LATTICE DYNAMICS OF ZnGeP 2 AND AgGaS 2 IN A TENSOR-CHARGE MODEL V. E.

Vladimirov,

A. V. K o p y t o v ,

a n d A. S. P o p l a v n o i

UDC 5 3 7 . 3 1 1 . 3 3 : 5 3 1 . 3

A tensor-charge model is developed for the lattice dynamics of the ternary c o m p o u n d s A2B4C~ a n d A1B3C~. The l o n g - w a v e l e n g t h phonon frequencies are calculated f o r ZnGeP 2 a n d AgGaS 2 c r y s t a l s . The t e n s o r - c h a r g e parameters are determined by comparing the theoretical and experimental values of the infrared intensities of active frequencies. In the crystal ZnGeP2, t h e tensor charges of the zinc and phosphorus are found to be close to the isotropic charges of the point-ion model, while the tensor charge of germanium is very different from the point-ion charge. I n t h e AgGaS 2 crystal, the tensor charges of all the atoms differ appreciably from the point-ion charges. The results are discussed from the point of view of the chemical bond.

Experimental investigations into the optical vibrations of ternary diamond-like semic o n d u c t o r s o f t h e t y p e A1B3C26 a n d A2B4C5 h a v e b e e n made b y many p e o p l e ; a considerable n u m b e r o f s t u d i e s h a v e a l s o b e e n made f o r t h e c o m p o u n d s ZnGeP 2 a n d AgGaS 2. Measurements of polarized reflection s p e c t r a w e r e made i n [1, 2] f o r ZnGeP 2 c r y s t a l s , a n d i n [1] t h e spectra of single-phonon absorption in unpolarized light were also measured. Raman s c a t t e r i n g was i n v e s t i g a t e d i n [3, 4] a n d t h e s p e c t r a o f i n f r a r e d absorption in the two-phonon region in [5-7]. I n AgGaS 2 c r y s t a l s , m e a s u r e m e n t s w e r e made o f t h e s p e c t r a o f Raman scattering [ 8 ] , Raman s c a t t e r i n g and infrared reflection [9] i n t h e s i n g l e - p h o n o n region, a n d Raman s c a t t e r i n g of first and second order in [10]. In [11], the temperature dep e n d e n c e o f t h e Raman s c a t t e r i n g s p e c t r a i n AgGaS 2 c r y s t a l s was a n a l y z e d w i t h a v i e w t o identifying the lines of first and second order. In [12], a rigid-ion model with tensor interaction of the first n e i g h b o r s was d e v e l oped for semiconductors with chalcopyrite lattice, a n d t h e p h o n o n s p e c t r a o f Z n S i P 2, CdGeP2, a n d CuA1S 2 c r y s t a l s were calculated. A somewhat simplified variant of the model o f [12] was u s e d i n [13] t o c a l c u l a t e the long-wavelength phonon frequencies o f ZnGeP 2 a n d AgGaS 2. I n [ 1 4 ] , t h e K e a t i n g m o d e l was u s e d t o c a l c u l a t e the long-wavelength phonon frequencies o f some t e r n a r y p h o s p h i d e s a n d i n [15] some t e r n a r y s u l p h i d e s . In the theoretical p a p e r s [ 1 2 - 1 5 ] , t h e Coulomb i n t e r a c t i o n is described by a simple model of point ions. However, as i s shown i n t h e g e n e r a l t h e o r y of t h e v i b r a t i o n s of nonconducting crystals [16], the dynamical matrix is determined by the expression

D~ (ss'/g) is the w h e r e D.~v a n, ( s s ' / g ) o f t h e wave v e c t o r g i n tensor, V i s t h e vo-lume the effective dynamical Cartesian indices. The

