Local-field and excitonic effects in the optical response of $\alpha$-alumina

This paper introduces key ingredients of the dielectric response of $\alpha$-alumina that go beyond an independent-particle (IP) treatment of the valence-electron excitations. The optical-response functions were calculated from first- principles both at the Bethe–Salpeter and the random-phase approximation (RPA) levels. Excitonic effects obtained within the Bethe–Salpeter framework were found essential for reproducing the low-energy part of the experimental spectra (below 15 eV) and the bound exciton in particular. For higher energies, local-field effects introduced through the RPA modified considerably the IP results and provided a satisfactory account of the reflectivity spectra and of the position and shape of the dominant bulk plasmon resonance in the electron energy-loss spectra.

Figure 1

DFT band structure of $\alpha$-alumina within the LDA. The roman numbers label the valence manifolds.

Figure 2

Isosurfaces of the MLWFs in $\alpha$-alumina corresponding to the manifolds in Fig. 1: magenta and cyan indicate opposite isovalues. O atoms are in red, Al atoms in gray. Corresponding to manifold I there are six $s$-type MLWFs, each center at one of the six oxygens in the unit cell and slightly polarized toward the bonded Al. Corresponding to manifold II there are 18 $p$-type MLWFs pointing in three orthogonal directions (a,b,c) and centered at the six oxygens. For clarity only one MLWF per type is shown.

Figure 3

QP corrections to the DFT eigenvalues for 12 valence and 12 conduction bands at high-symmetry points $\Gamma$ (square), A (dot), D (plus), Z (x) in the first BZ. Dashed lines indicate the valence-band maximum (VBM) and conduction band minimum (CBM).

Figure 4

Reflectivity spectra for light polarizations (a) perpendicular and (b) parallel to the $c$ axis: The calculated IP (magenta dotted-dashed), RPA (black dashed), and BS (solid blue) curves are compared with experimental data from Ref. 5 (black circles) and Ref. 9 (red stars).

Figure 5

Real part, ${\epsilon }_1$, and imaginary part, ${\epsilon }_2$, of the macroscopic dielectric function, ${\epsilon }_M$, for light polarization perpendicular to the $c$ axis. The calculated RPA (black dashed) and BS (solid blue) curves are compared with experimental data from Ref. 6 (black circles) and Ref. 9 (red stars).

Figure 6

Real part, ${\epsilon }_1$, and imaginary part, ${\epsilon }_2$, of the macroscopic dielectric function, ${\epsilon }_M$, for light polarization parallel to the $c$ axis. The calculated RPA (black dashed) and BS (solid blue) curves are compared with experimental data from Ref. 6 (black circles).

Figure 7

Index of refraction $n$ and extinction coefficient $\kappa$ for light polarization perpendicular to the $c$ axis. The calculated RPA (black dashed) and BS (solid blue) curves are compared with experimental data from Ref. 6 (black circles) and Ref. 9 (red stars).

Figure 8

Index of refraction $n$ and extinction coefficient $\kappa$ for light polarization parallel to the $c$ axis. The calculated RPA (black dashed) and BS (solid blue) curves are compared with experimental data from Ref. 6 (black circles).

Figure 9

Electron energy-loss function in the vanishing q limit for q oriented perpendicular to the $c$ axis. The calculated RPA (black dashed) and BS (solid blue) curves are compared with experimental data obtained from Kramers–Kronig analysis of VUV reflectivity spectra from Ref. 6 (black circles) and from Ref. 9 (red stars), and with experimental data from electron energy-loss spectra from Ref. 9 (magenta plusses).

Figure 10

Electron energy-loss function in the vanishing q limit for q oriented parallel to the $c$ axis. The calculated RPA (black dashed) and BS (solid blue) curves are compared with experimental data obtained from Kramers-Kronig analysis of VUV reflectivity spectra from Ref. 6 (black circles).