15. Optical Processes and Excitons

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 kγ<< G for γ in IR to UV regions. → Onlyε(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, … XPS

3. Optical Reflectance Consider the reflection of light at normal incidence on a single crystal. Reflectivity coefficient 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 = χE. → →

5. Conditions on α for satisfying the Kronig-Kramer relation: • All poles of α(ω) are in the lower complex ω plane. • C d ωα /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 (ωj = 0 ): KKR → Consider the Ampere-Maxwell eq. Treating the e-gas as a pure metal: → Treating the e-gas as a pure dielectric: Fourier components: →  pole at ω = 0

7. Electronic Interband Transitions R & Iabs seemingly featureless. R Selection rule allows transitions  k  B.Z. dR/dλ → Not much info can be obtained from them? Saving graces: Modulation spectroscopy: dnR/dxn, where x = λ, E, T, P, σ, … Critical points where provide sharp features in dnR/dxn which can be easily calculated by pseudo-potential method (accuracy 0.1eV) Electroreflectance: d3R/dE3

8. Excitons Non-defect optical features below EG → 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 vg , i.e. at any critical points

10. 3 ways to measure Eex : • Optical absorption. • Recombination luminescence. • Photo-ionization of excitons • (high conc of excitons required). GaAs at 21K I = I0 exp(–αx) Eex = 3.4meV

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

12. The translational states of Frenkel excitons are Bloch functions. Consider a linear crystal of N non-interacting atoms. Ground state of crystal is uj = ground state of jth atom. If only 1 atom, say j , is excited: (N-fold degenerated) In the presence of interaction, φj is no longer an eigenstate. For the case of nearest neighbor interaction T : j = 1, …, N Consider the ansatz  ψk 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 (Eg ~ 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 where n = 1, 2, 3, … For Cu2O, 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. Cu2O at 77K absorption peaks Eg = 2.17eV = 17,508 cm–1

16. Exciton Condensation into Electron-Hole Drops (EHD) Ge: For sufficiently high exciton conc. ( e.g., 1013 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. EHD obs. by e-h recomb. lumin. Ge at 3.04K FE @ 714 meV : Doppler broadened. EHD @ 709 meV : Fermi gas n = 21017 cm−3.

17. Unstrained Si

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

19. 1st order Raman λinc = 5145A K  0 GaP at 20K. ωLO = 404 cm−1. ωTO = 366 cm−1 . 1st order: Largest doublet. 2nd 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. 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).  ωp 18eV. 5s Ag: εF = 0

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 =