CHAPTER 7: Dielectrics … What is a capacitor ? a passive two-terminal electrical component used to store energy in an electric field M etal term in atio n M e ta l ele ctro d e C eram ic Epoxy Leads (b) M ultilayerceramiccapacitor (stackedceram iclayers) (a) Singlelayerceram iccapacitor (e.g.diskcapacitors) Singleandm ultilayerdielectriccapacitors (b ) A lm etallizatio n Polym er film (a ) Two polym ertapesin (a)eachwitham etallized film electrodeonthe surface(offsetfrom each other)can berolled together(like aSw issroll- cake)to obtain apolym erfilm capacitorasin (b).A sthetw o separate m etalfilm sarelined atopposeedges,electroding isdoneoverthew hole side surface.
Transcript
CHAPTER 7: Dielectrics …What is a capacitor ?a passive two-terminal electrical component used to store energy in an
(a) Single layer ceramic capacitor(e.g. disk capacitors)
Single and multilayer dielectric capacitors
(b )
A l m e ta l l i z a t io n
P o l y m e r f i lm
(a )
Two polymer tapes in (a) each with a metallized film electrode on thesurface (offset from each other) can be rolled together (like a Swiss roll-cake) to obtain a polymer film capacitor as in (b). As the two separatemetal films are lined at oppose edges, electroding is done over the wholeside surface.
CapacitorsWhat is a dielectric material ?an electrical insulator that can be polarized by an applied electric field
What are the ways of increasing capacitance?
d
AεεC . . .
dV
dQC ro
Co+Qo Qo
E
V
+Q QC
E
V
Dielectric
i (t)
V
PolarizationPolarization
… polar molecules vs. non-polar molecules
Can non-polar molecules become polar ?
aQp
The origin of electronic polarization.
(a ) A n e u tra l a to m in E = 0 . (b ) In d u c e d d ip o le m o m e n t in a fie ld
E le c tro n c lo u d
A to m icn u c le u s
p in d u c e d
E
C e n te r o f n e g a tiv ec h a rg e
xC O
The definition of electric dipole moment.
Electric Dipole Moment,
PolarizationHow do dielectrics increase the capacitance of a
capacitor ?
AQ
P P
Co+Qo Qo
E
V
+Q QC
E
V
Dielectric
i (t)
V
+Q E Q
V
Area = A ptotal
P QP +QP
d
(c)
QP +QP
Bound polarizationcharges on the surfaces
Polarization,
Polarization
Susceptibility, Χe: A proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. (dimesnionless)
Permittivity: A measure of how an electric field affects, and is affected by, a dielectric medium. (F/m)
polarizability … dielectric constant … are material properties.In a way they are equivalent to what mobility is for conductors and
semiconductors.
AQ
P P EεχP oe re ε1χ
αEpinduced
Polarization Mechanism
Dependence of ε’r & ε”r frequency depends on polarization mechanism:
a. Electronicb. Ionicc. Interfaciald. Orientational
POLARIZATION MECHANISMS
Ionic:polarization caused by relative displacements between positive and negative ions in ionic crystals
Electronic:the stretching of atoms/electronic clouds under an applied E-field (in covalent solids)
(a) Valence electrons in covalent bonds in the absence of an applied field. (b) When an electric field is applied to a covalent solid, the valence electrons in the covalent bonds are shifted very easily with respect to the positive ionic cores. The whole solid becomes polarized due to the collective shift in the negative charge distribution of the valence electrons.
POLARIZATION MECHANISMS
Interfacial:charge accumulation at defective interfaces (2 material or 2 regions of same material) leads to the formation of a net polarization vector
E
A c c u m u la te d c h a rg e
(b )
E le c tro d eD ie le c tr ic
E le c tro d e
F ix ed c h a rg eM o b ile c h a rg e(a )
E
G ra in b o u n d a ry o r in te r fa ce
(c )
(a) A crystal with equal number of mobile positive ions and fixed negativeions. In the basence of a field there is no net separation between all thepositive charges and all the negative charges. (b) In the presence of anapplied field the mobile positive ions migrate towards the negativeelectrode and accumulate there. There is now an overall separationbetween the negative charges and positive charges in the dielectric. Thedielectric therefore exhibits interfacial polarization. (c) Grain boundariesand interfaces between different materials frequently give rise to interfacialpolarization.
