The theoretical direct-band-gap optical gain of Germanium nanowires.

We calculate the electronic structures of Germanium nanowires by taking the effective-mass theory. The electron and hole states at the Γ-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 diameters 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 4 × 10¹⁹ cm^-3 when the doping concentration is zero, and a remarkable optical gain can be obtained when the injected carrier density is close to 1 × 10²⁰ 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.