licts a cross-over point of the hot-spot and second-breakdown loci, which was observed experimentally [114], and provides a basis for explaining the experimentally observed thermal hysteresis of the transistor current gain [115].

The model is based upon the concept of a stability factor [116] which accounts for the thermal-electrical feedback mechanisms of transistors. The expression for the stability factor which determines initiation of thermal instability is derived assuming that the current distribution throughout the device is uniform prior to instability and that

Ic is related to the emitter-base voltage,

V

C

BE'

by the Ebers-Moll equation [117]. Once instability has occurred, the model accounts for two phenomena which occur as the current constricts to the hot-spot area. First, the current density becomes large enough that current crowding (to the edges of the emitter fingers) occurs within the hot spot, and second, base widening occurs due to the large current densities involved. The effect of these two phenomena is to modify the expression relating I, to V, and thus also to BE modify the expression for the stability factor. The net result is that the device can restabilize in a stable hot-spot mode of operation and exhibit thermal hysteresis.

The quantitative predictions of device behavior in the stable hot-spot mode and the onset of second breakdown are not as accurate as those for the initiation of thermal instability primarily because additional simplifying assumptions for current crowding and base widening are contained in the model. However, while this portion of the model may require some refinements to give a more exact description of device parameters and characteristics in the stable hot spot, it does appear to provide a satisfactory picture of the physical behavior of the device for these conditions.

The newly automated infrared microradiometer proved to be extremely valuable in this study. In this automated facility, an interactive data acquisition system controls the steppermotor-driven X-Y stage upon which the device under test rests, acquires the output of the microradiometer through a digital voltmeter, and sends the results to a printer to be printed out in hard copy and paper tape. The thermal measurements were used to demonstrate the surprising fact that current crowding occurs within the hot spot. This was accomplished by noting that the temperatures along the emitter finger edges were higher than at their centers which can only be accounted for by current densities being higher at the edges than at the centers.