Early History of Metal Theory

1
Chapter 1 The Free Electron Fermi Gas

• Early History of Metal Theory
• 1900-1930 (Drude, Lorentz, Fermi, Dirac, Pauli,
  Sommerfeld, Bloch, )
• The Basic Hamiltonian
• Approximations Assumptions
• The Ground State (T 0)
• Wave-functions, allowed states, Fermi sphere,
  density of states
• Thermal Properties
• Expectation values, energy, specific heat
• Electrical Transport Properties
• DC and AC conductivities
• Magnetic Properties
• Classical Hall effect, Pauli paramagnetism,
  Landau quantization, the A-B phase, cyclotron
  resonance, the quantum Hall effect

2
Section 1.7 Magnetic Properties of a Free
Electron Fermi Gas

1. The Classical Hall Effect
2. Pauli Paramagnetism
3. Landau Quantization
4. The Aharonov-Bohm Phase
5. Cyclotron Resonance
6. The Quantum Hall Effect

3
The Hall Effect
z
x
y
Lorentz force
Balance equation
RH is independent of t and m ? An excellent
method for determining n
4
The Hall Effect
More formal derivation
magneto-resistivity tensor
magneto-conductivity tensor
Jy 0
5
Density within the Drude Model
rm kg/m3 mass density A kg atomic mass
(mass of one mole)
rm/A moles atoms per m3
NArm/A atoms per m3, NA 6.02 1023
n NArmZ/A electrons per m3, Z of valence
electrons
For Li, rm 0.542 103, A 6.941 10-3, Z 1
n 4.70 1028 m-3
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Comparison with Experiments
For Li, rm 0.542 103, A 6.941 10-3, Z 1
n 4.70 1028 m-3
RH ?1.33 10-10 m3/C
good
RH(exp) ?1.7 10-10 m3/C
For Zn, rm 7.13 103, A 65.38 10-3, Z 2
n 1.31 1029 m-3
RH ?4.77 10-11 m3/C
bad
RH(exp) 3 10-11 m3/C
Positive Hall coefficient!
7
Cyclotron Frequency and the Hall Angle
8
Deviation from the Classical Hall Effect
9
How Difficult is wct gt 1 ?
me 10000 cm2/Vs ? B gt 1 Tesla me 1000 cm2/Vs
? B gt 10 Tesla me 100 cm2/Vs ? B gt 100 Tesla
10
Section 1.7 Magnetic Properties of a Free
Electron Fermi Gas

1. The Classical Hall Effect
2. Pauli Paramagnetism
3. Landau quantization
4. The Aharonov-Bohm Phase
5. Cyclotron Resonance
6. The Quantum Hall Effect

11
Paulis Spin Matrices
Lets concentrate on electronic spins
12
Zeeman Effect
quantization of angular momentum

• B shifts the energy of each state by U
• ml magnetic quantum number

13
Electron Spin

• Anomalous Zeeman splitting
• Stern-Gerlach experiment (1922) ? splitting into
  an even number of components (should be 2l 1)
• Goudsmidt and Uhlenbeck (1925) spinning on its
  axis
• Diracs theory (1928) existence of spin angular
  momentum

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Spin Angular Momentum
ms spin quantum number
ms 1/2 spin up and ms -1/2 spin down
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Gyromagnetic Ratio and the Electron g-factor
gyromagnetic ratio
g-factor
Quantum Electrodymanics (QED)
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Spin is Purely Quantum Mechanical
Orbital angular momentum
As h ? 0, we can keep L non-zero by increasing
the size of l to infinity
Spin angular momentum
As h ? 0, S ? 0
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Magnetic Susceptibility
Total field T or Wb/m2
Applied field A/m
Induced field A/m
4p 10-7 T-m/A
magnetization curve
c gt 0 paramagnetic c lt 0 diamagnetic
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Calculate Spin c Classically
spin down
spin up
Consider N electrons in volume V at temperature T
in a magnetic field H, and calculate the total
magnetic moment M
Too large, and temperature dependent
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Paulis Spin Susceptibility
g?(e)
2mBm0H
e
spin imbalance
magnetic moment per electron
g?(e)
Net magnetic moment per m3