The solid solution Ba Sr TiO (BSTO) displays dielectric response that is highly tunable, while also exhibiting low losses in a broad frequency regime, including the microwave band. Therefore, there is a need for a better understanding of the influence of the BSTO microstructure on its relaxor properties and performance in a variety of technological applications. Since the local polarization in BSTO is strongly dependent on composition, so is its response to an applied AC field. In this work, we have adopted a phase field method to study the frequency-dependent dielectric response of this system while accounting for the local fluctuations in the solid-solution composition. By utilizing a thermodynamic potential that includes spatial dependence on the averaged Sr content, we connected relaxor-like features in the dielectric dispersion to local spatial inhomogeneities, such as average size of Sr- or Ba-rich regions, across a wide range of temperatures. These results show that the adopted simple coarse-grained approach to the relaxor problem is sensitive enough to reveal correlations between the frequency and temperature dependence of the dielectric response and modulations in the material morphology and microstructure.
REFERENCES
1.
L. E.
Cross
, “Relaxor ferroelectrics
,” Ferroelectrics
76
, 241
(1987
). 2.
A. A.
Bokov
and Z.-G.
Ye
, “Recent progress in relaxor ferroelectrics with perovskite structure
,” J. Mater. Sci.
41
, 31
(2006
). 3.
N.
Ortega
, A.
Kumar
, J. F.
Scott
, D. B.
Chrisey
, M.
Tomazawa
, S.
Kumari
, D. G. B.
Diestra
, and R. S.
Katiyar
, “Relaxor-ferroelectric superlattices: High energy density capacitors
,” J. Phys.: Condens. Matter
24
, 445901
(2012
). 4.
P.
Zhao
, H.
Wang
, W.
Longwen
, L.
Chen
, C.
Ziming
, L.
Li
, and X.
Wang
, “High-performance relaxor ferroelectric materials for energy storage applications
,” Adv. Energy Mater.
9
, 1803048
(2019
). 5.
S.
Guru
, M. W.
Cole
, N. X.
Sun
, T. S.
Kalkur
, N. M.
Sbrockey
, G. S.
Tompa
, X.
Guo
, C.
Chen
, S. P.
Alpay
, G. A.
Rossetti
, K.
Dayal
, L.-Q.
Chen
, and D. G.
Schlom
, “Challenges and opportunities for multi-functional oxide thin films for voltage tunable radio frequency/microwave components
,” J. Appl. Phys.
114
, 191301
(2013
). 6.
L.
Yang
, H.
Huang
, Z.
Xi
, L.
Zheng
, S.
Xu
, G.
Tian
, Y.
Zhai
, F.
Guo
, L.
Kong
, Y.
Wang
, W.
Lü
, L.
Yuan
, M.
Zhao
, H.
Zheng
, and G.
Liu
, “Simultaneously achieving giant piezoelectricity and record coercive field enhancement in relaxor-based ferroelectric crystals
,” Nat. Commun.
13
, 2444
(2022
). 7.
Z.
Zhu
, G.
Rui
, Q.
Li
, E.
Allahyraov
, R.
Li
, T.
Soulestin
, F. D.
Dos Santos
, H.
He
, P. L.
Taylor
, and L.
Zhu
, “Electrostriction-enhanced giant piezoelectricity via relaxor-like secondary crystals in extended-chain ferroelectric polymers
,” Matter
4
, 3696
(2021
). 8.
E. J.
Marksz
, A. M.
Hagerstrom
, X.
Zhang
, N.
Al Hasan
, J.
Pearson
, J. A.
Drisko
, J. C.
Booth
, C. J.
Long
, I.
Takeuchi
, and N. D.
Orloff
, “Broadband, high-frequency permittivity characterization for epitaxial Ba
Sr
TiO
composition-spread thin films
,” Phys. Rev. Appl.
15
, 064061
(2021
). 9.
A.
Vorobiev
, P.
Rundqvist
, K.
Khamchane
, and S.
