Transition energies and direct-indirect band gap crossing in zinc-blende AlxGa1−xN

Abstract

The electronic and optical properties of zinc-blende (zb) Al${}_{x}$Ga${}_{1\ensuremath{-}x}$N over the whole alloy composition range are presented in a joint theoretical and experimental study. Because zb-GaN is a direct (${\ensuremath{\Gamma}}_{v}\ensuremath{\rightarrow}{\ensuremath{\Gamma}}_{c}$) semiconductor and zb-AlN shows an indirect (${\ensuremath{\Gamma}}_{v}\ensuremath{\rightarrow}{X}_{c}$) fundamental band gap, the ternary alloy exhibits a concentration-dependent direct-indirect band gap crossing point the position of which is highly controversial. The dielectric functions of zb-Al${}_{x}$Ga${}_{1\ensuremath{-}x}$N alloys are measured employing synchrotron-based ellipsometry in an energy range between 1 and 20 eV. The experimentally determined fundamental energy transitions originating from the $\ensuremath{\Gamma}$, $X$, and $L$ points are identified by comparison to theoretical band-to-band transition energies. In order to determine the direct-indirect band gap crossing point, the measured transition energies at the $X$ point have to be aligned by the calculated position of the highest valence state. Thereby density-functional theory (DFT) based approaches to the electronic structure, ranging from the standard (semi)local generalized gradient approximation (GGA), self-energy corrected local density approximation (LDA-1/2), and meta-GGA DFT (TB-mBJLDA) to hybrid functional DFT and many-body perturbation theory in the $GW$ approximation, are applied to random and special quasirandom structure models of zb-Al${}_{x}$Ga${}_{1\ensuremath{-}x}$N. This study provides interesting insights into the accuracy of the various numerical approaches and contains reliable ab initio data on the electronic structure and fundamental alloy band gaps of zb-Al${}_{x}$Ga${}_{1\ensuremath{-}x}$N. Nonlocal Heyd-Scuseria-Ernzerhof-type hybrid-functional DFT calculations or, alternatively, $GW$ quasiparticle calculations are required to reproduce prominent features of the electronic structure. The direct-indirect band gap crossing point of zb-Al${}_{x}$Ga${}_{1\ensuremath{-}x}$N is found in the Al rich composition range at an Al content between $x=0.64$ and $0.69$ in hybrid functional DFT, which is in good agreement with $x=0.71$ determined from the aligned experimental transition energies. Thus our study solves the long-standing debate on the nature of the fundamental zb-Al${}_{x}$Ga${}_{1\ensuremath{-}x}$N alloy band gap.