v.2020.JAN B. Dielectric Properties of Materials

mostly organic (PET, PTFE, PP, PS …) • Materials strong covalent interchain, weak bonds intrachains – ceramics / polymers electronic ceramics crystalline inorganic (Al2O3, BaTiO3, glasses …) structural ceramics strong ionic bond (bricks…) – solids in which outer electrons are unable to move through structure • Functions • Properties – energy storage – large C (freq dependent)

– insulation – high r, br – new apps: capacitive sensing, … • Objectives – understand underlying energy storage mechanisms – understand insulation breakdown mechanisms – select proper dielectrics (from chips to high-power cables)

Reference: S.O. Kasap (4th Ed.) Chap. 7; RJD Tilley (Understanding solids, 2nd Ed) Chap. 11.

Dielectric Properties 2102308 1 Dielectric Materials

• introduction

e r – relative permittivity (er), polarizability (a) • polarization mechanisms

– types: electronic, ionic, dipolar, interfacial er () – frequency dependency

• br (electric field strength, breakdown field) – gas, liquid, solid • capacitors – ceramic, polymer, electrolytic • nonlinear dielectrics – piezo-, ferro-, pyro-electricity • special cases for EE – see brochure for EE ceramics (self-study)

Dielectric Properties 2102308 2 also called “dielectric constant” 7.1 Relative permittivity e r - dielectric is the working material (active component) in capacitors. The simplest structure is the

parallel plate capacitor (Fig. 7.1). Without the dielectric (a), the stored charge is Qo. With the dielectric (c), the stored charge increase to Q, or by a factor of er , the relative permittivity. - under electric field E, the constituents of the dielectric (ions, atoms, molecules) become polarized (Fig. 11.3). Internal electric dipole moment (p) induced by E, resulting in observable polarization (P).

Qo e o A e re o A Q C Co = = C = ; e r = = V d d Qo Co

* i(t) is a displacement current, not conduction current. 1 V Potential Energy:U = CV 2 = Q 2 2

Dielectric Properties 2102308 3 - electric dipole moment (p) or dipole moment (or just dipole) is charge balanced, see A - a mechanism that gives rise to such dipole is electronic polarization, see B - the resulting dipole is proportional to the electric field strength E, the proportionality constant is termed (electronic) polarizability, see C

- the relative permittivity (er) is related to the polarizability (a) of materials, see D A. dipole moment: separation of –ve and +ve charges (equal magnitude, charge balance)

p can interacts with external  Ex. H+-Cl−, p ~ 3.6×10−30 C.m (unit: C·m)

B. electronic polarization (all atoms) E C x O Electron cloud Q = 0 Qnet = 0 net p = 0  p  0  Atomic a  0 a = 0 nucleus 

pinduced = ae Center of negative ( unit: C·m = [F·m2][V/m] ) charge (a: polarizability)

(ae: electronic polarizability)

pinduced (a) A neutral atom in E = 0. (b) Induced dipole moment in a field

Dielectric Properties 2102308 4 The origin of electronic polarization. 30 C. polarizability ae fo 10 f with electric field  o x1015 Hz Rn Hook’s Xe Ze = x (restoring force) Ar Kr a ~ Z0.99 Z 2e2 1 e  pe = Qa = (Ze)x =  ae ae Ne  x10-40 F m2 without  He Z  → large electron cloud → further from nucleus → 2 can be shifted easily → a  d x 0.1  x = meZ 2 dt 1 10 100 2 2 Atomic number Z meZ d x 1 d x  x = 2 = 2 2  dt 0 dt Electronic polarizability and its resonance frequency vs. the number of electrons in the atom (Z). The dashed line is the best simple harmonic equation: fit line.

1/ 2       ~constant o     Zme  Ze2 , ae = 2 meo

Key message: The heavier the element ( Z ), the higher the electronic polarizability (ae)

Dielectric Properties 2102308 5 Before insertion: Qo D. e r &ae V Q Q  = = o = o → d Cod e o A

Bound polarization Qo = e o A  charges on the surfaces bound -Q +Q P P charges After insertion: Qp +Q -Q free E charges (b) ptotal = Qpd → p * p  N (Ad ) Q = total = induced p d d Area = A ptotal Qp = (Nae )A  (c) * ptot = (p / molecule) × (molecule / volume) × (volume) (a) -Q P +Q V P P

