Doped HfO2 thin films exhibit robust ferroelectric properties even for nanometric thicknesses, are compatible with current Si technology, and thus have great potential for the revival of integrated ferroelectrics. Phase control and reliability are core issues for their applications. Here, we show that, in (111)-oriented 5%La:HfO2 (HLO) epitaxial thin films deposited on (La0.3Sr0.7)(Al0.65Ta0.35)O3 substrates, the flexoelectric effect, arising from the strain gradient along the film's normal, induces a rhombohedral distortion in the otherwise Pca21 orthorhombic structure. Density functional calculations reveal that the distorted structure is indeed more stable than the pure Pca21 structure, when applying an electric field mimicking the flexoelectric field. This rhombohedral distortion greatly improves the fatigue endurance of HLO thin films by further stabilizing the metastable ferroelectric phase against the transition to the thermodynamically stable non-polar monoclinic phase during repetitive cycling. Our results demonstrate that the flexoelectric effect, though negligibly weak in bulk, is crucial to optimize the structure and properties of doped HfO2 thin films with nanometric thicknesses for integrated ferroelectric applications.

1.
T. S.
Böscke
,
J.
Mller
,
D.
Bräuhaus
,
U.
Schröder
, and
U.
Böttger
, “
Ferroelectricity in hafnium oxide thin films
,”
Appl. Phys. Lett.
99
,
102903
(
2011
).
2.
J.
Müller
,
P.
Polakowski
,
S.
Mueller
, and
T.
Mikolajick
, “
Ferroelectric hafnium oxide based materials and devices: Assessment of current status and future prospects
,”
ECS J. Solid State Sci. Technol.
4
,
N30
(
2015
).
3.
S. S.
Cheema
,
D.
Kwon
,
N.
Shanker
,
R.
Reis
,
S.
Hsu
,
J.
Xiao
,
H.
Zhang
,
R.
Wagner
,
A.
Datar
,
M. R.
McCarter
,
C. R.
Serrao
,
A. K.
Yadav
,
G.
Karbasian
,
C.
Hsu
,
A. J.
Tan
,
L.
Wang
,
V.
Thakare
,
X.
Zhang
,
A.
Mehta
,
E.
Karapetrova
,
R.
Chopdekar
,
P.
Shafer
,
E.
Arenholz
,
C.
Hu
,
R.
Proksch
,
R.
Ramesh
,
J.
Ciston
, and
S.
Salahuddin
, “
Enhanced ferroelectricity in ultrathin films grown directly on silicon
,”
Nature
580
,
478
482
(
2020
).
4.
H.
Lee
,
M.
Lee
,
K.
Lee
,
J.
Jo
,
H.
Yang
,
Y.
Kim
,
S.
Chae
,
U.
Waghmare
, and
J.
Lee
, “
Scale-free ferroelectricity induced by flat phonon bands in HfO2
,”
Science
369
,
1343
(
2020
).
5.
S. S.
Cheema
,
N.
Shanker
,
L.
Wang
,
C.
Hsu
,
S.
Hsu
,
Y.
Liao
,
M.
Jose
,
J.
Gomez
,
W.
Chakraborty
,
W.
Li
,
J.
Bae
,
S. K.
Volkman
,
D.
Kwon
,
Y.
Rho
,
G.
Pinelli
,
R.
Rastogi
,
D.
Pipitone
,
C.
Stull
,
M.
Cook
,
B.
Tyrrell
,
V. A.
Stoica
,
Z.
Zhang
,
J. W.
Freeland
,
C. J.
Tassone
,
A.
Mehta
,
G.
Saheli
,
D.
Thompson
,
D.
Suh
,
W.
Koo
,
K.
Nam
,
D.
Jung
,
W.
Song
,
C.
Lin
,
S.
Nam
,
J.
Heo
,
N.
Parihar
,
C. P.
Grigoropoulos
,
P.
Shafer
,
P.
Fay
,
R.
Ramesh
,
S.
Mahapatra
,
J.
Ciston
,
S.
Datta
,
M.
Mohamed
,
C.
Hu
, and
S.
