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16. Dielectrics and Ferroelectrics Maxwell Equations Polarization Macroscopic Electric Field Depolarization Field, E1 Local Electric Field at An atom Lorentz Field, E2 Field of Dipoles Inside Cavity, E3 Dielectric Constant And Polarizability Electronic Polarizability Classical Theory Examples Structural Phase Transitions Ferroelectric Crystals Classification of Ferroelectric Crystals Displacive Transitions Soft Optical Phonons Landau theory of the Phase Transition Second-Order Transition First-Order Transition Antiferroelectricity Ferroelectric Domains Piezoelectricity Maxwell Equations B 0 E 4 B 4 D E H J c t c c t Polarization Polarization P dipole moment per unit volume. Total dipole moment p qn rn n For a neutral system, p is independent of the choice of the coordinate origin. H2 O 3p r r r 2 p Dipole field: E r r5 Macroscopic Electric Field E0 = external (applied) field e(r) = microscopic field 1 Macroscopic field E r0 dV e r VC = volume of crystal cell VC E due to a volume of uniform P is equal to that due to a surface charge density nP ˆ For any point between the plates & far from the edges E1 4 4 P E E0 E1 E0 4 P z ˆ E1 = field due to ζ = n P. Depolarization Field, E1 If P is uniform, then E E0 E1 E1 = depolarization field Inside an ellipsoidal body, P uniform → E1 uniform Along the principal axes E1 j N j Pj Nj = depolarization factor Nj j 4 ellipsoid of revolution PE χ = dielectric susceptibility Along a principal axis of an ellipsoid: E E0 E1 E0 N P → P E0 N P E0 1 N Local Electric Field at an Atom In general, E Eloc. Consider a simple cubic crystal of spherical shape. 4 The macroscopic field in a sphere is E E0 E1 E0 P 3 If all dipoles are equal to p p z ˆ the dipole field at the center of the sphere is 3z 2 rj2 2 z 2 x2 y 2 Edipole p p j j j j j rj5 j rj5 x2 y2 z2 Cubic symmetry → r j j 5 j r j 5 j r j 5 → E dipole 0 → Eloc E0 E j j j For an abitrary symmetry Eloc E0 E1 E2 E3 E2 = Lorentz cavity field (due to charges on surface of cavity) E3 = field of atoms inside cavity E1 + E2 = field of body with hole. E1 + E2 + E3= field of all other atoms at one atom. 3 p j r j r j rj2 p j E1 E2 E3 j 0 rj5 Sites 10a (~50A) away can be replaced by 2 surface integrals: 1 over the outer ellipsoidal surface, the other over the cavity defining E2 . Lorentz Field, E2 P cos E2 2 asin a d 2 cos 0 a 4 P E1 3 E1 E2 0 Field of Dipoles inside Cavity, E3 E3 is only field that depends on crystal structure. For cubic crystals E3 0 4 4 → Eloc E0 E1 P E P Lorentz relation 3 3 Dielectric Constant and Polarizability E 4 P For an isotropic / cubic medium, ε is a scalar: 1 4 E P DE 1 E 4 E 4 For a non-cubic medium, ε & χ are tensors: P E 4 Polarizability α of an atom: p Eloc α is in general a tensor Polarization: P N j p j N j j Eloc j j j 4 For cubic medium, Lorentz relation applies : P j N j j E P 3 N j j 1 4 N j 4 → j Clausius-Mossotti relation N 2 3 j 1 j j j 3 j Electronic Polarizability Dipolar: re-orientation of molecules with permanent dipoles Ionic: ion-ion displacement Electronic: e-nucleus displacement In heterogeneous materials, there is also an interfacial polarization. At high frequencies, electronic contribution dominates. 1 4 n2 1 e.g., optical range: 2 3 N j j j electronic 2 n 2 Classical Theory of Electronic Polarizability Bounded e subject to static Eloc : m x 0 x eEloc 2 e Eloc Steady state: x m 0 2 p e x e2 Static electronic polarizability: el Eloc Eloc m 02 Bounded e subject to oscillatory Eloc : m x 0 x eEloc sin t 2 e Eloc Oscillatory solution : x x0 sin t x0 m 2 0 2 e2 Electronic polarizability: el m 0 2 2 e2 fi j Quantum theory: el m j 2 2 f i j = oscillator strength of dipole ij transition between states i & j. Structural Phase Transitions At T = 0, stable structure A has lowest free energy F = U for a given P. High P favors close-packing structures which tend to be metallic. E.g., H & Xe becom metallic under high P. Let B has a softer (lower ω) phonon spectrum than A. → SB > SA due to greater phonon occupancy for B. → TC s.t. FB = UB –T SB > FA = UA –T SA T > TC ( phase transition A → B unless TC > Tmelt ) FB (TC ) = FA (TC ) Near TC , transition can be highly stress sensitive. Ferroelectrics: spontaneous P. • Unusual ε(T). • Piezoelectric effect. • Pyroelectric effect. • Electro-optical