Phonon Dispersion and Thermodynamics Properties of CaF 2 via Shell Model Molecular Dynamics Simulations

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1 Commun. Theor. Phys. (Beijing, China) 51 (29) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 5, May 15, 29 Phonon Dispersion and Thermodynamics Properties of CaF 2 via Shell Model Molecular Dynamics Simulations CHENG Yan, 1 HU Cui-E, 1,2 ZENG Zhao-Yi, 1 GONG Min, 1, and GOU Qing-Quan 2 1 College of Physical Science and Technology, Sichuan University, Chengdu 6164, China 2 Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 6165, China (Received June 23, 28) Abstract The phonon and thermodynamics properties of face-centered cubic CaF 2 at high pressure and high temperature are investigated by using the shell model interatomic pair potential within General Utility Lattice Program (GULP). The phonon dispersion curves and the corresponding density of state (PDOS) in this work are consistent with the experimental data and other theoretical results. The transverse optical (TO) and longitudinal optical (LO) mode splitting as well as heat capacity at constant volume C V and entropy S versus pressure and temperature are also obtained. PACS numbers: 21.6.Cs, Qg, Dj, 65.4.Gr Key words: shell model, molecular dynamics, phonon dispersion, thermodynamic property, CaF 2 1 Introduction CaF 2, fluorite, is a prime candidate for a replacement of SiO 2 in metal-oxide-semiconductor (MOS) technology, due to its excellent lattice matching to silicon, with only.6% mismatch at room temperature. [1] Meanwhile, it is also proposed to have an excellent internal pressure calibration in moderate high-pressure and high-temperature x-ray diffraction experiments. [2 4] In addition, CaF 2 is a prototype ionic conductor showing a strong increase of the conductivity with temperature that saturates at 142 K. [5] The pressure-induced phase transformations of CaF 2 from face-centered cubic fluorite structure (with space group Fm3m) to an orthorhombic PbCl 2 -type structure (with space group Pnma) occurs at 8 1 GPa. [6 9] The melting temperature of face-centered cubic structure CaF 2 is 1633 K [1] or 1691 K. [11] Phonons are the primary excitations, which influence the thermodynamic behavior. Therefore, a systematic characterization of the phonon density of states (PDOS) and dispersion relations of alkaline-earth fluorides is highly desirable. There are many experimental and theoretical characterizations about the phonon characteristic of CaF 2. [12 16] As early as 197, Elcombe and Pryor [12] obtained the phonon spectrum from the inelastic neutron scattering on a triple-axis spectrometer experiment. Merawa et al. [13] evaluated the IR and Raman central zone phonon frequencies of CaF 2 by using the periodic ab initio linear combination of atomic orbital method in several different approximations. Schmalzl et al. [14] reported the phonon frequencies (dispersion and density of state) both experimentally and theoretically. Later, Verstraete and Gonze [15] studied the dielectric, and vibrational properties of CaF 2 from first principles using density functional theory. Recently, Ricci et al. [16] showed that the phonon confinement has been pointed as the main cause of the broadening and the blue-shift of the Raman peak in CaF 2. In this paper, we focus on the phonon properties, including the phonon density of state and phonon dispersion curves, of fluorite at high pressure and high temperature by using shell model with interatomic pair potential. Since the phonon properties of solids provide an important link with thermodynamic behaviors of crystals, thus we have also obtained some thermodynamic properties of CaF 2. All calculations are implemented through the General Utility Lattice Program (GULP) Code, [17] by which we have successfully investigated the elastic properties of GaN with hexagonal wurtzite structure. [18] The results obtained are well consistent with the available experimental data and other theoretical results. In Sec. 2, we make a brief review of the theoretical method. The results and some discussion are presented in Sec Theoretical Method The atomistic simulation method is based on the Born model of solids, where interatomic potential functions are defined to model the long-range attractive and short-range repulsive forces acting between atoms or ions in the solid. The contributions of the overlap repulsion and dispersion forces, between ions i and j at separation r ij, are described by a short-range Buckingham potential, which includes an exponential repulsive term and an attractive dispersion term: ( rij ) V ij = A ij exp C ij ρ ij Rij 6. (1) On the other hand, long-range electrostatic interactions The project supported by the National Natural Science Foundation of China under Grant No Corresponding author, mgong@scu.edu.cn

