Long wavelength absorption in crystalline silicon

Sub-gap absorption in c–Si arises from two distinct mechanisms. The first is electronic bandgap narrowing (BGN), denoted by ΔE_g (in eV). This allows c–Si to absorb sub-gap photons with energies less than E_g but above (E_g − ΔE_g), where E_g is the bandgap energy (in eV) of unperturbed c–Si. In Sentaurus, ΔE_g is estimated using Schenk’s model⁴⁰ and leads to a slight drop in V_OC. The second mechanism allows c–Si to absorb photons with energies less than (E_g − ΔE_g) and originates from the exponentially decaying Urbach tail below the continuum band edge^41,42. For non-crystalline solids, static disorder contributes to an exponential band tail of localized states below the electronic band edge. In c–Si, a similar tail of phonon-assisted optical absorption gives rise to mobile electron-hole pairs^43,44. The sub-gap absorption is characterized by an exponential of the form: α(ν) ~ exp[{hν − E_G(T)/E₀(T)}], where ν is the optical frequency, E_G(T) is the downshift of the continuum band edge corresponding to BGN and E₀(T) is the Urbach slope.

In order to model the sub-gap absorption, we take the frequency dependent dielectric constant of Si over the 1000–1200 nm wavelength range from⁵² and fit it to a sum of Lorentz oscillator terms: ε(ω)=ε∞+∑jΔεjωpj2(ωpj2−2iωγj−ω2). The fitting parameters ε_∞, ω_pj, Δε_j and γ_j, given in Table 6, are obtained using an open MATLAB program⁵³. We compare the absorption length λ/4πk (where k is the imaginary part of the refractive index) calculated form our fit and that obtained from⁵² in Fig. 13. The measured value of Urbach slope of c–Si at 300K is 8.5 ± 1.0 meV⁵⁴. In comparison to this experimental data, microscopic modeling of the optical-absorption edge due to acoustic and optical phonons yields a slope of 8.6 meV^43,44. The inset shows that the experimental data from⁵² exhibits an Urbach slope of 8.6 meV over the 1160–1190 nm wavelength range.

Fitting parameters for experimental Si dispersion data of⁵².

Fitting of absorption length of c–Si in 1000–1200 nm wavelength range with experimental data obtained from⁵². Fitting parameters are given in Table 6. Inset: Urbach slope exhibited by the experimental absorption coefficient.