1. 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? Dielectrics

2. Microscopic viewpoint Atomic polarisation in E field Change in charge density when field is applied Electric Polarisation E  (r) Change in electronic charge density Note dipolar character r No E field E field on - +  (r) Electronic charge density

3. 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

4. Uniform Polarisation Polarisation P, dipole moment p per unit volume Cm/m 3 = Cm -2 Mesoscopic averaging: P is a constant field for uniformly polarised medium Macroscopic charges are induced with areal density  p Cm -2 Electric Polarisation p E P E P - + E

5. 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 Electric Polarisation -- ++ E P -- --

6. Apply Gauss’ Law to right and left ends of polarised dielectric E Dep = ‘Depolarising field’ Macroscopic electric field E Mac = E + E Dep = E - P  o E + 2dA =  + dA  o E + =  +  o E - =  -  o E Dep = E + + E - = (  +  -  o E Dep = -P/  o P =  + =  - -- E P ++ E+E+ E-E-

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

8. Electric Polarisation Polarisation charge density Charge entering xz face at y = 0: P y=0  x  z Cm -2 m 2 = C Charge leaving xz face at y =  y: P y=  y  x  z = (P y=0 + ∂P y /∂y  y)  x  z Net charge entering cube via xz faces: (P y=0 - P y=  y )  x  z = -∂P y /∂y  x  y  z Charge entering cube via all faces: - (∂P x /∂x + ∂P y /∂y + ∂P z /∂z)  x  y  z = Q pol  pol = lim (  x  y  z)→0 Q pol /(  x  y  z) - . P =  pol xx zz yy z y x P y=  y P y=0

9. Electric Polarisation Differentiate - .P =  pol wrt time .∂P/∂t + ∂  pol /∂t = 0 Compare to continuity equation .j + ∂  /∂t = 0 ∂P/∂t = j pol 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  tot =  pol +  free

10. Dielectric Susceptibility Dielectric susceptibility  (dimensionless) defined through P =  o  E Mac E Mac = E – P/  o  o E =  o E Mac + P  o E =  o E Mac +  o  E Mac =  o (1 +  )E Mac =  o  E Mac Define dielectric constant (relative permittivity)  = 1 +  E Mac = E /  E =  E Mac Typical static values (  = 0) for  : silicon 11.4, diamond 5.6, vacuum 1 Metal:  →  Insulator:   (electronic part) small, ~5, lattice part up to 20

11. 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 M ion k m electron k M ion Si ion Bound electron pair Resonance frequency  o ~ (k/M) 1/2 or ~ (k/m) 1/2 Ions: heavy, resonance in infra-red ~10 13 Hz Bound electrons: light, resonance in visible ~10 15 Hz Free electrons: no restoring force, no resonance

12. Dielectric Susceptibility Bound charges Resonance model for uncoupled electron pairs M ion k m electron k M ion

13. Dielectric Susceptibility Bound charges In and out of phase components of x(t) relative to E o cos(  t) M ion k m electron k M ion in phase out of phase

14. Dielectric Susceptibility Bound charges Connection to  and       Im{  } Re{  }

15. Dielectric Susceptibility Free charges Let   → 0 in  and  j pol = ∂P/∂t     Im{  }  Re{  } Drude ‘tail’

16. Displacement Field Rewrite E Mac = E – P/  o as  o E Mac + P =  o E LHS contains only fields inside matter, RHS fields outside Displacement field, D D =  o E Mac + P =  o  E Mac =  o E Displacement field defined in terms of E Mac (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

17. Displacement Field Inside matter .E = .E mac =  tot /  o = (  pol +  free )/  o Total (averaged) electric field is the macroscopic field - .P =  pol .(  o E + P) =  free .D =  free 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 =  o E + P Limited validity: Expressions involving  and  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  )

19. Boundary conditions on D and E D and E fields at matter/vacuum interface matter vacuum D L =  o  L E L =  o E L + P L D R =  o  R E R =  o E R  R = 1 No free charges hence .D = 0 D y = D z = 0 ∂ D x / ∂ x = 0 everywhere D xL =  o  L E xL = D xR =  o E xR E xL = E xR /  L D xL = D xR E discontinuous D continuous

20. Boundary conditions on D and E Non-normal D and E fields at matter/vacuum interface .D =  free Differential form ∫ D.dS =  free, enclosed Integral form ∫ D.dS = 0 No free charges at interface D L =  o  L E L D R =  o  R E R dSRdSR dSLdSL LL RR -D L cos  L dS L + D R cos  R dS R = 0 D L cos  L = D R cos  R D ┴ L = D ┴ R No interface free charges D ┴ L - D ┴ R =  free Interface free charges

21. 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) E L.dℓ L + E R.dℓ R = 0 -E L sin  L dℓ L + E R sin  R dℓ R = 0 E L sin  L = E R sin  R E || L = E || R E || continuous D ┴ L = D ┴ R No interface free charges D ┴ L - D ┴ R =  free Interface free charges ELEL ERER LL RR dℓLdℓL dℓRdℓR

22. Boundary conditions on D and E D L =  o  L E L D R =  o  R E R dSRdSR dSLdSL LL RR