Infrared spectroscopy and lattice dynamical calculations of Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 single crystals

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1 Journal of Physics and Chemistry of Solids 64 (2003) Infrared spectroscopy and lattice dynamical calculations of Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 single crystals K. Papagelis, S. Ves* Department of Physics, Aristotle University of Thessaloniki, GR Thessaloniki, Greece Received 23 April 2002; revised 14 August 2002; accepted 21 August 2002 Abstract The infrared (IR) reflectance spectra of Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 single crystals are studied for the first time. At room temperature, 15 infrared active modes for Lu 3 Al 5 O 12 and 14 for Gd 3 Al 5 O 12,Tb 3 Al 5 O 12, out of the 17 theoretically predicted, have been experimentally recorded. The complex dielectric function the refractive index, the absorption coefficient as well as the longitudinal (v LO ) and transverse (v TO ) frequencies of the long-wavelength T 1u modes are determined by the Kramers-Kronig transformation of the reflectance spectra. The experimental data are compared and discussed with the theoretical results obtained by the rigid ion model. Our theoretical analysis reveals that the bonds in the tetrahedra exhibit a covalent character while those in the dodecahedra almost ionic character, which is in accordance with the results for other materials of this crystal family. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Garnets; C. Infrared spectroscopy; C. Raman spectroscopy; D. Optical properties; D. Lattice dynamics 1. Introduction The rare earth aluminum garnets Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 have attracted considerable attention as host crystals for near-infrared solid-state lasers [1] as well as for optoelectronics devices, including computer memories, microwave optical elements and as laser active media with applications in medical surgery, optical communications and coherent laser radar [2,3]. Single crystal IR spectra for Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 have not been reported yet. Single crystal IR reflectance data have been measured only for Y 3 Al 5 O 12 [4, 5] and very recently for Tm 3 Al 5 O 12 and Yb 3 Al 5 O 12 [6]. In the past, McDevitt [7] studied the infrared powder absorption spectra for a pleiad of garnet compounds and concluded that the high frequency modes are directly associated with the unit cell volume of the garnet crystals. Raman spectra at normal conditions of these compounds * Corresponding author. Tel.: þ ; fax: þ address: ves@auth.gr (S. Ves). have been studied in the past [8 10] and recently [11,12]. Although polarization Raman spectra for various rare earth aluminum garnets at low temperatures are available [8,9], some of the Raman active modes have not been resolved experimentally, most probably due to accidental degeneracies and/or low scattering efficiencies and to a lesser extent to the presence of electronic Raman scattering in this class of materials. Recently, by using the diamond anvil cell technique we have investigated the pressure response of the Raman active modes of the Lu 3 Al 5 O 12 [11] (up to 16 GPa) and Tb 3 Al 5 O 12 [12] (up to 26 GPa). From the pressure evolution of the corresponding Raman active phonons there is no evidence of pressure induced phase transitions in the pressure region under investigation, but all observed Raman peaks exhibit the usual expected blue shift with decreasing volume. Very recently we have carried out lattice dynamical calculations based on the rigid ion model (RIM) and using a Born-Mayer type potential, for Y 3 Al 5 O 12, Dy 3 Al 5 O 12, Tm 3 Al 5 O 12 and Yb 3 Al 5 O 12 compounds [13]. Due to the structural complexity, (80 atoms in the unit cell) this kind of calculations is a highly complex task. Finally, various microscopic and macroscopic phonon properties of /03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S (02)

