1. Free electron Fermi gas (Sommerfeld, 1928)
M.C. Chang
Dept of Phys
Free electron Fermi gas (Sommerfeld, 1928)
counting of states
Fermi energy, Fermi surface
thermal property: specific heat
transport property
electrical conductivity, Hall effect
thermal conductivity
In the free electron model, there are neither lattice, nor electron-electron interaction, but it gives good result on electron specific heat, electric and thermal conductivities… etc.
Free electron model is most accurate for alkali metals.

2. Energy levels in an empty 1-dim box
Plane wave solution
“Box” BC
Periodic BC
k = ±2π/L, ±4π/L, ±6π/L…
k = π/L, 2π/L, 3π/L…

3. Energy levels in an empty 3-dim box
box BC periodic BC
Each point can have 2 electrons (because of spin). After filling in N electrons, the result is a spherical sea of electrons called the Fermi sphere. Its radius is called the Fermi wave vector, and the energy of the outermost electron is called the Fermi energy.
Different BCs give the same Fermi wave vector and the same energy
box BC
periodic BC

4. Connection between electron density and Fermi energy
For K, the electron density N/V=1.4×1028 m-3, therefore
F is roughly the order of the atomic binding energy.
kFa is of the order of 1.
The Fermi temperature defined by EF=kTF is of the order of K
Fermi energy and fermi wave vector for typical metals

5. Density of states (for electrons)
The number of states within the shell bounded by the constant energy surfaces with energy E and E+dE is D(E)dE
What’s the DOS for free electron gas in 3 dim?
Free electron DOS and dimension E
D(E) 3D 2D 1D

6. Fermi energy, Fermi surface thermal property: specific heat
counting of states
Fermi energy, Fermi surface
thermal property: specific heat
transport property
electrical conductivity, Hall effect
thermal conductivity

7. Thermal distribution of electrons (fermions)
Combine DOS D(E) and thermal dist f(E,T)
Hotel rooms
tourists
money

8. Electronic specific heat (heuristic argument, see Kittel p
Electronic specific heat (heuristic argument, see Kittel p.152 for details)
Only the electrons near the Fermi surface are excited by thermal energy kT. The number of excited electrons are roughly of the order of N’ = N（kT/EF）
The energy absorbed by the electrons is U(T)-U(0)  NA (kT)2/EF
specific heat Ce  dU/dT
= 2R kT/EF
= 2R T/TF
a factor of T/TF smaller than classical result
T/TF 0.01 Therefore usually electron specific is much smaller than phonon specific heat
In general C = Ce + Cp
= γT + AT3
Ce becomes more important at very low T

10. Fermi energy, Fermi surface thermal property: specific heat
counting of states
Fermi energy, Fermi surface
thermal property: specific heat
transport property
electrical conductivity, Hall effect
thermal conductivity

11. Electrical transport Classical view Current density conductivity
The source of resistance comes from electron scattering with defects and phonons.
If these two types of scatterings are not related, then
Current density
conductivity

12. Semi-classical view
On the average, the center of the Fermi sphere is shifted by k=eE. One can show that when k<<kF, V沒重疊/V重疊3/2(k/kF). Therefore, the number of electrons being perturbed away from equilibrium is only about (vd/vF) Ne
Semiclassical vs classical
The results are the same. But the microscopic pictures are very different!
vF vs vd
(vd/vF) N vs N

13. Calculating the scattering time from measured resistivity 
At room temp The electron density τ= m/ne2 = 2.5×10-14 s
Fermi velocity of copper mean free path  = F  = 40 nm
For a very pure Cu crystal at 4K, the resistivity reduces by a factor of , which means  increases by the same amount ( = 0.4 cm). This cannot be explained using the classical theory.
For a crystal without any defect, the only resistance comes from phonon. Therefore, at very low T, the electron mean free path theoretically can be infinite
dirty
clean K
Residual resistance

14. Classical view: (consider only 2-dim motion)
Hall effect (1879)
Classical view: (consider only 2-dim motion)
|H| B

15. Positive Hall coefficient?
藉著測量霍爾係數可以推算自由電子濃度。
Positive Hall coefficient?
Can’t be explained by free electron theory. Band theory is required.

16. Quantum Hall effect (von Klitzing, 1979)
h/e2= (95) 
offers one of the most accurate way to determine the Planck constant
Classical prediction
1985
Also, an accurate and stable resistance standard (1990)

17. Thermal conduction in metal
Both electron and phonon can carry thermal energy (Electrons are dominant in metals).
Similar to electric conduction, only the electrons near the Fermi energy can contribute thermal current.
Wiedemann-Franz law, 1853
Lorentz number: K/σT=2.45×10-8 watt-ohm/deg2