Sidorkin, Domain Structure in Ferroelectrics and Related
Sidorkin, Domain Structure in Ferroelectrics and Related
Sidorkin, Domain Structure in Ferroelectrics and Related
Domain structure
Alexander Stepanovich Sidorkin is a Doctor of Physical and Mathematical
Sciences, Professor, the Director of the Research and Education Center
“Wave processes in inhomogeneous and non-linear media”, Head of the in ferroelectrics
and
Experimental Department of the Voronezh State University, a member of
the Scientific Council of the Russian Academy of Sciences on Physics of
Ferroelectrics and Dielectrics. Dr. Sidorkin has been awarded a medal by
the International Academy of Sciences of Nature and Society.
related materials
His scientific interests lie in the area of physics of non-linear polar
dielectrics (ferroelectrics) and related materials, physics of solid-state
emission phenomena. The principal scientific results have been obtained
in the area of exploration of domain structure formation and its relaxation,
including the fine-domain structure. Other significant results include the
description of domain – defect interaction, the structure and dynamics
of domain and interphase boundaries in defect-free ferroelectrics and in
imperfect materials, investigation of switching processes and dispersion
of dielectric permittivity in polydomain ferroelectrics, explanation of the
phenomenon of ‘freezing’ of the domain structure, influence of tunneling
of ferroactive particles on the structure and mobility of domain walls.
Special investigations carried out by Dr Sidorkin include the illumination
mechanism and the nature of electron emission stimulated by a change of
the macroscopic polarization of ferroelectrics.
A S Sidorkin
Sidorkin
Cambridge International Science Publishing Ltd.
7 Meadow Walk, Great Abington
Cambridge CB1 6AZ
United Kingdom
www.cisp-publishing.com
i
ii
DOMAIN STRUCTURE IN
FERROELECTRICS
AND
RELATED MATERIALS
A.S. Sidorkin
iii
Published by
© A.S. Sidorkin
© Cambridge International Science Publishing
Conditions of sale
All rights reserved. No part of this publication may be reproduced or transmitted
in any form or by any means, electronic or mechanical, including photo-
copy, recording, or any information storage and retrieval system, without
permission in writing from the publisher
iv
Contents
Introduction .................................................................................................. v
Chapter 1
Formation of a domain structure as a result of the loss of stability
of the crystalline lattice in ferroelectric and ferroelastic crystals of
finite dimensions ................................................................................. 1
Chapter 2
Structure of domain and interphase boundaries in defect-free
ferroelectrics and ferroelastics ........................................................ 28
v
Chapter 3
Discussion of the microscopic structure of the domain boundaries
in ferroelectrics ................................................................................. 58
Chapter 4
Interaction of domain boundaries with crystalline lattice defects
91
Chapter 5
Structure of domain boundaries in real ferroactive materials .. 121
vi
fluctuations of its profile .................................................................. 128
5.3. Effective width of the domain wall in real ferroelectrics ................. 131
5.4 Effective width of the domain wall in ferroelastic with defects ...... 139
Chapter 6
Mobility of domain boundaries in crystals with different barrier
height in a lattice potential relief .................................................. 143
Chapter 7
Natural and forced dynamics of boundaries in crystals of ferro-
electrics and ferroelastics .............................................................. 170
vii
viii
Introduction
ix
structure. Therefore, in order to study the nature of these properties
and their possible applications in practice, it is crucial to find out
the regularities that control the processes of origination of the domain
structure and the ways of its change with time.
It is well known that the change of macroscopic polarization
in ferroelectrics takes place by means of displacement of boundaries
between domains. These boundaries are called domain walls. Therefore,
studies of properties of domain structures cannot be separated
from the investigation of processes of domain boundaries motion.
Ferroelastics are closely related to ferroelectrics as far as their
properties are concerned. They are substances in which spontaneous
deformation of elementary cells takes place at certain temperatures.
The spontaneous deformation in ferroelastics as well as polarization
in ferroelectrics occur at structural phase transformations. This also
determines the likeness of methods of the theoretical description
of these materials. These methods involve symmetry-related principles,
studies of the properties of the corresponding thermodynamic functions,
etc. That is why it is quite natural to consider simultaneously the
properties of the mentioned ferroactive materials, the patterns of
domain structure and its dynamics, wherever it is possible.
The ferroelectric materials possess a lot of useful applied properties.
The presence of sustained polarization that lasts without the action
of a field, for example, makes it possible to use them for recording
and retrieving information. At the same time, the density of information
storage in ferroelectrics is much higher as compared to magnetic
media due to the significantly thinner transient layer (domain wall)
between domains, which makes their utilization preferable from
the point of view of at least this factor. Recently, a discussion was
started about the possibility of utilization of the periodicity of the
arrangement of domain walls for the generation of laser radiation
with the required wavelength, etc.
Thus, the studies of the domain structure of ferroelectrics
represent both fundamental and applied interest. In reality, considerable
attention is devoted to this problem, which is reflected by the
number of articles in magazines on general physics and by numerous
scientific conferences, both international and Russian, etc. At the
same time in contrast to ferromagnetics, for example, there is practically
no monograph literature which would be devoted solely to that
problem. The parts of the books devoted to the general properties
of ferroelectrics and dealing with this problem are usually too brief
and deal only with the experimental description of the domain
structure [1–28]. Despite the analogy to the ferromagnetics a simple
x
transfer of the results obtained for the ferromagnetics to the
ferroelectrics is not possible. The characteristics of the domains
and domain boundaries in ferroelectrics are controlled by the
interactions that differ from the ones in ferromagnetics. That fact
brings certain specifics. Namely, the width of the domain boundaries
in ferroelectrics is several orders of magnitude smaller and,
consequently, their interaction with the crystalline lattice and its
defects is very strong . Their bending displacements are controlled
not by surface tension but rather by long-range fields. The screening
effect and its influence on the domain structure and on the domain
walls motion do not have any analogues in ferromagnetics, etc.
This book represents an attempt to bridge the gap. It is devoted
to the description of the main characteristic parameters of the
domain structure and domain boundaries in ferroelectrics and related
materials.
As probably in any publication, the problems considered in
the book reflect, of course, certain preferences of the author. For
example, the first chapter deals with the mechanisms of formation
of the domain structure. The formation of the domain structure
is studied most thoroughly in the framework of the mechanism
of loss of initial phase stability in the finite size material. Particular
attention is devoted to the equilibrium domain structure and the
so-called fine domain structure. A hypothesis is analyzed that the
origination of the fine domain structure is connected with the transition
to a new phase under the conditions of inhomogeneous cooling
of only a thin layer of a ferroelectric material. It is shown that
this hypothesis can explain the fact of the onset of a periodical
domain structure in a ferroelastic with a free surface.
In the second chapter, the structure of domain and interphase
boundaries in defect-free ferroelectrics and related materials is
considered within the framework of the phenomenological description
of materials with different types of phase transitions. The influence
of the concentration of charge carriers, material surface, etc. on
the boundaries under consideration is examined. The problems
of stability of different types of domain boundaries are discussed.
The third chapter presents the results of the microscopic
description of the structure of domain boundaries in ferroelectrics.
Ferroelectric crystals of barium titanate and of the potassium
dihydrophosphate group are taken as an example. The results of
the microscopic and phenomenological descriptions are compared.
The limits of validity of the phenomenological way of description
of the problem under consideration are assessed.
xi
The fourth chapter deals with the description of the interaction
of domain boundaries in ferroelectrics and ferroelastics with different
types of crystalline lattice defects. The processes of interaction
of domain boundaries in ferroelectrics and ferroelastics with various
types of crystalline lattice defects are studied. This includes charged
defects, dilatation centers, non-ferroelectric inclusions, dislocations
with different orientation of Burgers’ vector with respect to the
direction of spontaneous shear and the domain wall plane.
The fifth chapter deals with the problems of stability of the
shape of inclined domain boundaries, and also with the structure
of domain boundaries in real ferroactive materials. The concept
of the effective width of a domain wall is introduced. It is shown
that the deformation of a domain wall shape in materials with
defects can be the reason for domain wall widening in real materials.
In the sixth chapter, the influence of lattice potential relief
on the mobility of domain walls is studied. The thermofluctuational
mechanism of the motion of domain walls is considered, parameters
and the probability of appearance of a critical nucleus on a domain
wall is calculated. On the basis of the results of the given consideration
the explanation to the effect of domain structure ‘freezing’ in
ferroelectrics of the potassium dihydrophosphate group is given.
The influence of the proton tunneling effect on hydrogen bonds
on the structure and mobility of domain boundaries in ferroelectrics
containing hydrogen is studied.
In the seventh chapter, the proper and forced dynamics of
domain boundaries in ferroelectric and ferroelastic crystals are
considered. Bending and translational dynamics of domain boundaries
in ferroelectric crystals are studied; the contribution of domain
boundaries to the dielectric properties of ferroelectrics and elastic
properties of ferroelastics is investigated. The experimental data
on the dielectric properties of ferroelectrics with different types
and concentration of defects are analyzed. In the final part of this
chapter the non-linear dielectric properties of ferroelectrics, associated
with the motion of domain boundaries and processes of ageing
and degradation of ferroelectric materials, are briefly considered.
Finally, I would like to thank very much Messrs. S.Kamshilin
for helping with the proofreading and correction of the translation
and K.Penskoy for his help in typesetting the book.
xii
1. Formation of a Domain Structure
Chapter 1
1
Domain Structure in Ferroelectrics and Related Materials
2
1. Formation of a Domain Structure
π P 2 L [d + − d − ]2 (1.9)
+ 0 .
εc 2d
As shown by further investigations, at any equilibrium size d
(1.14), the formation of a unipolar structure, i.e. the structure with
d + ≠ d – , increases Φ. The minimum of Φ corresponds to the overall
unpolarized structure. In this case
∞
16 P02 d 1
Φ=
π (1 + ε cε a
2
)
∑ (2n − 1)
n =1
3
. (1.10)
3
Domain Structure in Ferroelectrics and Related Materials
4
1. Formation of a Domain Structure
5
Domain Structure in Ferroelectrics and Related Materials
∂ϕ II ∂ϕ I ∂ϕ II ∂ϕ
ε = , ε = −ε z III . (2.4)
∂z ∂z L
z = +Δ ∂z ∂z z=
L
2 2
The simultaneous equations (2.4) allow to find the ratios between the
unknown coefficients in the expressions for the potentials (2.3). The
condition of the solvability of these equations, i.e. the equality to zero
of the determinant, compiled from the coefficients of the quantities A,
B, C, D, produces an equation determining the dependence of
coefficient α z on the wave vector k modified taking into account the
effect of correlation and electrostatic interaction of bound charges on
the surface of the crystal.
Substitution of the distribution (2.3) into (2.4) and notation of the
given determinant
⎛L ⎞ ⎛L ⎞ ⎛L ⎞
− k ⎜ +Δ ⎟ − k ⎜ +Δ ⎟ k ⎜ +Δ ⎟
e ⎝2 ⎠
−e ⎝2 ⎠
−e ⎝2 ⎠
0
⎛L ⎞ ⎛L ⎞ ⎛L ⎞
− k ⎜ +Δ ⎟ − k ⎜ +Δ ⎟ k ⎜ +Δ ⎟
−e ⎝2 ⎠
εe ⎝2 ⎠
−εe ⎝2 ⎠
0
L L
= 0.
−k k
0 e 2 e 2 − sin(tL / 2)
−k
L
k
L
εx /εx (2.5)
0 e 2 −e 2 − cos(tL / 2)
ε
t = k εx /εx
6
1. Formation of a Domain Structure
7
Domain Structure in Ferroelectrics and Related Materials
Fig. 1.4. Dependence of the critical values of the wave vector on the thickness of
the surface non-ferroelectric layer.
8
1. Formation of a Domain Structure
9
Domain Structure in Ferroelectrics and Related Materials
∂ 2 Pz ∂ϕ ⎛ ∂ϕ ∂ϕ ⎞
−α z Pz − = Ex′ = − = −⎜ sinψ + cosψ ⎟ . (3.5)
∂x′ 2
∂z′ ⎝ ∂x ∂z ⎠
Supplementing equations (3.4), (3.5) with the electrostatic equation
Δϕ = 4π ∇P (3.6)
yields a complete set of equations for determination of the
components of the vectors P = (P x , P z ) and E = (E x ,E z ).
Taking into account the linearity of equations of the set (3.4-3.6)
with respect to the components of the polarization vector P and the
potential ϕ , the solution of this set in our case may be presented
in the form
ϕ = ϕ 0 exp(ikx)exp(itz), (3.7)
where k is the real wave vector and t is a parameter determining
the variation of ϕ in the thickness of the plate. It may be complex
in general.
Substituting (3.7) into (3.4) and (3.5) we find
i
Px = (−k cosψ + λ sinψ )ϕ0 exp(ikx)exp(itz ), (3.8)
α x
10
1. Formation of a Domain Structure
i (k sinψ + t cosψ )
Pz = ϕ0 exp(ikx)exp(itz ). (3.9)
α z − k 2 cos 2 ψ
Substituting subsequently (3.8) and (3.9) into (3.6) taking into
account (3.3) in the approximation ψ < 1, we obtain the following
expression for the relationship between t and k:
⎛ 4π ⎞ ⎛ 4π ⎞
4π sinψ cosψ (α x + α z ) ± α xα z ⎜ − 1⎟ ⎜ + 1⎟
⎝ αz ⎠⎝ α x ⎠
t1,2 = k,
⎡ ⎛ sin 2
ψ cos 2
ψ ⎞ ⎤ (3.10)
α x α z ⎢1 + 4π ⎜ − ⎟⎥
⎢⎣ ⎝ αx α x ⎠ ⎥⎦
where
α z = α − k 2 cos 2 ψ .
z
(3.11)
In a special case of ψ = 0 ratio (3.10) yields formula (2.5)
εx
t=± k (3.12)
εz
of the previous section where
4π 4π
εx =1+ , − 1. εz = (3.13)
αx α z
The solution of simultaneous equations of the equilibrium for
polarization with the potential (3.7), noted down with consideration
of two roots in (3.10), in the form
L
ϕ = A exp(ikx)exp(it1 z ) − B exp(ikx)exp(it2 z ), z < , (3.14)
2
L
ϕ = C exp(ikx) exp(−kz ), , z >
(3.15)
2
must satisfy the boundary conditions on the surface of the plate at
z = ±L/2:
ϕ ( x , z ) x = L − 0 = ϕ ( x, z ) x = L + 0 , (3.16)
2 2
∂ϕ ∂ϕ
L − L = 4π P L , (3.17)
∂z z = −0
2
∂z z = +0
2
z = −0
2
11
Domain Structure in Ferroelectrics and Related Materials
4π L
εx tg(t1 − t2 ) = 2. (3.18)
α z 2
Since in the vicinity of the phase transition point the value of α z
is small, equation (3.18) may be rewritten in the form
L
( t1 − t2 )
= π n, (3.19)
2
where n is any integer. On the basis of (3.10) we find from (3.19)
−2π 3/ 2 n cos 2 ψ
ε xα z = . (3.20)
kL
Analysis of (3.20) shows that here the value n = 0 corresponds to
the onset of the homogeneous state of the ferroelectric phase,
whereas n = –1 corresponds to the stable state of the formed
heterogeneous ferroelectric phase. From (3.20) the required
dependence of α z on k in the approximation of kL/2>>1 is
4π 3 cos 4 ψ
α z (k ) = −α z + k 2 cos 2 ψ +. (3.21)
ε x k 2 L2
The minimum of the dependence (3.21) corresponds to the value
2π 3 4
km = cos1 2 ψ , (3.22)
(ε x ) L
14 12
12
1. Formation of a Domain Structure
a b c
Fig. 1.5. Dependence on the average width of the domain d of the surface density
of the depolarizing field (1), surface density of the energy of domain walls (2) and
the sum of these energies for the following cases: (a) – no screening, (b) – weak
screening, (c) – strong screening.
13
Domain Structure in Ferroelectrics and Related Materials
14
1. Formation of a Domain Structure
15
Domain Structure in Ferroelectrics and Related Materials
⎛ k 2ε + (1/ λ 2 ) ⎞
ϕIII = D sin ⎜
x
z⎟ (4.5)
⎜ εz ⎟
⎝ ⎠
with the Debye screening length
kT
λ= (4.6)
4π e 2 n0
in the case when the crystal contains a dopant of mainly one type
with the concentration of ionised centres equal to n 0 . In this case
the condition, determining the dependence of α z on the wave
vector k for Δ = 0 is rewritten as follows:
[k 2ε x + (1/ λ 2 )]ε z ⎛ [k 2ε + (1/ λ 2 )] L ⎞
ctg ⎜ x
⎟ =1+ 1 .
k ⎜ εz 2⎟ kΛ (4.7)
⎝ ⎠
At Λ, λ ≠ 0 the dependence α z (k) in this case is determined by the
equation
π 3 [ε x + (1/ k 2 λ 2 )]
α z (k ) = α 0 (T − Tc ) + k 2 + . (4.8)
(ε x kL / 2 + L / 2k λ 2 + 1/ k Λ ) 2
At Λ → ∞, this equation transforms into the relation
π3 4λ 2
α z (k ) = α 0 (T − Tc ) + k 2 + 2 2 ⋅ 2 . (4.9)
k λ εx +1 L
As a result of stability loss, the system will transform to the state
with the wave vector k corresponding to the condition ∂ α z /∂k = 0.
In the presence of screening by only free charges in the bulk of
the crystal, according to (4.9) the corresponding value of k is
determined by the expression [44]
2π π 1
k2 =− . (4.10)
ε x L ε x λ 2
Equation (4.10) shows clearly that at
L
λ2 = (4.11)
2π πε x
i.e. at
π kT ε x
n0 = (4.12)
2 e2 L
the period of the onsetting structure tends to infinity, which
corresponds to transition to the monodomain state.
The estimates of critical concentration n 0 from (4.12) at
T ~ 300 K, ~ a 2 ~ 10 –14 cm 2 , L ~ 10 –1 cm yield the value of
n 0 ~ 10 13 cm –3 . The corresponding shift of T c in comparison with the
16
1. Formation of a Domain Structure
π 3ε x
α z (k ) = α 0 (T − Tc ) + k 2 + . (4.14)
(ε x kL / 2 + 1/ k Λ ) 2
It differs from (2.8) in the following: electrostatic contribution in
α z (k) is no longer a monotonically dropping function k, but passes
through a maximum and tends to zero due to the efficiency of
screening in equilibrium at low k. Consequently, the overall
dependence α z (k) in the general case will have absolute maximum
at k = 0 and under certain ratio of the parameters it will have a
local minimum at k ≠ 0. The extrema of this dependence are
determined by the equation
⎡
⎢
− 2
(
k − 1 ⎤
2
⎥ ) 2 ε x LΛ 2
⎥ = 0, k = 2 k .
)
k⎢ 3
⎢π ε xΛ
⎣
2
(
k + 1 ⎥
⎦
(4.15)
17
Domain Structure in Ferroelectrics and Related Materials
Fig. 1.6. Behaviour of the dependence α z (k) in the vicinity of the local minimum
for ferroelectrics with charges on surface states. 1, 2, 3, 4, 5, 6 – Λ –1 = 1· 108 ;
1.2· 108 ; 1.5· 108 ; 2· 108 ; 3· 108 ; 4· 108 ; Δ = 0.
Fig. 1.7. Dependence of α z (k) in the vicinity of the local minimum at Λ –1 = 4· 108
and various Δ: 1,2,3,4,5,6,7,8 – Δ = 1.5· 10–8 ; 1.3· 10–8 ; 1· 10–8 ; 8· 10–9 ; 6· 10–9 ;
6· 10–9 ; 4· 10–9 ; 0.0.
18
1. Formation of a Domain Structure
The real conditions of transition to the polar state usually imply the
presence of a temperature gradient in a specimen being rapidly
cooled which, as shown below, has a significant effect on the
period of the resultant structure.
The result of the influence of inhomogeneous cooling on the domain
structure may easily be predicted if it is noted that in a
inhomogeneously cooled specimen the volume of the part of the
material, undergoing phase transition at the moment of nucleation
of the domain structure, decreases. From the viewpoint of
calculations, this means that while estimating the width of the domain
the equation (1.14) should include the thickness of the layer
undergoing phase transition and not the thickness of the specimen
L (Fig. 1.8). Since the former is evidently smaller than the thickness
of the specimen and decreases with increasing temperature gradient,
19
Domain Structure in Ferroelectrics and Related Materials
20
1. Formation of a Domain Structure
13
38
α11 2 ⎛ε k ⎞
k1 = k , z1 = z , z3 = ⎜ x 1 ⎟ z2 ,
α11 4 ⎝ 4π ⎠
14
α (5.3)
z2 = − z
+ z1 + k12
α
12 14
1
equation (5.2) is reduced to the form
∂ 2ϕ 1 ∂ϕ
− − z3ϕ = 0. (5.4)
∂z32 z3 ∂z3
Its solution has the form
⎛2 ⎞
ϕ ( z3 ) = z3 Z 2 3 ⎜ z33 2 ⎟ , (5.5)
⎝3 ⎠
where Z 2/3 (x) is any solution of the Bessell equation of the order
of 2/3.
Taking into account the fact that the value of the potential at
infinity should convert to zero, we select the Bessell function with
the corresponding asymptotic. Consequently, solution (5.5) of
equation (5.4) becomes more specific as shown below:
⎛2 ⎞ π z3 ⎛ 2 ⎞
ϕ ( z3 ) = z3 K 2 3 ⎜ z33 2 ⎟ exp ⎜ − z33 2 ⎟ . (5.6)
⎝ 3 ⎠ 2 ⎝ 3 ⎠
The equation for determination of the domain structure
parameters is found from the condition of equality to zero of the
field on the cooled surface
∂ϕ
z = 0 = 0. (5.7)
∂z
Substituting (5.6) into (5.7) and taking into account (5.3) we obtain
the dependence
13
⎛ πα ⎞
α z (k ) = k + ⎜⎜ 12 ⎟⎟ .
2
(5.8)
⎝ ε xk ⎠
The transition to the polar phase takes place in the state
corresponding to the condition ∂ α z /∂k = 0, i.e. in the state with
18
⎛ πα 2 ⎞
km = ⎜ 1 ⎟ . (5.9)
⎝ ε x ⎠
Equation (5.9) shows that the appearing structure is refined with
an increase of the temperature gradient. The result is completely
clear because in the presence of the gradient the transition is
observed not in the entire volume of the material but only in the
layer of the material which, according to (2.9), should lead to
(instead of L – the thickness of the layer in which the
transformation takes place) reduction of the resultant structure.
