degeneracy. For a further discussion of this problem, see the article by Warren(6) and the references cited therein. TABLE IX CHARACTER TABLE FOR THE GROUP Go(F) _ E 8C 3C2 6S4 6 6 nri) I1 1 1 1 1 1 0 "2 1 1 1 -1 -1 0 rl2 2 -1 2 0 0 0 Els 3 0 -l -l 1 2 r 51 3 5 0 -1 1 -1 0 Tr T 6 0 -2 -2 2 This concludes the discussion of the symmetry of the dynamical matrix at various symmetry points in the Brillouin zone. The manner in which the representations change as we move in reciprocal space is indicated in Table X taken from the work of Parmenter. Using this table and the results deduced above, we sketch schematically in Figure 6 the dispersion relations for the zinc-blende structure. In some directions and points there is more than one way of doing the labelling, and only one of the possibilities has been indicated in the figure. For example, at X, the top Al branch could go over to either X1 or X3 while the bottom Al goes over to either X3 or X1. We indicate one of the two possible choices. Similar considerations *The behavior of the optic modes at small values of q is a topic in itself.(l0,ll)

42 L, WI W2 W3 W4 W2 W3 L A A x z w r K X Figure 6. Schematic plot of the blende lattice. At point X, the and is model dependent. Similar dispersion relations for the zinclabelling of X1 and X3 is ambiguous ambiguities exist at Z and W.

43 TABLE X COMPATIBILITY TABLES CONNECTING REPRESENTATIONS AT DIFFERENT SYMMETRY POINTS r + A A -+ X X+ Z Z+W rF A1 A1 X1, X X1 Z1 Z1 W1,W2 rF A2 A X2, X4 X2 Z1 Z2 W3,W4 r'2 A1 + A2 A3X3 Z X5 "15 A1 + AS + A4 A4 X4 Z2 r25 A2 + A3 + A4 X5 Z + Z2 r +A r ++ S ri A1 Fi r 1 X1 S1 r2 A2 r2 72 X2 S2 r12 A3 r12 L1 + 2 X3 S1 F15 Al + A3 rl5 2Z1 + Z2 X4 S2 r25 A2 + A3 25 Z1 + 2Z2 Xs S1 + S2 have been used at Z, W. As noted earlier this ambiguity can be reduced only by explicit numerical calculations of the dispersion curves based on suitable force constant models. It is also of some interest to compare the representations of the zinc-blende lattice with those of two related lattice structures, viz., face center lattice (space group Oh) and the diamond lattice (space group OhI). Such a comparison has been made by Parmenter and his results are reproduced in Table XI.

TABLE XI COMPATABILITY TABLES FOR THE REPRESENTATIONS OF THE SINGLE GROUPS CONNECTING THE ZINC BLENDE (Td) WITH THE FACE-CENTERED-CUBIC (O0) AND THE DIAMOND (O0) STRUCTURES Td h5 d 0 d h d h Fr r2 215 25 ri F1 F2 F12 l 5 2 5 or F2 or FL t or F12 or F25 or r15, rl F1 Fl 2 r12 r15 r25 Fr r2 rl5 F25 or F2 or Fi or F12' or r25 or r15t Al A2 A3 A4 A1 or A2, A2 or A1, As A1 A2 A34 A4 } AI or A2, A2 or A1? As A1 A1 A2 A2 A3 A3 3 3n A1 A1 A2 2 A3 A3 E1 E1 or E3 ES2 E2 r E4 Z1 Z1 or Z2 Z2 Z3 or Z4 X1 X1 or X2 X2 X2 or X' X3 X3 or X4' X4 X4 or X3t X5 X5 or X51 E1 E1 or E3 TS2 E2 or E4 Z1) Z Z2S X1 ) X, Az X2 X3 or X4 W1 W2 w3 } W4 W1 W1 W3 or W2 or W2 wl} Ws w2\ W4 W1 W2

