  A-type antiferromagnetism, with an in-plane ferromagnetic order and the
interlayer antiferromagnetic coupling, owns inborn advantages for electrical
manipulations but is naturally rare in real materials except in those
artificial antiferromagnetic heterostructures. Here, a robust layered
antiferromagnetism with a high N\'eel temperature is predicted in a MXene
Cr$_2$CCl$_2$ monolayer, which provides an ideal platform as a magnetoelectric
field effect transistor. Based on first-principles calculations, we demonstrate
that an electric field can induce the band splitting between spin-up and
spin-down channels. Although no net magnetization is generated, the inversion
symmetry between the lower Cr layer and the upper Cr layer is broken via
electronic cloud distortions. Moreover, this electric field can be replaced by
a proximate ferroelectric layer for nonvolatility. The magneto-optic Kerr
effect can be used to detect this magnetoelectricity, even if it is a collinear
antiferromagnet with zero magnetization.
