When a system has a dopant, how much does the Fermi level shift?

For example, say a finite concentration of substitutional dopants replace some bulk atoms, and each has one extra electron. Ignore any changes in geometry. Say we do some DFT calculations that show the charge transfer to be 0.1 between the impurity and the bulk. Will the Fermi energy shift according to the new total number of electrons in the system, or as 0.1*number of electrons?

I am thinking the first one, as EF should be related to the total number of electrons in the system. However, with a small charge transfer could we not regard the remaining 0.9 of charge "bound" to the dopant?


Sometimes a dopant band is drawn in a band diagram to indicate that there are energy levels that can be filled by electrons, though they are bound levels. This full picture shows all the electrons, and so you could indicate somehow in the picture that 90% of the donors' electrons remain in their homes (side note: in order for this to occur, the donor band would lie mainly below the Fermi level). Sometimes the dopant band is not drawn and so those 90% of donor electrons are not shown.

Regardless, the figure "0.1 electron charge transfer" from a donor dopant means that the number of electrons in the conduction band has increased by 0.1 times the number of dopants. You can calculate the shift in energy levels based on that fact alone, because the energy difference between the conduction band minimum and the Fermi level is directly related to the number of carriers in the conduction band $n_C$:

$n_C = N_C \exp((E_F - E_C)/kT)$, where $N_C$ is a constant for a given material and temperature, at low doping levels

At high doping levels things get fairly complicated since the dopants do not quite act independently.


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