Locally Redistributing Charge.pdf
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31
Locally Redistributing Charge
CHAPTER PREVIEW
In this chapter we describe ceramic dielectrics. A dielectric is by definition an electrical insula-
tor (ρ is high and
E
g
is large). That means that dielectric behavior is a property associated with
certain ceramics and polymers but not a property associated with metals. We begin with a
background section. Some of this material may have been covered before but perhaps not spe-
cifically in terms of ceramics.
Dielectrics in the context of this chapter are more than just passive insulators. For example,
in BaTiO
3
and related perovskites structural changes create permanent electric dipoles that
cause the material to become polarized. Among other things, polarization allows the material
to store large amounts of charge: this is a prerequisite for a capacitor. Without dielectrics,
computers cannot function; some of today’s greatest challenges for the electronics industry
concern dielectrics more than semiconductors.
The following key topics are discussed in this chapter:
Dielectrics are polarizable: the separated charges cause an electric field that we characterize
by the dielectric constant.
Dielectrics can be self-polarizing: this is the ferroelectric effect. These ceramics are used
in capacitors because of their high dielectric constant.
The dimensions of a dielectric may change when it is polarized: this is the piezoelectric
effect and is used in microelectromechanical systems (MEMS), sonar, and medical ultra-
sound imaging.
The spontaneous polarization of a dielectric depends strongly on
T
; this is the pyroelectric
effect that we use for infrared (IR) detection (e.g., intruder alarms and thermal imaging).
31.1 BACKGROUND ON DIELECTRICS
Table 31.1 lists the important parameters discussed in
this chapter and their units.
All materials contain electrically charged particles. At a
minimum these are the electrons and protons that are part
of the constituent atoms. Many ceramics also contain ions,
which are charged. In a dielectric, charges have a limited
mobility and they will move only when they have enough
energy to overcome their inertia. When an insulator
receives a charge, it retains that charge, confining it within
the localized region in which it was introduced. However,
a conductor allows charge to flow freely and redistribute
itself within the material. The distinction between conduc-
tors and nonconductors (and it is not always a clear one)
arises from the relative mobility of charge within the
material.
The terms “dielectric,” “nonconductor,” and “insula-
tor” are often used interchangeably. However, we often
specify dielectrics as materials that are not only electri-
cally insulating but also have a high dielectric constant,
κ
Polarization Mechanisms
Even though no charge is transferred when a dielectric is
placed in a n elect r ic field t here is a red ist r ibution of cha rge,
which occurs by the formation and movement of electric
dipoles. There is an associated dipole moment,
μ
, having
both magnitude and direction
μ=
qd
(31.1)
where
d
is the separation of the positive and negative ends
of the dipole. The dipole direction is, by convention, taken
to point from the negative end to the positive end.
When a dielectric material is placed in an electric
field the induced dipoles, and any permanent dipoles,
become aligned. The material is now polarized and
.
556
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LOCALLY REDISTRIBUTING CHARGE
TABLE 31.1 Terms and Units Used to Describe Dielectric Behavior
Parameter
Defi nition
Units/value
Conversion factor
C
Capacitance
F, farads
1 F
=
1 C/V
=
1 A
2
s
4
kg
−
1
m
−
2
ε
0
Permittivity of a vacuum
8.85
×
10
−
12
F/m
ε
Permittivity
F/m
ε
r
Relative permittivity (
ε
/
ε
o
)
Dimensionless
κ
(same as
ε
r
)
Dielectric constant
Dimensionless
P
Polarization
C/m
2
Q
Charge
C, coulombs
1 C
=
1 A s
μ
Dipole moment
C · m
V
Voltage
V
q
or
e
Electron charge
0.16 aC
D
Dielectric displacement
C/m
2
D
=
Q/A
θ
c
Curie temperature
K
0 K
=−
273°C
T
cw
Curie–Weiss temperature
K
E
c
Coercive fi eld
V/m
χ
Dielectric susceptibility
Dimensionless
ξ
Electric fi eld strength
V/m
C
Curie constant
K
the polarization (or dipole
moment per unit volume)
is given by
and may also change the
overall dimensions of the
material. The dipole
moment is usually small
because, once again, the
displacements involved are
very small. Typically the
ion displacements are only
10 –100 am.
POLARIZATION MECHANISMS
A note
: In some texts you will find that the polarization
mechanism occurring in BaTiO
3
is described as dipolar
and in others as ionic. We prefer the former because
BaTiO
3
contains permanent dipoles (a condition of
dipolar polarization) that are being oriented in an elec-
tric fi eld. Although the permanent dipoles in BaTiO
3
are
the result of ion displacements, the term ionic polariza-
tion refers to the movement of any ions in an electric
fi eld (whether the material has a permanent dipole or
not).
