Giant Magnetic Anisotropy Induced by Ligand $LS$ Coupling in Layered Cr Compounds

We propose a novel origin of magnetic anisotropy to explain the unusual magnetic behaviors of layered ferromagnetic Cr compounds ($3d^3$) wherein the anisotropy field varies from $\lesssim 0.01$ to $\sim 3\mathrm{T}$ on changing the ligand atom in a common hexagonal structure. The effect of the ligand $p$ orbital spin-orbit ($LS$) coupling on the magnetic anisotropy is explored by using four-site full multiplet cluster model calculations for energies involving the superexchange interaction at different spin axes. Our calculation shows that the anisotropy energy, which is the energy difference for different spin axes, is strongly affected not only by the $LS$ coupling strength but also by the degree of $p\text{−}d$ covalency in the layered geometry. This anisotropy energy involving the superexchange appears to dominate the magnetic anisotropy and even explains the giant magnetic anisotropy as large as 3 T observed in ${\mathrm{CrI}}_3$.

Figure 1

(a) Top views of ${\mathrm{CrI}}_3$ (left) and ${\mathrm{CrGeTe}}_3$ (right) layers. Cr, I, Ge, and Te are represented by blue, purple, red, and yellow spheres, respectively. (b) $M\text{−}H$ curves corresponding to $H\parallel c$ and $H\parallel ab$ and (c) $M\text{−}T$ curves measured on ${\mathrm{CrGeTe}}_3$. Field cooling (FC) was processed under $H=0.5\mathrm{T}$.

Figure 2

Anisotropy fields $\mathrm{\Delta }{H}^S$ and magnetic anisotropy energies $\mathrm{\Delta }{E}_K$ of various layered Cr compounds, ${\mathrm{CrCl}}_3$ [18, 22], ${\mathrm{CrBr}}_3$ [21], ${\mathrm{CrI}}_3$ [19], ${\mathrm{CrGeTe}}_3$ [29], and ${\mathrm{CrSiTe}}_3$ [32], as a function of the ligand $p$ spin-orbit coupling strength ${\xi }_p$.

Figure 3

Cr $d_{yz}\text{−}\mathrm{L}$ $p_{j_{1/2}}\text{−}\mathrm{Cr}$ $d_{3x^2-r^2}$ superexchange for (a) $\mathit{M}\parallel \widehat{y}$ and (b) $\mathit{M}\parallel \widehat{z}$. The spin states are indicated by red ($+{\sigma }_{y,z}$) and blue ($-{\sigma }_{y,z}$) colors. (c) Two equivalent 90° superexchange paths through ligands ($L1$ and $L2$) between two adjacent Cr atoms. (d) Cr-L-Cr local network (red dashed line) in the crystal coordinates ($a$ and $c$).

Figure 4

Four-site full multiplet cluster model calculation results for $\mathrm{\Delta }{E}_{\mathrm{ME}}$ as functions of (a) ${\xi }_p$ for various $\delta \theta$ values at $\mathrm{\Delta }=1\mathrm{eV}$ and of (b) $\mathrm{\Delta }$ for various ${\xi }_p$ values at $\delta \theta =2°$.

Figure 5

Partial density of state of Te $5p$ orbitals determined from wannierization on ${\mathrm{CrGeTe}}_3$ and ${\mathrm{CrSiTe}}_3$. Strong Si(Ge)-Te covalent bonding effectively pushes down the occupied bonding states by about 1.5 eV (weighted average) and transfers about 30% Te $5p$ weight to the unoccupied antibonding states.