( )

part of the dynamical matrix that does not depend on the direction the long-wavelength limit, Ea6(oo) i s t h e h i g h - f r e q u e n c y permittivity of the unit cell, AZ(s)~, a n d ms a r e , r e s p e c t i v e l y , the tensor of charge and the mass of ion s in the unit cell; ~, ~, a, ~ a r e t e n s o r s AZ ( s ) h a v e t h e f o l l o w i n g s y m m e t r y p r o p e r t i e s [17]: A

A

A

A

A (~) ---- A -~ AZ r(A,~) A,

(2)

where F(A, s) is the number of the particle into which particle s is carried by the symmetry t r a n s f o r m a t i o n g = (~/t), A is the matrix of a point transformation, and t is an admissible translation. It follows from the condition of electrical neutrality of the State University, Kemerovo. T r a n s l a t e d from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 40-44, September, 1980. Original article submitted April 2, 1979; revision s u b m i t t e d June 12, 1979.

784

0038-5697/80/2309-0784507.50

9

198~Plenum

Publishing

Corporation

unit cell that

(3)

.~AZ (~) = O. 8

In crystals with a high degree of point symmetry of the sites of the crystal lattice, it follows in accordance with (2) that the tensor-charge matrix is a multiple of the identity matrix, and the expression (i) for this case is equal to the corresponding expression for the point-ion model. For crystals with chalcopyrite lattice A

5Z ( s ) = ~.=1,3

A

cs

as 0

,

0

0 b~

AZ ~-~

,f

b 0

(4)

,

where as, bs, ca, a, b, c, d, ~ are the tensor-charge parameters, which satisfy the relation ( 3 ) , and the atoms in the unit cell are numbered in accordance with [12]. The tensor charges for the remaining atoms in the unit cell are obtained from the relation (2). The displacement of the ions corresponding to the polar optical vibration leads to the appearance of an electric dipole moment, which can be r e p r e s e n t e d in the form [17] M x ( j ) = ~"~

hZ(S) ~ l~(s/j)(Vm~)-1. '2,

(5)

,$I~.

where ~qk(j) is the projection onto the axis ~ of the vector of the dipole moment resulting from vibration j; l~(s/j) is the projection onto the axis ~ of the p o l a r i z a t i o n vector of atom s of the vibration branch j. In the long-wavelength limit (g § 0), the polarization vectors can be written as

1~.(s/rzFoa) -~ Zanm (I-',,) Q~ (s/mFoU),

(6)

ill

w h e r e Fa i s t h e n u m b e r o f t h e i r r e d u c i b l e representation of the symmetry group of the crystal, n labels the eigenvalues of the dynamical matrix c o r r e s p o n d i n g to the symmetry Fo, ~ labels the eigenvectors in the case of multiple frequencies, anm(Fo) are constants determined from the solution of the secular equation, and Q~(s/mF~a) are the basis vectors of the vibrational representation of chalcopyrite. As the vectors Q~(s/mF(~a), one can use either the vectors determined by the group-theoretical methods in [18] or the basis vectors proposed in [13]. As is shown in [17], the only n o n v a n i s h i n g components of the vectors M(j) determined by the expression (5) are Mz(nF4) and M x ( n F 5 x ) = My(nF5Y ). The p e r m i t t i v i t y tensor in the phonon region can be written as N~

s~ (n)

~ (o0 = ~,~ ( ~ ) - t - ~

,~,~r(n) -

,,'~ '

(7)

rt=l

where

S~ (n) = 4uM~ (n, F4); S~ (n) = Sy (n) = 4~M~ (aF~x), and ~ T ( n )

are the eigenvalues

N z

= 3; N~ = Ny = 6,

(8)

of the analytic part of the dynamical matrix.

