Hysteretic phenomena and switching effects under phase transitions in external field
The field dependence of the order parameter is investigated theoretically, using a statistical approach within the framework of the Ginzburg-Landau model with respect to the ferroelectric phase transition. It is found that, along with the existence of ordinary non-switching hysteresis in not too hig...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
1999
|
| Назва видання: | Физика низких температур |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/129713 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Hysteretic phenomena and switching effects under phase transitions in external field / L.I. Stefanovich // Физика низких температур. — 1999. — Т. 25, № 1. — С. 33-37. — Бібліогр.: 5 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-129713 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1297132018-01-28T03:02:39Z Hysteretic phenomena and switching effects under phase transitions in external field Stefanovich, L.I. Низкотемпеpатуpный магнетизм The field dependence of the order parameter is investigated theoretically, using a statistical approach within the framework of the Ginzburg-Landau model with respect to the ferroelectric phase transition. It is found that, along with the existence of ordinary non-switching hysteresis in not too high fields ε(ε<<εc, where εc is the coercive field), hysteresis of a switching type can be observed at the fields in excess of the coercive one. 1999 Article Hysteretic phenomena and switching effects under phase transitions in external field / L.I. Stefanovich // Физика низких температур. — 1999. — Т. 25, № 1. — С. 33-37. — Бібліогр.: 5 назв. — англ. 0132-6414 PACS: 77.80.Dj, 77.80.F http://dspace.nbuv.gov.ua/handle/123456789/129713 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
| spellingShingle |
Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Stefanovich, L.I. Hysteretic phenomena and switching effects under phase transitions in external field Физика низких температур |
| description |
The field dependence of the order parameter is investigated theoretically, using a statistical approach within the framework of the Ginzburg-Landau model with respect to the ferroelectric phase transition. It is found that, along with the existence of ordinary non-switching hysteresis in not too high fields ε(ε<<εc, where εc is the coercive field), hysteresis of a switching type can be observed at the fields in excess of the coercive one. |
| format |
Article |
| author |
Stefanovich, L.I. |
| author_facet |
Stefanovich, L.I. |
| author_sort |
Stefanovich, L.I. |
| title |
Hysteretic phenomena and switching effects under phase transitions in external field |
| title_short |
Hysteretic phenomena and switching effects under phase transitions in external field |
| title_full |
Hysteretic phenomena and switching effects under phase transitions in external field |
| title_fullStr |
Hysteretic phenomena and switching effects under phase transitions in external field |
| title_full_unstemmed |
Hysteretic phenomena and switching effects under phase transitions in external field |
| title_sort |
hysteretic phenomena and switching effects under phase transitions in external field |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
1999 |
| topic_facet |
Низкотемпеpатуpный магнетизм |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/129713 |
| citation_txt |
Hysteretic phenomena and switching effects under phase transitions in external field / L.I. Stefanovich // Физика низких температур. — 1999. — Т. 25, № 1. — С. 33-37. — Бібліогр.: 5 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT stefanovichli hystereticphenomenaandswitchingeffectsunderphasetransitionsinexternalfield |
| first_indexed |
2025-07-09T12:02:02Z |
| last_indexed |
2025-07-09T12:02:02Z |
| _version_ |
1837170713886195712 |
| fulltext |
Fizika Nizkikh Temperatur, 1999, v. 25, No 1, p. 33–37L. I. StefanovichHysteretic phenomena and switching effects under phase
transitions in external fieldL. I. StefanovichHysteretic phenomena and switching effects under phase transitions in external field33
Hysteretic phenomena and switching effects under
phase transitions in external field
L. I. Stefanovich
A. Galkin Physics and Engineering Institute, National Academy of Sciences of Ukraine,
72 R. Luxemburg St., 340114, Donetsk, Ukraine
E-mail: listef@host.dipt.donetsk.ua
Received July 13, 1998
The field dependence of the order parameter is investigated theoretically, using a statistical approach
within the framework of the Ginzburg-Landau model with respect to the ferroelectric phase transition.
It is found that, along with the existence of ordinary non-switching hysteresis in not too high fields ε
(ε << εc , where εc is the coercive field), hysteresis of a switching type can be observed at the fields in
excess of the coercive one.
PACS: 77.80.Dj, 77.80.Fm
The influence of a quasi-stationary external field
on the formation and growth dynamics of 180°-do-
mains was investigated in [1] within the framework
of the Ginzburg-Landau model, using a statistical
approach.
