Magnetic properties of bilayer ${\mathbf{Sr}}_3{\mathbf{Ir}}_2{\mathbf{O}}_7$: Role of epitaxial strain and oxygen vacancies

Using ab initio methods, we investigate the modification of the magnetic properties of the $m=2$ member of the strontium iridates Ruddlesden–Popper series ${\mathrm{Sr}}_{m+1}{\mathrm{Ir}}_m{\mathrm{O}}_{3m+1}$, bilayer ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$, induced by epitaxial strain and oxygen vacancies. Unlike the single-layer compound ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$, which exhibits a robust in-plane magnetic order, the energy difference between in-plane and out-of-plane magnetic orderings in ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$ is much smaller and it is expected that small external perturbations could induce magnetic transitions. Our results indicate that epitaxial strain yields a spin-flop transition, which is driven by the crossover between the intralayer $J_1$ and interlayer $J_2$ magnetic exchange interactions upon compressive strain. While $J_1$ is essentially insensitive to strain effects, the strength of $J_2$ changes by one order of magnitude for tensile strains $\ge 3\%$. In addition, our study clarifies that the unusual in-plane magnetic response observed in ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$ upon the application of an external magnetic field originates from the canting of the local magnetic moments due to oxygen vacancies, which locally destroy the octahedral networks, thereby allowing for noncollinear spin configurations.

Figure 1

Structure and vacancy positions of ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$. (a) Crystal structure of ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$. ${\mathrm{IrO}}_6$ octahedra are denoted with polyhedra, and each O and Sr ion with small (red) and large (gray) spheres, respectively. (b) Side view and (c) top view of RP perovskite bilayer with $\sqrt{2}\times \sqrt{2}2\mathrm{D}$ unit cell. Each oxygen vacancy (Ov) position is marked with a dashed circle. (d) Schematic description of the Ir–O–Ir bond angle and Ir–O bond length of both apical (${\alpha }_{OP}$, $d_{OP}$) and planar (${\alpha }_{IP}$, $d_{IP}$) directions.

Figure 2

(a) Schematic magnetic structures of the two OP orderings, type A and B, proposed in the literature [15], and the IP one. According to our calculations, the B-type order corresponds to the ground-state phase and is only 0.1 meV/atom more stable than the A-type one. (b) The energy difference between collinear OP magnetic order and noncollinear IP magnetic order ($\mathrm{\Delta }E=E_{OP}-E_{IP}$) for both the pristine sample and the sample with oxygen vacancy. The overall decrease of the $\mathrm{\Delta }E$ for $U=1\mathrm{eV}$ is due to decrease of the Ir magnetic moment. The inset shows the zoom of the energy difference between OP A and OP B.

Figure 3

(a) Energy difference $\mathrm{\Delta }E$ between the IP and OP magnetic orders as a function of the epitaxial strain. A switch between the IP and OP magnetic orders is found at around $-3\%$ strain. The inset shows a zoom of the energy difference between OP A and OP B. (b) Calculated $J$ parameters upon strain. (inset) Schematic view of the bilayer lattice. $J_1$ and $J_2$ are the planar and apical exchange interactions, respectively.

Figure 4

(a) Formation energy of an oxygen vacancy ($E_{\text{form}}$). NM, NM SC, IP SC, and OP SC refer to nonmagnetic unit cell, nonmagnetic supercell, supercell with IP magnetic order, and supercell with OP magnetic order (OP SC), respectively. (b) Strain dependency of $E_{\text{form}}$.

Figure 5

(a) Total DOS of pristine ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$ structure. (b)–(d) Total DOS of ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$ with an oxygen vacancy at position 1 (Ov1), 2 (Ov2), and 3 (Ov3). Insets show magnification of total DOS around the Fermi level. (e) Ir-$d$ projected DOS for the pristine case. (f)–(h) Ir-$d$ projected DOS of Ir sites A, B, and C for the energetically more favorable Ov2 case [see Figs. 1 and 1]. Band structure for (i) pristine case and (j) Ov2 case.

Figure 6

Charge-density plot of IP layer for the (a) pristine case and (b) Ov2 case which contains Ov site. Energy range is from $-2.0$ to 0.0 eV with respect to the Fermi level. (a) The octahedron formed from ${\mathrm{IrO}}_6$ is shown for the pristine case with local axis ($x$ and $z$). (b) The square pyramid formed from ${\mathrm{IrO}}_5$ at Ir-A site is shown. (c) Charge-density difference between panels (a) and (b). (d) The charge-density plot for the in-gap states just below the Fermi level ($-0.12$ to 0.0 eV). (e)–(h) The structural change due to the oxygen vacancies. Side view of the octahedral network perpendicular the (e) $b$ direction and (f) $a$ direction, where pristine and Ov2 cases are compared. ${\mathrm{IrO}}_6$ is shown with octahedron. Clear octahedral tilting can be seen after introduction of Ov2. The local magnetic moment is canted off from the $c$ axis (OP direction) as seen from the (g) $b$ axis and (h) $a$ axis. The size and direction of each magnetic moment is denoted with the length and direction of the red arrow.