Achieving Desired Crystal Structures and Properties by Gradient-Based Inverse Problem Solving

In the last decade, machine learning has made significant contributions to material design. Recently, several studies have successfully adapted conditional generative models[1], known for their success in computer vision, to material design by using desired physical properties as conditions. However, conditional generative models may fail to generate materials with the desired properties and cannot measure the properties of the materials they generate. Moreover, there is no guarantee that the generated materials maintain charge neutrality. Furthermore, it is not possible to generate crystal structures that have not been included in the data set.<br/>In this study, we propose a method for optimizing crystal structures towards desired properties using a gradient-based inverse problem-solving approach[2][3] that optimizes inputs (crystal structures) directly through backpropagation. Specifically, we employ a deep learning model to predict the material properties and evaluate the discrepancy between target and predicted properties. We then optimize the crystal vectors, atomic coordinates, and atomic types based on the value of the gradient to achieve the desired physical properties. Optimization is conducted based on the predictions from the model, thereby guaranteeing the properties of candidate materials within the model's accuracy. Furthermore, since atomic types are not directly differentiable, we utilize a method that transforms them into a differentiable atomic distribution for optimization.<br/>This method offers the advantage of adaptively applying various conditions during optimization. For example, when optimizing typical perovskite crystal structures to achieve a specific band gap, it is essential to maintain the charges at sites A, B, and X in a 1:2:-1 ratio, while the angles between crystal vectors should remain at 90 degrees. In this scenario, random perovskite structures are used as the initial configuration. The crystal vectors' lengths, atomic coordinates, and atomic distribution are optimized while the angles between the crystal axes are fixed at 90 degrees. The atomic distribution at each site is composed exclusively of atoms with permissible charges, such as only monovalent cations at site A. Unlike conditional generative models, this method not only readily meets electrical neutrality conditions but also enables the optimization of the crystal structure for desired properties while maintaining the perovskite structure. It is noteworthy that the perovskite structure is preserved without the need for retraining on a dataset composed exclusively of perovskite materials.<br/>We applied our methodology to the perovskite structure to achieve target band gap values of 1.00±0.02 eV and 3.00±0.02 eV. The pre-trained Crystalformer[4] model was used to predict the band gap. This model was trained on the Materials Project dataset, which includes various types of crystal structures in the training set. Our methodology reveals two candidate materials, $¥mathrm{BaCeOS_2}$ and $¥mathrm{CsYbF_2Cl}$, for which the Crystalformer model predicts band gap values of 1.02 eV and 3.02 eV, respectively. Both materials possess perovskite structures and charge neutrality is maintained, showcasing the capabilities of our methodology.<br/>Our method paves the way for practical material design by incorporating specific crystal structures and properties. For instance, our approach allows the design of lead-free perovskite structures with specific band gaps without requiring extensive retraining or fine-tuning of the model. It ensures that the final materials not only meet the desired physical characteristics but also retain structural integrity and charge neutrality.<br/>[1] Zeni, C., et al. arXiv: 2312.03687 (2023). [2]Fujii, A. et al. arXiv: 2304.13860 (2023) [3]Fujii, A. et al. arXiv: 2403.13627 (2024) [4] Taniai, T., et al. arXiv:2403.11686 (2024).