21
Domain Structure in Ferroelectrics and Related Materials
22
1. Formation of a Domain Structure
(α − k ) k u + c ∂∂zu = 0.
2
2 2
2 (6.4)
The solution of equation (6.4) has to meet specific conditions at
the boundary of the material. In the present case they can be
represented by
∂u
u z =0 = 0, z = L = 0, (6.5)
∂z
i.e. it is assumed that at z = 0 there is a contact with the absolutely
rigid material, and the second boundary of the material z = L is
assumed to be free. The following function meets equation (6.4)
and conditions (6.5):
π
u = B sin
z. (6.6)
2L
Substituting (6.6) into (6.5) we obtain the dependence
2
⎛ π ⎞
α = k2 + c⎜
⎟ . (6.7)
⎝ 2kL ⎠
The value of k corresponding to the minimum of dependence
α(k)
14
⎛ cπ 2 ⎞
km = ⎜ ⎟ . (6.8)
⎝ 4 L ⎠
At usual c ~ 10 10 erg· cm–3 , ~ c · a 2, a 2 ~ 10 –15 cm 2, L ~ 10 –1 cm,
the period of the resultant structure d = π /k m has the order of 10 –4 cm
which corresponds to the experimentally observed domain dimen-
sions [2, 12, 16]. The shift of the phase transition temperature in
relation to T c of an infinite crystal is
π ( c)1 2
ΔT = . (6.9)
α0 L
23
Domain Structure in Ferroelectrics and Related Materials
24
1. Formation of a Domain Structure
25
Domain Structure in Ferroelectrics and Related Materials
γ = 2P0 E d d, (7.1)
26
1. Formation of a Domain Structure
27
Chapter 2
28
2. Structure of Domain and Interphase Boundaries
29
Domain Structure in Ferroelectrics and Related Materials
⎡ dP d ( δ P ) ⎤
2 2
⎡d
⎢ ( P + δ P ) ⎤⎥ = ⎢ + ⎥ =
2 ⎣ dx ⎦ 2 ⎣ dx dx ⎦
dP d (δ P ) ⎡ d ( δ P ) ⎤
2
2
⎛ dP ⎞ (1.3)
= ⎜ ⎟ + + ⎢ ⎥ .
2 ⎝ dx ⎠ dx dx 2 ⎣ dx ⎦
Then
⎧
⎪ ⎡ d ( P + δ P) ⎤ ⎪
2
⎫
Φ ( P + δ P ) = ∫ ⎨ϕ ( P + δ P ) + ⎢ ⎥ ⎬ dx =
⎪ 2⎣ dx ⎦ ⎪
⎩ ⎭
⎧
⎪
2
⎛ dP ⎞ ⎪ ⎫
= ∫ ⎨ϕ ( P ) + ⎜ ⎟ ⎬ dx +
⎩⎪ 2 ⎝ dx ⎠ ⎪⎭
⎧ϕ ' ( P ) dP d ( δ P ) ⎫
+∫ ⎨ δ P + ⎬ dx + (1.4)
⎩ 1! dx dx ⎭
⎧⎪ϕ "( P ) ⎡ d (δ P ) ⎤ ⎪
2
⎫
+∫ ⎨ (δ P ) + ⎢ ⎥ ⎬ dx ≡ Φ ( P ) + δ Φ +δ Φ.
2 2
⎪ 2! 2 ⎣ dx ⎦ ⎪
⎩ ⎭
The first term in the right-hand part of (1.4) describes the
thermodynamic potential of the optimum distribution P(x), with
respect to which the variation is performed. It coincides with
expression (1.1). The following terms represent respectively the first
and second variations of the potential (1.1).
The equality to zero of the first variation δΦ = 0 enables us to
find the distribution P(x), corresponding to the minimum Φ. Taking
into account integration by parts
⎧ dP d ( δ P ) ⎫ ⎧ d 2P ⎫
∫ ⎨⎩ϕ ' ( P ) δ P + ⎬ dx = ∫ ⎩⎨ϕ ' ( P ) − ⎬ δ Pdx. (1.5)
dx dx ⎭ dx 2 ⎭
Since the variation δP is an arbitrary small function, the identical
equality to zero of the integral is possible only if the expression in
the braces is equal to zero. From this we find the equation
describing distribution of polarization in the boundary:
d 2 P dϕ
2
= = −α P + β P 3 . (1.6)
dx dP
The sign of the second variation makes it possible to evaluate
the stability of the corresponding solution. Similarly as in (1.5)
30
2. Structure of Domain and Interphase Boundaries
⎧⎪ϕ "( P ) ⎡ d (δ P ) ⎤ ⎪
2
⎫
δ Φ = ∫⎨ (δ P ) + ⎢
2
⎥ ⎬ dx =
2
⎪ 2! 2 ⎣ dx ⎦ ⎪
⎩ ⎭
1⎪ ⎧d ϕ
2
d (δ P ) ⎪
2
⎫
= ∫ ⎨ 2 δ P − 2 ⎬ δ Pdx =
2⎪ ⎩ dP dx ⎪ ⎭ (1.7)
⎧ d 2
1 d ϕ⎫
2
= ∫δ P ⎨ 2
+ ⎬ δ Pdx = ∫ δ PLˆδ Pdx,
⎩ 2 dx 2 dP 2 ⎭
V ( x) =
1
2
( −α + 3β P 2 ( x ) ). (1.10)
In this case, the eigenvalue λ n and eigenfunctions ψ n play the
role of the eigenvalues of the energy of the particle and its wave
functions, respectively.
From the general theorems of quantum mechanics it is known
that λ n is the increasing function of number n. Therefore, it turns
out that in order to judge the stability of the corresponding solution,
it is sufficient to find out the sign of the minimum eigenvalue λ0 of
the operator (1.8). Let us show this. The arbitrary variation δP(x)
can always be expanded into a series in respect of the eigen-
functions ψ n (x) of the operator L :
δ P ( x ) = ∑ Anψ n ( x ) , (1.11)
n
31
Domain Structure in Ferroelectrics and Related Materials
∫ψ ( x )ψ ( x ) dx = δ
*
n m nm , (1.12)
and taking also into account the determination of the eigenvalue of
the operator L̂ (1.9), we obtain
∞
δ 2 Φ = ∑ λn An .
2
(1.13)
n=0
32
2. Structure of Domain and Interphase Boundaries
dP dx
∫ ϕ ⎡⎣ P ( x ) ⎤⎦ − ϕ [ P0 ]
=∫
/2
.
(1.17)
1 ⎛ dP dP ⎞
β P0 ∫ ⎜⎝ P + P + P − P ⎟⎠ =
0 0
1 P0 + P 2 (1.20)
= ln = ( x − U ).
β P0 P0 − P
33
Domain Structure in Ferroelectrics and Related Materials
∞
⎡ ⎛ dP ⎞ ⎤
2
γ 0 = ∫ ⎢ϕ ( P ) − ϕ ( P0 ) + ⎜ ⎟ ⎥ dx =
−∞ ⎢
⎣ 2 ⎝ dx ⎠ ⎥⎦
∞ ∞ 2
β ⎛ x⎞
= 2 ∫ ⎡⎣ϕ ( P ) − ϕ ( P0 ) ⎤⎦ dx = P04 ∫ ⎜1 − th 2 ⎟ dx =
−∞
2 −∞ ⎝ δ⎠ (1.22)
∞
αP δ
2
dx / δ 2 4
= ∫ ch = α P02δ = P02 .
0
2 −∞
4
x /δ 3 3 δ
αγ β⎛ ⎞
P0 =
⎜⎜ 1 + 2 − 1⎟⎟ . (1.24)
γ⎝ β ⎠
The first integration of equation (1.23) taking into account (1.24)
dP
and the boundary conditions = 0 yields
dx ±∞
2
⎛ dP ⎞
⎜ ⎟ = ϕ ⎡⎣ P ( x ) ⎤⎦ − ϕ [ P0 ] =
2 ⎝ dx ⎠
2 ⎡γ β γ ⎤ (1.25)
= ( P02 − P 2 ) ⎢ P 2 + + P02 ⎥ .
⎣6 4 3 ⎦
Integration of (1.25) leads to the following distribution of
polarization in the 180º boundary in the ferroelectric with a phase
transition of the first order
34
2. Structure of Domain and Interphase Boundaries
sh ( x / δ )
P ( x ) = P0 ,
ch 2 ( x / δ ) + ε
2γP02
δ= , ε= . (1.26)
β 4γP02 + 3β
γP +
0
4
P
0
2
2
The corresponding distribution is shown in Fig. 2.2 which shows
that at the temperatures close to the phase transition point, instead
of the distribution 2, analogous to the case of the crystal with the
phase transition of the second order, practically two independent
distributions in the sections of alteration of polarization –1<P/P 0<0
and 0<P/P 0 <1 are implemented here. This behaviour of polarization
in the transition layer is related to the presence of a metastable
state at P = 0 as an intermediate state between the polar states –
P 0 and P 0 (Fig. 2.3).
35
Domain Structure in Ferroelectrics and Related Materials
36
2. Structure of Domain and Interphase Boundaries
37
Domain Structure in Ferroelectrics and Related Materials
⎡⎛ dPx ⎞ 2 ⎛ dPz ⎞ 2 ⎤
Φ= ⎢⎜ ⎟ +⎜ ⎟ ⎥−
2 ⎣⎢⎝ dx ⎠ ⎝ dx ⎠ ⎦⎥
α β
−
2
(P x
2
+ Pz2 ) +
4
(P x
4
+ Pz4 ) +
(2.1)
β1 γ γ1
+
2
Px2 Pz2 +
6
(P x
6
+ Pz6 ) +
2
Px2 Pz2 ( Px2 + Pz2 ) .
38
2. Structure of Domain and Interphase Boundaries
d 2 Pz
= −α Pz + β Pz3 + β1 Px2 Pz + γPz5 +
dx 2
+ γ1 Px2 Pz ( Px2 + 2 Pz2 ) ,
d 2 Px
= −α Px + β Px3 + β1 Px Pz2 + γPx5 +
dx 2 (2.6)
⎛ P ⎞
+ γ1 Px Pz2 ( 2 Px2 + Pz2 ) + 4π ⎜ Px − 0 ⎟ .
⎝ 2⎠
P0 P0 sh ( x / δ )
Px = , Pz = ,
2 2 ch 2 ( x / δ ) + ε
γP02 (2.8)
δ =2 , ε= .
γP04 + β 2 P02 2γP02 + 3β 2
39
Domain Structure in Ferroelectrics and Related Materials
40
2. Structure of Domain and Interphase Boundaries
Fig.2.5. The variation of the polarization vector in the 90º domain boundary: 1,2 —
polarization components P z and P x in relation to the position in the boundary.
41
Domain Structure in Ferroelectrics and Related Materials
42
2. Structure of Domain and Interphase Boundaries
α ( P ± P0 )
2
d 2 P ∂ϕ
2α ( P − P0 ( x ) ) − = ,
dx 2 ∂z
∂ 2ϕ ∂ 2ϕ dP (3.1)
+ 2 = 4π .
∂x 2
∂z dz
In (3.1), as well as in section 1.1, P 0 (x) is the odd periodic
function, determined in the period (–d,d) as
⎧1, 0 < x < d ,
P0signx = P0 ⎨ (3.2)
⎩ −1, − d < x < 0.
(the 180º domain structure is discussed) and to simplify
considerations it is assumed that ε x = 1.
Let us find the solution of the system (3.1)–(3.2) in the form of
expansion into a Fourier series in respect of axes x where the
expansion coefficients depend on the coordinate along the polar axis
z. Taking into account the symmetry of the problem, we obtain
∞
P ( x, z ) = ∑ P2 n +1 ( z ) sin
( 2n + 1) πx ,
n =0 d
∞
ϕ ( x, z ) = ∑ ϕ 2 n +1 ( z ) sin
( 2n + 1) πx. (3.3)
n=0 d
From the first equation of system (3.1), the relation between the
Fourier coefficients of expansion of polarization and the potential
has the form of
8P0α
P2 n +1 ( z ) = −
π ( 2n +1) ⎡ 2α + ( 2n + 1) π 2 d 2 ⎤
2
⎣ ⎦
1 ∂ϕ2 n +1 (3.4)
− .
⎡ 2α + ( 2n + 1) π d ⎤ ∂z
2 2 2
⎣ ⎦
On the basis of Laplace’s equation, the distribution of the
potential in the crystal and outside it is as follows:
43
Domain Structure in Ferroelectrics and Related Materials
⎡
k2 n +1 z ⎤
ϕ2 n +1 ( z ≥ 0 ) = Aexp ⎢ − ⎥,
⎢⎣ ε z ( k ) ⎥⎦
ϕ2 n +1 ( z ≤ 0 ) = B exp [ k2 n +1 z ] ,
k2 n +1 = ( 2n + 1) π / d , (3.5)
4π
εz (k ) =1+ .
2α + ( 2n + 1) π 2 / d 2
2
Now, on the basis of (3.4) and (3.7) the final expression for the
Fourier coefficients of polarization expansion on the surface of the
ferroelectric is:
P2 n +1 ( 0 ) =
8 P0α ( ε z (k ) +1 ) .
π 2 ( 2n + 1) ⎡⎢( 2α + k22n +1 )
⎣ ( )
ε z ( k ) + 1 + 4π ⎤⎥
⎦
(3.8)
or approximately
8P0α ε z ( k ) 4 πα P0
P2n+1 ( 0) = .
4π ( 2n + 1)
2
π2 ( 2n + 1) 2α + k22n+1
(3.9)
Consequently
∞
4 P0α π π ( 2n + 1) x
P ( x,0 ) = ∑ sin . (3.10)
n=0 π 2 ( 2n + 1) 2α + k22n +1 d
To determine the width δ of the domain boundary in the vicinity
of the surface of the ferroelectric crystal let us determine the value
P(x,0) in the middle of the domain and the value of the derivative
∂P/∂x in the centre of the domain wall. Polarization on the surface
of the crystal in the middle of the distance between the domain
boundaries is:
44
2. Structure of Domain and Interphase Boundaries
4 P0 α
P ( d 2,0 ) . (3.11)
π 2π
Similarly, the derivative
∂P ( 0,0 ) 4 α P0 ∞
1 2α P0
∂x
= ∑
π π
.
2π d π ( 2n + 1)
2 2
n=0
(3.12)
1+
2d d2
Then the width of the domain wall in the vicinity of the surface
of the ferroelectric is
P ( d 2,0 ) 2
δ= = .
∂P ( 0,0 ) ∂x α (3.13)
45
Domain Structure in Ferroelectrics and Related Materials
46
2. Structure of Domain and Interphase Boundaries
∞ 2
⎛ dP ⎞ 2
∫−∞ ⎜⎝ dx ⎟⎠ dx = 4δ P0 ,
γ0 = (4.8)
dϕ δ Φ ⎛ dP ⎞ α (Tc ) 2
2
− =
dz δ P
, Φ= ⎜ ⎟ +
2 ⎝ dz ⎠ 2 P04
( P0 − P 2 ) P 2 ,
d 2ϕ dP ⎛ eϕ ⎞ 8πn0 e 2 ϕ (4.9)
− + 4π = 4π ⎜ −2 n esh ⎟ − ϕ=− 2.
dz 2
dz ⎝
0
kT ⎠ kT λ
As indicated by (4.9), we obtain different results depending on
the degree of screening. In the absence of screening, i.e. at n 0 →0,
λ →∞, the value of the depolarizing field according to (4.9) is equal
to E = –4πP, i.e. to the value considerably higher than the
thermodynamic coercive field. In this field, the polarization that
creates the field is unstable, i.e. it should spontaneously reverse.
Therefore, there is no sense in discussion of the structure of the
flat interphase boundary perpendicular to the polarization vector in
the absence of such screening.
Let us assume that a sufficient degree of screening is present
47
Domain Structure in Ferroelectrics and Related Materials
P0
P ( x) = ,
exp ⎡ −2 x δ + δ ⎤ +1
2 2
⎣ C
⎦ D
4πλ 2 (4.12)
δC = , δD = .
α ( Tc ) α (Tc )
48
2. Structure of Domain and Interphase Boundaries
49
Domain Structure in Ferroelectrics and Related Materials
⎧⎪ 1 ⎡⎛ dq1 ⎞ 2 ⎛ dq2 ⎞ 2 ⎤ 1
Φ= ∫ ⎨ ⎟ +⎜ ⎟ ⎥ − α ( q1 + q2 ) +
2 2
⎢⎜
⎩⎪ 2 ⎣⎢ ⎝ dx ⎠ ⎝ dx ⎠ ⎦⎥ 2
1 2⎫ (5.1)
+ β ( q14 + q24 ) + γ ' q12 q22 + ξq1q2 P +
1 1
P ⎬ dx.
4 2 2χ0 ⎭
d 2 q1
= −α q1 + β q13 + γ q1q22 ,
dx 2
d 2 q2 (5.4)
= −α q2 + β q23 + γ q12 q2 .
dx 2
In a homogeneous state at – β < γ < β , the following states are
stable (Fig. 2.7) [61]:
50
2. Structure of Domain and Interphase Boundaries
I, q1 = q2 = q0 = α ( β + γ ) , P = − x0ξ q02 = − P0 ,
II , −q1 = q2 = q0 , P = P0 ,
III , −q1 = −q2 = q0 , P = − P0 , (5.5)
IV , q1 = − q2 = q0 , P = − P0 .
At γ > β the pattern of the stable states is shown in Fig. 2.8.
I , q1 = q0 = α / β , q2 = 0,
II , q1 = 0, q2 = q0 ,
III , q1 = − q0 , q2 = 0, (5.6)
IV , q1 = 0, q2 = − q0 , PI = PII = PIII = PIV = 0.
The stability of the states (5.5) or (5.6) is determined by
comparing the values of the thermodynamic potential in points
⎛ α2 ⎞ ⎛ α2 ⎞
A ⎜⎜ Φ = − ⎟⎟ and points B ⎜ Φ = − ⎟ , respectively.
⎝ 2( β + γ) ⎠ ⎝ 4β ⎠
The transition from one stable state (domain) to another within
the limits of each of the diagrams (5.5) or (5.6) represents domain
walls in the material under consideration. These walls correspond
to the lines (the sides, the diagonals of the square or other curves)
on the graphs. As indicated by the distribution (5.5), all consecutive
transitions A I ⇔A II , A II ⇔A III, A III ⇔A IV , A IV ⇔A I represent ferro-
electric domain walls whereas the transitions A I ⇔A III , A I ⇔A IV are
antiphase boundaries. For the distribution (5.6), none of the stable
states is linked with the formation of polarization and, consequently,
all the transitions between them (both the sides and diagonals of the
square, and the other curves in Fig. 2.8) represent only antiphase
boundaries.
51
Domain Structure in Ferroelectrics and Related Materials
Using the Green function, the solution of the above equation can
be written as follows
γq0 ∞
exp ( −2 x − x ' δ )
δ q1 ( x ) = − ∫ dx ' δ . (5.11)
2β −∞
ch 2 ( x ' δ )
52
2. Structure of Domain and Interphase Boundaries
Φ ( x) = P2 + P4 + ⎜ ⎟ + ⎜ 2 ⎟ + P ⎜ ⎟ . (5.14)
2 4 2 ⎝ dx ⎠ 2 ⎝ dx ⎠ 2 ⎝ dx ⎠
Assuming for simplicity that η = 0, the equation for the
distribution of the polarization vector from the variation of the
functional ∫ Φ ( x ) dx is written in the form
d 4P d 2P
σ4
− 2
+ α P + β P 3 = 0. (5.15)
dx dx
The numerical solution of this equation with the boundary
conditions P = ± P 0 at x → ±∞ in the area of stability of the
commensurate ferroelectric phase leads to the distribution P(x) in
the boundary shown in Fig.2.9.
The characteristic distinguishing feature of this distribution is the
approach of polarization to the equilibrium value after oscillation as
53
Domain Structure in Ferroelectrics and Related Materials
54
2. Structure of Domain and Interphase Boundaries
Φ = ∫⎨ 1 ⎜ ⎟ + ⎜ ⎟ − η +
⎩ 2 ⎝ dx ⎠ 2 ⎝ dx ⎠ 2
β α β γ ⎫ (6.1)
+ 1 η 4 − 2 ϕ 2 + 2 ϕ 4 + ϕ 2η 2 ⎬ dx.
4 2 4 2 ⎭
Depending on the ratio between the coefficients, this potential
permits four homogeneous phases (Fig. 2.10).
In phase I, η = ϕ = 0. In phase II, η ≠ 0, ϕ = 0. This phase
exists in the area from – α 1 < 0 to – α 1 < α 2 β 1 / γ . In phase III
η = 0, ϕ ≠ 0. This phase exists in the area from – α 2 < 0 to
– α 1 > – α 2 γ / β 2 . And, finally phase IV, where η ≠ 0, ϕ ≠ 0 exists
between the lines 1 and 2, i.e. from – α 1 <– α 2 γ / β 2 to – α 1 <– α 2 β 1 / γ .
The distribution of the order parameters in sections with
heterogeneous η and ϕ , in particular, in the region of the domain
boundaries is described by the set of equations which, as usual, is
obtained by varying the potential (6.1) in respect of η and ϕ :
d 2η
1 2
= −α1η + β1η 3 + γηϕ 2 ,
dx
d 2ϕ (6.2)
2 2 = −α 2ϕ + β 2ϕ 3 + γη 2ϕ .
dx
For phase II, the set of equation (6.2) has a conventional solution
describing the single-parameter domain wall:
55
Domain Structure in Ferroelectrics and Related Materials
x α1
η ( x ) = η0 th , η0 = ,
δ β1
2 1 (6.3)
δ= , ϕ ( x ) = 0.
α1
As in the case of (1.9), the problem of determination of the
stability of this solution is reduced to investigation of the spectrum
of the eigenvalues of the set of equations
d 2ψ n′
− 1 + ( −α1 + 3β1η 2 + γϕ 2 )ψ n′ = ε n ψ n′ ,
dx 2
d 2ψ n (6.4)
− 2 + ( −α 2 + 3β 2ϕ 2 + γη 2 )ψ n = λnψ n .
dx 2
The substitution of the solution under investigation (6.3) into
(6.4) makes it possible to present each of the equations of set (6.4)
in the form of a Schrödinger equation with the potential of
V(x)~ch –2 (x/ δ ):
d 2ψ n′ ⎡ 3 ⎤
− 1 + α1 ⎢ 2 − 2 ⎥ψ n′ = ε nψ n′ ,
dx 2
⎢⎣ ch ( x δ ) ⎥⎦
d 2ψ n γη02 (6.5)
− 2
dx 2
+ ( 2 0 ) n ch 2 ( x / δ ) = λnψ n .
