Unformatted text preview: 8 Dielectric properties
D = ε0( + P (8.1) D: dielectric displacement P= P: polarization 1
V ∑p
i i (8.2) pi: electrical dipole moments
ε0 = 8.854 × 1012 As/Vm
P = χε0( , ⇒ vacuum permittivity χ: dielectric susceptibility D = ε0(1+χ)( = ε0ε( (8.3) (8.4) ε: dielectric function, ε(ω) ⇒ optical properties
in general 2nd rank tensor,
scalar for isotropic media and cubic crystals.
Insulators: P due to localized dipoles.
Metals and semiconductors: also displacement of mobile
electrons. Dielectric solids
81 a) Displacement of electrons with respect to atomic cores
Always present but can be
masked by other polarizations.
Fig. 8.1 Illustration of polarization due to
displacement of electrons in atoms with
respect to atomic cores. b) Additional contribution in ionic crystal: relative
displacement of cations and anions.
Paraelectric solids
Contain permanent electric dipoles (asymmetric polar
molecules, e.g., H2O)
( = 0: dipoles oriented statistically
⇒ no net polarization
( ≠ 0: orientation of dipoles in field against thermal agitation
⇒ P = P((/T) analogous to M = M(B/T) in
paramagnetism. Ferroelectric solids
82 Spontaneous electric polarizatin below the ferroelectric Curie
temperature
Ferroelectric domains and hysteresis loop P(E) analogous to
ferromagnetism.
Physical origin of ferroelectricity: see Fig. 8.2 Fig. 8.2 (a) The crystal structure of barium titanate. The prototype crystal is calcium titanate
(perovskite). The structure is cubic, with Ba2+ ions at the cube corners, O2 ions at the face
centers, and a Ti4+ ion at the body center. (b) below the Curie temperature the structure is
slightly deformed, with Ba++ and Ti4+ ions displaced relative to the O2 ions, thereby developing
a dipole moment. The upper and lower oxygen ions may move downward slightly. Antiferroelectric solids: analogous to antiferromagnetic
solids. Local electric field
83 Gases: electric polarization of an atom is not affected by fields
of polarized neighboring atoms.
⇒ P = ε0⋅nv α( (8.5) for identical molecules of polarizability α
nv: number of molecules / volume
⇒ ε = 1 + nvα, i.e., χ = nvα (8.6) Solids: local field at an atom is influenced by dipoles of
neighbours.
Local field at an atom:
(local = (ext + (sample
P = ε0nvα(local (8.7)
(8.8) Calculation of (local
84 a) Field EI acting on atom at the center of an imaginary
sphere
(I depends on crystal structure.
(I = 0 if atom sites have cubic symetry, e.g. for: sc, fcc,
bcc, NaCl, and CsCl structure
(see practical course for derivation) b) Imaginare sphere is cut out of the solid
Choise of radius R of imaginary sphere:
• 2R << λ of electromagnetic field ⇒ (I = 0 for cubic
symmetry.
• R large enough to describe polarization in center due to
atoms outside sphere by quasicontinuous dipole distribution,
i.e., macroscopic polarization P
⇒ Field due to P described by polarization charges ρP
on surface (see Fig. 8.3). 85 ρp = Pn (8.9) Pn: normal component of P
Note: unit normal points inside
hollow sphere ⇒ ″″ sign.
Cf.: sphere with homogeneous
surface charge density ρ ⇒
( = (n = ρ/ε0
(Rigorous proof of eq. (8.9): see, e.g.
