Second Harmonic Generation Susceptibilities from Symmetry Adapted Wannier Functions

Elucidating the orbital level origin of second harmonic generation (SHG) in materials and identifying the local contributions is a long-standing challenge. We report a first principles approach for the SHG where the contributions from individual orbitals or atoms can be evaluated via symmetry adapted Wannier functions without semiempirical parameters. We apply this method to the common SHG materials ${\mathrm{KBe}}_2{\mathrm{BO}}_3{\mathrm{F}}_2$, ${\mathrm{KCaCO}}_3\mathrm{F}$, and $\beta \text{−}{\mathrm{BaB}}_2{\mathrm{O}}_4$, and show that the orbitals on noncentrosymmetric sublattices are responsible for SHG effect and the energies of these orbitals control the magnitude.

Figure 1

Band structures of KBBF, ${\mathrm{KCaCO}}_3\mathrm{F}$, and $\beta \text{−}\mathrm{BBO}$. Solid lines: DFT bands; dots: Wannier-interpolated bands.

Figure 2

$s{p}^2$-3-like Wannier orbitals of oxygens (red) in (a) KBBF, (b) ${\mathrm{KCaCO}}_3\mathrm{F}$, and (c) some terminal oxygens of $\beta \text{−}\mathrm{BBO}$. The triangles represent $C_3$ symmetries and the triangle with horns in (a) represent that there is a $3_1$ screw axis. The sublattice with $R\overline{3}m$ or$P6/mmm$ symmetries consisting of only F (gray) and Be (gray green) in (d) KBBF, as well as F and Ca (gray blue) in (e) ${\mathrm{KCaCO}}_3\mathrm{F}$. The colors of remaining atoms are B (green), C (black), K (purple), Ba (bright green).

Figure 3

Contribution to the SHG coefficients of Bloch states of KBBF, ${\mathrm{KCaCO}}_3\mathrm{F}$, and $\beta \text{−}\mathrm{BBO}$. Here energy is relative to the band edge.

Figure 4

Artificial crystal structures of (a) $\mathrm{Ca}{\mathrm{CO}}_3$, as well as (d) $\mathrm{Al}{\mathrm{PO}}_4$ and the $s{p}^2$-1- as well as $s{p}^2$-2-like orbitals in (b) $\mathrm{Ca}{\mathrm{CO}}_3\mathrm{F}$ and (c) artificial ${\mathrm{CaCO}}_3$. Dark blue and pink represent aluminum and phosphorus, respectively.