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Abstract
Terahertz time-domain spectroscopy (THz-TDS) has been utilized extensively to characterize materials in a non-destructive way. However, when materials are characterized with THz-TDS, there are many extensive steps for analyzing the acquired terahertz signals to extract the material information. In this work, we present a significantly effective, steady, and rapid solution to obtain the conductivity of nanowire-based conducting thin films by utilizing the power of artificial intelligence (AI) techniques with THz-TDS to minimize the analyzing steps by training neural networks with time domain waveform as an input data instead of a frequency domain spectrum. For this purpose, Al-doped and undoped ZnO nanowires (NWs) on sapphire substrates and silver nanowires (AgNWs) on polyethylene terephthalate (PET) and polyimide (PI) substrates have been measured for dataset creation via THz-TDS. After training and testing a shallow neural network (SSN) and a deep neural network (DNN) to obtain the optimum model, we calculated conductivity in a conventional way, and the prediction based on our models matched successfully. This study revealed that users could determine a sample's conductivity without fast Fourier transform and conventional conductivity calculation steps within seconds after obtaining its THz-TDS waveform, demonstrating that AI techniques have great potential in terahertz technology.
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1. Introduction
Terahertz (THz) technology has been considered one of the key technologies applied to non-destructive and contact-free conductivity measurements of various metals, including 2D materials, semiconductors, and superconductors utilizing terahertz time-domain spectroscopy (THz-TDS). Electromagnetic waves with frequencies ranging from 0.1 to 10 THz and wavelengths ranging from 0.03 and 3 mm are called the Terahertz gap [1,2]. This gap is striking for various applications containing high-speed electronics [3], communications [4], security monitoring [5], biomedical imaging [6], chemical and biological sensing [7], and quality control applications [8]. When comparing the terahertz with conventional nondestructive methods, it can detect objects under low-radiation, nonionizing, and real-time evaluation conditions with high accuracy and no coupling besides contact-free measurements [9]. Hence, those features signalize that terahertz will contribute significantly to the future development of nondestructive measurements.
Terahertz time-domain spectroscopy (THz-TDS) is a spectroscopic technique that has developed into an emerging powerful tool and measurement method, e.g., characterization of carrier dynamics at high frequencies. It may open the door for a better understanding of high-frequency optoelectronics’ characteristics and many other materials’ properties in a non-destructive way. THz-TDS has been facilitated to determine conductivity, carrier concentrations, and other intrinsic parameters of various materials’ carrier dynamics, including nanostructures and nanowires without any electrical contact [10]. Thus, measuring the electrical properties of any type of nanostructures and nanomaterials becomes considerably simple. For example, conventional I-V measurement requires wires and probes, which are available. Still, it is extremely difficult for nanomaterials and nanowires to obtain good ohmic contact, especially with the patterning required. Also, when measuring the conductivity of the sample with bending and stretching for flexibility, probing or wire-bonding that requires extra work should apply to the sample. Therefore, these steps make measuring electrical conductivity more complicated than using THz-TDS. The electrical properties of various thin conducting films, including silver [11–13], hybrid graphene-silver [14], gold [12,15,16], aluminum [12], GaN and ZnO [17] have been reported via probing by THz-TDS.
Although measuring the conductivity of nanostructure and nanowires with THz-TDS is non-destructive and non-contact, there are many stages to obtaining conductivity of sample because of time-domain and frequency-domain signal processing, Fast Fourier Transform (FFT) and applying conventional conductivity equations after measurements. Therefore, a new approach to real-time processing is needed to simplify and hasten the analysis method. The artificial intelligence technique, a neural network, will be a great candidate for getting rid of time-consuming and complex post-processing. An artificial neural network (ANN) evolves by mimicking the human neural system as a learning model. By employing a shallow neural network (SNN) and deep neural network (DNN) as an ANN, extremely challenging human-like tasks can be completed by trained systems [18–20]. Recent developments in the performance and implementation of ANNs have enabled analysis of the information in acquired THz signals at a given task, such as spectrum classification and image processing in quality inspection of agricultural products, disease diagnosis, security inspection applications [21,22]. Recently, an efficient neural network based on the regression methods was proposed by Klokkou et al. [23] and Zhou et al. [24] for estimating the refractive index of materials in place of the conventional fitting function using frequency domain data as an input for their ANNs. To our knowledge, the use of THz-TDS with an artificial neural network for THz non-destructive conductivity measurement using time domain waveform as input data which is not required Fast Fourier Transform has yet to be reported.
