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Dielectric Materials
Chemistry 754
Solid State Chemistry
Lecture #27
June 4, 2002
References
A.R. West – “Solid State Chemistry and it’s
Applications”, Wiley (1984)
R.H. Mitchell – “Perovskites: Modern & Ancient ”,
Almaz Press, (www.almazpress.com) (2002)
P. Shiv Halasyamani & K.R. Poeppelmeier – “Noncentrosymmetric Oxides”, Chem. Mater. 10, 27532769 (1998).
M. Kunz & I.D. Brown – “Out-of-center Distortions
around Octahedrally Coordinated d0 Transition
Metals”, J. Solid State Chem. 115, 395-406 (1995).
A. Safari, R.K. Panda, V.F. Janas (Dept. of
Ceramics, Rutgers University)
http://www.rci.rutgers.edu/~ecerg/projects/ferroelectric.html
Dielectric Constant
If you apply an electric field, E, across a material the charges in the material
will respond in such a way as to reduce (shield) the field experienced within
the material, D (electric displacement)
D = eE = e0E + P = e0E + e0ceE = e0(1+ce)E
where e0 is the dielctric permitivity of free space (8.85 x 1012 C2/N-m2), P is
the polarization of the material, and ce is the electric susceptibility. The
relative permitivity or dielectric constant of a material is defined as:
er = e/e0 = 1+ce
When evaluating the dielectric properties of materials it is this quantity we
will use to quantify the response of a material to an applied electric field.
Contributions to Polarizability
a = a e + ai + ad + as
1. Electronic Polarizability (ae)
Polarization of localized electrons
2. Ionic Polarizability (ai)
Displacement of ions
3. Dipolar Polarizability (ad)
Reorientation of polar molecules
4. Space Charge Polarizability (as)
Long range charge migration
Polarizability
(a) increases
Response Time
Increases
(slower
response)
Frequency Dependence
Reorientation of the dipoles in response to an electric field is characterized
by a relaxation time, t. The relaxation time varies for each of the various
contributions to the polarizability:
1. Electronic Polarizability (ae)
Response is fast, t is small
2. Ionic Polarizability (ai)
Response is slower
3. Dipolar Polarizability (ad)
Response is still slower
4. Space Charge Polarizability (as)
Response is quite slow, t is large
Audiofrequencies (~ 103 Hz)
a = ae+ai+ad+as
Radiofrequencies (~ 106 Hz)
(as  0)
a = ae+ai+ad
Microwave frequencies (~ 109 Hz) (as, ad  0)
a = ae+ai
Visible/UV frequencies (~ 1012 Hz) (as, ad, ai  0) a = ae
Dielectric Loss
When the relaxation time is much faster than the
frequency of the applied electric field, polarization
occurs instantaneously.
When the relaxation time is much slower than the
frequency of the applied electric field, no polarization
(of that type) occurs.
When the relaxation time and the frequency of the
applied field are similar, a phase lag occurs and energy
is absorbed. This is called dielectric loss, it is normally
quantified by the relationship
tan d = er”/er’
where er’ is the real part of the dielectric constant and er” is the
imaginary part of the dielectric constant.
Frequency Dependence
e(w)
ad+ai+ae
er (Dielectric
Const.)
ae only
IR
Microwaves
ai+ae
tan d
(Loss)
UV
e0
e
log(w)
Ionic Polarization and Ferroelectricity
Most dielectric materials are insulating (no conductivity of either
electrons or ions) dense solids (no molecules that can reorient).
Therefore, the polarizability must come from either ionic and
electronic polarizability. Of these two ionic polarizability can make
the largest contribution, particularly in a class of solids called
ferroelectrics. The ionic polarizability will be large, and a
ferroelectric material will result, when the following two conditions
are met:
1.
Certain ions in the structure displace in response to the
application of an external electric field. Typically this
requires the presence of certain types of ions such as d0
or s2p0 cations.
2. The displacements line up in the same direction (or at
least they do not cancel out). This cannot happen if the
crystal structure has an inversion center.
3. The displacements do not disappear when the electric
field is removed.
What is a Ferroelectric
A ferroelectric material develops a spontaneous polarization
(builds up a charge) in response to an external electric field.
•The polarization does not
go away when the
external field is removed.
•The direction of the
polarization is reversible.
Applications of Ferroelectric Materials
•Multilayer capacitors
•Non-volatile FRAM (Ferroelectric Random Access Memory)
2nd Order Jahn-Teller Distortions
Occurs when the HOMO-LUMO gap is small and there is a symmetry allowed
distortion which gives rise to mixing between the two. This distortion is
favored because it stabilizes the HOMO, while destabilizing the LUMO.
Second order Jahn-Teller Distortions are typically observed for two classes
of cations.
1. Octahedrally coordinated high valent d0 cations (i.e. Ti4+, Nb5+,
W6+, Mo6+).
– BaTiO3, KNbO3, WO3
– Increasingly favored as the HOMO-LUMO splitting
decreases (covalency of the M-O bonds increases)
2. Cations containing filled valence s shells (Sn2+, Sb3+, Pb2+, Bi3+)
– Red PbO, TlI, SnO, Bi4Ti3O12, Ba3Bi2TeO9
– SOJT Distortion leads to development of a stereoactive
electron-lone pair.
Octahedral d0 Cation
G point
(kx=ky=0)
non-bonding
In the cubic perovskite structure the
bottom of the conduction band is nonbonding M t2g, and the top of the
valence band is non-bonding O 2p. If
the symmetry is lowered the two
states can mix, lowering the energy of
the occupied VB states and raising
the energy of the empty CB states.
This is a 2nd order JT dist.
2nd Order JT Distortion
Band Picture
M t2g(p*)
Overlap at G is
non-bonding by
symmetry
EF
DOS
M t2g(p*) Overlap at G is
slightly
EF
antibonding in the
CB & slightly
bonding in the VB.
O 2p
DOS
The 2nd order JT distortion reduces the
symmetry and widens the band gap. It is the
driving force for stabilizing ionic shifts. The
stabilization disappears by the time you get
to a d1 TM ion. Hence, ReO3 is cubic.
See Wheeler et al. J. Amer. Chem. Soc. 108, 2222 (1986), and/or
T. Hughbanks, J. Am. Chem. Soc. 107, 6851-6859 (1985).
What Determines the Orientation
of the Cation Displacements?
d=1.83Å
s = 0.96
d=2.21Å
s = 0.34
Tetragonal
BaTiO3
d=1.67Å
s = 1.90
d=2.33Å
s = 0.32
d=1.95Å
s = 0.90
The 2nd Order JT effect at the
metal only dictates that a distortion
should occur. It doesn’t tell how the
displacements will order. That
depends upon:
(i) the valence requirements at
the anion (i.e. 2 short or 2 long
bonds to same anion is unfavorable),
(ii) cation-cation repulsions (high
oxidation state cations prefer to
move away from each other)
MoO3
See Kunz & Brown J. Solid State Chem. 115, 395-406 (1995).
Why is BaTiO3 Ferroelectric
•Ba2+ is larger than the vacancy in the
octahedral network tolerance factor > 1.
•This expands the octahedron, which leads to
a shift of Ti4+ toward one of the corners of
the octahedron.
•The direction of the shift can be altered
through application of an electric field.
BaTiO3 Phase
Transitions
Cubic (Pm3m)
T > 393 K
Ti-O Distances (Å)
In the cubic structure BaTiO3 is
paraelectric. That is to say that the
orientations of the ionic
displacements are not ordered and
dynamic.
62.00
Tetragonal (P4mm)
273 K < T < 393 K
Ti-O Distances (Å)
1.83, 42.00, 2.21
Toward a corner
Below 393 K BaTiO3 becomes
ferroelectric and the displacement
4+ ions progressively
of
the
Ti
Orthorhombic (Amm2)
displace upon cooling.
183 K < T < 273 K
Ti-O Distances (Å)
21.87, 22.00, 22.17
Toward an edge
Rhombohedral (R3m)
183 K < T < 273 K
Ti-O Distances (Å)
31.88, 32.13
Toward a face
See Kwei et al. J. Phys. Chem. 97, 2368 (1993),
Structure, Bonding and Properties
Given what you know about 2nd order JT distortions and ferroelectric
distortions can you explain the following physical properties.
BaTiO3 : Ferroelectric (TC ~ 130°C, er > 1000)
–
SrTiO3 : Insulator, Normal dielectric (er ~ x)
–
PbTiO3 : Ferroelectric (TC ~ 490°C)
–
BaSnO3 : Insulator, Normal dielectric (er ~ x)
–
KNbO3 : Ferroelectric (TC ~ x)
–
KTaO3 : Insulator, Normal dielectric (er ~ x)
–
Structure, Bonding and Properties
BaTiO3 : Ferroelectric (TC ~ 130°C, er > 1000)
– Ba2+ ion stretches the octahedra (Ti-O dist. ~ 2.00Å), this lowers energy
of CB (LUMO) and stabilizes SOJT dist.
SrTiO3 : Insulator, Normal dielectric (er ~ x)
– Sr2+ ion is a good fit (Ti-O dist. ~ 1.95Å), this compound is close to a
ferroelectric instability and is called a quantum paraelectric.
PbTiO3 : Ferroelectric (TC ~ 490°C)
– Displacements of both Ti4+ and Pb2+ (6s26p0 cation) stabilize
ferroelectricity
BaSnO3 : Insulator, Normal dielectric (er ~ x)
– Main group Sn4+ has no low lying t2g orbitals and no tendency toward SOJT
dist.
KNbO3 : Ferroelectric (TC ~ x)
– Behavior is very similar to BaTiO3
KTaO3 : Insulator, Normal dielectric (er ~ x)
– Ta 5d orbitals are more electropositive and have a larger spatial extent
than Nb 4d orbitals (greater spatial overlap with O 2p), both effects raise
the energy of the t2g LUMO, diminishing the driving force for a SOJT dist.
2nd Order Jahn-Teller Distortions
with s2p0 Main Group Cations
Fact: Main group cations that retain 2 valence electrons (i.e. Tl+, Pb2+, Bi3+,
Sn2+, Sb3+, Te4+, Ge2+, As3+, Se4+, ect.) tend to prefer distorted environments.
M-X Bonding: The occupied cation s orbitals have an antibonding interaction
with the surrounding ligands.
Symmetric Coordination: The occupied M s and empty M p orbitals are not
allowed by symmetry to mix.
Distorted Coordination: The lower symmetry allows mixing of s and at least
one p orbital on the metal. This lowers the energy of the occupied orbital,
which now forms an orbital which is largely non-bonding and has strong mixed
sp character. It is generally referred to as a stereoactive electron lone pair
(for example as seen in NH3).
Tetrahedral Coordination (Td): s-orbital = a1, p-orbitals = t2
Trigonal Pyramidal Coord. (C3v): s-orbital = a1, p-orbitals = e,a1
Octahedral Coordination (Oh): s-orbital = a1g, p-orbitals = t1u
Square Pyramidal Coord. (C4v): s-orbital = a1, p-orbitals = e,a1
Some Examples s2p0 SOJT
Red PbO
Distorted CsCl
CsGeBr3
Distorted
Perovskite
SbCl3
Trig. Pyramidal
Sb3+
Cooperative SOJT Distortions
Tetragonal BaTiO3
TC = 120°C
Ti displacement = 0.125 Å
Ti-O short = 1.83 Å
Ti-O long = 2.21 Å
Ba2+ displacement = 0.067 Å
Tetragonal PbTiO3
TC = 490°C
Ti displacement = 0.323 Å
Ti-O short = 1.77 Å
Ti-O long = 2.39 Å
Pb2+ displacement = 0.48 Å
Related Dielectric Phenomena
Pyroelectricity – Similar to ferroelectricity, but the
ionic shifts which give rise to spontaneous polarization
cannot be reversed by an external field (i.e. ZnO).
Called a pyroelectric because the polarization changes
gradually as you increase the temperature.
Antiferroelectricty – Each ion which shifts in a given
direction is accompanied by a shift of an ion of the
same type in the opposite direction (i.e. PbZrO3)
Piezoelectricity – A spontaneous polarization develops
under the application of a mechanical stress, and viceversa (i.e. quartz)
PZT Phase Diagram
Pb(Zr1-xTix)O3 (PZT) is probably the most important piezoelectric
material. The piezoelectric properties are optimal near x = 0.5, This
composition is near the morphotropic phase boundary, which separates
the tetragonal and rhombohedral phases.
