Phase front and domains during the paraelectric-ferroelectric transition in KD2PO4: Optical observation and dielectric contributions

Article abstract of DOI:10.1103/PhysRevB.60.3806

Observations of the phase front and the domains during the paraelectric-ferroelectric KD₂PO₄ phase have been performed in very good thermal homogeneity conditions (thermal gradient equal to 2×10^-3 K mm^-1). Simultaneously, dielectric and loss constants have also been measured. The phase fronts exhibit a quasiplanar shape perpendicular to the c ferroelectric axis. The ferroelectric region is quasimonodomain when its c dimension is lower than 100 μm. The dielectric constant is demonstrated as the sum of three contributions: a monodomain ε′_mono, a contribution of domains ε′_d proportional to ε′_mono with a coefficient depending on the domain texture, and a contribution due to the existence of the phase front during the phase coexistence.

Full text of DOI:10.1103/PhysRevB.60.3806

PHYSICAL REVIEW B
VOLUME 60, NUMBER 6
1 AUGUST 1999-II
Phase front and domains during the paraelectric-ferroelectric transition in KD PO :
2
4
Optical observation and dielectric contributions
Jean Bornarel
Laboratoire de Spectrom e´ trie Physique (UMR 5588), Universit e´ Joseph Fourier, Bo ˆı te Postale 87,
3
8402 Saint-Martin-d’H e` res Cedex, France
Ryszard Cach
Institute of Experimental Physics, University of Wroclaw, Plac Maksa Borna, 9-50-205 Wroclaw, Poland
͑Received 11 January 1999͒
Observations of the phase front and the domains during the paraelectric-ferroelectric KD PO phase have
2
4
Ϫ3
Ϫ1
been performed in very good thermal homogeneity conditions ͑thermal gradient equal to 2ϫ10 K mm ).
Simultaneously, dielectric and loss constants have also been measured. The phase fronts exhibit a quasiplanar
shape perpendicular to the c ferroelectric axis. The ferroelectric region is quasimonodomain when its c dimen-
sion is lower than 100 ␮m. The dielectric constant is demonstrated as the sum of three contributions: a
monodomain ␧Јmono , a contribution of domains ␧Ј proportional to ␧Ј
with a coefficient depending on the
d
mono
domain texture, and a contribution due to the existence of the phase front during the phase coexistence.
͓
S0163-1829͑99͒10829-4͔
I. INTRODUCTION
A
B
C
2
4
6
⌽
ϭ⌽ ϩ ͑TϪT ͒P Ϫ P ϩ P ϩ¯
͑1͒
0
0
2
4
6
The KD PO ͑DKDP͒ crystal, the deuterated isomorph of
KH PO ͑KDP͒ undergoes a first-order transition between a
tetragonal paraelectric and paraelastic phase (42m) which is
the high-temperature phase, and an orthorhombic ferroelec-
tric and ferroelastic one (mm2). The polarization which is
2
4
with A, B, and C positive constants in temperature. Three
characteristic temperatures, important for the first-order tran-
sition, can be concluded from the above expression. T , the
lowest temperature for which the nonpolar phase can exist as
a metastable state, T*ϭT ϩB /4AC, the highest tempera-
ture for which the zero field ferroelectric phase can exist
2
4
¯
0
2
0
considered as the order parameter belongs to the B repre-
2
sentation as does the shear strain uxy due to the piezoelectric
behavior of DKDP. Moreover the shear strain appearance in
the plane perpendicular to the ferroelectric axis c, it is pos-
sible to notice the values of normal lattice deformation with
greater than uxx and uyy ͑about 1ϫ10 ).
In the low-temperature phase the ferroelectric-ferroelastic
2
metastably, and T ϭT ϩ3B /16AC, the temperature at
c
0
which both polar and nonpolar states correspond to the same
value. In the temperature range between T and T* the
polar and nonpolar phases can exist. Then B /4AC is the
⌽
0
2
Ϫ4
Ϫ4
uzzϭ6.5ϫ10
maximum magnitude of a possible thermal hysteresis of the
phase transition. Taking into consideration the A, B, and C
1
values given in the literature1
2,13
it can be concluded that the
domain structure exists with permissible walls in ͑100͒ and
010͒ tetragonal planes. These domains are also mechanical
͑
maximum range of the possible phase coexistence is of about
1° ͑much larger than for KH PO ). However, B /4AC is
2
twins and the importance of the mechanical energy in the2
domain structure KDP type crystals has long been known.
Similarly the linear dependence of the transition temperature
with the deuteration degree changing from 121 to 228 K is
known and explained.3
2
4
only the possible theoretical temperature range for the ther-
mal hysteresis. While the thermal hysteresis has been ob-
served by different authors, it is interesting that when the
temperature was changed very slowly with a good thermal
homogeneity for the crystal, this hysteresis could not be ob-
served practically.15 The present paper will clarify this phe-
nomenon.
