Insight into strain and electronic correlation dependent magnetism in monolayer $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$

$1T$-phase $\mathrm{Cr}{\mathrm{Te}}_2$ $\left(1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2\right)$ has received considerable interest recently due to its high Curie temperature $\left(T_C\right)$, which is desirable for practical spintronics applications. However, various magnetic behaviors of $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$ have been reported in recent experimental and theoretical studies when its thickness reduces to ultrathin limit. In this work, the magnetic diagram of monolayer (ML) $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$ with respect to in-plane biaxial strain and on-site Coulomb repulsion $U$ is obtained based on first-principles calculations. Our results indicate that the magnetic order of ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$ can vary among ferromagnets and antiferromagnets with strain and electronic correlation. We show that the large exchange anisotropy and higher-order biquadratic interactions are crucial to accurately describe the spin energies in ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$. The perplexing dependencies of the magnetocrystalline anisotropy on strain and $U$ are then well explained. Our work not only gives insight into the fundamental understanding of the unusual magnetic properties of ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$, which is helpful to understand the diverse observations on the magnetic order in $\mathrm{Cr}{\mathrm{Te}}_2$, but also sheds light on engineering their performance for spintronic devices.

Figure 1

(a) Top and side view of ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$. (b) Spin-polarized projected band structures of Te $p$ orbitals with $U=3\mathrm{eV}$. Green (black) dots represent the spin-up (down) channel. (c) Spin-polarized projected band structures of Cr $d$ orbitals with $U=3\mathrm{eV}$. Magenta (black) dots represent spin-up (down) channel.

Figure 2

Strain and electronic correlation effect on magnetism of ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$. (a) Top view of four magnetic configurations. Black arrows denote first-, second-, and third-nearest-neighbor exchange interactions $J_1, J_2$, and $J_3$. Positive (negative) values represent AFM (FM) interaction. Only Cr atoms are visible here for simplicity. (b) Diagram of DFT-calculated magnetic ground state with varied lattice constant and $U$. Red, green, and yellow represent AFM-$Z$, sAFM-AABB, and FM magnetic ground state, respectively. Phase boundaries indicated by black solid lines are just for eye guidance. Gray dot lines label lattice constant of $a_{\mathrm{DFT}}$ and $a_{\exp }$. Black stars [1]$\sim$ [3] mark lattice constant and $U$ adopted in Ref. [37], Ref. [13], and Ref. [19], respectively. (c) Exchange parameters as function of biaxial strain with $U=3\mathrm{eV}$, and (d) exchange parameters as function of $U$ with lattice constant of $a_{\mathrm{DFT}}$. Dashed black lines are fitted lines. Inset tables list first-order derivatives of exchange parameters with respect to strain or $U$.

Figure 3

Relative change of hopping term $t/t_0$ with (a) strain and (b) $U$. Only main $dd$ and $pd$ hopping channels are shown for simplicity. Wannier orbitals of corresponding hopping channels are shown as insets. Cr-Cr distance ($d_{\mathrm{Cr}\text{−}\mathrm{Cr}}$), Cr-Te distance ($d_{\mathrm{Cr}\text{−}\mathrm{Te}}$), and Cr-Te-Cr angle (${\theta }_{\mathrm{Cr}\text{−}\mathrm{Te}\text{−}\mathrm{Cr}}$) with respect to (c) strain and (d) $U$. (e) Schematic representation of (near-) ${90}^{\circ }$ Cr1 $d$ - Te $p$ - Cr2 $d$ superexchange interaction under 4% compressive strain (left) and 4% tensile strain (right). Panel (f) is similar to panel (e) but for $U=0\mathrm{eV}$ (left) and $U=5\mathrm{eV}$ (right).

Figure 4

Energetics of magnetic states for orthogonal cell of ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$ with two Cr atoms parallel aligned. Relative total energy dependence on magnetic propagation vector $q$ at (a) 0% and (c) 5% tensile strain with $U=3\mathrm{eV}$, respectively. Energy of FM state is chosen as reference. Dependence of spin energies on $q$ based on LSWT at (b) 0% and (d) 5% tensile strain with $U=3\mathrm{eV}$, respectively.

Figure 5

(a) MCA diagram of ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$ with respect to lattice constant and $U$. Black stars [1]$\sim$ [6] mark lattice constant and $U$ used in Ref. [20], Ref. [17], Ref. [19], Ref. [22], Ref. [46], and Ref. [22], respectively. Orbital-resolved MCA of (b) Cr $d$ and (c) Te $p$ orbitals of ML $1T\text{−}\mathrm{Cr}{\mathrm{Te}}_2$ with lattice constant of $a_{\mathrm{DFT}}$ and $U=3\mathrm{eV}$. Orbital-resolved MCA of Te $p$ orbitals as function of (d) strain with $U=3\mathrm{eV}$ and (e) $U$ at lattice constant of $a_{\mathrm{DFT}}$, for which only orbital pairs mostly affected by strain and $U$ are presented.