Computational Solid State Physics ??????? ?9?

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Computational Solid State Physics ??????? ?9?

• 9. Transport properties I
• Diffusive transport

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Electron transport properties
le mean free path of electrons lf phase
coherence length ?F Fermi wavelength
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Examples of quantum transport
key quantities
le mean free path of electrons lf phase
coherence length ?F Fermi wavelength
single electron charging
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Point contact ballistic
quantum conductance
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Aharanov Bohm effect phase coherent
quantum magnetic flux
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Quantum dot single electron charging
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Shubnikov-de Haas oscillations and quantum Hall
effect
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Diffusive transport
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Equation of motion for electrons
k wave vecot of Bloch electron
Scattering Rate
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How to solve equations of motion for electrons
with scattering?

• Relaxation time approximation for scattering
• Direct numerical solution
• Monte Carlo simulation
• Boltzmann equation for distribution function of
  electrons

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Relaxation time approximation
m effective mass
equation of motion
E electric field B magnetic field
current density
n electron concentration
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Drude model B0
drift velocity
conductivity
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Drude model steady state solution in magnetic
field
cyclotron frequency
B is assumed parallel to z.
drift velocity
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Conductivity tensor in magnetic field
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Hall effect
Hall effect
no transverse magneto-resistance
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Monte Carlo simulation for electron motion
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Drift velocity as a function of time
current
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Boltzmann equation
Motion of electrons in r-k space during
infinitesimal time Interval ?t
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Equation of motion for distribution function
equation of motion for electron distribution
function fk(r,t).
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Boltzmann equation
Steady state
Boltzmann equation
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Electron scattering
detailed balance condition for transition
probability
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Scattering term
assume elastic scattering, spherical symmetry
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Transport scattering time
Contribution of forward scattering is not
efficient. Contribution of backward scattering is
efficient.
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Linearized Boltzmann equation
Fermi sphere is shifted by electric field.
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Current density and conductivity
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Electron mobility in GaAs
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Energy flux and thermal conductivity
thermal conductivity
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Problems 9

• Calculate both the conductivity and the
  resistivity tensors in the static magnetic
  fields, by solving the equation of motion in the
  relaxation time approximation.
• Study the temperature dependence of electron
  mobility in n-type Si.
• Calculate the electron mobility in n-type silicon
  for both impurity scattering and acoustic phonon
  scattering mechanisms, by using the linearized
  Boltzmann equation.