The first information on phase coexistence in DKDP came
from lattice deformations by neutron diffractometry.
of the authors suggests, with the intention of interpreting
the measurements, that paraelectric and ferroelectric stripes
perpendicular to the c ferroelectric axis alternate in the
sample. The validity of this prediction was confirmed by
,4
The DKDP transition has been studied by means of clas-
sical macroscopic electric measurements. The double hyster-
esis loops, characteristic for first-order transitions, were ob-
served above the phase transition temperature.5 The
spontaneous polarization appears in leaps at the transition,
7
,15
One
Ϫ2
2 6
16
reaching a value of about 4.2ϫ10 C/m . A corresponding
jump of the shear angle uxy has been found, 24Ј ͑Ref. 7͒ or
3
3Ј ͑Ref. 8͒, by neutron studies. The character of the phase
transition was also confirmed by calorimetric studies.
The existence of a thermal hysteresis at the transition needs
to be clarified. As in other ferroic crystals the properties of
9
–11
1
7
direct optical observations along a tetragonal axis and by
x-ray topography using a synchrotron radiation.1
8–21
As in
DKP family crystals are frequently described within the
both cases, the results are projections in a given direction
therefore it was not possible to make any conclusion on the
phase front shape. More recently the phase coexistence of
DKDP has been systematically studied. Dielectric measure-
frame of Landau-Devonshire theory.1
2–15
In this approach
the thermodynamical potential ͑for example the elastic Gibbs
energy ⌽͒ is written versus the polarization P
0
163-1829/99/60͑6͒/3806͑9͒/$15.00
PRB 60
3806
©1999 The American Physical Society

PRB 60
PHASE FRONT AND DOMAINS DURING THE . . .
3807
ments and observations along the three tetragonal axes were
simultaneously performed giving a good knowledge of the
phase coexistence phenomena and of the phase front
centration of 90%. The weak tapering angle of the crystal ͑a
few degrees of arc͒, the room temperature optical studies,
and the dielectric properties lead to the conclusion of good
crystal quality. The sample was cut with a wire saw. The
orientations of sample faces were verified with x-ray Bragg
diffraction ͑accuracy of a minute of arc͒ and each face was
polished on a silk cloth with diamond paste. The sample
2
2–24
shape.
The great importance of an external thermal gra-
Ϫ1
dient G has been demonstrated: for G between 10 and
e
e
Ϫ2
Ϫ1
10
K mm , the phase front shape and orientation change
with G orientation. When G is parallel to the ferroelectric c
e
e
2
5
axis the phase front is quasiplanar and near ͑001͒ plane.
dimensions were a1ϭ5.5 mm, a2ϭ4.4 mm, and
c
When G is perpendicular to the c axis the phase front ap-
ϭ7.4 mm. Semitransparent gold electrodes were evaporated
on the c faces.
e
pears as a factory roof the section of which in a ͑or a )
1
2
tetragonal plane has a zig-zag shape.2
2,26
In all cases no part
of the phase front makes an angle greater than 25° with the
001͒ plane except the particular case where the zig-zag
The cryostat employed with a helium-gas exchange cham-
ber allows optical observations and measurements along
three perpendicular axes simultaneously with dielectric mea-
surements. In the present case these three axes correspond to
the tetragonal axes of the crystal. It has been demonstrated in
͑
height equals the sample thickness. Then the energies of the
zig-zag edges can decrease by relaxation in the sample bou-
daries and their number increases.27 When the angle ␣ value
26
a previous paper that the sample boundary conditions have
between G and the c axis changes between 0° and 90° of
arc, the phase front orientation changes too with the appear-
great importance for the temperature of the sample. It was
noted that the thermal conductivity of DKDP is about 20
times higher than that of the helium gas and similar to the
thermal conductivity of window glass. Detailed experimental
studies showed the correlation between the thermal distribu-
e
ance of the zig-zag shape for a critical ␣ value ͑60° of arc
Ϫ2
Ϫ1
with G equal 10 K mm ). A theoretical model explains
e
these phenomena as a competition between the mechanical
and chemical energies, with a quasinegligible effect of the
electrostatic energy at zero applied electric field.28 The inter-
action between the phase front shape and the domain textures
has been sometimes observed as in dagger processes with
tion inside the sample, the thermal boundary conditions, and
25,26
the temperature rate.
The results presented here have
been obtained in the following conditions. The sample was
hung in the helium gas chamber with the help of two thin
copper wires which were also used for the electrical contacts.