Q = Qo + Qp

d  Q Qp Nae (a) When a dilectric is placed in an electric field, bound polarization e r  =1+ =1+  charges appear on the opposite surfaces. (b) The origin of these Qo Qo e o polarization charges is the polarization of the molecules of the medium. (c) We can represent the whole dielectric in terms of its More detail analysis (solid polarized by a local surface polarization charges +QP and -QP. electric field) yields Clausius-Mossotti relation: e −1 Na r =  e r + 2 3e o - solid must be homogeneous isotropic (no note: a is atomic-level , e ismaterial-level parameters e r permanent dipoles, dipolar molecules) microscopic macroscopic − a includes other polarization mechanisms

Dielectric Properties 2102308 6 Bound polarization charges on the surfaces -QP +QP

Bound polarization charges on the surfaces +Q -Q -Q +Q E P P Polarization P (C/cm(b) 2) definition +Q -Q E (b) total dipole moment Area = A p Polarization  total volume e r (c) Area = A p Q d Q total ptotal p p (a) -Q P +Q = = = V P P volume Ad A (c)

(a) -Q P +Q Vd P P (a) When a dilectric is placed in an electric field, bound polarization charges appear on the opposite surfaasce sa. (functionb) The or iofgi nexternalof these electric field polarization charges is the polarization of the molecules surfaceof the d medium. (c) We can represe(nat) tWhehwenhaoldeildeicetlreicctriiscpilnacterdmchargeins oafnietsl ectric field, bound polarization surface polarization chargesch+aQrgPesanadpp-eQaPr.on the opposit(storage)e surfaces. (b) The origin of these polarization charges is the polarization of the molecules of the medium. (c) We can represent the wholedensitydielectric in tmaterialerms of its responds surface polarization charges +QP and -QP. Q   P = p = (Na ) = e (e −1) A e o r

polarizability excite (atomic/molecular)

Dielectric Properties 2102308 7 Ex. 7.2

-40 2 The electronic polarizability of the Ar atom is ae = 1.710 F m . What is the static dielectric 3 constant er of solid Ar (below 84K) if its density is 1.8 g/cm and atomic mass is 39.95.

Origin of charge/energy storage:

  material develops polarization (P)  pinduced (=a) binds or draws more charges (Qp) to the electrodes. These charges can be released to do work. (Think of stretched strings)

Dielectric Properties 2102308 8 7.2 (electronic) polarization in covalent solids (semiconductors)

• to shift electrons in ionic cores  need ~ 10 eV (difficult) er EG(eV) • to shift electrons in covalent bonds  need ~ 1-2 eV (easy) Ge 16 0.67 • stronger bonds (EG ↑)  smaller shifts (x, ae, er) Si 11.9 1.12 C 5.7 5.5 GaAs ? 1.43

SiO2 3.9 9.00

= 13.1, but why?

Nae e r =1+ e o

ae :ae−core +ae−valence ✓ ✓

(a) Valence electrons in covalent bonds in the absence of an applied field. (b) When an electric field is applied to a covalent solid, the valence electrons in the covalent bonds are shifted very easily with respect to the positive ionic cores. The whole solid becomes polarized due to the collective shift in the negative charge distribution of the valence electrons.

Q) why concern with er of “semiconductors”? A1) (device) In depletion layer of p-n junctions, dielectric property (er) is more important than electrical property (s ). A2) (circuit & system) er  C  CR  BW

Dielectric Properties 2102308 9 7.3 Polarization Mechanisms

7.3.0 Electronic polarization: (for all neutral atoms) displacement of electrons

7.3.1 Ionic polarization: (for charged ions) displacement of ions. Example: NaCl (below)

p = a = Qa p+ p-

large (a) x

Ð Clausius-Mossotti relation: Cl Na+ 3e  e −1  o  r  ai =   N  e r + 2  p' p' + - usually ai 10ae

(b)

E

(a) NaCl chain in the NaCl crystal without an applied field. Average or net dipole moment per ion = 0. (b) In the presence of an applied field the ions become slightly displaced which leads to a net average dipole moment per ion.