Kittel)
Fig. 8.3 In a continuum model the field at the
center of a hollow sphere due to the
polarization of the atoms outside the sphere
can be described in termens of polarization
charges on the surface of the sphere. Charge in ring at angle θ: 14
42 3 dq = − P cosθ ⋅ (2πa sin θ ) adθ
Pn (8.10)
Contribution to field in center:
d( = − 1 dq
cosθ
4πε 0 a 2 <( ⊥ P> = 0, (8.11) (  P =( cosθ ⇒ polarization field in center of hollow sphere
(″Lorentz field″)
π P
1
(L =
cos 2 θ sin θdθ =
P
2ε 0 ∫
3ε 0
0
86 (8.12) c) Account for depolarization Fig. 8.4 The depolarization field is opposite to the polarization and is due to the net surface
charges originating from the polarization of a body in an external field. The polarization is
homogeneous for an elipsoid of revolution. In this case the depolarizing field (N can be
described in terms of a geometrydependent depolarization factor N. Depolarizing field for homogeneous polarization (only for
ellipsoid of revolution) (N =NP / ε0 (8.13) Field inside the sample (″macroscopic field″): ( = ( ext − 1
N P,
ε0 P = ε 0 χ( = N: depolarization factor 87 χ
ε 0 ( ext
1 + Nχ (8.13a) Depolarization factor for special geometries:
Sphere: N = 1/3, hollow sphere: N = 1/3 (see eq. 8.12) Long rod  ( : N = 0
⇒ for (I = 0 :
(local = (ext + (N + (L (8.14) Spherical sample: (N = (L ⇒ (local = (ext (8.15) 1 1 2 Thin slap ⊥ ( : ( local = ( ext − ε P + 3ε P = ( ext − 3ε P (8.16)
0
0
0 1
( local = ( ext +
P
Long rod  ( : (N = 0 ⇒
3ε
0 88 (8.17) ClausiusMosotti equation
Solid with atom sites of cubic symetry ⇒ (I = 0
(local = ( + (L ,
( = (ext + (N , (8.18) macroscopic field inside sample P = ε0nv α(local (8.8)
1 ⇒ P = ε0nv α(( + 3ε P )
0 nvα (
1
1 − nvα ⇒
3 P = ε0 χ( P = ε0
⇒ χ= (8.3) nvα
1
1 − nvα
3 ε = 1+ χ = 1+ ⇒ (8.19) (8.20)
n vα
1
1 − n vα
3 ε −1
1
n vα =
3
ε +2 (8.21) ClausiusMosotti equation (8.22) Measurement of ε ⇒ molecular polarizability α. 89 Lorentz’s oscillator model for the
electronic polarizability
Small displacement x of an electron in a lattice atom from it’s
equilibrium position.
⇒  linear restoring force
 dipole moment ex
Applied electromagnetic field ⇒ ( = ( local e − iωt ⇒ forced oscillation.
0 Oscillating dipole emits adiation ⇒ damping ∝dx/dt
Equation of motion: d 2x
dx
2
0
m 2 + mβ
+ mω 0 x = −e(local e iωt
dt
dt (8.23) β: damping constant,
ω0 = f / m : frequency of undamped vibration, Restoring force: F = fx
Analogous to forced mechanical oscillation ⇒
stationary state approached with relaxation time
τ = 1 /β (8.24) 810 Complex amplitude: x=− 1
e
(
iω local
m ω2 −ω2 −
0
τ p = ex = αelε0(local Dipole moment: e2
1
α el (ω ) =
Electrical polarizability:
ε 0 m ω 2 − ω 2 − iω
0
τ (8.25) (8.26) (8.27) (8.27) in (8.21) ⇒
nv e 2
ε (ω ) = 1 +
ε 0m 1
1
nv e 2
= 1+
2
ω1
e
ε 0m ω 2 − ω 2 − i ω