POLARIZATION MECHANISMS
Orientational (Dipolar):in “rigid polarized molecule” materials, an applied field aligns the permanent dipoles to yield a net polarization vector
Cl H+
po
(a)
(b)
pav =0
q
+Q
Q
F=Q E
F
po = aQ
t
E
pav¹ 0 E
(c)
(d)
POLARIZATION MECHANISMS
DIELECTRIC LOSS
The dielectric constant is frequency dependent … why ?
The polarization process is not instantaneous … i.e. it takes a finite amount of time for the molecules to align themselves.
If the applied field is changing so fast that the molecules cannot respond to it at all … then the polarization is … zero!
Therefore ε is frequency dependent …
''r
'rr jεεε
DIELECTRIC LOSS
The imaginary part represents dielectric “losses” due to “slow” polarization.
v = V o s int
P = P o s in (t - )
E = E o s int
(a )
r ''
r '
r (0 )
1
1 / 1 0 /1 0 0 /0 .0 1 / 0 .1 /
r' andr''
(b ) ''r
'rr jεεε
t
t
E
E o
E
p
d (0 )E
pÐd(0 )E
d (0 )E o
0
The dc field is suddenly changed from Eo to E at time t = 0. The
induced dipole moment p has to decrease from d(0)Eo to a final value
of d(0)E. The decrease is achieved by random collisions of molecules
in the gas.
DIELECTRIC LOSS''r
'rr jεεε
d
εωAε
d
εAεjω
d
)Ajε(εεjωCjωY
''ro
'ro
''r
'ro
'r
''r
ε
ε tanδ... tangent loss
v = V o s int
P = P o s in (t -)
C
C ond uctance = G p = 1 /R p
v = V os int
tanδεεωEεωεd
V
dA
1
R
VW '
ro2"
ro2
22
Pvol Dielectric Loss
per unit volume,
DIELECTRIC LOSS
f
ε
10
MHz
0
ε’r
ε”r
1
2
1. For the dielectric material as in the figure, calculate the loss tangent at 10 MHz frequency.
If the parallel plate capacitor is formed with the material, where the separation is 1mm, plate area is 1mm2 and 1V is applied across…2. Calculate the value of resistance and capacitance.3. Calculate the power loss per unit volume.
'r
''r
ε
ε tanδ tangent loss 1.
''ro
P εωAε
dR 2.
d
εAεC
'ro
P
tanδεεωEεωεd
V W3. '
ro2"
ro2
2
vol
=0.5
=1.8x106 ohm =1.7x10-14 F
=556 Wcm-3
Matter Polarization & Permittivity
C
C
Q
Qε
00r
aQp ˆ
The definition of electric dipole moment.Electric Dipole Moment,
The origin of electronic polarization.
(a ) A n e u tra l a to m in E = 0 . (b ) In d u c e d d ip o le m o m e n t in a fie ld
E le c tro n c lo u d
A to m icn u c le u s
p in d u c e d
E
C e n te r o f n e g a tiv ec h a rg e
xC O
αE pinduced
β x Fr Ze EFe
)Eβ
eZ((Ze)xpe
22
Induced Electric Dipole Moment,
Springaction
Matter Polarization & Permittivity
The origin of electronic polarization.
