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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 ?

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dielectrics
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
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

Note dipolar character

r

- +

electric polarisation1
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 polarisation2

E

E

E

p

P

P

+

-

Electric Polarisation

Uniform Polarisation

  • Polarisation P, dipole moment p per unit volume Cm/m3 = Cm-2
  • Mesoscopic averaging: P is a constant field for uniformlypolarisedmedium
  • Macroscopic charges are induced with areal density spCm-2
electric polarisation3

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
electric polarisation4

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-

electric polarisation5

E

+

-

+

-

P

+

-

Electric Polarisation

Non-uniform Polarisation

  • Uniform polarisation  induced surface charges only
  • Non-uniform polarisation  induced bulk charges also

Displacements of positive charges Accumulated charges

electric polarisation6
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/∂yDy) DxDz

Net charge entering cube via xz faces: (Py=0 -Py=Dy)DxDz = -∂Py/∂yDxDyDz

z

Charge entering cube via all faces:

-(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z)DxDyDz = Qpol

rpol= lim (DxDyDz)→0Qpol/(DxDyDz)

-. P = rpol

Dz

Py=0

Py=Dy

y

Dy

Dx

x

electric polarisation7
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

Dielectric susceptibility c (dimensionless) defined 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

Typicalstatic values (w = 0) for e: silicon 11.4, diamond 5.6, vacuum 1

Metal: e →

Insulator: e (electronic part) small, ~5, lattice part up to 20

dielectric susceptibility1

Mion k melectron kMion

Si ion

Bound electron pair

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

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 susceptibility2

Mion k melectron kMion

Dielectric Susceptibility

Bound charges

Resonance model for uncoupled electron pairs

dielectric susceptibility3

Mion k melectron kMion

Dielectric Susceptibility

Bound charges

In and out of phase components of x(t) relative to Eo cos(wt)

in phase out of phase

dielectric susceptibility4
Dielectric Susceptibility

Bound charges

Connection to c and e

e(w)

Im{e(w)}

w = wo

w/wo

Re{e(w)}

dielectric susceptibility5

s(w)

Re{e(w)}

wo = 0

Drude ‘tail’

w

Im{s(w)}

Dielectric Susceptibility

Free charges

Let wo→ 0 in c and e jpol = ∂P/∂t

displacement field
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 e) 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

e contains the microscopic physics

displacement field1
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
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)
boundary conditions on d and e
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

boundary conditions on d and e1

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 = rfreeDifferential form∫ D.dS = sfree, 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

boundary conditions on d and e2

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