HAL Id: jpa-00216301https://hal.archives-ouvertes.fr/jpa-00216301

Submitted on 1 Jan 1975

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

LATTICE DYNAMICS OF AIIBIVCV2 ANDAIBIIICVI2 SEMICONDUCTORS WITH

CHALCOPYRITE LATTICE IN A RIGID-IONMODEL

A. Poplavnoi, V. Tjuterev

To cite this version:A. Poplavnoi, V. Tjuterev. LATTICE DYNAMICS OF AIIBIVCV2 AND AIBIIICVI2 SEMICON-DUCTORS WITH CHALCOPYRITE LATTICE IN A RIGID-ION MODEL. Journal de PhysiqueColloques, 1975, 36 (C3), pp.C3-169-C3-176. �10.1051/jphyscol:1975331�. �jpa-00216301�

JOURNAL DE PHYSIQUE Colloque C3, suppliment au no 9, Tome 36, Septembre 1975, page C3-169

LATTICE DYNAMICS OF A"BIVCX AND A1B"'C3j' SEMICONDUCTORS WITH CHALCOPYRITE LATTICE IN A RIGID-ION MODEL

A. S. POPLAVNOI

Kemerovo State University, Kemerovo, USSR

V. G. TJUTEREV

Siberian Physico-Technical Institute, Tomsk, USSR

Rbumk. - Un modele d'ions ponctuels non polarisables relies par des forces A courte distance a leurs plus proches voisins a Cte applique I'ktude de la dynamique du rkseau et des propriktes optiques infrarouges associees des composCs chalcopyrites ZnSiPz, CdGePz et CuAIS2.

La distribution des charges effectives entre les ions est en accord qualitatif avec celle qui derive de la theorie dielectrique de I'ionicite de Phillips.

Les frequences actives en IR et les intensites d'oscillateurs sont en bon accord avec l'experience. La caracteristique la plus jnthressante des spectres calculCs est la presence de bandcs interdites.

Abstract. - A model of unpolarizable point ions linked by short-range forces with their nearest neighbours is applied to the investigation of lattice dynamics and related optical properties in the infrared of chalcopyrite. Compounds ZnSiPz, CdGeP2 and CuAIS2.

The distribution of effective charges between the ions agrees qualitatively with that which follows from Phillips' dielectric theory of ionicity.

IR-active frequencies and the oscillator strengths are in good agreement with the experiment. The most interesting feature of the calculated spectra is the presence of forbidden bands.

A t the present time there is a considerable amount I ?

of data on the phonon spectra of A " B ' ~ C ~ and A'B"'C;' semiconductors which crystallize in the chalcopyrite lattice. The reflectivity and absorption in the infrared (IR) [ l , 21, Raman scattering (RS) of polarized light [3, 41 were measured with ZnSiP, crystals. IR-reflectivity measurements were performed with CdGeP, in references [S, 61. For CuAIS,, I R and RS measurements were reported in reference 171. There are many experimental data on I R and RS spectra of other ternary semiconductors with chalco- pyrite lattice. A group-theoretical analysis and some qualitative conclusions on the phonon spectra of crystals with chalcopyrite lattice were made in [8].

A unit cell of chalcopyrite lattice contains 8 particles (Fig. 1) yielding 24 branches of phonon spectra. Because of the complexity of spectrum it is necessary to begin the investigation of lattice dynamics with the simplest models of bonding forces between the ions. The model of unpolarizable point-charge ions, linked by a short-range forces seems to be perspective in this respect. This model turned out to be successful in the application to the ion-covalent A"'BV and A1'BV' semiconductors [9-121, which possess a type

1 111 V' of chemical bond related to A " B ' ~ C ~ ~ and A B C2 compounds. We can expect such a model to be sui- table for a description of lattice dynamics of ternary compounds under consideration.

FIG. 1. -The chalwpyrite lattice.

