Domain walls of ferroelectric BaTiO3 within the Ginzburg-Landau-Devonshire phenomenological model

0.77 MB·English

by  P. Marton

Domain walls of ferroelectric BaTiO within the Ginzburg-Landau-Devonshire
3
phenomenological model
P. Marton, I. Rychetsky and J. Hlinka
Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Praha 8, Czech Republic
(Dated: January 11, 2010)
Mechanically compatible and electrically neutral domain walls in tetragonal, orthorhombic and
rhombohedralferroelectricphasesofBaTiO aresystematicallyinvestigatedintheframeworkofthe
3
phenomenological Ginzburg-Landau-Devonshire (GLD) model with parameters of Ref.[Hlinka and
0 Marton, Phys. Rev. 74, 104104 (2006)]. Polarization and strain profiles within domain walls are
1 calculated numerically and within an approximation leading to the quasi-one-dimensional analytic
0 solutions applied previously to the ferroelectric walls of the tetragonal phase [W. Cao and L.E.
2 Cross,Phys. Rev. 44,5(1991)]. Domainwallthicknessesandenergydensitiesareestimatedforall

n mechanicallycompatibleandelectricallyneutraldomainwallspeciesintheentiretemperaturerange
a of ferroelectric phases. The model suggests that thelowest energy walls in theorthorhombic phase
J of BaTiO are the 90-degree and 60-degree walls. In the rhombohedral phase, the lowest energy
3
1 walls are the71-degree and 109-degree walls. Alltheseferroelastic walls havethicknessbelow 1nm
1 except for the 90-degree wall in the tetragonal phase and the 60-degree S-wall in theorthorhombic
phase, for which the larger thickness of the order of 5nm was found. The antiparallel walls of the

] rhombohedral phase have the largest energy and thus they are unlikely to occur. The calculation
ci indicatesthat thelowest energystructureof the109-degree wall andfew otherdomain walls in the
s orthorhombic and rhombohedral phases resemble Bloch-like walls known from magnetism.
-
l
r PACSnumbers: 77.80.-e,77.80.Dj,77.84.Dy
t
m
. I. INTRODUCTION mechanical models.
t
a
m BaTiO3 represents a typical ferroelectric material
Domain structure is an important ingredient in func- which undergoes a sequence of phase transitions from
-
d tionality of ferroelectric materials. Among others, it high-temperature paraelectric cubic Pm3m (Oh) to
n has impact on their nonlinear optical properties, dielec- the ferroelectric tetragonal P4mm (C4v), orthorhombic
o tric permittivity and polarization switching phenomena. Amm2 (C2v) and rhombohedral R3m (C3v) phase. En-
c
Sincedomainboundariesinferroelectricperovskitemate- ergetically equivalent directions of spontaneous polar-
[
rials can simultaneously play the role of the ferroelectric ization vector, identifying possible ferroelectric domain

1 and ferroelastic walls, such domain walls also strongly statesinaparticularferroelectricphase,aredisplayedin
v influencetheelectromechanicalmaterialproperties: they Fig.1. Domainboundaries separatingtwo domainstates
6
facilitateswitchingofspontaneouspolarizationandspon- arecharacterizedbytherotationalangleneededtomatch
7
taneous deformation, thus giving rise to a large extrinsic spontaneous polarizations on both sides of the bound-
3
1 contributiontoe.g. piezoelectricconstants,whichmakes ary. Forexample,theboundaryseparatingdomainswith
. ferroelectric materials extremely attractive for applica- mutuallyperpendicularspontaneouspolarizationiscom-
1
tions. The domainstructure alsoprovides additionalde- monly called 90◦ wall, while the one between antiparal-
0
0 greeoffreedomfortuningofmaterialproperties. Ingen- lel spontaneous polarization regions is called 180◦ wall.
1 eral, further development of domain engineering strate- Otheranglesarepossibleinorthorhombicandrhombohe-
v: gies requires deeper understanding of the physics of fer- dral phases of BaTiO3, where the spontaneous polariza-
roelectric domain wall itself. tion is oriented along the cubic face diagonals and body
i
X diagonals, respectively.
The Ginzburg-Landau-Devonshire (GLD) theory pro-
r
a vides a feasible tool for such a purpose. Landau- The aim ofthis paper is to calculate basic characteris-
Devonshire model describes phase-transition properties tics of all electrically neutral and mechanically compat-
of single-domain crystal using a limited number of pa- ible domain walls in all ferroelectric phases of BaTiO .
3
rameters, which are determined experimentally (or re- For a better comparison of domain wall properties like
cently also using ab-initio methods). Introduction of their thickness or energy density, we employ an Ising-
Ginzburg gradient term to the free energy functional like approximation leading to previously proposed an-
enables addressing nonhomogeneous multi-domain fer- alytically solvable one-dimensional solutions.1 The pa-
roelectric state. The GLD model was previously used per is organized as follows. In Section II. we give an
for computation of domain wall properties in ferroelec- overview of the different kinds of mechanically compati-
tric materials (e.g. Refs.1–5) and in phase-field com- ble domain walls in the three ferroelectric phases. It fol-
puter modeling of domain formation and evolution.6–9 lows from general theory about macroscopic mechanical
The GLD model can be regardedas a bridge model cov- compatibility of adjacent domain states.10,11 The GLD
ering length-scales inaccessible by ab-initio and micro- parametersusedforcalculationofthe domainwallprop-
2
FIG. 1: Directions of spontaneous polarization in the (a)
tetragonal, (b) orthorhombic and (c) rhombohedral phase.
