Dopant levels in large nanocrystals using stochastic optimally tuned range-separated hybrid density functional theory

We apply a stochastic version of an optimally tuned range-separated hybrid functional to provide insight on the electronic properties of P- and B-doped Si nanocrystals of experimentally relevant sizes. We show that we can use the range-separation parameter for undoped systems to calculate accurate results for dopant activation energies. We apply this strategy for tuning functionals to study doped nanocrystals up to 2.5 nm in diameter at the hybrid functional level. In this confinement regime, the P and B dopants have large activation energies and have strongly localized states that lie deep within the energy gaps. Structural relaxation plays a greater role for B-substituted dopants and contributes to the increase in activation energy when the B dopant is near the nanocrystal surface.

Figure 1

QP gap comparisons for Si NCs ranging from 1 to 2.5 nm. The sGW results are from Ref. [20]. The dashed lines show the bulk values for BNL and LDA. Inset: The values of $\gamma$ at different NC sizes are best fit to the line $0.403N_{\text{Si}}^{-1/3}$. The dashed line shows the reverse-engineered value of $\gamma$ that reproduces the experimental bulk value for Si [58].

Figure 2

Tuning $\gamma$ for the undoped and doped structures in the $D=1.0$ nm and $D=1.4$ nm NCs. The optimal $\gamma$ values are given by the intersect with the $y$ axis (dotted line). Upper panel: A partial charge is removed from the HOMO, which is the dopant level for a P donor. Lower panel: A partial charge is added to the LUMO, which is the dopant level for a B acceptor.

Figure 3

Dopant activation energies for Si NCs with P (upper panels) and B (lower panels) substituted at the center (left panels) and surface (right panels). Component IPs and EAs are also shown. The plotted $\mathrm{\Delta }\text{SCF}$ results were performed with the LDA functional.

Figure 4

Charge densities for dopant level in $D=1.4$ nm nanocrystal for (left to right) P center, P surface, B center, and B surface configurations. The isosurface is plotted at $20\%–25\%$ of the maximum value.