Gevorgian
, “Silicon substrate integrated high Q-factor parallel-plate ferroelectric varactors for microwave/millimeterwave applications
,” Appl. Phys. Lett.
83
, 3144
(2003
). 10.
C. J. G.
Meyers
, C. R.
Freeze
, S.
Stemmer
, and R. A.
York
, “Effect of BST film thickness on the performance of tunable interdigital capacitors grown by MBE
,” Appl. Phys. Lett.
111
, 262903
(2017
). 11.
A.
Crunteanu
, V.
Muzzupapa
, A.
Ghalem
, L.
Huitema
, D.
Passerieux
, C.
Borderon
, R.
Renoud
, and H. W.
Gundel
, “Characterization and performance analysis of BST-based ferroelectric varactors in the millimeter-wave domain
,” Crystals
11
, 277
(2021
). 12.
J.
Hlinka
, “Do we need the ether of polar nanoregions?
,” J. Adv. Dielectr.
2
, 1241006
(2012
). 13.
D.
Viehland
, M.
Wuttig
, and L. E.
Cross
, “The glassy behavior of relaxor ferroelectrics
,” Ferroelectrics
120
, 71
(1991
). 14.
G.
Geneste
, L.
Bellaiche
, and J.-M.
Kiat
, “Simulating the radio-frequency dielectric response of relaxor ferroelectrics: Combination of coarse-grained Hamiltonians and kinetic Monte Carlo simulations
,” Phys. Rev. Lett.
116
, 247601
(2016
). 15.
H.
Takenaka
, I.
Grinberg
, and A. M.
Rappe
, “Anisotropic local correlations and dynamics in a relaxor ferroelectric
,” Phys. Rev. Lett.
110
, 147602
(2013
). 16.
H.
Takenaka
, I.
Grinberg
, S.
Liu
, and A. M.
Rappe
, “Slush-like polar structures in single-crystal relaxors
,” Nature
546
, 391
(2017
). 17.
J.
Kim
, A.
Kumar
, Y.
Qi
, H.
Takenaka
, P. J.
Ryan
, D.
Meyers
, J.-W.
Kim
, A.
Fernandez
, Z.
Tian
, A. M.
Rappe
, J. M.
LeBeau
, and L. W.
Martin
, “Coupled polarization and nanodomain evolution underpins large electromechanical responses in relaxors
,” Nat. Phys.
18
, 1502
(2022
). 18.
H.
Tao
, H.
Wu
, Y.
Liu
, Y.
Zhang
, J.
Wu
, F.
Li
, X.
Lyu
, C.
Zhao
, D.
Xiao
, J.
Zhu
, and S. J.
Pennycook
, “Ultrahigh performance in lead-free piezoceramics utilizing a relaxor slush polar state with multiphase coexistence
,” J. Am. Chem. Soc.
141
, 13987
(2019
). 19.
A.
Jonscher
, “Dielectric relaxation in solids
,” J. Phys. D: Appl. Phys.
32
, R57
(1999
). 20.
21.
22.
K. S.
Cole
and R. H.
Cole
, “Dispersion and absorption in dielectrics I. Alternating current characteristics
,” J. Chem. Phys.
9
, 341
(1941
). 23.
S.
Havriliak
and S.
Negami
, “A complex plane analysis of
-dispersions in some polymer system
,” J. Polym. Sci. Part C
14
, 99
(1966
). 24.
S.
Havriliak
and S.
Negami
, “A complex plane representation of dielectric and mechanical relaxation processes in some polymers
,” Polymer
8
, 161
(1967
). 25.
R.
Hilfer
, “Experimental evidence for fractional time evolution in glass forming materials
,” Chem. Phys.
284
, 399
(2002
). 26.
D.
Guyomar
, B.
Ducharne
, and G.
Sébald
, “Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives
,” J. Appl. Phys. D: Appl. Phys.
40
, 6048
(2007
). 27.
Y. H.
Huang
, J. J.
Wang
, T. N.
Yang
, Y. J.
Wu
, and L.-Q.
Chen
, “A thermodynamic potential, energy storage performances, and electrocaloric effects of Ba
Sr
TiO
single crystals
,” Appl. Phys. Lett.
112
, 102901
(2018
). 28.
S.
Mukherjee
, A.
Nag
, V.
Kocevski
, P. K.
Santra
, M.
Balasubramanian
, S.
Chattopadhay
, T.
Shibata
, F.
Schaefers
, J.
Rusz
, C.
Gerard
, O.
Eriksson
, C. U.
Segre
, and D. D.
Sarma
, “Microscopic description of the evolution of the local structure and an evaluation of the chemical pressure concept in a solid solution
,” Phys. Rev. B
89
, 224105
(2014
). 29.
T.
Ishidate
, S.
Abe
, H.
Takahashi
, and N.
Mori
, “Vogel-Fulcher relationship for the dielectric permittivity of relaxor ferroelectrics
,” Phys. Rev. Lett.
78
, eid 2397
(1997
). 30.
P.
Marton
and J.
Hlinka
, “Phenomenological model of a
domain wall in BaTiO
-type ferroelectrics
,” Phys. Rev. B
74
, 104104
(2006
). 31.
O.
Diéguez
and M.
Stengel
, “Translational covariance of flexoelectricity at ferroelectric domain walls
,” Phys. Rev. X
12
, 031002
(2022
). 32.
Q.
Meng
, M.-G.
Han
, J.
Tao
, G.
Xu
, D. O.
Welch
, and Y.
Zhu
, “Velocity of domain-wall motion during polarization reversal in ferroelectric thin films: Beyond Merz’s law
,” Phys. Rev. B
91
, 054104
(2015
). 33.
J.
Hlinka
, “Mobility of ferroelastic domain walls in barium titanate
,” Ferroelectrics
349
, 49
(2007
). 34.
H.
Vogt
, J. A.
Sanjurjo
, and G.
Rossbroich
, “Soft-mode spectroscopy in cubic BaTiO
by hyper-Raman scattering
,” Phys. Rev. B
26
, 5904
(1982
). 35.
J.
Petzelt
, G. V.
Kozlov
, and A. A.
Volkov
, “Dielectric spectroscopy of paraelectric soft modes
,” Ferroelectrics
73
, 101
(1987
). 36.
J.
Mangeri
, Y.
Espinal
, A.
Jokisaari
, S.
Pamir Alpay
, S.
Nakhmanson
, and O.
Heinonen
, “Topological phase transformations and intrinsic size effects in ferroelectric nanoparticles
,” Nanoscale
9
, 1616
(2017
). 37.
C. J.
Permann
, D. R.
Gaston
, D.
Andrš
, R. W.
Carlsen
, F.
Kong
, A. D.
Lindsay
, J. M.
Miller
, J. W.
Peterson
, A. E.
Slaughter
, R. H.
Stogner
, and R. C.
Martineau
, “MOOSE: Enabling massively parallel multiphysics simulation
,” SoftwareX
11
, 100430
(2020
). 38.
P.
Ondrejkovič
, “Studies of relaxor ferroelectrics with spontaneous polar nanoregions,” Ph.D. thesis (Charles University, Prague, 2017).39.
G.
Van Rossum
and F. L.
Drake
, Python 3 Reference Manual
(CreateSpace
, Scotts Valley, CA
, 2009
).40.
P. Virtanen, R. Gommers, T. E. Oliphant et al.
, “SciPy 1.0: fundamental algorithms for scientific computing in Python,” Nat. Methods
17
, 261 (2020). 41.
N. I. M.
Gould
, C.
Sainvitu
, and P. L.
Toint
, “A filter-trust-region method for unconstrained optimization
,” SIAM J. Optim.
16
, 341
(2005
). 42.
N. I. M.
Gould
, S.
Lucidi
, M.
Roma
, and P. L.
Toint
, “Solving the trust-region subproblem using the Lanczos method
,” SIAM J. Optim.
9
, 504
(1999
). 43.
A. R.
Conn
, N. I. M.
