why beta drops at higher current for a bipolar

Beta drops off at high
currents because the electron concentration in the base-collector depletion region
becomes comparable to the background dopant ion concentration, leading to
a dramatic increase in the effective width of the base. This is called the Kirk
effect or base pushout. As a result, the base resistance is current dependent.
Another effect is emitter crowding, which comes about because of the distributed
nature of parasitic resistance at the base contact, causing the base-emitter voltage
to be higher close to the base contact. This results in the highest current density
at the edge of the emitter. In the other extreme, at low currents,Beta may be
reduced due to the excess current resulting from recombination in the emitterbase
depletion region.