210 likes | 216 Views
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.
E N D
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:
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)
E k “Empty Lattice” V=0, but apply lattice periodicity, V(r+a)=V(r) E k
E k “Empty Lattice”: Reduced Zone V=0, translated to First Brillouin Zone E k
Kronig-Penny Potential a V(x) x ψ(x)
E k Band Gaps V≠0 lifts degeneracy at band crossings 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)
E k Electron States in Band Number electron states/unit length in band = [π/a – (-π/a)]/(2π) = 1/a Eg E Eg k Δk=2π
Photon Momentum vs. Crystal Momentum Photon momentum is small compared to electron crystal momentum
E k Optical Band Transitions Momentum conservation implies optical transitions in band are nearly vertical Eg E Eg k
Effective Mass Approximation E k Near minimum: 0 m*=effective mass
“Hole” Approximation E Vacancy k Band energy = Filled band – electron vacancy Hole effective mass =mh* <0
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)