Solid State Communications 198 (2014) 7–12

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Solid State Communications

0038-10http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/ssc

Strain-controlled nonvolatile magnetization switching

S. Geprägs a,n, A. Brandlmaier a, M.S. Brandt b, R. Gross a,c, S.T.B. Goennenwein a

a Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germanyb Walter Schottky Institut, Technische Universität München, Am Coulombwall 4, 85748 Garching, Germanyc Physik-Department, Technische Universität München, 85748 Garching, Germany

a r t i c l e i n f o

Article history:Received 12 July 2013Accepted 20 July 2013

by G.E.W. Bauer

film/piezoelectric actuator hybrids. The piezoelectric actuator induces a voltage-controllable strainalong different crystalline directions of the ferromagnetic thin film, resulting in modifications of its

Available online 26 July 2013

Keywords:A. Multiferroic hybridsD. MagnetostrictionE. Ferromagnetic resonanceE. SQUID magnetometry

98/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.ssc.2013.07.019

esponding author. Tel.: +49 892 891 4255.ail address: stephan.gepraegs@wmi.badw.de (

a b s t r a c t

We investigate different approaches towards a nonvolatile switching of the remanent magnetizationin single-crystalline ferromagnets at room temperature via elastic strain using ferromagnetic thin

magnetization by converse magnetoelastic effects. We quantify the magnetization changes in the hybridsvia ferromagnetic resonance spectroscopy and superconducting quantum interference device magneto-metry. These measurements demonstrate a significant strain-induced change of the magnetization,limited by an inefficient strain transfer and domain formation in the particular system studied. Toovercome these obstacles, we address practicable engineering concepts and use a model to demonstratethat a strain-controlled, nonvolatile magnetization switching should be possible in appropriatelyengineered ferromagnetic/piezoelectric actuator hybrids.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In magnetoelectric multiferroics, where the ferromagnetic andferroelectric order parameters are coupled, an electric-field controlof the magnetic properties becomes possible [1–3]. This opens theway for appealing novel magnetization control schemes in futurespintronic devices [4]. Unfortunately, single-phase multiferroicswith strong magnetoelectric coupling remain rare [5,2]. Attractivealternatives are composite material systems made from ferro-electric and ferromagnetic compounds [6–9]. In such systems, anelectric-field control of magnetism can be realized using electricfield effects in carrier-mediated ferromagnets [10,11], or exchangecoupling at ferromagnetic/multiferroic interfaces [12,13]. A third,powerful approach relies on strain-mediated, indirect magneto-electric coupling in ferromagnetic/ferroelectric hybrid systems.In recent years, these hybrids were mostly fabricated by depositingferromagnetic thin films on ferroelectric substrates [14–25].Another approach to realize a strain-mediated control of themagnetization is to fabricate ferromagnetic thin film/piezoelectricactuator hybrids by either depositing or cementing ferromagneticthin films onto commercially available PbðZrxTi1�xÞO3 (PZT) multi-layer piezoelectric actuator stacks [cf. Fig. 1(a)] [26–31]. In thesehybrids, the application of a voltage to the piezoelectric actuatorresults in a deformation, which is transferred to the overlaying

ll rights reserved.

S. Geprägs).

ferromagnetic thin film, changing its magnetic anisotropy due tothe converse magnetoelastic effect.

In this paper, we report on two different experimental approachestowards a strain-mediated, nonvolatile, voltage-controlled magnetiza-tion switching in the complete absence of magnetic fields. Theyare based on ferromagnetic thin film/piezoelectric actuator hybridsusing Fe3O4 as the ferromagnet. Our experiments show that asignificant modification of the magnetic anisotropy is possible viavoltage-controlled strain. This work extends our previous studies onferromagnetic/ferroelectric hybrids [26,27,30,20], where we achieved areversible reorientation of the magnetization by up to 901 in Ni basedhybrids. However, a true switching of the magnetization between two(or more) remanent states solely by means of an electric field inducedstrain has not been realized experimentally up to now [32–37].

