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Strong topological insulators (TIs) support topological surfaces states on any crystal surface. In contrast, a weak, time-reversal-symmetry-driven TI with at least one non-zero  v  1 ,  v  2 ,  v  3  ℤ 2  index should host spin-locked topological surface states on the surfaces that are not parallel to the crystal plane with Miller indices ( v  1   v  2   v  3 ). On the other hand, mirror symmetry can protect an even number of topological states on the surfaces that are perpendicular to a mirror plane. Various symmetries in a bulk material with a band inversion can independently preordain distinct crystal planes for realization of topological states. Here we demonstrate the first instance of coexistence of both phenomena in the weak 3D TI Bi 2 TeI which ( v  1   v  2   v  3 ) surface hosts a gapless spin-split surface state protected by the crystal mirror-symmetry. The observed topological state has an even number of crossing points in thedirections of the 2D Brillouin zone due to a non-TRIM bulk-band inversion. Our findings shed light on hitherto uncharted features of the electronic structure of weak topological insulators and open up new vistas for applications of these materials in spintronics.

Mirror-symmetry protected non-TRIM surface state in the weak topological insulator Bi2TeI