The radius of short-range forces was restricted by second neighbours in references [9-121. I t turned out that first-neighbours force constants are of one order higher in magnitude than second-neighbours parameters. In the present paper the rigid-ion model

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975331

C3-170 A. S. POPLAVNOI AND V. G. TJUTEREV

is used for calculating phonon spectra of ternary inequivalent bonds A"-CV(A1-CV') B'~-C~(B"'-C~') . compounds in the first-neighbours approximation. The remaining bonds have turned out to be equivalent

In our model the dynamical matrix can be written to above two and the corresponding force-constant as a sum of Coulomb and non-Coulomb terms [13]. matrices can be derived with the help of symmetry The non-Coulomb interaction in the first-neighbours relations. After imposing the necessary restrictions approximation in the chalcopyrite is determined by which follow from the space homogeneity and iso- two force-constant matrices which correspond to tropy and also by using the Huang relation [l31 the

matrices of inequivalent bonds take the form

Here y = c/a ; c, a are lattice constants, and A , X, y, z, u where the corresponding unit cell is an undistorted are force constants. To obtain eq. (1) we have neglected sphalerite cell taken four times. In that case Cou- terms above the first order in powers of

lomb sums Qij ( ) determined in reference (131 6 ( X / - 0.25). satisfy the relationship

The latter describes the shift of the anions from their idealtetrahedral positions and it is generally small [14]. Matrices v,, and p,, are given in appendix A. The labelling of particles corresponds to figure 1. Substi-

which makes possible an additional check of compu- tuting y = 1, (5 = y = z = u = 0 in eq. (1) we get

tation accuracy. Here s and S' denote particles in the equivalent bonds and corresponding force-constants

sphalerite unit cell (S = 1, 2). p, v are the numbers of matrices are similar to that describing the first-neigh-

particles in chalcopyrite which satisfy the condition bours interaction in sphalerite and AIY [9-121. Three constants A , X, y (6 = z = u = 0, y = 1) givc us two R(p) - R(v) = l + R(s) - R(sf) ,

different matrices "'(E) and q N (i) each 1 being the sphalerite lattice vector which connects of them preserve the form of sphalerite bond matrix. inequivalent particles in the chalcopyrite unit cell. Hence parameters z and u describe asymnletry of In the 10% wave limit k + 0 (k is the wave vector) sphalerite bonds caused by the transition to the the calculated Coulomb sums take the form chalcopyrite lattice.

The Coulomb part of dynamical matrix was calcu- lated according to the Evalc17s method with the help Q ) = ( a p = 1, ..., 8 of a computer [13]. Following the rigid-ions model O O b we suppose

* e1 = e2 = e:(I) , e3 = e4 = ~IV( , , , )

e5 = e6 = e7 = e, = e,*(,,, ,

where e, is the charge of pth ion. The use of electrical neutrality condition for the unit cell

e:(,) + e;(III) + 2 e;(vl) = 0 (2)

gives two charge parameters of the model in addition to five force constants. In the calculation of the Coulomb part of dynamical matrix the real structure of a crystal was exactly taken into account contrary to the short-range part where the expansion in powers of 6 was used.

To check a computer programme it is convenient to use the chalcopyrite lattice with 6 = 0, y = 1,

LATTICE DYNAMICS OF AIIB'VC; AND A I B ~ I I C ~ ' SEMICONDUCTORS WITH CHALCOPYRITE LATTICE C3-171

establish their correspondence with the experimental frequencies. Such a qualitative analysis can serve as a basis for deriving of force constants T, X , y, z, u and ion charges from the least-squares fitting to the

n

= ij. experimental spectrum. Parameters of the model were found with the help of a computer as the values of the variables which correspond to the minimum of a function

The remaining matrices can be derived using the symmetry relations. The numerical values of quanti-

f = 1 { ,YP~ - miheor(A, X, Y , Z, U, ~ ; J ; ( I ) , ~V*(VI)) (5) ties expressed in letters are listed in table I. Coulomb f matrices which correspond to the values y and 6 for the real crystals differ from (4) by the small quantities of order 6 and (1 - y). For the sake of accuracy the calculations were performed using a different values of Evald's breaking parameter (sf [13]).

