Electronic and Optical Properties of Materials

Brendon LovettTrinity Term 2008

Thursday, Friday 9am Weeks 2 to 5

Part IIIPiezo-, Pyro- and Ferro-electricity

•Piezo-electricity• applied stress and polarization.

•Pyro-electricity• temperature dependent polarization

•Ferro-electricity• pyro-electrics where direction of polarization can be changed by applying a

field.• Domains and hysteresis

• Applications

Books

• Bleaney and Bleaney, Electricity and Magnetism, OUP, 21 BLE

• Kittel, Introduction to Solid State Physics, Wiley, 22 KIT/1

• Rose, et al., Structure and Properties of Materials, Wiley 50 WUL

• Solymar and Walsh, Lectures on Electrical Properties of Materials

• Moulson and Herbert, Electroceramics, Chapman & Hall 44 MOU

Pyro-Electricity• The unit cell of some crystals can posses a dipole moment, even in the absence of an applied electric field. Such crystals are pyroelectric.

• The point group of pyroelectrics must be non-centrosymmetric and polar. They are solids of low symmetry, which can support a directionality.

• Well known example is this phase of Barium Titanate.

• In pyro-electric, the polarization is often shielded by surface charges from the atmosphere. On heating a cooling, the polarization changes such that the charges can’t keep up - hence ‘pyro-electric’.

(Image missing)

This is Centro-Symmetric!

NaCl structurefcc lattice

(0,0,0) Na+

(1/2,0,0) Cl-

coordination number, z=6

Piezo-Electricity• Non-polar point groups that are again non-centrosymmetric can be made to be polar using mechanical stress.

• When this occurs, a dipole moment can be induced. This is the phenomenon of piezoelectricity.

P = qZ + !"0EPolarization

Stress

Piezoelectric coefficient

Electric field

Susceptibility

• To first order the induced moment is proportional to the stress.• Note that, when the field is zero a polarization is still possible.

• We had P = qZ + !"0E

• Now, conversely

e = sZ + qE

StrainElastic

compliance

• Therefore, an applied field will produce a strain, even in the absence of an applied stress.

Symmetry Summary

32 crystallographic point groups

of which 21 non-centrosymmetric and 11 centrosymmetric

of which 20 piezo-electric and 1 too symmetric

of which 10 polar (pyroelectric) 10 non-polar

of which some ferro-electric some not ferro-electric

Ferro-Electricity• Pyro-electrics, in which the direction of poarization can be switched by applying a electric field.

• As with ferromagnets, there exists a temperature (the ferro-electric Curie temperature) above which the material is not ferro-electric.

•High temperature phase has a higher symmetry, for example Barium titanate.

Above 393 K 278 to 393 K

(Image missing) (Image missing)

How Does Spontaneous Polarization Occur?

• Hard problem!

• One indication is the ‘ferro-electric catastrophe’.

•Two ions ‘A’ and ‘B’. A has dipole moment that causes a field at B.p E =2p

4!"0a3

• Leads to an induced dipole at B of

p!=

2p!

4"#0a3

• The polarization diverges upwards if or p

!> p

! > 2"#0a3

Domains and Hysteresis• A domain is a region where the spontaneous polarization is uniform. Adjacent domains can have different polarization, much like in ferromagnetism.

•Domains are separated by domain walls.

•A sample will polarize as domains that align with the field tend to grow, and those against it decrease in size.

•Hysteresis is possible, again like in ferromagnetic materials.•There are some important differences between ferromagnetic domains and ferroelectric domains.

No Polarization Rotation

• Discrete number of possible polarization directions fixed by crystal structure.

•e.g. Barium titanate 3 directions.•Rochelle salt 2 directions (see figure)

Differences: Ferroelectric and Ferromagnets

• Domain growth favoured in the ‘forward’ direction - long domains grow longer. Implies ‘head to tail’ coupling stronger then ‘sideways’.•Domains and domain walls tend to be narrow.

Weak “Sideways” Coupling

Hystersis - Nomenclature

• First polarization - OAB traced to saturation (residual slope: para-electric).

•Reduce field BAC’ is straight, and OC is the remanent polarization.

• Field OD is the coercive field needed to reduce the polarization to zero.

•The hysteresis loop area is a measure of the energy loss per cycle under AC conditions.

• Example values:•Barium titanate

•Remanent polarization OC: 6.1012 Cm-2

•Coercive field -OD: 8.104 Vm-1 (c.f. 15.104 Vm-1 for liquid crystal display - see later)

Uses• Piezo-Electrics

• Transducers: convert mechanical energy into electrical energy (or vice versa)• e.g. Mechanical to electrical: record player, strain gauge, cigarette lighter.• e.g. Electrical to mechanical: production of ultrasonic waves.

•Oscillators: example: a Quartz piezo-electric crystal of length L. Has a series of mechanical resonances corresponding to frequencies

f =nvsound

2L

if the polarization is driven at this frequency by an applied field, there is a strong, stable resonance. It dominates the dynamics of the system and effectively corrects for fluctuations in the applied field. Hence use in quartz oscillator or clock.

More Uses

• Pyro-Electrics

• Temperature sensing: not good for absolute measurements due to uncertainty in charge cancelling effects. Good however, for relative measurements. For example, infra red camera. (Sensitivity e.g. 0.5 K at 300 lines per frame).

• Figure of merit here is rate of change of polarization with temperature.

P =

dP

dt

More Uses

• Ferro-Electrics

• Computer memory elements. In uniaxial materials: 0 and 1 are + and - polarizations. No sideways cross talk means they can be closely spaced.

100 µm

electrodes