1. Fermi surfaces and metals
construction of Fermi surface
semiclassical electron dynamics
de Haas-van Alphen effect
experimental determination of Fermi surface
(the Sec on “Calculation of energy bands” will be skipped)
M.C. Chang
Dept of Phys

2. 1 Higher Brillouin zones (for square lattice) Reduced zone scheme 23
Every Brillouin zone has the same area
At zone boundary, k points to the plane bi-secting the G vector, thus satisfying the Laue condition
Bragg reflection at zone boundaries produce energy gaps (Peierls, 1930)

3. Beyond the 1st Brillouin zone
BCC crystal FCC crystal

4. Fermi surface for (2D) empty lattice
For a monovalent element, the Fermi wave vector 3 2 1
For a divalent element
For a trivalent element
Distortion due to lattice potential
1st BZ
2nd BZ

5. A larger Fermi "sphere " (empty lattice)
Extended zone scheme
Reduced zone scheme
Periodic zone scheme
Again if we turn on the lattice potential, then the corners will be smoothed.

6. Fermi surface of alkali metals (monovalent, BCC lattice)
kF = (3π2n)1/3
n = 2/a3
→ kF = (3/4π)1/3(2π/a)
ΓN=(2π/a)[(1/2)2+(1/2)2]1/2
∴ kF = ΓN

7. Fermi spheres of alkali metals
Percent deviation of k from the free electron value (1st octant)
4π/a

8. Fermi surface of noble metals (monovalent, FCC lattice)
Band structure (empty lattice)
kF = (3π2n)1/3,
n = 4/a3
→ kF = (3/2π)1/3(2π/a)
ΓL= ___
kF = ___ ΓL
Fermi surface (a cross-section)

9. Fermi surfaces of noble metals
Periodic zone scheme

10. Fermi surface of Al (trivalent, FCC lattice)
1st BZ nd BZ
Empty lattice approximation
Actual Fermi surface

11. Fermi surfaces and metals
construction of Fermi surface
semiclassical electron dynamics
de Haas-van Alphen effect
experimental determination of Fermi surface

12. Semiclassical electron dynamics (Kittel, Chap 8, p.192)
important
Semiclassical electron dynamics (Kittel, Chap 8, p.192)
Consider a wave packet with average location r and wave vector k, then
Derivation neglected here
Notice that E is the external field, which does not include the lattice field. The effect of lattice is hidden in εn(k) !
Range of validity
This looks like the usual Lorentz force eq. But It is valid only when
Interband transitions can be neglected. (One band approximation)
May not be valid in small gap or heavily doped semiconductors, but
“never close to being violated in a metal”
E and B can be non-uniform in space, but they have to be much smoother than the lattice potential.
E and B can be oscillating in time, but with the condition

13. k Bloch electron in an uniform electric field (Kittel, p.197)
Energy dispersion (periodic zone scheme, 1D)
ε(k) k
v(k)
-π/a
π/a
In a DC electric field, the electrons decelerate and reverse its motion at the BZ boundary.
A DC bias produces an AC current ! (called Bloch oscillation)

14. Partially filled band without scattering
Partially filled band with scattering time 
Current density

15. Why the oscillation is not observed in ordinary crystals?
To complete a cycle (a is the lattice constant),
eET/ = 2π/a → T=h/eEa
For E=104 V/cm, and a=1 A, T=10-10 sec.
But electron collisions take only about sec.
∴ a Bloch electron cannot get to the zone boundary without de-phasing.
To observe it, one needs
a stronger E field → but only up to about 106 V/cm (for semicond)
a larger a → use superlattice (eg. a = 100 A)
reduce collision time → use crystals with high quality
(Mendez et al, PRL, 1988)
Bloch oscillators generate THz microwave:
frequency ～ 1012~13,
wave length λ ～ 0.01 mm - 0.1mm
(Waschke et al, PRL, 1993)

16. Bloch electron in an uniform magnetic field
important
Bloch electron in an uniform magnetic field
Therefore, 1. Change of k is perpendicular to the B field,
k|| does not change
and 2. ε(k) is a constant of motion
For a spherical FS, it just gives the usual cyclotron orbit.
For a connected FS, there might be open orbits.
This determines uniquely an electron orbit on the FS: B

17. Cyclotron orbit in real space
The above analysis gives us the orbit in k-space. What about the orbit in r-space?
r-orbit
k-orbit ⊙
r-orbit is rotated by 90 degrees from the k-orbit and scaled by c/eB ≡ λB2
magnetic length λB = 256 A at B = 1 Tesla

18. Fermi surfaces and metals
construction of Fermi surface
semiclassical electron dynamics
de Haas-van Alphen effect
experimental determination of Fermi surface

19. De Haas-van Alphen effect (1930)
Silver
In a high magnetic field, the magnetization of a crystal oscillates as the magnetic field increases.
Similar oscillations are observed in other physical quantities, such as
Resistance in Ga
magnetoresistivity (Shubnikov-de Haas effect, 1930)
specific heat
sound attenuation
… etc
Basically, they are all due to the quantization of electron energy levels in a magnetic field (Landau levels, 1930)

20. Quantization of the cyclotron orbits
In the discussion earlier, the radius of the cyclotron orbit can be varied continuously, but due to their wave nature, the orbits are in fact quantized.
Bohr-Sommerfeld quantization rule (Onsager, 1952)
for a closed cyclotron orbit,
Why (q/c)A is momentum of field? See Kittel App. G.
Gauge dependence prob? Not worse than the gauge dependence in qV.
also
the flux through an r -orbit is quantized in units of
flux quantum: hc/e≡Φ0=4.14·10-7 gauss．cm2

21. Mansuripur’s Paradox Kirk T. McDonald

22. Energy of the orbit (for a spherical FS)
important
Since a k-orbit (circling an area S) is closely related to a r-orbit (circling an area A), the orbits in k-space are also quantized (Onsager, 1952)
B=0
Energy of the orbit (for a spherical FS)
Landau levels
Degeneracy of the Landau level
spin
Highly degenerate
B≠0
Notice that, in 3D, the kz direction is not quantized
In the presence of B, the Fermi sphere becomes a stack of cylinders.

23. Fermi energy ～ 1 eV, cyclotron energy ～ 0.1 meV (for B = 1 T)
Remarks:
Fermi energy ～ 1 eV, cyclotron energy ～ 0.1 meV (for B = 1 T)
∴ the number of cylinders usually ～ 10000
need low T and high B to observe the fine structure.
Radius of cylinders , so they expand as we increase B.
The orbits are pushed out of the FS one by one.
Cyclotron orbits FS E EF B
Landau levels
larger level separation, and larger degeneracy (both B)
Successive B’s that produce orbits with the same area:
Sn = (n+1/2) 2πe/c B
Sn-1'= (n-1/2) 2πe/c B' (B' > B)
equal increment of 1/B reproduces similar orbits

24. Quantitative details (2D)
filled
partially filled
N=50
filled
partially filled

25. Oscillation of the DOS at Fermi energy (3D)
Number of states in dε are proportional to areas of cylinders in an energy shell.
The number of states at EF are highly enhanced when there are extremal orbits on the FS.
There are extremal orbits at regular interval of 1/B.
This oscillation in 1/B can be detected in any physical quantity that depends on the DOS .
Two extremal orbits

26. Determination of FS
In the dHvA experiment of silver, the two different periods of oscillation are due two different extremal orbits.
Therefore, from the two periods we can determine the ratio between the sizes of the "neck" and the "belly“.
A111(belly)/A111(neck)=27
A111(belly)/A111(neck)=51
A111(belly)/A111(neck)=29 B