Lattice Vibration Spectra. LX. Lattice Dynamical Calculations on Spinel Type MCr 2 S 4 (M = Mn, Fe, Cd)

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1 Lattice Vibration Spectra. LX. Lattice Dynamical Calculations on Spinel Type MCr 2 S 4 (M = Mn, e, Cd) H. D. Lutz, J. H i m m r i c h, a n d H. Haeuseler Universität Siegen, Anorganische Chemie I, Siegen Z. Naturforsch. 45a, (1990); received April 9, 1990 Lattice dynamical calculations of the spinel type MCr 2 S 4 (M = Mn, e, Cd) have been done using various potential models (short-range, rigid-ion, polarizable-ion). The main results are that (i) the vibrational modes (eigenvectors) and potential energy distributions of the Raman and IR allowed phonon modes of the three chromium sulfides are very similar, (ii) the A-X and B-X short-range force constants (referring to AB 2 X 4 ) strongly depend on the structural parameter u, i.e., the tetrahedral A-X force constants are smaller than the respective octahedral B-X ones opposite to previous calculations on the basis of an ideal spinel structure with u = 0.25, (iii) bending force constants (X-A-X and X-B-X), but not X-X and B-B repulsive forces, are negligible, (iv) in the case of the breathing mode of the tetrahedral AX 4 unit (species A l g ) the demand on B-X and X-X (stretching and repulsion) forces is larger than that on the A-X force, and (v) the effective dynamic charges of the bivalent metal ions are nearly zero. Key words: Chromium sulfides, Spinels, Lattice dynamics, orce constant calculation, Vibrational modes. 1. Introduction A m o n g the t e r n a r y chalcogenides A, I B I 2 n X 4, spinel type c o m p o u n d s are of considerable experimental a n d theoretical interest. T h u s, m a g n e t i c a n d electric p r o p erties, structural features, such as structure m a p s a n d cation distributions with regard to the tetrahedral a n d o c t a h e d r a l sites, a n d vibrational b e h a v i o u r in terms of bonding, ionicity, a n d free carrier c o n t r i b u t i o n have been studied t h o r o u g h l y in the last four decades. In o r d e r to get m o r e detailed i n f o r m a t i o n on b o n d ing, structure, a n d d y n a m i c s of the spinel structure, lattice dynamical calculations should be a valuable tool. Such calculations, which are p e r f o r m e d since the early seventies [ 1-8 ], are mostly based on relatively crude models a n d suffer f r o m the lack of complete experimental IR a n d R a m a n d a t a. Thus, a p a r t f r o m one rigid-ion m o d e l ( R I M ) calculation [5] (see also [9]) simple s h o r t - r a n g e models (SRM), e.g., valence force fields, were used until very recently [10]. u r t h e r m o r e, most calculations were p e r f o r m e d o n the basis of the ideal spinel structure neglecting the structural p a r a m eter u of the real crystal structure. Very recently single crystal R a m a n d a t a [11] as well as the transversal a n d longitudinal optical zone centre p h o n o n frequencies of the IR allowed m o d e s [12,13] Reprint requests to Prof. Dr. H. D. Lutz, Anorganische Chemie I, Universität-Gesamthochschule, D-5900 Siegen. became available for several chalcogenide spinels. Using these d a t a, we carried out a n i m p r o v e d t r e a t m e n t of the lattice dynamics of spinel type A n C r 2 S 4 (A" = M n, e, Cd). In order to assess the physical m e a n i n g of the results o b t a i n e d we employed several force field models, viz., an improved S h i m a n o u c h i S R M [14] as well as a R I M [15] a n d a polarizable-ion model ( P I M ) [16]. P I M calculations on spinels were performed for the first time. In the case of M n C r 2 S 4 a n d e C r 2 S 4 n o lattice d y n a m i c a l calculations using IR a n d R a m a n d a t a were yet reported. The c o m p o u n d s under discussion are so-called n o r mal spinels where A11 a n d c h r o m i u m occupy only tetrahedral a n d o c t a h e d r a l sites, respectively. Lattice dynamical calculations o n inverse spinels such as A n I n 2 S 4 (A11 = e, etc.) are m o r e complicated owing to the lack of full translational s y m m e t r y in these comp o u n d s (see the discussion given in [11]). 2. Structure D a t a, S y m m e t r y Coordinates, and Phonon requencies N o r m a l spinels A B 2 X 4 crystallize in the space g r o u p d 3 m - O h. A part of the spinel structure is shown in igure 1. T h e primitive r h o m b o h e d r a l unit cell contains t w o f o r m u l a units. T h e fractional coordinates of the respective a t o m s given in Table 1 refer to / 90 / $ 01.30/0. - Please order a reprint rather than making your own copy. Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung in Zusammenarbeit mit der Max-Planck-Gesellschaft zur örderung der Wissenschaften e.v. digitalisiert und unter folgender Lizenz veröffentlicht: Creative Commons Namensnennung 4.0 Lizenz. This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution 4.0 International License.