Dielectric Properties 2102308 10 7.3.2 Orientational polarization (or dipolar polarization): re-orienting of molecules with permanent dipole moments. - certain molecule has “permanent” dipole moment due to bonding (see Examples in Box) - similar to ionic but |p+|  |p-| for each molecule - under electric fields, molecules are re-oriented such that p align along E as much as possible (atoms/molecules in liquid/solid phase are not free to move)

Examples (materials): note strong Polar liquids – water (), temperature acetone, alcohol, electrolyte dependency Polar gases – steam, gaseous HCl () po = Qa Polar solids – glasses p 2 a = o Examples (values): d 3kT



p ~ 3.6×10−30 C.m



p ~ 6.2×10−30 C.m

Dielectric Properties 2102308 11 7.3.3 Interfacial polarization (or space charge polarization): build-up of mobile charges - certain dielectric has mobile charges (electrons, holes, ions) - though they move under electric fields, they cannot leave dielectrics, but pile-up at grain boundaries (polycrystals), at equilibrium, internal field block further charge movement

− aif not significant in most cases, except at low frequencies

Dielectric Properties 2102308 12 7.3.4 Total polarization 7.3.0 7.3.1 7.3.2 a ae +ai +ad (average)

(aif is location specific) 7.3.3

1

dipolar 2

Dipolar solid general trend: 1. semiconductor: concerned with parasitic capacitance ae  ai  a d 2. insulator: more concerned with breakdown field (br) except those related to valence electrons Dielectric Properties 2102308 13 7.4 Frequency dependency of polarisability and relative permittivity - capacitors are used in applications throughout the frequency spectrum: from low, power line frequencies (50/60 Hz), to high, communications frequencies (MHz/GHz) - dielectric materials may or maynot have time to respond to the excitation (ac frequency), this depends on the dominant polarization mechanism(s) and the ac frequency - generally, the mechanisms which involves heavy masses are slowest, light are fastest (Fig. 11.5) 6 − aif (charge switch positions at grain boundaries), upto 10 Hz 9 − ad (dipoles of molecules rotate in medium), upto 10 Hz 12 − ai (ions stretch/compress), upto 10 Hz 16 17 − ae (electron cloud shifts around nucleus), upto 10 -10 Hz (see slide #5) - frequency dependency of polarisability:

atotal = aif + ad + ai + ae a a() = dc 1+ j

Dielectric Properties 2102308 14 - when dipoles respond to electric field, there’s a delay (in the case of step function, Fig. 7.12) or phase lag (sinusoidal function) because: 1. ions/molecules have to rotate in a viscous material (liquids, polymers, solids), this transfers energy to medium (working principle of microwave oven, 2.45 GHz), causing energy loss 2. thermal agitation tries to randomize dipole orientation - the dielectric response to electric field (delay, loss) is best described using a complex dielectric function (): the real part (☺) signifies energy storage, the imaginary part () signifies loss. Datasheet for dielectric usually state loss tangent () at frequencies of interest - frequency dependency of complex relative permittivity in materials with Case A: one polarization mechanism (simplest) Case B: four polarization mechanisms (complex, hypothetical) ☺   er = er − jer

e  tan = r er

Dielectric Properties 2102308 15 case A: material has one polarization mechanism Na a from e =1+ and a() = dc the frequency dependency of real and imaginary parts r e 1+ j o of the relative permittivity follows curves in (b)

P = Posin(t - ) out of phase

E = Eosint er' and er''   er' e r = e r − je r e (0) r Dielectric resonance: storage (C) • energy storage by field () = • energy transfer to random collisions (1/)

er''

1 loss (G) 

0.01/ 0.1 1/ 10 100/  /  /  v = Vosint (a) (b) (a) An ac field is applied to a dipolar medium. The polarization P (P = Np) is out of phase with the ac field. The relative permittivity is a complex number with real (er') and imaginary (er'') parts that exhibit frequency dependence.

Dielectric Properties 2102308 16 case B: material has many polarization mechanisms

e r −1 Na a dc from = ; atotal = aif +a d +ai +ae ; a() = e r + 2 3e o 1+ j

Interfacial and space charge

storage Orientational, e' r Dipolar

Ionic Electronic er'' loss er' = 1 a a if a e d a i ƒ 102 1 102 104 106 108 1010 1012 1014 1016 Radio Infrared Ultraviolet light

The frequency dependence of the real and imaginary parts of the dielectric constant in the presence of interfacial, orientational, ionic and electronic polarization mechanisms.

