Radiation Diagram Control of Graphene Dipoles by Chemical Potential Gabriel ilva Pinto Dept. of Electrical Engineering Federal University of Para Belem, Brazil Email: gabrielp@ufpa.br Abstract This work presents a method of controlling the radiation diagram of a graphene dipole by the chemical potential. The dipole analyzed has rectangular geometry with power supply by voltage source in the center, where each arm of the dipole is maintained to a different chemical potential. The geometry is analyzed by the two-dimensional moments method in MATLAB software, with an equivalent surface impedance. Calculations of the main radiative properties of the antenna are presented as a function of the chemical potential. The results show that the greater the difference between the chemical potentials, the greater the displacement of the main lobe of the radiation diagram. Keywords Graphene antenna, terahertz, chemical potential, control of radiation pattern, method of moments. I. INTRODUCTION Like graphite and diamond, graphene is also an allotrope of carbon. Graphene is formed by a planar monolayer of densely packed carbon atoms in a two-dimensional hexagonal structure (2D) [1]. In the last decade, graphene has attracted attention from researchers due to its remarkable electronic, optical and mechanical properties. As examples of graphene applications, we can mention graphene antennas [2], high-speed transistors [3] and high-efficiency solar cells [4]. Many articles have performed the chemical potential variation to adjust dynamically the conductivity of graphene in the terahertz and infrared frequencies. In [5] the authors presented a theoretical study to show that graphene can serve as a platform for infrared metamaterials and optical transformation devices. The variation of the chemical potential in [5] was done with an electrostatic field, which allowed obtaining different sections of the same graphene sheet with different conductivities. Another article in 2013 [6], proposed a terahertz phase shift for an arrangement of graphene antennas based on integrally-gated graphene parallel-plate waveguide. In [7], a dynamic control of the radiation diagram of a graphene dipole is presented, where parasite elements with different chemical potentials are positioned close to the dipole. A reconfigurable antenna with double layer of graphene is presented [8]. In [8], it has been demonstrated that the input impedance can be adjusted over a wide range of frequencies through the chemical potential. Other examples of control of the properties of graphene antennas by chemical potential are presented in [9]. The present article theoretically analyzes a graphene dipole with radiation diagram that can be controlled by the chemical Karlo Queiroz da Costa Dept. of Electrical Engineering Federal University of Para Belem, Brazil Email: karlo@ufpa.br potential. We show a parametric analysis to find the largest displacement of the diagram in relation to normal. The antenna has flat rectangular geometry (L is selected using the resonance condition and W to maximize η r), with a THz continuous-wave (CW) photomixer in the middle of the dipole [11]. The photomixer excites the graphene which enables radiation. The analysis is done in the terahertz range of 0.5-2.0 THz. The twodimensional moments method implemented in MATLAB software, with surface impedance of graphene, was used for theoretical analysis [10]. Calculations of the following parameters are presented: input impedance, current distribution, radiation pattern and normalized gain as a function of different values of the chemical potentials in each dipole arm. II. THEORETICAL DEVELOPMENT A. Antenna Geometry The geometry of the analyzed graphene dipole is shown in Fig. 1a, where the dimensions are: L = 17 µm, W = 10 µm and g = 2 µm, where these data and source are the same as those used in [11]. The dipole arms have potential chemical µ c1 and µ c2. The equivalent effective permittivity of the medium is ε r = 2.4, which is approximately the average between the permittivity of the substrate 3.8 (z<0) and air 1.0 (z>0) [11]. It will be shown in the following sections that different values of µ c1 e µ c2 will cause an asymmetry in the diagram (Fig. 1b). Fig. 1. Geometry of planar rectangular graphene dipole. Radiation diagram control illustration for µc1 µc2.