The c ferroelectric axis corresponded to an horizontal optical
axis and was perpendicular to the thermal gradient Ge . The
thermal gradient Ge in the helium gas chamber was con-
2
4
notable G values but the problem remains open. For ex-
e
ample, the observation of quasimonodomain states in small
volumes of the ferroelectric phase inside the paraelectric one
requires confirmation.2
4,25
Ϫ1
Measurements of the dielectric constant ␧Ј and the loss
trolled with an accuracy of 5 mK mm with the help of two
c
platinum resistors placed just above and below the sample.
The temperature T reported further on is that of the low
constant ␧Љ of the KDP family crystals have already been
c
made versus different parameters: the amplitude and the fre-
2
9–31
platinum resistor which was measured with a precision of
quency of the ac measuring field,
an external dc bias
Ϫ3
field in the c direction,3
2,33
the irradiation conditions with ␥
2
ϫ10 K. The dielectric measurements and optical obser-
vations of the phase front were done while cooling and heat-
rays and neutrons, the thermal history, etc. Optical observa-
tions of the domain texture performed simultaneously with
dielectric measurements allowed us to demonstrate the do-
main contribution in the dielectric properties at low
Ϫ2
Ϫ1
ing rates were lower than 10 K min . Only one comple-
mentary experiment corresponding to Fig. 7 was performed
with other thermal conditions which are described in the text.
The sample capacity and dissipation factor were measured
using an HP 4274 A impedance meter with a measuring field
2
9,34,35
33,36
temperatures
as nearer the transition temperature.
This effect is known even if the variation law of this dielec-
tric constant contribution of the domains versus the tempera-
ture must be clarified near the transition. Earlier studies dur-
ing the phase coexistence of DKDP carried out under an
Ϫ1
of 0.5 V cm in amplitude and 4 kHz in frequency allowing
us to calculate ␧Ј and ␧Љ with a relative accuracy of 3
c
c
Ϫ3
Ϫ2
ϫ10 and 1ϫ10 , respectively. As already described in a
previous paper the observations along the a1 and a2 axes
Ϫ1
24
external thermal gradient of about 0.2 K mm confirm this
contribution of the domains in the dielectric properties2
and suggest also that the presence of the phase front in the
crystal leads to an increase of its electric permittivity value.
In the present paper precise studies of the DKDP phase
coexistence under a controlled thermal gradient of a value
close to zero are presented. Dielectric measurements were
performed simultaneously with optical observations of the
domain structure and the phase front. They allow for a better
understanding of the phase coexistence phenomena and a
detailed discussion of the dielectric properties within this
temperature region.
5,26
allow us to rebuild the phase front shape while the observa-
tion along the c axis gives information on the domain tex-
ture. The diffraction of a laser beam propagating in the c
direction37 has also been used to detect the presence of the
domains.
III. RESULTS
The results obtained simmultaneously concerning the
phase front shape, the domain texture, and the dielectric ␧cЈ
and ␧Љ are presented for clarity successively. The correla-
c
tions between the different phenomena are obviously noted.
II. EXPERIMENTAL PROCEDURES
A. Phase front
The DKDP crystals were grown by slow cooling of a
supersaturated solution of KDP and heavy water. The ob-
served transition at 218.6 K corresponds to a deuterion con-
In the experimental conditions previously described as a
very small thermal grandient G ͑equal to or lower than 5
e

3808
JEAN BORNAREL AND RYSZARD CACH
PRB 60
FIG. 2. Photographs during a PF transition of the phase fronts in
a section a and of the domain texture in a c section.
1
B. Domain structure
Existence and texture of the domains are others interest-
ing phenomena worth studying. At the beginning of the PF
phase transition only a few domain walls can be observed
͑
Fig. 2͒. It has already been noticed that domains were not
2
4,25
observable in the initial stage of the PF phase transition.
But those observations have been made on a crystal under an
Ϫ1
FIG. 1. Evolution of the phase front during a DKDP phase tran-
external thermal gradient G equal to 0.2 K mm . From the
e
Ϫ1
sition in homogeneous thermal conditions (G ϳ0.002 K mm ).
i
present experiment with a very small thermal gradient, G
equal to 5ϫ10 K mm , which corresponds roughly to G
e
Schematic illustration of the phase coexistence ͑PF or FP cycles͒
Ϫ3
Ϫ1
i
and photographs of the phase front in a section a ͑FP cycle͒.
Ϫ3
Ϫ1
26
1
equal to 2ϫ10 K mm inside the crystal, it can be con-
cluded that the ferroelectric phase occupies a considerable
volume of the sample whereas there are still only a few do-
mains. Rapid formation of the domain structure begins when
fronts meet in the sample center and the paraelectric phase
ϫ10Ϫ3 K mmϪ1), the evolution of the phase front shape is
similar during a transition from the paraelectric to ferroelec-
tric phase ͑PF͒ and for the ferroelectric to paraelectric phase
stays only a dagger decreasing in height in the a direction
2
͑
FP͒. When the transition begins the phase front appears at
͓Figs. 2͑c͒–2͑e͔͒. At this moment, the creation of the domain
wall traversing the whole sample in the c direction is pos-
sible. When phase fronts disappear there is a fairly dense
domain texture which is dominated by one of the two do-
main species. Such a domain structure seems to be quite
stable, but some arrangements happen in the temperature in-
terval of a few degrees below the transition temperature as
noted in KDP studies.3
The FP phase transition behaves in a similar way. The
paraelectric phase nuclei appear in the sample corners ͓Fig.
