Analytical model for extracting optical properties from absorptance of femtosecond-laser structured hyperdoped silicon

We set up an analytical optical model to emulate the absorptance spectra of light scattering, sulfur-hyperdoped silicon that we fabricate by using femtosecond laser pulses. The model allows us to distinguish between contributions to the absolute sub-bandgap absorptance from the path length enhancement of photons due to laser-induced surface roughness, on the one hand, and from the actual hyperdoped layer, on the other hand. Both effects are quantified via the two free parameters of the model. By varying the laser fluence and the areal pulse density, we create a range from almost planar to heavily structured hyperdoped Si samples that we show to behave almost like a Lambertian scatterer. The optical depth a1, i.e., the product of the absorption coefficient close to the Si bandgap energy and the effective thickness of the hyperdoped layer, scales with the surface area enhancement, which we identify as the main driving force for large sub-bandgap absorptances of this material type. It reaches maximum values of nearly a1 = 0.4, which refers to an absolute absorptance of 82% at a wavelength of 1450 nm. We furthermore discuss, quantify, and reduce possible error sources when determining the absorptance of such optically rough, hyperdoped samples with a spectrophotometer.