Lecture 16 February 10, 2010 BaTiO3, Woodward-Hoffmann Rules

1. Lecture 16 February 10, 2010 BaTiO3, Woodward-Hoffmann Rules Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy William A. Goddard, III, wag@wag.caltech.edu 316 Beckman Institute, x3093 Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Teaching Assistants: Wei-Guang Liu <wgliu@wag.caltech.edu> Ted Yu <tedhyu@wag.caltech.edu>

2. Course schedule Wednesday Feb. 10, 2pm L16 TODAY(caught up) Midterm was given out on Friday. Feb. 5, due today Wed. Feb. 10 Friday Feb. 12, postpone lecture from 2pm to 3pm Friday Feb. 12, at normal time, 2pm

3. Last time

4. Ionic bonding (chapter 9) Consider the covalent bond of Na to Cl. There Is very little contragradience, leading to an extremely weak bond. Alternatively, consider transferring the charge from Na to Cl to form Na+ and Cl-

5. Electronegativity Based on M++

6. The NaCl or B1 crystal All alkali halides have this structure except CsCl, CsBr, CsI (they have the B2 structure)

7. 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

8. Ionic radii, main group Fitted to various crystals. Assumes O2- is 1.40A NaCl R=1.02+1.81 = 2.84, exper is 2.84 From R. D. Shannon, Acta Cryst. A32, 751 (1976)

9. Role of ionic sizes in determining crystal structures Assume that the anions are large and packed so that they contact, so that 2RA < L, where L is the distance between anions Assume that the anion and cation are in contact. Calculate the smallest cation consistent with 2RA < L. RA+RC = (√3)L/2 > (√3) RA Thus RC/RA > 0.732 RA+RC = L/√2 > √2 RA Thus RC/RA > 0.414 Thus for 0.414 < (RC/RA ) < 0.732 we expect B1 For (RC/RA ) > 0.732 either is ok. For (RC/RA ) < 0.414 must be some other structure

10. Radius 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)

11. Wurtzite or B4 structure

12. Sphalerite or Zincblende or B3 structure GaAs

13. 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

14. Structures for II-VI compounds B3 for 0.20 < (RC/RA) < 0.55 B1 for 0.36 < (RC/RA) < 0.96

15. 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

16. Rutile (TiO2) or Cassiterite (SnO2) structure Related to NaCl with half the cations missing Find for RC/RA < 0.67

17. CaF2 rutile CaF2 rutile

18. 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 environments 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=4/4=1. Thus each O2- must have just two Si neighbors

19. Olivine. Mg2SiO4. example 5 electrostatic balance 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 O = Blue atoms (closest packed) Si = magenta (4 coord) cap voids in zigzag chains of Mg Mg = yellow (6 coord)

20. 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 TiO2 structures , we expect the Ti to be in an octahedron of O2-, STiO = 2/3. 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. 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.

21. Prediction of BaTiO3 structure : Ba coordination Since nOBa* SBa = 2/3, the missing charge for the O, we have only a few possibilities: 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.

22. BaTiO3 structure (Perovskite)

23. 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)

24. 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; • K. Rappé and W. A. Goddard III; J. Phys. Chem. 95, 3358 (1991) (2) (3)

25. Charge dependence of the energy (eV) of an atom E=12.967 E=0 E=-3.615 Cl+ Cl Cl- Q=+1 Q=0 Q=-1 Harmonic fit Get minimum at Q=-0.887 Emin = -3.676 = 8.291 = 9.352

26. QEq parameters

27. 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

28. 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 Using RC=0.759a0 And l = 0.5

29. QEq results for alkali halides

30. QEq for Ala-His-Ala Amber charges in parentheses

31. QEq for deoxy adenosine Amber charges in parentheses

32. Ferroelectrics 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 ferromagnet which contains magnetic dipoles. 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.

33. Phases of BaTiO3 <111> polarized rhombohedral <110> polarized orthorhombic <100> polarized tetragonal Non-polar cubic Temperature 120oC -90oC 5oC Different phases of BaTiO3 Ba2+/Pb2+ Ti4+ O2- c Domains separated by domain walls a Non-polar cubic above Tc Six variants at room temperature <100> tetragonal below Tc

34. 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 Ferroelectric Actuators 10 8 s h a p e m e m o r y a l l o y 90o domain switching 10 7 s o l i d - l i q u i d f a t i g u e d S M A p n e u m a t i c t h e r m o - 10 6 P Z T 10 5 e l e c t r o m a g n e t i c ( E M ) Work per volume (J/m3) m u s c l e 10 4 e l e c t r o s t a t i c ( E S ) E M 1 0 3 E S Z n O m i c r o b u b b l e 1 0 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 2 3 4 5 6 7 C y c l i n g F r e q u e n c y ( H z ) Characteristics of common actuator materials Tetragonal perovskites: 1% (BaTiO3), 6.5% (PbTiO3)) P. Krulevitch et al, MEMS 5 (1996) 270-282

