Lecture 3: Optical Properties of Bulk and Nano 5 nm
The Previous Lecture Origin frequency dependence of χ in real materials Lorentz model (harmonic oscillator model) 0 e - n( ) n' n '' n ' = 1 + Nucleus Today optical properties of materials Insulators (Lattice absorption, color centers ) Semiconductors (Energy bands, Urbach tail, excitons ) 0 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: H+ E Two atoms: bond formation H +? + H Every electron contributes one state Equilibrium distance d (after reaction)
Classification Matter ~ 1 ev Pauli principle: Only electrons in the same electronic state (one spin & one spin )
Atoms with many electrons Classification Matter Empty outer orbitals Partly filled valence orbitals Outermost electrons interact Form bands Energy Filled Inner shells Electrons in inner shells do not interact 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 Refractive index: n 3.0.0 1.0 0.1 1.0 10 λ (μm)
Color Centers Insulators with a large E GAP 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 E C Phonon Photon E V Phonon
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 = -3.4 ev n = 3-1.51 ev Binding energy electron: E B 4 me e 13.6 = = ev, n= 1,,3,... ( 4πε 0hn) n h Where: m e = Electron mass, ε 0 = 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/ = 1/ e + 1/ m m m h Correct for the relative dielectric constant of Si, ε r,si (screening). e - ε r,si h - Binding energy electron: E Typical value for semiconductors: B * m = 1 13.6 ev, n= 1,,3,... m ε e E = 10meV 100meV B 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: Equation of motion electron in a metal: E { } ( t) = Re E( ) exp( it) d x dv v m = = m ee dt dt τ relaxation time ~ 10-14 s { } Look for a steady state solution: v( t) = Re v( ) exp( it) Substitution v into Eq. of motion: i mv( ) This can be manipulated into: The current density is defined as: J( ) = ne It thus follows: J( ) ( ) mv = ee( ) τ e v( ) = E( ) m 1 τ i ( ) v Electron density = ( ne m) ( 1 τ i) E ( )
Determination conductivity From the last page: Optical Properties of Metals J ( ) = ( ne m) ( 1 τ i) E ( ) = σ ( ) E( ) ( ) ( ) σ ( ne m) = = σ ( 1τ i) ( 1 iτ) Definition conductivity: J ne τ where: σ 0 = m Both bound electrons and conduction electrons contribute to ε From the curl Eq.: For a time varying field: () t () D ε B E t H = + J = + J t t E { } ( t) = Re E( ) exp( it) ε E σ H = + J = iε E + σ E = ε ε E ( ) ( ) ( ) ( ) i ( ) ( ) B B 0 B t iε 0 ε EFF ( ) Currents induced by ac-fields modeled by ε EFF σ For a time varying field: ε ε ε iε Bound electrons EFF = B = B + i σ ε 0 0 Conduction electrons 0 ( )
Optical Properties of Metals Dielectric constant at visible σ σ0 ( 1+ iτ 0 ) σ0 σ0 Since vis τ >> 1: σ ( ) = = + i 1 τ 1+ τ τ It follows that: ( i ) ( τ ) σ σ σ ε = ε + i = ε + i ε ε τ ε τ 0 0 EFF B B 3 0 0 0 σ 0 ne p 10eV for metals ετ 0 ε0m Define: ( ) ε p p EFF = εb + i 3 τ 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 8 8 6 6 ε p p EFF = εb + i 3 τ Dielectric functions 4 0 - -4-6 -8-10 ε ε n n Aluminum ħ p 4 0 - -4-6 -8-10 Only conduction e s contribute to: ε ε ε B 1 p p EFF, Al 1 + i 3 Agrees with: ε ( ) ε ' EFF τ ε '' ε ' = 1 0 5 10 15 Photon energy (ev) 0
Reflectance (%) Ag: effects of Interband Transitions 100 60 40 0 10 Ag 6 4 0 4 6 8 10 1 Photon energy (ev) Ag show interesting feature in reflection Both conduction and bound e s contribute to ε EFF 6 4 ε ib Feature caused by interband transitions Excitation bound electrons ε 0 ε EFF - -4 p ε f ' = 1-6 0 4 6 8 10 1 For Ag: ε = ε 0 ε B interband ib p p EFF = εib + i 3 τ Photon energy (ev)
Summary Light interaction with small objects (d < λ) Light scattering due to harmonically driven dipole oscillator Nanoparticles Insulators Rayleigh Scattering (blue sky) Semiconductors...Resonance absorption at ħ E GAP, size dependent fluorescence ) Metals Resonance absorption at surface plasmon frequency, no light emission) Microparticles Particles with dimensions on the order of λ Enhanced forward scattering λ-independent (white clouds) Microspheres with diameters much larger than λ Intuitive ray-picture useful Rainbows due to dispersion H 0 Applications: resonators, lasers, etc