Large effects of subtle electronic correlations on the energetics of vacancies in $\alpha$-Fe

We study the effect of electronic Coulomb correlations on the vacancy formation energy in paramagnetic $\alpha \text{−}\mathrm{Fe}$ within ab initio dynamical mean-field theory. The calculated value for the formation energy is substantially lower than in standard density-functional calculations and in excellent agreement with experiment. The reduction is caused by an enhancement of electronic correlations at the nearest neighbors of the vacancy. This effect is explained by subtle changes in the corresponding spectral function of the $d$ electrons. The local lattice relaxations around the vacancy are substantially increased by many-body effects.

Figure 1

The $2\times 2\times 2$ and $3\times 3\times 3$ supercells with the vacancy in the corner. Different colors indicate the atom nearest to the vacancy (red), the second nearest (purple), and the furthest (the central atom, yellow). Generated with xcrysden [52].

Figure 2

Vacancy formation energies calculated by different methods (GGA and DFT+DMFT) in the different setups: small and large supercell, relaxed or not, nonmagnetic, paramagnetic (PM) or ferromagnetic (FM). Calculations are performed at the equilibrium volume for the relevant setup. The average of experimental values is shown for comparison [36, 37, 38, 39, 40].

Figure 3

Difference in the interaction energy per atom, before and after adding a vacancy. Inset: Fe $3d$ charge in the cell with vacancy.

Figure 4

Imaginary part of the Matsubara self-energies for (a) the vacancy nearest neighbor and (b) central atom in the $3\times 3\times 3$ supercell with a vacancy present (red, full) or without it (blue, dashed). Correlations become stronger on the atom nearest to the vacancy, while the difference between $e_g$ and $t_{2g}$ is strongly reduced. (c) LDA density of states around the Fermi level and (d) hybridization function on the Matsubara axis for the nearest neighbors (blue, full) to the vacancy, and for the central atom (black, dashed). The full Fe $3d$ DOS is shown in the Supplemental Material [35].

Figure 5

Total energy vs distance to the vacancy (as a fraction of the lattice parameter) for the first nearest neighbor (blue) and the second nearest neighbor (red), in DFT+DMFT (full line) and GGA (dashed line). The black arrows show the position of the atoms in the unrelaxed bcc supercell.