Inflation rules, band structure, and localization of electronic states in a two-dimensional Penrose lattice

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Abstract
We study the center model of Penrose lattices with a fivefold rotational symmetric axis in the framework of a tight-binding Hamiltonian. The inflation (or production) rules of ‘‘finite Penrose patterns’’ generated by repeated application of deflation and rescaling are found, which show a definite hierarchical structure of the finite Penrose patterns. A similarity transformation is introduced to reduce the Hamiltonian, and the degeneracies of eigenstates are analytically determined. It is found that two-thirds of the energy states are doubly degenerate and the remaining one-third nondegenerate. The energy spectra for finite Penrose patterns of the first six generations and the density of states for three samples are presented. The Household and improved Dean method for solving energy eigenvalues and eigenstates are used to examine the localization of electronic states. By use of several different methods and criteria, three kinds of wave-function behavior (extended, localized, and intermediate states) are clearly observed.

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