I've written about these effects before, here. But not directly, just as an aside. These were discussed in older literature as "Region I," "Region II," and "Region III." Region I is pretty much guided by additional factors related primarily to the base current and Region III results from changes in the collector current.

"[On the Variation of Junction-Transistor Current-Amplification Factor with Emitter Current][1]," by W. M. Webster, Proc. IRE, Vol. 42, pp. 914-920, June 1954, is the seminal paper on this topic.


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In Region I, the decline in \$\beta\$ is due to three components which can be ignored in the other regions but cannot be ignored with low currents involved. These are:

 1. The formation of emitter-base surface channels (which can be reduced by the careful application of processing/manufacturing); and,
 2. the recombination of surface carriers (which *also* can be reduced by the careful application of processing/manufacturing, but still remains a dominant part of the problem); and,
 3. the recombination of carriers in the emitter-base space-charge layer.

All three of these have similar variations with the base-emitter voltage so that you wind up with something akin to the following typical component equations:

$$\begin{align*}
I_{B_{channel}}&=I_{SAT_{channel}}\cdot\left(e^\frac{V_{BE}}{4\cdot V_T}-1\right)\\\\
I_{B_{surface}}&=I_{SAT_{surface}}\cdot\left(e^\frac{V_{BE}}{2\cdot V_T}-1\right)\\\\
I_{B_{space-charge}}&=I_{SAT_{space-charge}}\cdot\left(e^\frac{V_{BE}}{2\cdot V_T}-1\right)
\end{align*}$$

Although summed exponentials are not exactly equivalent to any single resulting equivalent exponential, it is practical (and done) to combine the above into a single modeled exponential that uses \$\eta_{EL}\$ values often close to 2:

$$\begin{align*}
I_{B_{summed}}&=I_{SAT_{summed}}\cdot\left(e^\frac{V_{BE}}{\eta_{EL}\cdot V_T}-1\right)
\end{align*}$$

For most BJTs, the above equation can be made to approximate the reality well enough for practical purposes (and it sums into the usual current equations.)


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In Region III, the injection of minority carriers into the base region starts becoming increasingly important in comparison against the majority carrier concentrations. Because the space-charge neutrality is maintained in the base, the majority concentration has to increase by the same amount.

The finding is:

$$\begin{align*}
I_{C_{high-I_C}}&\propto e^\frac{V_{BE}}{2\cdot V_T}
\end{align*}$$

The other factor in Region III is, of course, an 'Ohmic resistance' and is already modeled as \$r_c\$ so it isn't included above.

A model constant is usually applied to the above equation and the resulting term then appears in the divisor used for the usual model saturation current equation.

  [1]: http://ieeexplore.ieee.org/abstract/document/4051728/