电子元器件工艺导论第5章Ch5-notes

Ch.5 Introduction to Dielectric Nonlinearity,Definition Nonlinear: (from TheFreeDictionary) Not in a straight line. Mathematics Occurring as a result of an operation that is not linear. Containing a variable with an exponent other than one. Used of an equation. Of or relating to a system of equations whose effects are not proportional to their causes. Such a set of equations can be chaotic. Of or relating to a device whose behavior is described by a set of nonlinear equations and whose output is not proportional to its input. Of or relating to the output of such a device.,Nonlinear effect of dielectric response: dielectric constant has a nonlinear relationship with displacement, or polarization in an electric field. These dielectrics are called strong dielectrics, or high permittivity (dielectric constant) dielectrics ( 1,000), or ferroelectrics. Examples: BaTiO3, Pb(ZrTi)O3, Pb(Mn1/3Nb2/3)O3-PbTiO3. 2. Review on famous ferroelectric materials BaTiO3 is a perovskite ferroelectrics, it is a representative of various ferroelectrics.,a. History 1942-46: Dielectric anomaly by Wainer and Solomon in USA (42); Ogawa in Japan (44); Wul Goldman in USSR (46) 1944-46: Ferroelectricity by von Hippel (M.I.T.) (44); Wul (46) 1945: Phase transition by Megaw 1945-46: Piezoelectricity by Gary (46, patent, Erie Resistor); Howatt (45) 1948: Domain observation by polarized light 1949: Phenomenological theory by Devonshire 1951: Microdomainpolar region (10-100 nm) 1953: Phase diagram of BaO-TiO2 (D. E. Rase and R. Roy) 1954: Surface layer in very small BTO particles 1954: Positive temperature coefficientPTC effect in doped BTO by Haayman (Philips) 1955: Surface layer in BTO crystal 1956: Crystal growth by flux method 1956: Pyroelectricity 1957: Electrostriction 1958: Birefrigence 1059/61: Soft mode theory by Cochran and Anderson 1962: PTC thermistor commercialized by Murata 1964: Linear electro-optic effect 1964: Second harmonic generation 1967: Grain boundary capacitor by Waku,Now, BaTiO3 is still widely used in high dielectric constant capacitors, such as X7R.,b. Dielectric behaviors in an electric field. 4 phases: Rhombohedral Orthorhombic Tetragonal Cubic Thermal hysteresis Permittivity differs along a and c axis,5.1,Polarization versus temperature for BaTiO3 4 phases: Rhombohedral Orthorhombic Tetragonal Cubic Typical 1st order phase transition Coercive field changes with phase transition,5.2,W. J. Merz, Phys. Rev. 91, 513 (1953) L. E. Cross, Phil. Mag. 44, 1161 (1953) M. E. Drougard, J. Appl. Phys. 27, 1559 (1956),Polarization versus electric field above Curie temperature double hysteresis loops,5.3,c. Domain structure of 90 and 180 domains,5.4,d. Applications Capacitors BaTiO3, SrTiO3, Ba(SnTi)O3, Ba(ZrTi)O3, PMN-PT, Sensors & Heaters (BaPb)TiO3 + La2O3, Y2O3, Nb2O5 + SiO2 (BaSr)TiO3 + La2O3, Y2O3, Nb2O5 + SiO2 Memories BaTi0.91(Hf0.5,Zr0.5)0.09O3: low Ec, high Pr, etc Pyroelectric devices (BaSr)TiO3 RF and MW devices (BaSr)TiO3,3. Nonlinear behaviors in ferroelectrics 3.1. Shift of Curie temperature in a high field,Fig. 5.5. Permittivity as a function of temperature and d.c. bias field at 1 kHz for a heating circle. EDC=0, 10, 20, 30, 40, 50 and 100 MV/m. Inset shows the perk temperature change versus d.c. bias field.,Considering the Gibbs energy (1) When the ferroelectric is placed in an electric field, then one has . (2),And,. (3) For higher fields, the nonlinear term is dominant, the -1 may be written as . (4) For a first-order phase transition, the ferroelectric phase may be induced by E, above Tc. The shift of Tc may be calculated from the Clausius-Claperon equation, . (5) Considering S=-1/2D2, and letting D=D, then . (6) or . (7) For a second-order phase transition, differentiating Eqs. (2) and (3), we have . (8),And . (9) At T=Tpeak, -1 is an extreme, then . (10) Substituting it into Eq. (2), . (11) For both the 1st and 2nd order phase transitions, the shift of permittivity peak temperature is proportional to E2/3. 