[Pw_forum] fermi level (XQ Wang)

[Lazaro Calderin]

Dear XQ Wang,

> My brain doesn't work. Any help to remove my understanding to fermi energy will be deeply appreciated.

May be mine neither ;-) ... but this is my understanding of the Fermi
energy .. which could be wrong of course. 

Usually the Fermi energy is introduced in the context of the free
electron (gas) and nearly free electron models and it is defined as the
highest occupied electron state at T=0K. Based on those models some
electronic properties are calculated at non-zero T and what should be
called the chemical potential is often still called Fermi energy. 

The chemical potential (\mu) is that number that constrains the integral
over the energy (E) of the Fermi-Dirac distribution (f) at a given
temperature (T) times the density of states (n), to be equal to the
number of electron (N) in the system. That is:

\int_{-\infty}^\infty n(E) f(E,T,\mu) dE = N                (1)

The chemical potential goes to the Fermi energy as the temperature
approaches 0K. But also, for metals the chemical potential usually
equals the Fermi energy at non-zero temperature because there are
abundant unoccupied states just above and all the way from the Fermi
energy up. I guess that is why we still call Fermi energy to the
chemical potential.

In semiconductors there are no empty states just above the highest
occupied one (top of the valence band). Therefore at non-zero
temperature the chemical potential is displaced to energies higher than
the top of the valence band, taking values usually in the energy gap and
therefore it can not be equal to the energy of any allowed electron
state. As for metals, we often call Fermi energy to what should be
called chemical potential. But while for metal it is fine to interchange
the terms chemical potential and Fermi energy, for semiconductors it is
not.

I think it would be better to keep the definition of Fermi energy
separated from the definition of chemical potential. If we stick to the
definition of the Fermi energy as the energy of the highest occupied
electron state at T=0K then there is a Fermi energy in semiconductors,
that is the top of the valence band, and a chemical potential somewhere
in the gap depending on the temperature. If the Fermi energy is defined
as a value of the energy that divides occupied and empty states (as in
Ashcroft & Mermin I think) then any value in the gap could be taken as a
Fermi energy.

Hope this helps, Lazaro


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Lazaro Calderin                        
Department of Physics, Queen's University, Kingston, ON, Canada
E-mail: calderin at physics.queensu.ca