Novel room temperature multiferroics on the base of single-phase nanostructuredperovskitesMaya D. Glinchuk, Eugene A. Eliseev, and Anna N. Morozovska Citation: Journal of Applied Physics 116, 054101 (2014); doi: 10.1063/1.4891459 View online: http://dx.doi.org/10.1063/1.4891459 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Charge control of antiferromagnetism at PbZr0.52Ti0.48O3/La0.67Sr0.33MnO3 interface Appl. Phys. Lett. 104, 132905 (2014); 10.1063/1.4870507 Structural, magnetic, and nanoscale switching properties of BiFeO3 thin films grown by pulsed electrondeposition J. Vac. Sci. Technol. B 31, 032801 (2013); 10.1116/1.4802924 New multiferroics based on EuxSr1−xTiO3 nanotubes and nanowires J. Appl. Phys. 113, 024107 (2013); 10.1063/1.4774208 Stability of the various crystallographic phases of the multiferroic ( 1 − x ) BiFeO 3 – xPbTiO 3 system as afunction of composition and temperature J. Appl. Phys. 107, 124112 (2010); 10.1063/1.3437396 Structural and physical properties of room temperature stable multiferroic properties of single-phase ( Bi 0.9 La0.1 ) FeO 3 – Pb ( Fe 0.5 Nb 0.5 ) O 3 solid solution systems J. Appl. Phys. 105, 07D919 (2009); 10.1063/1.3072034

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Novel room temperature multiferroics on the base of single-phasenanostructured perovskites

Maya D. Glinchuk,1 Eugene A. Eliseev,1 and Anna N. Morozovska1,2,a)

1Institute for Problems of Materials Science, NAS of Ukraine, Krjijanovskogo 3, 03142 Kyiv, Ukraine2Institute of Physics, NAS of Ukraine, 46, pr. Nauki, 03028 Kyiv, Ukraine

(Received 16 April 2014; accepted 15 July 2014; published online 1 August 2014)

The theoretical description of the nanostructured Pb(Fe1/2Ta1/2)x(Zr0.53Ti0.47)1�xO3 (PFTx-PZT(1�x))

and Pb(Fe1/2Nb1/2)x(Zr0.53Ti0.47)1�xO3 (PFNx-PZT(1�x)) intriguing ferromagnetic, ferroelectric,

and magnetoelectric properties at temperatures higher than 100 K are absent to date. The goal of this

work is to propose the theoretical description of the physical nature and the mechanisms of the

aforementioned properties, including room temperature ferromagnetism, phase diagram dependence

on the composition x with a special attention to the multiferroic properties at room temperature,

including anomalous large value of magnetoelectric coefficient. The comparison of the developed

theory with phase diagrams allow establishing the boundaries between paraelectric, ferroelectric,

paramagnetic, antiferromagnetic, ferromagnetic, and magnetoelectric phases, as well as the

characteristic features of ferroelectric domain switching by magnetic field are performed and

discussed. The experimentally established absence of ferromagnetic phase in PFN, PFT and in the

solid solution of PFN with PbTiO3 (PFNx-PT(1�x)) was explained in the framework of the

proposed theory. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4891459]

I. INTRODUCTION

The search of room temperature magnetoelectric multi-

ferroics is known to be a hot topic for researchers and engi-

neers working in the field of novel functional devices

fabrication.1–5 For the majority of these devices operation at

room temperature and significant magnetoelectric coupling

are especially vital. Until recently, such characteristics were

demonstrated on multiferroic heterostructures.6–11 The dis-

covery of single-phase room temperature magnetoelectrics

on the basic of solid solutions of ferroelectric antiferromag-

nets Pb(Fe1/2Ta1/2)O3 (PFT) and Pb(Fe1/2Nb1/2)O3 (PFN)

with Pb(Zr0.53Ti0.47)O3 (PZT) seems to be very impor-

tant.6,8–11 Since we will pay attention to the solid solutions

Pb(Fe1/2Ta1/2)x(Zr0.53Ti0.47)1�xO3 (PFTx-PZT(1�x)) and

Pb(Fe1/2Nb1/2)x(Zr0.53Ti0.47)1�xO3 (PFNx-PZT(1�x)) in this

paper, let us discuss briefly their properties at T � 100 K.

PFN is an antiferromagnet with G-type spin ordering

below at T<TNeel, where TNeel¼ 143–170 K.12–14 Also it

is conventional ferroelectric at temperatures T<TCurie.

Ferroelectric phase transition (as reported in different

works) appears in the range TCurie¼ 379–393 K.13–16 PFN

has biquadratic magnetoelectric (ME) coupling constant

2.2� 10�22 sm/(VA) at 140 K.17 PFT is an antiferromagnet

with Neel temperature TNeel¼ 133–180 K (Refs. 13, 18–20)

and ferroelectric with the phase transition at TCurie

� 250 K,21 at that the value is slightly dependent on the

external field frequency. PFT has biquadratic ME coupling

constant of the same order as PFN. PZT is nonmagnetic

and conventional ferroelectric with transition temperature

varied in the range 666–690 K depending on the sample

preparation.22

PFTx-PZT(1�x) and PFNx-PZT(1�x) were studied

at composition x¼ 0.1�0.4.6,8,9 At x¼ 0.1, ferromagnetism

is faint, while at x¼ 0.2–0.4, PFTx-PZT(1�x) exhibits

saturated square-like magnetic hysteresis loops with magnet-

ization 0.1 emu/g at 295 K and pronounced saturated ferro-

electric hysteresis with saturation polarization 25 lC/cm2,

which actually increases to 40 lC/cm2 in the high tempera-

ture tetragonal phase, representing an exciting new room

temperature oxide multiferroic.8,9 Giant effective ME coeffi-

cient of PFTx-PZT(1�x) was reported as 1.3� 10�7 s/m for

x¼ 0.4;8 however, it appeared to be a nonlinear effect.

