  Recent observation of intrinsic ferromagnetism in two-dimensional (2D)
CrI$_3$ is associated with the large magnetic anisotropy due to strong
spin-orbit coupling (SOC) 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 CrX$_3$ (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 antiferromagnetic state occurs. Unlike in CrCl$_3$ and CrBr$_3$,
the electronic band gap in 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 CrI$_3$, while an opposite
trend is observed in the other two compounds. In particular, the MAE of CrI$_3$
can be increased by 47\% with a compressive strain of $\varepsilon$ = 5\%.
