Tin dioxide from first principles: Quasiparticle electronic states and optical properties

The structural, electronic, and optical properties of the semiconducting oxide SnO${}_2$ are investigated using first-principles calculations. We employ the $G_0{W}_0$ formalism based on hybrid-functional calculations to compute the quasiparticle band structure and density of states for which we find good agreement with results from photoemission and two-photon absorption experiments. We also address open questions regarding the band ordering and band symmetries. In a second step we use our electronic structure as a starting point to calculate optical spectra by solving the Bethe-Salpeter equation including the electron-hole interaction. The dielectric tensor is predicted for a wide range of photon energies. Our results resolve the long-standing discrepancy between theory and experiment on the highly anisotropic onsets of absorption. The anisotropy can be explained in terms of dipole-allowed direct transitions in the vicinity of the valence-band maximum without having to invoke lower-lying valence bands.

Figure 1

Bonding geometry of the tetragonal unit cell of SnO${}_2$ in the rutile structure with the associated Brillouin zone. The special points $R$, $X$, and the zone center $\Gamma$ are highlighted.

Figure 2

Quasiparticle band structure and density of states of rutile SnO${}_2$ in $\text{HSE}03+G_0{W}_0$ (dotted red lines) and $\text{LDA}+U+\Delta$ (solid black lines). The valence-band maximum has been chosen as the common zero of energy.

Figure 3

Plot of the $\text{HSE}03+G_0{W}_0$ quasiparticle eigenvalues together with allowed optical transitions at the $\Gamma$ and the $R$ points of the Brillouin zone. Dashed lines indicate a twofold degeneracy of the respective level. Solid black (dotted red) arrows show dipole-allowed optical transitions for light polarized perpendicular (parallel) to the $c$ axis of the rutile lattice.

Figure 4

Calculated density of states of rutile SnO${}_2$ compared to experimental photoemission spectra. Open circles denote the XPS data by Nagata et al. (Ref. 68). The plot includes the $\text{HSE}03+G_0{W}_0$ DOS without broadening (gray shaded region), as well as with Gaussian and Lorentzian broadening (FWHM $\sigma =0.8$ eV), as explained in the text. The unbroadened DOS has been scaled by a factor of 0.5 to allow for visual comparison of all features of the spectrum. The zero of energy is chosen at the theoretical VBM, and the red lines indicate band-edge extrapolations. The inset shows the agreement between theory and XPS for the lower-lying VB complex derived from Sn $4d$ and O $2s$ states.

Figure 5

Imaginary part of the dielectric function in the IQPA (gray, light red) and from the solution of the BSE (black, red). We show the results for light polarization perpendicular (a) and parallel (b) to the $c$ axis of the rutile crystal structure. The anisotropy in the vicinity of the absorption edge becomes more pronounced when excitonic effects are included.

Figure 6

Absorption coefficient calculated from the BSE for perpendicular (a) and parallel (b) light polarization.

Figure 7

Absorption coefficient calculated from the BSE for SnO${}_2$ (solid lines) plotted alongside the experimental results at $T=7$ K (circles) (Ref. 24) for perpendicular (black curves on the left) and parallel (red curves on the right) light polarization. Lorentzian broadening of 0.001 eV has been added to the calculated results to simulate the lifetime and instrumental broadening inherent to experiment. The broadening artifacts below the absorption edge have been subtracted.

Figure 8

Absolute value of the optical transition-matrix elements (in units of $\hslash /a_B$) along two high-symmetry directions in the Brillouin zone for light polarized perpendicular ($\left|p_{\perp }\right|$, solid black) as well as parallel ($\left|p_{\parallel }\right|$, dashed red) to the tetragonal $c$ axis.