Semiconductor Band Gap: Fermi Momentum & Dispersion Relation

[andrewm]

In the first diagram on http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html" , the band gap in a semiconductor is shown.

What is the corresponding Fermi momentum? If you plug E_F into the dispersion relation, you get no solution for k_F, right?

I ask because an equation is given with a sum of all k such that k < k_F, and there is a band gap. Does one sum over all k?

 

[snapback]

late response ;-)

I do not know which equation you mean, but generally if you sum all k in a full band you end up with zero net momentum. This is the same as saying: a full band is not contributing to electrical conduction. In the bandgap, there are no electrons thus there is no contribution to the total momentum. At T=0, there are no elevtrons in the conduction band, so also no net momentum there

as far as I can see it, you can only use the concept of the Fermi momentum if the Fermi energy is at least few kT inside a band. This is the situation, you mya encounterin metals or degenerated (=very heavily doped) semiconductors

cheers

 

[charng]

The Fermi momentum in a semiconductor refers to the momentum of the electrons at the Fermi level, which is the highest occupied energy level at absolute zero temperature. It is given by the equation p_F = sqrt(2mE_F), where m is the effective mass of the electron in the semiconductor and E_F is the Fermi energy.

In the dispersion relation for a semiconductor, the energy is a function of the momentum, and the Fermi momentum is the value of momentum at which the energy is equal to the Fermi energy. Therefore, plugging E_F into the dispersion relation would indeed yield no solution for k_F, as the Fermi energy is not directly related to the momentum.

However, the equation given for the sum of all k < k_F is not directly related to the dispersion relation. This equation is known as the Fermi-Dirac distribution, which describes the probability of finding an electron at a certain energy level at a given temperature. The sum over all k represents the total number of electrons at energy levels below the Fermi energy.

In the case of a band gap in a semiconductor, the Fermi energy lies within the band gap, meaning there are no electrons at that energy level. Therefore, the sum over all k would only include the energy levels below the Fermi energy, and not the entire range of k values.

In conclusion, the Fermi momentum and the dispersion relation are not directly related in a semiconductor, but the Fermi-Dirac distribution can be used to understand the behavior of electrons at different energy levels and their contribution to the overall electronic structure of the material.