The polarization and strain response of ferroelectric materials at electric fields below the macroscopic coercive field is of paramount importance for the operation of many electronic devices. The response of real ferroelectric and related materials is, in general, complex and difficult to interpret. The reason for this is that many processes in a ferroelectric material contribute to its properties, often concurrently. Examples include the motion of ferroelectric and ferroelastic domains, the presence of domains within domains, the dynamics of different types of polar nano-entities, the interaction of polar nano-entities (e.g., polar nanoregions in relaxors) with the strain and polarization within domains, motion of defects, and rearrangement of defect clusters and their interaction with polarization and strain. One signature of these processes is nonlinearity of the strain and polarization. Most ferroelectrics exhibit nonlinear response at all practical field levels, meaning that the apparent material coefficients depend on the amplitude of the driving excitation. In this paper, we show that an investigation of nonlinear behavior is a sensitive way to study various mechanisms operating in dielectric and piezoelectric materials. We review the basic formalism of the nonlinear description of polarization and strain, give a physical interpretation of different terms, and illustrate this approach on numerous examples of relaxors, relaxor ferroelectrics, hard and soft ferroelectrics, and morphotropic phase boundary compositions. An experimental approach based on a lock-in technique that is well suited for such studies is also discussed.

1.
A.
Grigoriev
,
R.
Sichel
,
H. N.
Lee
,
E. C.
Landahl
,
B.
Adams
,
E. M.
Dufresne
, and
P. G.
Evans
,
Phys. Rev. Lett.
100
,
027604
(
2008
).
2.
D. A.
Hall
,
J. Mater. Sci.
36
,
4575
(
2001
).
3.
A.
Pramanick
,
D.
Damjanovic
,
J. C.
Nino
, and
J. L.
Jones
,
J. Am. Ceram. Soc.
92
,
2291
(
2009
).
4.
L.
Jin
,
F.
Li
, and
S.
Zhang
,
J. Am. Ceram. Soc.
97
,
1
(
2014
).
5.
K.
Kuramoto
and
E.
Nakamura
,
Ferroelectrics
157
,
57
(
1994
).
6.
L. M.
Riemer
,
K.
Chu
,
Y.
Li
,
H.
Uršič
,
A. J.
Bell
,
B.
Dkhil
, and
D.
Damjanovic
,
Appl. Phys. Lett.
117
,
102901
(
2020
).
7.
M.
Otoničar
,
A.
Bradeško
,
L.
Fulanović
,
T.
Kos
,
H.
Uršič
,
A.
Benčan
,
M. J.
Cabral
,
A.
Henriques
,
J. L.
Jones
,
L.
Riemer
,
D.
Damjanovic
,
G.
Dražić
,
B.
Malič
, and
T.
Rojac
,
Adv. Funct. Mater.
,
30
,
2006823
(
2020
).
8.
N.
Bassiri-Gharb
,
I.
Fujii
,
E.
Hong
,
S.
Trolier-McKinstry
,
D. V.
Taylor
, and
D.
Damjanovic
,
J. Electroceram.
19
,
49
(
2007
).
9.
F.
Li
,
S.
Zhang
,
Z.
Xu
,
X.
Wei
,
J.
Luo
, and
T. R.
Shrout
,
Appl. Phys. Lett.
96
,
192903
(
2010
).
10.
K.
Tamm
,
Ber. Bunsen. Phys. Chem.
84
,
1190
(
1980
).
11.
D.
Damjanovic
, in
The Science of Hysteresis
(
Academic Press
,
Oxford
,
2006
), Vol. III, pp.
337
465
.
12.
G. H.
Haertling
,
J. Am. Ceram. Soc.
82
,
797
(
1999
).
13.
D.
Sundararajan
,
Fourier Analysis—A Signal Processing Approach
(
Springer
,
Singapore
,
2018)
.
14.
T.
Furukawa
,
M.
Date
, and
E.
Fukada
,
J. Appl. Phys.
51
,
1135
(
1980
).
15.
M.
Morozov
,
D.
Damjanovic
, and
N.
Setter
,
J. Eur. Ceram. Soc.
25
,
2483
(
2005
).
16.
J. F.
Nye
,
Physical Properties of Crystals: Their Representation by Tensors and Matrices
(
Oxford University Press
,
Oxford
,
1984
).
17.
F.
Li
,
L.
Jin
,
Z.
Xu
, and
S.
Zhang
,
Appl. Phys. Rev.
1
,
011103
(
2014
).
18.
R.
Pirc
,
R.
Blinc
, and
V. S.
Vikhnin
,
Phys. Rev. B
74
, (
2006
).
19.
J.
Dec
,
W.
Kleemann
,
S.
Miga
,
C.
Filipic
,
A.