CHAPTER V SELECTION RULES IN OPTICAL SPECTRA In this chapter, we shall discuss selection rules for two phonon absorption in zinc-blende type crystals. Several methods(3, 12 13) for obtaining these have been reported in the literature. We follow the techniques of MV as they lead not only to the selection rules but to the complete structure of the dipole moment operator as well. Similar techniques for obtaining the structure of the polarizability tensor which controls Raman scattering are also discussed. A. INFRARED ABSORPTION Among the methods frequently employed for the study of phonons in crystals are infrared absorption and Raman scattering. In an infrared experiment, one examines the absorption of a thin slice of the crystal as a function of the frequency of the incident radiation, and looks for resonances in the absorption. These resonances arise due to the interaction between the photons and phonons, and are subject to the conservation laws: Energy of photon fw = algebraic sum of energies of all phonons contributing to the resonance. Momentum of photon IQ O (for infrared radiation) = fx(vector sum of wave vectors involved in the resonance). of all phonons In a one phonon resonance, for example, we will have tUJ = Li.)( O%) (68) Since in ZnS type crystals there are only two nonzero Figure 6), one phonon resonance does not lead to much other hand, for two phonon processes the conservation frequencies at q z 0 (see information. On the equations take the form

46 (sum mode) t LA t, A-) ( S 4 -' Li, " (%,) (difference mode) = t cJ (I ) - - ('V) Q::;, t 4 (69) and clearly many combinations are possible. Therefore it should be possible to say more about the phonon spectrum from a study of two phonon absorption than from one phonon absorption. Figure 7 shows the absorption curve for ZnS as observed by Deutsch.(14) We see at least six prominent transmission dips, and the question naturally arises as to which phonons are responsible for these. To answer this question, it is necessary to consider briefly the cross section for photon absorption. This is proportional to <I /J3x^) M_ c)X4)dX / (^E-t-r, (70) Here Xm(x) and Xn(x) denote the crystal vibrational wave functions in the vibrational states m and n respectively, and M is the dipole moment operator. Expand M in a power series of nuclear displacements as follows: M (x) = (0) (I) (Z) (X) + 4 (X) -+ M (XK) 4 9 * * _V _ x - (71) where (2. ) c<( M[v(i )?I / aL.2.g H (Se ) I)U, ( e ) ~~~~~~~~~~'d~~~~~~. (72)

Wave number - (crT'l 950 850 750 650 550 450 100 C u L. 0m 11 I - ~QZnS No. 5 Thickness 0-239 cm / 77~K / I - Oq A 10 0* E 0 Jc 1 tC Z- nSNo.6 Thickness 0-040cm 77~K K I I I II I I I I -I I I I a I -- - - - 10 12 14 16 18 20 dik di 1 I. 22 24 Wavelength (microns) (14) Figure 7. Transmission of zinc sulphide. Data of Deutsch. ( (Reproduced from Proc. Internat. Conf. on the Physics of Semiconductors, Exeter, 1962.)

and so on, with U ) denoting the displacement of the atom at ( ). In two phonon absorption, we are interested in the part M2(x) of M(). On account of the translational periodicity of the crystal,'\^ (e ^ yx - ( 0 A I - (73) Using this fact and the familiar normal coordinate expansion for U( )^( ^-; K ( K) = (N e ) where N is the number of primitive coordinate appropriate to the mode G(SfK~;, )d( ~ cells in the lattice, and Q(q.3 is the normal (q,j), we can express M2 as ^.1 ~a (2-) - Ijq t' I I3 1./Z: h-1) Q 2..5~0 Hm 01 G1 - Q f 4p-' (PG4, (74a) = -i! -g t, ( n i) In Eqs. ( 74a) and ( fib) above, jl I I i'j (. In Eqs. (74a) and (74b) above, Q (iJ)Q( g). I fo - 2y 1 II Q C (K, 3 -- y X I "i'fh)' (*'~~rp i ~CI)-,I f~ (75)