In practice: one mechanism dominates at operating frequency

Dielectric Properties 2102308 17 - the storage and loss characters of dielectric (in parallel plates) appear in the equivalent circuit () as an ideal (lossless) capacitor of capacitance C (☺) in parallel with a conductance G () - this can be derived from the basic definition of admittance () - important design parameters are loss per unit volume () and loss per unit capacitance ()

 Admittance of a parallel plate capacitor:

e e A e =e  − je  e eA e eA Y  j o r ⎯ ⎯r ⎯r ⎯r → j o r + o r = jC + G ideal capacitor (lossless): jC d d d e eA e εA real capacitor (lossy): jC + G where C = o r ; G = o r d d

2 2 V P = P sin(t -) Conductance = G = 1/R o p p W = GV = RP  e  r

W 2 C W = =  =  e e tan ☺   vol o r e r dA

W W = =  = V 2 tan v = V sint v = V sint  cap o o C

EE basic definitions: quantity DC ac quantity real imaginary I/V, i/v conductance (G) admittance (Y)  Y = conductance (G) + susceptance (jB) V/I, v/i resistance (R) impedance (Z) Z = resistance (R) + reactance (jX)

Dielectric Properties 2102308 18 Ex. 7.6-7.7

At a given voltage, which dielectric will have the lowest power dissipation per unit capacitance at 60Hz? Is this also true at 1MHz?

3 Calculate the heat generated per second due to dielectric losses per cm of XLPE (power cable insulator) and Al2O3 at 60Hz and 1MHz at a field of 100kV/cm. (from Kasap 3rd Ed., table 7.4 p.611)

f = 60 Hz f = 1 MHz

Loss/Volume Loss/Volume -1 -1 Material er' tan  -3 er' tan  -3 k (W cm K ) (mW cm ) (W cm )

-4 -4 XLPE 2.3 3 x 10 0.230 2.3 4 x 10 5.12 0.005

-3 -3 Alumina 8.5 1 x 10 2.84 8.5 1 x 10 47.3 0.33

Dielectric Properties 2102308 19 7.6 Dielectric strength and breakdown 7.6.1 Dielectric strength • dielectric materials used as insulator between two conductors at different voltages to prevent the ionization of air • at high fields, all dielectrics fail (breakdown, become conducting)

• critical field is called dielectric strength (br)

arcing

corona discharge

Types of pole insulators: - glass - ceramic - glass-ceramic

Dielectric Properties 2102308 20 The nature of breakdown: • breakdowns in liquid and gaseous dielectrics are temporary; in solids, permanent (conducting channels physically created)

• dielectric strength (br) depends on frequency: different for DC and AC fields • br depends on molecular structure, impurities, defects (esp. voids), geometry, T, humidity, ...

Dielectric strength (br) of typical insulators

gas (@ 1 atm)

liquid

glass solid polymer

Key: air (N2/O2, lower limit), glass (SiO2, upper limit)

Dielectric Properties 2102308 21 7.6.2 Dielectric breakdown in gases Mechanisms: * corona discharge -- partial breakdown of air around curved electrodes, see fig. (a) * partial discharge -- see figs. (b,c) Crack (or defect) at dielectric- High voltage conductor Void in dielectric electrode interface

e1 Gas e 2

Ground (a) (b) Partial Discharge (c) (does not connect the electrodes) Corona and Partial Discharges: (a) The field is greatest on the surface of the cylindrical conductor facing the ground. If the voltage is sufficiently large this field gives rise to a corona discharge. (b) The field in a void within a solid can easily cause partial discharge. (c) The field in the crack at the solid-metal interface can also lead to a partial discharge.

Gauss: e11 = e22 → voids & cracks reduce br Dielectric Properties 2102308 22 Mechanism (physical origin): electron avalanche effect (similar to reverse biassed p-n junction [2102385, C5]) • always a few free electrons (due to cosmic rays) • under high fields, these electrons can accelerate and impact ionize neutral gas atoms, giving electrons and ions which conduct • process repeat → avalanche “Spacing” matter:

Remedies: * increase conductor spacing d (decrease electric field, E = V/d) * increase air pressure: pressure  → mfp & mft  → average kinetic energy  → breakdown 

* replace dielectric: air → SF6

Dielectric Properties 2102308 23 7.6.3 Dielectric breakdown in liquids

• mechanisms not clear, but probably due to: • conductive particles bridging electrodes (in impure liquids) • gas bubbles in liquids (after partial discharge  → local temperature  → bubble size ) • electrode injection (see next slide) 7.6.4 Dielectric breakdown in solids Caused by intrinsic properties of dielectrics and environmental factors. Five main mechanisms: 1. Intrinsic (electronic) breakdown (by avalanche) • initial electrons: - pre-exist in CB of dielectrics - electrode injection (see next slide)

