Continuum Schroedinger Operators for Sharply Terminated Graphene-Like Structures

We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on L²(R²) : Hedgeλ=-Δ+λ2V♯, with a potential V_♯ given by a sum of translates an atomic potential well, V, of depth λ², centered on a subset of the vertices of a discrete honeycomb structure with a zigzag edge. We give a complete analysis of the low-lying energy spectrum of Hedgeλ in the strong binding regime (λ large). In particular, we prove scaled resolvent convergence of Hedgeλ acting on L²(R²) , to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in l²(N; C²). We also prove the existence of edge states: solutions of the eigenvalue problem for Hedgeλ which are localized transverse to the edge and pseudo-periodic plane-wave like parallel to the edge. These edge states arise from a “flat-band” of eigenstates of the tight-binding model.