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• 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
Dielectrics
E
Dr(r) Change in electronic charge density
Note dipolar character
r
No E fieldE field on
- +
r(r) Electronic charge density
Electrostatic potential of point dipole• +/- charges, equal magnitude, q, separation a• axially symmetric potential (z axis)
-or1
r1
4
q)(
r
potential dipole point''
r4
cos p
r
cos
4
qa
cos2r
a
r
1
cos r
a
2r
a1
r
1
r
1
cos r
a
2r
a1r
cos r a2
arr
2o
2o
2
21
2
22
222
r
a/2
a/2
r+
r-rq+
q-
x
z
p
q
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
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
Electric Polarisation
Electric field in model 1-D crystal with lattice spacing ‘a’
r(x)
x5 1 0 1 5 2 0
0 .2
0 .4
0 .6
0 .8
1 .0
a
symmetry by b a
0 electrons) (nucleidensity average a
a
x2nsin b
a
x2ncos a a(x)
(x)density charge of expansion series Fourier
nn
o
nno
00
Electric Polarisation
Expand electric field Ex in same way (Ey, Ez = 0 by symmetry)
r(x)
x5 1 0 1 5 2 0
0 .2
0 .4
0 .6
0 .8
1 .0
a
1n 2n
aa d
a d
a
2n
a
x2ncos
a
a
x2ncos d
a
2n
dx
dE(x).
d symmetry by c field ic)(macroscop average c
a
x2nsin d
a
x2ncos c cE(x)
nn
nn
n
n
nno
nno
oooo
el
00
E
Electric PolarisationApply external electric field and polarise charge density
r(x)
x5 1 0 1 5 2 0
0 .2
0 .4
0 .6
0 .8
1 .0
a
- - - - - - -
b c.f. b
aa 0 aa
a
x2nsin b
a
x2ncos a a(x)
charge dunpolarise forsymmetry c.f.
symmetry has charge onPolarisati
n'n
n'no
'o
'n
'n
'o
00
'
even
odd
E
Electric PolarisationApply external electric field and polarise charge density
r(x)
x5 1 0 1 5 2 0
0 .2
0 .4
0 .6
0 .8
1 .0
a
- - - - - - -
fieldc macroscopi the provide charges surface These
ends the at charges onpolarisati inducednow are there But
equation sPoisson' fromc a Since
dd c c.f. c
a
x2nsin d
a
x2ncos c c(x)E
beforesymmetry c.f.symmetry has charge onPolarisati
'o
'o
n'nn
'n
'n
'n
'o
0,0
00
'
evenodd
E
• Polarisation P, dipole moment p per unit volume Cm/m3 = Cm-2
• Mesoscopic averaging: P is a constant vector field for a uniformly polarised medium
• Macroscopic charges are induced with areal density sp Cm-2 in a uniformly polarised medium
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 PolarisationDefine dimensionless dielectric susceptibility c 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 values for e: silicon 11.8, diamond 5.6, vacuum 1Metal: e →Insulator: e (electronic part) small, ~5, lattice part up to 20
Electric PolarisationRewrite 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
Non-uniform polarisation
• Uniform polarisation induced surface charges only
• Non-uniform polarisation induced bulk charges also
Displacements of positive charges Accumulated charges
+ +- -
P- + E
Non-uniform polarisation
Charge entering xz face at y = 0: Px=0DyDz
Charge leaving xz face at y = Dy: Px=DxDyDz
= (Px=0 + ∂Px/∂x Dx) DyDz
Net charge entering cube: (Px=0 - Px=Dx ) DyDz = -∂Px/∂x DxDyDz
Dx
Dz
Dy
z
y
x
Charge entering cube via all faces:
-(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z) DxDyDz = Qpol
rpol = lim (DxDyDz)→0 Qpol /(DxDyDz)
-.P = rpol
Px=DxPx=0
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
.(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 = rfree, 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