B. Graphene urface Conductivity A sheet of graphene can be represented by a planar surface of very thin thickness. It is the size of a carbon atom. The planar dimensions of the dipole, these should be in the order of micrometers, to be able to radiate electromagnetic waves (EM) in the frequency range of terahertz [12],[13]. Through experimental results presented in [14], is demonstrated that the edge effect on the surface conductivity of graphene only appears in structures with lateral dimensions W and L (Fig. 1) considerably smaller than 100 nm. Therefore, the edge effect will be disregarded in this work, which in turn will enable the use of Kubo formalism to calculate the surface conductivity of an infinite sheet of graphene [15],[16]. In the frequency range considered here of 0.5-2.0 THz, the intraband contribution of the surface conductivity of the graphene predominates. 2 2e k B T μ C j σ ( ω) = ln 2 cosh 2 1 2 (1) πh k BT ω jτ where 10 s is the relaxation time, e is the charge of the electron, k B is the Boltzmann constant, the reduced Planck constant, T=300 K is the temperature and is the chemical potential. Fig. 2 shows the variation of conductivity as a function of frequency and chemical potential. the surface current density and Z = 1/σ is the surface impedance of graphene. The field E is given by: jkr jkr e 1 e E = j ωμ 0 J ds' + J d' (3) 4πR jωε 4πR where j is the imaginary unit, k = ω(µ 0ε) 1/2, ω is the angular frequency (rad/s), ε = ε rε 0 the relative permittivity of the medium, µ 0 e ε 0 are the magnetic permeability and electrical permittivity, respectively, in free space, and R is the distance between the source and observation points, both on the antenna surface. The numerical solution of (3) by MoM approximates the surface current of the antenna by a summation in a given set of base functions. In addition, the conventional test procedure is performed with a given set of test functions [11]. ubstituting (3) into (2), the resulting integral equation is transformed into an algebraic linear system, which is solved to obtain the current J. From this current, the input impedance and radiation diagram results can be obtained. III. REULT OF THREE EXAMPLE OF DIPOLE WITH DIFFERENT CHEMICAL POTENTIAL This section compares the radiative properties of three dipoles with different configurations of chemical potentials in their arms, which are presented in Table 1. TABLE 1. Chemical potential µc1 and µc2 of the three examples of graphene dipoles considered. Case 1 Case 2 Case 3 µ c1 0.08 0.14 0.08 µ c2 0.08 0.14 0.14 A. Input Impedance Fig. 3 shows the input impedances of the antennas of Table 1. It is observed that when the chemical potentials in the two arms increase from case 1 to case 2, the resonances of the Z in curve are shifted to the right, that is, to high frequencies. For example, the second resonance of cases 1 and 2 are approximately f 1=1.11 THz e f 2=1.42 THz, respectively. Fig. 2. urface conductivity of graphene versus frequency for different values of chemical potential µc. C. Method of Moments For the solution of the radiation problem of Fig. 1a by the method of moments [10], apply the condition of impedance contour on the surface of the antenna and obtain the following integral equation of the electric field in the frequency domain with temporal dependence exp(jωt) [( E + Ei) at] at = Z J (2) where E (V/m) is the scattered electric field of the antenna, E i (V/m) is the incident electric field due to the voltage source, a t is the tangential unit vector of the antenna surface, J (A/m) is Fig. 3. Input impedance versus frequency for the graphene dipoles of Table 1 with different chemical potentials.
