Electronic structure calculations of vacancy parameters in transition metals: Impact on the bcc self-diffusion anomaly

Abstract

ABSTRACT:The group dependence, i.e., the variation with the number of d valence electrons, of vacancy parameters in transition metals with the body-centered cubic (bcc) structure is investigated via a combination of electronic structure calculation techniques. A semiempirical tight-binding d-band approach is proposed that shows that the position of the Fermi level with respect to the pseudogap governs the sharp variations along a transition metal series of (i) the formation energy, (ii) the relaxation energy, (iii) the migration energy, and (iv) the electronic contribution to the formation and migration entropies. These predicted trends are confirmed by first-principles calculations in 5d bcc metals (bcc-Hf, Ta, and W) including structural relaxations within plane-wave pseudopotential computations performed on supercells containing up to 54 sites. The agreement with available experimental data is very conclusive. A fast version of the full potential linear muffin-tin orbital method is then used to show the weak influence of the method within density functional theory in the local density approximation and to generalize these results-without relaxation-to the 3d and 4d series. This data base allows us to test the validity for vacancy studies of spd tight-binding models proposed in the literature.

Bond-order potential for simulations of extended defects in tungsten

“…The convergence of the vacancy formation energy with the system size has been verified for supercells of sizes 4a ϫ 4a ϫ 4a and 5a ϫ 5a ϫ 5a, giving almost identical results. The atomic configuration was fully relaxed under the condition of constant volume but we found, in accordance with DFT first-principles calculations, 101 that the structural relaxation decreases the total energy only marginally. The vacancy formation energies calculated by FS, BOP, and BOP 0 are presented in Table V. In the case of the FS potential the vacancy formation energy is in fact a fitted quantity and thus it agrees closely with both the results of first-principles calculations and experimental data.…”

“…This suggests that predictions of the vacancy formation energy can vary significantly even for closely related models. It has been shown 101 that in simple TB schemes the vacancy formation energy depends to first order only on the ratio between the magnitude of the bond integrals and the magnitude of the repulsive pair interactions and can be easily adjusted. In more complicated schemes, such as spd-TB or screened BOP, the vacancy formation energy is determined from a subtle interplay of more contributions and a larger scatter of values is therefore likely unless the defect is included in the fitting database.…”

Impact of Electronic Structure Calculations on the Study of Diffusion in Metals

“…The defect parameters are calculated with the supercell method, using periodically repeated cells containing 53 atoms and one vacancy. [22,23] A very high accuracy is required in the Brillouin-zone integration: [21,24] the summation over reciprocal space vectors was performed using a dense mesh of ten special points and the Hermite±Gauss broadening scheme was employed to avoid broadening-width effects. [25] …”

“…[19] More precisely, it appears that the calculated values of the unrelaxed vacancy formation energy are systematically larger than the experimental values, the overestimate being larger in group V than in group VI. [21] The results presented below were obtained using the pseudo-potential technique. One advantage of this technique is that it is easy to compute the forces on the atoms and therefore to perform structural relaxations.…”

“…The simplest version of this model for transition metals consists in considering only the d valence electrons and to set the parameters describing the hopping integrals and the short-range repulsion to canonical values. [29] For a given value of the number of d electrons, N d , the number of independent parameters can be reduced to only two [21] Ðone each for the energy and length scalesÐas in the simplest empirical potentials such as the Lennard-Jones potential. The TB d band model is known to be quite robust and to account for most trends along the transition metal series such as the sequence of structural stability (hcp±bcc± hcp±fcc) or the non-parabolic behavior with band-filling of the elastic constants or phonon frequencies.…”