2. The spin-mechanics concept

The orientation of a well-defined homogeneous magnetizationin a ferromagnet depends on external mechanical stress dueto magnetostriction [38,39]. We exploit this so-called spin-mechanics scheme to control the magnetic anisotropy in Fe3O4

thin films cemented on PbðZrxTi1�xÞO3 (PZT) multilayer piezo-electric actuator stacks [cf. Fig. 1(a)]. In particular, we comparedifferent hybrids fabricated by cementing Fe3O4 thin filmswith different angles α between the crystallographic axes fx; ygof the film and the principal elongation axes fx′; y′g of the actuatorwith z∥z′.

x

y

zz

[001]

x[100]

y[010]

x[100]

y[010]

z[001]

MH

Vp

dominantelongation

axis

Fig. 1. (Colour online) (a) Schematic illustration of a Fe3O4 thin film/piezoelectricactuator hybrid. The coordinate system of the thin film and the actuator enclosingan angle α are denoted by fx; y; zg and fx′;y′; z′g, respectively. (b) Orientation of themagnetic field HðH; θ;ϕÞ and the magnetization MðMs ;Θ;ΦÞ with respect to thecrystallographic axes ⟨100⟩ of the Fe3O4 thin film.

S. Geprägs et al. / Solid State Communications 198 (2014) 7–128

The application of a voltage Vp40 ðVpo0Þ to the piezoelectricactuator causes an elongation ϵ′240 (contraction ϵ′2o0) along theactuator's dominant elongation axis y′, which is due to elasticityaccompanied by a contraction (elongation) along the two ortho-gonal directions x′ and z′ [cf. Fig. 1(a)]. This leads to a change ofthe strain state ϵ of the Fe3O4 thin film elastically clamped onto thepiezoelectric actuator. This causes a modification of the magneticanisotropy, and thus alters the direction of the magnetization M.In a macrospin model, the magnetization M of the Fe3O4 thin filmdescribed by MðMs;Θ;ΦÞ ¼MsmðΘ;ΦÞ aligns in such a way thatthe free energy density F takes its minimum value in equilibrium.Here, mx ¼ sin Θ sin Φ, my ¼ cos Θ, and mz ¼ sin Θ cos Φ [cf.Fig. 1(b)] are directional cosines and Ms the saturation magnetiza-tion. The orientation of the magnetization mðΘ;ΦÞ can be calcu-lated in the framework of a single domain model by using aphenomenological thermodynamic model based on the free energydensity

F ¼ FZeeman þ F001u;eff þ Fmc þ Fel þ Fme ð1Þ

with the Zeeman energy density FZeeman ¼�μ0MsmðΘ;ΦÞHhðθ;ϕÞ,the effective uniaxial anisotropy contribution F001u;eff ¼ 1

2 μ0M2sm

2zþ

K001u m2

z , which comprises the demagnetization contribution and the

uniaxial contribution K001u resulting from the pseudomorphic

growth of the ferromagnetic thin film, the first-order magnetocrys-talline anisotropy contribution Fmc ¼ Kcðm2

xm2y þm2

ym2z þm2

zm2x Þ

with the cubic anisotropy constant Kc, the elastic energy density[40] Fel ¼ 1

2 c11 ϵ21 þ ϵ22 þ ϵ23� �þ c12 ϵ1ϵ2 þ ϵ2ϵ3 þ ϵ1ϵ3ð Þ þ 1

2 c44 ϵ24þ�

ϵ25 þ ϵ26Þ, and the magnetoelastic contribution

Fme ¼ B1 ϵ1 m2x�

13

� �þ ϵ2 m2

y�13

� �þ ϵ3 m2

z�13

� �� �

þB2ðϵ4mymz þ ϵ5mxmz þ ϵ6mxmyÞ: ð2ÞThe magnetoelastic coupling coefficients B1 and B2 can bewritten asa function of the magnetostrictive constants λ100 and λ111, whichyields B1 ¼� 3

2 λ100 c11�c12ð Þ and B2 ¼�3λ111c44, respectively. Herewe use bulk values for the magnetostrictive constants (λ100 ¼�19:5� 10�6 and λ111 ¼ þ 77:6� 10�6) as well as for the elastic

stiffness constants cij (c11 ¼ 27:2� 1010 N=m2, c12 ¼ 17:8� 1010

N=m2, and c44 ¼ 6:1� 1010 N=m2) [41–43].To determine the modification of the magnetic anisotropy

caused by strain effects induced by the piezoelectric actuator, wefirst derive the strain tensor ϵ of the Fe3O4 thin film. In thefx′; y′; z′g coordinate system, the strain components ϵ′4, ϵ

