In a two-dimensional boron structure, the ordered high-concentration vacancy distribution can enhance structural stability and significantly modulates material properties. Based on recent experimental progress, herein we particularly focus on the two-dimensional boron structures with a striped distribution of hexagonal vacancies, in order to explore the formation of long-period boron structures.

Utilizing the structures of alloy generation and recognition (SAGAR) program developed by our group, we eliminate duplicate structures according to the structural symmetry to reduce computational cost. An effective model system is proposed to investigate the effect of vacancy distribution on the stability of the system, where the interactions between vacancies are utilized for estimating the total energy. By selecting structures with appropriate concentrations and combining first-principles calculations, the parameters in the model are fitted for different vacancy neighbor interactions, which can be further used to predict stable structures at various vacancy concentrations. The feasibility of model analysis is emphasized for structural screening, showing the good agreement between the parameterized model and the first-principles calculations.

Interestingly, under the same vacancy concentration, stable boron structures with different cell sizes exhibit distinct vacancy distributions, indicating a trend of long-period distribution for ground state structures. To address this phenomenon, when the stable candidate structures from the 1/6 series are dominant in number within the computable range and the changes in neighbor statistics can be clearly seen, we select the structures from this concentration series for detailed calculations.

The calculation results indicate that the convergence of the average energy is primarily influenced by the interaction between the fourth nearest neighbor and the sixth nearest neighbor. When considering only these two neighbors, the system energy changes with the increase of cell size as follows: the average energy of structures with a cell size being an even multiple of the minimum cell size keeps unchanged, while the average energy of structure with a cell size being an odd multiple of the minimum cell size gradually decreases, eventually converging to a stable value. When including the interactions between the ninth nearest neighbor and the tenth nearest neighbor, the average energy of structures with a cell size being an even times the minimum cell size also decreases gradually. The average energy decreases with oscillations, with the magnitude gradually diminishing and eventually stabilizing. This discovery reveals that the enhanced stability of long-period structures is attributed to the competitive interactions among different neighboring vacancies.