35. Bulk Ferroelectric Actuation s s V 0 V s s Strains, BT~1%, PT~5.5% • Apply constant stress and cyclic voltage • Measure strain and charge • In-situ polarized domain observation US Patent # 6,437, 586 (2002) Eric Burcsu, 2001

36. Ferroelectric Model MEMS Actuator [100] [010] • BaTiO3-PbTiO3 (Barium Titanate (BT)-Lead Titanate (PT) • Perovskite pseudo-single crystals (biaxially textured thin films) MEMS TestBed

37. Application: Ferroelectric Actuators Must understand role of domain walls in mediate switching 1.0 E 2 Experiments in BaTiO3 Strain (%) Domain walls lower the energy barrierby enabling nucleation and growth 0 1 -10,000 0 10,000 90° domain wall Electric field (V/cm) Switching gives large strain, … but energy barrier is extremely high! Essential questions: Are domain walls mobile? Do they damage the material? In polycrystals? In thin films? Use MD with ReaxFF

38. Some New and old material

39. Nature of the phase transitions <111> polarized rhombohedral <110> polarized orthorhombic <100> polarized tetragonal Non-polar cubic Temperature 120oC -90oC 5oC Displacive model Assume that the atoms prefer to distort toward a face or edge or vertex of the octahedron Increasing Temperature Different phases of BaTiO3 face edge vertex center

40. Nature of the phase transitions Displacive model Assume that the atoms prefer to distort toward a face or edge or vertex of the octahedron Increasing Temperature Order-disorder

41. Comparison to experiment Cubic Tetra. Ortho. Rhomb. Displacive  small latent heat This agrees with experiment R  O: T= 183K, DS = 0.17±0.04 J/mol O  T: T= 278K, DS = 0.32±0.06 J/mol T  C: T= 393K, DS = 0.52±0.05 J/mol Diffuse xray scattering Expect some disorder, agrees with experiment

42. Problem displacive model: EXAFS & Raman observations d (001) α (111) 42 • EXAFS of Tetragonal Phase[1] • Ti distorted from the center of oxygen octahedral in tetragonal phase. • The angle between the displacement vector and (111) is α= 11.7°. Raman Spectroscopy of Cubic Phase[2] A strong Raman spectrum in cubic phase is found in experiments. But displacive model  atoms at center of octahedron: no Raman • B. Ravel et al, Ferroelectrics, 206, 407 (1998) • A. M. Quittet et al, Solid State Comm., 12, 1053 (1973)

43. QM calculations The ferroelectric and cubic phases in BaTiO3 ferroelectrics are also antiferroelectric Zhang QS, Cagin T, Goddard WA Proc. Nat. Acad. Sci. USA, 103 (40): 14695-14700 (2006) Even for the cubic phase, it is lower energy for the Ti to distort toward the face of each octahedron. How do we get cubic symmetry? Combine 8 cells together into a 2x2x2 new unit cell, each has displacement toward one of the 8 faces, but they alternate in the x, y, and z directions to get an overall cubic symmetry

44. QM results explain EXAFS & Raman observations d (001) α (111) 44 • EXAFS of Tetragonal Phase[1] • Ti distorted from the center of oxygen octahedral in tetragonal phase. • 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. • B. Ravel et al, Ferroelectrics, 206, 407 (1998) • A. M. Quittet et al, Solid State Comm., 12, 1053 (1973)

45. 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.

46. Phase Transition at 0 GPa Thermodynamic Functions Transition Temperatures and Entropy Change FE-AFE Vibrations important to include

47. Polarizable QEq Proper description of Electrostatics is critical 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) Four universal parameters for each element: Get from QM

48. Validation • H. F. Kay and P. Vousden, Philosophical Magazine 40, 1019 (1949) • 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) • G.H. Kwei, A. C. Lawson, S. J. L. Billinge, and S.-W. Cheong, J. Phys. Chem. 97,2368 • M. Uludogan, T. Cagin, and W. A. Goddard, Materials Research Society Proceedings (2002), vol. 718, p. D10.11.

49. QM Phase Transitions at 0 GPa, FE-AFE R O T C 1. G. Shirane and A. Takeda, J. Phys. Soc. Jpn., 7(1):1, 1952

50. 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 Velocity Spectrum System Partition Function Thermodynamic Functions: Energy, Entropy, Enthalpy, Free Energy