Our goal in this paper is to follow the evolution
of the domain structure which has already been
formed in ferroelectrics depending on the magnitude
and the direction of the external field.
To characterize quantitatively the change of a
system state passing through the critical tempera-
ture point Tc of a phase transition, one or more
values (called order parameters) are introduced. In
the case of the ferroelectric phase transition the
projection of the polarization vector on a certain
crystallographic direction is used as a long-range
order parameter η(r, t).
To describe the relaxation processes taking place
in the system undergoing the phase transition, there
is no need to know the spatial distribution of order
parameter η(r, t) in detail over the total macro-
scopic specimen. Therefore, in the subsequent dis-
cussion, as before [1], we shall deal with the search
of the main physically significant characteristics of
this function, such, for example, as the average
(over the crystalline grain) value of the order pa-
rameter 〈η(r, t)〉 ≡ η
__
(t) and two-point correlation
function
〈ξ(r, t) ξ(r′, t)〉 ≡ K(s, t) , s = r − r′ , (1)
where we have introduced the centered order pa-
rameter ξ(r, t) ≡ η(r, t) − η
__
(t). The equations des-
cribing the temporal evolution of η
__
(t) and K(s, t) on
all stages of ordering were introduced in [1] on the
base of Ginzburg–Landau functional for nonequili-
brium order parameter.
Our interest in this work is with the behavior of
the order-disorder system for times which are
greater in comparison with the time of forming
(τd), but much less than the time of coalescence
(τc) of the domain structure (τd << t << τc). There-
fore, in the subsequent analysis for field dependence
of the order parameter η
__
= η
__
(ε) we shall use the
asymptotic system of equation obtained in Ref. 1
dη
__
dt
=
1
2
[(α − 3D(t) − η
__
2) η
__
+ ε] ,
dD(t)
dt
= (α − D(t) − 3η
__
2) D(t) ,
(2)
where D(t) ≡ K(0, t); α = (Tc − T)/Tc is a dimen-
sionless parameter which characterizes the proxi-
mity of temperature T, up to that the specimen
cooled, to the ordering temperature Tc ; ε = E/Tc is
an external field in corresponding units.
The singular (stationary) points of the system
(2) correspond to the thermodynamic equilibrium
states of ordering system and can be found from the
conditions of vanishing the first derivatives with
respect to time on the left sides of the Eqs. (2) as
t → ∞. Hence it follows that the system of two
© L. I. Stefanovich, 1999
algebraic equations in variables η
__
and D takes the
form
ε + αη
__
− 3Dη
__
− η
__
3 = 0 ,
(α − D − 3η
__
2)D = 0 .
(3)
The roots of the above-mentioned system yield the
coordinates of singular points in the plane (η
__
, D).
In this work we shall consider only the case
T < Tc , i.e., α > 0, when there is a whole system
of singular points. However, only the points located
in the upper half-plane of the phase pattern (in
variables η
__
, D) will have a physical sense. Recall
that the positions of these singular points, in the
case of weak field (ε << α3/2), were found by us in
Ref. 1.
The first singular point I [η
__
= −(ε/α), D = 0],
which corresponds to a homogeneous disordered
state, is an unstable node (Fig. 1).
The second point II (η
__
= √α + (ε/2α), D = 0)
corresponds to a homogeneous ordered state «alig-
ned with the external field» and is a stable node
(Fig. 1).
The third point III (η
__
= −√α + (ε/2α), D = 0)
corresponds to homogeneous ordering, but it is
«opposite to the field». This point is also a stable
node (Fig. 1).
Both the second and the third singular points
correspond to single-domain type of the specimen
ordering.
All the rest of the singular points (with D ≠ 0)
correspond to inhomogeneous ordering, i.e., to one
or another polydomain structures.
The fourth point IV [η
__
= ε/2α; D =α −(3ε2/4α2)]
(Fig. 1) is a stable node and corresponds to the
possibility of a polydomain structure realization.
The fact that η ≠ 0 in this state corresponds to some
non-equivalency of domains of two types. However,
in weak fields this distinction is small (to the extent
of the ratio ε/α).
Finally, there are another two singular points of the
saddle type. One of them V («right saddle») with
coordinates [η
__
= √α/2 − ε/4α; D = α/4+3ε/(4√α)]
(Fig. 1) corresponds to a possibility for realizing such
a quasi-stationary polydomain structure, where the
volume fraction of domains of the same type (e.g.,
with polarization vector, aligned with field) substan-
tially exceeds the volume fraction of the other type
domains (opposed to the field).
