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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics