# New trends in the nanophysics of ferroics, relaxors and multiferroics

M. D. Glinchuk  $|$ , L. P. Yurchenko  $|$ , E. A. Eliseev  $|$ Corresponding
author<EMAIL_ADDRESS>
(Received June 30, 2022, in final form October 30, 2022)

###### Abstract

The review covers the theoretical and experimental results obtained in the
recent years by the scientists with the help of comprehensive investigation of
nanoferroics and multiferroics. The main attention will be paid to spontaneous
flexoeffects and reentrant phase in nanoferroics as well as to a recently
discovered giant magnetoelectric effect in multiferroics.

Key words: ferroics, multiferroics, phase transitions, magnetoelectricity

###### Abstract

Îãëÿä îõîïëþ òåîðåòèí òà åêñïåðèìåíòàëüí ðåçóëüòàòè, îòðèìàí çà îñòàíí ðîêè
àâòîðàìè çà äîïîìîãîþ êîìïëåêñíîãî äîñëäæåííÿ íàíîôåðî¿êâ, ðåëàêñîðâ òà
ìóëüòèôåðî¿êâ. Îñíîâíà óâàãà ïðèäëåíà ñïîíòàííîìó ôëåêñîåëåêòðèíîìó åôåêòó
ðååíòðàíò-ôàç â íàíîôåðî¿êàõ, à òàêîæ íåùîäàâíî âäêðèòîìó ããàíòñüêîìó
ìàãíòîåëåêòðèíîìó åôåêòó â ìóëüòèôåðî¿êàõ.

Ключов слова: ôåðî¿êè, ìóëüòèôåðî¿êè, ôàçîâ ïåðåõîäè, ìàãíòîåëåêòðèêà

##  1 Introduction

The review covers the theoretical and experimental results obtained in the
recent years with the help of comprehensive investigation of nanoferroics and
multiferroics. The main attention will be paid to spontaneous flexoeffects and
reentrant phase in nanoferroics and to a recently discovered giant
magnetoelectric effect in multiferroics. Being characteristic of all nanoes,
geometrical confinement generates many physical effects, which are absent in
the bulk ferroic samples. These phenomena are scrupulously described in
Chapter 4 of reference [1]. In particular, the authors of [1] considered the
appearance of ferromagnetism at room temperature in nanoparticles and thin
films of undoped CeO2, HfO2, SnO2, Al2O3 and other nonmagnetic in
corresponding bulk oxides. Keeping in mind the detailed description of this
phenomenon in Chapter 4 of reference [1] and references given there, we
excluded this problem from our review. It appeared that in the recent years,
different kinds of flexo-coupling (flexoelectric, flexomagnetic, flexoelastic)
in ferroic nanosamples have been the most popular due to strong geometrical
confinement and the influence of the defects.

The flexoelectric effect existing for any solid bodies is known to be the most
studied. It was predicted by Mashkevich and Tolpygo [2]. Later on, theoretical
study of the flexoelectric effect in bulk crystals was performed by Kogan and
Tagantsev [3, 4, 5], experimental measurements of flexoelectric tensor
components were carried out by Ma and Cross [6, 7, 8] and Zubko et al. [9].
Renovation of the theoretical description for the flexoelectric response of
different nanostructures starts from the papers by Catalan and co-workers [10,
11], while recent achievements are presented in the papers by Majdoub et al.
[12], Kalinin and Meunier [13] and Lee et al. [14]. In the latter paper, a
giant flexoelectric effect (6–7 orders of magnitude larger than the typical
value reported for bulk oxides) was discovered in ferroelectric HoMnO3
epitaxial thin film on Pt/Al2O3 substrate. In these papers the flexoelectric
effect was considered as a coupling between intrinsic polar properties (e.g.,
polarization) and the extrinsic factors like the misfit strain relaxation [10,
11] or the system bending by external forces [12, 13]. The influence of
flexoelectric coupling on the properties of low-dimensional transition metal
dichalcogenides was studied by Morozovska et al. [15]. Recent trends and
achievements in flexoelectricity research and applications could be found in
[16] and in references therein. The coupling between intrinsic parameters,
namely, spontaneous polarization gradient inherent to nanosystems and strain,
was considered in [17]. The crucial role of the surface in all physical
properties of nanosystems including the strong order parameter gradients in
ferroic nanostructures [18] inevitably leads to the noticeable influence of
flexocoupling, almost negligible for bulk materials, since the order
parameters are usually homogeneous in this case.

Flexomagnetic effect is much less studied in comparison with flexoelectric
effect, and only a few relevant papers exist [19, 20]. Partly this can be
related to the fact that since the inversion of time must be included in
consideration, there is symmetry restriction for the existence of
flexomagnetic effect. In particular, the existence of time inversion or space
inversion uncoupled with other symmetry operations excludes the flexomagnetic
effect in para- and diamagnetics. The existence of the symmetry operations
coupling or the absence of time inversion (e.g., in antiferromagnetics) makes
it possible to get flexomagnetic effect. To find out the non-zero
flexomagnetic tensor components, one should consider 90 magnetic classes and
perform their symmetry consideration similarly to piezomagnetic and
magnetoelectric effects (see Section 4.3.7 in the book [1]).

In what follows we will consider the strong influence of flexoelectric effect
in nanoferroics, namely the critical size disappearance at size induced phase
transitions and reentrant phase occurrence in nanoferroics. Stabilization of
ferroelectric phase with nanoparticles sizes decrease as it was observed
earlier in tetragonal BaTiO3 nanospheres of radii 5–50 nm stayed unexplained
until the appearance of our paper [21]. Our calculations have shown that the
physical mechanism of this exciting phenomenon can be the flexo-chemo effect,
being a synergy of the flexoelectric stresses and the chemical pressure
induced by ion vacancies via Vegard effect. The coexistence of ferroelectric
and relaxor phases due to oxygen vacancies is considered in [22]. The results
of our calculations and comparison of the developed theory with experiment can
be found in sections 2, 3 and 4; a giant magnetoelectric effect in
multiferroics is presented in section 4.2 and was published in our papers [23,
24].

