Point defect modeling in materials: Coupling ab initio and elasticity approaches

Modeling point defects at an atomic scale requires careful treatment of the long-range atomic relaxations. This elastic field can strongly affect point defect properties calculated in atomistic simulations because of the finite size of the system under study. This is an important restriction for ab initio methods which are limited to a few hundred atoms. We propose an original approach coupling ab initio calculations and linear elasticity theory to obtain the properties of an isolated point defect for reduced supercell sizes. The reliability and benefit of our approach are demonstrated for three problematic cases: the self-interstitial in zirconium, clusters of self-interstitials in iron, and the neutral vacancy in silicon.

Figure 1

Structures of the stable SIA configurations in hcp Zr: octahedral (O), basal octahedral (BO), split dumbbell (S), basal split dumbbell (BS), crowdion (C), and buckled basal crowdion (BC’). PS and PS’ are obtained by a rotation of angle $\varphi ={30}^{\circ }$ and ${50}^{\circ }$ of S in the prismatic plane.[29]

Figure 2

Formation energies $E^{\mathrm{f}}$ of the four most stable SIAs in Zr versus supercell size. Solid symbols refer to ab initio uncorrected results and open symbols to the results corrected by the elastic model. The periodicity vectors of the supercell have been kept either fixed (squares) or relaxed (triangles) in the ab initio calculations.

Figure 3

(a) Uncorrected and (b) corrected Zr SIA formation energies $E^{\mathrm{f}}$ of the stable configurations versus the number of atoms for $\varepsilon =0$ calculations.

Figure 4

Migration pathways of Zr SIA calculated with the NEB method, between the BO and BC’ configurations and between the BS, BO, and O configurations: (a, c) uncorrected and (b, d) corrected results.

Figure 5

Formation energy of a SIA cluster containing eight interstitials in bcc iron calculated for fixed periodicity vectors ($\varepsilon =0$) or at zero stress ($\sigma =0$) for different sizes of the simulation cell: (a) C15 aggregate and (b) parallel-dumbbell configuration with a $\langle 111\rangle$ orientation. Atomistic simulations are performed either with the M07 empirical potential[7] (EAM) or with ab initio calculations (GGA). Solid symbols refer to uncorrected results and open symbols to the results corrected by the elastic model.

Figure 6

Vacancy formation energy $E^{\mathrm{f}}$ in silicon calculated with the LDA and HSE06 functionals, either for fixed periodicity vectors ($\varepsilon =0$) or at zero stress ($\sigma =0$). Solid symbols refer to the ab initio uncorrected results and open symbols to the results corrected by the elastic model. The vacancy configuration is displayed in the inset: the white sphere corresponds to the empty lattice site and the purple spheres represent its first nearest neighbors.