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Domain theory of polarization echoes in ferroelectrics
U. Kh. Kopvillem, S.V. Prants
To cite this version:
U. Kh. Kopvillem, S.V. Prants. Domain theory of polarization echoes in ferroelectrics. Journal de Physique, 1982, 43 (3), pp.567-574. �10.1051/jphys:01982004303056700�. �jpa-00209427�
Domain theory of polarization echoes in ferroelectrics
U. Kh. Kopvillem and S. V. Prants
Institute of Oceanology, Radio St. 7, Vladivostok 32, 690032, U.S.S.R.
(Reçu le 11 septembre 1980, révisé le 12 octobre 1981, accepté le 24 novembre 1981)
Résumé. 2014 Une classification des signaux d’écho est proposée. Pour ce faire, il est nécessaire de connaître à la fois la nature physique des particules ou micro-objets qui conservent la mémoire de la phase et peuvent engendrer l’écho, la nature physique des impulsions d’excitation et de réponse, la mobilité des particules qui produisent l’écho
et enfin quelle est la transformation qui permet aux signaux dans l’échantillon d’être détectés dans le récepteur.
Une telle identification a été faite pour les échos de polarisation dans les monocristaux ferroélectriques.
Abstract 2014 A classification of echo type signals is proposed. It is shown that to classify an echo one must know the following : the physical nature of the particles or the objects that possess phase memory and generate the echo signal, the physical nature of the exciting pulses and of the response signal, the mobility aspects of the particles
that generate the echo and the real correspondence between the physical nature of the echo signal and the signal
in a receiver. After all of these questions have been answered the nature and type of the echo may be said to be identified These rules are illustrated by an example of the polarization echoes in ferroelectric monocrystals.
Classification Physics Abstracts
77 . 80D
Introduction. - Many echo signals of various phy-
sical nature have been discussed in the literature. It
seems that many other echo-type processes will be found in solids, liquids, plasmas and gases over the
range from zero up to intranuclear frequencies.
A controversy exists in the literature concerning the question whether a newly discovered echo signal is really a new phenomenon or a known effect in a new
substance. The complexity of the problem is caused by the fact that contrary to the echo spin case, where
no one doubts that the echo is generated by spins,
in non-spin cases it is very difficult to identify the
nature of the objects that generate the echo. Some authors do not deal with this question and investigate
echo processes from the point of view of practical
and technical values. Nevertheless no complete investi- gation may be performed without the knowledge of
the nature of the objects that generate the echo signals.
In our paper this question will be discussed and as an
example the nature of polarization echoes in ferro- electrics will be considered
All echo signals are caused by coherent spontaneous emission of radiation and so quantum mechanics is needed for their self-consistent description. The other
reasons for such a description are the simplicity of
the Heisenberg picture of quantum mechanics and the
possibility of incorporating microscopical constants
into the theory in a straightforward way. Also spon- taneous quantum mechanical noise of pure quantum
origin is automatically present in this formulation.
It is not clear from the beginning if the centres that give the signals of polarization echoes are so large
that quantum effects may be ignored. Nevertheless a
classical description of echo effects is possible but in
our opinion it is less convenient than the quantum statistical formalism and we will not further discuss the classical theory of echo phenomena.
1. Echo classification principles. - An echo is the
signal spontaneously generated by a system as a result of a process of operational « thinking » over an infor-
mation introduced before into the system by means
of external perturbations. In the most simple case
two pulses of an arbitrary physical field are applied
to a sample at times t = 0 and t = i. After each
pulse the sample is found to radiate a signal of the free spin induction type. At the times t = mi, m = 2, 3, ...
the sample coherently radiates the echo signals.
The following types of echoes are known in physics : a) temporal echoes and b) spatial-temporal echoes.
The former is of the impulsive type and so the external perturbations, the process of operational thinking
and the echo generation take place in the same spatial region but in various time moments. The latter is of
impulsive or continuous type but the external pertur- bations, the process of operational thinking and the
echo generation take place in various spatial regions
and time moments.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004303056700
568
The main echo characteristic is the physical nature
of the objects performing the process of operational thinking. It is reasonably to introduce other echo characteristics : the physical nature of the external
perturbations, of the operational thinking and of the
echo signals. For the first time these characteristic classificational principles were discussed in [1] and
have been developed in successive publications [2-4].
