Lecture 3: Optical Properties of Bulk and Nano 5 nm Course Info First H/W#1 is due Sept. 10 The Previous Lecture Origin frequency dependence of χ in real materials • Lorentz model (harmonic oscillator model) e- n(ω) n' n'' n'1= ω0 + Nucleus ω0 ω Today optical properties of materials • Insulators (Lattice absorption, color centers…) • Semiconductors (Energy bands, Urbach tail, excitons …) • Metals (Response due to bound and free electrons, plasma oscillations.. ) Optical properties of molecules, nanoparticles, and microparticles Classification Matter: Insulators, Semiconductors, Metals Bonds and bands • One atom, e.g. H. Schrödinger equation: E H+ • Two atoms: bond formation ? H+ +H Every electron contributes one state • Equilibrium distance d (after reaction) Classification Matter ~ 1 eV • Pauli principle: Only 2 electrons in the same electronic state (one spin & one spin ) Classification Matter Atoms with many electrons Empty outer orbitals Outermost electrons interact Form bands Partly filled valence orbitals Filled Energy Inner Electrons in inner shells do not interact shells Do not form bands Distance between atoms Classification Matter Insulators, semiconductors, and metals • Classification based on bandstructure Dispersion and Absorption in Insulators Electronic transitions No transitions Atomic vibrations Refractive Index Various Materials 3.4 3.0 2.0 Refractive index: n’ index: Refractive 1.0 0.1 1.0 10 λ (µm) Color Centers • Insulators with a large EGAP should not show absorption…..or ? • Ion beam irradiation or x-ray exposure result in beautiful colors! • Due to formation of color (absorption) centers….(Homework assignment) Absorption Processes in Semiconductors Absorption spectrum of a typical semiconductor E EC Phonon Photon EV ωPhonon Excitons: Electron and Hole Bound by Coulomb Analogy with H-atom • Electron orbit around a hole is similar to the electron orbit around a H-core • 1913 Niels Bohr: Electron restricted to well-defined orbits n = 1 + -13.6 eV n = 2 n = 3 -3.4 eV -1.51 eV me4 13.6 • Binding energy electron: =−=−=e EB 2 2 eV, n 1,2,3,... 24( πε 0n) n Where: me = Electron mass, ε0 = permittivity of vacuum, = Planck’s constant n = energy quantum number/orbit identifier Binding Energy of an Electron to Hole Electron orbit “around” a hole • Electron orbit is expected to be qualitatively similar to a H-atom. • Use reduced effective mass instead of m : * e 1/m= 1/ mmeh + 1/ • Correct for the relative dielectric constant of Si, εr,Si (screening). e- εr,Si h- m* 1 Binding energy electron: = = EB 2213.6eV , n 1,2,3,... mne ε • Typical value for semiconductors: EB =10 meV − 100 meV • Note: Exciton Bohr radius ~ 5 nm (many lattice constants) Optical Properties of Metals (determine ε) Current induced by a time varying field • Consider a time varying field: EE(t) = Re{ (ωω) exp(−it)} dd2xv v • Equation of motion electron in a metal: m==−− meE dt2 dt τ relaxation time ~ 10-14 s • Look for a steady state solution: vv(t) = Re{ (ωω) exp(−it)} mv(ω) • Substitution v into Eq. of motion: −imωωvE( ) =−−e(ω) τ −e • This can be manipulated into: vE(ωω) = ( ) mi(1 τω− ) • The current density is defined as: Jv(ω) = −ne Electron density (ne2 m) • It thus follows: JE(ωω) = ( ) (1 τω− i ) Optical Properties of Metals Determination conductivity (ne2 m) • From the last page: JE(ωω) = ( ) (ne2 m) σ (1 τω− i ) σω( ) = = 0 (11τ−−ii ω) ( ωτ ) • Definition conductivity: JE(ω) = σω( ) ( ω) ne2τ where: σ = 0 m Both bound electrons and conduction electrons contribute to ε ∂D ∂ε E (t) • From the curl Eq.: ∇×HJ = + =B + J ∂∂tt • For a time varying field: EE(t) = Re{ (ωω) exp(−it)} ∂εB E (t) σω( ) ∇×=+=−+=−−H JiiωεωBB( ) EE( ωσω) ( ) ( ω) ωεεω0 ( ) E(ω) ∂t iεω0 εω Currents induced by ac-fields modeled by εEFF EFF ( ) σσ • For a time varying field: εεEFF=−=+ B εB i iεω00 εω Bound electrons Conduction electrons Optical Properties of Metals Dielectric constant at ω ≈ ωvisible σ+ ωτ σ0 0 (1 i ) σσ00 • Since ωvisτ >> 1: σω( ) = = ≈+i (1− iωτ ) (1+ωτ22) ω22 τ ωτ σ σσ • It follows that:εε=+=+ ε 00 − EFF B iiB 32 2 εω0 εωτ 00 εωτ 2 2 σ 0 ne • Define: ω p = =( ≈10eV for metals) ετ00 εm ωω22 εε=−+ppi EFF B ω23 ωτ Bound electrons Free electrons • What does this look like for a real metal? Optical Properties of Aluminum (simple case) 0 5 10 15 10 10 ωω22 8 8 εε=−+ppi EFF B ω23 ωτ 6 6 4 ε’’ 4 • Only conduction e’s 2 2 contribute to: ε EFF ħωp 0 0 ε B ≈1 -2 -2 ωω22 Aluminum pp ε EFF, Al ≈−1 +i -4 -4 ω23 ωτ Dielectric functions ε -6 ’ -6 • Agrees with: -8 -8 εω( ) -10 -10 ε ' n’ n’’ ε '' ε '1= 0 5 10 15 Photon energy (eV) ω0 ω Ag: effects of Interband Transitions 100 60 Ag • Ag show interesting feature in reflection 40 20 • Both conduction and bound e’s contribute 10 to ε EFF 6 Reflectance (%) 4 2 0 2 4 6 8 10 12 Photon energy (eV) 6 • Feature caused by interband transitions εib 4 Excitation bound electrons 2 interband ’ ε 0 • For Ag: εεB= ib ≠ 0 ε EFF -2 2 ω p 22 ε '1= − ωωpp -4 f ω 2 εε=−+i EFF ib ω23 ωτ -6 0 2 4 6 8 10 12 Photon energy (eV).