Electric polarisation Electric susceptibility Displacement field in matter Boundary conditions on...

• 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

Microscopic viewpoint

Atomic polarisation in E field

Change in charge density when field is applied

Electric Polarisation

E

Dr(r) Change in electronic charge density

Note dipolar character

r

No E fieldE field on

- +

r(r) Electronic charge density

Electric PolarisationDipole Moments of AtomsTotal electronic charge per atom

Z = atomic number

Total nuclear charge per atom

Centre of mass of electric or nuclear charge

Dipole moment p = Zea

space all

el )d( Ze rr

0 if d )(

d )()( Ze a Ze

nucspace all

el

space all

elnucelnuc

rrrr

rrrrrr

space all

nuc )d( Ze rr

space all

el/nuc

space all

el/nuc

el/nuc )d(

d )(

rr

rrr

r

Uniform Polarisation

• Polarisation P, dipole moment p per unit volume Cm/m3 = Cm-2

• Mesoscopic averaging: P is a constant field for uniformly polarised medium

• Macroscopic charges are induced with areal density sp Cm-2

Electric Polarisation

p E

P E

P- + E

P.n

• 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

s- s+

E

P

s- s-

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+/2o

E- = s-/2o

EDep = E+ + E- = (s++ s-)/2o

EDep = -P/o P = s+ = s-

s-

E

P s+

E+E-

Electric PolarisationNon-uniform Polarisation

• Uniform polarisation induced surface charges only

• Non-uniform polarisation induced bulk charges also

Displacements of positive charges Accumulated charges

+ +- -

P- + E

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/∂y Dy) DxDz

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

Charge entering cube via all faces:

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

rpol = lim (DxDyDz)→0 Qpol /(DxDyDz)

-. P = rpol

Dx

Dz

Dy

z

y

x

Py=DyPy=0

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 SusceptibilityDielectric susceptibility c (dimensionless) defined through

P = o c EMac

EMac = E – P/o

o E = o EMac + Po E = o EMac + o c EMac = o (1 + c)EMac = oEMac

Define dielectric constant (relative permittivity) = 1 + c

EMac = E / E = e EMac

Typical static 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 Susceptibility

Bound chargesAll valence electrons in insulators (materials with a ‘band gap’)Bound valence electrons in metals or semiconductors (band gap absent/small )

Free chargesConduction electrons in metals or semiconductors

Mion k melectron k MionSi ionBound electron pair

Resonance frequency wo ~ (k/M)1/2 or ~ (k/m)1/2 Ions: heavy, resonance in infra-red ~1013HzBound electrons: light, resonance in visible ~1015HzFree electrons: no restoring force, no resonance

Dielectric Susceptibility

Bound chargesResonance model for uncoupled electron pairs

Mion k melectron k Mion

tt

t

t

t

t

e Em

qe )A(

m

k

e Em

qx(t)

m

k

hereafter) assumed (Re{} x(t)(t)x x(t)(t)x

solution trial }e )Re{A(x(t)

}Re{e Em

qx

m

kxx

}Re{e qEkxxmxm

o

o

o

o

ii

i

i

i

i

i

i

i

2

2

2

Dielectric Susceptibility

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

Mion k melectron k Mion

22222222

22

22222222

22

22

222

oo

o

oo

o

o

oo

-i-i-i

i

i

t)sin(t)cos(

m

qE

})}Im{eIm{A(})}Re{eRe{A(})eRe{A( x(t)

m

qE )}Im{A(

m

qE )}Re{A(

1

m

qE )A(

m

k1

m

qE )A(

o

oo

o

o

ttt

in phase out of phase

Dielectric Susceptibility

Bound chargesConnection to c and e

function dielectric model

Vm

q1)( 1 )(

Vm

q )}(Im{

Vm

q )}(Re{

(t)eERemV

q(t)

qx(t)/V volume unit per moment dipole onPolarisati

2

22

o

2t

2222

22

22222222

22

22222222

22

o

o

o

ooo

o

o

o

oo

o

i

i -i EP

1 2 3 4

4

2

2

4

6 ( )e w

/w wo

= w wo

Im{ ( )e w }

Re{ ( )e w }

Dielectric Susceptibility

Free chargesLet wo → 0 in c and e jpol = ∂P/∂t

tyconductivi Drude

1

V

1N qe

m

Ne

mV

q)(

mV

q

mV

q

mV

q)(

LeteVm

q

t

(t)(t)

eVm

q

t

(t)(t)

e1

Vm

q(t)

tyconductivi (t)(t)density Current

2222

free

222

free

o

2

free

o

2

pol

o

2

t

t

t

0

0

2224

23

2

2

22

22

ii

i

i

i

i

i

i

i

oo

o

ooo

ooo

-i

-i

-i

EP

j

EP

j

EP

Ej

1 2 3 4

4

2

2

4

6

w

wo = 0

Im{ ( )s w }

( )s w

Re{ ( )e w }

Drude ‘tail’

Displacement FieldRewrite 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 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

• Always valid: Gauss’ Law for E, P and Drelation D = eoE + P

• Limited validity: Expressions involving e and

• Have assumed that is a simple number: P = eo Eonly true in LIH media:

• Linear: independent of magnitude of E interesting media “non-linear”: P = eoE + 2

eoEE + ….

• 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

D and E fields at matter/vacuum interface

matter vacuum

DL = oLEL = oEL + PL DR = oRER = oER R = 1

No free charges hence .D = 0

Dy = Dz = 0 ∂Dx/∂x = 0 everywhere

DxL = oLExL = DxR = oExR

ExL = ExR/L

DxL = DxR 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.dS = sfree, enclosed Integral form

∫ D.dS = 0 No free charges at interface

DL = oLEL

DR = oRER

dSR

dSL

qL

qR

-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 E

Non-normal D and E fields at matter/vacuum interface

Boundary conditions on E from ∫ 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

EL

ER

qL

qRdℓL

dℓR

Boundary conditions on D and E

DL = oLEL

DR = oRER

dSR

dSL

qL

qR

interface at charges free of absence in tan

tan

cos E

sinE

cos E

sinE

cos D

sinE

cos D cos D

sinE sinE

R

L

R

L

RRR

RR

LLL

LL

LL

LL

L/RL/RL/R

RRLL

RRLL

oo

oED