Topological phase transformations and intrinsic size effects in ferroelectric nanoparticles

3 MB·

by  John Mangeri

Topological phase transformations and intrinsic size effects in ferroelectric
nanoparticles
John Mangeri∗
Department of Physics, University of Connecticut
Yomery Espinal
Department of Materials Science and Engineering, University of Connecticut
Andrea Jokisaari
7 Center of Hierarchical Materials Design, Northwestern-Argonne Institute
1 for Materials Science and Engineering, Northwestern University
0
2 S. Pamir Alpay and Serge Nakhmanson
n Department of Materials Science and Engineering, University of Connecticut and
a Department of Physics, University of Connecticut
J
0 Olle Heinonen†
1 Center of Hierarchical Materials Design, Northwestern-Argonne Institute
for Materials Science and Engineering, Northwestern University and

] Material Science Division, Argonne National Laboratory
l
l (Dated: January 11, 2017)
a
h Composite materials comprised of ferroelectric nanoparticles in a dielectric matrix are being ac-
- tivelyinvestigatedforavarietyoffunctionalpropertiesattractiveforawiderangeofnovelelectronic
s
e and energy harvesting devices. However, the dependence of these functionalities on shapes, sizes,
m orientation and mutual arrangement of ferroelectric particles is currently not fully understood. In
this study, we utilize a time-dependent Ginzburg-Landau approach combined with coupled-physics
.
t finite-element-methodbasedsimulationstoelucidatethebehaviorofpolarizationinisolatedspher-
a
icalPbTiO orBaTiO nanoparticlesembeddedinadielectricmedium,includingair. Theequilib-
m 3 3
riumpolarizationtopologyisstronglyaffectedbyparticlediameter,aswellasthechoiceofinclusion
- andmatrixmaterials,withmonodomain,vortex-likeandmultidomainpatternsemergingforvarious
d
combinationsofsizeandmaterialsparameters. Thisleadstoradicallydifferentpolarizationvselec-
n
tric field responses, resulting in highly tunable size-dependent dielectric properties that should be
o
possibletoobserveexperimentally. Ourcalculationsshowthatthereisacriticalparticlesizebelow
c
[ which ferroelectricity vanishes. For the PbTiO3 particle, this size is 2 and 3.4 nm, respectively, for
high-andlow-permittivitymedia. FortheBaTiO particle,itis∼3.6nmregardlessofthemedium
3
1 dielectric strength.
v
3
1 I. INTRODUCTION persed within a dielectric matrix, that may be, e.g., of
6
polymeric,[6, 8, 9] ferromagnetic,[10] or oxide origin.[11]
2
0 With the end of Moore’s law in sight for silicon- On the other hand, synthetic processes governing the
. based device technology, new paradigms utilizing alter- formation of FE nanoparticles can produce a wide va-
1
native material families are being probed with increas- riety of shapes, including cuboidal [12–15], ellipsoidal or
0
7 ing intensity. Ferroelectric (FE) materials constitute spherical [13, 15–17], and core-shell[17–19] geometries.
1 one such family that is highly attractive for a broad These inclusions can be incorporated into the composite
: range of next-generation technological applications. FE in either aggregated,[6, 12] or dispersed,[9, 15] irregular
v
materials are already used in non-volatile random ac- arrangements, providing precise control of its dielectric
i
X cess memories[1–5] and are now being investigated more properties.
r broadly as possible components in a variety of new elec-
a However,studiesoftheinfluenceoffeaturesizeonuse-
tronic devices, as well as for energy-storage and battery-
ful FE properties, and specifically the intrinsic limit for
related technologies.[6–8]
FEresponse,havebeenlargelyfocusedonbulkceramics,
A particularly versatile approach for infusing all of
thin films[20, 21] and bi-/multilayers,[22–25] with rela-
these applications with FE functionalities involves cre-
tively few investigations dedicated to nanoparticles and
ation of composites consisting of small FE particles dis-
other nanostructures.[14, 26, 27] Tuning the particle size
and the material parameters, as well as those of the sur-
rounding environment, directly influences the strength
∗ [email protected] of competing energy interactions within the system, in-
† [email protected] cluding long-range electrostatics, short-range ferroelec-
2
tric ordering and electrostrictive coupling between polar allows us to establish a critical size for a transition into
andelasticdegreesoffreedom. Withsomeorallofthese the paraelectric state.