2 600 K. Papagelis, S. Ves / Journal of Physics and Chemistry of Solids 64 (2003) alumilosilicate garnets have been investigated by using the shell-model and semi-empirical transferable interatomic potentials [14,15]. In this study, we report single-crystal infrared reflectance spectra for Gd 3 Al 5 O 12, Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12. Kramers-Kronig analysis reveals 15 and 14 well resolved IR peaks for Lu 3 Al 5 O 12 Gd 3 Al 5 O 12 and Tb 3 Al 5 O 12, respectively. Furthermore, the new single crystal IR experimental data allow us to extent our RIM theoretical studies also in the cases of Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12, getting in this way a better insight in the lattice dynamics behavior of this interesting family of materials. All the missing IR and Raman modes have been calculated by the RIM model. Additionally, the effective charges of the ions along with the force constants of the bonds between the cations and oxygens in the basic polyhedra of the garnet structure have been calculated. 2. Experimental details The single crystal Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 samples were grown by the Czochralski technique. FTIR measurements were carried out at room temperature with a Bruker vacuum spectrometer, IFS113v. All the spectra from unoriented, polished single crystal samples were collected in reflectance mode in the spectral range between 100 and 5000 cm 21. The resolution was 2 cm 21 and for each spectrum 64 consecutive scans were recorded. 3. Results and discussion The crystal structure of the rare earth aluminum garnets is quite complicate since it contains 80 atoms/primitive cell. Details of the crystal structure of the RE 3 Al 5 O 12 described in Ref. [13]. In brief, we can say that the garnet structure Fig. 1. Single crystal reflectance spectra for Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12. The inset depicts the spectra in the whole region investigated.

3 K. Papagelis, S. Ves / Journal of Physics and Chemistry of Solids 64 (2003) consists of a three dimensional network of polyhedra (tetrahedra, octahedra and dodecahedra) having a proper cation at their centers (Al ion-tetrahedra, Al ion-octahedra, RE ion-dodecahedra) and surrounded by the proper number of oxygen ions [5]. Each oxygen belongs to two dodecahedra, one tetrahedron and one octahedron. The tetrahedral Al O distance is,1.77 Å, the octahedral Al O one is,1.94 Å while the dodecahedral RE O is,2.30 Å. Group theoretical analysis predicts that the Raman and IR active modes can be classified according to the following irreducible representations G ¼ 18T 1u þ 3A 1g þ 8E g þ 14T 2g The A 1g,E g and T 2g modes are Raman active while T 1u modes are IR active (one infrared inactive acoustic mode is included in T 1u symmetry species). Fig. 1 displays the measured reflectance spectra for Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 in the range from 100 to 950 cm 21. The insets show the corresponding IR spectra in the whole energy region investigated. The experimental spectra show 15 welldefined reflectance peaks for Lu 3 Al 5 O 12 and 14 for Gd 3 Al 5 O 12 and Tb 3 Al 5 O 12. The IR spectra are quite similar to the ones of Y 3 Al 5 O 12,Tm 3 Al 5 O 12 and Yb 3 Al 5 O 12 [4 6]. Nevertheless, some differences are clearly noticeable among the IR spectra of the investigated compounds. For example, ð1þ while the broad reflection peak at cm 21 (see Fig. 1) is well resolved in two peaks in Lu 3 Al 5 O 12, these peaks are hardly resolved in the case of Tb 3 Al 5 O 12 and almost unresolved in Gd 3 Al 5 O 12. The missing IR peaks are probably very weak and/or obscured by strong nearby lying peaks. As in the case of Y 3 Al 5 O 12,Tm 3 Al 5 O 12 and Yb 3 Al 5 O 12 [4 6] there appears a set of three bands in the frequency region cm 21, well separated by a reflection minimum from the rest reflectance peaks. Furthermore, the Raman spectra of the rare earth aluminum garnets [11 13] do not show any mode in the frequency region cm 21. Both these experimental observations are in agreement with our theoretical one-phonon density of states (1-DOS) for a number of rare earth aluminum garnets [13], which show a frequency gap in the region cm 21. Finally, all the measured compounds exhibit a strong reflectance maximum in the cm 21 region while the stronger absorption peaks in the garnet lattice correspond to the modes in the cm 21 and cm 21 regions (see also Fig. 4). In Figs. 2 4 are displayed, the real, 1 1 (v ), and imaginary, 1 2 (v ), part of the dielectric function, the real, n 1 (v ), and the imaginary k 1 (v ), part of the refractive index as well as the absorption coefficient as obtained by the Fig. 2. The real (1 1 ) and imaginary (1 2 ) part of the dielectric function derived from Kramers-Kronig analysis for the Gd 3 Al 5 O 12, Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 single crystals. Fig. 3. The real (n ) and imaginary (k ) part of the refractive index derived from Kramers-Kronig analysis for the Gd 3 Al 5 O 12, Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 single crystals.