• under br, electron move a distance l gains an energy of ebrl

• ebrl > EG → impact ionization (break valence bond)

• Ex: EG ~ 5 eV, l = 50nm → br ~ 1 MV/cm

• upper theoretical limits: occur only in high purity dielectric—e.g., SiO2 in MOSFET

Dielectric Properties 2102308 24 Electrode Injection = Enormous increase in the injected electrons from metal electrodes Mechanism: electron tunnelling through thin potential barrier (Fowler-Nordheim) Possible at: metal-air, metal-liquid, metal-solid interfaces PE(x) E +  F eff Vo Grid or Anode - e Cathode E EF F E

0 x 0 x = 0 x = x xF F HV V Metal Vacuum

(a) (dielectric) (b) (c) (a) Field emission is the tunneling of an electron at an energy EF through the narrow PE barrier induced by a large applied field. (b) For simplicity we take the barrier to be rectangular. (c) A sharp point cathode has the maximumfield at the tip where the field-emission of electrons occurs.

Dielectric Properties 2102308 25  Thermal breakdown  Electromechanical breakdown • e.g., in ceramics and glasses • e.g., in polyethylene and polyisobutylene • Two heat sources: • Positive feedback: 1. s. finite conduction @ LF (Joules heat = s2) F → d → C → Q → F  mechanical runaway 2 2. er. dielectric loss @ HF (V  tan ) • Positive feedback: T  → (1,2)   thermal runaway  eA Q1Q2  C = Q = CV F = k 2  (Metals, r  T. Insulators, s  T)  d d  Two indicators/signatures of thermal breakdown • End results: - plastic flow (viscous deformation) 1. br depends on field duration due to heat capacity (thermal lag) - cracks (electrofracture) Example: Pyrex @ 70C - thermal breakdown (since  ) - pulse 1ms → br = 9 MV/cm - cont. @ 30s → br = 2.5 MV/cm +Q •Q 2. br depends on temperature: V polyethylene-based  = polymeric insulation F F d

d V

An exaggerated schematic illustration of a soft dielectric medium experiencing strong compressive forces due to the applied voltage.

Dielectric Properties 2102308 26  Internal discharge Origin • microvoids (manufacturing defects) → partial discharge (see 7.6.2) → erosion of local, internal surfaces, then … • voids propagate → tree branches (see next slide) (hollow volumes in which gaseous discharge takes place and forms a conducting channel) Remedy: prevent microvoids by improving manufacturing process Example: Power Coax polymer M-S-I-S-M M-I-M • semi-PE surface has no microvoids due to PE manufacturing (extrusion process draws sheaths and PE at the same time) • semiconducting → equipotential (V) → reduce local high field PVC regions () → no tree Two types of dielectric: 1. PE, polyethylene, to maximize voltage branches 2. PVC, polyvinyl chloride, to protect cable (a) A schematic illustration of electrical treeing breakdown in a high voltage coaxial cable which was initiated by a partial discharge in the void at the inner conductor - dielectric interface. (b) A schematic diagram of a typical high voltage coaxial cable with semiconducting polymer layers around the inner conductor and around the outer surface of the dielectric.

Dielectric Properties 2102308 27 Dielectric Properties 2102308 28  Insulation ageing Sources: * physical: temp and mechanical stress variations → structural defects such as microcracks * chemical: radiation, ambient, oxidation → deteriorate chemical structure * electrical: dc fields dissociate & transport ion → structural change ac fields → treeing • in moist environment → microscopic voids containing water (or aqueous electrolyte)

Dielectric Properties 2102308 29 E br • Breakdown mechanism can change, depending on operating conditions 10 SiO2, dc  Intrinsic • It is not possible to clearly identify a specific Electronic Electro- breakdown mechanism for a material mechanical 1 MV cm-1

  Internal discharges and electrical trees Thermal 100  Water trees Air, 60 Hz 10

1 kV cm-1

1 ns 1 µs 1 ms 1 s 1 min 1 hr1 day 1 mo1 yr10 yrs Time to breakdown

Time to breakdown and the field at breakdown, br, are interrelated and depend on the mechanism that causes the insulation breakdown. External discharges have been excluded (based on L.A. Dissado and J.C. Fothergill, Electrical Degradation and Breakdown in Polymers, Peter Peregrinus Ltd. for IEE, UK, © 1992, p. 63)