terms being close in magnitude, the system may become
highlysensitivetochangesincontrolparameters,sothat
small external stimuli can generate large responses. II. METHODS
A detailed understanding how the particle size, shape
and morphology, as well as the elastic and electrostatic A general real-space finite-element approach in three
influenceofthesurroundingmediumaffectFEproperties dimensions (3D) is employed to track the evolution of
is currently lacking. In this study, we focus on spherical coupled polarization density, electrostatic potential and
particles,suchastheonesalreadysynthesizedbyanum- elastic displacement fields P, Φ and u in the system
ber of experimental groups.[13, 16, 17] We then attempt (boldfacefontmarksvectorfields). Thisapproachispar-
to elucidate connections between the size of the isolated ticularly well-suited for the studies of complex systems
FE nanoparticle immersed in a dielectric medium (in- at mesoscale, i.e., for characteristic lengths that range
cludingair)andthetopologicalfeaturesofanequilibrium from a few to 100s, or even 1000s of nm, which makes a
arrangementofpolardipoleswithinit—whatwereferto uniform treatment of all the possible system sizes with
below as its polarization texture or pattern. This model atomistic techniques impractical. All the numerical sim-
canbeconsideredasequivalenttoahighlydispersedpar- ulations presented here have been done with the code
ticle arrangement within the matrix, where electrostatic packageFerret,[39]whichisbeingdevelopedbytheau-
interactions among individual particles are negligible. It thors and is based on the Multiphysics Object-Oriented
can also be regarded as a first step towards construct- Simulation Environment (MOOSE) framework.[40] The
ing and evaluating more complex models for composite finite-element-based model consists of a spherical FE in-
ferroelectric-matrixsystems. Inourinvestigation,weaim clusion,ΩFE,embeddedinadielectric-mediumcube,ΩM,
to determine specific parameter combinations, such as withthewholesystemmeshedusinganunstructuredgrid
particle diameter and medium dielectric strength, when oftetrahedrons. Theinterfacebetweentheinclusionand
equilibrium polarization patters may become unstable. the matrix is assumed to be coherent.
A topological transformation between different polariza- The following expression, adopted from thermody-
tion texture morphologies can then be triggered in the namic Landau-Ginzburg-Devonshire (LGD) theory, is
vicinity of such an unstable state by an applied electric utilized to represent the total free energy of the system
or elastic field, producing the desired property response. in the domain ΩFE:
We use two archetypical FE materials, BaTiO (BT) (cid:90)
3 F = [f +f +f +f +f ]d3r.
and PbTiO (PT), to represent the properties of the bulk wall elastic elec coupled
3
inclusion. At room temperature, PT has weak elec- ΩFE
trostrictive coupling between ferroelectric polarization (1)
and elastic strain, but a large spontaneous polarization Here,fbulkisthebulkferroelectricenergydensity,fwallis
(PPT = 0.75 C/m2), whereas BT has a lower polariza- the energy density that arises from local gradients in P,
s
tion (PsBT = 0.26 C/m2) but is much more sensitive to felastic is the linear elastic energy density, fcoupled is the
applied strain. We also employ three material choices energy density due to electrostrictive coupling between
with radically different dielectric and elastic properties thelocalFEpolarizationdensityandthestrain,andfelec
for the dielectric matrix: SrTiO (ST, high dielectric is the electrostatic energy density. Detailed expressions
3
permittivity), amorphous silica (a-SiO , low dielectric for all of the free-energy densities are provided in the
2
permittivity), and vacuum (described by vacuum per- Supplemental Material.
mittivity (cid:15) ). Our findings suggest that, for a certain The evolution of the polarization density field P
0
range of particle sizes and materials parameters, vortex- is described by the time-dependent Landau-Ginzburg-
like polarization patterns are energetically favored over Devonshire equation (TDLGD)
mono- or multidomain geometries. These patterns are
∂P δ (cid:90)
not true topological vortices and therefore a ‘-like’ suffix −γ = d3rf(P), (2)
∂t δP
isusedtodescribethem. Essentiallysimilarpolarization
motifshavealreadybeenobservedexperimentally[28–32] ΩFE
and predicted theoretically[33–38] in some spatially con- that drives the system towards an equilibrium state by
fined FE nanostructures. We demonstrate that particles reducingitsfreeenergyuntilitreachesalocalminimum.
