POLINGER Victor

VIBRONIC ORIGIN OF
FERROELECTRICITY
Victor Polinger,
University of Washington, Seattle, USA
Pablo Garcia‐Fernandez,
Universidad de Cantabria, Spain
Isaac B. Bersuker,
University of Texas @ Austin, USA

• Introduction
Outline
• The Instability Theorem
• Ferroelectricity in cubic perovskites (BaTiO 3)
• Ab‐Initio Calculations of Instability
• Non‐Vibronic Approaches
• Conclusions

Introduction: Milestones in the theory of
Ferroelectricity
(understanding the mechanism)
• Electrostatic theory (order/disorder of dipoles): Slater (1950)
• Phenomenological approach (the concept of soft‐mode):
Landau &Ginzburg (1946), Devonshire (1949);
Anderson (1959); Cochran (1959)
• Electron polarizability approach: Jaynes& Wigner (1950)
• The vibronic theory (pseudo JT effect): Bersuker (1966)
• One‐band anharmonicity (a long‐range theory): Aizy (1966)
• Two‐band long‐range theories:
Kristoffel&Konsin(1967); Girshberg&Tamarchenko (1976)
• Combined long‐range/short‐range approaches:
Girshberg&Yacoby (1997); Konsin&Sorkin (2009)

DIPOLAR INSTABILITY: Estimating Polarizability
Lattice
Polarization, pj���ijEj j
Electron
polarizability:
polarizability: � � latt latt
ij
latt
i j
n En�E0 �
�
latt
el
(per one elementary cell)
� �
Its (under‐estimated value) is :
�
E � E ~ 0.01 eV, dQ ~ 4 D, � ~ 900 D
n
el el
��ij � �
el �
n , n �,
0
n 0 d0 �el
0 1 latt
d n n d
, � , ~ 1 eV, ~ 5 D, ~ 20 D
d n n d QQ
3
D As a rule, next to lattice polarizability,
900
� ~ 50 3
20 D
Its (over‐erestimated) value is:
(per one elementary cell)
electron polarizability can be neglected.
3
3

THE INSTABILITY THEOREM:
Picture the Problem. T → 0 K, , Q 0.
At T → 0 K, the free energy, F = � �(Q) –TS,
transforms into � �(Q), the ground‐state APES
��H� HJT � � Q
�Q
�
� �
VQ n
��H �
�1 �Q��� � �0, � V � 0�
� n
, �, �
n n
At a small vicinity of Q = 0,
perturbation theory applies: H = H 0 + H JT
The perturbation: Ψ 0 is frozen Ψ 1 is Q‐dependent:
n n
0 � Q �0
0
�Q�� �0� � �Q� � � �
0 1
Hybridization
The greater the energy gap, � � – � �, the better the perturbation theory applies.
At no coupling, V n = 0, the electron wave function remains frozen.
Induced by the (pseudo) JT coupling, Q‐dependent hybridization
provides electrons the flexibibility to adjust and adiabatically follow
nuclei.

THE INSTABILITY THEOREM:
The Strategy of the Proof
Second‐order perturbation theory gives: K � K0 � Kv
K
2
��H� 0 � � 2
Q
�
���0 0 0
is positive?
V
2
K ��� n �
v
, �,
n n 0
is negative!
Since K 0 > 0, then K 0 + K v < 0 only due to the dominant contribution of K v< 0.
Theorem (Bersuker 1980): Since K o> 0, only the
vibronic contribution, K v< 0, can make lattice
unstable.
0.

ProvingK 0> 0
Since H = H el‐nucl + H nucl‐nucl, we have K 0 = K el‐nucl + K nucl‐nucl
•Two‐step rigorous proof for cubic crystals (Bersuker, 1988): [BaTiO 3 is cubic]
� �
2 4
2
el-nucl n n el-nucl 3
n
n n
� �
K � C 0 � H 0 � �e C � n �0
(a) (b)
•The general case (Polinger, 2010, unpublished):
As
Knucl-nucl
K
�2
10 , neglect Knucl‐nucl. Then
el-nucl
Proving K el‐nucl> 0:
K � K
~ 0 el-nucl
� � �eZ
K 0 H Q 0 0 0
Q Q Q
2 2 2
n
el-nucl � 2 el-nucl � � � 2��0 � � �
mn , rm �Rn �an
In the integrals, the shift substitution, ρm � rm �anQ, rm �ρm �anQ
transfers the Q dependence into the electron wave function:
�r,0 � � � , Q�� � � � �Q� � � � � � ��
2 2
�
�
As �� , � 0 � � � 0 � � 0.
��
�Q
Variational Principle: �Q� ~ Q H 2 � � H 2 � �
0
K �
0
nucl-nucl