As follows from the calculations of [12, 13], the main contribution to the values of the optical frequencies in A2B4C~ and AIB3c~ crystals is made by the parameters of the short-range forces in the analytical part of the dynamical matrix. The Coulomb part is responsible for the n o n a n a l y t i c behavior of the frequencies in the long-wavelength limit, and it also determines the oscillator strengths S~(n) of the optically active modes (Eqs. (5) and (8)). In the present paper, the matrix elements of the tensor charges for the crystals ZnGeP 2 and AgGaS 2 are determined as follows. As the parameters of the s h o r t - r a n g e forces~ we take the quantities determined in [13] by fitting the optical frequencies of the crystals using the model of rigid point ions. In the same approximation, we calculate the analytic part of the dynamical matrix, since the Coulomb contribution to this part is a few percent. The tensor nature of the charge is taken into account only in the nonanalytic part of the dynamical matrix, and also in the calculation of the vectors M(j) (5). The tensor-charge parameters are determined directly by comparing the experimental values of S~(n) with the theoretical values calculated in accordance w i t h (8) and using

785

TABLE 1. T e n s o r - C h a r g e AgGaS 2 i n U n i t s o f l e l Compound 9

Zn(Ag)

at ZnGeP, AgGaS2

P a r a m e t e r s o f ZnGeP 2 a n d (e i s t h e E l e c t r o n C h a r g e )

l

0,607 1,240

b,

Ge(Ga)

[

C,

0,615 0,770

0,128 0,292

a3 0,436

ba

]

0,388 0,880

0,595

ca 0,269 --0,234

p (s)

Compound ZnGeP~ AgGaS2

(5)

a --0,592 --0,655

I

h --0,45l --1,180

I

~--0,501 --9.$30

]

,

r.~ 0~0~? --0.1.57

[

f --0..1%5 0,023

and (6).

