Determination of the conduction band non-parabolicity parameter of heavily doped germanium epsilon-near-zero thin films

The conduction band non-parabolicity parameter C plays a crucial role in the third-order optical nonlinearity in epsilon-near-zero (ENZ) materials. While experimental C values have been reported for transparent conducting oxides (TCOs), heavily doped germanium as an important ENZ material for mid-infrared applications lacks experimental values, with theoretically calculated values available for undoped Ge. We experimentally determined C for n-type Ge thin films by combining Hall measurement, reflection spectroscopy, and literature data spanning carrier concentrations from 2.0×1018 to 2.6×1020 cm−3. Unlike previous studies of TCOs that used electron effective mass at the Fermi level m*(Ef), we employed the average effective mass mavg* over the electron distribution in the conduction band for fitting experimental data and extracted C=0.867±0.301 eV−1. This value is significantly larger than theoretical predictions for undoped Ge (≈0.3 eV−1), which we attribute to doping-induced bandgap narrowing and epitaxial strain. Our work provides essential data for investigating nonlinear optical responses and designing mid-infrared ENZ devices based on heavily doped germanium.