Relationship between Electron Momentum and Fermi Momentum

[ian2012]

How can an electron's momentum be less than the Fermi momentum? Since the Fermi momentum (energy) is measured at absolute zero.

 

[Physics Monkey]

Hi ian2012,

Let's think about an ideal gas of fermions. At zero temperature the Fermi momentum can be defined as the momentum of the highest occupied state. Thus by definition all electrons in the system have momentum less than or equal to the Fermi momentum. The Fermi momentum is just telling you where the electrons have filled up to. Does this make sense?

 

[ian2012]

That makes perfect sense. But what is the nature of the lower momentum, it is QM isn't it. How would electrons have momentum (move) within occupied states? What would it look like intuitively?

 

[Physics Monkey]

The smallest momentum is set by the size of the system. Think back to the particle in a box problem or the particle on a ring problem. The allowed values of momentum are basically quantized in units of one over linear system size. For a big system system there is a large separation in scale between this smallest momentum (set by the system size) and the Fermi momentum (set by the particle density).

However, I think your question may be slightly different. There is a sense in which the electrons deep within the Fermi surface are frozen, meaning they don't contribute to heat capacity, for example. Similarly, you may have encountered the statement that filled bands don't contribute to conductivity, and in some sense this is because the electrons have "no where to go".

 

[BANG!]

How can an electron's momentum be less than the Fermi momentum? Since the Fermi momentum (energy) is measured at absolute zero.

At absolute zero the fermi momentum is the highest momentum an electron can have in the system. Most electrons sit below this energy. As you increase the temperature you are able to thermally excite electrons close to the fermi surface to energies above P_f and E_f. The probability of this exciatation is given by the fermi-dirac distribution.

How would electrons have momentum (move) within occupied states? What would it look like intuitively?

Who said they didn't move? A bound state does not mean that you glued the electron to the side of an ion. The point is that most electrons sit deep within the fermi sea and therefore it is nearly impossible for them to be excited to even the lowest unoccupied state. Thus they do not contribute to heat capacity, etc.

it is QM isn't it.

Yes, the story goes something like this.

Fermi Gas: Treat electrons as just a bunch of particles in a box and solve with QM. The Pauli exclussion principle ball parks correctly a lot of quantities because only electrons near the fermi surface are important, but you don't get any band structure, everything is a metal.

Next approximation is to include interactions with the lattice. Add a repeating potential, (delta fn, step, whatever). Solve with QM and your band structure pops out. Now you have metals, insulators, etc.

Next approximation is to include electron electron interactions, Fermi Liquid: Use the Fermi Gas Hamiltonian as the unperturbed Hamiltonian and treat electron electron interactions with perturbation theory.
This is just a rough sketch, there is plenty left unsaid here.

BANG!