Case 3, with different chemical potentials in each arm, has the Zin curves approximately with the same characteristics of cases 1 and 2 simultaneously. That is, it behaves roughly as a superposition of the resonances of cases 1 and 2. B. Current Distribution To analyze the resonant behavior of the current distribution of the antennas of Table 1, Fig. 4 shows the module (A/m) and phase (rad) of the current component Jx in the antennas of cases 1, 2 and 3, in frequencies F1=1.11 THz, F2=1.42 THz and F3=1.31 THz, respectively. It is observed that cases 1 and 2 have symmetrical distribution in the dipole arms, while case 3 is asymmetrical. This asymmetry is due to the dipole arms having different values of surface conductivity. According to Fig. 2, the dipole arm of case 3 with lower chemical potential has a more inductive surface impedance than that of the other arm. for the antenna. Fig. 5 shows the variation of Gz and Gn as a function of the frequency for the antennas of Table 1. We observe in this figure that the gains maxima of cases 1 and 2 occur approximately in the frequencies F1=1.11 THz and F2=1.42 THz of the second resonance frequency, where the current distributions were presented in Fig. 4. For case 3, the gain has two peaks of peaks close to the peaks of cases 1 and 2, and a minimum between these peaks, close to F3=1.31 THz, where the current distribution is shown in Fig. 4. This minimum point of Gz is related to the maximum displacement Δθ of the radiation diagram in relation to normal, according to Fig. 2. This can also be verified by calculating the minimum normalized gain curve Gn, which is also presented in Fig. 5. This displacement Δθ can be observed in the gain radiation diagrams in the plane xz shown in Fig. 6. Fig. 5. Absolute (Gz) and normalized (Gn) gain in the z direction versus frequency for the antennas of Table 1. Fig. 4. Distribution of the module and phase of the current component Jx of the antennas of cases 1, 2 and 3 in the frequencies F1=1.11; F2=1.42 and F3=1.31 THz, respectively, which correspond to the second resonance of the antennas. C. Gain and Radiation Diagram We observed in the previous section that different chemical potentials in each arm of the dipole causes an asymmetry in the module and phase of the surface current of the antenna. This asymmetry will also exist in the far field of the antenna, where the radiation pattern in the second resonance should have a displacement Δθ in relation to the z axis (Fig. 1). According to Fig. 1, we define the gains in the directions z, maximum and normalized respectively by Gz=Uz/4πPin, Gm=Um/4πPin and Gn=Gz/Gm, where Uz is the radiation intensity in the z (θ=0), Um is the intensity of radiation towards the maximum (θ=δθ, φ=0) and Pin the power the source provides (c) Fig. 6. Radiation gain diagrams in the xz plane for the antennas of Table 1. Case 1 - F1=1.11 THz. Case 2 - F2=1.42 THz. (c) Case 3 - F3=1.31 THz.
TABLE 2. Minimum normalized gain G n=g z/g m and corresponding frequency F (THz) as a function of chemical potentials µ c1 e µ c2 (ev). Gn / F µc2=0.00 µc2=0.02 µc2=0.04 µc2=0.06 µc2=0.08 µc2=0.10 µc2=0.12 µc2=0.14 µc2=0.16 µc2=0.18 µc2=0.20 µc1=0.00 1 / 0.82 0.99 / 0.8 0.98 / 0.83 0.96 / 0.86 0.94 / 0.89 0.92 / 0.89 0.92 / 0.92 0.91 / 0.92 0.9 / 0.92 0.89 / 0.92 0.89 / 0.92 µc1=0.02 0.99 / 0.8 1 / 0.83 0.99 / 0.87 0.96 / 0.9 0.94 / 0.93 0.92 / 0.94 0.9 / 0.95 0.89 / 0.96 0.89 / 0.96 0.88 / 0.96 0.87 / 0.97 µc1=0.04 0.98 / 0.83 0.99 / 0.87 1 / 0.92 0.98 / 0.98 0.95 / 1.02 0.91 / 1.04 0.89 / 1.05 0.87 / 1.06 0.85 / 1.07 0.84 / 1.08 0.84 / 1.08 µc1=0.06 0.96 / 0.86 0.96 / 0.9 0.98 / 0.98 1 / 1.05 0.97 / 1.11 0.93 / 1.15 0.89 / 1.17 0.85 / 1.19 0.83 / 1.2 0.81 / 1.21 0.79 / 1.22 µc1=0.08 0.94 / 0.89 0.94 / 0.93 0.95 / 1.02 0.97 / 1.11 1 / 1.18 0.98 / 1.24 0.93 / 1.30 0.87 / 1.31 0.83 / 1.33 0.79 / 1.34 