′5, and ϵ′6

vanish, since no shear strains are present. Furthermore, as the thinfilm is clamped to the piezoelectric actuator, the in-plane strains ϵ′1and ϵ′2 are not independent. Due to the actuator's elastic proper-ties, these strain components are related via the Poisson ratioν¼ 0:45 according to ϵ′1 ¼�νϵ′2. To obtain the strain tensor ϵ in thecoordinate system of the Fe3O4 thin film fx; y; zg, we apply a tensortransformation as described in detail in Refs. [38,44]. The straincomponents ϵi ði¼ 3; 4; 5Þ can then be deduced according to themechanical equilibrium condition si ¼ ∂F=∂ϵi ¼ 0 (i¼3, 4, 5). Withthis relation, we finally obtain ϵ as a function of ϵ′2, neglectingcomparably small magnetoelastic terms

ϵ¼

�12�1þ νþ 1þ νð Þ cos 2αð Þ½ �ϵ′2

121�νþ 1þ νð Þ cos 2αð Þ½ �ϵ′2

�c12c11

1�νð Þϵ′200

ð1þ νÞ sin ð2αÞϵ′2

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCA

: ð3Þ

Now we are in a position to derive the magnetization orienta-tion mðΘ;ΦÞ by tracing the minimum of the total free energydensity F as a function of ϵ′2, which can be controlled by Vp. Thecorresponding evolution is calculated by minimizing Eq. (1) withrespect to the orientation of the magnetization Θ. Since the straininduced in the ferromagnetic thin film is of the order of 10�3 inour hybrid structures, the magnetoelastic energy contribution Fme

will not overcome the demagnetization energy in Fe3O4 thin films.Thus, the magnetization remains in-plane in case of zero magneticfield, which results in Φ¼ 901.

To illustrate the concept of a strain-induced, nonvolatile mag-netization switching in zero magnetic and electric fields, Fig. 2exemplary shows free energy density FðΘ; ϵ′2Þ contours within thefilm plane for α¼ 01 [Fig. 2(b)–(d)] and α¼ 451 [Fig. 2(f)–(h)]. Inboth cases, the induced uniaxial strain is symmetric with respectto the crystallographic axes of the cubic ferromagnetic thin film.This results in two energetically equivalent minima in the freeenergy density F, which forces domain formation. To lift thisdegeneracy a small uniaxial magnetic anisotropy contributionin the film plane is introduced in the simulations given byF ipu ¼ K ip

u ðmx sin Θu þmy cos ΘuÞ2 with the uniaxial anisotropyconstant K ip

u . For illustration purposes, we here use Θu ¼ 101 andK ipu 40 with jK ip

u =Kcj ¼ 1=15. To meet the experimental conditionsof Fe3O4 thin films, we choose K001

u 40 and Kco0.In case of α¼ 01, the ferromagnetic thin film is elongated and

contracted along the cubic axes (x′∥x and y′∥y) [cf. Fig. 2(a)] andthus no shear strains appear ðϵ6 ¼ 0Þ [cf. Eq. (3)]. Starting atϵ′2 ¼ 0 ðVp ¼ 0 VÞ [cf. black line in Fig. 2(b)], the magnetizationorientation Θ is aligned along a magnetically easy axis, e.g., atΘ¼ 471 (point A). Upon increasing ϵ′2 ðVp40 VÞ, the magneticallyeasy axis and thus the magnetization orientation Θ continu-ously rotates towards Θ¼ 981 (point B). The corresponding freeenergy density contour for ϵ′2 ¼ þ ϵmax [cf. red line in Fig. 2(b)]is calculated assuming B1ϵmax=Kc ¼ 3=5, which corresponds toϵmax ¼ 1� 10�3 in case of Fe3O4 thin films. By decreasing ϵ′2 backto 0, the magnetization orientation continuously rotates to theenergetically stable direction Θ¼ 1331 (point C) at ϵ′2 ¼ 0. Thisdemonstrates that a reorientation of the magnetization by about861 is feasible. To check the possibility to reorient the magnetiza-tion orientation to the initial configuration (point A), ϵ′2 is inverted.Fig. 2(c) discloses that the easy axis gradually rotates fromΘ¼ 1331 (point C) to Θ¼ 1651 (point D) by inducing ϵ′2 ¼�ϵmax.However, upon reducing ϵ′2 back to 0, the easy axis rotates backto Θ¼ 1331 (point C). Thus, the magnetization remains in point C

