Physics of Materials 9 Dielectric Properties

H. S. Leipner: Physics of materials

1. Polarization mechanisms

2. Dielectric constant

3. Piezoelectricity

4. Ferroelectrics

5. Dielectric breakdown

6. High and low-k dielectrics

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

9. Dielectric properties

hsl 2006 – Physics of materials 9 – Dielectric properties 21:01 2

+++++++

−−−− −−−

Dielectric

Introduction

Electrical conductivity very small; insulators They are, however, affected by an electric field. Capacitance of a parallel plate capacitor

+++ +++

−−− −−−

d

Parallel plate capacitor with platesof area A and separation d. When adielectric material is placedbetween the plates, the dielectricbecomes polarized.

(vacuum/air) (with dielectric) Stored energy


Design of ceramic capacitors

Examples of ceramic capacitors. Single-layer ceramic capacitor (disk capacitors)and multilayer ceramic capacitor (stacked ceramic layers).

[Askeland 1994/Kasap 1997]


8.1 Polarization mechanisms

Effect of an electric field on an isolated atom

The electron orbits of an isolated atom in an electric field are distorted. Thecentroids of charge are separated by a distance d.

d

G+ G−

E


Dipole moment

Dipole moment of one atom induced by the external electric fieldPa = qd

Polarization of a dielectric crystal (dipole moment per unit volume)P = Nqd

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+ − + − + − + − + − + −

+

+

+

+

+

+

+

−

−

−

−

−

−

−

+−

Dielectric crystal as an ensembleof atomic dipoles. As a result ofthe application of the externalelectric field, a surface chargeappears.

E


Polarizability

Macroscopic polarization (surface charge density)P = ε0χE

(χ = εr – 1, electric susceptibility)

Similarly, the atomic dipole is proportional to the field,Pa = αpEloc

If the differences in the local field for surface atoms are not takeninto account, we get

P = NαpEloc

The polarizability αp can be expressed with the dielectric constant εr,taken into account the relation between the external and local electricfield

Local electric field


Other polarization mechanisms

In ionic crystals, the applied electric field pulls the kations andanions in different directions.

The result is an ionic contribution to the polarization, Pi.

Na+ Na+ Na+

Cl− Cl−

Cl− Cl−

Cl−

Cl−

Na+ Na+ Na+

+ −

di

The effect of applying an electricfield to the positions of the ionsin a NaCl crystal

Ptot = P + Pi

Pi = Nqidi

diE


Polar molecular crystals

Materials with molecules of a permanent dipole moment Dipole moment much larger than the induced dipole moment in

nonpolar molecules → much higher dielectric constant Normally in a solid the molecules are fixed and cannot follow the

electric field, but there are exceptions, e. g. hydrogen chloride.


Temperature variation of dielectric constant

Dielectric constant HCl

The HCl molecule and variation of the dielectric constant εr with temperature T for HCl.The abrupt change corresponds to the temperature at which the molecules are no longerable to align themselves with the external electric field.

[Turton 2000]

εr

e−

e−

e−e−

e−

e−e−

e−

Cl− H+


8.2 Dielectric constant

For a slowly varying field E, the polarization P is expected to vary at the samefrequency.

Polarization vs time

Time

P

E


High-frequency dielectric constant

High frequencies (optical range, ν ≈ 5·1014 Hz)

Dielectric constants from electromagnetic theory Speed of light

Thus, or εopt = n2 (n refractive index)

(vacuum) (inside the dielectric)

ˆ ˆ


Dielectric constant at optical frequencies

Material εs εopt

Diamond 5.68 5.66NaCl 5.90 2.34LiCl 11.95 2.78TiO2 94.00 6.80Quartz 3.85 2.13

Values of the dielectric constant in a static electric field,εr = εs, and at optical frequencies, εr = εopt

[Turton 2000]


Dielectric constant

Usually, the static dielectric constant is higher. Electrons can quickly follow the alternating electric field. However, the ionic contribution to the polarization becomes much

smaller. The ions cannot follow the quick change of the field. At high frequencies, only the induced polarization remains.


Survey of polarization mechanisms

Polarization mechanisms inmaterials: (a) electronic, (b)ionic, (c) high-frequencydipolar or orientation(present in ferroelectrics), (d)low-frequency dipolar(present in linear dielectricsand glasses), (e) interfacialspace charge at electrodes,and (f) interfacial spacecharge at heterogeneitiessuch as grain boundaries.E

E

E

E

E

E

d

(c) (f)

[Askeland 1994/Hench, West 1990]

+++++

–––––


Frequency dependence

Frequency dependence of polarization mechanisms

104 108 1012 1016 1020

Frequency (Hz)

Pol

ariz

atio

n Molecular

Ionic

Electronic


8.3 Piezoelectricity

In certain dielectric materials, the application of an external stressproduces electrical charges on the surface.

