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Inducing Lifshitz transition by extrinsic doping of surface bands in topological crystalline insulator Pb$_{1-x}$Sn$_{x}$Se

Narrow gap semiconductor Pb$_{1-x}$Sn$_{x}$Se was investigated for topologically protected surface states in its rock-salt structural phase for x=0.45, 0.23, 0.15, and 0. Angle-resolved photoelectron spectroscopy of intrinsically p-doped samples showed clear indication of two Dirac cones, eccentric about the time-reversal invariant point X of the surface Brillouin zone for all but the x=0 sample. Adsorption of alkalies gradually filled the surface bands with electrons, driving the x>0 topological crystalline insulator systems through Lifshitz transitions, and from a hole- to electron-like Fermi surface. The electron-doped bands in x>0 samples exhibited the full configuration of the Dirac cones, also confirming electron-hole symmetry of the surface bands.