Bloch oscillations are associated with weakly scattered transport in a periodic band structure. Most periodic structures have band gaps at the reciprocal lattice boundaries. Mathematically, this is because the two states at the edge of the (ID) Brillouin zone can have equal energy only by means of an accidental degeneracy. Helical quantum-wire structures are an exception to the general rule: although they are periodic in one dimension, the additional glide symmetry makes the boundary states degenerate, so the band gap vanishes. Their periodic band structure resembles qualitatively that of a free electron represented in the repeated-zone scheme. The Hamiltonian of a helical quantum wire with parabolic potential cross section can be represented by means of creation and annihilation operators, and this formalism has been used in a numerical study of helical quantum wire band structure.
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