Dielectric Properties 2102308 30 7.7 Dielectric Materials Fundamental trade-offs: • selection criteria: C, f, max. volt, acceptable loss C-Vmax • large C more easily obtained at low frequency (interface & dipolar polarizations) freq-loss

electrolytic

ceramic

polymer

ceramic

polymer

electrolytic

Dielectric Properties 2102308 31 source: source: Wiki

Dielectric Properties 2102308 32 Non-polar Ceramic capacitors (high-er)

Class 1 (low loss) er =10 d =10m Metal termination 2 resonant, tuning Epoxy A =1cm Ceramic → C = 885 pF

Leads

Metal electrode source: Wiki

A → C 100F

(a) Single layer ceramic capacitor (b) Multilayer ceramic capacitor (e.g. disk capacitors) (stacked ceramic layers) Class 2 (volume efficiency) Single and multilayer dielectric capacitors buffer, coupling MLCC status ferroelectric (doped) 2005: 100s F, 1,400 layers BaTiO3 (er ~ 100s – 1,000s) 2013: d = 0.5 m

Dielectric Properties 2102308 33 Polymer film capacitors (low-er) er  2−3 Polymers: low er but wide frequency response (low tan )

Al metallization slightly offset to provide means for connections (a)

Polymer film

Polymers

(b) Market share: er @ 1 kHz: 50% (PP) 2.2 40% (PET) 3.3 (PEN) 3.0 (PPS) 2.0 Two polymer tapes in (a) each with a metallized film electrode on the surface (offset from each other) can be rolled together (like a Swiss roll- cake) to obtain a polymer film capacitor as in (b). As the two separate metal films are lined at oppose edges, electroding is done over the whole source: Wiki side surface.

Dielectric Properties 2102308 34 Electrolytic capacitors: (high-C) electrolyte: ionic conducting liquid/solid Liquids Solids e r  9 by electrolysis * paper-soaked er = 28 grown electrolytically, * conducting solid electrolyte thin (0.1m), * makes good Epoxy responsible for large C contact with Al O 2 3 SAgilve paster paint

Ta2O5 Graphite Electrolyte Ta MnO2 Silver paste Al2O3 Ta Al foils Anode Cathode

m Al Al



100 -

Al case Lads 50

(a) (b)

(a) (b) Solid electrolyte tantalum capacitor. (a) A cross section without Capacitive behaviour due to fine detail. (b) An enlarged section through the Ta capacitor.

Al/Al2O3/electrolyte etched to make surface porous Polarity is important because Al/Al2O3 and Ta/Ta2O5 are rectifying contacts → need to be reverse biased; otherwise, (A) before forming Al2O3  : liquids dry the structures conduct! (no longer insulate / store energy)

Dielectric Properties 2102308 35 Comparison of dielectrics for capacitor applications low C high C 7.7.2 high frequency low-medium frequency

Capacitor name Polypropylene Polyester Mica Aluminum, Tantalum, High-K ceramic electrolytic electrolytic, solid (C3H6)n (C10H8O4)n KAl3Si3O10(OH)2 (ferroelectric)

Dielectric Polymer film Polymer film Mica Anodized Al2O3 Anodized X7R film Ta2O5 film BaTiO3 base (see 7.8.3)

er 2.2 – 2.3 3.2 – 3.3 6.9 8.5 27 2000 tan 4  10-4 4  10-3 2  10-4 0.05 - 0.1 0.01 0.01

-1 Ebr (kV mm ) DC 100 - 350 100 - 300 50 - 300 400 - 1000 300 - 600 10

d (typical minimum) 3 - 4 µm 1 µm 2 - 3 µm 0.1 µm 0.1 m 10 µm

-3  Cvol (µF cm ) 2 30 15 7,500a 24,000a 180

Rp = 1/Gp; C = 1 F; 400 kW 40 kW 800 kW 1.5 - 3 kW 16 kW 16 kW 1000 Hz

-3  Evol (mJ cm )b 10 15 8 1000 1200 100 Polarization Electronic Electronic and Ionic Ionic Ionic Large ionic Dipolar displacement

