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T. D. SOKOLOVSKII:honon Spectrum of Boron Nitride with Wurtzite Structure 493phys. stat. sol. (b) 118, 493 1983)Subject classification: 6; 8; 22Institute of Physics of Solids and Semiconductors,A d e m y of Sciences of the Belorussian S S R , Min sk1 )Phonon Spectrum of Boron Nitride with Wurtzite StructureBYT. D. SOKOLOVSKII

Using the quantum mechanics methods the interatomic force constants for the rigid-ion model ofboron nitride with wurtzite structure are calculated. An analysis of the symmetry of the fundamen-tal vibrations of the latt ice is performed by construction of the projection operators. The calcula-tions of the phonon dispersion relations along seven directions of high symmetry of this latticeare carried out in the harmonic approximation. In terms of this model the phonon sta te densityand the specific heat temperature dependence are determined for boron nitride with wurtzitestructure, the results being in good agreement with the experimental data on the specific heat.MeTOHaMH KBaHTOBOR MeXaHHKH HaBAeHbI IIapaMeTpbI MeHGITOMHOIO B3aHMOgefiCTBHH

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penremtn. Ha ocHoBe TO^ x e Monenx onpeneneHbI ~ Y H K W ~ R IJIOTHOCTH C O C T O H H H ~ ~

1. IntroductionIt is known [ l ,21 th at the information about the energy of phonons is necessary for theinvestigation of many important properties of crystals.The dynamical theory of the crystal latt ice of Born [ l ] allows to investigate theenergy characteristics of phonons in perfect crystals. Merten [3] used th is theory forstudying wurtzite-type crystals. To define such crystal properties the knowledge of t heinteratomic force parameters is needed. I n some cases these parameters can be deter-mined from the experimental data. But it is not always possible, due to the lack ofknowledge of the initial macroparameters. This, in particular, is the case for boronnitride with wurtzite structure.

I n thi s paper the interatomic force parameters of boron nitride are calculated allow-ing to determine the dispersion relations of the phonons a t the points and lines of highsymmetry for wurtzi te as well as phonon stat e density and specific hea t of boron nitridewith wurtzite structure. The results are obtained in the harmonic approximation t ak -ing into account both central and angular interactions [4 ] of rigid ions of boron andnitrogen, which are at th e nearest distances. The contribution of t he long-range Cou-lomb forces [ 5 ] is also considered within this model, The above-mentioned ion in ter-actions are described by eight interatomic force parameters, obtained in this paperby quantum-mechanical methods.2. Interatomic Force Parameters

Here the interatomic force parameters of pair interaction are dealt with. Each ion ofthe wurtzite lattice, as well as that of sphalerite, has four nearest ionsof opposite polari-l Prospekt Brovki 17, Minsk, USSR.

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Phonon Spectrum of Boron Nitride with Wurtzite Structure 495convenient t o use the method of th e symmetric orthonormalization [13]. The elementsof the overlap matrix S for the nearest boron and nitrogen ions were calculated. Thenthe new wave functions y i , p i were defined by the formula

This transformation causes the minimum deformation of the initial functions yl, p2Now it is assumed that t he functionsq,, p2are orthonormalized in relations 3 ) and 4).Then the electronic populations of the wave functions y1 and y 2 according to 4) reA2 and B2.The nucleus and two 1s-electrons of each of boron and nitrogen atom canbe considered as cores with charges 3e+ and 5e+, respectively. Therefore, the effectivecharges of the boron and nitrogen ions are equal to eT Z,e+ 3 A2) e+, e z =2,e- ( B 2 ) e-, where e+, e - are the proton and electron charges. It follows thateT + eg 0, i.e. the electroneutrality principle is fulfilled in the BN crystal.The effective charge value of a nitrogen ion equal to 0.379e- was chosen so that the

maximum energy of the transverse optical phonon in the infinitely long wave limit,was in agreement with the I R spectra data [14] of 3.36 x l O I s-l. I n this way the radialfunction F a ] I as been calculated, allowing to determine th e central P, and angularQ , components [4] of the interatomic force parameters : ax ?,,yl,a F,, a; F; accordingto the formulae

For boron nitride with wurtzite-type structure these parameters appeared to be0 1 ~ 0,7787; 16 1.817; y1 6, 0.3671; E, 0.9085; a; == a ; E; = 1.947. Theseparameter values are given in units of lo2N/m .The calculation showed that , not allseven interatomic force parameters are different,.

