1. 9. Fermi Surfaces and Metals
Construction of Fermi Surfaces
Electron Orbits, Hole Orbits, and Open Orbits
Calculation of Energy Bands
Experimental Methods in Fermi Surface Studies

2. Fermi Surface : surface of ε =εF in k-space
Separates filled & unfilled states at T = 0.
Close to a sphere in extended zone scheme.
Looks horrible in reduced zone scheme.
2nd zone nearly half-filled

3. Reduced Zone Scheme Reduced Zone Scheme: k  1st BZ.
k is outside 1st BZ.
k = k + G is inside.
Both
&
are lattice-periodic.
So is →
is a Bloch function

4. 1-D Free Electrons / Empty Lattice
Reduced zone scheme:
εk is multi-valued function of k.
Each branch of εk forms an energy band
Bloch functions need band index:
PWE

5. Periodic Zone Scheme εk single-valued εk multi-valued
εnk single-valued
εnk = εnk+G periodic
E.g., s.c. lattice, TBA

6. Construction of Fermi Surfaces
Zone boundary:

7. 3rd zone: periodic zone scheme

8. Harrison construction of free electron Fermi surfaces
Points lying within at least n spheres are in the nth zone.

9. Nearly free electrons:
Energy gaps near zone boundaries → Fermi surface edges “rounded”.
Fermi surfaces & zone boundaries are always orthogonal.

10. Electron Orbits, Hole Orbits, and Open Orbits
Electrons in static B field move on intersect of plane  B & Fermi surface.

11. Nearly filled corners:
P.Z.S.
Simple cubic
TBM
P.Z.S.

12. Calculation of Energy Bands
Tight Binding Method for Energy Bands
Wigner-Seitz Method
Cohesive Energy
Pseudopotential Methods

13. Tight Binding Method for Energy Bands
2 neutral H atoms
Ground state of H2
Excited state of H2
1s band of 20 H atoms ring.

14. TBM / LCAO approximation
Good for valence bands, less so for conduction bands.
α = s, p, d, …
j runs over the basis atoms
Bravais lattice , s-orbital only:
ψk is a Bloch function since
1st order energy:
Keep only on site & nearest neighbor terms:

15. For 2 H atoms ρ apart:
Simple cubic lattice: 6 n.n. at 
Band width = 12 
2 N orbitals in B.Z.
 =  surface. 1 e per unit cell.
periodic zone scheme

16. Fcc lattice: 12 n.n. at
surface
Band width = 24 

17. Band Structure

18. Wigner-Seitz Method Bloch function: Schrodinger eq.:
For k = 0, we have
u0 is periodic in R l .
is a Bloch function;
can serve as an approximate solution of the Schrodinger eq. for k  0.

19. d  /d r = 0 at cell boundaries. 
Prob 8
Wigner-Seitz B.C.:
d  /d r = 0 at cell boundaries.
Wigner-Seitz result for 3s electrons in Na.
Table 3.9, p.70  ionic r = 1.91A
r0 of primitive cell = 2.08A  n.n. r = 1.86A
  is constant over 7/8 vol of cell.

20. Cohesive Energy linear chain Na Table 6.1, p.139: F ~ 3.1 eV.
K.E. ~ 0.6 F ~ 1.9 eV.
 5.15 eV for free atom.
0 ~ 8.2 eV for u0 .
 +2.7 eV for k at zone boundary.
 ~  ~ 6.3 eV
Cohesive energy ~  ~ 1.1 eV
exp: eV

21. Pseudopotential Methods
Conduction electron ψ plane wave like except near core region.
Reason: ψ must be orthogonal to core electron atomic-like wave functions.
Pseudopotential: replace core with effective potential that gives true ψ outside core.
Empty core model for Na
(see Chap 10)
Rc = 1.66 a0 .
U ~ –50.4 ~ 200 Ups at r = 0.15
With Thomas-Fermi screening.

22. Typical reciprocal space Ups
(see Chap 14)
Empirical Pseudopotential Method
Cohen

23. Experimental Methods in Fermi Surface Studies
Quantization of Orbits in a Magnetic Field
De Haas-van Alphen Effect
Extremal Orbits
Fermi Surface of Copper
Example: Fermi Surface of Gold
Magnetic Breakdown

24. Experimental methods for determining Fermi surfaces:
Magnetoresistance
Anomalous skin effect
Cyclotron resonance
Magneto-acoustic geometric effects
Shubnikov-de Haas effect
de Haas-van Alphen effect
Experimental methods for determining momentum distributions:
Positron annihilation
Compton scattering
Kohn effect
Metal in uniform B field → 1/B periodicity

25. Quantization of Orbits in a Magnetic Field
q = –e for electrons
Bohr-Sommerfeld quantization rule:
Phase corrector γ = ½ for free electrons
B = const →
Flux quantization
Dirac flux quantum

26. →
For Δr  B :
Let A = Area in r-space, S = Area in k-space. →
Hence
Area of orbit in k-space is quantized If
then
Properties that depend on S are periodic functions of 1/B.

27. De Haas-van Alphen Effect
dHvA effect: M of a pure metal at low T in strong B is a periodic function of 1/B.
2-D e-gas:
PW in  (B) dir.
# of states in each Landau level =
(spin neglected)
See Landau & Lifshitz, “QM: Non-Rel Theory”, §112.
Allowed levels
B = 0
B  0

28. Number of e = 48
D = 16
D = 19
D = 24
For the sake of clarity, n of the occupied states in the circle diagrams is 1 less than that in the level diagrams.

29. Critical field (No partially filled level at T = 0):
s = highest completely filled level
Black lines are plots of n = s ρ B,
n = N = 50 at B = Bs .
Red lines are plots of n = s N / ( N / ρ B ),
n = N = 50 at N / ρ B = s .

30. Total energy in fully occupied levels:→
for
Total energy in partially occupied level s + 1:
for
for
where

31. S = extremal area of Fermi surface  B
for
where
S = extremal area of Fermi surface  B
Section AA is extremal.
Its contribution dominates due to phase cancellation effect.

32. Fermi Surface of Copper
Cu / Au
Monovalent fcc metal: n = 4 / a3
Shortest distance across BZ = distance between hexagonal faces
Band gap at zone boundaries → band energy there lowered → necks
Distance between square faces  12.57/a : necking not expected

33. Example: Fermi Surface of Gold
dHvA in Au with B // [110]: Dogbone
μ has period 210–9 gauss–1 for most directions →
Table 6.1: →
Period along [111] is 610–8 gauss–1 →
→ neck
Dogbone area ~ 0.4 of belly area

34. Multiply-connected hole surface of Mg in bands 1 & 2

35. Magnetic Breakdown → breakdown Change of connectivity
~ free electron-like
Affected quantities ~ sensitive to connectivity : magnetoresistance
E.g., hcp metals with zero (small if spin-orbit effect included) gap at hexagonal zone boundary
Mg: Eg ~ 10–3 eV, εF ~ 10 eV, breakdown :