# Theory of Edge Effects and Conductance for Applications in Graphene-Based Nanoantennas
## 1 Introduction
It corresponds to a point or contour at the surface of the body on which the normal vector is usually undefined (half-plane, wedge, cone, etc.). Taking the edge position into consideration allows us to simplify the boundary conditions and employ certain analytical techniques such as separation of variables, using for example special orthogonal coordinate systems [22] or the Wiener-Hopf method [23]. Such analytical solutions are especially attractive for the qualitative analysis of novel types of problems involving interacting fields of different physical origin (for example, plasmonics and optoelectrofluidics [24]). However, such idealization may lead to the loss of uniqueness of the problem statement. The reason for this is related to the fact that placing any point singularity at the edge generally does not violate the governing wave equation as well as the corresponding boundary and radiation conditions. The analytic solutions thus obtained, are correct from the mathematical point of view. However, they may turn to be physically incorrect because due to lack of uniqueness since they may correspond to another source of the field. In order to obtain the corresponding of physically unique solution, one must enforce the condition of finite (bounded) energy over an arbitrary area (finite extend) around the edge (Meixner condition) [25]. Such an approach usually results in the formation of field singularities at the vicinity of the edge. Note that an edge-like configuration generally does not affect the constitutive properties of the medium (e.g., a perfect conductor or a perfect insulator in classical electrodynamics). The edge for the different types of nanostructures corresponds to the spatial area in the vicinity of the sample boundary which size is rather large compared with the inter-atomic distance. However, the mechanism of electronic transport at the edge is dramatically different from the corresponding properties of the bulk region. One of the reasons for this disparity is the existence of new types of quantum states strongly confined to the vicinity of the edge, which are able to produce novel physical properties such as topological order (so called, edge states [17; 18; 26]). Such transport mechanisms manifest themselves in special optical and optomechanical properties and different types of metamaterials (e.g., carbon-based nanostructures). The edge areas play the main role in forming the emission in classical radio-frequency antennas [27] and optical nanoantennas [1; 2; 3; 4; 5; 6; 28] as well as interconnects [29], modulators [10], and filters [30]. One of the promising types of nanoantennas that operate in the THz and optical frequency range are based on carbon-based nanostructures (graphene, CNTs) [31; 32; 33; 34]. The physical mechanism underlying the radiation pattern is based on the existence of a strongly retarded surface plasmon predicted in [31; 35], experimentally observed in [36] and used for interpretation of the intriguing measurements of the THz conductivity peak [37]. These works use different models for the charge transport [31; 35; 37; 38; 39], but all of them are not self-consistent. In another words, the boundary value problem for the Maxwell equations is formulated for the real geometry of the object, while the value of conductivity is introduced as a phenomenological parameter which is determined by using a model corresponding to an infinitely large structure. Of course, such an approach appears to be attractive since it allows reaching simplification due to the separation of the EM field and charge transport equations. Such an approach keeps the self-consistent requirement for planar structures with the Fresnel transmission-reflection condition of EM fields (graphene films, semitransparent mirrors, etc.) [40; 41; 42; 43]. However, it may not be adequate for the detailed description of EM-field scattering and antenna emission in the general case.