Meanwhile, Ref. 9 reports about much smaller value of the

linear ME coefficient, 1.3� 10�11 s/m. The reason of the

strong discrepancy between the ME coefficients is not clear,

one of the possible reasons is that the magnetoelectric cou-

pling coefficient was measured in a different way in Ref. 9

and on different ceramic samples. The possible difference in

the samples microstructure is more important for our study.

PFNx-PZT(1�x) demonstrates loops of magnetization

vs. applied magnetic field at room temperature for the com-

position x between 0.1 and 0.4; however, an improvement in

ferromagnetic properties was observed for x¼ 0.2 and

x¼ 0.3, while a notable deterioration of these properties was

observed for x¼ 0.1 and 0.4.6,9 Saturated and low loss ferro-

electric hysteresis curves with a remanent polarization of

about 20–30 lC/cm2 were observed in Refs. 6, 9, and 23.

Note that Scott5 stressed that the magnetoelectric

switching of single-phase nanocrystals in PFTx-PZT(1�x)

was reported by Evans et al.8 Actually, PFTx-PZT(1�x)

samples studied that there have a lamellar structure with pro-

nounced nanodomains of about 10–50 nm average diameter.

These are known to be slightly Fe-rich nanoregions, but not

a)Author to whom correspondence should be addressed. Electronic mail:

anna.n.morozovska@gmail.com

0021-8979/2014/116(5)/054101/8/$30.00 VC 2014 AIP Publishing LLC116, 054101-1

JOURNAL OF APPLIED PHYSICS 116, 054101 (2014)

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a different phase in PFTx-PZT(1�x).5 Sanchez et al.9 stated

that Fe spin clustering plays a key role in the room-

temperature magnetoelectricity of these materials. In such

case, it is useful to have evidence that they may be consid-

ered as single-phase crystals rather than nanocomposites.

Moreover, Sanchez et al.6 revealed that the local phonon

mode A1g corresponds to ordered nanodomains in PFNx-

PZT(1�x) and it is attributed to the vibration of oxygen ion

in the oxygen octahedra. These facts speak in favour that the

presence of nanostructure can play an important role in the

description of solid solutions PFTx-PZT(1�x) and PFNx-

PZT(1�x) physical properties, including primary the room

temperature ferromagnetism. So that it seems prospective to

consider nanostructured PFNx-PZT(1�x) and PFTx-

PZT(1�x), where the term "nanostructured" suppose either

Fe-richer nanoregions, nanocrystals or nanosized lamellas,

or by extension, artificially created nanograined ceramics. In

what follows we will call any of them nanoregions.

To the best of our knowledge, there are no published

papers devoted to the theoretical description of the nano-

structured PFNx-PZT(1�x) and PFTx-PZT(1�x) intriguing

ferromagnetic, ferroelectric, and magnetoelectric properties

at temperatures T � 100 K. Therefore, the main goal of this

work is to propose the theoretical description of the physical

nature and mechanisms of the aforementioned properties,

including room temperature ferromagnetism, phase diagram

dependence on the composition x with a special attention to

the multiferroic properties at room temperature. The compar-

ison of the developed theory with experiments establishing

the boundaries between paraelectric (PE), paramagnetic

(PM), antiferromagnetic (AFM), ferroelectric (FE), ferro-

magnetic (FM), and magnetoelectric (ME) phases, as well as

characteristic features of the ferroelectric domain switching

by magnetic field are performed and discussed.

II. LANDAU-GINZBURG (L-G) POTENTIAL

The polarization and structural parts of the 2–4-power

homogeneous bulk density of L-G potential is the sum of

polarization (gP), antimagnetization (gL), magnetization

(gM), elastic (gel), and magnetoelectric (gME) parts24

GPM ¼ gP þ gL þ gM þ gel þ gME: (1)

The densities gP ¼ aP

2P2

i þbPij

4P2

i P2j þ q

ðeÞijkluijPkPl, gL¼ aL

2L2

i

þbLij

4L2

i L2j þq

ðlÞijkluijLkLl, gM¼ aM

2M2

i þbMij

4M2

i M2j þq

ðmÞijkl uijMkMl,

gel ¼ cijkl

2uijukl þ

Aijklmn

2uijuklPmPn þ Bijklmn

2uijuklLmLn þ Cijklmn

2uij

uklMmMn. P is the polarization, Li ¼ ðMai � MbiÞ=2 is the

components of antimagnetization vector of two equivalent

sub-lattices a and b, and Mi ¼ ðMai þ MbiÞ=2 is the magnet-

ization vector components; uij is elastic strain tensor; qðeÞijkl,

qðlÞijkl, and q

ðmÞijkl are the bulk electrostriction, antimagnetostric-

tion, and magnetostriction coefficients correspondingly, cijkl

are elastic stiffness.

The linear and quadratic magnetoelectric (ME) energy is

gME ¼ lijMiPj þgFM

ijkl

2MiMjPkPl þ

gAFMijkl

2LiLjPkPl; (2a)

lij is the bilinear ME coupling term, gFMijkl and gAFM

ijkl are the

components of the biquadratic ME coupling term. Tensors

gFMijkl and gAFM

ijkl have electro- and magneto-striction contribu-

tions as it were shown earlier25

gFMijkl Rð Þ ¼ � gijkl þ q

eð Þijmnsmnspq

mð Þspkl

�� ~Aijspg

mð Þksn g

mð Þlpn þ ~Bijspg

eð Þksng

eð Þlpn

� ��

� 1þ Rl1

Rþ Rl2

R

� �2 !

