Dielectrics and Polarization

Dielectric and Optical Properties • • • As with conductivity, we will start with macroscopic property and connect to the microscopic All aspects of free electrons have been covered: only bound electrons left Capacitance, Optical properties --> ε,n --> molecules and atoms Review of capacitance and connection to dielectric constant First, no material in capacitor to Vo Q = CV dq dV =i=C dt dt V C i Vo V t 1 The Capacitor + + + + + + -- I=0 always - in capacitor -t d/2 d/2 +V ρ ∇⋅E = E= E ρ εo −d 2 ∫ −d −t 2 V= ρ ρt Q = dx = εo ε o Aε o d 2 ∫ Edx = −d 2 V Qd Aε o Q Qd =V = C Aε o C= εo A d 2 The Capacitor • • • The air-gap can store energy! If we can move charge temporarily without current flow, can store even more Bound charge around ion cores in a material can lead to dielectric properties •Two kinds of charge can create plate charge: •surface charge •dipole polarization in the volume •Gauss’ law can not tell the difference (only depends on charge per unit area) Image from Wikimedia Commons, http://commons.wikimedia.org 3 Material Polarization + + -+++ E’ E' = P E D = ε o E + P = εE ε = ε rε o ε r = 1+ P = 1+ χ εoE + ++ ++ + + - - - C= ε rε o A d P is the Polarization D is the Electric flux density or the Dielectric displacement χ is the dielectric or electric susceptibility All detail of material response is in εr and therefore P 4 Origin of Polarization • • • We are interested in the true dipoles creating polarization in materials (not surface effect) As with the free electrons, what is the response of these various dipole mechanisms to various E-field frequencies? When do we have to worry about controlling – molecular polarization (molecule may have non-uniform electron density) – ionic polarization (E-field may distort ion positions and temporarily create dipoles) – electronic polarization (bound electrons around ion cores could distort and lead to polarization) • • Except for the electronic polarization, we might expect the other mechanisms to operate at lower frequencies, since the units are much more massive What are the applications that use waves in materials for frequencies below the visible? 5 Application for different E-M Frequencies Courtesy of the Opensource Handbook of Nanoscience and Nanotechnology, http://en.wikibooks.org/wiki/Nanotechnology In communications, many E-M waves travel in insulating materials: What is the response of the material (εr) to these waves? 6 Wave Eqn. With Insulating Material and Polarization r r ∂B ∇xE = − ∂t r r r r r r ∂D nonmag r r ∂ ε o E + P insulating r ∂E ∇xH = J + ⎯⎯⎯→ ∇xB = J + ⎯⎯ ⎯ ⎯→ ∇xB = ε ∂t ∂t ∂t r r r r (D = ε 0 E + P = εE) ( ) ∂2E εr ∂2E ∇ E = μ 0ε 0ε r 2 = 2 2 ∂t c ∂t 2 E = E0 e i(k •r −ϖt ) = E0 eik •r e −iϖt = E(r)e −iϖt ∇ E(r) = − 2 ω = 2 ω= c2 εr c ω 2ε r E(r) c2 k2 c ⎯⎯→ k k ⎯optical n εr So polarization slows down the velocity of the wave in the material 7 Compare Optical (index of refraction) and Electrical Measurements of ε Material Optical, n2 Electrical, ε diamond 5.66 5.68 Only electrical polariztion NaCl 2.25 5.9 Electrical and ionic polariztion H 2O 1.77 80.4 Electrical, ionic, and molecular polariztion Polarization that is active depends on material and frequency 8 Microscopic Frequency Response of Materials • • • Bound charge can create dipole through charge displacement Hydrodynamic equation (Newtonian representation) will now have a restoring force Review of dipole physics: -q r d - r r Dipole moment: p = qd + +q r p Applied E-field rotates dipole to align with field: r r Torque τ = pxE r r r r r Potential Energy U = − p ⋅ E = p E cos θ 9 Microscopic Frequency Response of Materials • For a material with many dipoles: r r r r r P = Np = NαE ( p = αE ) (polarization=(#/vol)*dipole polarization) α=polarizability χ= r r p = αE r P r , so χ = Nα εo E Actually works well only for low density of dipoles, i.e. gases: little screening For solids where there can be a high density: local field Eext Eloc For a spherical volume inside (theory of local field), r r r P Eloc = E ext + 3ε o 10 Microscopic Frequency Response of Materials • We now need to derive a new relationship between the dielectric constant and the polarizability D = ε r ε o Eext = ε o Eext + P P = ε r ε o Eext − ε o Eext ⎛ 2 + εr ⎞ Eloc = Eext ⎜ ⎟ ⎝ 3 ⎠ Plugging into P=NαEloc: ε r ε o Eext − ε o Eext = N α (ε r + 2 ) E 3 ext (ε r − 1)ε o = Nα (ε r + 2 ) 3 Clausius-Mosotti Relation: ε r − 1 Nα α = = ε r + 2 3ε o 3νε o Macro Micro Where v is the volume per dipole (1/N) 11 Different Types of Polarizability Highest natural frequency • • • Lowest natural frequency Atomic or electronic,αe Displacement or ionic, αi Orientational or dipolar, αo Lightest mass Heaviest mass α = αe + αi + αo As with free e-, we want to look at the time dependence of the E-field: E = Eo e −iωt ∂ 2 x m ∂x m 2 = − eE − Kx ∂t τ ∂t Restoring Force Response Driving Force Drag m&x& = −eE − Kx x = xo e −iωt m(−ω 2 )xo = −eEo − Kxo xo = ωo = eEo K⎞ ⎛ m⎜ ω 2 − ⎟ m⎠ ⎝ K m = eEo m ω 2 − ω o2 ( ) So lighter mass will have a higher critical frequency 12 Classical Model for Electronic Polarizability • Electron shell around atom is attached to nucleus via springs + + r E K K Zi electrons, mass Zim r E r p r Z i m&r& = −Kr − Z i eEloc , assume r = ro e −iωt 13 Electronic Polarizability ro = ro = eEo K ω ; = oe m ω 2 − ω oe2 mZ i ( ) p = qd = −Z i er; p = po e po = 2 ) Zie2 αe = m ω 2 − ω oe2 ) ( ω >> ω oe , α e = 0 ω << ω oe , α e = 2 Zie mω oe2 ⎛ K ⎞ ⎟⎟ m⎜⎜ ω 2 − mZ i ⎠ ⎝ If no Clausius-Mosotti, ε r = 1+ −iωt Zie E o = α e Eo m ω 2 − ω o2 ( eEo Nα e εo NZ i e 2 = 1+ = n2 2 2 ε o m ω − ω oe ( ) εr NZ i e 2 1+ ε o m ω oe2 ( ) 1 ωoe ω 14 QM Electronic Polarizability • • • At the atomic electron level, QM expected: electron waves QM gives same answer qualitatively QM exact answer very difficult: many-bodied problem E1 E0 e2 f10 E1 − E0 = α e (ω ) = ; ω 10 h m ω102 − ω 2 f10 is the oscillator strength of the transition (ψ1 couples to ψo by E-field) For an atom with multiple electrons in multiple levels: f E − E0 e2 α e (ω ) = ∑ j≠0 2 j 0 2 ; ω j 0 = j h m j10 − ω 15 Ionic Polarizability • • • Problem reduces to one similar to the electronic polarizability Critical frequency will be less than electronic since ions are more massive The restoring force between ion positions is the interatomic potential E(R) Parabolic at bottom near Ro Nucleii repulsion k(R − Ro ) 2 E= 2 ∂E F= = k(R − Ro ) ∂R Ro - + R Electron bonding in between ions F = kx ⇒ σ ij = Cijkl ε kl 16 Ionic Polarizability Eloc •2 coupled differential eqn’s •1 for + ions •1 for - ions + - + - && = u&&+ − u&&− w = u+ − u− , w M= p u- u+ Ionic materials always have ionic and electronic polarization, so: α tot e2 = αi + αe = α + + α − + M ω oi2 − ω 2 ( 1 1 1 + M+ M− && = −2Kw + eEloc Mw Eloc = Eo e −iωt , w = wo e −iωt wo = ) eEo 2K ω = , oi M ω oi2 − ω 2 M ( ) po = ewo = α i Eo e2 αi = M ω oi2 − ω 2 ( ) 17 Ionic Polarizability • Usually Clausius-Mosotti necessary due to high density of dipoles ⎤ ε r −1 Nα tot 1 ⎡ e2 = = + + α α − ⎢ + ⎥ ε r + 2 3ε o 3ε o v ⎣ M (ω oi2 − ω 2 )⎦ By convention, things are abbreviated by using εs and ε∞: ε s −1 1 ⎡ e2 ⎤ α+ +α− + = ω << ω oi , ⎥ ε s + 2 3ε o v ⎢⎣ M (ω oi2 )⎦ ε ∞ −1 n 2 −1 1 ω >> ω oi , = 2 = [α + + α − ] ε ∞ + 2 n + 2 3ε o v εr εs n2=ε∞ ∴ εr = ε∞ + ε∞ − εs 2 2 ⎛ ε∞ + 2 ⎞ ⎜⎜ ⎟⎟ , ω ω = T oi ω2 ε 2 + ⎝ s ⎠ −1 2 ωT ωT ω 18 Orientational Polarizability • No restoring force: analogous to conductivity O - p H + θ O O p=0 Analogous to conductivity, the molecules collide after a certain time t, giving: CDα = oα τωi−1 +q C H -q For a group of many molecules at some temperature: f =e −U k bT =e pE cosθ k bT After averaging over the polarization of the ensemble molecules (valid for low E-fields): α DC p2 ~ 3kbT 19