##  2 Spontaneous flexoeffect

The most general definition of the direct flexoeffect is the appearance of
either polarization $P$ or magnetization $M$ in response to inhomogeneous
mechanical impact, i.e., strain gradient $\partial{u_{ij}}/\partial{x_{l}}$.
The converse flexoeffect corresponds to the appearance of mechanical strain in
response to the gradient of either polarization or magnetization,
respectively. Therefore, the form of flexoelectric and flexomagnetic effects
can be written as:

${\eta_{i}}={f_{ijkl}}\frac{{\partial{u_{ij}}}}{{\partial{x_{l}}}},$ (2.1)
${u_{ij}}={f^{\prime}_{ijkl}}\frac{{\partial{\eta_{k}}}}{{\partial{x_{l}}}}.$
(2.2)

Here, $\eta=P$ and $M$ stand for flexoelectric and flexomagnetic effects,
respectively.

For flexoelastic direct and converse effects
${\sigma_{im}}={f_{imjkl}}\left({\partial{u_{jk}}/\partial{x_{l}}}\right)$ and
${u_{jk}}={f^{\prime}_{imjkl}}\left({\partial{\sigma_{im}}/\partial{x_{l}}}\right)$,
respectively, so that they are defined by 5th rank tensors, while
flexoelectric and flexomagnetic effects are defined by 4th rank tensors. In
what follows we will pay attention mainly to flexoelectric and flexomagnetic
effects in nanostructures. Let us underline that flexocoupling affects both
the system response to the external impact and the intrinsic gradient of the
order parameters.

Let us note that spontaneous flexocoupling was introduced in [17], physical
mechanism of reentrant phase appearance was considered in [21], the appearance
of morphotropic phase in the relaxor due to oxygen vacancies influence [22],
mechanism of giant magnetoelectric coupling appearance as well as the
materials of this phenomenon observation [23, 24] were proposed by the authors
of this review.

###  2.1 Analytical theory and comparison of the theory with experiment

Landau–Ginzburg–Devonshire (LGD) functional bulk (b) and surface (s) densities
have a relatively simple form for a nanoparticle with uniaxial ferroelectric
polarization $\vec{P}=\left(0,0,P_{3}\right)$ [17]:

$\displaystyle\Phi_{b}=\int\limits_{V}{{G}{\mathrm{d}^{3}}r},$ (2.3)
$\displaystyle{G}=\,$
$\displaystyle{\alpha_{b}}(T)\frac{{P_{3}^{2}}}{2}+\beta\frac{{P_{3}^{4}}}{4}+\gamma\frac{{P_{3}^{6}}}{6}+\frac{{{g_{ij33}}}}{2}\left({\frac{{\partial\,{P_{3}}}}{{\partial{x_{i}}}}\frac{{\partial{P_{3}}}}{{\partial{x_{j}}}}}\right)-\frac{{{F_{3ijk}}}}{2}\left({{P_{3}}\frac{{\partial\sigma_{ij}}}{{\partial{x_{k}}}}-\sigma_{ij}\frac{{\partial{P_{3}}}}{{\partial{x_{k}}}}}\right)$
$\displaystyle-{P_{3}}\left({\frac{{E_{3}^{d}}}{2}+E}\right)-{Q_{ij33}}{\sigma_{ij}}P_{3}^{2}-\frac{{{s_{ijkl}}}}{2}\sigma_{ij}\sigma_{kl}-{W_{ij}}{\sigma_{ij}}\delta
N,$ (2.4)
$\displaystyle{\Phi_{S}}=\int\limits_{S}{{\mathrm{d}^{2}}r}\left({\frac{{\alpha^{S}}}{2}P_{3}^{2}+\frac{{\beta^{S}}}{4}P_{3}^{4}+\mu_{\alpha\beta}^{S}u_{\alpha\beta}+...}\right).$
(2.5)

The coefficient $\alpha_{b}(T)$ typically depends on the temperature $T$.
Here, we assume the linear dependence, $\alpha_{b}(T)=\alpha_{T}(T-T_{\rm
C})$, where $T_{\rm C}$ is a Curie temperature. Coefficient $\beta$ sign
depends on the ferroelectric transition order, $\gamma>0$. $E_{3}^{d}$ is
depolarization field, $Q_{ijkl}$ are the bulk electrostriction tensor
coefficients, $g_{ijkl}$ is the gradient coefficients tensor, $F_{ijkl}$ is
the flexoelectric strain tensor, $\sigma_{ij}$ is the stress tensor, $W_{ij}$
is the elastic dipole (or Vegard strain) tensor, that is regarded diagonal
hereinafter, i.e., $W_{ij}=W\delta_{ij}$ ($\delta_{ij}$ is delta Kroneker
symbol). $\delta N=N\left({\vec{r}}\right)-N_{e}$ is the difference between
the concentration of defects $N(r)$ in the point $r$ and their equilibrium
(average) concentration $N_{e}$. Surface energy coefficients $\alpha^{S}$ and
$\beta^{S}$ are supposed to be positive and temperature independent,
$\mu_{\alpha,\beta}^{S}$ is the surface stress tensor [2, 25], $u_{ij}$ is the
strain tensor.

Figure 1 (a) shows the best fit of our theoretical results (solid curve)
obtained with the help of equations (2.3), (2.5), which allowed to fit all
physical properties with two fitting parameters (see [21] for details) to
experimental results on tetragonality [26] (symbols with error bars). The
latter is obtained from X-ray diffraction data on the size and temperature
dependence of the lattice constants. Our aim was not to fit well the
tetragonality for the smallest particle of radius 2.5 nm, because here the
phenomenological continuous approach may be invalid.

Figure 1: (Colour online) Room temperature tetragonality ratio $c/a$ (a),
spontaneous polarization (b), and transition temperature (c) vs. the particle
radius $R$. Plot (a) shows the best fit of our theory (solid curve) to
experimental data [26] (symbols). Temperature $T=293$ K. Adapted from [21].