Δl

Converse piezoelectric effect: A piezoelectric material becomesstrained when placed in an electric field.

E

F F


Dielectric effects

When any material undergoes polarization, its ions and electronicclouds are displaced, causing the development of a mechanical strainin the material. This effect is seen in all materials subjected to anelectric field and is known as the electrostriction.

Piezoelectrics – materials that develop voltage upon the applicationof a stress and develop strain when an electric field is applied.

Pyroelectric – the ability of a material to spontaneously polarize andproduce a voltage due to changes in temperature.

Ferroelectric – a material that shows spontaneous and reversibledielectric polarization.


Piezoelectric crystals

Inversion center

3+

−

− −

3+

−

− −

When a stress is applied to a crystal structure with three-fold symmetry and zeropolarization, the symmetry is altered and the material acquires a non-zeropolarization even without an electric field.


Piezoelectric coefficients

Material d (pC/N) g (mV m/N)

Quartz 2.3 50BaTiO3 190 12PZT 268 to 480 12 to 35PbNb2O6 80PbTiO3 47LiNbO3 6LiTaO3 5.7

[Askeland 1996]

~ ~


Design of a spark igniter

Example: A PZT spark igniter is made using a disk that has a 5 mm diameter and 20 mm height. Calculate the voltage generated if the g coefficient for PZT used is 35 mV m/N. Assume thata compressive force of 10 kN is applied on the circular face.

Solution:

Definition of the g coefficientStress σ = F/A = 5.09 MN/m2

Therefore, the electric field is E = gσ =1.78·105 V/mU = E d = 3565 V.

~

~

~


8.4 Ferroelectrics

Material with a finite polarization even in the absence of anyexternal electric field or applied stress

Name misleading: nothing to do with iron; properties resemblesthose of ferromagnetic solids

Hysteresis of polarization


Hysteresis

E

PPs

Pr

Ec

Hysteresis curve as the polarization versus the electric field. Characteristic values arethe saturation polarization Ps, the remanent polarization Pr, and the coercive field Ec.


Microscopic properties of ferroelectrics

Neighboring dipoles are aligned in mutual interaction. Arranged dipoles form domains

(compare the magnetic domains in ferromagnets). Characteristic transition temperature, where the alignment of the

dipoles is destroyed (ferroelectric Curie temperature) Typically, material with complicated crystal structure of low

symmetry Finite polarization even without external electric field All ferroelectrics are also piezoelectric materials Ferroelectric materials have a very high dielectric constant

(as large as several thousand). Permanent polarization can be used to store non-volatile digital

information.


Ferroelectric domains

Ferroelectric domains in BaTiO3 shown by imaging with crossed polarizers


Alignment of dipoles

The ferroelectric hysteresis loopand the alignment of the dipoles

[Askeland 1996]


Hysteresis loop of barium titanate

Ferroelectric hysteresis loop of single-crystalline BaTiO3 and a BTO polycrystal (right)[Askeland 1996]

P

P

E E


Dielectric constant vs. temperature

The effect of temperature and grain size on thedielectric constant in barium titanate. Abovethe Curie temperature, the spontaneouspolarization is lost due to a change in crystalstructure and barium titanate is in theparaelectric state. The grain size dependenceshows that the dielectric constant is amicrostructure sensitive property.

[Askeland 1996/Moulson, Herbert 1990]


Barium titanate structure

Ti4+

O2−

Ba2+

Compared to the ideal cubicarrangement the positive andnegative ions are displacedby a distance d ≈ 0.01 nm.

d


Ferroelectric perovskite phase

Tetragonal perovskite phase:each titanium atom is bonded tosix nearest-neighbor oxygenatoms, but is not exactly in thecentre of the octahedron. Thisoffset means that tetragonalBaTiO3 is ferroelectric. Theelectrical polarization may bereversed by applying an externalelectric field.

Single crystals of BaTiO3 generallycontain many domains,corresponding to differentdirections of Ti off-centering. Thenet effect of the different domainorientations is to cancel out anymacroscopic polarisation.


