  Double perovskite Sr$_2$CrOsO$_6$ is (or is very close to) a realization of a
spin-asymmetric semimetallic compensated ferrimagnet, according to first
principles calculations. This type of near-half metallic antiferromagnet is an
unusual occurrence, and more so in this compound because the zero gap is
accidental rather than being symmetry determined. The large spin-orbit coupling
(SOC) of osmium upsets the spin balance (no net spin moment without SOC): it
reduces the Os spin moment by 0.27 $\mu_B$ and induces an Os orbital moment of
0.17 $\mu_B$ in the opposite direction. The effects combine (with small oxygen
contributions) to give a net total moment of 0.54 $\mu_B$ per cell in \scoo,
reflecting a large impact of SOC in this compound. This value is in moderately
good agreement with the measured saturation moment of 0.75 $\mu_B$. The value
of the net moment on the Os ion obtained from neutron diffraction (0.73 $\mu_B$
at low temperature) differs from the calculated value (1.14 $\mu_B$). Rather
surprisingly, in isovalent Sr$_2$CrRuO$_6$ the smaller SOC-induced spin changes
and orbital moments (mostly on Ru) almost exactly cancel. This makes
Sr$_2$CrRuO$_6$ a "half (semi)metallic antiferromagnet" (practically vanishing
net total moment) even when SOC is included, with the metallic channel being a
small-band-overlap semimetal. Fixed spin moment (FSM) calculations are
presented for each compound, illustrating how they provide different
information than in the case of a nonmagnetic material. These FSM results
indicate that the Cr moment is an order of magnitude stiffer against
longitudinal fluctuations than is the Os moment.
