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Lecture 18, November 5, 2009 Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday William A. Goddard, III, [email protected] WCU Professor at EEWS-KAIST and Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Senior Assistant: Dr. Hyungjun Kim: [email protected] Manager of Center for Materials Simulation and Design (CMSD) Teaching Assistant: Ms. Ga In Lee: [email protected] Special assistant: Tod Pascal:[email protected] EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 1 Schedule changes TODAY Nov. 5, Thursday, 9am, L18, as scheduled Nov. 9-13 wag lecturing in Stockholm, Sweden; no lectures, Nov. 17, Tuesday, 9am, L19, as scheduled Nov. 18, Wednesday, 1pm, L20, additional lecture room 101 Nov. 19, Thursday, 9am, L21, as scheduled Nov. 24, Tuesday, 9am, L22, as scheduled Nov. 26, Thursday, 9am, L23, as scheduled Dec. 1, Tuesday, 9am, L24, as scheduled Dec. 2, Wednesday, 3pm, L25, additional lecture, room 101 Dec. 3, Thursday, 9am, L26, as scheduled Dec. 7-10 wag lecturing Seattle and Pasadena; no lectures, Dec. 11, Friday, 2pm, L27, additional lecture, room 101 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 2 Last time EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 3 The configuration for C2 Si2 has this configuration 1 1 2 4 4 4 1 2 3 2 2 2 From 1930-1962 the 3Pu was thought to be the ground state 2 2 1S + is ground state EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved Now 4 Ground state of C2 MO configuration Have two strong p bonds, but sigma system looks just like Be2 which leads to a bond of ~ 1 kcal/mol The lobe pair on each Be is activated to form the sigma bond. The net result is no net contribution to bond from sigma electrons. It is as if we started with HCCH and cut off the Hs EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 5 Low-lying states of C2 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 6 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 7 London Dispersion The universal attractive term postulated by van der Waals was explained in terms of QM by Fritz London in 1930 The idea is that even for spherically symmetric atoms such as He, Ne, Ar, Kr, Xe, Rn the QM description will have instantaneous fluctuations in the electron positions that will lead to fluctuating dipole moments that average out to zero. The field due to a dipole falls off as 1/R3 , but since the average dipole is zero the first nonzero contribution is from 2nd order perturbation theory, which scales like -C/R6 (with higher order terms like 1/R8 and 1/R10) Consequently it is common to fit the interaction potentials to functional froms with a long range 1/R6 attraction to account for London dispersion (usually refered to as van der Waals attraction) plus a short range repulsive term to acount for short Range Pauli Repulsion) EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 8 Noble gas dimers s Ar2 Re De EEWS-90.502-Goddard-L15 LJ 12-6 E=A/R12 –B/R6 = De[r-12 – 2r-6] = 4 De[t-12 – t-6] r= R/Re t= R/s where s = Re(1/2)1/6 =0.89 Re © copyright 2009 William A. Goddard III, all rights reserved 9 Remove an electron from He2 Ψ(He2) = A[(sga)(sgb)(sua)(sub)]= (sg)2(su)2 Two bonding and two antibonding BO= 0 Ψ(He2+) = A[(sga)(sgb)(sua)]= (sg)2(su) BO = ½ Get 2Su+ symmetry. Bond energy and bond distance similar to H2+, also BO = ½ EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 10 The ionic limit At R=∞ the cost of forming Na+ and Clis IP(Na) = 5.139 eV minus EA(Cl) = 3.615 eV = 1.524 eV But as R is decreased the electrostatic energy drops as DE(eV) = - 14.4/R(A) or DE (kcal/mol) = -332.06/R(A) Thus this ionic curve crosses the covalent curve at R=14.4/1.524=9.45 A Using the bond distance of NaCl=2.42A leads a coulomb energy of 6.1eV leacing to a bond of6.1-1.5=4.6 eV The exper De = 4.23 eV Showing that ionic character dominates EEWS-90.502-Goddard-L15 E(eV) © copyright 2009 William A. Goddard III, all rights reserved R(A) 11 GVB orbitals of NaCL Dipole moment = 9.001 Debye Pure ionic 11.34 Debye Thus Dq=0.79 e EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 12 electronegativity To provide a measure to estimate polarity in bonds, Linus Pauling developed a scale of electronegativity where the atom that gains charge is more electronegative and the one that loses is more electropositive He arbitrary assigned χ=4 for F, 3.5 for O, 3.0 for N, 2.5 