New multiferroics based on EuxSr1xTiO3 nanotubes and nanowiresEugene A. Eliseev, Maya D. Glinchuk, Victoria V. Khist, Chan-Woo Lee, Chaitanya S. Deo, Rakesh K. Behera,

and Anna N. Morozovska

Citation: Journal of Applied Physics 113, 024107 (2013); doi: 10.1063/1.4774208 View online: http://dx.doi.org/10.1063/1.4774208 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/113/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Coexistence of four resistance states and exchange bias in La0.6Sr0.4MnO3/BiFeO3/La0.6Sr0.4MnO3multiferroic tunnel junction Appl. Phys. Lett. 104, 043507 (2014); 10.1063/1.4863741 Mn substitution-modified polar phase in the Bi1xNdxFeO3 multiferroics J. Appl. Phys. 113, 214112 (2013); 10.1063/1.4810764 Structure and multiferroic properties of Bi(1-x)DyxFe0.90Mg0.05Ti0.05O3 solid solution J. Appl. Phys. 113, 054102 (2013); 10.1063/1.4790326 The origin of magnetism in perovskite ferroelectric ABO3 nanoparticles (A=K,Li; B=Ta,Nb or A=Ba,Sr,Pb; B=Ti) J. Appl. Phys. 112, 053907 (2012); 10.1063/1.4748319 Room temperature ferroelectric and ferromagnetic properties of multiferroics x La 0.7 Sr 0.3 Mn O 3 – ( 1 x ) ErMn O 3 (weight percent x = 0.1 , 0.2) composites Appl. Phys. Lett. 90, 162510 (2007); 10.1063/1.2723198

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New multiferroics based on EuxSr12xTiO3 nanotubes and nanowires

Eugene A. Eliseev,1 Maya D. Glinchuk,1 Victoria V. Khist,1 Chan-Woo Lee,2

Chaitanya S. Deo,3 Rakesh K. Behera,3,a) and Anna N. Morozovska4,a)

1Institute for Problems of Materials Science, NAS of Ukraine, Krjijanovskogo 3, 03142 Kiev, Ukraine2The Makineni Theoretical Laboratories, Department of Chemistry, University of Pennsylvania, Philadelphia,Pennsylvania 19104, USA3Nuclear and Radiological Engineering Program, George W. Woodruff School of Mechanical Engineering,Georgia Institute of Technology, Atlanta, Georgia 30332, USA4Institute of Physics, NAS of Ukraine, 46, Pr. Nauki, 03028 Kiev, Ukraine

(Received 30 October 2012; accepted 17 December 2012; published online 10 January 2013)

Using Landau-Ginzburg-Devonshire theory, we have addressed the complex interplay between

structural antiferrodistortive order parameter (oxygen octahedron rotations), polarization and

magnetization in EuxSr1�xTiO3 nanosystems. We have calculated the phase diagrams of

EuxSr1�xTiO3 bulk, nanotubes and nanowires, which include the antiferrodistortive, ferroelectric,

ferromagnetic, and antiferromagnetic phases. For EuxSr1�xTiO3 nanosystems, our calculations

show the presence of antiferrodistortive-ferroelectric-ferromagnetic phase or the triple phase at low

temperatures (�10 K). The polarization and magnetization values in the triple phase are calculated

to be relatively high (�50 lC/cm2 and �0.5 MA/m). Therefore, the strong coupling between

structural distortions, polarization, and magnetization suggests the EuxSr1�xTiO3 nanosystems as

strong candidates for possible multiferroic applications. VC 2013 American Institute of Physics.

[http://dx.doi.org/10.1063/1.4774208]

I. INTRODUCTION

The search for new multiferroic materials with large

magnetoelectric (ME) coupling is very interesting for funda-

mental studies and important for applications based on the

magnetic field control of the material dielectric permittivity,

information recording by electric field, and non-destructive

readout by magnetic field.1,2 Solid solutions of different

quantum paraelectrics (such as EuxSr1�xTiO3 or EuxCa1�x

TiO3) subjected to elastic strains can be promising for multi-

ferroic applications.

Bulk SrTiO3 is nonmagnetic quantum paraelectric at all

temperatures.3 Below 105 K, bulk SrTiO3 has antiferrodistor-

tive (AFD) structural order,4–6 characterized by spontaneous

oxygen octahedron rotation angles (or “tilts”), which can be

described by an axial vector Ui (i¼ 1, 2, 3).7 However,

SrTiO3thin films under misfit strain could be ferroelectric up

to 400 K.8

Bulk quantum paraelectric EuTiO3 is AFD below about

281 K,9–13 antiferromagnetic (AFM) at temperatures lower

than 5.5 K, and paraelectric at all other temperatures.1,2

Using first principles calculations, Fennie and Rabe14 pre-

dicted the presence of simultaneous ferromagnetic (FM) and

ferroelectric (FE) phases in (001) EuTiO3 thin films under

compressive epitaxial strains exceeding 1.2%. They demon-

strated that strain relaxation to the values lower than 1%

eliminates FE-FM phase appearance in EuTiO3 thin film.14

Lee et al.15 demonstrated experimentally that EuTiO3 thin

films with thickness of 22 nm on DyScO3 substrate become

FM at temperatures lower than 4.24 K and FE at tempera-

tures lower than 250 K under the application of more than

1% tensile misfit strain.

