  We report a comprehensive magneto-transport study on single crystalline
p-type topological crystalline insulator (TCI) SnTe, across the
cubic-to-rhombohedral (R3m) transition which occurs as a function of
temperature. The electrical resistivity of a well-characterized SnTe crystal
shows evidence for the cubic-to-rhombohedral structural transition at T$_s$
$\sim$64,K and a carrier density of $\sim$1.8$\times$10$^{20}$ at 77,K. As a
function of applied magnetic field perpendicular to the (100) plane, SnTe
exhibits a large unsaturated linear magnetoresistance (LMR) reaching a value of
42% at 5K and 8T. LMR is found to have a direct dependence on the mobility and
a detailed analysis shows that it follows the classical Parish-Littlewood model
of conductivity fluctuations arising from macroscopic inhomogeneity of
tellurium interstitial atoms. We also observe SdH oscillations in the
rhombohedral (R3m) phase with a Berry phase of $\pi$ and significantly lower
carrier density $\sim$5.32$\times$10$^{11}$cm$^{-2}$ at 2K, which provides
direct evidence of protected topological surface states in the (R3m) phase. The
Hall conductivity shows a transformation from one band to two-band behavior
across the structural transition, thus providing experimental evidence for the
degeneracy lifting of bulk valence bands below the cubic symmetry breaking
point which is consistent with recent band structure calculations. The overall
results indicate that magneto-transport studies can distinctly probe the
surface and bulk sensitive properties of SnTe, and can also track the
band-splitting of degenerate bands at the Fermi level across the
cubic-to-rhombohedral (R3m) transition.
