1. William A. Goddard, III, wag@wag.caltech.edu
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,
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
Ted Yu

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= = 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 = L/√2 > √2 RA
Thus RC/RA > 0.414
RA+RC = (√3)L/2 > (√3) RA
Thus RC/RA > 0.732
Thus for < (RC/RA ) < we expect B1
For (RC/RA ) > either is ok.
For (RC/RA ) < 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. Thus B3,B4 should be the stable structures for
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) = (RC + RA)
Thus RA < (RC + RA) or RC/RA > 0.225
Thus B3,B4 should be the stable structures for
0.225 < (RC/RA) <

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 = esu A
Where m(D) = 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 atomCl
Cl-
Cl+
Q=0
Q=-1
Q=+1
Harmonic fit
Get minimum at Q=-0.887
Emin =
= 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. Amber charges in parentheses
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 Different phases of BaTiO3 c
Temperature
120oC
5oC
-90oC
<111> polarized rhombohedral
<110> polarized orthorhombic
<100> polarized tetragonal
Non-polar cubic
Different phases of BaTiO3
Six variants at room temperature
Domains separated by domain walls
above Tc
<100> tetragonal
below Tc
O2-
Ba2+/Pb2+
Ti4+
Ferroelectric crystals are non-polar (paraelectric) above its Curie temperature, but spontaneously polarized (ferroelectric) below the Curie temperature. Along with the spontaneous polarization, large distortion also occurs. As an example, BaTiO3, the most extensively investigated ferroelectric material, is cubic and non-polar above its Curie temperature, and is tetragonal and spontaneously polarized along <100> direction in room temperature. Upon further cooling, <110> polarized orthorhombic phase is formed at 278K, and <110> polarized rhomboredral phase round 183K.
Typically, there is also a reduction of crystal symmetry along with transformation. The symmetry-related and crystallographically-equivalent variants can coexist as domains separated by domain walls.

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

35. Bulk Ferroelectric Actuation
Strains, BT~1%, PT~5.5%
0 V s V
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
BaTiO3-PbTiO3 (Barium Titanate (BT)-Lead Titanate (PT)
Perovskite pseudo-single crystals (biaxially textured thin films)
[010]
[100]
MEMS Test Bed

37. Application: Ferroelectric Actuators Must understand role of domain walls in mediate switching
Switching gives large strain,
… but energy barrier is extremely high!
Experiments in BaTiO3 1 2
10,000
-10,000
1.0
Electric field (V/cm)
Strain (%) E
90° domain wall
Domain walls lower the energy barrier by enabling nucleation and growth
Tahir
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
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
Temperature
120oC
5oC
-90oC
<111> polarized rhombohedral
<110> polarized orthorhombic
<100> polarized tetragonal
Non-polar cubic
face
edge
vertex
center
1960
Cochran
Soft Mode Theory(Displacive Model)

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
1960
Cochran
Soft Mode Theory(Displacive Model)
Order-disorder
1966
Bersuker
Eight Site Model
1968
Comes
Order-Disorder Model (Diffuse X-ray Scattering)

41. Comparison to experiment
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
Cubic
Tetra.
Diffuse xray scattering
Expect some disorder, agrees with experiment
Ortho.
Rhomb.

42. Problem displacive model: EXAFS & Raman observations42
(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°.
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): (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 observations44
(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.
Model
Inversion symmetry in Cubic Phase
Raman Active
Displacive
Yes No
FE/AFE
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 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
Phase Eo
(kJ/mol)
ZPE
Eo+ZPE R O T C
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 Phase Properties EXP QMd P-QEq Cubic (Pm3m) a=b=c (A)
B(GPa) εo
4.012a
6.05e
4.007
167.64
4.0002
159
4.83
Tetra.
(P4mm)
a=b(A)
c(A)
Pz(uC/cm2)
3.99c
4.03c
15 to 26b
3.9759
4.1722
98.60
3.9997
4.0469
17.15
135
Ortho.
(Amm2)
γ(degree)
Px=Py(uC/cm2)
B(Gpa)
4.02c
3.98c
89.82c
15 to 31b
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)
Px=Py=Pz(uC/cm2)
4.00c
89.84c
14 to 33b
4.0421
89.77
4.0286
89.56
12.97
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-AFEC
Transition
Experiment [1]
This Study
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
1. G. Shirane and A. Takeda, J. Phys. Soc. Jpn., 7(1):1, 1952

50. 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

51. 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 J/mol (R to O)
378 K and J/mol (O to T)
778 K and 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)

52. 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
Diffuse X-ray diffraction
Distorted structure in EXAFS
Intense Raman in Cubic Phase

53. Space Group & Phonon DOS
Phase
Displacive Model
FE/AFE Model (This Study)
Symmetry 1
atoms
Symmetry 2 C
Pm3m 5
I-43m 40 T
P4mm
I4cm O
Amm2
Pmn21 10 R
R3m

54. 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

55. X-Ray Diffuse Scattering
Photon K’
Phonon Q
Photon K
Cross Section
Scattering function
Dynamic structure factor
Debye-Waller factor

56. Diffuse X-ray diffraction predicted for the BaTiO3 FE-AFE phases.
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).

57. FE-AFE Explains X-Ray Diffuse Scattering
Experimental
Cubic Phase
(001) Diffraction Zone
Tetra. Phase
(010) Diffraction Zone
Cubic
Tetra.
Ortho.
Rhomb.
(100)
(010)
Strong
(100)
(001)
Weak
Strong
Rhomb. Phase
(001) Diffraction Zone
Ortho. Phase
(010) Diffraction Zone
(100)
(001)
Strong
Weak
(100)
(010)
Very weak
experimental
Diffuse X diffraction of BaTiO3 and KNbO3,
R. Comes et al, Acta Crystal. A., 26, 244, 1970

58. 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
Diffuse X-ray diffraction
Distorted structure in EXAFS
Intense Raman in Cubic Phase 58

59. Domain Walls Tetragonal Phase of BaTiO3 3 cases
Charge and polarization distributions at the 90 degrees domain wall in barium titanate ferroelectric; Zhang QS, Goddard WA Appl. Phys. Let., 89 (18): Art. No (2006)
CASE I
CASE II
CASE III
experimental P
E=0 E
Polarized light optical micrographs of domain patterns in barium titanate (E. Burscu, 2001)
Open-circuit
Surface charge not neutralized
Domain stucture
Short-circuit
Surface charge neutralized
Open-circuit
Surface charge not neutralized 59

60. 180° Domain Wall of BaTiO3 – Energy vs length60 y z o Ly
Type I
Type II
Type III
Type I
L>64a(256Å)
Type II
4a(16Å)<L<32a(128Å)
Type III
L=2a(8Å)

61. 180° Domain Wall – Type I, developed61
Ly = 2048 Å =204.8 nm y z o
Displacement dY C A B D
Zoom out
Displace away from domain wall
Displacement dZ A B C D
Displacement reduced near domain wall
Zoom out
Wall center
Transition layer
Domain structure 61

62. 180° Domain Wall – Type I, developed62 y z o
L = 2048 Å
Polarization P
Free charge ρf
Wall center: expansion, polarization switch, positively charged
Transition layer: contraction, polarization relaxed, negatively charged
Domain structure: constant lattice spacing, polarization and charge density 62

63. 180° Domain Wall – Type II, underdeveloped63 y z o
L = 128 Å A C B D
Polarization P
Displacement dY
Displacement dZ
Free charge ρf A B C D
Wall center: expanded, polarization switches, positively charged
Transition layer: contracted, polarization relaxes, negatively charged 63

64. 180° Domain Wall – Type III, antiferroelectric64 y z o
L= 8 Å
Displacement dZ
Polarization P
Wall center: polarization switch 64

65. 180° Domain Wall of BaTiO3 – Energy vs length65 y z o Ly
Type I
Type II
Type III
Type I
L>64a(256Å)
Type II
4a(16Å)<L<32a(128Å)
Type III
L=2a(8Å)

66. 90° Domain Wall of BaTiO3 z L o y L=724 Å (N=128)
Wall energy is 0.68 erg/cm2
Stable only for L362 Å (N64)
Wall center
Transition Layer
Domain Structure 66

67. 90° Domain Wall of BaTiO3 L z y o L=724 Å (N=128) Displacement dY
Displacement dZ
Free Charge Density
Wall center: Orthorhombic phase, Neutral
Transition Layer: Opposite charged
Domain Structure

68. 90° Domain Wall of BaTiO3 L L=724 Å (N=128) z y o
Polarization Charge Density
Free Charge Density
Electric Field
Electric Potential