The intrinsic surface stress can induce ferroelectricity,

ferromagnetism and increase corresponding phase transition

temperatures in conventional ferroelectrics and quantum

paraelectric nanorods, nanowires,16–20 and binary oxides.21

The surface stress is inversely proportional to the surface

curvature radius and directly proportional to the surface

stress tensor (similar to Laplace surface tension). Thus, the

intrinsic surface stress should depend on both the growth

conditions and the surface termination morphology.22,23 Sur-

face reconstruction should affect the surface tension value or

even be responsible for the appearance of surface

stresses.24,25 Using Landau-Ginzburg-Devonshire (LGD)

theory, Morozovska et al.26 predicted the FE-FM multifer-

roic properties of EuTiO3 nanowires originated from the

intrinsic surface stress. However, the important impact of the

structural AFD order parameter (oxygen octahedron rota-

tions) in EuTiO3 has not been considered so far. Since the

AFD order parameter strongly influences the phase dia-

grams, polar and pyroelectric properties of quantum para-

electric SrTiO3,27–30 similar influence is expected for

EuTiO3 (Ref. 31) and EuxSr1�xTiO3. Therefore, a fundamen-

tal study of the possible appearance of the polar, magnetic,

and multiferroic phases in AFD EuxSr1�xTiO3 solid solution

system seems necessary. Recently, the transition from para-

electrics cubic phase to AFD phase in solid solution

EuxSr1�xTiO3 has been studied by means of electron para-

magnetic resonance.32

In this work, we study the possibility of inducing simul-

taneous ferroelectricity and ferromagnetism in EuxSr1�xTiO3

nanosystems within conventional LGD theory allowing

structural ordering. Figure 1 illustrates the nanosystems

a)Authors to whom correspondence should be addressed. Electronic

addresses: rkbehera@ufl.edu and morozo@i.com.ua.

0021-8979/2013/113(2)/024107/9/$30.00 VC 2013 American Institute of Physics113, 024107-1

JOURNAL OF APPLIED PHYSICS 113, 024107 (2013)

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considered in this study, (a) nanotubes clamped to the rigid

core, where the outer sidewall of the tube is mechanically

free and electrically open, i.e., non-electroded (Fig. 1(a)) and

freestanding nanowires (Fig. 1(b)). In the nanotube cases,

technologically convenient materials for a rigid core can be

ZnO, Si, SiC ultra-thin nanowires. Perovskite-type cores like

LaAlO3, LaAlSrO3, DyScO3, KTaO3, or NdScO3 are more

sophisticated to design. Since the lattice constants of

EuTiO3, SrTiO3, and EuxSr1�xTiO3 in the cubic phase are

similar (�3.905 A), the misfit strains due to the rigid core for

EuxSr1�xTiO3 nanotubes will be similar to EuTiO3 nano-

tubes. Therefore, the misfit strains appeared at the

EuxSr1�xTiO3 tube-core interface are approximately �6%

for ZnO, �1.7% for Si, 10% for SiC, �4% for LaAlSrO3,

�3% for LaAlO3, þ0.9% for DyScO3, þ2.1% for KTaO3,

and þ2.6% for NdScO3 core. In this study, we considered

the axial polarization P3 directed along the tube axis z, while

the radial polarization Pq perpendicular to the surface of the

tube is neglected due to the strong depolarization field Edq /

�Pq=e0eb that appears for the component in the case of non-

electroded tube sidewalls.26

The application of the continuum media LGD theory and

intrinsic surface stress conception to the description of nano-

sized particles polar and magnetic properties requires justifica-

tion due to the small size of the object of study. The

continuum media theory was successfully used for the analy-

sis of elastic properties of metallic, semiconductor, dielectric,

or polymeric nanowires and nanotubes,33–39 and the piezo-

electric response.40 For nanosized ferroics, the applicability of

the continuous media phenomenological theory is corrobo-

rated by the fact that the critical sizes (�2–10 lattice con-

stants) of the appearance of long-range order calculated from

atomistic41,42 and phenomenological theories18–20 are in good

agreement with each other39,43–46 as well as with experimental

results.47 Generally, the long-range order appears for sizes

larger than the critical sizes (details can be found in Refs. 18–

21, 26). Once the long-range order is established, it is possible

to apply the mean field LGD theory.48 Thus, the agreement

between the magnitudes of the critical sizes calculated from

LGD and atomistic theories is extremely important.

This paper is organized as follows: the basic equations

of LGD-theory and material parameters of EuxSr1�xTiO3

used in the calculations are listed in Sec. II. The predicted

phase diagrams of EuxSr1�xTiO3 bulk, nanowires, and nano-

tubes are presented and analyzed in Sec. III. The variation of

spontaneous polarization and magnetization in EuxSr1�xTiO3

nanosystems is presented in Sec. IV. The results are dis-

cussed in Sec. V and the details of the fitting procedure to

obtain the parameters for the calculations are given in the

supplementary materials.71

II. LANDAU-GINZBURG-DEVONSHIRE THEORY FOREUXSR12XTIO3

The LGD free energy density G of EuxSr1�xTiO3 solid

solution depends on the polarization vector P, oxygen octa-

hedra tilt vector U, magnetization vector M, and antimagne-

tization vector L as

G ¼ Ggrad þ GS þ Gelastic þ GME þ GM þ GPU; (1)

where Ggrad is the gradient energy, GS is the surface energy,

Gelastic is the elastic energy, GME is the biquadratic ME

energy, GM is magnetization-dependent energy, and GPU is

polarization-and-tilt-dependent energy.

The form of Ggrad þ GS is the same as listed in Ref. 26.

The elastic energy is given as Gelastic ¼ �sijklrijrkl=2, where

elastic compliances sijklðxÞ ¼ xsEuTiO3

ijkl þ ð1� xÞsSrTiO3

ijkl ; rij is

the elastic stress tensor. The biquadratic ME coupling energy

density (GME) is given as

GME ¼ðV

d3rP2

3

2ðcFMM2 þ cAFML2Þ: (2)

Here P3 is FE polarization, M2 ¼ M21 þM2

2 þM23 is the

square of FM magnetization, and L2 ¼ L21 þ L2

2 þ L23 is the

square of AFM order parameter vector.

