Dielectrics • • • • Electric polarisation Electric susceptibility Displacement field in matter Boundary conditions on fields at interfaces • What is the macroscopic (average) electric field inside matter when an external E field is applied? • How is charge displaced when an electric field is applied? i.e. what are induced currents and densities • How do we relate these properties to quantum mechanical treatments of electrons in matter? Electric Polarisation Microscopic viewpoint Atomic polarisation in E field Change in charge density when field is applied r(r) Electronic charge density E No E field E field on Dr(r) Change in electronic charge density r - + Note dipolar character Electric Polarisation Dipole Moments of Atoms Total electronic charge per atom Z = atomic number Ze Total nuclear charge per atom Ze r (r )dr el all space r nuc all space Centre of mass of electric or nuclear charge r (r )dr el/nuc r el/nuc all space r (r ) r dr el/nuc all space Dipole moment p = Zea Ze a Ze r nuc r el r nuc (r )dr (r ) r el (r ) r dr all space r el all space (r ) r dr if r nuc 0 Electric Polarisation Uniform Polarisation • Polarisation P, dipole moment p per unit volume Cm/m3 = Cm-2 p E • Mesoscopic averaging: P is a constant field for uniformly polarised medium E P s P.n • Macroscopic charges are induced with areal density sp Cm-2 - P + E Electric Polarisation • Contrast charged metal plate to polarised dielectric s- P s+ E • Polarised dielectric: fields due to surface charges reinforce inside the dielectric and cancel outside s- s- • Charged conductor: fields due to surface charges cancel inside the metal and reinforce outside Electric Polarisation • Apply Gauss’ Law to right and left ends of polarised dielectric s- s+ P E- E+ • EDep = ‘Depolarising field’ • Macroscopic electric field E E+2dA = s+dA/o E+ = s+/2o E- = s-/2o EDep = E+ + E- = (s+ s-/2o EDep = -P/o P = s+ = s- EMac= E + EDep = E - P/o Electric Polarisation Non-uniform Polarisation • Uniform polarisation induced surface charges only P - + E • Non-uniform polarisation induced bulk charges also + - + Displacements of positive charges Accumulated charges - Electric Polarisation Polarisation charge density Charge entering xz face at y = 0: Py=0DxDz Cm-2 m2 = C Charge leaving xz face at y = Dy: Py=DyDxDz = (Py=0 + ∂Py/∂y Dy) DxDz Net charge entering cube via xz faces: (Py=0 - Py=Dy ) DxDz = -∂Py/∂y DxDyDz z Py=0 -(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z) DxDyDz = Qpol Dz Dy Dx x Charge entering cube via all faces: Py=Dy y rpol = lim (DxDyDz)→0 Qpol /(DxDyDz) -. P = rpol Electric Polarisation Differentiate -.P = rpol wrt time .∂P/∂t + ∂rpol/∂t = 0 Compare to continuity equation .j + ∂r/∂t = 0 ∂P/∂t = jpol Rate of change of polarisation is the polarisation-current density Suppose that charges in matter can be divided into ‘bound’ or polarisation and ‘free’ or conduction charges rtot = rpol + rfree Dielectric Susceptibility Dielectric susceptibility c (dimensionless) defined through P = o c EMac EMac = E – P/o o E = o EMac + P o E = o EMac + o c EMac = o (1 + c)EMac = oEMac Define dielectric constant (relative permittivity) = 1 + c EMac = E / E = EMac Typical static values (w = 0) for : silicon 11.4, diamond 5.6, vacuum 1 Metal: → Insulator: (electronic part) small, ~5, lattice part up to 20 Dielectric Susceptibility Bound charges All valence electrons in insulators (materials with a ‘band gap’) Bound valence electrons in metals or semiconductors (band gap absent/small ) Free charges Conduction electrons in metals or semiconductors Si ion Bound electron pair Mion k melectron k Mion Resonance frequency wo ~ (k/M)1/2 or ~ (k/m)1/2 Ions: heavy, resonance in infra-red ~1013Hz Bound electrons: light, resonance in visible ~1015Hz Free electrons: no restoring force, no resonance Dielectric Susceptibility Bound charges Resonance model for uncoupled