3. Phonon SpectrumThe normal mode frequencies of the vibrations of crystal latt ice ions are described by

with the elenients WEF(q) of the Fourier-transformed dynaniical matrix in the givenmodel,where q is the phonon wave vector, k, lc are the indices, denoting the ion numbers inthe primitive cell of the crystal, i.e. for the wurtzite structure k, k 1 , 2, 3 , 4 ;x,y are the indices of the Cartesian coordinates, i.e. x,y 1, 2, 3 ; D f i ( q ) s the matrixelement of t he short-range forces; CB[(q) the matrix element of the Coulomb long-rangeforces. Thus, for wurtzite W q ) s a 12 x 12) matrix.It is important to t ake into account the symmetry of this dynainical matrix. I n thepresent paper group-theoretical methods were used for simplifying the s tructure of th edynamical matrix D ( 0 ) .With this aim in view the projection operators were deter-

mined by

W t r q ) n2v26,,6kv( 0 ,

W:: q) D:;(q) c:: Y) >

7)

8)

O N q) I zk?(q, A ) T(q,4 > 9)Awhere&Y(q, A ) s the complex conjugate matr ix element of the j-th irreducible multi-plier or weighted representation [15, 161 of the point group Go(q)of the wave vector q ;T q ,A ) th e reducible representation of t he group G o ( q ) ;A the orthogonal matrix

3 x 3) corresponding to the element of the group Go(q) .

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96 T.D. SOKOLOVSKIIThe application of projection operator techniques allowed us t o determine th e uni-tary transformations of the dynamical matrix at the point and lines of high symmetryof the first Brillouin zone of the wurtzite lat tice of boron nitride. Wi th the pointgroup of the wave vector q being isomorphic to the latt ice point group, the largest

effect of accounting the symmetry can be expected at this point. I n this case theunitary transformation is as follows:

U r =

0 0 0 00 0 0 0a , a2 a3 a40 0 0 00 0 0 00 0 0 00 0 0 0a, a2 -a3 -a40 0 0 00 0 0 0b, b, -b, -b4

bl 62 b3 b4

a5-ia ,0

-ib,0aa ,0

-ib,0

b5

b5

a a7 a7ia5 - ia , ia i0 0 0ib , - ib , ib ,0 0 0a5 a7 a7i a , - ia , i a ,0 0 0ib, - ib, ib ,0 0 0

b5 b, b,

b5 b, b,

as 9-ins in,

0 0

-ib, ib,0 0

-a9 -a,i u , - i a s0 0

-bs -bsib, -ib,0 0

bs bs

an-iall0

-ibll0

11i a ,0

61,

-b11ibll0

all0

ib ,0

-a11- a,

0-bll-ib,

0

ia11bll

Here Kovalevs notations [l6] are used for the irreducible multiplier representationz t 5 tll hich appear twice in the reducible representation T q ,A ) and are relatedto the corresponding columns of the unitary transformation U r . The U r matrix ele-ments a re the complex numbers a z ,bl I 1 , 2, 3,4, , 7 , 9 , l l ) as well as th e numbersi and 0. The symmetry conditions of the wurtzite lattice allow t o choose arbi-trari ly the numbers a l , bz,which, however, must not be equal to zero and the modulusof each vector-column of the matrix U r must be equal to unity. If we apply the trans-formation of U r to the dynamical matrix D(O) , hen it is possible to factorize it intosix matrices of second order located along the principal diagonal of th e matrix D 0).

I: M R A+-----G v) arb.units/

Fig. 1. Phonon spectrum of boron nitride with the wurtzite-type structure. a) Phonon state den-sity, b) phonon dispersion branches along seven directions: IK, KH, HA, A r , r M , ML, LA

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Phonon Spectrum of Boron Nitride with Wurtzite Structure 497Fig. 2. Temperature dependence of specific heat ofboron nitride with the wurtzite-type structure (dots:experiment [18], solid line: calculation)

70-? -Ul A I I 1 1 I I

L

0 2 4 6 8J N

For the sake of brevity the relation is given as

where the matrices Dl D , describe the longitudinal branches, while the matricesD3 D4describe the transverse branches of the normal mode vibrations of the wurtzitelattice a t point r,with both D,, D , giving one acoustic and one optical branches of thedispersion relations of phonons.A similar situation is observed for the wave vector q 0,0, nj2 Ic]) ,where c is thebasis vector directed along the hexagonal axis of the wurtzite lattice.However, a t other symmetry points of the first Brillouin zone, where the order of thepoint group of the wave vector is smaller than that a t the point, the symmetry

effect is less significant. Besides, when the Coulomb long-range forces are taken intoaccount, the symmetry of the dynamical matrix C ( q )decreases, resulting in t he split-ting of the energy of optical phonons. Thus, instead of six different values of the phononenergies a t I point, predicted on the basis of the wurtzite lattice symmetry there are,actual ly, seven of them. This is seen from Fig. 1 b where the dispersion curves along theprincipal directions and at the high symmetry points of the first Brillouin zone of boronnitride with wurtzi te struc ture are shown. The curves are calculated in terms of theabove-mentioned model. Along the ordinate axis the values of the normal mode fre-quencies of vibrations of the latt ice in units of 1013s-l are given. The abscissa axisexhibits the conventionally designated points through which the wave vector end runs.I n this case the relative values of the lengths of FK, KH, HA, A r , FM, ML, LAare 213 i@313 i@333 f q8 , i 3 3 respectively. Because of the lattice symmetryalong the FA direction the transverse branches of t he phonon dispersion curves arepairwise degenerated, though it is not the case for the r K direction. The calculationsconfirm thesepredictions. I n r M as well as in IK directions there is no degeneracyof the energy levels, though in Fig. 1 b for the accoustic branches due to the narrownessof these lines it is not shown.