with vortex-like and multidomain polarization patterns The total free energy is considered converged when its
exhibithighlytunablemulti-stageswitchingbehaviorun- relative change between consecutive time steps is below
der electric field cycling. Such effects should be easy to 0.1%. The time constant γ is related to polar domain-
detect in experiments and may also be of use for a vari- wall mobility.[41] It is set to unity in this investigation,
ety of device applications. Additionally, we evaluate the as we are interested not in the temporal evolution of the
stability of observed polarization textures depending on system, but rather only in its final (local) equilibrium
particle diameter, selection of materials parameters and state. As a starting guess for the P field, a random con-
thechoiceofpolargradientenergycoefficients,whichalso figuration (cid:104)P(cid:105)≈0, also known as a random paraelectric
3
initialcondition(RPEIC),isusedforalltheparticlesizes presented in the Supplemental Material.
considered in this project. This condition allows one to
avoid any initial bias or symmetry that may restrict the
pathofthesystemduringthetimeevolutionofP.[42,43] III. RESULTS AND DISCUSSION
OutsidetheFEinclusion,P≡0,andthebehaviorofthe
matrixisgovernedbytheequationsforthelinearelastic- Herewediscussthetopologicalfeaturesofpolarization
dielectric medium. textures in FE nanoparticles and follow the evolution of
The coupled electrostatic potential field Φ is obtained thesefeatureswithchangingparticlediameter, dielectric
from the Poisson equation permittivity of the surrounding matrix and applied elec-
tric field. For all of the considered systems, we can iden-
∇·((cid:15) ∇Φ)=−∇·P, (3)
α tify two distinct transitions as the diameter of the inclu-
sion increases: from a paraelectric or polar-monodomain
which, along with the condition for mechanical equilib-
to a vortex-like state, and from a vortex-like to a poly-
rium, ∇·σ = 0, has to be satisfied at each step in sys-
domain state (see Fig. 1(a) for examples). While these
tem time evolution governed by Eq. (2). This require-
transitions are common to the materials systems investi-
ment implies that characteristic relaxation times for the
gated, their details, as well as the particle sizes at which
electrostatic potential and the elastic displacement fields
they occur, depend on the specific materials parameters.
are much shorter than that of the polarization-density
field. Here, (cid:15) is the background dielectric constant of
α
the FE (α = b) that originates from polarization of the
coreelectrons[44]orthedielectricconstantofthematrix A. Paraelectric and ferroelectric states for small d
(α = m), while σ = C ∂u /∂x is the stress field.
ij ijkl k l
Material parameters for the dielectric susceptibilities (cid:15) The disappearance of the system FE polarization be-
b
and (cid:15) , and elastic coefficients C utilized in this in- low a certain critical diameter d is found to be strongly
m ijkl c
vestigation are provided in the Supplemental Material. dependent on the selection of materials parameters for
As an important aside, we point out that local sur- the inclusion and the dielectric matrix. In general, i.e.,
face terminations of nanoparticles may be complex and for both of the FE inclusion materials considered here,
dependent on a particular synthesis route, which could we obtain values of d , above which a non-zero polariza-
c
also affect properties, such as support for metal species tion distribution P may exist in some form, as spanning
and adsorption coordination.[45] Specifically for the per- from 2 to ∼ 3.6 nm. These critical lengths fall within an
ovskite materials considered here, terminations consist- approximate range identified by other research groups
ing of Pb(Ba)O or TiO layers would produce different for a variety of different FE structures, including thin
amounts of uncompensated surface charge and thus in- films.[14, 21, 46–48] They are also substantially smaller
fluence the polarization field distribution at the surface thantheFEcorrelationlength(cid:96) [36,49]—i.e.,atypical
C
ofthenanostructure. However,bystudyingawiderange size for supporting a single FE domain wall — which for
of dielectric constants of the surrounding matrix, (cid:15) , these nanoparticles we estimate as being ∼ 5–10 nm.
