Finite-size supercell correction scheme for charged defects in one-dimensional systems

We propose a finite-size correction scheme for the formation energy of charged defects and impurities in one-dimensional systems within density functional theory. The energy correction in a supercell geometry is obtained by solving the Poisson equation in a continuum model which is described by an anisotropic permittivity tensor, with the defect charge distribution derived from first-principles calculations. We implement our scheme to study impurities and dangling bonds in silicon nanowires and demonstrate that the formation energy of charged defects rapidly converges with the supercell size.

Figure 1

(a) The results of induced surface charge densities (circles) for model hexagonal nanowires with various dielectric permittivities ${\varepsilon }_{\perp }^{\mathrm{NW}}$ under the electric field along the $x$ axis and their best fit to Eq. (8) (dashed line). (b) The variation of ${\varepsilon }_{\perp }^{\mathrm{NW}}$ along the $x$ axis based on DFT calculations (red solid line) for a hexagonal SiNW, the averaged dielectric permittivity (blue dashed line), and its mapping onto the hexagonal cross section (inset).

Figure 2

The atomic structure of the H-terminated SiNW and the total charge density (red line) along the $x$ axis. Black line represents the nanowire boundary corresponding to the volume fraction of 0.125 in the supercell for $L_x=L_y=40$ Å.

Figure 3

The screened defect charge distribution of a ${\mathrm{B}}_{\mathrm{Si}}^{-}$ impurity in SiNW. (a) The cumulative defect charge along the radial direction after integrating along the wire axis. Dashed line denotes the radial distance where the defect charge becomes minimum. (b) The defect charge distribution along the axial direction (red solid line), which is obtained by integrating over the cross section, and its Gaussian fit for the model charge distribution (blue dashed line). (c) Comparison of the cumulative defect charges along the $z$ axis in the SiNW (red solid line) and bulk Si (blue solid line). In the NW, the integration over the cross section is performed up to the cutoff radius and the total charge is normalized to one for comparison (for details see the text).

Figure 4

Comparison of the model potential (blue dashed line) with the DFT difference potential (red solid line) for a ${\mathrm{B}}_{\mathrm{Si}}^{-}$ impurity positioned at the wire center. In (a) and (b), the potential is averaged over the $yz$ and $xy$ planes, respectively. Color-scale plots of (c) the model potential and (d) the DFT difference potential, which are averaged along the axis.

Figure 5

The uncorrected (empty symbols) and corrected (filled symbols) transition levels of the B (blue diamonds) and P (red circles) impurities are compared for various supercell sizes. In (a) and (b), the lateral and axial cell sizes are set to be 40 and 37.96 Å, respectively. Lines are a guide to the eye. In (c), the supercell size is expressed as a multiple of the unit size, $10\times 10\times 9.49$ Å${}^3$. The extrapolated values (dashed lines) of the uncorrected levels are obtained by fitting to Eq. (16), whereas the average values of the corrected transition levels are drawn by solid lines. The shaded regions represent the conduction and valance bands of the H-terminated SiNW.

Figure 6

Color-scale plots of (a) the model potential and (b) the DFT difference potential, which are averaged along the axis, for a surface DB defect. (c) The uncorrected (empty symbols) and corrected (filled symbols) transition levels of the DB for various supercell sizes, with the same sizes and notations as those in Fig. 5. The right panel shows the results of hybrid functional calculations for the supercell of $20\times 20\times 18.98$ ${\AA }^3$.