MTLE-6120: Advanced Electronic Properties of Materials Optical - - PowerPoint PPT Presentation
1 MTLE-6120: Advanced Electronic Properties of Materials Optical properties of Materials Reading: Kasap: 9.1 - 9.18 2 Maxwells equations in free media Linear response of materials described very generally by ( ) and ( )
◮ Kasap: 9.1 - 9.18
◮ Linear response of materials described very generally by ǫ(ω) and µ(ω) ◮ Maxwell’s equations in the absence of free charges and currents
◮ Substitute second equation in curl of third equation:
◮ Write Maxwell’s equation in linear media with no free charge or current as:
◮ This is exactly the equation of a wave with speed v
◮ Solution to the wave equation with |
k· r
k· r−ωt)
◮ EM wave propagating along x:
◮ Linearly combine to
◮ Photons: circular polarizations have spin angular momentum ±
◮ Electromagnetic waves satisfy dispersion relation ω = v|
◮ EM wave velocity v(ω) = 1/
◮ In vacuum, ǫ(ω) = ǫ0 and µ(ω) = µ0, so that speed of light in free space
◮ In materials, speed of light usually specified by refractive index
◮ Most materials are non-magnetic µ ≈ µ0 ◮ Therefore n2(ω) ≈ ǫr ≡ ǫ(ω)/ǫ0 ◮ Several contributions to relative permittivity:
e
e
e
e ◮ Frequency dependence (Drude-Lorentz model):
p
0α
0α − iγαω − ω2
◮ First Drude term proportional to free carrier density:
◮ Dominates for metals ◮ Present to varying degrees for doped semiconductors
◮ Remaining terms contribute in all materials
◮ Frequency dependence (Drude-Lorentz model):
0α
0α − iγαω − ω2
0 → ω2 p, ω0 → 0 and γ → 1/τ) ◮ Equivalently in terms of wavelength λ = 2πc/ω:
0α
◮ Can use a number of α in the empirical Sellmeier relation instead:
α
◮ The n2 we have been working with so far has been implicitly complex
◮ Customary to call complex refractive index N = n + iK
◮ Wave speed v = c N = c n+iK is complex ◮ Wave vector k = ω v = ω c (n + iK) = k0(n + iK) is complex ◮ What do the imaginary parts here mean? ◮ Wave function (fields) ∝ eikx = ei(nk0)x · e−(Kk0)x ◮ Wave intensity ∝ e−(2Kk0)x (proportional to |field|2) ◮ ⇒ Wave attenuates as e−αx with absorption coefficient α = 2Kk0
1 2 3 4 5 0.5 1 1.5 2 α [q2/k] ω [(k/m)1/2] Re(α) Im(α)
◮ Real and imaginary parts of ǫ(ω) (and hence n(ω)) not independent ◮ Causality ⇒ response functions (eg. χ(ω))
◮ Leads to the Kramers-Kronig relations:
−∞
−∞
◮ Absorption at ω0 affects n at all ω
◮ Free electrons: resistive loss in Drude term ◮ Bound electrons: absorption excites electronic transitions ◮ Ions: absorption excites phonons ◮ Dipoles: absorption excites molecular rotations ◮ Simple model so far (Lorentz) assumed single resonant frequency ω0 ◮ Each of the above processes have a band of absorption frequencies instead
◮ Optical phonons in ionic crystals: charge oscillations ⇒ interact with EM
◮ High DOS for transverse and longitudinal optical phonons ◮ Absorption peaks at corresponding wavelengths: Restrahlen band ◮ Typically in the mid-infrared part of the spectrum
◮ Simple model for inuslators so far: electrons bound by springs ◮ In reality: electrons delocalized in insulators / semiconductors too ◮ Only no net current in completely filled (or empty) bands ◮ Low frequency response is effectively like bound electrons (because vd = 0) ◮ Bound model obtained response of form:
0 − iγω − ω2 ◮ Interpretation of behavior near ω ∼ ω0 needs revision ◮ Energy absorbed from light must be taken up by the electrons ◮ How can you increase the electronic energy of insulators? ◮ Move an electron from the valence to conduction band (leave hole behind) ◮ What is the minimum frequency that can do this? ω = Eg ◮ Effectively ω0 now corresponds to Eg/
◮ Light excites electron in valence band (initial energy Ei < 0) to conduction
