magnetoelectric effect in bi-layered ferromagnetic structure

copy 2011 WILEY-VCH Verlag GmbH amp Co KGaA Weinheim

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aPhys Status Solidi C 8 No 11ndash12 3059ndash3061 (2011) DOI 101002pssc201000378

Magnetoelectric effect in bi-layered ferromagnetic structure Zukhra Gareeva and Rurik Doroshenko

Institute of Molecular and Crystal Physics Prospect Oktyabrya 151 450075 Ufa Russia

Received 3 June 2010 revised 2 November 2010 accepted 2 November 2010 Published online 24 June 2011

Keywords magnetoelectric effect magnetization processes layered structure magnetic non-homogeneity Corresponding author e-mail gzvanrbru

Magnetoelectric effect and the properties of electric po-larization appearing in a vicinity of magnetic inhomoge-neity in bi-layered ferromagnetic film have been theoreti-cally considered It has been shown that a variety of elec-tric polarization transformations is possible under the ac-tion of applied magnetic field

The magnetization processes and the behaviour of polariza-tion appear to be dependent on the relation between layers parameters Two situations of ferromagnetic and antiferro-magnetic interlayer coupling have been considered and ana-lysed

copy 2011 WILEY-VCH Verlag GmbH amp Co KGaA Weinheim

1 Introduction The immense scientific activity in re-cent years has been devoted to investigation of materials (crystalline and non-crystalline) possessing with both magnetic and electric properties This is the challenging area of condensed matter research promising for the new applications in spintronic non-violate memory devices and having a lot of routes in fundamental physics A wide range of materials performing magnetoelectric properties is known nowadays [1-9] Among them are multiferroics [1 2] hybrid magnetoelectric materials comprising piezoelec-tric and magnetic layers [3 4] materials containing mag-netic inhomogeneity [5-9] In 1983 Baryakhtar [5] has pointed out on the possibility of electric polarization ap-pearance in a vicinity of magnetic inhomogeneity pointing out that the symmetry of its spatial distribution is deter-mined by the symmetry of magnetic inhomogeneity These ideas have been developed further in a series of theoretical as well as experimental works [6-9] Magnetic inhomoge-neities of the kinds of domain walls magnetic vortices Bloch lines existing in magnetic domain walls has been considered as sources of electric polarization

In this paper we report on magnetoelectric effect in the properly ferromagnetic structure comprising two ferro-magnetic films differing by the kind of magnetic anisot-ropy We consider two cases of the coupling between magnetic layers ferromagnetic and antiferromagnetic ones Magnetic inhomogeneity arises during magnetization process it can be present in the case of antiferromagneti-cally coupled films in the absence of magnetic field as well

We analyze the properties of electric polarization appear-ing in a vicinity of magnetic inhomogeneity dependent on the material parameters the direction and the value of ap-plied magnetic field

2 General equations Let us consider exchange-coupled double films with the magnetic anisotropy of ldquoeasy-axisrdquo type (K1) in one film and magnetic anisotropy of ldquoeasy planerdquo (K2) type in another film The geometry of a problem is shown in a Fig 1

Figure 1 Geometry of a problem

The combination of both kinds of anisotropy in a mate-

rial is a cornerstone for this problem We suppose the sys-tem is placed in a magnetic field oriented along a normal to a surface The equilibrium magnetic configurations local polarization arising in a vicinity of magnetic inhomogene-ity can be found from the minimum condition of the free energy of a system The free energy E comprises the en-

3060 Z Gareeva and R Doroshenko Magnetoelectric effect in bi-layered ferromagnetic structure

copy 2011 WILEY-VCH Verlag GmbH amp Co KGaA Weinheim wwwpss-ccom

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ergy of non-uniform exchange interaction Eex the energy of magnetic anisotropy Ean Zeeman energy EH for each of the layers and the energy of interlayer exchange interac-tion EJ energy of non-uniform magnetoelectric (flexomag-netoelectric) [9] interaction Eme

E=Eex+Ean+EH+Eel+EJ (1)

where

22 2

exE Ax y z

⎡ ⎤⎛ ⎞part part part⎛ ⎞ ⎛ ⎞= + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟part part part⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

M M M

is the energy of non-uniform exchange interaction here A is the exchange stiffness

( ) ( )2 21 1 2 2anE K K= minus +M n M n

is the energy of magnetic anisotropy in the layers here K2 is the constant of magnetic anisotropy of easy plane type K1 is the constant of magnetic anisotropy of easy axis type

HE = minusMH is the Zeeman energy H is an applied magnetic field

1 2JE J= minus M M is the interlayer exchange energy J is the constant of inter-layer coupling if Jlt0 the coupling between the layers is antiferromagnetic if Jgt0 the coupling between the layers is ferromagnetic