− α + γη 2
ψ −
56
2. Structure of Domain and Interphase Boundaries
Fig.2.11. Transition from a flat domain wall with the alteration of the order parameter
in respect of the modulus (a) to the wall with the simultaneous alteration of the
modulus and orientation of the order parameter (b).
57
Domain Structure in Ferroelectrics and Related Materials
Chapter 3
58
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
Fig. 3.2. Alternation of the polarization vector for different configurations of the
narrow domain wall (a). The periodic dependence of the surface density of the domain
wall energy taking into account the discreteness of the crystal lattice (b): V 0 is the
lattice energy barrier for the domain wall.
59
Domain Structure in Ferroelectrics and Related Materials
α β
where, as previously, ϕ ( Pn ) = − Pn2 + Pn4 , and P n satisfies
2 4
equation (1.2).
Transition in (1.3) to the continuous limit yields quantity γ that
is independent of coordinate U, whereas direct calculations of the
lattice sum lead to the dependence of the surface density of the
energy of the domain wall on its position γ = γ (U).
The simplest way for assessment of the parameters of the lattice
relief is the so-called quasi-continuous approximation [71]. Taking
into account the periodicity of the dependence γ (U), we expand it
into a Fourier series
∞
γ (U ) = ∑ γ (U ) ⋅ e π
m =−∞
2 imU a
, (1.4)
60
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
( − n +1) a ∞
γm = ∑ ∫ f (U ′ ) ⋅ e −2πimU '/ a dU ' = ∫ f ( x ) ⋅ e −2πimx / a dx. (1.6)
n − na −∞
∞
β dx / δ 2
γ0 =
2
P04δ ∫ ch
−∞
4
= α P02δ ,
x /δ 3
β ∞
cos ( 2πmx / a )
γ m >0 =
2
P04δ ∫
−∞
ch 4 x / δ
dx / δ =
(1.8)
2 ⎛δ ⎞ 1 ⎡ π 2 m2δ 2 ⎤
= β P δ ⋅ π2m ⎜ ⎟
4
⋅ ⎢ + 1⎥ .
⎝ a ⎠ sh (π 2 mδ / a ) ⎣ a
0
3 ⎦
⎝a⎠
According to (1.9), the dependence γ (U) is periodic in fact with
the value of the barrier V 0 , which strongly depends on the relative
width of the domain wall (in comparison with the lattice constant).
At δ >>a, as in the case with, for example, ferromagnetics and
ferroelectrics in the vicinity of the Curie point, the barrier in the
dependence γ (U) almost completely disappears, whereas for narrow
domain boundaries with δ ~ a its presence as shown below has a
strong influence on the possibility of displacement of the domain
boundaries.
61
Domain Structure in Ferroelectrics and Related Materials
where
∂2 1
I ij = ⋅ , (2.3)
∂xi ∂x j r − r ′
is the so-called structural factor.
In specific structural calculations we have to separate the sub-
lattices of different dipoles. Giving them indices μ and ν , and also
taking into account the symmetry of the problem, i.e. the fact that
all dipole moments of the ν -th type Pν j ( xν′ ) at fixed coordinate xν′
are identical along the direction of the normal to the boundary and
having taken the above into consideration introducing the structural
factor for the dipole plane
∂2
I ij ( xμ − xv′ ) =
1
∑
yv′ , zv′ ∂xμi ∂xμ j
⋅
rμ − rv′
, (2.4)
the i-th component of the electric field in the location of the μ-th
type dipole can be presented in the form
62
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
63
Domain Structure in Ferroelectrics and Related Materials
− n2 +1 − n3 +1
1
I ( m2 , m3 ) = ∑ ∫ ∫ ×
n2 , n3 − n2 − n3 s + s + ( n1 − s
'2
2
'2
3 1)
' 2
(2.9)
(
×exp −2πi ( m2 s2' + m3 s3' ) ⋅ ds2' ds3' . )
Replacing the sum of the individual integrals by integrals in
infinite limits, we obtain
∞ ∞
1
I ( m2 , m3 ) = ∫∫ ×
s2'2 + s3'2 + ( n1 − s1' )
2
−∞ −∞
(2.10)
(
× exp −2πi ( m s + m s '
2 2
'
3 3 ) ) ⋅ ds ds ,
'
2
'
3
2π ∞
1
I ( m2 , m3 ) = ∫∫ ⋅ exp ( −2π im ρ cos ϕ ) ⋅ ρ d ρ dϕ =
0 0 ρ + d2
2
∞
I 0 ( 2π mρ ) 2π exp(−2π mα )
= ∫ 2π ⋅ ρd ρ = , (2.11)
0 ρ2 + d2 α
α = n1 − s1 .
(
dependence of (2.11) on m 2 , m 3 m = m2 + m3 , we finally have the
2 2
)
following equation for the structural factor of the dipole plane [72]:
∂2
( )
I ij s, n1 =
∂si ∂s j
∑ 2π
m2 , m3
1
n1 − s1
×
{
×exp −2π n1 − s1 m22 + m32 × } (2.12)
×cos ( 2π m2 s2 ) cos ( 2π m3 s3 ) .
64
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
factor in any case for the so-called ‘ own’ plane, i.e. the plane
passing through the dipole, in the location of which electric field is
sought. As expected, in this case, i.e. at n 1 =s 1 , a divergence
appears in equation (2.12).
The calculation of the structural factor for the dipole line is carried
out using the same procedure as in the case of the dipole plane, except
that in this case we use expansion into a one-dimensional Fourier
integral. For the line, oriented along axis z
∂2 1
I ij ( n1 , n2 , s ) = ∑ ⋅ =
n3 ∂si ∂s j n1 − s1
∂2 (2.13)
=
∂si ∂s j
∑ I ( m ) exp(2π im s ),
m3
3 3 3
1
⎛ 1 ⎞
I ( m3 ) = ∫ ⎜ ∑ ⎟⎟ ⋅ exp(−2π im3 s3 )ds3 =
⎜ n n −s
0⎝ 3 1 1 ⎠
∞
1
= ∫ ×
(2.14)
( n1 − s1 ) + ( n2 − s2 ) + s
2 2 2
−∞
3
( )
(2.15)
= 2 K 0 2π m3 ( n1 − s1 ) + ( n2 − s2 ) ,
2 2
∂2 1
I ij ( s, n1 , n2 ) = ⋅ 2π ∑ ×
∂si ∂s j m3 π
⎛ m ⎞ (2.16)
× K 0 ⎜ 2π 3 a12 ( n1 − s1 ) + a22 ( n2 − s2 ) ⎟ ⋅ cos ( 2π m3 s3 ) .
2 2
⎝ a3 ⎠
65
Domain Structure in Ferroelectrics and Related Materials
The lattice factor for ‘ own’ line, i.e. the line passing through the
point, at which the following field is sought
∞
Σ ′⋅ n
1
I =2 3
. (2.17)
− s3
n3 =−∞
3
The equations obtained above make it possible to estimate in the
first place the value of the correlation constant used in Chapter 2,
assuming that the main contribution comes from the dipole–dipole
interaction. For this purpose, let us consider the simplest cubic
lattice of the dipoles oriented along the single axis z and let us
assume that the alteration of their values depends only on single
coordinate x. That enables us to remove all the indices in equation
(2.5) and to write the volume density of the energy of the dipole–
dipole interaction in the form of the sum
Φ n = ∑ − I ( n − m ) ⋅ Pn Pm (2.18)
m
dPn 1 d 2 Pn 2
Pn +1 = Pn + a+ a + ...,
dx 2 dx 2
dP 1 d 2 Pn 2 (2.19)
Pn −1 = Pn − n ⋅ a + a + ...
dx 2 dx 2
According to (2.12), the lattice factor I(n–m) decreases
exponentially with the increase of the argument, which allows us
to retain in (2.18) the interaction only with the dipoles of the ‘ own’
and adjacent atomic planes. As the result, expression (2.18) taking
into account the cancellation of the number of the terms containing
the first derivatives, can be written in the form of
1 d 2P
Φ n = − ⎡⎣ I ( 0 ) + 2 I (1) ⎤⎦ ⋅Pn2 − a 2 ⋅ I (1) ⋅ Pn ⋅ 2n (2.20)
2 dx
or, after calculating sum ∑Φ
n
n → ∫ Φ( x) dx a using the results of
integration by parts, in the form of
66
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
2
1 ⎛ dP ⎞
Φ n = Φ n 0 + a 2 ⋅ 2 I (1) ⋅⎜ n ⎟ ,
2 ⎝ dx ⎠
(2.21)
Φ n 0 = − ⎡⎣ I ( 0 ) + 2 I (1) ⎤⎦ ⋅ Pn2 .
As indicated in (2.21), the second term in Φ n coincides here with
the ordinary correlation term in (1.1) at
= a 2 ⋅ 2 I (1) . (2.22)
For a simple cubic lattice from (2.16)
I (1) = − ( 2π )
3
∑m
m2 , m3
2
3 ( )
⋅ exp −2π m22 + m32 −0,926. (2.23)
67
Domain Structure in Ferroelectrics and Related Materials
{ }
En ( O1 ) = ∑ I 0 ( n − n ' ) ⋅Pn ' + I1/ 2 ( n − n' ) ⋅ Pn' ' ,
n'
(3.1)
where
I s3 ( n − n ' ) = δ n − n' ,0 ∑ '
2
3
−
n3 n3 − s3
− ( 2π ) ∑ (1 − δ )
⋅ δ n2 ,0 ×
3
n − n' ,0
n2
∞
2 ⎛ ⎞ (3.2)
×∑ m2 K 0 ⎜ 2πm 2πm ( n1 − n ' ) + n 2 ⎟ ⋅ cos ( 2πms3 ) .
2
m =1 π ⎝ ⎠
Here, I 0 and I 1/2 are structural factors for dipole planes compiled
from dipoles situated in the locations of the considered oxygen ions
O 1 and titanium ions, respectively. The indices 0 and 1/2 of these
factors correspond to the value s 3 , which should be used when
calculating the corresponding factors on the basis of equation (3.2).
68
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
When the ratio between Pn and P'n is available, we can write the
equation for the distribution of total polarization in the boundary. As
previously, let us use the value of the strength of the electric field
calculated on the basis of (3.1) and (3.2) this time in the area where
the titanium ion is located. According to (3.1) and (3.2)
E ( Ti ) = 0 Pn' + I 0 (1) ⋅ ( Pn'+1 − 2 Pn' + Pn'−1 ) +
( )
+ 1/ 2 ⋅Pn + I1/ 2 (1) ⋅ Pn +1 − 2 Pn + Pn −1 .
(3.5)
69
Domain Structure in Ferroelectrics and Related Materials
⎧ 0, n = 0 ⎧0.6 P0, n = 1
⎪ ⎪
⎪
I . Pn = ⎨0.8 ⋅ P0, n = 1 II . Pn = ⎨0.9 P0 , n = 2
(3.10)
⎪ ⎪
⎩ P0 , n > 1 ⎩ P0 , n > 2
⎪
Calculations of the energies for the given boundary configuration
show that at room temperature configuration II is stable or basic.
The value of γ 0 for it obtained from equation (1.3) by adding here
γ
the term 0 ( Pn6 − P06 ) turns out to be equal to 6.3 erg· cm–2 . On the
6
70
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
71
Domain Structure in Ferroelectrics and Related Materials
Fig. 3.5. Schematic image of the PO 4 tetrahedrons in the structure of the KDP
crystal with adjacent protons.
72
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
Fig. 3.6. The model of the domain boundary in the KH 2 PO 4 crystal (indicated by
the dashed line), consisting of Slater static configurations: (a) – the neutral boundary,
(b)– the polar boundary [78].
73
Domain Structure in Ferroelectrics and Related Materials
↑ ) = a∞ ↑ + b∞ ↓ , ↓ ) = a∞ ↓ + b∞ ↑ ,
(a 2
∞ + b∞2 = 1) .
(4.2)
74
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
75
Domain Structure in Ferroelectrics and Related Materials
76
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
⎛ 0 1/ 2 0 1/ 2 ⎞
⎜ ⎟
xμ v ⎜ 0 −1/ 2 0 ⎟
s1μ v = = ,
ax ⎜ 0 1/ 2 ⎟
⎜⎜ ⎟
⎝ 0 ⎟⎠
⎛ 0 1/ 2 1/ 2 0 ⎞
yμ v ⎜⎜ 0 0
⎟
−1/ 2 ⎟
s2 μ v = = ,
ay ⎜ 0 −1/ 2 ⎟
⎜⎜ ⎟
⎝ 0 ⎟⎠
(4.10)
⎛ 0 1/ 2 1/ 4 −1/ 4 ⎞
zμ v ⎜⎜ 0 −1/ 4 −3 / 4 ⎟
⎟
s3 μ v = = .
az ⎜ 0 −1/ 2 ⎟
⎜⎜ ⎟
⎝ 0 ⎟⎠
77
Domain Structure in Ferroelectrics and Related Materials
P ( m) =
P
4 ⎣ (
sign ( m ) ⋅⎡3 1 − 2a2
2 m −2
) (
+ 1 − 2a2 ⎤ ,
2m
⎦ )
(4.14)
m > 1, P ( −1) = P (1) = ⋅ ⎡⎣(1 − 2a22 ) ⎤⎦ ,
P
4
from which the surface density of the dipole energy for the saddle
configuration of the wall is
4P2
a
{
H 2 = 5 A + Ba 4 ,
2 + Ca
2 2 } (4.15)
where A = 9.6, B = 25.9, C =16.4.
Summation of H 1 /a 2 and H 2 , H 1 /a 2 and H 2 gives the total
energy of the main and saddle configurations of the domain wall
respectively. In this case, the form of the transition layer is
determined by the variation in respect of the introduced parameter.
The use of the energy values of the boundary configuration of
type 2 ε 0H = 64 K, and Takagi's defect W H = 680 K, Ω H = 86 K [80–
85], which enables to find the energy constants (4.6) and (4.9),
gives the following values of variation parameters determining the
structure of the wall: a 1 = 0.13, a 2 = 0.20, Q = 0.98, a1 = 0.13,
a2 = 0.28. In this case, the surface density of the energy of the wall
for the main and saddle configuration of the wall are equal to
γ 0 = 25 erg· cm–2 , γ = 53 erg· cm–2 [79].
78
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
Here Z i,n , X i,n , are the operators of quasi-spin of the i-th bond
belonging to the n-th plane, parallel to the domain boundary. The
mean value of the quasi-spin 〈Z n 〉 depends on the number n of the
plane (layer) and determines the degree of ordering (polarization)
in the given location of the crystal.
When writing (5.1) it is assumed that the dependence of the
constant of quasi-spin interaction ij on the numbers of interacting
quasi-spins is reduced to the dependence of the constant on the
direction of interaction. In (5.1) constant is the cumulative constant
of interaction of the quasi-spin with neighbours in the direction parallel
to the plane of the boundary, and constant A – with neighbours in the
direction perpendicular to the plane of the boundary.
It was shown in the previous section that the electric fields,
generated by different dipole planes, in the approximation of the
rigid bond of the proton subsystem and the system of heavy ions,
combining equation (2.2) and (4.11), can be written in the form of
the product ~I(n–m) 〈Z n 〉, where I(n–m) is the corresponding
structural factor. At same time the energy of interaction of the given
dipole with all dipoles of the m-th plane is ~I(n–m) 〈Z n 〉〈Z m〉, i.e.
it has the same structure as the short-range part of the interaction
of quasi-spins. Taking into account the short-range nature of the
electric field of the dipole plane, its exponential decrease with
79
Domain Structure in Ferroelectrics and Related Materials
A
q = Ω / , Sn = Z n + ( Z n+1 + Z n−1 ) ,
⎛ 2A ⎞ (5.3)
S∞ = ⎜ 1 + ⋅ Z , Zn ≡ Zn ,
⎝ ⎟⎠ ∞
where Z ∞ is the mean value of the quasi-spin away from the
boundary, and S is the area of the side surface of the elementary
cell parallel to the plane of the domain wall and falling onto a single
quasi-spin chain.
Self-congruent values Z n and Z ∞ are determined in the general
case from the minimality conditions ∂γ / ∂ Z n = 0 and ∂γ / ∂ Z ∞ = 0 and
comply with the following respective equations
80
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
⎛ q2 + S 2 ⎞
Z n q 2 + S n2 = S n ⋅ th ⎜ n
⎟,
⎜ T ⎟
⎝ ⎠
⎛ q2 + S 2 ⎞ (5.4)
Z ∞ ⋅ q 2 + S∞2 = S∞ ⋅ th ⎜ ∞
⎟.
⎜ T ⎟
⎝ ⎠
In direct calculations of the structure of boundary configurations,
as in the case of barium titanate (see section 3.3), the first of the
equations (5.4) can be used for the middle of the boundary layer
with n=0, ±1. For the remaining part of the boundary, the
dependence of Z n on |n| can be simulated by the expression
Z n = Z ∞ ⋅ ⎡⎣1 − A ⋅ exp ( − n ⋅ λ ) ⎤⎦ , (5.5)
where parameter λ is determined from the self-congruent condition,
based on the application of distribution (5.5) in the general equation
(5.4) for high n, which leads to the following equation for determination
of λ:
⎛ S ⋅Z
q 2 + S∞2 = ⎜ th q 2 + S∞2 − ∞ ∞ +
⎜ T q 2 + S ∞2
⎝
⋅S∞2 / T q 2 + S∞2 ⎞ ⎛ A ⎞ (5.6)
+ ⋅ ch 2 ⎟ ⋅ ⎜ 1 + 2 ch λ ⎟ .
q 2 + S∞2 T ⎟ ⎝ ⎠
⎠
Coefficient A in (5.5) is determined from the condition of joining of
solutions (5.4) and (5.5) at n = 1.
Using the calculated value Ω H = 72 K, Ω D = 0, one can find the
total value of the constants ( +2A) from the condition for the
Ω
transition temperature = th ( Ω / Tc ) , resulting from (5.4) at
( + 2 A)
Z n = Z n+1 = Z n –1 = Z ∞ . This gives + 2A H = 139 K and ( D + 2A D ) =
T Dc = 213 K. Assuming that the short-range local interaction is
symmetric and the contribution of the dipole–dipole interaction to
constants and A is determined by the ratio of the factors I(0) and
I(±1), we can also find the value of the individual constants, which
turn out to be as follows: H = 113 K, A H = 13 K, D = 173 K,
A D = 20 K.
Numerical calculations of the structure and surface density of
the energy of the boundary configurations a and b (Fig.3.1) denoted
81
Domain Structure in Ferroelectrics and Related Materials
Fig. 3.9. Temperature dependence of the surface density of the energy of boundary
configurations and the values of the lattice barrier in the KDP crystal. 1) Z 1I , γ I ,
2) Z 1II , γ II , 3) Z ∞, V 0 .
82
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
Fig. 3.11. Structure of the narrowest configurations of the domain wall with extreme
energy values.
83
Domain Structure in Ferroelectrics and Related Materials
possible change of the type of the main configuration for the same
reason it is pointed out, in particular, in the Frenkel–Kontorova model
[73].
γ = ∫ ⎨ ( Z − Z∞ ) + ( Z − Z∞ ) + ⎜ ⎟ ⎬ ,
2 4
(6.1)
S ⎩⎪ 2 4 2 ⎝ dx ⎠ ⎭⎪ a
where
( + 2 A)
2
Ω
α = ( + 2 A) − ⋅ th , (6.2)
Ω T
( + 2 A)
4
⎡ Tc Ω 1 ⎤
β= ⎢ th − 2 ⎥, (6.3)
2Tc ⋅Ω ⎣ Ω Tc ch Ω / Tc ⎦
⎡ ( + 2 A) Ω ⎤
= A ⋅ a2 ⎢2 th − 1⎥ ≡ a 2 ⋅ A. (6.4)
⎣ Ω Tc ⎦
When writing (6.2)–(6.4) it is assumed that only coefficient α
depends on temperature in an explicit manner, and the coefficients
α, β , are normalized in the corresponding manner, since, for
example, α /Sa has the dimensionality of the volume density of
energy.
In the vicinity of T c
84
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
( + 2 A)
2
1
α α 0 (T − T0 ) , α 0 = ⋅ , (6.5)
2
ch Ω / Tc Tc2
where the value of T c itself is determined by the conventional ratio
Ω/( +2A) = thΩ/T c . In this case, the structure and half width of
the domain wall have the form [89]:
x 2 A Tc ⋅chΩ / Tc
Z ( x ) = Z ∞ ⋅ th , δ = a ⋅ .
δ T − Tc ( + 2 A) (6.6)
At low Ω
2A 2A Ω2
δ a +a ⋅ . (6.7)
T − Tc T − Tc 2 ( + 2 A )2
The surface density of the energy of the domain wall in
compliance with (6.1) and (6.2)–(6.4) is
2 2 2 3 / 2 1/ 2 −1 −3
γ = α Z 2δ ⋅ a −3 = α β a =
3 3
A1/ 2 ( T − Tc ) Ω2
3/ 2
4 2
= ⋅ 2 ×
3 a ( + 2 A ) ⋅ Tc2 ⋅ ch 3Ω / Tc
−1
(6.8)
⎡T Ω 1 ⎤
× ⎢ c th − 2 ⎥ .
⎣ Ω Tc ch Ω / Tc ⎦
At low Ω
Ω
γ = 2 ⋅ΔT 3 / 2 ⋅ A1/ 2 ( + 2 A ) / a 2 ⋅ ch 3
−1
,
(6.9)
( + 2 A)
ΔTc = T − Tc .
In this case, the dimensionless order parameter in the volume of the
domain is
1/ 2
⎡ Ω2 ⎤
(T − Tc )
1/ 2
3 ⋅T
c
1/ 2
⎢1 − 2 ⎥
Z = α/β = ⎣ Tc ⎦ , (6.10)
+ 2A
and, consequently, the derivative characterizing the curvature of the
profile of the domain wall is:
dZ Z
=
α
=
(T − Tc ) ⋅ ⎡1 − Ω2 ⎤ .
⎢ 2 ⎥ (6.11)
dx δ 2β A ⋅ Tc ⋅ a ⎣ Tc ⎦
85
Domain Structure in Ferroelectrics and Related Materials
As it can be seen from Figs. 3.12 and 3.13, tunnelling that differs
from zero increases the width of the domain wall and reduces the
density of its surface energy.
