We predict the frequency-dependent bulk quadrupole contribution to second harmonic generation in silicon quantitatively from the linear susceptibility by means of a generalized classical anharmonic oscillator model and the simplified bond hyperpolarizability model. We show that in single-beam setups the main contribution is found for the silicon (111) surface, and only a minor contribution for the (001) and (011) facets. The dipole contribution obtained from our model is compared to literature values for SiC, AlAs and GaAs to demonstrate the viability of the method.
Even‐order nonlinear methods have proven useful tools for the analysis of centrosymmetric materials where the surface dipole contribution is often dominant. Quadrupole‐order contributions from the surface discontinuity and the bulk can be on the same order of magnitude however, yet have been largely ignored in modern numerical ab initio calculations. We find that the macroscopic bulk quadrupole term boldPfalse(2Qfalse)=χfalse(2Qfalse): E∇E can be related to the first order derivatives of the microscopic response function χfalse(2false)false(k,boldk′,boldk″false) at the Γ‐point and outline how a quantum‐mechanical expression can be derived for a periodic system.
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