Second harmonic generation in centrosymmetric multilayered structures: Theoretical approach for nonlinear boundary conditions

Second harmonic generation (SHG) is used to characterize the interfaces of centrosymmetric materials typically used in microelectronic/optoelectronic devices. For such applications, the materials are actually multi-layer stacks, and in this case, the SHG can be difficult to interpret and model. This paper presents the theory of the second harmonic light generated from multilayer structures. The focus is on describing the nonlinear boundary conditions at the interfaces between two different materials, taking into account the distinct contributions of bulk and interface regions. Using these conditions, it is possible to calculate the second harmonic signal from any stack of materials. In this paper, we address stacks containing silicon (100) because it is a material with numerous applications. The nonlinear polarization expressions of the surface and bulk of Si(100), according to Sipe et al. [Phys. Rev. B 35, 1129 (1987)], were integrated into nonlinear boundary conditions in order to determine transmitted and reflected second harmonic waves. An analytical validation was performed on the simple case of an air-silicon interface. For multilayered stacks, the model was compared with experimental results obtained on samples corresponding to pragmatic substrates widely used in microelectronic and optoelectronic applications.