Quantum confinement and heavy surface states of Dirac fermions in bismuth (111) films: An analytical approach

Abstract

Recent high-resolution angle-resolved photoemission spectroscopy experiments have given a reason to believe that pure bismuth is a topologically nontrivial semimetal. We derive an analytic theory of surface and size-quantized states of Dirac fermions in Bi(111) films taking into account the new data. The theory relies on a new phenomenological momentum-dependent boundary condition for the effective Dirac equation. The boundary condition is described by two real parameters that are expressed by a linear combination of the Dresselhaus and Rashba interface spin-orbit interaction parameters. In semi-infinite Bi(111), near the $\overline{\mathrm{M}}$ point the surface states possess anisotropical parabolic dispersion with very heavy effective mass in the $\overline{\mathrm{\ensuremath{\Gamma}}}\text{\ensuremath{-}}\overline{\mathrm{M}}$ direction order of ten free electron masses and light effective mass in the $\overline{\mathrm{M}}\text{\ensuremath{-}}\overline{\mathrm{K}}$ direction order of one hundredth of free electron mass. In Bi(111) films with equivalent surfaces, the surface states from top and bottom surfaces are not split. In such a symmetric film with arbitrary thickness, the bottom of the lowest quantum confinement subband in the conduction band coincides with the bottom of the bulk conduction band in the $\overline{\mathrm{M}}$ point.