We present angle-resolved photoemission spectroscopy measurements of the surface states on in-situ grown (111) oriented films of Pb${}_{1-x}$Sn${}_x$Se, a three-dimensional topological crystalline insulator. We observe surface states with Dirac-like dispersion at $\overline{\Gamma }$ and $\overline{M}$ in the surface Brillouin zone, supporting recent theoretical predictions for this family of materials. We study the parallel dispersion isotropy and Dirac-point binding energy of the surface states, and perform tight-binding calculations to support our findings. The relative simplicity of the growth technique is encouraging, and suggests a clear path for future investigations into the role of strain, vicinality, and alternative surface orientations in (Pb,Sn)Se solid solutions.

We study the nature of $\left(001\right)$ surface states in Pb${}_{0.73}$Sn${}_{0.27}$Se in the newly discovered topological-crystalline-insulator (TCI) phase as well as the corresponding topologically trivial state above the band-gap-inversion temperature. Our calculations predict not only metallic surface states with a nontrivial chiral spin structure for the TCI case, but also nonmetallic (gapped) surface states with nonzero spin polarization when the system is a normal insulator. For both phases, angle- and spin-resolved photoelectron spectroscopy measurements provide conclusive evidence for the formation of these $\left(001\right)$ surface states in Pb${}_{0.73}$Sn${}_{0.27}$Se, as well as for their chiral spin structure.

The recent discovery of a topological phase transition in IV-VI narrow-gap semiconductors has revitalized the decades-old interest in the bulk band inversion occurring in these materials. Here we systematically study the (001) surface states of ${\text{Pb}}_{1-x}{\text{Sn}}_x\text{Se}$ mixed crystals by means of angle-resolved photoelectron spectroscopy in the parameter space $0\le x\le 0.37$ and $300\text{K}\ge T\ge 9\text{K}$. Using the surface-state observations, we monitor directly the topological phase transition in this solid solution and gain valuable information on the evolution of the underlying fundamental band gap of the system. In contrast to common model expectations, the band-gap evolution appears to be nonlinear as a function of the studied parameters, resulting in the measuring of a discontinuous band-inversion process. This finding signifies that the anticipated gapless bulk state is in fact not a stable configuration and that the topological phase transition therefore exhibits features akin to a first-order transition.

Since the advent of topological insulators hosting Dirac surface states, efforts have been made to gap these states in a controllable way. A new route to accomplish this was opened up by the discovery of topological crystalline insulators where the topological states are protected by crystal symmetries and thus prone to gap formation by structural changes of the lattice. Here we show a temperature-driven gap opening in Dirac surface states within the topological crystalline insulator phase in (Pb,Sn)Se. By using angle-resolved photoelectron spectroscopy, the gap formation and mass acquisition is studied as a function of composition and temperature. The resulting observations lead to the addition of a temperature- and composition-dependent boundary between massless and massive Dirac states in the topological phase diagram for (Pb,Sn)Se (001). Overall, our results experimentally establish the possibility to tune between massless and massive topological states on the surface of a topological system.