1. The phase field model used in the present work
The paraelectric to ferroelectric phase transition occurs in a ferroelectric material when
temperature is lower than the Curie point. The polarization, 𝐏 = (𝑃1 , 𝑃2 , 𝑃3 ) is usually used as the
order parameter in the phase field simulation of ferroelectrics and the total free energy is a function
of the polarization, polarization gradient, strain (or stress) and electric field. The domain
configuration and polarization switching are a direct consequence of the minimization process of the
total free energy of a whole simulated system. The temporal evolution of polarization is described by
the time dependent Ginzburg-Landau (TDGL) equation,
𝜕𝑃𝑖 (𝒙 , t)
𝜕𝑡
= −𝐿
𝛿𝐹
𝛿𝑃𝑖 (𝒙 , t)
,
(𝑖 = 1 , 2 , 3)
(1.1)
where L is the kinetic coefficient, F is the total energy of the system, 𝛿𝐹/𝛿𝑃𝑖 (𝒙 , 𝑡) is the
thermodynamic driving force for the spatial and temporal evolution of 𝑃𝑖 (𝒙 , 𝑡), and x=(𝑥1 , 𝑥2 , 𝑥3 )
denotes the spatial vector. The total free energy of Eq. (1.1) can be expressed as
𝐹 = ∫ [𝑓𝐿𝑎𝑛𝑑 (𝑃𝑖 ) + 𝑓𝑒𝑙𝑎𝑠 (𝑃𝑖 , 𝜀𝑖𝑗 ) + 𝑓𝑔𝑟𝑎𝑑 (𝑃𝑖,𝑗 ) + 𝑓𝑒𝑙𝑒𝑐 (𝑃𝑖 , 𝐸𝑖𝑒𝑥 )]𝑑𝑉.
(1.2)
𝑉
In Eq. (1.2), the Landau free energy density 𝑓𝐿𝑎𝑛𝑑 is given by
𝑓𝐿𝑎𝑛𝑑 (𝑃𝑖 ) = 𝛼1 (𝑃12 + 𝑃22 + 𝑃32 ) + 𝛼11 (𝑃14 + 𝑃24 + 𝑃34 ) + 𝛼12 (𝑃12 𝑃22 + 𝑃22 𝑃32 + 𝑃12 𝑃32 )
+𝛼111 (𝑃16 + 𝑃26 + 𝑃36 ) + 𝛼112 [𝑃14 (𝑃22 + 𝑃32 ) + 𝑃24 (𝑃12 + 𝑃32 ) + 𝑃34 (𝑃12 + 𝑃22 )]
+𝛼123 𝑃12 𝑃22 𝑃32 ,
(1.3)
where 𝛼11 , 𝛼12 , 𝛼111 , 𝛼112 , 𝛼123 are constant coefficients and 𝛼1 = (𝑇 − 𝑇0 )/2𝜅0 𝐶0 , 𝑇 and 𝑇0
denote the temperature and Curie-Weiss temperature, respectively, 𝐶0 is the Curie constant and 𝜅0
1
is the dielectric constant of vacuum. The elastic energy density takes the form of
𝑓𝑒𝑙𝑎𝑠 =
1
𝑒𝑙𝑎𝑠 𝑒𝑙𝑎𝑠
𝑐𝑖𝑗𝑘𝑙 𝜀𝑖𝑗
𝜀𝑘𝑙 ,
2
(1.4)
𝑒𝑙𝑎𝑠
where 𝑐𝑖𝑗𝑘𝑙 are the elastic constants and 𝜀𝑖𝑗
are the total elastic strains. The total elastic strains
𝑝,𝑒𝑙𝑎𝑠
𝑝
0
include two parts. The first part are induced by polarizations and is given by 𝜀𝑖𝑗
= (𝜀𝑖𝑗
− 𝜀𝑖𝑗
),
𝑝
0
where 𝜀𝑖𝑗
are the total strains produced by polarizations only and 𝜀𝑖𝑗
are the spontaneous strains or
𝑑
the eigenstrains of polarizations. The other part of the elastic strains, 𝜀𝑖𝑗
, is produced by dislocations.
