Fermi level in a nonuniformly doped semiconductor

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TL;DR

How does Fermi level constancy play out in a nonuniformly doped semiconductor?

Fermi level is known to be constant in a equilibrium state. It is also known to vary according to the number of donors/acceptors. In a nonuniformly doped semiconductor that has varying number of donors/acceptors at different position, how is the fermi level decided? Is it the average number of donor/acceptors across the semiconductor that is taken into account?

Thanks in advance.

 

[Baluncore]

https://en.wikipedia.org/wiki/Fermi_level#Terminology_problems
"In fact, thermodynamic equilibrium guarantees that the Fermi level in a conductor is always fixed to be exactly equal to the Fermi level of the electrodes; only the band structure (not the Fermi level) can be changed by doping or the field effect".

 

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https://en.wikipedia.org/wiki/Fermi_level#Terminology_problems
"In fact, thermodynamic equilibrium guarantees that the Fermi level in a conductor is always fixed to be exactly equal to the Fermi level of the electrodes; only the band structure (not the Fermi level) can be changed by doping or the field effect".

Yes but I learned that Fermi level increases (as in, it gets closer to the conduction band energy) when the semiconductor is n-type doped (donor doped) and decreases when it's p-type doped (acceptor doped). What I'm curious about is, in the case of a nonuniformly doped semiconductor (that has different amount of donor doping at different position within the semiconductor) how is the Fermi level decided?

As you have stated, Fermi level is constant throughout a semiconductor in an equilibrium state. However, different doping levels at different positions means that the Fermi level should be different at varying positions. So here, is the Fermi level decided using the average doping level of the whole semiconductor? Or is there some other way to solve this problem?

Thanks.

 

[phyzguy]

It depends. The free carriers will move until the Fermi level is constant. If you have a charge neutral semiconductor, then the number of free carriers is equal to the number of fixed dopant atoms. As the carriers move around, the number of free carriers doesn't change. However, suppose you add or remove free carriers. Then the Fermi level will move. So the total number of free carriers will determine where the Fermi level is. So solving the problem requires integrating over the whole region where the Fermi level is constant.