2 894 H. D. Lutz et al. Lattice Vibration Spectra. LX 894 the cubic system (x, y, z). The lattice constants a and the structural parameters u of the spinels under discussion were taken from Hill et al. [17] (see Table 1). Group theoretical treatment of the optical zone centre ( /c =0) phonon modes [18] yields = A lg + E g + lg + 3 2g + 2A 2u + 2E u + 4 lu + 2 2u. There are five Raman-active (A lg, E g, 3 2g ) and four IR-active modes (4 lu ). The symmetry coordinates of the Raman and IR allowed modes [18] are listed in Table 2. The band frequencies used for the calculations reported in this work were taken from the literature (see [11-13,19] and further references cited therein) (see Table 4). 3. Potential Model ig. 1. Part of the spinel structure AB 2 X 4 (marking of the atoms and short-range force constants see Tables 1 and 3, respectively). Table 1. Lattice constants a and structural parameters «of spinel type MCr 2 S 4 [17] and fractional coordinates of the 14 atoms in the primitive rhombohedral unit cell of spinels (see igure 1). Atom X y z A, 3/8 7/8-1 3/8 A 2 1/8 1/8 1/8 B, 0 0 1/2 B 2 0 1/4 3/4-1 B 3 1/4 1/4 1/2 B 4 1/4 0 3/4-1 X, -«+ 1/2 -«+1/2 «X 2 -u + 1/2 «+3/4-1 «+1/4 x 3 M+ 3/4 1 «+3/4-1 M X 4 «+ 3/4 1 -«+1/2 «+1/4 Xs u ««X 6 u -«+1/4 -«+1/4 X 7 -u +1/4 -«+1/4 «X 8 -«+ 1/4 «-«+1/4 MnCr,S 4 : a = pm, «= ; ecr 2 S 4 : a = *pm, «= ; CdCr 2 S 4 : a pm, «= The lattice dynamical calculations were carried out according to the Wilson-G-matrix method on the basis of cartesian symmetry coordinates [2,14]. The potential model used is as described in the literature [20, 21], or details see also [19]. It is built up in three stages. The first part is a short-range valence force model (SRM). In the second part this model is extended to a RIM by addition of Coulomb forces. The third part results in the PIM where the polarizabilities of the ions are also taken into account. According to Yamamoto et al. [20], the dynamical matrix D is given by D = M( N + C + ' + M )M, where M is a diagonal matrix specifying the masses 112 m k of the atoms involved. The matrix N represents the non-coulombic short-range interactions with valence and repulsive force constants K { and j, bending force constants H and interaction force constants k t [1, 2, 5]. The Coulomb matrix c is given by the expression c = ZQZ. In this equation, Z is a diagonal matrix specifying the effective dynamical charges z k and Q is a matrix derived from the Coulomb coefficient matrix Q. Q, which is calculated by Ewald's method, represents the electrostatic interactions of the ions present. If electronic polarizability is taken into account, an additional Coulomb interaction matrix 1 due to induced