Angles between polarization direction ’0’ and its symmetry
equivalent ones are indicated.
ertiesinBaTiO arethe sameasinourprecedingwork,4
3
but for the sake of convenience, the definition and the
GLD model and its parameters are resumed in Section
III. Section IV. is devoted to the description of the com-
putational scheme and approximations applied here to
solve analytically the Euler-Lagrange equations. The
main result of our study - systematic numerical evalu-
ation of thicknesses, energies, polarization profiles and
other properties for different domain walls, is presented
inSectionV.SectionsVI.andVII.aredevotedtothedis-
cussion of validity of used approximations and the final
conclusion, respectively.
II. MECHANICALLY COMPATIBLE DOMAIN
WALLS IN BARIUM TITANATE
The energy-degeneracy of different directions of spon-
taneouspolarizationleadstotheappearanceofferroelec- FIG. 2: Set of mechanically compatible and electrically neu-
tric domain structure. Individual domains are separated traldomainwallsinthethreeferroelectric phasesofBaTiO .
3
by domain walls, where the polarization changes from In the case of 180◦ domain walls, where the orientation is
one state to another. Here, only planar domain walls not determined by symmetry,walls with the most important
are considered. Orientations of mechanically compatible crystallographic orientations are displayed.
domainwallsaredeterminedbytheequationformechan-
icallycompatibleinterfacesseparatingtwodomainswith
the strain tensors e ( ) and e ( ) : (and its orientation can be therefore dependent on tem-
ij ij
−∞ ∞ perature). For N = there exists infinite number of
∞
3 wall orientations, some of them may be preferred ener-
[e ( ) e ( )]x x =0 . (1) getically.
mn mn m n
∞ − −∞
mX,n=1 Further, the electrically neutral domain walls will
be considered.10 It implies that the difference P( )
Systematic analysis of this equation using symmetry ar- P( )betweenthespontaneouspolarizationsinth∞ead−-
guments has been done e.g. in Refs.10–12. In general, jac−en∞tdomainsisperpendiculartotheunitvectors,nor-
the number N of mechanically compatible domain walls mal to the domain wall:
separating two particular domain states can have only
one of the three values: N = 0, N = 2 or N = . (P( ) P( )) s=0. (2)
∞ ∞ − −∞ ·
In case of N = 2 there exist two mutually perpendic-
ular domain walls. Each of them is either a crystallo- Wealsodefineaunitvectorr (P( ) P( )), which
k ∞ − −∞
graphic (W -type) wall or non-crystallographic(S-type) identifies the componentofthe spontaneouspolarization
f
wall. Orientationofthe W -wallisfixedby symmetryof which reverses when crossing the wall. Then the charge
f
the crystal,while orientationof the S-wallis determined neutrality condition (2) can be expressed as r s = 0.
by components of the strain tensor in adjacent domains Finally, let us introduce a third base vector t =· r s,
×
3
which complements the symmetry-adapted orthonormal 109◦ or 71◦ (R109 or R71, resp.). Since only neutral
coordinate system (r, s, t). walls are discussed here, the S-type domain wall will be
BaTiO symmetry allows a variety of domain walls.12 referred to as O60 and W wall as O120.
3 f
Ferroelectric walls of BaTiO can be divided in two
3
groups - the non-ferroelastic11 walls separating domains
with antiparallel polarization (e ( ) e ( )=0, III. GLD MODEL FOR BARIUM TITANATE
mn mn
∞ − −∞
N = ) and the ferroelastic walls with other than
180◦ be∞tween polarization in the adjacent domain states Calculations presented in this paper are based on the
(N = 2). The r s = 0 condition implies that the neu- GLDmodel withanisotropicgradientterms,reviewedin
·
tralnon-ferroelasticwallsareparalleltothespontaneous Ref.4. The free energy F is expressed in terms of polar-
polarization, and the neutral ferroelastic walls realize a ization and strain field taken for primary and secondary
”head-to-tail”junction. The set of plausible neutral and order-parameter,resp.:
mechanicallycompatibledomainwalltypesareschemat-
ically shown in Fig.2. Domain walls are labeled by a F [ P ,P ,e ]= fdr , (4)
symbolcomposedoftheletterspecifyingtheferroelectric { i i,j ij} Z
phase(T,OorRstayingforthetetragonal,orthorhombic
orrhombohedral,resp.),number indicatingthe polariza- where the free energy density f consists of Landau, gra-
tion rotation angle (180, 120, 109, 90, 71 or 60 degrees) dient, elastic and electrostriction part
and,ifneeded,theorientationofthedomainwallnormal
with respect to the parent pseudo-cubic reference struc- f =fL(e){Pi}+fC{Pi,eij}+fq{Pi,eij}+fG{Pi,j}. (5)
ture.
Our choice of the base vectors r,s and of the sponta- The Landau potential considered here is expanded up
to the sixth order in components of polarization for the
neouspolarizationandstraincomponentsintheadjacent
cubic symmetry (O1):
domain pairs for each domain wall type shown in Fig.2 h
are summarized in TableI. Base vectors coincide with
special crystallographical directions, except for the O60 fL(e) = α1 P12+P22+P32
wallwherethesandtvectorsdependontheorthorhom- +α(cid:0)(e) P4+P4+(cid:1)P4
bic spontaneous strain (see TableI) as follows:5 11 1 2 3
+α(e)(cid:0)P2P2+P2P2(cid:1)+P2P2
12 1 2 2 3 1 3
1 1
rO60 = (cid:18)√2,0,√2(cid:19) +α111(cid:0)P16+P26+P36 (cid:1)
+α ((cid:0)P4(P2+P2)+(cid:1)P4(P2+P2)
e e 2e e e 112 1 2 3 2 1 3
sO60 = a− c, b, c− a +P4(P2+P2)).
