1. Today’s Objectives Briefly discuss other Fermi terms
Last Two Lecture Summary:
Drude: Applied simple model to understand current density and the Hall coefficient in simple metals. Required knowledge/determination of carrier density.
Sommerfeld: Showed that Fermi-Dirac statistics can be added to improve the Drude model and defined the Fermi energy.
Today’s Objectives
Briefly discuss other Fermi terms
Derive Fermi-Dirac distribution
Discuss density of states for electrons
Review heat capacity with Sommerfeld model
This class was about 15 minutes short. Think about what would be good to expand upon

2. Other Fermi Terms Fermi temperature EF=kBTF Fermi momentum
And Fermi velocity
These are the momentum and the velocity values of the electrons at the states on the Fermi surface of the Fermi sphere.
Fermi temperature when “acts like a classical gas.” I find this statement confusing because it looks pretty classical to me around room temperature or so. Likely means when the Fermi Dirac statistics approach Maxwell Boltzmann like curves. Note Fermi temperatures are super high (~10,000K). kz ky kx
Fermi surface kF

3. This temperature will come back up later.
Example: potassium metal
This temperature will come back up later.

4. Thermal Properties At T=0, all levels up to EF are filled.
What happens when T is greater than zero?
Do all the levels have an equal chance of excitation? No. You only have so much thermal energy. The states closest to the Fermi level are most likely to be effected.
T=0

5. It’s a little confusing. You don’t need to know how to derive.
Probability with Temperature of N electrons (see pages Ashcroft, Kittel just gives it)
The probability that a state consisting of all N electrons is occupied depends on the Boltzmann distribution:
More precisely
To calculate properties, we have to take the average over all possible N states
Partition function Z
We are deriving the Fermi-Dirac distribution.
There will be two different probabilities I will talk about, which makes this a little confusing. One is the probability PN of a whole system being at some total energy E and the other is the probability of a particular energy level i being filled, referred to as fi. We actually want fi because that’s the fermi dirac distribution, but we need to talk about the other to derive it.
PN is the weight for each sum energy state for a N particle system and we should average over all N-particles states
Does anyone recognize this denominator? It has a name. The partition function.
Can rewrite the partition function in terms of the free energy
Helmholtz free energy F
One possible state N
U=internal energy, S=entropy
EN=34

6. Or one minus states  with no electron in state i
Combining
Will use soon…
Rewrite the probability in terms of the new partition function
I haven’t done anything yet to the Maxwell Boltzmann distribution, just rewriting in more convenient terms
f, the Fermi dirac distribution, will be useful to know for later calculations
fiN is the probability of one of N electrons being in a particular electron level i at temp T.
By exclusion principle, it equals the sum of probabilities of any system of N electrons with an electron in that state
Or one minus states  with no electron in state i

7. Remove the one in question
That sounds complicated. Let’s use a trick. What if we add one more electron?
Add one here
We can build a 35 electron system (N+1) and remove whatever energy we want to consider
Remove the one in question
Or more generically…

8. Some algebra manipulation
Just plugging in for E
Almost looks like PN, but maybe for N+1, what would we need to make it look like that?
Chemical potential

9. Solving for fi
The probably should not change that much from one electron when N large

10. Temperature affect on probability
At the chemical potential, there is a 50% chance of finding an electron
Fermi energy is not the same as the chemical potential. µ (T=0)=EF. EF is defined at T=0.
Subtle difference and often confused. Often µ~EF
As T0, EF

11. But it’s not just f, it’s also related to D(E).
How does this help us understand the problem with Drude’s heat capacity?
When a metal is heated, electrons are transferred from below EF to above EF. The rest of the electrons deep inside the Fermi level are not effected.
But it’s not just f, it’s also related to D(E).

12. Finding the 3 dimensional density of states D(E)
g(k)
g(E)
g(k)
g(E)
Move this to next class if time does not allow us to discuss

13. Free electrons in 3D Group: Find D(E) = D(k) dk/dE
More than one way to approach
D(E)

14. Writing Density of States in terms of number of orbitals NV
(See hand written notes) Leads to easier solution for the density of states, which will allow us to use less constants, which is great because there are a lot.
Solve fermi energy for N. Then just multiple 3/2 N/fermi energy and turns out to be density of states. Much easier to work with than all of these constants!

15. The free electron gas at T < Fermi tempk
N(E,T) number of free electrons per unit energy range is just the area under N(E,T) graph. N
Big difference at high temps
Sorry, I’m using g instead of D for density of states. I need to fix.
You will often see Fermi energy used instead of chemical potential.
F (mu) is still defined the same way, ½ for a semiconductor even though in the gap. It’s just that f(mu) times the density of states changes as the density of states is 0 in the gap.

16. Fermi-Dirac distribution function is a symmetric function, meaning:
The shaded area shows the change in distribution between absolute zero and a finite temperature. N
Fermi-Dirac distribution function is a symmetric function, meaning:
At low temperatures, the same number of levels below EF is emptied and same number of levels above EF are filled by electrons.
T>0
T=0
N(E,T) E
g(E) EF

17. Heat capacity of the free electron gas
From the diagram of N(E,T) the change in the distribution of electrons can be resembled into triangles of height ½ g(EF) and a base of 2kBT so the area gives that ½ g(EF)kBT electrons increased their energy by kBT.
The difference in thermal energy from the value at T=0°K ~
For an exact calculation:

18. Differentiating with respect to T ~
Differentiating with respect to T
gives the heat capacity at constant volume:
From earlier:
While this works great for metals, we will find that it’s very different for other materials.
Heat capacity of
Free electron gas

19. Comparison with Data It does appear linear at medium temps
Phonons dominate at high temperatures but this works well for metals at medium temperatures
Classical Model suggests one constant value and same value for all materials

20. Why Specific Heat? Profile of Frances Hellman
Physics Professor at University of California, Berkeley
Previous chair of the physics department
“My research group is concerned with the properties of novel magnetic and superconducting materials especially in thin film form. We use specific heat, magnetic susceptibility, electrical resistivity, and other measurements as a function of temperature in order to test and develop models for materials which challenge our understanding of metallic behavior.
Current research includes: effects of spin on transport and tunneling, including studies of amorphous magnetic semiconductors and spin injection from ferromagnets into Si; finite size effects on magnetic and thermodynamic properties…”
In case you needed another reminder as to why we should care

21. Measuring specific heat on a budget Step 1: Set-up the calorimeter
Energy conserved: qmetal = qwater + qcoffeecup
qcup can be ignored cmetalmmetal Tsystem = cwatermwater Twater
Step 2: Boil the water containing metal, Pour
Step 3: Stir while measuring temperature

22. constant volume or 'bomb' calorimeter
Modern calorimetry works on the same principles, just looks more fancy.
constant volume or 'bomb' calorimeter