A reconfigurable graphene patch antenna inverse design at terahertz frequencies

This article investigates the inverse design of a reconfigurable multi-band patch antenna based on graphene for terahertz applications to operate frequency range (2–5THz). In the first step, this article evaluates the dependence of the antenna radiation characteristics on its geometric parameters and the graphene properties. The simulation results show that it is possible to achieve up to 8.8 dB gain, 13 frequency bands, and 360\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\circ$$\end{document}∘ beam steering. Then and due to the complexity of the design of graphene antenna, a deep neural network (DNN) is used to predict the antenna parameters by given inputs like desired realized gain, main lobe direction, half power beam width, and return loss in each resonance frequency. The trained DNN model predicts almost with 93% accuracy and 3% mean square error in the shortest time. Then, this network was used to design five-band and three-band antennas, and it has been shown that the desired antenna parameters are achieved with negligible errors. Therefore, the proposed antenna finds many potential applications in the THz frequency band.

model for the designing of nanophotonic structures. Then to overcome the limited generalization ability of 23 , a probabilistic graphic model introduced as an all-inclusive explanation for metamaterial design.
We have classified using the neural networks in antenna design into three approaches. First, NNs and ML enhance some radiation properties by optimizing the antenna parameters and can not control antenna radiation patterns in real-time 35 . Second, by giving the antenna dimensions as input, the antenna radiation will be estimated as output in real-time so DL and ML can speed up the antenna simulation directly 30,31 . And third, the most widely used method is the inverse design of the antenna using DL. The required radiation pattern characteristics are provided as input, and the DNN's output estimates the antenna parameters. In this case, depending on the circumstances, the antenna parameters may be fully adjustable or non-adjustable 28,32 . Although in 29 a VO 2 is used as a reconfigurable component in the antenna, the proposed DNN outputs geometrical antenna parameters.
In this article, an inverse design of reconfigurable graphene circular patch antenna at THz frequencies is proposed and surveyed to the realization of an intelligent antenna for 6G wireless communication. Also, for the first time we apply a chemical potential of graphene as a reconfigurable component in the output of DNN to control the radiation properties in real-time. The antenna parameters are divided into two groups, graphene and antenna parameters. After the analysis, we generated a data set and filtered it with two specific conditions. Then, a DNN model is presented, that can accurately predict the values of the graphene properties and antenna parameters for the desired S-parameters and radiation pattern of the antenna.
For this, in Sect. "Deep learning", the design and simulation of the antenna have been discussed. Then, the used deep learning method will be explained and the achieved results have been examined in Sect. "Evaluation of deep neural network". Finally, some conclusions are remarked.

Antenna design and simulation
The flow chart of the activity steps is plotted in Fig. 1. At first, the behavior and dependency of the antenna characteristics on its parameters are investigated by taking into account the antenna parameters are divided into two groups, graphene parameters and geometrical parameters. Then, the data needed for training the DNN model is generated by sweeping the graphene and geometrical parameters in antenna simulations (see Fig. 1a). Secondly, two filters are applied to extract features from S-parameters and E-field patterns, in order to gather proper data for use in the training data of the DNN model (see Fig. 1b). Finally, we present a DNN model, which can accurately estimate the values of the thickness of the substrate, τ , and µ c for desired inputs. In this work, Desired inputs of DNN comprise resonance frequencies, realized gain, null level, main lobe direction, and half-power beam width (see Fig. 1c). Based on the mentioned procedure, each section will be explained below.
Antenna structure definition. The structure of the antenna along with its parameters is plotted in Fig. 2.
As shown in this figure, the structure is comprised of three layers. At the top and bottom, a graphene layer with thickness H g = 0.08mm and temperature T k = 273K is deposited. For the middle layer, a silicon layer by the relative permeability ǫ r = 11.9 and conductivity σ = (25 * 10 −5 ) is used. The other characteristics and dimensions of layers are given in Table 1. It is worth mentioning that during the simulations, a 50-ohm port impedance is www.nature.com/scientificreports/ assumed for the excitation lumped port, and the antenna parameters are determined such that input matching is achieved. Due to the use of graphene in the patch and ground of the structure, a reconfigurable radiation pattern or a reconfigurable operating bandwidth is the inherent property of the structure. As shown in Fig. 3 , if the graphene chemical potential changes from 0.1 eV to 0.8 eV, the operation frequencies of the antenna have been shifted while the radiation pattern at the 2.34 THz frequency remains relatively constant. In this simulation, H s = 30 µ m , relaxation time = 0.1 ps, W f = 12 µ m , and R p = 18 µ m . Similarly, it is possible to change the beam direction by making changes in chemical potential as shown in Fig. 4 . It is obvious in this figure; for the changes of chemical potential from 0.1 eV to 1.3 eV, the return loss of the antenna remains relatively constant while the beam direction has been tilted about 30 degrees at the 3.21 THz frequency. In this simulation, H s = 20 µ m , relaxation time = 1 ps, W f = 6 µ m , and R p = 18 µ m . Therefore, the proposed antenna can be used as a reconfigurable antenna.

