Recent theoretical work has provided evidence that hybrid functionals, which include a fraction of exact (Hartree Fock) exchange in the density functional theory (DFT) exchange and correlation terms, significantly improve the description of band gaps of semiconductors compared with local and semilocal approximations. Based on a recently developed order-N method for calculating the exact exchange in extended insulating systems, we have implemented an efficient scheme to determine the hybrid functional band gap. We use this scheme to study the band gap and other electronic properties of the ternary compound In1−xGaxN using a 64-atom supercell model. The design of novel functional semiconductors with given values of the energy band gap is an area of intense research 1,2,3,4,5 . In particular, much attention is focused on the band gap engineering of group-III nitride semiconductors, whose remarkable optical properties are important for optoelectronic device applications 6,7 . To guide the search for compounds with tailored properties 1 , experimental studies are often accompanied by electronic structure calculations based on density functional theory (DFT)8 . For these calculations, the local-density(LDA) or generalized gradient approximations(GGA) are typically used. Due to the delocalization error of the LDA and GGA exchange and correlation functionals, however, these approaches severely underestimate the materials band gaps 9,10 .As shown by several recent studies 11 , a significant improvement in the description of semiconductor and insulator band gaps is generally obtained by using hybrid functionals 12 , in which some exact (Hartree-Fock) exchange is mixed into the exchange and correlation functional. This reduces the delocalization and derivative discontinuity errors of (semi)local functionals 2,9,10,11 . However, because of the considerable computational cost of evaluating the non-local exact exchange term, hybrid functionals have been mostly applied to systems with small unit cells 13 . For the modeling of systems where a large supercell is needed, an additional screened exchange approximation is usually made to relieve the computational burden 2,11 .
Recently Wu et al. (WSC)14 introduced an order-N method to calculate the exact exchange in extended insulating systems. The WSC method is based on a localized Wannier function representation of the occupied (valence) space, so that the exchange interaction between two orbitals decays rapidly with the distance between their centers. A truncation can thus be introduced, which greatly reduces the computational cost. The effectiveness of the WSC method was demonstrated by ground state electronic minimizations for crystalline silicon in supercells with 64 and 216 atoms.In this paper, we extend the WSC scheme to compute hybrid functional band gaps. To this end, the system's first (few) empty conduction state(s) is(are) determined starting from the ground state calculated via the WSC method. With hybrid functionals, this requires the computation of the pair exchange ...