First-principles investigations of orbital magnetic moments and electronic structures of the double perovskites<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">Sr</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">FeMoO</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">Sr</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">FeReO</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">Sr</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">CrWO</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

The electronic structures and magnetic properties of double perovskites ${\mathrm{Sr}}_{2}{\mathrm{FeMoO}}_{6},$ ${\mathrm{Sr}}_{2}{\mathrm{FeReO}}_{6},$ and ${\mathrm{Sr}}_{2}{\mathrm{CrWO}}_{6}$ have been studied by using the full-potential linear muffin-tin orbital method within the local-spin-density approximation (LSDA) and the generalized gradient approximation (GGA). The on-site Coulomb energy U has also been taken into account in both schemes $(\mathrm{LSDA}+U$ and $\mathrm{GGA}+U).$ The results predict a half-metallic ferrimagnetic band structure with total spin magnetic moment of $4{\ensuremath{\mu}}_{B},$ $3{\ensuremath{\mu}}_{B},$ and $2{\ensuremath{\mu}}_{B}$ per formula unit for ${\mathrm{Sr}}_{2}{\mathrm{FeMoO}}_{6},$ ${\mathrm{Sr}}_{2}{\mathrm{FeReO}}_{6},$ and ${\mathrm{Sr}}_{2}{\mathrm{CrWO}}_{6},$ respectively. By including the spin-orbit coupling in the self-consistent calculations, we find that the $5d$ transition-metal atoms W and Re exhibit large unquenched orbital magnetic moments because of significant spin-orbit interaction in $5d$ orbitals. On the other hand, in Fe and Cr $3d$ and Mo $4d$ orbitals, the orbital moments are all quenched even though the on-site Coulomb energy U is taken into account. This is in strong contrast to the enhanced large $3d$ orbital moments in CoO and NiO but similar to the quenched orbital moment in ${\mathrm{CrO}}_{2}.$