1͑a͔͒. Fronts moving towards the sample center ͓as in Figs.
1͑b͒–1͑d͔͒ shorten the domain walls on both sides. As illus-
trated in Fig. 3, the disappearance of the domains can be
observed when the phase fronts are at a distance of a few
tens of micrometers. This optical observation has been con-
firmed also by studying the diffraction spectrum of a He-Ne
the same time in all the sample corners, which is schemati-
cally presented in Fig. 1͑a͒. That proves the good thermal
homogeneity of the sample. The corners are undoubtedly
privileged places in the sample. First, the heat exchange be-
tween the sample and the surroundings is easier there. Sec-
ondly, in the sample corner a new phase is created with the
minimum area of the phase front. Appearing nuclei join in
such a way that from eight small phase fronts, two approxi-
mately flat fronts perpendicular to the c axis are created ͓Fig.
3,36
1
͑b͔͒.
Those fronts move towards the sample center ͓Figs. 1͑c͒
and 1͑d͔͒ where they form a wedge which disappears quickly
Fig. 1͑e͔͒. Both fronts do not always move at the same
͓
speed. In the marginal cases only one front moves while the
other remains still.

PRB 60
PHASE FRONT AND DOMAINS DURING THE . . .
3809
FIG. 3. Photographs during a FP transition of the phase fronts in
a section a and of the domain texture in a c section.
1
FIG. 5. Temperature variation of the reciprocal dielectric con-
Ϫ1
stant ␧Јc
during a PF-FP thermal cycle.
laser beam propagating along the c axis. This is clear evi-
dence that the domains cannot exist when the ferroelectric
region is too small.
phase near the phase transition, the Curie-Weiss law is well
fulfilled with a constant 4250Ϯ30 K. When the phase front
appears ͓temperature T in Fig. 4͑b͔͒ the dielectric constant
1
C. Dielectric properties
4
␧
Ј increases rapidly reaching a value of the order of 10 . The
c
As demonstrated previously, due to the geometry of the
phase front, the temperature gradient inside the sample is
very small. In these experimental conditions the reciprocal
dielectric constant for the PF phase transition as a function of
temperature is presented in Fig. 4͑a͒. In the paraelectric
maximum value of ␧Ј depends on the dynamics of the do-
main structure creation and changes for successive PF phase
transitions. The phase front withdrawal from the sample ͓at
c
temperature T in Fig. 4͑b͔͒ is signalized by the decreasing
2
of ␧Ј and a peak of ␧Љ . In the ferroelectric phase the varia-
c
c
tion of the reciprocal ␧Ј versus the temperature is very regu-
c
lar except when the domain structure rearrangements occur.
A domain rearrangement brings about first of all a rapid in-
crease of ␧Ј and then its slow decrease to a value lower than
c
the one at the rearrangement beginning ͓see Fig. 4͑a͔͒. Such
rearrangements always happen at a few degrees below the
transition temperature. This cannot be the case if the cooling
does not go too deeply into the ferroelectric phase as in the
example of Fig. 5. Then the temperature dependence of ␧cЈ
obtained during cooling and heating is similar. However, ␧cЈ
values during the cooling process are always greater than
those during the heating one. Figure 5 gives an important
result: temperatures at which the phase transitions begin are
practically the same for PF cycle (T ) and for FP cycle (T ),
1
2
1
5
in relatively good agreement with Zeyen’s result. It is dif-
ficult to comment in detail on the dielectric properties during
the phase coexistence for the PF transition ͑Fig. 4͒. For the
arrangements of the domain structure which happen at the
same time the modification of the phase front shape compli-
cate the phenomena. The situation is simpler during the FP
transition. Then the domain structure rearrangements are not
observed during heating if the crystal remained in the ferro-
electric phase long enough. Figure 6 presents the variation of
␧
Ј and ␧Љ versus the temperature of a sample with a regular
c
c
homogeneous domain structure ͑a structure containing only
one domain wall orientation͒. The sample was left for 12 h at
a temperature of 200 K. During heating, ␧Ј changes monoto-
c
nously in the ferroelectric phase to a temperature of T when
2
the phase front appears ͓marked ͑a͒ in Figs. 6 and 1͔. From
FIG. 4. Temperature variation during a PF transition of the re-
ciprocal dielectric constant ␧Јc and of the loss constant ␧cЉ .
that moment the sudden increase of ␧Ј and ␧Љ is observed.