; (2b)

gAFMijkl Rð Þ ¼ gijkl þ q

eð Þijmnsmnspq lð Þ

spkl

�� ~Cijspg

mð Þksn g

mð Þlpn þ ~Dijspg

eð Þksng

eð Þlpn

� ��

� 1þ Rl1

Rþ Rl2

R

� �2 !

: (2c)

Here, gijkl is the "bare" ME coupling tensor, smnsq are elastic

compliances, gðeÞijk and g

ðmÞijk are tensors of piezoelectric and

piezomagnetic effects, respectively. One can see from

Eqs. (2b) and (2c) that jgAFMijkl j 6¼ jgFM

ijkl j due to the difference

in electrostriction and (anti)magnetostriction coefficients

qðmÞijkl 6¼ q

ðlÞijkl and different higher coupling constants ~Aijkl 6¼

~Cijkl and ~Bijkl 6¼ ~Dijkl. Moreover, since the striction, piezo-

electric, and piezomagnetic tensors strongly depend on the

composition x, the ME coupling coefficients gFMijkl and gAFM

ijkl

can vary essentially for PFN and PFT. Hereinafter, R is the

average characteristic size of the nanoregion (e.g., radius of

the nanograin). Rl1 and Rl2 are some characteristic radii,

which physical meaning is the curvature radius determined

by the balance of the intrinsic surface stress, linear electro-

and magnetostriction, and nonlinear electro- and

magnetostriction.

The characteristic radii Rl1 and Rl2 values are propor-

tional to the product of the surface tension coefficient l,

electrostriction Qij, magnetostriction Zij, nonlinear electro-

and magnetostriction coefficients Aij and Bij correspondingly

in Voight notations. Namely, Rl1 / l QijBijþZijAij

QijZij

� �and Rl2 /

lffiffiffiffiffiffiffiffiAijBij

QijZij

q(see details in Ref. 25). The characteristic radii

values depend on the nanoregion shape. In particular case of

nanoregions observed in Ref. 8, it can be modeled by a wire

of radius R, radii Rl1 ¼ 2l s12

s11

Q11B33þZ11A33

Q11Z11

� �and Rl2 ¼ 2l s12

s11ffiffiffiffiffiffiffiffiffiffiA33B33

Q11Z11

q, where sij are elastic compliances in Voight notations

(see Eq. (9c) in Ref. 25). It was shown that Rl1 and Rl2 val-

ues can reach several hundreds of nm and so the contribution

ðRl1=RÞ þ ðRl2=RÞ2 in Eqs. (2b) and (2c) may increase the

ME coupling coefficient in 10–103 times for the average size

R / 20–2 nm (see Figure 3 in Ref. 25).

The coefficient aP linearly depends on temperature, i.e.,

aP ¼ aTY T � TCY 1� RPY

R

� �� �,25 where Y¼ “N” or “T”; “N”

corresponds to PFN and “T” corresponds to PFT. TCY is the

ferroelectric transition temperature of homogeneous bulk. In

054101-2 Glinchuk, Eliseev, and Morozovska J. Appl. Phys. 116, 054101 (2014)

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the work,24 the form of coefficients aL and aM were obtained

for any antiferromagnet with two sublattices a and b. It is a

common knowledge (see, e.g., Ref. 26) that with account of

an exchange interaction constant c between the sub-lattices

and the interaction constant inside sublattices b, two charac-

teristic temperatures have to be introduced: Neel tempera-

ture, TN ¼ c�bð Þ2aT

M

, that defines the magnetic susceptibility

behavior at T � TN , and Curie temperature, hC ¼ � cþbð Þ2aT

M,

that defines the magnetic susceptibility behavior at T > TN .

Because of this, one can rewrite the expressions for the coef-

ficients obtained in Refs. 21 and 25 for the case PFT and

PFN in the form aL ¼ aLT T � TNY 1� RLY

R

� �� �and aM ¼ aMT

T � hCY 1� RMY

R

� �� �. Temperatures TN

Y and hCY correspond to

the homogeneous bulk.

Similarly to the characteristic size Rl1 that govern the

size dependence of ME coefficients (2b) and (2c), the critical

sizes RPY , RLY , and RMY originate from the surface tension

effect coupled with electrostriction and magnetostriction. The

values of RPY and RLY , RMY are proportional to the product of

the surface tension coefficient and the electrostriction or mag-

netostriction tensor coefficients correspondingly, namely,

RPY / l Qij

aTY TCY

� �, RLY / l

~Zij

aLT TNY

� �, and RMY / l Zij

aMThCY

� �(see

details in Ref. 25). In particular case of a nanowire, the radii

RPY ¼ 2l Q12�Q11 s12=s11ð ÞaTY TC

Y

� �, RLY / l

~Z12� ~Z11 s12=s11ð ÞaLT TN

Y

� �, and RMY

¼ 2l Z12�Z11 s12=s11ð ÞaMThC

Y

� �(see Eqs. (6a) and (6b) in Ref. 25). The

critical sizes can be positive or negative depending on the

nanoregion shape, electro- and magnetostriction tensor coeffi-

cients sign, and surface stress direction (compressive or ten-

sile). Their typical values are 1–10 nm.25

Note, that all the quantities can depend on the composi-

tion x of the solid solutions PFNx-PZT(1�x) and PFTx-

PZT(1�x).