In obtaining them, we have made use of the results (10) eco (x, e ) e (e - i j Ob, gj Q Q(j-) = N - ) _ F b(,.;)~~W1 and exP L I X ) (76) Observe that the momentum conservation (%a+ q') = 0 expressedn: Eq (69) arises through the A function above. On introducing (74) into (70) and further expressing Q i(-j in termis of + j annihilation and. creation operators aqj and a. as ^ ^-+ j( "- [? r YZ 0 4- L).9~ui _,; (TT) and then using tie following properties, q j I (Q > _ O h) I )= 0 Ga j 1 ij 1 ) > ( ('))1 | nj ) - >i n, t)i o, 1 It I k-2in, (1) + I j ) - M. ti)> ki, ( i i.. -j tlwsi._t I Y>j 3)>= kl ( i) ( l )I( > GCf i' i 7= C, C. 7-o 0 1 ) " 1) 1) I)

50 3 9 j = i~ I a(^p-f) ) r, i ( ) LIIexP (k LI (.) (J87 p t"T / ft ) I (78) we find that the absorption cross section for the difference band, for example, is oc Zj IOC I (-fl I /< n' (o>L[<ni(z> s- /9 ( t ) >C< ), -- (x) C wj ( - /17 I W, (S) w I (b) Wj, (t) - WL)J j - / I~c /1 dS/ S(LO) - I (i)>} (x) i Uj/^UJ~Iti) I' V LO = WI (ji- W/,,b (79) The integration above is over a constant frequency surface. A similar expression may be written down for the summation band.(15) From Eq. (79) we see the following:

51 (i) A given pair of phonons (-q,j) and (q,j') can combine and contribute to a difference band only if there is at least one nonvanishing element of M( —. ). This is the statement of the selection rule. J j (ii) If there is a nonvanishing element in M( q,), then the two branches j and j' can combine for all values of q from the zone center to the zone boundary resulting in a continuum in the absorption spectrum. (iii) The continuous spectrum will have singularities whenever JVW@cloj^(q).) (q) In ZnS type crystals |VDj(q) I vanishes at r, L. X, and W(2) (see also Figure 6). Hence singularities in two phonon absorption spectra may be expected due to combination of the "critical" point phonons, i.e., those at r, L, X, and W. We shall now employ group theoretical considerations to determine the structure of M at these points. We note first that 1\1'^ (. ij ^/' ( -i 3 i/ ) 1^ (:t J ) (80) s result is easily obtained from the definition of M( (., ) as given in (75) and the properties quoted in (16). Next we observe that under a space up operationS m, the triad M. v( a,) transforms as (Xy l a, )x = ~ 5 9S S( Mrt (, ) (81) where Se ( I') - X = ) and CSt (^ / ) X ( />( 2 (82)

52 and the O's merely reinforce the fact that the sublattices aK and K2 go to K and', respectively. Also from (82) above we have = J -/ _Sd Kx() - X% (,) X (t) 4- S X (C) - X ( C,) (83) and a similar expression for X(e2). Therefore using (81) and (83), we can readily establish that I I -F / 4 xi/ 2a, cM~( i(-.) ( rmy. I f I _2_ )' ipk) e ( K ~b - ) ej ( -Lt,I) ^ex 0p-. ( )-X u (8) g ~~~~~~~~~~~~~~(84a) = ~ ( (x, )') r(fl ^x I) V ex s ~ ~r (t ) r, X 0r ( I ( \ 17: x r,(k* g e) 7 ^()cx Ii;~j& K K} (x) "61 li? w ( 8~~~(4b)

53 If 5 is not a member of the group of the wave vector, then from Eq. we have (4) whiz rh(el)lu) S or s) e (inth) )=f ywhich helps us to recast () above in the form which helps us to recast (84b) above in the form ef (X' j') (85) o I c t ) a or more compactly as Sh -m* )k> -S$.= 5. m, h CFO ( (86a) _M (-, ) = S M(- SX )s (86b) Compare this with (7) which is a similar relationship for the dynamical matrix. If S is a member of G (q) (in which we shall, as in Chapter I, denote it by(m), then (84b) leads to an expression somewhat different from (86). We recall from Chapter I that ( ) (R211 e ( i ) - erp Le~~~~~~~~~yrn~~~~t~~im~~~~1 __ -W 4s /s) / I 7e * (87) This helps us to write (84b) in the form