for C, 2.0 for B, 1.5 for Be, and 1.0 for Li and then used various experiments to estimate other cases . Current values are on the next slide Mulliken formulated an alternative scale such that χM= (IP+EA)/5.2 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 13 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 14 The NaCl or B1 crystal All alkali halides have this structure except CsCl, CsBr, Cs I (they have the B2 structure) EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 15 The CsCl or B2 crystal There is not yet a good understanding of the fundamental reasons why particular compound prefer particular structures. But for ionic crystals the consideration of ionic radii has proved useful EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 16 Ionic radii, main group Fitted to various crystals. Assumes O2- is 1.40A From R. D. Shannon, Acta©Cryst. 751 (1976) EEWS-90.502-Goddard-L15 copyrightA32, 2009 William A. Goddard III, all rights reserved 17 Ionic radii, transition metals EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 18 Role of ionic sizes Assume that the anions are large and packed so that they contact, so that 2RA < L, where L is the distance between then Assume that the anion and cation are in contact. Calculate the smallest cation consistent with 2RA < L. RA+RC = L/√2 > √2 RA RA+RC = (√3)L/2 > (√3) RA Thus RC/RA > 0.414 Thus RC/RA > 0.732 Thus for 0.414 < (RC/RA ) < 0.732 we expect B1 For (RC/RA ) > 0.732 either is ok. For (R /R ) < 0.732 must be2009 some structure © copyright William other A. Goddard III, all rights reserved EEWS-90.502-Goddard-L15 C A 19 Radiius Ratios of Alkali Halides and Noble metal halices Rules work ok B1: 0.35 to 1.26 B2: 0.76 to 0.92 Based on R. W. G. Wyckoff, Crystal Structures, 2nd edition. Volume 1 (1963) EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 20 Wurtzite or B4 structure EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 21 Sphalerite or Zincblende or B3 structure EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 22 Radius rations B3, B4 The height of the tetrahedron is (2/3)√3 a where a is the side of the circumscribed cube The midpoint of the tetrahedron (also the midpoint of the cube) is (1/2)√3 a from the vertex. Hence (RC + RA)/L = (½) √3 a / √2 a = √(3/8) = 0.612 Thus 2RA < L = √(8/3) (RC + RA) = 1.633 (RC + RA) Thus 1.225 RA < (RC + RA) or RC/RA > 0.225 Thus B3,B4 should be the stable structures for 0.225 < (RC/RA) < 0. 414 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 23 Structures for II-VI compounds B3 for 0.20 < (RC/RA) < 0.55 B1 for 0.36 < (RC/RA) < 0.96 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 24 CaF2 or fluorite structure Like GaAs but now have F at all tetrahedral sites Or like CsCl but with half the Cs missing Find for RC/RA > 0.71 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 25 Rutile (TiO2) or Cassiterite (SnO2) structure Related to NaCl with half the cations missing EEWS-90.502-Goddard-L15 Find for RC/RA < 0.67 © copyright 2009 William A. Goddard III, all rights reserved 26 CaF2 rutile CaF2 rutile EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 27 Electrostatic Balance Postulate For an ionic crystal the charges transferred from all cations must add up to the extra charges on all the anions. We can do this bond by bond, but in many systems the environments of the anions are all the same as are the enviroments of the cations. In this case the bond polarity (S) of each cation-anion pair is the same and we write S = zC/nC where zC is the net charge on the cation and nC is the coordination number Then zA = Si SI = Si zCi /ni Example1 : SiO2. in most phases each Si is in a tetrahedron of O2- leading to S=1. Thus each O2- must have just two Si neighbors EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 28 Some old some New material EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 29 More examples electrostatic balance Example 2. The stishovite phase of SiO2 has six coordinate Si, leading to S=2/3. Thus each O must have 3 Si neighbors Example 3: the rutile, anatase, and brookhite phases of TiO2 all have octahedral T. Thus S= 2/3 and each O must be coordinated to 3 Ti. Example 4. Corundum (a-Al2O3). Each Al3+ is in a distorted octahedron, leading to S=1/2. Thus each O2must be coordinated to 4 Al Example 5. Olivine. Mg2SiO4. Each Si has four O2- (S=1) and each Mg has six O2- (S=1/3). Thus each