III. PZT-PFT AND PZT-PFN PHASE DIAGRAMS

A. Analytical formalism

For the solid solutions, the ferroelectric Curie tempera-

ture can be modeled using a linear law22

TFEPFY�PZTðx;RÞ ¼ xTPFYðRÞ þ ð1� xÞTPZTðRÞ: (3)

Hereinafter, Y¼N for PFNx-PZT(1�x) or Y¼T for

PFTx-PZT(1�x). The temperatures TPZT ¼ 666–690 K,

TPFN ¼ 383–393 K, and TPFT ¼ 247–256 K are defined for

homogeneous bulk material with error margins depending on

the sample preparation. For nanostructured material, the

temperatures become R-size dependent as TPFY Rð Þ ¼ TCY

1� RPY

R

� �and TPZT Rð Þ ¼ TC

PZT 1� RPZT

R

� �.

In order to describe the x-composition dependence of

the AFM-PM and FM-PM phase transition temperatures, we

will use the approach27 based on the percolation theory.28

We assume a linear dependence of FM ordering on Fe con-

tent, x, above the percolation threshold, x ¼ xcr. The critical

concentration of percolation threshold xFcr � 0:09 (Ref. 28)

for the case of face-centered cubic sub-lattices of magnetic

ions. The percolation threshold is supposed to be essentially

higher for AFM ordering, xAcr � 0:48 (see, e.g., Refs. 29 and

30 and references therein). Note that superscripts “F” and

“A” in xF;Acr designate the critical concentrations related to

FM and AFM ordering, respectively. Thus, we assume that

nonlinear magnetization expansion coefficient b critically

depends on x. In particular, bLðxÞ ¼ bLðx� xAcrÞ=ð1� xA

crÞ,bMðxÞ ¼ bMðx� xF

crÞ=ð1� xFcrÞ at content xF;A

cr � x � 1;

while bLðxÞ ¼ 0 and bMðxÞ ¼ 0 at x < xA;Fcr . Here, we suggest

that magnetization has only one component and re-designate

bL11 � bL and bM11 � bM, which means that we neglected

possible magnetic anisotropy.

In contrast to coefficient b, one can assume that the

power expansion on x is valid for biquadratic ME coeffi-

cients gAFMðx;RÞ and gFMðx;RÞ, but they also tend to zero at

x � xFcr , since the solid solution becomes nonmagnetic at

x � xFcr . Hereinafter, we assume that biquadratic ME cou-

pling coefficients of FM and AFM have different signs, and,

in the most cases, gAFM > 0 and gFM < 0.31 The assumption

about different signs of gAFMðxÞ and gFMðxÞ agrees with

Smolenskii and Chupis,32 Katsufuji and Takagi,33 and Lee

et al.34 In particular, Smolenskii and Chupis,32 Katsufuji and

Takagi33 stated that it is natural to consider that the dielectric

constant is dominated by the pair correlation between the

nearest spins, which leads to the ME phenomenological term

gP2ðM2 � L2Þ.For the composition x > xA

cr, the temperature of the solid

solution transition from the PM into AFM-ordering state is

renormalized by the biquadratic ME coupling

TAFMPFY�PZT x;Rð Þ ¼ TN

Y Rð Þ x� xAcr

� �1� xA

cr

� �� gAFM xð ÞaLT

P2S x;Rð Þ: (4)

Measured N�eel temperatures for conventional bulk PFT and

PFN are (143–170) K and (133–180) K, correspondingly.

Thus, the values TNY are "bare," because they are, in fact,

shifted by the biquadratic ME coupling term gAFMP2S=aLT to

lower temperatures, since gAFM > 0. For nanostructured ma-

terial, the temperature depends on the size R as

TNPFY Rð Þ ¼ TN

Y 1� RLY

R

� �. Corresponding size-dependence of

polarization will be discussed below.

For the composition x > xFcr, the temperature of the solid

solution transition from the PM into FM-ordering state is

TFMPFY�PZT x;Rð Þ ¼ hC

Y Rð Þ x� xFcr

� �1� xF

cr

� �� gFM x;Rð ÞaMT

P2S x;Rð Þ: (5)

Since PFT and PFN are antiferromagnets with negative tem-

perature hPFYC , there should be strong enough biquadratic ME

coupling term, gFMP2S=aMT , that can strongly increases the

FM-temperature for the solid solution up to the room and

higher temperatures, since gFM < 0. For nanostructured

material, the temperature hCPFY is R-dependent as

hCPFY Rð Þ ¼ hC

Y 1� RMY

R

� �.

In Eqs. (4) and (5), the spontaneous polarization

squire is

054101-3 Glinchuk, Eliseev, and Morozovska J. Appl. Phys. 116, 054101 (2014)

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P2Sðx;RÞ � aTYðTFE

PFY�PZTðx;RÞ � TÞ=bPðxÞ; (6)

where TFEPFY�PZTðx;RÞ is given by Eq. (3) allowing for

R-dependence in the nanostructured case, and bPðxÞ depends

on the composition x as bPðxÞ ¼ bPð0Þð1þ bxxþ bxxx2

þ:::Þ. The power dependence of bPðxÞ on x is in agreement

with the well-known experimental results for the Pb-based

solid solutions (see, e.g., Ref. 22 for PbZrTiO3). Hereinafter,

we suggest that polarization has only one component and re-

designate bP11 � bP.

The substitution of P2Sðx;RÞ, gAFMðx;RÞ, and gFMðx;RÞ

into Eqs. (4) and (5) leads to the evident dependences of the

temperatures on the composition x at given average radius R.