Volume efficiency: Max energy per  Capacitance per unit volume  unit volume

Dielectric Properties 2102308 36 7.8 Nonlinear dielectrics  - normal dielectric materials are polarized (polarization P) under electric fields (field strength E): zero E, zero P (see ) - some dielectrics have non-zero P even at zero E  due to pressure (piezoelectric, ), or heat (pyroelectric, ) - some (pyro & ferroelectric) even have 7.8.1

permanent or spontaneous polarization Ps (no fields, pressure, heat required, )

- the direction of Ps in pyro cannot be switched  (), in ferro can be switched () by E   7.8.3 dielectric piezo pyro ferro 

7.8.2

electric electric field pressure heat

materials materials respondto: - - -

Dielectric Properties 2102308 37 7.8.1 Piezoelectricity - Piezoelectricity comprises a direct and a converse effect (Fig. 7.40) - piezoelectric crystals must be non-centro symmetric (Figs. 11.7, 7.42)

(direct) piezoelectric effect converse piezoelectric effect

(a) P = 0 (c) V

P = d T S = d E i ij j Force j ij i l F strain S = stress T = l A (d) (b) P V V

dij: piezoelectric coefficient (or piezoelectric modulus)

PTi =h einducedpiezoe polarizationlectric effe calongt. (a) iA piezoelecStrj i=c inducedcrystal wmechanicalith no strain Taj =p pappliedlied st rmechanicaless or fiel dstress. (b) Talonghe c rjystal isEsit=ra appliedined by electrican app fieldlied force which induces polarization in the crystal and generates surface charges. (c) An applied field causes the crystal to become strained. In this case the field compresses the crystal. (d) The strain changes direction when the field is reversed, and now the Dielectric Properties crystal is extended. The das2102308hed rectangle is the original sample 38 size in (a). - of the possible 32 point groups (Appendix), only 20 are non-centrosymmetric (Fig. 11.7) - when centrosymmetric crystals are under pressure, the dipole changes cancel out (Fig. 7.41) - when non-centrosymmetric crystals are under pressure, the dipole changes do not cancel, resulting in net polarization P (Fig. 7.42) - non-centrosymmetric requirements → only piezo crystals, not polycrystals or amorphous - examples of piezo crystals and associated coefficients (Table 7.8) - applications: (Fig. 7.43) direct and converse piezoelectric effects are complementary and often used together in a transducer

Dielectric Properties 2102308 39 Centrosymmetric crystals: CoM of –ve & +ve charges coincide, Force even with external forces

O P = 0 P = 0

(a) (b)

A cubic unit celAl has a center of symmetry. Non-centrosymmetric crystals: (a) In the absence of an appliyed force the centeArs' of mass for positive * CoM of –ve & +ve charges shifted and negative ions coincide. (b) This situation does not change when under stress → P the crystal is strained by an applied fxorce. * Direction of P can be different from O P = 0 P those of applied force B' B A''

(a) P = 0 (b)

P

B'' Dielectric Properties 2102308 (c) 40 1

0

PbZrO3 + PbTiO3

2

0. standard piezo crystals (large d but ceramics cannot be bent) 1. most widely used: a quartz (Figs. 11.9-11.10) 2. light/flexible: polymers (Fig. 11.12)

Dielectric Properties 2102308 41 a quartz

- chemically SiO2; structurally, not amorphous, not single crystal, but ...

− a helix: helices () of distorted corner connected SiO4 tetrahedra (Fig. 11.10a) - tetrahedral unit (Fig. 11.9): internal dipoles cancel, but when force F applied, net dipole p results - quartz unit cell (Fig. 11.10): internal dipoles cancel, but when force F applied, net dipole p results

helix double helix

 a helix

Dielectric Properties 2102308 42 polymers - rely on permanent dipoles on polymer chains: strong polar bonds (C—F, C—Cl, C—N), H-bonds - example units, chains: PVF (Fig. 11.12a, c); PVDF (Fig. 11.12b, d)—to form isotactic chains (maximum dipoles) the polymers must be poled during cooldown - without poling, polymer crystallizes into centrosymmetric form (Fig. 11.14a), thus nonpiezoelectric - with poling, polymer crystallizes into non-centrosymmetric form (Fig. 11.14b), thus piezoelectric

Dielectric Properties 2102308 43 Applications Applications involve (electrical  mechanical): ultrasonic transducer, microphone, oscillator,...