O2- must be coordinated to 1 Si and 3 Mg neighbors EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 30 Illustration, BaTiO3 A number of important oxides have the perovskite structure (CaTiO3) including BaTiO3, KNbO3, PbTiO3. Lets try to predict the structure without looking it up Based on the TiiO2 structures , we expect the Ti to be in an octahedron of O2-, STiO = 2/3. The question is how many Ti neighbors will each O have. It cannot be 3 since there would be no place for the Ba. It is likely not one since Ti does not make oxo bonds. Thus we expect each O to have two Ti neighbors, probably at 180º. This accounts for 2*(2/3)= 4/3 charge. The Ba must provide the other 2/3. Now we must consider how many O are around each Ba, nBa, leading to SBa = 2/nBa, and how many Ba around each O, nOBa. Since nOBa* SBa = 2/3, the missing charge for the O, we have 31 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved only a few possibilities: Prediction of BaTiO3 structure nBa= 3 leading to SBa = 2/nBa=2/3 leading to nOBa = 1 nBa= 6 leading to SBa = 2/nBa=1/3 leading to nOBa = 2 nBa= 9 leading to SBa = 2/nBa=2/9 leading to nOBa = 3 nBa= 12 leading to SBa = 2/nBa=1/6 leading to nOBa = 4 Each of these might lead to a possible structure. The last case is the correct one for BaTiO3 as shown. Each O has a Ti in the +z and –z directions plus four Ba forming a square in the xy plane The Each of these Ba sees 4 O in the xy plane, 4 in the xz plane and 4 in the yz plane. EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 32 BaTiO3 structure (Perovskite) EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 33 New material EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 34 How estimate charges? We saw that even for a material as ionic as NaCl diatomic, the dipole moment a net charge of +0.8 e on the Na and -0.8 e on the Cl. We need a method to estimate such charges in order to calculate properties of materials. First a bit more about units. In QM calculations the unit of charge is the magnitude of the charge on an electron and the unit of length is the bohr (a0) Thus QM calculations of dipole moment are in units of ea0 which we refer to as au. However the international standard for quoting dipole moment is the Debye = 10-10 esu A Where m(D) = 2.5418 m(au) EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 35 Fractional ionic character of diatomic molecules Obtained from the experimental dipole moment in Debye, m(D), and bond distance R(A) by dq = m(au)/R(a0) = C m(D)/R(A) where C=0.743470. Postive 36 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved that head of column is negative Charge Equilibration First consider how the energy of an atom depends on the net charge on the atom, E(Q) Including terms through 2nd order leads to Charge Equilibration for Molecular Dynamics Simulations; A. K. Rappé and W. A. Goddard III; J. Phys. Chem. 95, 3358 (1991) (2) EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved (3) 37 Charge dependence of the energy (eV) of an atom E=12.967 Harmonic fit E=0 E=-3.615 Cl+ Q=+1 Cl Q=0 EEWS-90.502-Goddard-L15 ClQ=-1 = 8.291 Get minimum at Q=-0.887 Emin = -3.676 = 9.352 © copyright 2009 William A. Goddard III, all rights reserved 38 QEq parameters EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 39 Interpretation of J, the hardness Define an atomic radius as RA0 Re(A2) Bond distance of homonuclear H 0.84 0.74 diatomic C 1.42 1.23 N 1.22 1.10 O 1.08 1.21 Si 2.20 2.35 S 1.60 1.63 Li 3.01 3.08 Thus J is related to the coulomb energy of a EEWS-90.502-Goddard-L15 2009 William A. Goddard III, all rights reserved charge the size ©ofcopyright the atom 40 The total energy of a molecular complex Consider now a distribution of charges over the atoms of a complex: QA, QB, etc Letting JAB(R) = the Coulomb potential of unit charges on the atoms, we can write Taking the derivative with respect to charge leads to the chemical potential, which is a function of the charges or The definition of equilibrium is for all chemical potentials to be equal. This leads to © copyright 2009 William A. Goddard III, all rights reserved EEWS-90.502-Goddard-L15 41 The QEq equations Adding to the N-1 conditions The condition that the total charged is fixed (say at 0) Leads to the condition Leads to a set of N linear equations for the N variables QA. We place some conditions on this. The harmonic fit of charge to the energy of an