Bipolar ferromagnetic semiconductors and doping-tuned room-temperature half-metallicity in monolayer $\mathrm{Mo}{X}_3$ $(X=\mathrm{Cl}, \mathrm{Br}, \mathrm{I})$: An HSE06 study

Two-dimensional magnetic materials have great potential applications in designing spintronics devices. Here, electronic structures and magnetic properties of monolayer molybdenum trihalides $\mathrm{Mo}{X}_3$ $(X=\mathrm{Cl}, \mathrm{Br}, \mathrm{I})$ are systematically investigated by using first-principles calculations with hybrid functional HSE06. Our calculations show that the magnetic ground state of the monolayer ${\mathrm{MoI}}_3$ is ferromagnetic (FM) with a Curie temperature of 24 K, while the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ and $\mathrm{Mo}{\mathrm{Br}}_3$ are $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ antiferromagnetic. The monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ and $\mathrm{Mo}{\mathrm{Br}}_3$ can be tuned into FM phase with a tensile strain. The monolayer ${\mathrm{MoI}}_3$ has a large magnetic anisotropy energy (MAE) of 1.051 meV/Mo with an out-of-plane easy axis. The Curie temperature and MAE of the monolayer ${\mathrm{MoI}}_3$ can be increased by about $50\%$ with a tensile strain of $3\%$. Remarkably, the monolayer ${\mathrm{MoI}}_3$ and the tensile strained $\mathrm{Mo}{\mathrm{Cl}}_3$ and $\mathrm{Mo}{\mathrm{Br}}_3$ are found to be bipolar ferromagnetic semiconductors. The Curie temperature of the monolayer ${\mathrm{MoI}}_3$ can be increased up to room temperature by carrier doping. For the monolayer $\mathrm{Mo}{\mathrm{Br}}_3$, a magnetic phase transition from the antiferromagnetic to FM can be triggered by an electron doping. In addition, the room-temperature half-metallic states can be achieved in the carrier-doped monolayer ${\mathrm{MoI}}_3$ and tensile strained $\mathrm{Mo}{\mathrm{Br}}_3$. The topological properties for the valence bands of the monolayer $\mathrm{Mo}{X}_3$ are also studied. Chern insulating states are obtained in the tensile strained monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ and $\mathrm{Mo}{\mathrm{Br}}_3$. Our paper shows that the monolayer $\mathrm{Mo}{X}_3$ $(X=\mathrm{Cl}, \mathrm{Br}, \mathrm{I})$ are promising candidates for exploring two-dimensional magnetism and spintronics in experiments.

Figure 1

(a) Top and side views of the monolayer $\mathrm{Mo}{X}_3$ ($X$ = Cl, Br, I) with lattice vectors ${\vec{a}}_1$ and ${\vec{a}}_2$. The rhombus denotes the unit cell. The inset shows the first Brillouin zone with the high-symmetry points. (b)–(d) The total energies of the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (b), $\mathrm{Mo}{\mathrm{Br}}_3$ (c), and ${\mathrm{MoI}}_3$ (d) with the variation of the lattice constant.

Figure 2

(a)–(c) Top and side views of atomic configuration snapshots from the ab initio molecular dynamic simulations after 3.0 ps at 500 K for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (a), $\mathrm{Mo}{\mathrm{Br}}_3$ (b), and ${\mathrm{MoI}}_3$ (c). (d)–(f) The calculated phonon spectra for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (d), $\mathrm{Mo}{\mathrm{Br}}_3$ (e), and ${\mathrm{MoI}}_3$ (f).

Figure 3

(a)–(d) Top views of the four different magnetic configurations considered for the monolayer $\mathrm{Mo}{X}_3$: FM (a), AF-$\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ (b), AF-zigzag (c), and AF-stripy (d). Up and down spins are denoted by the red up and blue down arrows, respectively. The Mo atoms form a honeycomb lattice. The three magnetic pair interactions are denoted with $J_1$, $J_2$, and $J_3$. (e)–(g) The energy difference ($\mathrm{\Delta }E$) between FM state and FM state, AF-$\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ state and FM state, AF-zigzag state and FM state, and AF-stripy state and FM state as a function of Hubbard U for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (e), $\mathrm{Mo}{\mathrm{Br}}_3$ (f), and ${\mathrm{MoI}}_3$ (g), obtained from the GGA (PBE)$+$U calculations. The magnetic structures with the lowest energies under different U values are indicated.