2 No. 5 Phonon Dispersion and Thermodynamics Properties of CaF 2 via Shell Model Molecular Dynamics Simulations 95 are described by the Coulomb law, Vij Coulomb (r ij ) = e2 Z i Z j, (2) r ij where r ij is the interatomic distance between ions i and j, which have effective charge, Z i and Z j, respectively, e is the electron charge, A ij, ρ ij, and C ij are empirical parameters used to describe the interaction between ions. To account for the ionic polarization, the charge on the atom is split into a core of charge X i and a massless shell of charge Y i. The sum of the core charge is q i = X i + Y i. The core and the shell of the ion i are coupled by a harmonic force with spring constant k i. Thus, the polarization energy is given by V core-shell ij = 1 2 k ir 2 ij, (3) where r ij is the relative displacement of the core and shell. Atoms must constantly be in motion as a consequence of the Heisenberg uncertainty principle and this is achieved through vibrations. In the case of an infinitely perfect 3D solid, there will be a corresponding infinite number of phonons. These phonons are described by calculating their values at points in reciprocal space, usually within the first Brillouin zone. There will be 3N phonons per k-point. The lowest three modes represent the socalled acoustic branch, which tend to values of zero at the centre of the Brillouin zone (k =,, ), known as the -point. At this point, the acoustic modes correspond to the pure translation of the crystal lattice, and thus they are modes of zero frequency. The phonon density of states was obtained by integration across the Brillouin zone taking into account its symmetry. For performing the integration, a standard scheme developed by Monkhorst and Pack [19] was used for choosing the grid points. This is based around three so-called shrinking factors, n 1, n 2, and n 3 one for each reciprocal lattice vector. The heat capacity at constant volume C V and entropy S are obtained from the total phonon density of state (PDOS) using the following equations: ( ω ) 2 exp( ω/kb T) C V (T)=RnN g(ω) dω, (4) k B T [exp( ω/k B T) 1] 2 S(T) = RnN T g(ω) 1 ( ω ) 2 T k B T exp( ω/k BT) dtdω, (5) [exp( ω/k B T) 1] where, k B, and R are the Planck, Boltzmann, and universal gas constants, respectively, n is the number of atoms in the formula unit, and N is the number of formula units in a cell. In our work, MD method is applied to investigate the lattice constant, phonon and thermodynamics properties of face-centered cubic CaF 2 in the range of 1 GPa and 3 16 K. We make a calculation for a super cell, which contains 768 atoms (256 Ca atoms and 512 F atoms), in the NPT ensemble (constants N, P, and T represent the number of particles, pressure and temperature, respectively). The results of shell-model MD simulations in the NPT ensemble depend on the initial arrangement of atoms, time steps, number of atoms, simulation time and pressure fluctuations. The influence of these parameters was carefully studied by carrying out test run at various temperatures and pressures. It was found that the correct results can normally be obtained with step time.1 ps, equilibration time.5 ps and production time.5 ps. 3 Results and Discussion To find out the most stable structure under GPa and 3 K, we firstly optimize the crystal structure parameters to yield the minimum total energy of the crystal. Note that the cubic calcium fluoride with space group Fm-3m is described by only one lattice constant a. The calculated lattice parameter a is Å, consistent with the experimental data Å [2] or Å [21] and theoretical value Å. [22] We present the necessary parameters for our calculation in Table 1. The normalized lattice constant a/a and volume V/V of CaF 2 at high pressure and high temperature are presented in Fig. 1. It is noted that as the pressure increases, the relative lattice constant