As numerical estimates show at ΔT~10 K and a ~ 10 –7 cm, the
width of the domain wall in DKDP is δ D 2· 10–7 cm, at the same
time the surface density of its energy is γ D 6· 10–2 erg· cm–2 . In
KDP crystal at the same distance from T c δ H = 2.5· 10–7 cm,
γ H 4· 10–2 erg· cm–2 , which is in good agreement with the results
of numerical calculations of the previous section. The mentioned
agreement is conditioned by the possibility of using here the continual
approximation, the transition to which gives the relative error of
Δ γ / γ a 2/2 δ 2 << 1, as indicated in particular by estimates for γ .
As we will see in chapter 5, a not too large increase of the width
of the domain wall at Ω ≠ 0 can result in an extremely large
increase of its mobility.
The link of the parameters of continual approximation to the
microscopic model, found in this section, in particular, in the
Fig. 3.12. Alternation of the order parameter in the boundary for different values
of the tunnelling constant. 1 — Ω = 0, 2 — Ω ≠ 0.
Fig. 3.13. Change of the width of the domain wall (a) and surface density of its
energy (b) in relation to the value of the tunnelling integral.
86
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
87
Domain Structure in Ferroelectrics and Related Materials
⎛ SH ⎞
x ⋅ H ⋅ SnH + (1 − x ) ⋅ D ⋅ S nD − x H th ⎜ H n ⎟ −
⎝ T ⎠
⎛ SH ⎞ ⎛ SH ⎞
− xAH th ⎜ H n +1 ⎟ − xAH th ⎜ H n −1 ⎟ −
⎝ T ⎠ ⎝ T ⎠
⎛ SD ⎞ ⎛ SD ⎞
− (1 − x ) D th ⎜ D n ⎟ − (1 − x ) AD th ⎜ D n +1 ⎟ − (7.3)
⎝ T ⎠ ⎝ T ⎠
⎛ SD ⎞
− (1 − x ) AD th ⎜ D n −1 ⎟ = 0
⎝ T ⎠
Tc =
(1 − x ) ⋅ I D2 + x ⋅ I H2
(1 − x ) ⋅ I D + x ⋅ I H (7.5)
88
3. Microscopic Structure of Domain Boundaries in Ferroelectrics
expression:
1 dx ⎧ ⎡ I H 2 AH 2 ⎤
S ∫ a ⎩ ⎢⎣ 2
γ= ⎨x Z + a Z ⋅ Z "⎥ +
2 ⎦
⎡I A ⎤
+ (1 − x ) ⋅ ⎢ D Z 2 + D a 2 Z ⋅ Z "⎥ −
⎣2 2 ⎦
⎛ I ⋅ Z + AH ⋅a ⋅ Z " ⎞
2
−T ⋅ x In 2 ch ⎜ H ⎟− (7.6)
⎝ T ⎠
⎛ I ⋅ Z + AD ⋅ a ⋅Z " ⎞ ⎪
2
⎫
−T ⋅ (1 − x ) ln 2 ch ⎜ D ⎟⎬ − γ ( Z∞ ).
⎝ T ⎠ ⎭⎪
1 dx ⎧ ⎡ I H 2 AH 2 ⎤
S ∫ a ⎩ ⎢⎣ 2
γ= ⎨x Z + a Z ⋅ Z "⎥ +
2 ⎦
⎡I A2 ⎤
+ (1 − x ) ⋅ ⎢ D Z 2 + D a 2 Z ⋅ Z "⎥ −
⎣2 2 ⎦
⎡ I2 I ⋅ A ⋅ a2 ⎤
−x ⎢ H ⋅ Z 2 − H H ⋅ Z "⎥ − (7.7)
⎣ 2T T ⎦
⎡ I2 I ⋅ A ⋅ a2 ⎤⎪ ⎫
− (1 − x ) ⎢ D ⋅ Z 2 − D D ⋅ Z "⎥ ⎬ .
⎣ 2T T ⎦ ⎭⎪
∞ ∞
∫ ∫ ( dZ dx ) dx,
2
Taking into account integration by parts ZZ " ∂x =
−∞ −∞
after collecting together the terms with Z 2 and (dZ/dx) 2 , the
expression for γ is written in the form:
1 dx ⎧⎪ ⎡ I H2 I D2 ⎤ Z 2
( ) ( )
S∫ a ⎪
γ= ⎨⎢ Hx ⋅ I + 1 − x ⋅ I D − x − 1 − x ⎥⋅ +
⎩ ⎣ T T ⎦ 2
1⎡ 2 A I a2
+ ⎢ − xAH ⋅a 2 − (1 − x ) AD ⋅ a 2 + x H H +
2⎣ T
(7.8)
A I a ⎤ ⎛ dZ ⎞ ⎫⎪
2
+ (1 − x ) 2 D D ⎥ ⎜ ⎟⎬ − γ ( Z∞ ).
T ⎦ ⎝ dx ⎠ ⎪ ⎭
89
Domain Structure in Ferroelectrics and Related Materials
1 ⎧⎪α 2 ⎛ dZ ⎞ ⎫⎪ dx
2
⎨ ( ∞)
S ∫ ⎪⎩ 2
γ= Z − Z 2
+ ⎜ ⎟ ⎬ , (7.9)
2 ⎝ dx ⎠ ⎭⎪ a
we can see that here
I H2 I2
α = x ⋅I H + (1 − x ) ⋅ I D − x
− (1 − x ) D (7.10)
T T
or, taking into account the expression for T c (7.5)
I H2 I2 I2 I2
α = x⋅ + (1 − x ) ⋅ D − x H − (1 − x ) D = α 0 (T − Tc ) , (7.11)
Tc Tc T T
x ⋅ I H2 + (1 − x ) I D2
α0 = , (7.12)
Tc2
2 2 x ⋅ I H ⋅ AH 2 (1 − x ) I D AD
= + − x ⋅ AH − (1 − x ) AD . (7.13)
a Tc Tc
Hence the width of the boundary [94] is:
1/ 2
⎧⎪ 4 xI A + 4 (1 − x ) I A − 2 A T − 2 (1 − x ) A T ⎫
⎪
δ = a⎨ H H D D H c D c
⋅ T ⎬ . (7.14)
⎩⎪ ( xI 2
H + ( 1 − x ) I 2
D ) ( T − Tc )
c
⎪
⎭
At x = 1 (I H ≡ I = T c , A H ≡ A) the expression (7.14) changes to
the previously derived expression (5.6), written for Ω = 0.
90
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
Chapter 4
91
Domain Structure in Ferroelectrics and Related Materials
92
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
∂ 2ϕ ⎛ ∂ 2ϕ ∂ 2ϕ ⎞
εc + εa ⎜ 2 + 2 ⎟ =
∂z 2
⎝ ∂y ∂x ⎠
∂U
= 8π P0 ⋅ δ ( x ) ⋅ − 4π Ze ⋅ δ ( r − rd ) ,
∂z (1.5)
⎛ ∂ 2U ∂ 2U ⎞ ∂ϕ
−γ ⎜ 2 + 2 ⎟ + 2 P0 = 0.
⎝ ∂z ∂y ⎠ ∂z x =0
dk
U ( y , z ) = ∫ U k ⋅ exp(ikp) ,
( 2π )
2
dk
ϕ ( x, y, z ) = ∫ ϕ k ( x ) ⋅ exp(ikp) ,
( 2π ) (1.6)
2
ρ = ( y, z ) .
93
Domain Structure in Ferroelectrics and Related Materials
4πP0ik zU k ⎛ εc 2 ⎞
ϕk ( x ) = − ⋅ exp ⎜ − x
⎜
k z + k y2 ⎟ +
⎟
εc 2 ⎝ εa ⎠
εa k z + k y2
εa
2πZe ⎛ εc 2 ⎞ (1.8)
+ ⋅ exp ⎜ − x − xd k z + k y2 ⎟ .
εc 2 ⎜ εa ⎟
εa k z + k y2 ⎝ ⎠
εa
⎛ εc 2 ⎞
−4π P0 Ze ⋅ ik z ⋅ exp ⎜⎜ − xd k z + k y2 ⎟⎟
⎝ εa ⎠.
U k =
⎡ εc 2 ⎤ (1.10)
⎢8π P0 k z + γ k ε a k z + k y2 ⎥
2 2 2
⎣ εa ⎦
Substitution of (1.10) into equation (1.8) gives the final
expression for the Fourier expansion of the potential [96, 97]:
2π Ze ⎛ εc 2 ⎞
ϕk = ⋅ exp ⎜ − x − xd
⎜
k z + k y2 ⎟ −
⎟
εc 2 ⎝ εa ⎠
εa k z + k y2
εa
⎛ εc 2 ⎞
exp ⎜⎜ − ( xd + x ) k z + k y2 ⎟⎟
( 4π P0 ) k ⋅ Ze εa
2 2
⎝ ⎠ (1.11)
− ⋅ z
.
εc 2 ⎡ εc 2 ⎤
εa k + k y2 ⎢8π P02 k z2 + γ k 2ε a k z + k y2 ⎥
εa z ⎣ εa ⎦
94
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
Fig 4.2. Variation of the electrostatic potential, induced by the bent wall in the
area of location of the charge defect, in relation to the distance between them (linear
approximation).
95
Domain Structure in Ferroelectrics and Related Materials
defect of any sign with the domain wall is always of the attraction
type.
The energy of interaction of the charge with the wall is the
difference of the quantities Ze ϕ ind (x d ) taken in the area of the
maximum interaction and away from the boundary at x d → ∞. The
divergence of the mentioned expression at x d → 0 is obviously
associated with the application of the structureless boundary model.
Restricting the minimum values of x d to the half width of the
domain wall δ we find the energy of interaction of the defect with
the boundary
Z 2 e2
U0 = . (1.13)
2 ε cε a ⋅ δ
For Z =1, ε c ~10 3 , ε a ~10, δ ~10 –7 cm the value of U 0 is of the
order of 10 –3 eV, and at ε c ε a ~10 2 it is an order of magnitude
higher.
It should be mentioned that equation (1.13) for U 0 can be written
immediately if it is taken into account that the interaction of the
charged defect with the bound charge induced by this defect on the
boundary is in fact the interaction with an image charge, and the
domain wall itself in accordance with (1.1) ignoring the surface
tension is the equipotential surface. The flow of the bound charges
on this surface as a result of a bending of its profile is similar to
the motion of free charges on a metallic surface.
When determining the specific form of the bending of the
boundary let us consider the case of x d = 0 and the most typical
situation when ε c >> ε a . At conventional γ~0.1÷1 erg/cm 2 , ε c ~10 3 ,
ε a ~10, P 0 ~10 4 CGSE units, the ratio λ = γ ε cε a 8π P02 is of the
order of 10 –8 ÷10 –7 cm, whereas the maximum is k z ~2π/δ~10 7 cm –1 .
Therefore, taking into account the smallness of k z λ ≤ 1, the Fourier
image (1.10) of the boundary displacement can be written as
follows
4π P0 Ze ik z
Uk = − ⋅ .
γ ε cε a (k 2
y k z + k z2 λ ) (1.14)
96
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
Ze π ⎡1 ⎛ p ⎞⎤ ⎛ p2 ⎞
U ( y, z ) = ⎢ − C ⎜ ⎟⎥ cos ⎜ ⎟+
2π P0 λ z 2 ⎣2 ⎝ 2π ⎠ ⎦ ⎝ 4 ⎠
⎡1 ⎛ p ⎞⎤ ⎛ p2 ⎞
+⎢ −S⎜ ⎟ ⎥ sin ⎜ ⎟ , (1.15)
⎣2 ⎝ 2π ⎠ ⎦ ⎝ 4 ⎠
p=y λz ,
Ze
U ( y, z ) = ×
2 P0
2πλ z
× . (1.18)
⎡ 3
3⎤
⎢ 2 ( 2πλ z ) + p y ⋅ 2πλ z + Qy 2πλ z + R y ⎥
2 2
⎣ ⎦
As expected, the displacement (1.18) is asymmetrical along the
polar axis z with respect to the position of the charged defect, and
possesses the characteristic law of decrease 1/ z along the polar
axis and dependence U~1/y 3 in the perpendicular direction. With
increasing z, the displacement of the boundary, remaining maximum
for y = 0, spreads along the y-direction decreasing in value
simultaneously. At the same time, the integral from U(y,z)dy
remains equal to a constant (otherwise, we would be faced with the
localization of the bound charge at the boundary in the vicinity of
the fixing point, i.e. not with the compensation of the point defect
by the bound charge at the boundary, but only with the redistribution
97
Domain Structure in Ferroelectrics and Related Materials
of the density of the latter with its general zero value). This is very
well illustrated, for example, when considering the Fourier image of
the boundary displacement. Integral from U(y,z) over dy yields
delta function δ(k y ). Then assuming that k y = 0 in (1.14), we obtain
Ze
∫ U ( y, z ) dy = 2P Θ ( z ) , where Θ(z) is the sign function of
0
z, i.e. we
obtain a constant value at any given z and independent of the
coefficient of the boundary surface tension.
For a 90 o domain wall, as it was shown in section 2.2, the
interaction of the charged carriers with the domain wall takes place
not only by way of distortion of its profile similar to the one
discussed above, but also by way of the interaction with an internal
electric field existing in such a boundary.
98
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
Fig.4.4. Different orientations of the domain wall in ferroelastics. (a) the domain
boundary coincides with the invariant plane, (b) the inclined wall, the displacement
of the wall depends on the coordinate in the direction of spontaneous shear, (c)
the inclined wall, its displacement changes in the direction perpendicular to spontaneous
shear.
99
Domain Structure in Ferroelectrics and Related Materials
100
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
1 ⎛ 1 ⎞
ε ij = ⎜⎜ σ ij − σ kk ⋅ δ ij ⎟⎟ , m = 2 ( λ + m ) / λ,
2μ ⎝ ( m + 1) ⎠
(2.10)
σ ij ,kk +
m
m +1
(σ kk ,ij − σ kk ,ll ⋅ δij ) −
(2.11)
− (σ jk ,ki + σ ki , jk + σ kl ,kl ⋅ δ ij ) = 2 μ ⋅ηij .
Using the equation of dynamics of the elastic medium
σ ij , j + f i = ρ ⋅ ui (2.12)
where f i is the corresponding projection of the volume density of
the external forces, ρ is the density of the medium, instead of
(2.11) we obtain
σ ij ,kk +
m
m +1
(σ kk ,ij − σ kk ,ll ⋅ δ ij ) −
(2.13)
− ⎡⎣( ρ uj ) ,i + ( ρ ui ) , j ⎤⎦ + ( ρ ul ) ,l ⋅δ ij = 2μ ⋅ηij .
ρ
σ ij ,kk +
m
m +1
(σ kk ,ij − σ kk ,ll ⋅ δ ij ) − σij +
μ
ρ (λ + μ )
+ ⋅ ⋅ σkk δ ij + f j ,i + fi , j − fl ,l ⋅ δ ij +
μ ( 3λ + 2μ ) (2.15)
∂ ∂
+ ρ ( jij + j ji ) − ρ jll ⋅ δ ij = 2 μ ⋅ηij .
∂t ∂t
The resultant equation is referred to as the Beltrami–Mitchell
dynamic equation [102]. It enables using the available sources of
101
Domain Structure in Ferroelectrics and Related Materials
∂ 2ϕ ⎛ ∂ 2ϕ ∂ 2ϕ ⎞
εc + ε a⎜ + 2 ⎟=
∂z 2 ⎝ ∂y
2
∂x ⎠
∂U
= 8π P0δ ( x ) ⋅ − 4π Ze ⋅ δ ( r − rd ) ,
∂z (3.1)
∂ϕ
−2 P0 x=0 +2ε0σ 12 x =0 = 0.
∂z
As it can be seen from (3.1), in this case the equation of the
boundary equilibrium also includes the term related to the pressure
of the boundary from the direction of the field of elastic stresses
(it is written in the form similar to the pressure from the direction
of the electric field). At the same time, in the equation of the
boundary equilibrium the surface tension is ignored, which as it will
be shown later, is considerably smaller here than the other terms
for all orientations of bending and all values of the wave vector k.
For the combined solution of equations of set (3.1) it is
necessary to find first of all the relation of the stresses, formed at
the bending of the boundary, to the magnitude and orientation of its
102
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
∂u12s ∂U
α 22 = e231 = 2ε0 δ ( x),
∂x3 ∂z
∂u12s ∂U (3.4)
α 32 = −e321 = −2ε0 δ ( x ).
∂x2 ∂y
Substituting (3.4) into (2.6) we obtain components of the tensor
of incompatibility differing from zero
1 ∂α 22 ∂ 2U
η12 = − = −ε0 2 δ ( x ) ,
2 ∂x3 ∂z
1 ∂α 32 ∂ 2U
η13 = − = ε0 δ ( x),
2 ∂x3 ∂y∂z
∂α 32 ∂U
η33 = − = −2ε0 δ ′( x), (3.5)
∂x1 ∂y
1 ∂α 22 ∂U
η23 = = ε0 δ ′ ( x ).
2 ∂x1 ∂z
103
Domain Structure in Ferroelectrics and Related Materials
dk
U ( y, z ) = ∫ U k ⋅ exp(ikp) ,
( 2π )
2
dk (3.6)
σ ij = ∫ σ ij ( x ) ⋅ exp(ikp) , ρ = ( y, z ) .
( 2π )
2
where β = m / ( m + 1) , k = k y + k z , σ = σ11 + σ 22 + σ 33 .
2 2 2
Whence
2 με0ik yU k
σ ( x ) = ⋅ exp( − x k ) ⋅ sign x.
( 2 β − 1) (3.11)
On the basis of (3.2) and (3.5) the equation for determining the
Fourier image σ12 ( x) has the form
σ12′′ − k 2σ12 + β ik yσ ′ = 2 με0 k z2δ ( x ) ⋅ U k . (3.12)
Using expansion (3.9) for σ ( x) and the identical expansion for
σ12 ( x) , on the basis of (3.12) we obtain
2 με0 k z2 β k y kx
σ 12k = −
x
Uk − σk .
(k 2
x +k 2
) (k 2
x + k2 ) x (3.13)
104
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
2με0 k z2 4 με0 β k y2 k x2
σ 12k = −
x
Uk − Uk .
(k 2
x + k2 ) ( 2 β − 1) ( k x2 + k 2 )
2 (3.14)
Hence
⎪⎧ με0 k z με0 β k y ⎫⎪
2 2
⎧ με0 ⎫
σ12 ( x = 0 ) = − ⎨U k ( k z2 + ω k y2 ) ⎬ ,
⎩ k ⎭
β m 2(λ + μ ) (3.16)
ω= = = .
2β − 1 m − 1 λ + 2μ
Now using the solution of the first of the equations of system
(3.1) in the form of (1.6), (1.8), found in section 4.1, for the present
case of the ferroelectric-ferroelastic crystal we obtain the following
Fourier image of the displacement of the boundary in the field of
the point charge defect
⎛ εc 2 ⎞
−4π P0 ⋅ Zeik z ⋅ exp ⎜⎜ − xd k z + k y2 ⎟⎟
⎝ εa ⎠
Uk = . (3.17)
⎧ ⎫
⎪ ⎪
εc 2 2 ⎪ 8πP0 k z2 2 με02 2 2 ⎪
εa k + ky ⋅ ⎨
εa z
+ ( kz + ωk y )⎬
⎪ε ε c k 2 + k 2 k ⎪
⎪ a εa z y
⎪
⎩ ⎭
Z 2 e2 γ 4π P02
U0 = ⋅ , γ = ,
2εδ ⎡(1 + γ ) + 1 + γ ⋅ ω ⎤ εμε02
⎣ ⎦ (3.18)
ω = 2 ( λ + μ ) / ( λ + 2μ )
As can be seen from (3.18), at ε0 → 0 , i.e. at γ → ∞ the energy
105
Domain Structure in Ferroelectrics and Related Materials
106
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
⎛ 2 ⎞
f ( r ) = − ⎜ λ + μ ⎟ ⋅ Ω0 ⋅ grad δ ( r − rd ) , (4.1)
⎝ 3 ⎠
where r d is the coordinate of the defect. In the cubic crystal or in
the isotropic medium, Ω 0 has a simple physical meaning. Its value
is equal to the change of the crystal volume caused by the
presence of a single defect in the crystal. For an internodal atom
Ω 0 >0 and for a vacancy, where displacement of the adjacent atoms
takes place in the direction of the defect, Ω 0 <0.
The simultaneous equations describing the interaction between
the centre of dilatation and the domain boundary in a ferroelastic
are represented by the following set of equations
σ ij ,kk +
m
m +1
(σ kk ,ij − σ kk ,ll ⋅ δij ) +
+ fi , j + f j ,i − f l ,l ⋅ δ ij = 2 μηij , (4.2)
σ 12 x =0 = 0.
The first of these equations is the static Beltrami equation in the
presence of external forces, and the second one is the condition of
equality to zero of the elastic stresses at any section of the bent
boundary, as the consequence of its equilibrium equation. As in the
previous section, it is assumed that the domain boundary in the
elastic separates the domains characterised by spontaneous
deformation + ε0 , − ε0 , and the plane of the nondisplaced domain wall
coincides with the zy coordinate plane.
Let us assume that the centre of dilatation is located at the point
with coordinates r = (x d , 0,0). Let us find the distribution of the
stresses in the system. As in the previous section on the basis of
the Fourier expansion (3.6) here
107
Domain Structure in Ferroelectrics and Related Materials
whence
⎛ 2 ⎞
⎜λ + μ ⎟ 4με0 k x k yU k
σ kx = ⎝
3 ⎠
⋅ Ω0 ⋅ exp(−ik x xd ) + . (4.5)
( 2β − 1) ( 2β − 1) ( k x2 + k 2 )
The Fourier image σ 12 is determined here by the equation
σ ′′ − k 2σ + β ik σ ′ + 2ik f = 2με k 2δ ( x )U ,
12 12 y (4.6)
y 1 0 z k
2με0U k ⎡ 2β k x2 k y2 ⎤
=− ⋅ ⎢ k 2
+ . ⎥+
( kx2 + k 2 ) ⎢⎣ z ( 2β − 1) ( kx2 + k 2 ) ⎥⎦
(4.7)
⎛ 2 ⎞
⎜ λ + μ ⎟ Ω0 k x k y ⎡ β ⎤
+⎝
3 ⎠
⋅ exp(ik x xd ) ⋅ ⎢ 2 − ⎥.