(cid:18) D1 D1 D1 (cid:19) 3 1 2
+α P2P2P2 (6)
t = −eb,ea−ec, eb , (3) 123 1 2 3
O60
(cid:18) D D D (cid:19)
2 2 2 with the three temperature-dependent coefficients
α ,α and α , as in Ref.13. This expansion produces
with D = √2D , D = (e e )2+2e2 and with 1 11 111
1 2 2 a− c b the 6 equivalent domain states in the tetragonal phase,
ea,eb, and ec defined in TabpleI. 12intheorthorhombic,and8intherhombohedralphase
Although only the neutral walls are discussed in the
(see Fig.1).
following,the Fig.2 is actually helpful in enumerationof
Dependence of the free energy on the strain is en-
allpossiblemechanicallycompatibledomainwallspecies
countered by including elastic and linear-quadratic elec-
inBaTiO . Inprinciple,mechanicalcompatibilityallows
3 trostriction functionals F and F , resp. Their corre-
180◦ W -type domain walls with an arbitrary orienta- C q
∞ sponding free energy densities are
tion of the domain wall in all three ferroelectric phases
(T180,O180,R180). Obviously,theyareelectricallyneu-
1
tral only if the domain wall normal is parallel with the fC = eρCρσeσ (7)
2
spontaneouspolarization. Ferroelasticwallsexistsinmu-
tuallyperpendicularpairs. Inthetetragonalphase,there and
exist 90◦ W -type domain walls (T90), either charged
f
(head-to-head or tail-to-tail) or neutral (head-to-tail). fq = qijkleijPkPl , (8)
−
The orthorhombic phase is more complex. In the case
of 60◦ angle between polarization directions, the N=2 where eij = 21(∂ui/∂xj +∂uj/∂xi) and Cαβ, qαβ are
pair is formedby a chargedW -type wallandneutral S- componentsofelasticandelectrostrictiontensorinVoigt
f
type wall. The case of 120◦ angle is similar but Wf-wall notation, C11 = C1111, C12 = C1122, C44 = C1212, q11 =
is neutral and S-wall is charged. In addition, there are q1111, q12 =q1122, but q44 =2q1122.
againchargedorneutral90◦ W -walls(O90). Therhom- The elastic and electrostriction terms result in re-
bohedralphasehaspairsofcharfgedandneutralWf-type normalization of the bar expansion coefficients α(1e1) and
domainwallswiththeanglebetweenpolarizationseither α(e) when minimizing the free energy with respect to
12
4
TABLE I:Cartesian components of switching vectors r, domain wall normals s,and boundaryconditions for polarization and
strain in adjacent domain states for the inspected domain walls. Vector s is defined in Eqn.(3), P stands for magnitude
O60 0
ofspontaneouspolarization. Asusual, spontaneousquantitiesarethoseminimizingGLD functional. Numericalvaluesusedin
this work are given in Sec˙V.
Wall r s P(−∞)/P P(∞)/P e(−∞) e(∞)
0 0
T180{001} (1,0,0) (0,0,1) (1,0,0) (−1,0,0) (e ,e ,e ,0,0,0) (e ,e ,e ,0,0,0)
T180{011} (1,0,0) (0, 1 , 1 ) (1,0,0) (−1,0,0) (ek,e⊥,e⊥,0,0,0) (ek,e⊥,e⊥,0,0,0)
TO91080{1¯10} ((√√1122,,√√1122,,00)) ((√√1122√,,2√√−−1122√,,200)) (√(121,,0√1,20,)0) ((√−012,,−−√112,,00)) ((eeakk,,eea⊥⊥,e,ce,⊥⊥0,,00,,02,e0b)) ((eeak⊥,e,ea⊥,ke,ce,⊥⊥0,,00,,02,e0b))
O180{001} (√12,√12,0) (0,0,1) (√12,√12,0) (√−12,−√12,0) (ea,ea,ec,0,0,2eb) (ea,ea,ec,0,0,2eb)
O90 (0,1,0) (1,0,0) (√12,√12,0) (√12,−√12,0) (ea,ea,ec,0,0,2eb) (ea,ea,ec,0,0,−2eb)
O60 (√12,0,√12) sO60 (√12,√12,0) (0,√12,√−12) (ea,ea,ec,0,0,2eb) (ec,ea,ea,−2eb,0,0)
O120 (√16,√26,√−16) (√12,0,√12) (√12,√12,0) (0,√−12,√12) (ea,ea,ec,0,0,2eb) (ec,ea,ea,−2eb,0,0)
R180{1¯10} (√13,√13,√13) (√12,√−12,0) (√13,√13,√13) (√−13,√−31,√−13) (ea,ea,ea,2eb,2eb,2eb) (ea,ea,ea,2eb,2eb,2eb)
R180{¯211} (√13,√13,√13) (√−26,√16,√16) (√13,√13,√13) (√−13,√−31,√−13) (ea,ea,ea,2eb,2eb,2eb) (ea,ea,ea,2eb,2eb,2eb)
R109 (√12,√12,0) (0,0,1) (√13,√13,√13) (√−13,√−31,√13) (ea,ea,ea,2eb,2eb,2eb) (ea,ea,ea,−2eb,−2eb,2eb)
R71 (0,1,0) (√12,0,√12) (√13,√13,√13) (√13,√−31,√13) (ea,ea,ea,2eb,2eb,2eb) (ea,ea,ea,−2eb,2eb,−2eb)
strains (in the homogeneous sample). The bar α(e), α(e) where T is absolute temperature.13 The phase transi-
11 12
and the relaxed α , α coefficients are related as:4 tions occur in this model at the temperatures 392.3K