Our study proposes a significantly efficient way to reduce the time and complex calculation steps of conductivity measurement of samples by training with a vast amount of experimental data of nanowires samples considering artificial neural networks (ANNs). By doing so, ANNs can learn the correlation between given input and output datasets based on the THz time-domain signal and the calculated conductivity of samples at desired frequencies for the designated network, then perform the calculation of the conductivity of a new sample directly from the results of the THz time-domain signal. To demonstrate this study, Al-doped and undoped ZnO nanowires (NWs) on sapphire substrates which have been used for many types of sensors, and silver nanowires (AgNWs) on polyethylene terephthalate (PET) and polyimide (PI) substrates which can be used for flexible devices, were used for training and testing the artificial neural networks. The findings from this work will show that researchers will reach their samples’ conductivity within seconds, besides only the one-time cost to train the ANN.
2. Theoretical background
2.1. Conductivity calculation of thin films
Conductivities of thin films can be obtained based on THz-TDS measurement by analyzing THz waves transmitted through plain reference signal (Ereference(t)) and each sample signal (Esample(t)) separately. Detailed expressions of traveling frequency-dependent THz waves in multiple mediums with interfaces have been reported elsewhere [13,17,25]. The complex spectra (E($\omega $)) for each transmitted THz electric field can be represented as a function of frequency by taking the Fast Fourier Transform (FFT) of time-domain signal spectra by applying time-domain signal processing to remove unwanted reflections from each measured THz waveform. After frequency domain processing, followed by a complex transmission ratio (transfer function), T($\omega $) is defined as the transmitted signal from the interesting material, Esample($\omega $), divided by the transmitted signal from the substrate material, Ereference($\omega $), as shown in Eq. (1) and (2);
2.2. Artificial neural network
An artificial neural network (ANN) is a learning model inspired by the human brain. This model has become a phenomenon modeling method for physical and non-physical systems based on mathematical or scientific principles [18–20,26,27]. An ANNs with only one hidden layer are referred to as a shallow neural network (SNN). Otherwise, an ANNs containing two or more hidden layers is known as a deep neural network (DNN) [28]. The fundamental layers of ANN are the input layer, the number of hidden layers, and the output layer. This structure is named a fully-connected layer when all neurons are connected with adjacent layers in a conventional ANN. So, the number of hidden layers is adjustable to maximize the ANN’s performance. Moreover, the hidden and output layers’ neurons accept multiple inputs from neurons from the previous layer. Suppose the input layer has X neurons and the output has Y neurons. The neuron output (a) is calculated by applying an activation function to a weighted input(z). Weighted input(z) is generated by input (Xi) multiplied by weights (wi) and adding the bias(b) value. Common activation functions are shared as follows [29–32];
- • sigmoid: $f(z )= \frac{1}{{1 + \textrm{exp}({ - z} )}}$
- • softmax: $f({{z_y}} )= \frac{{\textrm{exp}({{z_y}} )}}{{\mathrm{\Sigma }_{j = 1}^Y\textrm{exp}({{z_j}} )}}\; ,\; 1 \le y \le Y$
- • hyperbolic tangent: $f(z )= \tan h(z )$
- • rectified linear unit (ReLU): $f(z )= \left\{ {\begin{array}{c} {0,\; \; \; z < 0}\\ {z,\; \; \; z \ge 0} \end{array}} \right.$
Determining ANN parameters as weights and biases compose a learning process of ANN. During the process, the network updates the parameters depending on when the error function decreases at each iteration rather than calculating the optimizing parameters by solving the equations. To achieve the best result, minimize the error function by measuring its gradient for each parameter. Updating the parameter slightly in a negative gradient direction assures a reduced error function. That method of the gradient is named gradient descent [33,34]. Moreover, partial computations of the gradient from one layer are reused in the computation of the gradient for the previous layer through the output layer to the input layer, called backpropagation. As a result, various structured method for calculating gradients for all parameters has been developed, besides many derivatives of gradient descent and learning algorithms for faster learning of ANN [35–42].
Our study uses a multilayer feed-forward network for conforming regression from the machine learning and deep learning toolbox in MATLAB to model our SNN and DNN. A scaled conjugate gradient back-propagation learning algorithm, the memory requirements are relatively less and much faster than other learning algorithms in the toolbox, was executed to train our networks. The network’s input layer consists of 528 input neurons, where 526 inputs represent THz-TDS measurement time domain data of the desired sample; one input shows the refractive index of the selected sample’s substrate; another input represents the thickness of the selected sample. There are 17 output neurons expressing the conductivity of the sample in the THz frequency domain. A sigmoid was used to activate the neurons in the hidden layers. The schematic illustration of our artificial neural networks is shown in Figure 2. To determine the hidden layer of the shallow neural network, the number of neurons was compared by applying the range from 30 to 1000 neurons. The best performance was achieved on 30 neurons in the SNN. After achieving the best performance with 30 neurons in the hidden layer, we checked the deep neural network’s performance in up to five hidden layers by utilizing 30 neurons in each hidden layer for comparison in our study.