3 H. D. Lutz et al. Lattice Vibration Spectra. LX 895 Table 2. Symmetry coordinates of the Raman and IR allowed modes in terms of Cartesian coordinates x, y, z (for the silent modes see [18]). In the case of the double and triple degenerate species the coordinates q b and q 3 of [18] are given. Species Nr. x y z x y z x y z x y z A, A 2, q 3 c 0 0 -c 0 0 lu q 6 c 0 0 c 0 0 IU,U B, B 2 B 3 B 4 Gi e 0 0 e 0 0 e 0 0 e 0 0 Gs 0 d d 0 d -d 0 d -d 0 -d d XI x 2 x 3 x 4 Gi a a a a a a a a a a a a E,' Gi 0 b -b 0 -b b 0 - b -b 0 b b 'E GA d 0 0 d 0 0 d 0 0 d 0 0 ^ Gs 0 b b 0 -b -b 0 b -b 0 -b b, G9 d 0 0 d 0 0 d 0 0 d 0 0 IU G io 0 b b 0 -b - b 0 b -b 0 -b b x 5 x 6 X 7 x 8 i LU Gi a a a a a a a a a a a a Gi 0 -b b 0 b -b 0 b b 0 -b -b Ga. -d 0 0 -d 0 0 -d 0 0 -d 0 0 Gi 0 -b -b 0 b b 0 -b b 0 b -b G9 d 0 0 d 0 0 d 0 0 d 0 0 GIO 0 b b 0 -b -b 0 b -b 0 -b b a = 1/^/24; 6=1/4; c = 1/^2; d= l/^ft; «=1/2. orce constant 3 Internal coordinate No. b Interatomic distances (pm) and angles ( ) c MnCr 2 S 4 ecr 2 S 4 CdCr 2 S 4 (fi) A-X (fi) B-X k (h) AX-BX d e e c, (/ 3 ) B-B Hi (f 5 ) X-A-X h 2 (f 6 ) X-B-X X-X X-X f 4 X-X Table 3. Short-range force constants and respective interatomic distances. a b 0 d e Labelling of the short-range force constants given in parentheses correspond to those reported in [2]. Number of internal coordinates per primitive unit cell. Calculated from literature data [17] of a and u (see Table 1). Off-diagonal force constant representing the interaction between the tetrahedral and octahedral unit of the structure. Distance A-B. dipole interactions must be added,, = -ZQ(A 1 -Q)- 1 QZ, where A is a diagonal matrix containing the elements of the polarizability tensor. The matrix M is the macroscopic field matrix describing the TO/ splitting of the phonon modes. or mathematical reasons the maximum number of force field parameters which can be determined from a lattice dynamical calculation is given by the number of observed vibrational frequencies, which is ten (or fifteen, comprising the phonons) in the case of spinels if the translation T is included.therefore, the number of short-range valence force constants is limited to ten. In our calculation, we used six valence and repulsive force constants X 1? K 2, l, 2, 3, 4, two bending force constants H 2, and one interaction force constant k (see Table 3). In addition to these

4 H. D. Lutz et al. Lattice Vibration Spectra. LX 896 Table 4. Observed [11. 12, 19] (RQ) and calculated (RC) phonon frequencies (cm -1 ). Species MnCr 2 S 4 ecr 2 S 4 CdCr 2 S 4 RQ RC RQ RC RQ RC SRM RIM PIM SRM RIM PIM SRM RIM PIM Ai, E* ig (1) g (2) (3) lu (1) TO iu (2) lu (3) i» (4) i u (l) iu (2) lu (3) iu (4) igure of merit Calculation /= X(RQ(I)-RC(I)) 2 /W ; = l Table 5. Short range force constants (N/cm), effective dynamical charges (e), and electronic polarizabilities (10 6 pm 3 ). orce constant Internal coordinate MnCr 2 S '4 ecr 2 s 4 CdCr 2 S 4 SRM RIM PIM SRM RIM PIM SRM RIM PIM [5] A-X K 2 B-X k AX-BX \ B-B , X-X l X-X h X-X X-A-X H 2 X-B-X