The other point («left saddle») [η
__
= −√α/2 −
− ε/4α; D = α/4 − 3ε/(4√α)] (the point VI, Fig. 1)
corresponds to a possibility for realizing the quasi-
stationary polydomain structure as well. However,
in this situation the volume fraction of domains
with the polarization vector oriented opposite to the
field substantially exceeds a volume fraction of
domains aligned with the field.
Two separatrices, leaving the origin of coordi-
nates and passing through the «left» and «right»
saddle points, divide the phase pattern into the
three parts. The upper central sector (1) is the
«attraction region» of inhomogeneous (polydo-
main) state, the lower right-hand (2) and lower
left-hand sectors (3) correspond to two «attraction
regions» of homogeneous single-domain states.
In the context of this work our interest will not
be with the locations of singular points by itself on
a phase pattern at fixed value of external field, but
their evolution and bifurcation under changes of a
value and direction of external field ε. It is sug-
gested that the field changes slowly enough (quasi-
statically) in comparison with time of polydomain
structure formation (τε >> τd). Recall that for defi-
niteness we shall consider a phase transition in
ferroelectrics connected with the appearance of a
spontaneous polarization at T < Tc .
In order to follow the evolution of singular
points on the phase pattern depending on the field
ε we direct our attention to the system (3). From
the second equation of a system it follows that there
are solutions of it satisfying the condition D(t) = 0.
In this case the first equation of the system (3)
takes the form
Fig. 1. Phase pattern of order–disorder system. The singular
points of the system (2) at ε = 0 are marked by the crosses (××);
the singular points (I–VI) for the system (2) at ε ≠ 0 are
marked by circles (❍) and separatrices are indicated by dotted
lines. The arrows indicate the directions of motion for the sin-
gular points with increasing the external field. The bifurcation
points (at ε = εc) for the system (2) (ηc and Kc) are shown by
black circles (●).
L. I. Stefanovich
34 Fizika Nizkikh Temperatur, 1999, v. 25, No 1
ε = η
__
3 − αη
__
. (4)
The right-hand side of (4) has extremes in two
symmetric points with respect to the ordinate axis
η
__
c
(1,2) = ± √α/√3 . (5)
The positive value of parameter (η
__
c > 0) corre-
sponds to the minimum critical field
εc
min = −
2α3/2
3 √3
, (6)
and the negative value of parameter (η
__
c < 0) corre-
sponds the maximum critical field
εc
max =
2α3/2
3 √3
. (7)
Since, while writing down the thermodynamic po-
tential (1) in [1] we assumed the at ε > 0, then it
is natural to consider only the maximum critical
field (εc ≡ εc
max), which we shall call critical field
simply. To elucidate its physical sense, we consider
what will happen with the singular points of the
system (2) under smooth variation of external field
beginning from values ε << εc .
Studying a behavior of the algebraic system (3)
roots it is easy to verify that as the field ε is
increased, the singular points I, III and VI located
in the second quadrant begin to approach each other
gradually. Simultaneously, the mutual approach of
the singular points IV and V occurs. And only the
second singular point (II), corresponding to the
homogeneous (monodomain) ordering of the system
(crystallite) as a whole, will recede further and
further from the rest of the singular points, moving
towards the right along the abscissa axis.
The approach of the above-mentioned groups of
singular points will cause the sectors 1 and 3 to
decrease (Fig. 1). It testifies that the potentialities
both for the polydomain and for the monodomain
ordering «opposite to the field» will be decreased
gradually. The sector 2, on the contrary, grows to
extend the range of conceivable initial conditions,
beginning with these, the system will be ordered
homogeneously, «aligned with the field». That is,
the external field specifies the direction of the
preferential ordering.
When the critical value of the field ε = εc is
achieved, the confluence of the singular points I, III
and VI into a single point ηc will occur. Simultane-
ously, i.e., at the same value of critical field, the
singular points IV, V are merged into the singular
point Kc with the coordinates (1⁄2 √α/3; 3α/4). At
the same moment (i.e., in merging two groups of
singular points) the sectors 1 and 3 disappear on the
phase pattern (Fig. 1). In other words, at the
critical value of the external field bifurcation of the
ordering system trajectories takes place.
The further increasing of the external field up to
the values ε > εc gives all the singular points to
disappear, except point II. The fields higher than
critical ones should be considered to be strong.