11 12
(C T), 282.5K (T O), and 201.8K (O R).13 All
→ → →
phase transitions are of the first order, the local min-
1 qˆ2 qˆ2
α(e) = α + 11 +2 22 ima corresponding to the tetragonal, orthorhombic and
11 11 6(cid:20)Cˆ11 Cˆ22(cid:21) rhombohedral phase exist for this Landau potential
1 qˆ2 qˆ2 q2 between 237K and 393K, between 104K and 303K,
α(e) = α + 2 11 2 22 +3 44 (9)
12 12 6(cid:20) Cˆ11 − Cˆ22 C44(cid:21) ainnddepbeenlodwent25p6aKra,mreestpeersctiovfeltyh.eFGulLlDsetmoofdteelmrpeaedrast:4u,1r4e
with α = 3.230 108Jm5C−4, α = 4.470 109Jm9C−6,
12 112
α = 4.910× 109Jm9C−6, G = 51 ×10−11Jm3C−2,
Cˆ11 = C11+2C12 G123= 2 ×10−11Jm3C−2, G11 = 2 ×10−11Jm3C−2,
12 44
Cˆ12 = C11−C12 q11 = 1−4.2×0 × 109JmC−2, q12 = −0.×74 × 109JmC−2,
qˆ11 = q11+2q12 Cq44 ==171..9507 ×1100190JJmmC−3−2a,ndCC11 ==52.74.350 ×1011001J0mJ−m3−.3,
qˆ12 = q11 q12 . (10) 12 × 44 ×
−
The Ginzburg gradient term f is considered in the
G
form IV. STRAIGHT POLARIZATION PATH
APPROXIMATION
1
f = G (P2 +P2 +P2 )
G 2 11 1,1 2,2 3,3
Let us consider a single mechanically compatible and
+G (P P +P P +P P )
12 1,1 2,2 2,2 3,3 1,1 3,3 electrically neutral domain wall in a perfect infinite
1
+ G ((P +P )2+(P +P )2 stress-free crystal. Within the continuum GLD theory,
44 1,2 2,1 2,3 3,2
2 suchdomainwallisassociatedwithaplanarkinksolution
+(P +P )2) . (11) of the Euler-Lagrange equations of the GLD functional
3,1 1,3
for polarization vector and the strain tensor.1,2,4,16 Do-
Itwaspointedout4 that the tensorofgradientconstants mainwalltypeisspecifiedbyselectionofthewallnormal
ofBaTiO3 is highly anisotropicwith fundamentalconse- sandbythetwodomainstatesats= and+ . This
quencesonpredicteddomainwallproperties. Uptonow, ideal geometry implies that the polar−iz∞ation an∞d strain
the isotropic gradient tensor Gi,j was mostly employed vary only along normal to the wall s and domain wall
in the computations. canbethusconsideredasatrajectoryinorder-parameter
The material-specific coefficients in the model are as- space.
sumed being constant, except for the three Landau po- Even if the domain wall is neutral, in its central
tential coefficients part the local electric charges can occur due to the
position-dependentpolarization. Theadditionalassump-
α = 3.34 105(T 381)
1 × − tion P = 0 ensures the absence of charges in the
α11 = 4.69 106(T 393) 2.02 108 whole∇d·omain wall. Then the local electric field is zero
× − − ×
α = 5.52 107(T 393)+2.76 109 , (12) andthe electrostaticcontributionvanishes. Suchstrictly
111
− × − ×
5
charge-freesolutions16 were found to be an excellent ap-
proximationforidealdielectricmaterials.4Thecondition
P = 0 implies that the polarization vector variation
∇isr·estrictedtoaplaneperpendiculartos(the trajectory
is constrained to P =P ( ) plane).
s s
±∞
In fact, the polarization trajectories representing the
T180andT90wallscalculatedunder P=0constraint
∇·
werefoundtobethestraightlinesconnectingthebound-
ary values.1,2,4,16 This greatly simplifies the algebra and
the solution of the variational problem can be found an-
alytically. Therefore,wedecidedtoimposethecondition
of a direct, straightpolarizationtrajectoryfor the varia-
tionalproblemofalldomainwallspeciesofBaTiO . This
3
condition, further referred as straight polarization path
(SPP) approximation, implies that both P and P po- FIG. 3: Profile of the ”reversed” polarization component
s t
P (Eqn.20) for a domain wall with the shape factor A = 1
larization components are constant across the wall. We r
(solid) and A = 3 (dashed). Both profiles have the same
shall come back to the meaning and possible drawbacks
derivative for s/2ξ = 0 (dotted) and therefore also the same
of this approximation in Section VI.
thickness(indicatedbyverticallines)accordingtothedefini-
Polarization and strain in the mechanically compat-
tion in Eqn.(23).