atom is assumed to be valid only for filling the valence shell. Thus we restrict Q(Cl) to lie between +7 and -1 and for C to be between +4 and -4 Similarly Q(H) is between +1 and -1 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 42 The QEq Coulomb potential law We need now to choose a form for JAB(R) A plausible form is JAB(R) = 14.4/R, which is valid when the charge distributions for atom A and B do not overlap Clearly this form as the problem that JAB(R) ∞ as R 0 In fact the overlap of the orbitals leads to shielding The plot shows the shielding for C atoms using various Slater orbitals And l = 0.5 EEWS-90.502-Goddard-L15 Using RC=0.759a0 © copyright 2009 William A. Goddard III, all rights reserved 43 QEq results for alkali halides EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 44 QEq for Ala-His-Ala Amber charges in parentheses EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 45 QEq for deoxy adenosine Amber charges in parentheses EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 46 QEq for polymers Nylon 66 PEEK EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 47 Perovskites Perovskite (CaTiO3) first described in the 1830s by the geologist Gustav Rose, who named it after the famous Russian mineralogist Count Lev Aleksevich von Perovski crystal lattice appears cubic, but it is actually orthorhombic in symmetry due to a slight distortion of the structure. Characteristic chemical formula of a perovskite ceramic: ABO3, A atom has +2 charge. 12 coordinate at the corners of a cube. B atom has +4 charge. Octahedron of O ions on the faces of that cube centered on a B ions at the center of the cube. Together A and B form an FCC structure GODDARD - Ch120a Ferroelectrics Nov 21, 2005 48 The stability of the perovskite structure depends on the relative ionic radii: if the cations are too small for close packing with the oxygens, they may displace slightly. Since these ions carry electrical charges, such displacements can result in a net electric dipole moment (opposite charges separated by a small distance). The material is said to be a ferroelectric by analogy with a ferro-magnet which contains magnetic dipoles. Ferroelectrics At high temperature, the small green B-cations can "rattle around" in the larger holes between oxygen, maintaining cubic symmetry. A static displacement occurs when the structure is cooled below the transition temperature. We have illustrated a displacement along the z-axis, resulting in tetragonal symmetry (z remains a 4-fold symmetry axis), but at still lower temperatures the symmetry can be lowered further by additional displacements along the x- and y-axes. We have a dynamic 3D-drawing GODDARDtransition. - Ch120a Ferroelectrics Nov 21, 2005 of this ferro-electric 49 Phases BaTiO3 <111> polarized rhombohedral <110> polarized orthorhombic -90oC <100> polarized tetragonal 120oC 5oC Non-polar cubic Temperature Different phases of BaTiO3 Ba2+/Pb2+ c Ti4+ O2- a Non-polar cubic above Tc Six variants at room temperature c = 1.01 ~ 1.06 a <100> tetragonal below Tc Domains separated by domain walls GODDARD - Ch120a Ferroelectrics Nov 21, 2005 50 Ferroelectric Actuators • MEMS Actuator performance parameters: – Actuation strain – Work per unit volume – Frequency • Goal: – Obtain cyclic high actuations by 90o domain switching in ferroelectrics – Design thin film micro devices for large actuations Tetragonal perovskites: 1% (BaTiO3), 6.5% (PbTiO3)) GODDARD - Ch120a Characteristics of common actuator materials 108 shape memory alloy 90o domain switching 107 solid-liquid fatigued SMA therm o- pneum atic 106 PZT 105 electromagnetic (EM) 104 muscle electrostatic (ES) EM 10 3 ES ZnO microbubble 10 2 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Cycling Frequency (Hz) P. Krulevitch et al, MEMS 5 (1996) 270-282 Ferroelectrics Nov 21, 2005 51 10 7 Bulk Ferroelectric Actuation Strains, BT~1%, PT~5.5% – Apply constant stress and cyclic voltage – Measure strain and charge – In-situ polarized domain observation s s V 0V s US Patent # 6,437, 586 (2002) GODDARD - Ch120a Ferroelectrics 21, 2005 EricNov Burcsu, 2001 s 52 Ferroelectric Model MEMS Actuator •BaTiO3-PbTiO3 (Barium Titanate (BT)-Lead Titanate (PT) •Perovskite pseudo-single crystals (biaxially textured thin films) [010] [100] MEMS Test Bed GODDARD - Ch120a Ferroelectrics Nov 21, 2005 53 Application: Ferroelectric Actuators Must understand role of domain walls in mediate switching Switching gives large strain, E … but energy barrier is extremely high! 90° domain wall Experiments in BaTiO3 2 Strain (%) Domain walls lower the energy barrier by enabling nucleation and growth 1.0 1 0 -10,000 0 10,000 Electric field (V/cm) Essential questions: Are domain walls mobile? Do they damage the material? 