[ ]2 2 2

( ) ( )2

x y zel

e

P P PE γ

χ+ +

= minus + + minusPE P M M M Mnabla nabla

is the electric energy incorporating the energy of a system in an electric field the energy of electric polarization the flexomagnetoelectric energy E is the electric field P is the electric polarization χe is the dielectric susceptibility γ is the constant of non-uniform magnetoelectric interaction Minimization of the free energy (1) has been carried out numerically We have used polar coordinates

0

(sin cos sin sin cos )M

θ ϕ θ ϕ θ= =Mm

to determine the position of magnetization vector As a re-sult of the free energy minimization we have obtained the dependences of magnetization upon the system parameters and the relation between electric polarization and magneti-zation

1 2( )K K H a b A Jθ θ= (2)

[ ]( ) ( )e eχ γχ= minus minusP E M M M Mnabla nabla

Following the formulas (2) (3) the dependences of polari-zation upon the system parameters have been constructed We suppose the rotation of magnetization occurs in a XOZ plane In this case the only Px component of polarization vector and the polar angle θ characterizing magnetization rotation are sufficient to be considered

3 Results and discussion Hereinafter we analyze the dependences of the electric polarization upon system

parameters and make a comparison between two cases of interlayer coupling (ferromagnetic coupling (FMC) and an-tiferromagnetic coupling (AFC)) The non-homogeneous distribution of magnetization over the film is present in bi-layered structure comprising films with the different kinds of magnetic anisotropy Dependent on the character and the strength of interlayer coupling magnetic inhomogeneity can be either localized close to interface or distributed over the structure Electric polarization arises in a vicinity of magnetic inhomogeneity the sign of polarization vector is determined by the direction of magnetization rotation In a case of FMC the local polarization is concentrated close to interface it is of the same sign in the both layers Applied magnetic field spread polarization over the layers and de-creases the local polarization value In a case of AFC the local polarization has different signs in the layers it is lo-cated at interface in magnetically hard layer and it is spread over the soft magnetic layer In a case of equal val-ues of magnetic anisotropy constants the dependences of the local polarization on the coordinate z is of symmetrical form but the signs of polarization are different in the layers thus overall polarization is equal to zero

Such peculiarities of local polarization will affect the distribution of overall polarization over the sample The overall polarization will always present in a case of FMC it is localized in a vicinity of interface (in an absence of applied magnetic field) In a case of AFC overall polariza-tion exists in the magnetically soft layer it approaches to the zero value when absolute values of anisotropy con-stants are equal (K1=K2) Applied magnetic field affects the magnetization processes and therefore the magnitude of electric polarization Our calculations has shown that mag-netic field can enhance or shrink overall polarization high magnetic fields suppress polarization (the system becomes magnetically saturated) Overall polarization can be also suppressed in the low magnetic field Overall polarization can change continuously or in a jump-like way with a change of applied magnetic field However the character of electric polarization transformations will be different for the different relation between magnetic anisotropy con-stants and the sign of interlayer exchange parameter J In Figs 2 3 we show dependences of overall polarization on the value and direction of applied magnetic field for the four cases K1gtK2 Jgt0 (Fig 2a) K1 gtK2 Jlt0 (Fig 2b) K1ltK2 Jgt0 (Fig 3a) K1ltK2 Jlt0 (Fig 3b) The following physical parameters has been chosen A1=A2=107 ergcm M1=M2=90 G J=104 ergcm3 γ=10-9SGSM χe=102 SGSE (Fig2) M1=M2=200 G A=2middot10-7 ergcm J=09 ergcm2 (Fig 3) Below we compare the obtained results In Fig 2 3 one can see that the character of polarization change is quite different for the presented situations but the general features can be found out When the upper layer is mag-netically hard and the lower layer is magnetically soft (situation K1gtK2 Fig 2) the dependences P(H) has sym-metrical form relative to the H=0 line both for the FMC case and AFC case In the saturating fields and at the zero point (H=0) electric polarization vanishes One can see that

(3)

Phys Status Solidi C 8 No 11ndash12 (2011) 3061

wwwpss-ccom copy 2011 WILEY-VCH Verlag GmbH amp Co KGaA Weinheim

Contributed

Article

distinguished features can bee seen as well For the FMC case (Jgt0) electric polarization is of a positive sign for the AFC case (Jlt0) the polarization is of the negative sign in magnetic field applied along the normal to the surface and it is of the positive sign in magnetic fields applied in the opposite direction For FMC films the polarization changes continuously with the magnetic field for AFC films the polarization changes in a jump-like way

H Oe-9000 -6000 -3000 0 3000 6000 9000

P x S

GSE

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21

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a

H Oe-600 -400 -200 0 200 400

P x SG

SEp

-015

-010

-005

000

005

010

b

Figure 2 Dependence of the overall polarization on applied magnetic field H for a case K1gtK2 a) Jgt0 K2=2105 ergcm3 K1= 2104 ergcm3 b) Jlt0 K2=1104 ergcm3 K1=1103 ergcm3

