Strain effects on Phase-Filling Singularities in Highly Doped n-Type Ge

Recently, Chi Xu et al. predicted the phase-filling singularities (PFS) in the optical dielectric function (ODF) of the highly doped $n$-type Ge and confirmed in experiment the PFS associated $E_{1}+\Delta_{1}$ transition by advanced \textit{in situ} doping technology [Phys. Rev. Lett. 118, 267402 (2017)], but the strong overlap between $E_{1}$ and $E_{1}+\Delta_{1}$ optical transitions made the PFS associated $E_{1}$ transition that occurs at the high doping concentration unobservable in their measurement. In this work, we investigate the PFS of the highly doped n-type Ge in the presence of the uniaxial and biaxial tensile strain along [100], [110] and [111] crystal orientation. Compared with the relaxed bulk Ge, the tensile strain along [100] increases the energy separation between the $E_{1}$ and $E_{1}+\Delta_{1}$ transition, making it possible to reveal the PFS associated $E_{1}$ transition in optical measurement. Besides, the application of tensile strain along [110] and [111] offers the possibility of lowering the required doping concentration for the PFS to be observed, resulting in new additional features associated with $E_{1}+\Delta_{1}$ transition at inequivalent $L$-valleys. These theoretical predications with more distinguishable optical transition features in the presence of the uniaxial and biaxial tensile strain can be more conveniently observed in experiment, providing new insights into the excited states in heavily doped semiconductors.