Figures 1 (b) and (c) illustrate the dependences of the spontaneous
polarization and transition temperature on the spherical particle radius
calculated for the same fitting parameters as in the figure 1 (a). Hence, we
“reconstruct” the polarization and transition temperature from to the best fit
of the tetragonality measured experimentally [26]. The reconstructed
dependences clearly demonstrate a strong (more than 2 times) enhancement of
polarization and transition temperature for the particle radius less than 10
nm. The reentrant phase region appears for radii $R<10$ nm. One can see that
theoretical figure 1 (a) rather well describes the spontaneous polarization
and transition temperature reconstructed from experimental $c/a$ value.

It is important to underline that the obtained results are very important for
both fundamental physics and applications in modern electronic technique.

##  3 Relaxor ferroelectrics with perovskite structure

In this section we consider relaxor ferroelectrics with perovskite structure
having a broad application in modern electronic devices [27, 28]. More than 15
years ago the authors of [29] observed the appearance of ferroelectricity in
relaxors lead zinc niobate - lead lanthanum zirconium titanate (PZN–PLZT)
sintered in nitrogen atmosphere, which induced high concentration of oxygen
vacancies.

The recent paper [22] based on the phenomenological theory approach showed
that since the oxygen vacancies are known to be elastic dipoles, they affect
the elastic and electric fields due to Vegard and flexoelectric couplings. We
have shown that a negative Curie temperature $T_{\rm C}^{*}$ of a relaxor is
renormalized by the elastic dipoles due to the electrostriction coupling and
could become positive at some large enough concentration of the vacancies. A
positive renormalized temperature $T_{\rm C}^{R}=T_{\rm C}^{*}+\Delta T$ is
characteristic of the ferroelectric state. At $T<T_{\rm C}^{R}$, all the polar
properties could be calculated in the conventional way for ferroelectrics, but
the obtained experimental data favor the coexistence of the ferroelectric
phase with a relaxor state, i.e., the presence of a morphotropic region in
PZN–PLZT relaxor.

Therefore, both the spontaneous flexoelectric effect in nanoferroics appearing
due to the surface influence and the flexoelectric effect induced by the
elastic field of the defects can be the source of new physical properties
important for fundamental physics and for modern technical applications.

Oxygen vacancies in ABO3 ferroelectrics have a great impact on their physical
properties and the perovskite structure is capable of conserving the structure
stability even for a high concentration of oxygen vacancies. “B” cations are
usually shifted from central position in the neighborhood of the oxygen
vacancy, because in ABO3 structure the size of oxygen ions and so its
vacancies used to be larger than the cation ones. To compensate for the loss
of the oxygen negative charges, the equivalent amount of B4+ cations should be
in a B3+ state. The PZN–PLZT samples sintered in nitrogen atmosphere appeared
to be black and opaque, because off-central Ti4+ transforms into color center
Ti3+. Note that Ti3+ can create layers of the ordered dipoles at large
concentration of oxygen vacancies. The electrons necessary for Ti4+ into Ti3+
transformation can be created from ionization of neutral oxygen vacancy
${V_{\rm O}}\to V_{\rm O}^{\bullet}+e$, $V_{\rm O}^{\bullet}\to V_{\rm
O}^{\bullet\bullet}+e$, the $V_{\rm O}^{\bullet}$ and $V_{\rm
O}^{\bullet\bullet}$ being positively charged vacancies. Uncharged vacancy
$V_{\rm O}$ represents dilatational center which creates a local compressive
strain. Since the conductivity is mainly attributed to the electromigration of
oxygen vacancies in perovskite ferroelectrics, the measurements of $dc$
conductivity temperature dependence of the NS and OA specimens were carried
out in order to estimate the oxygen vacancies concentration and charge states.
Since this complex defect can be represented as $V_{\rm
O}^{\bullet\bullet}+2\textrm{Ti}^{3+}$, it can be observed in high temperature
region only. Therefore, we are faced with the existence $V_{\rm
O}^{\bullet\bullet}$ and $V_{\rm O}$ in NS sample with high concentration of
oxygen vacancies.

Note that vacancies tend to accumulate in the vicinity of any inhomogeneities,
surfaces and interfaces, since the energy of vacancies formation in such
places can be much smaller than in the homogeneous volume [1]. In the places
of vacancies accumulation they can create sufficiently strong fields, which in
turn can lead to the appearance of new phases in relaxors, for example, polar
(ferroelectric) ones. On the contrary, in the places where there are few
vacancies, the non-polar relaxor remains. Thus, the polar ferroelectric and
nonpolar relaxor states coexistence can be realized in this case. Let us
briefly consider possible mechanisms of ferroelectricity appearance in NS
samples of PZN–PLZT. As we discussed above, the oxygen vacancies in this
sample are uncharged $V_{\rm O}$, singly and doubly positively charged $V_{\rm
O}^{\bullet}$ and $V_{\rm O}^{\bullet\bullet}$, respectively. Because of the
necessity of loss oxygen negative charges compensation, approximately an
equivalent amount of Ti3+ off-central ions is another group of defects.

If one formally puts charges $+4P\mu$ and $-4P\mu$ into the unoccupied point
of Ti4+, the charge $+4e$ makes an ideal lattice and $-4e$ corresponds to the
defect in it. A similar formal operation with the addition of $+e$ and $-e$ to
Ti3+ position leads to the appearance of an ideal lattice defect
$-4P\mu+(\textrm{Ti}^{3+}+e)$, that is dipole $d_{1}=4es$, where $s$ is Ti3+
off-central shift. The existence of electric dipoles will lead to the
appearance of ferroelectric phase due to an indirect interaction of dipoles
via the soft optic mode, while the soft mode existence in the ferroelectric
relaxors will be discussed later.

Keeping in mind that all the electric dipoles in the regions with sizes of the
order of correlation radius $r_{c}$ must be oriented, one can write the
criterion of FE phase appearance as $Nr_{c}^{3}\geqslant 1$, where $N$ is
concentration of dipoles.

Another possible mechanism of ferroelectricity in the relaxors can originate
from inhomogeneous elastic field via flexoelectric effect, namely
${P_{i}}={f_{ijkl}}\partial{u_{kj}}/\partial{x_{l}}$, where $P_{i}$ is
electric polarization component, $\partial{u_{kj}}/\partial{x_{l}}$ is
mechanical strain gradient, $f_{ijkl}$ is the tensor components of
flexoelectric effect. Detailed consideration of this mechanism along with the
mechanical strain field originated from the oxygen vacancies (Vegard
mechanism) contributions to the appearance of ferroelectricity in the relaxors
will be given below.