P
=
Nqd
(31.2)
where
N
is the number of
dipoles.
There are four possible
polarization mechanisms
in a dielectric:
Dipolar
This mechan-
ism is generally uncom-
mon in ceramics because
most of the permanent dipoles cannot be reoriented without
Electronic
Ionic
Dipolar (also called molecular or orientation)
Interfacial (also called space charge)
These mechanisms are each illustrated in Figure 31.1.
Electronic
Electronic
When an electric field is applied to an
atom, there is a displacement of the electrons relative to
the nucleus. The electrons will concentrate on the side of
the nucleus near the positive end of the field. The atom acts
as a temporarily induced dipole. This effect occurs in all
materials (because all materials contain atoms), but the
magnitude is small because
d
is very small. Typical
displacements are
+
+
E
E=0
Ionic
+
+
E
E=0
10
−37
C·m.
Electronic polarization is the only possible mechanism in
pure materials that are covalently bonded and does not
contain permanent dipoles (e.g., diamond and silicon).
∼
1 am giving
μ∼
1.6
×
Dipolar
E
E=0
E
E=0
Interfacial
Ionic
This occurs when an ionically bonded material
is placed in an electr ic field; it is com mon in many ceramics
(e.g., MgO, Al
2
O
3
, NaCl). The bonds between the ions are
elastically deformed. Consequently the charge is minutely
redistributed. Depending on the direction of the field, the
cations and anions move either closer together or further
apart. These temporarily induced dipoles cause polarization
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
FIGURE 31.1
Illustration of the different polarization mechanisms
in a solid.
..............................................................................................................................
557
31.1 BACKGROUND ON DIELECTRICS
destroying their crystal structure. But there are some very
important exceptions and it is these materials that will
form a large part of this chapter. The prototypical example
is barium titanate. The structure is shown in Figure 7.2. At
room temperature the octahedrally coordinated Ti
4+
ion is
displaced slightly from its ideal symmetric position causing
the crystal structure to become tetragonal and permanently
polarized. When an alternating electric field is applied to
a crystal of barium titanate, the Ti
4+
ion moves back and
forth between its two allowable positions to ensure that the
polarization is aligned with the field.
TABLE 31.2 Dielectric Constants of Various Ceramics
k
at
k
at
Material
1 MHz
Material
1 MHz
Diamond
5.5–6.6
Al
2
O
3
8.8
SiO
2
3.7–3.8
MgO
9.6
NaCl
5.9
BaTiO
3
3000
Mica
5.4–8.7
Pyrex glass
4.0–6.0
Soda-lime glass
7.0–7.6
TiO
2
14 –110
Steatite
5.5–7.5
Forsterite
6.2
(SiO
2
+
MgO
+
Al
2
O
3
)
(2MgO · SiO
2
)
Cordierite
4.5–5.4
Mullite
6.6
(SiO
2
+
MgO
+
Al
2
O
3
)
High-lead glass
19
Vycor glass
3.9
Interfacial
A charge may develop at interfaces (such
as grain or phase boundaries and free surfaces) normally
as a result of the presence of impurities. The charge moves
on the surface when the material is placed in an electric
field. This type of polarization is not well understood,
although it has considerable practical interest because most
real materials and, in particular, many ceramics, are not
pure.
The total
P
for the material is then the sum of all the
individual contributions:
P
=
(
κ−
1)
ε
0
ξ=χε
0
ξ
(31.8)
where
is a measure of the ratio of the bound charge/free
charge (i.e.,
P
/
Q
). For dielectrics that polarize easily
χ
κ
will be large and, in turn, a large quantity of charge can
be stored.
Table 31.2 l ists
for a range of materials. Many ceram-
ics and glasses have
κ
P
=
P
electronic
+
P
ionic
+
P
dipolar
+
P
interfacial
(31.3)
in the range of 4–10. Polarization
is electronic only in covalent ceramics such as diamond
and is a combination of electronic and ionic in materials
such as MgO. Some ceramics, in particular BaTiO
3
and
other titanates and zirconates, have very large
κ
Relating
P
and
k
The dielectric constant is an important materials property
and is a measure of the ability of an insulating material
to store charge when subjected to an electric field; as you
might expect, it is directly related to
P
.
We can develop an equation relating
P
and
κ
due to
their permanent dipole moments.
Frequency Dependence of Polarization
When a dielectric is placed in an alternating electric field
the dipoles attempt to maintain alignment with the field.
