  Double perovskite Bi$_2$FeCrO$_6$, related with BiFeO$_3$, is very
interesting because strong ferroelectricity and high magnetic Curie temperature
beyond room temperature are observed in it. However, existing
density-functional-theory (DFT) studies, using pseudo-potentials, produce
metallic ground state under the local density approximation (LDA) and need
LDA+U method to yield needed nonmetallic ground state, resulting in low
magnetic Curie temperature (below 130 K). Here, we optimize its crystal
structure and then investigate its electronic structure and magnetic and
optical properties by combining the full-potential augmented plane wave method
with Monte Carlo simulation. Our optimized structure is a robust ferrimagnetic
semiconductor. This nonmetallic phase is formed due to crystal field splitting
and spin exchange splitting, in contrast to Mott-Hubbard states in previous DFT
studies. Spin exchange constants and optical properties are calculated. Our ab
initio magnetic Curie temperature is 450 K, much higher than previous DFT-based
value and consistent with experimental results. Our study and analysis reveals
that the main magnetic mechanism is an antiferromagnetic superexchange between
Fe and Cr over the intermediate O atom. These results are useful to
understanding such perovskite materials and exploring their potential
applications.
