Ab initio formation volume of charged defects

When the formation volume of charged defects is evaluated by a straightforward cell minimization, the obtained ab initio formation volume is affected by a bias. Furthermore, the error does not vanish with increasing supercell sizes. The quantity to be minimized with respect to volume is not the defective supercell energy, but rather the formation energy of the charged defect. The formation energy of a charged defects contains additional correction terms, which have a non-vanishing derivative with respect to the volume. Surprisingly, the usually predominant electrostatic correction is shown to have almost no effect on the formation volume, whereas the small potential alignment correction is demonstrated to yield a huge correction that does not vanish for infinitely large systems.

Figure 1

Formation volume of a tetrahedric self-interstitial (Si${}_{\mathrm{Tet}}^{}$, circle symbols), a split self-interstitial (Si${}_{\mathrm{split}\langle 110\rangle }^{}$, triangle symbols), and a vacancy (V${}_{\mathrm{Si}}^{}$, square symbols) as a function of charge state, obtained by the straight minimization of the stress in a $216\pm 1$ atom silicon supercell.

Figure 2

Convergence of the formation volume of a tetrahedric self-interstitial atom Si${}_{\mathrm{Tet}}^{}$ in silicon as a function of supercell size for charge state 2$+$ (squares), 1$+$ (circles), and 0 (diamonds). The open symbols report the uncorrected volumes, the striped symbols the volumes corrected for the electrostatics, and the full symbols the volumes corrected for the potential alignment.

Figure 3

Formation volume of a tetrahedric self-interstitial (Si${}_{\mathrm{Tet}}^{}$, circle symbols), a split self-interstitial (Si${}_{\mathrm{split}\langle 110\rangle }^{}$, triangle symbols), and a vacancy (V${}_{\mathrm{Si}}^{}$, square symbols) as a function of charge state, obtained by the minimization of the stress correcting for the potential alignment pressure in a $216\pm 1$ atom silicon supercell.