Spin-orbit coupled $j_{\mathrm{eff}}=1/2$ iridium moments on the geometrically frustrated fcc lattice

Motivated by experiments on the double perovskites ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$ and ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$, we study the magnetism of spin-orbit coupled $j_{\mathrm{eff}}=1/2$ iridium moments on the three-dimensional, geometrically frustrated, face-centered cubic lattice. The symmetry-allowed nearest-neighbor interaction includes Heisenberg, Kitaev, and symmetric off-diagonal exchange. A Luttinger-Tisza analysis shows a rich variety of orders, including collinear $A$-type antiferromagnetism, stripe order with moments along the $\left\{111\right\}$ direction, and incommensurate noncoplanar spirals, and we use Monte Carlo simulations to determine their magnetic ordering temperatures. We argue that existing thermodynamic data on these iridates underscores the presence of a dominant Kitaev exchange, and also suggest a resolution to the puzzle of why ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$, but not ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$, exhibits “weak” ferromagnetism.

Figure 1

Phase diagram of the nearest-neighbor spin Hamiltonian as a function of $J_K/J_H$ and $\mathrm{\Gamma }/J_H$, obtained using the Luttinger-Tisza (LT) method. The AFM states are $A$-type antiferromagnets, having ferromagnetic planes stacked antiferromagnetically along the third direction, with spins either pointing perpendicular to the FM plane ($A$-I AFM) or lying in the FM plane ($A$-II AFM). Stripe order at $(\pi /2,\pi /2,\pi /2)$ features moments pointing in the $\left\{111\right\}$ or symmetry related directions. IC-1, IC-2, and IC-3 are incommensurate noncoplanar spirals; beyond the LT analysis, they are multimode states.

Figure 2

Real space spin configurations in the layered $A$-type antiferromagnetic states AFM $A$-I and AFM $A$-II.

Figure 3

Real space spin configurations in the collinear stripe state, showing moments pointing along the diagonal $\left\{111\right\}$ and $\left\{\overline{1}\overline{1}\overline{1}\right\}$ directions for $(k_x,k_y,k_z)\equiv (\pi /2,\pi /2,\pi /2)$.

Figure 4

(a) Comparison of the ground state energy per spin $E_{gs}$ obtained within LT method (solid line) and simulated annealing (dots). (b),(c) Magnetic transition temperature $T_c$ of the classical model (in units of $J_H{S}^2$, for spin length $S$) vs $\mathrm{\Gamma }/J_H$, obtained using Monte Carlo simulations for cuts through the phase diagram (Fig. 1) at $J_K/J_H=-1.0,+1.5$. (d) Plot of the “frustration parameter,” the ratio of the ${\tilde{T}}_c\equiv T_c(1+1/S)$ to ${\mathrm{\Theta }}_{CW}$; the rescaling of $T_c$ by $(1+1/S)$ accounts for the classical $S^2$ being replaced by the quantum $S(S+1)$. The dark square shows the result at $J_K/J_H\to \infty$.

Figure 5

Conjectured alignment of staggered octahedral rotations and the ferromagnetic planes in the $A$-II AFM state for ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$ (top) and ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$ (bottom), with the spins shown on Ir octahedra (yellow). We have picked the $z$ axis as the direction along which the octahedral rotations are staggered for the Ir octahedra, and shown only $z=0,1$ planes. For ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$, the FM planes are the $xy$ planes stacked antiferromagnetically along $z$, leading to a net weak ferromagnetic moment along $-\widehat{y}$, while for ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$ the FM planes are $xz$ planes stacked antiferromagnetically along $y$ leading to no net ferromagnetic moment. The uniform Ir octahedral tilts are unimportant for this discussion and are not shown.