Dielectrics Electric polarisation Electric susceptibility Displacement field in

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 on Dr(r) Change in electronic charge density - + Note dipolar character r

Electric Polarisation Dipole Moments of Atoms Total electronic charge per atom Z = atomic number Total nuclear charge per atom Centre of mass of electric or nuclear charge Dipole moment p = Zea

Electric Polarisation Uniform Polarisation • Polarisation P, dipole moment p per unit volume Cm/m 3 = Cm-2 E p • Mesoscopic averaging: P is a constant field for uniformly polarised medium E P • 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+2 d. A = s+d. A/ o E+ = s+/2 o E- = s-/2 o EDep = E+ + E- = (s++ s-)/2 o 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 charge density Charge entering xz face at y = 0: Py=0 Dx. Dz Cm-2 m 2 = C Charge leaving xz face at y = Dy: Py=Dy. Dx. Dz = (Py=0 + ∂Py/∂y Dy) Dx. Dz Net charge entering cube via xz faces: (Py=0 - Py=Dy ) Dx. Dz = -∂Py/∂y Dx. Dy. Dz z Py=0 -(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z) Dx. Dy. Dz = Qpol Dz Dy Dx x Charge entering cube via all faces: Py=Dy y rpol = lim (Dx. Dy. Dz)→ 0 Qpol /(Dx. Dy. Dz) -. 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 (dimensionless) defined through P = o EMac = E – P/ o o E = o EMac + P o E = o EMac + o EMac = o (1 + )EMac = o EMac Define dielectric constant (relative permittivity) = 1 + 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 ~1013 Hz Bound electrons: light, resonance in visible ~1015 Hz Free electrons: no restoring force, no resonance

Dielectric Susceptibility Bound charges Resonance model for uncoupled electron pairs Mion k melectron k Mion

Dielectric Susceptibility Bound charges In and out of phase components of x(t) relative to Eo cos(wt) Mion k melectron k Mion in phase out of phase

Dielectric Susceptibility Bound charges Connection to and (w) Im{ (w)} w = wo w/wo Re{ (w)}

Dielectric Susceptibility Free charges Let wo → 0 in and jpol = ∂P/∂t s(w) Re{ (w)} wo = 0 Drude ‘tail’ Im{s(w)} w

Displacement Field Rewrite EMac = E – P/ o as o. EMac + P = o. E LHS contains only fields inside matter, RHS fields outside Displacement field, D D = o. EMac + P = o EMac = o. E 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 . ( o. E + P) = rfree . D = rfree Introduction of the displacement field, D, allows us to eliminate polarisation charges from any calculation

Validity of expressions • Always valid: • Limited validity: • Have assumed that is a simple number: P = o E only true in LIH media: • Linear: independent of magnitude of E interesting media “non-linear”: P = o. E + 2 o. EE + …. • Isotropic: independent of direction of E interesting media “anisotropic”: is a tensor (generates vector) • Homogeneous: uniform medium (spatially varying ) Gauss’ Law for E, P and D relation D = o. E + P Expressions involving and

Boundary conditions on D and E fields at matter/vacuum interface matter DL = o L E L = o E L + P L vacuum DR = o RER = o. ER R = 1 No free charges hence . D = 0 Dy = Dz = 0 ∂Dx/∂x = 0 everywhere Dx. L = o LEx. L = Dx. R = o. Ex. R Ex. L = Ex. R/ L Dx. L = Dx. R 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. d. S = sfree, enclosed Integral form ∫ D. d. S = 0 No free charges at interface d. SL q. L D L = o LE L q. R D R = o RE R d. SR -DL cosq. L d. SL + DR cosq. R d. SR = 0 DL cosq. L = DR cosq. R 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 q. L EL q. R dℓR ER EL. dℓL + ER. dℓR = 0 -ELsinq. LdℓL + ERsinq. R dℓR = 0 ELsinq. L = ERsinq. R 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 d. SL q. L D L = o LE L q. R D R = o RE R d. SR