54 5 "IN.. I m 6I M< ( CSQ1 f ). I /:b..L/ R.,C;A *'0\r C<^A (S) I (R^) 1) t ) (iz M I'% "A I Sl l.'Ax2 (88) Results (80) and (88) are applicable to all crystals, and of these the last one determines the interrelations among the 3 x (3n)2 elements of M -q,q) for a given q. Observe that (88) plays a role similar to Eq. (9a) which determines interrelations between the elements of D. We will now use Eq. (88) to establish tne structure of M at point X. Let us first write H $z (5 =X: S s sIaIs' ) in the form of the following 6 x 6 (3n x 3n the Hermitian property (80). ) a SaX r' a X L) array, where we have made use of Mlx x, X Xs- II 32 X2 Xz2. X,< L'^, i c^ QW2 a g ^ rS. C., X3 O.., O,., O,,,,,,, X51 z /, I,c,- 4-, Qo X2, L Ct.,-7 a o xs 2- 2.. o,..., qC............... 9.'I,....

55 Next consider one of the operations of GO(q) say C2(y). Using the 3 x 3 matrix for C2(y) as given in Eq. (27) and the irreducible representations found in Table III in Eq. (88) we may write H, (%=X, X xI ) = ac, - Z FC 2-( jj x (I )( 1) 1M,(_=x,x,)X,) - H =x, x,,) = 0 In a similar fashion M. s = ), X'I I Ct3 =- Z c.c)]S ^ L),xz.>(C) ) -C!T~jS ((cjQLTjc 2%7 (x,> M - ( r = X X,5 1 r X,5 1 I= n MX (i- X, fl, l Xri = In this way we finally determine the matrix Mx to be 0 ^X~ Y1152Xs 1 )sz1_ Ysz Xs2.. sl../a 0 I 0.0 X\> I 0 4<C X5 1 1 12 0 ) ^i o a;^ oq1o ^C 0 Q, C 521^ Q* O cs o Q* O

56 which we shall write as X, I, X5,, X,,I X, 22. X, O CA O b o C 3 o* o ct o e o yx1 o a o o 0 X 2 ~ es O ~ j S2Z C 0 0 0 ~ there being only suits for My and 9 nonvanishing elements Mz are presented below. instead of the 21 we started with. Re It should be noted that while the elements of My and Mz are related among themselves, they are not related to those of Mx. This is a reflection of the fact that in the group Go(X) there is no element which interchanges x components of q with y and z components whereas there are elements (e.g., oyz) which mix the y and z components. MI' x,, xX, X. X, z, x, o A -B o -C 0 X3 A* o 0- o -E XU( -B o O F o H Xs 12 O -D F O G 0 YXZ. -E HO ) ~ X.2.* o - H 0 tI O x/ o A o B o C xs I: o 1 o F GS 1 3 0 FP 0 H 0 X5sl OE E* H O T X^/ 0 & 0G^ H 0T ^2^-C'*' 0 & ~^.0

57 Results for the points L and W are given in Table XII. Similar but less detailed results obtained earlier by Birman(3) are summarized in Table XIII. It can be seen readily that his results are in agreement with ours if one notes that a combination is allowed (insofar as an unpolarized beam experiment is concerned) if there is at least one nonzero element corresponding to that combination in any of the three matrices Mx, My Mz. One point worthy of note here is that the structure of M is dependent on the form of the irreducible representation. For example, the elements of Mx, My, and Mz involving L3 depend upon the matrices chosen for that representation. However, the squares of the elements summed over the degenerate partners (which is what is relevant to an experiment) must be representation independent. Thus Z ) = E,' r-LL, l,L3l,)I|