( kx + k )
2 2
⎣⎢ ( 2β − 1) ⎦⎥
Uk =
( 3λ + 2μ ) ⋅ Ω0ik y k ⋅ exp(−k xd ) , ω = 2 ( λ + μ ) .
3 ( λ + 2μ ) ε0 ⎡⎣ k z2 + ω k y2 ⎤⎦ ( λ + 2μ ) (4.9)
108
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
1
U0 = − Ω0 ⋅ σ kkind . (4.10)
3
Substituting (4.8) into (4.5) and then into (4.10) shows that the
energy of interaction decreases with increasing distance x d in
proportion to 1 xd3 . At the same time the maximum energy of
interaction, represented by its value at x d = δ is equal to
1 ( 3λ + 2 μ )
2
1
U0 = ⋅ μΩ02 ⋅ 3 . (4.11)
18π ( λ + 2μ )
2 2
δ
(in calculations of U 0 it was assumed that ω1 in order to simplify
calculations).
Using in calculations the values of μ~10 10 CGSE units,
d ~ 10 –7 cm, where a 3 is the atomic volume and the value of a is
equal to approximately half the size of the elementary cell, whose
typical value is 10 –7 cm, we obtain U 0 ~0.02 eV.
The distribution of displacements of the boundary, interacting
with the defect at ω 1, is described by the function (Fig.4.5)
1 ( 3λ + 2 μ ) Ω0 y
U ( z, y ) = − ⋅ ⋅ .
3π ( λ + 2 μ ) ε0 ( y 2 + z 2 + x 2 )3 / 2 (4.12)
d
109
Domain Structure in Ferroelectrics and Related Materials
110
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
σ ij ,kk +
m
m +1
(σ kk ,ij − σ kk ,ll ⋅ δ ij ) = 2μηij ,
(5.1)
σ 12 x =0 = 0,
and the components of the tensor of incompatibility are determined
not only by the density of the twinning dislocations (3.4)
∂U
α 22 = 2ε0 ⋅ δ ( x),
∂z
∂U (5.2)
α 32 = −2ε0 ⋅ δ ( x),
∂y
formed at the boundary bending, but also by the densities of the
111
Domain Structure in Ferroelectrics and Related Materials
112
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
113
Domain Structure in Ferroelectrics and Related Materials
β ⋅ k x2 4 με0 β ⋅ k x3 k y
σ 22k =
x
⋅ σ kx = ⋅U k =
(k 2
x + k y2 ) ( 2β − 1) ( k x2 + k y2 )
2
−2 μ ik x3b
⋅ (1 − xd k y ) ⋅ exp(− xd k y ).
(5.15)
=
( 2β − 1) ( k x2 + k 2 2
y )
Fig.4.7. Displacement of the domain wall interacting with the edge dislocation. Thin
line shows the displacement of the wall at x d =0 and also the image dislocation.
114
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
115
Domain Structure in Ferroelectrics and Related Materials
Hence,
μ bik z
σ12 ( x = 0 ) = ⋅ exp(−k z xd ). (5.22)
2k z
The Fourier image of the stresses, induced by the boundary bending
taking into account the homogeneity of all values along the
dislocation axis, i.e., the y axis from (3.16) is equal to
σ12ind ( x = 0 ) = − μ ε0U k k z sign k z . (5.23)
Equating of sums (5.22) and (5.23) to zero yields the equation for
determining the Fourier image of displacement of the boundary,
whence
i ⋅b
Uk = ⋅ exp(−k z .xd ). (5.24)
2ε0 k z
116
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
117
Domain Structure in Ferroelectrics and Related Materials
1 ∂α110 b
η12 = = ⋅ δ ( y ) ⋅ δ ′( z ) ,
2 ∂z 2
1 ∂α11
0
b (6.2)
η13 = = − ⋅ δ ′ ( y ) ⋅ δ ( z ).
2 ∂y 2
Taking into account equation (6.2), from the Beltrami equation,
the Fourier image of the part of the shear stress σ 12 generated by
the initial dislocation, is
μ bik z
σ12 = .
(k 2
y + k z2 ) (6.3)
b λ + 2μ ⎛ λ z ⎞
U ( y, z ) = ⋅ ⋅ arctg ⎜ ⋅ ⎟. (6.6)
2πε0 λ ⎜ λ + 2μ z + y2
2 ⎟
⎝ ⎠
As can be seen from the symmetry of the problem for an infinite
dislocation in this case there is no specific position with respect to
the boundary, and consequently the Peach–Koehler force in this
case is equal to zero. Calculation of this force for the dislocation
of finite dimensions is difficult and, therefore, to determine the
energy of interaction of the boundary with the dislocation
perpendicular to it with at least one of the dimensions - either of
boundary or of dislocation – being finite, one should use a different
procedure. The value of this dimension can be conveniently found
118
4. Interaction of Domain Boundaries with Crystalline Lattice Defects
4 μ bik x k y k z ⋅ k ⋅ ω
σ11 = ,
(k + k 2 ) ⋅ ( k z2 + ω k y2 )
2 2
x
4 μ bik x k y k z ( k z2 + k x2 ) ⋅ ω
σ 22 = ,
k ( k x2 + k 2 ) ⋅ ( k z2 + ω k y2 )
2
4 μ bik x k y k z ⎡ ( k y2 + k x2 ) ⋅ ω ⎤
σ 33 = ⋅⎢ − 1⎥ ,
k ( k z2 + ω k y2 ) ⎢⎣ ( k x2 + k 2 ) ⎥⎦
2μ bik z ⎡ 2k y2 k x2 ⋅ ω ⎤ μ bik zδ ( k x )
σ12 = ⋅ ⎢ − k 2
− ⎥+ ,
k ( k x2 + k 2 )( k z2 + ω k y2 ) ⎢⎣ ( k x2 + k 2 ) ⎥⎦ ( k x2 + k 2 )
z
(6.7)
2 μ bik y k z2 ⎡ 2k x2 ⋅ ω ⎤ μ bik zδ ( k x )
σ13 = ⋅ ⎢1 − ⎥+ ,
k ( k x2 + k 2 )( k z2 + ω k y2 ) ⎢⎣ ( k x2 + k 2 ) ⎥⎦ ( k x2 + k 2 )
λ ⎤ dkdk x (6.8)
− ⋅ σ pp
2
⎥ .
( 3λ + 2μ ) ⎥⎦ ( 2π )
3
119
Domain Structure in Ferroelectrics and Related Materials
2 μ bik y k z2 ⎡ 2k y2 ⋅ ω ⎤
σ 23 = ⋅ ⎢1 − ⎥.
k ( k x2 + k 2 )( k z2 + ω k y2 ) ⎢⎣ ( k x2 + k 2 ) ⎥⎦
The first term here is the intrinsic elastic energy of the screw
dislocation, and the second one describes the decrease of the elastic
energy of the system as a result of the domain wall bending in the
elastic field of the dislocation, L is the dislocation length.
Taking into account that L1 b 4ε0 , and dividing the second term
in (6.9) by the dislocation length, we obtain in this case the
following equation for the mean linear density of the energy of
interaction of the boundary with the dislocation
L1 μb2
U0τ = ⋅ .
L 2π ⋅ ⎡1 + λ / ( λ + 2 μ ) ⎤ (6.10)
⎣ ⎦
For the edge dislocation, intersecting the domain boundary along
the perpendicular with Burger’s vector b parallel to either y or z
axis, there is no interaction with the boundary.
120
5. Structure of Domain Boundaries in Real Ferroactive Materials
Chapter 5
121
Domain Structure in Ferroelectrics and Related Materials
122
5. Structure of Domain Boundaries in Real Ferroactive Materials
∂Φ
= −α z Pz + β Pz3 − 4π sinψ ( Px cosψ − Pz sinψ ) = 0. (1.6)
∂Pz
Hence
4π sinψ cosψ
Px = Pz , (1.7)
α x + 4π cos 2 ψ
α z α ⋅ 4π sin 2 ψ
Pz20 = , α z = α z − x . (1.8)
β α x + 4π cos 2 ψ
For the case of α x , α z , ψ << 1 that is important in practice
Px = Pz ⋅ sinψ , α z = α z − α x sin 2 ψ . (1.9)
The formulas (1.8) and (1.9) show that at a relatively large
angle of inclination of the boundary ψ 0 > α z α x the paraphase
becomes advantageous from the thermodynamic viewpoint. Thus,
one of the possible channels of
decrease of the energy of the
inclined domain boundary is the
volume channel associated with
the instability of polarization of
the bulk of the crystal in the
field of the bound charge,
generated by the inclined domain
boundary.
It should be mentioned that
this process is restricted in the
best case by the layer with the
thickness of 2d in the vicinity of
the inclined domain wall, where d
is the width of the domains into
the system of which the material Fig.5.2. Formation of the substructure
is divided (Fig. 5.2) in order to of the domains in the vicinity of the inclined
reduce the energy of the depol- domain boundary.
123
Domain Structure in Ferroelectrics and Related Materials
124
5. Structure of Domain Boundaries in Real Ferroactive Materials
⎧ E 2 + E z2 ⎫ dz ( x )
Φ = ∫ ⎨Φ 0 ( P ) + x ⎬ dx dz + γ ∫ 1 + dx. (1.14)
⎩ 8π ⎭ dx
Here the first term describes the volume energy of the
ferroelectric and the second term the surface energy of the domain
wall whose form is represented by the functions z = z(x). For the
purpose of investigation of the loss of stability of the shape of the
flat boundary we assume that z(x) = x ctg ψ + U(x), where U(x) is
the small displacement of sections of the boundary from the average
position.
Let us expand equation (1.3) for Φ 0 (P) into a series in the
vicinity of P302 = α z β and restrict ourselves to the quadratic term
⎛ dz ( x ) ⎞
2
ε E 2 + ε z Ez2
Φ=∫ x x dx dz + γ ∫ 1 + ⎜ ⎟ dx, (1.16)
8π ⎝ dx ⎠
where in accordance with (1.15)
125
Domain Structure in Ferroelectrics and Related Materials
4π 4π
εx =1+ , εz =1+ .
αx 12πα x ⋅ sin 2 ψ
2α z − (1.17)
α x + 4π cos 2 ψ
Finally, for the analysis the
functional (1.16) should be written
using the boundary coordinate z(x).
This can be conveniently carried out
if we calculate the first term in
(1.16) not as the energy of the field
in the dielectric medium but as the
energy of interaction of bound
charges on the boundary. Taking
into account (Fig.5.4), that the
linear density of the bound charge at
Fig.5.4. Calculations of the linear
density of the charge on the inclined the section of the boundary with the
boundary. length dl is equal to
dx′
= P0 z ⋅ dx ′,
P0nz ⋅ dl = P0 z ⋅ sinψ ⋅
sinψ
(1.18)
the electrostatic potential of these charges in the point with the
coordinates (x,z) taking into account the anisotropy of the dielectric
properties of the ferroelectric can be written in the form:
⎡ ( x − x′ )2 ( z − z ( x′ ) )2 ⎤
2 P0 z
ε xε z ∫ ⎢ ε x
ϕ=− ⋅ ln ⎢ + ⎥ dx′.
εz ⎥ (1.19)
⎣ ⎦
Then, taking into account (1.19), equation for the functional
(1.16) can be presented in the form
2 P0 z ⎡ ( x − x′ ) 2 ( z ( x ) − z ( x′ ) ) 2 ⎤
⋅ ln ⎢ ⎥ dx dx′ +
ε xε z ∫ ∫ ⎢ ε x
Φ=− +
εz ⎥
⎣ ⎦
⎛ dz ( x ) ⎞
2
+γ 1 + ⎜ ⎟ dx. (1.20)
⎝ dx ⎠
126
5. Structure of Domain Boundaries in Real Ferroactive Materials
d ⎛ δΦ ⎞
p ( x) = ⎜− ⎟=
dx ⎝ δ U ⎠
⎡ U ( x ) − U ( x′ ) ⎤
⎢ c tgψ + ⎥ dx′
4P 2
⎣ x − x′ ⎦
ε xε z ∫
= 0z
⋅ +
⎡ ε ⎛ U ( x ) − U ( x ′ ) ⎞
2
⎤
( x − x′) ⎢ z + ⎜ c tgψ + ⎟ ⎥
⎢⎣ ε x ⎝ x − x′ ⎠ ⎥⎦
(1.21)
⎡ ⎤
⎢ ( c tgψ + U ′ ) ⎥
2
1
+γ ⎢ − 3⎥
⋅ U ′′.
⎢ 1 + ( c tgψ + U ′ ) 2
⎢⎣ ( 1 + (c tgψ + U ′) ) 2 ⎥
⎥⎦
⎛ εz ⎞
⎜ − c tg ψ ⎟
2
ε ⎠ U ( x ) − U ( x′ ) dx′ +
2
⋅ ⎝ x
4P
p ( x) = 0z
2 ∫
ε xε z ⎛ ε z ( x − x′ )
2
⎞
⎜ − c tg ψ ⎟
2
⎝ εx ⎠ (1.22)
3 d U ( x)
2
+γ sinψ .
dx 2
The flat domain wall losses the stability with regard to small
displacements under the condition of vanishing of the pressure,
acting on the wall. Taking into account the low value of the
derivative d 2 U/dx 2 at the moment of the loss of stability, it can be
seen that it takes place under the condition of vanishing of the first
term in (1.22), i.e. at
εz
ctg 2ψ c = . (1.23)
εx
Taking into account that, according to (1.17) at low ψ
4π
εz , from (1.23) we obtain the expression for the
2α z − 3α x sin 2 ψ
critical angle of the boundary inclination, at which it losses the
stability of its shape [123–131]:
127
Domain Structure in Ferroelectrics and Related Materials
ψ c2 = α z 2α x . (1.24)
For specific experimental conditions there can be a situation, in
which the inclination angle of the boundary ψ > ψ c. In this case, the
period of the resultant zig-zag structure depends on the extent by
which ψ is greater than ψ c . To determine it, let us go over to
Fourier components of the wall displacement and of the pressure
acting on the wall U = U 0 ⋅ eikx , p = p0 ⋅ eikx . Substitution of these into
(1.22) gives
⎛ εz ⎞
⎜ − c tg ψ ⎟
2
2
ε
⋅ ⎝ x ⎠ ⋅ U ⋅ π k − γ sinψ 3 k 2ξ .
4P
p0 = 0z
0 0
ε xε z ⎛ ε z
2
⎞ (1.25)
⎜ + c tg ψ ⎟
2
⎝ εx ⎠
The competition between the volume and surface energies leads
to the period of the boundary λ* for which the rate of the breaking
of its flat shape is maximum. It is found from the condition
dp 0 /dk = 0 and is represented by the following equation
2
γ ε xε z sinψ ⎛ ε z
3
2π ⎞
λ∗ = ∗
= ⎜ + c tg ψ ⎟ .
2
k ⎛ε ⎞ ε
P02z ⎜ z − c tg 2 ψ ⎟ ⎝ x ⎠ (1.26)
⎝ εx ⎠
For angles close to ψ we obtain
2πγ α z 1
λ∗ = ⋅ .
P0 z 2α x (ψ − ψ c )
2 2 (1.27)
Thus, with the increase of the angle between the plane of the
domain wall and the plane, corresponding to its critical inclination,
the period of the resultant zig-zag structure decreases.
128
5. Structure of Domain Boundaries in Real Ferroactive Materials
∂z x =0 (2.1)
⎩2 ⎭
The crystal will be assumed to be infinite and the polar direction
and the direction of the spontaneous shear coincide as usual with
the z and y directions respectively.
The first term in (2.1) describes the increase of the energy
linked with the increase of the area of the domain wall, the second
and third describe respectively the energy of the depolarizing field
of the bound charges and the elastic energy of the twinning
dislocations, formed at the domain wall bending.
The variation (2.1) δ Φ/ δU results in the equation of equilibrium
of the boundary
∂ϕ
−γ∇ 2U + 2 P0 − 2ε0σ 12 x =0 = 0. (2.2)
∂z x =0
⎧γ
Φ = S ∑ ⎨ U kU − k + 2 P0ik zϕk x=0 ⋅ U − k −
k ⎩2
(2.4)
−2ε0σ12 x=0 ⋅ U − k }.
As shown in the previous chapter (section 4.1 and 4.3,
respectively), the contribution to the Fourier component ϕ k | x=0 and
σ 12 | x=0 , associated with the bending displacement of the boundary,
in the approximation of small displacements of the boundary is
expressed in the linear form by the Fourier component of its
displacement U k :
4πP0ik zU k
ϕk x =0 =− 1/ 2
⎛ε ⎞
ε a ⎜ c k z2 + k y2 ⎟ (2.5)
⎝ εa ⎠
and similarly
129
Domain Structure in Ferroelectrics and Related Materials
με0 2(λ + μ )
σ12 x =0 =− ⋅ U k ( k z2 + ω k y2 ) , ω = . (2.6)
k λ + 2μ
Substituting into (2.4) ϕ k | x=0 in the form of (2.5) and σ 12 | x=0 in
the form of (2.6) taking into account the condition U − k = U k* , that
follows from the reality of displacements U, we have
⎧ ⎫
⎪ ⎪
⎪γ 8π P02 k z2 2 με0 ( k z + ω k y ) ⎪
2 2 2
Φ = S∑⎨ k2 + 1/ 2
− ⎬×
k ⎪2 ⎛ εc 2 2⎞
k ⎪
⎪ ε a ⎜ kz + k y ⎟ ⎪
⎩ ⎝ εa ⎠ ⎭ (2.7)
S
× U k ≡ ∑ U k ⋅ ϕk .
2 2
2 k
U 2 = ∑ Uk ,
2
(2.12)
k
130
5. Structure of Domain Boundaries in Real Ferroactive Materials
131
Domain Structure in Ferroelectrics and Related Materials
132
5. Structure of Domain Boundaries in Real Ferroactive Materials
⎧ ∂ 2ϕ ⎛ ∂ 2ϕ ∂ 2ϕ ⎞ ∂U
⎪ε c 2 + ε a ⎜ 2 + 2 ⎟ = 8π P0δ ( x ) ,
⎪ ∂z ⎝ ∂y ∂x ⎠ ∂z
⎨
⎪−γ ⎛ ∂ U + ∂ U ⎞ + 2 P ∂ϕ
2 2
(3.2)
⎪ ⎜ ⎟ x =0 = W ⋅ δ ( z, y ) .
⎩ ⎝ ∂z ∂y 2 ⎠ ∂z
2 0
4π P0ik zU k ⎧⎪ ⎛ εc 2
1/ 2
⎪⎫
2⎞
ϕk x =0 =− ⋅ exp ⎨ − x ⎜ k z + k y ⎟ ⎬ .
⎝ εa
1/ 2
⎛ε ⎞ ⎪⎩ ⎠ ⎪⎭ (3.4)
ε a ⎜ c k z2 + k y2 ⎟
⎝ εa ⎠
Expressing ϕ k (x=0) from (3.3) in terms U k and equating it to the
pre-exponential multiplier in (3.4), we obtain the equation for
determination of U k from which we find [141]:
133
Domain Structure in Ferroelectrics and Related Materials
1/ 2
⎛ε ⎞
W ⎜ c k z2 + k y2 ⎟
Uk = ⎝ εa ⎠ , k 2 = k z2 + k y2 .
⎡ 1/ 2
⎤
2 ⎛ εc 2⎞ (3.5)
⎢γ k ⎜ k z + k y ⎟ + 8π P02 k z2 ε a ⎥
2
⎢⎣ ⎝ εa ⎠ ⎥⎦
Using the inverse Fourier transformation from (3.5) for the most
typical situation ε c >> ε a we obtain
W λ
U ( y, z ) = ×
γ 2π z
⎧⎡ 1 ⎤ ⎛π ⎞ ⎡1 ⎤ ⎛π ⎞⎫
× ⎨ ⎢ − S ( p ) ⎥ cos ⎜ p 2 ⎟ − ⎢ − C ( p ) ⎥ sin ⎜ p 2 ⎟ ⎬ ,
⎩⎣ 2 ⎦ ⎝ 2 ⎠ ⎣2 ⎦ ⎝ 2 ⎠⎭ (3.6)
y γ ε cε a
p= , λ= .
2πλ z 8π P02
Here C(p), S(p) are Frenel integrals. When deriving (3.6) it was
considered that for all k z ≤1/ δ that are active in bending (it should
be remembered that in accordance with the notations used δ is the
thickness of the domain boundary), taking into account the link
λ = γ ε cε a 8π P02 = ε a / ε c δ , the condition k z λ << 1 is satisfied.
For further considerations, the ratio in the braces in (3.6) can
be conveniently presented in the form of a polynomial [98], then the
coordinate dependence of the boundary displacement (3.6) is written
in the form
U ( y, z ) =
W ε cε a
⋅
( 2πλ z + ay ) ,
8π P 0
2
( 4πλ z + by 2πλ z + cy 2 ) (3.7)
134
5. Structure of Domain Boundaries in Real Ferroactive Materials
U=
W
2πγ
(
ln l π ρ . ) (3.8)
135
Domain Structure in Ferroelectrics and Related Materials
136
5. Structure of Domain Boundaries in Real Ferroactive Materials
Wτ ε cε a l
U ( z) = ⋅ ln , (3.16)
8π P 0
2
2z
1
From the conditions ⋅ U maxWτ = U0τ , where U 0τ is the energy of
2
interaction with the unit of length of the defect, taking into account
(3.16), the linear density of the detachment force is
4π P0 U0τ
WτOTP = . (3.17)
( ε cε a )
1/ 4
ln l 2a
On the basis of the ratio l ⋅ U = ns−1 , where n s is the surface
density of linear defects and U = U max Wτ , ( )
P0
lef . (3.18)
ns ( ε cε a )
1/ 4
U0τ
and consequently
Wτ ε cε a ( ε cε a ) U0τ
1/ 4
lef = 2U (Wτ ) =
= .
4π 2 P02
(
ln P0 ns a ( ε cε a )
1/ 4
U0τ (3.19)
137
Domain Structure in Ferroelectrics and Related Materials
with the plane of the domain boundary. In this case, the pinning of
the boundary by defects is more similar to the case of point pinning
and it appears that equations (3.13), (3.14) are more suitable for
determining l ef .