The interesting peculiarities of polarization behaviour is observed in a case K1ltK2 As in the previous situation the polarization is absent in the saturating magnetic fields for the both types of interlayer coupling A range of mag-netic fields [Hc1 Hc2] at which polarization attains zero values exists in AFC case Analysis of magnetization proc-esses has shown that in this region the antiferromagnetic ordering is established over the structure ie magnetic moments in ldquomagnetically softrdquo film are oriented in the magnetic field direction magnetic moments in the other film are oriented in the opposite direction In the both cases one can observe the jump-like and continuous change of the polarization upon the magnetic field Overall polariza-tion is of the positive sign for FMC films and it is of the negative sign for AMC films The field dependences of electric polarization will be inverted in a case of reversal magnetization cycle The mentioned peculiarities are ex-plained by the magnetization processes occurring in the considered systems in applied magnetic fields

The electric field also affects the magnetization In a case of FMC electric field changes the distribution of mag-

netization in a vicinity of interface between the layers in a case of AFC electric field allocates the direction of mag-netization vector in soft magnetic film possessing with ani-sotropy of ldquoeasy planerdquo type

H Oe-2000 -1600 -1200 -800 -400 0 400 800 1200

P x SG

SEp

0

2

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H2H1

a

H Oe

-1500 -1200 -900 -600 -300 0 300 600

P x SG

SEp

-016

-014

-012

-010

-008

-006

-004

-002

000

002

Hc1 Hc2

b

Figure 3 Dependence of the overall polarization on applied magnetic field H for a case K1ltK2 a) Jgt0 K2=2104 ergcm3 K1= 2105 ergcm3 b) Jlt0 K1=5middot104 ergcm3 K2=5middot103 ergcm3

Concluding the present research shows the magneto-

electricity is present in bi-layered ferromagnetic structure Non-uniform distribution of magnetization vector induces electric polarization in composite magnetic structure The behavior of electric polarization differs in ferromagnetical-ly coupled films and antiferromagnetically coupled ferro-magnetic layers Applied magnetic field affects the electric polarization electric field in ist turn influences the distri-bution of magnetization in bi-layered ferromagnetic struc-ture

References [1] M Fiebig et al Nature (London) 419 818 (2002) [2] J Wang et al Science 299 1719 (2003) [3] KE Kamentzev et al Zh Tekh Fiz 77 50 (2007) [4] S X Dong et al Appl Phys Lett 89 252904 (2006) [5] V G Baryakhtar et al Pisma Zh EkspTeor Fiz 37 565

(1983) [6] EP Stefanovskii et al Sov J Low Temp Phys 12 478

(1986) [7] M Mostovoy Phys Rev Lett 96 067601 (2006) [8] AS Logginov et al Appl Phys Lett 93 182510 (2008) [9] AK Zvezdin and AP Pyatakov Usp Fiz Nauk 179 897

(2009)

3060 Z Gareeva and R Doroshenko Magnetoelectric effect in bi-layered ferromagnetic structure

copy 2011 WILEY-VCH Verlag GmbH amp Co KGaA Weinheim wwwpss-ccom

ph

ysic

ap s sstat

us

solid

i c

ergy of non-uniform exchange interaction Eex the energy of magnetic anisotropy Ean Zeeman energy EH for each of the layers and the energy of interlayer exchange interac-tion EJ energy of non-uniform magnetoelectric (flexomag-netoelectric) [9] interaction Eme

E=Eex+Ean+EH+Eel+EJ (1)

where

22 2

exE Ax y z

⎡ ⎤⎛ ⎞part part part⎛ ⎞ ⎛ ⎞= + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟part part part⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

M M M

is the energy of non-uniform exchange interaction here A is the exchange stiffness

( ) ( )2 21 1 2 2anE K K= minus +M n M n

is the energy of magnetic anisotropy in the layers here K2 is the constant of magnetic anisotropy of easy plane type K1 is the constant of magnetic anisotropy of easy axis type

HE = minusMH is the Zeeman energy H is an applied magnetic field

1 2JE J= minus M M is the interlayer exchange energy J is the constant of inter-layer coupling if Jlt0 the coupling between the layers is antiferromagnetic if Jgt0 the coupling between the layers is ferromagnetic

[ ]2 2 2

( ) ( )2

x y zel

e

P P PE γ

χ+ +

= minus + + minusPE P M M M Mnabla nabla

is the electric energy incorporating the energy of a system in an electric field the energy of electric polarization the flexomagnetoelectric energy E is the electric field P is the electric polarization χe is the dielectric susceptibility γ is the constant of non-uniform magnetoelectric interaction Minimization of the free energy (1) has been carried out numerically We have used polar coordinates