Salahuddin
, “
Ultrathin ferroic HfO2–ZrO2 superlattice gate stack for advanced transistors
,”
Nature
604
,
65
71
(
2022
).
6.
T.
Francois
,
J.
Coignus
,
A.
Makosiej
,
B.
Giraud
,
C.
Carabasse
,
J.
Barbot
,
S.
Martin
,
N.
Castellani
,
T.
Magis
,
H.
Grampeix
,
S.
Duijn
,
C.
Mounet
,
P.
Chiquet
,
U.
Schroeder
,
S.
Slesazeck
,
T.
Mikolajick
,
E.
Nowak
,
M.
Bocquet
,
N.
Barrett
,
F.
Andrieu
, and
L.
Grenouillet
, “
High-performance operation and solder reflow compatibility in BEOL-integrated 16-kb HfO2:Si-based 1T-1C FeRAM arrays
,”
IEEE Trans. Electron Devices
69
,
2108
(
2022
).
7.
M.
Hoffmann
,
F.
Fengler
,
M.
Herzig
,
T.
Mittmann
,
B.
Max
,
U.
Schroeder
,
R.
Negrea
,
P.
Lucian
,
S.
Slesazeck
, and
T.
Mikolajick
, “
Unveiling the double-well energy landscape in a ferroelectric layer
,”
Nature
565
,
464
467
(
2019
).
8.
K.
Kim
,
Y.
Kim
,
M.
Park
,
H.
Park
,
Y.
Kwon
,
Y.
Lee
,
H.
Kim
,
T.
Moon
,
Y.
Lee
,
S.
Hyun
,
B.
Kim
, and
C.
Hwang
, “
Transient negative capacitance effect in atomic-layer-deposited Al2O3/Hf0.3Zr0.7O2 bilayer thin film
,”
Adv. Funct. Mater.
29
,
1808228
(
2019
).
9.
K.
Lee
,
J.
Byun
,
K.
Park
,
S.
Kang
,
M.
Song
,
J.
Park
,
J.
Lee
, and
S.
Chae
, “
Giant tunneling electroresistance in epitaxial ferroelectric ultrathin films directly integrated on Si
,”
Appl. Mater. Today
26
,
101308
(
2022
).
10.
H.
Yoong
,
H.
Wu
,
J.
Zhao
,
H.
Wang
,
R.
Guo
,
J.
Xiao
,
B.
Zhang
,
P.
Yang
,
S.
Pennycook
,
N.
Deng
,
X.
Yan
, and
J.
Chen
, “
Epitaxial ferroelectric Hf0.5Zr0.5O2 thin films and their implementations in memristors for brain-inspired computing
,”
Adv. Funct. Mater.
28
,
1806037
(
2018
).
11.
G.
Li
,
X.
Li
,
X.
Liu
,
A.
Gao
,
J.
Zhao
,
F.
Yan
, and
Q.
Zhu
, “
Heteroepitaxy of Hf0.5Zr0.5O2 ferroelectric gate layer on AlGaN/GaN towards normally-off HEMTs
,”
Appl. Surf. Sci.
597
,
153709
(
2022
).
12.
Y.
Yun
,
P.
Buragohain
,
M.
Li
,
Z.
Ahmadi
,
Y.
Zhang
,
X.
Li
,
H.
Wang
,
J.
Li
,
P.
Lu
,
L.
Tao
,
H.
Wang
,
J. E.
Shield
,
E. Y.
Tsymbal
,
A.
Gruverman
, and
X.
Xu
, “
Intrinsic ferroelectricity in Y-doped HfO2 thin films
,”
Nat. Mater.
21
,
903
909
(
2022
).
13.
S.
Starschich
,
D.
Griesche
,
T.
Schneller
,
R.
Waser
, and
U.
Böttger
, “
Chemical solution deposition of ferroelectric yttrium-doped hafnium oxide films on platinum electrodes
,”
Appl. Phys. Lett.
104
,
202903
(
2014
).
14.
T.
Olsen
,
U.
Schröder
,
S.
Müller
,
A.
Krause
,
D.
Martin
,
A.
Singh
,
J.
Müller
,
M.
Geidel
, and
T.