Gould
, and P. L.
Toint
, “Global convergence of a class of trust region algorithms for optimization with simple bounds
,” SIAM J. Numer. Anal.
25
, 433
(1988
). 44.
T.
Ostapchuk
, J.
Petzelt
, P.
Kuzel
, S.
Vejko
, A.
Tkach
, P.
Vilarinho
, I.
Ponomareva
, L.
Bellaiche
, E.
Smirnova
, V.
Lemanov
, A.
Sotniko
, and M.
Weihnacht
, “Infrared and THz soft-mode spectroscopy of (Ba,Sr)TiO
ceramics
,” Ferroelectrics
367
, 139
(2008
). 45.
T.
Ostapchuk
, J.
Petzelt
, J.
Hlinka
, V.
Bovtun
, P.
Kužel
, I.
Ponomareva
, S.
Lisenkov
, L.
Bellaiche
, A.
Tkach
, and P.
Vilarinho
, “Broad-band dielectric spectroscopy and ferroelectric soft-mode response in the Ba
Sr
TiO
solid solution
,” J. Phys.: Condens. Matter.
21
, 474215
(2009
). 46.
A. K.
Tagantsev
, “Vogel-Fulcher relationship for the dielectric permittivity of relaxor ferroelectrics
,” Phys. Rev. Lett.
72
, 1100
(1994
). 47.
K. C.
Pitike
, J.
Mangeri
, H.
Whitelock
, T.
Patel
, P.
Dyer
, S. P.
Alpay
, and S.
Nakhmanson
, “Metastable vortex-like polarization textures in ferroelectric nanoparticles of different shapes and sizes
,” J. Appl. Phys.
124
, 064104
(2018
). 48.
J.
Mangeri
, S. P.
Alpay
, S.
Nakhmanson
, and O. G.
Heinonen
, “Electromechanical control of polarization vortex ordering in an interacting ferroelectric-dielectric composite dimer
,” Appl. Phys. Lett.
113
, 092901
(2018
). 49.
D.
Zhu
, J.
Mangeri
, R.
Wang
, and S.
Nakhmanson
, “Size, shape, and orientation dependence of the field-induced behavior in ferroelectric nanoparticles
,” J. Appl. Phys.
125
, 134102
(2019
). 50.
K.
Co
, S.
Pamir Alpay
, S.
Nakhmanson
, and J.
Mangeri
, “Surface charge mediated polar response in ferroelectric nanoparticles
,” Appl. Phys. Lett.
119
, 262903
(2021
). 51.
T.
Dietl
, K.
Sato
, T.
Fukushima
, A.
Bonanni
, M.
Jamet
, A.
Barski
, S.
Kuroda
, M.
Tanaka
, P. N.
Hai
, and H.
Katayama-Yoshida
, “Spinodal nanodecomposition in semiconductors doped with transition metals
,” Rev. Mod. Phys.
87
, 1311
(2015
). 52.
H.
Kizaki
, K.
Kusakabe
, S.
Nogami
, and H.
Katayama-Yoshida
, “Generation of nano-catalyst particles by spinodal nano-decomposition in perovskite
,” Appl. Phys. Express
1
, 104001
(2008
). 53.
D.
Fuks
, Y.
Mastrikov
, E.
Kotomin
, and J.
Maier
, “Ab initio thermodynamic study of (Ba,Sr)(Co,Fe)O
perovskite solid solutions for fuel cell applications
,” J. Mater. Chem. A
1
, 14320
(2013
). 54.
G.
Buse
, C.
Xin
, P.
Marchet
, A.
Borta-Boyon
, M.
Pham-Thi
, H.
Cabane
, E.
Veron
, M.
Josse
, M.
Velazquez
, M.
Lahaye
, E.
Lebraud
, M.
Maglione
, and P.
Veber
, “Spinodal decomposition in lead-free piezoelectric BaTiO
–CaTiO
–BaZrO
crystals
,” Cryst. Growth Des.
18
, 5874
(2018
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
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