1. 15. Optical Processes and Excitons Optical Reflectance Kramers-Kronig Relations Example: Conductivity of Collisionless Electron Gas Electronic Interband Transitions Excitons Frenkel Excitons Alkali Halides Molecular Crystals Weakly Bound (Mott-Wannier) Excitons Exciton Condensation into Electron-Hole Drops (Ehd) Raman Effect in Crystals Electron Spectroscopy with X-Rays Energy Loss of Fast Particles in a Solid

2. Optical Processes Raman scattering: Brillouin scattering for acoustic phonons. Polariton scattering for optical phonons. + phonon emission (Stokes process) – phonon absorption (anti-Stokes) 2-phonon creation XPS k γ << G for γ in IR to UV regions. → Only ε(ω) = ε(ω,0) need be considered. Theoretically, all responses of solid to EM fields are known if ε(ω,K) is known. ε is not directly measurable. Some measurable quantities: R, n, K, …

3. Optical Reflectance Reflectivity coefficient Consider the reflection of light at normal incidence on a single crystal. Let n(ω) be the refractive index and K(ω) be the extinction coefficient. → see Prob.3 Complex refractive index Let Reflectance(easily measured) θ is difficult to measure but can be calculated via the Kramer-Kronig relation. →

4. Kramers-Kronig Relations Re α(ω)  KKR → Im α(ω) α = linear response Equation of motion: (driven damped uncoupled oscillators) Fourier transform: Linear response:→ → Let α be the dielectric polarizability χ so that P = χ E. → →

5. Conditions on α for satisfying the Kronig-Kramer relation: All poles of α(ω) are in the lower complex ω plane.  C d ω α /ω = 0 if C = infinite semicircle in the upper-half complex ω plane. It suffices to have α → 0 as |ω | → . α(ω) is even and α  (ω) is odd w.r.t. real ω.

6. Example: Conductivity of Collisionless Electron Gas For a free e-gas with no collisions (ω j = 0 ): KKR Consider the Ampere-Maxwell eq. Treating the e-gas as a pure dielectric: Fourier components:→  pole at ω = 0 Treating the e-gas as a pure metal: → →

7. Electronic Interband Transitions R & I abs seemingly featureless. Selection rule allows transitions  k  B.Z. → Not much info can be obtained from them? Saving graces: Modulation spectroscopy: d n R/dx n, where x = λ, E, T, P, σ, … Critical points where provide sharp features in d n R/dx n which can be easily calculated by pseudo- potential method (accuracy 0.1eV) dR/dλ Electroreflectance: d 3 R/dE 3 R

8. Excitons Non-defect optical features below E G → e-h pairs (excitons). Frenkel exciton Mott-Wannier exciton Properties: Can be found in all non-metals. For indirect band gap materials, excitons near direct gaps may be unstable. All excitons are ultimately unstable against recombination. Exciton complexes (e.g., biexcitons) are possible.

9. Exciton can be formed if e & h have the same v g, i.e. at any critical points

10. GaAs at 21K I = I 0 exp(–α x) E ex = 3.4meV 3 ways to measure E ex : Optical absorption. Recombination luminescence. Photo-ionization of excitons (high conc of excitons required).

11. Frenkel Excitons Frenkel exciton: e,h excited states of same atom; moves by hopping. E.g., inert gas crystals. Kr at 20K Lowest atomic transition of Kr = 9.99eV. In crystal it’s 10.17eV. E g = 11.7eV → E ex = 1.5eV

12. The translational states of Frenkel excitons are Bloch functions. Consider a linear crystal of N non-interacting atoms. Ground state of crystal is u j = ground state of j th atom. If only 1 atom, say j, is excited: (N-fold degenerated) In the presence of interaction, φ j is no longer an eigenstate. For the case of nearest neighbor interaction T : j = 1, …, N Consider the ansatz  ψ k is an eigenstate with eigenvalue Periodic B.C. →

13. Alkali Halides The negative halogens have lower excitation levels → (Frenkel) excitons are localized around them. Pure AH crystals are transparent (E g ~ 10 eV) → strong excitonic absorption in the UV range. Prominent doublet structure for NaBr ( iso-electronic with Kr ) Splitting caused by spin-orbit coupling.

14. Molecular Crystals Molecular binding >> van der Waal binding → Frenkel excitons Excitations of molecules become excitons in crystal ( with energy shifts ). Davydov splitting introduces more structure in crystal (Prob 7).

15. Weakly Bound (Mott-Wannier) Excitons Bound states of e-h pair interacting via Coulomb potential are wheren = 1, 2, 3, … Cu 2 O at 77K absorption peaks E g = 2.17eV = 17,508 cm –1 For Cu 2 O, agreement with experiment is excellent except for n = 1 transition. Empirical shift for data fit gives With ε = 10, this gives μ = 0.7 m.

16. Exciton Condensation into Electron-Hole Drops (EHD) Ge: For sufficiently high exciton conc. ( e.g., 10 13 cm −3 at 2K ), an EHD is formed. → τ ~ 40 µs ( ~ 600 µs in strained Ge ) Within EHD, excitons dissolve into metallic degenerate gas of e & h. Ge at 3.04K FE @ 714 meV : Doppler broadened. EHD @ 709 meV : Fermi gas n = 2  10 17 cm −3. EHD obs. by e-h recomb. lumin.

17. Unstrained Si

18. Raman Effect in Crystals 1 st order Raman effect (1 phonon ) Cause: strain-dependence of electronic polarizability α. u = phonon amplitudeLet Induced dipole: StokesAnti-Stokes App.C: →

19. 1 st order Raman λ inc = 5145A K  0 GaP at 20K. ω LO = 404 cm −1. ω TO = 366 cm −1. 1 st order: Largest doublet. 2 nd order: the rest. Si

20. Electron Spectroscopy with X-Rays XPS = X-ray Photoemission Spectroscopy UPS = Ultra-violet Photoemission Spectroscopy Monochromatic radiation on sample : KE of photoelectrons analyzed. → DOS of VB (resolution ~ 10meV) Only e up to ~ 50A below surface can escape. Ag: ε F = 0 5s 4d Excitations from deeper levels are often accompanied by plasmons. E.g., for Si, 2p pk ~99.2eV is replicated at 117eV (1 plasmon) and at 134.7eV (2 plasmons).  ω p  18eV.

21. Energy Loss of Fast Particles in a Solid Energy loss of charged particles measures Im( 1/ε ). Power dissipation density by dielectric loss: EM wave: Particle of charge e & velocity v : Isotropic medium:

22. Energy loss function =