Screening of point charges in Si quantum dots

The screening of a point charge in hydrogenated Si quantum dots ranging in diameter from $10\mathrm{\AA }\text{to}26\mathrm{\AA }$ has been studied using first-principles density-functional theory in the local density approximation. We find that the main contribution to the screening function originates from the electrostatic field set up by the polarization charges at the surface of the nanocrystals. We show that this contribution is well described by a classical electrostatics model of dielectric screening.

Figure 1

(a) shows the electronic charge enclosed in a sphere of radius $r$ around a positive unit charge $(q=+e)$ located at the center of ${\mathrm{Si}}_{275}{\mathrm{H}}_{172}$ (solid line) and ${\mathrm{Si}}_{275}{\mathrm{H}}_{112}$ (dashed line) nanocrystals. (b) and (c) show the spherically averaged screening function $\overline{\epsilon }\left(r\right)$ as a function of the distance $r$ from a positive (b) or negative (c) point charge. The vertical arrows denote the radius of the nanocrystal $(R=11.0\mathrm{\AA })$. Note that the screening functions of the ${\mathrm{Si}}_{275}{\mathrm{H}}_{172}$ and ${\mathrm{Si}}_{275}{\mathrm{H}}_{112}$ nanocrystals are almost indistinguishable.

Figure 2

The spherically averaged screening function $\overline{\epsilon }(r∕R)$ of several Si nanocrystals as a function of the scaled distance $(r∕R)$ from the test charge $(q=-e)$. Also shown (solid line) is the effective screening function calculated from Eq. (1) using ${\epsilon }_{\mathit{in}}=11.9$ (Ref. 10) and ${\epsilon }_{\mathit{out}}=1$.

Figure 3

A comparison between the LDA-derived screening function $\overline{\epsilon }\left(r\right)$ (solid lines) and the model screening function ${\overline{\epsilon }}_{\text{model}}\left(r\right)$ (dashed lines) for ${\mathrm{Si}}_{87}{\mathrm{H}}_{76}$ and ${\mathrm{Si}}_{465}{\mathrm{H}}_{228}$ nanocrystals.

Figure 4

Interaction energy between a positive unit charge at the center of the nanocrystal and an electron in the lowest unoccupied state, calculated using the LDA screening function $\overline{\epsilon }(\mathit{r},{\mathit{R}}_0)$ (solid line) and the model screening function ${\overline{\epsilon }}_{\text{model}}(\mathit{r},{\mathit{R}}_0)$ (dashed line).