Fig. 2. An illustration architecture of (a) Shallow Neural Network (SNN) and (b) Deep Neural Network (DNN).
3. Experimental approach
The silver (Ag) and Zinc Oxide (ZnO) nanowires (NWs) samples are used in this study. AgNWs flexible electrodes were fabricated using the spray method by adding varying concentrations of AgNW suspension solution on flexible 50 um thin PI and 100 um thick PET substrates. The details of the fabrication processes were reported elsewhere [13]. The diameter, length, and thickness estimation of AgNWs are 25-35 nm,10-20 um, and about 100 nm, respectively. An SEM image of the AgNW/PET sample is shown in Figure 3(a). ZnO NW samples in this study were grown on sapphire (Al2O3) substrate at 900°C in a 3-zone tube furnace. For additional samples, aluminum (Al) was chosen as a dopant in the growth process. Detailed information on the growth of ZnO NWs was described elsewhere [43,44]. The length of ZnO NWs has altered from 5 to 20 um based on the duration of growth. Both undoped and Al-doped ZnO NWs were grown vertically, as shown in Figure 3(b).
Fig. 3. Scanning Electron Microscope images of (a) AgNWs on PET and (b) vertically aligned undoped ZnO NWs have grown on the sapphire substrate.
The schematic illustration of the in-house transmission mode THz-TDS is shown in Figure 4. Detailed system configuration was reported elsewhere. [13,17] Our system consists of the photoconductive antenna from LT-GaAs as a source and ZnTe electro-optic crystal with balanced photodiodes as a detection system. A 790 nm mode-locked laser with 110 fs pulses generates broadband terahertz radiation from 0.2 to 2.0 THz. Two polyethylene lenses are being used for focusing the collimated radiation to a diameter of 0.5 mm on the sample and re-collimate the radiation into the detection array. THz-TDS is measured at room temperature in low humidity (about 21%), releasing nitrogen gas to purge moisture to prevent high absorption of waves in the THz frequency range.
4. Results and discussion
Conductivity calculation of nanowires via SNN and DNN procedure consists of three main steps; 1) performing non-destructive THz-TDS measurement to collect data from prepared samples, 2) designing datasets with artificial neural networks, and 3) training-testing each network with each dataset. The first step collected a total of 123 sample spectra data from prepared AgNWs on PI (41), AgNWs on PET (58), undoped ZnO NWs on sapphire (22), and Al-doped ZnO NWs on sapphire (2) samples with THz spectroscopy under a nitrogen blanket in transmission mode. The measured signal length was set as seven picoseconds (ps) and kept the same for all measurements, as shown in Figure 5(a). Then, their conductivities are calculated in the desired frequency range from 0.4 THz to 1.2 THz as the interval is 0.05, as shown in Figure 5(b). It should be noted that the frequency range was chosen to be consistent in data and to avoid inconsistent water or system absorption peaks above 1.2THz.
Fig. 5. (a) Terahertz time-domain signals of nanowire samples for SNNs’ and DNNs’ input (b) Calculated real conductivity of nanowire samples for SNNs’ and DNNs’ output.
From the collected data in the first step, four datasets are created in the second step, as detailed in Table 1. While the first and second datasets include AgNW/ PI, AgNW/PET, and undoped ZnO NW/Sapphire samples, the third and fourth consist of only AgNWs samples. Moreover, the first and third datasets are created using only experimental THz time-domain waveform data; the second and fourth ones were enhanced by attenuating the time-domain waveform of each measurement from 95% to 50%, as demonstrated in Figure 6. Enhanced datasets are created so SNN and DNN training benefit from the more extensive data sets. Data augmentation is a popular and convenient way to extend the dataset without preparing extra samples, which requires extra time and cost [45]. Therefore, after extracting the additional 12 test samples from the total sample collection, the only measurement datasets comprised 111 and 91 samples, respectively; the augmented datasets have 1221 and 1001 samples, accordingly.
