Abstract
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 $\boldsymbol{\chi}(-2\omega;\omega,\omega)$. 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 $\epsilon_{v}(\omega)=1$, 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 $\epsilon_{\ell}(\omega)$, and it is in this layer where the SSHG takes place. We consider the position of $\boldsymbol{\chi}(-2\omega;\omega,\omega)$ within this surface layer. The third layer is the bulk region denoted by $b$ and characterized by $\epsilon_{b}(\omega)$. 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 $\boldsymbol{\chi}(-2\omega;\omega,\omega)$ for the Si(111)(1$\times$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 \emph{ab initio}.
Highlights
Surface second-harmonic generation (SSHG) has been shown to be an effective, non-destructive, and non-invasive probe to study surface and interface properties (Chen et al, 1981; Shen, 1989; McGilp et al, 1994; Bloembergen, 1999; Lüpke, 1999; McGilp, 1999; Downer et al, 2001a,b)
This coefficient accounts for the multiple (M) reflections of the fundamental field that depends on the thickness d of the layer l included in the phase φ = 4π(d/λ0)wl(ω), where λ0 is the wavelength of the incoming light, wl(ω) = (εl(ω)−sin2θ0)1/2, θ0 is the angle of incidence, and nl = (εl(ω))1/2
To better view the effects of the z-dependence of χ(zn) on the secondharmonic generation (SHG) yield, we will apply our formulation on a test surface
Summary
Surface second-harmonic generation (SSHG) has been shown to be an effective, non-destructive, and non-invasive probe to study surface and interface properties (Chen et al, 1981; Shen, 1989; McGilp et al, 1994; Bloembergen, 1999; Lüpke, 1999; McGilp, 1999; Downer et al, 2001a,b). On the basis of this approach for the calculation of χ(−2ω; ω, ω), in this article, we generalize the “three-layer model” for the SH radiation from the surface of a centrosymmetric material (Anderson and Mendoza, 2016) This model considers that the SH conversion takes place in a thin layer just below the surface of the material that lies under the vacuum region and above the bulk of the material. It is the three-layer model that allows us to integrate the effects of multiple reflections for both the SH and fundamental fields into the SSHG yield.
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