Second-harmonic-generation (SHG) spectroscopy can be used as an all-optical probe of space-charge regions (SCR) in semiconductors such as $c \text{-} {\rm{Si}}$ by exploiting the electric-field-induced second-harmonic-generation (EFISH) effect. To do so accurately, a thorough understanding of the EFISH effect is needed, and here a detailed model is presented, taking into account the so-far neglected spatial extent of the SCR on the spectral shape of the EFISH contribution. This shows that the spatial extent causes a significant phase shift in the EFISH contribution ranging from $0.1 \pi \; {\rm rad}$ to $0.4 \pi \; {\rm rad}$ with decreasing SCR strength. This predicted phase shift was verified by dedicated measurements of the spectral SHG phase and intensity of a series of samples covering the typical range of SCR strengths. The spectral SHG response of these samples indeed showed the predicted phase shift in the EFISH contribution and demonstrated that the extent of the SCR cannot be neglected. The obtained insights can result in improved accuracy in the determination of built-in charges near a $c \text{-} {\rm{Si}}$ interface, which is of interest in the context of semiconductor devices using SHG spectroscopy. Moreover, this refined understanding of the EFISH effect is beneficial in SHG experiments on 2D materials with $c \text{-} {\rm{Si}}$ serving as a substrate of Si-based photonic devices.

中文翻译：

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c -Si中空间电荷区的空间范围对电场诱导的二次谐波产生效应的影响

二次谐波 (SHG) 光谱可用作半导体中空间电荷区 (SCR) 的全光学探针，例如$c \text{-} {\rm{Si}}$通过利用电-场致二次谐波产生 (EFISH) 效应。为了准确地做到这一点，需要对 EFISH 效应有一个透彻的了解，这里提出了一个详细的模型，考虑到迄今为止被忽略的 SCR 对 EFISH 贡献的光谱形状的空间范围。这表明空间范围导致 EFISH 贡献的显着相移，范围从$0.1 \pi \; {\rm rad}$到$0.4 \pi \; {\rm rad}$随着 SCR 强度的降低。这种预测的相移通过对光谱 SHG 相位和一系列样本的强度进行专门测量来验证，这些样本涵盖了典型的 SCR 强度范围。这些样品的光谱 SHG 响应确实显示了 EFISH 贡献的预测相移，并表明 SCR 的范围不能被忽略。获得的见解可以提高确定$c \text{-} {\rm{Si}}$界面附近的内置电荷的准确性，这在使用 SHG 光谱的半导体器件的背景下很有趣。此外，这种对 EFISH 效应的精细理解有利于对二维材料进行 SHG 实验，其中$c \text{-} {\rm{Si}}$作为 Si 基光子器件的基板。