Thus, the total elastic strains are given by
𝑝
𝑒𝑙𝑎𝑠
0
𝑑
𝜀𝑖𝑗
= 𝜀𝑖𝑗
− 𝜀𝑖𝑗
+ 𝜀𝑖𝑗
.
(1.5)
0
The spontaneous strains have the form as 𝜀𝑖𝑗
= 𝑄𝑖𝑗𝑘𝑙 𝑃𝑘 𝑃𝑙 , where 𝑄𝑖𝑗𝑘𝑙 are the electrostrictive
coefficients. The spontaneous strains are similar to the thermal expansion strains in thermal stress
analysis. In the present study, the elastic solution is separated to dislocation-related and
polarization-related boundary value problems. In each iteration step of the evolution of stress and
polarization distribution, the stress field is analyzed with a given polarization distribution. With such
methodology, the stress field is solved by linear elasticity. Once the stress field is determined, the
stress field will be taken into the analysis of the polarization field in the next iteration step of the
evolution by solving the time dependent Ginzburg-Landau (TDGL) equation. When the stable
polarization distribution is achieved through the iteration at an external loading, the elastic field
approaches the stable solution. For the dislocation-related boundary value problem, elastic field is
calculated separately though Stroh’s formalism.
2
With the periodic boundary condition, the general solution of the total displacement field
induced by polarizations is given in Fourier space by16,17
𝑢𝑖𝑝 (𝝃) = 𝑋𝑗 𝑁𝑖𝑗 (𝝃)/𝐷(𝝃),
(1.6)
0
where 𝑋𝑖 = −𝑖𝑐𝑖𝑗𝑘𝑙 𝜀𝑘𝑙
𝜉𝑗 , 𝑖 = √−1, 𝑁𝑖𝑗 (𝝃) are cofactors of a 33 matrix 𝑲(𝝃),
𝐾11
𝑲(𝝃) = [𝐾21
𝐾31
𝐾12
𝐾22
𝐾32
𝐾13
𝐾23 ],
𝐾33
(1.7)
and 𝐷(𝝃) is the determinant of matrix 𝑲(𝝃). Note that 𝐾𝑘𝑖 (𝝃) = 𝑐𝑘𝑗𝑖𝑙 𝜉𝑗 𝜉𝑙 , in which  i are the
coordinates in Fourier space. The corresponding strains are obtained from
𝑝
𝑝
𝜀𝑖𝑗
1 𝜕𝑢𝑖𝑝 𝜕𝑢𝑗
= {
+
}.
2 𝜕𝑥𝑗
𝜕𝑥𝑖
(1.8)
The dislocation-induced displacement ud is obtained from the Stroh formalism18 for the pure
dislocation problem without any polarizations, which is given in the Section 2 in this supplementary
𝑑
material. The elastic strain induced by dislocations, 𝜀𝑖𝑗
, is obtained from
𝑑
𝜀𝑖𝑗
1 𝜕𝑢𝑖𝑑 𝜕𝑢𝑗𝑑
= {
+
}.
2 𝜕𝑥𝑗
𝜕𝑥𝑖
(1.9)
𝑝
Both the polarization-induced total strains, 𝜀𝑖𝑗
, and the elastic strain induced by dislocations,
𝑑
𝜀𝑖𝑗
,
are calculated from the corresponding displacements. In the present work, the
dislocation-induced strains are added to the polarization-induced total strains in Eq. (1.5). Therefore,
the polarization-induced total strains and dislocation-induced strains are compatible and the
displacements are also compatible.
