1. Fermi Level and Potential An evaluation with the Sommerfeld expansion

2. ‡ E. Fermi, Rend. Lincei 3 , 145-9 (1926) P.A.M. Dirac, Proceedings of the Royal Society, Series A 112 , 661-77 (1926) The electron spin quantum number is represented by a half-integer ( ± 1/2), for this reason electrons are part of a bigger family of particles known as fermions ‡  = I- ½ >  =  I + ½ >

3. + + - - Ψ ( α ) = - Ψ ( β ) Energy level Ψ q Unlike bosons, the antisymmetric nature of fermions wavefunctions prevents them from occupying the same quantum states Ψ (q) = Ψ (q+L) L ∞ ∞

4. The Fermi distribution μ is linked to the Helmoltz free energy F F=U-TS and calculated over the density of states

5. The Fermi sea of electrons T = 0 K METALS

6. E F The Fermi sea of electrons T = 0 K METALS

7. E F The Fermi sea of electrons T = 0 K METALS

8. E F The Fermi sea of electrons T = 0 K METALS

9. E F The Fermi sea of electrons T = 0 K E> μ ; f(E)=0 E < μ ; f(E)=1 E= μ ; f(E)=undefined METALS

10. E F The Fermi sea of electrons f E T = 0 K μ =E F f E 1.0 E> μ ; f(E)=0 E < μ ; f(E)=1 E= μ ; f(E)=undefined METALS

11. E F The Fermi sea of electrons f E E F Δ E≈k B T 1.0 T = 0.01 K METALS

12. E F The Fermi sea of electrons f E μ 1.0 METALS

13. E F The Fermi sea of electrons f E μ 1.0 Holes Electrons above E F METALS

14. Metals at ordinary temperatures are electronically highly degenerate i.e. kT<<E F

15. kT Integration by parts

16. 1 Taylor series:

17. Taylor series:

18. E μ =E F f E 1.0 n

21. ≈

22. 0

23. 1.0 0.5 μ 4 μ 3 μ 1 ≈ E F μ 2 T 4 > T 3 > T 2 >T 1 >0 Classical distribution ->