m
the aggregate effects of charge compensation at differ- Simulations conducted for systems with particles hav-
ent nanoparticle surface terminations can be effectively ing d (cid:47) d provided the following insights into the
c
captured. We also note that in cases when surface ter- specifics of their evolution towards an equilibrium con-
minations may vary locally on the particle surface, elec- figuration. At the beginning of the simulation, the f
wall
trostatic and elastic fields arising from such variations energytermislargeduetoinhomogeneitiesinthePdis-
should rapidly average out. tribution produced by the RPEIC. In a small nanopar-
The size of the cubic computational domain contain- ticle with d (cid:28) (cid:96) , P initially evolves towards a mon-
C
ing the inclusion Ω and the surrounding dielectric Ω odomainstate,whichproducesanon-zerosurfacecharge
FE M
is taken to be large enough for the elastic-displacement density q = P·nˆ, where nˆ is the surface normal vec-
S
field u and the stresses σ arising from elastic mismatch tor. In turn, the presence of uncompensated q leads
ij S
at the interface between the inclusion and the matrix to a sharp increase in the f energy term, unless the
elec
to vanish at the domain boundaries. Dirichlet bound- dielectric permittivity of the matrix is large enough to
ary conditions u = 0 and Φ = 0 are used at the [±100], screen out the surface charge. In order to reduce f ,
elec
[0±10], and [00±1] boundaries of the computational the magnitude of P is uniformly diminished until P≈0
domain. We can also introduce an external electrostatic withinnumericalprecisionofthesimulation, resultingin
field by applying a Dirichlet boundary condition, Φ(cid:54)=0, theparaelectricstateofthesystem. Weverifiedthatthe
to the [001] surface. Consistency checks were performed robustnessoftheparaelectricsolutionford<d doesnot
c
to ensure that both the internal Φ and u, in fact, do dependontheinitialconditionsspecificsorfinite-element
vanish at the boundaries of the computational domain mesh spacing if the interfacial region between the inclu-
(intheabsenceofappliedexternalfields)throughoutthe sion and the matrix is sufficiently resolved.[42, 43]
range of all the investigated inclusion diameters d. All ForaPTinclusion,amonodomainFEstateisobserved
simulations presented here are done at room tempera- ford<3.4nmfor(cid:15) (cid:39)300(i.e.,STmatrix),withauni-
m
ture. Further details of the computational method are form distribution of local polarization. The polarization
4
FIG. 1. (a) Various polarization-field textures observed in the PT/ST system with increasing nanoparticle diameter d: (i)
monodomain, (ii) vortex, and (iii) - (iv) multidomain. Local directions of P are depicted by arrows, whose color represents
the field magnitude. (b) FE bulk energy F , normalized by volume and bulk spontaneous polarization P , for PT/ST and
bulk s
BT/ST composites plotted as a function of d. Monodomain-to-vortex-like phase transition occurs in the PT/ST system at
critical diameter d (cid:39)3.4 nm accompanied by an increase in energy. The BT/ST system does not form a monodomain state;
v
instead its P≈0 for d≤3.6 nm. Above that diameter value, it transforms into a vortex-like state, which is accompanied by
anenergydecrease. Thereasonbehindtheenergychangeduringthemonodomain-orparaelectric-to-vortex-liketransitioncan
beunderstoodfrompanel(c),whichshowstheradialprofileof|P|alongalineperpendiculartothevortexcorein4and7nm
wideinclusionswithvortex-likepolarizationtextures. InbothPT/STandBT/STsystems,thevaluesof|P|inthevortex-like
phasearedepressed,comparedtothoseinthemonodomainphase,orrespectiveP . Theinfluenceofthedielectricpermittivity
s
of the surrounding medium on the polarization values is also presented in the panel, with curves corresponding to different
dielectric materials shown for all the considered systems.
magnitude is somewhat reduced, compared to PPT, as size is increased is the one that has a vortex-like charac-
s
showninsketch(i)ofFig.1(a). Forlowervaluesof(cid:15) ,a ter.
m
paraelectric state is found for the PT inclusion, while in
theBTsystemsuchstatepersistsforallthevaluesof(cid:15) ,
m
includingonecorrespondingtotheSTmatrix. Thatmay B. Transition into a vortex-like state
be surprising, considering that PBT (cid:28)PPT, which must
s s
reduce qS and therefore reduce the penalty originating The inception of the vortex-like state in both PT and
from the felec energy term for the monodomain state of BT nanoparticles can be detected by following the de-
the BT system. However, due to the shallower bulk en- pendence of their normalized bulk free energy,
ergy minimum and stronger electrostrictive couplings in
theBTsystem,theenergyincreasesarisingfromthef 1 (cid:90)
bulk F = f d3r, (4)
andf termsarerelativelyminor,andthusthesys- bulk Ω P3 bulk
coupled FE s
tem evolution towards the equilibrium is still dominated ΩFE
by the influence of the f term and minimization of
elec on d, as shown in Fig. 1(b) for the ST dielectric ma-
q . Therefore, for all of the investigated particle/matrix
S trix. For the PT/ST system, the transition occurs at a
material combinations except PT/ST, the first FE state
critical diameter d (cid:39) 3.4 nm and is accompanied by a
with non-zero local P that is encountered as the particle v
sharpincrease ofF ,comparedtoitsvalueinthemon-
bulk
5
odomainstate. Ontheotherhand,intheBT/STsystem, tive coupling is responsible for the softening of the do-
F ,whichiszerobydefinitionintheparaelectricstate, main walls, which produces more rounded textures, such
bulk
sharply decreases upon the formation of the vortex-like as the ones that were observed or predicted in other ex-
state with non-zero polarization at d ≡d (cid:39) 3.5 nm. perimental and theoretical studies.[28, 35, 38] Represen-
v c
While the physics underpinning the energy change in tative images of polarization textures formed with and
the BT/ST system may be self-evident, understanding without the electrostrictive coupling in a PT/ST system
the behavior of the PT/ST system requires a detailed are shown in Supplemental Fig. 2.
