Theory of Edge Effects and Conductance for Applications in Graphene-based Nanoantennas

In this paper, we develop a theory of edge effects in graphene for its applications to nanoantennas in the terahertz, infrared, and visible frequency ranges. Its characteristic feature is selfconsistence reached due the formulation in terms of dynamical conductance instead of ordinary used surface conductivity. The physical model of edge effects is based on using the concept of Dirac fermions. The surface conductance is considered as a general susceptibility and is calculated via the Kubo approach. In contrast with earlier models, the surface conductance becomes nonhomogeneous and nonlocal. The spatial behavior of the surface conductance depends on the length of the sheet and the electrochemical potential. Results of numerical simulations are presented for lengths in the range of 2.1-800 nm and electrochemical potentials ranging between 0.1-1.0 eV. It is shown that if the length exceeded 800 nm, our model agrees with the classical Drude conductivity model with a relatively high degree of accuracy. For rather short lengths, the conductance usually exhibits spatial oscillations, which absent in conductivity and strongly affect the properties of graphene based antennas. The period and amplitude of such spatial oscillations, strongly depend on the electrochemical potential. The new theory opens the way for realizing electrically controlled nanoantennas by changing the electrochemical potential may of the gate voltage. The obtained results may be applicable for the design of carbon based nanodevices in modern quantum technologies.

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 will show, exactly optical conductance will be suitable parameter for formulation of the effective boundary conditions for electromagnetic field in nanoantennas. In contrast with conductivity, it is nonhomogeneous and not coupled via simple relation with optical conductivity (which is homogeneous value). It is described by the Kubo approach instead of Boltzmann's transport equation.
The edge areas play the main role in forming the emission in classical radio-frequency antennas [26] and optical nanoantennas [27]. One of the promising types of nanoantennas that operate in the THz and optical frequency range, are based on carbon-based nanostructures (graphene, CNTs) [28][29][30][31]. The physical mechanism of their radiation, is based on the existence of a strongly retarded surface Plasmon predicted in [28,32], experimentally observed in [33] and used for interpretation of the intriguing measurements of the THz conductivity peak [34]. These works use different models for the charge transport [28,32,34,35], 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 to reach, 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.) [37][38][39][40]. However, it may not be adequate for the detailed description of EM-field scattering and antenna emission in the general case. One can clearly expect that the error of such a non-consistent approach may be ignored as being very small, providing the size of the system is rather large. However, it means, that the effect of the Fresnel transmission-reflection dominates the field forming, whereas the antenna efficiency decreases. Nevertheless, it is impossible to say a priori what is the validity bound of such a simplification.
This problem becomes especially relevant for graphene-based structures, because of the corresponding large geometrical size disparity for applications in different graphene devices. The advanced graphene synthesis methods make it possible to grow graphene samples from 2.1nm to few centimeters [6,[41][42][43][44][45][46]. The detail description of optical graphene conductivity requires a self-consistent analysis. Such a self-consistent approach determines the field acting on electrons produced over their motion, by taking into consideration the finite-size configuration and using an appropriate microscopic model for the edge. The formulation of such a self-consistent analysis is one of the main contributions of this paper. One of the important results of this paper, is analyzing the effect of the edges (shape of finite extend) and determine the corresponding error when self-consistency is not prevailed. It is important to note; that edge effects do not only add up to the quantitative differences in the value of conductivity. Edge effects are also able to dramatically affect the special physical mechanism of charge transport. It makes required the formulation of the theory in terms of optical conductance, which is important for applications in antenna design.
The paper is organized as follows: models of the edge of a graphene ribbon based on the Dirac-fermion concept, are first discussed in Section 2. Thereafter, based on using the Kubo approach and the concept of general susceptibility [47], we analytically obtain an expression for the surface conductance of a terminated (finite) graphene sheet. Results of numerical simulations together with a discussion are presented in Section 3. Conclusion and outlook are finally given in Section 4.

