Free electron Fermi gas (Sommerfeld, 1928)

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Free electron Fermi gas (Sommerfeld, 1928)
1926 Schrodinger eq., FD statistics

• counting of states
• Fermi energy, Fermi surface
• thermal property specific heat
• transport property
• electrical conductivity, Hall effect
• thermal conductivity

• In the free electron model, there is neither
  lattice, nor electron-electron interaction, but
  it gives good result on electron specific heat,
  electric and thermal conductivities etc.
• Free electron model is most accurate for alkali
  metals.

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L. Hoddeson et al, Out of the crystal maze, p.104
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Quantization of k in a 1-dim box

• Plane wave solution

Box BC
Periodic BC (PBC)
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Free electron in a 3-dim box
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Quantization of k in a 3-dim box
box BC periodic BC

• Each point can have 2 electrons (because of
  spin). After filling in N electrons, the result
  is a spherical sea of electrons called the Fermi
  sphere. Its radius is called the Fermi wave
  vector, and the energy of the outermost electron
  is called the Fermi energy.

• Different BCs give the same Fermi wave vector
  and the same energy

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Connection between electron density and Fermi
energy

• For K, the electron density n1.41028 m-3,
  therefore

• eF is of the order of the atomic energy levels.
• kF is of the order of a-1.

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Fermi temperature and Fermi velocity

• The Fermi temperature is of the order of 104 K

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important
Density of states D(e) (DOS, ???)

• D(e)de is the number of states within the energy
  surfaces of e and ede

• For a 3D Fermi sphere,

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• Free electron DOS (per volume) in 1D, 2D, and 3D

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• counting of states
• Fermi energy, Fermi surface
• thermal property specific heat
• transport property
• electrical conductivity, Hall effect
• thermal conductivity

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important

• Thermal distribution of electrons (fermions)

• Combine DOS D(E) and thermal dist f(E,T)

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Electronic specific heat, heuristic argument (see
Kittel p.142 for details)

• Only the electrons near the Fermi surface are
  excited by thermal energy kT. The number of
  excited electrons are roughly of the order of N
  N(kT /EF)

• The energy absorbed by the electrons is
  U(T)-U(0) NA (kT)2/EF
• specific heat Ce dU/dT
• 2R kT/EF
• 2R T/TF
• a factor of T/TF smaller than classical result

• T/TF 0.01 Therefore usually electron specific
  is much smaller than phonon specific heat

• In general C Ce Cp
• ?T AT3

Ce is important only at very low T.
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• counting of states
• Fermi energy, Fermi surface
• thermal property specific heat
• transport property
• electrical conductivity, Hall effect
• thermal conductivity

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Electrical transport
Classical view
Relaxation time

• Electric resistance comes from electron
  scattering with defects and phonons.

• If these two types of scatterings are not
  related, then
• scattering rate

• Current density (n is electron density)

Electric conductivity
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Semi-classical view

• The center of the Fermi sphere is shifted by ?k
  -eEt.
• One can show that when ?kltltkF,
  V???/V??3/2(?k/kF).
• Therefore, the number of electrons being
  perturbed away from equilibrium is only about
  (?k/kF)Ne, or (vd/vF)Ne

• Semiclassical vs classical

• vF vs vd (differ by 109 !)
• (vd /vF) Ne vs Ne

The results are the same.
But the microscopic pictures are very
different.
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Calculating the scattering time t from measured
resistivity ?

• At room temp . The electron density
  ?t m/?ne2 2.510-14 s

• Fermi velocity of copper ? mean
  free path ? vFt 40 nm.

• For a very pure Cu crystal at 4K, the
  resistivity reduces by a factor of 105, which
  means ? increases by the same amount (? 0.4
  cm!). This cannot be explained using
  classical physics.

• For a crystal without any defect, the only
  resistance comes from phonon. Therefore, at very
  low T, the electron mean free path theoretically
  can be infinite.

Residual resistance at T0
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Hall effect (1879)
Classical view (consider only 2-dim motion)
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(?H)
???????????????????
Positive Hall coefficient? Cant be explained by
free electron theory. Band theory (next
chap) is required.
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optional
Quantum Hall effect (von Klitzing, 1979)
quantum
classical
1985

• h/e225812.807572(95) O
• offers one of the most accurate way to determine
  the Planck constant.

• Rxy deviates from (h/e2)/C1 by less than 3 ppm
  on the very first report.
• This result is independent of the shape/size of
  sample.

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optional
An accurate and stable resistance standard (1990)
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Thermal conduction in metal

• Both electron and phonon can carry thermal
  energy (Electrons are dominant in metals).
• Similar to electric conduction, only the
  electrons near the Fermi energy can contribute
  thermal current.

Heat capacity per unit volume

• Wiedemann-Franz law (1853) for a metal, thermal
  conductivity is closely related to electric
  conductivity.

Lorentz number K/sT2.4510-8
watt-ohm/deg2
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Thermal conduction in metal
Both electron and phonon can carry thermal energy
In a metal, electrons are dominant

• heat current density (classical theory)

Wiedemann-Franz law (1853)
Thermal conductivity
Classical
Semi-classical
Only electrons near FS contribute to ?
Lorentz number