A New Viewpoint in Computing Crystal Frequencies

Nierenberg, William A.

doi:10.1063/1.1748317

Cited by 10 publications

(9 citation statements)

References 1 publication

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“…Accordingly, in terms of the components k x and k y , the EDR is given by [12,41,42] E k x , k y = ε − λ (I) 2 cos k x a + cos k y a − λ (I I) 4 cos k x a cos k y a,…”

Section: Square Latticementioning

confidence: 99%

Algebraic Theory of Crystal Vibrations: Localization Properties of Wave Functions in Two-Dimensional Lattices

2017

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Abstract:The localization properties of the wave functions of vibrations in two-dimensional (2D) crystals are studied numerically for square and hexagonal lattices within the framework of an algebraic model. The wave functions of 2D lattices have remarkable localization properties, especially at the van Hove singularities (vHs). Finite-size sheets with a hexagonal lattice (graphene-like materials), in addition, exhibit at zero energy a localization of the wave functions at zigzag edges, so-called edge states. The striped structure of the wave functions at a vHs is particularly noteworthy. We have investigated its stability and that of the edge states with respect to perturbations in the lattice structure, and the effect of the boundary shape on the localization properties. We find that the stripes disappear instantaneously at the vHs in a square lattice when turning on the perturbation, whereas they broaden but persist at the vHss in a hexagonal lattice. For one of them, they eventually merge into edge states with increasing coupling, which, in contrast to the zero-energy edge states, are localized at armchair edges. The results are corroborated based on participation ratios, obtained under various conditions.

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“…Accordingly, in terms of the components k x and k y , the EDR is given by [12,41,42] E k x , k y = ε − λ (I) 2 cos k x a + cos k y a − λ (I I) 4 cos k x a cos k y a,…”

Section: Square Latticementioning

confidence: 99%

Algebraic Theory of Crystal Vibrations: Localization Properties of Wave Functions in Two-Dimensional Lattices

2017

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“…Introduction.-The focus of the experiments with microwave photonic crystals [1] reported in this Letter were the spectral properties of finite-size graphene sheets [2,3], particularly in the vicinity of the van Hove singularities [4] (vHSs) exhibited by the density of states (DOS) [3,5,6]. Graphene, a monoatomic layer of carbon atoms arranged on a honeycomb lattice, has exceptional electronic properties that stem from the shapes of its conduction and its valence band.…”

mentioning

confidence: 99%

Spectral Properties of Dirac Billiards at the van Hove Singularities

et al. 2016

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We study distributions of the ratios of level spacings of rectangular and Africa-shaped superconducting microwave resonators containing circular scatterers on a triangular grid, so-called Dirac billiards (DBs). The high-precision measurements allowed the determination of, respectively, all 1651 and 1823 eigenfrequencies in the first two bands. The resonance densities are similar to that of graphene. They exhibit two sharp peaks at the van Hove singularities which separate the band structure into regions with a linear and a quadratic dispersion relation, respectively. In the vicinity of the van Hove singularities we observe rapid changes in, e.g., the wave function structure. Accordingly, we question whether the spectral properties are there still determined by the shapes of the DBs. The commonly used statistical measures are no longer applicable; however, we demonstrate in this Letter that the ratio distributions provide suitable measures.

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“…These properties were first noticed by Nierenberg [31] for lines and further exploited and generalized for lines and rings in Refs. [29,32].…”

Section: The Fundamental Vibration V=1mentioning

confidence: 92%

“…This result follows from the mapping of the function f (x, y) defined over an area onto a function defined over a line, as discussed by Nierenberg [31].…”

Section: The Fundamental Vibration V=1mentioning

confidence: 99%

“…The matrices representative of these operators are of the type discussed previously and thus can be diagonalized analytically. The solution for the fundamental vibration of a square lattice was first given by Bowers and Rosenstock [33] in terms of functions over an area and subsequently by Nierenberg [31] in terms of functions over a line. In Ref.…”

Section: The Fundamental Vibration V=1mentioning

confidence: 99%

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Algebraic theory of crystal vibrations: Singularities and zeros in vibrations of one- and two-dimensional lattices

et al. 2015

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A novel method for the calculation of the energy dispersion relation (EDR) and density of states (DOS) in one (1D) and two (2D) dimensions is introduced and applied to linear lattices (1D) and square and hexagonal lattices (2D). The (van Hove) singularities and (Dirac) zeros of the DOS are discussed. Results for the 2D hexagonal lattice (graphene-like materials) are compared with experimental data in microwave photonic crystals.

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A New Viewpoint in Computing Crystal Frequencies

Cited by 10 publications

References 1 publication

Algebraic Theory of Crystal Vibrations: Localization Properties of Wave Functions in Two-Dimensional Lattices

Algebraic Theory of Crystal Vibrations: Localization Properties of Wave Functions in Two-Dimensional Lattices

Spectral Properties of Dirac Billiards at the van Hove Singularities

Algebraic theory of crystal vibrations: Singularities and zeros in vibrations of one- and two-dimensional lattices

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