I n t h e phonOn s p e c t r u m of t h e t e r n a r y compounds t h e r e a r e t h r e e o p t i c a l frequencies w i t h t h e s y m m e t r y ~4 a n d s i x f r e q u e n c i e s w i t h t h e s y m m e t r y F 5. T h u s , we h a v e n i n e e q u a tions s~he~ = s~XP(n); in conjunction with the equation of electrical neutrality we have four equations for the frequencies with symmetry F4 and seven equations for the F 5 frequencies. The s y s t e m o f e q u a t i o n s i s d e c o u p l e d , t h e e q u a t i o n s f o r t h e s y m m e t r y F4 containing only the tensor-charge parameters bl, b3, c, and f, the remaining parameters occurring in the system of equations f o r t h e s y m m e t r y F 5. Altogether, t h e r e a r e 11 e q u a tions for determining 11 u n k n o w n s . There is however an arbitrariness in the equations associated with the free choice of the phase factor in front of the polarization vectors /~(s/j), w h i c h o c c u r i n Eq. ( 5 ) . We h a v e r e s t r i c t e d ourselves to considering only real polarization vectors (phase factors • a n d t h i s l e d u s t o w r i t i n g down e i g h t v a r i a n t s of the equations f o r t h e s y m m e t r y ~4 ( o f w h i c h o n l y f o u r a r e i n e q u i v a l e n t ) a n d 64 v a r i a n t s of the equations f o r t h e s y m m e t r y F 5 (32 i n e q u i v a l e n t ) . All variants of the equations were programmed for computer solution. T h e v a l u e s o f S a ( n ) f o r ZnGe92 w e r e d e t e r m i n e d e x p e r i m e n t a l l y i n [2] a n d f o r AgGaS 2 i n [9] f r o m t h e ~ o l a r i z e d reflection spectra in the infrared region. For the frequencies o f t h e s y m m e t r i e s F 4 a n d F 5, w h i c h a r e n o t p r e s e n t i n t h e r e f l e c t i o n spectra, we t o o k S~XP(n) = 0. Using the experimental v a l u e s o f S~ [ 2 , 9 ] , we f o u n d a l l v a r i a n t s of solutions for the tensor-charge parameters. The o b t a i n e d v a l u e s w e r e t h e n u s e d t o c a l c u l a t e t h e d y n a m i c a l m a t r i x (1) and f i n d t h e s p e c t r u m o f t h e l o n g - w a v e l e n g t h p h o n o n f r e q u e n c i e s . F o r b o t h c o m p o u n d s ZnGeP 2 a n d AgGaS 2 o n l y o n e v a r i a n t o f s o l u t i o n was f o u n d i n w h i c h t h e tensor-charge parameters were relatively close to the point-charge parameters in these compounds [12-15]. The v a l u e s o f t h e t e n s o r - c h a r g e parameters corresponding to this variant a r e g i v e n i n T a b l e 1. The p h o n o n f r e q u e n c i e s o f ZnGeP2 a n d AgGaS 2 c a l c u l a t e d with the tensor~charge p a r a m e t e r s a g r e e t o w i t h i n ~10% w i t h t h e r e s u l t s of the calculations in [13]. The v a l u e s o f t h e L 0 - T 0 s p l i t t i n g s calculated i n t h e p r e s e n t w o r k a n d a l s o i n [13] are given in Table 2 together with the experimentally measured values. It can be seen that the splittings calculated i n t h e model o f an u n p o l a r i z e d point ion and in the tensorcharge model are in good agreement with one another and the experiment, the agreement with the experiment being better f o r ZnGeP 2 t h a n f o r AgGaS 2. This last circumstance is due to the fact that the model of lattice d y n a m i c s d e v e l o p e d i n [13] g i v e s a b e t t e r description of compounds that have the closest isoelectron a n a l o g s among t h e b i n a r y s e m i c o n d u c t o r s with sphalerite lattice. We r e c a l l t h a t i n [13] t h e v a l u e s o f t h e p o i n t c h a r g e s w e r e determined from the experimental frequencies a n d L0-TO s p l i t t i n g s , whereas in the present paper the tensor-charge parameters were determined from the experimental values of Sa(n). It follows that the intensity of the infrared absorption is intimately related to the L0-TO s p l i t t i n g of the modes. Let us donsider the tensor-charge p a r a m e t e r s g i v e n i n T a b l e 1. For the zinc and p h o s p h o r u s a t o m s i n ZnGeP 2, t h e n o n d i a g o n a l m a t r i x e l e m e n t s o f t h e t e n s o r c h a r g e a r e c 1, d , f ~ 0 . 1 , w h e r e a s t h e d i a g o n a l e l e m e n t s a r e a , b , a l , b 1, c ~ 0 . 5 . The a r i t h m e t i c mean o f t h e d i a g o n a l e l e m e n t s f o r t h e z i n c a t o m , e q u a l t o 0 . 6 1 , i s c l o s e t o t h e v a l u e s 0 . 7 [13] a n d 0 . 8 5 [14] o f t h e p o i n t - i o n charge of zinc. Por phosphorus, the arithmetic mean o f t h e d i a g o n a l e l e m e n t s i s - - 0 . 5 2 , w h i c h i s a l s o c l o s e t o t h e p o i n t - i o n charge:

786

TABLE 2. C a l c u l a t e d and Experimental Values Splitting in ZnGeP 2 a n d AgGaS 2 C r y s t a l s

Symmetry type

]

chalco- sphalerite I pyrite [

I':,

I'r

X~, W~ W~ )(:, W~ 11:, W.: W., j:,

ZnGeP2 theory here I [|31

exp. [21

Of the LO-TO

[

Ag GAS._,

I

theory I. exp. here [ [13] I [9]