Mikolajick
, “
Co-sputtering yttrium into hafnium oxide thin films to produce ferroelectric properties
,”
Appl. Phys. Lett.
101
,
082905
(
2012
).
15.
P.
Nukala
,
M.
Ahmadi
,
Y.
Wei
,
S.
Graaf
,
E.
Stylianidis
,
T.
Chakrabortty
,
S.
Matzen
,
H. W.
Zandbergen
,
A.
Björling
,
D.
Mannix
,
D.
Carbone
,
B.
Kooi
, and
B.
Noheda
, “
Reversible oxygen migration and phase transitions in hafnia-based ferroelectric devices
,”
Science
372
,
630
635
(
2021
).
16.
T.
Schenk
,
C. M.
Fancher
,
M.
Park
,
C.
Richter
,
C.
Künneth
,
A.
Kersch
,
J. L.
Jones
,
T.
Mikolajick
, and
U.
Schroeder
, “
On the origin of the large remanent polarization in La:HfO2
,”
Adv. Electron. Mater.
5
,
1900303
(
2019
).
17.
T.
Shiraishi
,
S.
Choi
,
T.
Kiguchi
, and
T. J.
Konno
, “
Structural evolution of epitaxial CeO2-HfO2 thin films using atomic-scale observation: Formation of ferroelectric phase and domain structure
,”
Acta Mater.
235
,
118091
(
2022
).
18.
D.
Martin
,
J.
Müller
,
T.
Schenk
,
T. M.
Arruda
,
A.
Kumar
,
E.
Strelcov
,
E.
Yurchuk
,
S.
Müller
,
D.
Pohl
,
U.
Schröder
,
S. V.
Kalinin
, and
T.
Mikolajick
, “
Ferroelectricity in Si-Doped HfO2 revealed: A binary lead-free ferroelectric
,”
Adv. Mater.
26
,
8198
8202
(
2014
).
19.
S.
Mueller
,
J.
Mueller
,
A.
Singh
,
S.
Riedel
,
J.
Sundqvist
,
U.
Schroeder
, and
T.
Mikolajick
, “
Incipient ferroelectricity in Al-doped HfO2 thin films
,”
Adv. Funct. Mater.
22
,
2412
2417
(
2012
).
20.
J.
Müller
,
U.
Schröder
,
T. S.
Böscke
,
I.
Müller
,
U.
Böttger
,
L.
Wilde
,
J.
Sundqvist
,
M.
Lemberger
,
P.
Kücher
,
T.
Mikolajick
, and
L.
Frey
, “
Ferroelectricity in yttrium-doped hafnium oxide
,”
J. Appl. Phys.
110
,
114113
(
2011
).
21.
Z.
Zhang
,
S.
Hsu
,
V. A.
Stoica
,
H.
Paik
,
E.
Parsonnet
,
A.
Qualls
,
J.
Wang
,
L.
Xie
,
M.
Kumari
,
S.
Das
,
Z.
Leng
,
M.
McBriarty
,
R.
Proksch
,
A.
Gruverman
,
D. G.
Schlom
,
L.
Chen
,
S.
Salahuddin
,
L. W.
Martin
, and
R.
Ramesh
, “
Epitaxial ferroelectric Hf0.5Zr0.5O2 with metallic pyrochlore oxide electrodes
,”
Adv. Mater.
33
,
2006089
(
2021
).
22.
L.
Bégon-Lours
,
M.
Mulder
,
P.
Nukala
,
S.
Graaf
,
Y. A.
Birkhölzer
,
B.
Kooi
,
B.
Noheda
,
G.
Koster
, and
G.
Rijnders
, “
Stabilization of phase-pure rhombohedral HfZrO4 in pulsed laser deposited thin films
,”
Phys. Rev. Mater.
4
,
043401
(
2020
).
23.
J.
Lyu
,
I.
Fina
,
R.
Bachelet
,
G.
Saint-Girons
,
S.
Estandía
,
J.
Gázquez
,
J.
Fontcuberta
, and
F.
Sánchez
, “
Enhanced ferroelectricity in epitaxial Hf0.5Zr0.5O2 thin films integrated with Si(001) using SrTiO3 templates
,”
Appl. Phys. Lett.