The calculated dynamical matrix can be reduced to the quasi-diagonal form with the help of symmetry coordinates given in [g]. However, representations r,, r, and I', yield secular determinants of 3th, 4th and 7th order respectively which cannot be solved in an analytical way. Thus the frequencies were cal- culated by diagonalization of the whole dynamical matrix. Polarization vectors were calculated in the long-wave limit and were used for the symmetry analysis of the calculated frequencies.

Some qualitative information can be obtained by putting xf = 0.25, y = z = u = 0. Parameters A, X, e,* can be taken from the calculations in referen- ces [9-1 l ] for A"'BY, A"BY1 - nearest analogues

11 IY Y 1 I11 YI of A B C2 and A B C compounds. Specific fea- tures of chalcopyrite are taken into account in such assumptions only through the difference of cation masses and lattice compression in the direction of tetrahedral axis. Even such a crude approximation leads to a qualitative accordance of calculated phonon spectrum with frequencies observed in optical experi- ments. Analysis of polarization vectors allowes us to determine the longitudinal and transversal modes - which can be IR-active for the light polarized parallel and perpendicular to the tetrahedral c-axis and to

where mj'"'' are the frequencies E, B, found in 1R experiments [l , 21, [5, 71, m:he0r are the longitudinal (L) and transversal (T) frequencies of symmetry I', and f, obtained by the diagonalization of the dynamical matrix in our model taking into consideration the real structure of crystals. A condition of the degeneracy of (L) and (T) modes was introduced into T, and I', frequencies which were not observed in the experi- ments.

The above method was applied to the investigation of ZnSiP,, CdGeP, and CuAlS, crystals. The values of the parameters in the nearest binary analogues [9-121 (neglecting the interaction of second neighbours) were adopted as initial parameters of chalcopyrite for the minimization procedure. IR-active frequencies for ZnSiP, and CdGeP, from the latest works [ l ] and [6] were taken as oj""". The minimization off for CuAIS, was performed on the basis of reference [7] in two versions. The version a) used only IR-active frequen- cies as cog"". In the addition to a) we took into account two low-frequency modes T, and I', observed in RS as ogxP' in the version b). The results of minimization are presented in tables 11, 111. In the square brackets in table I1 we quote works which report on lattice parameters a, c, xf. The value fmi, and the root-mean- square deviations of calculated frequencies from the experimental -

Values of Coulonzb sums for the chalcopyrite lattice (y = 1, 6 = 0)

- corn- fmin AW A X 10-3 X X 10-3 y X 10-3 z X 10-3 pound a A c A xf cm-2 n cm-l dyn/cm dyn/crn dynlcm dynlcm - - - - - - - - - -

ZnSiPz 5.399 0 [l81 5.217 5 [l81 0.269 1 [l81 599 12 5.8 51.44 41.46 8.78 0.10 CdGePz 5.740 [l91 5.388 5 [l91 0.283 [l91 94 14 2.6 45.75 39.90 7.42 1.41 CuAIS2

a) 5.334 [20] 5.222 [20] 0.275 [20] 183 12 3.9 38.98 32.17 15.80 - 1.12 b) - - 411 16 5.1 37.94 40.15 16.36 1.62

A. S. POPLAVNOI AND V. G. TJUTEREV

Ion charges (in units of electron charge)