Parameters
Range Finally, the number of simulations is equal to 2880. It is worth mentioning that the simulations are done in the frequency range of [1][2][3][4][5] THz and S 11 , radiation pattern in E-plane, and realized gain is extracted in 101 equally spaced frequency points in the mentioned bandwidth. Then, the number and directions of main beams are extracted from the realized gain data based on the condition G realized > 1dB. Furthermore, the number of frequency bands is determined on the condition that |S 11 | > 10dB. In the next step, the dependency of the antenna characteristics on the parameter values will be investigated. Fig. 5a,b, the number of frequency bands is studied as a function of the relaxation time and the chemical potential for H s = 20 µ m and H s = 30 µ m , respectively. As shown in this figure, higher chemical potential results into higher frequency bands. Also, more frequency band belongs to higher substrate thickness. Furthermore, changing the relaxation time will not affect the number of frequency bands for the zero chemical potential. It is worth mentioning that in Fig. 5, the color bar is devoted to relaxation time to make more clearance. Similarly, in

Deep learning
Artificial neural networks are machine learning techniques inspired by the human nervous system. The neural network consists of interconnected neurons, and by changing the weight of the interconnected neurons, that can learn the relationship between inputs and outputs by adjusting the weights of these connections, allowing them to generalize based on empirical knowledge. Figure 6 shows the schematic of a neural network. In this figure, n is the number of inputs and x i denotes the value of each input. Each input is associated with a specific weight, w i , which is multiplied by the input and summed up. After summation, an activation function φ is applied to estimate the output, which can be biased with an initial value b i . In equation (1), the output and input of the neural network are presented. Neural networks typically consist of an input layer, one or more hidden layers and an output layer, with each layer containning a specific number of neurons. Also, the weights of each neuron will change during a back propagation process to learn the input and output relationship pattern. In general, as the number of neurons and hidden layers increases, the ANN network becomes more DNN, which will cause the complexity of the model.
In this research, the inputs of the deep neural network include the number of bands, the resonance frequencies of the bands, the beam direction, the half-power beam width, the gain, and the depth value of S 11 at each  www.nature.com/scientificreports/ resonance frequency of the antenna. After the presented DNN has been learned, the required antenna parameters, such as the dimensions of the antenna and the properties of the required graphene, are estimated. As mentioned, the maximum number of bands achieved is 13, while the desired parameters total 5 in total. Therefore, each www.nature.com/scientificreports/ input vector is a row matrix of order 65, including the vector for each of the 13 frequency bands. In the proposed learning procedure, 60% of the data has been used for learning while the others are used in the test sequence. The output of the neural network includes a third-order vector comprises the thickness of the substrate, relaxation time, and chemical potential. Since the goal of this research is to accurately estimate the continuous values, the ReLU activation function has been used in each network layer. It should be noted that the number of layers and neurons was optimized to achieve the best performance in the proposed network. In the proposed model, the batch size and learning rate are 512 and 0.001, respectively, and the number of epochs is set to 5000. Also, the Adam optimization algorithm has been employed to determine the values of the weights in the model. Furthermore, the utility cost function MSE (Mean Squared Error) has been used to calculate the difference between the real value and the value estimated by the model, as given in equation (2), where y i and f i represents the real and estimated values, respectively.
To achieve the best performance of the DNN, we conducted several tests on the combination of layers and their output shape value. Table 2 displays the optimal structure of the DNN for learning the relationship between inputs and outputs.
The loss and accuracy diagrams of the proposed model have been sketched in Fig. 7a,b, respectively. As can be seen in this figure, the values of accuracy and loss achieve 91.5% and 0.03, respectively. Furthermore, it can be seen that the validation graph closely aligns with the training graph, indicating successful avoidance of overfitting in the model. In this procedure, the training time took about 262 seconds and the model can estimate the output in less than 0.05 seconds which was obtained on a system equipped with an Intel Core i7-10750H processor and 16GB RAM.

Evaluation of deep neural network
For evaluation of the proposed model, we have provided two arbitrary samples whose specifications are given in Figure 8. For this, the desired values are fed to the proposed network and the antenna parameters are estimated. Subsequentlly, a full wave analysis was conducted to simulate the performance of the antenna. As shown in Fig. 8a,b, the gain and return loss of a five-band and a three-band antenna are plotted versus frequency, respectively. In these figures, the desired values are also shown using colored circles and are in good agreement with the simulation results. In the same way, the simulation results and the desired values of the half-power beam width and the main lobe direction for five-band and three-band antennas are plotted in Fig. 8c, d, respectively. The small differences between the simulated and expected values are also evident in these figures. The estimated values of the substrate thickness, chemical potential, and relaxation time are presented in Table 3. Furthermore, the MSE of the resonance frequencies, half-power beam width, gain and return loss are shown in table 3. According to  www.nature.com/scientificreports/   www.nature.com/scientificreports/ these results, it can be claimed that negligble errors in achieving the desired antenna parameters are obtained through the estimation of antenna thickness and graphene properties.

Discussion
In this article, a planar graphene antenna was investigated and its radiation characteristics for different substrate thicknesses and graphene characteristics including chemical potential and relaxation time have been extracted by full-wave FEM simulations. Then, these parameters were examined as the design inputs and it was shown that in the proposed structure, by choosing the appropriate values for the input, the number of bands and their resonance frequencies, antenna gain and main lobe directions can be set. The simulations have been performed in the [2][3][4][5] THz frequency band and it has been shown that the maximum gain of 8.8dB and up to 13 frequency bands can be achieved. Due to the complexity of the design and in the following, a deep neural network has been used to provide a design solution. This network is trained based on the categorized simulation results and the network parameters are determined in such a way that the most accurate matching between the estimated and simulated parameters has been achieved for minimum input data. In the end, it has been shown that the optimal radiation parameters can be estimated with an error of less than 3%. This feature can be used to design antennas in various applications. In the simulations, we encountered one-to-many issues. In fact, the analysis of the simulation results shows that for the antenna parameters of [ W s , L s , R p ], relatively the same results for S 11 and E-fields may be generated. Furthermore, if all the structural antenna parameters are assumed as the outputs, errors would arise in estimating the input parameters. However, we considered this category of samples in order to maintain the number of neural network training dataset samples and improve the accuracy of the model, three parameters of [ µ c , τ , H s ] are estimated as the output.

Data availibility
The data that support the findings of this study are available from A.N. but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of A.N. (email: a_nooramin@ iust.ac.ir).