The gradient energy density can be expressed as
3
𝑓𝑔𝑟𝑎𝑑 =
1
P P
𝑔𝑖𝑗𝑘𝑙 i k ,
2
x j xl
(1.10)
where 𝑔𝑖𝑗𝑘𝑙 are the gradient energy coefficients. The gradient energy gives the energy penalty for
spatially inhomogeneous polarization. The electrical energy density includes the self-electrostatic
energy density and the energy induced by the external electric field, which can be expressed as
1
𝑓𝑒𝑙𝑒𝑐 = − 𝐸𝑖 𝑃𝑖  Eiex Pi ,
2
(1.11)
where 𝐸𝑖 and Eiex are the self-electrostatic electric field and external applied electric field,
respectively. The self-electrostatic field is the negative gradient of the electrostatic potential 𝜙, i.e.,
𝐸𝑖 = −𝜕𝜙/𝜕𝑥𝑖 . The electrostatic potential is obtained by solving the following electrostatic
equilibrium equation,
𝜅11 𝜕 2 Φ 𝜅22 𝜕 2 Φ 𝜅33 𝜕 2 Φ
𝜕𝑃1 𝜕𝑃2 𝜕𝑃3
𝜅0 (
2 +
2 +
2 ) = 𝜕𝑥 + 𝜕𝑥 + 𝜕𝑥
𝜕𝑥1
𝜕𝑥2
𝜕𝑥3
1
2
3
,
(1.12)
where κij are the relative dielectric constants of the material. With the periodic boundary condition,
the Equation (1.12) is analytically solved in Fourier space, which is similar to the method used in
solving the mechanical equilibrium equation.
For convenience, the following normalized variables and coefficients are employed in the
present study,17
∗
𝒙∗ = √|𝛼1 |/𝐺110 𝒙 , 𝑡 ∗ = |𝛼1 |𝐿𝑡 , 𝐏 ∗ = 𝐏/𝑃0 , 𝜅∗0 = 𝜅0 |𝛼1 | , 𝛼1∗ = 𝛼1 /|𝛼1 | , 𝛼11
= 𝛼11 𝑃02 /|𝛼1 | ,
∗
∗
∗
∗
∗
𝛼12
= 𝛼12 𝑃02 /|𝛼1 | , 𝛼111
= 𝛼111 𝑃04 /|𝛼1 | , 𝛼112
= 𝛼112 𝑃04 /|𝛼1 | , 𝛼123
= 𝛼123 𝑃04 /|𝛼1 | , 𝑄11
= 𝑄11 𝑃02 ,
∗
∗
∗
∗
∗
𝑄12
= 𝑄12 𝑃02 , 𝑄44
= 𝑄44 𝑃02 , 𝑐11
= 𝑐11 /(|𝛼1 |𝑃02 ) , 𝑐12
= 𝑐12 /(|𝛼1 |𝑃02 ) , 𝑐44
= 𝑐44 /(|𝛼1 |𝑃02 ) ,
4
∗
∗
∗
′∗
′
𝐺11
= 𝐺11 /𝐺110 , 𝐺12
= 𝐺12 /𝐺110 , 𝐺44
= 𝐺44 /𝐺110 , 𝐺44
= 𝐺44
/𝐺110 , 𝐸 ex,∗ = 𝐸 ex /(|𝛼1 |𝑃0 ) , (1.13)
where 𝑃0 = |𝑃0 | = 0.757C/m2 is the magnitude of the spontaneous polarization at room
𝑇−𝑇0
temperature, 𝛼1 = 2𝜀
0 𝐶0
= (25 − 479) × 3.8 × 105 m2N/C2, and 𝐺110 = 1.73 × 10−10 m4N/C2 is
a reference value of the gradient energy coefficients. The Voigt notations are used for the elastic
stiffness tensors. The values of the normalized material coefficients of PbTiO3 used in the
simulations are listed in the Table I.
TABLE I. Values of the normalized coefficients used in the simulation.