114
,
222901
(
2019
).
24.
P.
Nukala
,
Y.
Wei
,
V.
Haas
,
Q.
Guo
,
J.
Antoja-Lleonart
, and
B.
Noheda
, “
Guidelines for the stabilization of a polar rhombohedral phase in epitaxial Hf0.5Zr0.5O2 thin films
,”
Ferroelectrics
569
,
148
163
(
2020
).
25.
S.
Estandía
,
J.
Gàzquez
,
M.
Varela
,
N.
Dix
,
M.
Qian
,
R.
Solanas
,
I.
Fina
, and
F.
Sánchez
, “
Critical effect of the bottom electrode on the ferroelectricity of epitaxial Hf0.5Zr0.5O2 thin films
,”
J. Mater. Chem. C
9
,
3486
3492
(
2021
).
26.
S.
Estandía
,
T.
Cao
,
R.
Mishra
,
I.
Fina
,
F.
Sánchez
, and
J.
Gazquez
, “
Insights into the atomic structure of the interface of ferroelectric Hf0.5Zr0.5O2 grown epitaxially on La2/3Sr1/3MnO3
,”
Phys. Rev. Mater.
5
,
074410
(
2021
).
27.
Y.
Qi
,
S.
Singh
,
C.
Lau
,
F.
Huang
,
X.
Xu
,
F. J.
Walker
,
C. H.
Ahn
,
S.
Cheong
, and
K. M.
Rabe
, “
Stabilization of competing ferroelectric phases of HfO2 under epitaxial strain
,”
Phys. Rev. Lett.
125
,
257603
(
2020
).
28.
Y.
Cheng
,
Z.
Gao
,
K.
Ye
,
H.
Park
,
Y.
Zheng
,
Y.
Zheng
,
J.
Gao
,
M.
Park
,
J.
Choi
,
K.
Xue
,
C.
Hwang
, and
H.
Lyu
, “
Reversible transition between the polar and antipolar phases and its implications for wake-up and fatigue in HfO2-based ferroelectric thin film
,”
Nat. Commun.
13
,
645
(
2022
).
29.
Y.
Wei
,
P.
Nukala
,
M.
Salverda
,
S.
Matzen
,
H.
Zhao
,
J.
Momand
,  
G.
Everhardt
,
G. R.
Agnus
,
P.
Blake
,
B. J.
Lecoeur
,
J.
Kooi
,
B.
Íñiguez
,
B.
Dkhil
, and
Noheda
,
A. S.
,  “
A rhombohedral ferroelectric phase in epitaxially strained Hf0.5Zr0.5O2 thin films
,”
Nat. Mater.
17
,
1095
1100
(
2018
).
30.
M.
Pešić
,
F.
Fengler
,
L.
Larcher
,
A.
Padovani
,
T.
Schenk
,
E. D.
Grimley
,
X.
Sang
,
J. M.
LeBeau
,
S.
Slesazeck
,
U.
Schroeder
, and
T.
Mikolajick
, “
Physical mechanisms behind the field-cycling behavior of HfO2-based ferroelectric capacitors
,”
Adv. Funct. Mater.
26
,
4601
4612
(
2016
).
31.
T.
Mikolajick
,
S.
Slesazeck
,
M. H.
Park
, and
U.
Schroeder
, “
Ferroelectric hafnium oxide for ferroelectric random-access memories and ferroelectric field-effect transistors
,”
MRS Bull.
43
,
340
346
(
2018
).
32.
M. H.
Park
,
Y. H.
Lee
,
T.
Mikolajick
,
U.
Schroeder
, and
C.
Hwang
, “
Review and perspective on ferroelectric HfO2-based thin films for memory applications
,”
MRS Commun.
8
,
795
808
(
2018
).
33.
J.
Lyu
,
I.
Fina
,
R.
Solanas
,
J.
Fontcuberta
, and
F.
Sánchez
, “
Robust ferroelectricity in epitaxial Hf1/2Zr1/2O2 thin films
,”
Appl. Phys. Lett.
113
,
082902
(
2018
).
34.
S. S.
Fields
,
S. W.
Smith
,
P. J.
Ryan
,
S. T.
Jaszewski
,
I. A.
Brummel
,
A.
