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Non-collinear vs collinear description of the Ir-based one-$t_{2g}$ -hole perovskite-related compounds: SrIrO$_3$ and Sr$_2$IrO$_4$

We present an analysis of the electronic structure of perovskite-related iridates, 5d electron compounds where a subtle interplay between spin-orbit coupling, tetragonal distortions and electron correlations determines the electronic structure properties. We suggest via electronic structure calculations that a non-collinear calculation is required to obtain solutions close to the usually quoted $j_{eff}$ = 1/2 state to describe the $t_{2g}$ hole in the $Ir^{4+} :d^5$ cation, while a collinear calculation yields a different solution, the hole is in a simpler xz/yz complex combination with a smaller $L_z /S_z$ ratio. We describe what the implications of this are in terms of the electronic structure; surprisingly, both solutions barely differ in terms of their band structure, and are similar to the one obtained by a tight binding model involving $t_{2g}$ orbitals with mean field interactions. We also analyze how the electronic structure and magnetism evolve with strain, with the spin-orbit coupling strength and with the on-site Coulomb repulsion, suggest the way the band structure gets modified and draw some comparisons with available experimental observations.