Electrical and Optical Properties of Materials
Electrical and Optical Properties of Materials
Electrical and Optical Properties of Materials
a. Relative permittivity, r
c. Breakdown field
The capacitance increased with this insulator in place, such that we can
define the new capacitance C = r C0 , where C0 is the capacitance of the par-
allel plates when filled with a vacuum. The factor by which the capacitance
increases is thus the relative permittivity, r .
The electric displacement field D arises from the combination of the ap-
plied electric field E and the polarisation of the material P , in the relation:
D = 0 E + P (2.1)
D = 0 E + χe 0 E = 0 (1 + χe )E = 0 r E (2.2)
1
2. Dielectric properties of materials
Thus, the induced polarisation serves to increase the apparent electric field
by a factor r , which we can now express as:
P
r = +1 (2.3)
0 E
IR V0 /R 1
tan δ = = = (2.5)
IC CV0 ω ωCR
The power lost W = V I is:
1 T
Z Z 2π/ω
ω V0 sin ωt
W = VI = V0 sin ωt + CV0 ω cos ωt dt (2.6)
T 0 2π 0 R
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Electrical and optical properties of materials JJL Morton
2π/ω
ωV02
1 − cos 2ωt
Z
W = + Cω sin ωt cos ωt dt (2.7)
2π 0 2R
V02 1 1
W = = ωV02 C tan δ = ωV02 C0 r tan δ (2.8)
2R 2 2
So there is power dissipation proportional to the loss tangent (tan δ) as
well as relative permittivity r . We sometimes use the loss factor (r tan δ)
to compare dielectric materials by their power dissipation.
Another way of thinking about this is allowing a complex relative permit-
tivity which incorporates this loss tangent. The imaginary part of r is then
directly responsible for the effective resistance. The impedance of a capacitor
C is (C0 is the capacitance of the device were it filled with vacuum):
1 1
Z= = (2.9)
iωC iωr C0
For a complex r = Re(r ) + iIm(r ):
1 Re(r ) Im(r )
Z= = 2
− (2.10)
iωC0 [Re(r ) + iIm(r )] iωC0 |r | ωC0 |r |2
The first of these terms is imaginary and so still looks like an ideal capacitor,
with actual capacitance C 0 , while the second is real and so looks like a resistor
(Z = R), as defined below:
The loss tangent is then nothing more than the ratio of the imaginary and
real parts of the relative permittivity. Looking back at Eq. 2.3 we see the
relationship between the relative permittivity and the polarisation induced in
the dielectric. If the polarisation change is in phase with the applied electric
field, the material appears purely capacitive. If there is a lag (for reasons
we shall discuss in the coming section), the relative permittivity acquires
some imaginary component, the material acquires ‘resistive’ character and a
non-zero loss tangent.
3
2. Dielectric properties of materials
e e
- -
+ +
E=0 E
+ ! + ! + + ! + ! +
! + ! + ! ! + ! + !
+ ! + ! + + ! + ! +
! + ! + ! ! + ! + !
E=0 E
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Electrical and optical properties of materials JJL Morton
E=0 E
5
2. Dielectric properties of materials
+ ! + ! + + ! + ! +
! 2+ ! + ! ! 2+ ! + !
+ ! ! ! + ! ! + ! +
! + ! + ! ! + ! + !
E=0 E
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Electrical and optical properties of materials JJL Morton
Ez Pz z
– – ––
– – rδθ
δA – r
δθ
θ θ
r y
φ
+
+ δφ
+ + r sinθ δφ
+ + + x
7
2. Dielectric properties of materials
P
Es = (2.19)
30
The total electric field at some point within the material is the sum of this
contribution from the surrounding polarisation, and the applied electric field
E:
P
Eloc = E + (2.20)
30
(Note: this derivation assumed cubic symmetry, and the constant prefactors
in the above change when moving to other symmetries.) By combining this
result with Eqs 2.3 and 2.12 we obtain1 the Clausius-Mossotti relation:
r − 1 nα
= (2.21)
r + 2 30
This relation reveals how a microscopic property of a material, the polaris-
ability α, may be obtained from a measurable quantity r .
p
θ
Eloc
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Electrical and optical properties of materials JJL Morton
R
C is a normalising constant which ensures that dn = n. Our particle
at angle θ to Eloc contributes p cos θ to the overall polarisation (random
components perpendicular to Eloc will cancel out on average). Hence, the
volume polarisation is: R
p cos θ dn
P =n R (2.23)
dn
Rπ
pEloc cos θ
0
p cos θ C exp kB T
dU
P =n Rπ (2.24)
pEloc cos θ
0
C exp kB T
dU
Differentiating U with respect to θ tells us: dU = pEloc sin θ dθ, and so:
Rπ
pEloc cos θ
0
p cos θ C exp kB T
pEloc sin θ dθ
P =n Rπ (2.25)
0
C exp pEloc cos θ
kB T
pEloc sin θ dθ
This can be tackled by first cancelling the factor pEloc C from top and bottom
and then making some handy substitutions pEloc /kB T = y and cos θ = x
(which means dx = − sin θ dθ):
R −1
x exp(xy) dx
P = n R1 −1 (2.26)
1
exp(xy) dx
L(y) is known as the Langevin function and is plotted in Figure 2.9. In the
limit of small y (small fields and/or high temperatures), L(y) → y/3, i.e.
np2
P = Eloc (2.28)
3kB T
Looking back at Eq. 2.12, we now see that in this limit, the polarisability α
has a 1/T dependence with temperature, and goes with the dipole squared.
At the other limit (very high fields and/or low temperatures), all the dipoles
add up and the polarisation becomes bounded to P = np. (Note that in
this limit there is no further electric field dependence and Eq. 2.12 no longer
holds).
