The beam reconfigurable antennas have attracted growing interests, the beam directivities and the radiation patterns of which can be dynamically changed according to demand. Such antennas have the potentials to avoid noise sources, provide larger coverage by steering the angle of the main lobe, and improve the beam steering capability ^[1,2]. The reconfigurability of beam directivity is a frontier subject in the field of antennas. The beam reconfigurable antennas can be exploited in the multiple-input-multiple-output （MIMO） systems to increase the system capacity ^[3]. And such antennas are compatible with micro-strip-based monolithic-microwave integrated-circuit （MMIC）, which are useful to improve the performance of MMIC. Traditionally, the ways to realize reconfigurable antennas are mostly to load high-frequency electronic devices and use mechanical methods to adjust the antenna structures, which are complicated in structure, frequency limited, and inefficient ^[1]. Graphene, as the first truly two-dimensional material, has unique thermal, mechanical, optical, and electrical properties ^[4,5,6,7]. Among all these interesting properties, the bias-voltage-tunable optical conductivity covering the THz band is very useful for providing an effective way to realize real-time electrically reconfigurable THz antennas.

In recent years, many investigations have been performed in the research of graphene-based reconfigurable THz antennas and a series of progress has been achieved. Dragoman et al. designed a graphene-based THz dipole antenna and confirmed that the radiation characteristics can be controlled by applying a bias voltage to graphene^[8]. Llatser et al. proposed a graphene nano-patch antenna and systematically analyzed its THz radiation characteristics^[9]. Huang et al. proposed a switchable high-impedance surface （HIS） based on single-layer graphene and designed a radiation pattern reconfigurable THz antenna based on such a switchable HIS ^[10]. Zheng et al. proposed a reconfigurable MIMO system using graphene nano-patch antennas^[11]. Wu et al. designed a graphene-based Yagi-Uda antenna with reconfigurable radiation patterns^[12]. However, these antennas are not very good at steering the angle of the THz beam due to the limited steering angle and the low radiation efficiency.

The Yagi-Uda antenna was first reported in 1928^[13]. Because of the feature characteristics, such as end-fire radiating pattern and simple structure, the Yagi-Uda antennas have been used to design the beam reconfigurable antennas ^{[14,15,16,17]}. Several approaches for achieving the reconfigurability of antenna are through micro-strip Yagi-Uda antennas ^[18], which we refer to as quasi-Yagi-Uda （QYU） antennas. In general, a QYU antenna is composed of several patches （one driven element, one reflector, and one or more directors） that are electromagnetically coupled to one another to steer the angle of the main lobe.

In this paper, a type of graphene-based beam reconfigurable QYU antennas working at THz frequencies is proposed. Graphene patches are used to replace the metallic parasitic elements in the conventional Yagi-Uda antenna, and the patches are divided into three groups, each of which is as a director. Because the complex conductivity of graphene can be tuned by applying different bias voltages, by switching the states of different group of graphene-patch directors, the graphene patches can be operated as directors or not, and the function of sending （receiving） signals only toward （from） the target direction is achieved. When bias voltages are applied to different graphene-patch directors, the beam direction of the proposed device can be scanned in the wide-angle （azimuth angle φ） range of 30°-150° in zenith angle θ=60° plane. Moreover, it has very fast modulation speed, low return loss, and high radiation efficiency.

Graphene is a two-dimensional material composed of one layer of carbon atoms with a two-dimensional honeycomb lattice structure, which was first obtained in 2004 by Geim and Novoselov ^[19]. For a single-layer graphene, due to the unique electric band structure of linear energy-momentum dispersion relation near the Dirac cones at the K and K’ points in the first Brillouin zone, the optical conductivity can be written as the sum of the intra-band conductivity σ_intra and the inter-band conductivity σ_inter^[20,22,23],

σintra=2e2ΚΒΤπℏΓ-jωln-EF2ΚΒΤ+EF2ΚΒΤ

and

σinter=-je24πℏln2μc-ω-j2Γℏ2|μc+ω-j2Γℏ

where e is the electron charge, ω is the angular frequency, K_B is the Boltzmann constant, ℏis the reduced Plank constant, T is the temperature in Kelvin, E_F is the Fermi energy, µ_c is the chemical potential, and Γ=1/2τ is the carrier scattering rate with τ the carrier scattering time.