The dielectric and loss constants reach their maximum val-
c
c
Ϫ1

3810
JEAN BORNAREL AND RYSZARD CACH
PRB 60
FIG. 6. Temperature variation of ␧Ј and ␧Љ during a FP transi-
c
c
tion.
FIG. 7. The continuous line ͑c͒ corresponds to the temperature
of the platinum resistor below the sample. Curves ͑a͒ and ͑b͒ give
the temperature of the sample upper face during FP and PF transi-
tions, respectively.
ues when both fronts come off the crystal surfaces ͓marked
b͒ in Figs. 6 and 1͔ and the loss constant ␧Љ decreases
͑
c
sharply just after that. A shifting of the phase fronts towards
the sample center occurs, between ͑b͒ and ͑c͒ in Fig. 6 ͓͑b͒
and ͑d͒ in Fig. 1͔. That denotes that the length of the domain
walls in the c direction decreases, and as a consequence the
whole area of the domain wall decreases too. Between ͑c͒
and ͑d͒ in Fig. 6 in the narrow ferroelectric region a disap-
pearance of the domains takes place as shown in Fig. 3. At
parallel25 or perpendicular26 to the c-ferroelectric axis. In the
case presented the quality of the temperature homogeneity
and the results allow the three following assumptions.
The thermal gradient in the crystal can be neglected. The
main task of this paper was an observation of the phase tran-
sition ͑phase front and domain structure creation͒ as well as
the measurement of the dielectric permittivity in the condi-
tions near the thermal equilibrium of the crystal. The tem-
͑d͒ the phase front disappears which indicates the end of the
phase transition. After the withdrawal of the phase front
there is a rapid decrease of the ␧Ј value and then it changes
c
according to the Curie-Weiss law.
perature gradient G in the helium gas exchange chamber
e
To help the discussion of the results an additional experi-
ment has been carried out. The sample was put on a glass
window near the platinum resistor. Furthermore the welds of
a cupper-constantan thermocouple were glued with a silver
paste on this platinum resistor and on the upper face of the
sample. Then it was possible to measure the temperature
evolution of the platinum resistor ͓curve ͑c͔͒ and that of the
crystal ͓curves ͑b͒ and ͑a͒ for PF and FP transitions, respec-
tively, as shown in Fig. 7͔. Even in the present case with a
relatively high temperature rate ͑a few 10 K mm ), it
clearly appears that the crystal temperature stays approxi-
mately constant during the phase coexistence. It is necessary
to change the external temperature to bring thermal energy
variation for the phase transformation. As during this time
the temperature of the sample remains quasistable, a sharp
change is observed when the phase front appears or disap-
pears ͓Fig. 1͑a͒ and 1͑e͒ or ͑a͒ and ͑d͒ of Fig. 6͔. Then it is
was controlled and regulated. Its value did not exceed 0.005
K/mm. Taking into consideration the earlier discussion refer-
ring to the temperature distribution in the DKDP sample26
one can assume that the thermal gradient in the central part
of the sample was not larger than 0.002 K/mm. While chang-
ing the sample temperature with a rate of 10 mK/min it was
observed that the phase front position was a temperature
function rather than a time one. The phase front movement
ceased when the temperature was stabilized in the phase co-
existence region. One can suppose that in the described con-
ditions of the experiment a sample efficaciously changed
heat with the surrounding gas and at every moment was in
the state close to the thermal equilibrium.
Ϫ2
Ϫ1
The phase transition begins at temperature T at which
c
both phases are absolutely stable. This assumption has ex-
perimental confirmations. As earlier studies of DKDP crys-
tals have already proved, the phase transition can undergo
without thermal hysteresis.9
,15,38
Some of our observations
easier to understand why during a FP transition the ␧Ј value
c
changes abruptly just after the phase front withdrawal ͓below
also confirm such a character of the phase transition. First,
there is a slight difference between temperatures at which the
transition begins, regardless of the phase transition direction.
Secondly, a difference between T0 ͑determined from the
Curie-Weiss law͒ and Tc at which the phase transition starts
remains constant. The difference can be considered as a hys-
teresis measure. If there is no thermal hysteresis of transition,
then Tc denotes the phase equilibrium temperature hence
͑
d͒ in curve ␧Ј(T) in Fig. 6͔: the ␧Ј jump is in agreement
c
c
with the variation obtained by extrapolation of the paraelec-
tric curve until the temperature of the phase front appearance
͓
temperature T of situation ͑a͔͒.
2
IV. DISCUSSION
First let us explain the dielectric constant values already
2
⌬TϭT ϪT ϭ3B /16AC. In all our experiments, regardless
c
0
studied in DKDP crystals under notable thermal gradient
of the transition direction, the temperature difference was

PRB 60
PHASE FRONT AND DOMAINS DURING THE . . .