4 602 K. Papagelis, S. Ves / Journal of Physics and Chemistry of Solids 64 (2003) Kramers-Kronig transformation of the reflectance spectrum, for the studied compounds. The experimental frequency values of the transverse (v TO ) and longitudinal (v LO ) modes derived from the peak positions of 1 2 (v ) and the Imð1= 1ðvÞÞ; ~ respectively, are listed in Table 1. As mentioned earlier the new IR experimental data combined with our Raman measurements [11,12] allowed us to extent our RIM calculations also for Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12. Details of the RIM used here are described elsewhere [13]. We just mention briefly that the short-range forces, up to fourth neighbor, are described by a Born- Mayer type potential, V SR ðlrðlkþ 2 rðl 0 k 0 ÞlÞ ¼V 0 e 2ðlrðlkÞ2rðl0 k 0 ÞlÞ=ðs k þs k 0 Þ ð2þ Fig. 4. The absorption coefficient in the infrared region for the Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 single crystals. where rðlkþ is the position vector of the k atom in the lth primitive cell while V 0 and s k þ s k 0 are parameters which describe the strength and the range of the repulsive interactions, respectively. The long-range electrostatic forces are calculated by the Ewald method [16]. The interactions in the Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 crystals are described by a set of eight adjustable parameters. For the various kind of atoms, the parameters s Al(oct), s RE, s Al(tetr), s OX express the range of the repulsive interactions, the Z Al(oct), Z RE, Z Al(tetr) and Z OX correspond to the ionic effective charges and V 0 describes a mean value of the interaction strength in the compounds under study. The value of the Z OX parameter has been calculated from the charge neutrality condition in the primitive cell. Table 1 Experimental (room temperature) and calculated frequencies (by using the RIM) for the infrared active T 1u modes of Gd 3 Al 5 O 12,Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 single crystals. The v LO theoretical frequencies have been calculated with macroscopic field at the k100l direction of the Brillouin zone. The experimental error value is ^2cm 21 Gd 3 Al 5 O 12 Tb 3 Al 5 O 12 Lu 3 Al 5 O 12 Experiment Experiment Theory Experiment Theory v TO (cm 21 ) v LO (cm 21 ) v TO (cm 21 ) v LO (cm 21 ) v TO (cm 21 ) v LO (cm 21 ) v TO (cm 21 ) v LO (cm 21 ) v TO (cm 21 ) v LO (cm 21 )

5 K. Papagelis, S. Ves / Journal of Physics and Chemistry of Solids 64 (2003) The corresponding fitting parameter values for Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12, as determined by a least square fitting to the available experimental data, are listed in Table 2. In addition, the first neighbor bond stretching L ¼ d2 V SR ðrþ dr 2 and bond bending T ¼ 1 r dv SR ðrþ dr r¼lrðlkþ2rðl 0 k 0 Þl r¼lrðlkþ2rðl 0 k 0 Þl force constants for the Al O bonds of the tetrahedra and the octahedra as well as for the RE O bonds of the dodecahedra are given. Finally, we mention that the applied model has not been tested for producing an energy minimum for this crystal structure, as the available experimental database for the studied compounds are very limited for a reliable test. As it can be seen from Table 2, the parameter values for the Tb and Lu are quite similar indicating that they could be transferable between these two compounds (and probably between the other family members of these compounds [13]). However, the use of identical or of nearly identical model parameters suffers from the luck of a broad database for lattice dynamical properties, e.g. phonon dispersions, phonon density of states, elastic constants etc. In the case of RE 