  We predict that monolayer FeCl$_2$ is a two-dimensional piezoelectric
ferromagnet (PFM) with easy-axis magnetism and a Curie temperature of 260 K.
Our ab-initio calculations combined with data mining reveal 2H-FeCl$_2$ as the
only easy-axis 2D monolayer PFM, and that its magnetic anisotropy increases
many-fold with moderate hole doping. We develop an analysis based on magnetic
anisotropy energy densities that explain the magnetic and doping-dependent
behavior of FeCl$_2$, as well as VSe$_2$ and CrI$_3$, and can enable the design
of future 2D magnetically ordered materials.