3811
approximately stable and was equal to ⌬Tϭ(0.954
Ϯ0.007) K. This means that if there is a thermal hysteresis
of the phase transition, it is not higher than 0.01 K.
During phase coexistence the crystal remains at tempera-
ture T . It has been known from previous literature that the
c
1
5
phase transition in DKDP crystal is an isothermal one. In
Zeyen’s experiment during the phase coexistence, the crystal
temperature was constant at the same level T , upon cooling
and heating, confirming the absence of thermal hysteresis.
One can assume that the phase transitions ͑both PF and FP͒
FIG. 8. Schematic illustration of phase fronts and domains in
paraelectric ͑P͒ and ferroelectric phase ͑F͒ and during phase coex-
istence ͑PF or FP transitions͒.
c
begin at T and the crystal remains at that temperature until
c
the end of the transition, despite the changes of external tem-
perature as illustrated in Fig. 7. After the transition, the crys-
tal reaches isotherm of the surroundings.
The electric permittivity of a single domain crystal in an
isothermal state can be calculated based on the Landau
As from the present measurements one has obtained (Tc
ϪT )ϭ(0.954Ϯ0.007) K,
thus
⌰ϭ(1.353Ϯ0.015)
0
8
ϫ10 V m/C. This ⌰ value corresponds precisely to values
1
0
5
3
13
9
5
with Bϭ4.5ϫ10 V m /C and Cϭ1.5ϫ10 V m /C
known from previous studies.1
2,13
theory. Comparing calculated and measured values of ␧Ј we
Geometry of phase distribution during the phase coexist-
ence is illustrated in Fig. 8. Taking as first approximation a
model of plate capacitors, the electric permittivity of the
sample can be written as follows:
c
can obtain some information about domain structure and
phase front effect on the dielectric properties of the crystal.25
In the paraelectric phase the reciprocal electric permittivity is
a linear function of temperature ͑the Curie-Weiss law͒
1
V /V V /V
F
P
ϭ
ϩ
,
͑7͒
1
␧
Ј
␧Ј
␧Ј
F
P
ϭA͑TϪT ͒.
͑2͒
0
␧
Ј
P
where V , V denote volumes of ferroelectric and paraelec-
F
P
7
For the studied crystal A(ϭ2.66Ϯ0.01)ϫ10 Vm/C K has
been found.
tric phases and V is the total volume of the sample. ␧Ј and
F
␧ЈP stand for permittivity of the ferroelectric and paraelectric
In the ferroelectric phase the electric permittivity changes
with temperature according to the following equation:
phase at T , respectively,
c
1
1
3 B2
B2
B
ϭ
ϭ
ϭ4A͑T ϪT ͒,
c 0
1
T͒ϭ Ϫ4A͑TϪT ͒ϩ ͱB2Ϫ4AC͑TϪT ͒
␧Ј ␧Ј͑T ͒ 4 C
F
F
c
͑
0
0
␧
Ј
C
C
F
1
1
3 B2
2
A1/2
C1/2
B
ϭ
ϭ
ϭA͑T ϪT ͒.
͑8͒
͑T*ϪT͒1/2
c
0
ϭ4A͑T*ϪT͒ϩ
.
͑3͒
␧Ј ␧Ј͑T ͒ 16 C
P P
c
At the constant rate of cooling or heating, volumes VF and
Usually to trace that dependence we must know the values of
25
V change, more or less linearly with external temperature.
P
B and C coefficients apart from values A and T . Strictly
0
Hence T denotes the border of phase coexistence from the
2
1
speaking, we need ⌰ϭB /C value because Eq. ͑3͒ can be
paraelectric phase, and T from the ferroelectric side thus
2
written as follows:
VF ͑T ϪT͒
VP ͑TϪT2͒
1
1
ϭ
,
ϭ
,
͑9͒
2
2
͑
T͒ϭ⌰ Ϫ4A͑TϪT ͒ϩ
ͱ
⌰ Ϫ4A⌰͑TϪT ͒, ͑4͒
V
͑T1ϪT2͒
V
͑T1ϪT2͒
0
0
␧
Ј
F
and
the ⌰ parameter can be found knowing phase transition tem-
perature T and temperature T as
c
0
1
͑T ϪT͒͑1/␧Ј͒ϩ͑TϪT ͒͑1/␧Ј ͒
1
F
2
P
͑
T͒ϭ
.
͑10͒
B2 16
␧Ј
͑T ϪT ͒
1
2
⌰
ϭ
ϭ
A͑T ϪT ͒.