◮ Energy conservation: Ei + ω = Ej ⇒ ω = (Ej − Ei)/ > Eg/ ◮ Momentum conservation ki + ω/c = kj ⇒ kj − ki = ω/c = 2π/λ ◮ Typical Eg ∼ 1 − 10 eV with λ∼ 100 nm - 1 µm (NIR, visible, UV) ◮ Typical ki, kj ∼ 2π/a with a < 1 nm ◮ ⇒ photon momentum is negligible in electronic processes ◮ Effectively ki = kj (vertical transitions) for optical absorption
◮ Direct band-gap semiconductors (eg. GaAs): VBM and CBM at same k ◮ Possible to have Ej − Ei = Eg with ki = kj ◮ Direct absorption: light produces electron-hole pair ◮ Allowed here for ω ≥ Eg/
Phonon Photon
◮ Indirect band-gap semiconductors (eg. Si): VBM and CBM at different k ◮ Not possible to have Ej − Ei = Eg with ki = kj ◮ Indirect absorption: light produces electron-hole pair ± phonon ◮ Conservation: Ej − Ei ± ωph = ω, kj ± kph = ki ◮ Negligible energy in phonon, negligible momentum in photon ◮ Direct absorption becomes possible above interband threshold Et
◮ Direct interband transitions from d-bands to Fermi level ◮ Indirect transitions possible in two ways:
◮ Get momentum from phonons (lattice vibrations) ◮ Geometry effect in nanostructures: uncertainty principle / momentum from
◮ Additionally resistive losses at all frequencies (dominates at low frequency)
◮ Photons absorbed to produce electron-hole pairs for ω ≥ Eg ◮ This assumes electrons and holes don’t interact with each other ◮ Instead they attract with potential V (r) ∼ −e2/(rǫr) ◮ Corresponding binding energy Eb = meff mǫ2
r Ryd (typically ∼ 0.01 − 0.1 eV)
◮ Here meff ∼ m∗ em∗ h/(m∗ e + m∗ h) is reduced mass of electron-hole pair ◮ Exciton: bound pair of electron and hole ◮ Can absorb photons when ω ≥ Eg − Eb (slightly smaller than band gap)
◮ Velocity so far is phase velocity v(ω) = ω k = c n(ω) ◮ Wavepackets travel with group velocity vg = ∂ω ∂k ◮ Same concept for all waves; only difference: dispersion relation ω(k)
◮ Group velocity
◮ Correspondingly group index:
◮ For a non-dispersive medium (constant n), vg = v and ng = n
◮ Energy density in electric field = ǫE2 2 ◮ Energy density in magnetic field = B2 2µ ◮ Maxwell’s equation ∇ ×
| B| √ǫµ ◮ Therefore energy densities in
◮ Net energy density = ǫE2 ◮ Wave moves with velocity v, so power flux is vǫE2
µEB = EH ◮ More generally, power flux given by Poynting vector
◮ In material with index n1, light incident towards interface with n2 ◮ Wavevectors: incident
1 and transmitted
◮ At boundary,
◮ Frequencies: common ω for all three (phases match for all t) ◮ Phase of fields must match at all x and y ◮ y-matching implies all three wavevectors in same plane ◮ x-matching implies that for all x:
1x = eik2 sin θ2x
1
◮ Therefore θ1 = θ′ 1 and
◮ Matching conditions:
1 sin θ′ 1) = ǫ2E2 sin θ2
1 cos θ′ 1 = E2 cos θ2
◮ Solve for E′ 1 and E2:
1
ǫ2 sin θ2 ǫ1 sin θ1 cos θ1 − cos θ2 ǫ2 sin θ2 ǫ1 sin θ′
1 cos θ′
1 + cos θ2
◮ Matching conditions:
1 cos θ′ 1) = 1
1 = E2
◮ Solve for E′ 1 and E2:
◮ Power flow = vǫ|E|2 = (c/n)n2|E|2 ∝ n|E|2 ◮ Therefore, R = |r|2 and T = n|t|2 (for each of and ⊥)
◮ R⊥ increases monotonically with incidence angle ◮ R goes through zero at Brewster angle θB = tan−1 n ◮ R = 1 beyond θc = sin−1 n when n < 1 (total internal reflection) ◮ Structures on wavelength scale: interference effects
◮ Index change due to impurity, defect or particles
◮ Scattered light output in all directions
◮ Contributes to attenuation α (along with absorption) ◮ Rayleigh scattering: scatterer size ≪ λ (wave limit) ◮ Scatterng cross section ∝ λ−4 ◮ Angular distribution ∝ (1 + cos2 θ) ◮ Mie theory: general case of spherical particles <, ∼, > λ ◮ Used to determine particle sizes from diffraction measurements ◮ Scatter size ≫ λ (particle limit) ◮ Scattering cross section ∼ πr2 (constant in λ) ◮ Why is the sky blue, but clouds white/grey?
