Magnetic ground state of CrOCl: A first-principles study

By first-principles calculations, this paper reveals that the antiferromagnetic (AFM) ground state of CrOCl with an orthorhombic structure observed in experiment is mainly due to the competition between the neighboring magnetic exchange interactions ($J_i$'s). The AFM ground state requires that the first and second nearest neighbors ($J_1$ and $J_2$) are AFM couplings with comparable magnitudes, and that the seventh nearest neighbor $\left(J_7\right)$ has a certain size of AFM coupling. By using Liechtenstein's density functional theory $(\mathrm{DFT}+U)$ method and appropriately adjusting the on-site exchange $\left(\mathcal{J}\right)$ parameter, the $J_i$'s can satisfy these conditions and lead to the experimentally observed AFM ground state (AFM-Exp). The $\mathcal{J}$ parameter changes the energy levels of the Cr-$3d$ orbitals and consequently the energy differences between the orbitals, which in turn impacts the sign and magnitude of $J_i$'s. By decomposing the $J_i$'s into orbital contributions, it is shown that introducing the $\mathcal{J}$ parameter significantly reduces the ferromagnetic parts of $J_1$ and $J_2$ and converts them to AFM coupling, which leads to the AFM-Exp ground state.

Figure 1

(a) A top and side view of the CrOCl crystal structure. (b) Four AFM spin orders are considered. Blue, cyan, red, and green spheres represent the Cr atom in the upper sublayer, Cr atom in the lower sublayer, O atom, and Cl atom, respectively. The first-principles phase diagrams of the CrOCl (c) bulk and (d) monolayer, where the green dashed lines are phase boundaries. (e) Magnetic exchange paths.

Figure 2

Electronic band structure and projected density of states (PDOS) of CrOCl bulk. Since the CrOCl bulk is an AFM-Exp structure, the spin-up and spin-down bands are identical and the PDOS only shows the spin-up part.

Figure 3

The magnetic phase diagram of the Heisenberg model (2) in the text as a function of (a) $J_1$ and $J_2$, and (b) $J_1/J_7$ and $J_2/J_7$. Since our calculation shows that $J_7$ is AFM, only $J_7<0$ is considered. The black circle and triangle represent the points calculated by the FS method and MF theory, respectively, at $U=3$ eV and $\mathcal{J}=0$ eV, and the green circle and triangle represent the points calculated by the FS method and MF theory, respectively, at $U=3$ eV and $\mathcal{J}=1.5$ eV.

Figure 4

The $3d$ orbital splitting from an ideal octahedron with $O_h$ symmetry to a distorted octahedron with $C_{2v}$ symmetry, and $d^3$ electrons occupying $b_2$ and $e$ orbitals.

Figure 5

For $U=3$ eV and $\mathcal{J}=0$ eV, the decomposition of (a) $J_1$, (b) $J_2$, and (c) $J_7$ to the contributions of different $3d$ orbitals in CrOCl. Similarly, (d)–(f) are the orbital decompositions of $J_1$, $J_2$, and $J_7$, respectively, with $U=3$ eV and $\mathcal{J}=1.5$ eV as the parameters.

Figure 6

Hopping channels for $J_1$: (a) $e\text{−}e$, (b) $b_2\text{−}e$, and (c) $e\text{−}{a}_1$; $J_2$: (d) $b_2\text{−}{b}_2$, (e) $e\text{−}e$, and (f) $b_2\text{−}{b}_1$. Based on the octahedron of ${\mathrm{CrO}}_4{\mathrm{Cl}}_2$, the coordinate system of the orbital is defined as follows: $\vec{x}=\frac{1}{\sqrt{2}}(\frac{\vec{b}}{|\vec{b}|}+\frac{\vec{c}}{|\vec{c}|})$, $\vec{y}=\frac{1}{\sqrt{2}}(-\frac{\vec{b}}{|\vec{b}|}+\frac{\vec{c}}{|\vec{c}|})$, $\vec{z}=\frac{\vec{a}}{|\vec{a}|}$.