From the mathematical point of view the classi-
fication is based on the theory of abstract quantum dynamics [2-4]. So we call the most important features
of the task in question, namely, the fundamental characteristics of an abstract dynamics are the fol- lowing : the symmetry properties, the type of the energy spectrum, of the free evolution of the system, of the response to the external perturbations. All observed
echo phenomena are the concrete realizations of a
certain dynamics which is concerned with the quantum statistical formula for the calculation of a response
where po and U(t) are the equilibrium density matrix
and the evolution operator. A set of operators B1, B2, ..., Bm, generated by po, U and Q may be extended to Lie algebra L,, = { Bi } (i = 1, 2, ..., n ; n >, m)
which is the main mathematical characteristic of the
dynamics.
If the equation (1) may be written in the terms of
trigonometric and hyperbolic functions we shall say it describes the echoes of spin and oscillator type, res- pectively. An oscillator echo dynamics is comparable
. to that of an extended spring and is characterized by
the effects of unlimited amplification and of accumu-
lation. A periodicity and a saturation are inherent to
spin dynamics. A real echo dynamics contains the combinations of these ideal properties [5].
A common theory of echo effects enables not only
the known phenomena to be classified but predicts
the ways in which to search for new echo phenomena.
Let us consider a system in an equilibrium state with
the temperature T and the energy W. If we choose the energy diagonal representation, the operator po will be also diagonal. It is a consequence of the quantum mechanical superposition principle that the physical variable ( Q(t) may generate a coherent time-perio-
dic response if a suitable external perturbation trans-
forms the system into a new state where non-diagonal
matrix elements of the operator Q will be diagonal even
after the external field is cut off. So a possibility exists
to detect a signal of the free spin induction type [6]
when the usual resonance or non-resonance absorp-
tion processes have been observed An additional condition is necessary to observe an echo process,
namely, the Lie algebra, generated by po Q and U,
must be irresoluble. In practice, this reduces to the existence of angular moment dynamics describing the system in question.
Let a quantum system be assumed to consist of N
identical particles each possessing a variety of physical multipoles p’ simultaneously U = 1, 2, ..., N). If a particle has a momentpq it may interact with a physical
field of the q-th type and generate this field by means
of its own energy. If a particle has multipole moments pa andpb it may be excited by a field o a » and generate
a field b >>. At least three relaxation times T1, T2
and T!q are concerned with the pq which are named as energetic, reversible and irreversible phase relaxation
times. Strictly speaking, only the following times may be observed
An echo can be observed under the following condi-
tions
where At is the exciting pulse duration.
Two new oscillator echo representatives have been
discovered in powders (the powder echoes [7]) and
in monocrystals (the polarization echoes, PE [8-12]).
In the powder echo case the exciting pulses and radiat-
ing echo signals are of RF magnetic field nature.
The PE and the exciting pulses are of electric nature and of frequency - 1010 Hz. In [13] a new powder
echo representative has been observed in ferroelectric
powders and polycrystals in the RF range. The echo
signal and the exciting pulses were of electric nature.
After having repeatedly renamed the phenomenon,
the authors of [13] chose the name « electro-acoustic echo ». While in [8] a clear analogy between a cyclo-
tron echo and a PE was found and the problem to
discover the radiating centres was stated, in the
reference [13] the physical nature of radiating centres, of exciting pulses and of echo signals remained uncer-
tain or was not discussed.
The problem is basically very complicated owing
to the followihg circumstances :
a) a variety of possible echo objects, b) an uncer- tainty about the physical nature of exciting pulses
caused by its transformations at the sample ends, c) an uncertainty about the physical nature of the
echo signals for the same reason.
If is difficult to distinguish experimentally even the types of temporal and spatial-temporal echoes. Let
an echo and the exciting pulses be of electric nature then there are the following hardly distinguishable
variants :
1) Electric temporal echoes : the actual exciting pulses are electrical, the echo objects are fixed electric
multipoles, the actual echo signal is electrical and a
receiver detects an electric signal.
2) Acoustic temporal echoes : the actual exciting pulses are acoustica( echo objects are fixed elastic
multipoles, the actual echo signals are forward or
backward sound waves and, in the both cases, a
receiver detects an electrical signal generated by
sound at a sample end (1).
3) Electro-acoustical temporal echoes : the actual first pulse is acoustical and the actual second one is
electrical, echo objects are elastic and electric multi-
poles bound to the same active centre, the actual echo
signal is acoustical or elastic and a receiver detects
an electric signal in both cases.
4) Spatial-temporal , echoes or phonon echoes : in this case the collisions of phonons with matter may be regarded as the actual exciting pulses that act on the phonons, echo objects are phonons, an actual echo
signal is the restoration of phonon coherence or of
wave front at the first sample end, a receiver detects
an electric signal generated at the first sample end by a
sound pulse.