B a "x C b «X b oo Szigeti charge obtained from the TO/ splittings [12], High frequency dielectric constant obtained by classical oscillator-fit calculation [12]. short-range parameters, two effective dynamical charges (the third one is given by the electroneutrality condition) have to be added in the RIM, and moreover three polarizabilities in the PIM. The lattice dynamical calculations were carried out on a DIGITAL VAX 8820 system and an IBM-PC/AT compatible computer. or a detailed description of the self-made program see [19]. The input parameters are the fractional coordinates (.x, y, z), the unit cell dimensions a, the structural parameters u, the masses of the atoms m k, the symmetry coordinates q n, and the phonon frequencies coj. The short-range valence

5 H. D. Lutz et al. Lattice Vibration Spectra. LX 897 Table 6. Eigenvectors of the Raman and IR allowed phonon modes with respect to the symmetry coordinates q x q 5 and qb ql0 given in Table 2 obtained by polarizable-ion model (PIM) calculations. Compound Species Ii MnCr 2 S 4 A E8 ecr 2 S 4 (1) 2 g (2) 2 S (3) A LE Eg CdCr,S 4 2g 2* 2S A.f Eg <?2 <?3 <74 <7s (1) (2) (3) * 0 ) 2* (2) 2g (3) Compound Species MnCr 2 S 4 l u (1) TO l u (2) TO l u (3) TO,u (4) TO ecr 2 S 4 TO LU(2) TO (3) TO LU IU (4) TO lu (1) TO.u (2) TO (3) TO.U (4) TO.U LU(1) CdCr 2 S 4 <76 <?7 <78 <79 <7. o forces (Kt, i f,, t, /c;), the effective d y n a m i c a l charges zk, a n d the polarizabilities ock are treated as variable p a r a m e t e r s to give the best fit of the experimental frequencies. 4. Results T h e observed and calculated frequencies are given in Table 4. T h e force field p a r a m e t e r s (short-range force constants, effective dynamical charges, a n d p o larizabilities) are listed in Table 5. T h e eigenvectors a n d potential energy distributions ( P E D ) are given in Tables 6 a n d 7. T h e p h o n o n frequencies a n d the P E D s of the silent m o d e s (species A 2 u, E u, l g, 2 u ), calculated with the force field parameters, are given in Tables 8 a n d 9. T h e vibrational m o d e s of the zone-centre p h o n o n s obtained for M n C r 2 S 4 calculated with the P I M are shown in igs. 2 a n d 3, respectively. T h e vibrational m o d e s of e C r 2 S 4 a n d C d C r 2 S 4 are similar. As m a i n results of the lattice dynamical calculations the following points are emphasized: (i) T h e sequence of the short-range stretching force c o n s t a n t s K l (tetrahedron) a n d K2 (octahedron) is reversed (i.e. K 2 > X j ) c o m p a r e d to the results r e p o r t e d in the literature [1, 2, 8]. (ii) T h e bending force constants a n d H2 are negligible, but not the repulsive force c o n s t a n t s x, 2, 3, and 4. (iii) T h e effective d y n a m i c charges of the bivalent metals on the tetrahedral sites are nearly zero, those of the sulfide ions resemble