Understanding Fermi Sphere: Deriving Fermi Energy

[carbon9]

Hi all,

I'm studying on the current transfer at quantum level and I have a point that is not so much clear. While reading the Fermi sphere from the book "Current at the nanoscale", I could not understand the expression:

The number of electrons in the conductor, N, is the ratio of the total volume of the Fermi sphere to the volume per state

Is this sentence right, and what is its meaning? For clarity, I attached the page of the book where the author derives Fermi energy in terms of number of electrons and the volume of the conductor.

Thanks for all,
Regards

 

[genneth]

It's perfectly right. The Fermi sphere is a ball in momentum space, inside which are the occupied electron states. You can calculate the number of states in k-space by assuming that your electrons are in a box of size L, with periodic or fixed boundaries. You can figure out the size of the sphere with the further information of the number of electrons in that box. You will find that the size of the sphere only depend on the density of electrons, and independent of the size of the box.

 

[TheWye]

Just imagine that you want to fill up little boxes into a huge ball. The ball has a radius of kF while the little boxes has the sides (width, height and depth) of length equals to the periodic boundary. So the boxes will have a total volume of (periodic boundary length)^3 while the sphere will have the volume of 4/3 (pi) (kF)^3.

Both the ball and the boxes exist inside the k-space.

I attached a picture to provide the pictorial explanation. I hope this helps~