This process requires a finite time that is different for each
polarization mechanism. At the relaxation frequency the
dipoles will only just be able to reorient themselves in time
with the applied field. At this frequency the dielectic is
“lossy” and energy is lost in the form of heat. The dielec-
tric loss is at a maximum when the frequency of the
external field coincides with the relaxation frequency of a
given polarization mechanism. This is the principle behind
the microwave oven. It operates at the relaxation frequency
of water molecules and the heat generated warms the
food.
At frequencies above the relaxation frequency the
dipoles will no longer be able to keep up with changes in
the applied field and the contributing polarization mecha-
nism becomes effectively “frozen” and no longer contrib-
utes. Figure 31.2 shows the variation of polarization with
frequency for a hypothetical material that exhibits all four
of the polarization mechanisms.
by
beginning with a simple parallel plate capacitor. From
electromagnetic theory we know that the total charge
per unit area of a capacitor plate,
D
0
, is proportional to
the applied electric field
κ
ξ
. The constant of proportionality
is
ε
0
:
D
0
=
Q
/
A
=ε
0
ξ
(31.4)
If we now place a dielectric between the parallel plates we
write
D
=εξ
(31.5)
D
is also known as the dielectric displacement and repre-
sents the extra charge that can be stored because of the
presence of the dielectric. So we can rewrite Eq. 31.5 as
D
=ε
0
ξ+
P
(31.6)
By substituting Eq. 31.5 into Eq. 31.6 we obtain
At optical frequencies only electronic polarization is
operative.
εξ = ε
0
ξ+
P
(31.7)
Dipolar and ionic contributions are small at high fre-
quencies because of the inertia of the molecules and
By simple rearrangement we can write
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LOCALLY REDISTRIBUTING CHARGE
TABLE 31.4 Dielectric Strengths for Various Ceramics
Material
Dielectric strength (MV/cm at 25°C)
Al
2
O
3
(99.5%)
0.18
P
Al
2
O
3
(94.0%)
0.26
α
interfacial charge
High-voltage porcelain
0.15
Steatite porcelain
0.10
IR
UV
Lead glass
0.25
Lime glass
2.5
α
dipolar
Borosilicate glass
5.8
Fused quartz
6.6
α
ionic
Quartz crystal
6.0
NaCl [100], [111], [110]
2.5, 2.2, 2.0
Muscovite mica
10.1
α
electronic
RF to
μ
wave
10
0
10
4
10
8
10
12
10
16
f (Hz)
tric strengths are important in applications in which the
thickness of the material is going to be small, e.g., in
capacitors. Values of
dielectric strength for
several ceramics are given
in Table 31.4. Note the very
high value of mica, which
is one of the reasons it was
used in early ceramic disk
capacitors.
FIGURE 31.2
Frequency dependence of polarization.
ions. The peaks occur-
ring at
REAL AND IMAGINARY COMPONENTS OF
e
The permittivity under an alternating field can be rep-
resented mathematically as the sum of real (
∼
10
13
and
∼
10
15
Hz are due to res-
onance effects where
the external field is
alternating at the
natural vibrational fre-
quency of the bound
ions
ε′
) and
imaginary (
ε′′
) parts:
ε=ε′−
j
ε′′
(Box 31.1)
In an alternating electric field the phase angle of the
electric fl ux density lags behind that of the electric field
due to the finite speed of polarization. The delay angle
δ
or
electrons,
respectively.
Nonlinear Dielectrics
Nonlinear dielectrics have
permanent dipoles that
interact to give a polariza-
tion in the absence of an
applied electric field. These
materials are the ferroelec-
trics. The topic shares
many similarities with fer-
romagnetism described in Chapter 33. For example, above
a critical temperature, the Curie temperature
is
Dielectric Strength
A dielectric will be able to
withstand a certain applied
electric field strength
before it breaks down and
current flows. High dielec-
tan
δ=ε′′
/
ε′
(Box 31.2)
The electric power loss per unit time (also called the
dielectric loss) is proportional to tan
δ
. Typical values
are given in Table 31.3.
θ
c
, the spon-
taneous polarization is destroyed by thermal disorder. A
plot of
P
versus
TABLE 31.3 Dielectric Loss for Some Ceramics and
Glasses at 25°C and 1 MHz
is shown in Figure 31.3 and demon-
strates hysteresis. This behavior is similar to that produced
by a ferromagnet when it is cycled through an alternating
magnetic field. The description is based on the domain
structure of ferroelectrics.