Polarization

Example: Determine the magnitude of the remanent polarization in BaTiO3. The molar volume is 3.8·10−5 m3.

Solution:

1 mol of BaTiO3 consists ofNA ions Ba2+, NA ions Ti4+, 3NA ions O2−

The total charge associated with 1 mol is ±6NAe

The dipole moment for 1 mol is:

Pm = 6NAed

Polarization is dipole moment per unit volume


BaTiO3 phase transitions

Different phases of barium titanate as a function of the temperature

[Moulson, Herbert 1990]


BTO polymorphs

The polymorphs of barium titanate at different temperatures

Cubic Tetragonal Orthorhombic Rhombohedral130 °C 0 °C –90 °C

Ps Ps Ps


Magnitude of polarization

Magnitude of the polarization of the different polymorphs of bariumtitanate at different temperatures. Tc is the Curie temperature.

[Moulson, Herbert 1990]

Tc

Ps


Dielectric constant

Dielectric constant of barium titanate as a function of the temperature[Moulson, Herbert 1990]


Design of a multilayer capacitor

Example: A multi-layer capacitor is to be designed using a BaTiO3-based formulation containing SrTiO3. The dielectric constant of the material is 3000. Calculate the capacitance of a multi-layer capacitor consisting of 100 layers connected in parallel using Ni electrodes. The sides of the capacitor are 10 mm × 5 mm and the thickness of each layer is 10 µm. What is the role of SrTiO3? What processing technique will be used to make these?

Solution:

Capacitance

Capacitance of one layer: 13.27 µFTotal capacitance of 100 layers connected in parallel:

1327 µF


8.5 Dielectric breakdown

At a critical (high) field, the dielectric material becomes a conductor. Increasing number of electrons by an avalanche process

Con

duct

ivity

Electrical fieldEb

Conductivity against the applied electricfield. If the field is below the breakdownfield then no current flows.

e−

e− e−

e− e− e− e−

e− e− e− e− e− e− e− e−

Scheme of the avalanche process. Theaccelerated electrons are able to exciteother electrons into the conductionband. Each of these electrons thenexcites another electron, and so on.


8.6 High and low-k dielectrics

Materials with certain dielectrical properties are important inmicroelectronics

Gate dielectrics with shrinking dimensions require high-k materials(i. e. high dielectric constant εr)

Static power consumption of a transistor depends on the leakagecurrent

International Technology Roadmap for Semiconductors:effective oxide thickness < 1 nm, gate leakage current < 103 A/cm2

at 100 °C (2007) in MPUs Standard SiO2 cannot support these requirements (high tunneling

probability through thin layer)


Variation of the DK with the band gap

Variation of the dielectric constant with the band gap in binary oxides[Bersuker:2004]

Dielectric constant

Band

gap

(eV)


Low-k dielectric materials

Increase in the speed of integrated circuits by reducing the size(feature size 1 µm to 90 nm in the last decades)

Not all IC components work faster when decreased in size:Interconnections work slower

Figure of merit is the product(resistance × capacitance, in units of time)

Shrinking cross section of wire → increase in the resistance,reduction in wire distance → increase in the capacitance

As a result delay increases


Interconnections

Embedded Cu lines with interline capacitance[Shamiryan:2004]

Cross section ofinterconnections indamascence technology

[www.tecchannel.de]


Dual damascence technology

1. Etch of trenches for contacts and interconnections

2. Deposition of insulating film (barrier layer) to prevent the in-diffusion

of copper in the semiconductor

3. Evaporation of a thin metal layer where the contacts and the

interconnects grow galvanically (dual damascence process)

4. When the trenches are filled the excess copper has to be removed

by chemo–mechanical polishing (similar to the intarsia in damascence

swords).

5. Deposition of a insulating oxide layer

6. Deposition of up to six additional interconnect layers


Reduction of the dielectric constant

Decreased polarizability with less polar bonds Decreased density by using of porous material

Low-k materials

Non-Si Si-based

PolymersAmorphous carbon

Silica-based SSQ-based

HSSQMSSQ

SiOFSiOCH


Summary

A dielectric placed in a electric field becomes polarized. 3 factors which can contribute to polarization: induced

polarization, ionic, and permanent dipole contributions Specific frequency dependence of these contributions Requirement to control the dielectric constant in modern devices

Read about this lecture:R. Turton: The physics of solids. Oxford University Press 2000,chapter 10.

References to pictures and data used are given in the fileReferences.pdf

Physics of Materials 9 Dielectric Properties

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