effects such as optical frequency doubling. Ferroelectric Crystals Ferroelectric TC → Paraelectric Ferroelectric : P vs E plot shows hysteresis. Pyroelectric effects (P T ) are often found in ferroelectrics where P is not affected by E less than the breakdown field. E.g., LiNbO3 is pyroelectric at 300K. High TC = 1480K. Large saturation P = 50 μC/cm2 . Can be “poled” (given remanent P by E at T >1400K). PbTiO3 Classification of Ferroelectric Crystals 2 main classes of ferroelectrics: • order-disorder: soft (lowest ωTO ) modes diffusive at transition. e.g., system with H-bonds: KH2PO4 . Most are in between • displacive: soft modes can propagate at transition. e.g., ionic crytsls with perovskite, or ilmenite structure. Order-disorder TC nearly doubled on H→D. Due to quantum effect involving mass-dependent de Broglie wavelength. n-diffraction → for T < TC , H+ distribution along H-bond asymmetric. Displacive T > TC T < TC : displaced BaTiO3 At 300K, PS = 8104 esu cm–2 . → p 510–18 esu cm VC = (4 10–8 )3 = 64 10–24 cm3. Moving Ba2+ & Ti4+ w.r.t. O2– by δ = 0.1A gives p /cell = 6e δ 310–18 esu cm In LiNbO3, δ is 0.9A for Li & 0.5A for Nb → larger p. Displacive Transitions 2 viewpoints on displacive transitions: • Polarization catastrophe ( Eloc caused by u is larger than elastic restoring force ). • Condensation of TO phonon (t-indep displacement of finite amplitude) Happens when ωTO = 0 for some q 0. ωLO > ωTO & need not be considered . In perovskite structures, environment of O2– ions is not cubic → large Eloc. → displacive transition to ferro- or antiferro-electrics favorable. Catastophe theory: Let Eloc = E + 4 π P / 3 at all atoms. In a 2nd order phase transition, there is no latent heat. The order parameter (P) is continuous at TC . 8 1 3 N j j 3 Nj j j C-M relation: 4 Catastophe condition: 4 1 3 N j j j j 8 4 1 3 N j j 3 6s 1 N j j 1 3s j → 4 for s → 0 3 j 1 3 N j j j 3s s T TC s → (paraelectric) T TC Soft Optical Phonons TO 2 ωTO → 0 ε(0) → LST relation LO 0 2 no restoring force: crystal unstable E.g., ferroelectric BaTiO3 at 24C has ωTO = 12 cm–1 . 1 TO T TC T TC → if ωLO is indep of T 2 Near TC , 0 SrTiO3 SbSI from n scatt from Raman scatt Landau Theory of the Phase Transition Landau free energy density: 1 F P; T , E E P g 0 g 2 j P 2 j E P g 1 g P 2 1 g P 4 1 g P 6 j 1 2j 0 2 2 4 4 6 6 Comments: • Assumption that odd power terms vanish is valid if crystal has center of inversion. • Power series expansion often fails near transition (non-analytic terms prevail) . e.g., Cp of KH2PO4 has a log singularity at TC . The Helmholtz free energy F(T, E) is defined by 0 F P; T , E E g2P g4P3 g6P5 Transition to ferroelectric is facilitated by setting g2 T T0 0 , T0 TC (This T dependence can be explained by thermal expansion & other anharmonic effects ) g2 ~ 0+ → lattice is soft & close to instability. g2 < 0 → unpolarized lattice is unstable. Second-Order Transition 0 E T T0 P g4P3 g6P5 For g4 > 0, terms g6 or higher bring no new features & can be neglected. E=0 → 0 T T0 P g4P3 → PS = 0 or PS2 T0 T g4 Since γ , g4 > 0, the only real solution when T > T0 , is PS = 0 (paraelectric phase). This also identifies T 0 with TC . For T < T0 , PS T0 T minimizes F ( T, 0 ) (ferroelectric phase). g4 LiTaO3 First-Order Transition 0 E T T0 P g4P3 g6P5 For g4 < 0, the transition is 1st order and term g6 must be retained. E=0 → 0 T T0 P g4 P3 g6P5 1 → PS = 0 or PS2 g4 g 4 4 g 6 T T0 2 2 g6 BaTiO3 (calculated) For E 0 & T > TC , g4 & higher terms can be neglected: E T T0 P 4 P 4 1 1 T0 = TC for 2nd order trans. E T T0 T0 < TC for 1st order trans. Fundamental types of structural phase transitions from a centrosymmetric prototpe Perovskite Lead zirconate-lead titanate (PZT) system Widely used as ceramic piezoelectrics. Ferroelectric Domains Atomic displacements Domains with of oppositely polarized 180 walls domains. BaTiO face c axis. Ea // c axis. Piezoelectricity Ferroelectricity → Piezoelectricity (not vice versa) BaTiO3 : d = 10−5 cm/statvolt Unstressed Pd E ζ = stress (tensor) e d = piezoelectric constant (tensor) di e sd E χ = dielectric susceptibility Ei e = elastic compliance constant (tensor) α = 1,∙∙∙, 6 A+3 B3− PiezoE not FerroE e.g., SiO2 d 10−7 cm/statvolt Unstressed: 3-fold symmetry PVF2 films are flexible & often used as ultrsonic transducers