Efficient Search for Stacking Patterns in van der Waals Heterostructures Using Bayesian Optimization

Graphene, hexagonal boron nitride (hBN), and transition metal dichalcogenides (TMDC) are atomic layer materials that have garnered significant attention. These materials exhibit unique physical properties in both single and multiple layers. Recently, these materials have been studied more actively as materials that connect 2D and 3D, also known as 2.5D materials [1]. The fabrication of 2.5D materials is achieved through the stacking of various atomic layers, which allows for the design of an almost infinite number of structural patterns. This process results in the emergence of various physical properties. Extensive research has been conducted on these materials through experiment, theory, and simulation. In particular, density functional theory (DFT) has been heavily used for individual material analysis because of its compatibility with low-dimensional materials. Although there have been reports of building a DFT database of monolayer or homo-bilayer materials [2, 3], there is a limit to the conventional intuition-based search for promising materials because of the nearly infinite number of possibilities when hetero-stacking is taken into account. Therefore, due to the large number of candidates being researched, an inclusive approach using data science should be a powerful method.

In this study, we investigate an efficient method for searching optimal stacking structures of 2D materials using a combination of DFT and Bayesian optimization. This study verified the effectiveness of Bayesian optimization on 10 types of homo- or hetero-bilayer structures, including MoS₂, WS₂, MoSe₂, and WSe₂, which are representative 2D materials. We set 420 structures as our search space. This incorporates 42 patterns of stacking displacement for each of the 10 different stacking bilayers, 42 multiplied by 10 yields 420 structures. After optimizing each initial structure, the band gap and Berry curvature at K point were calculated. The density functional theory (DFT) calculations were conducted using the Vienna Ab initio Simulation Package (VASP) code, with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional and Grimme-D3 for van der Waals interaction correction. We compared the number of trials required to find a best stacking structure with the desired electronic properties using random search and Bayesian optimization. Three stacking pattern searches were conducted: (1) maximum band gap, (2) minimum band gap, and (3) maximum Berry curvature at K point. To ensure objectivity in the evaluation of effectiveness, each process was conducted 20 times independently to prevent the effects of randomness. The average number of searches was then counted.

As a result, Bayesian optimization was found to be about three times more efficient than random search in the maximum and minimum bandgap search. The explanatory variables for Bayesian optimization are quite simple, consisting only of the material names of the first and second layers and the stacking displacement. The result that a model with a relatively simple set of explanatory variables achieved an efficiency improvement of approximately threefold is regarded as substantial evidence in support of the utility of Bayesian optimization. Furthermore, Bayesian optimization was about twice as efficient as random search in focusing on Berry curvature, a more sophisticated and complex physical quantity than the band gap. The outcomes of this study illustrate the efficacy of Bayesian optimization in the pursuit of optimal stacking patterns of 2D materials and suggest that highly efficient searches within multilayer structures are also feasible.

[1] H. Ago et al, STAM 23, 275-299 (2022).
[2] S. Haastrup et al, 2D Materials, 5, 042002 (2018)
[3] M. N. Gjerding et al, 2D Materials, 8, 044002 (2021)