͑5͒
c
0
C
3
All parameters needed for the calculations are obtained from
the present measurements: A and T from temperature de-
0
This means that the electric permittivity for the ferroelectric
phase can be calculated as a function of temperature when
coefficient A value and temperatures T and T are known.
pendence of ␧Ј in the paraelectric phase and T is a tempera-
c
c
ture at which the phase transition begins (T ϭT for PF and
c
1
0
c
0
T ϭT for FP transitions͒.
c
2
The dependence is the following:
In Fig. 9͑a͒ the experiment results are compared with the
values obtained from the above equations (␧Јmono). At tem-
perature T , both ␧Ј ͑measured͒ and ␧Ј
change abruptly.
1
16
͑
T͒ϭ A͑T ϪT ͒Ϫ4A͑TϪT ͒
c
0
0
1
mono
␧
Ј
3
F
It can be explained by a rapid change of the crystal tempera-
2
ture from T ϭT to T which is the actual temperature in the
2
c
1
1
6
64
3
2
ϩ
ͱ
ͫ
A͑T ϪT ͒
ͬ
Ϫ
A ͑T ϪT ͒͑TϪT ͒.
chamber. It seems to prove the hypothesis about the isother-
mal character of the phase transition. Figure 9͑b͒ represents
the difference (␧ЈϪ␧Јmono) which can be treated as a contri-
c
0
c
0
0
3
͑6͒

3812
JEAN BORNAREL AND RYSZARD CACH
PRB 60
FIG. 9. ͑a͒ Temperature variation of the measured ␧Ј and the
FIG. 10. ͑a͒ Comparison between the ␧Ј(T) curve and the cal-
c
c
calculated monodomain value ␧Јmono ͑continuous line͒ during a FP
transition. ͑b͒ Temperature variation of the difference ␧ЈϪ␧Ј
culated curve ␧Јmonoϩ␧Ј versus T. ͑b͒ Calculated contribution in the
dielectric constant of the phase front existence versus T during a FP
d
.
mono
c
T and T are the temperatures where the phase fronts appear and
transition.
2
1
disappear, respectively.
with temperature in Fig. 9͑b͒ does not result from change in
domain structure geometry. The ratio between the measured
bution of the domain structure and the phase front existence
to the electric permittivity. Figure 10 shows how the electric
permittivity should change if in the phase coexistence region
␧
cЈ and ␧mЈ ono calculated according to the Landau theory is
presented in Fig. 11. It is natural that the ratio in the
paraelectric phase equals 1. This simply means that the
Curie-Weiss law is fulfilled. However, unexpectedly, for the
FP cycle in the ferroelectric phase, the ratio is practically
constant having the value 3.8. One can expect that the value
␧
Ј were the same as measured at temperature T . The solid
F
2
line in Fig. 10͑a͒ represents permittivity taking into account
the contribution of the domain structure (␧Јmonoϩ␧Ј). Figure
d
1
0͑b͒ reflects the difference ␧ЈϪ(␧Јmonoϩ␧Ј), showing dis-
d
tinctly that permittivity in the phase coexistence region is
higher than could be expected.
of ␧Ј/␧Ј
depends on domain structure geometry, which is
c
mono
confirmed by the results presented in Fig. 12. In this case of
At this stage it is difficult to form unique interpretation.
However comments are possible. First let us discuss the ␧cЈ
values in the ferroelectric phase. It is generally accepted that
in a case of a single domain state, the dielectric constant ␧cЈ
is well described by Eq. ͑3͒. Furthermore, it has already been
the PF cycle, the results of Fig. 4 are used. Again ␧Ј/␧Ј is
c
mono
suggested to add an additional term ␧ which is the domain
d
3
3
structure contribution. This term depends on the density of
2
,33,36,39
the domain walls
and clearly on temperature as illus-
trated in Fig. 9. It has been well known for a long time that
the domain contribution ␧Ј is always greater during a cool-
d
3
3,34
ing cycle than during a heating one:
the density of the
domain walls is greater in the cooling cycle. In addition, this
property has been used to obtain regular domain textures at a
given temperature with different domain wall density using
different temperature cycles.34 But if this result is well
known, what is the dependence between ␧Ј and the density
d
of domain walls? Because during the FP cycle the domain
FIG. 11. Temperature variation of the ratio ␧Ј/␧Ј during a FP
c mono
structure has not undergone visible changes, ␧Ј variation
transition.
d

PRB 60
PHASE FRONT AND DOMAINS DURING THE . . .
