Dielectrics
description
Transcript of Dielectrics
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• 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
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Microscopic viewpointAtomic 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
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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
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rr
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r
r
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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.ns
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• 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 Polarisations- s+
E
P
s- s-
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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-
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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
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Electric PolarisationPolarisation charge density
Charge entering xz face at y = 0: Py=0DxDz Cm-2 m2 = CCharge 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
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Electric PolarisationDifferentiate -.P = rpol wrt time.∂P/∂t + ∂rpol/∂t = 0Compare 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’ orpolarisation and ‘free’ or conduction charges
rtot = rpol + rfree
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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 = EMac
Typical static values (w = 0) for : silicon 11.4, diamond 5.6, vacuum 1Metal: →Insulator: (electronic part) small, ~5, lattice part up to 20
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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
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Dielectric Susceptibility
Bound chargesResonance model for uncoupled electron pairs
Mion k melectron k Mion
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Dielectric Susceptibility
Bound chargesIn and out of phase components of x(t) relative to Eo cos(wt)
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Dielectric SusceptibilityBound chargesConnection to c and
function dielectric model
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Vmq )}(Im{
Vmq )}(Re{
(t)eERemVq(t)
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Dielectric SusceptibilityFree chargesLet wo → 0 in c and jpol = ∂P/∂t
tyconductivi Drude
1 V1N qe
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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 ) and E (in vacuum, relative permittivity 1). Define
D = o E
where is the relative permittivity and E is the electric fieldThis is one of two constitutive relations contains the microscopic physics
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Displacement FieldInside 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 eliminatepolarisation charges from any calculation
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Validity of expressions
• Always valid: Gauss’ Law for E, P and Drelation D = oE + P
• Limited validity: Expressions involving and c
• Have assumed that c is a simple number: P = o c Eonly true in LIH media:
• Linear: c independent of magnitude of E interesting media “non-linear”: P = c oE + c2
oEE + ….
• Isotropic: c independent of direction of E interesting media “anisotropic”: c is a tensor (generates vector)
• Homogeneous: uniform medium (spatially varying )
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Boundary conditions on D and ED 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 everywhereDxL = oLExL = DxR = oExR
ExL = ExR/L DxL = DxR
E discontinuousD continuous
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Boundary conditions on D and ENon-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
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Boundary conditions on D and ENon-normal D and E fields at matter/vacuum interfaceBoundary 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|| continuousD┴L = D┴R No interface free chargesD┴L - D┴R = sfree Interface free charges
EL
ER
qL
qRdℓL
dℓR
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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 DsinE sinE
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