B. The impact of the size effect on the origin of theferromagnetism in the solid solution PZT-PFT andPZT-PFN

The principal question is about the impact of the average

size R of the nanoregion on the possible origin of the ferro-

magnetism in the studied solid solution with one nonmag-

netic component (PZT) and antiferromagnetic, one (PFT or

PFN) with hCY < 0. To answer the question, let us analyze

Eq. (5). For the case x � xFcr , any ferromagnetism is absent,

since bMðxÞ ¼ 0. For the case of x increase and x > xFcr, the

first term hCPFY Rð Þ x�xF

crð Þ1�xF

crð Þ decreases the Curie temperature

TFMPFY�PZTðx;RÞ because hC

PFY < 0, but the second termgFM x;Rð Þ

aMTP2

S x;Rð Þ can increase it, since gFMðx;RÞ < 0 and the

spontaneous polarization of PZT is 0.5 C/m2 and it less than

0.10 C/m2 for pure PFN at room temperature. PFT is in PE

phase at room. Since the electro, magnetostriction, piezo-

electric, and piezomagnetic tensors strongly depend on the

composition x, the ME coupling coefficients gFM and gAFM

strongly vary with the composition x and size R in agreement

with Eqs. (2b) and (2c). Moreover, it is maximal for interme-

diate concentration x, where electrostriction,

magnetostriction, and piezoelectric coefficients are high and

piezomagnetic coefficients are nonzero.

In result of the P2S and gFM x-dependences, the term

gFMP2S=aMT can be 10 times higher for PFNx-PZT(1�x) and

PFTx-PZT(1�x) than that for PFN or PFT and, for negative

gFM, it makes the sum hCPFY Rð Þ x�xF

crð Þ1�xF

crð Þ �gFM x;Rð Þ

aMTP2

S x;Rð Þ� �

positive for intermediate x-compositions xFcr � x < xA

cr.

Immediately, it leads to the positive FM Curie temperature

in accordance with Eq. (5). So, negative gFM and high

enough ratio jgFMP2S=aMT j can give rise to the FM phase at

intermediate x-compositions xFcr � x < xAFM

cr and lead to its

disappearance with x increase at fixed R value. It is worth to

underline that for the case x¼ 1, it is possible to consider the

limit R!1, when there is only the first negative term in

Eq. (5), so that FM phase is absent for PFT and PFN in

agreement with experiment.

For arbitrary x value, the phase diagrams of solid solu-

tions PFNx-PZT(1�x) and PFTx-PZT(1�x) are shown in

coordinates composition x—temperature T in Figures 1(a)

and 1(b), correspondingly. Some of material parameters are

taken from Ref. 35.

The PE-FE boundary (empty magenta boxes) was fitted

by the linear dependence, the data are taken from experimen-

tally measured Curie temperature of the solid solution for

x¼ 0.2–0.8 from Figure 5 in Ref. 9 and well-known tabulated

data for PZT (x¼ 0) taken from Ref. 22 and PFT or PFN

(x¼ 1) taken from Refs. 12–15. The experimental data from

Figure 5 in Ref. 9 are in a reasonable agreement with the

maximum of dielectric susceptibility from Figure 4 in Ref. 6.

Available experimental points for the FM-FE phase are

not the boundary, but just indicate the region, where it exists.

So the FM boundary that has a mountain-like form can reach

higher temperatures. Appeared that the temperature “height”

of the FM-FE boundary is defined by the values of the nega-

tive ratio cF ¼ gFMaPT=ðaMTbPÞ and “virtual” temperature

hCY . The higher is the value jcFj the higher and wider is the

x-composition region of FM boundary. The smaller is the

FIG. 1. Phase diagrams in coordinates composition—temperature for solid solutions (a) Pb(Fe1/2Nb1/2)x(Zr0.53Ti0.47)1�xO3 and (b) Pb(Fe1/2Ta1/2)x

(Zr0.53Ti0.47)1�xO3. Different symbols are experimental data collected from Refs. 9 and 22. Boxes are data for PE-FE phase transition from Refs. 9 and 22, tri-

angles are AFM-PM boundary,9 diamonds show the points, where FM behaviour is observed.9 Solid curves are theoretical modelling for the percolation thresh-

old concentrations xFcr � 0:09, xA

cr ¼ 0:48, TCPZT ¼ 690 K, PSð0Þ ¼ 0:5 C/m2 corresponds to PZT. Other parameters: (a) TN

PFN ¼ 140 K, cA ¼ 1:5, cF ¼ �46,

hCPFN ¼ � 550 K and (b) TN

PFT ¼ 180 K, cA ¼ 1, cF ¼ �44, hCPFT ¼ �500 K.

054101-4 Glinchuk, Eliseev, and Morozovska J. Appl. Phys. 116, 054101 (2014)

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absolute value of the negative hCY , the wider is the FM bound-

ary. The fitting values of TYN appeared quite realistic; hC

Y are

negative and its value is high enough that is also in agree-

ment with general knowledge.36 The dimensionless ratio

cA ¼ gAFMaTY=ðaLTbPÞ is positive and is about unity. The ra-

tio cF is negative and high enough (<�30). As it follows

from Figure 1, the theory given by solid lines describes the

experimental points pretty good.