M orientation

0

0

0

- max

max

0° 90° 180°0° 90° 180°M orientation

0

0

0

- max

max

F (a

rb. u

nits

)

0° 90° 180°

F (a

rb. u

nits

)

0° 90° 180°

F (a

rb. u

nits

)

M orientation M orientation

F (a

rb. u

nits

)

Vp > 0 ( 2>0)

x[100]

y[010]

Vp < 0 ( 2<0)Vp > 0 ( 2>0)

x[100]

y[010]

45°

Vp < 0 ( 2<0)x

y

A

C

B

AC

D

CAB

C

D

2=+ max

2=+ max/2

2=0

2=- max/22=- max

2=0

C

A

B

D

C

A

B

DA

2=+ max

2=0

2=- max

2=0

C

Fig. 2. (Colour online) (a)–(d) Stress applied to a cubic thin film along the in-planecrystallographic 100h i axes ðα¼ 01Þ. (b)–(d) Corresponding free energy densitycontours FðΘ; ϵ′2Þ, with capital letters indicating the equilibrium magnetizationorientations. The full yellow line in (d) traces the minimum of F. Forward switchingoccurs ðA-CÞ, while a back switching ðC-AÞ is not possible. (e)–(h) Deformation ofa cubic crystal along the in-plane ⟨110⟩ axes ðα¼ 451Þ. Both a forward and a backswitching is feasible. The green arrows illustrate the discontinuous change of themagnetization orientation.

250

300

H orientation θ

250

300

μ 0H

res

(mT)

μ 0H

res

(mT)

0

4

F/M

s (m

T)F/

Ms

(mT)

0° 90° 180° 0° 90° 180°

-4

0

M orientation Θ

-30V

0V

+90V

hybrid <110>

-30V

+150V

0V

+60V

hybrid <100> hybrid <100>

hybrid <110>

-30V0V

+90V

+60V+150V

-30V0V

Fig. 3. (Colour online) FMR fields μ0Hres for a rotation of the magnetic field in thefilm plane hðθ;ϕ¼ 901Þ as a function of Vp (symbols) for the hybrid ⟨100⟩ (a) and thehybrid ⟨110⟩ (c). The lines depict the simulated FMR fields. (b), (d) Calculated freeenergy density contours as a function of the magnetization orientation Θ in the filmplane at zero external magnetic field.

S. Geprägs et al. / Solid State Communications 198 (2014) 7–12 9

and a further strain-induced switching process is not possible.Consequently, the configuration α¼ 01 allows for a single, irrever-sible, and nonvolatile magnetization switching. The whole reor-ientation process of the magnetization for α¼ 01 is shown in Fig. 2(d), which displays the free energy density surface FðΘ; ϵ′2Þ. The fullyellow line traces the minimum of F. By applying the sequence0-þ ϵmax-0-�ϵmax-0 for ϵ′2, the remanent magnetizationaligns along A-B-C-D-C.

In contrast to α¼ 01, the configuration with α¼ 451 leads toa finite shear strain component ϵ6≠0, since the piezoelectricactuator exerts stress along the in-plane ⟨110⟩ directions of theferromagnetic thin film [cf. Fig. 2(e)]. For simplicity, we assumejB1ϵmax=Kcj ¼ jB2ϵmax=Kcj. At the beginning ðϵ′2 ¼ 0Þ [cf. black linein 2(f)], the magnetization orientation Θ is aligned along 1331(point A). While increasing ϵ′2, the easy axis basically retains itsinitial orientation. However, the free energy density minimumgradually transforms into a maximum. Upon a certain criticalinduced strain ϵsw, the easy axis changes discontinuously to

Θ¼ 461 (point B), indicating an abrupt magnetization switching[cf. green arrow in 2(f)]. The orientation of the easy axis essentiallystays along Θ¼ 461 while reducing ϵ′2 back to 0 (point C). Sub-sequently, we continuously increase the inverted induced strainϵ′2o0 [Fig. 2(g)]. Starting from point C the easy axis abruptlyrotates to Θ¼ 1331 (point D). This magnetization orientationremains unchanged, while increasing ϵ′2 back to zero again. Thus,in case of α¼ 451, a reorientation of the magnetization back to theinitial state is possible, which demonstrates that a reversible,nonvolatile magnetization switching in the absence of a magneticfield is possible. The switching of the magnetization from point Ato point C and back to point A upon applying the strain sequence0-þ ϵmax-0-�ϵmax-0 is further illustrated in Fig. 2(h).