Numerical estimates of the value l ef at P 0 ~10 4 , U 0 ~1 eV,
a~10 –7 cm, U 0τ~U 0 /a~10 –5 (a is the size of the elementary cell),
γ~1 erg/cm 2 give the following results. For a point defect with
−7
n~10 18 cm –3 , lef 4 ⋅ 10 cm in the absence of compensation of long-
range forces and l ef ~10 –6 cm in the presence of such a
compensation. For linear defects with n s ~10 8 cm –2 in the case when
their axes are perpendicilar to the vector of spontaneous
⊥ −6
polarization, lef 10 cm, otherwise at the same value n s ,
lef ≤ 10 −4 cm. These estimates show that for all types of defects
at their real concentration, the value of l ef is greater or considerably
greater than δ . This allows us to assume that the observation in
experiments of wide domain walls with the thickness considerably
greater than δ can be attributed to the interaction of domain
boundaries with crystal defects.
This is also proved by the fact that temperature dependence of
l ef differs in comparison with the prediction of the standard
thermodynamic theory (equation (1.21) in chapter 2). The tem-
perature dependence of l ef is closer, for example, to the
experimental results, obtained by measurements of the thickness of
the domain wall in triglycine sulphate crystal [134] where the
decrease of the domain wall thickness at T→T c (Fig 5.7) is
observed instead of its increase.
Fig. 5.7. Qualitatively different temperature behaviour (a) of the effective and
(b) local thickness of the domain wall.
138
5. Structure of Domain Boundaries in Real Ferroactive Materials
⎧ ⎛ ∂ 2U ∂ 2U ⎞
⎪−γ ⎜ 2 + 2 ⎟ − 2ε0σ 12 x = 0 = W δ ( z , y ) ,
⎪ ⎝ ∂z ∂y ⎠
⎨
⎪ (4.1)
⎪
⎩
σ ij ,kk +
m
m +1
(σ kk ,ij − σ kk ,llδ ij ) = 2μηij .
As previously, the considered material is assumed to be isotropic
in respect of elasticity.
The connection of the components of the tensor of elastic
stresses with the displacement of the domain wall determined by
the equation of incompatibility of the strain in (4.1) by the
dependence of tensor η ij = η ij (U) on the wall displacement naturally
turns out to be the same as in the previous problems (section 4.3,
5.2) that dealt with the bending displacement of the walls in elastics.
In particular, the Fourier image
με0
σ12 ( x = 0 ) = − U k ( k z2 + ω k y2 ) ,
k (4.2)
ω = 2 ( λ + μ ) ( λ + 2μ ) .
The equation of the boundary equilibrium (4.1) in the Fourier space
139
Domain Structure in Ferroelectrics and Related Materials
W z2 + y2
U ( z, y ) = ⋅ , (4.6)
4πμε02 ⎡⎣ z 2ω + y 2 ⎤⎦
i.e. it possesses the characteristic law of decrease ~1/ ρ .
Comparison of displacement (4.6) with displacement of the
boundary (4.3), determined only by surface tension, shows that the
latter in fact determines displacement of the boundary only at
(
ρ < γ 2με02 ⋅ ln l )
π a , i.e. almost beyond the limits of applicability
of consideration of ρ > a carried out here.
To determine the effective width of the boundary l ef let us first
of all find the value of W at which the boundary detaches itself from
defects. On the basis of the previously mentioned condition
1
⋅ U max ⋅ W = U0 ( U 0 is the energy of interaction of the boundary
2
with the defect) and of equation (4.6) we have
W = 2ε0 2πμ U0 a . (4.7)
The average displacement of the boundary is
W
U= . (4.8)
2 π με02 ⋅ l
140
5. Structure of Domain Boundaries in Real Ferroactive Materials
1/ 4
1 ⎛ 8πμε02 a ⎞
l= ⋅⎜ ⎟ (4.9)
2 n ⎝ U0 ⎠
Then, taking into account (4.7)–(4.9), the effective width of the
domain wall lef = 2U (W ) turns out to be the following:
3/ 4
2 ⎛ U0 ⎞
⋅ ⎜ 2 ⎟ ⋅ n1/ 2 ( 8π a ) ⋅
1/ 4
lef = (4.10)
π ⎝ με0 ⎠
In conclusion of the consideration of the deformed profile of the
domain wall in crystals with defects, it is important to note the
following. As it follows from the linearity of the equations used in
this case, the magnitude of the maximum displacement of the wall
in the region of bending increases linearly with the increase of the
force acting on the wall. At the same time, the bending itself being
controlled by the long-range electrical or elastic fields both in the
case of the ferroelectric and the ferroelastic is extremely localized
in the vicinity of pinning of the bent wall (Fig.5.8). Consequently,
if the displacement of the domain wall counted from the location
of the defect is discussed (which is natural, for example, in the
problem of displacement of a pinned domain wall in the external
field), then for not so high concentration of the defects the average
displacement of the wall coincides almost completely with its
maximum value. This means that the quantity U is also proportional
to W. Introducing the proportionality coefficient between U = U max
and W from the condition W= ϑ U max on the basis of expressions
(4.3) and (4.6) we obtain the effective coefficients of the quasi-
elastic force, acting on the boundary displaced with regard to the
defect, which is pinning it. For a 'pure' ferroelectric
Fig.5.8. Localization of the region of bending in the vicinity of pinning the domain
wall in the case of (a) ferroelectric and (b) ferroelastic. The closed line shows the
lines of the equal displacements of the domain wall.
141
Domain Structure in Ferroelectrics and Related Materials
4 2π P0 γ a
ϑ= . (4.11)
ε
For a 'pure' ferroelastic
ϑ = 4πμε02 a. (4.12)
At that the domain wall being displaced now is regarded already
as a flat one that evidently greatly simplifies further consideration.
In the case of the ferroelectric–ferroelastic, the Fourier image
of the boundary displacement is obtained by adding the term
2πμε02 ( k z2 + ω k y2 ) k to the denominator of the expression for U k
(3.5) of the ‘ pure’ ferroelectric. As the result, the structure of
displacement of the wall turns out to be qualitatively similar to the
case of ‘ pure’ ferroelastic (4.6), i.e. U~1/ ρ , and the effective
coefficient of the quasi-elastic force is:
4 πμ ε0 P0 a
ϑ= . (4.13)
ε
142
6. Mobility of Domain Boundaries in Crystals
Chapter 6
143
Domain Structure in Ferroelectrics and Related Materials
∞
Φ= ∫ ( Φ + T ) dx =
−∞
⎧ μ ⎛ ∂P ⎞ 2 ⎛ ∂P ⎞ 2 α 2 β 4 ⎪
⎪ ⎫ (1.2)
= ∫⎨ ⎜ ⎟ + ⎜ ⎟ − P + P ⎬ dx.
⎩⎪ 2 ⎝ ∂t ⎠ 2 ⎝ ∂x ⎠ 2 4 ⎪ ⎭
On the basis of (1.2), the equation of motion for polarization in
the absence of dissipation and the external effects can be written
in the form of
∂2 P ∂2P
μ − − α P + β P 3 = 0. (1.3)
∂t 2
∂x 2
∂2 P
= −α P + β P 3 ,
∂x '2
= − μ v 2 = (1 − v 2 / c02 ) ,
(1.4)
c =
2
.
μ
0
viscosity and of the external field the domain wall can freely move
with a permanent velocity, which assumes in magnitude arbitrary
144
6. Mobility of Domain Boundaries in Crystals
145
Domain Structure in Ferroelectrics and Related Materials
−α P + β P 3 = E. (1.11)
At E ≠ 0, these roots P 01 , P 02 , P 03 (see Fig 6.1) no longer have
those ratios of symmetry P 01 = –P 02 = α / β , P 03 =0 which exist in the
crystal in the absence of the external field. In the given case,
P 01 > P 0 , |P 02 | < P 0 and P 03 ≠0. In accordance with the definition of
the domain boundary, in one limit polarization in the boundary should
have the value of P 01 , and in the other limit the value of P 02 , not
equal to the former one in magnitude.
To form the solution of equation (1.10) with the mentioned
asymptotics, it is convenient to write the equation using the
dimensionless variables first p = P / P0 , ξ = 2 ( x − υ t ) / δ 1 − υ 2 / c02 .
Dividing both parts of (1.10) by α P 0 we obtain
1 δ 2 ∂ 2 p Γ ∂p 1 2 ∂2 p
⋅ + − p + p 3
− δ = E ′, (1.12)
2 c02 ∂t 2 α ∂t 2 ∂x 2
where E'=E/ α P 0 . Therefore, taking into account the relationship
∂2 p ∂2 p 2υ 2
= ⋅ ,
∂t 2 ∂ξ 2 δ 2 (1 − υ 2 / c02 )
∂2 p ∂2 p 2 (1.13)
= 2⋅ 2 ,
∂x 2
∂ξ δ (1 − υ 2 / c02 )
146
6. Mobility of Domain Boundaries in Crystals
2 ⋅Γ
υ = − ⋅υ ⋅ (1.15)
α ⋅ δ 1 − υ 2 c02
To write the solution of equation (1.14) we use the solution of
this equation at υ , E'=0 (ratio (1.21) in chapter 2) written in the
asymmetrical form
p x exp ( 2 x / δ ) − 1 2
= th = =1− .
p0 δ exp ( 2 x / δ ) + 1 exp ( 2 x / δ ) + 1 (1.16)
p (ξ ) = a +
(b − a ) ,
⎡1 + exp ( b − a ) ξ 2 ⎤⎦ (1.17)
⎣
where a=p 01 /p 0 , b=p 02 /p 0 , c=p 03 /p 0 are the dimensionless roots of
the polynomial
p 3 − p − E ′ = ( p − a )( p − b )( p − c ) . (1.18)
Substitution of (1.17) into (1.14) shows that function (1.17) is the
solution of equation (1.14) at
υ =
( a + b − 2c ) = − 3c .
(1.19)
2 2
The last ratio follows from the condition a+b+c=0, which is
satisfied by the roots of the polynomial (1.18) because of the
absence of the quadratic term in it.
If we know the root, taking into account (1.15) and (1.19), we
obtain an implicit dependence of the velocity of the domain wall υ on
the magnitude of the applied external electric field [146–148]. At low
velocities of the wall, this dependence can be written in the explicit
form. In this case, as it can be seen in Fig.6.2, c=p 03 /p 0 E/ α p 0 ,
147
Domain Structure in Ferroelectrics and Related Materials
The expressions for the domain wall velocity and its mobility
derived above are applicable to relatively wide domain boundaries
formed in the vicinity of T c , for which the influence of the lattice
relief on their motion can be ignored. For conventional domain walls
that are usually narrow the presence of the lattice relief, connected
to the coordinate dependence of their energy, almost completely
prevents their motion as a unit in relatively weak external fields.
In fact, the achievement of the activationless domain wall motion
mode is determined by the condition when the external pressure on
the domain wall from the direction of the electric field E cr exceeds
the pressure from the direction of the Peierls' force ∂ γ /∂U| max ,
where γ (U) is the dependence of the energy of the domain
boundary on its displacement. For the extremely narrow domain
wall with zero thickness the change of the electrostatic energy of
the dipole subsystem of the crystal in the external electric field E,
resulting from the displacement of the domain wall, is equal to
δΦ=2(P0 E)δU, where δU is the displacement of the wall. Hence,
148
6. Mobility of Domain Boundaries in Crystals
the pressure on the wall from the direction of the external field is
δΦ
p= = 2 ( P0 , E ) . (2.1)
δU
Equating the pressure (2.1) to the pressure from the direction of
the Peierls' force
∂γ 2V0
2 ( P0 , Ecr ) = , (2.2)
∂U max a
where V 0 is the magnitude of the barrier in the lattice relief, a is
the size of the elementary cell, we determine the strength of the
critical field
V0
Ecr . (2.3)
P0 a
Calculations of V 0 for certain ferroelectrics, presented in chapter
3 show that, in particular, even for a crystal with a highly mobile
domain structure – potassium dihydrophosphate, the values of V 0 at
(T c –T) equal to several degrees are equal to the order of several
hundredths of erg· cm–2 . At these values of V 0 and P 0 ~10 4 of CGSE
units, a~10 –7 cm, E cr is of the order of ~1 kV· cm–1 . In crystals with
a less mobile domain structure the value of E cr is expected to be
approximately by an order of magnitude greater.
The mentioned estimates correspond to the results of a large
number of experiments carried out to determine the inverse
switching time of the ferroelectric crystal. As can be seen, in
particular, in Fig.6.3, which shows this dependence for the crystal
of triglycine sulphate, the curve of switching current can be
qualitatively divided into two
sections. In section I, the inverse
switching time and, consequently,
the velocity of the domain
boundaries motion follows by the
exponential law 1/t s=1/t ∞· exp(–δ /E),
υ = υ ∞· exp(–δ /E). In section II, this
dependence follows the linear law:
1/t s=const·E (v=const·E). The speci-
fic value of the critical field E cr ,
separating these sections, for the
crystal of triglycine sulphate is
Fig. 6.3. Dependence of the inverse
switching time on the field for TGS
~20 kV· cm–1 .
crystal [16]. In fields E>E cr, the motion of the
149
Domain Structure in Ferroelectrics and Related Materials
150
6. Mobility of Domain Boundaries in Crystals
151
Domain Structure in Ferroelectrics and Related Materials
a⎡ 2 (z − Z )⎤ 4 P02 a 2
U (z) = ⎢1 − arctg ⎥ , λ = . (2.8)
2⎣ π λ1 ⎦ 1
π V0 ε cε a
Here Z is the coordinate of the middle of the charged lateral
wall of the nucleus, λ 1 is its width. At P 0~10 4 , a~10 –7 cm, ε c ~10 3 ,
ε a ~10, V 0 ~10 –2 ÷10 –1 erg· cm–2 we obtain λ 1 ~10 –6 ÷10 –7 cm, which
justifies the possibility of use of the continual consideration in this
case. The condition λ 1 >>a of the small incline of the nucleus wall
in relation to the nondisplaced boundary, makes it possible to place,
when writing equations (2.5)–(2.8), the bound charge on the
boundary into the plane of the nondisplaced boundary.
The energy of the charged wall of the nucleus consists of the
energy of misalignment of the boundary with the minimum of the
potential relief γ (U) and the electrostatic energy of the bound
charges in it.
The linear density of misalignment energy is
∞ ∞ ∞
d γ dU
Wυ = ∫ (γ (U ) − γ ) dz = − ∫ zd γ (U ) = − ∫ z dU dz dz.
−∞
0
−∞ −∞
(2.9)
152
6. Mobility of Domain Boundaries in Crystals
4 P02 a 2
Wq = ln ( λ1 a ) . (2.14)
ε cε a
The structure of the uncharged wall of the nucleus is determined
by the equation similar to (2.7), in which the left hand part is
substituted by the Laplace pressure:
d 2U π V0 ⎛ 2π U ( y ) ⎞
γ0 2
= sin ⎜ ⎟. (2.15)
dy a ⎝ a ⎠
Integration of the equation, using the previous boundary conditions
U(∞)=0, U(–∞)=a (the period of the function γ(U) in the direction
y is assumed also to be equal to a to simplify consideration) gives
[100]:
2
γ 0 ⎛ dU ⎞
⎜ ⎟ = γ (U ) − γ 0 (2.16)
2 ⎝ dy ⎠
and the equation for determination of the coordinate dependence
U(y):
γ0 dU
2∫
= y. (2.17)
V0 sin (π U a )
Hence, the distribution of the displacements in the uncharged wall
of the nucleus is
a⎡ 2 ⎤ γ0
U ( y) = ⎢1 − arctg ( exp ( −π x λ2 ) ) ⎥ , λ2 = a . (2.18)
2⎣ π ⎦ 2V0
The linear density of the energy of the uncharged wall of the
nucleus, linked with the increase of the total surface of the domain
wall transient into the adjacent x=const plane
∞ 2
γ0
⎛ dU ⎞
Wγ = ∫ ⎜ ⎟ dy = ∫ γ (U ) dy = Wυ (2.19)
2 −∞ ⎝ dy ⎠
is equal to the linear density of the misalignment energy. Therefore,
the total density of the energy of the uncharged wall of the nucleus
is
∞
2a
Wγ + Wυ = 2 ∫ ( γ (U ) − γ 0 ) dy = 2γ 0V0 . (2.20)
−∞ π
The width of the uncharged wall at V 0 ~0,1 γ 0 , γ 0 ~1 erg/cm 2 is
~2.5a. The energy of the charged wall, related to the unit of its
153
Domain Structure in Ferroelectrics and Related Materials
2 γ P0 a
υ2 = E. (2.22)
ηV0
Then
4 2 P02 a 2
υ1 υ 2 = λ1 λ2 = = γ1 γ 2 . (2.23)
π ε cε a γ V0
As shown below, the ratio of the dimensions of the critical
nucleus is zmax ymax = 2 γ 1 γ 2 . Since, in the ratio of the velocities
there is a higher degree of the ratio γ 1 / γ 2 , then in the ratio of the
dimensions of the nucleus and, as a rule γ 1 / γ 2 >>1, then in the
154
6. Mobility of Domain Boundaries in Crystals
m 2V0
m2∗ = , m = ρ a3 . (2.27)
a γ0
6.3. Velocity of the lateral motion of a domain wall of a
ferroelectric under the conditions of thermofluctuation
formation and growth of nuclei of inverse domains
∫ γ (ϕ ) dl − 2P0 Ea ∫ dS . (3.1)
∏=
Here γ ( ϕ ) is the linear density of
the energy of the lateral wall of a flat
nucleus as a function of its orientation,
the angle ϕ is determined by the ratio
tg ϕ = y', where y=y(z) is the coord-
inate dependence of the curve describ-
155
Domain Structure in Ferroelectrics and Related Materials
156
6. Mobility of Domain Boundaries in Crystals
hence
1⎛ dγ ⎞
y= ⎜ γ cos ϕ − sin ϕ ⎟ . (3.8)
L⎝ dϕ ⎠
Substituting in (3.6) and (3.8) γ ≡ γ ( ϕ ) from (3.2), we obtain
the equation for the boundary of nucleus in parametric form
⎧ 1
⎪ z = − 2 P a ⎣⎡γ 1 sin ϕ + γ 2 sin ϕ + 2γ 1 sin ϕ cos ϕ ⎦⎤
3 2
⎪ 0
⎨
⎪ y = 1 ⎡ −γ sin 2 ϕ cos ϕ + γ cos ϕ ⎤ . (3.9)
⎪ 2 P0 a ⎣ ⎦
1 2
⎩
Analysis of the relations (3.9) shows that depending on the ratio
between γ 2 and γ 1 , the form of the critical nucleus can change
qualitatively. In order to illustrate this, let us consider a section of
the wall of the nucleus, resting on a unit base perpendicular with
regard to the polar axis and forming angle ϕ with it. The density
of its energy is
γ (ϕ )
∏= . (3.10)
sin ϕ
The minimality condition Π Π ( )
= 0 has the form
γ sin ϕ0 = γ cos ϕ0 . (3.11)
Substitution of γ in the form of (3.2) into (3.11) makes it
possible to find the optimum orientation of the considered wall of
the nucleus from the ratio
sin 2 ϕ0 = γ 2 γ 1 . (3.12)
Equation (3.12) shows that an oval nucleus is stable only at
γ 2 ≥ γ 1 , at γ 1 > γ 2 the oval form becomes unstable and the nucleus
becomes lenticular with the angle ϕ 0 between the surfaces forming
it in the area of their intersection (Fig. 6.6).
Condition (3.11) exactly corresponds to the conversion of
coordinate y to zero. Taking this into account, from the expression
for z (3.9) we obtain zmax = γ 1γ 2 P0 Ea . The maximum value of y
is equal to ymax = γ 2 2 P0 Ea . Thus, their ratio zmax ymax = 2 γ 1 / γ 2 is
determined by the ratio of the linear densities of the energy of the
charged and uncharged walls of the nucleus and in accordance with
the actual relation between γ1 and γ 2 indicates the elongation of the
critical nucleus along the polar axis (Fig. 6.6).
To obtain the energy of the critical nucleus, let us write the
157
Domain Structure in Ferroelectrics and Related Materials
Fig.6.6. The form of a critical nucleus in the plane of the domain wall.
2ϕ 0 is the angle between the forming surfaces in the area of the sharp
tip of the lense.
Substituting here γ and γ , on the basis of (3.2) for the arbitrary
ratio between γ 1 and γ 2 we obtain
1 γ 2 ⎛ γ 2 ⎞ ⎧ γ 12 7 ⎫
∏∗ = ⎜ 1 − ⎟ ⎨ + γ 2γ 1 ⎬ +
P0 Ea γ 1 ⎝ γ 1 ⎠ ⎩ 8 4 ⎭
1 γ ⎧ γ2 ⎫ (3.16)
+ arcsin 2 ⎨ − 1 + γ 1γ 2 + γ 22 ⎬ .
P0 Ea γ1 ⎩ 8 ⎭
If γ 1 >> γ 2
8
∏∗ γ 1γ 23 , (3.17)
3P0 Ea
in the inverse limiting case γ 1 <<γ 2
π
∏∗ γ 22 , (3.18)
2P0 Ea
158
6. Mobility of Domain Boundaries in Crystals
at the moment of the alteration of the nucleus form from the oval
to lenticular, when 1– γ 2 / γ 1 <<1
π
∏∗ ⎡γ 1γ 2 + γ 22 ⎤⎦ .
2P0 Ea ⎣ (3.19)
159
Domain Structure in Ferroelectrics and Related Materials
8 γ 1γ 23
δ= (3.24)
3P0 aT
in the case of weak and strong fields and field δ /3 in the case of
intermediate fields. Substitution of the specific values γ1 and γ2 into
(2.24) shows that value δ decreases at T → T c. To obtain the law
of decrease of δ it is necessary to known the temperature
dependence of barrier V 0 . If the temperature dependence of V 0 is
the same as that of the energy of the domain wall γ, then δ
proves to be ~ΔT 3/2 .
Let us give another result obtained from the analysis of the
dependence υ (E) in various velocity regimes. As shown in
equations (3.20) and (3.23) for υ in the case of weak and strong
fields, and the expression for s * , the pre-exponential multiplier in
the dependence of υ (E) is proportional to E 2 in the case of weak
fields and independent of the field in the case of strong fields. The
mentioned alteration of the pre-exponential multiplier in the
expression υ(E) in the assumption that it doesn't depend on the field
can be interpreted also as some increase of activation field δ with
the increase of the strength of the applied field E that was
experimentally observed. It should be noted that the result is
obtained here while considering the nuclei with the thickness of the
constant of the elementary cell and, consequently, does not require
taking into account the multilayer nuclei, which was proposed in
[153,154].