0 0 3

O 1 2

--2

1 0 9

0 0 4

1

3

2

14

9

3 15 0 13 7

7 13 1 9 14

5 17 -12 9

24 22 1 30 25

10 93 229 22

0 0 13 6.2 27.7 28.3 -30.4 18.2

--0.55 [13] and --0.57 [14]. Thus, the parameters of the Coulomb interaction in the pointion and tensor-charge models for the zinc and phosphorus atoms in ZnGeP 2 are close to each other, which serves as a justification for the point-ion model for these atoms. In the case of germanium, the arithmetic mean of the diagonal elements of the tensor charge, 0.42, is also close to the values 0.4 [13] and 0.3 [14] of the point-ion charge, but the nondiagonal m a t r i x element c 3 has the same order of magnitude as the diagonal elements. For this atom, the point-ion approximation is less justified. This may be due to the circumstance that for lattice deformations of the A2B4C~ compounds the atom of the fourth group requires a t e t r a h e d r a l environment, so that the electrostatic forces due to this a t o m are not well described in the point-ion model. In the case of the crystal AgGaS2, it is only for the sulfur atom that the nondiagonal matrix elements of the tensor charge are appreciably smaller than the diagonal elements, and the arithmetic mean of the diagonal elements (--0.89) is close to the pointion charge (--0.8) [13]; however, the anisotropy of the charge (the difference b e t w e e n the diagonal m a t r i x elements) is here greater than for the phosphorus a t o m in ZnGeP 2. For the silver and gallium atoms, the nondiagonal matrix elements are large and the diagonal matrix elements differ considerably. Thus, the point-ion model appears less well justified for AgGaS 2 than it does for ZnGeP 2. One of the reasons for the strong departures from the point-ion model in crystals of the type AIB3c~ could be the strong polarizability of the electron d shells of the atoms of noble metals [19]. LITERATURE

1. 2. 3. 4. 5. 6. 7. 8. 9. i0. Ii. 12. 13. 14. 15. 16. 17. 18.

CITED

Yu. F . ~ { a r k o v , V. S . G r i g o r ' e v a , B. S . Z a d o k h i n , a n d T . V. R y b a k o v a , O p t . S p e k t r o s k . , 3_66, 1 6 3 ( 1 9 7 4 ) . A. M i l l e r , G. D. H o l a h , a n d W. C. C l a r k , J . P h y s . Chem. S o l i d s , 35, 685 (1974). M. Bettini and A. Miller, Phys. Status Solidi B: 66, 579 (1974). I. S. Gorban', V. A. Gorynya, V. I. Lugovoi, and I. I. Tychina, Fiz. Tverd. Tela (Leningrad), 17, 2631 (1975). I. S. Gorban', V. A. Gorynya, V. I. Lugovoi, and I. I. Tychina, Ukr. Fiz. Zh., 20, 1428 (1975). S. Jsomura and K. ~asumoto, Phys. Status Solidi A: 6, K139 (1971). G. C. Bhar and R. C. Smith, Phys. Status Solidi A: i_33, 157 (1972). J. P. van der Ziel, A. E. ~eixner, H. M. Kosper, and J. P. Ditzenberger, Phys. Rev. B: 9, 4286 (1974). G. D. Holah, J. S. Webb, and H. ~|ontgomery, J. Phys. C: 7, 3875 (1974). D. J. Lockwood and H. Montgomery, J. Phys. C: 8, 3241 (1975). D. J. Lockwood, Ternary Compound. Third Intern. Cond. on Ternary Compounds: The Institute of Physics Bristol and London. Conference Series, No. 35 (1977), p. 97. A. S. Poplavnoi and V. G. Tjuterev, J. Phys. (France), 36, C3-169 (1975). A. V. K o p y t o v and A. S. Poplavnoi, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 5, 3 (1980). ~. Bettini, Phys. Status Solidi B: 699, 291 (1975). W. H. Koschel and M. Bettini, Phys. Status Solidi B: 7_22, 729 (1975). L. J. Sham, Phys. Ray., 188, 1431 (1969). A. S. Poplavnoi and V. G. Tyuterev, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 6, 39 (1978). G. F. Karavaev, A. S. Poplavnoi, and V. G. Tyuterev, Izv. Vyssh. Uchebn. Zaved. Fiz.,

787

19.