3. Towards a nonvolatile magnetization switching via strainin experiment

As described in the previous section, a nonvolatile magnetiza-tion switching is theoretically possible in Fe3O4 thin film/piezo-electric actuator hybrid structures. In the following, we discusstwo hybrids corresponding to the configurations discussed inSection 2. The hybrids are based on the same (001)-oriented,44 nm thick Fe3O4 film grown on a MgO (001) substrate by laser-MBE. After the deposition the thin film sample was cut into twopieces, which were cemented onto the piezoelectric actuators insuch a way that stress is either exerted along the ⟨100⟩ crystal axesðα¼ 01Þ or along ⟨110⟩ ðα¼ 451Þ. The fabrication process of the thinfilm/piezoelectric actuator hybrid structure is described in detail inRef. [26]. The samples thus obtained are referred to as hybrid ⟨100⟩and hybrid ⟨110⟩, respectively. The magnetic anisotropy of theFe3O4 thin film was determined by angular-dependent ferromag-netic resonance (FMR) spectroscopy at constant actuator voltagesVp with the magnetic field applied in the film plane hðθ;ϕ¼ 901Þat room temperature [26].

For the hybrid ⟨100⟩ ðα¼ 01Þ, the evolution of the obtained FMRfields μ0HresðθÞ as a function of the external magnetic fieldorientation θ reveals a superposition of a cubic magnetic aniso-tropy with a uniaxial one [cf. Fig. 3(a)] [26]. For a quantitativesimulation of the experimental data, the FMR angular dependenceis simulated according to Eq. (1) [45,26]. The best agreementbetween the FMR fields for Vp¼0 V observed in experiment [cf.black symbols in Fig. 3(a)] and simulation [cf. black line in Fig. 3

-200

0

200

M (k

A/m

)

50°

60°

1

2

θ max

Mm

ax (102kA

/m) M

max (10

2kA/m

)

0° 180°

-200

0

200

M (k

A/m

)

H orientation θ0 100

120°

130°

1

2

θ max

Vp (V)360°

hybrid <100>

hybrid <110>

-30V

+150V+60V

0V

-30V

+90V0V

Fig. 4. (Colour online) SQUID magnetometry measurements as a function of θ withϕ¼ 901 at different voltages Vp using the hybrid ⟨100⟩ (a), (b) and the hybrid ⟨110⟩(c), (d). All measurements were carried out at μ0H¼ 0 mT. The symbols representthe experimental data and the lines denote fits to cosine functions. (b),(d) Orientation θmax, which denotes the angle of the maximum value of MðθÞ(black squares), and the corresponding magnitude Mmax (green circles) as afunction of Vp.

S. Geprägs et al. / Solid State Communications 198 (2014) 7–1210

(a)] was obtained by using the voltage-independent anisotropyfields K001

u;eff=Ms ¼ 80:2 mT, Kc=Ms ¼�14:9 mT, and K ipu =Ms ¼ 3:2

mT. An additional uniaxial contribution in the film plane F ipu withθu ¼ 01, which is not observed in the as-grown Fe3O4 thin film andcaused by an anisotropic thermal expansion during the curingprocess, has to be included in the free energy density F. ForVp≠0 V, i.e., ϵ′2≠0, the FMR fields μ0Hres are modeled by using ϵ′2 asfit parameter [cf. red and blue lines in Fig. 3(a)]. The derived strainΔϵ′2 ¼ ϵ′2ðþ90 VÞ�ϵ′2ð�30 VÞ ¼ 0:23� 10�3 induced in the ferro-magnetic thin film amounts to only about 27% of the nominalstroke of Δϵideal2 ¼ 0:87� 10�3 of the piezoelectric actuator [46].This is most likely caused by an imperfect strain transmissionbetween the piezoelectric actuator and the Fe3O4 thin film, whichcan be described by Δϵ′2 ¼ χ100Δϵideal2 with χ100 ¼ 0:27.