Kubo approach for optical conductance of graphene
In the following we will use the Kubo approach for conductance calculation as a universal technique which couples the generalized forces of arbitrary physical origin with the responses of correspondent origin via general susceptibilities [47]. Towards this goal it is convenient to define the forces as tangential components of electric field and responses as components of current densities. Thus, the general susceptibilities are defined by a 22  matrix where ;, ) similar to the dc conductance of graphene in [16] . The next step is the transformation of Equation (2) [37][38][39][40]. This potential vanishes for a perfectly clean graphene at zero temperature [16] and may be effectively controlled via the gate voltage [37][38][39][40].
Because of the Hermittivity nature of the current operator, we have where the upper asterisk denotes complex conjugate. Therefore, one can also write Equation (11) in the . compact form

Zigzag edges
In this Subsection we will apply the general result to the edge of a zigzag type. The electronic properties of the ribbon are described by the model of Dirac fermions corresponding to a tight-binding model on a two-dimensional honeycomb lattice [16,25,48]. We will use the zigzag-type of boundary conditions for Dirac fermions on a terminated (finite) lattice derived in [16,25,48], since it was demonstrated that this type of boundary condition can be generally applied to a terminated honeycomb lattice in the case of electron-hole symmetry [25]). The A and B atoms are coupled in every spinor modes. From physical point of view, the interpretation of electronic transport may be considered as a motion from A atoms to B ones and vice versa, while A-A and B-B motions are forbidden. The electron transport for a zigzag edge in valleys K and K' is independent and will be considered separately. The wave function is expressed as a linear combination of the eigen 4D spinors 7 The pseudo-spinor modes for valley K have the following form; The functions     , u x v x in the valley K (and in K' respectively) take the following form for bulk and edge states where l represents the normalization length in the y-direction (which goes to infinity in the final result), This dispersion relation has an infinite number of real roots with a single point of concentration at infinity (confined modes) and one imaginary root that corresponds to the edge state [16]. Transformation to the edge state from confined modes in Equations (14a)-(14e) and characteristic equation may be done via exchange px   .The two signs in (14) correspond to electrons and holes respectively. As one can see Eq. (14) satisfies the Dirac equation and is subject to the following boundary conditions; .The components u(x) and v(x) are separately orthogonal, while non-orthogonal mutually due to their coupling over electron motion between the atoms of A and B sub-lattices. Since the expression for the ac-conductivity with a zigzag edge is isotropic, we have Here represents the matrix element of the current operator, σ denotes the xy-vector of the Pauli matrices,  is the energy of the s-th state and , F ev correspond to the electron charge and Fermi velocity respectively.
The general susceptibility may be presented here as the sum of two components of different origin (for details of deriving Equation (16) see Appendix A). The explicit expression for the conductivity given in (16) is spatially inhomogeneous (x-depended) and is a manifestation of edge effect and incorporating a self-consistent description.

Approximation for zero temperature.
Let us next consider the important case of the conductance at zero temperature, which opens the way for further simplification of Equation (16). In this case the Fermi distribution may be replaced by a step function and its derivative in (16) can be transformed into a Dirac delta function  , which allows an integration of (16). Performing the integration in The values of yn k depend on the electrochemical potential and satisfy the characteristic Equation (17 (for details see Appendix B).

The limit of an infinite sheet.
In this Subsection we show that our model reduces to the familiar Drude relation for the conductivity in infinitely wide sheet. Taking the limit L , and using the following transformation  (25) If the observation point x (of under the assumption of large L), is placed rather far from the edges, the arguments of Bessel functions become large, so that the asymptotic relations [49] . In this case, the last two terms in (25) become indefinitely small. The first term, which is exactly equal to the Drude conductivity , becomes dominate over the whole area of the ribbon excluding the narrow vicinities of the edges. Thus, it is demonstrated that in this limit our model asymptotically reduces to the classical expression for the Drude conductivity.

