Combined LDA and LDA-1/2 method to obtain defect formation energies in large silicon supercells

A source of uncertainty in the state of the art calculations of defect levels is the inaccurate prediction of band-gap energies. Several approaches were developed to surpass this problem. However, another source of uncertainty remains: the small number of clustered atoms imposed by the computational restrictions. In this work, the LDA-1/2 method is explored in an attempt to overcome both problems with a small computational cost. We considered the self-interstitial defects in silicon as a benchmark for calculating defect states and charge-transition levels of point defects in semiconductors. We found neutral formation energies, including reaction barriers, of 4.65, 4.49, and 4.87 eV, for hexagonal, split $\langle 110\rangle$ and $C_{3v}$ configurations, respectively, in good agreement with most experimental results.

Figure 1

Difference between the total local potential of the defect-containing and the defect-free supercell along the diagonal line of a cubic 216-atom supercell, for a tetrahedral silicon self-interstitial defect with charge 2+. Far away from the defect atom, the local potential difference becomes constant and its value 0.1681 eV gives the correspondent $\Delta V$ correction.

Figure 2

Scheme of sequential electron addition from the 2$+$ to neutral charge states. $E_{\mathrm{rel}}$(q) is the relaxation energy for two different geometric configurations with same charge $q$, and $A(q\to q-1)$ is the electron addition energy from charge state $q$ to $q-1$ in the same geometric configuration.

Figure 3

Ball-and-stick models of the different geometric configurations of silicon self-interstitial defects (orange balls) used in this work.

Figure 4

Formation energies as a function of Fermi level for Si${}_i$ defect, 217-atom supercell, LDA-1/2 (solid lines) and LDA (dashed lines) results. The Fermi level is varied throughout the LDA-1/2 band gap, and vertical dashed lines indicate the LDA band-gap limits. The slopes of the segments indicate the charge states.