Gibbs potential density of relaxor ferroelectric materials having some hidden
soft phonon polar mode has the following form [22]

$\displaystyle
G=\frac{{{a_{ij}}\left(T\right)}}{2}P_{i}P_{j}+\ldots+\frac{{g_{ijkl}}}{2}\frac{{\partial{P_{i}}}}{{\partial{x_{j}}}}\frac{{\partial{P_{k}}}}{{\partial{x_{l}}}}+\frac{{{F_{ijkl}}}}{2}\left({\sigma_{kl}\frac{{\partial{P_{i}}}}{{\partial{x_{j}}}}-{P_{i}}\frac{{\partial\sigma_{kl}}}{{\partial{x_{j}}}}}\right)-Q_{ijkl}\sigma_{ij}{P_{k}}{P_{l}}$
$\displaystyle-\frac{{s_{ijkl}}}{2}\sigma_{ij}\sigma_{kl}-{P_{i}}E_{i}\left({\bf{r}}\right)-u_{ij}^{W}\left[{\delta{N_{d}}\left({\bf{r}}\right)}\right]{\sigma_{ij}}+{k_{\rm
B}}TS\left({N_{d},N_{d}^{+}}\right),$ (3.1)

where $P_{i}$ are the components of polarization vector ($i=$1, 2, 3) and
$\sigma_{ij}$ is the elastic stress tensor. The summation is performed over
all repeated indices. Dielectric stiffness expansion coefficients $a_{ij}(T)$
are positive, because the intrinsic ferroelectricity is absent, but it depends
on temperature reflecting the fact that the hidden phonon mode could soften at
negative absolute temperatures. This statement follows from the above
mentioned fact, that in PZN, a soft mode was observed at $T=20$ K, so that its
frequency could be zero at a negative temperature. Note that extrapolation of
PMN soft mode frequency to zero leads to $T_{\rm C}\approx-150$ K.

Matrix of the gradient coefficients $g_{ijkl}$ is positively defined.
$Q_{ijkl}$ is the electrostriction tensor, $s_{ijkl}$ is the elastic
compliances tensor, $F_{ijkl}$ is the forth-rank tensor of flexoelectric
coupling. In equation (3.1), $E_{i}(r)$ denotes the random electric field. The
configuration entropy function $S(x,y)$ is taken as
$S\left({x,y}\right)=y\ln\left({y/x}\right)-y$ in the Boltzmann–Planck–Nernst
approximation; $k_{\rm B}=1.3807\times 10^{-23}$ J/K, where $T$ is the
absolute temperature. Equations of state $\partial
G/\partial{\sigma_{ij}}=-{u_{ij}}$ determine the strains $u_{ij}$.
Euler–Lagrange equations $\partial G/\partial{\sigma_{ij}}=0$ determine the
polarization components.

Equation (3.1) includes Vegard-type concentration-deformation energy,
$u_{ij}^{W}\left[{\delta{N_{d}}\left(r\right)}\right]{\sigma_{ij}}$, which is
determined by the random defects with concentration of
$\delta{N_{d}}\left(r\right)\sim\sum_{k}\delta\left({r-{r_{k}}}\right)-{\bar{N}_{d}}$
(e.g., charged or electroneutral oxygen vacancies). The equilibrium
concentration of defects is ${\bar{N}_{d}}\ll 2.25\times{10^{28}}$ m-3. The
average distance between the defect centres $2R$ should be associated with the
average volume per inclusion and so it is defined from the relation
$({{4\piup}}/{3}){R^{3}}=1/{\bar{N}_{d}}$. The defect size $r_{0}$ is much
smaller than the average distance $R$, e.g., $r_{0}$ is ionic radius
$\sim(0.1-1)$ Å3 (see figure 2).

Figure 2: (Colour online) Schematics of the spherical defects with radius
$r_{0}$ embedded into the matrix. The distance between defects “$i$” and “$j$”
is $R_{ij}$. The average distance between defects is $2R$. The average volume
per one defect is $V=({{4\piup}}/{3}){R^{3}}$. Adapted from [22].

The results [22] showed, that at some concentration of oxygen vacancies, their
contribution can be larger than the negative value of temperature $T_{\rm
C}^{*}$ of a relaxor, and so we obtained a positive transition temperature
characteristic of a ferroelectric. To transform the relaxors into
ferroelectrics, one needs a large enough concentration of oxygen vacancies and
Vegard tensor amplitude. Unfortunately, the exact value of negative Curie
temperature $T_{\rm C}^{*}$ is not known and we should discuss some
estimations only. It is obvious that even a large enough value of $T_{\rm
C}^{*}$ can be overcome by special choice of oxygen vacancies concentrations
and parameters. In such a case, at $T<T_{\rm C}^{R}$ (ferroelectric phase),
all the properties can be written by conventional way on the base of free
energy $\Phi=\frac{1}{2}\alpha(T-T_{\rm
C}^{R}){P^{2}}+\frac{1}{4}\gamma{P^{4}}$, so that, e. g., polarization
${P^{2}}=\alpha(T_{\rm C}^{R}-T)/\gamma$, while at $T>T_{\rm C}^{R}$, $P=0$,
and we again have a relaxor. For this case, we consider local polarization and
electric field induced by Vegard and flexoelectric effects (flexo-chemical
coupling). The dependence of the mean square variation of polarization and
electric field on concentration of oxygen vacancies showed that flexo-chemical
coupling essentially contributes to local polarization and internal electric
field.

Keeping in mind the accumulation of oxygen vacancies in the vicinity of
different inhomogeneities, we came to the conclusion regarding the oxygen
vacancies concentration inhomogeneity. In such a case one can expect
coexistence of relaxor state and ferroelectricity Therefore, at some
concentration of oxygen vacancies and $T<T_{\rm C}^{R}$, we are faced with
morphotrophic region in PZN–PLZT.

