Magic angle effect driven by deep learning: Giant electromechanical coupling in four-layer twisted lithium niobite

Abstract

The electromechanical coupling coefficient (k_t^2) is a key factor in determining the bandwidth of acoustic resonators. Most piezoelectric materials exhibit relatively low coupling coefficients, making it challenging to achieve broader frequency ranges. In this study, we used a deep learning model to explore the magic angle effect in four-layer twisted lithium niobate (LN) and significantly improved k_t^2. By leveraging deep learning, we can optimize the material’s thickness and twist angles, and accurately predict the resonant and anti-resonant frequencies across different configurations to enhance k_t^2. Theoretical results demonstrate that, at specific thicknesses and twist angles, the four-layer LN structure achieves a coupling coefficient of up to 90%, outperforming traditional single-crystal and dual-layer twisted LN. This study provides a new perspective at the intersection of materials science and deep learning, highlighting the potential of AI to optimize complex multilayer structures for improved k_t^2.

Graphical abstract

Keywords

deep learning / magic angle / electromechanical coupling coefficient / lithium niobate

1 Introduction

Surface acoustic wave (SAW) and bulk acoustic wave (BAW) resonators are essential components in the construction of SAW (BAW) filters. An important indicator of the performance of acoustic resonators is the electromechanical coupling coefficient [1], which represents the efficiency of the conversion between mechanical vibrations and electrical signals [2]. It directly impacts the efficiency [3], response characteristics [4], and frequency stability of the resonator [5]. However, enhancing the electromechanical coupling coefficient remains a challenge due to material limitations, as traditional piezoelectric materials such as lithium tantalite (LT) and aluminum nitride exhibit relatively low electromechanical coupling coefficients [6], as well as structural constraints such as acoustic leakage [7], mode conversion [8], and fabrication complexities [9]. Recently, two-dimensional stacked materials with relative twist angles (referred to as magic angles) which can lead to some novel physical phenomena have attracted attention [10−13]. This phenomenon was first discovered in bilayer graphene [14] and resulted in novel and unconventional flat bands, it has since been gradually extended to the fields of quantum mechanics [15−19] and photonics [20−24]. Inspired by this, the magic angle concept was eventually introduced into bilayer twisted LN [25], where the relative positions of the two twisted layers of LN are adjusted in a specific geometric configuration, resulting in unique properties, including enhanced electromechanical coupling. This has led to an extraordinary electromechanical coupling coefficient of 85.5%. Advances in geometric design and material property control have provided new ideas for addressing this issue.

While these advancements in material design offer promising improvements, their practical implementation in resonator design requires efficient computational approaches to fully harness their potential. The traditional design of SAW (BAW) resonators relies on finite element simulations [26, 27], which are essential for accurately modeling the behavior of these systems. However, performing multi-frequency, multi-parameter calculations within this framework requires significant computational resources and time. Furthermore, finite element-based design methods often involve the coupling of multiple physical fields, such as electrical, acoustic, and mechanical fields, which significantly increases the design complexity compared to models that only consider a single physical field. Moreover, traditional methods typically require manual parameter tuning for optimization, and when faced with high-dimensional parameter spaces, they struggle with global optimization, relying on trial and error, which lengthens the design cycle. Fortunately, the advent of artificial intelligence technologies presents a promising solution to these challenges. In recent years, deep learning, a subset of artificial intelligence, has made remarkable progress, achieving notable success in fields such as speech signal processing [28], image processing [29], and natural language processing [30]. In the field of physics, thanks to deep learning models that can effectively approximate the solutions of physical field equations, they have achieved remarkable results in many fields, such as invisibility [31−35], metasurface designs [36−42], and photonic device designs [43−49]. In acoustic filter design, deep learning’s powerful nonlinear fitting capabilities enable it to map complex admittance functions without the need for extensive physical knowledge [50−57]. Furthermore, it can efficiently explore large parameter spaces, offering substantial potential for optimizing material properties. By integrating deep learning with physical modeling, we can predict and optimize material responses with significant accuracy and speed.

In this work, we propose an inverse four-layer twisted lithium niobite (abbreviated as IFTLN) structure with twisting characteristics and utilize deep learning to optimize the material thickness and twist angles, thereby altering the resonant and anti-resonant frequencies of each mode. Among these modes, the electromechanical coupling coefficient of the fourth-order mode is significantly enhanced, reaching 90%, which far surpasses that of single-crystal LN and is 4.5% higher than that of dual-layer twisted LN. This study not only provides deep insights into the behavior of twisted LN structures but also demonstrates the immense potential of deep learning in advancing the design of high-performance electromechanical systems.