54 GODDARD Ch120a Nov 21, 2005 ReaxFF Use MD with In polycrystals? In- thin films? Ferroelectrics Simulating the Role of Nano/Meso Structure in Tunability and Losses of Ceramics FerroElectrics Experimental observations show the importance of mechanical constraints on the electrical response of ferroelectric ceramics at microwave frequencies, in particular on the tunability and losses. These constraints arise from the interaction with other grains as well as with the substrate. FerroElectric Thin Film Substrate Overall Goal : Full-field atomistically informed mechanical coupled mechanical/electrical mesoscale simulations at system level Thin Film + Substrate Hystersis is significantly influenced by the constraints around the grain. E. Baucsu, etc., JMPS, 52, 2004 December 04: multiscale simulations of highly and loosely constrained ferroelectric ceramics Polycrystal ceramics HIGHLY CONSTRAINED ceramics embedded in soft matrix. LOOSELY CONSTRAINED GODDARD - Ch120a Grain size effects are due to internal constraints inside a grain. M.P. McNeal, etc., J. Appl. Phy., 83, 1997 Ferroelectrics Nov 21, 2005 55 Ti atom distortions and polarizations determined from QM calculations. Ti distortions are shown in the FE-AFE fundamental unit cells. Yellow and red strips represent individual Ti-O chains with positive and negative polarizations, respectively. Low temperature R phase has FE coupling in all three directions, leading to a polarization along <111> direction. It undergoes a series of FE to AFE transitions with increasing temperature, leading to a total polarization that switches from <111> to <011> to <001> and then vanishes. Microscopic Polarization Ti atom distortions Cubic I-43m = + z o x Pz Py Px = + FE / AFE y Tetragonal I4cm Macroscopic Polarization + = + = FE / AFE Teperature Orthorhombic Pmn21 = + + = FE / AFE Rhombohedral R3m GODDARD - Ch120a = +Nov 21, +2005 Ferroelectrics = 56 Space Group & Phonon DOS Phase Displacive Model FE/AFE Model (This Study) Symmetry 1 atoms Symmetry 2 atoms C Pm3m 5 I-43m 40 T P4mm 5 I4cm 40 O Amm2 5 Pmn21 10 R R3m 5 R3m 5 GODDARD - Ch120a Ferroelectrics Nov 21, 2005 57 Phase Transition at 0 GPa Thermodynamic Functions ZPE = Transition Temperatures and Entropy Change FE-AFE 1 (q, v) 2 q ,v (q, v) 1 E = Eo (q, v) coth 2 q ,v 2 k BT Phas e (q, v) F = Eo k BT ln 2 sinh q ,v 2 k BT R 0 22.78106 0 O 0.06508 22.73829 0.02231 T 0.13068 22.70065 0.05023 C 0.19308 22.66848 0.08050 S= 1 2T (q, v) ( q , v ) coth 2 k T q ,v B (q, v) - k B ln 2 sinh q ,v 2 k BT Eo (kJ/mol) ZPE (kJ/mol) Eo+ZPE (kJ/mol) Vibrations important to include GODDARD - Ch120a Ferroelectrics Nov 21, 2005 58 Phase Transitions at 0 GPa, FE-AFE R Experiment [1] Transition O T C This Study T(K) ΔS (J/mol) T(K) ΔS (J/mol) R to O 183 0.17±0.04 228 0.132 O to T 278 0.32±0.06 280 0.138 T to C 393 0.52±0.05 301 0.145 - Ch120a Ferroelectrics 1.GODDARD G. Shirane and A. Takeda, J. Phys. Soc. Jpn., 7(1):1, 1952Nov 21, 2005 59 EXAFS & Raman observations d (001) EXAFS of Tetragonal Phase[1] •Ti distorted from the center of oxygen octahedral in tetragonal phase. α (111) •The angle between the displacement vector and (111) is α= 11.7°. PQEq with FE/AFE model gives α=5.63° Raman Spectroscopy of Cubic Phase[2] A strong Raman spectrum in cubic phase is found in experiments. 1. 2. Model Inversion symmetry in Cubic Phase Raman Active Displacive Yes No FE/AFE No Yes B. Ravel et al, Ferroelectrics, 206, 407 (1998) GODDARD Ch120a Ferroelectrics A. M. Quittet et- al, Solid State Comm., 12, 1053 (1973) Nov 21, 2005 60 60 Polarizable QEq Proper description of Electrostatics is critical E = E Coulomb E vdW Allow each atom to have two charges: A fixed core charge (+4 for Ti) with a Gaussian shape A variable shell charge with a Gaussian shape but subject to displacement and charge transfer Electrostatic interactions between all charges, including the core and shell on same atom, includes Shielding as charges overlap Allow Shell to move with respect to core, to describe atomic polarizability Self-consistent charge equilibration (QEq) c 2 ic 3 2 c c r (r ) = ( p ) Qi exp( -i | r - ri | ) s 2 is 3 2 s s s ri (r ) = ( p ) Qi exp( -i | r - ri | ) c i Four universal parameters for each element: Get from QM io , J io , Ric , Ris & qic GODDARD - Ch120a Ferroelectrics Nov 21, 2005 61 Validation Phase Properties EXP QMd P-QEq Cubic (Pm3m) a=b=c (A) B(GPa) εo 4.012a 4.007 167.64 4.0002 159 4.83 a=b(A) c(A) Pz(uC/cm2) B(GPa) 3.99c 4.03c 15 to 26b 3.9759 4.1722 3.9997 4.0469 17.15 135 a=b(A) c(A) γ(degree) Px=Py(uC/cm2) B(Gpa) 4.02c 3.98c 89.82c 15 to 31b 4.0791 3.9703 89.61 a=b=c(A) α=β=γ(degree) Px=Py=Pz(uC/cm2) B(GPa) 4.00c 89.84c 14 to 33b 4.0421 89.77 Tetra. (P4mm) Ortho. (Amm2) Rhomb. (R3m) 6.05e 98.60 97.54 97.54 4.0363 3.9988 89.42 14.66 120 4.0286 89.56 12.97 120 a. H. F. Kay and P. Vousden, Philosophical Magazine 40, 1019 (1949) b. H. F. Kay and P. Vousden, Philosophical Magazine 40, 1019 (1949) ;W. J. Merz, Phys. Rev. 76, 1221 (1949); W. J. Merz, Phys. Rev. 91, 513 (1955); H. H. Wieder, Phys. Rev. 99,1161 (1955) c. d. G.H. Kwei, A. C. Lawson, S. J. L. Billinge, and S.-W. Cheong, J. Phys. Chem. 97,2368 GODDARD - Ch120a Ferroelectrics Nov 21, 2005 M. Uludogan, T. Cagin, and W. A. Goddard, Materials Research Society Proceedings (2002), vol. 718, p. D10.11. 62 Free energies for Phase Transitions Common Alternative free energy from Vibrational states at 0K We use 2PT-VAC: free energy from MD at 300K Velocity Auto-Correlation Function C vv = V (0) V (t ) c = dV (0) V (t ) r () U ({ri , i = 1...3 N }) = U o ( ri o , i = 1...3 N ) 1 3 N 2U 2 i , j =1 ri r j v Dri Dr j Rio , r jo Velocity Spectrum ~ C vv (v) = - dte 2pivt C vv (t ) 3N ~ S (v) = 2 b m j C vv (v) j =1 System Partition Function Q= dvS(v) ln Q(v) 0 63 GODDARD Ch120a Ferroelectrics Nov 21, 2005 Thermodynamic Functions: Energy, Entropy, Enthalpy, Free Energy Free energies predicted for BaTiO3 FE-AFE phase structures. AFE coupling has higher energy and larger entropy than FE coupling. Get a series of phase transitions with transition temperatures and entropies Theory (based on low temperature structure) 233 K and 0.677 J/mol (R to O) 378 K and 0.592 J/mol (O to T) 778 K and 0.496 J/mol (T to C) Free Energy (J/mol) Experiment (actual structures at each T) 183 K and 0.17 J/mol (R to O) 278 K and 0.32 J/mol (O to T) 393 K and 0.52 J/mol (T to C) GODDARD - Ch120a Ferroelectrics Nov 21, 2005(K) Temperature 64 Nature of the phase transitions Displacive 1960 Cochran Soft Mode Theory(Displacive Model) Order-disorder 1966 Bersuker Eight Site Model 1968 Comes Order-Disorder Model (Diffuse X-ray Scattering) Develop model to explain all the following experiments (FE-AFE) EXP Displacive Order-Disorder FE-AFE (new) Small Latent Heat Yes No Yes Diffuse X-ray diffraction Yes Yes Yes Distorted structure in No EXAFS Yes Yes Intense Raman in Cubic Phase Yes Yes No GODDARD - Ch120a Ferroelectrics Nov 21, 2005 65 Frozen Phonon Structure-Pm3m(C) Phase - Displacive Pm3m Phase Frozen Phonon of BaTiO3 Pm3m phase Brillouin Zone Γ (0,0,0) X1 (1/2, 0, 0) X2 (0, 1/2, 0) X3 (0, 0, 1/2) M1 (0,1/2,1/2) M2 (1/2,0,1/2) M3 (1/2,1/2,0) R (1/2,1/2,1/2) GODDARD - Ch120a 15 Phonon Braches (labeled at T from X3): TO(8) LO(4) TA(2) LA(1) PROBLEM: Unstable TO phonons at BZ edge centers: M1(1), M2(1), M3(1) Ferroelectrics Nov 21, 2005 66 Frozen Phonon Structure – Displacive model P4mm (T) Phase Unstable TO phonons: Amm2 (O) Phase Unstable TO phonons: M1(1), M2(1) GODDARD - Ch120a M3(1) R3m (R) Phase NO UNSTABLE PHONONS Ferroelectrics Nov 21, 2005 67 Next Challenge: Explain X-Ray Diffuse Scattering Cubic Tetra. Ortho. Rhomb. Diffuse X diffraction of BaTiO3 and KNbO3, R. Comes et al, Acta Crystal. A., 26, 244, 1970 GODDARD - Ch120a Ferroelectrics Nov 21, 2005 68 X-Ray Diffuse Scattering Photon K’ Phonon Q Photon K Cross Section Scattering function Dynamic structure factor Debye-Waller factor GODDARD - Ch120a s 1 K' =N S1 (Q) K 1 (n(Q, v) ) 2 2 S1 (Q) = F (Q, v) 1 (Q, v) v F1 (Q, v) = i fi exp - Wi Q iQ ri Q e*i Q, v Mi 1 2 n(q, v) Q ei (q, v) 2 Wi (Q) = 2MN m q ,v (q, v) Ferroelectrics Nov 21, 2005 69 Diffuse X-ray diffraction predicted for the BaTiO3 FE-AFE phases. Qx Qx -4 -3 -2 -1 0 1 2 3 4 -5 5 5 3 3 2 2 1 1 Qz 4 4 0 -4 -3 -2 -1 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -4 -3 -2 -1 0 2 3 4 5 3 4 5 Qx 1 2 3 4 5 -5 5 4 3 3 2 2 1 1 Qz 4 0 -1 -2 -2 -3 -3 -4 -4 -5 -5 O (250K) -4 -3 -2 -1 0 1 2 0 -1 GODDARD - Ch120a 1 T (350K) Qx -5 5 0 0 C (450K) Qz The partial differential cross sections (arbitrary unit) of X-ray thermal scattering were calculated in the reciprocal plane with polarization vector along [001] for T, [110] for O and [111] for R. The AFE Soft phonon modes cause strong inelastic diffraction, leading to diffuse lines in the pattern (vertical and horizontal for C, vertical for T, horizontal for O, and none for R), in excellent agreement with experiment (25). Qz -5 5 R (150K) Ferroelectrics Nov 21, 2005 70 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Summary Phase Structures and Transitions •Phonon structures •FE/AFE transition Agree with experiment? EXP Displacive Order-Disorder FE/AFE(This Study) Small Latent Heat Yes No Yes Diffuse X-ray diffraction Yes Yes Yes Distorted structure in EXAFS No Yes Yes Intense Raman in Cubic Phase No Yes Yes GODDARD - Ch120a Ferroelectrics Nov 21, 2005 71 71 Domain Walls Tetragonal Phase of BaTiO3 Consider 3 cases experimental Polarized light optical micrographs of domain patterns in barium titanate (E. Burscu, 2001) CASE I CASE II +++++++++++++++ CASE III ++++ ---- ++++ P P P P P ----------------- +++++++++++++++ E=0 ---- P - - - - +++ + - - - ++++ ---- ++++ ++++ ---- E ----------------- - - - - +++ + - - - - ++++ •Open-circuit •Short-circuit •Open-circuit •Surface charge not neutralized •Surface charge neutralized •Surface charge not neutralized •Domain stucture E cs = E el E vdw GODDARD - Ch120a E cs = E el E vdw E cs = E el E vdw Nov 21, 2005 72 Ferroelectrics 72 E dw E surface - PE 180° Domain Wall of BaTiO3 – Energy vs length (001) z (00 1 ) o Ly y Type I Type II Type III GODDARD - Ch120a Type I L>64a(256Å) Type II 4a(16Å)<L<32a(128Å) Type III L=2a(8Å) Ferroelectrics Nov 21, 2005 73 73 180° Domain Wall – Type I, developed Displacement dY (001) (00 1 ) Ly = 2048 Å =204.8 nm z C o Zoom out A A D B A B C Displacement dZ y Displace away from domain wall D Displacement reduced near domain wall Zoom out A B C D Wall center - Ch120a Transition layer Domain structure GODDARD Ferroelectrics Nov 21, 2005 74 74 180° Domain Wall – Type I, developed (001) z L = 2048 Å Polarization P (00 1 ) Free charge ρf o Wall center: expansion, polarization switch, positively charged Transition layer: contraction, polarization relaxed, negatively charged Domain structure: constant lattice spacing, polarization and charge density GODDARD - Ch120a Ferroelectrics Nov 21, 2005 75 75 y 180° Domain Wall – Type II, underdeveloped (001) (00 1 ) z L = 128 Å A C Displacement dY B A B Displacement dZ Free charge ρf D C o y Polarization P D Wall center: expanded, polarization switches, positively charged Transition layer: contracted, polarization relaxes, negatively charged GODDARD - Ch120a Ferroelectrics Nov 21, 2005 76 76 180° Domain Wall – Type III, antiferroelectric (001) (00 1 ) z L= 8 Å o Polarization P Displacement dZ Wall center: polarization switch GODDARD - Ch120a Ferroelectrics Nov 21, 2005 77 77 y 180° Domain Wall of BaTiO3 – Energy vs length (001) z (00 1 ) o Ly y Type I Type II Type III GODDARD - Ch120a Type I L>64a(256Å) Type II 4a(16Å)<L<32a(128Å) Type III L=2a(8Å) Ferroelectrics Nov 21, 2005 78 78 90° Domain Wall of BaTiO3 (010) ( 001) L z 2 2N 2 2 o y L=724 Å (N=128) •Wall energy is 0.68 erg/cm2 •Stable only for L362 Å (N64) Wall center Transition Layer Domain Structure GODDARD - Ch120a Ferroelectrics Nov 21, 2005 79 79 90° Domain Wall of BaTiO3 (010) ( 001) L L=724 Å (N=128) Displacement dY z o Displacement dZ y Free Charge Density Wall center: Orthorhombic phase, Neutral Transition Layer: Opposite charged Domain Structure GODDARD - Ch120a Ferroelectrics Nov 21, 2005 80 90° Wall – Connection to Continuum Model 3-D Poisson’s Equation r 2 U = o r = r p r f r p = - P 1-D Poisson’s Equation d 2U r = 2 o dy r = r p r f r = - dPy p dy y 1 y Solution U ( y ) = Py d - r f dd c y 0 o o C is determined by the periodic boundary condition: U (0) = U ( 2 L) GODDARD - Ch120a Ferroelectrics Nov 21, 2005 81 90° Domain Wall of BaTiO3 (010) L=724 Å (N=128) Polarization Charge Density Electric Field GODDARD - Ch120a ( 001) L Free Charge Density z o y Electric Potential Ferroelectrics Nov 21, 2005 82 Summary III (Domain Walls) 180° domain wall •Three types – developed, underdeveloped and AFE •Polarization switches abruptly across the wall •Slightly charged symmetrically 90° domain wall •Only stable for L36 nm •Three layers – Center, Transition & Domain •Center layer is like orthorhombic phase •Strong charged – Bipolar structure – Point Defects and Carrier injection GODDARD - Ch120a Ferroelectrics Nov 21, 2005 83 83 Mystery: Origin of Oxygen Vacancy Trees! 