Piezoelectricity and ferroelectricity Phenomena and properties

Prof.Mgr.Ji ří Erhart, Ph.D. Department of Physics FP TUL What is the phenomenon about?

• Metallic membrane with „something strange“ • LEDs flash, why?

J.Erhart: Demonstrujeme piezoelektrický jev, Matematika, fyzika, informatika 20 (2010) 106-109

FPM - Piezoelectricity 1 2 History

1880, 1881 - Pierre a Jacques Curie, discovery of piezoelectricity, tourmaline and quartz 1917 – A.Langevin – ultrasound generation, sonar 1921 – ferroelectricity - J.Valasek: Piezoelectricity and allied phenomena in Salt, Phys.Rev. 17 (1921) 475 1926 – W.Cady – frequency stabilization of oscillator circuit by the quartz resonator

1944-1946 – USA, SSSR, Japonsko – BaTiO 3 ferroelectric ceramics 1954 – B.Jaffe et al – PZT ceramics

60. léta – LiNbO 3 and LiTaO 3 single crystals 70.léta – ferroelectric polymer PVDF 80.léta – piezoelectric composites 90.léta – domain engineering in PZN-PT, PMN-PT single crystals

FPM – Piezoelectricity 1 3 Predecessors of piezoelectricity

Pyroelectricity tourmaline = lapis electricus

(Na,Ca)(Mg,Fe) 3B3Al 6Si 6(O,OH,F) 31

Franz Ulrich Theodor Aepinus Tourmaline crystal. Carl Linnaeus (1707 – 1778) (1724 – 1802) – polar phenomenon

David Brewster – “pyroelectricity” 1824 pyros=ohe ň

David Brewster (1781 – 1868) FPM - Piezoelectricity 1 4 Pyroelectricity

A.C.Becquerel William Thomson (Lord Kelvin) first pyroelectricity measurement first pyroelectricity theory 1828 1878, 1893

∆ === ⋅⋅⋅ ∆θ PS p

William Thomson, Antoine César Lord Kelvin Becquerel (1824 – 1907) (1788 –1878)

FPM - Piezoelectricity 1 5 Piezoelectricity discovery

Tourmaline crystal, 1880 8.4.1880 Société minéralogique de France 24.8.1880 Académie des Sciences ∆P = d ⋅⋅⋅T

Paul-Jacques Pierre Curie Curie (1859 – 1906) (1856 – 1941)

Curie J, Curie P (1880) Développement, par pression, de l’électricité polaire dans les cristaux hémièdres à faces inclinées. Comptes rendus de l’Académie des Sciences 91: 294; 383. Curie J, Curie P (1881) Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées. Comptes rendus de l’Académie des Sciences 93: 1137-1140.

FPM - Piezoelectricity 1 6 Phenomenon properties

• Linear phenomenon, charge is independent from crystal length, it depends on the electrode area • Phenomenon exists due to crystal anisotropy (polar axes), amorphous materials are not piezoelectric Curie’s principle Symmetry elements of external field must be included in the phenomenon symmetry.

Crystal under the influence of external field exhibits only symmetry elements common to the symmetry of crystal itself and of the external field.

Example: mechanical pressure along [111] direction exerted on the cubic crystal with m3m symmetry causes symmetry reduction of deformed crystal to 3m symmetry class

FPM - Piezoelectricity 1 7 Piezoelectricity

Direct phenomenon mechanical pressure → electrical charge Converse phenomenon electric field → mechanical deformation

It is limited to certain crystallographic symmetry classes (20 from 32 classes) ,1 2, m, 222 , mm2, 4, 4, 422 , 4mm, 42m, 3, 32 , 3m, 6, 6, 622 , 6mm, 62m, 43m, 23

Example: SiO 2 (quartz) symmetry 32

FPM - Piezoelectricity 1 8 Measurement technique

Direct phenomenon (Brothers Curie) • Thompson’s electrometer • Null method ∆Q=V∆C (Daniel’s cell – 1.12V, 1836)

FPM - Piezoelectricity 1 9 Measurement technique

Converse phenomenon (Brothers Curie) • Holtz’s machine (induction electricity, HV) • Strain measured by the direct effect, optically