examinationandcomparisonofthepolarizationpatterns
before and after the transition, i.e., sketches (i) and (ii)
depicted in Fig. 1(a). As can be seen from sketch (ii),
even though local values of |P| ∼ PPT near the surface
s
of the inclusion, they become strongly suppressed close
to its center, forming a weakly polar or even completely
paraelectric core region of the vortex. In all of the PT
systems, this core region is cylindrical in shape and pen-
etratesthesphericalinclusioncompletelyfromitsnorth-
erntosouthernpole,asshowninSupplementalFig.1(a).
In the BT systems, such a region is also present, but
its shape may be twisted or bent, as illustrated in Sup-
plemental Fig. 1(b), and its final conformation exhibits
dependence on the RPEIC.
Thechangeinthevalueof|P|alongthedirectionper-
pendiculartothevortexcoreaxisispresentedinFig.1(c)
FIG. 2. Normalized gradient energy F [54] as a function
for particles of two different sizes for both PT and BT wall
of d for the PT/ST and BT/ST systems. F is zero for
systems coupled with all the considered dielectric matri- wall
d<d , but increases rapidly in the vortex-like state because
v
ces. These data show that |P| in the core region may
ofsub-optimalarrangementsofthelocalpolarizationvectors.
be reduced by a factor of 3–5, compared to its surface It then levels off, i.e., becomes bulk-like, in the multidomain
value (in PT), or even disappear completely (in PT and state,withthetransitionpoint(markedbyverticallines)de-
always in BT). This tendency is mostly unaffected by pending rather sensitively on the choice of G parameters
ijkl
the dielectric strength of the surrounding matrix. Such for PT.
behavior is in sharp contrast with that of ferromagnetic
vortices, where, at temperatures well below T , magne-
C
tization density at the core is constrained to a constant
magnitude.[50, 51] C. Subsequent transition into multidomain state
Fig. 1(c) also describes the effect of the surrounding
matrix on the value of |P| at the surface of the inclu- Forparticlediameters,d>dv,themostimportanten-
sion. In the PT system, surface polarization is ∼ 20% ergyterminfluencefurtherpolarizationtextureevolution
largerwhenitiscoupledwithahighdielectricpermittiv- is the normalized gradient energy,
itymedium,suchasST.Incontrast,surfacepolarization
1 (cid:90)
of the BT system is not affected by the strength of the F = f d3r. (5)
dielectric screening provided by the matrix. wall ΩFEPs3 wall
Topological features of polarization textures, summa- ΩFE
rizedinFig.1, areingeneralagreementwiththeanalyt- In the case of BT, the well-known parameterization of
ical work of Levanyuk and Blinc.[36] The observed simi- Hlinka and Marton[44] can be used for the gradient en-
laritiesincludethedependenceof|P|onthesurrounding ergy tensor G . However, a number of various param-
ijkl
medium, its suppression at the core of the vortex-like eterizations for G exist for PT. In this investigation,
ijkl
phase,aswellastransitorynatureofmonodomainstates we considered three different sets — attributed to Li et
in low dielectric permittivity medium. However, the in- al.[42, 43] (set I), Wang et al.[55] (set II) and Hong et
vestigation of Levanyuk and Blinc did not consider cou- al.[56] (set III). The values of the G coefficients used
ijkl
plingbetweenferroelectricandelasticdegreesoffreedom. in all of these sets, as well as for BT, are listed in the
Althoughtheresultspresentedhereincludetheeffectsof Supplemental Material.
electrostrictive coupling, we have also performed a se- The dependence of F on d is shown in Fig. 2 for
wall
ries of simulations with the electrostrictive tensor set to PT/ST and BT/ST systems. The gradient energy is
zero in order to examine its influence on the behavior zero in monodomain and paraelectric states for d < d .
