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
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
r
-
+
Note dipolar character
Electric Polarisation
Dipole Moments of Atoms
Total electronic charge per atom
Z = atomic number
 Ze 
Total nuclear charge per atom
 Ze 
r
(r )dr
el
all space
r
nuc
all space
Centre of mass of electric or nuclear charge
r
(r )dr
el/nuc

r
el/nuc
all space
r
(r ) r dr
el/nuc
all space
Dipole moment p = Zea

Ze a  Ze r
nuc
 r
el
   r
nuc
(r )dr
(r )  r el (r ) r dr
all space

r
el
all space
(r ) r dr if r
nuc
0
Electric Polarisation
Uniform Polarisation
• Polarisation P, dipole moment p per unit volume Cm/m3 = Cm-2
p
E
• Mesoscopic averaging: P is a constant field for uniformly polarised medium
E
P
s  P.n
• Macroscopic charges are induced with areal density sp Cm-2
-
P
+
E
Electric Polarisation
• Contrast charged metal plate to polarised dielectric
s-
P
s+
E
• Polarised dielectric: fields due to surface charges reinforce inside the
dielectric and cancel outside
s-
s-
• Charged conductor: fields due to surface charges cancel inside the metal
and reinforce outside
Electric Polarisation
• Apply Gauss’ Law to right and left ends of polarised dielectric
s-
s+
P
E-
E+
• EDep = ‘Depolarising field’
• Macroscopic electric field
E
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-
EMac= E + EDep = E - P/o
Electric Polarisation
Non-uniform Polarisation
• Uniform polarisation  induced surface charges only
P
-
+
E
• Non-uniform polarisation  induced bulk charges also
+
-
+
Displacements of positive charges Accumulated charges
-
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
z
Py=0
-(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z) DxDyDz = Qpol
Dz
Dy
Dx
x
Charge entering cube via all faces:
Py=Dy
y
rpol = lim (DxDyDz)→0 Qpol /(DxDyDz)
-. P = rpol
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 c (dimensionless) defined through
P = o c EMac
EMac = E – P/o
o E = o EMac + P
o E = o EMac + o c EMac = o (1 + c)EMac = oEMac
Define dielectric constant (relative permittivity)  = 1 + c
EMac = E / E =  EMac
Typical static values (w = 0) for : silicon 11.4, diamond 5.6, vacuum 1
Metal:  →
Insulator:  (electronic part) small, ~5, lattice part up to 20
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
Si ion
Bound electron pair
Mion k melectron k Mion
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 Susceptibility
Bound charges
Resonance model for uncoupled electron pairs
mx  mx  kx  qEo Re{e iwt }
x  x 
Mion k melectron k Mion
k
q
x  Eo Re{e iwt }
m
m
x(t)  Re{A(w ) e iwt } trial solution
x (t)  iw x(t)
x(t)  w 2 x(t)
(Re{} assumed hereafter)
k
q

2
iwt
  w  iw   x(t)  Eo e
m
m

k
q

2
iwt
 Eo e iwt
  w  iw   A(w ) e
m
m

Dielectric Susceptibility
Bound charges
In and out of phase components of x(t) relative to Eo cos(wt)
A(w ) 
A(w ) 
qEo
1
m  w 2  iw  wo2

qEo
1
m wo2  w 2  iw

wo2 

Mion k melectron k Mion

wo2  w 2
qEo
Re{A(w )} 
m wo2  w 2 2  w 2

k
m


Im{A( w )} 
qEo
w
m wo2  w 2 2  w 2

x(t)  Re{A(w )e -iwt }  Re{A(w )}Re{e -iwt }  Im{A( w )}Im{e -iwt }
qEo

m




 w 2  w 2 cos(wt)
w sin(wt)

 2o
2
2 2
2
2 2
wo  w  w 2
 wo  w  w 
in phase
 

out of phase






Dielectric Susceptibility
Bound charges
Connection to c and 
Polarisati on  dipole moment per unit volume  qx(t)/V





wo2  w 2
q2
i w
-i wt 
P(t) 
Re 

Eo e    o c E(t)
2
2
2
2
2
2
2
2
mV
wo  w  w  
 wo  w  w 

wo2  w 2
q2
q2
w
Re{ c (w )} 
Im{
c
(
w
)}

m o V wo2  w 2 2  w 2
m o V wo2  w 2 2  w 2



 


wo2  w 2  iw
q2
 (w )  1  c (w )  1
m o V wo2  w 2 2  w 2






6
w

Im{w}
w  wo
4
2
model dielectric function
w/wo
1
2
4

2
Re{w}
3
4
Dielectric Susceptibility
Free charges
sw
Let wo → 0 in c and  jpol = ∂P/∂t
Current density j(t)  s E(t) s  conductivi ty
6
4
q2
1
-iwt
P(t)   o
E
e
o
m o V wo2  w 2  iw
2
Re{w}
wo  0
Drude ‘tail’
1
P(t)
q2
 iw
-iwt
jpol (t) 
 o
E
e
o
t
m o V wo2  w 2  iw
2
Im{sw}
4
P(t)
q2
 iw
-iwt
jf ree(t) 
 o
E
e
Let wo  0
o
2
t
m  o V  w  i w


q2
iw
q2 iw 3   w 2
q 2 iw  
s f ree(w ) 


2
2
4
mV w  iw mV w  w  mV w 2   2
q2
Ne 2
s f ree(0) 

mV 
m
Drude conductivi ty
e q
2
2
1
N
V
1


2
w
3
4
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 ) 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
 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 D
relation D = oE + P
Expressions involving  and c
•
Limited validity:
•
Have assumed that c is a simple number: P = o c E
only 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 )
Boundary conditions on D and E
D and E fields at matter/vacuum interface
matter
DL = oLEL = oEL + PL
vacuum
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 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
dSL
qL
DL = oLEL
qR
DR = oRER
dSR
-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)
dℓL
qL
EL
qR
dℓR
ER
EL.dℓL + ER.dℓR = 0
-ELsinqLdℓL + ERsinqR dℓR = 0
ELsinqL = ERsinqR
E||L = E||R
D┴L = D┴R
D┴L - D┴R = sfree
E|| continuous
No interface free charges
Interface free charges
Boundary conditions on D and E
dSL
qL
DL = oLEL
qR
DR = oRER
dSR
EL sinqL  ER sinqR
DL cosqL  DR cosqR
DL/R   L/R oEL/R
EL sinqL
EL sinqL
ER sinqR


DL cosqL  L oEL cosqL  R oER cosqR
tan qL

 L
tan qR  R
in absence of free charges at interface