Lumped models for planar, graded-base transistors

Lumped models of the type proposed by Linvill to represent graded-base transistors are particularly useful because they portray the relationship between terminal behavior and physical processes, and because they are applicable in both small- and large-signal situations. Previous treatments of lumped models for graded base transistors have used an exponential approximation for the impurity distribution in the base, although in fact most present-day high-performance planar transistors have a gaussian distribution. The exponential approximation has previously been employed because it leads to a constant electric field in the base region, and the resulting equations for carrier behavior are easy to solve. In this paper we present a method for obtaining lumped models for the cases in which the impurity profiles are either gaussian or complementary error function. The models are systematically derived from a consideration of the geometrical properties and the dominant physical processes of the device. Charts are presented which enable one to determine the element values and weighting factors for a model of any desired complexity for a transistor of arbitrary base width whose base region is located at an arbitrary depth below the planar surface. A chart is also given for determining the alpha cutoff frequency as a function of base width and depth below the surface.