# Fermi gases and holes

It is a well known fact that at $$T\to0$$ the energy of the Fermi level for a noninteracting electron gas is given by $$E_{F}=\frac{\hbar^{2}k_{F}^{2}}{2m},\,\,k_{F}=(3\pi^{2}n)^{1/3},$$ where $$k_{F}$$ is the Fermi wavevector, and $$n=N/V$$ is the electronic density. This represents the maximum energy of electrons at that temperature.

For the case of semiconductors, one of the most important effects to consider is the creation of pairs electron-hole, where one finds the usually found in textbooks parabolic curves illustrating the band width.

In general, for some reason, one does not consider very often this effect in the simplest case, that is, the free Fermi gas, which is possible as well.

I'm interested in the curve that one would obtain for this case, in a scenario in which an electron with $$E promotes to a level with $$E>E_{F}$$. Of course, since $$k=k(E)$$, a somehow similar result should be obtained, but here we are not considering effective masses or bands. Intuitively, the curves should depend on $$k_{F}$$, at least when $$\Delta E=0$$.

EDIT: The analogy I’m making with semiconductors involves the dispersion relation obtained near the top and bottom of the bands taking into account the effective mass. This does not make sense when the gas if of free electrons. I’m tying to find the values of momenta and energies (which are of course related to each other) in the transition described above.

Any ideas?

• Please specify what the "usually found in textbooks parabolic curves illustrating the band width" are. Are you talking about the dispersion relations $E_n(\vec{k})$ with $n$ being a band index and $\vec{k}$ the wave vector? – flaudemus Mar 5 at 8:38
• @flaudemus Yes, in particular the analogy I’m making with semiconductors involves the dispersion relation $E_{n}(\textbf{k})$. That can’t be the case for a gas of free electrons. Now suppose that at T->0 formation of electron-hole pairs occurs, following the transition specified in the post. How would that translate into a region of possible (E, k) values? – KernelPanic Mar 5 at 8:53

In a semiconductor, the creation of an electron-hole pair involves two states, i.e., $$E_\mathrm{VB}(\vec{k})$$ and $$E_\mathrm{CB}(\vec{k}')$$, where the subscripts indicate valence band (VB) and conduction band (CB). An electron is removed from the valence band at wave vector $$\vec{k}$$ and excited into the conduction band at wave vector $$\vec{k}'$$. Momentum conservation implies that the particle creating this excitation (e.g. a phonon, a photon, or another (high energy) electron) has to provide the wave vector difference $$\vec{k}'-\vec{k}$$, and energy conservation implies that the particle creating the excitation has to provide the energy difference between initial and final state.
For a free electron gas there are no bands and therefore no excitations of electron-hole pairs between valence and conduction bands. However, there can be excitations across the Fermi energy. The dispersion relation is $$E(\vec{k}) = \frac{\hbar^2k^2}{2m}$$ with $$m$$ being the free electron mass. An electron-hole excitation created in the Fermi-sea takes an electron from $$E(\vec{k}) to $$E(\vec{k}')>E_\mathrm{F}$$. Like in the case of semiconductors, energy conservation and momentum conservation apply. How these two conservation laws can be fulfilled depends on the exact nature of the excitation process considered.
• Thanks! If one only considers the possibility that one electron $E<E_{F} is promoted to$E>E_{F}$, is there any closed formula that describes the possible pairs$(\Delta E, \Delta k)$for this process? That’s the question I was investigating initially. – KernelPanic Mar 5 at 12:01 • Such a formula would depend on the details of the excitation mechanism, i.e., on the interaction that causes the excitation. Unless specified, no such formula can be derived beyond the standard conservation laws. – flaudemus Mar 5 at 12:25 • I see. In any case I’m having trouble with the implementation of the standard conservation laws in this case. What exactly needs to be compared before and after the transition? – KernelPanic Mar 5 at 12:56 • I’m guessing the part I’m missing is the gap energy. – KernelPanic Mar 5 at 12:58 • Energy conservation: take the difference in energy between final and initial state of the electron; this energy difference has to be brought up by the excitation mechanism (e.g., if a photon or phonon is absorbed, this difference needs to be equal to$\hbar\omega$). Momentum conservation: take the difference between the momentum in the final state and that in the initial state of the electron. This difference has to be brought up by the excitation mechanism (i.e., in case of a photon or phonon,$\hbar\vec{\kappa}$, where$\vec{\kappa}\$ is the wave vector). – flaudemus Mar 5 at 13:01