Current transport over parabolic potential barriers in semiconductor devices

Current transport over the potential barrier of an n⁺-p-n⁺ structure is studied using the diffusion theory and the thermionic-diffusion theory of current transport. Thermionic emission over the barrier is shown to be the asymptotic isothermal diffusion current. The J-V characteristics are derived for both the diffusion and thermionic-diffusion models. In particular, when the Bethe and the thermionic-diffusion (T-D) models are compared, both with and without backscattering effects, it is seen that the T-D model with backscattering is preferable to the Bethe approach and requires a relatively lower dopant concentration to be applicable. It is shown that two characteristic velocities are needed for the transport analysis: an effective collection velocity to terminate the region in which current is driven by diffusion and an emission velocity associated with carrier injection beyond the potential energy maximum. For a typical situation, the emission velocity can be as much as a factor of four greater than the collection velocity, showing that the velocity of injected carriers beyond the maximum can appreciably exceed the scatter-limited velocity