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The effective mass

The effective mass. Conductivity effective mass – determines mobility. Density of states effective mass – determines N C Cyclotron effective mass – can be measured directly. Electron in a periodic potential. Why does the semiconductor industry use single crystal material (when possible) ?.

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The effective mass

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  1. The effective mass • Conductivity effective mass – determines mobility. • Density of states effective mass – determines NC • Cyclotron effective mass – can be measured directly

  2. Electron in a periodic potential • Why does the semiconductor industry use single crystal material (when possible) ?

  3. Electrons are not scattered by a periodic potential – move with a constant velocity as in vacuum !

  4. Electron in vacuum

  5. Dispersion

  6. Electron in vacuum

  7. Electron in a periodic potential

  8. Electron in a 3D periodic potential

  9. primative reciprocal lattice basis vector respectively. and .

  10. Expansion of E(kx,ky,kz) near a minimum value E0= E(kx0,ky0,kz0)

  11. Expansion of E(kx,ky,kz) near a minimum value E0= E(kx0,ky0,kz0)

  12. In the coordinate system in which the effective mass tensor is diagonal

  13. Acceleration due to an electric filed (F)

  14. Effective mass tensor – valid near E(k) minima and maxima only

  15. Constant energy surfaces in crystal momentum space cookies

  16. Constant energy surfaces near a minimum are ellipsoids

  17. Constant energy surfaces in Si and Ge near a minimum are ellipsoids of revolution

  18. Acceleration of an electron near an energy minimum in silicon

  19. Electron transport effective mass in silicon and germanium

  20. Homework competition – find a shorter way to prove this equation for germanium than given in last year’s home exam

  21. The spherical case - • Electrons in GaAs • Holes in Si, Ge, GaAs

  22. Cyclotron resonance effective mass

  23. Cyclotron resonance effective mass –to be shown in the tutorial

  24. Cyclotron resonance effective mass –to be shown in the tutorial

  25. Density of states effective mass

  26. Density of states

  27. E g(E) Example: density of states of hydrogen gas

  28. E g(E) Density of states of solids

  29. E E f g(E) Density of electrons in an energy band

  30. E E f g(E) Density of holes in an energy band

  31. Approximation for the Fermi Dirac distribution for E-Ef>3KT

  32. E E f g(E) Density of electrons in an energy band

  33. Density of states of solids in K space

  34. The density of states in an energy interval is proportional to the volume in K space between two constant energy surfaces

  35. Constant energy surfaces in crystal momentum space cookies

  36. Volume of an ellipsoid

  37. Volume of a constant energy (E’) ellipsoid

  38. Density of states near a conduction band minimum or valence band maximum, and the definition of the density of states effective mass

  39. Conduction band density of states effective mass

  40. Valence band density of states effective mass

  41. Summary • Conductivity effective mass – determines mobility. • Density of states effective mass – determines NC • Cyclotron effective mass – can be measured directly

  42. k hh lh E Elective home exercise • Derive conductivity and density of states effective mass for holes.

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