Quantum Confinement and Electronic Properties of Silicon Nanowires

We investigate the structural, electronic, and optical properties of hydrogen-passivated silicon nanowires along [110] and [111] directions with diameter $d$ up to 4.2 nm from first principles. The size and orientation dependence of the band gap is investigated and the local-density gap is corrected with the $GW$ approximation. Quantum confinement becomes significant for $d<2.2\mathrm{nm}$, where the dielectric function exhibits strong anisotropy and new low-energy absorption peaks start to appear in the imaginary part of the dielectric function for polarization along the wire axis.

Figure 1

Ball-and-stick models of silicon nanowires (SiNWs) along the [110] (right) and [111] (left) directions viewed from the top (upper panels) and the side (lower panels). Filled circles stand for Si atoms, and open circles for H atoms.

Figure 2

LDA band gaps calculated for [110] (empty squares) and [111] (empty circles) wires, and the $GW$-corrected gaps (filled squares) for the two thinnest [110] wires and bulk Si, compared with the measured gaps for [112] wires (+) and a [110] wire (*). The dotted, dashed, and long-dashed lines are fitted to the data points (see text). The LDA band gap of bulk Si is indicated by the solid line, and the bulk $GW$ gap is marked around $d=8.0\mathrm{nm}$ for convenience.

Figure 3

Electronic charge distribution in the $xy$ plane (integrated along the wire axis) for the [110] wire with $d=1.20\mathrm{nm}$. The band indices are shown in each figure. The top panels marked with $a$, $b$, and $c$ are the results of a simple model for bands 1, 2, and 3, respectively (see text). The high (low) density region is indicated by red/dark (blue/light).

Figure 4

Imaginary dielectric functions ${ϵ}_2(\omega )$ polarized along the $z$ direction [${ϵ}_2^{\parallel }(\omega )$, solid lines] and in the $xy$ plane [${ϵ}_2^{\perp }(\omega )$, dotted lines], for silicon [110] wires with $d=1.2$, 1.6, and 2.2 nm, respectively. Shown in the bottom panel is ${ϵ}_2(\omega )$ for bulk $c\mathrm{\text{−}}\mathrm{Si}$. The dielectric functions are calculated with the scissor operator to fix the band gap at the $GW$ values ($E_g^{GW}$) and the energy zero is set at the top of the valence band. The arrows and vertical dashed lines mark $E_g^{GW}$ and the optical absorption edges, respectively. The black dots indicate the original absorption peak in bulk Si, and the inverted triangles show the new peaks developed in nanowires.