0

(sin cos sin sin cos )M

θ ϕ θ ϕ θ= =Mm

to determine the position of magnetization vector As a re-sult of the free energy minimization we have obtained the dependences of magnetization upon the system parameters and the relation between electric polarization and magneti-zation

1 2( )K K H a b A Jθ θ= (2)

[ ]( ) ( )e eχ γχ= minus minusP E M M M Mnabla nabla

Following the formulas (2) (3) the dependences of polari-zation upon the system parameters have been constructed We suppose the rotation of magnetization occurs in a XOZ plane In this case the only Px component of polarization vector and the polar angle θ characterizing magnetization rotation are sufficient to be considered

3 Results and discussion Hereinafter we analyze the dependences of the electric polarization upon system

parameters and make a comparison between two cases of interlayer coupling (ferromagnetic coupling (FMC) and an-tiferromagnetic coupling (AFC)) The non-homogeneous distribution of magnetization over the film is present in bi-layered structure comprising films with the different kinds of magnetic anisotropy Dependent on the character and the strength of interlayer coupling magnetic inhomogeneity can be either localized close to interface or distributed over the structure Electric polarization arises in a vicinity of magnetic inhomogeneity the sign of polarization vector is determined by the direction of magnetization rotation In a case of FMC the local polarization is concentrated close to interface it is of the same sign in the both layers Applied magnetic field spread polarization over the layers and de-creases the local polarization value In a case of AFC the local polarization has different signs in the layers it is lo-cated at interface in magnetically hard layer and it is spread over the soft magnetic layer In a case of equal val-ues of magnetic anisotropy constants the dependences of the local polarization on the coordinate z is of symmetrical form but the signs of polarization are different in the layers thus overall polarization is equal to zero

Such peculiarities of local polarization will affect the distribution of overall polarization over the sample The overall polarization will always present in a case of FMC it is localized in a vicinity of interface (in an absence of applied magnetic field) In a case of AFC overall polariza-tion exists in the magnetically soft layer it approaches to the zero value when absolute values of anisotropy con-stants are equal (K1=K2) Applied magnetic field affects the magnetization processes and therefore the magnitude of electric polarization Our calculations has shown that mag-netic field can enhance or shrink overall polarization high magnetic fields suppress polarization (the system becomes magnetically saturated) Overall polarization can be also suppressed in the low magnetic field Overall polarization can change continuously or in a jump-like way with a change of applied magnetic field However the character of electric polarization transformations will be different for the different relation between magnetic anisotropy con-stants and the sign of interlayer exchange parameter J In Figs 2 3 we show dependences of overall polarization on the value and direction of applied magnetic field for the four cases K1gtK2 Jgt0 (Fig 2a) K1 gtK2 Jlt0 (Fig 2b) K1ltK2 Jgt0 (Fig 3a) K1ltK2 Jlt0 (Fig 3b) The following physical parameters has been chosen A1=A2=107 ergcm M1=M2=90 G J=104 ergcm3 γ=10-9SGSM χe=102 SGSE (Fig2) M1=M2=200 G A=2middot10-7 ergcm J=09 ergcm2 (Fig 3) Below we compare the obtained results In Fig 2 3 one can see that the character of polarization change is quite different for the presented situations but the general features can be found out When the upper layer is mag-netically hard and the lower layer is magnetically soft (situation K1gtK2 Fig 2) the dependences P(H) has sym-metrical form relative to the H=0 line both for the FMC case and AFC case In the saturating fields and at the zero point (H=0) electric polarization vanishes One can see that

(3)



Contributed

Article


H Oe-9000 -6000 -3000 0 3000 6000 9000

P x S

GSE

p

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07

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21

28

35

42

a

H Oe-600 -400 -200 0 200 400

P x SG

SEp

-015

-010

-005

000

005

010

b





H Oe-2000 -1600 -1200 -800 -400 0 400 800 1200

P x SG

SEp

0

2

4

6

8

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14

H2H1

a

H Oe

-1500 -1200 -900 -600 -300 0 300 600

P x SG

SEp

-016

-014

-012

-010

-008

-006

-004

-002

000

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Hc1 Hc2

b







(2009)



Contributed

Article


H Oe-9000 -6000 -3000 0 3000 6000 9000

P x S

GSE

p

00

07

14

21

28

35

42

a

H Oe-600 -400 -200 0 200 400

P x SG

SEp

-015

-010

-005

000

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010

b





H Oe-2000 -1600 -1200 -800 -400 0 400 800 1200

P x SG

SEp

0

2

4

6

8

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14

H2H1

a

H Oe

-1500 -1200 -900 -600 -300 0 300 600

P x SG

SEp

-016

-014

-012

-010

-008

-006

-004

-002

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Hc1 Hc2

b







(2009)

magnetoelectric effect in bi-layered ferromagnetic structure

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