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

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

3. 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

4. Point contact: ballistic quantum conductance

5. Aharanov Bohm effect: phase coherent quantum magnetic flux

6. Quantum dot: single electron charging

7. Shubnikov-de Haas oscillations and quantum Hall effect

8. Diffusive transport

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

10. 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

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

12. Drude model: B=0 conductivity : drift velocity

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

14. Conductivity tensor in magnetic field

15. no transverse magneto-resistance Hall effect

16. Monte Carlo simulation for electron motion Drift Scattering Drift Scattering

17. : current Drift velocity as a function of time

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

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

20. Boltzmann equation Steady state Boltzmann equation

21. Electron scattering detailed balance condition for transition probability

22. Scattering term assume: elastic scattering, spherical symmetry

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

24. Linearized Boltzmann equation Fermi sphere is shifted by electric field.

25. Current density and conductivity

26. Electron mobility in GaAs

27. Energy flux and thermal conductivity thermal conductivity

28. 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.