𝛼1∗
∗
𝛼11
∗
𝛼12
∗
𝛼111
∗
∗
𝛼112
𝛼123
∗
𝑄11
∗
𝑄12
∗
𝑄44
∗
𝑐11
∗
𝑐12
∗
𝑐44
∗
𝐺11
∗
∗
𝐺12
𝐺44
′∗
𝐺44
-1
-0.24
2.5
0.49
1.2
0.05
-0.015
0.019
1766
802
1124
1.6
0
0.8
-7.0
0.8
With the dimensionless variables and Eq. (1.1) and Eq. (1.2), we rewrite the time-dependent
Ginzburg-Landau equation as
  [ f land ( Pi* )  f elas ( Pi* ,  kl* )  f elec ( Pi* , Ei* , Eiex ,* )]dV *   f grad ( Pi*,j )dV *
Pi* ( x * , t * )


.
t *
Pi*
Pi*
(1.14)
In Fourier space, Eq. (1.14) takes the form
𝜕
𝑃̂ (𝝃 , 𝑡 ∗ ) = −{𝑓̂(𝑃𝑖∗ )}𝝃 − 𝐺𝑖 𝑃̂𝑖 (𝝃 , 𝑡 ∗ ) ,
𝜕𝑡 ∗ 𝑖
where
𝑃̂𝑖 (𝝃 , 𝑡 ∗ )
and
{𝑓̂(𝑃𝑖∗ )}𝝃
are
∗
)+𝑓𝑒𝑙𝑒𝑐 (𝑃𝑖∗ ,𝐸𝑖𝑒𝑥∗ )]𝑑𝑉 ∗
𝛿 ∫[𝑓𝐿𝑎𝑛𝑑 (𝑃𝑖∗ )+𝑓𝑒𝑙𝑎𝑠 (𝑃𝑖∗ ,𝜀𝑘𝑙
𝛿𝑃𝑖∗
the
Fourier
transformations
(1.15)
of
𝑃𝑖∗ (𝒙∗ , 𝑡 ∗ )
and
, respectively, 𝐺𝑖 are the gradient operators correspond to
the ith-component of the polarization field, which are defined as follows,
∗ 2
∗
′∗
𝐺1 = 𝐺11
𝜉1 + (𝐺44
+ 𝐺44
)(𝜉22 + 𝜉32 ) ,
5
∗ 2
∗
′∗ )(𝜉 2
2
𝐺2 = 𝐺11
𝜉2 + (𝐺44
+ 𝐺44
1 + 𝜉3 ) ,
(1.16)
∗ 2
∗
′∗
𝐺3 = 𝐺11
𝜉3 + (𝐺44
+ 𝐺44
)(𝜉22 + 𝜉12 ) ,
The semi-implicit Fourier-spectral method was employed to solve the partial differential equation
(1.15) in the present work.
6
2. The stress/strain field of dislocation walls
In two-dimensional anisotropic linear elasticity, Stroh’s formalism18 gives the general solutions,
ud= 𝐀𝐟(𝑧𝛼 ) + ̅̅̅̅̅̅̅̅̅
𝐀𝐟(𝑧𝛼 ) ,
(2.1a)
𝜓 = 𝐁𝐟(𝑧𝛼 ) + ̅̅̅̅̅̅̅̅̅
𝐁𝐟(𝑧𝛼 ) ,
(2.1b)
where ud and 𝜓 are the displacement and stress function vectors, respectively, A and B are the
eigenvector matrices determined by the elastic constants and orientation of the ferroelectric crystal,
𝐟(𝑧𝛼 ) = [𝑓1 (𝑧1 ) , 𝑓2 (𝑧2 ) , 𝑓3 (𝑧3 )]𝑇 is an analytic vector of 𝑧𝛼 = 𝑥1 + 𝑝𝛼 𝑥2 (𝛼 = 1 , 2 , 3), and 𝑝𝛼
with Im(𝑝𝛼 )> 0 is the eigenvalue of the eigen-equation
𝐍𝜂 = 𝑝𝛼 𝜂.