Hysteresis Loops in PbZr1-xTixO3
PbTiO3
Ferroelectric
Tetragonal
PbZr1-xTixO3
x ~ 0.3
Ferroelectric
Rhombohedral
PbZr1-xTixO3
Paraelectric
Cubic
PbZrO3
Antiferroelectric
Monoclinic
An antiferroelectic material does not polarize much for low applied fields, but
higher applied fields can lead to a polarization loop reminiscent of a
ferroelectric. The combination gives split hysteresis loops as shown above.
What is Piezoelectricity
A piezoelectric material converts mechanical (strain)
energy to electrical energy and vice-versa.
Voltage In
Mechanical Signal In
Mechanical Signal Out
Voltage Out
i.e. Speaker
i.e. Microphone
Applications of Piezoelectrics
 Piezo-ignition systems
 Pressure gauges and transducers
 Ultrasonic imaging Ceramic phonographic
cartridge
 Small, sensitive microphones
 Piezoelectric actuators for precisely
controlling movements (as in an AFM)
 Powerful sonar
Symmetry Constraints and
Dielectric Properties
Dielectric properties can only be found with certain crystal symmetries
Piezoelectric
Do not posses an inversion center (noncentrosymmetric)
Ferroelectric/Pyroelectric
Do not posses an inversion center (noncentrosymmetric)
Posses a Unique Polar Axis
The 32 point groups can be divided up in the following manner (color
coded according to crystal system: triclinic, monoclinic, etc.).
Piezoelectric
1, 2, m, 222, mm2, 4, -4, 422, 4mm, 42m, 3, 32, 3m,
6, -6, 622, 6mm, 6m2, 23, 43m
Ferroelectric/Pyroelectric
1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm
Centrosymmetric (Neither)
-1, 2/m, mmm, 4/m, 4/mmm, -3, 3/m, 6/m, 6/mmm, m3, m3m
Electronic Polarizability
Let’s limit our discussion to insulating extended solids. In the
absence of charge carriers (ions or electrons) or molecules, we
only need to consider the electronic and ionic polarizabilities.
E
without
field
with
field
-q
x
+q
The presence of an electric field polarizes the electron
distribution about an atom creating a dipole moment,
m=qx
The dipole moment per unit volume, P, is then given by
P = nmm
where nm is the number of atoms per unit volume.
Microwave Dielectrics
Were not talking microwave ovens here, rather
communication systems which operate in the microwave
region:
– Ultra high frequency TV (470-870 MHz)
– Satellite TV (4 GHz)
– Mobile (Cellular) Phones (900-1800 MHz)
All such systems depend upon a bandpass filter that
selects a narrow frequency range and blocks all others.
These filters are constructed from ceramics with
desirable dielectric properties.
Microwave Dielectrics-Properties
The following dielectric properties are intimately related to it’s
performance
Dielectric Constant (Permitivity)
– A high dielectric constant allows components to be
miniaturized
Dielectric Loss
– A low dielectric loss is needed to prevent energy
dissipation and minimize the bandpass of the filter
Temperature Coefficient
– For device stability the dielectric properties should be
relatively insensitive to temperature
Microwave Dielectrics
Materials by Design
The the required properties it is possible to apply some concepts of
rational design to the search for materials.
High Dielectric Constant
– High electron density (dense structure type, polarizable cations, i.e.
Ta5+).
Low Dielectric Loss
– Ionic polarizability comes with large losses in the microwave region.
Therefore, one needs to avoid ferroelectrics, disorder and
impurities. Ions should not be able to rattle around too much.
Temperature Coefficient
– Very sensitive to rotations of polyhedra, vibrations of atoms, as well
as thermal expansion. In perovskites the temperature coefficient is
linked to octahedral tilting distortions. Tolerance factors just below
1 tend to have very low temperature coefficients.
Commercial Microwave Dielectrics
See Dr. Rick Ubic’s (University of Sheffield) site for a more detailed
treatment of microwave dielectrics.
http://www.qmul.ac.uk/~ugez644/index.html#microwave