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 – “Non centrosymmetric Oxides”, Chem. Mater. 10, 2753 2769 (1998).

M. Kunz & I.D. Brown – “Out-of-center Distortions around Octahedrally Coordinated d 0 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 =

e

E =

e 0

E + P =

e 0

E +

e 0 c e

E =

e 0

(1+

c e

)E

where e

0

is the dielctric permitivity of free space (8.85 x 10 the polarization of the material, and c

e

12 C 2 /N-m 2 ), P is is the electric susceptibility. The relative permitivity or dielectric constant of a material is defined as: e r

=

e/e 0

= 1+

c e 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 +

a

i +

a

d +

a

s

1. Electronic Polarizability ( a e ) Polarization of localized electrons Polarizability ( a ) increases 2. Ionic Polarizability ( a i ) Displacement of ions Response Time Increases (slower response) 3. Dipolar Polarizability ( a d ) Reorientation of polar molecules 4. Space Charge Polarizability ( a s ) Long range charge migration

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 (

a

e )

Response is fast, t is small

2. Ionic Polarizability (

a

i )

Response is slower

3. Dipolar Polarizability (

a

d )

Response is still slower

4. Space Charge Polarizability (

a

s )

Response is quite slow, t is large Audiofrequencies (~ 10 3 Radiofrequencies (~ 10 6 Hz) Hz) Microwave frequencies (~ 10 9 Visible/UV frequencies (~ 10 12 Hz) Hz) ( ( a

s

a

s



,

a 0)

d

 ( a

s ,

a

d ,

a

i

0)  0) a a a a

=

a

e +

a

i +

a

d +

a

s =

a

e +

a

i +

a

d =

a

e +

a

i =

a

e

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

=

e

r ”/

e

r ’

where e r ’ is the real part of the dielectric constant and imaginary part of the dielectric constant.

e r ” is the

Frequency Dependence

e

(

w

)

e

0

a

d

+a

i

+a

e

a

i

+a

e

e

r (Dielectric Const.)

a

e only

e  tan d (Loss)

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 d 0 or s 2 p 0 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)

2

nd

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 d 0 W 6+ , Mo 6+ ). cations (i.e. Ti 4+ , Nb 5+ – – BaTiO 3 , KNbO 3 , WO 3 Increasingly favored as the HOMO-LUMO splitting decreases (covalency of the M-O bonds increases) 2. Cations containing filled valence s shells (Sn 2+ , Sb 3+ , Pb 2+ , Bi 3+ ) – – Red PbO, TlI, SnO, Bi 4 Ti 3 O 12 , Ba 3 Bi 2 TeO 9 SOJT Distortion leads to development of a stereoactive electron-lone pair.

,

Octahedral d

0

Cation

G

point (k x =k y =0) non-bonding

In the cubic perovskite structure the bottom of the conduction band is non bonding M t 2g , 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 2 nd order JT dist.

2

nd

Order JT Distortion Band Picture

Overlap at G symmetry is non-bonding by

E M t F 2g (

p

*) E M t 2g (

p

*) F O 2p

Overlap at G is slightly antibonding in the CB & slightly bonding in the VB.

DOS DOS

The 2 nd 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 d 1 TM ion. Hence, ReO 3 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

d=1.67Å s = 1.90

d=1.95Å s = 0.90

d=2.33Å s = 0.32

Tetragonal BaTiO 3

The 2 nd 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

oxidation state cations prefer to move away from each other) (high

MoO 3

See Kunz & Brown J. Solid State Chem. 115, 395-406 (1995).

Why is BaTiO

3

Ferroelectric

•Ba 2+ is larger than the vacancy in the octahedral network tolerance factor > 1.

•This expands the octahedron, which leads to a shift of Ti 4+ toward one of the corners of the octahedron.

•The direction of the shift can be altered through application of an electric field.

Cubic (Pm3m) T > 393 K Ti-O Distances (Å) 6



2.00

Tetragonal (P4mm) 273 K < T < 393 K Ti-O Distances (Å) 1.83, 4



2.00, 2.21

Toward a corner

Orthorhombic (Amm2) 183 K < T < 273 K Ti-O Distances (Å) 2



1.87, 2



2.00, 2



2.17

Toward an edge

BaTiO

3

Phase Transitions

In the cubic structure BaTiO paraelectric. That is to say that the orientations of the ionic dynamic.

