Kushwaha, A K

Kushwaha, S S

English

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Indian J. Phys. 81(3) 343-349 (2007) Lattice dynamical calculations for selenide spinels A K Kushwaha* and S Sf Kushwaha Department of Physics, K N Government P.G. College, Gyanpur, Bhadohi-221 304, Uttar Pradesh, India E-mail : akkphys_bu@yahoo.com Received 14 July 2006, accepted 1 December 2006 Abstract: The infrared and Raman phonon modes in the normal cubic MCr2Se4 (M = Hg, Zn and Cd) spinels are calculated at the first Brillouin zone-centre using rigid-ion model. The interatomic interaction between Cr-Se dominates over the other interatomic interactions. The tetrahedrat MSe4 units are less ionic than the octahedral CrSee groupings. Good agreement is found with the available experimental infrared and Raman spectra for chromium selenide spinels Keywords : Selenide spinel, lattice dynamics, infrared and Raman phonon modes. PACS No.: 63.20.-e 1. Introduction Contrary to collective bonding features, such as cohesive energies, enthalpies of formation, band structures etc., determination of the strengths of individual bonds in solids is not straightforward. One of the few procedures suitable is lattice dynamical calculations. The spinel structure is one of the most important and most comprehensive studies of all structure types. It is observed that wide band gaps in these structures offer attractive photo electronic and optical applications [1]. Studies of spinels also have numerous applications in geophysics and magnetism [2, 3]. For instance, the spinel transition of olivine, which is a major constituent of Earth's mantle, is widely accepted as the origin of the near 410 km seismic discontinuity [4J. The ferromagnetic properties of some spinels are also of great importance in the solid-state chemistry. Detailed analysis of the vibrational spectra of spinels is complicated by order/disorder phenomena that occur in these structures. In order to get more detailed information on bonding, structure, and dynamics of the spinel structure, lattice dynamical calculations should be a valuable tool. Such calculations, which are performed since the early seventies [5-12], are mostly based on relatively crude models and suffer from lack of complete experimental infrared and Raman data. Thus apart from rigid-ion model (ROW) calculations * Corresponding Author O 2007 IACS 344 A K Kushwaha and S S Kushwaha (9,13], simple short-range models (SRM) [14] were used. Furthermore, most calculations were performed on the basis of the ideal spinel structure neglecting the structure parameter . JJ of the real crystal structure. Recently ab-initio calculations have been performed to calculate the Raman and infra* red-phonon modes at the zone-centre by Pascal and Gervais [15] and Wijs et al [16] for MgAI204 and by Fang et al [17] for cubic Si3N4. In the present paper, we have carried out an improved treatment of the lattice dynamical calculations of the spinel type MCr2Se4 (M = Hg, Zn, Cd) by employing rigid-ion model and compare our results with the experimental and other theoretical results available in the literature. 2. Crystal structure and potential model The compounds under discussion are so-called normal spinels. It is commonly accepted that normal spinels exhibits a cubic structure, with space group Fd3m(o£) and 56 atoms per unit cell. Fourfold-(special 8a Wyckoff position) and eightfold-(special 16d Wyckoff position) coordinated cationic sites are located in an oxygen (general 32e Wyckoff posi-tion) close-packed pseudo-face-centred-cubic sublattice. To complete the description of the anion position, an additional parameter, u and generally known as the internal parameter, is introduced. In most spinels, u lies between 0.24 and 0.275 if the origin of the unit cell is taken at the centre of inversion. For a particular value u = 0.25, the anions form an exactly cubic close-packed lattice and define a regular tetrahedron and octahedron with the cation as centre. Only two formula units are present in the primitive unit cell, leading to 42 normal modes. In addition to three acoustic modes of F1o symmetry, 39 optic modes are distributed among the following symmetries at the Brillouin zone-centre [18-20] as T « A + E + 2£ + 2A + F + 3F„ + 4F + 2Fo (1) \g g u 2u \g 2g \u 2u x ' There are five Raman active (A , E, 3F„) and four infrared active (4F ) modes and rest 1y g 2g' y 1ir ate Raman inactive (F ) and infrared inactive (2A , 2E and 2F0) modes. The lattice dynamical calculations are carried out by using the rigid-ion model proposed by Kushwaha et at [21]. In this model the potential energy of the spinel structure com-pounds are expressed as the summation of short-range non-Coulombic part and long-rang Coulombic part as • - # *+# c . (2) In view of eq. (2), It is convenient to express the dynamical matrix in terms of dimensionless frequency tensor T^fakk*), D^kk') = ~ K r o * r % ( Q , Wr'), (3) where V is the volume of the unit cell, e is the electronic charge. mk and mk are the masses of the atoms involved, kk' denote the number of atoms per unit celt, and a and Lattice dynamical calculations for selenide spinels 345 p represent the coordinate axis x, y, and z and the frequency tensor 7^(g,toc') is given as the sum of two contributions: T^qM') = T£(qtkk') + T&(q,kk'). (4) In the present investigation, a rigid-ion model [21 f has been applied to study the zone-centre phonons of selenide spinels MCr2Se4 (M = Hjj, Zn, and Cd). The set of analytical expressions are obtained by solving the dynamical ^ matrix at the zone-centre. In these expressions, Ay% A2 and A3 are bond-stretching Iprce constants for the interatomic interactions M-Se, Cr-Se and Se-Se, respectively whereas B^, B2 and B3 are the corresponding bond-bending force constants. 