the periodic crystal potential which causes bands to form
that electrons are fermions which obey the Fermi-Dirac Statistics and they also obey the Pauli Exclusion Principle
This lecture we will be carry on ignoring the first problem however will we be covering the second one
10.2 The Free Electron Fermi Gas
Figure 10.1 - Chunk of Material, Side Length L
The plane wavefunctions of the solid is
Y (x,y,z)=e
c(Kxx+Kyy+K(z)z)
To satisfy the periodic boundary conditions we use
Y (0)=Y (L)
so we say say that nl =L for a travelling wave so that
Kx=
2p
l
=
2p n
L
This implies that the allowed K vectors form a grid in K space which are spaced 2p/L in all three dimensions.
Figure 10.2 - A Picturial Representation of the Free Electron Fermi Gas
In figure 10.2 the dot spacings are very close together when compared with the reciprical lattice vectors.
10.3 The Fermi Sphere
Consider a K space sphere of radius KF , the Fermi Wavevector
Figure 10.3 - An Example of the Fermi Wavevector
The volume of the sphere in K space is 4/3p KF3m-3 . Each K state is associated with a K volume which is related to (2p/L)3 .
The sphere holds
4
3
p KF3
æ ç ç è
2p
L
ö ÷ ÷ ø
3
K-States
Each K state can hold electrons, the first with spin up and the second with spin down. Therefore we conclude that the sphere holds the following number of electron levels ( N )
N=L3
1
3p2
KF3
If we divide this by L3 then we obtain the electron density ( n )
KF=(3p2n)
1
3
We conclude theat the states are full (for when T=0 ) up to this wavevector.
10.4 Fermi Energy
EF=
2KF2
2m0
where EF is the Fermi Energy and m0 is (for now) the free electron mass. With straight substitution we get
EF=
2
2m0
(3p2n)
2
3
typical valus for the Fermi Energy is of the order 10eV , this is very big.
10.5 Fermi Temperature
TF=
EF
K
where TF is the Fermi Temperature which is typically in the order of 104-105K .
Unless the actual temperature is more than the Fermi Temperature then the electron behaviour will be highly non-classical.
10.6 Free Electron Fermi Gas at a Finite Temperature
The Fermi-Dirac distribution is
f(E)=
1
e
E-EF
kT
+1
Figure 10.4 - The Fermi-Dirac Distribution
At a finite temperature the edge of the Fermi Sphere is blurred over a range of K , which corresponds to an energy range of about kT . Not until the actual temperature is more than the Fermi Temperature do you recover the classical Boltzmann behaviour ( 3/2kT of kinetic energy per particle, etc). In general only the electrons within kT of an empty state can move (ie. scatter from one state to another). The rest don't count!
