  This paper studies short-range order (SRO) in the semiconductor alloy
(GaN)$_{1-x}$(ZnO)$_x$. Monte Carlo simulations performed on a density
functional theory (DFT)-based cluster expansion model show that the
heterovalent alloys exhibit strong SRO because of the energetic preference for
the valence-matched nearest-neighbor Ga-N and Zn-O pairs. To represent the
SRO-related structural correlations, we introduce the concept of Special
Quasi-ordered Structure (SQoS). Subsequent DFT calculations reveal dramatic
influence of SRO on the atomic, electronic and vibrational properties of the
(GaN)$_{1-x}$(ZnO)$_x$ alloy. Due to the enhanced statistical presence of the
energetically unfavored Zn-N bonds with the strong Zn3$d$-N2$p$ repulsion, the
disordered alloys exhibit much larger lattice bowing and band-gap reduction
than those of the short-range ordered alloys. Inclusion of lattice vibrations
stabilizes the disordered alloy.
