  A two-dimensional (2D) material system with both piezoelectricity and
ferromagnetic (FM) order, referred to as a 2D piezoelectric ferromagnetism
(PFM), may open up unprecedented opportunities for intriguing physics. Inspired
by experimentally synthesized Janus monolayer MoSSe from $\mathrm{MoS_2}$, in
this work, the Janus monolayer $\mathrm{CrBr_{1.5}I_{1.5}}$ with dynamic,
mechanical and thermal stabilities is predicted, which is constructed from
synthesized ferromagnetic $\mathrm{CrI_3}$ monolayer by replacing the top I
atomic layer with Br atoms. Calculated results show that monolayer
$\mathrm{CrBr_{1.5}I_{1.5}}$ is an intrinsic FM half semiconductor with valence
and conduction bands being fully spin-polarized in the same spin direction.
Furthermore, monolayer $\mathrm{CrBr_{1.5}I_{1.5}}$ possesses a sizable
magnetic anisotropy energy (MAE). By symmetry analysis, it is found that both
in-plane and out-of-plane piezoelectric polarizations can be induced by a
uniaxial strain in the basal plane. The calculated in-plane $d_{22}$ value of
0.557 pm/V is small. However, more excitingly, the out-of-plane $d_{31}$ is as
high as 1.138 pm/V, which is obviously higher compared with ones of other 2D
known materials. The strong out of-plane piezoelectricity is highly desirable
for ultrathin piezoelectric devices. Moreover, strain engineering is used to
tune piezoelectricity of monolayer $\mathrm{CrBr_{1.5}I_{1.5}}$. It is found
that compressive strain can improve the $d_{22}$, and tensile strain can
enhance the $d_{31}$. A FM order to antiferromagnetic (AFM) order phase
transition can be induced by compressive strain, and the critical point is
about 0.95 strain. That is to say that a 2D piezoelectric antiferromagnetism
(PAFM) can be achieved by compressive strain, and the corresponding $d_{22}$
and $d_{31}$ are 0.677 pm/V and 0.999 pm/V at 0.94 strain, respectively.
