Transcript Materials Considertations in Semiconductor Detectors

Materials Considerations in
Semiconductor Detectors
S W McKnight and
C A DiMarzio
Electrons in Solids: Schrodinger’s
Equation
Kinetic energy
Potential energy
Total energy
= electron wave function
= probability of finding
electron between x and x+dx
at time between t and t+dt
Normalization: Integral of ΨΨ* over all space and time=1
Wavefunction and Physical
Observables
Momentum:
Energy:
(Planck’s constant)
Time-independent Schrodinger’s
Equation
Separation of variables:
Solutions to Schrodinger’s
Equation
Free particle: V=0
Solution:
Wave traveling to left or right with:
E
Free Particle
k
Periodic Potentials
a
V(x)
x
where:
= “crystal momentum”
Bloch Theorem
Since this holds for any x+a, adding or subtracting any
number of reciprocal lattice vectors (2π/a) from crystal
momentum does not change wavefunction. Can
describe all electron states by considering k to lie in
interval (π/a > k > -π/a) (first Brillouin zone)
Physical Interpretation: electron can exchange
momentum with lattice in quanta of (2π/a)
“Empty Lattice”
V=0, but apply lattice periodicity, V(r+a)=V(r)
k 
2
a
E
k  

2
a


a
k

a
2
a
2
a
“Empty Lattice”: Reduced Zone
E
V=0, translated to First Brillouin Zone


a
k

a
Kronig-Penny Potential
a
V(x)
x
ψ(x)
Band Gaps
V≠0 lifts degeneracy at band crossings
E
Eg
Eg


a
k

a
Electron States in Band
Electron state “phase space” volume: ΔpxΔx=h
Number of electron states per unit length (per
spin) with –kf<k<kf = 2 kf / (2π)
Electron States in Band
Number electron states/unit length in band = [π/a – (-π/a)]/(2π)
= 1/a
E
Eg
Eg


a
k
Δk=2π

a
Photon Momentum vs. Crystal
Momentum
Photon momentum is small
compared to electron crystal
momentum
Optical Band Transitions
Momentum conservation implies optical
transitions in band are nearly vertical
E
Eg
Eg


a
k

a
Effective Mass Approximation
E
Near minimum:
0
k
m*=effective mass
“Hole” Approximation
E
Vacancy
k
Band energy = Filled band – electron vacancy
Hole effective mass =mh* <0
Semiconductor Band Structures
Semiconductor Band Structures
Direct and Indirect Gaps
• Direct-gap semiconductors
– Electrons and holes at same k
– Ge, GaAs, CdTe
– Strong coupling with light, Δk≈0
• Indirect-gap semiconductors
– Electrons at different k than holes
– Si
– Weak coupling with light, Δk≠0
•
•
•
•
Need phonon to conserve momentum
Multistep process: photon + electron(E, k) →
electron (E+hν, k) + phonon →
electron(E+hν, k+Δk)