◮ Higher n core surrounded by
◮ Channel light using total
◮ Pulse shape distortion due to
◮ Rectify / minimize using
◮ Attenuation: scattering at low
◮ Alternate glasses like heavy
◮ In crystals, n2 = ǫ is a tensor ◮ In general, values n1, n2, n3 along principal directions ◮ Isotropic: n1 = n2 = n3 eg. cubic ◮ Uniaxial: n1 = n2 = n3 eg. tetragonal, hexagonal ◮ Biaxial: n1 = n2 = n3 eg. orthorhombic, triclinic ◮ Optic axis: direction along which n⊥ is isotropic ◮ Uniaxial: single optic axis (direction 3 above) ◮ Biaxial: two optic axes (not along 1, 2 or 3) ◮ Consequence: two polarizations have different n for same direction ◮ Refracted waves in different directions (birefringence) ◮ Commonly used material: calcite (rhombohedral) ◮ Polarizers, half-wave and quarter-wave plates (Read 9.15 - 9.17)
◮ Reverse of electronic absorption: electron de-excitation emits light
◮ Electrons in excited states for multiple reasons:
◮ Optical excitation: photoluminescence (PL) ◮ Electronic excitation: cathodoluminescence ◮ Thermal excitation: thermoluminescence
◮ Includes fluorescence and phosphorescence
◮ Phosphors: materials used for their luminescence ◮ Example: Cr3+ in corundrum Al2O3 emits in red (ruby) ◮ Traps / dopants in semiconductors: radiative recombination ◮ Applications:
◮ Defunct display technologies (CRT) ◮ Down-conversion: white LEDs generate red components from blue ◮ Time-resolved PL: analysis of trap levels and lifetimes
◮ Consider two level system with populations N1 and N2 ◮ Incident light with intensity I: absorb to excite 1 → 2
◮ Emission in presence of light:
stimulated ◮ Einstein’s theorem: B12 = B21 for detailed balance / microscopic
◮ Basically, matrix element for forward and reverse process ◮ Net result:
◮ Important: stimulated emission in same quantum state as incident
◮ Basic idea: a synchronized
◮ Population inversion (N2 > N1)
◮ Need long-lived metastable state
◮ Resonant cavity: light reflects
◮ Stimulated emission enhances
◮ Light output through slightly
◮ Coherence: light with single
◮ Nonlinearity of ǫ and hence n with respect to
Pockel
◮ Kerr effect: present in all materials to varying degrees ◮ Pockel’s effect: present only in materials that are non-centosymmetric
◮ Note: n must be tensorial for such materials ◮ Similarly, magneto-optic effect: nonlinearity in
◮ Faraday rotation: magnetic field along propagation direction rotates
◮ Relative permeability of iron ∼ 2 × 105 ⇒ n = √ǫµ ∼ 450 ◮ Relative permeability of ferrite ∼ 640 ⇒ n = √ǫµ ∼ 25 ◮ Much higher than any dielectric: why are these not used in optics?