Compound -

ZnSi P,

CdGeP,

Ion - Zn Si P

Our model Dielectric theory -

+ 0.86 - + 0.496 + 0.48 + 0.267 i 0.67 - 0.380

are listed too, n being the number of experimental frequencies used for fitting (degenerate modes consi- dered as distinct ones). Calculated force constants turned out to be close to the parameters of the nearest- neighbours interaction in binary analogues. We illus- trate this closeness giving the example of ZnSiP,. The parameter u influences only the position of frequency which is TR-inactive and was not used for the para- meter fitting. Therefore hereafter u = 0 for all com- pounds. As it is seen from table I1 the second asym- metry parameter z is small in comparison with A , X, y. If we neglccted this clsymmetry of bonds together with the corrections on the tetrahedral compression and the shift of anions from the ideal positions (that is put y = 1, 6 = u = z = 0) we would get sphalerite type matrices for the bonds. The quantities ( A - y) and (X - y) for Zn-P bond, ( A + y) and (X + y) for the Si-P bond would play the role of sphalerite parameters a and ,4 (sf [9-121). The Zn-P bond ( A - 11 = 42.66, .r - j) = 32.68, all quantities here and below are given in units 103 dynlcm) is to be compared with the bond in Gap (a = 44.48, /l = 33.00 191). The Si-P bond ( A + y = 60.22, X + y = 50.24) is close to that in Si (a = 55.5, /l = 37.34 [12]). Such a closeness of force constants takes place also for other compounds under consi- deration, supporting the idea about the similarity of type of chemical bond in binary and ternary semicon- ductors. The distribution of charges between the ions as a whole agrees with that calculated according to the dielectric theory of ionicity [15, 161. It is worth- noting that simultaneous reversal of charge signs on the cation and anion sublattices does not affect the spectrum and therefore the correct sign of charge associated with any sublattice cannot be determined by optical experiments only.

The model parameters resulted from the minimi- zation were used for the calculation of phonon spectra of crystals under consideration. Results are shown in figures 2-4. In the central part of figures is represent- ed the dependence of frequencies in the longwave limit (k + 0) against the angles between the phonon wave vector and the tetrahedral c-axis. The symmetry

Binary analogues

- 0.559 Gap [9] 0.66 InP [22]

of frequencies and the type of vibrations (L, T) were established by analysis of polarisation vectors. It is seen from figures that frequencies with symmetry T4 and r, show nonanalytic behaviour. The upper mode o,, has the symmetry T4 when k // c and it appears to be of symmetry T5 when k I c. For the other orien- tations this mode has a symmetry f4 @ T, . Vibra- tions with frequency are longitudinal for any orientations of k. The next two modes ~ 0 1 9 , ~ ~ are transversal ones for each directions of k, o19 being of r, symmetry for k // c, of T4 for k I c and of f4 @ f 5 symmetry for any other directions of k. The mode m,, doesn't change its symmetry f,. The remaining modes with symmetry T, are degene- rate and transversal when k // c. For k I c they split into the longitudinal and transversal components without changing their T5 symmetry. The highest modes C O ~ ~ , , ~ , , ~ are IR-active mainly because of displacement of lighter cation with respect to the anions. Midfrequency group o, ,,,,,, , ( a ,,,, ,,,, in ZnSiP,) has a dipole character mainly owing to the displacement of a heavier cation with respect to anions. The essential contribution in the above vibra- tions (up to 50 0/,) belongs to the symmetry coordi- nates with zero dipole moment (sf 181). 1R-activity of the remaining frequencies is mainly caused by the mutual displacement of cations. Experimental data are shown in the same (central) part of the figures. Heavy dots and squares in figure 2 show experimental values of IR-active frequencies taken from [3, 41, crosses represent IR-frequencies from [2] and triangles are the frequencies from [l], the latter being used as oYP' in the minimizing off. In figure 3 squares and triangles denote IR-frequencies from references [5, 61, the latter being used to fit parameters. In figure 4 triangles exhibit values of IR and RS fre- quencies from [7]. The symmetry interpretation according to references [l-71 is shown near the experi- mental points in figures 2-4. The calculated phonon spectra in points of high symmetry N, T and in high- symmetry directions A and R of the Brillouin zone are shown in the same figures. The symmetry of frequencies was established from the compatibility

