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Crystals 2017, 7(8), 246;

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

School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou 730000, China
Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520-8120, USA
Institute of Scientific Instruments, The Czech Academy of Sciences, Královopolská 147, Brno 61264, Czech Republic
Author to whom correspondence should be addressed.
Academic Editors: Cristina E. Giusca and Spyros Yannopoulos
Received: 14 July 2017 / Revised: 2 August 2017 / Accepted: 3 August 2017 / Published: 7 August 2017
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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. View Full-Text
Keywords: algebraic models; graphene-like materials; striped structures; photonic crystals algebraic models; graphene-like materials; striped structures; photonic crystals

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Dietz, B.; Iachello, F.; Macek, M. Algebraic Theory of Crystal Vibrations: Localization Properties of Wave Functions in Two-Dimensional Lattices. Crystals 2017, 7, 246.

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