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$\alpha$-Sn is an elemental topological material, whose topological phases can be tuned by strain and magnetic field. Such tunability offers a substantial potential for topological electronics. However, InSb substrates, commonly used to stabilize $\alpha$-Sn allotrope, suffer from parallel conduction, restricting transport investigations and potential applications. Here, the successful MBE growth of high-quality $\alpha$-Sn layers on insulating, hybrid CdTe/GaAs(001) substrates, with bulk electron mobility approaching 20000 cm$^2$V$^{-1}$s$^{-1}$ is reported. The electronic properties of the samples are systematically investigated by independent complementary techniques, enabling thorough characterization of the 3D Dirac (DSM) and Weyl (WSM) semimetal phases induced by the strains and magnetic field, respectively. Magneto-optical experiments, corroborated with band structure modeling, provide an exhaustive description of the bulk states in the DSM phase. The modeled electronic structure is directly observed in angle-resolved photoemission spectroscopy, which reveals linearly dispersing bands near the Fermi level. The first detailed study of negative longitudinal magnetoresistance relates this effect to the chiral anomaly and, consequently, to the presence of WSM. Observation of the $\pi$ Berry phase in Shubnikov-de Haas oscillations agrees with the topologically non-trivial nature of the investigated samples. Our findings establish $\alpha$-Sn as an attractive topological material for exploring relativistic physics and future applications., Comment: Accepted version; Main text: 37 pages, 7 figures; Supplementary Materials: 29 pages, 13 figures

The helical spin texture on the surface of topological crystalline insulators (TCI) makes these materials attractive for application in spintronics. In this work, spin-polarization and electronic structure of surface states of (111)-oriented Pb$_{1-x}$Sn$_x$Se TCI epitaxial films are examined by angle -- as well as spin-resolved photoemission spectroscopy (SR-ARPES). High-quality epilayers with various Sn content are grown by the molecular beam epitaxy (MBE) method. Topological-normal insulator transition manifesting itself as band gap opening is observed. It is shown that the gap opening can be induced not only by changing the Sn content of the epilayer but also depositing a transition metal (TM) on its surface. In the latter case, the observed gaping of the surface states is caused by change in surface composition and not by magnetism. We also show that helical spin polarization is present not only for samples of topological composition but also for trivial ones (with an open band gap). The observed spin polarization reaches a value of 30 % for the in-plane spin component and is almost absent for the out-of-plane one. We believe that our work will pave the way for the application of surface states not only of topological but also normal insulators based on lead-tin chalcogenides in spin-charge conversion devices., Comment: 9 pages, 7 figures; Accepted version