Salanova
,
G.
Esteves
,
S. L.
Wolfley
,
M. D.
Henry
,
P. S.
Davids
, and
J. F.
Ihlefeld
, “
Phase-exchange-driven wake-up and fatigue in ferroelectric hafnium zirconium oxide films
,”
ACS Appl. Mater. Interfaces
12
,
26577
26585
(
2020
).
35.
J.
Yuan
,
G.
Mao
,
K.
Xue
,
N.
Bai
,
C.
Wang
,
Y.
Cheng
,
H.
Lyu
,
H.
Sun
,
X.
Wang
, and
X.
Miao
, “
Ferroelectricity in HfO2 from a coordination number perspective
,”
Chem. Mater.
35
,
94
103
(
2023
).
36.
W.
Hamouda
,
C.
Lubin
,
S.
Ueda
,
Y.
Yamashita
,
O.
Renault
,
F.
Mehmood
,
T.
Mikolajick
,
U.
Schroeder
,
R.
Negrea
, and
N.
Barrett
, “
Interface chemistry of pristine TiN/La:Hf0:5Zr0:5O2 capacitors
,”
Appl. Phys. Lett.
116
,
252903
(
2020
).
37.
F.
Mehmood
,
M.
Hoffmann
,
P. D.
Lomenzo
,
C.
Richter
,
M.
Materano
,
T.
Mikolajick
, and
U.
Schroeder
, “
Bulk depolarization fields as a major contributor to the ferroelectric reliability performance in lanthanum doped Hf0.5Zr0.5O2 capacitors
,”
Adv. Mater. Interfaces
6
,
1901180
(
2019
).
38.
A. G.
Chernikova
,
M. G.
Kozodaev
,
D. V.
Negrov
,
E. V.
Korostylev
,
M.
Park
,
U.
Schroeder
,
C.
Hwang
, and
A. M.
Markeev
, “
Improved ferroelectric switching endurance of La-doped Hf0.5Zr0.5O2 thin films
,”
ACS Appl. Mater. Interfaces
10
,
2701
2708
(
2018
).
39.
P.
Jiao
,
J.
Li
,
Z.
Xi
,
X.
Zhang
,
J.
Wang
,
Y.
Yang
,
Y.
Deng
, and
D.
Wu
, “
Ferroelectric Hf0.5Zr0.5O2 thin films deposited epitaxially on (110)-oriented SrTiO3
,”
Appl. Phys. Lett.
119
,
252901
(
2021
).
40.
A.
Demkov
and
A.
Navrotsky
,
Materials Fundamentals of Gate Dielectrics
(
Springer
,
Dordrecht
,
2005
).
41.
S.
Einfeldt
,
V.
Kirchner
,
H.
Heinke
,
M.
Dießelberg
,
S.
Figge
,
K.
Vogeler
, and
D.
Hommel
, “
Strain relaxation in AlGaN under tensile plane stress
,”
J. Appl. Phys.
88
(
12
),
7029
(
2000
).
42.
S.
Park
,
T. S.
Cho
,
S. J.
Doh
, and
J. H.
Je
, “
Structural evolution of ZnO/sapphire (001) heteroepitaxy studied by real time synchrotron x-ray scattering
,”
Appl. Phys. Lett
77
(
3
),
349
(
2000
).
43.
B.
Voigtländer
and
A.
Zinner
, “
Simultaneous molecular beam epitaxy growth and scanning tunneling microscopy imaging during Ge/Si epitaxy
,”
Appl. Phys. Lett.
63
,
3055
(
1993
).
44.
M.
Materano
,
P. D.
Lomenzo
,
A.
Kersch
,
M. H.
Park
,
T.
Mikolajick
, and
U.
Schroeder
, “
Interplay between oxygen defects and dopants: Effect on structure and performance of HfO2-based ferroelectrics
,”
Inorg. Chem. Front.
8
,
2650
(
2021
).
45.
R.
Materlik
,
C.
Kunneth
,
M.
Falkowski
,
T.
Mikolajick
, and
A.