In the microwave, THz, and far-infrared spectral bands, theoretical and experimental investigations show that the optical conductivity of graphene satisfies the widely used Drude model ^{[21,24,25,26,27,28]}. And in the THz band, inter-band transitions are prohibited due to insufficient incident photon energy, and intra-band transitions are in the dominant position. Within the framework of Boltzmann transport theory, the intra-band Drude conductivity for single-layer graphene in THz band can be written as ^[24,26,29],

σintra=je2|EF|/ℏ2πω+jΓ

The Fermi energy E_F is a function of carrier concentration N,

EF=±ℏvFπΝ1/2

where v_F is the Fermi velocity, and the positive （negative） sign in the formula represents the electron （hole） doping.

From equations （3） and （4）, the intra-band optical conductivity can be controlled by adjusting the Fermi energy （carrier concentration） in graphene. The bipolar electrical doping effect of graphene allows its electron （hole） concentration to be continuously changed up to 10¹³ cm^-2[20,30]. Therefore, a bias voltage can be applied to effectively adjust the carrier concentration and then the optical conductivity of graphene.

Recently, beam reconfigurable antennas have attracted many interests in wireless communication ^[31]. In some cases, in order to vary the resonant frequencies, the lengths of dipolar graphene antennas can be dynamically changed ^[32]. In other scenarios, by phasing different elements of an antenna array, the reconfiguration features can be realized. In this paper, as shown in Fig. 1, a QYU antenna working at 500 GHz is proposed to realize the beam reconfigurable feature.

The features of the proposed antenna are numerically simulated with the ANSYS High Frequency Structure Simulator （HFSS） software ^[33]. The antenna working at 500 GHz and operating frequency range of the antenna is from 470 GHz to 530 GHz. The wave port excitation mode is selected and the input impendence is set to 50 Ω. The layered impedance boundary condition is used for the thin graphene patches. Each graphene patch has two states. If a high bias voltage （Fermi energy µ_c=1 eV, denoted as “1”） is applied to the graphene patches, the conductivity of the graphene patches will significantly increase. This will result in the metal-like behavior of graphene patches, and the graphene patches work as directors as in the conventional Yagi-Uda antenna. On the contrary, once a low or no bias voltage （Fermi energy µ_c=0 eV, denoted as “0”） is applied to the graphene patches, they will behave as thin dielectric layers, which have no essential influence on the behavior of the patch antenna. Depending on the different bias voltages applied to the graphene patches, the antenna can work in two different states （state 1 for µ_c=0 eV and state 2 for µ_c=1eV）.

Fig. 1（a） and （b） show the schematic of the antenna. The substrate is a high resistivity silicon wafer with the size of L_s× L_s and L_s=1.8 mm the length and width. There is a copper ground layer with thickness T on the high resistivity silicon substrate. A 30-μm-thick （t_s） SiO₂ dielectric cavity layer is on the metallic ground layer. The relative permittivity of SiO₂ is ε_r =4. The size of the micro-strip fed line is l_m×w_m, the driven patch is L×W, and the thickness of both of the micro-strip fed line and the driven patch is T. The size of the reflector patches is l_r×w_r, the distance between the reflector patch and the driven patch is given by l_rs. The graphene patches are used as the director in the Yagi-Uda antenna, the size of which is given by l_d×w_d, and the gap between the two graphene patches is given by g. The p-doped silicon strips beneath the graphene patches are used as electrical connections between the metallic gating pads and the graphene patches, and the thickness of the doped Si strip is given by t_k. The dimension of the driven copper patch is p×p. As reported in Ref. [34], We can calculate the theoretical value of the length （L） and width （W） of the driven patch. Further, L, W and other parameters of the antenna are optimized by simulation and the results are listed in Table 1.

Fig. 1（c） depicts the reflection coefficients S11 for the antenna working in state 1 and state 2 for the proposed antenna. The antenna is resonating at 500 GHz, and the -10 dB return loss bandwidth is about 3% （495 GHz - 510 GHz）. When the antenna is in state 1 and state 2 respectively, the simulated radiation pattern in E-plane （where the electric field vector and the maximum radiation direction are located） of the proposed antenna is shown in Fig. 1（d）. It can be observed that in E-plane when the antenna is in state 1, the antenna behaves as a conventional patch antenna. When the antenna is in state 2, the angle of the main lobe veers round to along the graphene patches. This confirms the effect of the graphene director on the radiation direction of the antenna.

