Tunable magneto-optical effect, anomalous Hall effect, and anomalous Nernst effect in the two-dimensional room-temperature ferromagnet $1T-{\mathrm{CrTe}}_2$

Utilizing first-principles density functional theory calculations together with group theory analyses, we systematically investigate the spin-order-dependent magneto-optical effect (MOE), anomalous Hall effect (AHE), and anomalous Nernst effect (ANE) in the recently discovered two-dimensional room-temperature ferromagnet $1T\text{−}{\mathrm{CrTe}}_2$. We find that the spin prefers an in-plane direction by the magnetocrystalline anisotropy energy calculations. The MOE, AHE, and ANE display a period of $2\pi /3$ when the spin rotates within the atomic plane, and they are forbidden if a mirror plane perpendicular to the spin direction exists. By reorienting the spin from the in-plane to out-of-plane direction, the MOE, AHE, and ANE are enhanced by around one order of magnitude. Moreover, we establish the layer-dependent magnetic properties of multilayer $1T\text{−}{\mathrm{CrTe}}_2$ and predict antiferromagnetism and ferromagnetism for bilayer and trilayer $1T\text{−}{\mathrm{CrTe}}_2$, respectively. The MOE, AHE, and ANE are prohibited in antiferromagnetic bilayer $1T\text{−}{\mathrm{CrTe}}_2$ due to the existence of the space-time inversion symmetry, whereas all of them are activated in ferromagnetic trilayer $1T\text{−}{\mathrm{CrTe}}_2$ and are significantly enhanced compared to those of monolayer $1T\text{−}{\mathrm{CrTe}}_2$. Our results show that the magneto-optical and anomalous transports proprieties of $1T\text{−}{\mathrm{CrTe}}_2$ can be effectively modulated by altering spin direction and layer number.

Figure 1

(a) and (b) Top and side views of monolayer $1T-{\mathrm{CrTe}}_2$. The blue spheres represent Cr atoms, whereas dark gray and silver-white spheres represent Te atoms in the upper and lower sublayers. The pink dashed lines draw the 2D primitive cell, and the red arrows indicate the directions of spin magnetic moments. The top and bottom panels in (b) present the spin directions along the $y$ and $z$ axes, respectively. (c) and (d) The magnetocrystalline anisotropy energy of monolayer $1T-{\mathrm{CrTe}}_2$ by rotating the spin magnetic moment within the $yz$ and $xy$ planes. The spin along the $y$ axis is set to be the reference state.

Figure 2

(a) The spin-polarized band structure and density of states (in units of states/eV per cell) for monolayer $1T\text{−}{\mathrm{CrTe}}_2$. (b) The relativistic band structure and orbital-decomposed density of states (in units of states/eV per cell) for monolayer $1T\text{−}{\mathrm{CrTe}}_2$ when the spin is along the $y$ axis.

Figure 3

(a) and (e) The real diagonal, (b) and (f) imaginary diagonal, (c) and (g) real off-diagonal, and (d) and (h) imaginary off-diagonal elements of optical conductivity for monolayer $1T\text{−}{\mathrm{CrTe}}_2$ with in-plane and out-of-plane magnetization, respectively. For a better comparison, the curves when spin points along the $y$ axis ($S\parallel y$) are replotted in (e)–(h).

Figure 4

(a) The Kerr rotation angle ${\theta }_{\mathrm{K}}$, (b) Kerr ellipticity ${\varepsilon }_{\mathrm{K}}$, (c) Faraday rotation angle ${\theta }_{\mathrm{F}}$, and (d) Faraday ellipticity ${\varepsilon }_{\mathrm{F}}$ for monolayer $1T\text{−}{\mathrm{CrTe}}_2$ with in-plane magnetization. (e)–(h) The Kerr and Faraday rotation angles and ellipticities as a function of azimuthal angle $\phi$ at selected photon energies.

Figure 5

(a) The Kerr rotation angle ${\theta }_{\mathrm{K}}$, (b) Kerr ellipticity ${\varepsilon }_{\mathrm{K}}$, (c) Faraday rotation angle ${\theta }_{\mathrm{F}}$, and (d) Faraday ellipticity ${\varepsilon }_{\mathrm{F}}$ for monolayer $1T\text{−}{\mathrm{CrTe}}_2$ with out-of-plane magnetization. For a better comparison, the curves for the spin pointing along the $y$ axis ($S\parallel y$) are also plotted in (a)–(d). (e)–(h) The Kerr and Faraday rotation angles and ellipticities as a function of polar angle $\theta$ at selected photon energies.

Figure 6

(a) and (d) The intrinsic anomalous Hall (${\sigma }_{xy}^{\mathrm{A}}$) and anomalous Nernst (${\alpha }_{xy}^{\mathrm{A}}$) conductivities for monolayer $1T\text{−}{\mathrm{CrTe}}_2$ as a function of the Fermi energy when the spin points along the $x$, $y$, and $z$ axes. ${\alpha }_{xy}^{\mathrm{A}}$ is calculated at a temperature of 300 K. The vertical dashed lines mark the realistic doping region of 0.31–0.70 eV. The green and red arrows indicate the energies selected for plotting ${\sigma }_{xy}^{\mathrm{A}}$ and ${\alpha }_{xy}^{\mathrm{A}}$ as a function of $\phi$ or $\theta$, respectively. (b) and (e) ${\sigma }_{xy}^{\mathrm{A}}$ and ${\alpha }_{xy}^{\mathrm{A}}$ as a function of azimuthal angle $\phi$ when the Fermi energies are set to be $-0.30$ and $-0.26$ eV, respectively. (c) and (f) ${\sigma }_{xy}^{\mathrm{A}}$ and ${\alpha }_{xy}^{\mathrm{A}}$ as a function of polar angle $\theta$ when the Fermi energies are set to be 0.50 and 0.10 eV, respectively.

Figure 7

Side views of (a)–(d) bilayer and (e)–(h) trilayer $1T\text{−}{\mathrm{CrTe}}_2$ with in-plane and out-of-plane ferromagnetic and antiferromagnetic configurations. The red arrows label the spin directions.

Figure 8

Relativistic band structures of bilayer and trilayer $1T\text{−}{\mathrm{CrTe}}_2$ when the spin is along the $y$ axis.

Figure 9

(a) The Kerr and (b) Faraday rotation angles of bilayer and trilayer $1T\text{−}{\mathrm{CrTe}}_2$. (c) The anomalous Hall and (d) anomalous Nernst conductivities of bilayer and trilayer $1T\text{−}{\mathrm{CrTe}}_2$ as a function of the Fermi energy. The magnetization direction is along the $y$ axis. For a better comparison, the curves of monolayer $1T\text{−}{\mathrm{CrTe}}_2$ are also plotted. The vertical dashed lines in (c) and (d) mark the realistic doping region of $-0.31$ to 0.70 eV.