Fig. 6. Demonstration of data augmentation utilizing attenuations of original sample (AgNW on PI substrate) signal for enhanced datasets
Table 1. Artificial Neural Network Dataset Designs
In addition, all datasets purposely exclude the Al-doped ZnO NWs on sapphire samples during the training of SNNs or DNNs. Datasets are randomly divided into 70% for the training set, 15% for the validation set, and 15% for the testing during the training section of SNNs and DNNs. Then, we trained and tested our SNNs and DNNs by utilizing four datasets separately in the third step. To increase training effectiveness, standardization is applied to our datasets. The validation set stops the training of the network when the network begins to overfit the data. After training and testing the network, the estimation performance was measured using the R (regression) value and the mean-square error (MSE).
In the first dataset, which used only measured waveform data of samples, the best validation performance of the R-value was 0.995, and MSE was 8.41 × 10−3 for SNN with 30 neurons in the hidden layer. When increasing the hidden layer’s neuron numbers for SNNs, the network performance was getting lower to 0.864 for R-value, and the error is getting higher up to 3.17 × 10−1 at 1000 neurons in the hidden layer. However, applying the DNN model by adding 30 neurons to each additional hidden layer showed better performance than SNN. While R-value reached 0.999, MSE reduced at 2.77 × 10−5 in DNN.
Figure 7 compares the calculated and prediction results of the unseen sample test set for both models with various neuron numbers and layers. It is evident that the prediction of our models is remarkably well-matched with the exact conductivity value of each sample, especially for the DNN model for the first dataset. We have also tested up to five hidden layers in the DNN model. It had good regression prediction ability when working on the first dataset, which was limited only measured samples. When enhanced the second dataset via the data augmentation technique used for training and testing our artificial neural networks, SNN using 30 neurons in the hidden layer and DNN models have similar performance as shown in Figure 8. As expected, with training more samples R-value has reached 1 for both models, and MSE decreased strikingly to 6.70 × 10−6 for SNN and 1.65 × 10−6 for DNN. That performance proved that with enhancing datasets, users might not need to use more complex neural network designs such as DNN.
Fig. 7. Prediction based on the first dataset: SNN model with various hidden layer’s neuron numbers conductivity predictions based on the first dataset for (a) AgNW/PI sample, (b) AgNW/PET sample, (c) Undoped ZnO NW/Sapphire sample. DNN model with up to five hidden layers, which includes 30 neurons in each hidden layer, conductivity predictions built on the first dataset for (d) AgNW/PI sample, (e) AgNW/PET, (f) Undoped ZnO NW/Sapphire sample. All prediction results are compared with calculated conductivity for accuracy.
Fig. 8. Prediction based on the second dataset: SNN model with various hidden layer’s neuron numbers conductivity predictions based on the second dataset for (a) AgNW/PI sample, (b) AgNW/PET sample, (c) Undoped ZnO NW/Sapphire sample. DNN model with up to five hidden layers, which includes 30 neurons in each hidden layer, conductivity predictions built on the second dataset for (d) AgNW/PI sample, (e) AgNW/PET, (f) Undoped ZnO NW/Sapphire sample. All prediction results are compared with calculated conductivity for accuracy.
To confirm the efficacy and accuracy of our models, we tested our networks with a new, untrained sample, Al-doped ZnO NWs on the sapphire substrate that is never used in the training section of our SNNs and DNNs. Since Al-doped ZnO NWs will have different, possibly higher conductivity, it is the ideal sample to test the functionality of the network. Figure 9 shows the THz-TDS signature of Al-doped ZnO NWs, which is used in the input for trained networks. Our trained networks are predicted with unseen and untrained Al-doped ZnO NWs on the sapphire substrate for the first and second datasets, as shown in Figure 10.
Fig. 10. Conductivity prediction of untrained and unseen Al-doped ZnO NW/Sapphire sample with comparing calculated one for SNNs model prediction based on (a) the first dataset training process and (b) the second dataset training process. For DNNs model prediction built on (c) the first dataset training and (d) the second dataset training.
Our network showed outstanding results in testing new and untrained samples, closely matched with manually calculated conductivity results. SNNs and DNNs models follow the same behavior as seen on trained sample testing results for the unknown sample testing. So, the performance of our models better predicts the conductivity of new (untrained) nanowire samples by utilizing an augmented dataset as observed for testing results. DNNs performed better for the first dataset, even for a new sample.