Optical conductance for the edge with infinite-mass boundary condition
This problem is of special interest in connection with a suspended sheet. The interaction between the edges of the sheet with the electrodes, results in the creation of electrostatic potential. Following [13], the electron-hole symmetry, which is generally restricted for boundary conditions of a zigzag or armchair types is broken. It may be considered as a manifestation of a staggered potential at zigzag boundaries, which may change the nature of the boundary condition. For, infinitely large value of potential it leads to an "infinite-mass" boundary condition, which may be written as  The corresponding eigen pseudospins are then given by Equation (14)  In order to calculate the conductance, we use the Kubo approach with the pseudo spinors defined in (26) and (27). The difference from the zigzag edge formulation, is due to the lack of orthogonality inf (26) and (27), which leads to a non-local conductance (spatial dispersion). The conductance operator does not add up to the convolution form because of the non-homogeneity of the finite-length structure. In the case of a rather wide sheet the nonlocal component becomes relatively small and may be usually ignored. In this particular case the conductance relates to Equation (16) with the pseudospins given given in (26) and (27).

Numerical results and discussion
The optical properties of graphene are generally defined by the geometrical size of the sample and the value of the electrochemical potential. These parameters may vary over a wide range and thus are of practical interest to nanoantenna design. The recent advances in graphene technology make it possible to produce graphene samples from few nanometers to few centimeters [6,7,21]. The electrochemical potential may also vary over the interval 0 1.0   eV, by doping the sample or by applying gate voltage [6,7,21]. In this Section we will present some numerical results of conductivity simulations for a wide range of physical parameters, based on the theory developed above. One of the main results of the present study according to Equations (16) and (19), is that the non-homogeneity of conductance is mainly due to the edge effects. The conductance distribution is controlled by the parameter , which defines the number of modes supported by the conductance value. Figures 3-5 present the normalized conductance distribution for a rather large length 800 L  nm and for different values of the electrochemical potential (increasing  corresponds to increasing number of modes). The qualitative behavior of the conductance is the same in all these Figures. The conductance oscillates with respect to the spatial variable and decreases near the edges. The amplitude and period of oscillations decrease with increasing of the electrochemical potential  . The average value of the conductance (to a high degree of accuracy), corresponds to the classical Drude model of conductivity, as discussed in Section 2.4. One can anticipate that such small oscillations are unable to manifest themselves in the scattering of electromagnetic waves, due to the smallness of their period compared with the wavelength. Thus, we may conclude that the Drude model can be applied for this range of parameters.
The considered scenario changes dramatically with shortening of the sheet, as shown in   Figure corresponds to a different value of the length for the same value of electrochemical potential). One can clearly see the enhancement of oscillations, which makes the Drude model invalid for such values of parameters. In fact, it becomes impossible to introduce the conductivity concept in its ordinary meaning. As it was mentioned above, the value defined by Equation (16) (29) is x-dependent coefficient, in which the configuration of the sample manifests itself. The situation is rather similar to the electron transport in graphene in dc field (the concepts of conductance and conductivity discussed and compared in [16,50]).
The physical mechanism for the conductivity oscillations is the interference between the pseudospin modes due to the reflection from the sheet boundaries. The important features are demonstrated in the single-mode conductance (Figures 6 (a),(b)). For the rather small electrochemical potential the active mode is an edge type. The sign of the conductance is negative, which corresponds to its inductive origin. The increase in the electrochemical potential leads to the transformation from inductive to capacitive one (sign exchange) due to the transformation from the edge mode to the bulk one.
As one can see, the conductivity of a graphene sheet changes its qualitative behavior for rather small values of length. However, graphene antennas generally exhibit a resonate behavior at much lower frequencies as well as their metallic counterparts, which is experimentally implemented in the THz range [43,45,46,51]. Thus one can effectively exploit the electrically tunable conductance of graphene exactly for such small sizes, where the conventional models of conductivity become invalid due to the importance of edge effects. In 13 summary, it is important to note that including edge effects in the physical modeling, opens a new way for electrical controlling of resonant graphene antennas via the overturned electrochemical potential by means of the gate voltage. In some cases where the electrochemical potential varies adiabatically slow in time, it will also produce a modulation of the THz emission.
Their using leads to modification of integral equations of antenna theory and methods of their solution.
14  The considered scenario changes dramatically with shortening of the sheet, as shown in Figures 6-8 (each Figure corresponds to a different value of the length for the same value of electrochemical potential). One can clearly see the enhancement of oscillations, which makes the Drude model invalid for such values of parameters. In fact, it becomes impossible to introduce the conductivity concept in its ordinary meaning. The value defined by Equation (16), has the meaning of an "effective" conductivity, which strongly depend on the geometrical size of the sheet. The situation is rather similar to attempt to describe the optical properties of semiconductor quantum dots via the dielectric function in the limit of weak conferment [50] (the "effective" dielectric function strongly depends on the sample configuration). The physical mechanism for the conductivity oscillations, is the interference between the pseudospin modes due to the reflection from the sheet boundaries. The important features are demonstrated in the single-mode conductivity (Figures 6 (a), (b)). For the rather small electrochemical potential the active mode is an edge type. The sign of the conductivity is negative, which corresponds to its inductive origin. The increase in the electrochemical potential leads to the transformation from inductive to capacitive one (sign exchange), due to the transformation from the edge mode to the bulk one. (c) 18