3.2. C-V characteristics For unstressed crystals, the free energy density can be expressed in terms of even powers of the displacement (D) developed by the external field, i.e. . (12),where G(T, 0) - zero-field free energy density, T - absolute temperature, C and T0 are Curie-Weiss constant and temperature B(T) - phenomenological constant 0 - the permittivity of vacuum. The electric field is then given by . (13) The incremental dielectric constant r(T, E) is approximately given by . (14) where r(T, 0) = C/(T-T0) is the incremental dielectric constant in the absence of a d.c. electric field. In Eq. (14), it is assumed the r(T, E) 1, which is the case for most ferroelectric materials. For a second-order phase transition, since B is positive, then when the electric field increases, the dielectric constant decreases, which is consistent with the experimental results.,Using Eqs. (13) and (14), Johnson proposed an expression for the dielectric constant in terms of the electric field. Considering first the case of weak electric fields, then the terms in D4 or greater can be neglected. Since D = 0rE, then Eq. (14) can be written as . (15) Eq. (14) can be further expressed as . (16) Because for a r(T,E)/0(E,0) = 0.58, the numerator has a maximum deviation from unity by only 15%, so we can further assume that the numerator to be unity, therefore Eq. (16) becomes . (17) 3.3. Discussion (a) High field situation Eq. (17) was obtained under an assumption that the ferroelectric was applied with a low electric field. In a high field, although the dielectric constant would go as E1/3, the,dipoles would not only align but could also distort, without saturation, so Eq. (17) seemed quite suitable to represent the dielectric nonlinearity. (b) r-E relations at different temperatures Assuming that B(T) is temperature independent, then an increase in temperature reduces r(T, 0) according to the Curie-Weiss law. Consequently, the width of the function in Eq. (17) is increased, so reducing the tunability in the practical range of electric fields. (c) Why BST has a large tunability In general, the phenomenological coefficient B(T) has a small variation from material to material. For bulk BST, it is 3.7-4.2 10-18 (cmV-1)2 at 77 K, 4.4 10-18 (cmV-1)2 at room temperature, while for PbTiO3, is 2.0 10-18 (cmV-1)2 at room temperature. Outzourhit found that B(T) had the same order of magnitude for both BST (Sr=0.90) thin film and BST bulk material. Hence, the tunability will be dominated by the zero bias dielectric constant. Lets look at the room temperature dielectric constant of different materials as a comparison. BaTiO3 crystal, 4,000 (a-axis); BaTiO3 ceramic, 1500; SrTiO3 crystal, 13,000; BST ceramic (Sr=0.3), 5,000; BST ceramic (Sr=0.4), 5,000; BST ceramic (Sr=0.5), 2,500 5,000; PZT ceramic, 200. One can see that because the BST system has a large zero bias dielectric constant, so they have a big tunability compared with other materials. Moreover, usually the dielectric constant has a maximum value at the phase transition temperature, so the tunability also has a maximum value at this point.,S. G. Lu, et al. Appl. Phys. Lett. 82 (17), 2877 (203),Fig. 5.7 Tunability as a function of temperature at 1 kHz and at 50 and 100 MV/m for P(VDF-TrFE) 55/45 copolymer and at 50 MV/m for P(VDF-TrFE-CFE) 64.3/27.6/8.1 terpolymer. S. G. Lu et al. Appl. Phys. Lett. 93, 042905 (2008),5.6,Fig.5.8 Permittivity as a function of electric field for P(VDF-TrFE) 55/45 mol% copolymer at 1 kHz. Eq. 17 was used for fitting.,4. Domain wall movementRayleigh law 4.1. Dielectric and piezoelectric properties: Intrinsic: ferroelectric domains Extrinsic: domain wall motion Phase boundary movement