Continuum Schroedinger Operators for Sharply Terminated Graphene-Like Structures

• Published in: Communications in mathematical physics Vol. 380; no. 2; pp. 853 - 945
• Main Authors: Fefferman, C. L., Weinstein, M. I.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-12-2020; Springer Nature B.V
• Subjects: Apexes; Binding; Classical and Quantum Gravitation; Complex Systems; Eigenvalues; Eigenvectors; Energy spectra; Graph theory; Graphene; Honeycomb structures; Mathematical and Computational Physics; Mathematical Physics; Operators (mathematics); Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Single electrons; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03868-0

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
(
R
2
)
:
H
edge
λ
=
-
Δ
+
λ
2
V
♯
, with a potential
V
♯
given by a sum of translates an atomic potential well,
V
0
, of depth
λ
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
edge
λ
in the strong binding regime (
λ
large). In particular, we prove scaled resolvent convergence of
H
edge
λ
acting on
L
2
(
R
2
)
, to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in
l
2
(
N
0
;
C
2
)
. We also prove the existence of
edge states
: solutions of the eigenvalue problem for
H
edge
λ
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.