The antenna structure shown in Fig. 1 confirms the guiding effect of the graphene director on the beam direction in E-plane. But it cannot achieve the beam shift in thezenith angle direction. To overcome this deficiency, we design an improved antenna structure shown in Fig. 2（a）. Compared with the former antenna structure, the graphene-patch director is divided into two groups, and each of the two groups is similar to the director in the conventional Yagi-Uda antenna. The gap between the graphene patches along the y-axis is g=8 µm and the widths of all the gaps are the same. All the graphene patches have the same size, given by l_d2=129 µm and w_d2=94 µm. To realize strong coupling between the driven patches and the two neighboring graphene patches of the director, a very small value of s=0.5 µm is chosen. In order to keep strong couplings between the neighboring graphene patches within the same group, the parameters l_d2 and g are optimized, and the parameters h₁=39 µm and h₂=76 µm are also optimized to realize a higher gain. The two groups of graphene directors are labeled as D₁ and D₂, respectively. Depending on the different bias voltages applied to the graphene patches, the antenna can work in three different states, （state 1: （1,0） with µ_c=1 eV for graphene patches in D₁ and µ_c=0 eV for graphene patches in D₂; state 2: （0,1）; and state 3: （1,1））.

Fig. 2（b） shows the reflection coefficient S₁₁ of the proposed antenna. The resonant frequency of the antenna is about 500 GHz, and for different working states of the antenna, the resonant frequency reveals a tiny shift. It is because that the effective impedance of the graphene directors and the driven patch is a little different when the antenna works in different states. The -10 dB return loss bandwidth is about 2.5%.

Fig. 2（d） shows the radiation pattern of the antenna in θ=60° plane. It can be observed that, when the antenna is in states 1, 2, and 3, the angles of the main lobe in θ=60° plane are about 120°, 60°, and 90°, respectively. Thus, the angle of the main lobe for the antenna in θ=60° plane can be swept from 60° to 120°. While in the E-plane, the angle of the main lobe is fixed at about θ=60°, as shown in Fig. 2（c）.

For the antenna structure mentioned in subsection 2.2, the main lobe can be swept from 60° to 120° in θ=60° plane. In order to increase the steering angle, a QYU antenna with three groups of graphene-patch directors is considered. In this case, we find that a rectangle driven metallic patch does not work well. Therefore, the driven metallic patch is changed from a rectangle patch to a half-round patch, and the structure of the improved antenna is shown in Fig. 3（a）.

Fig. 3（a） shows the structure of the antenna with three groups of graphene-patch directors. Different from the former two antennas, the driven patch is a half-round patch. The graphene patches are divided into three groups, and each of them is similar to the director in the conventional Yagi-Uda antenna. The diameter of the driven patch is W₁=370 µm, and the distance between the reflector patch and the driven patch is l_rs=175 µm. The gap between two neighboring graphene patches in the first and the third group is given by g₁=15 µm, and the gap between two neighboring graphene patches in the second group is given by g=20 µm. The angle of graphene patches in each group is given by α₁=58°, α₂=40.3°, and α₃=30.2° respectively, and the width of the graphene patch is given by l_d3=100 µm. The three groups of graphene directors are labeled as D₁, D_2, and D₃,respectively. Depending on the different bias voltages applied to the graphene patches, the antenna can work in six different states, （state 1: （1,0,0） with µ_c=1 eV for graphene patches in D₁ and µ_c=0 eV for graphene patches in D₂ and D₃; state 2: （1,1,0）; state 3: （0,1,0）; state 4: （0,1,1）; state 5: （0,0,1） and state 6: （1,1,1））.

Fig.3（b） shows the simulated reflection coefficient S₁₁ of the antenna. The resonant frequency of this antenna is around 500 GHz. In all working states, good impedance matching with the return loss S₁₁<-30 dB is obtained around the resonant frequency. The -10 dB return loss bandwidth is about 8%.