0.1μm Oxgen deficient dendrites in LiTaO3 (Bursill et al, Ferroelectrics, 70:191, 1986) GODDARD - Ch120a Ferroelectrics Nov 21, 2005 84 Aging Effects and Oxygen Vacancies Problems •Fatigue – decrease of ferroelectric polarization upon continuous large signal cycling •Retention loss – decrease of remnant polarization with time •Imprint – preference of one polarization state over the other. •Aging – preference to relax to its pre-poled state Vz Pz c Vx Three types of oxygen vacancies in BaTiO3 tetragonal phase: - Ch120a Vx, Vy &GODDARD Vz Vy a Ferroelectrics Nov 21, 2005 85 Oxygen Vacancy Structure (Vz) P O O Ti 1 domain Ti O 2.12Å 1.93Å O 2.12Å O O Ti No defect O 2.12Å O 1.84Å O O 2.12Å Remove Oz O Ti 4.41Å Ti O O O 1.85Å O O O 2.10Å 1.93Å 2.12Å O O Ti O O Ti Leads to Ferroelectric Fatigue GODDARD - Ch120a O Ti 2.12Å 1.93Å Ti O defect leads to domain wall 1.93Å 2.12Å O Ti O O O O 1.93Å Ti Ti O O 1.93Å O P P Ferroelectrics Nov 21, 2005 O P 86 Single Oxygen Vacancy TSxz(1.020eV) TSxy(0.960eV) Vy(0eV) Vx(0eV) TSxz(0.011eV) oa2 DE exp( ) Diffusivity D = 2 k BT Mobility Dq * m= k BT GODDARD - Ch120a Ferroelectrics Nov 21, 2005 87 Divacancy in the x-y plane •V1 is a fixed Vx oxygen vacancy. •V2 is a neighboring oxygen vancancy of type Vx or Vy. •Interaction energy in eV.. Vacancy Interaction O O Ti 1. Short range attraction due to charge redistribution. 2. Anisotropic: vacancy pair prefers to break two parallel chains (due to coherent local relaxation) O O Ti O O O O Ti O O Ti Ti O O O O Ti O O O y z O GODDARD - Ch120a O z Ti O Ferroelectrics Nov 21, 2005 Ti O O 88 Vacancy Clusters Vx cluster in y-z plane: z y 0.1μ m 0.335eV 0.360 eV Best 1D 0.456 eV 0.636 eV 0.669 eV Best branch 0.650 eV Dendritic •Prefer 1-D structure •If get branch then grow linearly from branch •get dendritic structure GODDARD - Ch120a Ferroelectrics Nov 21, 2005 •n-type conductivity, leads to breakdown 1.878 eV 2D Bad 89 Summary Oxygen Vacancy •Vacancies trap domain boundary– Polarization Fatigue •Single Vacancy energy and transition barrier rates • Di-vacany interactions: lead to short range ordering •Vacancy Cluster: Prefer 1-D over 2-D structures that favor Dielectric Breakdown GODDARD - Ch120a Ferroelectrics Nov 21, 2005 90 FE-AFE Explains X-Ray Diffuse Scattering Experimental Cubic Ortho. Cubic Phase Tetra. Phase (001) Diffraction Zone (010) Diffraction Zone (100) (010) (100) (001) Strong Strong Weak Strong Tetra. Rhomb. Ortho. Phase Rhomb. Phase (010) Diffraction Zone (001) Diffraction Zone (100) (001) (100) (010) Strong Weak Very weak Very weak experimental Diffuse X diffraction of BaTiO3 and KNbO3, - Ch120a Ferroelectrics Nov 21, 2005 R. Comes etGODDARD al, Acta Crystal. A., 26, 244, 1970 91 Hysterisis Loop of BaTiO3 at 300K, 25GHz by MD E = E el E vdw Dz (V/A) 2pP P PD VV 3 o O Electric Displacement Correction Apply Dz at f=25GHz (T=40ps). T=300K. Applied Field (25 GHz) Dipole Correction Monitor Pz vs. Dz. Time (ps) Polarization (mC/cm2) Pr D-P E= Ec o Get Pz vs. Ez. Ec = 0.05 V/A at f=25 GHz. GODDARD - Ch120a Applied Field (V/A) Ferroelectrics Nov 21, 2005 92 92 O Vacancy Jump When Applying Strain O atom O vacancy site z y o x X-direction strain induces x-site O vacancies (i.e., neighboring Ti’s in x direction) to y or z-sites. GODDARD - Ch120a Ferroelectrics Nov 21, 2005 93 93 Effect of O Vacancy on the Hystersis Loop Supercell: 2x32x2 Total Atoms: 640/639 Pr Ec Perfect Crystal without O vacancy Crystal without 1 O vacancy. O Vacancy jumps when domain wall sweeps. •Introducing O Vacancy reduces both Pr & Ec. •O Vacancy jumps when domain wall sweeps. GODDARD - Ch120a Novfrom 21, 2005 Can look at bipolar case whereFerroelectrics switch domains x to y 94 94 Summary Ferroelectrics 1. The P-QEq first-principles self-consistent polarizable charge equilibration force field explains FE properties of BaTiO3 2. BaTiO3 phases have the FE/AFE ordering. Explains phase structures and transitions 3. Characterized 90º and 180º domain walls: Get layered structures with spatial charges 4. The Oxygen vacancy leads to linearly ordered structures dendritic patterns. Should dominate ferroelectric fatigue and dielectric breakdown GODDARD - Ch120a Ferroelectrics Nov 21, 2005 95