Out-of-plane ferromagnetism in two-dimensional $1T\text{−}{\mathrm{RhO}}_2$

Using the combined theoretical approaches, the structural, electronic, and magnetic properties of the two-dimensional (2D) $1T$ phase of monolayer ${\mathrm{RhO}}_2$ were studied. Density-functional theory plus $U$ calculation indicate that $1T\text{−}{\mathrm{RhO}}_2$ favors a ferromagnetic metallic phase with an out-of-plane magnetization, and it can be achieved by exfoliation from the layered bulk oxides containing monolayer $1T\text{−}{\mathrm{RhO}}_2$. Monte Carlo simulation shows that the ferromagnetic phase is stable below the Curie temperature of 73.9 K based on the Heisenberg Hamiltonian model. Magnetic anisotropy energy and its cause were further investigated to understand the microscopic origin of the 2D magnetization. Our results indicate that spin-orbit coupling interaction between Rh $4d$ stabilizes perpendicular magnetic anisotropy, resulting in the out-of-plane magnetization, over in-plane magnetic anisotropy. In addition, mechanical tensile strain can strengthen perpendicular magnetic anisotropy by increasing positive contribution of the interaction to the magnetic anisotropy energy.

Figure 1

Crystal structure and magnetic configurations of $1T\text{−}{\mathrm{RhO}}_2$. (a) A three-atom unit cell, (b)–(d) ferromagnetic (FM), (e) ${120}^{\circ }\mathrm{N}\acute{\text{e}}\mathrm{el}$ configuration where the angle between spin moments is ${120}^{\circ }$, and (f)–(h) antiferromagnetic (AFM) configurations. The arrows show the directions of the spin moments on the Rh atoms along the $x$, $y$ (in-plane), or $z$ (out-of-plane) direction. The crystal structures are visualized using the vesta code [15].

Figure 2

Electronic structure of $1T\text{−}{\mathrm{RhO}}_2$ and bulk ${\mathrm{NaRhO}}_2$. The highest electron occupied state was set as zero. Namely, the Fermi level and the VBM are the reference in $1T\text{−}{\mathrm{RhO}}_2$ and ${\mathrm{NaRhO}}_2$, respectively. In bulk ${\mathrm{NaRhO}}_2$, the DOS of the Na atom is negligibly small in the energy range between $-4$ and 4 eV.

Figure 3

Magnetization ($m$) as a function of the temperature ($T$), obtained using Monte Carlo simulation. A convergence test was performed and then a large enough supercell that reduces the tail above $T_{\mathrm{C}}$ and provides the saturated $T_{\mathrm{C}}$ was utilized.

Figure 4

Strain dependence of $E_{\mathrm{MA}}$. $E_{\mathrm{MA}}^X$ denotes the contribution of each atom ($X=\text{Rh}$ and O) to $E_{\mathrm{MA}}$.

Figure 5

The Rh $4d$ orbital resolved $E_{\mathrm{MA}}$ ($E_{\mathrm{MA}}^{4d}$) under biaxial strain. Positive and negative value of $E_{\mathrm{MA}}$ favor perpendicular magnetic anisotropy (PMA) and in-plane magnetic anisotropy (IMA), respectively. Two kinds of contributions of the $d\otimes d$ interactions, $d_{xy}\otimes d_{x^2-y^2}$ and $d_{yz}\otimes d_{z^2}$ interactions, are significant.

Figure 6

The partial density of states (PDOS) of Rh $4d$ orbitals at $\varepsilon =-4\%$, $0\%$, and $+4\%$ strains. The SOC effect was not included explicitly in the $\mathrm{LDA}+U$ calculation. $o^{\uparrow }$, $o^{\downarrow }$, $u^{\uparrow }$, and $u^{\downarrow }$ are responsible for the occupied spin-up, the occupied spin-down, the unoccupied spin-up, and the unoccupied spin-down states, respectively. Symbols represent $d\otimes d$ interactions (see the text and Table 3). The vertical dashed line denotes the Fermi level. The PDOS of the $d_{xz}$ orbital was not shown because its contribution to $E_{\mathrm{MA}}^{4d}$ is negligible.