a/a and volume V/V decreases and the effect of temperature is opposite to that of the pressure. Table 1 Interatomic pair potential parameters for CaF 2. Atom types A (ev) ρ (Å) C (ev/å 6 ) k (ev/ Å 2 ) Y (e) Cut-off (Å) Ca c F s F s F s F c F s The quantification of the charges associated with atoms/ions is one of the most problematic tasks in theoretical chemistry and physics. There are many different definitions of charge, the most famous of which is the Born effective charge. The Born effective charges are also especially important for phonon properties. Because of the cubic symmetry, Born effective charges for a given atom are all diagonal and have the same value along x, y, and z. The calculated effective charge for Ca ion is Schmalzl et al. [14] reported the effective charge by Hartwigsen Goedecker Hutter pseudopotential calculation within ABINIT. Verstraete et al. [15] ob-

3 96 CHENG Yan, HU Cui-E, ZENG Zhao-Yi, GONG Min, and GOU Qing-Quan tained the Born effective charge 2.18 using the implements density-functional perturbation theory within ABINIT. All the theoretical values of the effective charge are larger than the nominal ionic charge of ZCa = 24. It seems to indicate that there is an electronic antiscreening rather than a screening effect for the Γ (LO) and Γ (TO) vibrations. The pressure and temperature dependence of effective charges are presented in Fig. 2. The pressure-induced reduction of the dynamical ion charges indicates charge redistribution from the fluorin ion to the calcium ion in comparison with the pressure-free situation. When temperature increases, electron transfer occurs from calcium ion to fluorin ion. Vol. 51 observation of all phonon branches along the main symmetry directions. We present the observed phonon dispersion curves and corresponding PDOS at GPa and 3 K in Fig. 3. All calculated branches of the phonon spectrum yield positive frequencies, which indicates the stability of the potential model. The Raman-mode frequency (ωraman ) is slightly larger than that of the infrared TO mode (ωto ). This is indeed observed. The ωlo, ωraman, and ωto at Γ point in this work are mev, mev and mev respectively. Compared with other theoretical results[14,15] and the experimental data,[12,14,23 26] the agreement is very satisfactory. The main disagreement occurs at Γ (Raman) for the first optical mode. All these results are listed in Table 2. Fig. 2 The Born effective charge versus pressure at 3 K (a) and temperature at GPa (b). Fig. 1 Calculated ratios a/a, V /V versus pressure P at 3 K (a) and temperature T at GPa (b). With three atoms per unit cell, we can obtain nine phonon branches. The phonon calculations along several high-symmetry directions Γ-M (Σh11i), M-R ( h1i) and R-Γ (Λh111i) connect the high-symmetry points Γ(,, ), M (.5,.5, ), and R (.5,.5,.5) of the primitive cubic Brillouin zone. The orientation allows the We think that the small difference in frequencies is caused by two main factors: the lattice parameter and the ionic polarization. The phonon frequencies are very sensitive to the lattice parameter a. The calculated lattice constant is A, which has a slight difference with the experimental values. This is because when we optimized to obtain the stability structure, the zero point energy is neglected in energy minimization. The other main source of discrepancy is the ionic polarization. We use a relative simple spring potential to describe the polarization energy, but in fact, the interaction core and shell is very complex. Table 2 Comparison of our calculation of point frequencies at GPa and 3 K with the experiments and other calculations. Present calc. hωlo (mev) hωraman (mev) hωto (mev) Other calc. Expt (LDA)[14] 58.65[14] (LDA) / (HGH)[15] [23] (LDA)/38.42 (GGA)[14] 39.6,[14] [12] (LDA)/37.58 (HGH)[15] ,[24] 4.63[25] (LDA)/29.99 (GGA)[14] 32.28,[14] [25] (LDA)/ (HGH)[15] ,[12] [26]