To resume, the transition to a ferroelectric phase can be induced in a relaxor
by the influence of oxygen vacancies being elastic dipoles due to the joint
action of electrostrictive and Vegard couplings at some large enough
concentration of the vacancies. In the regions where the concentration of
vacancies is low, the local polarization and electric field could be induced
by the flexo-chemical coupling in dependence on the concentration of oxygen
vacancies. Because of inhomogeneity of the vacancies concentration, the
coexistence of ferroelectricity and relaxor state can be expected.

##  4 Giant magnetoelectric coupling at room temperature in multiferroics

Nowadays, magnetoelectric coupling is a very important problem for scientists
and engineers (see e.g., Feibig et al. [30]). That is why we decided to give a
separate introduction and a list of references for the convenience of readers.
Moreover, this part is divided into two parts, namely subsection 4.1 based on
reference [23] and subsection 4.2 based on reference [24]. The subsections 4.1
and 4.2 differ from one another by the choice of the materials, although the
preparation of samples, structural characterization and experimental methods
will be the same. The concluding remarks were given as a separate section.

###  4.1 Room-temperature ferroelectricity, superparamagnetism and large
magnetoelectricity in solid solution PbFe1/2Ta1/2O3 with
(PbMg1/3Nb2/3O3)0.7(PbTiO3)0.3

A strong magnetoelectric (ME) coupling existing at room temperature is
especially vital for novel functional device fabrication [31, 32, 33, 34, 35].
A straightforward way of increasing the magnitude of the ME response is to
choose the components with large magnetostrictive and piezoelectric
coefficients. For a long time, mainly composite multiferroics on the base of
PbZr1-xTixO3 (PZT) or (PbMg1/3Nb2/3O3)0.7(PbTiO3)0.3 (PMN–PT) has been used.
The latter component produces the largest ME effect and has been often used in
the multilayer multiphase (ferroelectric/ferromagnetic) structures. The reason
is the large piezoelectricity of PMN–PT in the morphotropic region with the
coexistence of the relaxor and ferroelectric phases. For a single crystal, it
is 7 times larger than the piezoelectricity of PZT [30]. For ceramic
materials, the piezoelectricity in PMN–PT is 2 times larger than that in PZT
[36].

In the recent years, considerable attention of scientists and engineers was
paid to ferroelectric antiferromagnets PbFe1/2Ta1/2O3 (PFT), $T_{N}\approx
130$–180 K, and PbFe1/2Nb1/2O3 (PFN), $T_{N}\approx 140$ K [37, 38] and their
solid solutions with PbZr0.53Ti0.47O3 [39, 40, 41, 42, 43, 44, 45, 46]. Some
of these solid solutions exhibit room-temperature multiferroism and large
enough ME coupling, which includes a mixture of linear and biquadratic
contributions. The ME coupling was theoretically analyzed in references [47,
48] and was described by second and fourth rank tensors, namely,
$\mu_{ij}P_{i}M_{j}$ and $\xi_{ijkl}P_{i}P_{j}M_{k}M_{l}$ ($P$ is polarization
and $M$ is magnetization). The authors of papers [47, 48] have shown that
large ME effect and the appearance of magnetization originate from the
nanostructure of the considered materials. Another mechanism of the appearance
of magnetization in chemically disordered antiferromagnetic multiferroics was
considered in works [49, 50]. It was shown that antiferromagnetically
interacting Fe3+ ions may form superstructures having a ferrimagnetic ground
state. Such a structure has a different number of nonequivalent Fe positions
in a unit cell. As a result, the ground state magnetization may reach several
B per Fe spin. Experimental indications on the ferrimagnetic superstructure
formation were reported in reference [51] for PbFe1/2Sb1/2O3. The F-center
exchange mechanism was proposed in [52]. It provides an explanation for the
existence of ferromagnetism in oxides due to the presence of oxygen vacancies.

Recently, the attention of scientists and engineers was attracted to the
paramagnetoelectric (PME) effect introduced by Hou and Blombergen [53]
described by the term $\lambda_{ijk}P_{i}M_{j}M_{k}$. The results of
experimental and theoretical studies of this effect in PFN and its solid
solution with PbTiO3 were published in references [54, 55].

It was not excluded that the replacement of PZT by PMN–PT in solid solution
with PFT could lead to an essential increase of ME effect due to a larger
piezoelectric coefficient in PMN–PT. Surprisingly, to the best of our
knowledge, there is no information about the synthesis of the
(PFT)x(PMN–PT)1-x solid solution. Probably the more complex characteristics of
PMN–PT than PZT, and thus the more complicated mechanism of ME effect, were
the main reason for the fact. On the other hand, both components of the
(PFT)x(PMN–PT)1-x solid solution were broadly studied (see e. g., [56, 57, 58,
59, 60, 61, 62]). In particular, it was shown that multiferroic PFT is
antiferromagnetic-ferroelectric with ferroelectric phase transition at $T_{\rm
C}\approx 250$ K [36]. The considered PMN–PT is non-magnetic with a maximum of
dielectric permittivity at $T_{m}\approx 410$–420 K [61, 63].

The aim of this study is to briefly describe the synthesis methodology of the
novel single-phase multiferroic, to characterize the obtained samples and to
present the results of experimental and theoretical investigation of its
properties, namely, polar, magnetic and magnetoelectric characteristics.

As seen in figure 3, the temperature dependence of the inverse dielectric
permittivity, measured for both samples at 1 kHz does not follow the linear
behavior predicted by the Curie–Weiss law for a certain temperature range
above $T_{m}$. In this case, the modified Curie–Weiss law can be used to
describe the temperature dependence of the dielectric permittivity [64]:

$\displaystyle\frac{1}{\varepsilon}=\frac{1}{{{\varepsilon_{m}}}}+{\left({\frac{{T-{T_{m}}}}{C}}\right)^{\gamma}},\quad({\varepsilon_{m}}={\varepsilon^{\prime}_{\max}}).$
(4.1)

Parameter $\gamma$ characterizes the degree of phase transition diffuseness,
and its values lie in the region from 1 (for ferroelectrics) to 2 (for
relaxors). Fitting of the experimental data gives the $\gamma$ values equal to
$\sim$1.85 for $x=0.4$ and $\sim$1.8 for $x=0.5$. As shown in [65], the
diffuse phase transition could be considered as an intermediate state in which
both ferroelectric and relaxor phases can be presented simultaneously. This
allows us to suggest that both relaxor and normal ferroelectric phases coexist
in the studied compounds, and an increase in the PFT content leads to an
increase in the ferroelectric phase contribution.

