# Theory of Edge Effects and Conductance for Applications in Graphene-Based Nanoantennas
## 1 Introduction  
Innovative electromagnetic nanoantennas, which generally function at wide range of frequencies (from THz until video frequencies), play a vital role in the emerging field of photonics and plasmonics [1; 2; 3; 4; 5; 6]. These antennas can be used as promising tools for transforming near-field light into far-field and vice versa. Their excellent capabilities are usually utilized in a wide scope of applications. Among them, are traditional applications for basic elements in electronics, high-speed communications, informatics, and quantum computing [7] (in particular quantum nanomechanical qubit on carbon nanotube [8]). As far as commercial and industrial applications of carbon-based nanostructures [9; 10], one can list some novel applications such as: (i) high-resolved spectroscopy [11]; (ii) high-speed communication [12]; (iii) light emission and detection [10]; (iv) identification of biomolecules and medical diagnostic applications [13; 14]. The design of the devices mentioned above, require taking into account edge effects and their correct physical description. It should cover a wide frequency range from microwave until optical. These recent innovations have stimulated huge interest in the theory of nanoantennas. Microwave standards become invalid starting from the THz region, as the size of the antennas is miniaturized to micrometer scale [6]. Therefore, the common model of perfect electric conductor, which is widely used in microwaves, is not suitable in the range from THz to optical frequencies. By manipulating the different types of finite-size (edge) effects [15; 16; 17], the conventionalantenna configurations can be also used in the range from microwave to optical frequencies [1; 2; 3; 4; 5; 6]. However, antenna devices which operate in the terahertz range (0.1-10 THz), have some fundamental limitations on their applicability in practical devices [6]. The extensive recent progress in nanotechnologies, has led to the discovery of novel artificial types of condensed matter, such as graphene [18], carbon nanotubes (CNTs) [7], Weyl and Dirac semimetals [19], and topological insulators [20; 21]. Recent studies have noted that graphene is one of the best candidates for overcoming the limitations mentioned above [6]. Consequently, a great number of fundamental physical problems were reconsidered. The electrical properties of macroscopic structures may be described both in terms of conductivity and conductance [17], which are equivalent and coupled via the simple constant coefficient defined by the size values of the sample. As it was noted in [17], the conductivity for nanomaterials (including graphene) is a well-defined value only when the sample is enough large. It is clear from an intuitive point of view that the electric current becomes homogeneous and insensitive to boundaries of the sample. When the sample's size is reduced, the current becomes non-homogeneous and non-local with respect to the field variations in the material. Then, the concept of conductivity loses its meaning. The behavior of the electrons in nanoantennas becomes sensitive to the feeding lines, detectors, modulators, and the edges due to the quantum-mechanical interference. As we show, exact optical conductance is suitable parameter for formulation of the effective boundary conditions for electromagnetic field in nanoantennas. In contrast with conductivity, it is non-homogeneous and not coupled via simple relation with optical conductivity (which is homogeneous value). It is described by the Kubo approach instead of [PERSON]'s transport equation. A nanoantenna in its theoretical analysis is generally considered as a system terminated by a well-defined edge. The edge geometry is one of the widely used idealizations in modern physics. For example, in all branches of classical mechanics and physics (e.g., elasticity, hydrodynamics, acoustics, electrodynamics), the edge has a configurational character.