Download Ferroelectrics Nov 21, 2005 GODDARD - Ch120a

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]
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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
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Last time
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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
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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
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Low-lying states of C2
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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)
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Noble gas dimers
s
Ar2
Re
De
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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
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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
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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
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E(eV)
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R(A)
11
GVB orbitals
of NaCL
Dipole moment
= 9.001 Debye
Pure ionic
11.34 Debye
Thus
Dq=0.79 e
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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
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The NaCl or B1 crystal
All alkali halides
have this
structure except
CsCl, CsBr, Cs I
(they have the B2
structure)
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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
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Ionic radii, main group
Fitted to various crystals. Assumes O2- is 1.40A
From
R. D. Shannon, Acta©Cryst.
751 (1976)
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A. Goddard III, all rights reserved
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Ionic radii, transition metals
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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
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William other
A. Goddard
III, all rights reserved
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C
A
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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)
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Wurtzite or B4 structure
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Sphalerite or Zincblende or B3 structure
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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
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Structures for II-VI compounds
B3 for 0.20 < (RC/RA) < 0.55
B1 for 0.36 < (RC/RA) < 0.96
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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
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Rutile (TiO2) or Cassiterite (SnO2) structure
Related to NaCl
with half the
cations missing
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Find for RC/RA < 0.67
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CaF2
rutile
CaF2
rutile
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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
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Some old some New material
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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
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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
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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.
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BaTiO3 structure (Perovskite)
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New material
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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)
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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
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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)
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(3)
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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
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ClQ=-1
= 8.291
Get minimum at Q=-0.887
Emin = -3.676
= 9.352
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38
QEq parameters
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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
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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
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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
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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
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Using RC=0.759a0
© copyright 2009 William A. Goddard III, all rights reserved
43
QEq results for alkali halides
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44
QEq for Ala-His-Ala
Amber
charges in
parentheses
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45
QEq for deoxy adenosine
Amber
charges in
parentheses
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46
QEq for polymers
Nylon 66
PEEK
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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
=
 dV (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
- PE
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 L362 Å
(N64)
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  dd   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 L36 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
PD

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