69. Summary III (Domain Walls)
Three types – developed, underdeveloped and AFE
Polarization switches abruptly across the wall
Slightly charged symmetrically
90° domain wall
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 69

70. New material

71. Mystery: Origin of Oxygen Vacancy Trees!
Oxgen deficient dendrites in LiTaO3 (Bursill et al, Ferroelectrics, 70:191, 1986)

72. 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 c a Vz Vy Vx Pz
Three types of oxygen vacancies in BaTiO3 tetragonal phase:
Vx, Vy & Vz

73. Oxygen Vacancy Structure (Vz)P P P Ti O
4.41Å
2.12Å
1.85Å
1.84Å
2.10Å Ti O
2.12Å
1.93Å O O Ti
1 domain
No defect
defect leads to domain
wall
1.93Å O
2.12Å O O
Remove Oz Ti
1.93Å O
2.12Å O O Ti
1.93Å O
2.12Å O O Ti P
Leads to Ferroelectric Fatigue

74. Single Oxygen Vacancy Diffusivity Mobility Vy(0eV) Vx(0eV)
TSxz(1.020eV)
TSxz(0.011eV)
TSxy(0.960eV)
Diffusivity
Mobility

75. 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
Short range attraction due to charge redistribution.
Anisotropic: vacancy pair prefers to break two parallel chains (due to coherent local relaxation) Ti O O O O Ti Ti O O O y O Ti Ti O z O O z

76. Vacancy Clusters z Vx cluster in y-z plane: y Best 1D Best branch 2D
0.1μm z
Vx cluster in y-z plane: y
0.335eV
0.360 eV
0.456 eV
0.636 eV
0.669 eV
0.650 eV
1.878 eV
Best 1D
Best
branch 2D
Dendritic
Bad
Prefer 1-D structure
If get branch then grow linearly from branch
get dendritic structure
n-type conductivity, leads to breakdown

77. 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

78. Hysterisis Loop of BaTiO3 at 300K, 25GHz by MD
Dz (V/A)
Time (ps)
Applied Field (25 GHz)
Electric Displacement
Correction
Dipole Correction
Apply Dz at f=25GHz (T=40ps).
T=300K.
Monitor Pz vs. Dz.
Applied Field (V/A)
Polarization (mC/cm2) Pr Ec
Get Pz vs. Ez.
Ec = 0.05 V/A at f=25 GHz. 78

79. O Vacancy Jump When Applying Strain
O atom
O vacancy site x z y o
X-direction strain induces x-site O vacancies (i.e., neighboring Ti’s in x direction) to y or z-sites. 79

80. Effect of O Vacancy on the Hystersis Loop
Supercell: 2x32x2
Total Atoms: 640/639 Ec Pr
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.
Can look at bipolar case where switch domains from x to y 80

81. Summary Ferroelectrics
The P-QEq first-principles self-consistent polarizable charge equilibration force field explains FE properties of BaTiO3
BaTiO3 phases have the FE/AFE ordering. Explains phase structures and transitions
Characterized 90º and 180º domain walls: Get layered structures with spatial charges
The Oxygen vacancy leads to linearly ordered structures  dendritic patterns. Should dominate ferroelectric fatigue and dielectric breakdown

82. Woodward-Hoffmann rules orbital symmetry rules Frontier Orbital rules
Roald Hoffmann
Certain cycloadditions occur but not others
2s+2s
2s+4s
4s+4s

83. Woodward-Hoffmann rules orbital symmetry rules Frontier Orbital rules
Certain cyclizations occur but not others
conrotatory
disrotatory
disrotatory
conrotatory

84. 2+2 cycloaddition – Orbital correlation diagramGS
Allowed
Forbidden ES

85. WH rules – 2 + 4 Ground State
Allowed

86. WH rules – 2 + 4 Excited State
Forbidden A S S

87. Summary WH rules cycloaddition
2n + 2m
n+m odd: Thermal allowed
Photochemical forbidden
n+m even: Thermal forbidden
Photochemical allowed
n=1, m=1: ethene + ethene
n=1, m=2: ethene + butadience (Diels-Alder)

88. S
WH rules – cyclization-GS A A A S A A S
Forbidden
Allowed A S S A S A S S
Rotation, C2
Reflection, s

89. Summary WH rules cyclization
n odd: thermal disrotatory
Photochemical conrotatory
n even: Thermal conrotatory
Photochemical disrotatory
n=2  butadiene
n=3  hexatriene

90. GVB view reactions Reactant HD+T H D T Product H+DT
Goddard and Ladner, JACS (1971)

91. GVB view reactions Reactant HD+T H D T
During reaction, bonding orbital on D stays on D,
Bonding orbital on H keeps its overlap with the orbital on D but delocalizes over H and T in the TS and localizes on T in the product. Thus highly overlapping bond for whole reaction
Nonbonding Orbital on free T of reactant becomes partially antibonding in TS and localizes on free H of product, but it changes sign
Product H+DT

92. GVB view reactions Reactant HD+T H D T
Bond pair keeps high overlap while flipping from reactant to product
Transition state
nonbond orbital keeps orthogonal, hence changes sign
Product H+DT H D T

93. GVB analysis of cyclization (4 e case)
Move AB bond; Ignore D; C changes phase as it moves from 3 to 1
4 VB orbitals: A,B,C,D reactant φB φA φB φA φB φC 2 3 4 φD 1 φC φA φD φC 2 3 φD 4 1 φB φA 2 3
Now ask how the CH2 groups 1 and 4 must rotate so that C and D retain positive overlap.
Clearly 4n is conrotatory φC φD 1 4

94. GVB analysis of cyclization (6 e case)1 2 φC φD 4 3

95. Apply GVB model to 2 + 2 φB φA φA φB φC φD φD φB φD φA φC φC
4 VB orbitals:A,B,C,D reactant
Transition state: ignore C φB φA φA φB φC φD φD φB φD
Nodal plane
4 VB orbitals product φA φC \ φC

96. Transition state for 2 + 2 φB φA φD φC 4 3 1 2 2 1 3 4
Transition state: ignore C
Orbitals A on 1 and B on 2 keep high overlap as the bond moves from 12 to 23 with B staying on 2 and A moving from 1 to 3 φB 2 1 φA
Orbital D must move from 3 to 1 but must remain orthogonal to the AB bond. Thus it gets a nodal plane
The overlap of D and C goes from positive in reactant to negative in product, hence going through 0. thus break CD bond. 3 4 φD
Nodal plane φC
Reaction Forbidden

97. GVB model fast analysis 2 + 2
4 VB orbitals:A,B,C,D reactant
Move A from 1 to 3 keeping overlap with B
Simultaneously D moves from 3 to 1 but must change sign since must remain orthogonal to A and B 2 1 φA φB φC φD 3 4 \ φB φA φD φC
C and D start with positive overlap and end with negative overlap. Thus break bond  forbidden

98. Next examine 2+4

99. 1. Move AB bond; Ignore D; C changes phase as it moves from 3 to 1
GVB 2+4 φA φB φA φB φC φD 1 2 4 3 φF φE 5 6 φD φC 2 3 1 4 6 5 φE φF
1. Move AB bond; Ignore D; C changes phase as it moves from 3 to 1

100. 2. Move EF bond; C changes phase again as it moves from 1 to 5
GVB 2+4
2. Move EF bond; C changes phase again as it moves from 1 to 5 φA φB φD φC 2 3 1 4 φA φB φD 2 3 6 5 1 4 φE φF φE φC
3. Now examine overlap of D with C. It is positive. Thus can retain bond CD as AB and EF migrate 6 5 φF
Reaction Allowed

101. GVB 2+4 φA φB φC φB φD φC φD φA φA φB φD φE φF φE φF φE φC φF
2. Move EF bond; C changes phase again as it moves from 1 to 5 φA φB φA φB φC φD 1 2 4 3 φF φE 5 6 φD φC 2 3 1 4 φA φB φD 2 3 6 5 1 4 φE φF
3. Examine final overlap of D with C. It is positive. Thus can retain bond CD as AB and EF migrate φE
1. Move AB bond; Ignore D; C changes phase as it moves from 3 to 1 φC 6 5 φF
Reaction Allowed

102. Transition metals

103. Transition metals Aufbau (4s,3d) Sc---Cu (5s,4d) Y-- Ag
(6s,5d) (La or Lu), Ce-Au

104. Ground states of neutral atomsSc
(4s)2(3d)1 Ti
(4s)2(3d)2 V
(4s)2(3d)3 Cr
(4s)1(3d)5 Mn
(4s)2(3d)5 Fe
(4s)2(3d)6 Co
(4s)2(3d)7 Ni
(4s)2(3d)8 Cu
(4s)1(3d)10

105. stop