1 MTLE-6120: Advanced Electronic Properties of Materials Optical properties of Materials Reading: ◮ Kasap: 9.1 - 9.18
2 Maxwell’s equations in free media ◮ Linear response of materials described very generally by ǫ ( ω ) and µ ( ω ) ◮ Maxwell’s equations in the absence of free charges and currents ∇ · ( ǫ ( ω ) � E ) = 0 ∇ × � E = iω � B � B µ ( ω ) = − iω ( ǫ ( ω ) � ∇ × E ) ∇ · � B = 0 ◮ Substitute second equation in curl of third equation: ∇ × ( ∇ × � B ) = − iωǫ ( ω ) ∇ × � E = ω 2 ǫ ( ω ) � B µ ( ω ) −∇ 2 � B = ω 2 ǫ ( ω ) µ ( ω ) � using ∇ · � B B = 0
3 Electromagnetic waves ◮ Write Maxwell’s equation in linear media with no free charge or current as: v 2 ( ω ) ∇ 2 � B = − ω 2 � B v 2 ( ω ) ∇ 2 � E = − ω 2 � E � where v ( ω ) ≡ 1 / ǫ ( ω ) µ ( ω ) ◮ This is exactly the equation of a wave with speed v v 2 ∇ 2 f = − ω 2 f v 2 ∇ 2 f = ∂ 2 f ∂t 2 where f is either � E or � B ◮ Solution to the wave equation with | � k | = ω/v : r ) = fe i� k · � r f ( � r, t ) = fe i ( � k · � r − ωt ) f ( �
4 Polarization ◮ EM wave propagating along x : � � ye i ( kx − ωt ) ze i ( kx − ωt ) E ( � r, t ) = E 0 ˆ or E ( � r, t ) = E 0 ˆ two field directions ⇒ two independent linear polarizations ◮ Linearly combine to y + i ˆ ˆ z ˆ y − i ˆ z � 2 e i ( kx − ωt ) � 2 e i ( kx − ωt ) E ( � r, t ) = E 0 √ or E ( � r, t ) = E 0 √ ⇒ two independent circular polarizations (left/right) ◮ Photons: circular polarizations have spin angular momentum ± � along propagation direction (photons are spin s = 1 particles)
5 Wave speed and refractive index ◮ Electromagnetic waves satisfy dispersion relation ω = v | � k | � ◮ EM wave velocity v ( ω ) = 1 / ǫ ( ω ) µ ( ω ) ◮ In vacuum, ǫ ( ω ) = ǫ 0 and µ ( ω ) = µ 0 , so that speed of light in free space 1 c = √ ǫ 0 µ 0 ◮ In materials, speed of light usually specified by refractive index � c ǫ ( ω ) µ ( ω ) n ( ω ) ≡ v ( ω ) = ǫ 0 µ 0
6 Refractive index: frequency dependence ◮ Most materials are non-magnetic µ ≈ µ 0 ◮ Therefore n 2 ( ω ) ≈ ǫ r ≡ ǫ ( ω ) /ǫ 0 ◮ Several contributions to relative permittivity: n 2 ≈ ǫ r = 1 + χ (free e − ) + χ (bound e − ) + χ (ions) + χ (dipoles) e e e e ◮ Frequency dependence (Drude-Lorentz model): ω 2 χ 0 α ω 2 � p n 2 ( ω ) = 1 − 0 α ω 2 + iω/τ + ω 2 0 α − iγ α ω − ω 2 α where α includes bound e − , ions and dipoles ◮ First Drude term proportional to free carrier density: ◮ Dominates for metals ◮ Present to varying degrees for doped semiconductors ◮ Remaining terms contribute in all materials
7 Refractive index: wavelength dependence ◮ Frequency dependence (Drude-Lorentz model): χ 0 α ω 2 � 0 α n 2 ( ω ) = 1 + ω 2 0 α − iγ α ω − ω 2 α (Drude term captured by setting χ 0 ω 2 0 → ω 2 p , ω 0 → 0 and γ → 1 /τ ) ◮ Equivalently in terms of wavelength λ = 2 πc/ω : χ 0 α λ 2 � n 2 ( λ ) = 1 + λ 2 − iδ α λ − λ 2 0 α α where λ 0 = 2 πc/ω 0 and δ = 2 πcγ α /ω 2 0 ◮ Can use a number of α in the empirical Sellmeier relation instead: A α λ 2 � n 2 ( λ ) = 1 + λ 2 − iδ α λ − λ 2 α α
8 Complex refractive index ◮ The n 2 we have been working with so far has been implicitly complex (due to the τ and γ ’s) ◮ Customary to call complex refractive index N = n + iK (where n and K are real) ◮ Wave speed v = c c N = n + iK is complex ◮ Wave vector k = ω v = ω c ( n + iK ) = k 0 ( n + iK ) is complex ◮ What do the imaginary parts here mean? ◮ Wave function (fields) ∝ e ikx = e i ( nk 0 ) x · e − ( Kk 0 ) x ◮ Wave intensity ∝ e − (2 Kk 0 ) x (proportional to | field | 2 ) ◮ ⇒ Wave attenuates as e − αx with absorption coefficient α = 2 Kk 0
9 Kramers-Kronig relations 5 Re( α ) Im( α ) 4 3 2 α [ q 2 / k ] 1 0 -1 -2 -3 0 0.5 1 1.5 2 ω [( k/m ) 1/2 ] ◮ Real and imaginary parts of ǫ ( ω ) (and hence n ( ω ) ) not independent ◮ Causality ⇒ response functions (eg. χ ( ω ) ) must be complex analytic functions ( χ ( z ) ) ◮ Leads to the Kramers-Kronig relations: � + ∞ � + ∞ Re χ ( ω ) = 1 d ω ′ Im χ ( ω ) Im χ ( ω ) = − 1 d ω ′ Re χ ( ω ) π P ω ′ − ω , π P ω ′ − ω −∞ −∞ ◮ Absorption at ω 0 affects n at all ω
