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 n
c
in 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
x
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.