2 Mode analysis in single-layer LN and IFTLN

To examine the field distribution in both single-layer LN and the proposed IFTLN, we have plotted the field distribution and corresponding admittance curves, as shown in Fig. 1. First, we analyze the existing modes in the single-layer X-cut LN. The orientation of LN is determined based on the rotation coordinate system (α, β, γ). In Fig. 1(a), uppercase letters denote the Cartesian coordinate system, while lowercase letters represent the material coordinate system. By rotating the material coordinates, LN is aligned in the X-cut orientation. The field consists of two perpendicular polarization waves, with red indicating α-polarization and blue representing β-polarization. These waves are excited by a longitudinal electric field generated by the top and bottom electrodes (not shown in the figure). The admittance response curve for this configuration is shown in Fig. 1(c). According to Ref. [25], only odd-order modes are present in this orientation, with two mutually perpendicular polarization states. In the admittance curve, red and blue dots mark the first- and third-order α- and β-polarized modes, respectively. Figure 1(b) illustrates the schematic of the IFTLN, composed of four single-crystal LN layers, where the x-axes of every two adjacent layers are oriented oppositely. By tuning the thickness of the first three layers and the rotation angles of θ₁, θ₂, and θ₃ (which are defined relative to the fourth layer of lithium niobate) around the z-axis, the admittance response can be effectively tuned. Due to the increased complexity of the field modes in the IFTLN, only the field distribution of the first layer is depicted. The admittance response of the four-layer structure is shown in Fig. 1(d). Compared to a single-layer configuration, the field distribution in the IFTLN is significantly more intricate, with both odd- and even-order polarized modes present. In Fig. 1(d), only the field patterns of two mutually perpendicular fourth-order modes are displayed.

3 Deep learning-assisted realization of ultra-high electromechanical coupling coefficient

3.1 Design method

The proposed methodology, as illustrated in Fig. 2(a), integrates a deep learning model for forward prediction with a Bayesian optimization algorithm to facilitate the inverse design of IFTLN. This process begins by defining the structural parameters of the IFTLN, where each layer is characterized by its thickness (h_n) and rotation angle (θ_n). The total thickness of the four layers remains at 1000 μm. These parameters govern the overall acoustic response of the structure and significantly influence the admittance spectrum. For forward prediction, a neural network [represented as a sequence of yellow nodes in Fig. 2(a)] is employed to establish a high-dimensional functional mapping between the input structural parameters and the admittance response. The neural network is trained using a pre-simulated dataset of admittance spectra, enabling it to accurately predict the resonance and anti-resonance frequencies of any given structural configuration. The forward prediction process, indicated by the green arrow, provides a rapid assessment of design performance `without the need for direct numerical simulations, which are typically computationally expensive. It is important to note that the dataset obtained from direct simulations contains numerous high-Q peaks, making direct training on the raw dataset extremely challenging. To address this issue, we proposed a preprocessing technique to enhance prediction accuracy (see Appendix A for details). Once the forward model has been trained, it can be integrated with a Bayesian optimization algorithm for inverse design, as shown by the red path in Fig. 2(a). This optimization framework iteratively refines the input parameters to achieve a target spectral response by minimizing the discrepancy between the predicted and desired admittance spectra. Since resonance and anti-resonance frequencies always appear in pairs in different admittance responses, the extracted resonance and anti-resonance frequencies from a given response can be represented as a vector space F = [f_r1, f_a1, f_r2, f_a2, ···, f_rn, f_an]. The primary objective of this study is to maximize the electromechanical coupling coefficient for a specific mode. According to the equation k_t^2=\frac{{\pi }^2}{8}\frac{(f_a^2-f_r^2)}{f_r^2}, the objective function can be formulated as f_o = max(k_t1², k_t2², k_t3², ···, k_tn²). Additionally, it should be noted that the directly predicted admittance spectra often exhibit non-smooth characteristics with numerous parasitic modes. To eliminate the interference of these parasitic modes with the primary mode, a post-processing step involving Gaussian filtering is applied to the predicted admittance spectra. This ensures a more accurate and robust prediction of the desired admittance response. By combining deep learning-based forward modeling with Bayesian optimization, the proposed framework provides an efficient and accurate approach for the inverse design of IFTLN structures. This methodology significantly reduces the computational burden associated with direct numerical simulations while ensuring optimal structural configurations that maximize k_t².