10 Absorption mechanisms Every polarizability term has a corresponding absorption mechanism: ◮ 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 ω 0 ◮ Each of the above processes have a band of absorption frequencies instead
11 Lattice absorption: Restrahlen band ◮ Optical phonons in ionic crystals: charge oscillations ⇒ interact with EM (hence the name ‘optical’ phonons) ◮ High DOS for transverse and longitudinal optical phonons ◮ Absorption peaks at corresponding wavelengths: Restrahlen band ◮ Typically in the mid-infrared part of the spectrum
12 Electronic absorption: bound model ◮ 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 v d = 0 ) ◮ Bound model obtained response of form: χ 0 ω 2 0 χ ( ω ) = ω 2 0 − iγω − ω 2 ◮ Interpretation of behavior near ω ∼ ω 0 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 = E g ◮ Effectively ω 0 now corresponds to E g / �
13 Optical excitation of electrons ◮ Light excites electron in valence band (initial energy E i < 0 ) to conduction band (final energy E j > E g ) ◮ Energy conservation: E i + � ω = E j ⇒ ω = ( E j − E i ) / � > E g / � ◮ Momentum conservation � k i + � ω/c = � k j ⇒ k j − k i = ω/c = 2 π/λ ◮ Typical E g ∼ 1 − 10 eV with λ ∼ 100 nm - 1 µ m (NIR, visible, UV) ◮ Typical k i , k j ∼ 2 π/a with a < 1 nm ◮ ⇒ photon momentum is negligible in electronic processes ◮ Effectively k i = k j (vertical transitions) for optical absorption
14 Direct absorption ◮ Direct band-gap semiconductors (eg. GaAs): VBM and CBM at same k ◮ Possible to have E j − E i = E g with k i = k j ◮ Direct absorption: light produces electron-hole pair ◮ Allowed here for ω ≥ E g / �
15 Indirect absorption 6 Photon 0 E in eV Phonon − 6 − 12 L Λ Γ Δ Χ Σ Γ ◮ Indirect band-gap semiconductors (eg. Si): VBM and CBM at different k ◮ Not possible to have E j − E i = E g with k i = k j ◮ Indirect absorption: light produces electron-hole pair ± phonon ◮ Conservation: E j − E i ± � ω ph = � ω , k j ± k ph = k i ◮ Negligible energy in phonon, negligible momentum in photon ◮ Direct absorption becomes possible above interband threshold E t
16 Absorption in metals (plasmon decay) ◮ 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 surface ◮ Additionally resistive losses at all frequencies (dominates at low frequency)
17 Excitons ◮ Photons absorbed to produce electron-hole pairs for � ω ≥ E g ◮ This assumes electrons and holes don’t interact with each other ◮ Instead they attract with potential V ( r ) ∼ − e 2 / ( rǫ r ) ◮ Corresponding binding energy E b = m eff r Ryd (typically ∼ 0 . 01 − 0 . 1 eV) mǫ 2 (similar to the donor/acceptor level estimate) ◮ Here m eff ∼ m ∗ e m ∗ h / ( m ∗ e + m ∗ h ) is reduced mass of electron-hole pair ◮ Exciton: bound pair of electron and hole ◮ Can absorb photons when � ω ≥ E g − E b (slightly smaller than band gap)
18 Wave packets x 0 = ω ’( k 0 ) t e i ( k 0 x - ω ( k 0 ) t ) ◮ Velocity so far is phase velocity v ( ω ) = ω c k = n ( ω ) ◮ Wavepackets travel with group velocity v g = ∂ω ∂k ◮ Same concept for all waves; only difference: dispersion relation ω ( k )
19 Group velocity and group index ◮ Group velocity v g = ∂ω 1 ∂k = ∂k/∂ω 1 = ∂ ( ωn ( ω ) /c ) /∂ω c = n ( ω ) + ωn ′ ( ω ) c since ω ∝ λ − 1 = n ( λ ) − λn ′ ( λ ) ◮ Correspondingly group index: n g ≡ c/v g = n ( ω ) + ωn ′ ( ω ) = n ( λ ) − λn ′ ( λ ) ◮ For a non-dispersive medium (constant n ), v g = v and n g = n
20 Optical energy density and power flux ◮ Energy density in electric field = ǫE 2 2 ◮ Energy density in magnetic field = B 2 2 µ | � ◮ Maxwell’s equation ∇ × � E = − ∂ � B/∂t ⇒ | � E | = v | � B | B | = √ ǫµ ◮ Therefore energy densities in � E and � B are equal ◮ Net energy density = ǫE 2 ◮ Wave moves with velocity v , so power flux is vǫE 2 = v 2 ǫEB = 1 µ EB = EH ◮ More generally, power flux given by Poynting vector � E × � H
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