Tuning the magnetic anisotropy and topological phase with electronic correlation in single-layer $H\text{−}{\mathrm{FeBr}}_2$

Electronic correlation can strongly influence the electronic properties of two-dimensional (2D) materials with open $d$ or $f$ orbitals. Herein, by taking single-layer (SL) $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$ as a representative of the SL $H\text{−}\mathrm{Fe}{\mathrm{X}}_2$ (X = Cl, Br, I) family, we investigated the electronic correlation effects in the magnetic anisotropy and electronic topology of such a system based on first-principles calculations with the density functional theory $+U$ approach. Our result is that the magnetic anisotropy energy (MAE) of SL $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$ shows a nonmonotonic evolution behavior with increasing electronic correlation strength, which is mainly due to the competition between different element-resolved MAEs of Fe and Br. Further investigations show that the evolution of element-resolved MAE arises from the variation of the spin-orbital coupling interaction between different orbitals in each atom. Moreover, tuning the strength of the electronic correlation can drive the occurrence of band inversions, causing the system to undergo multiple topological phase transitions and resulting in a quantum anomalous valley Hall effect. These exotic properties are universal for the SL $H\text{−}\mathrm{Fe}{\mathrm{X}}_2$ (X = Cl, Br, I) family. Our work sheds light on the role of electronic correlation effects in tuning magnetic and electronic structures in the SL $H\text{−}\mathrm{Fe}{\mathrm{X}}_2$ (X = Cl, Br, I) family, which could guide advances in the development of new spintronics and valleytronics devices based on these materials.

Figure 1

(a) Top and (b) side view of the atomic structure of SL $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$. (c) Splitting of the $3d$ orbitals of Fe atom under the trigonal prismatic crystal field. The trigonal prismatic crystal structure is also shown. (d) The Brillouin zone with high-symmetry points labeled.

Figure 2

(a) The schematic top views of the FM, stripy AFM, and zigzag AFM magnetic configurations. The solid and open circles represent spin-up and spin-down states, respectively. (b) The evolution of the energy difference between FM and stripy AFM (zigzag AFM) states with $U_{\mathrm{eff}}$. The energy difference is defined as $\mathrm{\Delta }E=E_{\mathrm{Stripy}/\mathrm{Zigzag}}-E_{\mathrm{FM}}$, which is labeled as blue (red) dots.

Figure 3

The evolution of (a) total MAE and (b) element-resolved MAE in a unit cell with $U_{\mathrm{eff}}$.

Figure 4

(a) The difference in the orbital-resolved MAE of each Br atom between $U_{\mathrm{eff}}=0.0\mathrm{eV}$ and $U_{\mathrm{eff}}=2.0\mathrm{eV}$, which is defined as $\mathrm{\Delta }\text{MAE}={\text{MAE}}_{U_{\mathrm{eff}}=2.0\mathrm{eV}}-{\text{MAE}}_{U_{\mathrm{eff}}=0.0\mathrm{eV}}$. The spin-polarized DOS of Br-$p_x$ and Br-$p_y$ orbitals at (b) $U_{\mathrm{eff}}=0.0\mathrm{eV}$ and (c) $U_{\mathrm{eff}}=2.0\mathrm{eV}$ is given. (d) The difference in the orbital-resolved MAE of each Fe atom between $U_{\mathrm{eff}}=0.8\mathrm{eV}$ and $U_{\mathrm{eff}}=2.0\mathrm{eV}$, which is defined as $\mathrm{\Delta }\text{MAE}={\text{MAE}}_{U_{\mathrm{eff}}=2.0\mathrm{eV}}-{\text{MAE}}_{U_{\mathrm{eff}}=0.8\mathrm{eV}}$. The spin-polarized DOSs of the $\mathrm{Fe}-d_{xy}$ and Fe -$d_{x^2-y^2}$ orbitals at (e) $U_{\mathrm{eff}}=0.8\mathrm{eV}$ and (f) $U_{\mathrm{eff}}=2.0\mathrm{eV}$ are given.

Figure 5

The band structure of ferromagnetic SL $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$ calculated with (a) spin-polarized case without including SOC, (b) out-of-plane magnetism with SOC, and (c) in-plane magnetization magnetism with SOC. In (b) and (c), the red, blue, and green dots represent Fe-$A{}_1$, Fe-$E{}_1$, and Fe-$E{}_2$ orbitals, respectively.

Figure 6

The band structure of SL $H\text{−}{\mathrm{FeBr}}_2$ calculated at (a) $U_{\mathrm{eff}}=0.0\mathrm{eV}$, (b) $U_{\mathrm{eff}}=0.5\mathrm{eV}$, (c) $U_{\mathrm{eff}}=0.6\mathrm{eV}$, (d) $U_{\mathrm{eff}}=0.8\mathrm{eV}$, (e) $U_{\mathrm{eff}}=0.9\mathrm{eV}$, and (f) $U_{\mathrm{eff}}=1.2\mathrm{eV}$. The red, blue, and green dots represent Fe-$A{}_1$, Fe-$E{}_1$, and Fe-$E{}_2$ orbitals, respectively.

Figure 7

The Berry curvature of SL $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$ in the 2D Brillouin zone calculated at (a) $U_{\mathrm{eff}}=0.0\mathrm{eV}$ and (b) $U_{\mathrm{eff}}=0.8$ eV; both units of Berry curvature in (a) and (b) are ${\mathrm{Bohr}}^2$. (c) Topological edge state of SL $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$ along the (100) direction calculated at $U_{\mathrm{eff}}=0.8$ eV; (d) topological phase diagram of SL $H\text{−}{\mathrm{FeBr}}_2$ with varied $U_{\mathrm{eff}}$.

Figure 8

Evolution of total MAE of (a) SL $H\text{−}\mathrm{Fe}{\mathrm{Cl}}_2$, (b) SL $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$, and (c) SL $H\text{−}\mathrm{Fe}{\mathrm{I}}_2$. The evolution of element-resolved MAE of (d) SL $H\text{−}\mathrm{Fe}{\mathrm{Cl}}_2$, (e) SL $H\text{−}\mathrm{Fe}{\mathrm{Br}}_2$, and (f) SL $H\text{−}\mathrm{Fe}{\mathrm{I}}_2$ are also given.