Abstract
In this chapter, we apply the spectral approach to the Anderson localization problem in 2D lattices of various geometries. The numerical simulations shown suggest that extended states exist for the disordered honeycomb, triangular, and square crystals. This observation stands in contrast to the predictions of scaling theory and aligns with experiments in photonic lattices and electron systems. A comparison of the results for the three geometries indicates that the triangular and honeycomb lattices experience transition in the transport behavior for similar levels of disorder, which is to be expected from the planar duality of the lattices. This provides justification for the use of artificially-prepared triangular lattices as analogues for honeycomb materials, such as graphene. The analysis also shows that the transition in the honeycomb case happens more abruptly compared to the other two geometries, which can be attributed to the number of nearest neighbors.
This chapter published as: E. G. Kostadinova, K. Busse, N. Ellis, J. Padgett, C. D. Liaw, L. S. Matthews, & T. W. Hyde (2017). Delocalization in infinite disordered 2D lattices of different geometry. ArXiv preprint: 1706.02800. (recommended for publication in Phys. Rev. B)
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Notes
- 1.
Note that in the spectral approach an exponential decay of the distance plots to a finite nonzero value corresponds to delocalization. This should not be confused with the exponential decay of the electron wavefunction in scaling theory, which corresponds to localization. The D value is a mathematical construct that is used to test for cyclicity, not a physical wavefunction.
- 2.
Note that there are two distinct errors in the discussion. The error estimates obtained from the spread of the random realizations for each disorder (the ones shown in Tables I and II) indicate the certainty with which we can determine the limiting behavior of the distance values. The root mean squared error shows the goodness of the fit.
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Kostadinova, E.G. (2018). Delocalization in 2D Lattices of Various Geometries. In: Spectral Approach to Transport Problems in Two-Dimensional Disordered Lattices. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-02212-9_4
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DOI: https://doi.org/10.1007/978-3-030-02212-9_4
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