Figure 4

The magnetic phase diagrams of the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (a), $\mathrm{Mo}{\mathrm{Br}}_3$ (b), and ${\mathrm{MoI}}_3$ (c) with respect to strain, obtained by using the HSE06 calculations. The energy difference ($\mathrm{\Delta }E$) between FM state and FM state, AF-$\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ state and FM state, AF-zigzag state and FM state, and AF-stripy state and FM state as a function of strain are displayed. The magnetic structures with the lowest energies under different strains are illustrated in different colors.

Figure 5

(a) The $xyz$ coordinate built with respect to the crystal structure of the monolayer $\mathrm{Mo}{X}_3$. The lattice vectors ${\vec{a}}_1$ and ${\vec{a}}_2$ lie in the O-$xy$ plane, ${\vec{a}}_1$ is along the $x$ axis, and the $z$ axis is perpendicular to ${\vec{a}}_1$ and ${\vec{a}}_2$. (b) The calculated DOSs of the monolayer ${\mathrm{MoI}}_3$ based on the coordinate O-$xyz$ system in (a). (c) Schematic diagram of the evolution of ${\mathrm{Mo}}^{3+}$ $d$ orbitals in monolayer ${\mathrm{MoI}}_3$ based on the coordinate O-$xyz$ system in (a). (d1) The $x^{\prime }{y}^{\prime }{z}^{\prime }$ coordinate built with respect to the iodine octahedron in the monolayer $\mathrm{Mo}{X}_3$. The $\pm x$, $\pm y$, and $\pm z$ axes are almost along the six Mo-I connecting direction. (d2) Schematic plot of the direct exchange coupling of Mo-Mo. (d3) Schematic plot of the superexchange coupling of Mo-I-Mo. (e) The calculated DOS of monolayer $\mathrm{Mo}{X}_3$ based on the coordinate O-$x^{\prime }{y}^{\prime }{z}^{\prime }$ system in (d1). (f) Schematic diagram of the evolution of ${\mathrm{Mo}}^{3+}$ $d$ orbitals in monolayer ${\mathrm{MoI}}_3$ based on the coordinate O-$x^{\prime }{y}^{\prime }{z}^{\prime }$ system in (d1).

Figure 6

The MAEs with respect to strain for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$, $\mathrm{Mo}{\mathrm{Br}}_3$, and ${\mathrm{MoI}}_3$.

Figure 7

(a), (b) Special heat capacity with respect to the temperature for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (a) and $\mathrm{Mo}{\mathrm{Br}}_3$ (b) calculated by Monte Carlo simulations. (c) Monte Carlo simulations on the average magnetic moment per Mo atom and the special heat capacity for the monolayer ${\mathrm{MoI}}_3$. (d) The critical temperature $T_c$ as a function of strain for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$, $\mathrm{Mo}{\mathrm{Br}}_3$, and ${\mathrm{MoI}}_3$ obtained from Monte Carlo simulations.

Figure 8

(a)–(e) Spin-polarized band structures of the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (a), monolayer $\mathrm{Mo}{\mathrm{Br}}_3$ (b), $5\%$ tensile strained monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (c), $4\%$ tensile strained monolayer $\mathrm{Mo}{\mathrm{Br}}_3$ (d), and monolayer ${\mathrm{MoI}}_3$ (e). (f, g) Spin-polarized band structures for hole doped (f) and electron doped (g) monolayer ${\mathrm{MoI}}_3$ with the carrier-doping concentrations of 0.5 hole and 0.5 electron per unit cell, respectively. (h) Band structure for the electron doped monolayer $\mathrm{Mo}{\mathrm{Br}}_3$ with the carrier-doping concentration of 0.5 electron per unit cell. The red and blue curves denote the spin-up and spin-down bands, respectively. All the band structures are calculated by using the HSE06 functional.

Figure 9

(a) The schematic spin-resolved DOS of the $\mathrm{Mo}{X}_3$ with the BFMS behavior. The spin-up and spin-down states are denoted in red and blue, respectively. The three energy-gap parameters ($\mathrm{\Delta }1$, $\mathrm{\Delta }2$, and $\mathrm{\Delta }3$) of the BFMS are labeled in (a). (b) The schematic spin-resolved DOS of the BFMS when the $E_F$ is tuned into the $\mathrm{\Delta }1$ gap by the application of an appropriate negative gate voltage ($V_G<0$). (c) The schematic spin-resolved DOS of the BFMS when the $E_F$ is tuned into the $\mathrm{\Delta }3$ gap by the application of an appropriate positive gate voltage ($V_G>0$). (d) The schematic plot of the spin field effect transistor of the $\mathrm{Mo}{X}_3$, built based on the BFMS behavior. The spin orientations in the conducting channel can be controlled by the applied gate voltage $V_G$. (e) The schematic plot of spin-polarized current with respect to the gate voltage.