The next important issue is to analyze the fitting param-

eters used in Figure 1 and to understand if they can relate to

conventional or nanostructured material, and to what typical

sizes R they correspond in the latter case. Appeared that the

fitting values of cF require the factor 1þ Rl1

R þRl2

R

� �2� �

to

be at least 10 times or higher for the coefficient gFM to be the

same order as the realistic value of 2� 10�22 s m/(VA)

measured experimentally for PFN.17 The condition leads to

the inequality on the average size R < 0:1Rl1;2, i.e., can be

true for nanostructured solid solution, but not for the homo-

geneous bulk. Finally, the inequalities Rl1;2 � 50� 100 nm

and R � 5� 10 nm should be valid, at the same time RPY ,

RLY , and RMY should be smaller than 1�2 nm in order not to

shift essentially the temperatures TPFY Rð Þ ¼ TCY 1� RPY

R

� �,

TPFYN Rð Þ ¼ TC

Y 1� RLY

R

� �, hC

PFY Rð Þ ¼ hCY 1� RMY

R

� �, and polar-

ization P2Sðx;RÞ from their values for homogeneous bulk.

The small values of RPY , RLY , and RMY are required to pro-

vide high enough values of P2Sðx;RÞ. Since the numerical

values of Rl1;2 and RPY , RLY , RMY are defined by different pa-

rameters, e.g., Rl1 / l QijBijþZijAij

QijZij

� �, Rl2 / l

ffiffiffiffiffiffiffiffiAijBij

QijZij

qand

RPY / l Qij

aTY TCY

� �, RLY / l

~Zij

aLT TNY

� �, RMY / l Zij

aMThCY

� �, to have,

respectively, "big" value (Rl1;2 � 50� 100 nm) and "small"

ones (RP;L;MY < 1� 2 nm) are quite possible. At that the

"optimal" nanoregion size should vary in the range

5RY � R < 0:5Rl1;2. For example, the sizes R ¼ 10 nm,

Rl2 / Rl1 � 100 nm, RPY , RLY , RMY about 1 nm satisfy all

the conditions, since 5 nm< 10 nm< 50 nm.

C. Absence of the ferromagnetism in conventionalPFN, PFT, and in the solid solution PFNx-PT(12x)

It was shown experimentally that ferromagnetism is

absent in conventional PFN, PFT without any nanostructural

elements as well as in the conventional solid solution PFNx-

PT(1�x),35 in contrast to nanostructured PFNx-PZT(1�x).

First, let us make estimations based on Eq. (5) to show

that effective Curie temperature TFMPFY ¼ hC

Y �gFM

aMTP2

S is nega-

tive. Using the parameters hCY ¼ �ð450� 500Þ K (from

experiment35 and our fitting in Figure 1), PS � (0.1–0.2) C/m2,

gFM � 2� 10�22 s m/(VA) (Ref. 37), and ðaMTÞ�1 / CM

�(0.2–0.25) K,17 we estimate the value CMgFMP2S as

(40–250)K, so that the temperature TFMPFY < �200 K. So, the

ferromagnetism is absent in pure PFN and PFT.

The possible reason of the ferromagnetism absence in

the solid solution PFNx-PT(1�x) is that PbTiO3 (PT) has

essentially smaller piezoelectric coefficients and dielectric

permittivity at room temperature than PZT near the morpho-

tropic boundary, meanwhile its spontaneous polarization can

be even about 50% higher than that of PZT. Biquadratic ME

coupling coefficient is absent for PT (x¼ 0), but increases

with x increase up to the value g � 2� 10�22 s m/(VA)

for PFN. So, the biquadratic ME coupling coefficient is

expected to be the same or even smaller than that for PFN.

Using the parameters hCN ¼ �500 K, “average” polarization

PS � 0.5 C/m2, permittivity e¼ 103, “average” value ~g� 10�22 s m/(VA), and ðaMTÞ�1 / CM � 0.25 K, we esti-

mated the CMgFMP2S as �62.5 K making the temperature

TFMPFY < �390 K. So, the ferromagnetism is absent in

PFN0.5�PT0.5 in agreement with experiment.35

Therefore, these estimations lead to the conclusion that

the biquadratic ME coefficients gFMðxÞ can be much smaller

for PFNx-PT(1�x) than the ones for PFNx-PZT(1�x). Even

higher polarization P2S cannot make the term gFMP2

S=aMT for

PFNx-PT(1�x) as high as for PFTx-PZT(1�x). In result, the

term appeared not enough to make the sum positive and the

conventional solution PFNx-PT(1�x) exhibit only AFM

region below 100 K on the phase diagram.35

IV. ESTIMATIONS OF EFFECTIVE ME COUPLINGCOEFFICIENTS

A. Estimation of linear ME coupling coefficients forconventional material at low temperatures

Linear ME coupling coefficients estimations are based

on our previous results,25,38 namely, the bilinear coupling

term is

lij / dðeÞjqs d

ðmÞiqs : (7a)

Here, dðeÞijk and d

ðmÞijk are coupling tensors of piezoelectric and

piezomagnetic effects, respectively.

In order to estimate the piezoeffects contribution to the

bilinear coupling term of the solid solution, we should esti-

mate the product, dðeÞjqs d

ðmÞiqs . In order to do this, we take into

account that the corresponding strain is uðmÞij / s�1

ijpsqðmÞpsklMk

Ml / sijkldðmÞqkl Mq; sijkl are components of the elastic compli-

ances tensor.Using of the values sijkl / 5� 10�12 Pa�1, max-

imal piezoelectric coefficient corresponding to morphotropic

boundary of Pb(Zr0.53Ti0.47)O3, gðeÞijk / 0:1 Vm/N at room

temperature, high enough spontaneous magnetization,

M� 5� 103 A/m (M� 0.5 emu/g� 0.5 Am2/kg, mass den-

sity 9.68� 103 kg/m3) and essential magnetostrictive strain

value, uðmÞ / qðmÞM2=s � 10�6, gives us dðmÞ / uðmÞ=ðsMÞ� 102 Pam/A. Thus, dðeÞdðmÞ � 10 V/A and so lij / 10 V/A.