v
of the system. The resulting polarization textures have It then grows rapidly upon transition to the vortex-like
sharp 90◦ domain walls, resembling Landau flux-closure state, which can be construed as consisting of a large
patternsfoundinsomemagneticmicrostructures.[52,53] number of domain walls separating small polar regions
Thus, we conclude that the presence of the electrostric- that have energetically sub-optimal mutual polarization
6
arrangements (as opposed to optimal ones of 90◦ and rated monodomain configuration in the particle with the
180◦). As the inclusion diameter increases beyond d , polarization aligned with the field. After the saturated
v
F gradually recedes until it saturates at a constant configurationisestablished,thefieldisincreasedinsmall
wall
non-zero value. Such leveling off indicates the formation steps to +E zˆ and then decreased back to −E zˆ,
max max
of a‘bulk-like’ multidomainstatecomprised ofrelatively thus completing the poling loop. At each step, the aver-
large areas of correlated P divided by domain walls that agedprojectionontheCartesianzˆ-axis, P¯ ,iscomputed
z
are similar to their 90◦ and 180◦ bulk variants. This for the converged polarization field P. Note that initial
transition happens at d ≡ d (cid:39) 17 nm for the BT/ST poling curves starting from zero applied field and zero
m
system, while in PT/ST its arrival is quite sensitive to polarization are not simulated with this approach. For
thechoiceoftheG -coefficientsetandcoverstherange PT, we use the G parameterization set I[42, 43] in all
ijkl ijkl
of 13 to 21 nm. However, the equilibrium topologies of calculations. Furtherdetailsofthepolingmethodimple-
multidomain states obtained for PT/ST do not seem to mentation are provided in the Supplemental Material.
be strongly affected by the choice of Gijkl parameteri- InFig.3(a)weshowfieldinducedpolarizationresponse
zations. We speculate that by measuring certain exper- in PT/ST structures of four different sizes, exhibiting
imentally observable quantities linked to this transition monodomain [case (i), d = 4 nm], vortex-like [case (ii),
— e.g., electric field response that is discussed next — d = 10 nm] and multidomain [d = 16 nm and case (iii),
it may be possible to obtain better gradient energy esti- d = 35 nm] zero-field equilibrium polarization textures,
mates for PT. someofwhicharealsovisualizedinpanel(c)ofthesame
Representative images of multidomain polarization figure. In the monodomain case, the tetragonal crystal-
texturesareshownascases(iii–iv)inFig.1(a),aswellas lographic axis, or the “easy” polarization axis, is aligned
case (iii) in Fig. 3(c). We find that domains always tend with the zˆ direction. The associated poling loop is sim-
to orient their polarization tangentially to the surface of ilar to that of a generic bulk FE, with abrupt switching
theinclusioninordertominimizetheelectrostaticenergy of P between −zˆand +zˆorientations at a distinct value
arising from qS. The non-polar or weakly polar vortex of coercive field Ec. A comparison of field induced po-
core area becomes unstable at d > dm, developing uni- larization responses from different crystallographic PT
form polarization and eventually splitting into multiple orientations is shown in Supplemental Fig. 4.
domains. The apparent vorticity of the polarization tex-
For particle diameters d > d , switching between
v
ture sharply decreases after the transition into the mul-
−zˆ and +zˆ monodomain orientations proceeds in two
tidomain state, but it does not disappear completely, as
stages. The initial monodomain configuration persists
localized vortices, marked by largely suppressed P, still
from E = −E to E (cid:39) +E , at which point it
z max z c
remain near some domain walls. Due to the curvature of
is replaced with a hybrid texture consisting of a mon-
the inclusion surface, the observed domain patterns can-
odomaincore polarizedalongthe +zˆdirectioncombined
not be directly partitioned into collections of low-energy
with a vortex-like closure pattern in the near-surface re-
90◦ or 180◦ variants in the near-surface region, which,
gion. Thisresultsintheresponsecurveexhibitingasmall
in combination with remaining vorticity, results in finite plateau at E < E < E with roughly constant P¯ ,
c z max z
saturation values of F at large d.
wall originatingfromthepolarizedinclusioncore,thatiscon-
Quantitative evaluations of the amount of vorticity
siderably smaller than the saturation polarization. As
present in a three-dimensional vector field can be con-
the field is increased further, the vortex-like texture is
ducted by computing of its Chern-Simons topological
abruptly expelled from the near-surface region and the
windingnumberdensity[57]n .InthecaseofP,n =
CS CS polarization aligns along the +zˆdirection everywhere in
(∇×P)·P. It should be close to zero for monodomain
the particle. Multistage switching processes similar to
and bulk-like multidomain patterns, and must have an
the one observed herehavebeen reported beforein some
extremum in the vortex-like phase that separates them,
nanopatterned FE[28, 37] and ferromagnetic[58, 59] sys-
asshowninSupplementalFig.3. Wefindthismeasureto
tems.
