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

Re-based double perovskites (DPs) have garnered substantial attention due to their high Curie temperatures (${T}_{C}$) and display of complex interplay of structural and metal-insulator transitions (MIT). Here we systematically study the ground-state electronic and structural properties for a family of Re-based DPs ${A}_{2}B{\mathrm{ReO}}_{6}$ ($A$=Sr, Ca and $B$=Cr, Fe), which are related by a common low-energy Hamiltonian, using density functional theory +$U$ calculations. We show that the on-site interaction $U$ of Re induces orbital ordering (denoted C-OO), with each Re site having an occupied ${d}_{xy}$ orbital and a C-type alternation among ${d}_{xz}/{d}_{yz}$, resulting in an insulating state consistent with experimentally determined insulators ${\mathrm{Sr}}_{2}{\mathrm{CrReO}}_{6}$, ${\mathrm{Ca}}_{2}{\mathrm{CrReO}}_{6}$, and ${\mathrm{Ca}}_{2}{\mathrm{FeReO}}_{6}$. The threshold value of ${U}_{Re}$ for orbital ordering is reduced by inducing ${E}_{g}$ octahedral distortions of the same C-type wavelength (denoted C-OD), which serves as a structural signature of the orbital ordering; octahedral tilting also reduces the threshold. The C-OO and the concomitant C-OD are a spontaneously broken symmetry for the Sr-based materials (i.e., ${a}^{0}{a}^{0}{c}^{\ensuremath{-}}$ tilt pattern), while not for the Ca-based systems (i.e., ${a}^{\ensuremath{-}}{a}^{\ensuremath{-}}{b}^{+}$ tilt pattern). Spin-orbit coupling does not qualitatively change the physics of the C-OO/C-OD, but can induce relevant quantitative changes. We prove that a single set of ${U}_{Cr},{U}_{Fe},{U}_{Re}$ capture the experimentally observed metallic state in ${\mathrm{Sr}}_{2}{\mathrm{FeReO}}_{6}$ and insulating states in other three systems. We predict that the C-OO is the origin of the insulating state in ${\mathrm{Sr}}_{2}{\mathrm{CrReO}}_{6}$, and that the concomitant C-OD may be experimentally observed at sufficiently low temperatures (i.e., space group $P{4}_{2}/m$) in pure samples. Additionally, given our prescribed values of $U$, we show that the C-OO induced insulating state in ${\mathrm{Ca}}_{2}{\mathrm{CrReO}}_{6}$ will survive even if the C-OD amplitude is suppressed (e.g., due to thermal fluctuations). The role of the C-OO/C-OD in the discontinuous, temperature driven MIT in ${\mathrm{Ca}}_{2}{\mathrm{FeReO}}_{6}$ is discussed.