BJT action is not easy to understand or express mathematically, so here is a short version.
The modus operandi is that of two p-n semiconductor junctions in *very close proximity* to one another. Let's use an npn device for this example. The base-emitter, or b-e junction gets forward biased, whereas the base-collector, or b-c junction gets reverse biased. Under these conditions an electric field exists in the b-c region such that positive charge in the said region would move from the collector to the base, and opposite for negative charge. In the b-e region, an electric field is also present such that positive charge will move from base to emitter and vice-versa for negative charge.
Because the collector is n-type material, and the base is p-type, very little collector current, only leakage Icbo, exists when the b-e junction is not forward biased, as the b-c jcn is reverse biased. When the b-e jcn is fwd biased, holes (positive charge) move from the base towards the emitter, and electrons (negative) move from the emitter towards the base. If the base region is very very thin, the following will occur. The electrons that have just been emitted by the emitter will enter the base region and encounter the strong electric field associated with the reverse biased b-c junction. The polarity of said field is such as to attract electrons from the base region into the collector. Thus emitter electrons cross the base region into the collector and become collector current.
If the forward bias on the b-e jcn is a time-varying, or *ac* signal, the emitter current will be time-varying as well. The motion of electrons from emitter towards base will vary in time with the ac signal at the b-e jcn. Likewise, these electrons will be attracted into the collector region as a function of time in accordance with the input signal. Thus the collector current is an amplified facsimile of the input signal.
What makes active devices (bjt, FET, vacuum tube, IGBT, etc.) so useful is that they provide both current and voltage gain. The signal variation at the b-e jcn consists of a small current and small voltage. By intentionally doping the p-type base with a much lower density of acceptor atoms vs. the n-type emitter doping with a high density of donor atoms, the hole density from base to emitter is quite small compared with the electron density from emitter to base, or base current Ib << emitter current Ie. Also, due to the logarithmic diode nature of the forward biased b-e p-n junction, the base to emitter voltage swing, Vbe, is very small compared with the collector terminal voltage swing. Thus a bjt provides a large value of both current gain and voltage gain.
To summarize, when the b-e jcn is forward biased, charges emitted from the emitter are drawn into the collector due to the strong attraction of the electric field present in the reverse biased b-c junction. By using light doping in the base, and heavy doping in the emitter, large values of current gain can be realized. Because of the diode junction formed at the b-e region, large current swings are accompanied by small voltage swings due to the logarithmic nature of the p-n junction. Large values of voltage gain are attainable as a result.
Claude