Metals: Free Electron Gas Metallic conductivity does not require an activation energy – certain electrons in a metal must be free to move, i.e. unbounded.
2 d 2 ( x) ( E V ( x)) ( x) 0 2m dx 2
Ψ Aei(k r ωt)
Free electron: V=0
2mE k 2
As with phonons, each k state occupies a volume of k space given by: 8 3 3 k V and so the density of states is the same as we found for phonons:
N (k ) V 8 3
Fermi Wave Vector Each k state can take two electrons, spin 1/2. If the electron gas contains a total of N electrons then 1/2N will be filled at absolute zero. There will be a maximum value of the wave vector corresponding to the highest occupied level – the Fermi vector, kF. N/2 must correspond to the volume in k space occupied multiplied by the density of states: 3 N 4k F V 2 3 8 3 giving
k F 3 2 n 3 where n is the free electron density N/V. The corresponding maximum kinetic energy for the electrons, the Fermi energy is 2 2kF EF . 2m
Density of States N(E) in a metal. Because of the simple relationship between E and k, any interval between k and k + dk corresponds to a spherical shell bounded by surfaces of constant energy E and E + dE. Assuming this shell contains dN electron states, then
dN V 3 4k 2 dk . 2 8
A change of variable yields,
dN V 2m 3 E 2 3
dE N ( E )dE
1 V 2m 3 E 2 2 3 is the density of states, which is the number of electron levels (spin included) per unit energy range available in the sample. Note the E dependence of N(E)!
Average Electron Energy
N ( E ) EdE
N ( E )dE
3 EF 5
Also we can show that, 3 n0 N (EF ) 2 EF where no is the valence of the metal. See Myers for typical values for all these quantities. Example: Aluminium, EF = 11.6eV, no = 3, N(EF) = 0.39/eV/atom = 2.34 x 1023 levels/eV/mole. The separation between levels is therefore of the order 4.3 x 10-24eV. Note that the average thermal energy at room temperature is ~ 0.025eV. The distribution within this energy band is effectively continuous. However, the Aufbau and Exclusion principles do apply.