FPM - Piezoelectricity 1 10 Piezoelectricity origin

Brothers Curie – molecular theory

Charges generated at the compression (a) and tension (b) of quartz crystal basic unit

FPM - Piezoelectricity 1 11 First theory and application of piezoelectricity

Tensor analysis developed for the crystal anisotropy description 1890 Woldemar Voigt: Lehrbuch der Kristallographie, Teubner Verlag 1910

First application – electrometer, radioactivity study (Maria Skłodowska-Curie)

Woldemar Voigt (1850 –1919)

Maria Skłodowska- Curie’s electrometer Curie (1867 – 1934)

FPM - Piezoelectricity 1 12 Coupled field phenomena

Heckmann’s diagram

Piezoelectricity

= E + Sλ sλµTµ diλ Ei Jonas Ferdinand Gabriel Lippmann = + ε T (1845 –1921) Di diµTµ ij E j = D + Sλ sλµT µ giλ Di = − + β T Ei giµTµ ij D j

FPM - Piezoelectricity 1 13 Ferroelectricity

Joseph Valasek, 1920, Rochelle Salt NaKC 4H4O6.4H 2O

Joseph Valasek (1897-1993)

Hysteresis loop of Rochelle Salt – dry crystal, 0oC

Valasek J (1921) Piezoelectricity and allied phenomena in Rochelle salt. Phys. Rev. 17: 475-481

FPM - Piezoelectricity 1 14 Electromechanical phenomena

Direct conversion between mechanical and electrical energy • Linear effects – Piezo- and Pyroelectricity • Nonlinear effects - Ferroelectricity, Electrostriction

Analogy in magnetic materials • Piezomagnetic properties, magnetostriction, magnetoelectric effect, etc.

FPM - Piezoelectricity 1 15 Crystallographic constraints for piezoelectricity

Noncentrosymmetric classes (except of 432 ) 20 piezoelectric crystallographic classes

• Polar classes (10) – singular polar axis 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm • Polar-neutral classes (10) – multiple polar axes 222, 4, 422, 42m, 32, 6, 622, 6m2, 43m, 23

FPM - Piezoelectricity 1 16 Equations of state

Example: piezoelectricity

= E + = D + Sµ sµν Tν d kµ Ek Sµ sµν Tν g kµ Dk = + εT = − + βT Di diνTν ik Ek Ei giνTν ik Dk

= E − = D − Tµ cµν Sν ekµ Ek Tµ cµν Sν hkµ Dk = + ε S = − + βS Di eiν Sν ik Ek Ei hiν Sν ik Dk

FPM - Piezoelectricity 1 17 Piezoelectric coefficients

Different coefficients due to the choice of independent variables

FPM - Piezoelectricity 1 18 Material property anisotropy

d33 surface

PS[001] T

PbTiO3 – 4mm Maximum 3 2 2 d33 l3 +(d 31 +d 15 )l 3(l 1 +l 2 ) 83.7pC/N for θ = 0 o, i.e. [001] l1=sin( θ)cos( φ), l 2=sin( θ)sin( φ), l 3=cos( θ) C d33 = 83.7, d 31 = -27.2, d 15 = 60.2 [pC/N] FPM - Piezoelectricity 1 19 Material property anisotropy

d33 surface

PS[001] T

BaTiO3 – 4mm Maximum o 3 2 2 224pC/N for θ = 51 d33 l3 +(d 31 +d 15 )l 3(l 1 +l 2 ) l =sin( θ)cos( φ), l =sin( θ)sin( φ), l =cos( θ) 1 2 3 90pC/N for [001] C d33 = 90, d 31 = -33.4, d 15 = 564 [pC/N] 221pC/N for [111] C

FPM - Piezoelectricity 1 20 Pyroelectricity

Direct effect Temperature change → electric charge Converse effect (electrocaloric effect) Electric field → heat generation or absorption Anisotropy

Example:

Lithium tetraborate Li 2B4O7, symmetry 4mm

FPM - Piezoelectricity 1 21 Crystallographic constraints for pyroelectricity Polar symmetry classes (10) – singular polar axis 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm

Pyroelectric polarization of material (dipole moment) – polar axis direction

FPM - Piezoelectricity 1 22 Electrostriction

• Nonlinear effect Deformation proportional to the square of electric field

• No constraints on the material symmetry, effect exists in all materials

FPM - Piezoelectricity 1 23 Electrostriction

Nonlinear equations of state = + Sij dkij Ek Qijkl Pk Pl

Without crystallographic limits! All materials exhibit electrostrictive properties

FPM - Piezoelectricity 1 24 Electrostriction in cubic materials

Electrostrictive coefficients Q11 , Q12 , Q44

 S   Q Q Q 0 0 0   2   1   11 12 12   P1   2   S2  Q12 Q11 Q12 0 0 0  P2       S Q Q Q 0 0 0  2  3  ===  12 12 11   P3     S4   0 0 0 Q44 0 0  P P      2 3   S5   0 0 0 0 Q44 0   P1P3         S6   0 0 0 0 0 Q44   P1P2 

FPM - Piezoelectricity 1 25 Ferroelectricity

Spontaneous dipole moments = pyroelectricity with switchable polarization Electric analogy of permanent magnets

Characteristic properties • Ferroelectric domains and domain walls • Hysteresis loop D-E (S-E)

FPM - Piezoelectricity 1 26 Hierarchy of electromechanical phenomena

Piezo- SiO 2, GaPO 4, AlPO 4, ...

Pyro- Li 2B4O7, ...

BaTiO , PbTiO , PZT, Ferro- 3 3 KNbO 3, TGS, KDP, ...

FPM - Piezoelectricity 1 27 Ferroelectricity

Spontaneous existence of polarization (and spontaneous strain at the same time) - structural phase transition Ferroelectric/ferroelastic domains - orientation domain states Domain walls Hysteresis

FPM - Piezoelectricity 1 28 BaTiO3 – paraelectric phase

Perovskite structure

A2+

B4+ O2-

FPM - Piezoelectricity 1 29 BaTiO3 – ferroelectric phase

Several orientation states exist for the spontaneous polarization

m3m

4mm mm2 3m

FPM - Piezoelectricity 1 30 Domains, domain walls

Domain – space continuous region with the same orientation of the spontaneous dipole moment (polarization) Domain walls – interfaces between domains • Charged walls • Neutral walls

Ferroelectric domains exist in ferroelectric phase

Phase transition – Curie temperature TC

FPM - Piezoelectricity 1 31 D-E a S-E hysteresis loops

T. Liu, C. S. Lynch: Domain engineered relaxor ferroelectric single crystals Continuum Mech. Thermodyn. 18 (2006) 119–135

FPM - Piezoelectricity 1 32 Remanent polarization

PS, Pr, EC coercive field

FPM - Piezoelectricity 1 33 Mechanism of domain reorientation

FPM - Piezoelectricity 1 34 Remanent deformation

Ferroelasticity in ferroelectric materials

FPM - Piezoelectricity 1 35 → Domains in BaTiO3 ceramics m3m 4mm

Domain structure evolution during heating of BaTiO 3 ceramics over the Curie temperature (a) Temperature gradient parallel to the domain walls (b) Temperature gradient perpendicular to the domain walls Sang-Beom Kim, Doh-Yeon Kim: J. Am. Ceram. Soc., 83 [6] 1495–98 (2000)

FPM - Piezoelectricity 1 36 → Domains in BaTiO3 ceramics m3m 4mm

Etched surface of BaTiO 3 ceramics a) Herringbone and chessboard pattern b) Band structure of DW’s, domains continuously cross the grain boundaries G.Arlt, P.Sasko: J.Appl.Phys. 51 (1980) 4956-4960

FPM - Piezoelectricity 1 37 Ferroic phases

Structural phase transitions Parent phase (e.g. paraelectric) → ferroic phase Feroelectrics Ferroelastics LiNbO 3m → 3m 3 AgNbO → 3 KNbO3 m3m mm2 → NaNbO3 m3m mmm → BaTiO3 m3m 4mm → Pb 3(PO4)2 3m 2 / m Pb 5Ge 3O11 6 → 3 → KIO3 3m m

FPM - Piezoelectricity 1 38 Material property anomalies

Dielectric permittivity, spontaneous polarization, etc.