Magnetic properties are observed in EuTiO3 and are

absent in SrTiO3. Therefore, composition dependence of the

biquadratic ME coupling coefficients cFMðxÞ and cAFMðxÞshould be included in Eq. (2). Here, we assume a linear de-

pendence on Eu content (x) above percolation threshold (xcr)

(see, e.g., Ref. 49), namely cAFMðxÞ¼cEuTiO3

AFM ðx�xAcrÞ=ð1�xA

crÞand cFMðxÞ¼cSrTiO3

AFM ðx�xFcrÞ=ð1�xF

crÞ at content xF;Acr �x�1;

while cAFMðxÞ¼0 and cFMðxÞ¼0 at x<xA;Fcr . The percola-

tion threshold concentration xcr can be estimated from the

percolation theory.49 For the simple cubic sub-lattice of

magnetic ions (Eu) xFcr�0:24,49 while the percolation

threshold is supposed to be higher for AFM ordering, xAcr�

0:48 (see, e.g., (Ref. 50)). Note that superscripts “F” and

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

FM and AFM ordering, respectively. Following Lee

et al.,15 we can regard that gEuTiO3

AFM ��gEuTiO3

FM >0 for nu-

merical calculations, as anticipated for equivalent magnetic

Eu ions with antiparallel spin ordering in a bulk EuTiO3.

The magnetization-dependent part of the free energy

is21,26

GM ¼ðV

d3raM

2M2 þ aL

2L2 þ bM

4M4 þ bL

4L4 þ k

2L2M2

�

�rmnðZmnklMkMl þ ~ZmnklLkLlÞ�; (3)

FIG. 1. (a) Schematics of a nanotube clamped on a rigid core. Mismatch

strain uc can exist at the tube-core interface. The tube outer radius is Re, the

inner radius is Ri, q is the polar radius. (b) Schematics of a nanowire.

024107-2 Eliseev et al. J. Appl. Phys. 113, 024107 (2013)

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where coefficient aMðT; xÞ ¼ aCðT � TCðxÞÞ, T is absolute

temperature, TCðxÞ ¼ T0Cðx� xF

crÞ=ð1� xFcrÞ is the solid solu-

tion FM Curie temperature defined at xFcr � x � 1. T0

C is the

Curie temperature for bulk EuTiO3. Note that aMðT; x ¼ 0Þ¼ aCðT � TCÞ determines the experimentally observed

inverse magnetic susceptibility in paramagnetic phase of

EuTiO3 (Ref. 1, 2, and 15). Also, coefficient aLðT; xÞ¼ aN

�T � TNðxÞ

�, where N�eel temperature TNðxÞ ¼

T0Nðx� xA

crÞ=ð1� xAcrÞ is defined at xA

cr � x � 1. T0N is the

N�eel temperature for bulk EuTiO3. The magnetic Curie and

N�eel temperatures are zero at x < xF;Acr . For equivalent

amount of magnetic Eu ions with antiparallel spin order-

ing it can be assumed that aC � aN . The positive cou-

pling term k2

L2M2 prevents the appearance of FM as well

as ferrimagnetic (FiM) phases at low temperatures

(T < TC) under the conditionffiffiffiffiffiffiffiffiffiffiffibMbL

p< k. Coefficients

bM, bL, k are regarded x-independent. Zmnkl and ~Zmnkl rep-

resent magnetostriction and antimagnetostriction tensors,

respectively.

The polarization and structural parts of the free energy

bulk density for cubic m3m parent phase is

GPU ¼ðV

d3raP

2P2

3 þbP

4P4

3 � Qij33rijP23 þ

aU

2U2

3 þbU

4U4

3

�

�RijklrijUkUl þgi3

2U2

i P23

�: ð4Þ

Here, Pi is the polarization vector and Ui is the structural order

parameter (rotation angle of oxygen octahedron measured as

the displacement of oxygen ion). The biquadratic coupling

between the structural order parameter Ui and polarization

components Pi is defined by the tensor gij.27,51 The biquadratic

coupling term was later regarded as Houchmandazeh-Laizero-

wicz-Salje (HLS) coupling.52 The coupling was considered as

the reason of magnetization appearance inside the ferroelectric

domain wall in non-ferromagnetic media.53 Biquadratic cou-

pling tensor and higher order expansion coefficients

are regarded composition dependent: bP;UðxÞ ¼ xbEuTiO3

p;U

þð1� xÞbSrTiO3

p;U , gijðxÞ ¼ xgEuTiO3

ij þð1� xÞgSrTiO3

ij . QijklðxÞ¼ xQEuTiO3

ijkl þ ð1� xÞQSrTiO3

ijkl and RijklðxÞ ¼ xREuTiO3

ijkl þ ð1� xÞRSrTiO3

ijkl are the electrostriction and rotostriction tensor com-

ponents, respectively, which also depend linearly on the

composition x. Note that recently Zurab Guguchia et al.32

experimentally observed a nonlinear composition depend-

ence of temperature of transition from cubic non-AFD to tet-

ragonal AFD phase. Coefficients aPðT; xÞ and aUðT; xÞdepend on temperature in accordance with Barrett law54

and composition x of EuxSr1�xTiO3 solid solution as

aPðT;xÞ ¼ xaEuTiO3

P ðTÞþ ð1� xÞaSrTiO3

P ðTÞ and aUðT;xÞ � aUT�T�TSðxÞ

�, where TSðxÞ � 113:33þ 390:84x� 621:21x2

þ398:87x3 in accordance with Refs. 32 and 55. Coefficient

amðTÞ ¼ ðTmq =2ÞðcothðTm

q =2TÞ� cothðTmq =2Tm

c ÞÞ, where sub-

and superscript m¼ P;U. Temperatures Tmq are so called

quantum vibration temperatures for SrTiO3 and EuTiO3,

respectively, related with either polar (P) or oxygen octahe-

dron rotations (U) modes, Tmc are the “effective” Curie tem-

peratures corresponding to polar soft modes in bulk EuTiO3

and SrTiO3.

