  The advancement of two-dimensional polar metals tends to be limited by the
incompatibility between electric polarity and metallicity as well as dimension
reduction. Here, we report polar and metallic Janus monolayers of MoSi$_2$N$_4$
family by breaking the out-of-plane (OOP) structural symmetry through Z (P/As)
substitution of N. Despite the semiconducting nature of MoSi$_2$X$_4$
(X=N/P/As), four Janus MoSi$_2$N$_{x}$Z$_{4-x}$ monolayers are found to be
polar metals owing to the weak coupling between the conducting electrons and
electric polarity. The metallicity is originated from the Z substitution
induced delocalization of occupied electrons in Mo-d orbitals. The OOP electric
polarizations around 10$-$203 pC/m are determined by the asymmetric OOP charge
distribution due to the non-centrosymmetric Janus structure. The corresponding
OOP piezoelectricity is further revealed as high as 39$-$153 pC/m and
0.10$-$0.31 pm/V for piezoelectric strain and stress coefficients,
respectively. The results demonstrate polar metallicity and high OOP
piezoelectricity in Janus MoSi$_2$N$_{x}$Z$_{4-x}$ monolayers and open new
vistas for exploiting unusual coexisting properties in monolayers derived from
MoSi$_2$N$_4$ family.
