1. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 1 Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy William A. Goddard, III, wag@kaist.ac.kr WCU Professor at EEWS-KAIST and Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday Senior Assistant: Dr. Hyungjun Kim: linus16@kaist.ac.kr Manager of Center for Materials Simulation and Design (CMSD) Teaching Assistant: Ms. Ga In Lee: leeandgain@kaist.ac.krleeandgain@kaist.ac.kr Special assistant: Tod Pascal:tpascal@wag.caltech.edu Lecture 18, November 5, 2009

2. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 2 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

3. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 3 Last time

4. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 4 2 2 2 2 2 2 4 4 2 The configuration for C2 4 1 1 3 1 From 1930-1962 the 3  u was thought to be the ground state Now 1  g + is ground state Si 2 has this configuration

5. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 5 Ground state of C 2 MO configuration Have two strong  bonds, but sigma system looks just like Be 2 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

6. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 6 Low-lying states of C2

7. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 7

8. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 8 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/R 3, but since the average dipole is zero the first nonzero contribution is from 2 nd order perturbation theory, which scales like -C/R 6 (with higher order terms like 1/R 8 and 1/R 10 ) Consequently it is common to fit the interaction potentials to functional froms with a long range 1/R 6 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)

9. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 9 Noble gas dimers Ar 2 ReRe DeDe  LJ 12-6 E=A/R 12 –B/R 6 = D e [   12 – 2   6 ]  D e [   12 –   6 ]  = R/Re  = R/  where  = R e (1/2) 1/6 =0.89 R e

10. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 10 Remove an electron from He 2 Ψ(He 2 ) = A [(  g  )(  g  )(  u  )(  u  )]= (  g ) 2 (  u ) 2 Two bonding and two antibonding  BO= 0 Ψ(He 2 + ) = A [(  g  )(  g  )(  u  )]= (  g ) 2 (  u )  BO = ½ Get 2  u + symmetry. Bond energy and bond distance similar to H 2 +, also BO = ½

11. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 11 The ionic limit At R=∞ the cost of forming Na + and Cl - is IP(Na) = 5.139 eV minus EA(Cl) = 3.615 eV = 1.524 eV But as R is decreased the electrostatic energy drops as  E(eV) = - 14.4/R(A) or  E (kcal/mol) = -332.06/R(A) Thus this ionic curve crosses the covalent curve at R=14.4/1.524=9.45 A R(A) E(eV) 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

12. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 12 GVB orbitals of NaCL Dipole moment = 9.001 Debye Pure ionic  11.34 Debye Thus  q=0.79 e

13. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 13 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 χ M = (IP+EA)/5.2

14. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 14

15. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 15 The NaCl or B1 crystal All alkali halides have this structure except CsCl, CsBr, Cs I (they have the B2 structure)

16. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 16 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

17. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 17 Ionic radii, main group From R. D. Shannon, Acta Cryst. A32, 751 (1976) Fitted to various crystals. Assumes O 2- is 1.40A

18. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 18 Ionic radii, transition metals

19. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 19 Role of ionic sizes Assume that the anions are large and packed so that they contact, so that 2R A < L, where L is the distance between then Assume that the anion and cation are in contact. Calculate the smallest cation consistent with 2R A < L. R A +R C = L/√2 > √2 R A Thus R C /R A > 0.414 R A +R C = (√3)L/2 > (√3) R A Thus R C /R A > 0.732 Thus for 0.414 < (R C /R A ) < 0.732 we expect B1 For (R C /R A ) > 0.732 either is ok. For (R C /R A ) < 0.732 must be some other structure

20. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 20 Radiius Ratios of Alkali Halides and Noble metal halices Based on R. W. G. Wyckoff, Crystal Structures, 2 nd edition. Volume 1 (1963) Rules work ok B1: 0.35 to 1.26 B2: 0.76 to 0.92

21. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 21 Wurtzite or B4 structure

22. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 22 Sphalerite or Zincblende or B3 structure

23. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 23 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 (R C + R A )/L = (½) √3 a / √2 a = √(3/8) = 0.612 Thus 2R A < L = √(8/3) (R C + R A ) = 1.633 (R C + R A ) Thus 1.225 R A 0.225 Thus B3,B4 should be the stable structures for 0.225 < (R C /R A ) < 0. 414

24. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 24 Structures for II-VI compounds B3 for 0.20 < (R C /R A ) < 0.55 B1 for 0.36 < (R C /R A ) < 0.96

25. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 25 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 R C /R A > 0.71

26. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 26 Rutile (TiO 2 ) or Cassiterite (SnO 2 ) structure Related to NaCl with half the cations missing Find for R C /R A < 0.67

27. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 27 rutile CaF 2 rutile CaF 2

28. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 28 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 = z C / C where z C is the net charge on the cation and C is the coordination number Then z A =  i S I =  i z Ci / i Example1 : SiO 2. in most phases each Si is in a tetrahedron of O 2- leading to S=1. Thus each O 2- must have just two Si neighbors

29. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 29 Some old some New material

30. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 30 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 (  -Al2O3). Each Al 3+ is in a distorted octahedron, leading to S=1/2. Thus each O 2- must be coordinated to 4 Al Example 5. Olivine. Mg 2 SiO 4. Each Si has four O 2- (S=1) and each Mg has six O 2- (S=1/3). Thus each O 2- must be coordinated to 1 Si and 3 Mg neighbors

31. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 31 Illustration, BaTiO 3 A number of important oxides have the perovskite structure (CaTiO 3 ) 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 O 2-, S TiO = 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, Ba, leading to S Ba = 2/ Ba, and how many Ba around each O, OBa. Since OBa * S Ba = 2/3, the missing charge for the O, we have only a few possibilities:

32. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 32 Prediction of BaTiO3 structure Ba = 3 leading to S Ba = 2/ Ba =2/3 leading to OBa = 1 Ba = 6 leading to S Ba = 2/ Ba =1/3 leading to OBa = 2 Ba = 9 leading to S Ba = 2/ Ba =2/9 leading to OBa = 3 Ba = 12 leading to S Ba = 2/ Ba =1/6 leading to OBa = 4 Each of these might lead to a possible structure. The last case is the correct one for BaTiO 3 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.

33. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 33 BaTiO3 structure (Perovskite)

34. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 34 New material

35. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 35 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 (a 0 ) Thus QM calculations of dipole moment are in units of ea 0 which we refer to as au. However the international standard for quoting dipole moment is the Debye = 10 -10 esu A Where  (D) = 2.5418  (au)

36. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 36 Fractional ionic character of diatomic molecules Obtained from the experimental dipole moment in Debye,  (D), and bond distance R(A) by  q =  (au)/R(a 0 ) = C  (D)/R(A) where C=0.743470. Postive  that head of column is negative

37. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 37 Charge Equilibration Charge Equilibration for Molecular Dynamics Simulations; A.K. Rappé and W. A. Goddard III; J. Phys. Chem. 95, 3358 (1991) First consider how the energy of an atom depends on the net charge on the atom, E(Q) Including terms through 2 nd order leads to (2)(3)

38. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 38 Charge dependence of the energy (eV) of an atom E=0 E=-3.615 E=12.967 ClCl - Cl + Q=0Q=-1Q=+1 Harmonic fit = 8.291= 9.352 Get minimum at Q=-0.887 Emin = -3.676

39. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 39 QEq parameters

40. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 40 Interpretation of J, the hardness Define an atomic radius as H0.840.74 C1.421.23 N1.221.10 O1.081.21 Si2.202.35 S1.601.63 Li3.013.08 RA0RA0 R e (A 2 ) Bond distance of homonuclear diatomic Thus J is related to the coulomb energy of a charge the size of the atom

41. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 41 The total energy of a molecular complex Consider now a distribution of charges over the atoms of a complex: Q A, Q B, etc Letting J AB (R) = the Coulomb potential of unit charges on the atoms, we can write or Taking the derivative with respect to charge leads to the chemical potential, which is a function of the charges The definition of equilibrium is for all chemical potentials to be equal. This leads to

42. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 42 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 Q A. 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

43. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 43 The QEq Coulomb potential law We need now to choose a form for J AB (R) A plausible form is J AB (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 J AB (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 = 0.5 Using R C =0.759a 0

44. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 44 QEq results for alkali halides

45. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 45 QEq for Ala-His-Ala Amber charges in parentheses

46. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 46 QEq for deoxy adenosine Amber charges in parentheses

47. © copyright 2009 William A. Goddard III, all rights reservedEEWS-90.502-Goddard-L15 47 QEq for polymers Nylon 66 PEEK

48. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 48 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: ABO 3, 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

49. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 49 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 ferro- electric by analogy with a ferro-magnet 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. 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 of this ferro-electric transition.

50. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 50 c a Temperature 120 o C 5oC5oC-90 o C polarized rhombohedral polarized orthorhombic polarized tetragonal Non-polar cubic Different phases of BaTiO 3 Six variants at room temperature Domains separated by domain walls Non-polar cubic above Tc tetragonal below Tc O 2- Ba 2+ /Pb 2 + Ti 4+ Phases BaTiO3

51. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 51 Ferroelectric Actuators MEMS Actuator performance parameters: –Actuation strain –Work per unit volume –Frequency Goal: –Obtain cyclic high actuations by 90 o domain switching in ferroelectrics –Design thin film micro devices for large actuations Characteristics of common actuator materials Tetragonal perovskites: 1% (BaTiO 3 ), 6.5% (PbTiO 3 )) P. Krulevitch et al, MEMS 5 (1996) 270-282

52. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 52 Bulk Ferroelectric Actuation –Apply constant stress and cyclic voltage –Measure strain and charge –In-situ polarized domain observation 0 V   V   US Patent # 6,437, 586 (2002) Eric Burcsu, 2001 Strains, BT~1%, PT~5.5%

53. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 53 Ferroelectric Model MEMS Actuator BaTiO 3 -PbTiO 3 (Barium Titanate (BT)-Lead Titanate (PT) Perovskite pseudo-single crystals (biaxially textured thin films) MEMS Test Bed [010] [100]

54. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 54 Application: Ferroelectric Actuators Must understand role of domain walls in mediate switching Switching gives large strain, … but energy barrier is extremely high! E 90° domain wall Domain walls lower the energy barrier by enabling nucleation and growth Essential questions: Are domain walls mobile? Do they damage the material? In polycrystals? In thin films? Experiments in BaTiO 3 1 2 010,000-10,000 0 1.0 Electric field (V/cm) Strain (%) Use MD with ReaxFF

55. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 55 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. Hystersis is significantly influenced by the constraints around the grain. E. Baucsu, etc., JMPS, 52, 2004 Grain size effects are due to internal constraints inside a grain. M.P. McNeal, etc., J. Appl. Phy., 83, 1997 Overall Goal : Full-field atomistically informed mechanical coupled mechanical/electrical mesoscale simulations at system level Thin Film + Substrate Substrate FerroElectric Thin Film Polycrystal ceramics HIGHLY CONSTRAINED ceramics embedded in soft matrix. LOOSELY CONSTRAINED December 04: multiscale simulations of highly and loosely constrained ferroelectric ceramics

56. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 56 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 direction. It undergoes a series of FE to AFE transitions with increasing temperature, leading to a total polarization that switches from to to and then vanishes.

57. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 57 Space Group & Phonon DOS PhaseDisplacive ModelFE/AFE Model (This Study) Symmetry 1atomsSymmetry 2atoms CPm3m5I-43m40 TP4mm5I4cm40 OAmm25Pmn2110 RR3m5 5

58. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 58 Phase Transition at 0 GPa Thermodynamic Functions Transition Temperatures and Entropy Change FE-AFE Phas e E o (kJ/mol) ZPE (kJ/mol) E o +ZPE (kJ/mol) R022.781060 O0.0650822.738290.02231 T0.1306822.700650.05023 C0.1930822.668480.08050 Vibrations important to include

59. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 59 Phase Transitions at 0 GPa, FE-AFE TransitionExperiment [1] This Study T(K)ΔS (J/mol)T(K)ΔS (J/mol) R to O1830.17±0.042280.132 O to T2780.32±0.062800.138 T to C3930.52±0.053010.145 1. G. Shirane and A. Takeda, J. Phys. Soc. Jpn., 7(1):1, 1952 ROTC

60. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 60 EXAFS & Raman observations 60 (001) (111) d α 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. 1.B. Ravel et al, Ferroelectrics, 206, 407 (1998) 2.A. M. Quittet et al, Solid State Comm., 12, 1053 (1973) ModelInversion symmetry in Cubic Phase Raman Active DisplaciveYesNo FE/AFENoYes

61. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 61 Four universal parameters for each element: Get from QM Polarizable QEq 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) Proper description of Electrostatics is critical

62. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 62 Validation 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.G.H. Kwei, A. C. Lawson, S. J. L. Billinge, and S.-W. Cheong, J. Phys. Chem. 97,2368 d.M. Uludogan, T. Cagin, and W. A. Goddard, Materials Research Society Proceedings (2002), vol. 718, p. D10.11. PhasePropertiesEXPQM d P-QEq Cubic (Pm3m) a=b=c (A) B(GPa) ε o 4.012 a 6.05 e 4.007 167.64 4.0002 159 4.83 Tetra. (P4mm) a=b(A) c(A) Pz(uC/cm2) B(GPa) 3.99 c 4.03 c 15 to 26 b 3.9759 4.1722 98.60 3.9997 4.0469 17.15 135 Ortho. (Amm2) a=b(A) c(A) γ(degree) Px=Py(uC/cm2) B(Gpa) 4.02 c 3.98 c 89.82 c 15 to 31 b 4.0791 3.9703 89.61 97.54 4.0363 3.9988 89.42 14.66 120 Rhomb. (R3m) a=b=c(A) α=β=γ(degree) P x =P y =P z (uC/cm2) B(GPa) 4.00 c 89.84 c 14 to 33 b 4.0421 89.77 97.54 4.0286 89.56 12.97 120

63. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 63 Free energies for Phase Transitions Velocity Auto-Correlation Function Velocity Spectrum System Partition Function Thermodynamic Functions: Energy, Entropy, Enthalpy, Free Energy We use 2PT-VAC: free energy from MD at 300K Common Alternative free energy from Vibrational states at 0K

64. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 64 AFE coupling has higher energy and larger entropy than FE coupling. Get a series of phase transitions with transition temperatures and entropies Free energies predicted for BaTiO3 FE-AFE phase structures. 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) 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) Free Energy (J/mol) Temperature (K)

65. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 65 Nature of the phase transitions 1960CochranSoft Mode Theory(Displacive Model) EXPDisplaciveOrder-DisorderFE-AFE (new) Small Latent HeatYesNoYes Diffuse X-ray diffraction Yes Distorted structure in EXAFS NoYes Intense Raman in Cubic Phase NoYes Develop model to explain all the following experiments (FE-AFE) Displacive Order-disorder 1966BersukerEight Site Model 1968ComesOrder-Disorder Model (Diffuse X-ray Scattering)

66. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 66 Frozen Phonon Structure-Pm3m(C) Phase - Displacive Brillouin Zone Frozen Phonon of BaTiO3 Pm3m phasePm3m Phase 15 Phonon Braches (labeled at T from X3): TO(8) LO(4) TA(2) LA(1) PROBLEM: Unstable TO phonons at BZ edge centers: M 1 (1), M 2 (1), M 3 (1) Γ(0,0,0) X1X1 (1/2, 0, 0) X2X2 (0, 1/2, 0) X3X3 (0, 0, 1/2) M1M1 (0,1/2,1/2) M2M2 (1/2,0,1/2) M3M3 (1/2,1/2,0) R(1/2,1/2,1/2)

67. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 67 Frozen Phonon Structure – Displacive model Unstable TO phonons: M 1 (1), M 2 (1) Unstable TO phonons: M 3 (1) P4mm (T) PhaseAmm2 (O) PhaseR3m (R) Phase NO UNSTABLE PHONONS

68. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 68 Next Challenge: Explain X-Ray Diffuse Scattering CubicTetra. Ortho.Rhomb. Diffuse X diffraction of BaTiO 3 and KNbO 3, R. Comes et al, Acta Crystal. A., 26, 244, 1970

69. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 69 X-Ray Diffuse Scattering Photon K Phonon Q Cross Section Scattering function Dynamic structure factor Debye-Waller factor Photon K’

70. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 70 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). Diffuse X-ray diffraction predicted for the BaTiO3 FE-AFE phases.

71. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 71 Summary Phase Structures and Transitions Phonon structures FE/AFE transition EXPDisplaciveOrder-DisorderFE/AFE(This Study) Small Latent HeatYesNoYes Diffuse X-ray diffraction Yes Distorted structure in EXAFS NoYes Intense Raman in Cubic Phase NoYes Agree with experiment?

72. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 72 experimental Domain Walls Tetragonal Phase of BaTiO 3 Consider 3 cases 72 Short-circuit Surface charge neutralized PP + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - E=0 E + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - + + + + - - - - - - - - + + + + P P P P + + + + - - - - - - - - + + + + Open-circuit Surface charge not neutralized Open-circuit Surface charge not neutralized Domain stucture CASE ICASE IICASE III Polarized light optical micrographs of domain patterns in barium titanate (E. Burscu, 2001)

73. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 73 180° Domain Wall of BaTiO 3 – Energy vs length yz o 73 LyLy Type I Type II Type III Type IL>64a(256Å) Type II4a(16Å)<L<32a(128Å) Type IIIL=2a(8Å)

74. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 74 180° Domain Wall – Type I, developed 74 Displacement dY Displacement dZ Wall centerTransition layer Domain structure C A A B D ABCD ABCD L y = 2048 Å =204.8 nm Zoom out y z o 74 Displace away from domain wall Displacement reduced near domain wall

75. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 75 Polarization PFree charge ρ f L = 2048 Å Wall center: expansion, polarization switch, positively charged Transition layer: contraction, polarization relaxed, negatively charged Domain structure: constant lattice spacing, polarization and charge density y z o 75 180° Domain Wall – Type I, developed

76. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 76 180° Domain Wall – Type II, underdeveloped 76 Displacement dYDisplacement dZ Polarization P ABCD Wall center: expanded, polarization switches, positively charged Transition layer: contracted, polarization relaxes, negatively charged AC BD Free charge ρ f L = 128 Å 76 y z o

77. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 77 180° Domain Wall – Type III, antiferroelectric 77 Displacement dZPolarization P Wall center: polarization switch L= 8 Å 77 y z o

78. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 78 180° Domain Wall of BaTiO 3 – Energy vs length yz o 78 LyLy Type I Type II Type III Type IL>64a(256Å) Type II4a(16Å)<L<32a(128Å) Type IIIL=2a(8Å)

79. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 79 90° Domain Wall of BaTiO 3 79 z y o Wall center Transition Layer Domain Structure Wall energy is 0.68 erg/cm 2 Stable only for L  362 Å (N  64) L=724 Å (N=128) L

80. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 80 90° Domain Wall of BaTiO 3 Wall center: Orthorhombic phase, Neutral Transition Layer: Opposite charged Domain Structure Displacement dYDisplacement dZFree Charge Density L z y o L=724 Å (N=128)

81. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 81 90° Wall – Connection to Continuum Model 1-D Poisson’s Equation C is determined by the periodic boundary condition: Solution 3-D Poisson’s Equation

82. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 82 90° Domain Wall of BaTiO 3 Polarization Charge Density Free Charge Density Electric FieldElectric Potential L z y o L=724 Å (N=128)

83. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 83 Summary III (Domain Walls) 83 Three types – developed, underdeveloped and AFE Polarization switches abruptly across the wall Slightly charged symmetrically Only stable for L  36 nm Three layers – Center, Transition & Domain Center layer is like orthorhombic phase Strong charged – Bipolar structure – Point Defects and Carrier injection 180° domain wall 90° domain wall

84. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 84 Mystery: Origin of Oxygen Vacancy Trees! Oxgen deficient dendrites in LiTaO 3 (Bursill et al, Ferroelectrics, 70:191, 1986) 0.1μm

85. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 85 c a VzVz VyVy VxVx 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 Three types of oxygen vacancies in BaTiO 3 tetragonal phase: V x, V y & V z Pz

86. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 86 Oxygen Vacancy Structure (Vz) Ti O O O O O O O OO O O 2.12Å 1.93Å Ti O O O O O O OO OO 4.41Å 2.12Å 1.85Å 1.84Å 2.10Å Remove O z Ti O O O O O OO OO OO 2.12Å 1.93Å PP P P Leads to Ferroelectric Fatigue 1 domain No defect defect leads to domain wall

87. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 87 Single Oxygen Vacancy V y (0eV)V x (0eV) TS xz (1.020eV) TS xz (0.011eV) TS xy (0.960eV) Diffusivity Mobility

88. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 88 Divacancy in the x-y plane V 1 is a fixed V x oxygen vacancy. V 2 is a neighboring oxygen vancancy of type V x or V y. Interaction energy in eV.. 1.Short range attraction due to charge redistribution. 2.Anisotropic: vacancy pair prefers to break two parallel chains (due to coherent local relaxation) Vacancy Interaction Ti O O O O O O O OO O O O O O O O O OO O O O z y z

89. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 89 Vacancy Clusters Prefer 1-D structure If get branch then grow linearly from branch get dendritic structure n-type conductivity, leads to breakdown V x cluster in y-z plane: 0.335eV0.360 eV0.456 eV0.636 eV0.669 eV0.650 eV1.878 eV y z 1D2D Dendritic Best branch 0.1μ m Bad

90. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 90 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

91. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 91 Diffuse X diffraction of BaTiO 3 and KNbO 3, R. Comes et al, Acta Crystal. A., 26, 244, 1970 FE-AFE Explains X-Ray Diffuse Scattering CubicTetra. Ortho.Rhomb. Experimental (100)(010) Strong Cubic Phase (001) Diffraction Zone (100)(001) WeakStrong Tetra. Phase (010) Diffraction Zone Ortho. Phase (010) Diffraction Zone (100)(001) StrongWeak Rhomb. Phase (001) Diffraction Zone (100)(010) Very weak experimental

92. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 92 Hysterisis Loop of BaTiO3 at 300K, 25GHz by MD 92 Apply Dz at f=25GHz (T=40ps). T=300K. Monitor Pz vs. Dz. Get Pz vs. Ez. Ec = 0.05 V/A at f=25 GHz. D z (V/A) Time (ps) Applied Field (25 GHz) Applied Field (V/A) Polarization (  C/cm 2 ) Dipole Correction Electric Displacement Correction EcEc PrPr

93. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 93 O Vacancy Jump When Applying Strain 93 X-direction strain induces x-site O vacancies (i.e., neighboring Ti’s in x direction) to y or z-sites. x z y x z y o O atom O vacancy site

94. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 94 Effect of O Vacancy on the Hystersis Loop 94 Introducing O Vacancy reduces both P r & E c. O Vacancy jumps when domain wall sweeps. Perfect Crystal without O vacancy Crystal without 1 O vacancy. O Vacancy jumps when domain wall sweeps. Supercell: 2x32x2 Total Atoms: 640/639 Can look at bipolar case where switch domains from x to y EcEc PrPr

95. Ferroelectrics Nov 21, 2005 GODDARD - Ch120a 95 Summary Ferroelectrics 1.The P-QEq first-principles self-consistent polarizable charge equilibration force field explains FE properties of BaTiO 3 2.BaTiO 3 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