Fermi level is electron chemical potential at 0K and we can obtain it from DFT calculation. However, the chemical potential is dependent on temperature. But I only find the equation for gas or simple compounds. Is there any way to estimate the chemical potential of electrons in solid crystals at finite temperature?

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    $\begingroup$ Do you have a solid crystal structure, or does that also need to be first estimated? If you have a structure I think phonon analysis via plane wave DFT can get you the free energy. David Mobley has worked on this using alchemical free energy calculations via molecular mechanics, but it is brutally expensive and not as accurate as one would like. $\endgroup$
    – B. Kelly
    May 2, 2022 at 12:43
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    $\begingroup$ Do you mean the chemical potential of the electrons, or the chemical potential of the atoms (or, for molecular crystals, molecules)? These are two completely unrelated quantities; the first one is related to the Fermi level, while the second one is related to cohesive (free) energies, vapor pressures, etc. If you found equations for "gas or simple compounds", they are probably equations for the chemical potential of molecules, not the chemical potential of electrons. $\endgroup$
    – wzkchem5
    May 11, 2022 at 18:30
  • $\begingroup$ @wzkchem5 yes, I mean for electron. Do you know how? $\endgroup$
    – Binh Thien
    May 12, 2022 at 0:42

1 Answer 1


I guess the you are working on doping. Suppose in the following we consider only semiconductors.

I disagree with "Fermi level is electron chemical potential at 0K and we can obtain it from DFT calculation." First, I would not be bothered by the textbook to consider "Fermi level is electron chemical potential at 0K". Instead, I simply think "Charge is exchanged with a reservoir of electrons, the energy of which is the electron chemical potential, in other words, the Fermi energy" (see the review paper "First-principles calculations for point defects in solids"). Second, DFT calculations only give an artificial Fermi level for 0K calculation of a pure system.

Usually, the electron temperature equals the ionic temperature. At finite temperature, the Fermi level is determined by self-consistent solution of the charge neutral condition (see the review paper). A special case is when your system does not contain a defect, then the Fermi level can be obtained by n=p (still the charge neutral condition), where n is free electron concentration (you can sum the occupations in the conduction bands), and p is the free hole concentration (you can sum the unoccupations in the valence band). You can adjust the Fermi level position, this will mimic free-carrrier doping, i.e., the so-called rigid-band approximation in the literature.


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