When the dipoles in a crystal are randomly oriented
there is no net
P
. When a field is applied, the dipoles begin
to line up with the electric field. The total dipole moment
changes either by the movement of the walls between
domains or by the nucleation of new domains. Eventually
the fi eld aligns all of the dipoles and
P
s
is obtained. When
all the dipoles are aligned in the same direction the mate-
rial is “poled.”
When the field is subsequently removed a remnant
polarization
P
r
exists due to the coupling between adja-
cent dipoles. The material is permanently polarized in the
ξ
Material
Tan
d
LiF
0.0002
MgO
0.0003
KBr
0.0002
NaCl
0.0002
TiO
2
(
c
)
0.0016
TiO
2
(
a
,
b
)
0.0002
Al
2
O
3
(
c
)
0.0010
Al
2
O
3
(
a
,
b
)
0.0010
BaO
0.0010
KCl
0.0001
Diamond
0.0002
Mg
2
SiO
4
(forsterite)
0.0003
Fused silica glass
0.0001
Vycor (96 SiO
2
–4B
2
O
3
) glass
0.0008
Soda-lime silica glass
0.0100
High-lead silica glass
0.0057
..............................................................................................................................
559
31.1 BACKGROUND ON DIELECTRICS
P
TABLE 31.5 Noncentrosymmetric Crystals
P
s
Noncentrosymmetric
P
r
Crystal system
point groups
Piezoelectric
Pyroelectric
Triclinic
1
Yes
Yes
Monoclinic
2
Yes
Yes
m
Yes
Yes
Orthorhombic
mm2
Yes
Yes
222
Yes
No
Tetragonal
4
Yes
Yes
4
Yes
No
422
Yes
No
4 mm
Yes
Yes
-E
c
E
c
42 m
Yes
No
0
E
Trigonal
3
Yes
Yes
32
Yes
No
3 m
Yes
Yes
Hexagonal
6
Yes
Yes
6
Yes
No
622
Yes
No
6 mm
Yes
Yes
6m2
Yes
No
Cubic
23
Yes
No
432
No
No
43 m
Yes
No
FIGURE 31.3
Hysteresis curve for a typical ferroelectric.
31.2 FERROELECTRICITY
absence of an electric field. This property is the key to
ferroelectricity.
When the direction of
Ferroelectrics exhibit an
electric dipole moment in
the absence of an external
electric field. The direction
of the dipole moment may
be switched by the applica-
tion of an alternating field. This property of polarization
reversal and remanence cannot be predicted by looking
only at the structure of a material; it must be determined
experimentally.
Ferroelectricity is a property that is associated not only
with ceramics. Certain polymers such as polyvinylidene
fl uoride (PVDF) and copolymers between PVDF and tri-
fluoroethylene are ferroelectric. PVDF is a semicrystalline
polymer. The crystalline conformation has an orthorhom-
bic unit cell (mm2).
A ferroelectric crystal consists of regions called
domains. Within each domain the polarization is in a
common direction, but in adjacent domains the polariza-
tion is in a different direction as illustrated in Figure 31.4.
The net polarization then depends on the difference in
volumes of the two domain orientations. If the volumes
are equal the material will not exhibit a net polarization.
By etching in a suitable chemical we can see the domain
structure. This is analogous to the process we described
in Section 12.3 to reveal dislocations.
Domain walls separate adjacent domains and are
transition regions in which the direction of polarization
FERROELECTRICS
Ferroelectrics do not contain iron. The term comes from
the analogy with ferromagnetism, which also does not
require iron.
is reversed the dipole
orientation switches to
become aligned with the
new field direction. As the
strength of the reverse field
is increased,
P
s
will eventually occur with the opposite
polarization. As the field alternates a hysteresis loop is
produced. The area contained within the loop is related to
the energy required to cause the polarization to switch
directions. Linear dielectrics (which is most of them) do
not show significant hysteresis in an alternating electric
field.
There is a structural requirement for ferroelectricity.
There are a total of 32 different symmetry point groups,
21 of which do not possess a center of symmetry. Ferro-
electrics are part of a small subgroup of noncentrosym-
metric crystals. Related properties are piezoelectricity and
pyroelectricity. Dielectrics belonging to all but one of the
groups of noncentrosymmetric crystals are piezoelectric.
Pyroelectric crystals form a further subgroup of 10 types
of crystal having especially low symmetry as shown in
Table 31.5.
ξ
All ferroelectrics are pyroelectric and piezoelectric.
All pyroelectrics are piezoelectric.
All piezoelectrics are not pyroelectric.
All pyroelectrics are not ferroelectrics.
560
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LOCALLY REDISTRIBUTING CHARGE
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