TABLE XII THE FORM OF THE MATRICES Mc(qsaks'a'X') FOR POINTS L AND W. THE MATRICES FOR POINT A HAVE THE SAME FORM AS FOR POINT L M L11 L12 L311 L312 L321 L322 x L11 al a5 a7 / a7 a9 a9 L1 2 a5 a2 a1o \a1 o a3 a L311 a7 alo a3 0 a6 0 L312 N/a* /a"o 0 a3 0 a6 L321 a9 a8 a6 0 a4 0 L^22 w/3aa O_ a __ 0 a a4 M Ll L12 L311 L312 L321 L322 11 al a5 a7 -5/a7 ag -5ac L12 a* a2 alo -\3alo as.Jae L311 a7 a_ a3 0 a6 0 L312 -a7 -/alo 0 a3 0 a6 L321 a9 a8 a6 0 a4 0 L3 22 -aF _-4a8 O a* ~ a4 - a. 9 - a..0. a.. 0 a_ M L11 L12 L311 L312 L321 L322 L11 al a5 -2a7 0 -2ag 0 L12 a* a2 -2aio 0 -2a8 0 L311 -2a7 -2a*o a3 0 a6 0 L312 0 0 0 a3 0 a6 L321 -2a| -2a a a* 0 a4 0 L.22 0 O O a9 0 a4 M W W21 W22 W31 W32 W4 x................... W1 bl b4 b6 W21 0 b7 b2 b5 W22 __b b8 b3 W31 bi b7 bg W32 b b b8 W4 b6 bg b ___.___ M W1 W21 W22 W31 W32 W4 W1 0 C1 C2 W21 C* 0 0 0 W2 C* 0 0_________ W2 C2 W31 0 0 C3 W22 0 0 0 C4 I'c___W__IC* C_ o0

59 TABLE XIII SUMMARY OF TWO-PHONON INFRARED ABSORPTION SELECTION RULES DEDUCED BY BIRMAN(3) Symmetry Point Species Type X i X5 TO(X) + LO(X); TO(X) + LA(X) Xs 3 X5 TA(X) + LO(X); TA(X) + LA(X) X5 X X5 2TO(X); 2TA(X); TO(X) + TA(X) W W1 3a W2 W1 n W3 W1 n W4 W2 9 W3 W2 W4 W3 W4 L L1 m L1 2LO(L); 2LA(L); LO(L) + LA(L) L3 m L3 2TO(L); 2TA(L); TO(L) + TA(L) L3 a L1 TO(L) + LO(L); TO(L) + LA(L); TA(L) + LO(L); TA(L) + LA(L) as is to be intuitively expected for the (111) direction, and this is precisely what one finds with another form of the representation L3 given below. 0~I 0 Li' ) [ \ o \1 V o cj'-A o by /\ # o )^ o )VAc o / _~ I. C)t.o\ -(- C 2S Given a knowledge of such selection rules and some idea of the phonon dispersion curves based on model calculations or experimental results for a similar material it is possible to assign features of an absorption curve (such as Figure 7) to various combination processes and thus to deduce the critical frequencies. For example, Johnson(l5) deduces the following critical frequencies for ZnS based on the curve in Figure 7.

6o r LO 366 cm TO 338 L TO 321 LO 272 LA 227 X TO 305 LO 275 LA 221 (89) B. RAMAN SCATTERING We consider now briefly two phonon processes in Raman scattering. We recall that in a Raman scattering experiment monochromatic light of frequency X and wave vector Q(ZO) is allowed to be scattered by the crystal and the scattered beam frequency analyzed. The scattered spectrum shows both lines of higher (anti-Stokes) and lower (Stokes) frequency due to interaction with phonons, and as in the case of infrared absorption, the conservation rules are Change in energy of photon = -Q = k2' -AcD = + (algebraic sum of phonon energies) (+ anti-Stokes process) (- Stokes process) Change in momentum of photon X 0 =4i (vector sum of wave vectors of participating phonons). For two phonon processes this becomes ----- -w = (LA)-iB w2 ~' +'F.i-._o a,- (97s _?-^' ^ 0

61 As in the case of infrared absorption, two phonon processes lead to a continuum with singularities due to combination of critical phonons. If radiation with an electric vector is Raman scattered, and if K denotes the polarization of the scattered radiation, then the cross section for scattering I(o1') s E2 i. h W),) s a ~., E- E? E (90) where PIa) ~= <(b^ n I PsI k K )C I p 1 )> M (n -,^ +)>((9 Aue, oA v vThe tensor P above is the polarizability tensor and as with M earlier, may be expanded in powers of nuclear displacements as follows: (a ( x P(u) (I) (~2) ( ^) 4^ jp -+ i )>-" (X,~~ — ( x )- ox R C~ j - (92)

