Self-consistent drift-diffusion model of nanoscale impurity profiles in semiconductor layers, quantum wires, and quantum dots

We propose and model an experiment where impurity profiles in low dimensional structures can be controlled (during heat treatment) by an external parabolic potential defined by a variety of gate arrangements. At high temperatures the impurities are ionized and are able to move relatively quickly. After a realistic equilibrium time of typically one hour, the profiles are rapidly cooled such that the impurities are frozen in place. The model, which takes the electronic distribution as well as the mobile impurities into account results in a nonlinear Poisson equation. Similar models are widely used in semiconductor device theory where doping profiles are fixed. A parabolic potential in one, two, and three dimensions is applied to a semiconductor layer, a cylindrical quantum wire, and a spherical quantum dot, respectively. The impurity profiles are typically Gaussian shaped, where the distribution broadens with increasing temperature. The results demonstrate that the profile can be widely altered by changing the temperature, the average doping density, the size (radius), and the parabolic potential constant. The effect of parabolic confinement dimensionality on the diffusion is also studied. The temperature effect is studied up to a theoretical zero-temperature limit for which an analytic solution for the impurity profile is derived. The impurity profiles are sharper as the parabolic constant increases and the processing temperature is lowered. The processing time, however, increases exponentially as the temperature is lowered, and this must be considered in the practical situation.