ω 02 − ω 2 − i − nv
1
τ
τ 3 ε 0m (8.28)
1
e2
with ω = ω − 3 nv ε m
0
2
1 2
0 Separation: ε(ω) =ε1(ω) + iε2(ω) (8.29) nv e 2
ω12 − ω 2
ε1 (ω ) = 1 +
ε 0 m (ω12 − ω 2 ) 2 + ω 2 / τ 2 (8.30) nv e 2
ω /τ
ε 2 (ω ) =
ε 0 m (ω12 − ω 2 ) 2 + ω 2 / τ 2 (8.31) 811 Maxwell equations for solid
divB = 0 ⇔ ∫∫ Bd A = 0 divD = ρ ⇔ ∫∫ Dd A = Q (no magnetic monopoles) (8.32)
(8.33) (charges are sources of dielectric displacement field)
&
curl ( =  B ⇔ &
∫ ( d r = − ∫∫ Bd A (8.34) (law of induction)
&
curl H = j + D ⇔ &
∫ H d r = I + ∫∫ Dd A (8.35) (current or charge in flux cause curled magnetic field)
&
(8.35) ⇒ curl H = j + D = σ ( + ε 0ε (& ⇒
& = − µ H = curl( ⇒ µ curl H = curl curl (
&
&
(8.34) ⇒ − B
0
0 − 1
curl curl ( = σ (& + ε 0ε (&&
23
µ 0 14 4
curl curl ( = ∇ × (∇ × ( ) = ∇ (∇( ) − ∇ (
2 ∇( = 0 if ρ = 0 inside the sample ⇒ 2
∇ ( = ε 0εµ 0(&& + µ 0σ (& Wave equation for nonmagnetic solid (µ ≈ 1) 812 (8.36) 2
∇ ( = ε 0εµ 0 (&& + µ 0σ (& (8.36) Solution by damped wave ansatz: ε = ε 0e ~
i ( k ⋅ r −ω t ) ~ 2 εω 2
ω
k = 2 + iσ
c
ε 0c 2 (8.37) in (8.37) ⇒
c = 1 / µ 0ε 0 , ⇒ (8.37) speed of light in vacuum ~ω
σ
ω~
k=
=n
ε +i
c
ε 0ω c ~
Complex refractive index n = n + iκ (8.38) (8.39) n: refractive index, κ: absorbtion coefficient ~ω
ω
k = n+i κ
⇒
c
c (8.40) Plane wave propagating in zdirection:
ω κω
i n z −ωt z
−
(8.37)
c c
⇒ ε = ε 0e e2 13
14 4
23
(8.40) damping in direction
plane wave
of propagation c in vacuum → c/n in solid 813 (8.41) Damping of amplitude by e
Damping of intencity by e
K= − − κω
z
c ⇒ 2κω
z
c 2κω
: absorbtion constant
c (8.42) Relation to ε(ω) =ε1(ω) + iε2(ω) ~
n 2 = ( n + iκ ) 2 = ε (ω ) = ε 1 (ω ) + iε 2 (ω ) (8.43) ε1(ω) = n2  κ2 = ε (8.44) σ
ε 0ω (8.49) ε 2 (ω ) = 2nκ = σ
~
n2 = ε +i
↑ (8.38) ⇒
ε 0ω 814 Fig. 8.5 General form of ε1(ω) and ε2(ω) for a dipole oscillator according to equations (8.30)
and (8.31). ε2(ω) ≈ 0 outside resonance maximum
Width of maximum: 1/τ
ε2(ω) ≈ 0 ⇒ n ≈ ε 1 (ω )
e2
1
Resonance frequency: ω1 = ω 0 − 3 nv ε m < ω 0
0 ω0: resonance frequency of αel(ω)
ω1 < ω0 due to effect of local field.
Classical harmonic oscillator: ω 0 = 815 f /m Quantum mechnics: resonances are absorbtion frequencies due
to electronic tansitions
⇒ always several resonance maxima in atom mainly in
1016 s1 range (UV).
Solid: also interband and intraband transitions
Intraband transitions: between filled and empty states of
conduction band in metals and
semiconductors.
ω < ω1 : ε1(ω) > 1 ⇒ n ≈ ε1 (ω ) > 1 lim ε 1 (ω ) = ε (0) ω →0 ε(0): statistic dielectric constant (already holds for
visible range).
ω > ω1 : ε1(ω) < 1 ⇒ n < 1
e.g. for Xrays lim ε1 (ω ) = 1 ω →∞ 816 ...
View
Full
Document
This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.
 Spring '02
 Adelung

Click to edit the document details