Ab initio investigation of lattice dynamics of fluoride scheelite
Abstract
We report on the phonon dynamics of LiYF4 obtained by direct method using first principle calculations. The agreement between experimental and calculated modes is satisfactory. An inversion between two Raman active modes is noticed compared to inelastic neutron scattering and Raman measurements. The atomic displacements corresponding to these modes are discussed. Multiple inversions between Raman and infrared active groups are present above 360 cm-1. The total and partial phonon density of state is also calculated and analyzed.
Introduction
Researches on YLiF4 crystals are strongly linked to laser technology. The first structural data obtained by Thomas et al. date from 1961[1], just one year after the demonstration of the first laser. At ambient pressure, the crystal cell of YLiF4 is tetragonal with space group I41/a (C4h6). This phase is commonly named the scheelite structure in reference of the CaWO4 crystal. Lithium ions (Li+) are in the center of tetrahedrons composed by 4 fluoride ions (F-). Yttrium ions (Y3+) are in the center of polyhedrons composed by 8 F-. Y3+ can be substituted by rare earth presenting an oxidation state of +3, such as Erbium (Er+3) [2] or Thulium (Tm+3) [3], providing good matrix for upconversion laser. The efficiency of this kind of laser relies on intraionic and interionic process of relaxation that strongly depends on the host matrix[4].
This relationship is evidenced particularly by the multiphonon relaxation process implying electron-phonon coupling[5]. Consequently, a fair knowledge of the structural and dynamics properties of the host matrix is crucial for the development of host matrix. To this end many studies have been carried out on the subject. Phonon frequencies were measured by Raman and IR spectra [6][7][8][9][10]. These methods give information at the center of the Brillouin zone. Inelastic neutron scattering measurement is needed to obtain the complete phonon dispersion curves that are essential to a good understanding of the global vibrational and relevant properties. This was done for LiYF4 by Salaün et al.[11]. Besides experimental work, numerical methods have been developed. Among them we can notice empirical methods, such as rigid ion model (RIM). Using this method, Salaün et al[11] and Sen et al.[12] performed lattice dynamical calculations on LiYF4 providing a large number of interesting results about lattice vibration. Obviously, the correctness and precision of this model is limited by the empirical parameters. Density functional theory is an empirical free parameter methods whose usefulness and predictive ability in different fields[13][14] are known since a long time. Recently, the association of DFT with different techniques such as linear response method[15][16] or direct methods[17][18][19] allows to evaluate phonon dispersion curves without empirical parameter. In particular Parlinsky et al.[20][21] developed a direct method where the force constant matrices are calculated via the Hellmann-Feynman theorem in total energy calculations.
In this work we present a first principle investigation of YLiF4 in its scheelite phase. DFT associated with projector augmented wave (PAW) and direct method were used. Cell parameters, phonon dispersion curve, phonon density of state are discussed and compared with previous experimental or numerical results. To our knowledge, this is the first ab initio calculation of LiYF4 lattice dynamics.