the Szigeti charges o b t a i n e d f r o m the T O / L O splittings [12] fairly well (see Table 5). (iv) T h e lacking R a m a n b a n d s (species 2 g ) of M n C r 2 S 4 a n d e C r 2 S 4 are at a b o u t 108 a n d 106 c m - 1, respectively (see Table 4). (v) T h e eigenvectors a n d P E D s of the respective T O a n d L O p h o n o n m o d e s differ only slightly (see Tables 6 a n d 7 a n d igure 2). (vi) T h e respective p h o n o n m o d e s of the various c h r o m i u m sulfide spinels under investigation are very similar with respect to b o t h eigenvectors (see Table 6) a n d P E D s (see Table 7). (vii) T h e contributions of the A-X ( K t ) a n d B-X {K2) force constants to the potential energy of the IR active vibrations (see Table 7) as well as the m o t i o n s of the various a t o m s in the vibrational m o d e s (see ig. 2) largely resemble the findings of the spectroscopic studies of b o t h the isostructural series A n B 2 n S 4 [11,12] a n d sulfide spinel solid solutions [23], viz., c o n t r i b u t i o n of K,: l u (4) > l u ( 3 ) > l u ( 2 ) > l u ( l ), K2: 1 u (2) > l u (1)!> l u (3) ~ l u (4), m o t i o n of A: l u ( 4 ) ^ l u ( 3 ) > l u ( 2 ) ~ l u ( l ), B: 1U (1) ~ LU (2) ~ l u ( 3 ) ^ > l u ( 4 ), and X: l u (1) ~ l u (2) ~ l u (3) > 1U (4). (viii) In opposition to conclusions f r o m simple Ram a n studies [11] the P E D of the total symmetric Ra-

6 Table 7. Potential energy distributions (%) of the Raman and IR allowed phonon modes. H. D. Lutz et al. Lattice Vibration Spectra. LX 898 Species orce Co- MnCr-,S 4 ecr,s 4 CdCr-,S 4 con- uruindic stant SRM RIM PIM SRM RIM PIM SRM RIM PIM A i g 3 X-X K 2 B-X K, A-X X-X E g K 2 B-X X-X X-X c LRC ,(l) K 2 B-X X-X X-X i, X-X g (2) K 2 B-X K A-X X-X X-X g (3) K 1 A-X KI B-X X-X I X-X iu (I) K 2 B-X (63) 67 (68) (58) 67 (66) (74) 71 (76) TO () i B-B (25) 29 (29) (29) 37 (37) (12) 14 (13) C LRC 2 ( -2) -12(- -12) 2 ( -2) 17 (--17) -4( -4) -10! (-9) 3 X-X 6 6 (5) 6 (5) 6 5 (5) 5 (5) 9 8 (8) 9 (8) lu (2) K 2 B-X (73) 93 (81) (73) 96 (85) (73) 101 (78) TO () c LRC -4 (8) -12 (3) -4 (8) -14 (2) -4 (9) -17 (3) HI X-B-X (9) 10 (9) (11) 9 (8) 12 6 (5) 10 (9) KX A-X (8) 8 (7) 9 9 (7) 8 (5) (19) 6 (9) iu (3) i B-B (29) 31 (32) (28) 30 (30) (29) 36 (36) TO () 3 X-X (21) 22 (22) (22) 23 (23) (20) 20 (20) K, A-X (23) 21 (23) (20) 20 (22) (21) 20 (21) I X-X 8 8 (8) 9 (9) 9 9 (9) 10 (10) 8 8 (9) 9 (8) lu (4) K L A-X (65) 66 (67) (70) 69 (69) (65) 66 (66) TO () 3 X-X (13) 12 (12) (10) 11 (11) (14) 13 (13) K 2 B-X 7 8 (7) 9 (8) 7 8 (7) 10 (9) 6 7 (6) 6 (5) I X-X 5 5 (5) 5 (5) 5 4 (4) 5 (5) 7 6 (6) 6 (6) Table 8. Phonon energies (cm *) of the silent modes. Species MnCr 2 S 4 ecr 2 S 4 CdCr 2 S 4 SRM RIM PIM SRM RIM PIM SRM RIM PIM [1] A 2u (l) A 2U (2) E u (1) E u (2) ig u (D u (2)

7

8 900 H. D. Lutz et al. Lattice Vibration Spectra. LX 900 (PIM) calculations.

