Analytic resolution of Poisson–Boltzmann equation in nanometric semiconductor junctions

The design rules in nanoscale integrated circuits have now reached the depletion regions widths in semiconductors devices. In order to describe the complete behavior of junctions, we present the analytic resolution of Poisson–Boltzmann equation in n–p and n–p–n junctions in thermal equilibrium. The main point of this resolution is the exact determination of the boundary conditions at each n–p interface. In n–p–n junction, it appears that under a few hundred of nanometers, the positive potential of the n-type region can overlap the negative potential of the p-type region. This result appears in n–p–n (or p–n–p) structures when the intermediate region is less doped in regard to the opposite regions. An illustration in the evolution of nanoscale metal–oxide–semiconductor field effect transistor (MOSFET) is presented.