By itself, the electronic structure is dependent on temperature is negligibly weak. For example, the value of the Fermi energy for a degenerate semiconductor or metal at approximately equal of the effective masses of electrons and holes varies with characteristic factor of 10 ^ -6 eV / deg. The situation changes radically, if the mass of the electrons and holes are very different. However, to get a significant dependense of EF(T) it is required to electron and hole masses differ by at least 100 times. This situation is rare.

Much more realistic is the situation when crystal structure is temperature dependent, and it already defines the electronic structure. For example, such a situation we have seen in the article, which is attached.

Article Crystal and electronic structure of high temperature superco...

By itself, the electronic structure is dependent on temperature is negligibly weak. For example, the value of the Fermi energy for a degenerate semiconductor or metal at approximately equal of the effective masses of electrons and holes varies with characteristic factor of 10 ^ -6 eV / deg. The situation changes radically, if the mass of the electrons and holes are very different. However, to get a significant dependense of EF(T) it is required to electron and hole masses differ by at least 100 times. This situation is rare.

Much more realistic is the situation when crystal structure is temperature dependent, and it already defines the electronic structure. For example, such a situation we have seen in the article, which is attached.

Article Crystal and electronic structure of high temperature superco...

It does, solids expand or even undergo phase transitions, the distance between atoms changes. At finite temperature, the solid would not be in its ground state anymore. So, the band structure is only the starting point.

But other factors might be more important. In semiconductors, for example, a tiny concentration of  dopants, one to million or even less, is major factors in the conductivity

Yes, the electronic structure changes with temperature. Temperature leads to phonon excitations. The population of phonon would couple with electronic states and renormalize the energy of electronic energy. Even at 0 K, the electronic structure would change somehow due to 1/2 population of phonon.(zero-point correction!)

In principle the electronic structure should depend on the temperature. Although in many cases, it might be marginal, the fact that in semiconductors the band gap sometimes changes quite appreciably with temperature, implying that the electronic structure changes with temperature. I illustrate the point with the following example.

Both PbTe and SnTe crystallize in the NaCl crystaI structure and are degenerate semiconductors. In case of PbTe the band gap increases from about 0.19 eV at 0K to about 0.32 eV at about 400K.  The band gap in SnTe decreases from about 0.3 eV at 0K to 0.28 eV at about 400K. While the change in the band gap is marginal in case of SnTe, it is not so in case of PbTe.

There are two effects contributing to the change of band gap due to temperature:

( 1 ) thermal expansion of the lattice, contributing to the change in the electronic energy via volume effect

( 2 ) changes the electron-phonon interaction brings to the electronic structure.

The effect of the electron-phonon (EP) interaction, furthermore, implies the calculation of two types of processes- the Debye-Waller (DW) factor and the self- energy (SE) terms.

Ref :  Y. W. Tsang and M. L. Cohen, Phys. Rev. B 3, 1254 (1971).

The zero kelvin is also called the ground state of condensed matter. At this temperature, all atoms are arranged very orderly and formed lattice. Since then, we can apply band theory because period boundary condition can be used. 

When the temperature arises,  the electrons close to Fermi level excited.  And Fermi statistics can be  used as usual.

Finite temperature Fermi statistics is the minimum which happens. But good and interesting examples have been given above. Many phase transitions cause fundamental changes to the electronic band structure (ferromagnetism leads to lifting of spin degeneracy, Fermi surface nesting and/or Peierls distortion can induce gaps at low temperature which vanish above the critical temperature etc.) Yet another example would be heavy fermion systems with 4f bands at low(est) temperature transitioning to 4f states with localized character (no phase relations between 4f lattice sites) (well) above the coherence temperature.

Dear Kai Fauth,

All your examples of phase transitions, excluding heavy fermions, include the participation of the lattice. Indeed, in the limit of low temperatures freeze-phonon transitions the only reason can be the electronic subsystem. Outside this area, the role of the electronic subsystem is not exclusive. Therefore, the state of the electron subsystem is not determined by its own characteristics, but the state of the lattice and electron-lattice interaction.

I don't entirely agree - in a ferromagnetic phase transition, participation of the lattice (via spin orbit coupling) is not essential to the fundamental change in electronic structure (lifting of spin degeneracy), e.g. within the simple Stoner picture [of course I do know that the full picture is generally not captured in this model].

Also, when considering solids with strong variationss in the density of states (doped semiconductors being an example), changes owing to occupation statistics alone may bring about dramatic changes in electronic properties such as conductivity without a fundamental change in the underlying DOS.

Otherwise, I do agree that it makes sense to distinguish cases with and without "participation" of the lattice [even though the lattice itself results from electronic interactions...].