Defect response 4.2. Contribution to dielectric constant from domain wall: Soft piezoelectric: 60 % Hard piezoelectric: 40 % Q. M. Zhang, W. Y. Pan, S. J. Jang, and L. E. Cross, J. Appl. Phys. 64, 6445 (1988). Rayleigh law was firstly proposed for formulizing the hysteresis loop of ferromagnetic materials at low fields. The mathematic formulae are expressed below , (18) , (19) where - susceptibility init - initial susceptibility - Rayleigh constant H - the magnetic field M magnetization H0 - magnitude if a.c. sinusoidal magnetic signal,Similarly, the Rayleigh law for ferroelectrics can be written as , (20) . (21) Here - permittivity init - initial permittivity - Rayleigh constant E - the electric field P polarization E0 - magnitude if a.c. sinusoidal electrical signal.,Fig.5.9 Permittivity as a function of a.c. field for 0, 1 and 3 at% Fe doped PZT 30/70 thin films ( 60 nm).,Fig. 5.10 Minor P-E loop for PZT 30/70 thin films ( 60 nm). Calculation is based on Eqs. (20) and (21).,For Rayleigh law, when applying a sinusoidal function E = E0sin(t), Eq. (21) can be expanded in Fourier series, yielding . (22) a) Amplitude of 3rd and 5th harmonic response E02 b) Phase angle between 3rd and 1st response 90,4.3 Polarization and phase angle versus a.c. electric field,Fig. 5.11 3rd harmonic polarization response as a function of a.c. field for 0, 1 and 3 at% Fe doped PZT 30/70 thin films ( 60 nm).,Fig. 5.12 1st, 2nd, and 3rd harmonic phase angles as a function of a.c. electric field for 0, 1 and 3 at% Fe doped PZT thin films.,4.4 Summary Definition of nonlinear effect in dielectrics. D2 + BaTiO3: history, phase transition, domain structure, high field p-E loops, and applications. Nonlinear effect in ferroelectrics C-V characteristics & tunability Tc E2/3 Domain wall motion & Rayleigh law,Ch.6 Phenomenological Theory of Ferroelectrics,1. Definition Phenomenological Theory. A theory which expresses mathematically the results of observed phenomena without paying detailed attention to their fundamental significance. Thewlis, J. (Ed.) (1973). Concise Dictionary of Physics. Oxford: Pergamon Press, p. 248. a) History. Landau theory of second order phase transitions (1936). He was motivated to suggest that the free energy of any system should obey two conditions: that the free energy is analytic, and that it obeys the symmetry of the Hamiltonian. Given these two conditions, one can write down (in the vicinity of the critical temperature, Tc) a phenomenological expression for the free energy as a Taylor expansion in the order parameter. For example, the Ising model free energy may be written as the following:,. (6.1) where the parameter (6.2) .,where the parameter,for physical reasons. The variable is the coarse-grained field of spins, known as the order parameter or the total magnetization. b) Devonshires theory on BaTiO3 (1949). Similar to Landaus approach, the free energy as a function of order parameter polarization and stress with the stresses equated to zero for barium titanate crystal can be expressed as (6.3) The first derivatives of G with respect to Px give the field-components for the free crystal (6.4) here =(T-T0), 110, and 0.,Further differentiating above equation, we have , (6.5) , (6.6) , (6.7) , (6.8) , (6.9) , (6.10),Basic assumptions. There are some features for ferroelectrics should be mentioned: Physical properties are associated with phase transition. Some physical parameters occur anomalies at the phase transition temperature, e.g. specific heat, dielectric constant, susceptibility, etc. Illustration is possible. Taylor expansion can be used to describe the phenomena. Similar to Landaus assumptions, two postulations were made, Relevant equation of state can be represented by a polynomial; and The same polynomial can be used above and below the phase transition anomaly.,2. General expression Assuming that Di=D is directed along one of the crystallographic axes only (i.e. that spontaneous polarization occurs along this direction and the applied field is restricted to this direction also), that all stresses are zero, and that the non-polar phase is centrosymmetric, the free energy G1 can be expressed (6.11) where energy is measured from the non-polar phase and we terminate the polynomial rather arbitrarily at D6 for mathematical simplicity. The coefficients =(T-Tc), and are temperature independent for 1st and 2nd order phase transition. 