For tetragonal ferroelectric, (anti)ferromagnetic, and

cubic elastic symmetry groups, coefficients a are renormal-

ized by the surface tension,16–18 misfit strains,56 and biqua-

dratic coupling with a structural order parameter.28–30 For

considered geometry, the renormalization is

aRPðT; xÞ ¼ aPðT; xÞ þ4Q12ðxÞl

Re� Q11ðxÞ þ Q12ðxÞ

s11ðxÞ þ s12ðxÞR2

i

R2e

uc

� g11ðxÞaUðT; xÞ

bU; ð5aÞ

aMRðT;R;xÞ�aC T� T0C�

W

aC

4lRe

��

þ Z11þZ12

aC

�s11ðxÞþs12ðxÞ

�R2i

R2e

uc

�x�xF

cr

1�xFcr

hðx�xFcrÞ�;

(5b)

aLRðT;R;xÞ � aN T� T0N�

~W

aC

4lRe

��

þ~Z11þ ~Z12

aC

�s11ðxÞþ s12ðxÞ

�R2i

R2e

uc

�x� xA

cr

1� xAcr

hðx� xAcrÞ�:

(5c)

In Eqs. (5), Re is the tube outer radius, Ri is the tube inner ra-

dius (see Fig. 1); l is the surface tension coefficient, which is

regarded as positive; and uc is misfit strain at the tube-core

interface. For the practically important case of the ferroelec-

tric tube deposited on a rigid dielectric core, the tube and

core lattices mismatch or the difference of their thermal

expansion coefficients determines uc value allowing for the

possible strain relaxation for thick tubes.

If the spontaneous (anti)magnetization is directed

along z-axes, the parameters in Eqs. (5) are ~W ¼ þ ~Z12;W ¼ þZ12, where Zij and ~Zij are the magnetostriction and

anti-magnetostriction coefficients. When the spontaneous

(anti)magnetization is within the {x,y} plane, the parameters

are ~W¼�ð ~Z12þ ~Z11Þ=2; W¼�ðZ12þZ11Þ=2.26 Function

hðx�xcrÞ is the Heaviside step-function,57 i.e., hðx�0Þ¼1

and hðx<0Þ¼0. Notice that it is possible to consider radial

magnetization, since the influence of demagnetization field

existing for such case is typically negligibly small.48

The terms in Eqs. (5) proportional to l=Re are originated

from the intrinsic surface stress, while the terms proportional

to ucR2i =R2

e are caused by the strains induced by the rigid

core. The size, misfit strain, and composition dependence of

the ordered phase stability can be obtained from the condi-

tion aRPðT; xÞ < 0. In particular, the term 4Q12ðxÞl=Re in

Eq. (5a) is negative because Q12ðxÞ < 0; so it leads to a

reduction in aRPðT; xÞ and thus favors FE phase appearance

for small Re. Since Q11ðxÞ þ Q12ðxÞ > 0, the term

��Q11ðxÞ þ Q12ðxÞ

�ðR2

i =R2eÞuc in Eq. (5a) leads to a reduc-

tion in aRPðT; xÞ and thus favors FE phase appearance for pos-

itive uc. Similarly, positive terms �4Z12l=Re and

�ðZ11 þ Z12Þ ðR2i =R2

eÞuc favor the appearance of FM phase.

024107-3 Eliseev et al. J. Appl. Phys. 113, 024107 (2013)

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The numerical values of material parameters (Eqs. (1)–

(5)) used in the LGD model are listed in Tables I and II. The

fitting procedures used to estimate the EuTiO3 material pa-

rameters are summarized in the supplemental material.71

To make size effects pronounced, nanosystem sizes

should vary from several lattice constants to several tens of

lattice constants (lc). Our previous analysis showed that size

effects can be neglected for nanosystems with sizes of more

than 100 lc.16–18,21,26 To illustrate typical results for our nu-

merical simulations, we have shown mainly the results of

three different EuxSr1�xTiO3 nanosystems: (i) nanotube with

small inner radius Ri ¼ 10 lc, thickness d ¼ 5 lc, which

results in an outer radius Re ¼ Ri þ d¼ 15 lc; (ii) nanotube

with high inner radius Ri ¼ 100 lc, thickness d ¼ 5 lc and

outer radius of 115 lc (this represents thin film with in-plane

polarization); and (iii) a special case of nanotube with

Ri ¼ 0 lc, thickness d ¼5 lc, i.e., a nanowire with radius

Re ¼ 5 lc. In addition, we have also calculated the bulk phase

diagram of EuxSr1�xTiO3 solid solution for comparison.

Following Ref. 15, modern epitaxial methods allow to vary

misfit strain uc in the range –5% to þ5%. Therefore, we will

consider the misfit strains within this range for our calcula-

tion. Due to the lack of experimental measurements of the

surface tension coefficients for EuxSr1�xTiO3, we have consid-

ered l¼ 30 N/m based on experimental data for ferroelectric

ABO3 perovskites (36.6 N/m (Ref. 62) or even �50 N/m

(Ref. 63) for PbTiO3, 2.6-10 N/m for PbTiO3 and BaTiO3

nanowires,64 9.4 N/m for Pb(Zr,Ti)O3 (Ref. 65)).

III. PHASE DIAGRAMS OF EUXSR12XTIO3 BULK ANDNANOSYSTEMS

Before discussing the phase diagrams for the

EuxSr1�xTiO3 nanosystems, it is necessary to establish the

phase diagram of bulk EuxSr1�xTiO3 system for comparison.

Figure 2(a) shows the predicted phases for EuxSr1�xTiO3

bulk solid solution. The phase diagram shows the presence

of 5 different phases: Para (paraelectric-paramagnetic),

AFD, AFD-FM, AFD-AFM, and AFD-FiM. The magnetic

phases AFD-FM, AFD-AFM, and AFD-FiM exist at temper-

atures lower than 10 K. This phase diagram shows a nonlin-

ear dependence of the AFD phase boundary as a function of

x. Note that there was no directional control of the polariza-

tion and magnetization direction for the bulk system, e.g.,

TABLE I. List of parameters for polarization and tilt dependent part of the free energy.