Methodology
Cell parameter and atomic positions of the initial structure were obtained from experimental results by E. Garcia and R.R Ryan[22]. All calculations were carried out with the VASP[23] code, based on DFT [24][25], as implemented within MEDEA[26] interface. Here the generalized gradient approximation (GGA) through the Perdew Wang 91 (PW91)[27] functional and projector augmented wave (PAW)[28] were employed for all calculations. Electronic occupancies were determined according to a Methfessel-Paxton scheme[29] with an energy smearing of 0.2 eV. The crystal structure was optimized without the constraints of the space group symmetry at 0 Gpa until the maximum force acting on each atom dropped below 0.002 eV/Å. The self consistent field (SCF) convergence criterion was set at 10-6 eV. High precision calculations, as defined in VASP terminology, were performed with a basis set of plane wave truncated at a kinetic energy of 700eV. The Pulay stress[30] obtained on the unit cell was -4 MPa and the convergence of the total energy was within 0.4 meV/atom compared to an energy of 750 eV. Brillouin zone integrations were performed by using a 3X3X3 k-points Monkorst-Pack[31] grid leading to a convergence of the total energy within 0.1 meV/cell compared to a 7x7x3 k-point mesh. PHONON code[19], based on the harmonic approximation, as implemented within MEDEA[26] was used to calculate the phonon dispersion. From the optimized crystal structure, a 2X2X1 supercell consisting of 96 atoms, was generated from the conventional cell to account for an interaction range of about 10 Å. The asymmetric atoms were displaced by +/- 0.03 Å leading to 14 new structures. The dynamical matrix was obtained from the forces calculated via the Hellmann-Feynman theorem. G point and medium precision, as defined in VASP terminology, were used for theses calculations. The error on the force can perturb the translation-rotational invariance condition. Consequently, this condition has to be enforced. A strength of enforcement of the translational invariance condition was fixed at 0.1 during the derivation of all force constants. The longitudinal optical (LO) and transversal optical mode (TO) splitting was not investigated in this work. Consequently, only TO modes at the G point were obtained.
Results and discussion:
Structural parameters.
Shows calculated and previous experimental or numerical structural properties of LiYF4. Compared to the most recent experimental data[32][33] our calculated volume is over-estimated. Nevertheless, the c/a axial ratio, whose evolution is significant in pressure induced transition phase, is close to experimental results. DFT results are strongly dependent on the approximation of the exchange correlation term. It's known that local density approximation (LDA) favorizes high electron densities resulting in short bonds prediction and so in low equilibrium volume. Results obtained by Li et al.[35] and Ching et al. [36] owing to LDA illustrate this behavior. GGA corrects and sometimes over-corrects the failures of the LDA. That's why cell parameters obtained using PW91 differ from experimental results. At least two reasons explain why our results are at variance with Li et al.[35]. The first one is due to the utilization of different parameters such as the energy cut off. The other one can be attributed to the difference of method to evaluate the equilibrium volume. Indeed, during a structure optimization the convergence criterion is set on the stress.
Lattice dynamic.
The phonon dispersion curves along several lines of high symmetry for LiYF4 structure at zero pressure are shown in Figure 1. To evaluate our calculated phonon dispersion curves, the acoustic branches will be first compared to results extracted from ultrasonic measurements and rigid ion models (RIM). Then the modes at the G point will be compared to experimental results obtained from Raman, IR or neutron scattering and RIM. Velocities of sound following different directions of propagation have been evaluated from the slopes of acoustic branches. Our results and experimental ultrasonic velocities at 4.2 K are presented in Table 3. The difference between calculated and measured velocities lies within 5% for 7 velocities out of 8. The 9% of discrepancy is obtained for the acoustic branch following the [001] direction. In this direction the longitudinal acoustic branches are non-degenerate although the modes at the Z point are degenerate. This behavior has been observed on the two phonon dispersion curves calculated with RIM but seems absent from neutron scattering experiments. Concerning the phonon modes, the spectrum contains 36 phonons modes at the G point as expected from the number of atoms per primitive cell.
Conclusion
This work presents at our knowledge, the first ab initio lattice dynamics calculation of fluoride scheelite. Concerning the phonon dispersion curves, satisfactory agreement with inelastic neutron scattering measurement was obtained. Discrepancies between sound velocities calculated from acoustic branches and ultrasonic measurement do not exceed 300m.s-1. Moreover, at the center of the Brillouin zone the error on Raman active modes calculated compared to experimental results does not exceed 9%, the most important error being 33 cm-1. One inversion between the last Bg mode and the fourth Eg mode was put in evidence in comparison with experimental results. Concerning infrared active modes error lies within 8%, the most important error being 25 cm-1. Below 360 cm-1, only one inversion can be notice compared to experimental results, which is less than in RIM calculations. Important differences between ab initio and RIM calculated DOS were put in evidence mainly above 500 cm-1.
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