62 The term of interest to us is the second order term given by 2(I) 2? Y <V-` j? P ()(] - -1 p'^(2/~ ^ y (^*^0R L^(^( t (93) Expressed in terms of normal coordinates, this becomes ^ (X)- f... - ( C; ( )XQ(A) (94) where p,% I3 P4y ( e ~e) ( i, )'a; P.?' (,) exp - X, L (_ 4) -<, t-.Q (95) Compare with Eq. (75). In arriving at (94), use has been made of translational invariance analogous to (73). *,(XI IX) P' )( )Y ~l lll, P,4e -?.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l~ / 0 Ld It / &,(, I - IP-.,,,;r S (-R (96) The cross section may now be evaluated in a straig-ht-forward manner by substituting for the normal coordinates in terms of annihilation and creation operators and evaluating the thermal average as before. As in the case of infrared absorption, we have summation and difference bands both in Stokes and anti-Stokes processes. For example, the cross section for difference band in anti-Stokes process is proportional to

63 2 S ill) a nr E r CE [,(: 7/) P) 2c I i' -l ) -1 -1 (Y) / dS I - 4L m ) ) - W3 (t) (97) The selection rules are thus determined by the structure of Po ( &-). CIP J 3 ) A relation connecting the different elements of P o3 may now we did earlier for M&. We observe that P I(1i a2) transforms UL Qp^/0^!~~K1K2 be obtained as under Sm as,*s- ) ~k S ( P&L AI X, YL_ tve s, S',r-, so. It ( le; Of ) K;,, h3 9 xt' -, E - -2 1 (98) With the help of (98), we may write down the analogue of (84), i.e.,?(. ) =. m M)< / VX () -EXp - c S - C2< ( x ( l9 I

64 (X) {< K5 (_ 1 ) e * (\,, - I ) c ^S I z I 7 (99) or r % I 2> 73, ) I,/ / (100) if~k~ 65-&( ~) & ForSm Go(q), we obtain 0.6. Fo<K I scX - A JgJl X1 - -f 1) - oA. Ix ) ~5 (tS';'r9] Lt' ('.)7..> -i 5 -S;/0A (101)

65 which is similar in content to Eq. (88, a::Jd may be employed to determine the structure of Pa.t gS'

REFERENCES 1. L. A. Feldkamp, G. Venkataraman, and J. S. King, to be published. 2. R. H. Parmenter, Phys. Rev. 100, 573 (1955). 3. J. L. Birman, Phys. Rev. 131, 1489 (1965). 4. After this work was completed, we received a paper by Yarnell, et al., in which similar but less complete results are derived. J. L. Yarnell, J. L. Warren, R. G. Wenzel, and P. J. Dean, International Atomic Energy Agency Symposium on Inelastic Neutron Scattering (1968). 5. A. A. Maradudin and S. H. Vosko, Rev. Mod. Phys. 40, 1 (1968). 6. J. L. Warren, Rev. Mod. Phys. 40, 38 (1968). 7. G. Ya.Lyubarski, The Application of Group Theory in Physics, (Pergamon Press, Inc., New York, 1960). 8. I. V. V. Raghavacharyulu, Can. J. Phys. 39, 830 (1961). 9. R. J. Elliott and M. F. Thorpe, Proc. Phys. Soc. 91, 903 (1967). 10. M. Born and K. Huang, Dynamical Theory of Crystal Lattices, (Oxford University Press, Oxford, England, 1954). 11. R. Loudon, Adv. Phys. 13, 425 (1964). 12. R. J. Elliott and R. London, J. Phys. Chem. Solids 15, 146 (1960). 13. M. Lax and J. Hopfield, Phys. Rev. 124, 115 (1961). 14. T. Deutsch in Proceedings of the International Conference on Semiconductors, Exeter, 1962 (The Institute of Physics and Physical Society, London, 1962). 15. F. A. Johnson in Progress in Semiconductors, (Temple Press Books Limited, London, 1965). 66

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