No. 10, 42 ( 1 9 7 0 ) . K. F i s c h e r , H. B i l z ,

a n d R. H a b e r k o r n ,

Phys.

Status

Solidi,

54,

285

(1972).

VARIATIONAL ESTI~.~TES OF THE POLARIZABILITIES OF INHOMOGENEOUS DIELECTRIC BODIES OF RANDOM SHAPE V. P. K a z a n t s e v

UDC 5 3 7 . 2 1

On the basis of the fact that the energy functional (represented in different forms) of the electric field in dielectrics is minimal on the solutions of the equations of the electrostatics of the dielectrics, methods are proposed for determining bounds for the polarizability tensors of inhomogeneous anisotropic dielectric bodies of random shape. The effectiveness of the method is demonstrated by examples.

The p o l a r i z a b i l i t y a homogeneous electric

tensor ~ relates t h e v e c t o r p o f t h e d i p o l e moment o f a b o d y i n field linearly to thefintensity vector E 0 of the field [1]: A

p

=

~Eo.

(i)

In the general case, a is a symmetric second-rank tensor [1], whose calculation requires solution of the corresponding problem of electrostatics, which for an inhomogeneous body of arbitrary shape is very complicated. The e x a c t v a l u e o f t h e p o l a r i z a b i l i t y tensor is known o n l y f o r t h r e e r e g u l a r s h a p e s : sphere, cylinder, ellipsoid. However, in practice, for example, in problems of scattering of electromagnetic waves by small particles such as dust particles, snowflakes, etc., i t i s n e c e s s a r y t o know how t o e s t i m a t e t h e i r p o l a r izabilities. I n t h e m o n o g r a p h [2] o f two w e l l - k n o w n A m e r i c a n m a t h e m a t i c i a n s , some v a r i a t i o n a l estimates of the polarizabilities of conducting bodies are considered and, in particular, i t i s shown t h a t t h e p o l a r i z a b i l i t y of a conductor is always less than the polarizability of any conductor that envelops it. Here, in contrast t o [ 2 ] , we c o n s i d e r t h e m o r e g e n eral problem of variational estimates of the polarizability tensor of dielectric particles, from Which t h e p r o b l e m o f t h e p o l a r i z a b i l i t y of conductors will follow as a special case for very large, in the limit infinite, values of the permittivity. To o b t a i n we u s e t h e f a c t

a l o w e r b o u n d o n ~ f o r some p a r t i c l e that the solution of the equation

which occupies

the

region

V of space,

A

rot [~-~ (E0 + rot A)] = 0 for the vector potential A, w h i c h i s terior of the particle by

related

to the

(2)

polarization

vector

P(r)

in the

in-

A(r)== S P(r'lX(r--r') dV', maximizes

(3)

the functional

iV(A) =

S [ rotA

(e -- ~-])Eo

~ ~-lrotA

] dV.

(4)

The i n t e g r a t i o n i n (3) i s o v e r t h e r e g i o n o f s p a c e o c c u p i e d b y t h e p a r t i c l e , a n d i n (4) it is over the whole of space. We a s s u m e t h a t t h e p e r m i t t i v i t y tensor ~ is symmetric and positive definite, and that it differs from the unit tensor e only within the particle. T h a t W(A) i s m a x i m a l o n t h e s o l u t i o n (2) i n t h e c l a s s of v e c t o r p o t e n t i a l s which can be represented i n t h e form (3) can be s e e n by n o t i n g t h a t t h e r i g h t - h a n d side of the readily verified identity ^

State University, Krasnoyarsk. Fizika, No. 9 , p p . 4 4 - 4 7 , S e p t e m b e r , v i s i o n s u b m i t t e d A u g u s t 2, 1979.

788

Translated from Izvestiya Vysshikh Uchebnykh Zavedeni~, 1980. Original article s u b m i t t e d A p r i l 2, 1979; re-

0038-5697/80/2309-0788507.50

9

1981 Plenum Publishing

Corporation