Strain-tunable magnetic anisotropy in monolayer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>CrCl</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>CrBr</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>CrI</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>

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 $\ensuremath{\varepsilon}$ have been carried out. It is found that all three compounds exhibit ferromagnetic ordering at the ground state (with $\ensuremath{\varepsilon}=0$), 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 $\ensuremath{\varepsilon}$ = 5%.