0.77 / 1.36 µc1=0.10 0.92 / 0.89 0.92 /0.94 0.91 / 1.04 0.93 / 1.15 0.98 / 1.24 1 / 1.31 0.97 / 1.37 0.91 / 1.41 0.85 / 1.44 0.8 / 1.46 0.76 / 1.47 µc1=0.12 0.92 / 0.92 0.9 / 0.95 0.89 / 1.05 0.89 / 1.17 0.93 / 1.30 0.97 / 1.37 1/1.43 0.97 / 1.49 0.91 / 1.52 0.84 / 1.55 0.79 / 1.57 µc1=0.14 0.91 / 0.92 0.89 / 0.96 0.87 / 1.06 0.85 / 1.19 0.87 / 1.31 0.91 / 1.41 0.97 / 1.49 1 / 1.54 0.97 / 1.6 0.91 / 1.63 0.84 / 1.66 µc1=0.16 0.9 / 0.92 0.89 / 0.96 0.85 / 1.07 0.83 / 1.2 0.83 / 1.33 0.85 / 1.44 0.91 / 1.52 0.97 / 1.6 1 / 1.65 0.97 / 1.7 0.9 / 1.73 µc1=0.18 0.89 / 0.92 0.88 / 0.96 0.84 / 1.08 0.81 / 1.21 0.79 / 1.34 0.8 / 1.46 0.84 / 1.55 0.91 / 1.63 0.97 / 1.7 1 / 1.74 0.97 / 1.79 µc1=0.20 0.89 / 0.92 0.87 / 0.97 0.84 / 1.08 0.79 / 1.22 0.77 / 1.36 0.76 / 1.47 0.79 / 1.57 0.84 / 1.66 0.9 / 1.73 0.97 / 1.79 1 / 1.85 IV. PARAMETRIC ANALYI As seen in Fig. 5, the normalized gain G n can be used as a parameter to determine the largest displacement Δθ. This occurs at the least of the G n curve versus frequency. However, the position of this minimum is a function of the values of µ c1 and µ c2. To obtain a more complete understanding of this dependence, a parametric analysis was performed by varying the values of the potentials as shown in Table 2, where for each value of µ c1 and µ c2 we present the minimum G n value and the frequency where it occurs. For example, for µ c1=0.08 ev and µ c2=0.14 ev (G n of Case 3 in Fig. 5), we obtain a normalized minimum gain of G n=0.87 in the frequency F=1.31 THz. From the results of Table 2 we obtain the following important conclusions. The smaller value between µ c1 and µ c2 controls the frequency where the minimum of G n occurs, whereas the greater value controls the minimum of G n, where for higher values of this we obtain smaller minima G n, and consequently a greater displacement Δθ of the diagram with respect to the z-axis. This dependence can be observed in the G n curves in Figs. 7, 8 e 9, for µ c1=0.12, 0.16 and 0.20 ev, respectively. We can cite as examples the cases µ c1=0.08 ev and µ c2=0.12 ev (Fig. 7); µ c1=0.08 ev and µ c2=0.16 ev (Fig. 8); µ c1=0.08 ev and µ c2=0.20 ev (Fig. 9), where the value of the lowest chemical potential is constant (µ c2=0.08 ev) and the position of the minima are in frequency near and equal to F=1.31, 1.33 and 1.36 THz, for µ c1=0.12, 0.16 and 0.20 ev, respectively. In addition, we also observe from these examples that for larger values of µ c1 lower values of G n are obtained, that is, larger displacements Δθ of the diagram with respect to the z-axis are obtained. To observe these results in the radiation diagram, Fig. 10 shows the radiation diagrams of these examples of gain G n in the xz plane. In this figure it is also presented case µ c2=0.08 ev e µ c1=0.10 ev. From this figure we can prove the previous result, where the higher the value of µ c1, the greater the displacement of the diagram in relation to normal. In these examples, the largest displacement obtained was approximately Δθ =25º for the case of Fig. 10d. Fig. 7. Variation of G n=g z/g m versus frequency for µ c1=0.12 ev and Fig. 8. Variation of G n=g z/g m versus frequency for µ c1=0.16 ev and
shifted relative to normal. Then a parametric analysis of the displacement of the diagram was made in relation to normal as a function of chemical potentials. The results showed that the frequency at which the greatest displacement occurs is more sensitive with the lower chemical potential, that is, the chemical potential with the lowest value controls the frequency at which the greatest displacement occurs. The higher value chemical potential controls the level of the displacement of the diagram, where for higher values of this, the greater the displacement of the diagram. In general, the results obtained can be used, for example, to design terahertz graphene