I n terms of theabove-described mode1,using the Blackman method [17], the calcula-tions of the s ta te density function G v ) of the phonons in boron nitride with wurtzitestructure were performed (Fig. 1a) , taking into account 24084 points of the first Bril-louin zone. This function is characterized by a series of extreme points related, ingeneral, t o the optical branches of the normal modes of the crystal vibrations and theacoustic vibrations make the main contribution to one of them.The knowledge of the G v ) function made it possible to calculate the temperaturedependence of the specific heat of boron nitride with wurtzite structure in accordance

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498 T. D. SOKOLOVSKII:honon Spectrum of Boron Nitride with Wurtzite Structurewith

These results are given in Fig. 2 by a solid curve, while dots show the experimentaldata [ZS]. It follows from 10) th at with the change of temperature the same energyvalue of the phonon affects differently the hea t capacity value. Therefore, th e agree-ment between the dat a obtained here and the calorimetric d at a [18] over a considerabletemperature range shows tha t all the frequencies are given sufficiently exact in G v ) .On the other hand, although there is a lack of experimental da ta on dispersion rela-tions of t he wurtzite modification of boron nitride in literature, th e optical phononenergy values in the centre of the Brillouin zone (Fig. 1 b) agree with the dat a on the IRspectra [14]. Thus, the method suggested, where minimum information is used, allowsto obtain important results.Acknozoledgement

The au thor would like to thank the Computation centre of the Academy of Sciencesof the BSSR for help with the calculations.References

[1] M. BORN nd K. HUANG, inamicheskaya teoriya kristallicheskikh reshetok, Izd. Inostr. Lit.,Moscow 1958 (p. 485) (in Russian).[2] J A. REISSLAND,izika kristallov, Mir, Moscow 1975 (p. 365) (in Russian).[3] L. MERTEN, . Naturf. 15a 512 (1960).[4] J DE LAUNAY,he Theory of Specific Heats and Lattice Vibrations, in: Solid State Physics,[5] E. W. KELLERMANN,hil. Trans. Roy. SOC.A238, 513 (1940).161 A. A. MARADUDIN nd S. H. VOSKO, ev. mod. Phys. 40, 1 (1968).171 L. PAULING, riroda khimicheskoi svyazi, Goskhimizdat, Moskow/Leningrad 1947 (p.440)LS] J. SLATER, lektronnaya struktura molekul, Mir, Moscow 1965 (p. 587) (in Russian).191 C. A. COULSON,F. R. S., L. B. REDEI, and D. STOCKER,roc. Roy. Soc. 270, 357 (1962).

Vol. 2, Academic Press, New York 1956 (p. 219).

(in Russian).

[lo] 0 MADELUNC,Fizika poluprovodnikovykh soedinenii elementov I11 i V grupp, Mir, Moscow[l I ] V. F BRATTSEV,ychislenie atomnykh volnovykh funktsii, in: Metodi rascheta elektronnoi1121 R . P. FEYNMAN,hys. Rev. 56, 340 (1939).1131 P.-0.LOWDIN,Adv. Quant. Chem. 5, 185 (1970).1141 T. Yu. GAVRONSKAYA,. V. BELOUSOV, . 0 GOMON, and A. A. SHULTIN,Kolebatelnyespektry razlichnykh modifikatsii nitrida bora, in: Zakonomernosti obrazovaniya elbora dlyaabrazivnykh i lezviinykh instrumentov i ikh primenenie v promyshlennosti Metallurgiya,Leningrad 1975 (p. 26).[15] G. YA. LYUBARSKII,Teoriya grupp i ee primenenie v fizike, Fizmetgiz, Moscow 19581161 0 V. KOVALEV, eprivodimye predstavleniya prostranstvennykh grupp, Izd. Akad. Nauk[17] M. BLACKMANN,roc. Roy. Soc. A148, 384 (1934); A149, 117 (1935).[18] N. N. SIROTA nd N. A. KOFMAN, okl. Akad. Nauk SSSR 230, 82 (1976); 249, 1346 (1979).(Received February 10, 1983)

1967 (p. 4 7 7 ) (in Russian).struktury atomov i molekul, Izd. Leningr. Universiteta, Leningrad 1976 (p. 1).

(p.354).SSSR, Kiev 1961 (p. 154).