All of the above echo-types represent quite different physical phenomena. Until the type of powder, electro-
acoustical or polarization echo signals is determined,
no definite conclusions can be made about its physical
nature.
Another main task is to determine the physical
nature of resonance frequencies of the echo objects.
The essence of powder echoes in our opinion, is the following : powder particles or polycrystals form acoustic, electro-acoustic or magneto-acoustic resona-
tors which determine the resonance frequencies of the
echo objects localized inside the resonators. In the
case of PE in monocrystals the resonators are formed
by some interior structures of the type of electric domains and domain walls, acoustic and electro- acoustic domains and walls, magnetic domains and
walls, defects and so on. Below we shall consider
possible physical mechanisms of PE generation in
ferroelectric monocrystals.
In this paper we shall propose a microscopical theory of PE in ferroelectric monocrystals. It will be
shown that this phenomenon is a new representative
of oscillator dynamics.
2. The motion of ferroelectric domain walls in elec- tric fields. - As was said in the previous section the echo is the general dynamical effect that occurs if the Hamiltonian of the system has certain symmetry properties and does not depend on the physical
nature of the constants in the Hamiltonian. In the case
of ferroelectrics, this feature of the problem allows
the first microscopical version of the theory of PE to be
worked out because no self consistent quantum theory
of ferroelectricity exists at the present time. So we do our best to show the physical nature of the constants
involved in our theory of echoes and their dynamical properties apart from the theory of echo dynamics and
(1) If the echo objects are spins bound to elastic multi- poles the effect may be called « acoustic spin echoes ».
This was predicted theoretically in [15] and was detected in [11,14].
of phase memory. A complete quantum theory involv- ing from the beginning all the relevant constants may be a further step in this investigation after more infor-
mation is available from PE experiments on the
systems that give these signals.
What we have tried to show is that an ad hoc
assumption of all phenomenological theories of PE in ferro- and piezoelectrics namely that the echo is the property of the whole sample described phenomeno- logically is incorrect and does not provide the possibi- lity of discriminating between quite different types of echo processes in a material. To get information from PE we must know the system giving the signal and this
is the question we shall try to answer in a concrete experimental situation.
We restrict ourselves to ferroelectric substances,
where adjacent domains have opposite polarization, aligned with ferroelectric axis z. Such domains are
separated by 1800 walls which lie parallel to i. We consider, for simplicity, planar walls normal to a
direction x (x -L z). So the polarization P and the
strain S are functions of x and t only and the thermo-
dynamic potential and kinetic energy densities of a ferroelectric crystal in an external electric field are the
following (2) :
In equation (3) the first and the second terms are the electric energy, the third term is the correlation energy caused by an inhomogeneity of P, the fourth and the fifth terms are the elastic and electrostrictive energies.
In equation (4) the first and the second terms are the elastic and ionic energies. The polarization is assumed
to arise from a displacement of a certain ion [16].
J1 = m no 1 e , P = no ez, where z, e, no and m are
the displacement, the effective charge, the number and the effective mass of the ion. The remaining
notations are the following : S = - Ox X is the coor-
dinate of the wall centre, c2 is the elastic modulus of the second order, q is the electrostrictive constant,
K is the correlation parameter and p is the mass density.
A variation of the Lagrangian density with respect
to P and X leads to the equations of motion
(2) Strictly speaking, equation (3) is valid for the centro- symmetric-in-paraphase ferroelectric crystals of the type of TGS, BaTi03, LiTaO3, LiNb03 . PE has been observed as
in LiTao3 and LiNb03 monocrystals [9, 10, 12] as in non- centrosymmetric-in-paraphase KDP and Rochelle Salt [8, 12]. In the both cases the echo dynamics has the same properties.
570
where T is the stress, Wr is the ionic damped frequency
and the damping term is introduced in the right hand
side of equation (5). In order to obtain the energy of the domain wall it is necessary to know the solutions of equation (5) and equation (6). In the case of a har-
monic external field E = Eo exp(iwt), these have a special form [17]
where P c( ç) and X c( ç) are the static solutions in the absence of E, ç = [x - X(t)] d -1 is the new dimen-
sionless variable, d characterizes the thickness of the wall and it is given by d2 = 21(;13-1 Po, here Po is the spontaneous polarization inside the domain.
