Volkov-Pankratov states in topological graphene nanoribbons
In topological systems, a modulation in the gap onset near interfaces can lead to the appearance of massive edge states, as were first described by Volkov and Pankratov. In this work, we study graphene nanoribbons in the presence of intrinsic spin-orbit coupling smoothly modulated near the system edges. We show that this space modulation leads to the appearance of Volkov-Pankratov states, in addition to the topologically protected ones. We obtain this result by means of two complementary methods, one based on the effective low-energy Dirac equation description and the other on a fully numerical tight-binding approach, finding excellent agreement between the two. We then show how transport measurements might reveal the presence of Volkov-Pankratov states, and discuss possible graphenelike structures in which such states might be observed. ; The work of T.B. and D.B. was supported by the Spanish Ministerio de Ciencia, Innovation y Universidades (MICINN) through Project No. FIS2017-82804-P, and by the Transnational Common Laboratory Quantum-ChemPhys. M.R.C. acknowledges funding from the Spanish Government through Project No. MAT2017-88377-C2-2-R (AEI/FEDER) and the Generalitat Valenciana through Grant No. Cidegent2018004.