0. The order of the transition depends on the sign of , 0 for 2nd order. 3. 2nd order phase transition Since both 0 and 0, the D6 may be omitted. G1 may be written as (6.12),Differentiating Eq. (6.12) with respect to D at constant temperature generates the dielectric equation of state (6.13) Where =(T-Tc), E is the electric field. Further one may obtain (6.14) Then when TTc, D=0, Eq. (6.14) becomes . (TTc) (6.15) This is the Curie-Weiss law. Curie constant C is 105 or 103. For instance, C=3.0 105 for Pb(ZrxTi1-x)O3 (0.0x0.6). When TTc, E=0, Eq. (6.13) becomes (6.16),Then (6.17),Fig. 6.1 Qualitative temperature dependence of the spontaneous polarization (Ps) near a 2nd-order ferroelectric paraelectric phase transition.,Fig. 6.2 Polarization versus temperature for P(VDF-TrFE) 55/45 mol% copolymers.,Substituting Eq. (6.17) into Eq. (6.14), we have . (TTc) (6.18),Question: How to obtain the phenomenological coefficients and ?,Fig. 6.3 Qualitative temperature dependence of the reciprocal isothermal permittivity () near a 2nd-order ferroelectric paraelectric phase transition.,Substituting Eq. (7) into Eq. (2), one may obtain (6.19),We can also qualitatively plot the G1 as a function of T under different conditions. Let us discuss Eq. (3). When T=Tc, Eq. (3) becomes (6.20),Fig. 6.4 Gibbs free energy as a function of temperature at TTc.,Fig. 6.5 Gibbs free energy versus displacement for different T.,It is a curve passing through the origin. At enough high E, the D will saturate. See below #2. When T Tc, according to Eq. (6.14), we have (6.21) D versus E is a straight line passing through the origin. At enough high E, the D will saturate. See below #3. When TTc, Eq. (6.13) has solutions which are centrosymmetric. They form the D-E loop that is sketched in below #1.,Fig. 6.6 Schematic drawing of polarization versus E for different T.,In higher electric fields, D versus T demonstrates characteristics which are different from Eq. (6.17) derived ones. Below figure shows an example. Finally, we will simply discuss the entropy and specific heat. In accordance with the definition, entropy S can be expressed as (6.22) Then the zero-field entropy can be obtained (6.23),Fig. 6.7 D versus T with different E fields for P(VDF-TrFE) 55/45 mol% copolymers.,Substituting Eq. (6.17) into Eq. (6.23), one can obtain , (TTc) (6.25) Since Ps goes continuously from zero, S=0 at Tc, no latent is involved. The corresponding contributions TdS/dT to zero-field specific heat are , (TTc) (6.27) It should be noticed that the first derivatives of the free energy (Ps, and S) are continuous, while second derivatives (e.g. c) are discontinuous, which conforms with the definition of a second-order phase transition.,It is noted that S is continuous at Tc, but that c is discontinuous with a jump c=1/2Tc/., Electrocaloric effect Electrocaloric effect is the entropy change and temperature change of a material which results from an adiabatic application of an electric field. Following two equations may be used to estimate the entropy change and temperature change (6.28) (6.29) Here, cE is the specific heat. For BaTiO3, b =6.7E+5 (K-1) 1, P=0.26 (C/m2) 2, then S = 2.26E+4 (J/m3K). Considering the specific heat cE=3.05E+6 J/m3K 3, Tc=107 C 2, then T = 2.8 C.,For Pb(ZrxTi1-x)O3 (0.0x