A new framework for atom-resolved decomposition of second-harmonic generation in nonlinear-optical crystals

Abstract

In this work, we develop a new framework for computing atom-resolved contributions to optical properties based on atoms-in-molecules (AIM) schemes. The formalism is independent of the specific AIM method and is made rigorous by partitioning momentum matrix elements into atomic contributions while exactly satisfying the relevant sum rules. We apply it to second-harmonic generation (SHG) in six representative UV and deep-UV nonlinear-optical crystals, namely $β$-\ce{BaB2O4} (BBO), \ce{LiB3O5} (LBO), \ce{CsB3O5} (CBO), \ce{CsLiB6O10} (CLBO), \ce{KBe2BO3F2} (KBBF), and \ce{LiCs2PO4} (LCPO). The atom-triplet decomposition reveals a clear hierarchy for the largest SHG component of each crystal. In general, two-center terms provide the leading contribution, one-center terms remain comparatively small, and fully three-center terms supply an important secondary contribution. A motif-triplet decomposition further indicates behavior dominated by the anionic framework in KBBF and LBO. In BBO, CBO, and CLBO, contributions from the anionic framework and the cation sublattice act cooperatively, although the cation contribution is crystal dependent. Moreover, cooperative contributions from the phosphate framework and the Cs sublattice are also observed in LCPO, where the O-Cs contribution is particularly significant. These results may provide a new perspective for understanding the microscopic origin of SHG in nonlinear-optical materials.

Figures (6)

• Figure 1: AIM charges of the symmetry-inequivalent atomic sites in BBO, KBBF, CBO, CLBO, LBO, and LCPO. Each marker corresponds to one symmetry-inequivalent site and is distinguished by element type.
• Figure 2: Percentage contributions of the unordered atom-triplet terms to the largest SHG component of each compound, obtained with scheme-N without explicit enforcement of Kleinman symmetry. The total response is decomposed into the one-center term $\chi^{abc}_\mathrm{1c}$, the two-center term $\chi^{abc}_\mathrm{2c}$, and the three-center term $\chi^{abc}_\mathrm{3c}$. The corresponding percentages, defined with respect to the largest SHG component, are denoted by $s^{abc}_\mathrm{1c}$, $s^{abc}_\mathrm{2c}$, and $s^{abc}_\mathrm{3c}$.
• Figure 3: Representative unordered atom-triplet contributions to the SHG coefficient of KBBF, obtained with scheme-N without explicit enforcement of Kleinman symmetry. Panel (a) shows the interatomic distance matrix in Å, and panel (b) shows the atom-triplet contributions to the largest component $\chi^{xxx}$ in pm/V. The dashed lines in both panels separate blocks associated with different atomic species. In panel (b), diagonal entries represent on-site $\{A,A,A\}$ terms, whereas off-diagonal entries correspond to off-site $\{A,A,B\}$ terms. Here, the "repeated atom $A$" denotes the atom that appears twice in $\{A,A,B\}$, whereas the "single atom $B$" denotes the atom that appears once.
• Figure 4: Representative unordered atom-triplet contributions to the SHG coefficient of BBO, obtained with scheme-N without explicit enforcement of Kleinman symmetry. Panel (a) shows the interatomic distance matrix in Å, and panel (b) shows the atom-triplet contributions to the largest SHG component $\chi^{xxy}$ in pm/V. The dashed lines in both panels separate blocks associated with different atomic species. In panel (b), diagonal entries represent on-site $\{A,A,A\}$ terms, whereas off-diagonal entries correspond to off-site $\{A,A,B\}$ terms. Here, the "repeated atom $A$" denotes the atom that appears twice in $\{A,A,B\}$, whereas the "single atom $B$" denotes the atom that appears once.
• Figure 5: Percentage contributions of the motif-triplet classes to the largest SHG component of each compound, obtained with scheme-N without explicit enforcement of Kleinman symmetry. The total response is decomposed into the one-element term $\chi^{abc}_\mathrm{1e}$, the two-element term $\chi^{abc}_\mathrm{2e}$, and the three-element term $\chi^{abc}_\mathrm{3e}$. The corresponding percentages, defined relative to the net SHG response of the tensor component under consideration, are denoted by $s^{abc}_\mathrm{1e}$, $s^{abc}_\mathrm{2e}$, and $s^{abc}_\mathrm{3e}$.