Electronic Properties of Graphene Nanoribbons

Виконав: Копинець Володимир Іванович IАН-104

Перевірив: Азнакаєв Е.Г.

Київ 2014

Recently, graphene, a single-layer hexagonal lattice of carbon atoms, has emerged as a fascinating system for fundamental studies in condensed matter physics, as well as a promising candidate material for future applications in nanoelectronics and molecular devices [1]. The honeycomb crystal structure of single-layer graphene consists of two non-equivalent sublattices and results in a unique band structure for the itinerant π-electrons near the Fermi energy that behave as massless Dirac fermions. The valence and conduction bands touch conically at two non-equivalent Dirac points, called K ₊ and K − points, which form a time-reversed pair, i.e. opposite chirality. The chirality and a Berry phase of π at the two Dirac points provide an environment for highly unconventional and fascinating two-dimensional electronic properties [2], such as the half-integer quantum Hall effect [3], the absence of backward scattering [4, 5], and π-phase shift of the Shubnikov–de Haas oscillations [6].

The successive miniaturization of graphene electronic devices inevitably demands clarification of the edge effects on the electronic structures and electronic transport properties of nanometer-sized graphene. The presence of edges in graphene has strong implications for the low-energy spectrum of the π-electrons [7]–[9]. There are two basic shapes of edges, armchair and zigzag, that determine the properties of graphene ribbons. It was shown that ribbons with zigzag edges (zigzag ribbons) possess localized edge states with energies close to the Fermi level [7]–[10]. These edge states correspond to non-bonding wave functions, where the amplitudes of the edge states reside on one sublattice only [7]. In contrast, edge states are completely absent for ribbons with armchair edges. Recent experiments support the evidence for edge-localized states [11, 12]. Also, graphene nanoribbons can experimentally be produced by using lithography techniques and chemical techniques [13]–[17].

The electronic transport through graphene nanoribbons shows a number of intriguing phenomena such as zero-conductance Fano resonances [18, 19], valley filtering [20], halfmetallic conduction [21], the spin Hall effect [22] and a perfectly conducting channel (PCC) [23]. Recent studies also clarify the unconventional transport through graphene junctions, quantum point contact and heterojunctions [19], [24]–[44]. It is also expected that the edge states play an important role in the magnetic properties in nanometer-sized graphite systems, because of their relatively large contribution to the density of states at the Fermi energy [7, 9], [45]–[56]. Recent studies explore the robustness of edge states to size and geometries [56]–[58], [60], and various edge structures and modifications [56, 59].

Since graphene nanoribbons and carbon nanotubes can be viewed as a new class of quantum wires, one might expect that random impurities inevitably cause Anderson localization, i.e. conductance decays exponentially with increasing system length L and eventually vanishes in the limit of L → ∞. However, it was shown that zigzag nanoribbons and armchair nanotubes subjected to long-range impurities (LRIs) possess a PCC [23, 61]. Recent studies show that PCCs can be stabilized in two standard universality classes. One is the symplectic universality class with an odd number of conducting channels [61]–[63], and the other is the unitary universality class with an imbalance between the numbers of conducting channels in two propagating directions [23, 64, 65]. The symplectic class consists of systems having time-reversal symmetry (TRS) without spin-rotation invariance, while the unitary class is characterized by the absence of time-reversal symmetry [66].

In this paper, we will give a brief overview of the electronic transport properties of disordered graphene nanoribbons. In zigzag nanoribbons, the edge states play an important role, since they appear as special modes with partially flat bands and, under certain conditions, lead to chiral modes separately in the two valleys. There is one such mode of opposite orientation in each of the two valleys of propagating modes, which are well separated in k -space. The key result of this study is that for disorder without inter-valley scattering a single PCC emerges introduced by the presence of these chiral modes. This effect disappears as soon as inter-valley scattering is possible. On the other hand, the low-energy spectrum of graphene nanoribbons with armchair edges (armchair nanoribbons) is described as the superposition of two non-equivalent Dirac points of graphene. In spite of the lack of two well separated valley structures, the single-channel transport subjected to LRIs is nearly perfectly conducting, where the backward scattering matrix elements in the lowest order vanish as a manifestation of internal phase structures of the wave function [67]. For the multi-channel energy regime, however, conventional exponential decay of the averaged conductance occurs. Symmetry considerations lead to the classification of disordered zigzag ribbons into the unitary class for LRIs, and the orthogonal class for short-range impurities (SRIs). Since inter-valley scattering is not completely absent, armchair nanoribbons can be classified into the orthogonal universality class irrespective of the range of impurities.

L

Figure 1. Structure of graphene nanoribbon with (a) armchair edges (armchair ribbon) and (b) zigzag edges (zigzag ribbon). The lattice constant is a and N defines the ribbon width. The circles with dashed lines indicate the missing carbon atoms for the edge boundary condition of the massless Dirac equation. The disordered region with randomly distributed impurities lies in the shaded region and has the length L (see text in section 3). Randomly distributed circles schematically represent the LRIs.