Influence of the Spatial Extent of the Space-Charge Region in c-Si on the Electric-Field Induced Second-Harmonic Generation Effect

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.