Second-order microscopic nonlinear susceptibility in a centrosymmetric material: application to imaging valence electron motion

TL;DR

This study demonstrates phase-matched nonlinear x-ray and optical sum-frequency generation in silicon, revealing microscopic charge and current responses, including second-order effects related to bond symmetry, supported by first-principles calculations.

Contribution

First measurement of second-order microscopic nonlinear susceptibility in a centrosymmetric material using combined x-ray and optical techniques.

Findings

First- and second-order sidebands observed at the 220 Bragg peak.

Efficiency peaks when the optical field aligns with the reciprocal lattice vector.

Second-order response shows anisotropy linked to bond symmetry.

Abstract

We report measurements of phase-matched nonlinear x-ray and optical sum-frequency generation from single-crystal silicon using sub-resonant 0.95 eV laser pulses and 9.5 keV hard x-ray pulses from the LCLS free-electron laser. The sum-frequency signal appears as energy and momentum sidebands to the elastic Bragg peak. It is proportional to the magnitude squared of the relevant temporal and spatial Fourier components of the optically induced microscopic charges/currents. We measure the first- and second-order sideband to the 220 Bragg peak and find that the efficiency is maximized when the applied field is along the reciprocal lattice vector. For an optical intensity of $\sim10^{12} \text{W}/\text{cm}^2$, we measure peak efficiencies of $3\times 10^{-7}$ and $3\times 10^{-10}$ for the first and second-order sideband respectively (relative to the elastic Bragg peak). The first-order sideband is consistent with induced microscopic currents along the applied electric field (consistent with an isotropic response). The second-order sideband depends nontrivially on the optical field orientation and is consistent with an anisotropic response originating from induced charges along the bonds with C$_{3v}$ site symmetry. The results agree well with first-principles Bloch-Floquet calculations.