Dielectrics

1. 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?

2. 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 E field on Dr(r) Change in electronic charge density Note dipolar character r - +

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

4. 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

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

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

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

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

9. E E E p P P + - Electric Polarisation • Polarisation P, dipole moment p per unit volume Cm/m3 = Cm-2 • Mesoscopic averaging: P is a constant vector field for a uniformlypolarised medium • Macroscopic charges are induced with areal density sp Cm-2 in a uniformly polarised medium

10. s- s+ E P s- s- Electric Polarisation • Contrast charged metal plate to polarised dielectric • Polarised dielectric: fields due to surface charges reinforce inside the dielectric and cancel outside • Charged conductor: fields due to surface charges cancel inside the metal and reinforce outside

11. E P s+ s- E- E+ Electric Polarisation • Apply Gauss’ Law to right and left ends of polarised dielectric • EDep = ‘Depolarising field’ • Macroscopic electric field EMac= E + EDep = E - P/o E+2dA = s+dA/o E+ = s+/2o E- = s-/2o EDep= E+ + E- = (s++ s-)/2o EDep= -P/o P = s+ = s-

12. Electric Polarisation Define dimensionless dielectric susceptibility c through P = ocEMac EMac = E– P/o oE = oEMac + P oE = oEMac + ocEMac = o (1 + c)EMac = oEMac Define dielectric constant (relative permittivity)  = 1 + c EMac = E/ E = eEMac Typical values for e: silicon 11.8, diamond 5.6, vacuum 1 Metal: e → Insulator: e (electronic part) small, ~5, lattice part up to 20

13. Electric Polarisation 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 e) and E (in vacuum, relative permittivity 1). Define D = oE where  is the relative permittivity and E is the electric field

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

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

16. 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

17. Non-uniform polarisation 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

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

19. Boundary conditions on D and E D and E fields at matter/vacuum interface matter vacuum DL = oLEL= oEL+PL DR = oRER= oERR=1 No free charges hence .D = 0 Dy = Dz = 0 ∂Dx/∂x = 0 everywhere DxL = oLExL= DxR = oExR ExL=ExR/L DxL= DxR E discontinuous D continuous

20. DR = oRER dSL qR qL dSR DL = oLEL Boundary conditions on D and E Non-normal D and E fields at matter/vacuum interface .D = rfree Differential form∫ D.dS = rfree, enclosed Integral form ∫ D.dS = 0 No free charges at interface -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

21. ER dℓL qR qL dℓR EL Boundary conditions on D and E Non-normal D and E fields at matter/vacuum interface Boundary conditions on Efrom∫ E.dℓ = 0(Electrostatic fields) EL.dℓL + ER.dℓR = 0 -ELsinqLdℓL + ERsinqR dℓR = 0 ELsinqL = ERsinqR E||L = E||R E|| continuous D┴L = D┴R No interface free charges D┴L - D┴R = sfree Interface free charges

22. DR = oRER dSL qR qL dSR DL = oLEL Boundary conditions on D and E