Materials Considerations in Semiconductor Detectors

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Presentation transcript:

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:

Free Particle

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 E E

“Empty Lattice”: Reduced Zone V=0, translated to First Brillouin Zone k E E k

Kronig-Penny Potential V(x) x ψ(x)

Band Gaps V≠0 lifts degeneracy at band crossings k E Eg E Eg k

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 k E Eg E Eg k Δk=2π

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 k E Eg E Eg k

Effective Mass Approximation k Near minimum: m*=effective mass

“Hole” Approximation E k Hole effective mass =mh* <0 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)