The corresponding free energy contours within the film planeF=MsðΘ;Φ¼ 901Þ in the absence of an external magnetic field areshown in Fig. 3(b). In agreement with Fig. 2(b)–(d), the contour forVp ¼ 0 V ðϵ′2 ¼ 0Þ exhibits a fourfold symmetry with a superim-posed magnetic hard axis, which is found to be along Θu ¼ 01in the experiment. Upon the application of Vp≠0 we observe achange of the relative strength of the magnetic hard axes, asevident from the different magnitudes of the maxima of the freeenergy, while they retain their orientation. The easy axes—i.e., thefree energy density minima—almost retain their strength, but theorientation Θ of the easy axes clearly is dependent on Vp. This isthe basis for the continuous and reversible rotation of M in thespin-mechanics scheme and confirms the simulations of Fig. 2.However, the free energy density contours in Fig. 3(a) reveal arotation of the easy axes by ΔΘ¼ 61 for �30 V≤Vp ≤þ 90 V atroom temperature. Hence, a continuous and reversible voltagecontrol of magnetization orientation is possible, but a voltage-controlled magnetization switching is out of reach in the presenthybrid, since the induced strain ϵ′2 is much lower than the nominalstrain of the piezoelectric actuator.

In case of α¼ 451, the experimentally obtained and simulatedFMR fields μ0HresðθÞ are shown in Fig. 3(c). In analogy to theconfiguration α¼ 01, the total free energy density F for the presentsample is composed of Eq. (1), with the additional, thermallyinduced uniaxial anisotropy F ipu in the film plane along Θu ¼ 51. Thesolid lines in Fig. 3(b) represent the numerically simulated FMRfields using the voltage-independent anisotropy fields K001

u;eff=Ms ¼75:3 mT, Kc=Ms ¼�14:5 mT, K ip

u =Ms ¼ 1:1 mT, and the strain ϵ′2 asfit parameter. From these values a non-ideal strain transfer ofχ110 ¼ 0:09 can be inferred. The corresponding calculated freeenergy density curves in the film plane F=MsðΘ;Φ¼ 901Þ aredepicted in Fig. 3(d). According to Section 2, upon the applicationof a voltage Vp, the energy minima mainly retain their orientation.More importantly, the relative strengths of the magnetic easy axesconsiderably change, as illustrated for the energy density mini-mum at Θ¼ 1331, which remarkably loses depth for Vp ¼ þ 150 Vand approaches transforming into a maximum. Due to the low χ110

value the strain-induced anisotropy is unfortunately not largeenough to cause an abrupt magnetization switching as shown inFig. 2(f) and (g). However, by optimizing the strain transmissionefficiency, magnetization switching should be possible for α¼ 451.

Fig. 3 demonstrates that angular-dependent FMR measure-ments allow to quantitatively determine the contributions to thetotal free energy density F. However, it does not directly measurethe remanent magnetization orientation. Moreover, Eq. (1) isapplicable only to homogeneously magnetized samples. This isvalid for FMR measurements, since the applied external fieldsuffices to fully saturate the magnetization for the present hybrids.As we are particularly aiming at a magnetization switching atvanishing external magnetic field, magnetic domain formationmight be important. Therefore, in the following, we utilize super-conducting quantum interference device (SQUID) magnetometry

measurements as a function of the in-plane magnetic fieldorientation θ to directly measure the remanent magnetizationas a function of Vp. For these angular-dependent magnetizationmeasurements, we magnetized the hybrid along a magneticallyeasy axis by applying μ0Hprep ¼ þ 1 T and then swept the mag-netic field to μ0H ¼ 0 T at a fixed strain state, i.e., at a fixed voltageVp. After the preparation of the magnetization, we recorded theprojection of the magnetization on the magnetic field direction asa function of the magnetic field orientation θ.