SrTiO3 EuTiO3

Parameter (SI units) Value Source and notes Value Source and notes

eb 43 Refs. 58 and 59 33 Fitting to Refs. 1 and 60

aðPÞT (� 106 m/(F K)) 0.75 4 and 27 1.95 Fitting to Refs. 1 and 60

TðPÞc (K) 30 4 and 27 �133.5 Fitting to Refs. 1 and 60

TðPÞq (K) 54 4 and 27 230 Fitting to Refs. 1 and 60

ar11 (� 109 m5/(C2F)) 1.724 4 and 27 1.6 Fitting to Ref. 14

Qij (m4/C2)

Q11¼ 0.046,

Q12¼�0.014

Recalculated from

Ref. 4 Q11¼ 0.10, Q12¼�0.015 Fitting to Ref. 15

aðUÞT (� 1026 J/(m5 K)) 18.2 4 3.91 Fitting to Refs. 9 and 10

TðUÞc (K) 105 4 270

Averaged value of exp. 220

(Ref. 10), 275 (Ref. 32), and

282 (Ref. 9)

TðUÞq (K) 145 4 205 Fitting to Ref. 10

bU (� 1050 J/m7) 6.76 4 1.74 Fitting to Refs. 9 and 10

Rij (� 1018 m�2)

R11¼ 8.82,

R12¼�7.77

Recalculated from

Ref. 4 R11¼ 5.46, R12¼� 2.35 Fitting to Ref. 10

g11 (� 1029 (F m)�1) 4.19 4 �4.45 Fitting to Refs. 1 and 60

sij (� 10�12m3/J)

s11¼ 3.52,

s12¼�0.85

Recalculated from

Refs. 4 and 27 s11¼ 3.65, s12¼�0.85 First-principles61

TABLE II. List of parameters for magnetic part of the free energy for EuTiO3.

LGD-coefficient aC � aN Henri/(m�K) 2p�10�6 1,15EXP

LGD-coefficient bM J m/A4 0.8� 10�16 Fitting results of Ref. 1

LGD-coefficient bL J m/A4 1.33� 10�16 Fitting results of Ref. 1

LGD-coefficient k J m/A4 1.0� 10�16 Fitting results of Ref. 1

Magnetostriction coefficients Zij (Voigt notation) m2/A2 Z12¼�(7.5 6 0.3)� 10�16, Z11¼ (11.9 6 0.3)� 10�16 Fitting results of Ref. 15

Magnetostriction coefficients ~Zij (Voigt notation) m2/A2 ~Z12¼�(8.7 6 0.2)� 10�16, ~Z11 ¼(9.2 6 0.2)� 10�16 Fitting results of Ref. 15

AFM Neel temperature TN K 5.5 1

FM Curie temperature TC K 3.5 6 0.3 1 and 2

Biquadratic ME coupling coefficient cAFM ¼ �cFM J m3/(C2 A2) 0.08� 10�3 Fitting results of Ref. 1

024107-4 Eliseev et al. J. Appl. Phys. 113, 024107 (2013)

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our calculations for the bulk solid solution do not differenti-

ate between in-plane, out-of-plane, or mixed- ferroelectric

phases.

Analyzing the phase diagrams predicted for the bulk

systems (Fig. 2(a)), we observed the unexpected appearance

of the ferromagnetic ordering (AFD-FM and AFD-FiM

phases) for Eu content x> 0.4. Therefore, we predict Sr-

diluted ferromagnetism for Eu composition x from 0.45 to

0.75 or ferrimagnetism for x � 0.8. The FM ordering may

originate from spin canting,66,67 especially if the energies

of different magnetic orderings (A-, C-, F-, and G-types)

are very close. Therefore, bulk solid solution EuxSr1�xTiO3

should be included to the multiferroic family. The results

presented in Figures 2(b)–2(d) clearly show that the ferro-

magnetic phase (AFD-FE-FM) is only present at high Eu

content and absent at Sr-rich EuxSr1�xTiO3. Therefore, the

increase in Sr content in EuxSr1�xTiO3 dilutes the spins

and reduces the overall magnetization. In addition, the

crossover of the AFD magnetic phases AFD-AFM !AFD-FiM ! AFD-FM originated from the magnetic perco-

lation model, namely due to the different percolation

thresholds for ferromagnetism xFcr � 0:24 and antiferromag-

netism xAcr � 0:48, which is in agreement with classical per-

colation theory.49 We hope that this prediction will be

verified either experimentally or from the first-principles

calculations.

For EuxSr1�xTiO3 nanotubes and nanowires, our calcu-

lations demonstrated that several ordered phases can be ther-

modynamically stable under tensile strain (see Figs. 2(b)–

2(d)), namely Para, FE, AFD, AFD-FE, and AFD-FE-FM.

Note that ferroelectric FE, AFD-FE, and AFD-FE-FM

phases are absent in the bulk EuxSr1�xTiO3, since ferroelec-

tric ordering appearance in incipient ferroelectrics is possible

for small sizes only.16–26

In accordance with our calculations, general conditions

of the multiferroic AFD-FE-FM stability in nanosystems are

(i) small thickness and radius, (ii) relatively low tempera-

tures (<20 K), (iii) high Eu content (x> 0.7), and (iv) posi-

tive tensile misfit strains uc > 0. For instance, the

temperature—composition phase diagram of EuxSr1�xTiO3

nanowires, calculated for tensile strains uc¼þ3% (Fig.

2(b)), demonstrates that magnetoelectric AFD-FE-FM phase

appears in the nanowires for Eu content x more than xc �0.85 and temperatures less than 20 K. Figures 2(c) and 2(d)

illustrate that AFD-FE-FM phase appears in EuxSr1�xTiO3

nanotubes for Eu concentration more than xc � 0.7, and tem-

peratures lower than 10 K. Increase of the Eu composition

from xc to 1 essentially enlarges the temperature interval of

AFD-FE-FM phase stability. The diagram of EuxSr1�xTiO3

nanowires contains smaller x-region of AFD-FE-FM phase

(namely, 0.85� x� 1, see Fig. 2(b)) in comparison to the

corresponding x-region of nanotubes (0.7� x� 1, see Figs.