beespeciallyusefulinelucidatingthetopologicalchanges
Astheparticlediameterincreasesfurther,theinterme-
ofpolarizationtexturesininclusionssubjectedtoapplied
diatevortex-liketextureoccurringinthethenear-surface
electric fields, as discussed below.
region at E (cid:39)+E is replaced with a multidomain tex-
z c
ture, such as the one shown as case (iii) in Fig. 3(c).
Similar to the zero-field configurations, domains at the
D. Intrinsic field dependence of polarization inclusionsurfaceprefertohavetangentialorientationsof
their P to minimize the electrostatic energy arising from
We have studied the electric-field induced topologi- q . As the magnitude of the field is increased, the tran-
S
cal changes for all of the polarization textures — mon- sition into the monodomain state occurs by a gradual
odomain,vortex-likeandmultidomain—thatweobserve alignment of the surface domain polarizations along the
in both PT and BT-based FE nanoparticles, thus prob- +zˆdirection. Thisswitchingmechanismproducesmulti-
ing their intrinsic response to applied fields. Such sim- pleshouldersintheresponsecurve,whichmergetogether
ulations are done by first applying an external electric smoothly for larger particles that contain many surface
field E ≡−E zˆto volume Ω , which induces a satu- domains [see, e.g., curve (iii) in Fig. 3(a)].
z max M
7
FIG.3. (a)AveragepolarizationP¯ and(b)polarization-scaledChern-Simonsnumberdensity|n¯ |·P¯ asfunctionsofapplied
z CS z
electricfieldE inPT/STstructuresofdifferentsizes. Panel(c)showspolarizationtexturesin(i)monodomain,d=4nm,(ii)
z
vortex-like, d=10 nm, and (iii) multidomain, d=35 nm, structures corresponding to the markings in panel (a).
InFig.3(b)wepresentthechangesinsystemvorticity sion sizes. Therefore, unlike in the PT/ST systems with
underthechangingelectricfield, whichisrepresentedby d < d < d , where vortex-like texture in the near-
v m
thesphere-averagedvalueof|n¯ |weightedbytheP¯ ,for surface region gets expelled abruptly upon transitioning
CS z
the same PT/ST structures as in Fig. 3(a). In case (i), into the monodomain state [see curve (ii) in Fig. 3(a-
monodomaintextureatzerofield,polarizationswitching b)], in BT/ST this texture disappears gradually, as local
occurs with |n¯ |≡0 everywhere throughout the poling polarization continuously rotates to align itself with the
CS
loop. For all the other cases at d > d , switching be- appliedfieldandthecoremonodomaingrowsoutward. A
v
tween monodomain states at −E zˆand +E zˆhap- vectormapofsuchanintermediatehybridstate,exhibit-
max max
pens through the formation of an intermediate vortex- ing both monodomain and vortex-like features, is shown
like state, as indicated by the non-zero value of averaged in Fig. 4(c).
n . In case (ii), vortex-like texture at zero field, vortic-
CS
itychangesabruptly,but,ascanbeseenfromcomparison All these results suggest that a wide variety of differ-
of the curves for d = 16 and 35 nm, this transition be- ent switching patterns and behaviors can be designed by
comes progressively more diffuse for zero-field multido- controllingthesizeoftheparticleaswellasthematerials
main textures at increasing d. However, even for large properties of the particle and the matrix. We note that
nanoparticles containing many domains, such as in case the dielectric response of an aggregate system consisting
(iii), |n¯ | remains non-zero during the switching due to of FE particles of varying sizes dispersed in a dielectric
CS
thepresenceofvortex-liketwistsofPalongdomainwalls. medium will depend strongly on the nature of the ap-
pliedmechanicalandelectricalboundaryconditions. For
Fig. 4 presents field induced variations in polariza- example, coherency (misfit) strains between the parti-
tion response [panel (a)] and vorticity [panel (b)] in the cle and the dielectric matrix, thermal and/or epitaxial
BT/ST systems with particle d = 9 to 30 nm. These stresses introduced into thin-film heterostructures dur-
curveslookmoreslim,incomparisonwiththeonesshown ing growth, specifics of the electric field application to
for PT/ST in Fig. 3(a), while the dependence of |n¯ | thestructure—suchasusageoftop-bottomorinterdig-
CS
on E suggests diffuse poling behavior in BT/ST pro- itated electrodes, or an atomic-force microscope tip —
z
ceeding through the formation of an intermediate state will all have a significant effect on the overall dielectric
with non-zero vorticity at all of the considered inclu- response of the composite.