Fig. 3（d） shows radiation patterns of the proposed antenna in the θ=60° plane operated at different states. One can see that in state 1, only D₁ works as a director, and the angle of the main lobe is 150°. In state 2, both D₁ and D₂ work as directors, and the angle of the main lobe is 110°. In state 3 only D₂ works as a director, and the angle of the main lobe is 90°. Because of the symmetry of the antenna, the radiation patterns of the antenna working in states 4 and 5 and working in states 1 and 2 will be symmetric about the y-axis. Thereby the beam in θ=60° plane of the antenna can be deflected from 30° to 150°. While the angle of the main lobe in E-plane is fixed at about θ=60°, as shown in Fig. 3（c）. The maximal values of gain are in the range of 6.5 dB - 8.7 dB. The angles of the main lobe and the maximal values of gain for the antenna in different working conditions are shown in table 2.

Fig. 4（a-1）-（a-4） and （b-1）-（b-4） show the 3D radiation patterns that are observed from different perspectives of the antenna. From the 3D radiation patterns, the beam reconfigurable ability can be seen clearly. Fig. 4（c-1）-（c-4） show the surface current distributions for the proposed antenna in different working conditions. It is easy to see that different group graphene patches of the antenna are resonating in different working conditions. For instance, in state 1, as shown in Fig. 4（c-1）, only D₁ is resonating, the surface current is excited only on D₁ and it works as a director, the other graphene patches with µ_c=0 eV have no effects on the behaviors of the antenna. While in state 2, only D₁ and D₂ are resonating. Because of the symmetry of the structure, the surface current distributions of the antenna working in states 4 and 5 and the antenna working in states 2 and 1 will be symmetric about the y-axis. The variations of radiation patterns are well-explained by the current distributions of the antenna working in different states. And the capacitive structure is used to apply bias voltages to graphene patches to achieve electrical regulation. Therefore, when a bias voltage is applied to graphene, there is no current distribution or voltage drop on the surface, and the voltage is evenly distributed.

In this Section, the effects of the relaxation time in graphene and the p-doped silicon strip are discussed. The graphene quality is primarily determined by the relaxation time. In the above simulations, the relaxation time 20 ps in graphene is selected. In Fig. 5（a）, several relaxation times in the range of 0.1-20 ps are set in the simulations. The numerical results show that the relaxation time has a negligible effect on the angle of the main lobe and very little effect on the gain of the antenna. The gain of the antenna increases with the increase in relaxation time.

We also study the effects of the p-doped silicon strips. As shown in Fig. 5（b）, the p-doped silicon strips have negligible effects on antenna performance. The electron plasma frequency is ^[35]

fr=n0e2/4π2meε0ε1/2

where n₀ is the electron density, e is the electron charge, m_e is the electron effective mass, ε₀ is the vacuum dielectric constant and ε is the dielectric permittivity.

The p-doped silicon, which we choose has a very low doping concentration （1e15 cm^-3）, from equation （5）, the electron plasma frequency of the p-doped silicon is 116 GHz, which is much less than the antenna working frequency （500GHz）. Thus, the p-doped silicon trips have negligible effects on antenna performance.

The magnitudes of bias voltages applied to the graphene patches are considered for the antenna mentioned in section 2.1, as shown in Fig. 5（c）. As the Fermi energy increase from µ_c=0 eV to µ_c=1 eV, the graphene patches change from dielectric layers to mental-like layers, and the angle of the main lobe in E-plane converts from θ=0° to θ=50°. Especially when the Fermi energy is lower than 0.4 eV, the graphene patches cannot be operated as directors, and when the Fermi energy is higher than 0.5 eV, the graphene patches gradually behave as metal-like layers and work as directors. Furthermore, if the Fermi energy µ_c=0 eV, as shown in Fig. 5（d）, the radiation patterns of the antenna are almost the same as those of the antenna without graphene patches. This indicates that the graphene patches with µ_c=0 eV mainly behave as dielectric layers.

In this work, reconfigurable QYU micro-strip antennas with graphene-patch directors are proposed. Because the complex conductivity of graphene can be tuned by changing the bias voltages, the graphene patches can be electrically switched between metallic state and dielectric state. Therefore, the graphene-patch directors can be electrically reconfigurable and the main lobe angle of the antenna can be electrically swept in a large range in θ=60° plane （60° to 120° for the configuration of two graphene-patch directors and 30° to 150° for the configuration of three graphene-patch directors）. The reconfigurable behaviors of the antenna are not sensitive to the scattering time of graphene. It is expected that such a graphene-based antenna is attractive for THz application in the future.