(2.2a)
𝐴
𝜂=( ),
𝐵
(2.2b)
In Eq. (2.2a),
is the eigenvector with A and B being the column vectors of A and B, respectively, and the matrix N
is given by
𝐍𝟏
𝐍=(
𝐍𝟑
𝐍𝟐
),
𝐍𝟏𝑇
(2.2c)
with
𝐍𝟏 = −𝐓 −1 𝐑𝑇 , 𝐍𝟐 = 𝐓 −1 = 𝐍2𝑇 , 𝐍𝟑 = 𝐑𝐓 −1 𝐑𝑇 − 𝐐 = 𝐍𝟑𝑇 ,
(2.2d)
where
𝑄𝑖𝑘 = 𝑐𝑖1𝑘1 , 𝑅𝑖𝑘 = 𝑐𝑖1𝑘2 , 𝑇𝑖𝑘 = 𝑐𝑖2𝑘2 .
(2.2e)
Matrices A and B have the following properties
7
𝑇 = 𝐁𝐁 𝑇 + ̅̅̅̅̅̅
̅̅̅̅̅̅
𝐀𝐀𝑇 + 𝐀𝐀
𝐁𝐁 𝑇 =0,
(2.3a)
𝑇 = 𝐈,
̅̅̅̅̅̅
𝐁𝐀𝑇 + ̅̅̅̅̅̅
𝐁𝐀𝑇 =A𝐁 𝑇 + 𝐀𝐁
(2.3b)
where I is the identify matrix. The stress field is calculated from the stress function vector 𝜓 as
σ𝑖2 = 𝜓𝑖,1 , σ𝑖1 = −𝜓𝑖,2 .
(2.4)
For a line dislocation located at z𝛼𝑑 in an infinite body, the two dimensional solution is given with
the analytic vector in the form of
𝐟(𝑧𝛼 ) = ⟨ln(𝑧𝛼 − z𝛼𝑑 )⟩
1 𝑇
𝐁 𝐛,
2𝜋𝑖
(2.5)
where the angle bracket denotes a diagonal matrix and 𝐛 = (𝑏1 , 𝑏2 , 𝑏3 )𝑇 is the burgers vector of the
dislocation.
If there is only a dislocation of Burgers vector 𝐛𝑖 located at 𝑥1,i and 𝑥2,i in the representative
cell, we use superposition and have the analytic vector

𝐟𝐢 = ⟨

  ln {(𝑥1 + 𝑝𝛼 𝑥2) − [(𝑚𝐿𝑥1 + 𝑝𝛼 𝑛𝐿𝑥 ) + (𝑥1,𝑖 + 𝑝𝛼 𝑥2,𝑖 )]}⟩ ×
2
m   n  
1 𝑇
𝑩 𝒃𝑖 ,
2 i
(2.6)
where 𝐿𝑥1 and 𝐿𝑥2 are the dimensions of the representative cell in the 𝑥1 and 𝑥2 directions,

respectively. Using the basic sum formula
1
 n  a  𝜋cot𝜋𝑎,
we can reduce Eq. (2.6) to the
n  
following form with an analytical solution along the 𝑥2 direction,

𝐟𝑖 = ⟨

  ln {sin[𝜋
m   n  
(𝑥1 + 𝑝𝛼 𝑥2 ) − (𝑥1,𝑖 + 𝑝𝛼 ) − 𝑚𝐿𝑥1
1 𝑇
]}⟩ ×
𝐁 𝐛i ,
−𝑝𝛼 𝐿𝑥2
2𝜋𝑖
(2.7𝑎)
or another type of analytical solution along the 𝑥1 direction,

𝐟𝑖 = ⟨

  ln {sin[𝜋
m   n  
(𝑥1 + 𝑝𝛼 𝑥2 ) − (𝑥1,𝑖 + 𝑝𝛼 ) − 𝑝𝛼 𝑛𝐿𝑥2
−𝐿𝑥1
]}⟩ ×
1 𝑇
𝐁 𝐛i .
2𝜋𝑖
(2.7b)
The analytical vector f for n dislocations in the representative cell is thus given by
8
n
𝐟=
f
i 1
i
.
(2.8)
Substituting Eq. (2.8) into Eq. (2.1a) yields the displacements generated by the dislocations.
Then, using the strain-displacement equation of Eq. (1.9), we have the dislocation strain field.
9