8
FIG.4. (a)AveragepolarizationP¯ and(b)polarization-scaledChern-Simonsnumberdensity|n¯ |·P¯ asfunctionsofapplied
z CS z
electric field E in BT/ST structures of different sizes. Panel (c) shows a BT/ST system with a 15 nm inclusion that is in the
z
process of nucleating a monodomain core (the blue region) as its polarization aligns with the external field. At zero applied
field, this particle has a vortex-like P texture. In the near-surface region, P is curling around, which results in nonzero value
of n .
CS
IV. CONCLUSIONS vspolarizationresponseloopontheparticlesize—anef-
fectthatcannotbedirectlyreproducedinbulkFEmate-
rials—wealsopredicthighintrinsicdielectrictunability
We have investigated the behavior of ferroelectric
of such FE nanoinclusions. This behavior is rooted in a
nanoparticles in dielectric media in a parametric space
multistageswitchingofthepolarizationthroughaninter-
wheredifferentinteractions,suchastheelectrostaticsre-
mediatestatewithnon-zeroChern-Simonsvorticitythat
sultingfromsurfaceandbulkcharges,aswellasdomain-
canemerge/dispersegraduallyorabruptly,dependingon
wall and electrostrictive energies, are of similar magni-
a particular choice of material parameters and particle
tudes and compete. As a consequence, the observed fer-
sizes. Field induced polarization response curves, such
roelectric behavior and response is complex and highly
as the ones presented in Figs. 3(a) and 4(a), can be ob-
tunable by the selection of materials parameters and/or
tainedexperimentally, e.g., usingpiezo-forcemicroscopy.
externalfields. Ourresultsshowthatahigh-permittivity
Therefore, it may be possible to utilize such measure-
dielectric medium, that compensates charges on the in-
ments to explore the predicted rearrangements of polar-
clusion surface, can stabilize non-zero polarization in
ization textures within the FE particles, as well as to
PT particles as small as 2 nm in diameter. In con-
evaluate the quality of the LGD G -parameterizations
trast, embedding in a low-permittivity medium results ijkl
for the gradient energy terms by observing and compar-
in large uncompensated surface charges that completely
ing the shapes of the response curves in samples with
suppress polarization in particles smaller than d (cid:39) 3.4–
different particle sizes.
3.6nm. Abovethatcriticalsize,theFEstateemergesas
a vortex-like texture, which minimizes the electrostatic
energy arising from the surface charges, while at even
largersizes,amultidomaintextureisformedasacompro- V. ACKNOWLEDGMENTS
mise between the electrostatic and polarization-gradient
energy contributions. The electrostrictive coupling be-
The authors are indebted to Dmitry Karpeyev for sig-
tween the elastic and polar degrees of freedom softens nificant contributions to the Ferret repository. J.M.
sharp 90◦ domain walls, producing rounded vortex-like
acknowledgesfundingsupportfromtheU.S.Department
textures with cylindrical cores that can penetrate all the
of Energy, Office of Science, Office of Workforce Devel-
way through the spherical particle.
opment for Teachers and Scientists, Office of Science
From an intricate dependence of the shape of the field Graduate Student Research (SCGSR) program. The
9
SCGSR program is administered by the Oak Ridge In- nology as part of the Center for Hierarchical Material
stituteforScienceandEducation(ORISE)fortheDOE. Design (CHiMaD). J.M. would also like to thank Can-
ORISE is managed by ORAU under contract number dost Akkaya for a helpful discussion.
de-sc0014664. The work by O.H. was funded by the US The authors also acknowledge the computing resource
Department of Energy, Office of Science, Basic Energy support provided on Blues, a high-performance comput-
Sciences, Division of Materials Science and Engineering. ing cluster operated by the Laboratory Computing Re-
The work of A.M.J. was performed under financial as- source Center at Argonne National Laboratory, and on
sistance award 70NANB14H012 from U.S. Department the Hornet cluster, hosted by the Taylor L. Booth Engi-
of Commerce, National Institute of Standards and Tech- neering Center for Advanced Technology, located at the
University of Connecticut at Storrs.
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