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12 UNIVERSITÄT OSNABRÜCK, FACHBEREICH PHYSIK energy surfaces determined for LiNbO3 look very similar to those of LiTaO3. As a consequence, the difference in the transition temperatures in these compound can be ascribed to the differences in their energetics at the zone boundary. Similar results were obtained later by Yu and Park [10]. More recently, Parlinski et al. [11] derived the dispersion relations for both paraelectric and ferroelectric phases of LiNbO3 using the direct method [12]. From their study, the phonon frequencies are also available, whereas the eigenvectors were not calculated. Ergebnisse Within the framework of density functional theory (DFT) [13], we optimized the equilibrium groundstate structure of paraelectric and ferroelectric structures of LiNbO3 using the full-potential linearized augmented plane wave method with the addition of the local orbital basis functions [14] as implemented in the WIEN97 code [15]. Besides the calculation of the theoretical lattice parameters, the structure optimization included also the relaxation of the atomic positions according to the symmetry constraint imposed by the crystal symmetry. The computational setup used in our calculation as well as the values of the lattice parameters and the relaxed atomic positions were reported in Ref. [16]. The ferroelectric instability in LiNbO3 was investigated with respect to its fully optimized paraelectric structure. The total-energy profile calculated for atomic displacements described by A2u symmetry showed that this phase is energetically unstable (with respect to the symmetry-lowering atomic displacements). This result agrees with those of Inbar and Cohen, but it was obtained without any a priori assumption regarding the experimental pattern of the ferroelectric transition in this material. Using the frozen-phonon method [17], we proceeded with the analysis of the zone-center lattice dynamics in the ferroelectric phase of LiNbO3. The Γ-TO phonons are split by symmetry into four A1, five A2 and nine E modes. The A1 and E modes are both Raman and infrared active, whereas the A2 modes are silent. The motivation of our study resides in a controversial attribution of the displacement patterns corresponding to different modes. As was emphasized in Ref. [18,19], the physical properties of LiNbO3 depend significantly on the presence of both intrinsic and extrinsic defects. Particularly, the Raman spectroscopy is very sensitive to the stoichiometry of LiNbO3 crystals, and this could result in the appearance of new lines or in the shift of some frequencies. Thus, only limited experimental information regarding the atomic displacements was available by means of isotope effect [20] or from the comparison of Raman spectra recorded for LiNbO3 and LiTaO3 [21]. The measured Raman frequencies [18,20,22] for congruent and nearly stoichiometric samples together with those calculated [11,23] for perfectly ordered LiNbO3 crystal are presented in Table I. Table I. Calculated and measured frequencies (cm -1 ) of four A1-TO modes in LiNbO3 calc. 7 LiNbO3 exp. 7 LiNbO3 calc. 6 LiNbO3 Exp. 6 LiNbO3 [23] [11] [20] [18] [22] [23] [20] TO1 208 239 256 251 251 208 256 TO2 279 320 275 273 273 299 289 TO3 344 381 332 331 331 344 TO4 583 607 632 631 631 583 637 The agreement of our calculations and experimental values is very good for TO2 and TO3, which thus are largely harmonic. The average difference of 45 cm -1 between the computed and experimental frequency for TO1 and of 50 cm -1 for TO4 clearly indicates that these modes are highly anharmonic. A detailed discussion of the corresponding displacement patterns is reported in Ref. [23]. Also, in this paper the frequencies and eigenvectors of A2 and E modes are presented. The calculated normal vibration coordinates for these modes can be used in constructing the reliable lattice dynamics models for this system doped with various impurity ions. In the case of LiTaO3, we investigated the origin of the experimentally observed hardness of the Γ-TO3 frequency in LiTaO3 by a comparison with those in LiNbO3. The calculated Γ-TO frequencies within frozenphonon approach and the experimental data are presented in Table II.
GRADUIERTENKOLLEG MIKROSTRUKTUR OXIDISCHER KRISTALLE 13 Table II. Calculated and measured frequencies (cm -1 ) of four A1-TO modes in LiTaO3 calc. LiTaO3 exp. LiTaO3 present [24] [25] [26] [21] TO1 194 206 203 206 201 TO2 242 253 252 253 253 TO3 360 356 356 356 356 TO4 599 600 597 597 597 One can see that the TO3 and TO4 vibration modes are harmonic, whereas the TO1 and TO2 exhibit a certain degree of anharmonicity. Most significantly, the calculated TO3 frequency in LiTaO3 is higher than those in LiNbO3. Since both modes are largely harmonic, one can infer from the corresponding atomic displacement patterns, that a substantial contribution of Li movement in the TO3 mode of LiTaO3 - while absent in LiNbO3 - , leads to the observed feature of their Raman spectra. The potential technological applications of LiNbO3 can be considerably extended by its doping with various impurity ions. A theoretical insight of the ground-state properties of the doped LiNbO3 can be gained through supercell calculations which are quite demanding because of the low symmetry of the studied materials. Moreover, the influence of the impurities on its optical properties can be estimated from the analysis of the impurity states localized in the optical band gap. The spin-polarized results obtained for a 40 atom supercell spanned by [-111], [111] and [111] with Fe on Nb site were previously reported [27]. The total density of states calculated for undoped LiNbO3 and the partial density of states corresponding to Fe 3d energy levels are shown in Figure I. a) b) Fig. I. a) Total density of states for undoped LiNbO3; b) Partial density of states for 3d energy levels of a Fe impurity Apart from the hybridization of the impurity 3d states with those of Nb and O over an energy range of 7 eV, one can observe that several groups of the Fe states are localized in the optical band gap. This situation is different from the Fe doped KNbO3 (see, e.g., [28]), where a small band gap preclude any analysis of its optical properties. However, a more realistic calculations must be based on the experimental observation that an impurity in an insulator matrix is charge compensated. Assuming a self-compensated pair CrNb-CrLi, such calculations for Cr impurities in a 2x2x2 supercell are now in progress. Literatur [1]. E. Krätzig, O. F. Schirmer: In Photorefractive Materials and Their Applications I, ed. by. P. Günter, J.-P. Huignard (Springer, Berlin, Heidelberg 1988) pp. 131. [2]. M. E. Lines, Phys. Rev. 177, 797 (1969). [3]. M. E. Lines, Phys. Rev. 177, 812 (1969). [4]. H. J. Bakker, S. Hunsche, and H. Kurz, Phys. Rev. B 48, 9331 (1993).
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