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Just worked my way through the temperature dependence of the carrier density in n-Si (doped with As(donor doping) and a dash of Ga (acceptor doping)).
The steep rise of the intrinsic carrier concentration and the extrinsic plateau where the donors are exhausted are straight forward. However, I noticed that several textbooks don't really go into detail why there is a kink in the freeze-out region under certain circumstances.
And That is where this thread comes in. Starting with an electrically neutral equilibrium state in a piece of homogenous semiconductor, I eliminated the Fermi energy and solved for the carrier concentration.
You can find more about the topic here if there is something you're not sure about.
As far as I understand the behavior of the system, the Fermi level lies in the center of the gap for high temperatures, drifting a little with 3/4 k*T ln(mh*/me*) as the conduction band becomes intrinsically populated, accounting for the band curvatures. For low temperatures, the Fermi level moves upwards, approaching the donor states. Now here's the tricky part: if no "minority doping" is present, the Fermi level comes to rest between the conduction band edge and the donor energy. With minority doping present at even some orders of magnitude less than the intentioned doping, the Fermi level is again pulled towards the donor level when the temperature approaches 0K because the acceptor states have stolen some electrons, leaving several donor levels empty even at low temperatures.
How do I calculate the Fermi energy over temperature? At some point, I made a Maxwell-Boltzmann approximation of the Fermi-Dirac distribution within the conduction band (this yields an analytical expression for the effective DOS depending on the effective conduction band mass and temperature) but I sense a violation of the mass action law in subtracting the intrinsic conduction band population from the total carrier density and deriving the Fermi energy from the degree of de-population of the donor states. Maybe it just works for lower temperatures?
Ok, finally, let's see some graphs. Here is what happens to the carrier density over temperature when the "minority dopant concentration" is varied. The exciting case is the 1E15 : 1E13 concentration as pointed out here . A kink can clearly be seen as the concentration passes the acceptor doping concentration. Now tell me, how does this happen in detail?
Oh, and here is another picture of the Fermi energy over temperature I am looking for - unfortunately, the "exciting case" is elegantly left unmentioned.
approaching compensation, the carrier concentration is lowered by an order of magnitude, the slope is steep (-Edonor / (k_B))
intermediate case, looking quite unspecific
note the kink in the freeze-out zone here:
the minority doping concentration is negligible here. Notice the flat slope ( -Ed/(2*k_B))
I attach the gnuplot script for the sake of completeness: ]compensation_effects.plt.txt[/file]
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Physics of doped Silicon: carrier density and fermi level