4 No. 5 Phonon Dispersion and Thermodynamics Properties of CaF2 via Shell Model Molecular Dynamics Simulations 97 Fig. 3 The phonon dispersion curves and corresponding PDOS at GPa and 3 K. ωto and the LO/TO splitting versus pressure and temperature. It is found that as pressure increases, the blue-shift occurs to ωlo and ωto, but the enhancive speed of ωto is slower than that of ωlo. When pressure increase from GPa to 1 GPa at 3 K, ωlo and ωto increase and mev, respectively. So the frequency of LO/TO splitting is decrescent versus the pressure. It means that the ionicity weakens versus the increase of pressure, as also shown in Fig.2. When the temperature increases, the red-shift can be seen. When temperature increase from 3 K to 16 K at GPa, ωlo and ωto increase and mev, respectively. Fig. 4 Calculated phonon frequency of LO, TO and LO-TO versus pressure at 3 K (a) and temperature at GPa (b). At the Γ point, there is an extra complication in the calculation of the phonons. In crystals the degeneracy of the transverse optical (TO) and longitudinal optical (LO) modes is broken due to the electric field that is generated during vibration. CaF2 is a typically ionic compound. So the polarity causes the intensive LO/TO splitting. The splitting also depends on the direction of approach to the -point in reciprocal space. In Fig. 4, we present the ωlo, Fig. 5 Calculated entropy S and heat capacity CV versus pressure at 3 K. The calculated CV and S at GPa and 3 K is J mol 1 K 1 and J mol 1 K 1. The calculated heat capacity at constant volume CV and entropy S versus pressure and temperature are represented in Figs. 5 and 6, respectively. It is obvious that when the pressure is enhanced, the calculated S and CV decreased, while they are enhanced with the increase of the temperature, which accord with the second law of thermodynamics. It is readily seen that, the CV approaches approximately to

5 98 CHENG Yan, HU Cui-E, ZENG Zhao-Yi, GONG Min, and GOU Qing-Quan Vol. 51 the Dulong Petit limit at higher temperatures. Fig. 6 Calculated heat capacity C V and entropy S versus temperature at GPa. In summary, we have investigated the phonon and thermodynamics properties of face-centered cubic CaF 2 at high pressure and high temperature by using the shell model interatomic pair potential within General Utility Lattice Program (GULP). The phonon dispersion curves and corresponding phonon density of state (PDOS) in present work are consistent with the experimental data and other theoretical results. The LO-TO splitting, heat capacity at constant volume C V and entropy S versus pressure and temperature are also obtained successfully. Acknowledgments The authors would like to thank Prof. J.D. Gale for his providing us the GULP code. References [1] J.E. Ortega, F.J. Garca de Abajo, P.M. Echenique, D. Ochs, S.L. Molodtsov, and A Rubio, Phys. Rev. B 58 (1998) [2] R.J. Angel, J. Phys.: Condens. Matter 5 (1993) L141. [3] R.J. Angel, D.R. Allan, R. Miletich, and L.W. Finger, J. Appl. Crystallogr. 3 (1997) 461. [4] R. Miletich, D.R. Allan, and W.F. Kuhs, Rev. Mineral. Geochem. 41 (21) 445. [5] J.B. Boyce and B.A. Huberman, Phys. Rep. 51 (1979) 189. [6] L. Gerward, J. Staun Olsen, S. Steenstrup, M. Malinowski, S. Asbrink, and A. Waskowska, J. Appl. Cryst. 25 (1992) 578. [7] S. Speziale and T.S. Duffy, Phys. Chem. Miner. 29 (22) 465. [8] V. Kanchana, G. Vaitheeswaran, and M. Rajagopalan, Physica B 328 (23) 283. [9] F.S. El kin, O.B. Tsiok, L.G. Khvostantsev, and V.V. Brazhkin, J. Exp. Theor. Phys. 1 (25) 971. [1] Y. Ida, Phys. Rev. 187 (1969) 951. [11] P.W. Mirwald, J. Phys. Chem. Sol. 39 (1978) 859. [12] M.M. Elcombe and A.W. Pryor, J. Phys. C 3 (197) 492. [13] M. Merawa, M. Llunell, R. Orlando, M. Gelize-Duvignau, and R. Dovesi, Chem. Phys. Lett. 368 (23) 7. [14] K. Schmalzl, D. Strauch, and H. Schober, Phys. Rev. B 68 (23) [15] M. Verstraete and X. Gonze, Phys. Rev. B 68 (23) [16] P.C. Ricci, A. Casu, G.De Giudici, P. Scardi, and A. Anedda, Chem. Phys. Lett. 444 (27) 135. [17] J.D. Gale, J. Chem. Soc. Faraday Trans. 93 (1997) 629. [18] Y. Cheng, Y.J. Tu, Z.Y. Zeng, and Q.Q. Gou, Commun. Theor. Phys. (Beijing, China) 5 (28) [19] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13 (1976) [2] X. Wu, Z.Y. Wu, L. Guo, C. Liu, J Liu, and X.D. Li, Solid State Commun. 135 (25) 78. [21] E.A. Zhurova, B.A. Simonov, and V.I. Sobolev, Kristallogr. 41 (1996) 438. [22] R. Khenata, B. Daoudi, M. Sahnoun, H. Baltache, M Rérat, A.H. Reshak, B. Bouhafs, H. Abid, and M. Driz, Eur. Phys. J. B 47 (25) 63. [23] W. Kaiser, W.G. Spitzer, R.H. Kaiser, and L.E. Howarth, Phys. Rev. 127 (1962) 195. [24] J.P. Russel, Proc. Phys. Soc. London 85 (1965) 194. [25] P. Denham, G.R. Field, P.L.R. Morse, and G.R. Wilkinson, Proc. R. Soc. London Ser. A 317 (197) 55. [26] R.P. Lowndes, J. Phys. C 3 (1971) 383.