Therewith, in the range of the strong field
(ε >> εc), the average value of order parameter in
the single-domain state does not depend on tem-
perature to a first approximation, and it is defined
only by the value of the field
η
__
~ ε1/3 . (8)
Let us consider the manner in which the charac-
ter of the ferroelectric domain structure will vary
under the influence of the external field. With this
in mind we shall trace the variation of the order
parameter average value η
__
in relation to the magni-
tude and the direction of the applied field ε, identi-
fying η
__
with the average polarization of the speci-
men and the field ε — with the uniform electric
field.
Analysis will be based on treating the behavior
of the solutions for the system of Eqs. (3) under
slow, continuous variation of the parameter ε, i.e.,
on the study of the influence of the external field on
the evolution of the ordering system singular points
on the phase pattern (Fig. 1).
Initially let us take up the manner in which the
ordering system which in the polydomain state
corresponds to the singular point IV on the phase
pattern (Fig. 1), behaves oneself with time in a
weak varying external field. If the external field is
changed slowly (quasi-statically) (i.e., τε >> α−1),
according to the phase diagram (Fig. 1) at the small
deviations of field magnitude from zero, the system
is limited by the polydomain state. The external
field brings into disbalance between volume frac-
tions of the phases: oriented «opposed to the field»
and «aligned with the field» in favour of the latter.
In this case under small changes of the field
(|ε| << εc) the order parameter η
__
turns out to be
proportional to the field
η
__
~ ε/α . (9)
The inhomogeneities of the order parameter are
therewith smoothed out by the external field
D ≅ α − 3ε2/4α2 . (10)
Decreasing the dispersion of the order parameter, in
accordance with (10), appears to be a second order
effect in the field.
Hysteretic phenomena and switching effects under phase transitions in external field
Fizika Nizkikh Temperatur, 1999, v. 25, No 1 35
On further increasing the external field, as soon
as the dependence η
__
= η
__
(ε) ceases to obey the linear
law (9), the order parameter η
__
in cyclic change of
the field will not follow the line OAOA’. This
change of the order parameter will have the form of
a hysteretic loop (dotted line in Fig. 2). The hyste-
resis involved (in the limits of the polydomain
states) has nothing to do with the switching one
and it will appear to be more significant if the
amplitude of varying of the external field is larger
(naturally, within the area |ε| < εc).
However such a behavior of the order parameter
η
__
as a function of the slowly varied external field
ε takes place only in the range of relatively weak
field, when |ε| << εc . As soon as the external field
reaches its critical value εc , the confluence of the
singular points IV and V on the phase pattern (Fig. 1)
into a single singular point Kc , with coordinates
(1⁄2 √α/3, 3α/4), occurs. Just at such a value of the
external field (ε = εc) any polydomain structure ap-
pears to be absolutely unstable with respect to
further growth of field. In stronger fields (ε > εc)
the singular points IV, V disappear at all, that is
any polydomain state becomes energetically dis-
advantageous and order-disorder system switches
sharply (practically by a leap) to the thermody-
namically stable monodomain state oriented «with
the field», with the average value of the order
parameter η
__
= 2√α/3 and with dispersion equal to
zero (D = 0). This single-domain state corresponds
to the singular point II on the phase pattern (Fig. 1).
Further increasing the field will cause polariza-
tion, in accordance with (8), i.e., the average value
of order parameter η
__
at the section of the curve
from B to C (Fig. 2) grows insignificantly (~ε1/3).
The field dependence of the order parameter
η
__
= η
__
(ε) follows the curve CB (Fig. 2) as the value
of the field decreases, i.e., decreasing the average
value of the order parameter up to the field value
ε = εc in a reversible way. However, when the field
ε becomes less than a critical one (ε < εc), the
average value of the order parameter does not re-
turn to the polydomain branch OA, i.e., the order-
ing character of a specimen remains homogeneous
(monodomain). Furthermore, as the field magni-
tude decreases, the average value of the order pa-
rameter will continue to decrease monotonely, fol-
lowing the curve BE (Fig. 2) and attaining the
residual magnitude √α in total switching off the
field (ε = 0). To relieve this residual value of the
order parameter η
__
, i.e., to switch the specimen into
the alternative ordering state, it is enough to
change the sign of the field ε. In increasing the field
of the opposite polarity in the range of values
−εc < ε < 0 the further decreasing of the average
order parameter to the value √α/3 (the curve ED
on Fig. 2) will proceed. The ordering character will
not be changed here until the critical value of the
field −εc has been attained. At the part of the curve
from E to D, when the external field ε and the
order parameter η
__
have the opposite polarity, the
states of the system are unstable thermodynami-
cally, it would be advantageously energetically for
the system, if the sign of the order parameter
coincided with the field sign. In other words, the
section ED (as well as E′D′) corresponds to the
metastable states of the order–disorder system.
