Nonlinear electronic polarization and optical response in borophosphate BPO 4

The electronic structure, nonlinear electronic polarization induced by a static external electric field, and frequency dependent second-harmonic susceptibility tensor of the borophosphate ${\mathrm{BPO}}_{4}$ are studied by a first-principles calculation based on density-functional theory. Our calculated results show that the borophosphate ${\mathrm{BPO}}_{4}$ has a large band gap $\ensuremath{\sim}10.4$ eV, which is larger than the band gap of the widely used nonlinear optical crystal ${\mathrm{KBe}}_{2}{\mathrm{BO}}_{3}{\mathrm{F}}_{2}$. However, ${\mathrm{BPO}}_{4}$ also has a nonlinear coefficient ${d}_{36}=0.92$ pm/V at static limit, which also is larger than the nonlinear coefficient ${d}_{11}=0.47$ pm/V of ${\mathrm{KBe}}_{2}{\mathrm{BO}}_{3}{\mathrm{F}}_{2}$. The unexpected larger nonlinear coefficient of ${\mathrm{BPO}}_{4}$ can be interpreted by the relatively strong $s\text{\ensuremath{-}}p$ hybridization in ${\mathrm{BPO}}_{4}$, which can enhance the inter-band Berry connections, while the O $2p$ orbitals dominating valence bands in ${\mathrm{KBe}}_{2}{\mathrm{BO}}_{3}{\mathrm{F}}_{2}$ are very flat, resulting from weak $s\text{\ensuremath{-}}p$ hybridization.