Transcript Lecture Notes 6 - Materials Considertations 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
• 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”
E
Eg
Eg


a
k

a
Metal Band Structure
Eg
Ef


a
k
2π

a
Eg
Metal Under Electric Field
Eg
Ef


a
k

a
Eg
Insulator/Semiconductor Band
Structure
Eg
Ef


a
k
2π

a
Eg
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
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
Fermi Function vs. T
1.2
Ef=1 eV
kT=26 meV (300K)
1
kT=52 meV (600K)
f(E)
0.8
T=0
0.6
0.4
T=300K
T=600K
0.2
0
0
0.2
0.4
0.6
0.8
1
E/E
1.2
1.4
1.6
1.8
2
“Density of States”: N(E)
N(E) dE = number of electron states between E and
E+dE
n = number of electrons per unit volume
E
Isotropic Parabolic Band
k
Density of States:
Isotropic, Parabolic Bands
NT(E) = number of states/unit volume with energy<E
= “k-space” volume/(2π)3 (per spin direction)
Density of States: N(E)
Isotropic Band Density of States
(2 spin states)
Electron-Hole Picture
E
Conduction
Band
Ec
Ef
Electron
Eg
Ev
Vacancy
= “Hole”
Valence
Band
Unoccupied state
k
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
Integration of f(E)N(E) over Band
Assume Ec-Ef >> kT → f(E) ≈ e -(E-Ef)/kT
(Boltzmann distribution approximation)
Integration of f(E)N(E) over Band
Use definite integral:
Intrinsic Carrier Concentration
Semiconductor Band Structures
Semiconductor Band Structures
Real Band Effects
Eg
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
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