ible and electrically neutral SPP walls can be cast
in the form: P = [P (s),P ( ),P ( )] and e =
r s t
±∞ ±∞
[e ( ),e (s),e ( ),2e (s),2e ( ),2e (s)], re-
rr ss tt st rt rs where (see Ref.4)
±∞ ±∞ ±∞
spectively. The s-dependent strain components are cal-
culated from the mechanical equilibrium condition f (p)=a p(s)2+a p(s)4+a p(s)6 . (18)
EL 1 11 111
3 3
∂σij = ∂ ∂f =0 . (13) The function fEL(p) is a double-well ”Euler-Lagrange”
Xj=1 ∂xj Xj=1 ∂xj (cid:18)∂eij(cid:19) potential with two minima ±p∞, where
Boundaryconditionsforstressandthefactthatallquan- a + a2 3a a
p2 = − 11 11− 111 1. (19)
tities vary only along direction s imply ∞ p3a
111
∂f
Cq +C =0 (14) The differential Eqn.(16) has the analytical solution
∂e
ij
sinh(s/ξ′)
for ij ss,st,rs with the integration constant C to be p(s)=p , (20)
∞
determ∈in{ed from}boundary conditions. A+sinh2(s/ξ′)
The Euler-Lagrange equation for polarization reduces q
to where
∂ ∂f ∂f
∂s∂P − ∂P =0 . (15) A= 3a111p2∞+a11 . (21)
r,s r 2a p2 +a
111 ∞ 11
wherethe straincomponentsfromEqn.(14)weresubsti-
and
tuted into f. Thus, the elastic field was eliminated. It
turns out that the resulting Euler-Lagrangeequation for ξ
P is possible to rewrite in the form ξ′ = . (22)
r √A
d2p(s)
g =2a p(s)+4a p3(s)+6a p5(s) , (16) QuantityA determines deviationofthe profile (20) from
ds2 1 11 111
the tanh profile,whichoccurs forthe 4th-orderpotential
where p(s) stands for Pr(s), and where the coefficients (i.e., a111 =0, A=1).
g,a ,a anda ,differentforeachdomaintype,depend The domain wall thickness (Fig.3) is defined as
1 11 111
only on the material tensors. The boundary values are
Pr( )=Pr( )=p∞. 2g
− ∞ −∞ 2ξ =p . (23)
Solution of the Euler-Lagrange equation (16) is well ∞rU
known.1,16–19 We shall follow the procedure of Ref.20.
Integrating Eqn. (16) one can obtain the equation: with U being the energy barrier between the domain
states:
g ∂p 2
=f (p) (17)
2(cid:18)∂s(cid:19) EL U =fEL(0)−fEL(p∞)=2a111p6∞+a11p4∞ . (24)
6
The surface energy density of the domain wall is
∞
Σ = (f (s) f (p ))ds
EL EL ∞
Z −
−∞
4
= p 2gU A5/2I(A) , (25)
∞
3
p h i
where
3 ∞ cosh2(h)dh
I(A)= . (26)
4Z (A+cosh2(h) 1)3
−∞
−
The domain wall characteristics depend on the coeffi-
cientsg, a , a , a throughEqs. (19,20,23,25). The
1 11 111
explicitexpressionsfor these coefficientsaresummarized
for various domain walls in TableII. The expressionsare
simplified using the notation inspired by Ref.1. For all
phases we are using:
1 qˆ2 1 qˆ2 (q +q )q′
ar = α 11 + 22 11 12 12 P2
1 1−(cid:20)3Cˆ 6Cˆ − 2C′ (cid:21) 0
11 22 11 FIG.4: Dependenceofcorrection factorintheexpressionfor
domain wall energy density (25) as a function of the shape
α(e) α(e) q′2 coefficient A. Full points indicate values of A for particular
ar = 11 + 12 12
11 2 4 − 2C′ domain walls considered in TableIII.
11
α(e) q′ q′ qˆ2
ars = 3α(e) 12 11 12 22
12 11 − 2 − C′ − 2Cˆ
11 22
1
a′ = (α +α )
111 4 111 112
1
a′ = (15α α ) ,
112 4 111− 112
for tetragonal and orthorhombic phases we are abbrevi-
ating
C +C +2C
C′ = 11 12 44
11 2
C +C 2C
C′ = 11 12− 44
12 2
C C
C′ = 11− 12 FIG. 5: Course of strain components along the domain wall
66 2 normalcoordinatesfortheT90domainwall. Indicesreferto
cubicaxes of theparent phase.
q +q +q
q′ = 11 12 44
11 2
q +q q
q′ = 11 12− 44 V. QUANTITATIVE RESULTS
12 2
q′ = q q ,
66 11− 12 Advantage of the GLD approach is that domain wall
and for the rhombohedral phase properties can be obtained at any temperature. In Ta-
bleIII we give numerical results for domain wall param-
C +C +2C
C′ = 11 12 44 eters at one particulartemperature for eachof the ferro-
33 2
electricphases: at298K,208K,and118Kforthetetrag-
q +2q +2q
q′ = 11 12 44 onal,orthorhombic,andrhombohedralphase,resp. Cor-
11 3
responding numerical values of the spontaneous quanti-
q +2q q
q1′3 = 11 312− 44 . (27) ties appearing in TableI are: P0 = 0.265Cm−2, ek =
Q P2 = 7.77 10−3 and e = Q P2 = 3.18 10−3
The expressions for the coefficients of the T180 and (th11e t0etragona×l phase, 298K⊥); P 1=2 00.331−Cm−2×, e =
0 a
T90 domain walls are equivalent to the previously pub- Q11+Q12P2 = 3.58 10−3, e = Q P2 = 4.96 10−3
lished expressions.1,4 Derivations for O60, O120, and 2 0 × c 12 0 − ×
R180 ¯211 wallsleadtocomplicatedformulas,andthere- and eb = Q444P02 = 0.79 × 10−3 (the orthorhombic
fore o{nly n}umerical results for g,a1,a11, and a111 coeffi- phase, 208K); P0 = 0.381Cm−2, ea = Q11+32Q12P02 =
cients are presented here. 0.97 10−3 and e = Q44P2 = 0.70 10−3 (the rhom-
× b 4 0 ×
7
TABLE II: Parameters characterizing various mechanically compatible and neutral domain wall species of BaTiO -like fer-
3
roelectrics within the SPP treatment described in the Section IV. Results for O60, O120, and R180{¯211} walls as well as a
few other parameters are omitted because the corresponding analytical expressions through the GLD model parameters and
spontaneous values of polarization and strain are too complicated.