Conclusion and outlook.
The main results of the paper can be summarized as follows: 1) We have developed a new theory of interaction of electromagnetic field with graphene sheet for nanoantenna applications in the THz, infrared and optical frequency ranges. The main characteristic feature of our theory is accounting for edge effects in a self-consistent manner. It is based on the concept of optical conductance considered as a general susceptibility and calculated by Kubo approach. The model is based on the concept of Dirac pseudo-spins founded via solving the boundary-value problem for the Dirac equation with the appropriate boundary conditions satisfying the physical model, including edge effects of the sheet; 2) The main manifestation of the importance of edge effects is demonstrated by the inhomogeneity of the optical conductance. The amplitude and period of its oscillations depend on the length of the sheet and on the electrochemical potential. It is defined by the number of pseudo-spin modes supporting the conductance; 3) The developed theory is applied for the simulation of the sheet conductance in a wide range of sample parameters ( length 2.1nm -800nm and electrochemical potential 0.1 -1.0 eV). It is shown, that for a length exceeding 800nm our model and the widely used Drude model of conductivity agree to a high degree of accuracy. However, for small geometric sizes (i.e., smaller than 50nm), the physical picture of conductivity with respect to the Drude model changes dramatically due to the influence of edge effects. This circumstance should be accounted for in the design of graphene-based resonant THz antennas and other types of photonic and plasmonic nanodevices; 4) It is shown, that the qualitative distribution of the conductivity along the sheet strongly depends on the electrochemical potential. Thus, it is possible to control the conductivity and performance of graphene nanoantennas, by means of varying the gate voltage; Our theory allows reformulation of the effective boundary conditions for the electromagnetic field at the surface of the graphene sheet with accounting of the edge effects. It requires the modification of integral equations of antenna theory and the methods of their solution. This should be one of the subjects of future research activity as well as their application to nanoantennas and other nanodevices.

Appendix A. Derivation of Equation (16).
In this Appendix we discuss the boundary-value problem for pseudo-spin defined by Equation (14). As it was mentioned above, the pseudo-spin satisfies the Dirac equation with the following boundary conditions; [25]. It may be also transformed into the Helmholtz equation with two special sets of boundary conditions. We have the Dirichlet condition at the left-hand side and the impedance condition   Starting from the xx-component and using the basis relation (11)    The above equation relates to the only existing spin-state (a real physical spin rather than a pseudospin). Therefore, the total conductivity must be doubled, which corresponds to the value of the conductivity given by Equation (16)  with the conductivity given explicitly in Equation (21).