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 diamagnetism, cyclotron resonance, the
  quantum Hall effect

2
Drude-Lorentz Model 1900-1904

• Ashcroft-Mermin Chapter 1
• Based on the discovery of electrons by J. J.
  Thomson (1897)
• Electrons as particles (Newtons equation)
• Electron gas (Maxwell-Boltzmann statistics)
• Worked well
• Explained Wiedemann-Franz law (1853) k/sT 2-3 x
  10-8 (W-ohm/K2) (by double mistakes)
• Could not account for
• T-dependence of k alone
• T-dependence of s alone
• Electronic specific heat ce (too big)
• Magnetic susceptibility cm (too big)

Paul Drude (1863-1906)
Hendrik Lorentz (1853-1928)
3
Fermi-Dirac Statistics 1926
m
Enrico Fermi (1901-1954)
Paul Dirac (1902-1984)
Shown by Fermi and Dirac independently
? Opened a way to a realistic theory of metals
4
Pauli Susceptibility 1927
First successful application of FD statistics to
metal theory
Pauli
Wolfgang Pauli (1900-1958)
26 meV at 300 K
5 eV
? Explains why the classical result is too big
5
Sommerfeld Model 1928

• Ashcroft-Mermin Chapter 2
• Systematically recast Drude-Lorentz theory in
  terms of FD statistics rather than MB statistics
• Wiedemann-Franz law (still) came out right
• Estimated specific heat right
• Difficulties remained
• Sign of Hall coefficient
• Magneto-resistance
• What determines the scattering time t?
• What determine the density n?
• Why are some elements non-metals?

Arnold Sommerfeld (1868-1951)
6
Bloch Theory 1928

• Ashcroft-Mermin Chapters 8-10
• A major breakthrough in solid state theory
• Took into account lattice periodic potential
• Still treated electrons independently
• Major accomplishments
• Meaning of t lattice imperfections (phonons,
  defects, impurities, dislocations)
• Concept of energy bands
• Distinction between metals and insulators (also
  by A. H. Wilson in 1931)
• Meaning of holes ? positive Hall coefficient

Felix Bloch (1905-1983)
7
The Birth of Fermiology 1930

• Ashcroft-Mermin Chapter 14
• Landau Landau levels (1930) ? predicted
  oscillations in c vs. H ? period has info on the
  shape of the Fermi surface
• Observations of Shubnikov-de Haas oscillations
  de Haas-van Alphen oscillations (1930)
• Cyclotron resonance
• Predicted by Dingle (1951)
• First observed by Dresselhaus et al. (1953)
• Theory refined by Luttinger Kohn (1955,56)

Lev Landau (1908-1968)
8
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 diamagnetism, cyclotron resonance, the
  quantum Hall effect

9
The Basic Hamiltonian
N nuclei (positive ions) Total charge
NZae
e-
e-
e-
e-
NZa electrons Total charge -NZae
e-
e-
e-
e-
e-
e-
e-
Za atomic number
N 1023
10
The Schrödinger Equation
N 1023
Massive many-body problem
Exact solutions cannot be expected
11
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 diamagnetism, cyclotron resonance, the
  quantum Hall effect

12
The Static Lattice Approximation
Constant potential ? 0
13
The Free Electron Approximation
Z valence electrons per nucleus ? free (or
conduction) electrons (Za Z) tightly-bound
electrons per nucleus
14
The Uniform-Background Approximation
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
ion
15
The Independent Electron Approximation
But, Requirement 1 Electrons are still
confined to a volume V ? allowed
states Requirement 2 Electrons obey FD
statistics ? ground state construction
16
The Simplified Hamiltonian
in the absence of an external field
Separable into NZ terms
neglect i
solve
17
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 diamagnetism, cyclotron resonance, the
  quantum Hall effect

18
Ground-State (T 0) Properties of
Non-interacting Electrons

• Procedure
• Find the energy levels of a single electron
• Fill these levels up in a manner consistent with
  the Pauli principle

19
Born-Von Karmans Periodic Boundary Conditions
Idea If the metal is sufficiently large, we
should expect its bulk properties to be
unaffected by the detailed configuration of its
surface.
20
Solution
21
Number of Allowed States
22
Construction of the Ground State the Fermi Sphere
23
Fermi Surface in a Real Metal
Fermiology The study of the shape of Fermi
surfaces in metals
24
Fermi Wave Vector, Momentum, Energy, Velocity,
and Temperature
The Fermi wave vector
The Fermi energy
The Fermi temperature
25
Density of States (DOS)
To calculate thermodynamic quantities, one needs
to carry out summations of the type
over allowed values, i.e.,
For mathematical convenience, want to convert
sums into integrals
k-space DOS Dk
26
Density of States (DOS)
In many cases, k-dependence appears only through
Namely,
So, want to write
D(?) Energy DOS the number of allowed states
having energies between ? and ? d?
Note
is the DOS per unit volume
27
Density of States (DOS)
Want to calculate D(?). Easier in spherical
coordinates
So,
28
Density of States (DOS)
? ? k2 ? ? k
g3D(?) ? ?1/2 ? ?2
g2D(?) constant ? ?
g1D(?) ? ?-1/2 constant
Exercise derive the exact form of DOS for each
case