Electron-velocity saturation at a BJT collector junction under low-level conditions

The consequences of electron-velocity saturation at the collector junction of an n-p-n biopolar junction transistor (BJT) are examined in a manner similar to that employed by Middlebrook twenty years ago. Dimensional shrinkage, especially in base thickness, that has occurred over this time interval causes a change in the prediction of this analysis from a negligible effect twenty years ago to a significant effect today, even in the low-level regime. However, a more detailed analysis considering the electron profile near the collector junction yields the somewhat surprising result that the linear portion of the base-region electron profile is quite unaffected by velocity saturation, in spite of the fact that the minimum electron density ncin the collector region can exceed by many orders of magnitude the equilibrium electron density in the base region. The reason for this is a scaling phenomenon wherein the linear-profile extrapolation rotates about a point on the x axis that is invariant with respect to current throughout the low-level range and approximately invariant with respect to base-region doping. Furthermore, this point on the axis is very close to the depletion-approximation boundary of the collector-junction space-charge layer. Hence, the classical assumption of vanishing electron density at the boundary of the collector space-charge region constitutes an excellent approximation for low-level conditions. Incidental to the detailed analysis are updated empirical expressions for electron velocity saturation, and an application of the general solution for step junctions that was offered recently.