Variational calculation for the direct-gap exciton in the Ge quantum well systems

The variational method based on the two-dimensional (2D), three-dimensional (3D), and anisotropic 3D exciton models is used to investigate the ground-level direct-gap electron-heavy-hole exciton behaviors in the Ge quantum well systems, including an infinite Ge well case and a Ge/SiGe finite well case. The exciton radius, binding energy, and oscillator strength are calculated for various well thicknesses and bias voltages. The three exciton models are compared indicating that the dimensionality in the exciton model is nearly 2D for the thin finite well and 3D with anisotropic exciton radiuses for the infinite well and thick finite well. The exciton radius minimum and oscillator strength maximum occur at 1.6 nm well for the finite well case, thus proving that this Ge/SiGe quantum well system possesses strong quantum confinement, even with a thin-well thickness. Also, the effect of the conduction-band nonparabolicity effect on the exciton behavior is discussed. The variational calculation agrees well with the experimental results and other theoretical calculations in the 10 nm finite well case.