  Stoichiometric SnTe is theoretically a small gap semiconductor that undergoes
a ferroelectric distortion on cooling. In reality however, crystals are always
non-stoichiometric and metallic; the ferroelectric transition is therefore more
accurately described as a polar structural transition. Here we study the Fermi
surface using quantum oscillations as a function of pressure. We find the
oscillation spectrum changes at high pressure, due to the suppression of the
polar transition and less than 10 kbar is sufficient to stabilize the
undistorted cubic lattice. This is accompanied by a large decrease in the Hall
and electrical resistivity. Combined with our density functional theory (DFT)
calculations and angle resolved photoemission spectroscopy (ARPES) measurements
this suggests the Fermi surface $L$-pockets have lower mobility than the
tubular Fermi surfaces that connect them. Also captured in our DFT calculations
is a small widening of the band gap and shift in density of states for the
polar phase. Additionally we find the unusual phenomenon of a linear
magnetoresistance that exists irrespective of the distortion that we attribute
to regions of the Fermi surface with high curvature.
