  The search for non-centrosymmetric superconductors that may exhibit unusual
physical properties and unconventional superconductivity has yielded the
synthesis of a non-centrosymmetric phosphide Mg$_2$Rh$_3$P with an
Al$_2$Mo$_3$C-type structure. Although stoichiometric Mg$_2$Rh$_3$P does not
exhibit superconductivity at temperatures above 2 K, we found that an Mg
deficiency of approximately 5 at.% in the Mg$_2$Rh$_3$P induced
superconductivity at 3.9 K. Physical properties such as the lattice parameter a
= 0.70881 nm, Sommerfeld constant $\gamma_n$ = 5.36 mJ mol$^{-1}$ K$^{-2}$,
specific heat jump $\Delta$C$_{el}$/$\gamma_n$Tc = 0.72, electron-phonon
coupling constant $\lambda$$_{e-p}$ = 0.58, upper critical field H$_{c2}$(0) =
24.3 kOe, and pressure effect dTc/dP = -0.34 K/GPa were measured for the
superconducting Mg$_{2-\delta}$Rh$_3$P ($\delta$ $\sim$ 0.1). Band-structure
calculations indicate that exotic fermions, which are not present in
high-energy physics, exist in Mg$_2$Rh$_3$P. Since Mg, Rh, and P are the first
elements used at each crystal site of Al$_2$Mo$_3$C-type compounds, the
discovery of Mg$_2$Rh$_3$P may guide the search for new related materials.
