Extended Tauc–Lorentz model (XTL) with log-normal distributed bandgap energies for optical permittivity in polycrystalline semiconductors

An extended Tauc–Lorentz model is proposed to incorporate the bandgap variation in different grains in the polycrystalline semiconductors. The probability of a certain bandgap in the Tauc–Lorentz model is assumed to follow a log-normal distribution. After a Kramer–Kronig transform, the real part of this model is suggested as well. A comparison between this model and the experimental data in polycrystalline Si is carried out to validate this model. The experimental variation of grain size in the polycrystalline Si thin film can be correlated with the width of log-normal distribution of bandgap energies.