3 Al 5 O 12 compounds, the only available experimental Table 2 Effective charges, the bond stretching (L ) and bond bending (T ) force constants for Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 calculated using the RIM. The first neighbor bond stretching and bending force constants are given for the Al O bonds in the tetrahedra (L tetr, T tetr ) and the octahedra (L oct, T oct ) as well as for the RE O bonds in the dodecahedra (L dod, T dod ). The masses and the electronic configuration of the involved rare earth ions are also included Parameter Tb 3 Al 5 O 12 Lu 3 Al 5 O 12 e 2 shells [Xe] 4f 9 6s 2 [Xe] 4f 14 5d 1 6s 2 m RE s Al(oct) (Å) s RE (Å) s Al(tetr) (Å) s OX (Å) V 0 (ev) Z Al(oct) (e) Z RE (e) Z Al(tetr) (e) Z OX (e) L tetr (N/m) L oct (N/m) L dod (N/m) T tetr (N/m) T oct (N/m) T dod (N/m) data to be tested by a lattice dynamical model calculation are IR and Raman experimental data. Concerning the values of the various model parameters estimated for Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12, they exhibit practically the general trend found for most compounds of this material class [13]. In particular, the RE ions have larger effective charges than the other cations and that the octahedral Al ions have larger effective charges than the tetrahedral Al ions. The different charge distribution in the tetrahedral and octahedral Al-sites could be understood as the influence of the local symmetry to the electronic charge distribution. Regarding the values of the Al-effective charges, they are in accordance with recent ab initio electronic structure calculation for Y 3 Al 5 O 12 by Xu and Ching [17], where also the difference in chemical bonding for Al tetr and Al oct is delineated. The short-range bond stretching force constants for the Al O bonds in the tetrahedra, (L tetr ), are almost double that of the L oct for the octahedral Al O bonds and the L dod for the RE O bonds (Table 2). Likewise, the bond bending force constants exhibit the same behavior, even though their values are an order of magnitude smaller than the stretching ones. Finally, the magnitude of the L and T force constants, in the octahedra (L tetr, T tetr ) and dodecahedra (L dod, T dod ), are very close. Although empirical models give only an approximate description of the interatomic forces and usually quantum mechanical approaches are necessary for a deeper understanding of the bonding, the repulsive Born-Mayer potential, originating from the Pauli principle due to the overlapping of the wavefunctions of the outer electrons, can be considered to express sufficiently the amount of overlapping of the bonding electrons. Therefore, the stronger the strength of the repulsive force constant the higher the degree of covalent character of the corresponding bond. Consequently, from Table 2 we infer that the tetrahedral Al O bonds exhibit mostly a covalent character, while the dodecahedral RE O bonds an almost ionic character. The IR eigenfrequencies are obtained (Table 1) employing the RIM. The v LO theoretical frequencies have been obtained by applying the macroscopic field in the k100l direction of the Brillouin zone. Similar calculations along the k110l reveal negligible differentiation. In the fitting procedure, we have used only the experimental v TO frequencies because of uncertainties in assigning the experimental v LO values to the corresponding branch. These uncertainties arise from the enhanced contributions from the zone center irreducible representations. The last ones are enabled because of a lowering of the wave vector symmetry along various Brillouin zone directions caused, in turn, by the application of the macroscopic field. The agreement between the experimental and theoretical values for both the v TO and v LO frequencies is very satisfactory. The only inconsistency is the overestimation of the LO TO splitting for the highest frequency mode (,70 cm 21 experimentally and,120 cm 21 theoretically). The reason