electron pairs mx mx kx qEo Re{e iwt } x x Mion k melectron k Mion k q x Eo Re{e iwt } m m x(t) Re{A(w ) e iwt } trial solution x (t) iw x(t) x(t) w 2 x(t) (Re{} assumed hereafter) k q 2 iwt w iw x(t) Eo e m m k q 2 iwt Eo e iwt w iw A(w ) e m m Dielectric Susceptibility Bound charges In and out of phase components of x(t) relative to Eo cos(wt) A(w ) A(w ) qEo 1 m w 2 iw wo2 qEo 1 m wo2 w 2 iw wo2 Mion k melectron k Mion wo2 w 2 qEo Re{A(w )} m wo2 w 2 2 w 2 k m Im{A( w )} qEo w m wo2 w 2 2 w 2 x(t) Re{A(w )e -iwt } Re{A(w )}Re{e -iwt } Im{A( w )}Im{e -iwt } qEo m w 2 w 2 cos(wt) w sin(wt) 2o 2 2 2 2 2 2 wo w w 2 wo w w in phase out of phase Dielectric Susceptibility Bound charges Connection to c and Polarisati on dipole moment per unit volume qx(t)/V wo2 w 2 q2 i w -i wt P(t) Re Eo e o c E(t) 2 2 2 2 2 2 2 2 mV wo w w wo w w wo2 w 2 q2 q2 w Re{ c (w )} Im{ c ( w )} m o V wo2 w 2 2 w 2 m o V wo2 w 2 2 w 2 wo2 w 2 iw q2 (w ) 1 c (w ) 1 m o V wo2 w 2 2 w 2 6 w Im{w} w wo 4 2 model dielectric function w/wo 1 2 4 2 Re{w} 3 4 Dielectric Susceptibility Free charges sw Let wo → 0 in c and jpol = ∂P/∂t Current density j(t) s E(t) s conductivi ty 6 4 q2 1 -iwt P(t) o E e o m o V wo2 w 2 iw 2 Re{w} wo 0 Drude ‘tail’ 1 P(t) q2 iw -iwt jpol (t) o E e o t m o V wo2 w 2 iw 2 Im{sw} 4 P(t) q2 iw -iwt jf ree(t) o E e Let wo 0 o 2 t m o V w i w q2 iw q2 iw 3 w 2 q 2 iw s f ree(w ) 2 2 4 mV w iw mV w w mV w 2 2 q2 Ne 2 s f ree(0) mV m Drude conductivi ty e q 2 2 1 N V 1 2 w 3 4 Displacement Field Rewrite EMac = E – P/o as oEMac + P = oE LHS contains only fields inside matter, RHS fields outside Displacement field, D D = oEMac + P = o EMac = oE Displacement field defined in terms of EMac (inside matter, relative permittivity ) and E (in vacuum, relative permittivity 1). Define D = o E where is the relative permittivity and E is the electric field This is one of two constitutive relations contains the microscopic physics Displacement Field Inside matter .E = .Emac = rtot/o = (rpol + rfree)/o Total (averaged) electric field is the macroscopic field -.P = rpol .(oE + P) = rfree .D = rfree Introduction of the displacement field, D, allows us to eliminate polarisation charges from any calculation Validity of expressions • Always valid: Gauss’ Law for E, P and D relation D = oE + P Expressions involving and c • Limited validity: • Have assumed that c is a simple number: P = o c E only true in LIH media: • Linear: c independent of magnitude of E interesting media “non-linear”: P = c oE + c2 oEE + …. • Isotropic: c independent of direction of E interesting media “anisotropic”: c is a tensor (generates vector) • Homogeneous: uniform medium (spatially varying ) Boundary conditions on D and E D and E fields at matter/vacuum interface matter DL = oLEL = oEL + PL vacuum DR = oRER = oER R = 1 No free charges hence .D = 0 Dy = Dz = 0 ∂Dx/∂x = 0 everywhere DxL = oLExL = DxR = oExR ExL = ExR/L DxL = DxR E discontinuous D continuous Boundary conditions on D and E Non-normal D and E fields at matter/vacuum interface .D = rfree Differential form ∫ D.dS = sfree, enclosed Integral form ∫ D.dS = 0 No free charges at interface dSL qL DL = oLEL qR DR = oRER dSR -DL cosqL dSL + DR cosqR dSR = 0 DL cosqL = DR cosqR D┴ L = D┴ R No interface free charges D┴L - D┴R = sfree Interface free charges Boundary conditions on D and E Non-normal D and E fields at matter/vacuum interface Boundary conditions on E from ∫ E.dℓ = 0 (Electrostatic fields) dℓL qL EL qR dℓR ER EL.dℓL + ER.dℓR = 0 -ELsinqLdℓL + ERsinqR dℓR = 0 ELsinqL = ERsinqR E||L = E||R D┴L = D┴R D┴L - D┴R = sfree E|| continuous No interface free charges Interface free charges Boundary conditions on D and E dSL qL DL = oLEL qR DR = oRER dSR EL sinqL ER sinqR DL cosqL DR cosqR DL/R L/R oEL/R EL sinqL EL sinqL ER sinqR DL cosqL L oEL cosqL R oER cosqR tan qL L tan qR R in absence of free charges at interface