3. Results and discussion By using the analytical expressions A1, E, and F^ and the corresponding measured phonon frequencies {5, 22-24] at the zone-centre, the bond stretching and bond-bending force constants are evaluated for the first three neighbours. These force constants are listed in Table 1. Table 1. The value of force constants (kdynes/cm) and effective dynamical charges for HgCr2Se4, ZnCr2Se4 and CdCr2Se4 Force parameters \>Cr*Se< ZnCr2Se4 CdCrSe, 2 4 B. A, B. 2 54.14 1.44 78.23 4.12 14.70 1.61 0.00 0.88 0.44 66.77 2.54 74^14 4.73 12.66 1.92 0.00 0.92 0.46 49.74 2.93 74.41 4.75 14.33 1.08 0.00 0.86 0.43 To evaluate the effective dynamical charges, we have used the equation derived by Scott [22], for ternary compounds A^BJO^ given as . r„v,-vz 4*C23>?£,-a>2n,) = An V nA{z\ef | nB{Z'tfif nc(Z'cef ">A mB m_ (5) where c=speed of light. n , n and n are stoichiometric coefficients and the condition of charge neutrality in a unit cell is given as n„2;+n8z; + nc2:=0. (6) In order to determine three unknowns Z^.Zg.z; from the eqs. (5) and (6), we have follow the method proposed by Gervais [23]. These charges are given in Table 1. 346 A K Kushwaha and SS Kushwaha Using above force parameters, we have calculated the zone-centre phonon frequencies of selenide spinels given in Table 2 (for HgCr2Se4), Table 3 (for ZnCr2Se4) and Table 4 (for CdCr2Se4) along with the available experimental and theoretical results in the literature. Table 2. Experimental [24-26] and calculated zone-centre phonon frequencies of HgCr2Se4 along with the results of Lauwers [5] Species Frequencies (cm-1) \ E g »V1> F„<2> v> FJ1XTO) FJ2> F, (3) F1 u <4 > FJDLO F, (2) F, (3) F,„<4> A*,(1> V2> E„<1> E.(2) F. , F*,<1> V) Experimental 237 157 210 168 65 286 268 169 55 291 282 172 58 Calculated 239 154 215 164 69 285 273 174 60 293 284 171 64 475 281 416 150 157 388 141 Lauwers et al 220 158 207 176 69 310 242 197 76 311 255 197 78 Table 3. Experimental [24] and calculated zone-centre phonon frequencies of ZnCr2Se4 along with the results of Lauwers [5] Species E 9 FJ2) i * ™ - - - i Experimental 240 152 230 182 Frequencies (cm*1) Calculated 244 150 229 185 Lauwers et al 235 153 230 186 Lattice dynamical calculations for selenide spinels 347 Table 3. (contd.) •V3> F,u(D(TO) F,u (2) FJ3> FJ4 ) FJ1>LO F,„<2) F,u (3) F,„<4> V 1 > AJ2) Eu<1> Eu(2) F„ FJ1> FJ 2 ) 112 297 274 198 85 302 288 201 87 , t I 115 302 275 197 88 304 291 201 90 462 283 406 202 152 379 133 Table 4. Experimental [5f24J and calculated zone-centre phonon frequencies of CdCr2Se4 along with the results of Lauwers [5]. Species \ E g %(D F*< 2 > V> F1u(D<TO) FJ2> Ft a « F,<4> F,„0>LO F, .<2> F,„<3) F,u<4> AJD Experimental 237 154 225 169 84 288 267 186 75 293 281 188 77 Frequencies (cm'1) Calculated 336 152 226 168 85 290 266 189 75 294 285 195 79 466 Lauwers and Herman[9] 226 152 218 178 92 299 253 207 86 299 277 207 89 101 296 266 211 88 298 286 211 90 348 A K Kushwaha andSS Kushwaha Table 4. (contd.) A ^ ) 280 Eu(1) 409 E„(2) 207 F 154 FJ1) 380 F, (2) 136 Results as obtained on the basis of the lattice dynamical calculations we may draw the following conclusions: (i) The stretching force constant A2 (CrSe6 octahedron) is larger than A1 (ZnSe4 tetrahedron), /.e.f second neighbour interaction is stronger than the first neighbour interac-tion, whereas the bending force constants are negligible. (ii) The effective dynamical charges of bivalent metal ions are nearly zero. Model parameter calculations showed that the force constants A1 and A2 greatly depend upon the distances between M-Se and Cr-Se ions. The model calculations also shows that the octahedral bonding is stronger than tetrahedral bonding, the reason being that the bonding between Cr-Se ions is more ionic than the bonding between M-Se ions. This is also conformed from the values of effective ionic charges. Altogether, the tonicity of sul-phide spinels [21] is larger than those of the corresponding selenide spinels, as it is expected from common chemical sense. In the case of Raman active modes, value of A mode differs by 0.4-0.8% and that of Eg mode differs by 1.3-1.9% from their respective experimental values. Values of F2^(1), F2 (2) and F2 (3) modes show deviations of the order of 0.4-2.4%, 0.6-2.4% and 1.2-6.1% from their corresponding observed values in the case of all selenide spinels. In the case of infrared active modes, values of F (1), F (2), F (3) and F (4) modes differ by amounts lu ' 1ir ' 1u ' It/ ' 0.4-1.7%, 0.4-1.9%, 0.5-3.1% and 0-9.1% from their respective measured values for trans-verse optic polarization and by amounts 0.3-0.7%, 0.7-1.4%, 00.0-3.7% and 2.6-10.3% for longitudinal optic polarizations. References [1] NUecto,T<>T>ata,NHiku^ [2] H W H Green Geophys. Res. 11817 (1984) [3] W Schiessl etai, Phys. Rev. B53 9143 (1996) [4J A E Ringwood and A Major Phys. 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Lattice Dynamical Calculations for Selenide Spinels

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Lattice Dynamical Calculations for Selenide Spinels

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