Low Energy Radiation Transport Applied to Secondary Electron Emission

1. Low Energy Radiation Transport Applied to Secondary Electron Emission H. P. Hjalmarson, R. P. Kensek, R. J. Magyar and R. J. Bondi Sandia National Labs International Conference on Transport Theory September 15-20, 2013 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000

2. Secondary Electron Emission (SEE): Basic Phenomena Energy relaxation processes Plasmon emission Impact ionization Phonon emission Emission from the solid

3. Outline • Single and multiple scattering • Classical and quantum • Distant collisions (density effects) • CSDA • Dielectric function • Boltzmann equation: Monte Carlo • Based on wavefunctions and bandstructure • SEE yield for low energy electrons • Proposed idea • Assume free particles but ignore multiple elastic scattering (50 eV to a few keV)

4. Scattering • Single Scattering • Multiple Scattering • Used in a bandstructure method (KKR method) • Becomes classical at high energies (interference terms washed out)

5. Two Types of Electron Collisions • Close • Classical Rutherford cross-section • Distant • Polarization of the target by the incident electron • Implementation • Continuous Slowing Down Approximation (CSDA) • Plasmon emission (one by one)

6. Dielectric Function Close and distant collision effects included in the dielectric function • Billiard ball collisions • Impact Ionization • Plasmon emission The key quantity in the low energy Monte Carlo calculations Depends on wavefunctions and bandstructure

7. Methods • Multiple classical scattering & CSDA • Elastic scattering, inelastic scattering and CSDA • Multiple classical scattering & dielectric function • Elastic scattering & the dielectric function • Multiple quantum scattering & dielectric function • Wavefunction includes all the elastic scattering • Bandstructure controls particle transport • Dielectric function, using this information, controls all scattering • Phonons included at very low energies

8. Evolution in Momentum Space (K-Space) Free particles: Bandstructure particles: Each electron or hole is a particle in momentum space E (eV) K. E. Kambour Thesis

9. Ensemble Monte Carlo (EMC) • Injection of energetic electrons at a constant rate • “Free flight” of electrons and holes in momentum space • Solve the equations of motion for momentum and position • Scattering (changes in momentum and energy) • Plasmon emission • Impact ionization • Phonon emission • Radiative recombination • Emission from the sample (with no return) • Collection or emission

10. Evolution in Momentum Space (K-Space) Each electron or hole is a particle in momentum space Jumps to a different point caused by scattering E (eV) K. E. Kambour Thesis

11. Free Flight: Newton’s Laws F = electric field, q = electron charge, m = effective mass

12. Scattering: Impact Ionization E(k) k Collisions must conserve energy and momentum

13. Scattering Physics • Mechanisms • Plasmon emission • Impact ionization • Phonon emission • Wavefunctions and bandstructure • Lowest energy bands control the effects • Higher energy bands not important • An oscillator-strength sum rule test makes this statement rigorous

14. Secondary Electron Emission (SEE): Basic Phenomena Energy relaxation processes Plasmon emission Impact ionization Phonon emission Emission from the solid

15. Calculations • Electrons are injected at a constant rate • The energetic electrons undergo transport and the various scattering processes • If their energy exceeds the plasmon energy, they can emit plasmons. Plasmon create electron-hole pairs. • If their energy exceed the bandgap energy, they create electron-hole pairs. • Below this energy, the electrons and holes emit phonons • The electrons and holes have three fates • Some are collected at a back contact • Some remain near the interface until they recombine. At very long times, the solid can become charged. • A small fraction are emitted as secondary electrons

16. Results: One Specific Calculation (3 eV incident electrons) The electrons are absorbed • 0.02 electron emitted for each incident electron • Most electrons are collected at the back end

17. Insights • Wavefunctions • The oscillator strength sum rule tells us that higher energy wavefunctions not important • Bandstructure energies • Free particle-like at high energies • New Idea • Free particles • Omit elastic scattering • Include plasmon emission (for changes in direction)

18. Proposed Approach in Future Work Ignore multiple classical scattering at intermediate energies • Assume bandstructure • Find details not important • Approximate as free particles Key bandstructure effect: A forbidden gap

19. Summary • Dielectric function • Contains all the electron scattering information • Bandstructure details not important at high energies • Secondary Electron Emission • Monte Carlo solution of the Boltzmann equation • Results show SEE yield is small • Proposal for future work • Bandstructure approach in spirit • Use free electrons for intermediate energies