Figure 3: (Colour online) (a) Temperature dependences of the reciprocal
dielectric permittivity of the (PFT)x(PMN–PT)1-x ($x=0.4$, 0.5) compounds; the
inset shows logarithmic dependence of $1/\varepsilon-1/\varepsilon_{m}$ on
$T-T_{m}$. Adapted from [23]. (b) Pristine nano-grained ceramics before
prepoling. The angles $\theta$ between the grain polarization $P$ and global
axis $3^{\prime}$ vary between 0 and 180 degrees. The static magnetic field
$H$ was applied to measure $M(H)$. (c) The ceramics after the pre-poling in a
strong electric field $E$. The field was applied and then removed. The angle
$\theta$ changes between 0 and 90 degrees after strong pre-poling. For $m3m$
parent symmetry, the local axes can be re-numbered in such a way that the axis
with minimal angle to the global axis 3 can be labeled as axis 3*. The low-
frequency magnetic field $H$ is applied after pre-poling to measure the ME
current.

The magnetic response of solid solutions (PFT)x(PMN–PT)1-x is due to the
presence of octahedrally coordinated Fe3+ ions having $3d^{5}$ electronic
$d$-shell configuration in $S$-state, spin $S_{\textrm{Fe}}=5/2$ and
$g$-factor $g\approx 2$. Their fraction in the formula unit is $x/2$. Figures
4 (a) and (b) show magnetization isotherms $M(H)$ at $T=293$ K for the
compositions $x=0.4$ (a) and $x=0.5$ (b). The $M(H)$ isotherms have a
qualitatively similar look for $x=0.4$ and $x=0.5$. Symbols are experimental
data [23]. For all considered temperatures, the curves are anhysteretic, i.e.,
reversible. We see that the measured magnetization may be presented as a sum
of a paramagnetic contribution that is proportional to the field
$M_{p}(H,T)=\chi_{p}(T)H$ and of superparamagnetic-like contribution
$M_{s}(H,T)$ that saturates at the field of the order of 1 kOe. The absence of
noticeable hysteresis for the observed curves allows us to consider
nonparamagnetic contributions as superparamagnetic ones registered at $T\gg
T_{b}$, $T_{b}$ being a blocking temperature [66, 67]. Therefore, the blocking
temperature is $T_{b}\ll 293$ K.

Figure 4: (Colour online) Magnetization dependence on magnetic field for solid
solution (PFT)x(PMN–PT)1-x for $x=0.4$ (a) and $x=0.5$ (b) at $T=293$ K. Small
red symbols are experimental data [23], solid and dashed curves are the
fitting using Langevin and Brillouin functions, respectively, for describing
the superparamagnetic contribution. Plots (c) and (d) show the ratio of the
superparamagnetic (SPM) to paramagnetic (PM) contributions to magnetization
for $x=0.4$ and 0.5, respectively.

Solid and dashed curves in figures 4 (a) and (b) are theoretical fitting using
Langevin and Brillouin functions, respectively, for the description of the
superparamagnetic contribution. Herein below we assume the following
dependence of the magnetization $M$ on applied quasi-static magnetic field
$H$:

$\displaystyle
M\left(H\right)={M_{PM}}\left(H\right)+{M_{SPM}}\left(H\right)\approx{\chi_{m}}H+\int\limits_{{}_{{\mu_{\min}}}}^{{}_{{\mu_{\max}}}}{F\left({\mu-{\mu_{S}}}\right){M_{P}}\left(\mu\right)f\left({\frac{{\mu
H}}{{{k_{\rm B}}T}}}\right)}\mathrm{d}\mu+\delta M,$ (4.2)

where the first term is a purely paramagnetic (PM) contribution of effective
“host-matrix”, and the second is the superparamagnetic (SPM) contribution of
the nanoregions. The value $\delta M$ can be non-zero reflecting a very small
device-related systematic error (shift).

_PM contribution_. Note that the linear approximation for PM contribution,

$\displaystyle M_{PM}\left(H\right)\approx{\chi_{m}}H,$ (4.3)

is valid for magnetic fields $\left|H\right|\ll({k_{\rm B}}T)/(g{\mu_{B}}S)$,
where $\mu_{B}$ is Bohr magneton, $k_{\rm B}$ is Boltzmann constant, and
factor $g$ is taken $\approx{2}$. At room temperature, the fields should be
much smaller than 800 kOe. The number of Fe3+ ions in the PM part of the
sample can be obtained from the Curie constant $C_{CW}$ value in the Curie-
Weiss law for the ${\chi_{m}}={C_{CW}}/(T-\theta)$. The temperature $\theta$
is negative in case of the antiferromagnetic interaction between the ionic
spins. Constant $C_{CW}={N_{PM}}\mu_{p}^{2}/3{k_{\rm B}}\approx
4{N_{PM}}\mu_{B}^{2}{g^{2}}S\left({S+1}\right)/3{k_{\rm B}}$, where $N_{PM}$
is the number of paramagnetic spin $S$ in a unit mass volume. Hence,
$N_{PM}=3{k_{\rm
B}}{C_{CW}}/\left[{4\mu_{B}^{2}{g^{2}}S\left({S+1}\right)}\right]$.

_SPM contribution_. Following Binder and Young [68], and Wiekhorst et al.
[67], the integration (or averaging) in the last term in equation (4.2)
reflects the fact that the number of elementary spins, which contribute to the
magnetic moment $\mu$ of a given SPM no-regions [and thus to its super-spin]
should be different for different inclusions. The magnetic moment can
fluctuate around the average value $\mu_{S}(T)$ in dependence on the sharpness
of its distribution function $F\left(\mu-\mu_{S}\right)$. The function is
defined as
$\mu_{S}\left(T\right)=\int_{{\mu_{\min}}}^{{\mu_{\max}}}{F\left({\mu-{\mu_{S}}}\right)\mu\mathrm{d}\mu}$.
The magnetization amplitude $M_{P}\left(\mu\right)$ of SPM contribution is
equal to $M_{P}={N_{S}}\mu$, where $N_{S}$ is the average number of super-
spins in one gram of the material having the average magnetic moment
$\mu_{S}(T)$.

