The Effects of the Electron-Phonon Interaction on the Vibrational Anomalies and Polymorphism in Titanium
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at. Res. Soc. Symp. Proc. Vol. 491 © 1998 Materials Research Society
the orthogonal tight-binding approach. We demonstrate the consistency of the phonon frequencies obtained by molecular dynamics and FP calculations. We also analyse how the phonon properties are coupled to the details of the electronic spectrum of the system. Calculation procedure We employ the frozen phonon method [4,5] for the calculation of phonon frequencies and third- and forth-order phonon force constants, and in the case of ct-Ti we compare these results with the frequencies obtained from (NVE) molecular dynamics simulations by evaluating the power spectrum of the projected velocity autocorrelation function [6]. Both methods assume Born-Oppenheimer (full adiabatic) approximation, in which the electronic density of states instantly follows the changes in atomic positions. For the energy and force calculation we use the semiempirical orthogonal tight-binding (TB) method within the kspace formalism and within the O(N) bond order potential (BOP) method implemented in the OXON code [7]. The parameters for the hopping integrals and the pairwise repulsive part were developed by Girschik and co-authors [8], and have been fitted to reproduce the equilibrium lattice and elastic constants for ct-Ti. Originally TB parameters were fitted with fixed number of moments within BOP formalism (9 moments in the electron DOS expansion) and finite electron temperature of 0.3 eV [8]. In order to control the effects of finite number of moments in the DOS evaluation, we have tested the parametrization within the k-space method and zero electron temperature, and found it to give similar cohesive energies and static geometries to BOP for
ct-,
0-, fcc, and w-phases of titanium. The static energies
give the following sequence in stability of titanium polymorphs: a (ground state), w (almost degenerate to ct), f cc, 3, which is in agreement with the full potential muffin-tin local density functional calculations. However, the calculated static lattice constant for 3-Ti, a6 = 3.20 Sis lower than the experimental value aXP = 3.28 A. This results in the predicted density of the 3-phase being the same as of the high-pressure w-phase, while experimentally the latter is higher. This discrepancy is due to the absence in the TB scheme of the explicit environmentally-dependent terms, and thus a poor account for the variation of interactions with a change in local coordination. Having these limitations, the TB parametrization developed in [8] reproduces reliably the static and, as we show in this paper, important dynamic properties of different titanium polymorphs. In the BOP method we have used 13 moments in the DOS expansion and an electron temperature 0.1 eV. We also show in this paper that this level of approximation has little effect on the accuracy of the evaluated phonon frequencies and higher order force constants. As the crystalline lattice distorts, so does the charge density, causing the changes in bonding energy. In tight-binding calculations this is reflected by a c
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