2(c) and 2(d)). This is because of the additional contribution

of misfit-related strain terms �ucðR2i =R2

eÞ (Eqs. (5)) originat-

ing from the tube-core lattice constants mismatch for nano-

tubes. Figure 2(d) presents the limit of the thin epitaxial

EuxSr1�xTiO3 film with in-plane polarization, where the

ADF-FE-FM phase appeared from misfit effect, correspond-

ing term isQ11ðxÞþQ12ðxÞs11ðxÞþs12ðxÞ

R2i

R2euc � Q11ðxÞþQ12ðxÞ

s11ðxÞþs12ðxÞ uc.

Therefore, it is important to emphasize that the misfit

strain existing between the nanotube-core interface allows

the possibility of controlling the phase diagram of the

EuxSr1�xTiO3 nanotubes. Such possibility is absent for nano-

wires. The misfit strain–composition phase diagrams of

EuxSr1�xTiO3 nanotubes with internal radius 10 lc, outer ra-

dius 15 lc, and thickness d¼ 5 lc are shown in Fig. 3 at two

different temperatures, at 4 K (low temperature), and at

300 K (room temperature). From Fig. 3(a), it is clear that the

FM properties of EuxSr1�xTiO3 nanotubes can appear at

about xc> 0.8, which is much higher than the percolation

threshold of xFcr � 0:24 at low temperatures T< 4 K and pos-

itive tensile strains (uc > 0). The region of AFD-FE-AFM

stability becomes narrower with the increase in temperature

and it disappears at higher temperatures. At low temperatures

FIG. 2. Predicted temperature-composition phase diagrams of (a) bulk

EuxSr1�xTiO3 system, where Para (paraelectric-paramagnetic), AFD, AFD-

FM, AFD-AFM, and AFD-FiM phases are present, (b) EuxSr1�xTiO3 wire

of radius 5 lc, (c) EuxSr1�xTiO3 nanotube of radius 10 lc, and (d)

EuxSr1�xTiO3 nanotube of radius 100 lc. The tubes of thickness for (c) and

(d) are 5 lc with a tensile misfit strain uc¼þ3%. The surface tension coeffi-

cient l¼ 30 N/m for nanowire and nanotubes. The existing phases in the

nanosystems are para, FE, AFD, AFD-FE, and AFD-FE-FM.

FIG. 3. The misfit strain–composition phase diagrams of EuxSr1�xTiO3

nanotube with internal radius 10 lc, outer radius 15 lc, and thickness d¼ 5 lcat (a) temperature T¼ 4 K and (b) at T¼ 300 K.

024107-5 Eliseev et al. J. Appl. Phys. 113, 024107 (2013)

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(�10 K), there are two stable multiferroic phases, namely

AFD-FE-AFM and AFD-FE-FM. Additional calculations

(data not shown) proved that only pure EuTiO3 can be AFM

for given sizes and strains at T> 10 K. At room temperature

(Fig. 3(b)), the disordered para phase appears at x< 0.8 and

uc<þ1%. Such enlarged region of the para phase occurs at

room temperature because of the absence of the axial ferro-

electric polarization P3 in the compressed nanotubes (similar

effect is reported for compressed ferroelectric films56). Due

to the strong depolarization field Edq � �Pq=e0eb, the ferro-

electric phase with radial polarization Pq perpendicular to

the surface of the tube/wire may appear at very high com-

pressive strains uc<�5% (Ref. 68).

In addition, the phase diagram in coordinates misfit

strain–temperature for different Eu compositions (x¼ 0, 0.5,

0.75, and 1) are presented for EuxSr1�xTiO3 nanotubes with

internal radius 10 lc, outer radius 15 lc, and thickness d¼ 5

lc (Fig. 4). The results show the transition between different

phases where 4 phases are present for x� 0.5 ((Para, FE

AFD, and AFD-FE), which transforms to 8 phases for

x¼ 0.75 (Para, AFD, FE, AFD-FE, AFD-AFM, AFD-FiM,

AFD-FM, and AFD-FE-FM) and then to 6 phases for x¼ 1

(Para, AFD, AFD-FE AFD-AFM, AFD-FE-AFM, and AFD-

FE-FM). It is evident from Fig. 4 that the phase boundaries

have relatively small horizontal slopes (i.e., these are weakly

misfit-dependent), while the content of Eu influences the

vertical position of the phase boundaries. These results are

consistent with the model assumptions, as the model did not

consider the gradient effects and stress relaxation via the

appearance of dislocations.

From Figs. 3 and 4, it is important to emphasize that the

multiferroic ADF-FE-FM phase in EuxSr1�xTiO3 nanosys-

tems can be stable only for tensile strain (uc > 0). The multi-

ferroic phases are absent for zero strain (uc ¼ 0) and

compressive strains (uc < 0). This is because the FE phases

with spontaneous polarization P3 parallel to the tube axis

become unstable for zero (uc ¼ 0) and negative (uc < 0)

strains. The phases with spontaneous polarization Pq perpen-

dicular to the tube surface q ¼ Re would become stable;

however, they appeared to be completely suppressed by the

strong depolarization field Edq � �Pq=e0eb, since we did not

impose any type of short-circuit conditions at the tube/wire

sidewalls. The effect of tensile strains can be readily

explained from Eqs. (5a) and (5b), where the terms,Q11þQ12

s11þs12

R2i

R2euc and Z11þZ12

s11þs12

R2i

R2euc, should become positive in order

to increase effective FE and FM Curie temperatures. SinceQ11þQ12

s11þs12> 0 and Z11þZ12

s11þs12> 0 in perovskites, the terms are posi-

tive under the condition uc > 0.