Domain wall p g a a a
1 11 111
∞
Tetragonal phase
TT118800{{000111}} PP00 GG4444 αα11−−ekeqk1q111−+e⊥CCq11′1212ek+q1CC211−12(2ekCC14′+14ee⊥⊥q)1q212 αα(1(1ee11))−− 2211CCqq112211′2211 αα111111
T90 √P02 G11−2G12 αr1+αr1s2P202 +α′112P404 αr11+α′112P202 α′111
Orthorhombic phase
O180{1¯10} P0 G11−2G12 α1−eb(q1′1−q1′2)−ecq12−ea(q1′1+q1′2) α(12e1) + α(14e2) − 2qC1′2′2 41(α111+α112)
+Cq1′′22P02 11
O180{001} P0 G44 α1−ea(q11+q12)−11ebq44+2eaq12CC1112 α(12e1) + α(14e2) − 2qC12121 41(α111+α112)
O90 √P02 G44 αr1+α(1e2)P202 +α112P404 α(1e1)+ 12hα112P02− Cq12121i α111
O120 √32P0 G11−G162+4G44 *** *** 2α111+211α01812+2α123
Rhombohedral phase
R180{1¯10} P0 G11−G312+G44 α1+ 6C1′ [C12(ea(q11+2q12−4q44)−6ebq44) α(1e1)+3α(1e2) − 2qC1′2′3 3α111+6α27112+α123
33 33
−C11(6ebq44+3eaq1′1)
−2C44(3ea(q11+2q12)+6ebq1′1)]
R180{¯211} P0 G11−G312+G44 *** *** 3α111+6α27112+α123
R109 √P03 G44 α1+ α(1e23)P02 + α1192P04 + 2q312C21P102 − q642C4P4402 α(12e1) + α(14e2) − 2qC12121 α111+4α112
−e (q +2q )−e q + 1 (2α +α )P2
a 11 12 b 44 12 112 123 0
R71 √P03 G44 α1+ 2α1(e23)P02 + 2α1192P04 + α1293P04 α(1e1)+ 2α1132P02 − 2qC123′23 α111
−ea(q11+2q12)+ q312C2P3′302 − q342C4P4402
TABLEIII:PredictedvaluesofthicknessandplanarenergydensityofdomainwallspeciesillustratedinFig.2togetherwiththe
determiningparametersappearingintheSPPtreatment. Resultsareevaluatedfrom theBaTiO -specificGLDmodelatthree
3
selected temperatures corresponding to thetetragonal, orthorhombic and rhombohedral phase, respectively. Domain walls for
which relaxing of the Pt component results in a lower-energy CPP solution (see in SectionVI.) are denoted by †. Numerical
values are in SI units (2ξ in nm; Σ in mJ/m2; U in MJ/m3; p in C/m2; g in 10−11 kgm5s−2C−2; a1 in 107 kgm3s−2C−2;
a11 in 108 kgm7s−2C−4; a111 in 109 kgm11s−2C−6). ∞
Domain wall 2ξ Σ U A p g a a a
1 11 111
∞
Tetragonal phase(298K)
T180{001} 0.63 5.9 6.41 1.43 0.265 2.0 -14.26 1.69 8.00
T180{011} 0.63 5.9 6.41 1.43 0.265 2.0 -14.26 1.69 8.00
T90 3.59 7.0 1.45 1.09 0.188 26.5 -7.86 9.53 3.12
Orthorhombicphase (208K)
O180{1¯10}† 2.66 31.0 8.20 1.70 0.331 26.5 -9.71 -2.75 4.36
O180{001} 0.70 8.9 8.95 1.64 0.331 2.0 -11.07 -2.13 4.36
O90 0.72 4.3 4.26 1.50 0.234 2.0 -11.62 -0.08 12.97
O60 3.62 5.3 1.09 1.08 0.166 26.1 -7.64 12.14 4.36
O120† 1.70 13.7 5.82 1.47 0.287 10.2 -10.82 0.50 4.92
Rhombohedralphase (118K)
R180{1¯10}† 2.13 36.0 11.81 1.83 0.381 18.3 -9.53 -3.64 3.17
R180{¯211}† 2.13 36.0 11.81 1.83 0.381 18.3 -9.53 -3.64 3.17
R109† 0.70 7.8 7.81 1.65 0.311 2.0 -10.84 -2.56 5.60
R71 0.74 3.7 3.52 1.58 0.220 2.0 -10.31 -2.42 17.94
8
FIG.6: Temperaturedependenceofthethickness(a)and energydensity(b)for variousmechanically compatible andneutral
domain walls of BaTiO estimated within SPP approximation. The SPP values for domain walls for which relaxing of the P
3 t
component results in a lower-energy CPP solution (see in SectionVI.) are shown by dashed lines. Thickness of O180{001},
O90 has avery close temperature dependence. Theresults for T180 as well as for R180 walls are not distinguished by specific
orientations of the wall normal, since for both cases the angular dependence of thickness and energy of SPP solutions on the
domain wall normal is negligible (see Tab.III). Vertical lines mark phase-transition temperatures.
