  We calculate the electronic structures of Germanium nanowires by taking the
effective-mass theory. The electron and hole states at the G-valley are studied
via the eight-band k.p theory. For the [111] L-valley, we expand the envelope
wave function using Bessel functions to calculate the energies of the electron
states for the first time. The results show that the energy dispersion curves
of electron states at the L-valley are almost parabolic irrespective of the
radius of Germanium nanowires. Based on the electronic structures, the density
of states of Germanium nanowires are also obtained, and we find that the
conduction band density of states mostly come from the electron states at the
L-valley because of the eight equivalent degenerate L points in Germanium.
Furthermore, the optical gain spectra of Germanium nanowires are investigated.
The calculations show that there are no optical gain along z direction even
though the injected carrier density is 4x1019 cm-3 when the doping
concentration is zero, and a remarkable optical gain can be obtained when the
injected carrier density is close to 1x1020 cm-3, since a large amount of
electrons will prefer to occupy the low-energy L-valley. In this case, the
negative optical gain will be encountered considering free-carrier absorption
loss as the increase of the diameter. We also investigate the optical gain
along z direction as functions of the doping concentration and injected carrier
density for the doped Germanium nanowires. When taking into account
free-carrier absorption loss, the calculated results show that a positive net
peak gain is most likely to occur in the heavily doped nanowires with smaller
diameters. Our theoretical studies are valuable in providing a guidance for the
applications of Germanium nanowires in the field of microelectronics and
optoelectronics.