To recalculate the value a in s/m, we can use that dgME

¼ lijMiPj ¼ aijHiEj and Pi ¼ e0eðeÞij Ej, Mi ¼ vðmÞij Hj along

with the values vðmÞij < 10�3 and eðeÞij � 1:5� 103. Thus, the

linear ME coefficient value is a/ e0eðeÞvðmÞl¼ 8:85�10�12

ðF=mÞ�1:5�103�10�3�10ðV=AÞ� 1:3�10�10 s/m.

The value is at least 3 orders of magnitude smaller than

the value of effective ME coefficient, a¼ 1.3� 10�7 s/m,

measured by Evans et al.8 for lamellar nanostructured mate-

rial, where the ME coupling behavior appeared to be nonlin-

ear. Actually, Evans et al.8 admitted that their effective

coupling coefficient does not adhere to the strict definition of

054101-5 Glinchuk, Eliseev, and Morozovska J. Appl. Phys. 116, 054101 (2014)

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the linear ME coupling. On the other hand for single-phase

multiferroics, linear ME coupling coefficients are typically

of the order of 10�10 s/m, while for heterostructures values

increase to of the order of 10�6 to 10�8 s/m.8 Note that one

order smaller piezoelectric coefficient gðeÞijk / 10�2 Vm/N

will lead the "standard" ME coefficient a� 10�11 s/m in

Eq. (7a).

B. Estimation of effective ME coupling coefficients fornanostructured material

Let us estimate the effective ME coefficient for lamellar

nanostructured material allowing for it possible nonlinearity

and size effects. For the case of ferroelectric domain wall

moving in external magnetic field H, Evans et al. calculated

the ME coefficient from the formulae, aef fij ¼ eEcoer

i =Hcritj ,

where the critical magnetic field Hcritj ¼ 3 kOe, coercive elec-

tric field and dielectric permittivity were taken from Sanchez

et al.6 for x¼ 0.4 as “bulk” coercive field Ecoeri ¼ 15 kV/cm

and bulk permittivity e¼ 1200. For the case of ferroelectric

domain wall moving in external magnetic field H, more com-

plex expression should be used for aef fij

aef fij ðRÞ / aij þ bijkðRÞe

ef fkl Ecoer

l þ cijkðRÞvklHcritl

þ gijklðRÞeef fkn Ecoer

n vlmHcritm : (7b)

Here, eef fkl is the local dielectric permittivity, vkl is the mag-

netic permittivity, the coefficients bijk Rð Þ / 1þ Rlb

R

� �,

cijk Rð Þ / 1þ Rlc

R

� �, and gijkl Rð Þ / 1þ Rl1

R þRl2

R

� �2� �

. For

the case of the nanowire, Rlb ¼ 2l s12

s11

A33

Q11, Rlc ¼ 2l s12

s11

B33

Z11,

Rl1 ¼ 2l s12

s11

Q11B33þZ11A33

Q11Z11

� �, and Rl2 ¼ 2l s12

s11

ffiffiffiffiffiffiffiffiffiffiA33B33

Q11Z11

q. Note

that the dependence on size R can increase the ratio aef fij =aij

up to 10 or even 103 times (more realistically in 10–100

times). Also one should take into account that the local

dielectric permittivity eef f in the immediate vicinity of the

ferroelectric domain wall can be much higher than the bulk

value e. Indeed in accordance with the thermodynamic

theory, the ratio eef fij =eij diverges at the wall plane, but in

reality it is finite (due to the presence of internal electric

field), but can reach rather high values. So we see real possi-

bilities to increase the ratio aef fij =aij up to 2–4 orders of mag-

nitude due to the size and local permittivity increase. Thus,

the high value aeff� (10�8�10�7)s/m is reachable for nano-

structured material in contrast to the conventional bulk value

a� (10�11�10�10) s/m. Note that since the coefficient

gFMijkl ðRÞ of nonlinear ME coupling has the same strong de-

pendence on R (see, e.g., Eq. (2b)), it is not excluded that

Evans et al.8 indeed observed quadratic ME effect.

V. FERROELECTRIC DOMAINS SWITCHING BY AMAGNETIC FIELD

The estimation of aij is required to explain the ferroelec-

tric domain structure switching by applied magnetic field

reported by Evans et al.8,10 The nature of the phenomena is

the following. Due to the presence of the strong bilinear ME

coupling, the magnetic field induces the electric field. For

the magnetically isotropic media corresponding "acting"

electric field is

EMEi ¼ lMi �

av mð ÞHi

e0e eð Þv mð Þ ¼aHi

e0e eð Þ ; (8a)

where M is magnetization, H is magnetic field, Mi ¼ vðmÞHi,

and the relation between l and a magnitude is a ¼ e0eðeÞvðmÞl.

Namely, variation of the thermodynamic potential (1) and (2a)

via polarization leads to the equation of state

aPPi þ bPijPiP2j þ q

ðeÞmnilumnPl ¼ �EME

i : (8b)

