Dimensionality-strain phase diagram of strontium iridates

The competition between spin-orbit coupling, bandwidth $\left(W\right)$, and electron-electron interaction $\left(U\right)$ makes iridates highly susceptible to small external perturbations, which can trigger the onset of novel types of electronic and magnetic states. Here we employ first principles calculations based on density functional theory and on the constrained random phase approximation to study how dimensionality and strain affect the strength of $U$ and $W$ in ${\left({\mathrm{SrIrO}}_3\right)}_m/\left({\mathrm{SrTiO}}_3\right)$ superlattices. The result is a phase diagram explaining two different types of controllable magnetic and electronic transitions, spin-flop and insulator-to-metal, connected with the disruption of the $J_{\mathrm{eff}}=1/2$ state which cannot be understood within a simplified local picture.

Figure 1

(a) Crystal structures of bilayer $({\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7,m=2)$ and of ${\left({\mathrm{SrIrO}}_3\right)}_m/\left({\mathrm{SrTiO}}_3\right)$ SL for different $m$. ${\mathrm{IrO}}_6$ and ${\mathrm{TiO}}_6$ octahedra are depicted in gray and light gray (blue), respectively. O and Sr atoms are not shown. (b) Schematic description of the IP and OP magnetic orderings for one single ${\mathrm{IrO}}_2$ layer. (c) 2D lattice parameters of typical substrates and corresponding lattice mismatch between ${\mathrm{SrTiO}}_3$ and RP strontium iridates.

Figure 2

Thin (red) lines denote PBE band structures of SLs with nonmagnetic setup, for different values of $m$. Wannier-interpolated bands are shown with thick (blue) lines. The shaded background (yellow) marks to the $t_{2g}$ bandwidths of the systems. Substrate strain is set to 0%.

Figure 3

Same as Fig. 2 but for $m=1$ for different strains.

Figure 4

Variation of $U$ and $W$ as a function of (a) dimensionality $m$ with no strain and (b) strain (in %) for $m=1$. (c) Electronic and magnetic phase diagram of ${\left({\mathrm{SrIrO}}_3\right)}_m/\left({\mathrm{SrTiO}}_3\right)$ SLs. IP and OP magnetic orderings are denoted with filled squares (red) and filled circle (blue), respectively, whereas the nonmagnetic phase is labeled with filled triangles (green). The insulating and metallic phases are highlighted with the gray and white background area. The arrows trace the IMT driven by dimensionality and strain. For $m=\infty$ $\left({\mathrm{SrIrO}}_3\right)$ the system remains a nonmagnetic metal at any strain [14, 25]. (d) Schematic band diagram showing the role of $U$ and $W$ in the IMT.

Figure 5

(a) Change of rotation/tilting angle upon strain. Solid/dotted lines refer to IP/OP cases. Note that ${\alpha }_{OP}$ is nonzero only for the $m=3$ case (right axis). (b) Change of the $d_{IP}/d_{OP}$ ratio upon strain. (c) Schematic description of the Ir-O-Ir bond angle and Ir-O bond length both in the $ab$ plane $({\alpha }_{IP},d_{IP})$ and in the out-of-plane $({\alpha }_{OP},d_{OP})$ direction. (d) Change of $J_1$ and $J_2$ as a function of strain. (Inset) Schematic structure of the Ir sublattice for $m=2$, showing the IP and OP exchange interactions $J_1$ and $J_2$, respectively [43].

Figure 6

Phase diagram for fixed $U$ parameters of 1 eV, 2 eV, and 3 eV. Due to the delicate balance of the $U$ with other energy scales, the overall metallicity depends highly on $U$ parameters. $U=1$ eV and $U=3$ eV give metallic and insulating phase for all $m$ values and strain ranges, respectively.

Figure 7

(a) ${\mu }_O/{\mu }_S$ and (b) $\delta$ as a function of epitaxial strain for $m=1$ SL [43]. Red and blue cross marks are for the value for bulk ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ and ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$ systems in their IP lattice parameter positions. (c) and (d) are the same for a local model. Inset in (c) describes the structure used for the local model. The $\delta =0$ and the ${\mu }_O/{\mu }_S=2$ points are marked by arrows.

Figure 8

Charge density plot in the ${\mathrm{IrO}}_2$ plane for (a) SL $m=1$ case and (b) local structure. Both plots are obtained from the 0% epitaxial strain.