The consideration above was based on the approximation of an
ideal defect-free material. In real crystals, as shown in experimental
observation [158–160], in addition to the Peierls relief the influence
of crystalline lattice defects has to be considered as well [161–163].
160
6. Mobility of Domain Boundaries in Crystals
⎛δ ⎞ ⎛ π 2δ ⎞
3
V0 = 8π γ ⎜ ⎟ exp ⎜ −
4
⎟. (4.2)
⎝a⎠ ⎝ a ⎠
Substituting (4.2) into (4.1) we obtain the following for the
velocity of the lateral motion of the domain wall:
⎧
⎪ ⎧ 3π 2 δ ⎫ ⎪⎫
υ = υ∞ exp ⎨ −W exp ⎨− ⎬⎬ ,
⎪
⎩ ⎩ 4 a ⎭ ⎪⎭
9/4
⎛δ ⎞ (4.3)
W = 2 (γ a 2 ) (ε cε a ) T −1 ( 8π 4γ )
3/ 4 −1/ 4 3/ 4
⎜ ⎟ E −1 .
⎝a⎠
The ratios (4.3) show that the velocity of the domain wall in the
given regime very strongly depends on its width (the functional
dependence is exponent in exponent). Therefore, even a relatively
small increase of the width of the wall δ as a result of the
tunnelling effect should result in a considerable increase of the
velocity of the wall v.
Evidently, this increase can be one of the reasons for the
increase of the mobility of the domain walls by six orders of
magnitude at once observed in the experiments [78] when replacing
deuterium by hydrogen in the structure of KD 2 PO 4 . In fact, at the
above values of the constants (J+2A) D213 K, (J+2A) H 140 K,
A D 20 K, Ω D 0 K, Ω H 86 K, a~10 –7 cm at ΔT~20 K, where
γ D 4.2· 10–2 erg.cm –2 , γ H 3.5· 10–2 erg.cm –2 and (δ a ) D 1, 4,
(δ a )H 2, ε cH ∼ 103 , ε cD ∼ 102 , ε aH ∼ 10, ε aD ∼ 5 at E~1 V· cm–1 the
value of the constant W for the deuterated and undeuterated
crystals is equal to respectively W D 5· 105 , W H 7.7· 105 .
Assuming hereinafter that the pre-exponential multiplier in (4.3) does
not change at substitution H → D , taking into account the obtained
values of W H and W D for the ratio of the velocity of the domain
boundaries in the deuterated and undeuterated crystals equidistant from
T c by the value ΔT~20 K, we have υ D/ υ H~10 –5 ÷10 –6.
In the regime of viscose motion of the domain walls with the
linear dependence of their velocity on the strength of the external
field
3 δ
υ= E (4.4)
2 P0 Γ
the value of the latter velocity as seen from (4.4), is inversely
proportional to the curvature of distribution of the order parameter
161
Domain Structure in Ferroelectrics and Related Materials
E A ⋅ Tc ⋅ a
υ= .
P0 ( T − Tc ) ⎡⎣1 − Ω2 / Tc2 ⎤⎦ Γ (4.5)
162
6. Mobility of Domain Boundaries in Crystals
Fig.6.7. Temperature dependences of ε (1,3) and tg δ (2,4) for crystals of RbH 2 AsO 4
(1,2) and KH 2 PO 4 (3,4). E ~ =1 V· cm–1 , f=1 kHz.
163
Domain Structure in Ferroelectrics and Related Materials
164
6. Mobility of Domain Boundaries in Crystals
165
Domain Structure in Ferroelectrics and Related Materials
Fig.6.10. Dependence of the shift of the 'freezing' temperature of the domain structure
(T f =T max tgδ) for a RbH 2 AsO 4 crystal on amplitude E ~ of the measuring field.
166
6. Mobility of Domain Boundaries in Crystals
167
Domain Structure in Ferroelectrics and Related Materials
τ=
U
υ
U
υ∞
(
⋅ exp δ E~ ) (5.3)
On the basis of the ratio (5.6) and the expression for δ (3.1),
the value of the lattice barrier, determining the location of the
maximum losses, is
V0 max = V03 / 4 ( T = T f ) =
3/ 4
(ε aε c )
3/ 2
⎛ 8π P0υ∞ ⎞
1/ 4
Tf ⎛ π ⎞ (5.7)
= ⎜⎜ ⎟⎟ ln ⎜ ⎟ E~ = A ⋅ E~ .
⎜ ω εε dE ⎟
16 ⎝ 2a 2γ ⎠ ⎝ 0 ~ ⎠
168
6. Mobility of Domain Boundaries in Crystals
169
Domain Structure in Ferroelectrics and Related Materials
Chapter 7
170
7. Natural and Forced Dynamics of Boundaries in Crystals
1 ⎛ ∂u j ∂uk ⎞
u jk = ⎜ + ⎟,
2 ⎜⎝ ∂xk ∂x j ⎟⎠ (1.5)
and the link to the strength of the electric field with potential
E i = – ∂ϕ / ∂ x i, we rewrite the set of equations (1.1)–(1.2) in the form
171
Domain Structure in Ferroelectrics and Related Materials
⎧ ∂ 2 uk ∂ 2ϕ
⎪ ρ
u = c − β ,
∂xl ∂x j ∂xk ∂x j
i ijkl ijk
⎪
⎪ ∂ 2u j
⎪ ∂ 2ϕ ∂U
−ε
⎨ ij − 4π β = 8πP0δ ( x ) ,
∂xi ∂x j ∂xk ∂xi ∂z
ijk
⎪
⎪ ∂ϕ (1.6)
⎪ mU = 2 P0 .
⎪
⎩ ∂z x = 0
the longitudinal sound wave. Taking this into account, instead of (1.8)
we obtain
ρ ui = −
( λ + μ ) ∂2 Λl ∂xl ∂xi
+
∂ 2 ui
− Λi .
ρ ( cl2 k 2 − ω 2 ) ∂xk2 (1.10)
172
7. Natural and Forced Dynamics of Boundaries in Crystals
1 ⎡
⎢ β
( cl2 − ct2 )
β
⎤
⎥ kx
u = − klj k j l i ϕ ,
kx
k k k k k k
i
(
ρ ct2 k 2 − ω 2 ⎢⎣ )
kij k j
cl2 k 2 − ω 2 ( ⎥
⎦ )
(1.12)
ct = μ ρ .
Similarly, from the electrostatic equation we have
ε ij ki k jϕ k + 4πβ ijk kk ki u kj = 8π P0ik zU .
x x
(1.13)
k
Substituting expression (1.12) into (1.13), instead of u j z , we find
the expression for ϕ kx from which
∂ϕ
− =
∂z x =0
dk x
∞ −8π P0 k z2U
2π
= ∫⎧ .
−∞
⎪ 4πβimk kk ki ⎡
⎢ β pmj k p k j −
( cl2 − ct2 ) ⎤ ⎪⎫ (1.14)
⋅ β plj k p k j kl km ⎥ ⎬
⎨ε ij ki k j +
⎪⎩ (
ρ ct2 k 2 − ω 2 ) ⎢
⎣ (
cl2 k 2 − ω 2 ) ⎥
⎦ ⎪⎭
173
Domain Structure in Ferroelectrics and Related Materials
∞
⎛ ∂ϕ ⎞ dk x
−⎜ ⎟
⎝ ∂z ⎠
= ∫ 2π
( −8π P0 k z2U ) ⎡⎣ε c k z2 + ε a ( k x2 + k y2 ) +
x =0 −∞
+
4π k
2
z {β k 2
3
2
z ⎣
2
⎦ }
+ ⎡( β 2 + β1 ) + 2 β1 β 3 ⎤ ( k y2 + k x2 ) + β1 ( k y2 + k x2 )
−
ρ 2 2
c k − ω2 ( t ) (1.16)
−1
4π ( c − c ) ⎡⎣ k
2 2 2
⎡ β k + ( β 2 + 2 β1 ) ( k + k ) ⎤ ⎤
2 2 2
l t z⎣ 3 z ⎦⎥ y x
− .
ρ ( 2 2 2 2 2
ct k − ω cl k − ω )( 2
) ⎥
⎥⎦
⎛ ⎞
⎜m + 5πP02 β32 k z6 ⎟ω 2 = 4πP02 k z2
.
⎜ ρε 2 cl4 ( k z2 + k y2 ) ⎟ εc 2
7/2
(1.18)
⎝ ⎠ εa kz + k y2
εa
The expression in the round brackets in front of ω2 can be interpreted
as the renormalised effective mass of the domain wall containing
the non-local term m* ~ 1/k, due to involvement in the motion of
the entire layer of the material with the thickness equal to 1/k as
a result of the piezoelectric effect. This layer surrounds the boundary
and starts to move with the motion of the domain wall.
For comparison of values m* and m let us present the maximum
value m* for the given modulus k for the case of the wave propagating
along the polar direction in the form
4πβ 2 5π P02 1 ⎛ 4πβ 2 ⎞ γ 1 ⎛ 4πβ 2 ⎞ 1 (1.19)
m∗ = ⎜ ⎟ =⎜ ⎟m ,
ρε cl2 4ε cl2 k ⎝ ρε cl2 ⎠ cl2 kδ ⎝ ρε cl2 ⎠ kδ
174
7. Natural and Forced Dynamics of Boundaries in Crystals
k* << k max = 1/ δ , where k max is the limiting value of the wave vector
k determined by the limit of applicability of the approximation of
the geometrical boundary. For the other orientations of k the value
of k* is evidently lower.
The right-hand part of equation (1.18) with the accuracy to
multiplier 1/k 2 represents the effective rigidity relative to bending
displacements of the domain wall. As it can be seen in the adopted
approximation (4 πβ 2 / ρε c 2 ) << 1 the contribution to it as a result of
piezoelectric interactions is negligible and is completely determined
by the electrostatic interaction of the charges on the bent boundary.
Analysing the law of dispersion of domain boundary vibrations
under consideration, from equation (1.18) we can easily see that in
the region of low k, where m* > m, we have ω ~ k and, consequently,
the velocity of propagation of the corresponding waves does not depend
here on the wave vector.
175
Domain Structure in Ferroelectrics and Related Materials
176
7. Natural and Forced Dynamics of Boundaries in Crystals
4) and equation (2.4) in this chapter, tensor jij has the unique component
differing from zero
∂U
j12 = −2ε δ ( x ) · (2.5)
dt
Let every element of the domain boundary make small harmonic
vibrations that propagate along the wall in the form of a wave
(
U = U 0 · exp( ikρ − ω t ) ,k = k y k, z ) ,ρ ( y z, ) (2.6)
Then
σ 12 = σ12 ( x ) ⋅ exp ( ikρ − ωt ) ,
σ kk ≡ σ = σ ( x ) ⋅ exp ( ikρ − ωt ) ,
ϕ = ϕ ( x ) ⋅ exp ( ikρ − ωt ) .
(2.7)
On the basis of the Beltrami equation, the expressions for the components
of the tensor η ij and component j 12 (2.5), expressed by means of
displacement of the wall U, and also by means of representations
(2.6)–(2.7) the equation for component σ 12 is as follows
ρ ∂ 2U
σ 12′′ − k 2σ 12 + β ik yσ ′ − σ12 = 2με0 k z2U + 2 ρε0δ ( x ) 2 . (2.8)
μ ∂t
The system of equations for σ 11 , σ 22 , σ 33 is
ρ ρ (λ + μ )
σ 11′′ − k 2σ 11 + β k 2σ − σ11 + σ = 0,
μ μ ( 3λ + 2μ )
ρ ρ (λ + μ )
′′ − k 2σ 22 + β (σ k z2 − σ ′′ ) −
σ 22 σ22 + σ = 0,
μ μ ( 3λ + 2μ )
ρ ρ (λ + μ )
σ 33′′ − k 2σ 33 + β (σ k y2 − σ ′′ ) − σ33 + σ = (2.9)
μ μ ( 3λ + 2 μ )
= −4 με0ik yU δ ' ( x ) .
177
Domain Structure in Ferroelectrics and Related Materials
we obtain
4 με0 k y k xU 0
σk = .
x
( 2β − 1) ⎡⎣ k
+ k − ρω 2 ( 3λ + 2 μ )( 2β − 1) ⎤⎦
2
x
2(2.12)
2με0U 0 ( k z2 − ρω 2 μ )
σ 12k = −
x
−
⎡⎣ k x2 + k 2 − ρω 2 μ ⎤⎦
4 με0 k y2 k x2 β U 0 (2.13)
− .
( 2β − 1) ⎡⎣ k 2
x + k − ρω
2 2
( 3λ + 2μ )( 2β − 1) ⎤⎦ ⎡⎣ k 2
x + k − ρω μ ⎤⎦
2 2
Hence
− με0U ( k z2 − ρω 2 μ )
σ 12 x =0
= −
k 2 − ρω 2 μ
2 με0U β k y2 (2.14)
− .
( 2β − 1) ⎡⎣ k − ρω μ + k − ρω
2 2 2 2
( λ + 2μ ) ⎤⎦
Similarly, on the basis of the Poisson equation (2.2) for the tetragonal
symmetry of tensor εij we determine the potential of the bound charges
on the boundary
−4πP0 ik zU ⎡ εc 2 ⎤
ϕ= exp ⎢ − x k z + k y2 ⎥
εc 2 ⎣⎢ εa ⎦⎥
εa k z + k y2 (2.15)
εa
and
∂ϕ 4π P0 k z2U
= .
∂z x=0 εc 2
εa k + k y2 (2.16)
εa z
Substituting (2.14) and (2.16) into (2.1) after cancelling the common
factor U we have
178
7. Natural and Forced Dynamics of Boundaries in Crystals
2 με02 ( k z2 − ρω 2 μ )
mω 2 = +
k 2 − ρω 2 μ
4με02 β k y2
+ +
( 2β − 1) ⎡⎣ k 2 − ρω 2 μ + k 2 − ρω 2 ( λ + 2 μ ) ⎤
⎦
(2.17)
8π P0 k z
2 2
+ .
εc 2
εa k z + k y2
εa
( cos 2
ϕ − υ 2 ct2 ) ⎛ c2
+ 4 ⎜1 − t2
⎞
⋅
sin 2 ϕ
+
⎟
1 − υ 2 ct2 ⎝ cl ⎠ ⎡ 1 − υ 2 ct2 + 1 − υ 2 ct2 ⎤
⎣ ⎦
γ ⋅ cos ϕ
2
+ = 0,
εc
cos 2 ϕ + sin 2 ϕ
εa (2.18)
γ = 4π P02 ε a με02 ,
where angle ϕ is counted from the polar direction or, after evident
transformations, as in [197, 198]
2
υ2 ⎛ υ2 ⎞ υ2 ⎛ υ2
υ2 ⎞
4 1 − 2 ⋅ 1 − 2 − ⎜ 2 − 2 ⎟ + 2 ⎜1 − 2 ⎟ ⋅ ctg ϕ +
2
ct cl ⎝ ct ⎠ ct ⎝ ct ⎠
υ2 υ2 (2.19)
+γ 2
1− 2
⋅ cos 2 ϕ = 0.
ct c t
179
Domain Structure in Ferroelectrics and Related Materials
without the last term ( γ = 0). It is clearly seen that for the given
specific direction, the velocity of propagation of the wave is constant
and independent of k. This velocity depends on the direction of the
propagation of the wave, i.e. υ = υ ( ϕ ). The orientation dependence
of υ is such that for the direction of spontaneous shear (ϕ = π /2)
it coincides with the velocity of the Rayleigh wave, which is determined
by the equation [99, 199]
2
⎛ υ2 ⎞ υ2 υ2
⎜ 2 − ⎟ = 4 1 − 1 − . (2.20)
⎝ ct2 ⎠ ct2 cl2
In the case of deviation from this direction, the wave is gradually
transformed and completely changes to a volume shear wave for
direction ϕ = 0 normal to the direction of spontaneous shear (Fig.
7.2).
In terms of twinning dislocations, the dynamics of corresponding
displacements is determined by the interaction of an ensemble of
moving dislocations. At ϕ = π/2 these are purely edge dislocations.
At ϕ < π /2 screw dislocations add up to them. With the appearance
of these dislocations, the velocity of the surface wave on the domain
wall increases and its localization decreases respectively, tending
to infinity for the volume shear wave (Fig. 7.2) at ϕ = 0.
In a defect-free ferroelectric–ferroelastic the limiting values for
the velocity of the boundary surface waves at ϕ = 0 and ϕ = π /2
remain the same. The value of υ for intermediate values of ϕ in
comparison with the pure ferroelastic is always higher. Evidently,
this is caused by the appearance here of the additional rigidity of
the boundary in relation to its bending displacements, and connected
to polarization.
As in the case of the pure ferroelectric, the direct proportionality
Fig. 7.2 (a) – Formation of twinning dislocations with the deviation of the domain
wall of the ferroelastic from the direction of spontaneous shear. The orientation
dependence of the velocity of the surface wave, localized on the domain boundary –
(b) and the depths of its penetration at fixed value of k in the material – (c) in
defect-free 1 – ferroelastic, 2 – ferroelectric–ferroelastic.
180
7. Natural and Forced Dynamics of Boundaries in Crystals
Fig. 7.3 Dependence of the effective mass of a domain wall on the wave vector in
pure ferroelectrics (excluding the case with k = 0), ferroelastics and ferroelectrics–
ferroelastics (a). Formation of an infinite mass (the involvement into movement
of all upper half-space) at translational (k = 0) displacement of the domain boundary
in a ferroelastic (b).
181
Domain Structure in Ferroelectrics and Related Materials
( cos 2
ϕ − υ 2 / ct2 ) ⎛ c2 ⎞
+ 4 ⎜ 1 − t2 ⎟ ⋅
sin 2 ϕ
+
1 − υ 2 / ct2 ⎝ cl ⎠ ⎡ 1 − υ 2 / ct2 + 1 − υ 2 / cl2 ⎤
⎣ ⎦
γ ⋅ sin ϕ
2 (2.23)
+ = 0.
cos ϕ + δ ⋅ sin 2 ϕ
2
182
7. Natural and Forced Dynamics of Boundaries in Crystals
Fig. 7.5 Orientation dependence of the velocity of the surface wave on the 90 o-domain
wall in a ferroelectric: γ1 < γ2 < γ3 .
183
Domain Structure in Ferroelectrics and Related Materials
4 2π P0 γ a
ϑ= , ε ≡ ε c = ε a , (3.1)
ε
4 πμ ε0 P0 a
ϑ= , (3.2)
ε
ϑ = 4πμε02 a. (3.3)
The influence of the defects on the boundary in this approximation
results in the appearance of additional pressure on the boundary in
the form of term KU in the right-hand part of the equations of the
boundary motion (1.6) and (2.1), where K = ϑ/l 2 and l is the average
distance between the defects pinning the boundary and, consequently,
it leads to the direct addition of the coefficient K to the right-hand
part of equations (1.17) and (2.18) describing the laws of dispersion
of the bending vibrations of the domain boundaries in pure ferroelectrics
and ferroelectrics–ferroelastics.
The result of the influence of the defects on the dependence ω (k)
under consideration is already clearly visible from the example of
pure ferroelectrics with the 180 o domain structure. In the presence
of defects, the equation describing the law of dispersion of the bending
vibrations of the domain boundaries has the form
⎛ ⎞
⎜m + 5π 2 P02 β12 k z6 ⎟ω 2 = 4π P02 k z2
+ K.
⎜
⎝ ρ ⋅ ε 2 4
cl ( k 2
x + k y )
2 72 ⎟
⎠ εa
εc 2
kx + k y2 (3.4)
εa
As it can be seen from equation (3.4) and Fig. 7.6, the presence
of defects results in a change of the effective coefficient of quasi-
Fig. 7.6. Dependence of the effective Fig. 7.7. The law of dispersion of bending
coefficient of the quasielastic force, acting vibrations of domain boundaries in a
on the boundary, on the wave vector in defective material (2) in comparison with
a defect-free (a) and defective (2) materials. a defect-free material (1).
184
7. Natural and Forced Dynamics of Boundaries in Crystals
elasticity, which is now determined by two terms. For the polar direction
in the range of low values of k, where the rigidity of the boundary
in the defect-free material decreases in proportion to k, the behaviour
of Kef is determined by the term K . In the range of high k, the situation
is opposite.
As the result, the linear dependence ω ~ k, which in the defect-
free material was observed both in the range of low and relatively
high k, in the crystal with defects will be implemented only at high
k. In the long-wave limit the dependence ω (k) is transformed to the
route dependence ω ~ k . For polar direction in particular it takes
place at k z = K ε a ε c 4πP02 . Taking into account the ratio of the
coefficient K with the individual coefficient ϑ and the direct expression
for the latter (3.1) we can find the value of the critical wavelength
λ, at which the change of the vibration modes takes place (see also
Fig. 7.7) [194, 195]:
2π P0l 2
λ= . (3.5)
( ε cε a ) γa
1/ 4
185
Domain Structure in Ferroelectrics and Related Materials
186
7. Natural and Forced Dynamics of Boundaries in Crystals
Fig. 7.8. The translational displacements of the domain boundaries in the ferroelectric
with the 180° domain structure in the quasi-acoustic (a) and quasi-optical branches
of vibrations of the domain structure (b).
187
Domain Structure in Ferroelectrics and Related Materials
188
7. Natural and Forced Dynamics of Boundaries in Crystals
Fig. 7.9. Equivalence of the resultant charge for equal displacement of the domain
boundaries for the following cases: initially non-compensated (a) and completely
compensated (b) charges of spontaneous polarization.
189
Domain Structure in Ferroelectrics and Related Materials
⎧ ∂ 2ϕ ∂ 2u j
−ε
⎪ ij − 4π β = 4πγ ( t ) δ ( x ) δ ( z ) ,
∂xi ∂x j ∂xk ∂xi
ijk
⎪
⎨
⎪ ∂ 2 uk ∂ 2ϕ (4.7.)
⎪ ρ i = cijkl
u − β ,
∂xl ∂x j ∂xk ∂x j
kij
⎩
where in accordance with the conditions of the problem
γ = γ 0 · exp(i ω t).
The further determination of the relation between potential ϕ and
displacement u i from (4.7) is similar to the case of bending
vibrations of the domain boundaries in a pure ferroelectric.