6 604 K. Papagelis, S. Ves / Journal of Physics and Chemistry of Solids 64 (2003) for this discrepancy is not clear at present, but probably it may be attributed to the fact that in the fitting procedure only the v TO experimental values have been used. The calculated frequencies for the two missing peaks in our IR experimental spectra for Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 are found to be about,100 and,380 cm 21 (see Table 1). These results are in agreement with the observations of Hofmeister and Campbell [5] for Y 3 Al 5 O 12 as discussed in detail in Ref. [6]. The complete calculated set along with the experimental Raman frequencies for Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 are presented in Table 3. In the fitting procedure we used as input data Raman frequencies with an unambiguous assignment (marked with a or b in Table 3). As already mentioned earlier, for the most RE 3 Al 5 O 12 compounds not only the complete set of the Raman mode frequency values has not been determined conclusively up today but even the symmetry assignment of certain Raman modes is still debated in the literature [8 10]. For example, Mace et al. [8], based on the similarity of the Raman spectra of the various rare earth garnets, proposed that for Tb 3 Al 5 O 12 the set of E g modes should contain modes at,310,,530,,540 cm 21 and a mode at,715 cm 21 in agreement with our theoretical results (Table 3). On the other hand, a considerable discrepancy exists for the T 2g symmetry modes. In particular, Mace et al. [8] proposed that the set of T 2g modes should contain two T 2g Raman modes with frequencies,100 cm 21 and a mode at,220 cm 21 while our model predicts T 2g modes frequencies at,335,,479 and,614 cm 21. Furthermore, Mace et al. [8] suggested a T 2g mode at,715 cm 21 in accordance with our theoretical results. In the case of Lu 3 Al 5 O 12 the mode assignment of Song et al. [9] is very close to our theoretical predictions. For instance, they proposed that the E g set should contain a mode at,520 cm 21 while the T 2g set a mode at,374 cm 21 in agreement with our calculations (Table 3). The only manifested discrepancy between our results and the ones of Song et al. is that they reported a mode at,443 cm 21 while our calculations predicts a Raman mode at,347 cm 21. The static 1 0 and the high frequency 1 1 dielectric function constants are related to the infrared frequencies through the Lydanne Sachs Teller (LST) relation [18, 19]: Y v i ðloþ 2 ¼ 1 0 : ð3þ v i ðtoþ 1 1 i The theoretical values of the ratio 1 0 /1 1 are 3.27 for Tb 3 Al 5 O 12 and 3.16 for Lu 3 Al 5 O 12, very close to the corresponding values of 3.87, 3.30 and 3.15 deduced from our experimental v TO - and v LO -frequencies for Tb 3 Al 5 O 12, Lu 3 Al 5 O 12 and Gd 3 Al 5 O 12, respectively. Based on the reasonable assumption, that the T 1u missing modes have weak oscillator strengths with negligible Table 3 Experimental and calculated (using the RIM) Raman mode frequencies for Tb 3 Al 5 O 12 and Lu 3 Al 5 O 12 compounds. The experimental frequencies marked with a or b are used as input data in the fitting procedure. For completeness, available experimental frequency values are included Symmetry Tb 3 Al 5 O 12 Lu 3 Al 5 O 12 v exp (cm 21 ) v calc (cm 21 ) v exp (cm 21 ) v calc (cm 21 ) A 1g 369 a b 