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. k E Eg

4. Metal Band Structure Eg Ef 2π

5. Metal Under Electric FieldEg Ef

6. Insulator/Semiconductor Band StructureEg 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
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