Kersch
, “
Al-, Y-, and La-doping effects favoring intrinsic and field induced ferroelectricity in HfO2: A first principles study
,”
J. Appl. Phys.
123
,
164101
(
2018
).
46.
Y.
Zhou
,
Y. K.
Zhang
,
Q.
Yang
,
J.
Jiang
,
P.
Fan
,
M.
Liao
, and
Y. C.
Zhou
, “
The effects of oxygen vacancies on ferroelectric phase transition of HfO2- based thin film from first-principle
,”
Comput. Mater. Sci.
167
,
143
(
2019
).
47.
N.
Kaiser
,
Y.
Song
,
T.
Vogel
,
E.
Piros
,
T.
Kim
,
P.
Schreyer
,
S.
Petzold
,
R.
Valentí
, and
L.
Alff
, “
Crystal and electronic structure of oxygen vacancy stabilized rhombohedral hafnium oxide
,”
ACS Appl. Electron. Mater.
5
,
754
763
(
2023
).
48.
V.
Lenzi
,
J. P. B.
Silva
,
B.
Šmíd
,
V.
Matolín
,
C. M.
Istrate
,
C.
Ghica
,
J. L.
MacManus-Driscoll
, and
L.
Marques
, “
Ferroelectricity induced by oxygen vacancies in rhombohedral ZrO2 thin films
,”
Energy Environ. Mater.
2023
,
e12500
.
49.
R.
Materlik
,
C.
Künneth
, and
A.
Kersch
, “
The origin of ferroelectricity in Hf1−xZrxO2: A computational investigation and a surface energy model
,”
J. Appl. Phys.
117
,
134109
(
2015
).
50.
X.
Sang
,
E. D.
Grimley
,
T.
Schenk
,
U.
Schroeder
, and
J. M.
LeBeau
, “
On the structural origins of ferroelectricity in HfO2 thin films
,”
Appl. Phys. Lett.
106
,
162905
(
2015
).
51.
M.
Park
,
H.
Kim
,
Y.
Kim
,
T.
Moon
, and
C.
Hwang
, “
The effects of crystallographic orientation and strain of thin Hf0.5Zr0.5O2 film on its ferroelectricity
,”
Appl. Phys. Lett.
104
,
072901
(
2014
).
52.
S.
Estandía
,
N.
Dix
,
M. F.
Chisholm
,
I.
Fina
, and
F.
Sánchez
, “
Domain-matching epitaxy of ferroelectric Hf0.5Zr0.5O2(111) on La2/3Sr1/3MnO3(001)
,”
Cryst. Growth Des.
20
,
3801
3806
(
2020
).
53.
C. V.
Ramana
,
K. K.
Bharathi
,
A.
Garcia
, and
A. L.
Campbell
, “
Growth behavior, lattice expansion, strain, and surface morphology of nanocrystalline, monoclinic HfO2 thin films
,”
J. Phys. Chem. C
116
,
9955
9960
(
2012
).
54.
M.
Royo
and
M.
Stengel
, “
Lattice-mediated bulk flexoelectricity from first principles
,”
Phys. Rev. B
105
,
064101
(
2022
).
55.
O.
Diéguez
and
M.
Stengel
, “
Translational covariance of flexoelectricity at ferroelectric domain walls
,”
Phys. Rev. X
12
,
031002
(
2022
).
56.
D.
Lee
,
A.
Yoon
,
S. Y.
Jang
,
J.-G.
Yoon
,
J.-S.
Chung
,
M.
Kim
,
J. F.
Scott
, and
T. W.
Noh
, “
Giant flexoelectric effect in ferroelectric epitaxial thin films
,”
Phys. Rev. Lett.
107
,
057602
(
2011
).
57.
Y.
Li
,
C.
Adamo
,
P.
Chen
,
P. G.
Evans
,
S. M.
Nakhmanson
,
W.
Parker
,
C. E.
Rowland
,
R. D.
Schaller
,
D. G.
Schlom
,
D. A.
Walko
,
H.
Wen
, and
Q.
Zhang
, “
Giant optical enhancement of strain gradient in ferroelectric BiFeO3 thin films and its physical origin
,”
Sci. Rep.
5
,
16650
(
2015
).
58.
