Electron-electron interaction

[Rzbs]

TL;DR Summary

Electron-electron interaction in metals vs semiconductors

What is the difference between Electron-electron interaction in metals and semiconductors? And for which one it is stonger?

 

[Astronuc]

Is one asking about e-e interactions of valence/conduction electrons, as opposed to bound electrons in the lower orbitals? Even in metals, there are variations depending on the element(s). Alloys, e.g., are very complex, and semiconductors can be very different, e.g., Si or Ge based, n-type vs p-type, or GaAs, or types of nitrides.

 

[Rzbs]

I think it is a general question, we can ask what is the conditions for considering e-e interaction significant in a solid? And then with these conditions is e-e interaction stronger in metals or semiconductors? For example if the strength of e-e interaction dependes on the density of electrons so it could be stronger in metals compered to semiconductors (for conduction electrons).

 

[dRic2]

The question should rather read: "when can e-e interactions be neglected"? As you can find in Ashcroft, Mermin page 152, there is no way to tell a priori whether the free (or almost free) electron model can be applied to conduction electrons. Experiment have shown that some metals in I, II, III an IV groups (mainly with s and p partially filled orbitals) can me modeled neglecting all sort of interactions and it is thought to be due to screening effects and the exclusion principle. I am not an expect in the field, but I don't think there is a sound way to tell in advance the importance of the interaction without getting your hands dirty with some calculations/experiments.

 

[Rzbs]

I think page 152 of ashcroft is about electron-ion interaction.But my question is about chapter 17 of this book.

 

[dRic2]

I think page 152 of ashcroft is about electron-ion interaction.But my question is about chapter 17 of this book.

You can read at page 152

Thera are two fundamentals reasons why the strong interaction of conduction electrons with each other and with the positive ions can have the net effect of a weak potentials...

Chapter 17 is about e-e interaction, it shows you how to deal with it. You should always try to include interactions in your model. Sometimes you can save yourself the trouble and neglect them, but that you can not say that in advance without proper evidence. It is true that in the high density limit the interaction potential in the hamiltonian of the electron gas becomes small and you can treat it using standard perturbation theory. That is however not a reason to fully neglect it: there could be a pretty big difference from a free theory and an interacting theory even if the interactions are small.

This is all I really have in mind, but I'm not an expert so...

 

[Rzbs]

I saw the page, you are right. I thought it was only about e-i interation.
Thanks so much.

But I didn't know how we consider e-e interaction in metals and semiconductor? Is the relation 17.50 in ashcroft use for both?
As in wikipedia:

The Thomas–Fermi wavevector (in Gaussian-cgs units) is[1]

${\displaystyle k_{0}^{2}=4\pi e^{2}{\frac {\partial n}{\partial \mu }}}$,
where μ is the chemical potential (Fermi level), n is the electron concentration and e is the elementary charge.

Under many circumstances, including semiconductors that are not too heavily doped, n∝eμ/kBT, where kB is Boltzmann constant and T is temperature. In this case,

${\displaystyle k_{0}^{2}={\frac {4\pi e^{2}n}{k_{\rm {B}}T}}}$,
i.e. 1/k0 is given by the familiar formula for Debye length. In the opposite extreme, in the low-temperature limit T=0, electrons behave as quantum particles (fermions). Such an approximation is valid for metals at room temperature, and the Thomas–Fermi screening wavevector kTF given in atomic units is

${\displaystyle k_{\rm {TF}}^{2}=4\left({\frac {3n}{\pi }}\right)^{1/3}}$.
If we restore the electron mass ${\displaystyle m_{e}}m_{e} and the Planck constant {\displaystyle \hbar }\hbar$, the screening wavevector in Gaussian units is ${\displaystyle k_{0}^{2}=k_{\rm {TF}}^{2}(m_{e}/\hbar ^{2})}$

 

[dRic2]

This is very much out of my knowledge, but I will venture a guess. The Thomas-Fermi approximation for e-e interactions is derived starting from the homogeneous electron gas model. In this model you don't really care about the structure of the lattice (i.e. you assume that electrons move through an homogenously distributed positive charge). This is a key point for me. The main difference between metals and semiconductors arises from the different crystal lattice, which you are totally ignoring here, so you have to be careful I think.
I'm sorry but I don't know the answer.

 

[Rzbs]

Thanks. Your answers help me to think about the question more precisely.