Ferroelectricity in BaTiO 3
In perovskites (e.g., BaTiO 3),
ferroelectricity is initiated by local PJT dipolar distortions
For a Ti 4+ center
in the octahedral environment
of six oxygen ions O2‐ the MO LCAO energy level
diagram is:
Nine states, t2g, t1u, and t2g*, are involved in the PJTE
under the Ti4+ off-center
(odd) displacements T1u, X, Y, and Z.

Ferroelectricity in BaTiO 3
Approximate solution of the pseudo JT problem
for a TM site in cubic perovskites
In the TiO 8‐
6 cluster, the linear PJTE is represented by a coupling
matrix 9 9 with respect to three‐fold degenerate T1udisplacements X, Y, and Z. The secular equation includes three constants, K0, Δ,and
z ��H� F � 2 p�(O) � � 3 dxz(Ti)
��X� Assuming weak covalency , it yields nine roots
ε(R). Populating the lowest six orbitals with the
12 electrons, gives the following ground state
APES:
0
� �
1 2 2 2 2 2 2 2 2 2 2 2 2 2
�R� 2 K0R 2 F �R X � F �R X � F �R X �
� � � � � � � � � � � � � �

Ferroelectricity in BaTiO 3
The pseudo JT effect is due to
additional covalency gained by distortion
In the undistorted configuration,
the overlap 3d �(Ti)‐p �(O) is zero.
By distortion, this overlap
becomes non‐zero producing
new covalence �‐bonding which,
however, deteriorates the �‐bonds.
Distortion is spontaneous when
the �‐covalency gain is larger than
the loss in � bonding.
The whole effect is critically dependent on the nature of mixing �‐orbitals
(the vibronic coupling constant, F, and the energy gap, Δ)
versus inner bonding �‐orbitals(the force constant, K 0)

The shape of the APES of the Ti 4+ ion in BaTiO 3 is extraordinary:
If 4F 2 /K0 >Δ (or EJT>Δ), the APES has
(a) A maximum at Qx=Qy=Qz=0 (meaning instability);
(b) Eight C3v minimaat |Qx|= |Qy|=|Qz|= along trigonal axes
at a depth of � = 3Δ(Y + Y ‐1 )with Q0 = 2F/ K0 and Y =K0Δ/(4F2 2
0
);
21 Q Y �
2
(c) Six C4v saddle points at Qp= Qq= 0, Qr= 2Q along tetragonal
0 1�Y
axesat a depth of �3=(2/3)�; (d) Twelve C2v saddle points at |Qp|= |Qq| 0
Qr= 0 (p, q, r = x, y, z), along orthorhombic
axeswith a depth �in between (b) and (c).
� �

Some Numerical Results of the Vibronic Theory
Applied to
ATiO 3 (A = Ca, Sr, Ba) and BaMO 3 (M = Ti, Zr, Hf)
Rough numerical estimates from band-structure calculations:
CaTiO3 SrTiO3 BaTiO3 BaZrO3 BaHfO3 Kv ‐1.51 ‐1.33 ‐1.15 ‐0.20 ‐0.17
K0 1.41 1.23 0.98 0.83 0.86
K ‐0.10 ‐0.10 ‐0.17 0.63 0.69
Regularly, SrTiO3 is an incipient ferroelectric, K 0.
Strong isotope effect:Replacing 16O →18O changes SrTiO3 into a
true ferroelectric with TC 24.5 K. (Itohet al. 1999)
With 16O, the effective mass is m* = 24 u. With 18O, m* = 25.4 u.
K 0
Since �0 �
, increasing m* reduces 0 and thus enhances the
instability
m *
See similar in Yacoby and Girshberg, 2008

Ferroelectric BaTiO 3 versus Non-Ferroelectric ABO 3
Why some cubic perovskites are not ferroelectric?
1 2 2 2 2
0�Q�� 2 K0Q �2� �F
Q
1 2 2 2 2 2 2 2
, , �Q�� K Q �2� �F Q � � �F
Q
0 2 0
Adding just one d electron reduces the PJT instability significantly.
In BaVO 3, long‐range elastic forces prevail over the weakened PJTE.

Phases, phase transitions, and disorder dimensionality
in BaTiO 3 and KNbO 3 type crystals
predicted by the PJT theory of ferroelectricity
Phases Rhombohedral Orthorhombic Tetragonal Cubic
Direction of
polarization
[111] [011] [001] ‐
Dimensionality
of disorder
0 1 2 3
Number of
minimum points
involved in disorder
2
‐ [111] and [111]
4 8
Temperature of
phase transition
Tc(I) = 183 K Tc(II) = 278 K Tc(III) = 393 K
Emerging from the pseudo JT coupling, the special shape of the
APES allows for a direct interpretation of all the phases observed
in BaTiO 3 and similar crystals

K = K 0 + K v
Nature of ferroelectric phase transitions:
displacive or order‐disorder?
Short‐range versus long‐range driving forces
��� Kv �� i, p
��Hel� ph �
0 � i,
q
� p
� � � �0
E0�Ei�p� ( � )
��Kv
�i, p�,
i,
p
(a) Assuming weak �‐covalency (neglected in the zero‐order approximation)
(b) Keeping �‐overlap of the wave functions for next‐neighbor atoms only
(c) Looking for coupling to boundary optical phononswith q = 0(which results in
p = 0), meaning mutual displacement of two sub‐lattices,
we come to the following relation:
K
��� v
2 2 2
��H � � el-latt
�V � � �V
�
0 � i, 0 i 0 i
q
� p �
� Q
� �
Q
�
� � �0 �� � � �
0 � � �0
� � �
� � � �
, �, , �, , �,
2
0, p
�p� �0� �0� i, p 0 i i, p 0 i i 0 i
Pseudo JT effect is a local property. There are no long range forces
in PJTE contribution to the instability. Long range forces are all
included in K 0>0.

Experimental evidence of local origin of distortions and order‐disorder
nature of phase transitions in "displacive" ferroelectrics
Authors, year Method System Main result
Comes, Lambert,
&Guinier, 1968
X‐ray diffuse scattering BaTiO 3 Qualitative confirmation of all the main
predictions of the vibronic theory
Quittetet al., 1973 Raman spectra BaTiO 3, KNbO 3 Polar distortions in cubic phase
Burns, Dacol, 1981 Index of refraction BaTiO 3 Non‐vanishing component in cubic phase
Gervais, 1984 Infraredreflectivity BaTiO 3 Qualitatively the same
Ehseset al., 1981 X‐ray BaTiO 3 Strong order‐disorder component in
cubic phase
Itohet al., 1985 X‐ray BaTiO 3 [111] displacement of Ti in cubic phase
up to 180 K above T C
Mülleret al., 1986,
1987, 1991, 2007
Hanske‐Petitpierreet
al., 1991
Dougherty et al.,
1992
Stern et al., 1994,
1998,
Blinc, Zalar, &Laguta,
2003, 2005
ESR with probing ions Mn 4+ :BaTiO 3 [111] shift of Ti in the R phase and
reorientations in O, T, and cubic phase
XAFS KNb xTa 1‐xO 3 [111] shift of Nbin all phases for any x > 0.08.
Femtosecond‐resolved
scattering of light
BaTiO 3, KNbO 3
XAFS BaTiO 3, PbTiO 3,
KNbO 3
No relaxation mode found that might rule out
the local distortion model
[111] shift of Ti, Pb, and Nb in all phases
including the cubic phase up to 200 K above T C
NMR BaTiO 3 [001] shift of Ti in the cubic phaseup to 35 K
above T C

NMR versus XAFX in BaTiO 3
(Zalar et al., 2003; Stern 2004)
[111]
[100]
[110]
NMR for 47,49 Ti: In the C‐phase, at
T< 420 K
(by 20 K above the T ‐ C phase
transition), the local cubic symmetry at
the Ti sites lowers to tetragonal, [100]
(Zalar, Laguta, &Blinc, 2003).
In XAFS, in the C‐phase, in the interval
35 K 10 ‐8 s.
In XAFS, the averaging time, � < 10 ‐15 s.
XAFS provides instantaneous snapshots
of nuclei.

X- and Q-Band EPR in Mn 4+ :BaTiO 3
(Völkel&Müller, 2007)
197 K,
T C (R → O)
Temperature dependence of the line
width at the X band
The measured line broadening is fitted well
by �B = �B 0 + a/(T 0 ‐ T) with T 0 � 233 K.
Fast
In the R‐phase, there are two processes with
different time scale. One is fast, � 1 � 6ps.It
forms a precursor to the O‐phase. Another
one is slow, � 2 � 1000 ps.
This line broadening is a direct indication that order‐disorder processes occur not only
in the cubic phase, as detected by NMR, but also in the rhombohedral phase.
The reorientation correlation time, � ‐‐1 � (T 0 ‐ T) with T 0 = 293 K, about 100 K higher than T C
Fast

Coexistence of Order-Disorder and Displacive
Dynamics in Ferroelectrics
(Bussmann-Holder et al., 2009)
Model 1D Hamiltonian: H = H latt + H ”el” + H ”el”-latt;
g 2is parameter of polarizing “elasticity” of electron shells.
Soft mode requires
longer time
Snapshots
The “soft‐mode” feature can be observed
at t > 1000 ps. At t < 1 ps, snapshots of the
system are observed.
Large clusters
Small clusters
The critical length scale as a function
of T/T C.
Both types of dynamics, order‐disorder and displacive, exist in parallel. Occurring in
different length and time scale, in a single experiment, just one of the two component
can be observed.

Arguing Against Criticism:
Is the Energy Gap Too Large?
In BaTiO 3, the populated O(2p) is separated from the empty Ti(3d) by a broad energy gap.
Experimentally, the gap is about 3.2 eV (M. Cardona, 1965, & W. Wemple,1970)
To induce instability the “pseudo‐degeneracy” condition must be satisfied: � < F 2 /K 0 .
It was argued that the energy gap of � = 3.2 eV is too large to satisfy the condition
of electron “pseudo‐degeneracy”.
However, many molecules present even larger excitation energies and a strong PJT
D 2d� D 2hrotation in ethylene involves a
5 eV excitation (P. Garcia‐Fernandez, I.B.
Bersuker et al. sent to PCCP 2010)
CaF 2
10.03 eV
SrF 2
9.17 eV
BaF 2
9.03 eV
The bending of CaX 2 (X=F, Cl, Br) molecules
involves excitations around 10 eV
(P. Garcia‐Fernandez, I.B. Bersuker et al. JCPA 2007)

Electronic Structure Model Calculations
First-Principles DFT calculations. Part 1: Small Clusters
(Garcia‐Fernandez, ( 2010, unpublished) )
Isolated complex Ti(OH) 6 8‐center complex: 2 2 2 cell
Geometry optimization results in a
polarized complex with trigonal
symmetry C 3v. The central Ti 4+ is
shifted off‐center in the direction
[111], by 0.11 .
The cluster includes H atoms at the surface, has a
central Ba 2+ , and 6 more Ba atoms at the faces of the
cluster. Geometry optimization:Ba atoms are fixed.
Ti 4+ ions are off‐center shifted by 0.03 in opposite
trigonal directions along the main diagonal.

Electronic Structure Model Calculations
First-Principles DFT calculations. Part 2: Large Cluster
(Garcia‐Fernandez, 2010, unpublished)
27‐center complex: 3 3 3 cell The cluster includes 253 atoms with H atoms at the
surface. Geometry optimization does not include Ba
atoms. They are fixed at the respective cubic‐lattice
experimental positions. In the optimization iteration
procedure, each step moves ions by 0.001 .
If all other Ti atoms are
in their cubic position,
for the central Ti, the
APES has just one very
shallow cubic minimum.
Inter‐site dipole‐dipole coupling results in a stripe‐type
ferro‐antiferro ordering. With a predetermined shifted
position of some surface ions of Ti , the analysis does not
support the assumption of dipole‐dipole coupling.
The central ion Ti 4+ is off‐center shifted by about 0.11
in one of the four trigonal directions along the main
diagonal.

4 cells
high
Electronic Structure Model Calculations
First-Principles DFT calculations. Part 3: Periodic calculations
(Garcia‐Fernandez, 2010, unpublished)
Unit cell
.
intermediate
ferro anti
x
y
z
Frequency calculations for ferroelectric instability
Cubic
phase
183i
183i
183i
Tetragonal
phase
155i
155i
175
Orthorhombic
phase
148i
160
160
Rhombohedral
phase
153
153
181
Calculations in ordered phases indicate that cubic, tetragonal
and orthorhombic phases are unstable and thus not
experimentally detectable
antiferro
intermediate
ferroelectric
Long‐range (dipole‐
dipole) theories for
ferroelectricity predict
that no unstable
antiferroelectric modes is
not possible
Non‐cooperative
modes become
unstable when the
lattice is expanded

Electronic Structure Model Calculations
First-Principles DFT calculations. Part 4: Periodic Cluster PbTiO 3
(Choi et al. 2006 )
Analysis of covalency at
different points of the
Brillouin zone
Jahn‐Teller
stabilization at
different
points of the
Brillouin zone
Electron density contour
plots:
(a) Total electron density
(b) 3d z2(Ti) –2p x(O) hybrid
states at the � point
(c) 3d z2(Ti) –2p x(O) hybrid
states at the X z point
(d) hybrid states 3d zxz(Ti) –
2p x(O) at the � point

Why Born Charges are So Big in Perovckite
Ferroelectrics?
( (Garcia‐Fernandez, , 2010, , unpublished) p )
Born charges measure the change of the total
polarization with the distortion
dP
Z*
�
dQ
Born charges are a consequence of PJT effect
P = ΣZ iR i+
The ground electronic state at the cubic phase is A 1g and it
mixes with T 1u during the distortion thus
Ψ = c A(Q)Ψ A+c T(Q)Ψ T
P = ΣZ iR i+2c Ac T
Thus all the electronic contribution to the Born Charge
depends exclusively on the PJT effect
In ferroelectrics Born charges are
much larger than (usual) static
charges
Charge
Nominal
Mulliken
Born
Ti
O
Ti
O
Ti
O(σ)
O(π)
BaTiO 3
+4
-2
2.23
-1.24
7.38
-5.85
-2.12

Some Comments about The Non-
Vibronic Approaches
(a)The Ginzburg‐Landau theory
(b)The electron polarization theory (used by non‐believers)
� Polarization of ionic electron shells, �E � 10 eV
(Exception: rare earth compounds)
� Polarizing effect of electron hybridization, �E � 1 eV
[Exception: electron (quasi) degenerate systems without inversion symmetry]
(a)The adiabatic theories (used by half‐believers)
� First‐principles calculation of the APES
� The �4 theories

Polarization Model of Vanderbilt
(Zhong, Vanderbilt, and Rabe, 1995)
� �
H � H � H �H
The Hamiltonian: Strain
n nm
n n, m
� �
H �� u ��u �� u u �u u �u
u
2 4 2 2 2 2 2 2
n 2 n n ny nz nx nz nx ny
From the DFT‐LDA calculation (all in hartees),
�2�0.0568, � �0.320, � ��0.473
At these parameter values, the APES has just one shallow minimum
at the origin, u nx = u nx = u nx = 0. Used by Zhong, Vanderbilt, and
Rabe, the � 4 this model is not flexible enough to capture the multi‐
minimum shape of the APES

Conclusions
• The vibronic theory of ferroelectricity is the only first‐principles‐based
approach that explains ferroelectricity by the specific character of
chemical bonding in the respective substances.
• Main predictions of the vibronic theory are supported by an
impressive number of experiments.
• In the vibronic theory, polar distortive forces have local nature. Local
distortions are due to a gain in covalency between the transition metal
and ligands.
• The vibronic eight‐site order‐disorder model explains ordered low‐
symmetry clusters and their partial order above TC. • The vibronic theory is a great tool in the search for new substances with
tunable properties (multiferroics, etc.)
• Eventually, the quiet truth of the vibronic theory of ferroelectricity
will win!

Acknowledgements
VP: Prof. Daniel Gamelin, Seattle, USA