Homogenisation for hexagonal lattices and honeycomb structures

Makwana, Mehul P.

doi:10.1093/qjmam/hbu019

Cited by 11 publications

(4 citation statements)

References 35 publications

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“…The low-lying dispersion surfaces of honeycomb Schrödinger operators in the strong binding regime, where the potential is the superposition of a general class of atomic potential wells, and its relation to the tight-binding limit, was studied in [29]. Tight binding models have been studied extensively in the fundamental and applied physics and mathematics communities; see, for example, [1,6,17,20].…”

Section: Connections To Previous Rigorous Analytical Workmentioning

confidence: 99%

Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene

Lee-Thorp

Weinstein

Zhu

2018

Arch Rational Mech Anal

111

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Consider electromagnetic waves in two-dimensional honeycomb structured media, whose constitutive laws have the symmetries of a hexagonal tiling of the plane. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operatordenotes the equilateral triangular lattice), and such that with respect to some origin of coordinates,where R is a 120 • rotation in the plane). A summary of our results is as follows: a) For generic honeycomb structured media, the band structure of L A has Dirac points, i.e. conical intersections between two adjacent Floquet-Bloch dispersion surfaces. b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose waveenvelope, on long time scales, is governed by an effective two-dimensional time-dependent system of massless Dirac equations. c) Dirac points are unstable to arbitrary small perturbations which break either C (complexconjugation) symmetry or P (inversion) symmetry. d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. e) These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. arXiv:1710.03389v2 [math-ph] 13 Sep 2018 PreliminariesIn this section, we outline the relevant spectral theory [22,50, 51,76] and introduce terminology and notation for discussing the symmetry properties of the (unperturbed) bulk operator L A . Fourier analysisLet {v 1 , v 2 } be a linearly independent set in R 2 . The lattice generated by {v 1 , v 2 } is the subset of R 2 :A choice of fundamental cell is the parallelogram: Ω = {θ 1 v 1 + θ 2 v 2 : 0 ≤ θ j ≤ 1, j = 1, 2}.

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Section: Connections To Previous Rigorous Analytical Workmentioning

confidence: 99%

Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene

Lee-Thorp

Weinstein

Zhu

2018

Arch Rational Mech Anal

111

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“…The stand of the mathematical understanding of graphene is comparably less developed. All available results are extremely recent and concern the modeling of transport properties of electrons in graphene sheets [3,6,13,16,25,34,35], homogenization [8,33], atomistic-to-continuum passage for nanotubes [14], geometry of monolayers under Gaussian perturbations [11], external charges [27] or magnetic fields [10], combinatorial description of graphene patches [22], and numerical simulation of dynamics via nonlocal elasticity theory [44]. Remarkably, the determination of the equilibrium shapes and the Wulff shapes of graphene samples and graphene nanostructures is still a challenging problem [1,5,19].…”

Section: Introductionmentioning

confidence: 99%

Wulff shape emergence in graphene

Davoli

Piovano

Stefanelli

2016

Math. Models Methods Appl. Sci.

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Graphene samples are identified as minimizers of configurational energies featuring both two- and three-body atomic-interaction terms. This variational viewpoint allows for a detailed description of ground-state geometries as connected subsets of a regular hexagonal lattice. We investigate here how these geometries evolve as the number [Formula: see text] of carbon atoms in the graphene sample increases. By means of an equivalent characterization of minimality via a discrete isoperimetric inequality, we prove that ground states converge to the ideal hexagonal Wulff shape as [Formula: see text]. Precisely, ground states deviate from such hexagonal Wulff shape by at most [Formula: see text] atoms, where both the constant [Formula: see text] and the rate [Formula: see text] are sharp.

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“…That the two-band tight-binding model gives an accurate approximation of the low-lying dispersion surfaces in the regime of strong binding was proved in [24]; see also Remark 1.9 below. Other results on Dirac points for Schroedinger operators on R 2 may be found in [1,2,14,27,38], coupled oscillator models [43] and on quantum graphs in [17,37].…”

Section: Introductionmentioning

confidence: 83%

Continuum Schroedinger operators for sharply terminated graphene-like structures

Fefferman

Weinstein

2018

Preprint

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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, H λ , with a potential given by a sum of translates of identical atomic potential wells of depth λ 2 , centered on a subset of a honeycomb structure with a zigzag edge. In the strong binding regime (λ large) there exist edge states, solutions of H λ Ψ = EΨ, which are localized transverse to the edge and are propagating (plane-wave like) parallel to the edge. Upon λ− dependent rescaling, these edge-states are well-approximated by, and converge (as λ ↑ ∞) to the flat-band of edge states of a limiting discrete " tight-binding " model .

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Homogenisation for hexagonal lattices and honeycomb structures

Cited by 11 publications

References 35 publications

Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene

Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene

Wulff shape emergence in graphene

Continuum Schroedinger operators for sharply terminated graphene-like structures

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