In case of the hybrid ⟨100⟩, the preparation of the magnetiza-tion was carried out at Vp ¼�30 V with the external magneticfield oriented along θprep ¼ 501, which corresponds to a minimumof the free energy density F [cf. Fig. 3(b)]. The results obtained bycarrying out angular-dependent magnetometry measurements areshown in Fig. 4(a). Since in the absence of an external magneticfield the magnetization preferably aligns along a magnetic easyaxis, maxima in the MðθÞ curves correspond to minima in the FðΘÞcontours. The respective maxima of the MðθÞ curves are evaluatedin Fig. 4(b), regarding their orientation θmax (black squares) andmagnitude Mmax (green circles) as a function of Vp. Mmax changesby only 1% in the voltage range �30 V≤Vp ≤þ 90 V and thus isalmost independent of Vp. This demonstrates that domain forma-tion plays only a negligible role in case of α¼ 01. However, θmax

changes by about 91. This proves that the macroscopic, homo-geneous remanent magnetization M rotates by about 91 in the filmplane for �30 V≤Vp ≤þ 90 V, which confirms the results obtainedby FMR measurements [cf. Fig. 3(b)].

We now turn to the hybrid ⟨110⟩. In a first set of experiments,the preparation field was applied along θprep ¼ 1331 [Fig. 4(c)]. Asthe free energy density at this orientation continuously evolvesfrom a deep minimum towards a shallow one with increasing ϵ′2,i.e., Vp [cf. Fig. 3(d)], this minimum will be referred to as localminimum in the following. The angle-dependent SQUID measure-ments reveal a qualitatively different behavior compared to themeasurements on the hybrid ⟨100⟩. In case of the hybrid ⟨110⟩,both the magnitude Mmax as well as the orientation θmax of themaximum significantly change as a function of Vp [cf. Fig. 4(d)].Upon increasing Vp from �30 V to +150 V, Mmax decreases by 49%of its initial value, while the orientation of Mmax rotates by 201towards the free energy density minimum at 471 [cf. Fig. 2(f)].The reduction of Mmax elucidates magnetic domain formation withincreasing Vp. After the magnetic preparation at Vp ¼�30 V,the Fe3O4 thin film exhibits a single-domain state with the

0.10120140

F (a

rb. u

nits

)

0° 90°

F (a

rb. u

nits

)

M orientation Θ-90° 0° 90° 180° 270°

M orientation Θ

χ=0.27

A

E

B

D

C

α=20°

C

AB

C

D

E

0

0

0

-εmax

+εmax

0

0

-εmax

+εmax

ε 2=+εmax

ε 2=+εmax/2

ε 2=0

ε 2=-εmax/2ε 2=-εmax

ε 2=0

α=20°

α=20°

S. Geprägs et al. / Solid State Communications 198 (2014) 7–12 11

homogeneous magnetization oriented along Θ¼ 1331. As theapplied voltage Vp increases, the local energy density minimumat Θ¼ 1331 looses depth and thus magnetically hardens, while theglobal minimum of the energy density at Θ¼ 471 magneticallysoftens, favoring domain formation [cf. Fig. 3(c)]. Thus, theangular-dependent magnetization measurements shown in Fig. 4(c) and (d) are not consistent with the single-domain free energydensity approach used to calculated the energy density contours inFig. 3(d).

In a second set of experiments, we repeated the SQUIDmeasurements with the magnetic preparation field Hprep appliedalong θprep ¼ 431, close to the global minimum of the free energydensity at Θ¼ 471. The experimental data coincide in good appro-ximation for different applied voltages Vp [not shown here]. Themagnetization M retains its orientation at Θ¼ 471 independentof Vp, while the magnitude of the magnetization at this orientationMmax changes by only 5% within the full voltage range. Consideringthe free energy density contours [cf. Fig. 3(d)], the energy barrierfor domain formation is much larger in this case, such that theFe3O4 thin film remains in a magnetically single domain state andcan be described by Eq. (1).

0 5 10 15 20 25 30 35 40 450.00

0.02

0.04

0.06

0.08

εsw (%

)

α (°)

020406080100

V

(V

)sw p

χ=1.0

Fig. 5. (Colour online) Approach to a nonvolatile, all-voltage controlled magnetiza-tion switching. (a)–(c) Calculated free energy density contours FðΘ; ϵ′2Þ for α¼ 201.The full yellow line in (c) traces the minimum of F. Discontinuous switchingprocesses of the magnetization by 901 from A-C-E while applying ϵsw withalternating sign are visible. (d) Calculated critical strain ϵswðαÞ values and thecorresponding switching voltages V sw

p ðαÞ at which a discontinuous switching of themagnetization, which is illustrated by the green arrows in (a) and (b), occurs. Theblue curve depicts perfect strain transmission ðχ ¼ 1:0Þ and the black curverepresents the experimentally realized strain in the hybrid ⟨100⟩ð χ ¼ 0:27Þ.

