Triangular Vacancy Structures in Hexagonal Boron Nitride: Size‑Dependent Stability, Magnetism, and Edge‑Pair Missing Effects

Abstract

Using density functional theory (DFT), we examined the atomic, electronic, and magnetic behavior of triangular vacancies in single‑layer hexagonal boron nitride (h‑BN). Our calculations reveal that the relaxed geometry of a triangular vacancy depends strongly on its size and that all stable configurations possess N‑terminated zigzag edges. When a boron‑nitrogen (BN) pair is removed from the vacancy edge, the resulting structure’s stability and magnetic moment vary with the missing‑pair location; the most favorable configuration occurs when the pair is removed from the face region, producing a minimal magnetic moment.

Background

Hexagonal boron nitride (h‑BN) is a two‑dimensional insulator analogous to graphene, but composed of alternating B and N atoms. During synthesis, h‑BN inevitably contains point defects—particularly vacancies—which alter its electronic and mechanical properties and, consequently, the performance of h‑BN‑based nanodevices ^1,2. Because the lattice contains two atom types, vacancy edges can be N‑terminated or B‑terminated; the former is energetically preferred and often adopts a zigzag motif ^3,4. Prior theoretical work has shown that triangular vacancy stability and magnetism depend on edge termination and vacancy size ^3–11. Experiments have confirmed triangular vacancy formation under electron‑beam irradiation, with vacancies growing while maintaining their triangular shape ^12–15.

In a recent study, we observed that h‑BN atoms are ejected in bundles rather than individually at vacancy edges, suggesting a cooperative mechanism in vacancy growth ¹⁵. This article expands on that observation by providing a comprehensive DFT analysis of triangular vacancy structures, their size‑dependent relaxation, and the influence of BN‑pair removal.

Computational Methods

All calculations were performed with VASP (Vienna Ab‑initio Simulation Package) using a plane‑wave cutoff of 400 eV and projector‑augmented wave (PAW) pseudopotentials ^18,19. Exchange‑correlation energies were treated within the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof, augmented by Grimme’s DFT‑D2 van der Waals correction to capture dispersion interactions ^20,21,22. Atomic geometries were relaxed until residual forces fell below 0.01 eV/Å; Brillouin‑zone sampling employed only the Γ‑point for the (9×9) and (15×15) supercells used. The calculated lattice constant of 2.56 Å matches the experimental value ²³.

Results and Discussion

Triangular Vacancies in h‑BN

We investigated several triangular vacancy sizes, focusing on N‑terminated edges because they are more stable than B‑terminated ones ^3,4. Two distinct relaxed geometries emerged for N‑terminated triangles: the symmetric N‑symm configuration, in which the vertex atoms remain largely unaltered, and the distorted NN‑bond configuration, where N–N homopolar bonds form at each vertex. For the smallest vacancy (V_1B), only the N‑symm structure is found, with a 2.66 Å N–N distance—slightly expanded relative to the pristine lattice (2.48 Å). As the vacancy size increases to V_3B+1N and V_6B+3N, both N‑symm and NN‑bond configurations are locally stable, in agreement with previous studies ⁶.

For larger vacancies (V_10B+6N and beyond), the NN‑bond configuration becomes the sole stable geometry. In this regime, the B–N bonds around the hole remain essentially unchanged, while N–N bonds persist at the vertices. Table 1 (not reproduced here) shows that the energy difference between N‑symm and NN‑bond structures diminishes with increasing vacancy size, reflecting the reduced influence of vertex reconstruction on the overall lattice.

The magnetic moments of these vacancies correlate with edge atom terminations. In N‑symm structures, each dangling N atom contributes one μ_B, yielding moments equal to the number of edge N atoms. In contrast, NN‑bond structures exhibit reduced or enhanced magnetism depending on whether the N–N bonds quench or preserve the dangling‑bond character. For example, V_3B+1N, V_6B+3N, and V_10B+6N in the NN‑bond state possess moments of 0, 3, and 6 μ_B, respectively.

BN‑Pair Removal at Vacancy Edges

Experimental observations indicate that vacancy growth often proceeds via the ejection of BN pairs from the edge ^14,15. To emulate this, we removed a BN pair from various edge sites of a large V_21B+15N vacancy (15×15 supercell). Six distinct missing positions were examined: two corners, two near‑corner faces, and two middle faces. After relaxation, the optimized geometries fall into three classes—corner‑missing, face‑missing, and middle‑missing—each exhibiting characteristic reconstruction.

Corner‑missing structures show minimal change except for a slightly distorted BN ring. In contrast, face‑missing structures form an N dimer with a pentagonal ring adjacent to the vacancy, and the N dimer significantly influences both stability and magnetism. The relative energies reveal that configurations with missing pairs closer to the triangle’s center are energetically favored (Table 2). Magnetic moments vary from 12 μ_B for corner‑missing to 9 μ_B for certain face‑missing cases, reflecting the redistribution of unpaired electrons.

Spin‑density maps (Figure 3) confirm that the unpaired spins are localized on the edge N atoms, particularly in the distorted BN rings and N dimers. Density‑of‑states analyses (Figure 4) show defect states within the band gap of pristine h‑BN, with spin‑asymmetric features around the Fermi level. The resulting band gaps for the three representative missing‑pair configurations are 0.35, 0.24, and 0.36 eV.

Conclusions

Our first‑principles study demonstrates that triangular vacancy stability and magnetism in h‑BN are governed by vacancy size, edge termination, and the precise location of BN‑pair removal. Large vacancies preferentially adopt an NN‑bond configuration at the vertices, which dictates their magnetic moments. When a BN pair is removed from the edge of a large triangular hole, the most stable outcome is a face‑missing structure that forms N–N bonds, with magnetic properties and local density of states sensitive to the missing position. These insights provide a microscopic foundation for tailoring defect‑engineered h‑BN nanostructures in electronic and spintronic devices.