H.
Zhong
,
M.
Li
,
Q.
Zhang
,
L.
Yang
,
R.
He
,
F.
Liu
,
Z.
Liu
,
G.
Li
,
Q.
Sun
,
D.
Xie
,
F.
Meng
,
Q.
Li
,
M.
He
,
E.
Guo
,
C.
Wang
,
Z.
Zhong
,
X.
Wang
,
L.
Gu
,
G.
Yang
,
K.
Jin
,
P.
Gao
, and
C.
Ge
, “
Large-scale Hf0.5Zr0.5O2 membranes with robust ferroelectricity
,”
Adv. Mater.
34
,
2109889
(
2022
).
59.
H.
Cao
,
V. C.
Lo
, and
Z.
Li
, “
Simulation of flexoelectricity effect on imprint behavior of ferroelectric thin films
,”
Solid State Commun.
138
,
404
(
2006
).
60.
B.
Xu
,
J.
Íñiguez
, and
L.
Bellaiche
, “
Designing lead-free antiferroelectrics for energy storage
,”
Nat. Commun.
8
,
15682
(
2017
).
61.
Y.
Qi
,
S.
Liu
,
A. M.
Lindenberg
, and
A. M.
Rappe
, “
Ultrafast electric field pulse control of giant temperature change in ferroelectrics
,”
Phys. Rev. Lett.
120
,
055901
(
2018
).
62.
R.
Ruh
and
P. W. R.
Corfield
, “
Crystal structure of monoclinic hafnia and comparison with monoclinic zirconia
,”
J. Am. Ceram. Soc.
53
,
126
129
(
1970
).
63.
K. B.
Lee
,
S.
Tirumala
, and
S. B.
Desu
, “
Highly c-axis oriented Pb(Zr,Ti)O3 thin films grown on Ir electrode barrier and their electrical properties
,”
Appl. Phys. Lett.
74
,
1484
(
1999
).
64.
J.
Degrieck
and
W. V.
Paepegem
, “
Fatigue damage modeling of fibre-reinforced composite materials: Review
,”
Appl. Mech. Rev.
54
,
4
(
2001
).
65.
H. T.
Hahn
and
R. Y.
Kim
, “
Fatigue behavior of composite laminate
,”
J. Compos. Mater.
10
,
156
180
(
1976
).
66.
M.
Krysko
,
J. Z.
Domagala
,
R.
Czernecki
, and
M.
Leszczynski
, “
Triclinic deformation of InGaN layers grown on vicinal surface of GaN (00.1) substrates
,”
J. Appl. Phys.
114
,
113512
(
2013
).
67.
G.
Kresse
and
J.
Furthmüller
, “
Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
,”
Phys. Rev. B
54
,
11169
(
1996
).
68.
P. E.
Blöchl
, “
Projector augmented-wave method
,”
Phys. Rev. B
50
,
17953
(
1994
).
69.
J. P.
Perdew
,
A.
Ruzsinszky
,
G. I.
Csonka
,
O. A.
Vydrov
,
G. E.
Scuseria
,
L. A.
Constantin
,
4. X.
Zhou
, and
K.
Burke
, “
Restoring the density-gradient expansion for exchange in solids and surfaces
,”
Phys. Rev. Lett.
100
,
136406
(
2008
).
70.
H.
Fu
and
L.
Bellaiche
, “
First-principles determination of electromechanical responses of solids under finite electric fields
,”
Phys. Rev. Lett.
91
,
057601
(
2003
).
71.
L.
Chen
,
C.
Xu
,
H.
Tian
,
H.
Xiang
,
J.
Íñiguez
,
Y.
Yang
, and
L.
Bellaiche
, “
Electric-field control of magnetization, Jahn–Teller distortion, and orbital ordering in ferroelectric ferromagnets
,”
Phys. Rev. Lett.
122
,
247701
(
2019
).
72.
R.
Resta
,
M.
Posternak
, and
A.
Baldereschi
, “
Towards a quantum theory of polarization in ferroelectrics: The case of KNbO3
,”
Phys. Rev. Lett.
70
,
1010
(
1993
).

Supplementary Material

You do not currently have access to this content.