Note that Eq. (8b) defines the dependence of EMEi and so aP

on x, R and T via the dependence of polarization on these

quantities. If the component of the EMEi conjugated with the

spontaneous polarization component Pi is higher than the

critical field (Ecri ) required for the domain wall motion,

the polarization Pi can be reversed by the field of appropriate

direction. So that applied magnetic field can act as the source

of ferroelectric domain structure triggering observed by

Evans et al.Note that the field Ecr

i is typically much smaller than the

thermodynamic coercive field. For PZT, the critical field was

measured as Ecri ¼ 120 kV/cm,39 but it appeared to be much

smaller for PFNx-PZT(1�x) or PFTx-PZT(1�x) due to

the composition disorder, namely, we will use the value

Ecri ¼ 15 kV/cm measured by Sanchez et al.6 Using the esti-

mation for aef fij � 10�7 s/m and e¼ 1200 in Eq. (8a), one

can lead to the conclusion that the critical magnetic field

Hi � 1:59� 105 A/m is required. The value is in a reasona-

ble agreement with the fields of 3 kOe (that is equal to 2.3

105A/m) applied by Evans et al.8

Figure 2 illustrates the dependence of the electric field

EME induced by the magnetic field H on bilinear ME cou-

pling coefficient aeff and H calculated from Eq. (8a). The

FIG. 2. Contour map of the “acting” electric field is EMEi in coordinates

effective ME coupling coefficient aeff–applied magnetic field H. The critical

electric field is taken as 15 kV/cm.

054101-6 Glinchuk, Eliseev, and Morozovska J. Appl. Phys. 116, 054101 (2014)

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contours of constant EME have hyperbolic shape in coordi-

nates ME coupling coefficient a–applied magnetic field H.

The higher is l, the smaller field H can induce the critical

electric field that, in turn, can move the ferroelectric domain

walls and switch their polarization. The contour EME ¼ Ecr

was calculated for the critical field of 15 kV/cm. For H and lvalues below the contour, the induced electric field EME

< Ecr and, thus, cannot change the domain structure. For Hand aeff values above the contour, the induced electric field

EME > Ecr and, thus, can change the domain structure.

VI. DISCUSSION AND CONCLUSION

The key point of our consideration was the calculation

of FM phase appearance in the nanostructured solid solution

of (PFY)x(PZT)1�x (Y¼Nb, Ta) at room temperature.

Multiferroics PFN and PFT are examples of ferroelectric

antiferromagnets with TC� 380 K, TN¼ 140 K and

TC� 250 K, TN¼ 180 K, respectively, and no FM transition

was observed.

Room temperature FM and magnetoelectric phase with

strong enough linear and biquadratic ME coupling recently

have been revealed in the PFN and PFT solid solution with

PZT. The way, which permitted us to find out the mechanism

of this wonderful phenomenon, was the following. Allowing

for we were looking for FM phase at T much higher than TN,

i.e., in paramagnetic phase, where susceptibility is described

as C/(T–hC) (hC< 0), it can be supposed that hC could be

considered as some seeding temperature for FM phase that

has to be renormalized to positive value by some special

mechanism. Since PZT is known to have strong piezoelectric

and ferroelectric properties, which can increase linear and

nonlinear ME couplings, we supposed that biquadratic ME

coupling strongly enhanced by size effect could be the mech-

anism we were looking for. However, in pure PFN and PFT

as well as in conventional (PFN)x(PT)1�x without any nano-

structure, the addition given by this mechanism appeared to

be not enough to make TFMC > 0 and so FM phase did not

appear. In the case of nanostructured solid solutions with

PZT high piezo- and electrostriction coefficients, gFM value

along with large enough polarization it appeared possible to

obtain FM phase even at room temperature. Note that there

was a discussion in literature (see, e.g., Refs. 40 and 41)

about existence of FM phase at low temperatures in different

structures like A2þFe1=2B5þ1=2

O3 (A2þ¼Ba, Sr, Ca, Pb;

B5þ¼Nb, Ta, W). To our mind, the measurements of field-

cooled susceptibility 1/vFC at high temperature region as it

was done in Ref. 35 for PFN will give the sign and value of

temperature that corresponds to 1/vFC¼ 0. Negative sign

speaks in favours of antiferromagnet, while positive sign cor-

responds to ferromagnetic material. The obtained value

hC� –(450–500) for PFN (Ref. 35) is the direct evidence of

FM phase absence in this material at any temperature

T� 0 K.

Our consideration performed for nanostructured PFTx-

PZT(1�x) and PFNx-PZT(1�x) made it possible to explain

main experimental results observed by Evans et al.8 and

Sanchez et al.6 Namely, the anomalously large effective ME

coupling coefficient aef fij was shown to originate from size

and local effects, which increase ME-coupling on 2–4 orders

in comparison with conventional ceramics. The effective

ME-coupling appeared to be close to quadratic ME effect,

rather than linear one. The calculated values of aef fij and criti-

cal magnetic fields necessary for ferroelectric domain walls

motion were shown to be in reasonable agreement with

experimentally observed quantities.

The quantitative estimations of linear ME coupling coef-

ficient a via piezocoefficients for conventional ceramics

(reported by Evans et al.8) had shown that for PFTx-

PZT(1�x) solid solution at x¼ 0.3 and 0.4, a is about

1.3� 10�10 s/m that is close to standard values measured by

different authors.6,9 The developed theory explains the ab-

sence of ferromagnetic phase in PFN, PFT, and in the solid

solution with PFN-PT.6,9,35 We proposed the theoretical

explanation of the solid solutions PFTx-PZT(1�x) and

PFNx-PZT(1�x) phase diagrams dependence on the compo-

sition x with a special attention to the ferromagnetism and

multiferroic properties for intermediate concentration x at

room temperature.

ACKNOWLEDGMENTS

We are very grateful to Professor Donald Evans and Dr.

Roman Kuzian for useful comments, suggestions, and

multiple discussions. Authors acknowledge the support via

bilateral SFFR-NSF Project (US National Science

Foundation under NSF-DMR-1210588 and State Fund of

Fundamental State Fund of Fundamental Research of

Ukraine, Grant No. UU48/002) and National Academy of

Sciences of Ukraine (Grant No. 35-02-14).

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