Depending on a spin, one can use Langevin (LF) or Brillouin (BF) functions for
the function $f(x)$ describing SPM contribution:

$\displaystyle
f\left(x\right)=\left\\{\begin{array}[]{l}\coth\left(x\right)-\frac{1}{x},\quad{\rm{LF}}{\rm{,}}\\\
\frac{{2S+1}}{{2S}}\coth\left({\frac{{2S+1}}{{2S}}x}\right)-\frac{1}{{2S}}\coth\left({{x}/{{2S}}}\right),\quad
S\gg 1,\quad{\rm{BF}}.\end{array}\right.$ (4.6)

As one can see, BF transforms into LF in the limit $S\gg 1$, since
$\coth\left[{x(2S+1)/2S}\right]\approx\coth\left(x\right)$ and
$\coth\left({x/2S}\right)\approx 2S/x$.

In the simplest assumption, when all SPM particles are formed by the same Fe3+
ions with $S=5/2$ and $g=2$, one can obtain that in average each of these
particles contains $n_{\rm{SPM}}={\mu_{S}}\left({T\to
0}\right)/[{2{\mu_{B}}\sqrt{S\left({S+1}\right)}}]$ ions and the number of
spins (ions) that is included in SPM ensemble is
$N_{\rm{SPM}}=n_{\rm{SPM}}{N_{S}}$. The ratio of the ion Fe3+, which belongs
to PM and SPM ensembles in the sample, is equal to:

$\displaystyle\frac{{{N_{{\rm{PM}}}}}}{{{N_{{\rm{SPM}}}}}}=\frac{{2{\mu_{B}}\sqrt{S\left({S+1}\right)}{N_{{\rm{PM}}}}}}{{{N_{S}}{\mu_{S}}\left({T\to
0}\right)}}\equiv\frac{{2{\mu_{B}}\sqrt{S\left({S+1}\right)}}}{{{N_{S}}{\mu_{S}}\left({T\to
0}\right)}}\cdot\frac{{3{k_{\rm
B}}{C_{CW}}}}{{4\mu_{B}^{2}{g^{2}}S\left({S+1}\right)}}\equiv\frac{{3{k_{\rm
B}}{C_{CW}}}}{{2\mu_{B}{g^{2}}\sqrt{S\left({S+1}\right)}{N_{S}}{\mu_{S}}}}.$
(4.7)

Figures 4 (c) and (d) show the ratio of SPM to PM contributions to
magnetization, i.e., the dimensionless parameter

$\displaystyle\alpha\left(H\right)=\frac{{{M_{P}}}}{{\chi_{m}H}}f\left({\frac{{{\mu_{S}}H}}{{{k_{\rm
B}}T}}}\right),$ (4.8)

for $x=0.4$ and 0.5, respectively.

Figure 5 shows ME current as a function of the applied $dc$ magnetic field in
a PFT–PMN–PT ceramics for the compositions $x=0.4$ (a) and $x=0.5$ (b).
Symbols are experimental data [23]. One can see that ME signal sharply
increases with an increase of the $dc$ magnetic field in the range of $\pm$300
Oe, then it saturates in value and decreases down to almost zero at fields
larger than $\pm$3000 Oe. Note, that the ME current does not change too much
with temperature lowering from 293 K down to 120 K [23]. Only the current peak
position moves from $\pm$300 Oe to $\pm$400 Oe when the temperature changes
from 293 K to 120 K. The measurements show that the main contribution to the
ME current is caused by the superparamagnetic phase while the contribution of
isolated Fe3+ spins from the paramagnetic phase is negligibly small due to
their much lower magnetic moment.

Figure 5: (Colour online) Experimental data [23] (symbols) for ME current in
(PFT)x(PMN–PT)1-x ceramics at $T=293$ K for $x=0.4$ (a) and $x=0.5$ (b).
Dotted, dashed and solids curves are fitted using LF. Plots (c) and (d) show
the ratio of SPM to PM contributions to ME current calculated for $x=0.4$ and
0.5, respectively.

Solid and dashed curves in figures 5 (a), (b) are theoretical fitting using
Langevin and Brillouin functions. Indeed, using equations (4.2) and (4.6), one
can suppose that the dependence of ME current on the magnetic field $H$ could
be described with the following equation

$\displaystyle{I_{ME}}$
$\displaystyle\approx{I_{0}}\left\langle{\frac{{\partial
M_{\rm{PM}}^{2}}}{{\partial
H}}}\right\rangle+{I_{1}}\left\langle{\frac{{\partial
M_{\rm{SPM}}^{2}}}{{\partial H}}}\right\rangle$ $\displaystyle\approx
2{I_{0}}\chi_{m}^{2}H+2{I_{1}}\int\limits_{{}_{{\mu_{\min}}}}^{{}_{{\mu_{\max}}}}{M_{P}^{2}f\left({\frac{{\mu
H}}{{{k_{\rm B}}T}}}\right)\frac{\partial}{{\partial H}}f\left({\frac{{\mu
H}}{{{k_{\rm B}}T}}}\right)F\left({\mu-{\mu_{S}}}\right)\mathrm{d}\mu}.$ (4.9)

Here, $f(x)$ is a LF or BF given by equations (4.6). Since the phenomena are
sample-dependent, the fitting parameters in equations (4.9) are $\chi_{m}$,
$M_{P}$, $\mu_{S}$, $I_{0}$, $I_{1}$, and the distribution function
$F\left(\mu-{\mu_{S}}\right)$, while the signs of $I_{0}$ and $I_{1}$ can be
arbitrary.