BaTiO3

FPM - Piezoelectricity 1 39 Material property anomalies

Piezoelectric coefficient

LiTaO3

FPM - Piezoelectricity 1 40 Landau-Ginzburg-Devonshire theory

Power expansion of thermodynamic potential 1 1 Φ (T,η) =Φ + α(T − T )η 2 + βη4 0 2 C 4 Equilibrium and stability ∂∂∂Φ ∂∂∂2Φ === 0, >>> 0 ∂∂∂η ∂∂∂η 2 Equilibrium phase transition parameter value >>> ∂∂∂Φ 3  0 T TC === α(T −−− T )η +++ βη === 0  ∂∂∂η C 0 0 η ===   α −−− 1 2 0 ±±± −−− (T TC ) <<<    T TC   β 

FPM - Piezoelectricity 1 41 2nd order phase transition

Without hysteresis

FPM - Piezoelectricity 1 42 1st order phase transition

Hysteresis

FPM - Piezoelectricity 1 43 Curie – Weiss law

Dielectric permittivity temperature dependence

FPM - Piezoelectricity 1 44 Spontaneous strain vs. spontaneous polarization

Generally (for normal ferroelectrics) === Skl Qijkl PiPj → Example for the ferroelectric species 4 / mmm mxy  S  Q Q Q 0 0 0   2   1   11 12 13   PS1  2  S2  Q12 Q11 Q13 0 0 0   P   S1   S  Q Q Q 0 0 0  2  3  =  31 31 33   PS3   S4   0 0 0 Q44 0 0   P P       S1 S3   S5   0 0 0 0 Q44 0   PS1PS3       2   S6   0 0 0 0 0 Q66   PS1  = PS (PS1,PS1,PS3) FPM - Piezoelectricity 1 45 Experimental characterization of spontaneous deformation Lattice constants measured by X-Ray diffraction

• In parent phase (extrapolation down to the ferroic phase)

• In ferroic phase

LB Tables III/16a

General formula for the components of strain tensor

J.L.Schlenker, G.V.Gibbs, M.B.Boisen, Jr.: Acta Cryst. A34 (1978) 52-54

FPM - Piezoelectricity 1 46 Bi 4Ti 3O12

→ cmon 4 / mmm mxy

bmon c tetr b tetr amon a 8 ferroelectric DS tetr 4 ferroelastic DS

Pa>>P c

FPM - Piezoelectricity 1 47 Bi 4Ti 3O12

Spontaneous deformation/polarization (components in the parent phase coordinate system)  S S S   11 12 13  1 1 1 1 S 1)( ===  S S −−− S  P I === ( P ,−−− P , P ) P II === (−−− P , P ,−−−P ) 12 11 13 2 a 2 a c 2 a 2 a c  −−−   S13 S13 S33   S −−− S S   11 12 13  1 1 1 1 2)( === −−− III === IV === −−− −−− −−− S  S12 S11 S13  P ( Pa , Pa , Pc ) P ( Pa , Pa , Pc )   2 2 2 2  S13 S13 S33   S S −−− S   11 12 13  1 1 1 1 S 3)( ===  S S S  PV === (−−− P , P , P ) PVI === ( P ,−−− P ,−−−P ) 12 11 13 2 a 2 a c 2 a 2 a c  −−−   S13 S13 S33   S −−− S −−− S   11 12 13  1 1 1 1 S 4)( ===  −−− S S −−− S  PVII === (−−− P ,−−− P , P ) PVIII === ( P , P ,−−−P ) 12 11 13 2 a 2 a c 2 a 2 a c  −−− −−−   S13 S13 S33 

FPM - Piezoelectricity 1 48 Domain wall orientation

(2) 2 − (1) 2 = (2) − (1) = (ds ) (ds ) (Sij Sij )ds i ds j 0

Example for Bi 4Ti 3O12 Domain state pair S (1) (P (1) , P (2) ) and S (2) (P (3) ,P (4) ) S − 13 = (ds 1 ds 3 )ds 2 0 S12 Two perpendicular domain walls Charged wall (010) W-wall S = − 13 Neutral wall (10K) S-wall K S12