(a ) A n e u tra l a to m in E = 0 . (b ) In d u c e d d ip o le m o m e n t in a fie ld
E le c tro n c lo u d
A to m icn u c le u s
p in d u c e d
E
C e n te r o f n e g a tiv ec h a rg e
xC O
)EeZ
((Ze)xpe
22
Induced Electric Dipole Moment,
Removal of applied E field cause the vibration with a resonant frequency…
2
2
dt
xdZmx e t)(ωxx(t) o 0cos
2
0
eZm
βω
Electronic polarization resonance frequency
2
20
2
ee mω
Ze
Electronic Polarizability
F=ma
Permittivity: Electronic Polarization
A re a = A p to ta l
P- QP + Q P
(c )
- QP + Q P
B o u n d p o la riz a tio nc h arg e s o n th e su rfac e s
(b )
d
+ Q E - Q
V(a )
(a) When a dilectric is placed in an electric field, bound polarizationcharges appear on the opposite surfaces. (b) The origin of thesepolarization charges is the polarization of the molecules of themedium. (c) We can represent the whole dielectric in terms of itssurface polarization charges +QP and -QP.
Total Polarization
A
Q
Ad
dQ
volume
pP pptotal
EP oep
ENNpP einduced
eo
e N
1
Also, total polarization
Total Polarization =surface polarization charge density!
Permittivity: Electronic Polarization
A re a = A p to ta l
P- QP + Q P
(c )
- QP + Q P
B o u n d p o la riz a tio nc h arg e s o n th e su rfac e s
(b )
d
+ Q E - Q
V(a )
(a) When a dilectric is placed in an electric field, bound polarizationcharges appear on the opposite surfaces. (b) The origin of thesepolarization charges is the polarization of the molecules of themedium. (c) We can represent the whole dielectric in terms of itssurface polarization charges +QP and -QP.
before insertion of dielectric medium
00
oo
o
o
A
Q
dC
Q
d
VE
free surface charge densityA
Qoo
after insertion of dielectric medium
po QQQ
dividing by the area, A
pE 0
Permittivity: Electronic Polarization
A re a = A p to ta l
P- QP + Q P
(c )
- QP + Q P
B o u n d p o la riz a tio nc h arg e s o n th e su rfac e s
(b )
d
+ Q E - Q
V(a )
(a) When a dilectric is placed in an electric field, bound polarizationcharges appear on the opposite surfaces. (b) The origin of thesepolarization charges is the polarization of the molecules of themedium. (c) We can represent the whole dielectric in terms of itssurface polarization charges +QP and -QP.
Substitution for surface charge density
Relative permittivityoo
r Q
Q
Ee 0)1(
er 1
0
1 e
r
N Relation between polarization
mechanism to relative permittivity
Clausius-Mossotti Equation
The bulk electric field assumption is not valid in the atomic level
Pε
EElocal03
1
01
1
e
r
r N
Relation between polarization
mechanism to relative permittivityClausius-Mossotti EquationFor electronic polarization
E lo c
x
E le c tr ic f ie ld a ta to m ic sca le
E = V /dE
E lo c
The electric field inside a polarized dielectric at the atomic scale isnot uniform. The local field is the actual field that acts on amolecules. It can be calculated by removing that molecules andevaluating the field at that point from the charges on the plates andthe dipoles surrounding the point.
Lorentz Field
Total Polarization
031
1
ii
r
r N
Clausius-Mossotti Equation
For ionic polarization
locdlociloceav EEEp
iieer
r NN
03
1
1
1
Total Polarization
Relative permittivity due to ionic and electronic polarization
Total Polarization7.9 Electronic and ionic polarization in KCl KCl has the FCC crystal structure. Lattice parameter is 0.629 nm. The ionic polarizability per ion pair (per K+-Cl- ion) is 4.58 10-40 F m2. The electronic polarizability of K+ is 1.264 10-40 F m2 and Cl- is 3.408 10-40 F m2. Calculate the dielectric constant under dc operation and at optical frequencies. Experimental values are 4.84 and 2.19.
FCC… 4 KCl ion pairs per unit cell. The number of ion pairs, or individual ions, per unit volume (N) is:
