Download Ferroelectrics from first principles Designing ferroelectrics

Ferroelectrics from first principles
Designing ferroelectrics the improper way
Nick Bristowe
Functional Materials Group, University of Kent, UK
CCP9 meeting, April 11th 2017
M
Ferro-electric/magnets, Multiferroics
Multiferroic:materialcombiningtwoormoreferroic parameters
Ferromagnetic:
●
N
N
N
S
S
S
N
N
N
S
S
S
Ferroelectric:
M
P
Ferroelectric:
P
+
-
+
-
+
+
-
-
+
+
-
-
●
Hysteresis:
Multifunctional magnetoelectrics
(Generalized) Magnetoelectric: cross coupled
response to electric and magnetic fields
Non-volatile data-storage!
Magnetoelectric:
-
+
Polarization, P
E
H
Claude Ederer
First principles studies of multiferroic materials
P
M
Magnetization, M
Possibleapplications:
i.e. control of the magnetic M (electric P) phase with an applied
electric E (magneticRAM:electricwrite/magneticread
H) field
- Magnetoelectric
- 4-statememory
Perovskites
e of A2+ Ti4+ O3 and A3+ Ti3+ O3 ([001])
ABX3
Widerangeofproperties
Duetocouplingdegreesoffreedom
P. Zubko et al. Annu. Rev. Condens. Matter Phys. 2, 141 (2011).
Why are most perovskites NOT FE?
ectricity as a lattice property
CaTiO3
Purely tilted ground state
PbTiO
3 as the
ce distortion, Q, has identical symmetry properties
polarization, i.e. Q∝ P, involve
smallFE
atomic
distortions
Pure
ground
state
Ti
paraelectric
e group Pm3m
neering Physics !
P
+
–
tetragonal-ferroelectric
Space group P4mm
42
Why are most perovskites NOT FE?
Structuralinstabilitiesfromsimplecubic
CaTiO3
Purely tilted ground state
PbTiO3
Pure FE ground state
lattice property
Ferroelectricity as a lattice property
entical symmetry properties as the
involve small atomic
FEdistortions
(ferroelectric)
polar distortion
P
Oxygen
M(f+)
FE lattice distortion, Q, has identical symmetry properties as the
FEatomic
(ferroelectric)
distortions
Octahedral Tilts polarization, i.e. Q∝ P, involve small
Pb
polar distortion
R(f-)
Ti
+
–
O
Phonons courtesy of Ph Ghosez
P
+
–
Why are most perovskites NOT FE?
Competing FE and AFD lattice instabilities
Energy landscape
1
1
2
E = A0 P + B0 P 4 + C12φ12φ22
2
4
1 2 1 4
+ A1φ1 + B1φ1 + C01φ12 P 2
2
4
1
1
2
+ A2φ2 + B2φ24 + C02φ22 P 2
2
4
FE (P)
AFD-M (f1)
AFD-R (f2)
Tilting often wins! €
FE and AFD usually competing
through bi-quadratic coupling
(C01,C02>0)
signiÆcance occurs in perovskites used for microwave dielectric applications. Colla et al. (1993) have shown that the sign
Why are most perovskites NOT FE?
2. Octahedral tilt sy
As previously noted, o
of the A-site cation co
change in A–X bond
ways in which the o
different coordination
Theyoftentiltinstead
The coordination num
nation sphere has b
common tilt systems (
A standard notation
hedral tilting distortio
alternative,
but equa
AFD (M-point) : f+
AFD (R-point)
: fAleksandrov (1976). F
Glazer notation throug
Figure 1
and Woodward Acta Cryst. B57 725 (2001)
a tilt system by rotati
Distribution of Lufaso
tilt systems
among known perovskites with a single
octahedral cation.
orthogonal Cartesian a
726
Lufaso and Woodward
✏
Prediction of the crystal structures
signiÆcance occurs in perovskites used for microwave dielectric applications. Colla et al. (1993) have shown that the sign
Why are most perovskites NOT FE?
2. Octahedral tilt sy
As previously noted, o
of the A-site cation co
change in A–X bond
ways in which the o
different coordination
Theyoftentiltinstead
The coordination num
nation sphere has b
common tilt systems (
A standard notation
hedral tilting distortio
alternative,
but equa
AFD (M-point) : f+
AFD (R-point)
: fAleksandrov (1976). F
Glazer notation throug
Figure 1
and Woodward Acta Cryst. B57 725 (2001)
a tilt system by rotati
Distribution of Lufaso
tilt systems
among known perovskites with a single
octahedral cation.
orthogonal Cartesian a
726
Lufaso
and Woodward
Detour:
Why is
Prediction
of the
crystal structures
Pnma
most
common?
✏
Pnma most stable
N Miao et al
structural phase
as a successive
ns in the cubic
.
fic Pnma ground
R+
4 instabilities
t is not a priori
ation of tilts is
the structures of
icted to M+
3 and
ation of tilts, the
er the symmetry
Figure 4. The calculated gains of energy, with respect to the ideal
Miao, Bristowe et al JPCM 26 035401 (2014)
cubic SrRuO3 phase
taken as reference, for different relaxed phases,
Pnma most stable
N Miao et al
structural phase
as a successive
ns in the cubic
Duetoother
motions
(nottilts!)
.
fic Pnma ground
R+
4 instabilities
t is not a priori
ation of tilts is
the structures of
icted to M+
3 and
ation of tilts, the
er the symmetry
->couplingsat
play?
Figure 4. The calculated gains of energy, with respect to the ideal
Miao, Bristowe et al JPCM 26 035401 (2014)
cubic SrRuO3 phase
taken as reference, for different relaxed phases,
Trilinear terms
N Miao et al
f structural phase
as a successive
rns in the cubic
).
ific Pnma ground
d R+
4 instabilities
it is not a priori
ination of tilts is
t the structures of
tricted to M+
3 and
nation of tilts, the
wer the symmetry
ystem will further
other modes that
ally understood as
d are rationalized
ical Goldschmidt
ubic perovskites
ation is small and
Miao, Bristowe et al JPCM 26 035401 (2014)
Figure 4. The calculated gains of energy, with respect to the ideal
+
cubic SrRuO
phase taken :asfreference,
for differentAnti-polar
relaxed phases,
AFD 3(M-point)
: X5z
labeled in terms of the compatible tilt pattern. Eoxygen corresponds to
the gain of energy that can be achieved from the relaxation of
oxygen atomic positions only. Ecation corresponds to the
supplemental gain of energy that can be achieved when allowing for
additional concomitant cation motions. In this latter case, the
oxygen distortions are modified through the coupling with cation
motions: the dashed line identifies the reduced gain of energy
0
(Eoxygen
) produced by pure oxygen motions in this fully relaxed
phase. All the previous calculations are done when keeping the unit
cell fixed. E
corresponds to the additional gain of energy when
AFD (R-point) : fx-
Can we cooperatively couple AFD with FE?
Turn anti-polar X mode to polar mode?
Considerdigitalsuperlattice
LayersofA andA’ inalternate(001)planes
canbegrownlayer-by-layer(e.g.PLDMBE)
ornaturallyordered(e.g.doubleperovskites)
(thesameconceptwillworkonthicker
superlattices,andotherlayeredmaterials
e.g.RP,DJ,Aurivilius)
Rotationally driven ferroelectricity
digitalsuperlattice
2
2
AFD motions
AFD motions
P4/mmm
structure
P4/mmm
structure
E = λφ xy− φ z+ Pxy
Pxy
OnlyingredientisPnma typerotationsandlayering
Breathing Jahn-Teller
Pxy
Charge and orbital
Bousquet et al Nature 452 732 (2008), Fukushima et al PCCP 13 12186 (2011), Rondinelli et al Adv Materials 24 1961 (2012)
(Hybrid) Improper Ferroelectricity
Shift the well to lower energy through the
coupling with other phonon modes
Hybrid improper ferroelectricity
“Hybrid” = requires two
independent order parameters
belonging to a different
subspaces
1
1
E = A0 P 2 + B0 P 4 + C12φ12φ22
2
4
1
1
2
+ A1φ1 + B1φ14 + C01φ12 P 2
2
4
1
1
+ A2φ22 + B2φ24 + C02φ22 P 2
2
4
Trilinear
+λφ1 φ2 P
coupling term
- lf1f2 acts as an effective field shifting P well to lower energy
- Switching P requires reversing either f1 or f2 (and perhaps M)
Figure courtesy of Ph Ghosez
Experimental signatures
ARTICLES
PUBLISHED ONLINE: 12 JANUARY 2015 | DOI: 10.1038/NMAT4168
Experimental demonstration of hybrid improper
ferroelectricity and the presence of abundant
charged walls in (Ca,Sr)3Ti2O7 crystals
Yoon Seok Oh1,2†, Xuan Luo3, Fei-Ting Huang1,2, Yazhong Wang1,2 and Sang-Wook Cheong1,2,3*
On the basis of successful first-principles predictions of new functional ferroelectric materials, a number of new ferroelectrics
PHY
I C A L rotation,
R E Vhybrid
I E Wimproper
LETTERS
PRL
114, 035701
(2015)
have been experimentally
discovered.
Using trilinear
coupling of two types
of Soctahedron
ferroelectricity has been theoretically predicted in ordered perovskites and the Ruddlesden–Popper compounds (Ca3 Ti2 O7 ,
Ca3 Mn2 O7 and (Ca/Sr/Ba)3 (Sn/Zr/Ge)2 O7 ). However, the ferroelectricity of these compounds has never been experimentally
confirmed and even their polar nature has been under debate. Here we provide the first experimental demonstration of roomtemperature switchable polarization in bulk crystals of Ca3 Ti2 O7 , as well as Sr-doped Ca3 Ti2 O7 . Furthermore, (Ca,Sr)3 Ti2 O7
is found to exhibit an intriguing ferroelectric domain structure resulting from orthorhombic twins and (switchable) planar
polarization. The planar domain structure accompanies abundant charged domain walls with conducting head-to-head and
insulating tail-to-tail configurations, which exhibit a conduction di�erence of two orders of magnitude. These discoveries
provide new research opportunities, not only for new stable ferroelectrics of Ruddlesden–Popper compounds, but also for
meandering conducting domain walls formed by planar
1,2,* polarization.
1
1
3
4
week ending
23 JANUARY 2015
Negative Thermal Expansion in Hybrid Improper Ferroelectric Ruddlesden-Popper
Perovskites by Symmetry Trapping
T
M. S. Senn,
1
A. Bombardi, C. A. Murray, C. Vecchini, A. Scherillo, X. Luo,5 and S. W. Cheong5,6
Diamond
Light Source
Limited,energy
Harwell
Science
and Innovation
Campus,
Didcot OX11 0DE, United Kingdom
here have been numerous attempts
at computational
ferroelectric
barriers
(for example,
20 meV for BaTiO
3
2
materials design based on first-principles calculations
for (ref. 18),of
30 Chemistry,
meV for PbTiOInorganic
25 meV for hexagonal
Department
Laboratory,
University of Oxford,
3 (ref. 19) andChemistry
Vol 452 | 10 April 2008 | doi:10.1038/nature06817
new functional materials1,2 . A large number of ferroelectric/ RMnO3 (R = rare earths) (ref. 20 and N. A. Spaldin, private
South
Parks
Road,
Oxford
OX1
3QR,
United
Kingdom
3–7
piezoelectric materials have been computationally predicted
, communication), the calculated switching barriers of 200 meV
3 5,8–13
Hampton
Road,
Teddington
TW11 0LW, United Kingdom
some of which have been experimentally confirmedNational
. For Physical
for Ca3 Mn2Laboratory,
O7 and ⇠100 meV
for (Ca/Ba)
3 (Sn/Zr)
2 O7 are still
15,17
4
example, the presence of ferroelectricity and strong coupling
too largeRutherford
to switch polarization
for this Didcot
reason, theOX11 0QX, United Kingdom
ISIS, STFC,
Appleton. Primarily
Laboratory,
between magnetism and ferroelectricity
were theoretically validity of hybrid improper ferroelectricity in A3 B2 O7 has been
5
21–23
Laboratory
Materials
and Department of Physics, Pohang University of Science and Technology,
predicted in EuTiO3 (ref. 3) and
FeTiO3 (ref. for
4). Pohang
The polar Emergent
hotly debated
.
transition of the compounds was experimentally confirmed8,9 .
Here we report the first experimental
of hybrid
Pohang demonstration
790-784, Korea
6
Some half-Heusler semiconductors are predicted
to be new improper ferroelectricity in bulk single crystals of (Ca,Sr)3 Ti2 O7 .
Rutgers
Center
for
Emergent
Materials
and
Department
of
Physics and Astronomy, Rutgers University,
7
piezoelectrics with large polarizations . The theoretical prediction Electric polarization versus electric field P(E) hysteresis loops
Piscataway,
New polarization
Jersey 08854,
of stabilizing ferroelectricity in strained Srn+1 Tin O3n+1 (n 3; ref. 6) clearly show the existence
of switchable
with a USA
was also experimentally confirmed in biaxially strained films, which unexpectedly
low switching
electric field.2014;
Moreover,
in-plane piezo(Received
9
September
published
22 January 2015)
exhibit switchable polarization at low temperatures10 ; Srn+1 Tin O3n+1 response force microscope (IP-PFM) images reveal intriguing
films with large n such as SrTiO3 , corresponding to the n = 1 ferroelectric domain structures comprising abundant meandering
We present newcharged
results
on the microscopic nature of the ferroelectricity mechanisms in Ca3 Mn2 O7 and
member, do show ferroelectricity at room temperature11–13 .
domain walls. To understand the origin of the unexpectedly
Geometric ferroelectrics are improper Ca
ferroelectrics
where
low
switching
electric field and
abundancethe
of charged
walls,
we
Ti
O
.
To
the
first
approximation,
wetheconfirm
hybrid
improper
ferroelectric mechanism recently
3 2 7
geometric structural constraints, rather than typical cation–anion propose a classification of eight types of ferroelectric and four types
by Benedek
and Fennie
for by
these
compounds.
However, in Ca Mn2 O7 we 1
2
paring, induce ferroelectric polarization14proposed
. Hybrid improper
of ferroelastic
domain1walls
meansRuddlesden-Popper
ofDawber
crystallographic
symmetry.
Eric Bousquet
*, Matthew
*{, Nicolas
Stucki2, Céline Lichtensteiger23, Patrick
Hermet , Stefano Gariglio2,
ferroelectricity, one kind of geometric ferroelectricity,
results
on this
classification, between
we
individual
2 suggest that
1 switching
find that there
is a Based
complex
competition
lattice
modes
of
different
symmetry
which
leads
to
a
phase
& Philippe
Ghosez
Jean-Marc
Triscone
from the combination of two or more non-ferroelectric structural of elementary
tilting
modes results
in the low
switching electric
LETTERS
Improper ferroelectricity in perovskite oxide artificial
superlattices
Alternative to tilts?
Tri-linearcouplingoflatticemodes:
E-field
E ∝ λ R1 R2 P
R1?
R2?
N
WantRtostronglycoupletoelectronic
degreesoffreedom:
S
orbital
magnetic
charge
Jahn-Teller distortion
Manganite systems
6
GOODENOUGH
LaMnO3/BiMnO3$
The“Q2”distortion:
Annu. Rev. Mater. Sci. 1998.28:1-27. Downloaded from www.annualreviews.org
Access provided by Imperial College London on 01/28/16. For personal use only.
Goodenough,Annu.Rev.Mater.Sci.2811998
YMnO3/BiMnO3$
YMnO3/LaMnO3$
Orthorhombicwithtwoshort,
mediumandlongbondlengths
A3+Mn3+O3/A’3+Mn3+O3
eg
3+ : d4
Mn
3+
e.g.Mn
Q
MJT
2 modes Q
Figure 2 The E vibrational
g
2
dx2-y2
and Q3 of an
octahedral-site complex.
dz2
complex is independent of ✓, which means that the
p ground state correspondsdz2
dx2-y2
to any point on the circle of radius ⇢ = = g/ C, where C is the stiffness
t2g constant associated with the vibrations. This situation corresponds to a dynamic coupling of the e electrons to the modes Q2 and Q3 and is referred to
as a dynamic J-T stabilization of vibronic states. In order to obtain a static J-T
deformation, it is necessary to introduce anharmonic terms into the potential
energy and/or
higher-order directly
coupling terms.connected
The ground state ofto
an octahedral
Bandgap
JT distortion
complex
then
becomes
Marcus Schmitz et al., unpublished
1E =
2
[C/2 + (A3
B3 ) cos 3✓],
5.
where A3 is generally positive (4). Although a point-charge calculation gives
B3 > 0, covalent considerations favor square-coplanar bonding, which makes
B3 < 0 and hence unambiguously favors a static deformation to tetragonal
!
Jahn-Teller distortion
Manganite systems
6
GOODENOUGH
LaMnO3/BiMnO3$
The“Q2”distortion:
Goodenough,Annu.Rev.Mater.Sci.2811998
R-point
M-point
Annu. Rev. Mater. Sci. 1998.28:1-27. Downloaded from www.annualreviews.org
Access provided by Imperial College London on 01/28/16. For personal use only.
MJT
YMnO3/BiMnO3$
YMnO3/LaMnO3$
Orthorhombicwithtwoshort,
mediumandlongbondlengths
RJT
A3+Mn3+O3/A’3+Mn3+O3
eg
3+ : d4
Mn
3+
e.g.Mn
Q
MJT
2 modes Q
Figure 2 The E vibrational
g
2
dx2-y2
and Q3 of an
octahedral-site complex.
dz2
complex is independent of ✓, which means that the
p ground state correspondsdz2
dx2-y2
to any point on the circle of radius ⇢ = = g/ C, where C is the stiffness
t2g constant associated with the vibrations. This situation corresponds to a dynamic coupling of the e electrons to the modes Q2 and Q3 and is referred to
as a dynamic J-T stabilization of vibronic states. In order to obtain a static J-T
deformation, it is necessary to introduce anharmonic terms into the potential
energy and/or
higher-order directly
coupling terms.connected
The ground state ofto
an octahedral
Bandgap
JT distortion
complex
then
becomes
Marcus Schmitz et al., unpublished
1E =
2
[C/2 + (A
B ) cos 3✓],
5.
3
3
Producesdifferentorbitalorderings,
where A3 is generally positive (4). Although a point-charge calculation gives
andhencespinorderings
B > 0, covalent considerations favor square-coplanar bonding, which makes
3
B3 < 0 and hence unambiguously favors a static deformation to tetragonal
!
Alternative to tilts – Jahn-Teller distortion
Tri-linearcouplingoflatticemodes:
E ∝ λ R1 R2 P
E-field
Alternative to tilts – Jahn-Teller distortion
pss
Phys. Status Solidi RRL 9, No. 1, 62–67 (2015) / DOI 10.1002/pssr.201409470
Tri-linearcouplingoflatticemodes:
High-temperature
ferroelectricity
www.pss-rapid.com
E-field
and strong magnetoelectric
effects
E ∝ R1 R2 P
in a hybrid organic–inorganic perovskite framework
λ
Ying Tian1, Alessandro Stroppa*, 2, Yi-Sheng Chai1, Paolo Barone2, Manuel Perez-Mato3, Silvia Picozzi2,
and Young Sun*, 1
N
S
1
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, P.R. China
2
CNR-SPIN, L’Aquila, Italy
3
Departamento de Fisica de la Materia Condensada, Facultad de Ciencia y Tecnologia, UPV/EHU, Bilbao, Spain
Received 10 October 2014, revised 3 November 2014, accepted 10 November 2014
Published online 18 November 2014
orbital
magnetic
Keywords metal-organic framework, multiferroic,
magnetoelectric effect, hydrogen bond
*
Corresponding authors: e-mail [email protected], [email protected]
Potentialapplications:magnetoelectrics,electrochromic,MITs,transistors….??
A Cu-based organic–inorganic perovskite framework exhibits
high-temperature ferroelectricity with strong magnetoelectric
effects. Both electric field control of magnetization and magnetic field control of polarization are realized. Theoretical
calculations suggest that a new mechanism of hybrid improper ferroelectricity arising from the Jahn–Teller distortions of
magnetic metal ions and tilting of the organic cations are responsible for the peculiar multiferroic behaviors.
SimilarmechanismproposedinrelatedMOFs
formultiferroic magnetoelectric applications
[1] Stroppa et al., Adv.Mater.25,2284(2013)
[2] Tian etal.,Phys.StatusSolidiRRL9,62(2015)
bulk modulus B = (c11 +2c12 )/3 of cubic SrRuO3
to be 172 GPa. No experimental data are a
comparison, but bulk modulus values of 200 a
have been reported in previous LSDA calculation
the equation of states [25], which likely overesti
MJT isallowed,andalwaysappears,inPnma perovskites [1,2]
to the typical overbinding tendency of the LSDA
Lattice-driven Jahn-Teller distortion
Figure 2. The calculated phonon dispersion curves and phonon density of states of cubic SrR
MJT
0–X–M–0–R–M
− of the cubic Brillouin zone. The total DOS and the projected DOS of oxyge
plotted using a solid line (in black), a dashed
line (in blue), a dottedinstabilities
line (in green) and
shor
4. Antiferrodistortive
anda dist
JT
xy
xy
frequencies indicate imaginary values.
E ∝ λM φ A
phases
Axy
Φxy-
As previously discussed, the cubic structure
points, attesting tha
exhibits strong antiferrodistortive instabilities, ass
exchange–correlatio
rotations of the oxygen octahedra. If, in line with
Finally, the ela
we restrict ourselves to the oxygen rotations ass
were computed wit
+
M+
that such r
3 and R4 modes and consider
method [37]. For c
appear along any of the three cubic directions,
elastic constants: c
six distinct basic tilt patterns (spanning the six-1
312.8 GPa, 101.8 G
M+
R+
of the Pm3̄m
3
4 reducible representation
to the Voigt–Reuss
can be combined to generate various tilted struc
bulk modulus B = (
Figure 3. A schematic illustration of relevant phonon modes of
group theory analysis, Howard and Stokes [49] d
to be 172 GPa. N
SrRuO3 . Black arrows indicate the atomic motions. Sr atoms (inQ2
the
existence
of
15
distinct
combinations o
Doesnothavetoappearasanelectronicinstability!
red) at the corners, Ru atoms (in green) at the centers and O atoms
comparison, but bu
patterns that, when condensed within the Pm3̄m
(in blue) at the face centers of the perovskite cubic cell. The
have been reported i
lower the symmetry to distinct subgroups. T
wavevectors are also given in parentheses for corresponding modes.
the equation of stat
HerewedefineaJahn-Tellerdistortion:
oxygen-tilted structures are usually specified us
to the typical overbi
bythesymmetryofthemode(Q2),whetheritiselectronicallyorlatticedriven
Glazer notations [48] such as a0 b+ c , in whi
literals refer to the three cubic directions and
[1] Carpenter & Howard, Acta Cryst. B 65, 134 (2009)
we also performed the phonon calculation with the LSDA
4. Antiferrodistor
superscripts refer to the condensation
of no
[2] Miao, Bristowe, Xu, Verstraete & Ghosez, JPCM 26 035401 (2014)
P1: psa/dpk
P2: ARK/plb
June 3, 1998
14:4
27. Downloaded from www.annualreviews.org
ge London on 01/28/16. For personal use only.
6
QC: ARK
Annual Reviews
AR059-01
GOODENOUGH
Figure 2 The Eg vibrational modes Q2 and Q3 of an octahedral-site complex.
complex is independent of ✓, which means that the
p ground state corresponds
to any point on the circle of radius ⇢ = = g/ C, where C is the stiffness
constant associated with the vibrations. This situation corresponds to a dynamic coupling of the e electrons to the modes Q2 and Q3 and is referred to
+
Highlight three P-JT couplings
Symmetryanalysissupportedbyfirstprinciplescalculations(PBEsol+U and/orB1WC)
1)Superlattices (d1-d0)
Titanates:ATiO3-RTiO3
2)Superlattices (d2-d2)
Vanadates:RVO3-R’VO3
3)Epitaxialbulk(alldfillings)
ferrites,titanates,manganites …
−
xy xy
E ∝ λ M JT φ P
Bristowe, Varignon, Fontaine, Bousquet & Ghosez,
Nat. Commun. 6, 6677 (2015)
E ∝ λ M JT Pz RJT
Varignon, Bristowe, Bousquet & Ghosez,
Sci Reports 5, 15364 (2015)
E ∝ λ M JT Pxy Axy
Varignon, Bristowe & Ghosez,
Phys. Rev. Lett 116, 057602 (2016)
Highlight three P-JT couplings
Symmetryanalysissupportedbyfirstprinciplescalculations(PBEsol+U and/orB1WC)
1)Superlattices (d1-d0)
Titanates:ATiO3-RTiO3
2)Superlattices (d2-d2)
Vanadates:RVO3-R’VO3
3)Epitaxialbulk(alldfillings)
ferrites,titanates,manganites …
−
xy xy
E ∝ λ M JT φ P
Bristowe, Varignon, Fontaine, Bousquet & Ghosez,
Nat. Commun. 6, 6677 (2015)
E ∝ λ M JT Pz RJT
Varignon, Bristowe, Bousquet & Ghosez,
Sci Reports 5, 15364 (2015)
E ∝ λ M JT Pxy Axy
Varignon, Bristowe & Ghosez,
Phys. Rev. Lett 116, 057602 (2016)
R is a trivalent
ferromagnetic and ferroelectric insulating ground state.
a rich variety of
The electronic structure exhibits
an intricate
orbital and
3
3
oped manganites
charge ordering which is argued to be at the heart of the
ype AFM metallic
observed ferromagnetism. A symmetry lowering struca CE-type AFM
tural distortion enabling this particular orbital ordering
Ground-state
2+ =Sr,Ba,(Ca)
A
with two different
(Inallcases!)
lumnar [11]), and 3.5+
” and “antiferro” Ti =Ti
TABLE I: Key quantities for a selection of ATiO3 -RTiO3 sud electronic phase R3+=La,Pr,Sm Y,
- MonoclinicP21symmetry
perlattices including amplitude Q (Å) of lattice distortions
n the A and R Tm,(Lu)
−
(in-phase Φ+
z and anti-phase Φxy AFD motions, polar- mode
Insulating
disordered (such
Pxy , Jahn-Teller mode appearing at the M -point of the cu- Ferroelectric
ch as for Ba and
bic Brillouin zone MJT , breathing Jahn-Teller BJT ), polari-
mpare the physics
of the half-doped
nd R cations are
the titanates, and
d Mott insulating
An exception has
of very small Aa rocksalt charget insulating phase
r hand, in layered
ATiO -RTiO superlattice
sation, P (µC/cm2 ), band gap, ∆ (eV), and gain of energy
- Ferromagnetic
for FM vs AFM solution (see Methods) per 20-atom formula
unit, ∆E (meV).
Symmetryadaptedmodeanalysis
R, A
Sm, Sr
Y, Sr
Tm, Sr
Sm, Ba
Y, Ba
Tm, Ba
Φ+
z
Φ−
xy
0.96
1.10
1.18
0.75
0.95
1.05
1.19
1.30
1.36
0.96
1.08
1.16
(a0a0c+)(a-a-c0)
Q
Pxy
0.56
0.66
0.72
0.48
0.59
0.65
BJT
0.10
0.11
0.11
0.13
0.14
0.16
MJT
0.04
0.04
0.03
0.07
0.07
0.07
P
∆
∆E
14.9
16.7
18.2
18.6
21.2
23.4
0.46
0.57
0.63
0.50
0.60
0.66
20.1
18.0
16.4
18.5
13.9
10.5
Pnma-like(a-a-c+)
groundstate
E = λφ xy− φ z+ Pxy
Ferroelectric:
AmplificationofP
throughdissimilarZ*
can display ferromagnetic (FM) or A-type AFM metallic
behaviour [5–7] or more commonly a CE-type AFM
Mott insulating phase [8, 9] associated with two different
charge orderings (Rocksalt [10] and Columnar [11]), and
two different orbital orderings (“ferro” and “antiferro”
Mn d eg orderings [12]). The preferred electronic phase
appears to be strongly dependent on the A and R
cation sizes and whether they appear disordered (such
as with Ca and La/Pr) or layered (such as for Ba and
La/Tb/Y [5, 10, 12]) in the crystal.
observed ferromagnetism. A symmetry lowering structural distortion enabling this particular orbital ordering
System remains insulating
In this regard, it is interesting to compare the physics
of the half-doped manganites, with that of the half-doped
titanates. At the bulk level, the A and R cations are
found to naturally disorder [13, 14] in the titanates, and
typically no charge and orbital ordered Mott insulating
phase is observed at half-doping [15]. An exception has
been recently discovered for the case of very small Acations, such as Ca0.5 Lu0.5 TiO3 , where a rocksalt chargeordered and dxy t2g orbital-ordered Mott insulating phase
was recently proposed [16]. On the other hand, in layered
TABLE I: Key quantities for a selection of ATiO3 -RTiO3 suz distortions
perlattices including amplitude Q (Å) of lattice
y
+
−
(in-phase Φz and anti-phase Φxy AFD motions, polar mode
Pxy , Jahn-Teller mode appearing at the M -point of the cux
bic Brillouin zone MJT , breathing Jahn-Teller BJT ), polarisation, P (µC/cm2 ), band gap, ∆ (eV), and gain of energy
for FM vs AFM solution (see Methods) per 20-atom formula
unit, ∆E (meV).
R, A
Sm, Sr
Y, Sr
Tm, Sr
Sm, Ba
Y, Ba
Tm, Ba
Φ+
z
Φ−
xy
0.96
1.10
1.18
0.75
0.95
1.05
1.19
1.30
1.36
0.96
1.08
1.16
Q
Pxy
0.56
0.66
0.72
0.48
0.59
0.65
BJT
0.10
0.11
0.11
0.13
0.14
0.16
MJT
0.04
0.04
0.03
0.07
0.07
0.07
P
∆
∆E
14.9
16.7
18.2
18.6
21.2
23.4
0.46
0.57
0.63
0.50
0.60
0.66
20.1
18.0
16.4
18.5
13.9
10.5
System remains insulating
z
y
x
d0
d1
d1
d0
d1
d0
d0
d1
Charge ordered (d1- d0) state associated with a Breathing distortion
LiftsthedegeneracybetweenneighboringTisites
Chargeorderreportedinsimilarsystems:
R. Pentcheva and W.E. Pickett, Phys. Rev. Lett. 99, 016802 (2007)
A.C. Komarek et al., arXiv:1109.0234 (2012)
Electronic structure – charge+ orbital ordering
Projected DOS
z
Total Density of States
O 2p
y
‘d1’
x
Ti t2g
-0.5
dxz
0.0
Total up
Total down
Ti1 dxz + Ti3 dyz
Ti2+4 dxz+yz
-4
-3
-2
-1
0
Energy (eV)
1
mixed
‘d0’
- Spin-polarizedsplit-offd1 band
- Nottheideald1–d0 occupancy
- d1 site:orbitalorderingdxz – dyz
- d0 site:orbitalmixing
‘d1’d
yz
nly a CE-type AFM
ated with two different
d Columnar [11]), and
ferro” and “antiferro”
erred electronic phase
nt on the A and R
pear disordered (such
d (such as for Ba and
l.
o compare the physics
that of the half-doped
A and R cations are
] in the titanates, and
€
dered Mott insulating
15]. An exception has
case of very small Ahere a rocksalt chargeMott insulating phase
other hand, in layered
tural distortion enabling this particular orbital ordering
Origin of orbital ordering
TABLE I:M
Keydistortionallowedbysymmetry
quantities for a selection of ATiO3 -RTiO3 suJT
perlattices including amplitude Q (Å) of lattice distortions
−
(in-phase (equivalenttothecouplinginPnma)
Φ+
z and anti-phase Φxy AFD motions, polar mode
Pxy , Jahn-Teller mode appearing at the M -point of the cu−
Jahn-Teller− BJT ), polaribic Brillouin zone MJT , +breathing
z gap,
xy ∆ (eV),
xy
JT
and xy
gain of energy
sation, P (µC/cm2 ),xyband
for FM vs AFM solution (see Methods) per 20-atom formula
unit, ∆E (meV).
F ∝P φ φ +P M φ
R, A
Sm, Sr
Y, Sr
Tm, Sr
Sm, Ba
Y, Ba
Tm, Ba
Φ+
z
0.96
1.10
1.18
0.75
0.95
1.05
Φ−
xy
1.19
1.30
1.36
0.96
1.08
1.16
Q
Pxy
0.56
0.66
0.72
0.48
0.59
0.65
BJT
0.10
0.11
0.11
0.13
0.14
0.16
MJT
0.04
0.04
0.03
0.07
0.07
0.07
P
∆
∆E
14.9
16.7
18.2
18.6
21.2
23.4
0.46
0.57
0.63
0.50
0.60
0.66
20.1
18.0
16.4
18.5
13.9
10.5
Origin of orbital ordering
MJT distortionsproducestheC-typeorbitalordering
F ∝ Pxyφ φ + Pxy M JT φ
+
z
MJT distortion
€
−
xy
−
xy
FM due to intrasite Hund’s
FM Hund’s
dxz!
dxz!
dyz!
pz!
dyz!
py!
AFM Pauli’s
dxy!
dxy!
px!
Ti1
“d1”
dxy!
py!
O1
Ti2
“d0”
O2
Ti3
“d1”
MJT crucial for FM
IfweartificiallysuppressAFDmotionsandhenceMJT
−
xy xy
E ∝ λ M JT φ P
- Noorbitalordering:dxy occupancyeverywhere
- AFMGS
- Pauli’sexclusionprinciple
FM Hund’s
WithMJT motions:
dxz!
dxz!
dyz!
pz!
dyz!
py!
AFM Pauli’s
WithoutMJT motions:
dxy!
dxy!
px!
Ti1
“d1”
dxy!
py!
O1
Ti2
“d0”
O2
Ti3
“d1”
Highlight three P-JT couplings
Symmetryanalysissupportedbyfirstprinciplescalculations(PBEsol+U and/orB1WC)
1)Superlattices (d1-d0)
Titanates:ATiO3-RTiO3
2)Superlattices (d2-d2)
Vanadates:RVO3-R’VO3
3)Epitaxialbulk(alldfillings)
ferrites,titanates,manganites …
−
xy xy
E ∝ λ M JT φ P
Bristowe, Varignon, Fontaine, Bousquet & Ghosez,
Nat. Commun. 6, 6677 (2015)
E ∝ λ M JT Pz RJT
Varignon, Bristowe, Bousquet & Ghosez,
Sci Reports 5, 15364 (2015)
E ∝ λ M JT Pxy Axy
Varignon, Bristowe & Ghosez,
Phys. Rev. Lett 116, 057602 (2016)
RVO3
Pnma
P21/c
G-o.o
Pnma
Pnma
P21/c
G-o.o
P21/c
G-o.o
G-type AFM
Pnma
C-o.o
P21/c
C-o.o
+
G-o.o
C-type AFM
Rare-earthvanadates Pnma atroomT,andwithdecreasingT
appearanceof:
- CandGtypeorbitalorderings
- GandCtypeAFMorderings
- StructuralphasetransitiontoP21/cforC-AFM
Sage et al, PRB 76 195102 (2007)
P
properties, including ferroelectricity, (anti)ferromagnetism
Φ+z
(AFM), superconductivity and magnetoresistance. This
3
diverse behaviour is appealing
for both3 fundamental and applied
P4/mmm
investigations, and has resulted in an intense global research
structure
P4/mmmref
effort over the past few decades. Many of these functional
AFMG Pb2and
AFMCPb
• Ground
state:
1m subtle
properties
manifest due to the complex
interplay
between spin, charge,
orbital and lattice
degrees of +7.76meV
freedom in
YVO/LaVO
0meV
perovskites1–4. Of the perovskites, the doped manganites have
PrVO/YVO
0meV
-0.87meV
Breathing oxygen
become a prototypical playground for the study of this interplay.
BOC
0meV
Just considering thePrVO/LaVO
case of half-doping,
that is, A20:5þ -3.72meV
R30:5þ MnO3 ,
3 þ is a
metal
ion
and
R
where A2 þ is a divalent alkaline earth
(“Pnma”)
(“P21/c”)
_
trivalent rare earth ion, manganites exhibit a rich variety
(“Pm3m”)
of electronic phases. For example, half-doped manganites can
display ferromagnetic
(FM) or A-type
AFM metallic
behaviour
• Symmetry
mode analysis
of ground
states
(Å) 5–7
8,9
or more commonly a CE-type- AFM +Mott insulating
phase
Φxycharge
Φzorderings
Φz- (rocksalt
RJT 10 and
MJT
Pz
Pxy
associated with two different
and
two different
columnar11)Pb2
YVO/LaVO
1.58 orbital
1.14 orderings (‘ferro’ and
0.12
0.77
1m AFMG
12). The preferred electronic phase
Figure 1 | Superlattice geometry, ma
‘antiferro’
Mn
d
e
orderings
g
PrVO/YVO
Pb AFMC
1.61
1.16
0.01
0.10
0.04
0.01
0.81
appears to be strongly dependent on the A2 þ and R3 þ cation resulting ferroelectric, charge and o
3 þ TiO digital
PrVO/LaVO
Pb AFMC
1.36disordered
0.94 (such
0.01 as with
0.10 Ca and
0.01 R0.00(4)
0.59 superlattice 10-atom
sizes and whether
they appear
3
-c0)
(a-afor
reference structure undergoes two m
a0c+) La/Tb/Y5,10,12) in the
La/Pr) or layered (such as
Ba(a0and
motions and a breathing oxygen cage
crystal.
“Pnma”-liketiltpattern
In this regard, it is interesting to compare the physics of the of large (blue) and small (grey) octahe
half-doped manganites, with that of the half-doped titanates. At are shown in the 20-atom cell. The A
3 þ cations are found to naturally
Pb
component,andcombinationofbothJahn-Tellers
through a unique anharmonic couplin
thephaseshaveadditionalP
bulk level, the A2 þ andz R
disorder13,14 in the titanates, and typically no charge and orbital- combination of the AFD motions and
RVO -R’VO Superlattices
R’VO3-RVO3 couplings
(RVO3)1/(R’VO3)1 superlattice expansion
[2]
• Pb21m (Pnma in bulk)
• Pb (P21/c in bulk)
[1]
New trilinear coupling identified
Out-of-plane polarization coupled to Jahn-Teller !
[1] Bousquet et al, Nature 452 (2008)
[2] Fukushima et al, Phys. Chem. Chem. Phys 13 (2011); Rondinelli et al, Adv. Materials 24 (2012)
Magnetoelectric application?
Electric field driven magnetic transition ?
YVO/LaVO: ΔE(AFMG-AFMC) = -7.76 meV
Ground-state: Pb21m (‘’Pnma’’) – AFMG phase:
Change of orbital and
AFM orderings !
Electric field driven magnetic transition
Finite electric field method (transition at 0.55 V / bilayer)
Pb21m
Pb
E-field directly controls RJT distortion amplitude!
In turn, this induces M transition
Highlight three P-JT couplings
Symmetryanalysissupportedbyfirstprinciplescalculations(PBEsol+U and/orB1WC)
1)Superlattices (d1-d0)
Titanates:ATiO3-RTiO3
2)Superlattices (d2-d2)
Vanadates:RVO3-R’VO3
3)Epitaxialbulk(alldfillings)
ferrites,titanates,manganites …
−
xy xy
E ∝ λ M JT φ P
Bristowe, Varignon, Fontaine, Bousquet & Ghosez,
Nat. Commun. 6, 6677 (2015)
E ∝ λ M JT Pz RJT
Varignon, Bristowe, Bousquet & Ghosez,
Sci Reports 5, 15364 (2015)
E ∝ λ M JT Pxy Axy
Varignon, Bristowe & Ghosez,
Phys. Rev. Lett 116, 057602 (2016)
Can couplings appear in general bulk ABO3?
Strainengineering?
TERS
Can couplings appear in general bulk ABO3?
Strainengineering?
PHYSICAL
UnusualPmc21phase(equivalentlyPb21m)undertensilestrain:
GroundstateforBiFeO3,PbTiO3,BaMnO3,EuTiO3,CaTiO3 (+?)atabout5%[1]
week ending
“Orbitalordering”observedforBiFeO
3 AUGUST 2012
3 [1]
(thoughnoJahn-Tellerdistortionmentioned)
Tri-linearcouplingfound[2]:
(butnotinvolvingJahn-Teller?)
E ∝ λ Axyφ z+ Pxy
Axy
(M5+)
[1]Yangetal.,Phys.Rev.Lett.109 057602(2012)
F1:1
[2]Yangetal.,Phys.Rev.Lett.112 057202(2014)
FIG. 1.
Schematic view of the main four lattice distort
Reinvestigate Pmc21 phase
PHYSICAL REVIEW LETTERS
Trulygeneral?Testonarangeofd-fillings
TABLE I. Epitaxial strain (%), magnetic ground state, amplitudes of distortions (Å), and electronic band gap value
(eV) for each material. The spontaneous polarization is also reported in μC cm−2 . Only the relevant distortions are
d0
d3
d4
d5
summarized in the present table [59].
Strain
Magnetism
P (Γ−5 )
þ
M
Qþ
2JT (M 3 )
A (M þ
5)
þ
ϕz (M þ
2)
Gap
(%)
(Å)
(μC cm−2 )
(Å)
(Å)
(Å)
(eV)
SrTiO3
BaMnO3
BiFeO3*
þ7.35 [61]
NM
0.615
76
0.232
0.558
0.640
3.02
þ6.1 [61]
FM
0.421
45
0.190
0.217
P H Y0.059
SICAL
0.28
þ5.8 [61]
AFMG
0.346
29
0.644
1.072
R E1.668
VIEW
1.88
YMnO3*
þ4.0 [61]
AFMG
0.753
7 [62]
0.737
0.940
L E T1.733
TERS
1.88
the authors reported
the existence of
*AlsodevelopΦ
xy
169
170
171
172
173
174
175
direction is along the [001] axis of the Pbnm structure).
Beyond around 5% tensile
strain, the four compounds,
th modefound:M
4
indeed, develop the desired Pb21 m ground state.JTStrained
BaMnO3 [ferromagnetic (FM)]
and YMnO3 [G-type antiVerylarge!
ferromagnetic (AFMG)] exhibit a different magnetic
ground state compared to the bulk (AFMG and E-type
antiferromagnetic, ↑↑↓↓ zigzag chains coupled antiferro-
3þ
the
Fe
3d orbitals,active
explained
from
expected since YMnO3 is known to be Jahn-Teller
in
210
polar
and
yield
the bulk. We emphasize, at this
stage,
thatthe
theantipolar
polar modemotion
in
211
[33].
This 212
orbita
BiFeO3 (and YMnO3 ) is not distortion
unstable, andpattern
therefore,
highly
in this ferroelectric
system, noinJahn-Telle
strained BiFeO3 appears assince,
an improper
213
3þ
contradiction to Refs. [18]form
anda [64].
the (Fe
214
Mott Computing
insulating state
a
2
phonons in the intermediatespin
strained
phase of both
215
t32g ePbnm
g configuration). A Jahn-Te
BiFeO3 and YMnO3 compounds
only one in
hybrid
is yetreveals
to be reported
the Pb2216
1m p
form a M
spin
configuration). A Jahn-Tell
spin t32g e2g
is yet to be reported in the Pb2
phb
1 m to
is yet
best of our knowledge. Frombest
ourofsyo
clearly demonstrate that, as this
Pb2de
clearly
1
aforementioned distortions (P,aforemen
A, and
+
Axy
Φz of Eq. (1) is automatically lowered
of Eq. thr
(1)
(M5+)
(M2+)
four
of a fourth lattice distortion:of aa Jahn
Therefore
Therefore, while it may not Teller
be unst
mo
Teller motion is forced into the
system
mechanis
mechanism arising from theresult
trilinea
cla
F1:1
FIG.+ 1. Schematic view of the
main
four
lattice
distortions
result clarifies the origin of displayed
the unu
in
the
Pb2
m
phase
of
perovskites
under
tensile
FIG. 1. Schematic view of theF1:2
mainxyinvolved
four
lattice
distortions
P Hby
Y
STI C
A
L
REV
I EitW
z xy
xy 1 JT xydisplayed
achieve
a
;
moreover,
pr
þ 3R
− EBiFeO
P
H
Y
S
I
C
A
L
R
E
V
I
E
W
L
T
E
S
epitaxial strain.
Polar distortion (irreps Γ5 ), (b) Q2 Jahninvolved in the Pb21 m phase F1:3
of perovskites
under(a) tensile
þ
bulkofpero
achieve an electric field control
th
þ
−
F1:4
Teller
distortion
(irreps
M
3 ), (c) antipolar A distortion (irreps
epitaxial strain. (a) Polar distortion (irreps
Γ
),
(b)
Q
Jahn2
5
þ
þ
the pr
a
The
F1:5
M 5 ), (d) a0 a0 cþ ϕþ
antiferrodistortive
motion
(irreps
M 2 ). the existence
þ
bulk
perovskites.
z
the
authors
reported
Teller distortion (irreps M 3 ), (c) antipolar A distortion (irreps
perovskit
thePb
F
3þ
þ
0 0 þ þ
The
predicted
highly
strained
the
Fe
3d
orbitals,
explained
fro
Mþ
),
(d)
a
a
c
ϕ
antiferrodistortive
motion
(irreps
M
).
z
2
5
to occur
a
polar
perovskites
is the
notperovskite
restricted motion
toinBiFeO
Starting from the reference
Pm3̄mand
cubic
86
polar
antipolar
yi
3
BaMn
Pxy
M
−
disto
phase, the condensation JT
of
the
mode
Ppattern
(irreps
Γ
87
to polar
occur
also in
some titanates
(CaTiO
distortion
[33].
This
orb
5)
pound
Tb
+
(GM
)
þ (M3 )
þ
5 3̄m cubic perovskite
Starting from the reference88Pm
since
and the JT mode Q2 (irreps Min3 ) BaMnO
lowers the
a no in
[33],system,
andtoeven
athen
Jaha
since,
in3 symmetry
this
Jahn-Te
are
−
phase, the condensation of the 89polarPb2
mode
P (irreps
Γ5 ) subgroup
form
a polar
ofTbMnO
Pbnm. We,
then,
3þ
1 m phase,
pound
[36].
The
highly
strai
between
form
a
Mott
insulating
state
(Fe
3
þ
þ
perform
free energytoexpansion
the reference
and the JT mode Q2 (irreps M3 90
) lowers
the asymmetry
a
spin
3 2ideal
order
to c
are[25]
then(around
an
playground
to
demo
spin
t
e
configuration).
A
Jahng
2g
structure)
termsthen,
of the lattice distortions allowed by
91
Pb21 m phase, a polar subgroup
of
Pbnm.in We,
yet
between the polarization and this
theisLette
Jahn
Invariants analysis
F1:1
F1:2
F1:3
F1:4
F1:5
86
87
88
89
E ∝ λA φ P + λA M P
t32g e2g
E-field control of gap via JT
YMnO3 largesteffectsinceitisJTactive
HYSICAL REVIEW LETTERS
+
4,5
0,4
0,2
1,5
-1
-0,5
0
0
0,05
0
-0,05
-0,1
0
-0,5
dz2
1
1,5
EFermi (eV)
2
0,5
0
-0,5
-1
2,5
0,3
0,2
SrTiO3
BaMnO3
YMnO3
BiFeO3
0,1
0
0,6
∆Gap (eV)
4
0,4
ΔQ∆Q
MJT (Å)
2
5
4
3
2
1
0
dxz
0,4
0,2
0
-0,2
0
10
5
15
-1
Electric field (MV.cm )
20
On-going/Future research in my group
Emergentphenomenaatperovskite
interfaces:
DFT and DFPT Phonon Calculations
Khang Le
Negativethermalexpansion
Innaturallylayeredperovskites
Fvib =
c
X ~!sq
sq
2
✓
+kB T ln 1
e
Strainengineeringstructuralphases
◆
~!sq
kB T
200
150
100
JordanCowell
-1
ω (cm )
ChrisAblitt
50
Grüneisen Analysis
0
AndrewWarwick
-50
γ
i
12
10
8
6
4
2
0
-2
-4
=
✓
@ ln (!i )
@ ln(V )
◆
P5 M5+ Γ5X3+ Λ5
-100
-150
Emergentferroic ordersatdomainwalls
Γ
Photoferroicity inlayeredhybridperovskites
7
Methodology
“Effectivepotential”forlatticedynamics
Wojdel etal.,JPCM25 305401(2013)
Energy changes around reference structure due to distortions:
(1) Energy change from atomic
displacements (p: phonons), with:
(2) Energy change
due to strain only
FiniteE:PbTiO3-SrTiO3
0.4
Polarization along z (C/m2)
Strain (from DFT cubic SRO)
0.014
0.012
KENNEDY, HUNTER, AND HESTER
0.008
0.3
Pz
Px
0.04
0.2
0.02
0.1
PHYSICAL REVIEW B 65 224103
0
0
-0.1
0.006
0.06
-0.02
-0.2
0.004
0.002
-0.04
-0.3
0
FIG. 1. Temperature dependence of the lattice parameters for
SrRuO3 . The open and closed symbols give the results of duplicate
measurements. The vertical dashed lines show the transitions between the three phases.
100 200 300 400 500 600 700 800 900
Temperature (K)
reflections indicating the presence of in-phase !!" and outof-phase !"" tilts, respectively.17 The data were well fitted to
9,10,12
FIG. 2. Portions of the synchrotron diffraction patterns recorded
at 363, 678, 883, and 948 K, showing the splitting indicative Pnma,
Imma, I4/mcm and Pm3 m phases, respectively. The indices of the
reflections in the cubic structure are indicated.
-0.4
-0.15 -0.1 -0.05 0 0.05 0.1
Electric field along z (GV/m)
lost while the peaks clearly show a tetragonal splitting and
the assignment of cubic symmetry is the point at which no
asymmetry of the diagnostic reflections remains.
The diffraction data clearly reveal the presence of two
-0.06
0.15
Polarization along x (C/m2)
FiniteT:SrRuO3
0.01
(3) Strain-phonon
coupling term
Collaborators
Theoretical Materials Physics, University of Liege, BELGIUM
Philippe Ghosez, Julien Varignon (now at CNRS/Thales), Eric Bousquet
CSIC, UAB, Spain
Massimiliano Stengel, Miguel Pruneda
Department of Materials, Imperial College London, UK
Arash Mostofi, Chris Ablitt
Department of Chemistry, University of Warwick, UK
Mark Senn
LIST, Luxembourg
Jorge Iniguez
Collaborators
Theoretical Materials Physics, University of Liege, BELGIUM
Philippe Ghosez
Eric Bousquet
Julien Varignon
NowatCNRS,Thales,France
Denis Fontaine