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Transport in the Two-Dimensional Honeycomb Lattice with Substitutional Disorder

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Spectral Approach to Transport Problems in Two-Dimensional Disordered Lattices

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Abstract

In this chapter, we examine the transport properties of the two-dimensional honeycomb lattice with substitutional disorder, using the site quantum percolation model. Like the Anderson localization, the quantum percolation problem is characterized by an Anderson-type Hamiltonian and can therefore be studied using the spectral approach. Here we use the discrete random Schrödinger operator (Eq. 1.3) with a (modified) bimodal probability distributions χ for the random variables ϵi, which is a realistic representation of a doped system where nearest-neighbor interactions dominate. The discussion starts with a brief introduction of basic definitions in the discrete percolation setup together with an overview of some currently established results (Sect. 5.1). The formulation of the quantum percolation problem describing the doped two-dimensional honeycomb lattice is presented in Sect. 5.2. Section 5.3 provides a theoretical and numerical justification for the choice of the (modified) probability distribution of random variables representing doping. Finally, the results from the spectral analysis of lattices with various concentration of doping are provided in Sect. 5.4.

This chapter published as: E. G. Kostadinova, A. Cameron, F. Guyton, Liaw, C. D., Matthews, L. S., & Hyde, T. W. Transport in the two-dimensional lattice with substitutional disorder (submitted for review in Phys. Rev. B)

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Notes

  1. 1.

    Alternatively, one can consider all vertices to be open and let the bonds be open or closed with a certain probability. This setup is called a bond percolation problem.

  2. 2.

    Note that the spectral approach only requires that v0 and v1 are any two (different) vectors in the Hilbert space of interest. The choice v0 = δ0 and v1 = δ1 ensures faster computation times and is not related to the generality of the spectral method.

  3. 3.

    Here nD stands for the concentration of the B-type doping material and is equal to the probability for a closed state 1 − p in the quantum percolation problem.

  4. 4.

    Here the subscript hc in Dhc stands for honeycomb.

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Authors and Affiliations

  1. Center for Astrophysics, Space Physics and Engineering Research, Baylor University, Waco, TX, USA

    Evdokiya Georgieva Kostadinova

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  1. Evdokiya Georgieva Kostadinova
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Kostadinova, E.G. (2018). Transport in the Two-Dimensional Honeycomb Lattice with Substitutional Disorder. 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_5

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