Figure 10

(a)–(c) The calculated energy differences ($\mathrm{\Delta }E$) of $E_{\text{FM}}-E_{\text{FM}}$, $E_{\text{AF-Néel}}-E_{\text{FM}}$, $E_{\text{AF-zigzag}}-E_{\text{FM}}$, and $E_{\text{AF-stripy}}-E_{\text{FM}}$ as a function of the carrier concentration for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (a), $\mathrm{Mo}{\mathrm{Br}}_3$ (b), and ${\mathrm{MoI}}_3$ (c), obtained by using the HSE06 calculations. (d)–(f) The exchange interactions $J_1$, $J_2$, and $J_3$ as a function of the carrier concentration calculated for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (d), $\mathrm{Mo}{\mathrm{Br}}_3$ (e), and ${\mathrm{MoI}}_3$ (f). (g)–(i) The calculated critical temperatures ($T_c$) as a function of the carrier concentration for the monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$ (g), $\mathrm{Mo}{\mathrm{Br}}_3$ (h), and ${\mathrm{MoI}}_3$ (i). The positive and negative values of all the horizontal axes are for electron and hole doping, respectively. The FM phases are denoted by light yellow backgrounds.

Figure 11

(a), (b) The calculated energy differences ($\mathrm{\Delta }E$) of $E_{\text{FM}}-E_{\text{FM}}$, $E_{\text{AF-Néel}}-E_{\text{FM}}$, $E_{\text{AF-zigzag}}-E_{\text{FM}}$, and $E_{\text{AF-stripy}}-E_{\text{FM}}$ as a function of the carrier concentration for the $5\%$ tensile strained $\mathrm{Mo}{\mathrm{Cl}}_3$ (a) and $1\%$ tensile strained $\mathrm{Mo}{\mathrm{Br}}_3$ (b), obtained by using the HSE06 calculations. (c), (d) The exchange interactions $J_1$, $J_2$, and $J_3$ as a function of the carrier concentration calculated for the $5\%$ tensile strained $\mathrm{Mo}{\mathrm{Cl}}_3$ (c) and $1\%$ tensile strained $\mathrm{Mo}{\mathrm{Br}}_3$ (d). (e), (f) The corresponding critical temperatures $T_c$. The positive and negative values of all the horizontal axes are for electron and hole doping, respectively. The FM phases are denoted by light yellow backgrounds.

Figure 12

(a) Magnification of the spin-polarized valence bands of the monolayer ${\mathrm{MoI}}_3$. The red and blue curves denote the spin-up and spin-down bands, respectively. (b) The corresponding magnified band structure of the monolayer ${\mathrm{MoI}}_3$ with the consideration of SOC. (c) The calculated Berry curvatures when the $E_F$ is tuned into the SOC induced gap of $\mathrm{\Delta }$ in (b). The inset in (c) shows the magnified Berry curvatures around the $\mathrm{\Gamma }$ point. (d) The calculated DOSs of the semi-infinite monolayer ${\mathrm{MoI}}_3$ system.

Figure 13

(a) Magnification of the spin-polarized valence bands for the $5\%$ tensile strained monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$. The red curves denote the spin-up bands. (b) The corresponding magnified band structure of (a) with SOC. (c) The calculated DOSs of the $5\%$ tensile strained semi-infinite monolayer $\mathrm{Mo}{\mathrm{Cl}}_3$. (d)–(f) The same as (a)–(c), except that the calculated system is the $5\%$ tensile strained monolayer $\mathrm{Mo}{\mathrm{Br}}_3$.

Figure 14

(a) The band structure of the $2\%$ compressive strained monolayer ${\mathrm{MoI}}_3$ by using HSE06 calculations. The red and blue curves denote the spin-up and spin-down bands, respectively. (b) Zooming in of the band structure in the rectangular dashed box in (a). (c) The corresponding band structure of (b) with the consideration of SOC. (d) The calculated DOSs of the $2\%$ compressive strained semi-infinite monolayer ${\mathrm{MoI}}_3$.