  A two-dimensional (2D) material with piezoelectricity, topological and
ferromagnetic (FM) orders, namely 2D piezoelectric quantum anomalous hall
insulator (PQAHI), may open new opportunities to realize novel physics and
applications. Here, by first-principles calculations, a family of 2D Janus
monolayer $\mathrm{Fe_2IX}$ (X=Cl and Br) with dynamic, mechanical and thermal
stabilities is predict to be room-temperature PQAHI. At the absence of
spin-orbit coupling (SOC), monolayer $\mathrm{Fe_2IX}$ (X=Cl and Br) is a half
Dirac semimetal state. When the SOC is included, these monolayers become
quantum anomalous hall (QAH) states with sizable gaps (more than two hundred
meV) and two chiral edge modes (Chern number C=2). It is also found that
monolayer $\mathrm{Fe_2IX}$ (X=Cl and Br) possesses robust QAH states against
biaxial strain. By symmetry analysis, it is found that only out-of-plane
piezoelectric response can be induced by a uniaxial strain in the basal plane.
The calculated out-of-plane $d_{31}$ of $\mathrm{Fe_2ICl}$ ($\mathrm{Fe_2IBr}$)
is 0.467 pm/V (0.384 pm/V), which is higher than or comparable with ones of
other 2D known materials. Meanwhile, using Monte Carlo (MC) simulations, the
Curie temperature $T_C$ is estimated to be 429/403 K for monolayer
$\mathrm{Fe_2ICl}$/$\mathrm{Fe_2IBr}$ at FM ground state, which is above room
temperature. Finally, the interplay of electronic correlations with nontrivial
band topology is studied to confirm the robustness of QAH state. The
combination of piezoelectricity, topological and FM orders makes monolayer
$\mathrm{Fe_2IX}$ (X=Cl and Br) become a potential platform for
multi-functional spintronic applications with large gap and high $T_C$. Our
works provide possibility to use the piezotronic effect to control QAH effects,
and can stimulate further experimental works.
