Second-harmonic generation in noncentrosymmetric phosphates

Motivated by the discovery of more and more phosphates with relatively strong nonlinear optic effect, we studied the mechanism of the second-harmonic generation (SHG) effect in several phosphates by band model and first-principles calculations. When the energy of an incident photon is much smaller than the band gap of material, the SHG is almost frequency independent and determined by the combination of Berry connection and a symmetric tensor. The SHG effect in phosphates can be enhanced by the enhancement of orbital hybridization or the reduction of charge-transfer energy, which results in widened bandwidth of occupied state and reduced band gap in the electronic structure, respectively. By the first-principles calculation on the electronic structures of several phosphates—${\mathrm{BPO}}_4, {\mathrm{LiCs}}_2{\mathrm{PO}}_4, \beta -{\mathrm{Li}}_3{\mathrm{VO}}_4$, and $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4$—we interpreted the relatively strong SHG effect in ${\mathrm{LiCs}}_2{\mathrm{PO}}_4$ and $\beta -{\mathrm{Li}}_3{\mathrm{VO}}_4$ as the consequence of the reduced charge-transfer energy compared to their parent $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4$, while the enhanced SHG in ${\mathrm{BPO}}_4$ is resulting from enhanced orbital hybridization.

Figure 1

Calculated band structures of two-band model with different hopping parameters.

Figure 2

The calculated ${\mathrm{\Gamma }}_{zx}$ on $k_y=0$ plane with parameters $t_x=t_y=t_z=0.8$ eV (a), and $t_x=t_y=t_z=1.0$ eV (b).

Figure 3

The calculated photon energy dependent SHG ${\chi }^2{\left(2\omega \right)}_{zxx}$ with $t_x=t_y=t_z=t$.

Figure 4

Calculated band structure of $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4$ by first-principles calculation.

Figure 5

The calculated partial DOS of (a)$s$ orbitals and (b) $p$ orbitals in $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4, {\mathrm{LiCs}}_2{\mathrm{PO}}_4$, and ${\mathrm{BPO}}_4$.

Figure 6

Calculated partial DOS of Li (a), O (b), and V (c) in $\beta -{\mathrm{Li}}_3{\mathrm{VO}}_4$.

Figure 7

Calculated SHG of $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4, {\mathrm{LiCs}}_2{\mathrm{PO}}_4$, and ${\mathrm{BPO}}_4$ with SOS approximation.