In general, we predict that tensile misfit strains are nec-

essary for the appearance of multiferroic phase in

EuxSr1�xTiO3 nanosystems. Similar to tensile misfit strains,

high positive surface tension coefficients (l) increase the

effective Curie temperatures for both nanotubes and nano-

wires.16 The effect of surface tension can be readily

explained from Eqs. (5), since for positive surface tension

coefficient l and negative electrostriction coefficient Q12,

the term 4Q12l=Re increases the effective Curie tempera-

tures for both nanotubes and nanowires.

We have also estimated the effect of wire radii and tube

thickness on the phase diagram of EuxSr1�xTiO3 nanosys-

tems. Figure 5 illustrates the phase diagrams with respect to

temperature-nanowire radius (Fig. 5(a)) and temperature-

tube thickness (Figs. 5(b) and 5(c)) for composition x¼ 0.9

and misfit strain uc¼þ3%. Since the multiferroic phases are

present at low temperatures, only T< 10 K are evaluated.

For the nanowire system, the multiferroic AFD-FE-FM

phase (light green region) is stable below 3 K (Fig. 5(a)).

Similar analysis on the nanotube systems shows that the

FIG. 4. Temperature–misfit strain phase diagrams of EuxSr1�xTiO3 tube

with internal radius 10 lc, outer radius 15 lc, and thickness d¼ 5 lc for dif-

ferent Eu compositions (a) x¼ 0, (b) x¼ 0.5, (c) x¼ 0.75, and (d) x¼ 1.

FIG. 5. Phase diagrams of EuxSr1�xTiO3

(a) nanowire with Ri¼ 0, (b) nanotube

with Ri¼ 10 lc, and (c) nanotube with

Ri¼ 100 lc in coordinates wire radius Re

or tube thickness d vs. temperature T cal-

culated for composition x¼ 0.9 and mis-

fit strain uc¼þ3%. Other parameters

and phase designations are the same as

in Fig. 2.

024107-6 Eliseev et al. J. Appl. Phys. 113, 024107 (2013)

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multiferroic phase is stable at T< 6 K for smaller nanotubes

(with Ri¼ 10 lc (Fig. 5(b)), and at T< 4 K for larger nano-

tubes (with Ri¼ 100 lc (Fig. 5(c))). Comparing the nanowire

and nanotube systems, the temperature stability of the multi-

ferroic phases is higher for the nanotube systems. This differ-

ence in temperature stability can be attributed to the misfit

strains present in nanotube systems. The EuxSr1�xTiO3 tube

with inner radius of 100 lc virtually presents the limit of the

thin epitaxial film with in-plane polarization, where the

AFD-FE-FM phase appeared from misfit effect only. This

independence of the AFD-FE and AFD-FE-FM phase boun-

daries on the tube thickness in Fig. 5(c) is due to the lack of

misfit dislocation and polarization gradient effect considera-

tion in the current model.

IV. SPONTANEOUS POLARIZATION ANDMAGNETIZATION

The spontaneous polarization and magnetization vs.

composition x of Eu in EuxSr1�xTiO3 nanowires and nano-

tubes are shown in Fig. 6 for fixed radii, tensile misfit strain,

and different temperatures (specified near the individual

curves). Note that spontaneous magnetization is absent at

compressive strains and thus the case with tensile misfit

strain of þ3% is considered. It is observed that spontaneous

polarization increases with the reduction in temperature. The

magnitude of spontaneous polarization increases with the

increase in Eu content for most of the temperatures. How-

ever, the trend is not followed above 280 K, which is the

temperature of the structural phase transition in bulk EuTiO3

(�280 K). Spontaneous magnetization abruptly appears with

Eu content more than the threshold value xc and at tempera-

tures less than 10 K. Such abrupt composition-induced FM

phase transition is of the first order. It is clear from Fig. 6

that FE phase exists at all x and temperatures less than

300 K. The jumps on spontaneous polarization values at low

temperatures (4 K data in Figs. 6(a) and 6(b)) match with the

simultaneous appearance of spontaneous magnetization

phases, i.e., they indicate magnetoelectric FE-FM phase

transition.

Spontaneous polarization and magnetization dependence

on the wire radii or the tube thickness of EuxSr1�xTiO3

nanosystem are shown in Fig. 7. The calculations are per-

formed for Eu content x¼ 0.9, tensile misfit strain þ3%, and

at different temperatures. For wire radius less than 10 lc, the

spontaneous polarization reaches rather high values

�20�100 lC/cm2 up to room temperatures (Fig. 7(a)). At

low temperatures (<10 K) and small tube thickness (<10 lc),

spontaneous polarization reaches a maximum value of

�50 lC/cm2 (Fig. 7(b)). Note, that spontaneous polarization

increases with the reduction in wire radii or tube thickness. It

is seen from the plots that FE polarization and FM magnet-

ization disappear when the tube thickness overcomes the

critical value. The tube critical thickness for spontaneous

polarization disappearance is temperature dependent; it

decreases with the increase in temperature from 20 lc at 3 K

to 7 lc at 300 K (Fig. 7(b)). The quantitative analysis of spon-

taneous magnetizations at 3 K are characterized to be �0.7

MA/m for nanowires of radii range 8 lc<Re< 15 lc (see Fig.

7(c)), while �0.5 MA/m for nanotubes of thickness less than

20 lc (see Fig. 7(d)). These results are in agreement with the

results presented in Figs. 5(a) and 5(b). For nanotubes, the

tube-on-core geometry seems more favorable for

ferromagnetism.

FIG. 6. Change in spontaneous polarization vs. composition x of

EuxSr1�xTiO3 (a) nanowire, and (b) nanotube at different temperatures.

Change in spontaneous magnetization vs. composition x of EuxSr1�xTiO3

(c) nanowire, and (d) nanotube at different temperatures. The nanowire is of

radius 5 lc while the nanotube is with internal radius 10 lc and thickness 5

lc. The results are shown for tensile misfit strain of þ3%. The temperature

values are specified near the curves.

FIG. 7. Change in spontaneous polarization with respect to (a) wire radii

and (b) tube thickness in EuxSr1�xTiO3 nanosystem calculated for x¼ 0.9,

and tensile misfit strain uc ¼ þ3%. Change in magnetization M and

antimagnetization L with respect to (c) wire radii, and (d) tube thickness.