9
2. Dielectric properties of materials
1 L(y)
0.75
0.5
y = pEloc/kBT
-10 -7.5 -5 -2.5 2.5 5 7.5 10
L(y) ~ y/3
-0.5
-0.75
-1
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Electrical and optical properties of materials JJL Morton
and then use the Clausius-Mossotti relation (Eq. 2.21) to obtain the relative
permittivity:
r − 1 nq 2 1
= (2.33)
r + 2 3m0 (ω02 − ω 2 ) + iωγ
We notice that r is therefore complex. The imaginary part is directly due
to the ‘drag’ factor γ and leads to absorption of energy by the system, as we
might expect.
nq 2 ωγ
Im(r ) = − (2.37)
m0 (ω0 − ω 2 )2 + ω 2 γ 2
2
These expressions can be simplified for the case where the ω is close to the
resonance frequency ω0 with the substitution2 : (ω02 − ω 2 ) = 2ω(ω0 − ω).
nq 2 (ω0 − ω)
Re(r ) = 1 + (2.38)
2mω0 (ω0 − ω)2 + γ 2 /4
nq 2 γ/2
Im(r ) = − (2.39)
2m0 (ω0 − ω)2 + γ 2 /4
These terms are plotted in Figure 2.10 and show the maximum absorption
(imaginary part of r ) right on resonance at ω0 , as expected. For the limits
where ω is small (ω → 0) or large (ω → ∞) we can go back to Eq. 2.35:
nq 2
If ω → 0, then r − 1 → (2.40)
m0 ω02
2
Let ω = ω0 + δ, and write down ω02 − ω 2 neglecting powers of δ 2
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2. Dielectric properties of materials
If ω → ∞, then r − 1 → 0 (2.41)
In both cases the relative permittivity is purely real. We can think of the
low frequency limit as that in which the polarisation can easily keep up with
the oscillating electric field, and so there is no loss. At the other extreme, i.e.
very high frequencies, the polarisation simply has no chance of following the
oscillating electric field and ignores it. There is therefore a drift in the ‘back-
ground’ (non-resonant) part of r to lower values as frequency is increased,
in addition to the resonance peak. We can also see from the figure that the
linewidth of the resonance feature is equal to γ.
0.5 1
Re[ε]-1
0.25 0.75
Re[ε]-1
Im[ε]
0 0.5
Im[ε]
-0.25 0.25
-0.5 0
-2 -1 0 +1 +2
(ω-ω0)/γ
Figure 2.10: The real and imaginary parts of the relative permittivity close
to resonance
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Electrical and optical properties of materials JJL Morton
Thus we can write down the typical form of the Debye Equation
r (ω) − r (∞) 1
= (2.48)
r (0) − r (∞) 1 + iωτ
r (0) − r (∞)
Re(r (ω)) = r (∞) + (2.49)
1 + ω2τ 2
r (0) − r (∞)
Im(r (ω)) = ωτ (2.50)
1 + ω2τ 2
These real and imaginary parts are sketched in Figure 2.11. We can see
similarities with the resonant case described above. For example, we see a
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2. Dielectric properties of materials
Re[ε]
ε'
Im[ε]
ε''
ωτ
0.1 1 10 100
Figure 2.11: The real and imaginary parts of the relative permittivity close
according to Debye relaxation
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Electrical and optical properties of materials JJL Morton
log10(Frequency (Hz))
0 2 4 6 8 10 12 14 16 18
Re[ε]
ε'
0 e
-
+ +
+
+ +
Ionic Electronic
Dipolar
Im[ε] molecular spectra atomic
ε'' spectra
rotation vibration
bands bands
permanent dipoles
00 2 4 6 8 10 12 14 16 18
radio MW IR UV X-ray
log10(Frequency (Hz))
15
2. Dielectric properties of materials
Take a look at Figure 2.13, where a large electric field is applied to a semicon-
ductor. Typically, an electron in the valence band lacks the energy to enter
the conduction band. However, in a strong electric field the energy bands
bend with distance and an electron can hop from the valence to conduction
band by changing its position. There is clearly an energy barrier to such
a jump, but as the field increases, the necessary distance ∆x decreases and
there is increasing likelihood that the electron tunnels successfully. Note that
it also leaves a hole behind, which will also conduct.
In even larger electric fields (a larger bandgap will require larger fields for
Zener breakdown) an electron, once in the conduction band, may acquire very
large amounts of kinetic energy in between collisions. It may be that when
it does experience a collision, sufficient energy can be given to a valence-
band electron to promote it into the conduction band. The original electron
loses some kinetic energy but stays in the conduction band. There are now
two electrons (and two holes) which can then generate more, producing an
‘avalanche’ effect.
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Electrical and optical properties of materials JJL Morton
Ec Ec
ba
nd
e-
ion
duct Ev Ev
con
Energy, E
Energy, E
d
e-
b an
nce h+
e
val
∆x
Position, x Position, x
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2. Dielectric properties of materials
cluded gas bubbles (e.g. interlayer air in some micas). The field experienced
by the gas is higher than that in the solid because of the continuity condi-
tion on the electric displacement field D = r 0 E (see Figure 2.14). Because
r (solid) is likely to be significantly greater than r (gas), a larger electric
field E will be present within the gas region. Thus, even if the breakdown
strength of the gas were to be greater than that of the solid, it is likely to
fail at a lower applied field because of the amplifying effect of the relative
permittivity of the solid.
D
solid
gas
ε0εr,s Es ε0εr,s Es
ε0εr,g Eg
Eg = Es (εr,s/εr,g)
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