Quantum transport in Sierpinski carpets

Edo van Veen, Shengjun Yuan, Mikhail I. Katsnelson, Marco Polini, and Andrea Tomadin

Phys. Rev. B 93, 115428 – Published 21 March 2016

Abstract

Recent progress in the design and fabrication of artificial two-dimensional (2D) materials paves the way for the experimental realization of electron systems moving on complex geometries, such as plane fractals. In this work, we calculate the quantum conductance of a 2D electron gas roaming on a Sierpinski carpet (SC), i.e., a plane fractal with Hausdorff dimension intermediate between 1 and 2. We find that the fluctuations of the quantum conductance are a function of energy with a fractal graph, whose dimension can be chosen by changing the geometry of the SC. This behavior is independent of the underlying lattice geometry.

Received 15 April 2015
Revised 26 January 2016

DOI:https://doi.org/10.1103/PhysRevB.93.115428

Physics Subject Headings (PhySH)

Transport phenomena

2-dimensional systems Fractals Two-dimensional electron gas

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Edo van Veen¹, Shengjun Yuan^1,*, Mikhail I. Katsnelson¹, Marco Polini², and Andrea Tomadin³

¹Institute for Molecules and Materials, Radboud University, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands
²Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
³NEST, Istituto Nanoscienze–CNR and Scuola Normale Superiore, I-56126 Pisa, Italy

^*s.yuan@science.ru.nl

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Vol. 93, Iss. 11 — 15 March 2016

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Images

Figure 1
(a) Top-down geometric construction of a Sierpinski carpet (SC). The black squares represent regions that are removed from the white sample. At the $m th$ iteration one removes $N$ copies of the regions removed at the $(m - 1) th$ iteration, after scaling them down in linear size by a factor $L$ . [In (a), $N = 8$ and $L = 3$ .] For a number of iterations $m ≫ 1$ one obtains an approximation of the SC, a plane fractal with dimension $d_{H} = {log}_{L} N [≃ 1.89$ in (a)]. (b)–(d) The triangular, square, and hexagonal underlying lattices considered in this work. The width of the sample is $n$ lattice cells. (e) and (f) Examples of delocalized and localized electronic states in a SC with an underlying square lattice. The square modulus $| ψ (r_{i}) |^{2}$ of the electron wave function is shown, with a color scale varying from white (zero) to red (maximum). Because of the fractal geometry of the sample, delocalized and localized states coexist in narrow energy ranges, producing fractal conductance fluctuations (see Fig. 2).
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Figure 2
(a) and (b) Energy dependence of the conductance $G (E)$ (in units of $e^{2} / h$ ) of a square-lattice SC with $N = 8, L = 3, m = 3$ , and $n = 54$ . Data in (a) and (b) refer to central and diagonal lead positions, respectively. (c) BC algorithm analysis of the conductance fluctuations for SCs with geometry as in (a) ( $+$ ) and (b) ( $\times$ ), for $m = 4$ , and $n = 162$ . The horizontal dashed lines represent the saturation value $N = N_{s}$ , with $N_{s} = 3 \times 10^{4}$ . The slope $d$ of the solid line has been set equal to the Hausdorff dimension $d_{H} ≃ 1.89$ of the SC. (d) BC dimension $d$ of the conductance fluctuations for square-lattice SCs with different dimensions for $m = 3, n = 54$ ( $\circ, ⋄$ ), and $m = 4, n = 162$ ( $□, ▵$ ). Results for both center ( $\circ, □$ ) and diagonal ( $⋄, ▵$ ) lead positions are shown. The solid line represents $d = d_{H}$ .
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Figure 3
(a) and (b) Energy dependence of the conductance $G (E)$ (in units of $e^{2} / h$ ) of a SC with $N = 8, L = 3, m = 3$ , and central lead positions. Data in (a) refer to a triangular lattice with $n = 101$ , while data in (b) refer to a hexagonal lattice with $n = 101$ . (c) BC algorithm analysis for a triangular-lattice SC with $N = 8, L = 3, m = 4$ , and $n = 284$ . (d) BC algorithm analysis for a hexagonal-lattice SC with $N = 8, L = 3, m = 4$ , and $n = 284$ . Results for both center ( $+$ ) and diagonal ( $\times$ ) leads are shown in (c) and (d). The slope $d$ of the solid line has been set equal to the dimension $d_{H} ≃ 1.89$ of the SC.
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Figure 4
(a) A Sierpinski gasket ( $N = 3, L = 2$ ), obtained with $m = 5$ iteration steps, with $n = 32$ lattice sites along the base. (b) A Vicsek fractal ( $N = 5, L = 3$ ), obtained with $m = 3$ iteration steps, with $n = 54$ lattice sites along the main cross arm. (c) and (d) BC analysis for a gasket with $m = 8$ and $n = 256$ and a Vicsek fractal with $m = 6$ and $n = 1458$ . The slope of the solid line has been set equal to the Hausdorff dimension of the samples. Results for two different lead locations are shown with different symbols in (c) and (d).
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