For two-dimensional layered hexagonal boron nitride (2D-hBN) materials, various defects are inevitably generated during experimental synthesis. Defects directly affect the electronic structure of the material and thus influence the optical properties of the material. The defect-related electronic states cause the red-shift of the linear absorption spectrum from the deep ultraviolet region of pristine 2D-hBN to the ultraviolet?visible region of defective 2D-hBN. Meanwhile, defects also have a significant effect on the nonlinear optical properties of the material. Recent experiments have shown that defects increase the nonlinear second-harmonic generation (SHG) coefficient of 2D-hBN by an order of magnitude. These findings suggest that defects can be used to tune the nonlinear optical properties of materials. To use defects to tail the optical properties of materials, it is of great significance to reveal their influence on optical properties from a microscopic view. In this study, we attempt to study the influence of point defects on the SHG coefficient of monolayer 2D-hBN (ML-BN). The sum-over-states method is used to study the microscopic mechanism of the enhancement of SHG coefficients and to reveal the relationship between the enhanced SHG coefficients and defect states.

We create three vacancy-related defective structures in the 5×5 supercell of ML-BN (Fig. 1). The PBE functional of generalized gradient approximation (PBE-GGA) combined with the norm-conserving pseudopotential plane-wave method is used to optimize all defective structures. We use a 3×3×1 k grid, a force threshold of 0.01 eV/?, and a pressure threshold of 0.02 GPa for optimization. The relaxation of the unit cell is included during the optimization process and a vacuum spacing greater than 15 ? is used to ensure negligible interlayer interactions. The PBE-GGA combined with the norm-conserving pseudopotential plane-wave method is used to calculate the energy band structure of three defective structures. A k-grid of 24×24×1 and a kinetic energy cutoff of 60 Ry are used in the calculations. The linear and nonlinear optical properties are calculated within the independent particle approximation (IPA). The linear one-photon absorption (OPA) ?₂(ω) and the SHG coefficient χ_zyx(-2ω;ω,ω) are calculated by the sum-over-states expressions Eqs. (1)?(3). A 24×24×1 k grid and 200 IPA empty states (300 states in total) are used to obtain the converged SHG spectra within 4 eV. Hybrid GGA-HSE06 calculations are performed to correct the energy band structure used in optical calculations.

The OPA of V_N almost overlaps with that of ML-BN in the input photon energy range from 6 eV to 8 eV and shows a characteristic absorption peak at about 3.3 eV (Fig. 4). The reason for the overlap is that the absorption peaks after 6 eV are mainly transitions between intrinsic states. The defects result in a characteristic absorption peak at about 3.3 eV. This characteristic absorption peak mainly originates from the electronic transition from the defect band 98 to the intrinsic conduction band (101, 102, and 103). The B-atom vacancy defect leads to three OPA characteristic absorption peaks (Fig. 5). These three characteristic absorption peaks are mainly formed by the transition of electrons from the valence band to three defect states. V_BN has two characteristic absorption peaks at the same position in two directions. The transition of electrons from the intrinsic valence band and the defect state at the valence band edge to the defect state 97 produces these two characteristic absorption peaks (Fig. 6). For both the real part Re[χxxx(2)(ω)] and the imaginary part Im[χxxx(2)(ω)] of V_N, pure interband transitions and intraband transitions have similar contributions with opposite signs, which leads to relatively small SHG coefficients in the entire 4.0 eV range except for the near-resonance peak (Fig. 7). The SHG enhancement peaks within 4 eV are caused by single-photon or two-photon resonance, or both. The two-photon resonance process related to the defect bands plays a major role in the enhancement of the SHG coefficient. For V_B and V_BN, contributions of defect bands to the enhancement of the SHG coefficient are also observed and the defect bands are located by tracing the sum-over-states process (Figs. 8 and 9). In the range of 1.0 to 4.0 eV, the defective structure has a significant enhancement in SHG coefficient relative to native ML-BN (Fig. 10). Although it is difficult to directly compare the theoretical and experimental results quantitatively, the enhancement in the SHG coefficient caused by the defect state is consistent qualitatively between the theory and the experiment.

The defect unit leads to the defect states in the energy band gap of native ML-BN, which causes a red shift of the OPA spectrum from the deep ultraviolet region to the ultraviolet?visible region. Tracing the sum-over-states process confirms the main contribution of defect states to the absorption peak in the UV-visible region. In the visible light region, the enhancement of the SHG coefficient is caused by single-photon or two-photon resonance, or both. Compared with the SHG of native ML-BN, the SHG of the defective structure has an obvious enhancement trend, in good agreement with recent experimental results. Our results are applicable to other defective structures such as doping and surface defects not discussed here because these defects have similar effects on the OPA and ultimately on the SHG.