Dielectric screening and vacancy formation for large neutral and charged ${\mathrm{Si}}_n{\mathrm{H}}_m$ $(n>1500)$ nanocrystals using real-space pseudopotentials

A commonly used procedure for computing the properties of defects in crystalline materials is to consider a large supercell that includes the defect of interest. This is a straightforward technique as standard energy band codes can be used for such computations. For neutral defects, the only impediment of such an approach is to avoid defect-defect interactions between adjoining cells. However, this procedure can be complex if the defect of interest is charged as the system at large contains Coulombic divergences. Moreover, some have recently argued that the conventional definition of formation energies for charged defects cannot be reconciled with statistical mechanics. Here, we focus on an alternative approach. We consider large nanocrystals wherein a charged defect can be placed. Since the system is confined, a charged defect within the nanocrystal does not result in a Coulombic divergence. The chief impediment is computational, i.e., while no defect-defect or Coulombic divergences are present, the nanocrystal must be sufficiently large to allow the system to properly replicate a bulklike configuration. With the development of new algorithms and hardware advances, computations for systems of sufficient size to address this issue are feasible. In particular, we solve the Kohn-Sham equation in real space using pseudopotential-density-functional theory for large silicon nanocrystals, which contain thousands of atoms. We focus on (i) the screening of a point charge and (ii) the formation of a charged vacancy in hydrogen-terminated silicon nanocrystals. This approach allows us to examine the role of quantum confinement in addition to exploring the bulk limit. Comparisons to other methods confirm the viability of this approach.

Figure 1

Ball and stick model [62] of a Si nanocrystal, ${\mathrm{Si}}_{1523}{\mathrm{H}}_{524}$, of radius 39 a.u. ($1\text{a.u.}=0.5292$ Å). The boundary of the bulk-terminated fragment is passivated by H-like atoms (light pink). When structural optimization is performed, the Si atoms at the surface (red) are not relaxed, whereas the interior Si (blue) atoms are.

Figure 2

Spherically averaged screening function vs normalized distance from center. (a) For the screening of a negative charge, we compare results for ${\mathrm{Si}}_{4403}{\mathrm{H}}_{1060}$ to previous work [25] and to a classical model. (b) The screening of a positive charge.

Figure 3

Spherically averaged change in electron density. The $y$ axes are in units of $e/\text{a.u.}$, $e$ being the fundamental charge. Please refer to the text for the definition for the symbols in the legends. The NC surface is at $\sim$54.2 a.u., indicated by the gray vertical line. Shown for (a) a negative charge and (b) a positive charge.

Figure 4

Atomic relaxation magnitudes in ${\mathrm{Si}}_{1522}{\mathrm{H}}_{524}$ (i.e., with vacancy) and in ${\mathrm{Si}}_{1523}{\mathrm{H}}_{524}$ (intrinsic).

Figure 5

Wave function squared using an isosurface (yellow) of value $5\times {10}^{-5}$ $e/{\text{Å}}^3$, for the (a) filled and (b) empty gap state of the neutral ${\mathrm{Si}}_{1522}{\mathrm{H}}_{524}$. The first (red) and second (blue) shell atoms are shown in their unrelaxed positions. Their relaxation movement vectors, scaled by a factor of 2.5, are represented by the arrows in (a). For the second shell atoms, only a few vectors are (barely) visible. The wave function in (b) is an average over doubly degenerate states.

Figure 6

Kohn-Sham energy levels in the gap for ${\mathrm{Si}}_{1522}{\mathrm{H}}_{524}$. We show the gap of ${\mathrm{Si}}_{1523}{\mathrm{H}}_{524}$ at the left column for easy comparison. Red, light green, and dark blue denote full, half, and empty occupancy, respectively. The left and right half of each column are for the spin-up and spin-down channels, respectively. Singly ($s$) and doubly ($d$) degenerate states are also labeled.

Figure 7

Formation energies for neutral and charged vacancy.