  Epitaxial thin films of (Sn$_{x}$Pb$_{1-x}$)$_{1-y}$In$_{y}$Te were
successfully grown by molecular-beam-epitaxy (MBE) in a broad range of
compositions (0 $\leq$ x $\leq$ 1, 0 $\leq$ y $\leq$ 0.23). We investigated
electronic phases of the films by the measurements of electrical transport and
optical second harmonic generation. In this system, one can control the
inversion of band gap, the electric polarization that breaks the inversion
symmetry, and the Fermi level position by tuning the Pb/Sn ratio and In
composition. A plethora of topological electronic phases are expected to
emerge, such as topological crystalline insulator, topological semimetal, and
superconductivity. For the samples with large Sn compositions (x > 0.5), hole
density increases with In composition (y), which results in the appearance of
superconductivity. On the other hand, for those with small Sn compositions (x <
0.5), increase in In composition reduces the hole density and changes the
carrier type from p-type to n-type. In a narrow region centered at (x, y) =
(0.16, 0.07) where the n-type carriers are slightly doped, charge transport
with high mobility exceeding 5,000 cm$^{2}$V$^{-1}$s$^{-1}$ shows up,
representing the possible semimetal states. In those samples, the optical
second harmonic generation measurement shows the breaking of inversion symmetry
along the out-of-plane [111] direction, which ensures the presence of polar
semimetal state. The thin films of (Sn$_{x}$Pb$_{1-x}$)$_{1-y}$In$_{y}$Te
materials systems with a variety of electronic states would become a promising
materials platform for the exploration of novel quantum phenomena.