FPM - Piezoelectricity 1 49 Domain wall orientations in Bi 4Ti 3O12

S(1) S(2) S(3) S(4) P(1) ,P (2) P(3) ,P (4) P(5) ,P (6) P(7) ,P (8) S(1) N/A (010) (1-10) (100) P(1) ,P (2) (10-K) (001) (01K) S(2) N/A (100) (110) P(3) ,P (4) (01-K) (001) S(3) N/A (010) P(5) ,P (6) (10K) S(4) N/A P(7) ,P (8)

FPM - Piezoelectricity 1 50 Domain wall types

W-walls

° W∞ - arbitrary wall orientation

° Wf – fixed crystallographic domain wall orientation S-walls (“strange” walls, W’-walls)

° S1 –PS direction

° S2 – bijk and Qijkl

° S3 –PS direction, b ijk and Qijkl

° S4 –PS direction and magnitude, b ijk and Qijkl

° S5 –PS magnitude, b ijk and Qijkl

FPM - Piezoelectricity 1 51 Permissible domain wall pairs

W∞ - arbitrary domain wall orientation

WfWf – fixed domain wall orientation

WfS – fixed and „strange“ walls SS – pair of „strange“ walls R – walls are not permissible

J.Fousek, V.Janovec: J.Appl.Phys. 40 (1969) 135 J.Sapriel: Phys.Rev. B12 (1975) 5128 J.Erhart: Phase Transitions 77 (2004) 989-1074

FPM - Piezoelectricity 1 52 Antiferroelectricity

Dipole moments are partially compensated Switching possible at higher fields

E

FPM - Piezoelectricity 1 53 Electromechanical coupling coefficient

Energy transfer efficiency between mechanical and electrical energy via piezoelectric effect

FPM - Piezoelectricity 1 54 Electromechanical coupling

d 2 k 2 = 33 33 E εT s33 33

W.Heywang, K.Lubitz, W.Wersing (edts.): Piezoelectricity, Springer Verlag 2008

FPM - Piezoelectricity 1 55 Electromechanical coupling coefficients

Different modes Transversal thickness-shear d 2 2 2 = 31 2 d15 k31 k = s E εT 15 E ε T 11 33 s55 11 thickness radial

e2 2d 2 k 2 = 33 k 2 = 31 t D ε S p ε T E + E c33 33 33 (s11 s12 )

FPM - Piezoelectricity 1 56 Dielectric losses

Permittivity – complex values ε '' ε = ε '− jε '' tan( δ ) = ε ' Dissipated power = energy loss during 1s

2 2 = ⋅ = 2 = C '' I − C'I P U I ZC I j ω(C'2 +C '' 2 ) ω(C'2 +C '' 2 ) S C = ()ε '− jε '' = C'− jC '' d − = 1 = C '' jC ' ZC jωC ω(C'2 +C '' 2 )

FPM - Piezoelectricity 1 57 Mechanical losses

Elastic properties – complex compliance s = s'− js ''

Mechanical quality Qm

1 Q === m π T 2 2 fm Z C keff

FPM - Piezoelectricity 1 58 Young’s modulus and elastic coefficients

Examples for 4mm symmetry

Mechanical pressure Mechanical deformation σ σ = 1 − 11 − 33 T = c S + c S + c S S11 T11 T22 T33 11 11 11 12 22 13 33 Y11 Y11 Y33 T = c S + c S + c S σ σ 22 12 11 11 22 13 33 = 1 − 11 − 33 = + + S22 T22 T11 T33 T33 c13 S11 c13 S22 c33 S33 Y11 Y11 Y33 σ σ = 1 − 11 − 11 S33 T33 T11 T11 Y33 Y11 Y11

FPM - Piezoelectricity 1 59 Young’s modulus

Matrix form s11 =1/ Y11 -1 c =s s33 =1/ Y33

2c2 Y =c − 13 33 33 + c11 c12

(c − c )[ c (c + c ) − 2c2 ] Y = 11 12 33 11 12 13 11 − 2 c11 c33 c13

FPM - Piezoelectricity 1 60 Thank you for your attention!

FPM - Piezoelectricity 1 61