LATTICE DYNAMICS OF A~BIVC; AND AIB"'C~' SEMICONDUCTORS WITH CHALCOPYRITE LATTICE C3-173

FIG. 2. -The phonon spectrum of ZnSiP2. 0 is the angle between the wave vector and c-axis in the long-wave limit. (A) and ( X ) are the experimental values of infrared-active frequencies taken from infrared reflectivity [ l ] and [2]. (e) and

(m) show Raman scattering frequencies from 13, 41. Light dots represent calculated frequencies.

relations here. In the direction A a unique classifica- tion can be given, at the point T one can identify the representation 7 , which splits in A-direction. Represen- tations ( 7 , @ 7,) and (7 , @3 7,) turn into ( A , + A,) and cannot be distinguished by compatibility relations. At the point N a double degenerate representation NI exists. In R-direction a possible symmetry notation of branches is presented. An interesting feature of the calculated spectra is the presence of forbidden bands. The calculated values of parameters enable us to calculate oscillator strengths for the dipole frequencies and components of static dielectric tensor. In the reference [l71 an expression for the low-fre- quency dielectric tensor of insulators was derived

where E ~ , ( C O ) is the high-frequency dielectric tensor

4 n AZ?~. AZ'"'? e Z ( s I j n ) e,,(s1 I j n ) = - C - -

v, SS,,",

. (7) J G G Here AZ~), is the effective charge matrix of sth ion as introduced in [17], m, being its mass. V , is the unit cell volume, 523, e,(s ( jn) are the eigenvalues and eigenvectors of analytical part of dynamical matrix ; n denotes eigenvectors which correspond to the same frequency ; A, ,U are the Cartesian components.

A. S. POPLAVNOI AND V. G. TJUTEREV

FIG. 3. -The phonon spectrum of CdGeP1. (B) and (A) are the frequencies measured in infrared reflectivity experiments [5, 61.

TABLE IV

IR-active frequencies and oscillator strengths

Symmetry of

modes

-

O5 cm-1 i Sri. X 10-5 cm-2

theor. ( exp. [ I ] I theor. I exp. [ l ]

(*) These frequencies are taken from RS-experiment. (**) Components SA,I which are not listed in the table vanish.

LATTICE DYNAMICS OF A~B'VC; AND A'B"'C;' SEMICONDUCTORS WITH CHALCOPYRITE LATTICE C3-175

FIG. 4. - The phonon spectrum of CuAlSz (Version b)). (A) represents the frequencies measured in infrared reflectivity and Raman scattering [7].

A comparison of expressions for the macroscopic electric field in our model with that in reference [l71 gives us

k being the phonon wave vector. Hence e,* can be considered as a screened ion charge. According to the symmetry one gets

S,,(j) can be considered as an oscillator strength associated with jth mode for the light polarized along

the R-axis. Results of calculations for ZnSiP,, CdGeP, and CuAIS, are listed in table IV. Good agreement between theoretical and experimental values of oscilla- tor strengths is worthnoting. Frequencies which were not observed in experimental works used as referen- ces in table IV should be discussed separately. In the reference [2] the peculiarity in the absorption on ZnSiP, was discovered as a sharp peak at a frequency w = 460 cm-'. In the reflection the peculiarity was found at the frequency w = 464 cm-'. These frequen- cies are quite close to the calculated o = 459 cm-'. Oscillator strengths are comparatively small for the remaining r, and r, frequencies not observed in IR experiments. Discrepancies between the calculated and measured values of oscillator strengths are pro- bably due to absence of phonon damping in our

A. S. POPLAVNOI AND V. G. TJUTEREV

The static dielectric tensor

Compound theor. expt. theor. expt. - - - - -

ZnSiP, 11.1 [l] 12.3 [3]

40) (unpolarized

light) -

CdGeP, 12.13 11.76 [6] 12.72 12.27 [6] 16.1 + 0.3 [5]