Levstik
,
R.
Pirc
,
T.
Granzow
, and
R.
Pankrath
,
Phys. Rev. B
68
,
092105
(
2003
).
20.
P. H.
Sydenham
and
R.
Thorn
,
Handbook of Measuring System Design
(
Wiley
,
Chichester
,
2005
).
21.
M.-M.
Yang
,
Z.-D.
Luo
,
Z.
Mi
,
J.
Zhao
,
S.P. E
,
and
M.
Alexe
,
Nature
584
,
377
(
2020
).
22.
M. I.
Morozov
and
D.
Damjanovic
,
J. Appl. Phys.
104
,
034107
(
2008
).
23.
L.
Néel
,
Cah. Phys.
12
,
1
(
1942
).
25.
G.
Bertotti
,
Hysteresis in Magnetism: For Physicists, Materials Scientists, and Engineers
(
Academic Press
,
San Diego
,
1998
).
26.
R.
Holland
,
IEEE Trans. Sonics Ultrason.
14
,
18
(
1967
).
27.
Q. M.
Zhang
,
H.
Wang
,
N.
Kim
, and
L. E.
Cross
,
J. Appl. Phys.
75
,
454
(
1994
).
28.
M. E.
Caspari
and
W. J.
Merz
,
Phys. Rev.
80
,
1082
(
1950
).
29.
R. E.
Newnham
,
V.
Sundar
,
R.
Yimnirun
,
J.
Su
, and
Q. M.
Zhang
,
J. Phys. Chem. B
101
,
10141
(
1997
).
30.
M.
Sjostrom
,
IEEE Trans. Magn.
35
,
2097
(
1999
).
31.
S.
Hashemizadeh
and
D.
Damjanovic
,
Appl. Phys. Lett.
110
,
192905
(
2017
).
33.
Lord
Rayleigh
,
Lond. Edinb. Dubl. Philos. Mag. J. Sci.
23
,
225
(
1887
).
34.
H.
Kronmüller
,
Z. Angew. Phys.
30
,
9
(
1970
).
35.
A.
Turik
,
Sov. Phys. Solid State
5
,
885
(
1963
).
36.
D.
Damjanovic
and
M.
Demartin
,
J. Phys. D Appl. Phys.
29
,
2057
(
1996
).
37.
M.
Algueró
,
B.
Jiménez
, and
L.
Pardo
,
Appl. Phys. Lett.
83
,
2641
(
2003
).
38.
S.
Kustov
,
E.
Cesari
,
I.
Liubimova
,
V.
Nikolaev
, and
E. K. H.
Salje
,
Scr. Mater.
134
,
24
(
2017
).
39.
D.
Damjanovic
,
J. Appl. Phys.
82
,
1788
(
1997
).
40.
A.
Pramanick
,
D.
Damjanovic
,
J. E.
Daniels
,
J. C.
Nino
, and
J. L.
Jones
,
J. Am. Ceram. Soc.
94
,
293
(
2011
).
41.
I.
Urbanavičiūtė
,
T. D.
Cornelissen
,
X.
Meng
,
R. P.
Sijbesma
, and
M.
Kemerink
,
Nat. Commun.
9
,
1
(
2018
).
42.
S. A.
Turik
,
L. A.
Reznitchenko
,
A. N.
Rybjanets
,
S. I.
Dudkina
,
A. V.
Turik
, and
A. A.
Yesis
,
J. Appl. Phys.
97
,
064102
(
2005
).
43.
I. D.
Mayergoyz
,
G.
Friedman
, and
C.
Salling
,
IEEE Trans. Magn.
25
,
3925
(
1989
).
44.
45.
46.
G.
Robert
,
D.
Damjanovic
,
N.
Setter
, and
A. V.
Turik
,
J. Appl. Phys.
89
,
5067
(
2001
).
47.
G.
Robert
,
D.
Damjanovic
, and
N.
Setter
,
J. Appl. Phys.
90
,
2459
(
2001
).
48.
Y.
Saito
,
Jpn. J. Appl. Phys.
34
,
5313
(
1995
).
49.
Q. M.
Zhang
,
W. Y.
Pan
,
S. J.
Jang
, and
L. E.
Cross
,
J. Appl. Phys.
64
,
6445
(
1988
).
50.
M.
Morozov
, “Softening and hardening transitions in ferroelectric Pb(Zr,Ti)O3 ceramics,”
Ph.D. thesis
,
3368
(
EPFL
,
2005
).
51.
D.
Berlincourt
and
H. H. A.
Krueger
,
J. Appl. Phys.
30
,
1804
(
1959
).
52.
S.
Li
,
W.
Cao
, and
L. E.
Cross
,
J. Appl. Phys.
69
,
7219
(
1991
).
53.
V.