9 H. D. Lutz et al. Lattice Vibration Spectra. LX 901 2u (1) 2u (2) ig. 3. Vibrational modes of the silent zone centre phonons of MnCr 2 S 4 calculated from the obtained force field parameters (polarizable-ion model calculations). work and, hence, is undoubtedly significant. The larger values of K { (compared to K 2 ) in previous works are obviously pretended by the wrong distances of the ideal (structural parameter u = 0.25) spinel structure used. rom the dynamic effective charges obtained the nearly zero charge of the metal ions on the tetrahedral sites is remarkable. The question arises, how far these results give evidence for the static effective charge of the tetrahedrally coordinated metal ions in spinels. The polarizabilities obtained are reasonable with respect to the literature data (see, e.g. [24] and further references cited therein). The vibrational modes of the zone centre phonons obtained (see ig. 2) reveal that division into vibrations of the tetrahedral AX 4 and the octahedral BX 6

10 902 H. D. Lutz et al. Lattice Vibration Spectra. LX 902 (or the cube B 4 X 4 ) units of the structure proposed in the older literature (see, for instance [25]) is not possible. Instead of this, all modes must be regarded as coupled vibrations of these structural units. Thus, 2g (1) can be described as a "combination" of <5 as (v 4 ) of a tetrahedral AX 4 and Ö (v 5 ) of an octahedral BX 6 unit, and lu (2) as a "combination" of v as (v 3 ) of a tetrahedral AX 4 and one of the two 2g modes of a cube-shaped B 4 X 4 unit. The energies calculated for the silent modes (and the respective potential energy distributions) differ somewhat with respect to the potential model used, especially in the case of E u (2) and 2u (2) (see Tables 8 and 9). The vibrational modes (and eigenvectors) obtained, however, are largely independent of the model used and, hence, are near to the true ones. The lattice vibration with the highest energy is A 2u (1), which can be described as coupled antiphase breathing vibrations of both the AX 4 tetrahedra and the B 4 X 4 cubes (see igure 3). [1] P. Briiesch and. D'Ambrogio, Phys. Stat. Sol. B 50, 513 (1972). [2] H. D. Lutz and H. Haeuseler, Ber. Bunsenges. Phys. Chem. 79, 604 (1975). [3] H. Shimizu, Y. Ohbayashi, K. Yamamoto, and K. Abe, J. Phys. Soc. Jpn. 38, 750 (1975). [4] S. I. Boldish and W. B. White, Rare Earths Mod. Sei. Technol., 13th, 607 (1978), C.A. 91, [5] H. A. Lauwers and M. A. Herman, J. Phys. Chem. Solids 41, 223 (1980). [6] M. A. Aldzhanov, A. M. Aliev, R. K. Veliev, K. K. Mamedov, M. I. Mekhtiev, and V. Ya. Shteinshraiber, Phys. Stat. Sol. B 115, K75 (1983). [7] M. Wakaki, Jpn. J. Appl. Phys., Part 1, 24, 1471 (1985). [8] K. Wakamura, H. Iwatani, and K. Takarabe, J. Phys. Chem. Solids 48, 857 (1987). [9] M. E. Striefler, and G. R. Barsch, J. Phys. Chem. Solids 33, 2229 (1972). [10] H. C. Gupta, G. Sood, A. Parashar, and B. B. Tripathi, J. Phys. Chem. Solids 50, 925 (1989). [11] H. D. Lutz, W. Becker, B. Müller, and M. Jung, J. Raman Spectrosc. 20, 99 (1989). [12] H. D. Lutz, G. Wäschenbach, G. Kliche, and H. Haeuseler, J. Solid State Chem. 48, 196 (1983). [13] K. Wakamura, Solid State Commun. 71, 1033 (1989). [14] T. Shimanouchi, M. Tsuboi, and T. Miyazawa, J. Chem. Phys. 35, 1597 (1961). [15] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford (1954). [16] J. R. Hardy, Phil. Mag. 4, 1278 (1959); J. R. Hardy and A. M. Karo, Phil. Mag. 5, 859 (1960). [17] R. J. Hill, J. R. Craig, and G. V. Gibbs, Phys. Chem. Miner. 4, 317 (1979). [18] H. D. Lutz, Z. Naturforsch. 24a, 1417 (1969). [19] J. Himmrich. Thesis, Univ. Siegen (1990). [20] A. Yamamoto, T. Utida, H. Murata, and Y. Shiro, J. Phys. Chem. Solids 37, 693 (1976). [21] V. Devarajan and W. E. Klee. Phys. Chem. Miner. 7, 35 (1981). [22] H. D. Lutz, and M. eher, Spectrochim. Acta, Part A, 27, 357 (1971). [23] Th. Schmidt and H. D. Lutz, J. Less.-Common Met. (in press). [24] J. Shanker, G. G. Agrawal, and N. Dutt, Phys. Stat. Sol. B 138, 9 (1986). [25] J. Preudhomme and P. Tarte, Spectrochim. Acta, Part A, 27, 845 (1971).