To demonstrate the robustness of our proposed method to obtain the conductivity of nanowires by combining artificial intelligent techniques with THz-TDS, it is critical to evaluate the reliability of our SNN and DNN models by removing undoped ZnO NWs on sapphire substrate samples from datasets. Thus, we have created third and fourth datasets by only adding AgNWs samples to train our neural networks and testing with both AgNWs and ZnO NWs samples. Even though the number of pieces for training SNN and DNN models decreased, the performance of models stayed the same, as shown in Figure 11 and Figure 13 for the third dataset, Figure 12, and Figure 14 for the fourth dataset. About the untrained new sample test, the DNN model and SNN model with 30 neurons in the hidden layer for the third and fourth datasets are pretty well matched with the calculated conductivity of nanowires. Based on the results of four datasets with two artificial neural network models, THz-TDS combined with AI demonstrated the rigorous and simplified way to obtain the sample’s conductivity by removing time-domain signal processing, Fast Fourier Transformation (FFT), and numerically solved equations.
Fig. 11. Prediction based on the third dataset: SNN model with various hidden layer’s neuron numbers conductivity predictions based on the third dataset for (a) AgNW/PI sample, (b) AgNW/PET sample. DNN model with up to five hidden layers, which includes 30 neurons in each hidden layer, conductivity predictions built on the third dataset for (c) AgNW/PI sample, (d) AgNW/PET. All prediction results are compared with calculated conductivity for accuracy.
Fig. 12. Prediction based on the fourth dataset: SNN model with various hidden layer’s neuron numbers conductivity predictions based on the fourth dataset for (a) AgNW/PI sample, (b) AgNW/PET sample. DNN model with up to five hidden layers, which includes 30 neurons in each hidden layer, conductivity predictions built on the fourth dataset for (c) AgNW/PI sample, (d) AgNW/PET. All prediction results are compared with calculated conductivity for accuracy.
Fig. 13. Conductivity prediction of untrained and unseen undoped ZnO NW/Sapphire sample with comparing calculated one for SNNs model prediction based on (a) the third dataset training process and (b) the fourth dataset training process. For DNNs model prediction built on (c) the third dataset training and (d) the fourth dataset training.
Fig. 14. Conductivity prediction of untrained and unseen Al-doped ZnO NW/Sapphire sample with comparing calculated one for SNNs model prediction based on (a) the third dataset training process and (b) the fourth dataset training process. For DNNs model prediction built on (c) the third dataset training and (d) the fourth dataset training.
All network performance parameters for four dataset training are shared in Table 2 for SNNs and Table 3 for DNNs, respectively. For SNNs, the MSE value increases when the number of neurons in the hidden layer increases, and the R-value decreases for all datasets. On the contrary, DNNs followed different behavior. When increasing the number of hidden layers on DNNs, they perform better, demonstrating lower MSE and higher R-value. Furthermore, we achieved optimum results by utilizing enhanced datasets for all network designs of SNN and DNN.
Table 2. Performance of Shallow Neural Networks (SNN)
Table 3. Performance of Deep Neural Network (DNN)
5. Conclusion
In summary, we demonstrated that both shallow neural networks (SNN) and deep neural networks (DNN) provide a significantly efficient and reliable way in terms of time and speed of data analysis with examples of determining the conductivity-vs.-frequency spectra of various nanowire based thin films by utilizing THz-TDS time domain waveform as direct input data to train and test the neural networks. Our proposed method eliminates not only the conventional conductivity calculation processes but also the need to carry out Fast Fourier Transform of the time-domain data. In doing so, we demonstrated a highly efficient process that yields the expected results. MSE and R-value were used for the main performance criteria of our systems analysis. The efficiency of the data augmentation method is tested in the experimental study. Moreover, to test the robustness of our systems, four different datasets are created and used for training SNN and DNN models to find an optimal model. The experimental results demonstrate that the AI-aided method can replace traditional calculation methods with high accuracy, speed, and a reduced number of steps involved in implementing the method. Obtaining the conductivity of desired samples would display a convenient and easy way within seconds to users. Our proposed method of training and predicting conductivity will significantly facilitate the process of fitting the conductivity data to the Drude model to extract physical parameters underlying carrier dynamics. Therefore, ANN will be further leveraged to predict these physical parameters directly from the time-domain data in our future work. It is also worth mentioning that, although our work reported here focused on real-part conductivity data, the method reported here will work as efficiently for the imaginary-part conductivity and any other types of conducting films. Besides the non-destructive evaluation of various samples using THz-TDS measurements, combining artificial intelligent techniques with THz technology has a strong potential in many future THz applications.
Acknowledgment
MZG acknowledges the Ministry of National Education of the Republic of Türkiye scholarship.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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