Dimension-dependent magnetic anisotropy for tunable anomalous Hall effect in transition metal dichalcogenides

Two-dimensional (2D) ferromagnetism at room temperature is highly desired in spintronic devices and applications. Recently, ${\mathrm{CrTe}}_2$ film has been successfully synthesized. Bulk ${\mathrm{CrTe}}_2$ has in-plane magnetic anisotropy (MA), and the Curie temperature reaches up to 300 K. When the dimensions are down to two, MA switches to out-of-plane. However, the mechanism of MA transition from in-plane to out-of-plane remains unknown. Here, we investigate the switching of MA in ${\mathrm{CrTe}}_2$ by second-order perturbation theory. The MA energy (MAE) originates from competition between $p_z/p_y$ and $p_x/p_y$ orbital hybridization, which contributes to out-of-plane and in-plane MA, respectively. The $p_x/p_y$ orbital hybridization, induced by van der Waals force, is absent in the monolayer. Moreover, the out-of-plane MA of monolayer ${\mathrm{CrTe}}_2$ breaks mirror symmetry, producing an anomalous Hall effect. Slight charge doping is significant to achieve a tunable anomalous Hall effect. Based on charge doping, we design a spintronic anomalous Hall device by taking advantage of the easily controllable properties of ${\mathrm{CrTe}}_2$. This highly thickness-dependent magnetism provides a perspective to control the magnetic properties of 2D materials.

Figure 1

(a) Crystal structure of bulk ${\mathrm{CrTe}}_2$, Blue and brown balls represent Cr and Te atoms, respectively. (b) Schematic diagrams of electron occupancy and (c) the exchange interaction in ${\mathrm{CrTe}}_2$; blue and red arrows represent spin-up and spin-down, respectively. (d) Exfoliation energy (blue line and dot) for ${\mathrm{CrTe}}_2$ as a function of its separation distance $d$ from the bulk (as illustrated in the inset).

Figure 2

The different magnetic configurations of (a) a ferromagnet (FM) and (b) an antiferromagnet (AFM) in $1T-{\mathrm{CrTe}}_2$. Red and yellow arrows represent spin up and spin down, respectively.

Figure 3

Atomic orbital-resolved spin-orbit coupling (SOC) energy difference $\mathrm{\Delta }{E}_{\mathrm{soc}}$ of (a)–(c) monolayer and (d)–(f) bulk ${\mathrm{CrTe}}_2$. Blue and red bars represent out-of-plane and in-plane contribution, respectively.

Figure 4

(a) Hubbard model vs traditional density functional theory (DFT) calculation, where up and down arrows represent spin-up and spin-down electrons, respectively. (b) Lattice constants, (c) magnetic moment, and (d) magnetic anisotropy energy (MAE) of monolayer and bulk ${\mathrm{CrTe}}_2$ as a function of Coulomb interaction $U_{\text{eff}}$.

Figure 5

The atom-resolved spin-orbit coupling (SOC) energy difference $\mathrm{\Delta }{E}_{\mathrm{soc}}$ of monolayer and bulk ${\mathrm{CrTe}}_2$ as a function of Coulomb interaction $U_{\text{eff}}$. Blue and red bars represent Te and Cr atoms, respectively.

Figure 6

(a) Magnetic anisotropy energy (MAE; blue line) and magnetic moment (red line) of monolayer ${\mathrm{CrTe}}_2$ as functions of strain. (b) The strength of $p_z/p_y$ and $p_x/p_y$ couplings for the ${\mathrm{CrTe}}_2$ monolayer under tensile strain. (c) State I and (d) II show the orbital resolved spin-orbit coupling (SOC) energy difference $\mathrm{\Delta }{E}_{\mathrm{soc}}$ under 0 and 6.7% strains, respectively.

Figure 7

(a) The side view of bilayer ${\mathrm{CrTe}}_2$, where the distance between Cr and Cr atomic layers is $d_{\mathrm{Cr}\text{−}\mathrm{Cr}}$. (b) The lattice constant and magnetic anisotropy energy (MAE) for bilayer ${\mathrm{CrTe}}_2$ under interlayer distance $d_{\mathrm{Cr}\text{−}\mathrm{Cr}}$. The blue and red lines represent lattice constant and MAE, respectively. The intrinsic layer distance of bilayer ${\mathrm{CrTe}}_2$ is marked with a dash line.

Figure 8

Atom-resolved spin-orbit coupling (SOC) energy difference $\mathrm{\Delta }{E}_{\mathrm{soc}}$ of (a) monolayer and (b) bulk $1T-{\mathrm{VSe}}_2$. Blue and red bars represent contributions of out-of-plane and in-plane, respectively.

Figure 9

The anomalous Hall conductivity of (a) monolayer and (b) bulk ${\mathrm{CrTe}}_2$, where dark and red lines represent in-plane and out-of-plane magnetization direction, respectively. (c) Magnetic anisotropy energy (MAE) of monolayer ${\mathrm{CrTe}}_2$ as a function of electron/hole doping concentration. (d) A spintronic device based on the anomalous Hall effect. Contacts S and D label the source and drain electrodes, respectively. Contacts ${\mathrm{V}}_1, {\mathrm{V}}_2, {\mathrm{V}}_3$, and ${\mathrm{V}}_4$ label the voltage probes.