Finite-Size Supercell Correction for Charged Defects at Surfaces and Interfaces

A finite-size supercell correction scheme is introduced for the formation energy of charged defects at surfaces and interfaces. The scheme combines classical electrostatics with the dielectric profile and the electrostatic potential extracted from the electronic-structure calculation. Spurious electrostatic interactions are removed while retaining the dielectric and quantum-mechanical features of the system of interest, which may have no interface (bulk), a single interface or surface, or two interfaces. A pertinent extrapolation scheme validates the proposed corrections. Applications to the charged Cl vacancy at the surface of NaCl and to the dangling bond at the Si(100) surface show that the corrected formation energies are largely independent of the supercell dimensions and of the size of the vacuum region.

Figure 1

(a) System geometry used for modeling a Cl vacancy in the $+1$ charge state at the surface of NaCl, with adopted notation. Uncorrected and corrected formation energies vs (b) lateral width of the supercell with fixed $w_{\mathrm{mat}}=2$ and $w_{\mathrm{vac}}=2$ and (c) size of the vacuum region with fixed $w_{\mathrm{mat}}=2$ and $L_x\times L_y=3\times 3$. Distances are given in units of $a_{\mathrm{lat}}=5.66\AA$.

Figure 2

Electrostatic energies for a Gaussian charge distribution at the surface (triangles) and in the middle (circles) of a material slab with dielectric constant ($\epsilon =4$), as obtained by uniformly scaling all dimensions in the supercell. The values corresponding to the isolated systems are obtained separately (squares) and can be compared to the linear extrapolations (dashed lines) fitted to the last four points. Scaling only the vacuum region (crosses) or only the lateral dimension (plusses) of the supercell leads to divergences.

Figure 3

(a) Plane-averaged distributions of the model charge and of the Cl vacancy defect charge as found in the DFT calculation compared for various defect positions in a NaCl slab. Inset: geometry of the slab and isosurface of the defect wave function. (b) Plane-averaged and unit-cell averaged dielectric constant profiles as obtained in the DFT calculation compared to the adopted model profile. (c) Electrostatic potentials from the DFT and from the model calculation along with their difference. Inset: potentials along the lateral direction.

Figure 4

Formation energies calculated within DFT without (circles, blue) and with (crosses, green) the correction as a function of inverse system size for: (a) A chlorine vacancy defect in the $+1$ charge state in the middle (dash-dotted) and at the surface (solid line) of a NaCl slab. (b) A Si dangling bond in the $-1$ charge state at the H-passivated $\mathrm{Si}(001)\mathrm{\text{−}}(2\times 1)$ surface. The supercell sizes used are given at the top. In all cases, half of the cell consists of vacuum. In (a), the result from a separate bulk calculation is also shown (square).

Figure 5

Uncorrected (circles, blue) and corrected formation energies as a function of the defect position within the slab. The uncorrected system corresponds to a slab in a periodic supercell. The applied electrostatic corrections depend on the targeted dielectric environment, which can be the bulk (crosses, green), a single surface (squares, red), or a slab with two surfaces (triangles, cyan).