Polarization is shown for two different temperatures, 3 K and 300 K.

024107-7 Eliseev et al. J. Appl. Phys. 113, 024107 (2013)

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It is seen from Fig. 7 that the size-induced ferroelectric

phase transition is of the second order at 300 K and of the

first order at 3 K, while the size-induced ferromagnetic phase

transition is of the first order at 3 K. Numerical simulations

proved that all the first order size-induced ferroelectric phase

transitions are the transitions between two AFM phases (e.g.,

AFD-AFM and AFD-FE-AFM), while the second order tran-

sitions correspond to Para-FE transition. All jumps at low

temperatures indicate magnetoelectric FE-FM transitions.

V. DISCUSSION

In this study, we have calculated the phase diagrams of

bulk EuxSr1�xTiO3 and EuxSr1�xTiO3 nanosystems (nano-

tubes and nanowires) using phenomenological LGD theory.

For bulk EuxSr1�xTiO3 solid solution, the FM phase is pre-

dicted to be stable at low temperatures for the concentration

0:4 < x < 0:8, while AFM phase is stable at 0:8 < x < 1.

Within the mesoscopic phenomenological approach, the

AFM to FM transition is guided by the difference in the criti-

cal percolation concentrations xFcr � 0:24 for FM state and

xAcr � 0:48 for AFM state. The AFM to FM critical percola-

tion concentration ratio (xAcr=xF

cr) of � 2 is in agreement with

general percolation theory as well as with the intuitively

clear fact that the percolation threshold for FM ordering

should be lower than the AFM ordering.49,50 Even though

our results only indicate to the possible microscopic origin

of the diluted magnetism in EuxSr1�xTiO3, we hope that our

results will stimulate further research to investigate the asso-

ciated mechanisms using low temperature experiments and

ab initio simulations.

From our calculations, nanosized EuxSr1�xTiO3 wires

and tubes under favorable conditions can be multiferroics.

Magnetoelectric AFD-FE-FM phase, which is the most im-

portant phase for multiferroic applications, can exist at Eu

content more than critical (xc � 0.75), at tensile strains

(uc�þ1%) and low temperatures <10 K. Increase of the Eu

composition x from xc to 1 essentially enlarges the region of

ADF-FE-FM phase stability at a fixed temperature. The max-

imal polarization (25–100 lC/cm2) and magnetization (0.5–1

MA/m) values (see Fig. 8) are high or comparable with

proper ferroelectrics BaTiO3 (26 lC/cm2),59 LiNbO3 (75 lC/

cm2),59 PbTiO3 (81 lC/cm2),69 and typical ferromagnetics.70

The existence of ME coupling manifests itself by jumps in

maximum polarization (Fig. 8(a)) at the content, x, of mag-

netization appearance (Fig. 8(b)). The temperature region of

magnetization existence depends on the size of the nanosys-

tems and the Eu composition x. Therefore, the appearance of

the multiferroic phase depends on the choice of the amount

of substitution (x), the rigid core material for misfit strain

(uc), the radius and thickness of the nanosystems, and

temperature.

All results presented in this article for EuxSr1�xTiO3 are

based on the parameters of EuTiO3 and SrTiO3 taken from

experiments,9,15 independent first principles calculations,14

and our first principles calculations (see Tables I and II, and

supplemental material71). Although the sources we collected

from the parameters are quite reliable, the collective use of

different parameter sets may result in some uncertainties in

the prediction. For example, the physical origin of some of

the predicted phenomena, such as the increase of FE phase

transition temperature with the increase in Eu content in

nanosystems and the appearance of FM phase with dilution

of bulk EuTiO3 by SrTiO3 require further investigation and

experimental verification. Actually, this conclusion may

seem counterintuitive since the extrapolated Curie tempera-

ture of the soft mode in bulk SrTiO3 (� þ30 K) is much

higher than that of bulk EuTiO3 (nearly �133 K). In order to

understand the increase in FE transition temperature with the

increase in Eu content, it is important to consider the cou-

pling of the structural order parameter. Following recent ex-

perimental data,10 the value of the structural order parameter

(corresponding to oxygen atom displacement) for bulk

EuTiO3 is more than twice compared to bulk SrTiO3. At the

same time the biquadratic coupling constants between the

polar and structural modes for SrTiO3 and EuTiO3 have sim-

ilar magnitudes with opposite signs (as we deduced from the

experimental results on dielectric permittivity,60 see Table I

and supplementary materials71 for details). These opposite

signs lead to the increase in “renormalized” FE transition

temperature with increase of the structural order parameter

caused by the addition of Eu. As anticipated, the resultant

“renormalized” FE transition temperature never reaches a

positive value for stress-free bulk EuTiO3. Under the influ-

ence of surface tension in cylindrical nanoparticles and misfit

strain, structural order parameter increases which results in

an increase in the renormalized FE transition temperature.

We hope that our predictions will stimulate experimen-

tal and computational studies of EuxSr1�xTiO3 nanosystems,

where the coupling between structural distortions, polariza-

tion, and magnetization can lead to the versatility and ten-

ability of the magnetoelectric multiferroic phases.

ACKNOWLEDGMENTS

The authors gratefully acknowledge multiple discus-

sions with Professor Annette Bussmann-Holder. E.A.E.,

M.D.G., and A.N.M. acknowledge Science and Technology

Center of Ukraine, Project No. STCU-5514.

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estimated from the condition 0 ¼ aPðT; xÞ þ lRe

�Q11ðxÞ þ Q12ðxÞ

�þ 2Q12ðxÞ

s11ðxÞþs12ðxÞR2

i

R2euc � g12ðxÞ

aUðT;xÞbUþ 1

e0eb:

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details of parameters fitting with experimental results.

024107-9 Eliseev et al. J. Appl. Phys. 113, 024107 (2013)

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