  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.
