TY - JOUR

T1 - Continuum Schroedinger Operators for Sharply Terminated Graphene-Like Structures

AU - Fefferman, C. L.

AU - Weinstein, M. I.

N1 - Funding Information:
The authors wish to thank Gian Michele Graf, Alexis Drouot and Jacob Shapiro for very stimulating discussions. We would also like to thank Bernard Helffer for correspondence concerning previous general results on tight-binding limits. Part of this research was done while MIW was Bergman Visiting Professor at Stanford University. CLF and MIW wish to thank the Department of Mathematics at Stanford University for its hospitality. We are grateful to the referees for their very careful reading of the paper. This research was supported in part by National Science Foundation Grants DMS-1265524 (CLF) and DMS-1412560, DMS-1908657, DMS-1620418 and Simons Foundation Math + X Investigator Award #376319 (MIW).
Funding Information:
The authors wish to thank Gian Michele Graf, Alexis Drouot and Jacob Shapiro for very stimulating discussions. We would also like to thank Bernard Helffer for correspondence concerning previous general results on tight-binding limits. Part of this research was done while MIW was Bergman Visiting Professor at Stanford University. CLF and MIW wish to thank the Department of Mathematics at Stanford University for its hospitality. We are grateful to the referees for their very careful reading of the paper. This research was supported in part by National Science Foundation Grants DMS-1265524 (CLF) and DMS-1412560, DMS-1908657, DMS-1620418 and Simons Foundation Math + X Investigator Award #376319 (MIW).
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/12/1

Y1 - 2020/12/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-020-03868-0

DO - 10.1007/s00220-020-03868-0

M3 - Article

AN - SCOPUS:85092939639

SN - 0010-3616

VL - 380

SP - 853

EP - 945

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 2

ER -