3.2 Results from deep learning

The training process and optimization progression are illustrated through two key aspects and the results are displayed in Figs. 2(b) and (c). The loss curves for both the training and validation sets in Fig. 2(b) show a steady decrease over 3000 iterations. This consistent reduction in loss for both sets suggests effective learning and a lack of neither overfitting nor underfitting (Appendix B presents the model’s validation performance on the test set). In addition, the changes in k_t² during the optimization process in Fig. 2(c) show that the performance is significantly improved with the refinement of parameter tuning, which further proves the effectiveness of the optimization method. To investigate deeper into the influence of structural parameters on admittance response and k_t², deep learning was employed to predict the admittance spectra across various parameter configurations. The results demonstrate the neural network model’s ability to capture the intricate relationship between these parameters and the admittance response, allowing for a comprehensive analysis of how different configurations impact the k_t².

After training the forward network, the neural network successfully predicted the admittance curves for three distinct sets of design parameters and the results are shown in Fig. 3(a), each curve corresponding to a unique structural configuration. In each case, the main resonant modes are marked and the corresponding k_t² is annotated at the peak position. It is worth noting that the coupling coefficient of the first configuration (blue curve) is relatively low, only 23.9%. In contrast, the coupling coefficient of the second configuration (yellow curve) is significantly higher, at 66.13%. The red curve corresponding to the third configuration shows that its k_t² reaches 88.2% (It is worth noting that this value is the current known maximum obtained after multiple optimizations). indicating a significant improvement in energy conversion efficiency. This progress proves that the deep learning model can identify the optimal structural configuration, thereby effectively improving the coupling efficiency. Thus, the findings provide valuable insights into the impact of structural parameters on performance and show the potential of deep learning techniques in optimizing design for superior energy conversion.

The structural deformations, stress distribution and polarization type corresponding to the three distinct admittance curves in Figs. 3(b)−(d) offer valuable insights into the physical behavior of each design. The stress distribution and the polarization direction marked by the white oblique arrow show that the optimized modes are of the same type. In Fig. 3(b), the structure with the lowest coupling coefficient shows relatively uniform deformation with minimal internal deformation, indicating weak localized energy. In contrast, the structure with a moderate coupling coefficient in Fig. 3(c) shows more obvious deformation and significantly improved energy localization. The structure with the highest coupling coefficient in Fig. 3(d) shows a more complex deformation pattern with significant interlayer deformation, which shows structural modulation effectively improves electromechanical performance. To validate the predictions made by the neural network, a numerical simulation of the optimized structure, corresponding to the highest coupling coefficient, was conducted and the result was shown in Fig. 3(e). The simulated admittance response closely matches the neural network’s prediction, with a peak coupling coefficient of approximately 90%. This strong agreement demonstrates the reliability of the deep learning model in capturing the nonlinear relationships between structural parameters and admittance behavior. The simulation results confirm that the identified structure is highly effective in maximizing k_t², making it a promising candidate for practical application.

A detailed summary of the structural parameters for the three cases is provided in Fig. 3(f), which shows the variations in layer thicknesses (h₁−h₄) and twist angles (θ₁, θ₂, θ₃) that influence the admittance response. Notably, the optimized structure (represented by the highest coupling coefficient) exhibits distinct twist angle variations, particularly in θ₁ and θ₃, compared to the other configurations. These differences emphasize the importance of precise control over the twist angles in achieving superior coupling efficiency, further supporting the conclusion that structural modulation plays a critical role in optimizing energy conversion performance.

3.3 Results from Finite element method

To further highlight the advantages of deep learning, we optimized the proposed structure using an alternative method. The optimization process conducted using COMSOL, where the optimization was controlled by MATLAB. Iteration curve graph of k_t² is shown in the Fig. 2(d) and the curve converged after 300 iterations. After the model converges, we also extract three different sets of parameters in the iteration process to dynamically demonstrate the improvement of its coupling performance. The results are shown in Fig. 4. Figure 4(a) shows the admittance spectra of three different parameter configurations, illustrating the frequency response of the structure under different conditions. The resonance peaks marked by red dots on each curve represent the electromechanical coupling coefficient of each mode. The blue curve represents a relatively weakly coupled system of 14.68%, while the yellow curve shows improved resonant behavior with a coupling efficiency of 42.6%. The red curve corresponds to the optimized design, with a coupling enhancement of up to 50.45%, and a significant improvement in both electromechanical performance and resonant characteristics to the maximum.

