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From what I've studied and read, it doesn't really clear my concept as to why the collector current increases exponentially, more specifically Beta times the base current, when increased.

From my understanding, upon providing a positive voltage at the emitter with respect to the base the base-emitter junction is forward biased, so a large number of electrons flow into the base, where some recombine, giving rise to one component of the base current. On the other hand, the majority carriers (holes) in the base, drift to the emitter region, contributing to the second component of the base current.

So if anything, increasing the base current would imply more recombination and more drifting of holes from base to emitter region, how does that, in any way, affect the collector current?

This is what I got from youtube videos:

For an NPN transistor:

In an NPN transistor, increasing the base current (consisting of electrons) does not directly cause more electrons to be injected from the emitter into the base region. Instead, it affects the operation of the transistor in the following way:

The base current flows through the base-emitter junction, which is forward-biased. This biasing arrangement allows the base-emitter junction to act as a diode, permitting current flow.

As the base current increases, it establishes a higher level of electron concentration in the base region compared to the equilibrium concentration.

This higher electron concentration in the base narrows the depletion region around the base-emitter junction.

The narrower depletion region reduces the resistance for electron injection from the emitter into the base region.

Consequently, more electrons are injected from the emitter into the base region, forming the emitter current.

The injected electrons combine with the available holes in the base region, which are the majority carriers in the base.

Some of the injected electrons recombine with holes, and the remaining electrons that did not recombine constitute the base current.

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  • \$\begingroup\$ One answer is that 'everything in nature' related to the results of large numbers of states of matter lead inevitably to exponential results after integration. It pops up everywhere and can be seen as the result of the fact that nature isn't analog, but digital, in the sense that interactions are individual 'events' and not from some continuum. Interactions 'sample' reality. In between samples (events), known reality 'decays' exponentially. So it's expected unless other reasons impinge. \$\endgroup\$
    – periblepsis
    Commented Jun 11, 2023 at 21:24
  • \$\begingroup\$ Another answer is this is diffusion through a barrier. Decreases in the potential barrier permit carriers to diffuse more readily. And the diffusion process depends exponentially on the potential barrier. Finally, classical mechanics tried to explain thermodynamics and gave a mess, no explanation. Boltzmann tried. So did Gibbs. They came close. No cigar. Quantum mechanics was needed because statistical thermodynamics (large-N pop. statistics of matter) depends sensitively on atoms. And the Boltzmann factor unavoidably plays a role with two different states of matter in contact. \$\endgroup\$
    – periblepsis
    Commented Jun 11, 2023 at 21:26
  • \$\begingroup\$ Most early books on microelectronics don't get deeply into the details, preferring to cover as much as possible in reasonable time. Statistical thermodynamics covers the partition function, Boltzmann's factor, entropy, and so on. I believe that's a starting point. The earliest papers on semiconductor behavior -- dating back to the 1940's and 1950's -- should go into better detail. But I've not read them so I can't help there. It would be where I'd start if looking for a detailed understanding at the atomic interaction level. \$\endgroup\$
    – periblepsis
    Commented Jun 11, 2023 at 21:35

1 Answer 1

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The NPN transistor base and emitter form a diode. The emitter (which is heavily doped) has a LOT of charge carriers, and the base (lightly doped) has few.

So, hole current into the base (i.e. electrons removed from the base into the circuit that is the base drive power supply) biases a diode, and creates net current (not the equilibrium situation with zero base bias voltage). Most of that current is from the emitter (it has many MORE charge carriers than the base, and the electric field draws both) and while base (p-type carriers) current goes to the emitter, much of the emitter current (n-type carriers) goes through the base without recombining, rather is 'collected' in the transistor collector's circuit connection.

Maintaining the bias so that a constant collector current is sustained, requires continuous base current, roughly proportional to the collector current available. That 'rough' proportionality is not present if the collector bias doesn't draw the emitter's current swiftly out of the base region, i.e. beta (ratio of collector to base current) drops when collector voltage is not kept high (the transistor is said to be saturated).

As the base current increases, it establishes a higher level of electron concentration in the base region compared to the equilibrium concentration.

That's confusing; NPN transistor base current injects HOLES into the base, not conduction electrons, and holes don't recombine there, but after being drawn to the emitter, recombine in the emitter (or in the metal wire attached to the emitter...). Recombination of excess holes in the base is mainly from emitter-region electrons drawn to the positive potential of the base region, by the usual charge-in-an-electric field electric force.

In saturation, though, those injected base holes will recombine in the emitter or the collector or the base, until equilibrium is reached.

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  • \$\begingroup\$ @ Whit3rd I notice that you always talk about "bias" without using the term "voltage". Why? What physical quantity do you mean with "bias"? For a pn-junction (diode) the exponential relationship between voltage and current is given by the Shockley equation. I think this equation is also valid for the bipolar transistor, isn't it? \$\endgroup\$
    – LvW
    Commented Jun 12, 2023 at 7:18
  • \$\begingroup\$ @LvW The usual bias conditions for transistor effect are the collector voltage (higher than Vbase, to avoid saturation) and the forward bias (out of the cutoff at zero or negative Vbe) of the base (Vbase greater than Vemitter). The Shockley equation only relates Vbe and Iemitter, leaving beta and base and collector current as unsolved variables. \$\endgroup\$
    – Whit3rd
    Commented Jun 12, 2023 at 19:37

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