CuAIS, 6.60 6.45 [7] 6.70 6.73 [7] -

model. Besides it should be noted that polarization vectors used in this calculation are reproduced with a greater errors than the frequencies themselves. Fre- quencies observed in RS but 1R-inactive were not used as myP' in the minimization procedure. Quantitative discrepancies between theoretical and experimental values attain 50 cm-' for such modes and are to be associated with the imperfection of the model. One should note that discrepancies between values of IR [l, 21 and RS [3, 41 modes for ZnSiP, are of the same order of magnitude. Calculated values of static

dielectric tensor are in the good agreement with the experimerit (Table V). However since the experimental values of E , were used in the calculations one should note the agreement between the theoretical and experimental differences of static and high-frequency dielectric tensors.

In conclusion it is to be noted that calculated phonon spectra of ZnSiP,, CdGeP, and CuAIS, are expected to be useful together with reference [21] in the analysis of inelastic neutron scattering in these crystals.

A A

Appendix A : Matrices p,, and p,,

References

[l] HOLAH, G. D., J. Phys. C (Solid State Phys.) 5 (1972) 1893. [2] ZLATKIN, L. B., MAHKOV, Yu. F., Opt. Spektrosk. 32 (1 972)

7642. [3] KAMINOW, I. P., BUEHLER, E., WERNICK, J. H., Phys. Rev. B.

2 (1970) 960. [4] NAHORY, R. JAGDEEP SHAH, LEITE, R. C. C., BUEHLRR,

E., WERNICK, J. H., Phys. Rev. B 1 (1970) 4677. [5] MARKOV, Yu. F., RESHETNJAK, N. B., Opt. Spektrosk. 33

(1 973) 520. [6] MILLER, A., HOLAH, G. D., CLARK, W. C., J. Phys. Chem.

Solids 35 (1974) 685. [7] KOSHEL, W. H., HOHLER, V., RAURER, A., BAARS, J., Solid

State Commun. 13 (1973) 101 1. [8] KARAVAEV, G. F., POPLAVNOI, A. S., TJUTEREV, V. G., IZV.

Vyssh. Uchebn. Zaved. Fiz. N l0 (1970) 42. [g] BANERIEE, R., VARSHNI, J. P., Can. J. P h y ~ . 47 (1969) 451.

[l01 BANERJEE, R., VARSHNI, J. P., J. Phys. SOC. Jap. 30 (1971) 1015.

[l]] TALWAR, D. N., AGRAWAL, Bal. K., Phys. S W . Sot. (b) 63 (1974) 441.

[l21 SIROTA, N. N., SOKOLOVSKY, T. D., Dokl. Akad. Nauk. SSSR 174 (1967) 797.

[l31 BORN, M., HUANG, K., Dynamical Theory of Crystal Lattices (Inostrannaja literatura, Moskva) 1958.

[l41 VAIPOLIN, A. A., Fiz. Tverd. Tela I 5 (1973) 1430. [l51 HURNER, K., Phys. Stat. Sol. (b) 52 (1972) K 33. [l61 HUBNER, K., UNGER, K., Phys. Stat. Sol. (b) 54 (1972) K 65. [l71 SHAM, L. J., Phys. Rev. 188 (1969) 1431. [l81 ABRAHAMS, S. C., BERNSTEIN, L. J., J. Chem. Phys. 52 (1970)

5607. [l91 Poluprovodniki AzB4Cz, Moskva, Sov. Radio, 1974. [20] SPIESS, H. W., HAERERLEN, U,, BRANDT, G., RAUBER, A.,

SCHNEIDER, J., Phys. Stat. Sol. (b) 62 (1974) 183. [21] POPLAVNOI, A. S., TJUTEHEV, V. G., IZV. Vyssh. Uchebn.

Zaved. Fiz. N 5 (1973) 133. 122) HUBNER, K., Phys. Stat. Sol. (b) 57 (1973) 627.