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Re-based double perovskites (DPs) exhibit a complex interplay of structuraland metal-insulator transitions. Here we systematically study the ground stateelectronic and structural properties for a family of Re-based DPs $A_2B$ReO$_6$($A$=Sr, Ca and $B$=Cr, Fe), which are related by a common low energyHamiltonian, using density functional theory + $U$ calculations. We show thatthe on-site interaction $U$ of Re induces orbital ordering (denoted COO), witheach Re site having an occupied $d_{xy}$ orbital and a C-type alternation among$d_{xz}/d_{yz}$, resulting in an insulating state consistent withexperimentally determined insulators Sr$_2$CrReO$_6$, Ca$_2$CrReO$_6$, andCa$_2$FeReO$_6$. The threshold value of $U_{Re}$ for orbital ordering isreduced by inducing $E_g$ octahedral distortions of the same C-type wavelength(denoted COD), which serves as a structural signature of the orbital ordering;octahedral tilting also reduces the threshold. The COO, and the concomitantCOD, are a spontaneously broken symmetry for the Sr based materials (i.e.$a^0a^0c^-$ tilt pattern), while not for the Ca based systems (i.e. $a^-a^-b^+$tilt pattern). Spin-orbit coupling does not qualitatively change the physics ofthe COO/COD, but can induce relevant quantitative changes. We prove that asingle set of $U_{Cr}$,$U_{Fe}$,$U_{Re}$ capture the experimentally observedmetallic state in Sr$_2$FeReO$_6$ and insulating states in the other threesystems. We predict that the COO is the origin of the insulating state inSr$_2$CrReO$_6$, and that the concomitant COD may be experimentally observed atsufficiently low temperatures (ie. space group P$4_2/m$). Additionally, givenour prescribed values of $U$, we show that the COO induced insulating state inCa$_2$CrReO$_6$ will survive even if the COD amplitude is suppressed (e.g. dueto thermal fluctuations).

Structural and metal-insulator transitions in rhenium based double perovskites via orbital ordering