Strain-tunable magnetic anisotropy in monolayer ${\mathrm{CrCl}}_3$, ${\mathrm{CrBr}}_3$, and ${\mathrm{CrI}}_3$

Recent observation of intrinsic ferromagnetism in two-dimensional (2D) ${\mathrm{CrI}}_3$ is associated with the large magnetic anisotropy due to strong spin-orbit coupling of I. Magnetic anisotropy energy (MAE) defines the stability of magnetization in a specific direction with respect to the crystal lattice and is an important parameter for nanoscale applications. In this work we apply the density functional theory to study the strain dependence of MAE in 2D monolayer chromium trihalides $\mathrm{Cr}{X}_3$ (with $X$ = Cl, Br, and I). Detailed calculations of their energetics, atomic structures, and electronic structures under the influence of a biaxial strain $\varepsilon$ have been carried out. It is found that all three compounds exhibit ferromagnetic ordering at the ground state (with $\varepsilon =0$), and upon applying a compressive strain, phase transition to the antiferromagnetic state occurs. Unlike in ${\mathrm{CrCl}}_3$ and ${\mathrm{CrBr}}_3$, the electronic band gap in ${\mathrm{CrI}}_3$ increases when a tensile strain is applied. The MAE also exhibits a strain dependence in the chromium trihalides: it increases when a compressive strain is applied in ${\mathrm{CrI}}_3$, while an opposite trend is observed in the other two compounds. In particular, the MAE of ${\mathrm{CrI}}_3$ can be increased by 47% with a compressive strain of $\varepsilon$ = 5%.

Figure 1

Atomic structure of monolayer ${\mathrm{CrX}}_3$ (X = Cl,Br,I). (a) Top view and (b) side view of a single layer. (c) Bonding between chromium and iodine atoms. The unit cell of ${\mathrm{CrX}}_3$ which includes two Cr and six X atoms has been indicated in (a). The bond length $l$ between Cr and an X atom, the bond angle ${\theta }_1$ between Cr and two X atoms in the same plane, and the axial angle ${\theta }_2$ are also shown in (c). (d) The two magnetic orders, namely, Néel antiferromagnetic (AFM) and ferromagnetic (FM)

Figure 2

Noncollinear spin-polarized electronic band dispersions for monolayer chromium trihalides. The effect of typical compressive and tensile strains is also shown for (a) ${\mathrm{CrCl}}_3$, (b) ${\mathrm{CrBr}}_3$, and (c) ${\mathrm{CrI}}_3$. The energy band gaps between the conduction band minimum and valence band maximum (VBM) are indicated using red arrows in each case. The VBM has been shifted to zero.

Figure 3

Energy difference between the FM and AFM phases for (a) ${\mathrm{CrCl}}_3$, (b) ${\mathrm{CrBr}}_3$, and (c) ${\mathrm{CrI}}_3$. The AFM phase region is highlighted in red. The calculated Curie temperature is also shown for each case.

Figure 4

Dependence of structural parameters on biaxial strain $\varepsilon$ for the chromium trihalides. (a) Bond length $l$. (b) Bond angle ${\theta }_1$. (c) Axial bond angle ${\theta }_2$.

Figure 5

The effect of strain on the magnetic anisotropic energy. Change in energy with respect to the magnetization angle ${\theta }_M$ for (a) ${\mathrm{CrCl}}_3$, (b) ${\mathrm{CrBr}}_3$, and (c) ${\mathrm{CrI}}_3$. The lines are the fitted results. Change in MAE with respect to strain in (d) ${\mathrm{CrCl}}_3$, (e) ${\mathrm{CrBr}}_3$, and (f) ${\mathrm{CrI}}_3$. The AFM phase region is highlighted in red. (g)–(i) The change in the fitted parameters ${\lambda }_1$ and ${\lambda }_2$.