372 A 1g 558 a A 1g 774 a b 768 E g 130 a b 121 E g b 314 E g 326 a b 317 E g 402 c b 374 E g E g b 561 E g b 693 E g 742 a b 783 T 2g 109 a b 120 T 2g 167 a b 175 T 2g 238 a b 237 T 2g 263 a b 247 T 2g 293 a d 312 T 2g T 2g 372 c T 2g 396 c d 404 T 2g d 488 T 2g 544 c d 573 T 2g d 624 T 2g 686 a b 711 T 2g b 744 T 2g 850 a b 881 a From Ref. [12]. b From Ref. [11]. c From Ref. [8]. d From Ref. [9]. LO TO separation the small discrepancy between the experimental and theoretical values of 1 0 /1 1 is mainly due to the overestimation of the LO TO splitting as mentioned earlier and the low limit restrictions in our measurements. The experimental values of 1 1 are 3.60, 3.25 and 3.37 for Lu 3 Al 5 O 12,Tb 3 Al 5 O 12 and Gd 3 Al 5 O 12, respectively, (see Fig. 2) differing only by a few percent from the corresponding values 3.40, 3.51 and 3.49 obtained by refraction index measurements (n 2, 1 1 ) [20] at visible wavelengths. Finally, the 1 0 values obtained by using the 1 0 /1 1 ratio agree well with those obtained from the Kramers-Kronig analysis of the dielectric function data (see Fig. 2). A value around 10, for 1 0, is generally found for other Y-garnets [5], which seem to be correct also in our case. The moderate

7 K. Papagelis, S. Ves / Journal of Physics and Chemistry of Solids 64 (2003) difference between the optic and static dielectric constants ( ), 6 indicates a rather mixed character in the bond strengths of the whole crystal. This may be related to the presence of various types of bonds and to the strongly coupled polyedra in the garnet structure. Acknowledgements We would like to thank L.G. VanUitert and A. Jayaraman for they kindly supply of the samples and Ms T. Zorba for her technical assistance in the IR measurements. References [1] R.C. Powell, Physics of Solid State Lasers Materials, AIP, New York, [2] Numerical Data and Functional Relationships in Science and Technology, Landolt-Börnstein Group III, vol. 12, Springer, Berlin, [3] W. Pandl, Th. Bruckl, in: S. Ghose, J.M.D. Coey, E. Salje (Eds.), Advances in Physical Geochemistry, Structural and Magnetic Phase Transitions in Minerals, vol. 7, Springer, Berlin, [4] J.P. Hurrell, P.S. Porto, I.F. Chang, S.S. Mitra, R.P. Bauman, Phys. Rev. 173 (1968) 851. [5] A.M. Hofmeister, K.R. Campbell, J. Appl. Phys. 72 (1992) 638. [6] K. Papagelis, G. Kanellis, T. Zorba, S. Ves, G.A. Kourouklis, J. Phys.: Condens. Matter 14 (2002) 915. [7] N.T. McDevitt, J. Opt. Soc. Am. 59 (1969) [8] G. Mace, G. Schaack, Ng. Toaning, J.A. Köningstein, Z. Phys. 117 (1969) [9] J.J. Song, P.B. Kein, R.L. Wadsack, M. Selders, S. Mroczkowski, K. Chang, J. Opt. Soc. Am. 63 (1973) [10] R.L. Wadsack, J.L. Lewis, B.E. Argyle, B.K. Chang, Phys. Rev. B 3 (1971) [11] K. Papagelis, J. Arvanitidis, G. Kanellis, G.A. Kourouklis, S. Ves, Phys. Status Solidi B 211 (1999) 301. [12] K. Papagelis, J. Arvanitidis, G. Kanellis, G.A. Kourouklis, S. Ves, Physica B 265 (1999) 277. [13] K. Papagelis, G. Kanellis, S. Ves, G.A. Kourouklis, Phys. Status Solidi B 233 (2002) 134. [14] R. Mittal, S.L. Chaplot, N. Choudhury, Phys. Rev. B 64 (2001) [15] R. Mittal, S.L. Chaplot, N. Choudhury, C.K. Long, Phys. Rev. B 61 (2000) [16] M. Bohr, K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, [17] Y.-N. Xu, W.Y. Ching, Phys. Rev. B 59 (1999) [18] R.H. Lyddane, R.G. Sachs, E. Teller, Phys. Rev. 59 (1941) 673. [19] W. Cochran, R.A. Cowley, J. Phys. Chem. Solids 23 (1962) 447. [20] C.B. Rubinstein, R.L. Barns, Am. Miner. 50 (1965) 782.C.

Study of vibrational modes in Cu x Ag 12x In 5 S 8 mixed crystals by infrared reflection measurements

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