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Materials Considerations in Semiconductor Detectors

Study the materials considerations in semiconductor detectors, including Schrodinger’s equation for electron wave functions, momentum, energy, and band structures in solids. Understand the concepts of effective mass approximation and optical band transitions. Differentiate between direct-gap and indirect-gap semiconductors in terms of electron and hole behavior and their coupling with light.

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Materials Considerations in Semiconductor Detectors

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  1. Materials Considerations in Semiconductor Detectors S W McKnight and C A DiMarzio

  2. 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

  3. Wavefunction and Physical Observables Momentum: Energy: (Planck’s constant)

  4. Time-independent Schrodinger’s Equation Separation of variables:

  5. Solutions to Schrodinger’s Equation Free particle: V=0 Solution: Wave traveling to left or right with:

  6. Free Particle

  7. Periodic Potentials a V(x) x where: = “crystal momentum”

  8. 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)

  9. E k “Empty Lattice” V=0, but apply lattice periodicity, V(r+a)=V(r) E k

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

  11. Kronig-Penny Potential a V(x) x ψ(x)

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

  13. 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)

  14. 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

  15. Photon Momentum vs. Crystal Momentum Photon momentum is small compared to electron crystal momentum

  16. E k Optical Band Transitions Momentum conservation implies optical transitions in band are nearly vertical Eg E Eg k

  17. Effective Mass Approximation E k Near minimum: 0 m*=effective mass

  18. “Hole” Approximation E Vacancy k Band energy = Filled band – electron vacancy Hole effective mass =mh* <0

  19. Semiconductor Band Structures

  20. Semiconductor Band Structures

  21. 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)

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