Excitonic effects in the optical properties of a SiC sheet and nanotubes

The quasiparticle band structure and optical properties of single-walled zigzag and armchair SiC nanotubes (SiC-NTs) as well as a single SiC sheet are investigated by ab initio many-body calculations using the GW and the GW plus Bethe-Salpeter equation approaches, respectively. Significant GW quasiparticle corrections, of more than 1.0 eV, to the Kohn-Sham band gaps from the local density approximation (LDA) calculations are found. The GW self-energy corrections transform the SiC sheet from an indirect LDA band gap to a direct band gap material. Furthermore, the quasiparticle band gaps of SiC-NTs with different chiralities behave very differently as a function of tube diameter, and this can be attributed to the difference in the curvature-induced orbital rehybridization among the different chiral nanotubes. The calculated optical absorption spectra are dominated by discrete exciton peaks due to exciton states with a high binding energy, up to 2.0 eV, in the SiC sheet and SiC-NTs. The formation of strongly bound excitons is attributed to the enhanced electron-hole interaction in these low-dimensional systems. Remarkably, the excited electron amplitude of the exciton wave function is found to peak on Si atoms near the hole position (which is on the C site) in zigzag SiC-NTs, indicating a charge transfer from an anion (hole) to its neighboring cations by photoexcitation. In contrast, this pronounced peak structure disappears in the exciton wave function in armchair SiC-NTs. Furthermore, in armchair SiC-NTs, the bound exciton wave functions are more localized and also strongly cylindrically asymmetric. The high excitation energy, $\sim$3.0 eV, of the first bright exciton, with no dark exciton below it, suggests that small-radius armchair SiC-NTs could be useful for optical devices working in the UV regime. On the other hand, zigzag SiC-NTs have many dark excitons below the first bright exciton and hence may have potential applications in tunable optoelectric devices ranging from infrared to UV frequencies by external perturbations.

Figure 1

Calculated quasiparticle band gap of a SiC sheet (open symbols) as a function of the inverse of the intersheet separation ($L_z$). The converged band gap, indicated by the solid line, is derived by employing the Coulomb truncation scheme. The dashed line linking the open circles is a guide for the eye only.

Figure 2

LDA band structure (left panel) and GW corrections to LDA eigenvalues (right panel) of the graphitic SiC sheet. Left: The $\pi$ and $\sigma$ bands are shown as red and blue lines, respectively. The LDA $E^{\pi -\pi *}$ gap is shown by the arrow. The valence band maximum is aligned at 0 eV. Right: GW corrections for the $\pi$ and $\sigma$ bands are represented by red circles and blue diamonds, respectively, whereas GW corrections of the NFE states are represented by green squares.

Figure 3

(a) Optical polarizability per unit area from both $\mathit{GW}+\mathit{RPA}$ (blue line) and $\mathit{GW}+\mathit{BSE}$ (red line) calculations and (b) $E_1^{\pi -\pi *}$ exciton wave function of the SiC sheet. (a) Theoretical spectra are broadened with a Gaussian smearing width of 0.15 eV. (b) The hole position (red sphere) is fixed at the top of a C atom (yellow sphere) and the squared electron amplitude [$n=|{\Phi }_S({\mathbf{r}}_e,{\mathbf{r}}_h=0){|}^2$ in arbitrary units (a.u.)] is mainly distributed on Si atoms (cyan spheres) next to the hole.

Figure 4

Fundamental band gaps of some zigzag ($n$,0) and armchair ($n$,$n$) SiC nanotubes as a function of the tube diameter from the GW (filled symbols) and LDA[17](open symbols) calculations. The band gap of the isolated SiC sheet obtained from the GW and LDA calculations is also displayed as the solid and the dashed line, respectively. Dotted lines between the symbols are a guide for the eye only.

Figure 5

Charge density contours ($\rho$ in ${10}^{-2}{\AA }^{-3}$) of states at the conduction band minimum (CBM) at the 1D Brillouin zone boundary ($Z$ point) of (3,3) and (5,5) SiC-NTs and also at the zone center ($\Gamma$ point) of (5,0) and (6,0) SiC-NTs. Si and C atoms are represented by cyan and yellow spheres, respectively.

Figure 6

Energy differences between the quasiparticle eigenvalues($E^{\mathit{QP}}$) and LDA eigenenergies ($E^{\mathit{LDA}}$) of the armchair (3,3) (upper panel) and zigzag (5,0) (lower panel) SiC-NTs. The corresponding LDA band structures are shown in the left panels for reference. Dashed lines denote the valence band maximum at 0 eV.

Figure 7

(a) Quasiparticle band structure and (b) optical absorption spectra of the armchair (3,3) SiC-NT. (b) Optical spectra from $\mathit{GW}+\mathit{RPA}$ and $\mathit{GW}+\mathit{BSE}$ calculations are shown as the dashed (blue) and solid (red) curves, respectively. Theoretical absorption spectra were broadened with a Gaussian of 0.15 eV. The lowest energy bright exciton ($I_1$) is formed by the mixture of four interband transitions as indicated by the red arrows in (a). In the inset in (b), the squared electron amplitude [$n=|{\Phi }_S({\mathbf{r}}_e,{\mathbf{r}}_h=0){|}^2$ in arbitrary units (a.u.)] of the $I_1$ exciton on a cross-sectional tubular plane with the hole fixed at the position of the red sphere (see text) is displayed. Si and C atoms are represented by cyan and yellow spheres, respectively.

Figure 8

(a) Quasiparticle band structure and (b) optical absorption spectrum (broadened with a Gaussian width of 0.15 eV) of the zigzag (5,0) SiC-NT. In (b), the first bright exciton, $I_1$, is mainly related to the interband transition indicated as the solid (red) arrow in (a), whereas the transition by the dashed (red) arrow in (a) corresponds to the third ($I_3$) bright exciton in (b). The inset in (b) shows the squared electron amplitude ($n$, in arbitrary units) of $I_1$ with the fixed hole position marked by the red sphere. Si and C atoms are represented by cyan and yellow spheres, respectively.

Figure 9

Isosurface plots of the electron distribution ${\left|\Phi \right|}^2$ of the $I_1$ exciton (see Figs. 7 and 8) with the fixed hole position [indicated by the (red) ”X”] in the armchair (3,3) and (5,5) as well as the zigzag (5,0) and (6,0) SiC-NTs. The corresponding integrated intensities, obtained by averaging out the coordinates perpendicular to the tube axis, are also shown. Here the hole position is set at 0. Si and C atoms are represented by cyan and yellow spheres, respectively.