Figures 5 (c), (d) show the ratio of SPM to PM contributions to ME current,
dimensionless parameter

$\displaystyle\gamma\left(H\right)=\frac{{{I_{1}}M_{P}^{2}}}{{{I_{0}}\chi_{m}^{2}H}}f\left({\frac{{{\mu_{S}}H}}{{{k_{\rm
B}}T}}}\right)\frac{\partial}{{\partial
H}}f\left({\frac{{{\mu_{S}}H}}{{{k_{\rm B}}T}}}\right),$ (4.10)

calculated from equations (4.10) for $x=0.4$ and 0.5, respectively.

Note that the strong scattering of ME effect values in PFT–PZT solid solutions
made it cumbersome to perform a direct comparison with our results obtained
for PFT–PMN–PT solid solution. Our study demonstrates that multiferroics with
superparamagnetic phase can be considered as promising materials for
applications along with composite multiphase (ferroelectric/ferromagnetic)
structures. Since the ME response in multiferroics with the superparamagnetic
phase is proportional to $\mathrm{d}M^{2}/\mathrm{d}H$, its value can be
amplified by many orders due to a sharp change of magnetization with the
field. The measurements and theoretical analysis show that the main
contribution to the ME current is caused by the superparamagnetic phase.

###  4.2 Giant magnetoelectric response in multiferroics with coexistence of
superparamagnetic and ferroelectric phase at room temperature

Another example of ME materials where the ME coupling is quite large is the
solid solution of PFT with Pb(ZrTi)O3 (PZT). In particular, the composition
0.4PFT–0.6PZT was studied. Figure 6 shows the dependence of the ME voltage as
a function of bias magnetic field at temperatures from 300 K down to 6 K. One
can see that the behavior of the ME response in the magnetic field at $T>100$
K is very similar to that of the 0.4PFT–0.6(PMN–PT) ceramics presented in
subsection 4.1. Namely, there is a strongly nonlinear ME response in small
magnetic fields and the linear part at $H>\pm 1$ kOe. This linear part of the
ME signal is related to the paramagnetoelectric contribution in the ME signal.
The paramagnetoelectric contribution strongly increases at the temperature
decrease. Its temperature dependence (the calculated ME coupling coefficient)
measured at the magnetic field of 10 kOe is shown in figure 7. It follows well
the temperature dependence of the magnetic susceptibility (or its square) in
the paramagnetic phase created by isolated Fe3+ ions. The ME coefficient in
this phase tends to infinity as the temperature approaches 0 K due to the
relation $\chi\sim 1/T$. However, its actual value is, of course, limited by a
saturated magnetization of spins in the magnetic field.

Figure 6: (Colour online) ME voltage as a function of the applied bias
magnetic field measured for 0.4PFT–0.6PZT ceramics at temperatures from 300 K
down to 6 K. Adapted from [24].

Figure 7: (Colour online) Temperature dependence of the PME coupling
coefficient of 0.4PFT–0.6PZT ceramics at the bias magnetic field of 10 kOe.
Square symbols and smooth line are measured and calculated data, respectively.
Adapted from [24].

Another part of the ME signal is related to superparamagnetism or rather weak
magnetism of the 0.4PFT–0.6PZT ceramics. Figure 8 shows this ME signal at
small magnetic fields together with magnetization. The magnetization
nonlinearly changes at $H<\pm 2$ kOe and shows a hysteresis with slim
magnetization loop and remanent magnetization $\approx 0.004$ emu/g. Similar
hysteresis is seen in the ME signal. Obviously, the superparamagnetic clusters
in 0.4PFT–0.6PZT ceramics are larger in volume than those in
0.4PFT–0.6(PMN–PT) and some interaction may also exist between them. This
leads to freezing (thermal blocking) of the superparamagnetic moments of the
clusters even at room temperature. Since the remanent magnetization is non-
zero, the linear ME effect is expected. Note that similar magnetic loops were
observed for the PFN–PZT and PFT–PZT ceramics. However, the remanent
magnetization ascribed to ferromagnetism was a few times bigger than that in
our ceramics.

Figure 8: Field dependence of the linear ME coefficient (a) and magnetization
(b) measured for 0.4PFT–0.6PZT ceramics at 296 K. Adapted from reference [24].

The results are valid for other multiferroics having superparamagnetic and
ferroelectric phases. The study performed by us demonstrated that
multiferroics having superparamagnetic and ferroelectric phases can be
considered as promising materials for various applications along with
composite multiphase (ferroelectric/ferromagnetic) layered structures. Since
the ME response in multiferroics having superparamagnetic phase is
proportional to $\mathrm{d}M^{2}/\mathrm{d}H$, its value can be amplified by
many orders due to a sharp change of magnetization with the field. The
discovery of physical mechanism responsible for a giant ME response opens the
way for obtaining novel multiferroics, both single-phase and composite,
necessary for modern electronic technological devices. In particular,
superparamagnetic magnetization obtained for a composite based on carbon
powder with nanopores filled with Ni might probably be obtained using some
ferroelectric material (e.g., KNbO3 with large region of FE phase) instead of
carbon. On the other hand, in single-phase multiferroic on the base of
nanosized particles of Fe3O4 having a superparamagnetic phase, the
ferroelectric phase can be induced by annealing in the nitrogen atmosphere,
producing oxygen vacancies, which due to the Vegard effect is capable of
introducing the FE phase.

##  5 Conclusion

We reviewed the recent trends of nanoferroics and multiferroics studies. The
main attention is paid to spontaneous flexoeffects and reentrant phase in
nanoferroics and to recently discovered giant magnetoelectric effect in
multiferroics. In particular, ME coefficient measured by the authors of
references [23, 24] at room temperature is equal to $\beta=0.54\cdot 10^{-15}$
s/Å for (PFT)x(PMN0.7–PT0.3)1x while its value for PFN–PT appeared to be much
smaller, namely $\beta\sim 10^{-18}$ s/Å and its value for BiFeO3 is
$\beta=2.54\cdot 10^{-19}$ s/Å. We have to note that both these phenomena (see
references [15, 18]) as well as the appearance of morphotropic phase in a
relaxor due to the influence of the oxygen vacancies (see [22]) were proposed
by the author of this review for the first time in the world science. The same
statement is also correct for the recently discovered giant magnetoelectric
effect in multiferroics (see [23]).

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