3 is displacements are not ordered and Below 393 K BaTiO ferroelectric and the displacement of the Ti 4+ 3 becomes ions progressively displace upon cooling.

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 2 nd order JT distortions and ferroelectric distortions can you explain the following physical properties.

BaTiO 3

–

SrTiO

–

PbTiO 3 3 :

–

BaSnO 3 : Ferroelectric (T : Ferroelectric (T C C ~ 130°C, ~ 490°C)

e

Insulator, Normal dielectric ( r : Insulator, Normal dielectric (

e

r

e

r

–

KNbO 3

–

KTaO 3

–

: : Ferroelectric (T C ~ x) Insulator, Normal dielectric (

e

r > 1000) ~ x) ~ x) ~ x)

Structure, Bonding and Properties

BaTiO

–

SrTiO 3

–

PbTiO 3

– –

3 : Ferroelectric (T C : : BaSnO 3 : ~ 130°C,

e

r Insulator, Normal dielectric ( Ferroelectric (T

Displacements of both Ti 4+ ferroelectricity

Insulator, Normal dielectric (

Main group Sn 4+ dist.

C ~ 490°C)

and Pb 2+ has no low lying t 2g e e

r r > 1000)

Ba 2+ ion stretches the octahedra (Ti-O dist. ~ 2.00Å), this lowers energy of CB (LUMO) and stabilizes SOJT dist.

(6s 2 6p 0

~ x)

Sr 2+ ion is a good fit (Ti-O dist. ~ 1.95Å), this compound is close to a ferroelectric instability and is called a quantum paraelectric.

cation) stabilize

~ x)

orbitals and no tendency toward SOJT

KNbO 3

–

KTaO 3

–

: Ferroelectric (T C

Behavior is very similar to BaTiO 3

: ~ x) Insulator, Normal dielectric (

e

r ~ 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 t 2g LUMO, diminishing the driving force for a SOJT dist.

2

nd

Order Jahn-Teller Distortions with s

2

p

0

Main Group Cations

Fact: Main group cations that retain 2 valence electrons (i.e. Tl + , Pb 2+ , Bi 3+ , Sn 2+ , Sb 3+ , Te 4+ , Ge 2+ , As 3+ , Se 4+ , ect.) tend to prefer distorted environments.

M-X Bonding:

The occupied cation s orbitals have an antibonding interaction with the surrounding ligands.

Symmetric Coordination

Distorted Coordination: : The occupied M s and empty M p orbitals are not allowed by symmetry to mix.

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 NH 3 ).

Tetrahedral Coordination (T d ): s-orbital = Trigonal Pyramidal Coord. (C 3v ): s-orbital = Octahedral Coordination (O h ): s-orbital = Square Pyramidal Coord. (C 4v ): s-orbital =

a a a 1

, p-orbitals =

t 2 a 1

, p-orbitals =

e, a 1 1g 1

, p-orbitals = , p-orbitals =

t e, 1u a 1

Some Examples s

2

p

0

SOJT

Red PbO Distorted CsCl CsGeBr 3 Distorted Perovskite SbCl Sb 3+ 3 Trig. Pyramidal

Cooperative SOJT Distortions

Tetragonal BaTiO 3 T C = 120°C Ti displacement = 0.125 Å Ti-O short = 1.83 Å Ti-O long = 2.21 Å Ba 2+ displacement = 0.067 Å Tetragonal PbTiO 3 T C = 490°C Ti displacement = 0.323 Å Ti-O short = 1.77 Å Ti-O long = 2.39 Å Pb 2+ 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. PbZrO 3 )

Piezoelectricity

– A spontaneous polarization develops under the application of a mechanical stress, and vice versa (i.e. quartz)

PZT Phase Diagram

Pb(Zr 1-x Ti x )O 3 (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 PbZr

1-x

Ti

x

O

3 PbTiO 3 Ferroelectric Tetragonal PbZr 1-x Ti x O 3 x ~ 0.3

Ferroelectric Rhombohedral PbZr 1-x Ti Cubic x O Paraelectric 3 PbZrO 3 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 Out i.e. Speaker Mechanical Signal In Voltage Out 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.

without field

E

-

q

with field +

q

The presence of an electric field polarizes the electron distribution about an atom creating a dipole moment,

x

m

=qx

The dipole moment per unit volume, P, is then given by

P = n m

m where n m 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. Ta 5+ ).

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