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A typical lattice structure (W/2 = 18, iW = 7). At the domain wall, the position of the alternating positive (red) and negative (blue) sublattice potentials is changed from A+/B− to A−/B+, where A and B denote the index of individual pairs. The AB pairs are chosen so as to be positioned along the axis of the ribbon. The unit cell is shown by the dashed line.
Band gap trends with increasing m for (top to bottom) gapless superlattice, potentially gapless (with even number of A–B pairs), and Dirac ribbons. Note the large dip with all ribbons when iW = 2, which is the smallest possible Dirac domain. Insets are for the same width of ribbon, but with a sinusoidally varying potential; we see that this does not alter the gap-closing condition, but instead decreases the integrated strength of the potential, and thus moves the gap-closing condition to larger values of m/t.
The mass dependence of the band gap at the Γ point, and . The behavior at the Γ point establishes the third universal classification of gapless superlattice “ribbons,” which are the only armchair superlattices to become gapless as . The other two points define the high-symmetry gapless points and band-inversion points, which are lifted for sufficiently large m. For each system, iW = 8.
The different classifications of armchair honeycomb nanoribbon band gaps Δ according to their widths W. δ is the variation in hopping near the edge of a hydrogen passivated ribbon, such that . The new classification, presented here, is appropriate in staggered sublattice potentials, an experimentally relevant, and tuneable quantity, in particular, contexts.
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