Electronic properties of zircon and hafnon from many-body perturbation theory

The electronic properties of zircon and hafnon, two wide-gap high-$\kappa$ materials, are investigated using many-body perturbation theory (MBPT) combined with the Wannier interpolation technique. For both materials, the calculated band structures differ from those obtained within density-functional theory and MBPT by (i) a slight displacement of the highest valence-band maximum from the $\Gamma$ point and (ii) an opening of the indirect band gap to 7.6 and 8.0 eV for zircon and hafnon, respectively. The introduction of vertex corrections in the many-body self-energy does not modify the results except for a global rigid shift of the many-body corrections.

Figure 1

(Color online) Global decay of the real-space representation of the DFT-LDA (blue circles) and $GW$ (red squares) Hamiltonians as a function of the lattice vector size $\left|\mathbf{R}\right|$ are measured by $\sqrt{V_{\mathbf{R}}^H}$ [see Eq. (10)] for all the valence bands.

Figure 2

(Color online) Calculated $GW$ band structure of (a) zircon and (b) hafnon. The corresponding Brillouin zone is illustrated in part (c), adopting the notation of Ref. 36. The various manifolds are labeled with Roman numbers starting from the lowest in energy.

Figure 3

(Color online) Isosurfaces of the maximally localized Wannier functions for zircon: Red and blue colors indicate opposite isovalues. The O atoms are colored in dark green, Zr atoms in gray, and Si atoms in yellow. The different manifolds are labeled with Roman numbers according to Fig. 2. Different views of the crystal are represented: (a) top view (perpendicular to the tetragonal axis, i.e., the [001] direction) and [(b)–(d)] side view (perpendicular to the [100] or [010] directions, which are equivalent). For the latter, the focus is on an O atom and its three nearest neighbors, two Zr atoms and one Si atom, all lying in the same plane.

Figure 4

(Color online) Band structure of zircon: comparison between (a) $G{W}_0$ (black) and DFT-LDA (red) (b) $G{W}_0$ (black) and $G{W}_0\Gamma$ (red).