The energy of a domain wall per 1 CM2 is calculated
by
where the second term is the energy density of the homogeneously polarized ferroelectric. Substituting
the solutions (7) and (8) in equation (9) and expanding
in a power series under the conditions
we obtain (3)
As a consequence of (11), we may produce the equation
of motion with the damping term
where the effective mass of the wall per 1 cm’ is given by
Thus far, the domain wall was considered as being
free except for ionic damping. Apart from the external pressure - 2 Po E one must also take into considera- tion the restoring force following from the change in
energy of the crystal as displacement of the wall from its original position gives rise to the energy of the electric field produced by the charge on the edge of the
domain or of the sample. In the usual form the energy contribution per 1 cm2 of wall is
where Ål and Å2 are the non-linear constants and fo
is the linear constant. The higher order terms can be dropped on the basis that they are small. The restoring
force Felar acting on unit area of wall is obtained from this expression
From the condition of equilibrium of the wall in the electric field
one can find the natural frequency of the domain wall
where xo is the electric susceptibility at the frequency m = 0.
Taking into account the restoring force, equation (12) may be generalized in the following way
In such a way the domain wall is considered as an ordinary anharmonic driven oscillator with the restoring
force Felas and damping. It would be possible to start ab initio from equation (14) but we would not have the microscopic expressions for the effective mass mw and natural frequency W’ in this case.
Finally, the total energy of the wall is
(3) We drop the surface strain energy of the wall per 1 cm’
WO = 4 Po K(3 d) -1 which is not essential for us.
where a is the domain length, M:ff f = mw ab, and the electric dipole moment of the wall is given by
Let us estimate the resonant frequencies and the
thickness of 1800 walls for a few ferroelectric
crystals (4) :
then o-)’/2 n = 0.15 x 10’0 s-’. In this case the flexing oscillations of walls are possible [19] owing to a large wall thickness (~ 10- 5 cm), wq/2 1t = 1011 s-1.
4) The Rochelle Salt. p = 10-25 s2 (OH- group),
b = 10-3 cm, then w’/2 7r = 0.3 x 101° s-1.
From the estimates it is easy to check if the condi- tions are satisfied (10) : co (oq and
All the resonance frequencies 1) - 4) lay in the range
( 109- 1010 Hz which gives a good fit to the experimen-
tal data. PE experiments have been carried out at
liquid helium temperatures, frequencies - 1010 Hz (5)
and electric pulse widths - 10- 8 s.
3. Are the domain walls and (or) domains the objects generating polarization echo signals in ferroelectrics ?
- What we mention about a domain wall (or domain)
motion in an external field really relates to the motion
(4) These estimations were carried out with an accuracy to the first term in the equation (13).
(1) In PE experiments the resonance frequencies lay in the
range from 8.9 GHz to 9.6 GHz for KH2PO4 [8] and 9.4 GHz for LiTa03 [9], LiNbo3 [12].
of ions of spontaneous polarization in the field. The domains and the walls serve as the resonators for the ions which are the actual microscopic objects generat-
ing PE signals in ferroelectric monocrystals.
So far as an actual motion of a domain wall is a
result of a variation of polarization in a whole domain volume, it must be considered, in some cases, as an
oscillating element with its own effective mass. A domain motion in an external field may be described
by the same equation (14) as for a domain wall but now
the coefficients of equation (14) have to be normalized
to a unit volume of a domain. Using the formulas for
a wall motion, resonance domain frequencies may be written as [21] ]
where M is the effective domain mass,
A Vd is a variation of the domain volume in an external electric field E.
A table may be constructed to study how the natural
frequencies of domains and walls depend on possible
dimensions of domains in BaTi03 crystals, for example.
Table I. - The natural frequencies of domain and wall
resonators.
The dipole moment is the same (Eq. (16)) whether a
domain wall or a whole domain is assumed to oscil- late.
In addition to PE experiments in microwave range
- 1010 Hz at 4.2 K [8-12] a few experiments have
been carried out in ferroelectric monocrystals in RF
range - 10’ Hz at 300 K [23]. In particular, PE have
been observed in Rochelle Salt monocrystals in the temperature range from + 22°C to - 15 OC at the
frequencies 12 MHz and 24 MHz and also in KH2P04 monocrystals (from - 143°C to - 160 OC) at the
same frequencies. Signals have been detected indepen- dently on the presence of a constant electric field, and
a temperature variation of relaxation time T2 was not
in accordance with the temperature dependence of the
sound attenuation. Indeed the time T2 diminishes
when the echo intensity grows. This will not be the
case if only the relaxation process becomes more intensive and the echo amplitude must drop because
of the loss of coherence by the echo objects. So pho-
nons are not the echo objects in this experiment. But if
electric dipoles are the echo objects, the growth of the
