Depth-Dependent Three-Layer Model for the Surface Second-Harmonic Generation Yield - Anderson, Sean M.; Mendoza, Bernardo S. (Frontiers in Materials, 2017)
We present a generalization of the three-layer model to calculate the surface second harmonic generation (SSGH) yield, that includes the depth dependence of the surface nonlinear second order susceptibility tensor $\boldsymbol{χ}(-2ω;ω,ω)$. This model considers that the surface is represented by three regions or layers. The first layer is a semi-infinite vacuum region with a dielectric function $ε_{v}(ω)=1$, from where the fundamental electric field impinges on the material. The second layer is a thin layer ($\ell$) of thickness $d$ characterized by a dielectric function $ε_{\ell}(ω)$, and it is in this layer where the SSHG takes place. We consider the position of $\boldsymbol{χ}(-2ω;ω,ω)$ within this surface layer. The third layer is the bulk region denoted by $b$ and characterized by $ε_{b}(ω)$. We include the effects caused by the multiple reflections of both the fundamental and the second-harmonic (SH) fields that take place within the thin layer $\ell$. {\color{red} As a test case, we calculate $\boldsymbol{χ}(-2ω;ω,ω)$ for the Si(111)(1$×$1):H surface, and present a layer-by-layer study of the susceptibility to elucidate the depth dependence of the SHG spectrum. We then use the depth dependent three-layer model to calculate the SSHG yield, and contrast the calculated spectra with experimental data. We produce improved results over previous published work, as this treatment can reproduce key spectral features, is computationally viable for many systems, and most importantly remains completely ab initio.