 # Electronic Structure of 1D and 2D Quasiperiodic Systems

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This Demonstration shows the electronic levels of 1D and 2D aperiodic systems compared with the related electronic density of states (DOS) and integrated DOS (IDOS). The latter are computed as histograms, counting the number of electronic levels in a given energy bin. The electronic structure is calculated using the tight-binding method .

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The 1D and 2D systems are the Fibonacci and octonacci chains and the square and labyrinth tilings considered in the previous Demonstration by the author: Labyrinth Tiling from Quasiperiodic Octonacci Chains. You can choose the DOS or the IDOS of a 1D or 2D system to see the differences between the two systems, either with Fibonacci or octonacci rule for the generation of the quasiperiodic sequence. Additionally, for the 2D systems the effects of a different topology in the tight-binding electronic Hamiltonian can be seen by selecting either the square or the labyrinth tiling for the nearest neighbor site connections, although the difference can be barely noticed in the DOS profile.

The nesting depth of the quasiperiodic sequence can be chosen by the corresponding control setter for the 1D or 2D system. Since 1D systems have lower density of energy levels than 2D systems, in this Demonstration the nesting depths for either dimensionality have been differentiated, thus allowing for deeper nesting depths when investigating 1D quasiperiodic systems. For the 1D Fibonacci quasiperiodic system you can verify that the number of computed energy levels for each value of the nesting setter n actually obeys the Fibonacci number sequence: 1, 1, 2, 3, 5, 8, 13, …

By lowering the value of the ratio, you can drive the system from periodic order ( ), in which the bond lengths in the systems are the same everywhere, to aperiodic order, in which there are both short and long bonds. At the same time, the systems experience a transition from a metallic to an insulating state. You can see this from the appearance of gaps in the electronic level spectra and DOS profile. In the IDOS plots, note the presence of plateaus in the profile, since there are no electronic levels to be counted in those energy ranges (see snapshot 6).

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Contributed by: Jessica Alfonsi (July 2013)
Open content licensed under CC BY-NC-SA

## Details

Snapshot 1: electronic density of states for the 1D Fibonacci system with periodic order ( )

Snapshot 2: electronic density of states for the 1D octonacci system with periodic order ( )

Snapshot 3: electronic density of states for the 2D Fibonacci system (square tiling) with aperiodic order ( )

Snapshot 4: integrated electronic density of states for the 1D Fibonacci system with periodic order ( )

Snapshot 5: integrated electronic density of states for the 1D octonacci system with periodic order ( )

Snapshot 6: integrated electronic density of states for the 2D Fibonacci system (square tiling) with aperiodic order ( )

Reference

 U. Grimm and M. Schreiber, "Energy Spectra and Eigenstates of Quasiperiodic Tight-Binding Hamiltonians," in Quasicrystals: Structure and Physical Properties (H.-R. Trebin, ed.), Weinheim, Germany: Wiley-VCH, 2003 pp. 210–235. arxiv.org/abs/cond-mat/0212140.

## Snapshots   ## Permanent Citation

Jessica Alfonsi

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