Electronic and dielectric properties of insulatingZr3N4

The electronic structures and dielectric functions of ${\mathrm{Zr}}_3{\mathrm{N}}_4$ were studied by a first-principles density-functional method. Three different structure models were used: (1) ZrN in a rocksalt structure with an ordered Zr vacancy, or ${\mathrm{Zr}}_3\square {\mathrm{N}}_4,$ (2) $c-{\mathrm{Zr}}_3{\mathrm{N}}_4$ in the cubic spinel structure and (3) $o-{\mathrm{Zr}}_3{\mathrm{N}}_4$ in the orthorhombic structure. All three structures of ${\mathrm{Zr}}_3{\mathrm{N}}_4$ are found to be insulators with small indirect band gaps. This appears to be consistent with experimental observation that ${\mathrm{Zr}}_3{\mathrm{N}}_4$ is an insulator. Total-energy (E) calculations show that ${\mathrm{Zr}}_3\square {\mathrm{N}}_4$ has an expanded lattice constant compared to ZrN by about 0.75%, and that $E\left({\mathrm{Zr}}_3\square {\mathrm{N}}_4\right)<E(o-{\mathrm{Zr}}_3{\mathrm{N}}_4)<E(c-{\mathrm{Zr}}_3{\mathrm{N}}_4)$ which indicates that the ordered defect model for ${\mathrm{Zr}}_3{\mathrm{N}}_4$ is valid. It is further demonstrated that past calculations showing ${\mathrm{Zr}}_3\square {\mathrm{N}}_4$ to be a metal were caused by the failure to relax the vacancy structure. The electronic structures and the optical dielectric functions for all three models were calculated and analyzed in detail. The ${\mathrm{Zr}}_3\square {\mathrm{N}}_4$ model again shows the best overall agreement with available experimental data. All three structures have relatively large optical dielectric constants at zero frequency. Possible implications from these calculations are discussed.