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 $$L^2(\mathbb {R}^2)$$ : $$H^\lambda _{\mathrm{edge}}=-\Delta +\lambda ^2 V_\sharp $$ , with a potential $$V_\sharp $$ given by a sum of translates an atomic potential well, $$V_0$$ , of depth $$\lambda ^2$$ , 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 $$H^\lambda _{\mathrm{edge}}$$ in the strong binding regime ( $$\lambda $$ large). In particular, we prove scaled resolvent convergence of $$H^\lambda _{\mathrm{edge}}$$ acting on $$L^2(\mathbb {R}^2)$$ , to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in $$l^2(\mathbb {N}_0;\mathbb {C}^2)$$ . We also prove the existence of edge states: solutions of the eigenvalue problem for $$H^\lambda _{\mathrm{edge}}$$ 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.