Dielectrics What is the macroscopic average electric field

Dielectrics • 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 • What is the electric energy density inside matter? • How do we relate these properties to quantum mechanical treatments of electrons in matter?

Dielectrics Microscopic picture of 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 - + r Note dipolar character

Electrostatic potential of point dipole • +/- charges, equal magnitude, q, separation a • axially symmetric potential (z axis) z q+ a/2 q p a/2 q- r+ r rx

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 distribution • Dipole moment p = Zea

Electric Polarisation Electric field in model 1 -D crystal with lattice spacing ‘a’ r(x) a x

Electric Polarisation Expand electric field Ex in same way (Ey, Ez = 0 by symmetry) r(x) a x

Electric Polarisation Apply external electric field and polarise charge density r(x) E a x

Electric Polarisation Apply external electric field and polarise charge density r(x) E a x

Electric Polarisation • Polarisation P, dipole moment p per unit volume Cm/m 3 = Cm-2 E p • Mesoscopic averaging: P is a constant vector field for a uniformly polarised medium E P • Macroscopic charges are induced with areal density sp Cm-2 in a uniformly polarised medium - 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 Define dimensionless dielectric susceptibility 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 values for : silicon 11. 8, diamond 5. 6, vacuum 1 Metal: → Insulator: (electronic part) small, ~5, lattice part up to 20

Electric Polarisation 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

Non-uniform polarisation • Uniform polarisation induced surface charges only - P E + • Non-uniform polarisation induced bulk charges also + - Displacements of positive charges Accumulated charges

Non-uniform polarisation Charge entering xz face at y = 0: Px=0 Dy. Dz Charge leaving xz face at y = Dy: Px=Dx. Dy. Dz = (Px=0 + ∂Px/∂x Dx) Dy. Dz Net charge entering cube: (Px=0 - Px=Dx ) Dy. Dz = -∂Px/∂x Dx. Dy. Dz Charge entering cube via all faces: z Px=0 Dz Dy Dx x -(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z) Dx. Dy. Dz = Qpol Px=Dx rpol = lim (Dx. Dy. Dz)→ 0 Qpol /(Dx. Dy. Dz) y -. P = rpol

Non-uniform 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

Non-uniform polarisation 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 = rfree, 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