International Scholarly Research Network ISRN Nanotechnology Volume 2011, Article ID 161849, 5 pages doi:10.5402/2011/161849 Research Article Electron Spectrum and Tunneling Current of the Toroidal and Helical Graphene Nanoribbon-Quantum Dots Contact Mikhail B. Belonenko, 1, 2 Nikolay G. Lebedev, 3 Alexander V. Zhukov, 4, 2 and Natalia N. Yanyushkina 3 1 Volgograd Institute of Business, Yuzhno-Ukrainskaya Street 2, 400048 Volgograd, Russia 2 Scientific Department, Entropique Inc., London, ON, Canada N6J 3S2 3 Volgograd State University, University Avenue 100, 400062 Volgograd, Russia 4 M 2 NeT Lab, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5 Correspondence should be addressed to Mikhail B. Belonenko, mbelonenko@yandx.ru Received 29 March 2011; Accepted 17 April 2011 Academic Editors: B. Coasne and S. Maksimenko Copyright © 2011 Mikhail B. Belonenko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the electron spectrum and the density of states of long-wave electrons in the curved graphene nanoribbon based on the Dirac equation in a curved space-time. The current-voltage characteristics for the contact of nanoribbon-quantum dot have been revealed. We also analyze the dependence of the specimen properties on its geometry. 1. Introduction The problem of modified graphene properties attracts a considerable attention of researchers [1, 2] because the “pure” graphene has no energy gap in the band structure and, therefore, the creation of different structures (e.g., analogs of transistors) is extremely difficult. However, the situation becomes more promising when various modifications of the specimen are introduced. As an example, we consider the modified graphene, for example, graphene nanoribbon, which have quantized electron energy spectrum due to the limited space in one dimension, which in turn can lead to the formation of the energy gap. Furthermore, we know that the graphene has a naturally wave-like curved surface due to the instability of the planar structure of its sheets [3, 4]. All of the above reasons have stimulated the study of different modifications of a curved graphene [5, 6]. The long-wave approximation, which is widely used to describe the properties of electrons in graphene, leads to an analog of the Dirac equation, which in turn makes it easy to produce generalization to the case when the graphene surface is curved. Note that in this case the degeneracy in the Dirac points is removed and therefore it becomes possible to create various structures with different band gaps. Consideration of the Dirac equation for curved graphene [5] also shows a change in the density of states of electrons and, therefore, it is possible to change the whole set of electrical characteristics of the graphene sample. Apparently, the easiest way to experimentally verify the changes in the density of states is to study the tunneling current [7], for example, through the contact with quantum dots. Quantum dots are still rather “young” objects of study, but their use in various fields of science and technology is obviously extremely promising (from the design of lasers and new generation displays to building qubits) [8]. 2. Basic Equations and Spectrum of Electrons We consider the graphene nanoribbon, which is curved along the toroidal and the helical surfaces, as presented in Figure 1. Properties of electrons in the graphene nanoribbons in the long-wave approximation in the vicinity of the Dirac point will be described on the basis of the Dirac equation generalized for the case of a curved space-time [9]: γ μ  ∂ μ − Ω μ  Ψ = 0. (1)