Figures 4(b)−(d) show the deformation, stress distribution, polarization type plots for each of the three configurations. These visual representations, derived from the finite element method (FEM), display how the structure deforms under different configurations. The deformation corresponding to the blue curve in Fig. 4(b) is relatively simple and the displacement distribution is relatively regular, which shows that its electromechanical coupling coefficient is low. Figure 4(c) shows the deformation mode corresponding to the yellow curve, which is more complex in comparison. This structural configuration improves the degree of electromechanical coupling. Figure 4(d) shows the deformation mode corresponding to the red curve, with a more complex displacement distribution, showing the optimized structural characteristics. It can be seen that parameter optimization has a significant impact on the mechanical response of the device.

We have presented the detailed design parameters for each configuration and the results are displayed in Fig. 4(d), including the thicknesses of the four layers (h₁, h₂, h₃, h₄) and angular orientations (θ₁, θ₂, θ₃) for the three parameter sets. These parameter sets illustrate the structural modifications required to achieve the optimized resonance and performance, demonstrating the utility of the FEM-based optimization in achieving targeted electromechanical characteristics.

Although the finite element method (FEM) can predict the admittance spectra of the resonator more accurately, providing more precise and reliable results for predicting the physical behavior of the system, its drawbacks are also quite evident. As shown in Fig. 2(d), the results from FEM are less efficient compared to those in Fig. 2(c), even under the same parameter space and iteration count. This is due to the need to solve a set of complex partial differential equations for each configuration, which is time-consuming and computationally expensive. Therefore, although FEM offers high accuracy, its computational inefficiency and the potential challenges it faces in handling large-scale optimization tasks make it less suitable for rapid design iterations when compared to deep learning models.

4 Conclusion

In summary, we propose a deep learning-driven approach that leverages the magic angle effect to enhance the electromechanical coupling coefficient in four-layer twisted LN structures. Our results demonstrate that the optimized structure achieves a coupling coefficient of up to 90%, which significantly surpasses traditional single-layer and double-layer LN designs. By combining deep learning with physical modeling, we effectively map the complex relationship between structural parameters and admittance response, thereby achieving fast and accurate predictions. The inverse design framework employs neural networks for forward modeling and utilizes Bayesian optimization for parameter tuning, offering substantial computational efficiency over conventional finite element methods (FEM). While FEM provides high precision, its time-consuming nature limits its applicability in large-scale optimization. In contrast, our deep learning model efficiently explores the parameter space and can facilitate the design of high-performance acoustic resonators. This work demonstrates the potential of AI-driven approaches to advance the design of electromechanical systems and provide new insights into optimizing twisted multilayer structures for improved acoustic performance. It is worth mentioning that although the electromechanical coupling coefficient of multilayer LN has been greatly improved, its production cost is relatively high. Compared to LN, LT is less expensive. Furthermore, since the properties of LT are similar to those of LN, and recent breakthroughs have been made in the design of LT devices [58, 59], LT could be a better option. Therefore, the method proposed in this paper can also be applied to the study of maximizing the k_t² in multilayer LT or in stacks of multilayer LN and LT.

5 Appendix A

Due to the presence of numerous high-Q spikes in the original dataset’s admittance spectra, the network struggled to learn these points effectively. To enhance learning and prediction accuracy, we preprocessed the dataset prior to the training. First, all admittance values in the spectra were normalized by dividing by 200. Next, spike values in the spectra were replaced with a median value, calculated as the average of the higher admittance value between the two adjacent points and the spike’s admittance value. During network training, we extracted the spike points separately and assigned them higher loss weights in the loss function to improve prediction accuracy. As shown in Fig. A1, the loss value for unprocessed data struggled to converge during training, whereas the preprocessed data’s loss value converged to 10⁻⁴ after 3000 iterations [see Fig. A1(d)]. In the test set validation, the unprocessed dataset failed to predict most of the spikes, as indicated by the red markers in Figs. A1(b) and (c). In contrast, the preprocessed data achieved higher prediction accuracy for the spikes, as illustrated in Figs. A1(e) and (f).

6 Appendix B

To validate the model’s fitting performance, we randomly selected four samples from the test set, which the model had not seen during training, for evaluation. The results are shown in Fig. A2. As observed in Figs. A2(a)−(d), the network’s predicted values closely match the theoretical values, confirming that the model exhibits strong robustness.

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