  Motivated by the observation of nonmetallic nature in double perovskite
Ca2CrReO6 and Ca2FeReO6 with high magnetic Curie temperatures of 360 and 522 K,
we systematically investigate the structural, electronic, and magnetic
properties of Ca2MReO6 (M=Cr,Fe) using the full-potential linear augmented
plane wave (FP-LAPW) method within the density functional theory. Our full
optimization confirms the stable ground-state structure with $P2_1/n$ symmetry.
The modified Becke-Johnson (mBJ) exchange potential is used for investigating
electronic structures. Our mBJ calculation shows that they are both
ferrimagnetic semiconductors with semiconductor gaps of 0.38 eV and 0.05 eV,
respectively, in contrast with wrong metallic phases from the generalized
gradient approximation (GGA). The origin of semiconductor gap is due to the
further distortion of ReO$_6$ octahedra caused by John-Teller effect, which
drives the three partially-occupied Re $t_{2g}$ bands split into two
fully-filled bands and one empty band in the minority-spin channel. With the
spin-orbit coupling (SOC) taken into account, the Ca2MReO6 (M=Cr,Fe) shows high
magneto-crystalline anisotropy (MCA) with the magnetic easy axis along
pseudocubic [010] direction, and the total magnetic moments increase by
0.209$\mu_B$ and 0.258$\mu_B$ per formula unit, respectively, due to the strong
SOC effect on Re ion. Although reducing to 0.31 and 0.03 eV, the semiconductor
gaps remain open in spite of the SOC broadening of the Re $t_{2g}$-related
bands. Therefore, our DFT investigation with mBJ has established the correct
ferrimagnetic semiconductor ground state for the double perovskites Ca2MReO6
(M=Cr,Fe). This mechanism, different from that in double perovskite Sr2CrOsO6,
can help understand physical properties of other similar compounds.