bohedral phase, 118K). The base vectors for the O60 can be easily moved. Moreover, in case of O60 wall, the
domain wall (see Eqn.3) at 208K are pinning is further suppressed due to ’incommensurate’
character of Miller indices of the wall normal.
r = (0.707,0,0.707)
O60
As follows from Eqn.(23), domain wall thickness is
s = (0.701,0.130, 0.701)
tOO6600 = ( 0.092,0.991−,0.092) . (28) tdieotneromfinthedeirbyvalquueasnitnitiTesabgl,eIUII,, awnedsepe∞t.hatBytheinsmpoesct-
−
important factor is the coefficient g. Indeed, the do-
The right four columns of the TableIII contain corre-
main walls R71, R109, O90, O180 001 , T180 with
spondingnumericalvaluesofthedomainwallcoefficients { }
g =2 10−11kgm5s−2C−2areallverynarrow(thickness
g, a1, a11, and a111 derived from GLD parameters and ×
below 1nm), and the thickness of various walls mono-
spontaneous order-parametervalues using above derived
tonically increases with increasing value of g. Let us
analyticalexpressions,mostlygivenexplicitlyinTableII.
stress that coefficients a , a , a , and g depend on
TheleftpartoftheTableIIIcontainsthekeydomainwall 1 11 111
the direction of the domain wall normal. For exam-
properties such as wall thickness 2ξ and energy density
ple, O180 1¯10 with g = 26.5 10−11 kgm5s−2C−2
Σ. { } ×
is almost four times broader than O180 001 with g =
Clearly, T90 and O60 domain walls are considerably 2 10−11 kgm5s−2C−2. { }
broader then others. T90 wall in the tetragonal phase is ×
predicted to be 3.59nm thick at 298K (as already cal- In the absence of other constraints, the probability of
culated in Ref.4) and S-type O60 domain wall in the appearanceofdomainwallspecies shouldbe determined
orthorhombic phase is predicted to be 3.62nm thick at by the surface energy density Σ. Therefore, in the or-
208K. Since the wall thickness is greater than the lat- thorhombic phase, the thinner O180 001 wall is more
tice spacing, the pinning of the walls is weak21 and they likely to occur than the 0180 1¯10 {one.}Interestingly,
{ }
9
theO180 001 wallhasalmostthesamethicknessasthe
O90wall,{whil}einthetetragonalphase,itisthe90◦ wall
whichis muchthickerthan 180◦ wall(3.59nm compared
to 0.63nm).
Ingeneral,thenormaloftheneutral180◦-domainwall
can take any direction perpendicular to the spontaneous
polarization of the adjacent domains. Therefore also Σ
dependsontheorientationofthewallnormal(inthes t
−
plane). We havecheckedinthe orthorhombicphasethat
the0180 001 and0180 1¯10 correspondtotheextremes
{ } { }
in the angular dependence of Σ, which is monotonous
between them. This strong angular dependence is corre-
lated with the anisotropy of the tensor G . However,
ijkl
g is independent of the wall direction in the tetragonal
and rhombohedral phase, and also the variation of coef-
ficients a , a , and a is insignificantly small, so that
1 11 111
the effective directional dependence of 180◦ domain wall
properties in these phases is negligible.
The values of the shape factor A (see TableIII) deter-
minesthedeviationoftheP (s)polarizationprofileofthe
r
domain walls from the simple tanh form. For all studied
cases the values of A range between 1 and 2, where the
correction factor A5/2I(A) appearing in the Eqn.(25) is
almostlinearfunctionofA,asitcanbeseeninFig.4. It
means that the shape deviations are much smaller than
those shown by broken line in Fig.3. Unfortunately, in
the case of broad walls T90 and O60, which are good
candidates for study of the structure of the wall central
part, A is almost one.
Eqn.(14)canbealsousedtoevaluatelocalstrainvari-
ation in the wall. In Fig.5 the profiles of strain tensor
FIG.7: Equipotential contoursof theEuler-Lagrange poten-
components for T90 domain wall are shown for illustra-
tials showing ELPS associated with the P =P (±∞) plane
s s
tion. Obviously, e ,e , and e strain components are
33 23 13 foraset ofBaTiO domain walls considered inTableIII.Po-
3
strictlyconstantandequaltotheirboundaryvaluesasit larization scale given for the bottom left inset (in Cm−2) is
follows from mechanical compatibility conditions. The equallyvalidforallshownELPS’s. Energetically most favor-
e11 and e22 components and polarization Pr vary be- able domain wall solutions with trajectories restricted to the
tween their spontaneous values. Let us stress that the Ps=Ps(±∞) plane are indicated with bold lines. The solu-
’re-entrant’ shear component e (it has the same value tions corresponding to a linear segment are denoted in the
12
inbothadjacentdomains)approachesthenon-zerovalue text as SPP walls, as opposed to the CPP solutions (see in
of about 5 10−4 in the middle of the domain wall. SectionVI.),whichhaveacurvedorder-parametertrajectory.
