Conductance quantization in graphene nanoribbons: Adiabatic approximation

A theory of electron states for graphene nanoribbons with a smoothly varying width is developed. It is demonstrated that the standard adiabatic approximation allowing to neglect the mixing of different standing waves is more restrictive for the massless Dirac fermions in graphene than for the conventional electron gas. In general, one can expect a well-pronounced conductance quantization only for highly excited states. This difference is related to the relativistic Zitterbewegung effect in graphene. PACS. 73.43.Cd Theory and modeling – 81.05.Uw Carbon, diamond, graphite – 03.65.Pm Relativistic wave equations The experimental discovery of a truly two-dimensional allotrope of carbon, graphene [1,2] and of the massless Dirac character of its electronic energy spectrum [3,4] has initiated an enormously growing interest in this field (for review, see Refs. [5,6]). One of the most exciting aspects of the problem is the hope to develop novel carbon-based electronics. Very recently, the experimental realization of quantum dots [5] and carbon nanoribbons [7,8] has been announced, the former demonstrating single-electron transistor (SET) effect [5]. The conductance quantization in the ballistic regime [9, 10,11,12,13] is one of the most important physical phenomena determining the functioning of such nanodevices. It was considered recently [14] for the case of confinement due to a smooth external electrostatic potential. The experimental situation [5,7,8] corresponds rather to the case of electron confinement due to a curvilinear shape of the graphene samples than to an external field. The description of the penetration of electron waves through constrictions in the nanoribbons requires a different theoretical approach. Numerical calculations of electronic transport in graphene nanoribbons demonstrating a very interesting “valley filter” effect have been recently published [15]. However, a general theoretical analysis of the situation is still absent. Here we present an analytical theory of conductance quantization in graphene nanoribbons based on the adiabatic approximation [10,12]. The latter means a separation of the electron motion in the directions perpendicular and along the stripe. For nonrelativistic electrons the adiabatic approximation requires only the smoothness of the shape of the stripe boundary and results in the quantization of the conductance. It will be shown that due to the relativistic Zitterbewegung effects in graphene [16] this theory is essentially different from that for nonrelativistic electrons and, in general, there is no reason to expect an adiabatic regime and well-pronounced conductance jumps for the lowest states of the ribbon. The two-component wave function (u, v) for charge carriers in graphene with wave vectors close to theK point is described by the Dirac equation