3813
transition. It was already clearly demonstrated in the KDP
family crystals that at low temperatures below freezing, ␧dЈ
and the corresponding losses ␧Љ were proportional to the
d
total domain wall area. That proved that there is no lateral
interaction of the domain walls at this temperature range.34
The domain freezing corresponds to a region where the do-
main tips begin to move longitudinally.29 For the above tem-
perature the natural domain texture is not often the equilib-
rium one. The reason is that during the cooling, the thermal
energy variation is not sufficient to create domain rearrange-
ment. Then, small thermal or electric hysteresis cycles allow
the simplification of the domain texture with the decrease of
␧
Ј and ␧Љ values. Domains are in strong lateral interactions
c
c
and as consequence great permittivity values at zero field as
show by the thermal noise method. Collective phenomena
3
6
between domains have been experimentally demonstrated
such as the nonlinear response in dielectric29 and piezoelec-
tric responses. The present results concern a temperature
range near the transition where domain rearrangements are
easier. The study should be continued to obtain the correla-
tion between the domain texture ͑density, arrangement͒ and
the ␧Ј and ␧Љ constants.
c
c
The Fig. 10 result is obtained with hypothesis a ␧Ј value
F
during the phase coexistence equal to ␧Ј(T ) given by the
F
c
relation ͑8͒. Our experimental results are in agreement with
the hypothesis of a quasiconstant temperature T in the crys-
c
tal during the transition obviously when the temperature rate
Ϫ2
Ϫ1
is small ͑lower than 10 K min ). But it is open to discus-
FIG. 12. Temperature variations of the ratio ␧Ј/␧Ј
PF transition.
during a
c
mono
sion that the ␧Ј contribution should be the same for a given
d
sample volume at the temperature T in both situations F and
c
FP of Fig. 8: in F the limits of the domains in a section ͑c͒
are the sample electrodes and in FP these limits are the phase
fronts. It can be suggested that the pinning of the quasidis-
locations inside the wall is not the same with possible differ-
ences in the domain density and in the lateral motions of the
domain walls under an ac electric field. Furthermore phase
fronts can be regions with space charges electric response of
which is difficult to predict. To progress on these problems,
experiments are actually performed to detect the possible in-
teraction between domains and phase fronts, and between the
phase fronts themselves. The action of a small dc electric
field on the phase front by different optical methods is also
studied in these good thermal conditions. One can see that
the phase front at the microscopic level is an open problem.
But the result demonstrated in the present paper is clear: the
existence of the phase front inside the sample creates a spe-
cial contribution in the permittivity value which can be
added to the monodomain contribution (␧Ј or ␧Ј) and the
practically constant in the region of a few degrees below the
phase transition where great domain structure arrangements
are not observed. When a notable rearrangement happens as
near 210 K, an abrupt increase of ␧Ј occurs but with time it
c
decreases and ␧Ј/␧Ј
reaches a new quasiconstant value
c
mono
lower that the value of the beginning. The rearrangements of
the domain texture always correspond to a decrease of the
domain wall density. It is now possible to describe the
known decrease of ␧Ј versus temperature with the following
c
expression:
␧
␧
Јc ϭ⌳ϫ␧Ј
mono
Јϭ␧Ј ϩ␧dЈ
͑11͒
͑12͒
c
mono
with
␧
dЈϭ͑⌳Ϫ1͒␧mЈ ono
,
F
P
extrapolated domain contribution ␧Ј for ferroelectric regions.
d
where the coefficient ⌳ grows with the domain wall density.
Further experiments will be necessary to obtain the de-
pendence between the ration ␧Ј/␧Ј
mains. For example, dielectric measurements may be simul-
and the density of do-
c
mono
V. CONCLUSION
taneously performed with diffraction of a laser beam to
Optical observations of phase fronts and domains during
the DKDP phase coexistence with a high homogeneity in the
sample temperature have been performed. They demonstrate
that usually the transition happens with the help of the cre-
ation and motion of two quasiplanar phase fronts only almost
perpendicular to the c-ferroelectric axis. The ferroelectric re-
gion remains quasimonodomain if the distance between a
attempt the mean domain width.3
6,37,40,41
Small electrical
hysteresis cycles at a given temperature will be useful to
ensure that the observed domain texture is the equilibrium
3
6
one. In the same way thermal cycles may be used to obtain
3
4
similar results. To conclude, quantitative information is ob-
tained on the domain structure contribution in ␧Ј near the
c

3814
JEAN BORNAREL AND RYSZARD CACH
PRB 60
phase front and the c sample boundary or between the phase
fronts themselves is lower than a few tens of micrometers. In
the other cases, domains appear in the ferroelectric region
often with walls parallel to one of both permissible orienta-
mains ␧Ј , as observed in the ferroelectric phase at lower
temperatures, and the contribution due to the phase front
d
existence during the phase coexistence. The ␧Ј contribution
d
follows in the temperature range of a few degrees below Tc
tions ͑100 or 010͒. The dielectric constant ␧Ј can be consid-
a simple law of variation versus the temperature ␧Јϭ(⌳
c
d
ered as the sum of three contributions: the contribution of a
Ϫ1)␧Ј with ⌳ function of the domain texture. ⌳ decreases
F
monodomain sample (␧Ј or ␧Ј), the contribution of the do-
with decreasing density of domains.
P
F
1
2
3
19
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