×
Thetemperaturedependence ofthe thicknessandsur-
faceenergydensityofthe12studieddomainwallspecies VI. CURVED POLARIZATION PATH
SOLUTIONS
is plotted in Fig.6a and Fig.6b, respectively. Although
some properties do vary considerably, e.g. thickness of
T90 wall in the vicinity of the paraelectric-ferroelectric So far we have investigated domain walls within the
phase transition, the sequence of the thickness values as SPP approximation, i.e. components Ps and Pt, which
well as the surface energy-density values of different do- are the same in both domain states, were kept constant
main wall types remain conserved within each phase. inside the whole wall. This is quite usual assumption
made for a ferroelectric domain wall. Nevertheless, the
The temperature dependence of domain wall thick- full variational problem, where all three components of
ness follows the trend given by Eqn.(23). It increases P could vary along s coordinate, leads in general to a
with increasing temperature due to the dependence on lower energy solution corresponding to a curved polar-
p (see dependence of U on p in Eqn.(24)). Such ization path (CPP) in the 3D primary order-parameter
∞ ∞
behavior is well known also from the experimental space. Appearance of the non-zero ’re-entrant’ compo-
observations.3,22,23 nents within a ferroelectric domain wall was considered
10
e.g. inworksofRefs.3,4,24,25. For180◦walls,thepolar-
izationprofileassociatedwithSPPissometimesdenoted
as the Ising-type wall. In contrast, the CPP solutions
with non-zero P , P are often considered as N´eel and
s t
Bloch-like,25 even though, in contrast with magnetism,
the modulus of P is far from being conserved along the
wall normal s.
We have previously considered non-constant P com-
s
ponentofpolarizationinthe90◦ wallwithexplicittreat-
ment of electrostatic interaction and realized that the
deviationsfromSPPapproximationarequitenegligible.4
In general, non-constant P would lead to non-vanishing
s
P and finite local charge density, which in a perfect
∇·
dielectric causes a severe energy penalty. The same sit-
uation is expected for all domain wall species. However,
thereisnosuchpenaltyfornon-constantP -solutions. It
t
was previously argued3 that the Bloch-like (with a con-
siderable magnitude of P at the domainwall center)so-
t
lutions could occur in orthorhombic BaTiO . Therefore,
3
it is interesting to systematically check for existence of
such solutions using our model.
InordertostudysuchBloch-likesolutions,wehavecal-
culated Euler-Lagrangepotential in the order-parameter
planeP =P ( )byintegratingEuler-Lagrangeequa-
s s
±∞
tions (Eqn.13 and 14) for all domain wall species from
TableIII similarly as e.g. in the Refs.1,3. Resulting 2D
Euler-Lagrange potential surfaces (ELPS) are displayed FIG.8: PredictedprofilesofpolarizationcomponentsforO60
in Fig.7. In each ELPS, the bold lines indicate numeri- domainwallinBaTiO atT =208K.Fulllinestandsforthe
3
cally obtained domain-wall solution with the lowest en- P component of polarization vector, broken line for the P
r t
ergy. The spatial step was chosen as 0.1nm, P and P one. Analytically obtained SPP solution (a) practically does
r t
were fixed to boundary conditions in sufficient distance not differ from the numerically obtained CPP solution (b)
from domain wall (6nm) and initial conditions for Pt with an unconstrained Pt component in this case.
were chosen so that the polarization path bypasses the
energy maximum of the ELPS, and the system was re-
laxed to the equilibrium.
Ising-like O60 and O120 walls are associated with the
Among the twelve treated wall species, there are six
fact that ELPS is not symmetric with respect to P =
cases where only the SPP solutions with P =constexist: t
t
T180 001 ,T180 011 ,T90,O180 1¯10 ,O90,andR71. Pt(±∞) mirror plane. In these cases not only the po-
{ } { } { } larization path and wall energies, but also domain wall
Thesesolutionsareclearly”Ising-like”. Inallthesecases,
thicknesses of the SPP and CPP counterpart solutions
the (P =0, P =0)point is the only saddle point ofthe
r t
are very similar. This is demonstrated in Fig.8, which
ELPS. In the other six cases – O180 001 , O60, O120,
R180 1¯10 ,R180 ¯211 ,andR109–th{eEL}PShasamax- shows polarization profiles of both SPP and CPP solu-
{ } { } tions for the O60 wall.
imumatthe(P =0,P =0),andthelowestenergysolu-
r t
tions correspondto curvedpolarizationpaths. This sug- Much more pronounced difference between domain
gests that the previously discussed SPP description may wallprofiles ofSPPandCPPsolutions arefound incase
not necessarily be the proper approximation for these of O180 001 ,R180 1¯10 , R180 ¯211 , and R109 Bloch-
walls. Nevertheless, in the case of O60 and O120 walls likewall{swhe}retheC{PP}trajecto{ries}bypassthe(P =0,
r
the deviations from the SPP model are marginal, and P = 0) maximum near the additional minima, which
t
only the remaining four solutions exhibit strong Bloch- originate from ’intermediate’ domain states either of the
like behavior. Moreover, the energy differences between same phase or even of the different ferroelectric phases.
SPPandCPPsolutionswerefound to be almostnegligi- In these cases, the inadequacy of SPP approximation is
ble, except for R180, where the CPP energy is by about obvious. Forexample,theCPPofR180 ¯211 wallseems
10% lowerin the entire temperature range of stability of to pass through a additional minimum c{orre}sponding to
the rhombohedral phase. Therefore, it is quite possible an’orthorhombic’polarizationstate(consultcorrespond-
that in the case of O180 001 , R180 1¯10 , R180 ¯211 , ing inset in the Fig.2). As expected, the profile of such
{ } { } { }
and R109 walls both Bloch-like and Ising-like solutions CPPsolutiondeviatesstronglyfromtanhshapeandeven
may be realized. definitionofthewallthicknesswouldbeproblematic(see
The deviations from SPP in the case of the almost Fig.9).