Lecture 8 – Light-Matter Interaction Part 2 Basic excitation and coupling EECS 598-002 Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku Schedule for the rest of the semester Introduction to light-matter interaction (1/26): How to determine ε(r)? The relationship to basic excitations. Basic excitations and measurement of ε(r). (1/31) Structure dependence of ε(r) overview (2/2) Surface effects (2/7 & 2/9): Surface EM wave Surface polaritons Size dependence Case studies (2/14 – 2/21): Quantum wells, wires, and dots Nanophotonics in microscopy Nanophotonics in plasmonics Dispersion engineering (2/23 – 3/9): Material dispersion Waveguide dispersion (photonic crystals) EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 2 Last time We learned: To determine ε(r), we need to study how the microscopic interaction between atoms/electrons with the light. This interaction is similar to coupling of two SHO’s. The only details we need to know are the interaction near resonances of basic excitations. The rest of the information needed to complete the calculation of ε(r) is through the Kramers-Kronig relation. EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 3 Today Basic excitations by photons Æ polaritons Plasmons (last time) Phonons Excitons, biexcitons, etc. Measurement of ε(r) Ref: D. L. Mills and E. Burstein, “Polaritons,” Rep. Prog. Phys., 37 (1974) 817. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, 2nd ed., Springer-Verlag (1999) chapters 6 and 7. EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 4 Review of the concept of polaritons In a dielectric medium: ⎡⎤ ⎢⎥ω2 Nq2 ( ) 1pn where 2 nn εω=− ε0 ⎢⎥∑ 22 ωpn ≡ ⎢⎥n basic ()ω −+ωγω0n0nnim ε excitations ⎣⎦⎢⎥or oscillators 2 ωp =εε∞ − 0 22 ()ωω−+0 i γω where the index denotes the n-th kind of basic excitation or SHO. QM analogue: 2 2 2 nq exˆ ωp =→ ε00m ε EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 5 Transverse and longitudinal vibrations Longitudinally vibrated SHO’s Vertically vibrated SHO’s EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 6 Transverse and longitudinal polaritons For homogeneous media: K ∇⋅D =0 KKK ⇒∇⋅()εεEkE =00 ⇒ ⋅ = K K ⇒⋅kE =0 or ε =0 Normally EM wave couples only to transverse SHO’s unless the dielectric constant vanishes. EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 7 Longitudinal polaritons k 2 ε ==2 0 ωµ0 EM wave can couple to the longitudinal vibration. Photons ω In free space 22 ω0 + ωp ω0 k EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 8 Phonons M m a vn un optical ⎧munnnnn=−αα()() uv −−+11 − uv − ω ⎨ ⎩Mvnnnnn=−αα()() vu −−+11 − vu − By periodicity: ⎪⎧uun =−exp[ inkat (2ω )] ⎨ acoustic ⎩⎪vvn =+−exp[] inkat (2( 1)ω ) ⎧ 2 ⎡⎤ika− ika ⎪⎣−=muωα ve() +− e2 u ⎦ ⇒ ⎨ k −=Mvωα2 ⎡⎤ ueeika +−− ika 2 v π / a ⎩⎪ ⎣⎦() 2 1/2 2 ⎛⎞⎛⎞mM++−⎡ mM2(1 cos ka )⎤ ⇒=ωα⎜⎟⎜⎟ ± α⎢ − ⎥ ⎝⎠⎝⎠mM⎣⎢ mM mM ⎦⎥ EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 9 Orders of magnitude 7 At optical frequencies, k=ω/c~10 . 10 For typical crystal lattice, π/a~10 . Only optical phonons couple to the light. k ≈ 0 For optical branch: u M ≈− Can generate the dipole moment v m For acoustic branch: u ≈ 1 v EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 10 Examples of phonon dispersion curves silicon GaAs Taken from P. Yu and M. Cardona. EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 11 Dispersion curve for phonon polaritons Free-space-photon-like Photons In free space ω 22 ω0 +=ωωpLO ω0 = ωTO k Photon-like but with strong phonon influence EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 12 Raman processes When light is not at the infrared frequency, phonons can still participate in the inelastic processes with light Æ Raman processes. The scattered light has a frequency shift w.r.t to the incident light due to its energy lost (or gain) to phonons. Energy lost: Stokes process Energy gain: Anti-Stokes process EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 13 Excitons (two-level system) el The Coulomb interaction b/w electron and hole makes the exciton. Exciton is like a hydrogen atom. hole Exciton absorption Coulomb enhancement Eg E EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 14 Hot carriers relaxation processes Carrier capture Phase relaxation T2~100fs - ps k Thermalization Recombination (T1~ns) EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 15 Size dependence (quantum confinement) g(E) = Density of states dot wire bulk sheet 3D 2D 1D 0D g(E) g(E) g(E) g(E) Eg E Eg E Eg E Eg E EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 16 Exciton absorption in low-dim structures Exciton binding energy: Exciton absorption: 23dd 2D: 2D: EEB,1nBn=== 4 ,1 1D: 1D: a0=exciton Bohr radius~100A EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 17 Excitons (three-level system) If we consider the polariton corresponding to exciton 1: EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 18 Electromagnetically induced transparency 1 α ωp= ω31 3 ω = ω s 21 n 2 EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 19 Measurement of ε(r) - ellipsometry ellipsometry Sensitive to: Need to know underlying 1. Film thickness composition of materials. 2. Surface roughness 3. Anisotropy EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 20.