bipolar transistor: Ic

[Martin Czech]

Gray/Meyer, Analysis and Design of Analog Integrated Circuits, Wiley &
Sons ISBN 0 471 59984 0, says (in my understanding):

the collector current is basically described as:

Ic=K*exp{Vbe/Vt}

where K=q*A*D*ni^2/Qb

q: charge of an electron
A: cross sectional emitter area
Dn: electron diffusion "constant" (dependend on impurities)
ni: intrinsic carrier concentration
Qb: number of doping atoms in the base per unit area of emitter

next comes Mr. Early:

Ic=K*(1+Vce/Va)*exp{Vbe/Vt}

Va depends inversely proportional on base width (which is a very very
sensitive parameter)

next comes substitution for K (emitter current) and additional collector
reverse current:

Ic=alpha*Ies*(1+Vce/Va)*exp{Vbe/Vt}+Icsa

alpha is the emitter efficiency or base recombination
term (depends heavily on defects, impurities)

it is said that second order effects dominate for the actual reverse bias
collector current Icsa, this leakage current is serveral orders of magnitude
larger then the Ies or Ics current which is used in the forward equation.
Needless to say that Icsa depends heavily on impurities, defects,
stress and encapsulation.

So , from this we can see that the actual collector current depends
a whole lot on actual geometry, impurities, encapsulation, stress and
doping, things that are unique for every single transistor out of the box.
And there will be aging effects.

These factors dominate the "real Ics" which we extract out of our
measurement data, and this "real Ics" was asked for.

If we believe this argumentation, the "real Ics" depends on a lot 
of statistical parameters, and common centroid matching is the only
way to get something reasonable, without discrete matching.
And a good encapsulation.

But this is all first order (or maybe second order due to Early)
theory. Someone should really do some measurements.

m.c.