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Materials Considerations in Semiconductor Detectors – II

Materials Considerations in Semiconductor Detectors – II. S W McKnight and C A DiMarzio. Band Filling Concept. Electron bands determined by lattice and ion core potentials Bands are filled by available conduction and valence electrons

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Materials Considerations in Semiconductor Detectors – II

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  1. Materials Considerations in Semiconductor Detectors–II S W McKnight and C A DiMarzio

  2. Band Filling Concept • Electron bands determined by lattice and ion core potentials • Bands are filled by available conduction and valence electrons • Pauli principle → only one electron of each spin in each (2π)3 volume in “k-space” • Bands filled up to “Fermi energy” • Fermi energy in band → metal • Fermi energy in gap → insulator/semiconductor • For T≠0, electron thermal energy distribution = “Fermi function”

  3. Eg E E Eg k k

  4. Metal Band Structure Eg Ef 2π

  5. Metal Under Electric Field Eg Ef

  6. Insulator/Semiconductor Band Structure Eg Ef 2π

  7. Temperature Effects “Occupancy” of state = PE,T = Probability that state at energy E will be occupied at temperature T = f(E) (“Fermi function) k=Boltzmann constant = 8.62 × 10-5 eV/K

  8. Fermi Function Limits • f(E)=0.5 for E=Ef • For kT<<Ef: E<Ef→ f(E) = 1 • For kT<<Ef: E>Ef→ f(E) = 0 • For (E-Ef)/kT >> 1: Boltzmann distribution

  9. Fermi Function vs. T Ef=1 eV kT=26 meV (300K) kT=52 meV (600K)

  10. “Density of States”: N(E) N(E) dE = number of electron states between E and E+dE n = number of electrons per unit volume

  11. Isotropic Parabolic Band

  12. Density of States: Isotropic, Parabolic Bands NT(E) = number of states/unit volume with energy<E = “k-space” volume/(2π)3 (per spin direction)

  13. Density of States: N(E)

  14. Isotropic Band Density of States (2 spin states)

  15. Electron-Hole Picture E Conduction Band Ec Eg k Ef Ev Electron Vacancy = “Hole” Valence Band Unoccupied state

  16. Electron-Hole Picture • n=number of electrons/(unit volume) in conduction band • p=number of vacancies (“holes”)/(unit volume) in valence band • For intrinsic (undoped) material: n=p=ni

  17. Integration of f(E)N(E) over Band Assume Ec-Ef >> kT → f(E) ≈ e -(E-Ef)/kT (Boltzmann distribution approximation)

  18. Integration of f(E)N(E) over Band Use definite integral:

  19. Intrinsic Carrier Concentration

  20. Semiconductor Band Structures

  21. Semiconductor Band Structures

  22. Real Band Effects Eg

  23. Real Band Effects • Thermal Eg≠ Optical Eg • Electron effective mass ≠ Hole effective mass • More than one electron/hole band • Multiple “pockets” • Overlapping bands • Anisotropic electron/hole pockets • Non-parabolic bands

  24. Effect of Real Band Effects • N(E)=sum of all bands and all pockets • md*=“density of states” mass • md*(electrons) ≠ md* (holes) • Fermi level for T≠0 moves toward band with smaller density of states (smaller md*) • ni=pi fixes position of Ef

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