electric polarisation electric susceptibility displacement field in matter boundary conditions on...
TRANSCRIPT
• 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