Mueller
and
Q. M.
Zhang
,
Appl. Phys. Lett.
72
,
2692
(
1998
).
54.
D. V.
Taylor
,
D.
Damjanovic
, and
N.
Setter
,
Ferroelectrics
224
,
299
(
1999
).
55.
D. V.
Taylor
, “Dielectric and piezoelectric properties of sol-gel derived Pb(Zr, Ti)O3 thin films,”
Ph.D. thesis
,
1949
(
EPFL
,
1999
).
56.
C.
Zhao
,
D.
Hou
,
C.-C.
Chung
,
H.
Zhou
,
A.
Kynast
,
E.
Hennig
,
W.
Liu
,
S.
Li
, and
J. L.
Jones
,
Acta Mater.
158
,
369
(
2018
).
57.
J.
Toulouse
,
Ferroelectrics
369
,
203
(
2008
).
58.
O.
Aktas
and
E. K. H.
Salje
,
Appl. Phys. Lett.
113
,
202901
(
2018
).
59.
A. E.
Glazounov
and
A. K.
Tagantsev
,
Ferroelectrics
221
,
57
(
1999
).
60.
W.
Kleemann
and
J.
Dec
,
Ferroelectrics
553
,
1
(
2019
).
61.
J.
Hlinka
,
J. Adv. Dielect.
02
,
1241006
(
2012
).
62.
M.
Eremenko
,
V.
Krayzman
,
A.
Bosak
,
H. Y.
Playford
,
K. W.
Chapman
,
J. C.
Woicik
,
B.
Ravel
, and
I.
Levin
,
Nat. Commun.
10
,
2728
(
2019
).
63.
S.
Prosandeev
,
B.
Xu
, and
L.
Bellaiche
,
Phys. Rev. B
98
,
024105
(
2018
).
64.
H.
Takenaka
,
I.
Grinberg
,
S.
Liu
, and
A. M.
Rappe
,
Nature
546
,
391
(
2017
).
65.
M.
Tyunina
and
J.
Levoska
,
Phys. Rev. B
72
,
104112
(
2005
).
66.
V. V.
Shvartsman
and
A. L.
Kholkin
,
Z. Kristallogr.
226
,
108
(
2011
).
67.
P. M.
Gehring
,
H.
Hiraka
,
C.
Stock
,
S.-H.
Lee
,
W.
Chen
,
Z.-G.
Ye
,
S. B.
Vakhrushev
, and
Z.
Chowdhuri
,
Phys. Rev. B
79
,
224109
(
2009
).
68.
A.
Bradeško
,
L.
Fulanović
,
M.
Vrabelj
,
M.
Otoničar
,
H.
Uršič
,
A.
Henriques
,
C.-C.
Chung
,
J. L.
Jones
,
B.
Malič
,
Z.
Kutnjak
, and
T.
Rojac
,
Acta Mater.
169
,
275
(
2019
).
69.
N.
Bassiri-Gharb
,
S.
Trolier-McKinstry
, and
D.
Damjanovic
,
J. Appl. Phys.
110
,
124104
(
2011
).
70.
L. E.
Cross
,
Ferroelectrics
76
,
241
(
1987
).
71.
A. E.
Glazounov
and
A. K.
Tagantsev
,
J. Phys. Condens. Matter
10
,
8863
(
1998
).
72.
F.
Li
,
S.
Zhang
,
T.
Yang
,
Z.
Xu
,
N.
Zhang
,
G.
Liu
,
J.
Wang
,
J.
Wang
,
Z.
Cheng
,
Z.-G.
Ye
,
J.
Luo
,
T. R.
Shrout
, and
L.-Q.
Chen
,
Nat. Commun.
7
,
13807
(
2016
).
73.
A.
Kumar
,
J. N.
Baker
,
P. C.
Bowes
,
M. J.
Cabral
,
S.
Zhang
,
E. C.
Dickey
,
D. L.
Irving
, and
J. M.
LeBeau
,
Nat. Mater.
20
,
62
(
2021
).
74.
G.
Xu
,
J.
Wen
,
C.
Stock
, and
P. M.
Gehring
,
Nat. Mater
7
,
562
(
2008
).
75.
T.
Sluka
,
A. K.
Tagantsev
,
D.
Damjanovic
,
M.
Gureev
, and
N.
Setter
,
Nat. Commun.
3
,
1
(
2012
).
76.
S.
Wada
,
S.
Suzuki
,
T.
Noma
,
T.
Suzuki
,
M.
Osada
,
M.
Kakihana
,
S.-E.
Park
,
L. E.
Cross
, and
T. R.
Shrout
,
Jpn. J. Appl. Phys.
38
,
5505
(
1999
).

Supplementary Material

You do not currently have access to this content.