The further increasing of the external field mag-
nitude will produce «the frustration» of the value
η
__
to the magnitude −2√α/3 (the point B′, Fig. 2).
This fact indicates the switching of the crystallite
from the monodomain state with one direction of
the spontaneous polarization into a state like this
but with opposite direction of polarization in re-
sponse to the electric field over the whole volume
simultaneously. Thus the field −εc is nothing but a
coercive field. In accordance with (7), the magni-
tude of this coercive field depends on temperature
through the parameter α = (Tc − T)/Tc , which cha-
Fig. 2. Field dependences of the order parameter average value
η
__
: the ordinary nonswitching hysteresis is indicated by a dotted
line; the hysteresis of switching type is marked by a solid line.
L. I. Stefanovich
36 Fizika Nizkikh Temperatur, 1999, v. 25, No 1
racterizes a proximity of the system considered to
the phase transition critical point. To evaluate the
coercive field εc obtained by us theoretically for
really achieved values of α, one should go to corres-
ponding dimensional notations in (7). The transi-
tion in triglycine sulphate crystal is one of the most
completely studied ferroelectric second order phase
transitions. The departures of behavior answering to
Landau theory are not observed in this ferroelectrics
up to α ~ 10−4–10−5 [2]. For these values of α it
turns out that the magnitude of coercive field Ec
found by us will achieve quite large values ~ 50–
100 kV/cm. However, the experimental investiga-
tions of ferroelectric hysteretic phenomena in the
fields such as these or even larger ones (E ~ 106 V/cm)
are known (see, e.g., [3,4]).
The analysis of the following order parameter
average value η
__
behavior depending on magnitude
and direction of the applied external field indicates
that it is described by the curve C′B′E′D′ (Fig. 2),
which is nothing but symmetric doubling of the line
CBED with respect to the origin of the coordinates.
When the field value εc has been reached, the
order-disorder system falls from the point D′ to the
point B′, i.e., it switches from one monodomain
state into another one. Thus, the large loop BDB′D′,
known as the switching hysteresis, is closed.
However it should be borne in mind that the
hysteresis pattern described theoretically above is
restricted to the idealized order-disorder system. It
is unlikely that «the repolarization» of the system
from one monodomain state to another monodomain
one (under the changes of an external field sign)
will proceed over the whole volume of the specimen
(crystallite) simultaneously. Actually in a volume
of monodomain crystal the energetically «advanta-
geous» regions with the opposite signs of the order
parameter occur in a fluctuational way and grow
rapidly [5]. In other words, along the way from one
monodomain state to another monodomain one the
system passes through a number of the intermediate
polydomain states. Besides, different defects and
imperfections of the crystalline structure give rise to
the obstacles to the free motion of the domain
boundaries and for their passing out of the volume.
All this will make the hysteresis loop to narrow and
«to smooth out», so that, for example, the coercive
field may turn out to be less than the value Ec
obtained theoretically.
The author thanks to Prof. E. P. Feldman for
fruitful discussions. This work was partially sup-
ported by the State Foundation of Fundamental
Investigation of Ukraine, Grant No. 2. 4/220–97.
1. L. I. Stefanovich, Fizika Nizkikh Temperatur 24, 856
(1998).
2. M. E. Lines and A. M. Glass, Principles and Application of
Ferroelectrics and Related Materials, Clarendon Press,
Oxford (1977).
3. O. E. Fesenko, R. V. Kolesova, and Y. G. Syndeyev, Fer-
roelectrics 20, 177 (1978).
4. S. Sawada, T. Yamaguchi, and H. Suzuki, Ferroelectrics 63,
3 (1985).
5. B. A. Strukov and A. P. Levanyuk, Physical Principles of
Ferroelectric Phenomena in Crystals [in Russian], Nauka,
Moscow (1995).
Hysteretic phenomena and switching effects under phase transitions in external field
Fizika Nizkikh Temperatur, 1999, v. 25, No 1 37
|