Computational Solid State Physics 計算物性学特論第９回 9. Transport properties I: Diffusive transport.

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Presentation transcript:

Computational Solid State Physics 計算物性学特論第９回 9. Transport properties I: Diffusive transport

Electron transport properties l e : mean free path of electrons l φ : phase coherence length λ F : Fermi wavelength

Examples of quantum transport l e : mean free path of electrons l φ : phase coherence length λ F : Fermi wavelength single electron charging key quantities

Point contact: ballistic quantum conductance

Aharanov Bohm effect: phase coherent quantum magnetic flux

Quantum dot: single electron charging

Shubnikov-de Haas oscillations and quantum Hall effect

Diffusive transport

Equation of motion for electrons Scattering Rate k: wave vecot of Bloch electron

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

Relaxation time approximation equation of motion current density m: effective mass n: electron concentration E: electric field B: magnetic field

Drude model: B=0 conductivity : drift velocity

Drude model: steady state solution in magnetic field : B is assumed parallel to z. : cyclotron frequency drift velocity

Conductivity tensor in magnetic field

no transverse magneto-resistance Hall effect

Monte Carlo simulation for electron motion Drift Scattering Drift Scattering

: current Drift velocity as a function of time

Boltzmann equation r k Motion of electrons in r-k space during infinitesimal time Interval Δt

Equation of motion for distribution function equation of motion for electron distribution function f k (r,t).

Boltzmann equation Steady state Boltzmann equation

Electron scattering detailed balance condition for transition probability

Scattering term assume: elastic scattering, spherical symmetry

Transport scattering time k k’ Θ kk’ Contribution of forward scattering is not efficient. Contribution of backward scattering is efficient.

Linearized Boltzmann equation Fermi sphere is shifted by electric field.

Current density and conductivity

Electron mobility in GaAs

Energy flux and thermal conductivity thermal conductivity

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.