Quantum conductance of achiral graphene ribbons and carbon tubes
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LECTRONIC PROPERTIES OF SOLIDS
Quantum Conductance of Achiral Graphene Ribbons and Carbon Tubes1 L. I. Malysheva* and A. I. Onipko** Bogolyubov Institute for Theoretical Physics 03680, Kiev, Ukraine *e-mail: [email protected] **e-mail: [email protected] Received July 6, 2008
Abstract—Explicit expressions of the band spectrum near the neutrality point are derived for armchair and zigzag graphene ribbons and carbon tubes. Several spectral features, which were previously observed only in numerical calculations, are given an adequate analytic description in terms of elementary functions. The obtained dispersion relations are used for a comparison of conductance ladders of graphene-based wires; these relations are also beneficial for many other applications. PACS numbers: 73.22.-f DOI: 10.1134/S1063776109010166 1
1. INTRODUCTION
2. BAND STRUCTURE
For electrons and holes, graphene ribbons and carbon tubes are one-dimensional wires made of one-atom thick material. In comparison with the two-dimensional electron gas counterparts in semiconductor heterostructures, the transport of charge carriers in graphene [1] (in particular, quantum conductance [2]) demonstrates a number of unusual properties. This paper gives a precise analytic description of the conductance of four basic graphene wires and specific features of each member of the wire family represented in Fig. 1, achiral graphene ribbons and carbon tubes. Formally, this problem can be considered already “solved” by finding two equations that describe the spectrum of a graphene sheet with two armchair- and two zigzag-shaped edges [3] (in the center of Fig. 1). But spectrum peculiarities near the Fermi energy [2–8] are far from being obvious from the general equations. Here, we show that (i) the band spectrum of a metallic armchair (zigzag) ribbon (tube) is not the same as for zigzag (armchair) ribbon (tube); this difference is given an accurate quantitative description; (ii) in moving away from the Fermi energy, the bottoms (tops) of conduction (valence) bands in zigzag (armchair) ribbons (tubes) are shifted towards larger wave vectors; and (iii) there exist three types of spectra (conductance ladders) with equal, irregular, and alternating band spacing (ladder step width). By expressing each of these features in elementary functions, the understanding achieved in previous studies is considerably improved.
Figure 1 illustrates the parent honeycomb N × lattice and its daughter wire-like structures, armchair and zigzag graphene ribbons and carbon tubes, henceforth abbreviated as GR and CT, respectively. The lattice label indicates that in the armchair direction, graphene contains N hexagons in polyparaphenylene-like chains, whereas in the zigzag direction, it has hexagons, forming polyacene-like chains. Hydrogen atoms along the edges are not shown and not taken into account in the nearest-neighbor tight-binding Hamiltonian [9, 10].
1 The
article is published in the original.
The π electron spectrum for this model is given by [3] ak 3ak 2 ak ± E = ± 1 + 4 cos -
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