antenna systems with controllable diagrams. Future work will investigate the possibilities of obtaining greater displacement in the diagram with four or more different chemical powers along the arms of the dipole. Fig. 9. Variation of G n=g z/g m versus frequency for µ c1=0.20 ev and (c) (d) Fig. 10. Radiation diagrams of normalized gain G n in the xz plane for dipoles with different values of µ c1 and µ c2. µ c2=0.08 ev and µ c1=0.10 ev in F=1.24 THz. µ c2=0.08 ev and µ c1=0.12 ev in F=1.31 THz. (c) µ c2=0.08 ev and µ c1=0.16 ev in F=1.33 THz. (d) µ c2=0.08 ev and µ c1=0.20 ev in F=1.36 THz. V. CONCLUION In this paper we present a graphene dipole with radiation diagram that can be controlled through different chemical potentials applied to the dipole arms. Graphene was modeled by a surface impedance and the method of moment was used for numerical analysis. Firstly, it has been demonstrated through examples that antennas with equal chemical potentials in each arm have diagrams of the first resonances, normal to the plane of the dipole, as expected. While antennas with different chemical potentials in each arm have asymmetric diagrams in relation to normal, that is, the maximum of the diagram is REFERENCE [1] A. Geim and K. Novoselov, The rise of graphene, Nature materials, vol. 6, nº 3, pp. 183-91, Mar 2007. [2] I. F. Akyildiz, and J. M. Jornet, Electromagnetic wireless nanosensor networks, Nano Communication Networks, vol. 1, pp. 3 19, 2010. [3] F. chwierz, Graphene transistors, Nature nanotechnology, vol. 5, nº 7, pp. 487-496, 2010. [4] Z. Fang et al, Graphene-antenna sandwich photodetector, Nano letters, vol. 12, nº 7, pp. 3808-3813, 2012. [5] A. Vakil and N. Engheta, Transformation optics using graphene, cience, vol. 332, nº 6035, pp. 1291-1294, 2011. [6] P.-Y. Chen, C. Argyropoulos and A. Alu, Terahertz antenna phase shifters using integrally-gated graphene transmission-lines, IEEE Transactions on Antennas and Propagation, vol. 61, nº 4, pp. 1528-1537, 2013. [7] Nilton R. N. M. Rodrigues; Rodrigo M.. de Oliveira; Victor Dmitriev, A terahertz graphene antenna with dynamical control of its radiation pattern, 2017 BMO/IEEE MTT- International Microwave and Optoelectronics Conference (IMOC), pp. 1-4, 2017. [8] M. Tamagnone, J.. Gómez-Dı az, J. R. Mosig, and J. Perruisseau- Carrier, Reconfigurable terahertz plasmonic antenna concept using a graphene stack, Appl. Phys. Lett., vol. 101, no. 21, p. 214102, 2012. [9] Correas-errano, Diego, and J. ebastian Gomez-Diaz. "Graphene-based antennas for terahertz systems: A review." arxiv preprint arxiv:1704.00371, 2017. [10] K. Q. da Costa, V. Dmitriev, Planar Monopole UWB Antennas with Cuts at the Edges and Parasitic Loops, InTech: Ultra Wideband Communications: Novel Trends - Antennas and Propagation, 1st ed., pp. 143-145, 2011. [11] M. Tamagnone, J.. Gómez-Díaz, J. R. Mosig and J. Perruisseau-Carrier, Analysis and Design of Terahertz Antennas Based on Plasmonic Resonant Graphene heets, Journal of Applied Physics, vol. 112, 2012. [12] J. M. Jornet and I. F. Akyildiz, Graphene-based nano-antennas for electromagnetic nanocommunications in the terahertz band, 2010. [13] I. L. Martí, C. Kremers, A. C. Aparicio, J. M. Jornet,. E. Alarcón and D. N. Chigrin, cattering of terahertz radiation on a graphene based nano antenna, AIP Conference Proceedings, vol. 1398, pp. 144-146, 2011. [14] M. Y. Han, B. Ozyilmaz, Y. Zhang and P. Kim, Energy band-gap engineering of graphene nanoribons, Physical Review Letters 98, May 2007. [15] L. A. Falkovsky,.. Pershoguba, Optical far-infrared properties of a graphene monolayer and multilayer. Physical Review B, v. 76, n. 15, p. 153410, 2007. [16] George W. Hanson, Dyadic Green's functions for an anisotropic, nonlocal model of biased graphene. IEEE Transactions on Antennas and Propagation, v. 56, n. 3, p. 747-757, 2008.