Seiko Astron 35SQ World's first Quartz watch December 1969

Elastic Mechanical waves in the vibrations solid Piezoelectric transducer

A B

Oscillator Oscilloscope

Dielectric Properties 2102308 44 Ex. 7.13

-12 Given piezoelectric coefficient d = 25010 m/V and er = 1000, piezoelectric cylinder has a length of 10mm and diameter of 3mm. Spark gap is in air and has a breakdown voltage of about 3.5kV. What is the force required to spark the gap? Is the force realistic?

soln F F P = dT = d  F A A Piezoelectric P ⎯ (⎯Q= AP⎯ )→Q ⎯ (⎯Q=CV⎯ )→V V F = L Piezoelectric Piezoelectric

F F (a) (b) The piezoelectric spark generator

note: J = W.s = N.m

Dielectric Properties 2102308 45 7.8.2 Ferroelectric

- ferroelectric crystals have spontaneous or permanent polarization (even without applied field)

4+ - example: perovskite BaTiO3 where small cation (Ti ) displaced from the center of unit cell, thus internal dipole, results in increased overall stability (Fig. 7.46, next slide) - origin of ferroelectricity comes from long-range dipolar interactions (vs short-range chemical bonds between atoms). If the interaction results in parallel internal dipoles, we have ferroelectric (Fig. 11.5), but if antiparallel we have antiferroelectric materials (Fig. 11.10), Table 11.2 - the long-range dipolar interactions (resulting polarization) is reduced by increasing temperature.

At the Curie temperature TC, the dipole orientations are random and the material becomes paraelectric (Fig. 11.21) - ferroelectric is a subset of piezoelectric: it responds to external force F which causes change of polarization P (Fig. 7.47) - ferroelectric is a subset of dielectric: it responds to electric field E which causes change of polarization P. The P-E characteristic is hysteresis (Fig. 18.35). (hysteresis = “to lag bebind”) - “ferro-” in analogy to ferromagnetic (such as Fe) that posses permanent magnetization - ferroelectric, ferromagnetic are ferroic materials (that exhibit hysteresis and domain structure)

Dielectric Properties 2102308 46 BaTiO3 (Curie temperature = 130C)

c/a = 1.01 a = 4Å

For BaTiO3: er along a axis = 4,100 c axis = 160

Perovskite (ABO3) think of cubic lattice with Ba at corners of cube (sc) O at every face center (fcc) Ti at body center (bcc)

- practical ceramic ferroelectrics are polycrystalline; internal dipoles in different domains sum to zero, hence no ferroelectricity unless they are poled - Poling: manufacturing process whereby electric field is applied during crystal cooling which leads to well-

defined polarization direction at T < TCurie

Dielectric Properties 2102308 47 ferroelectric antiferroelectric

ferroelectric paraelectric

Dielectric Properties 2102308 48 Dielectric Properties 2102308 49 Ferroelectric ⊆ Piezoelectric

y P

x P

Ferroelectric ⊆ Dielectric P-E characteristic p = a  P Np

field (V/m) polarizability (F.m2)

P dipole moment (C.m) is a trademark used herein license. under herein used trademark is a ™ molecules/volume (/m3) 2

Learning  polarization (C/m ) ©2003 Brooks/Cole, a division Thomson Inc. of Learning, Thomson division a Brooks/Cole, ©2003 P = a P = a a = e (e −1) Figure 18.35 Ferroelectric hysteresis loops 0 r

Dielectric Properties 2102308 50 LiTaO pyroelectric heat detector 7.8.3 Pyroelectricity 3 “Pyro” = fire, heat - pyroelectric crystals must be noncentro-symmetric + have unique polar axis (internal dipoles spontaneously lie parallel to this axis)

- Examples i) BaTiO3 under heat yields measurable voltage proportional to heat (Fig. 7.48), due 4+ to T → Ti shifted → Ps. ii) LiTaO3 operates similarly (above). iii) wurtzite ZnS (Fig. 11.8) - the sensitivity of pyroelectric crystals is reflected by the pyroelectric coefficient p: the ratio

between the change in permanent polarization Ps to the change in temperature T Temperature change = T P p = s T

Heat

P V

Dielectric Properties 2102308 51 pyroelectric material used in human/animal intruder detector systems

Ex. For PZT, how much voltage is generated over a 0.1mm gap when the temperature change is 1mK?

Dielectric Properties 2102308 52 self-study

https://th.mouser.com/new/Kemet- Electronics/kemet-pyroelectric-sensor-modules/

Dielectric Properties 2102308 53 Dielectric Properties 2102308 54