Continuum Schroedinger operators for sharply terminated graphene-like structures

Abstract

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 $L^2({\mathbb{R}}^2)$: $H_{\mathrm{edge}}^{\lambda }=-\Delta +{\lambda }^2{V}_{♯}$, with a potential $V_{♯}$ 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 $H_{\mathrm{edge}}^{\lambda }$ in the strong binding regime ($\lambda$ large). In particular, we prove scaled resolvent convergence of $H_{\mathrm{edge}}^{\lambda }$ 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 {\it edge states}: solutions of the eigenvalue problem for $H_{\mathrm{edge}}^{\lambda }$ which are localized transverse to the edge and pseudo-periodic (propagating or plane-wave like) parallel to the edge. These edge states arise from a "flat-band" of eigenstates the tight-binding Hamiltonian.

arXiv commentRevised version -- 89 pages, 2 figures; new title and abstract, revised introduction. In addition to a construction of the nearly flat band of edge states, the article now includes a proof of scaled resolvent convergence in a neighborhood of the low-lying spectrum

References in corpus (16)

• The electronic properties of graphene
• Topological Insulators
• Experimental Observation of Quantum Hall Effect and Berry's Phase in Graphene
• Photonic Floquet Topological Insulators
• Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry
• Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal
• Analogs of quantum Hall effect edge states in photonic crystals
• The Zak phase and the existence of edge states in graphene
• Observation of Photonic Topological Valley-Hall Edge States
• Photonic crystals for nano-light in moiré graphene superlattices
• Edge states and the bulk-boundary correspondence in Dirac Hamiltonians
• Bulk-edge correspondence for two-dimensional topological insulators
• Two-dimensional Mott-Hubbard electrons in an artificial honeycomb lattice
• Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene
• Characterization of edge states in perturbed honeycomb structures
• Photonic realization of topologically protected bound states in domain-wall waveguide arrays

Cited by in corpus (3)

• Characterization of edge states in perturbed honeycomb structures
• Wave packet dynamics in slowly modulated photonic graphene
• Differential Equations of Quantum Mechanics