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

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, …

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. →

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. → →

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.

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: → →

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

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.

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

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).

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

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. →

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.

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).

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.

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

Unstrained Si

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: →

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

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.

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:

Energy loss function =