Representing ϕ (x,z) and u i (x,z) in the form of Fourier expansion:
190
7. Natural and Forced Dynamics of Boundaries in Crystals
dk x dk z
ϕ ( x, z ) = ∫ ϕk ( t ) e−ik x e −ik z
x z
,
( 2π )
2
dk x dk z (4.8)
ui ( x, z ) = ∫ uik ( t ) e− ikx x e − ikz z ,
( 2π )
2
where ϕ k (t) and u ik (t) are proportional to exp(i ω t), for the material
isotropic in the elastic respect as in the considerations of section
7.1. for the normalized Fourier coefficient ϕk = ϕk γ , which
represents the Fourier image of the Green function of the equation
for the electric field taking the piezoelectric effect into account, we
obtain
⎧ 4πβijk kk ki
⎪
ϕk (t ) = 4π ⎨ε ij ki k j + ×
⎪⎩ (
ρ c 2 k 2 − ω 2
t )
−1
⎡ ( cl2 − ct2 ) ⋅ β k k k k ⎤⎥ ⎫⎪ ,
× ⎢ β pmj k p k j − 2 2
(4.9)
plj p j l m ⎬
⎢
⎣ (
cl k − ω 2 ) ⎥⎪
⎦⎭
where k 2 = k x2 + k z2 .
The interaction of the equilibrium distributed charges (4.5) with
each other determines in (4.2) only the constant term Φ 0 which is
not included in the equation of domain boundaries motion (4.4). The
next term in (4.2) is determined by the interaction of charges (4.6).
Therefore, it is natural here to restrict our considerations by the
calculation of the fields and interaction of only these charges in
particular. The volume density of the charge, corresponding to (4.6),
distributed on both surfaces of the ferroelectric plate is
⎡ ⎛ Lz ⎞ ⎛ Lz ⎞ ⎤
ρ ( x, z ) = ∑ γ nδ ( x − dn ) ⎢δ ⎜ z −
⎟ − δ ⎜ z + ⎟⎥ . (4.10)
n ⎣ ⎝ 2⎠ ⎝ 2 ⎠⎦
Then, in accordance with the properties of the Green function,
the potential of these charges is
dk − ik x ( x − dn ′ )
ϕ1 ( x, z ) = ∫ ρkϕk ⋅ e −ikρ ⋅ =∑ γ n′∫ ϕk e ×
( 2π )
2
n'
⎡ ⎛ ⎛ L ⎞⎞ ⎛ ⎛ L ⎞ ⎤ dk ⋅ dk (4.11)
× ⎢ exp ⎜ −ik z ⎜ z − z ⎟ ⎟ − exp ⎜ −ik z ⎜ z + z ⎟ ⎥ × x 2 z .
⎣ ⎝ ⎝ 2 ⎠⎠ ⎝ ⎝ 2 ⎠ ⎦ ( 2π )
The energy of their interaction
191
Domain Structure in Ferroelectrics and Related Materials
⎛ L ⎞ ⎛ L ⎞
Φ − Φ 0 = Ly ∫ σ 1 ⎜ z = z , x ⎟ ⋅ ϕ1 ⎜ z = z , x ⎟ dx =
⎝ 2 ⎠ ⎝ 2 ⎠
= Ly ∑∑ γ nγ n' ϕk ⋅ exp ( −ik x d ( n − n ') ) ×
n n'
(4.12)
dk x dk z
×[1 − exp(−ik z Lz )] .
( 2π )
2
Taking into account that γ =2P 0 U n , and γ n' = 2P 0 U n' (–1) n–n' ,
where the multiplier (–1) n–n' takes into account that at the same
signs of U n and U n' the charge, appearing on the surface of the
crystal, for the adjacent domain walls has the opposite sign, the last
equation may be written in the form
1
Φ − Φ0 = ∑ K ( n, n ')U nU n ' ,
2 n,n ' (4.13)
where
dk x dk z
×[1 − exp(−ik z Lz )] . (4.14)
( 2π )
2
8 P02 Ly ⎡ ⎛ ε L 1 ⎞ ⎤
2
( −1)
n−n '
= ⎢
ln 1 + ⎜ a z
⎟ ⎥. (4.16)
ε aε c ⎢ ⎜⎝ ε c d ( n − n ') ⎟⎠ ⎥
⎣ ⎦
Equation (4.16) cannot be used for the case of n'=n because the
method of calculation of the coefficients (4.16) does not foresee the
separation of the effect of the self-influence of the charges formed
in the region of the displaced domain wall which naturally leads to
an infinite increase of expression (4.16) at n=n'.
To calculate any of the coefficients K (n, n ') in sum (4.13) in
192
7. Natural and Forced Dynamics of Boundaries in Crystals
a b
Fig. 7.10. The change of the electric state of the polydomain ferroelectric with
the 180º domain structure when only one wall is displacement (a) and when all
other walls are displaced the same distance (b).
16 P02 Ly ⎧⎪ ∞ ⎡ ε a L2z 1 ⎤
K ( n, n ) ≡ K = ⎨∑ ln ⎢1 + ⎥−
ε cε a ⎪⎩ n =1 ⎣⎢ ε c d 2 ( 2n − 1) ⎦⎥
2
∞ ⎡ ε L2 1 ⎤ ⎫⎪
−∑ ln ⎢1 + a z2 ⎥ =
2 ⎬
n =1 ⎢⎣ ε c d ( 2n ) ⎥⎦ ⎭⎪ (4.17)
2
16 P Ly ⎛ π ε a Lz ⎞ ⎛ π ε a Lz ⎞
= ⋅ cth ⎜
0
ln ⎜ ⎟ ⎜ 2 ε d ⎟⎟
.
ε cε a ⎜⎝ 2 ε c d ⎟⎠ ⎝ c ⎠
Let us examine the spectrum of vibrations of the domain
structure in this case ( β =0). Let the wall coinciding with the origin
of the coordinates have the number n=0. As already mentioned, the
adjacent walls (in the notations used, these are the walls with even
193
Domain Structure in Ferroelectrics and Related Materials
−mω 2 + K 1 K 2
= 0,
K −mω 2 + K
2 1
∞
K 1 = K + 2∑ K ( 2n ) cos 2n kd ,
n =1 (4.20)
∞
K 2 = 2∑ K ( 2n − 1) cos ( 2n − 1) kd .
n =1
194
7. Natural and Forced Dynamics of Boundaries in Crystals
16 P02 Ly ⎧⎪ ∞ ⎡ ε a L2z 1 ⎤
ω ∓2 = ⋅ ⎨∑ ln ⎢1 + ⎥×
ε aε c m ⎪⎩ n =1 ⎣⎢ ε c d ( 2n − 1) ⎦⎥
2 2
×1 ⎡⎣1 ∓ cos ( 2n − 1) kd ⎤⎦ −
(4.22)
∞ ⎡ ε L2 1 ⎤ ⎫⎪
−∑ ln ⎢1 + a z2 ⎥ ⋅ [1 − cos 2 nkd ] ⎬ .
⎢⎣ ε c d ( 2n ) ⎥⎦
2
n =1
⎭⎪
195
Domain Structure in Ferroelectrics and Related Materials
32 P02 Ly ∞ ⎡ ε L2 1 ⎤
ω ( 0) =
2
+ ⋅ ∑ ln ⎢1 + a z2 ⎥=
ε cε a m n =1 ⎢⎣ ε c d ( 2n − 1) ⎥⎦
2
∞ ⎡ ⎤
32 P02 Ly ε L2 1
= ln ∏ ⎢1 + a z2 ⎥=
ε cε a m n = 0 ⎣⎢ ε c d ( 2n + 1) ⎦⎥
2
(4.24)
32 P Ly 2
⎛ π ε a Lz ⎞
=
0
ln ch ⎜
⎜ 2 ε d ⎟⎟
.
ε cε a m ⎝ c ⎠
ω ∓2 ( k = π 2d ) =
ω +2 ( k = 0 ) 32 P02 Ly ⎡ ε L2
∞
1 ⎤
=
2
− ∑ ln ⎢1 + a z2 2
ε cε a m n =1 ⎢⎣ ε c d 2 ( 2n − 1) ⎥⎦
2
⎥=
16 P02 Ly ⎡ ⎛ π ε a Lz ⎞ ⎛ π ε a Lz ⎞ ⎤
= ⋅ ⎢ ln ch ⎜ − 2ln ch ⎜
⎟ ⎜ 4 ε d ⎟⎟ ⎥
.
ε cε a m ⎢⎣ ⎜2 ε d ⎟
⎝ c ⎠ ⎝ c ⎠ ⎥⎦
(4.26)
At (π ε a Lz 2 ε c d 1 )
16 P02 ln 2
ω ∓2 ( k = π 2d ) . (4.27)
ε aε c Lz m
It is difficult to carry out in the general form the analytical
calculations of the spectrum of the domain structure vibrations with
the piezoelectric effect taken into account. However, this can be
196
7. Natural and Forced Dynamics of Boundaries in Crystals
197
Domain Structure in Ferroelectrics and Related Materials
area
( 4π ) P02 β 2 L2z Lx
2
m ∗
=ρ⋅ . (4.32)
s
ε c2 d 2 c 2
Since this mass is distributed between L x /d walls, for the effective
mass of the unit of area of a single wall we have [206, 207]
∗ 16π 2 P02 β 2 L2z
mdw = ⋅ , (4.33)
ε c2 ρ cl4 d
where c l is the velocity of the corresponding sound wave.
∗
The estimates of the value mdw at conventional P 0 ~10 4 (here, as
previously, we use CGSE units), β ~10 6 , L z ~1, d~10 –4 , ε c ~10 –3 ,
ρ ~5, c l ~10 5 CGSE units gives 3· 10–2 g· cm–2 , which is many orders
of magnitude higher than the conventional effective mass of the
domain walls [143], which is linked with the conversion of the
spontaneous polarization in the region of the moving domain wall
and which for the same values of the constants included in this
equation is equal to ~10 –11 g/cm 2 .
Evidently, the obtained increase of the effective mass of the
domain wall also greatly reduces the frequency of natural vibrations
of the domain walls (Fig.7.11). In particular, according to (4.25) for
limiting long-wave quasi-optical vibrations instead of the value
~10 10 Hz without piezoelectric effect taken into account, the value
of the corresponding frequency with the piezoelectric effect taken
into account decreases to ~10 6 Hz, i.e. to megahertz frequencies.
For the same reasons as in the case of ferroelectrics,
translational vibrations of the domain walls also occur in the
ferroelastics. For arbitrary values of the wave vector such vibrations
have not been studied but, for the case of k=0, the frequency of
the corresponding vibrations can be easily estimated, avoiding
labour-consuming calculations. For this purpose, let us first of all
calculate the increase of the elastic energy of the ferroelastic
associated with the equal displacements of the domain walls. The
average deformation in the material of each domain at displacement
of the domain boundaries by the value U is equal to u = 2ε0U / d .
Then, for a contact with an absolutely rigid material, where all
elastic fields are concentrated in the material of the ferroelastic,
this increase of elastic energy for a single domain is
1 2 1
Φ= μu V = με02U 2 Ly Lz d . (4.34)
2 2
Relating these values of Φ to the unit of area of the domain wall
198
7. Natural and Forced Dynamics of Boundaries in Crystals
199
Domain Structure in Ferroelectrics and Related Materials
200
7. Natural and Forced Dynamics of Boundaries in Crystals
∂U
F ⋅ U = ∫ n ( x, t ) dx =
∂t
∂U ∂U (5.1)
=∫ ⋅ U ⋅ n ( x, t ) dx = −U ∫ ⋅ n ( x, t ) dx.
∂U ∂x
Here U=U(x–U(t)) is the increase of the energy associated with
the bending of the wall from its equilibrium position (symmetric in
this case) in the system of points of its pinning by defects, i.e.
finally, linked with extra bending of the domain wall, U is the
coordinate of the plane of the average orientation of the domain
wall interacting with the defects, U is its velocity, n(x,t) is the
volume concentration of points of the boundary pinning by the
defects.
The time dependence of the distribution of pinning points n(x,t)
is described by kinetic equation with the single relaxation time:
dn n − n∞
=− , (5.2)
dt τ
where n ∞(x) is the equilibrium distribution of pinning points in the
given region of the crystal, which in accordance with the results of
chapter 5, can be regarded as having a stepped form:
n = n ⋅ Θ U − x .
∞ ( ) (5.3)
Here Θ(x) is Heaviside’s function, n is the volume concentration
of the defects, displacement U , as previously, characterizes the
maximum distance of the defect from the plane of the average
orientation of the boundary at which the boundary is still captured
by the defect.
The solution of (5.2) has the form
⎛ t −ξ ⎞ dξ
t
n ( x, t ) = ⎟ n∞ (ξ ) .
∫ exp ⎜⎝ − (5.4)
−∞ τ ⎠ τ
Substituting (5.4) into the expression for the pressure of the
force acting on the boundary from the direction of the defects,
which according to (5.1) is
∂U
n ( x, t ) dx
F = −∫ (5.5)
∂x
and replacing here the difference x–U(t) by x we obtain the
following [214]:
201
Domain Structure in Ferroelectrics and Related Materials
∂U ( x − U ( t ) ) t
⎛ t −ξ ⎞ dξ
F = −∫ ∫ exp ⎜⎝ − ⎟ n∞ ( x − U (ξ ) ) dx =
∂x −∞
τ ⎠ τ
⎛ t − ξ ⎞ ∂U ( x )
∞
dξ
t
= ∫−∞ exp ⎜⎝ − τ ⎟⎠−∞∫ ∂x n∞ ⎡⎣U ( t ) − U (ξ )⎤⎦ dx τ =
(5.6)
⎛ t −ξ ⎞ dξ
t
= ∫ exp ⎜ − ⎟ E ⎣⎡U ( t ) − U (ξ ) ⎤⎦ ,
−∞ ⎝ τ ⎠ τ
where
∞
∂U ( x )
E ⎡⎣U ( t ) − U (ξ ) ⎤⎦ = ∫ n∞ ⎡⎣U ( t ) − U (ξ ) ⎤⎦ dx. (5.7)
−∞ ∂x
When determining the motion of the domain boundaries in
harmonic approximation, the difference Δ≡U(t)–U(ξ) will be
regarded as small. Then taking into account the specific type of
n ∞(x) (5.3) we have
∂ε
ε ⎡⎣U ( t ) − U (ξ ) ⎤⎦ ≡ ε [ Δ ] = ε ( 0 ) + 0 ⋅Δ + ...
∂Δ
(5.8)
nϑU ⎡⎣U ( t ) − U (ξ ) ⎤⎦ ≡ K ⎡⎣U ( t ) − U (ξ ) ⎤⎦ ,
where ϑ and U for a pure ferroelectric are determined by the
expressions from chapter 5, and for a ferroelectric–ferroelastic by
the equations from this chapter, respectively.
Let us further consider the motion of the domain boundaries in
the external field. Since the strength of the later depends only on
time and is almost independent of the coordinates, the motion of all
domain walls in this field will be identical. Therefore, when writing
equations of motion of an arbitrary wall, the number of the
displaced wall can be omitted and, taking this into account, the
mentioned equation can be written in the form
m∗U + KU ( t ) + F [U ] = 2 P E ( t ) , (5.9)
0
⎛ π ε a Lz ⎞
⎜ 2 ε d ⎟⎟ ε cε a Lz
where K = 32 P02 ln ch ⎜ is the coefficient of the
⎝ c ⎠
quasi-elasticity of the domain wall in a defect-free crystal related
to the unit of area of the wall, which evidently is equal to the
coefficient of quasi-elasticity in the optical branch of the vibrations,
taken at k = 0 (4.24), force F[U] is specified by expression (5.6),
and term 2P 0 E describes the pressure on the domain wall in the
202
7. Natural and Forced Dynamics of Boundaries in Crystals
external field.
Replacing F[U] in (5.9) by the expression (5.6) where
E [U (t ) = U (ξ )] is determined by expression (5.8), we obtain
⎛ t −ξ ⎞ dξ
t
m∗U + K ⋅ U ( t ) + KU ( t ) − K ⎟ ⋅ U (ξ )
∫ exp ⎜⎝ − = 2 P0 E ( t ) . (5.10)
−∞ τ ⎠ τ
Let the external electric field change with time in accordance
with the harmonic law E=E 0exp(i ω t). Let us look for the solution
of equation (5.10) for steady motion in the form of U(t)=U 0
exp(i( ω t+ α )). Substituting it into (5.10) in general case we obtain
[194, 195]:
2 P0 E ( t )
U (t ) = .
⎣
∗ 2
( )
⎡ −m ω + K + K − K (1 + iωτ ) ⎤
⎦
(5.11)
2 P0 E0
U (t ) = ×
(K + K )
⎧⎪ ⎡ K K ⎤ ωτ c K K ⎫
⎪ (5.12)
× ⎨ ⎢1 + ⎥ cos ω t + sin ω t ⎬,
⎩⎪ ⎣⎢ (1 + iω τ
2 2
)
c ⎦⎥ (1 + iω τ
2 2
c ) ⎪
⎭
where
τc =τ ⋅
(K + K ). (5.13)
K
The motion of the domain wall, interacting with a system of
mobile defects, is described using the above approach, with the only
difference that the role of n(x,t) in all expressions here is
performed directly by the concentration of defects in the given
location in the crystal and therefore τ here is the relaxation time
of the defective atmosphere. Thus, in contrast to previous
discussion, devoted to consideration of the passage of the domain
wall through the system of stationary stoppers, where time τ
characterizes the relaxation properties of the domain wall,
determined primarily by the energy of its interaction with defects,
here time τ characterizes the mobility of defects, i.e. depends
mainly on the activation energy of its motion.
203
Domain Structure in Ferroelectrics and Related Materials
ε (ω ) = ε ∞ +
(ε0 − ε ∞ ) ,
(6.2)
1 + iωτ c
the real and imaginary parts of which are
(ε 0 − ε ∞ ) ( ε 0 − ε ∞ )ωτ c
ε ' = ε∞ + , ε '' = .
(1 + iω τ )
2 2
c
1 + iω 2τ c2 (6.3)
204
7. Natural and Forced Dynamics of Boundaries in Crystals
16π P02 π Lz ε cε a
ε0 = = ,
dK (
2d ln ch π ε a Lz 2 ε c d ) (6.4)
1/ 4
16πP02 4 2 ⎛ π 3ε P03 ⎞
ε∞ = ⎜ ⎟ .
(
d K+K ) nd ⎝ με02 U02 a 2 ⎠ (6.5)
1 nϑU 1 ⎛ με 3 ⎞
1/ 4
nd U0ε0 a
( tg δ )max = ⎜ ⎟ . (6.10)
2 K 4 ⎝ π3 ⎠ P0 P0
Equations (6.8) and (6.10) show that with the increase of the
concentration of defects, the initially linear growth of the height of
the maximum is subsequently replaced by a root dependence, i.e.
205
Domain Structure in Ferroelectrics and Related Materials
206
7. Natural and Forced Dynamics of Boundaries in Crystals
Fig.7.12. Dependence ε (T) for a RDA crystal measured after applying of a constant
field E to the crystal: 1, 2, 3, 4 – E = 0; 1; 1.5; 2 kV/cm; E ~ =1 V/cm.
Fig.7.13. Dependence ε (T) for a KDP crystal with different content of chromium
ions [233]: 1) nominally pure KDP: 2,3,4 – n=10 18 , 10 19 , 10 20 cm –3 ; E ~ =1 V/cm.
207
Domain Structure in Ferroelectrics and Related Materials
208
7. Natural and Forced Dynamics of Boundaries in Crystals
209
Domain Structure in Ferroelectrics and Related Materials
210
7. Natural and Forced Dynamics of Boundaries in Crystals
211
Domain Structure in Ferroelectrics and Related Materials
212
7. Natural and Forced Dynamics of Boundaries in Crystals
213
Domain Structure in Ferroelectrics and Related Materials
214
7. Natural and Forced Dynamics of Boundaries in Crystals
215
Domain Structure in Ferroelectrics and Related Materials
⎧ ⎛ U0 − ϑ x 2 / 2 ⎞
⎛ U ( x) ⎞ ⎪ 0
n exp ⎜ ⎟ , x < U ,
n∞ ( x ) = n0 exp ⎜ − ⎟=⎨ ⎝ T ⎠
⎝ T ⎠ ⎪ (9.4)
⎩n0 , x > U .
The condition determining the relation between n 0 and l 2 in our
case is
U
∫ n ( x ) dx = 1/ l .
2
(9.5)
0
π T
n0U ⎡⎣ e − t / τ + (1 − e − t / τ ) ⋅ e U0 / T ×
2 U0
(9.6)
× erf ( U0 T ) ⎤⎦ = 1/ l 2 .
216
7. Natural and Forced Dynamics of Boundaries in Crystals
217
Domain Structure in Ferroelectrics and Related Materials
218
7. Natural and Forced Dynamics of Boundaries in Crystals
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219
Domain Structure in Ferroelectrics and Related Materials
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Domain Structure in Ferroelectrics and Related Materials
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Domain Structure in Ferroelectrics and Related Materials
232
7. Natural and Forced Dynamics of Boundaries in Crystals
Index
233
Domain Structure in Ferroelectrics and Related Materials
K Q
KDP crystals 209 quasi-continuous approximation 60
Kittle domain structure 2 quasi-spin operator 74
Kronecker symbol 101 quasispin 79
Kröner incompatibility tensor 100
R
L
Rayleigh wave 184
Lame coefficient 101 renormalised effective mass of the domain
Laplace equation 3 wall 175
Laplace pressure 92
lattice barrier 143 S
lattice energy barrier 59, 160 Schrödinger equation 31, 56
linear density of the detachment force 137 screening 47
Lorenz reduction 144 screening length 48
M screening of polarization 47
skew cut 9
Macdonald function 65 Slater static configuration 73
Maxwell relaxation 189 spatial modulation 54
Maxwell's equations 102 spontaneous polarization 2, 92
misalignment energy 152 St-Venant condition 100
static dielectric permittivity 169
N strain tensor 100
non-critical elastic modulus 22 structure factor 62
non-ferroelectric inclusions 110 surface screening 18
non-twinning dislocations 111 T
O Takagi's defect 78
odd periodic function 43 tensor of correlation constant 29
tensor of dielectric permittivity 92
P tensor of dislocation density 99
tensor of elastic distortion 99
Peach–Koehler force 114, 115 tunnelling 160
Peierls' force 148 twinning dislocations 91
Peierls relief 59
perovskite 177 U
perturbation theory 39
phase transitions in the domain walls 54 unit antisymmetric tensor 100
piezoelectric deformation 198 V
point charge potential 94
point charged defects 91 vector of elastic displacement 173
polar’ defects 26 vector of electrostatic induction 92
polarization screening 13
polarization vector 38 Z
potassium dihydrophosphate 71 Zig-zag domain boundary 125
pure ferroelastics 139 zig-zag structure 128
234