4. Impact of different strain orientations

The experimental results discussed above show that an align-ment of the strain axes along crystallographic axes might favormagnetic domain formation. Therefore, we now discuss config-urations with an angle α in between 01 and 451. The correspond-ing concept is exemplarily illustrated for α¼ 201 in Fig. 5(a), (b),and (c).

Starting at ϵ′2 ¼ 0, we assume an initial magnetization orienta-tion along Θ¼ 1351 [point A in Fig. 5(a)]. Upon increasing ϵ′240 inthe thin film, the easy axis continuously rotates, until it switchesdiscontinuously to point B at a certain critical strain ϵ′;crit2 ¼ ϵswðαÞ[green arrow in Fig. 5(a)]. When the strain ϵ′2 is reduced back to 0,the easy axis rotates to point C at Θ¼ 451. Upon subsequentlyincreasing the strain ϵ′2o0 with opposite sign [Fig. 5(b)], thesituation appears qualitatively different from the situation illu-strated in Fig. 2(g), as we do not observe a back switching to theinitial orientation (point A), but a further switching process alongthe original direction of rotation via point D to point E at Θ¼�451[Fig. 5(b)]. Hence, iteratively applying ϵsw with alternating signprovides a concept to discontinuously rotate the equilibriummagnetization orientation by 901 via nonvolatile switching pro-cesses [37]. Such magnetization switching processes are “quasi-reversible”, since four consecutive switching processes (in a ferro-magnet with cubic symmetry) evidently restore the initial mag-netization orientation state [Fig. 5(c)]. Hence, this constitutes avery elegant voltage-control scheme of magnetization orientation.

Assuming a perfect strain transmission between the piezo-electric actuator and the ferromagnetic thin film ðχ ¼ 1Þ, ϵsw can bederived using the free energy density F given in Eq. (1) witha cubic anisotropy field Kc=Ms ¼�14:7 mT, which is the averagedvalue of the cubic anisotropy measured in hybrid ⟨100⟩ and hybrid⟨110⟩, as well as the elastic and magnetoelastic constants of Fe3O4.Fig. 5(d) shows that ϵsw required to induce a magnetizationswitching process significantly decreases with increasing angle α,exhibits a minimum at α¼ 331, and finally slightly increases with αapproaching 451. Overall, ϵsw has comparatively moderate valueslower than 10�3, which are experimentally achievable using theconcept described in Fig. 1(a). These values correspond to switch-ing voltages V sw

p ðαÞo150 V in our hybrid concept, which areexperimentally accessible [cf. Fig. 5(d)].

To furthermore lower the switching strain ϵsw, the propertiesof the ferromagnetic film itself must be fine-tuned, as ϵsw linearly

depends on the cubic anisotropy constant Kc and inversely dependson the magnetostriction constants λ100 and λ111. Most promisingcandidates regarding the realization of a magnetization switchingtherefore evidently are materials with a small cubic anisotropy andhigh magnetostriction constants.

5. Conclusion

In summary, we have investigated concepts for a voltage-controlled, nonvolatile 901 switching of the remanent magnet-ization in Fe3O4 thin film/piezoelectric actuator hybrids at roomtemperature. The possibility to induce strain along differentdirections in the film plane with respect to the crystallographicaxes depending on the cementing procedure, allows to investigatethe switching behavior and particularly to take advantage of themagnetostriction constants λ along different crystalline orienta-tions. We have discussed the qualitatively different switchingbehavior for two different configurations, namely strain exertedalong the in-plane crystalline Fe3O4 ⟨100⟩ and along the in-plane⟨110⟩ directions. The free energy density of the ferromagnetic thinfilms was determined by FMR spectroscopy, which allows to inferthe equilibrium magnetization orientation in a Stoner–Wohlfarthmodel. The results show a rotation of the easy axes by a fewdegrees and a significant modification of the relative strength ofthe easy axes, respectively. However, in combination with SQUID

S. Geprägs et al. / Solid State Communications 198 (2014) 7–1212

magnetometry measurements we find that the angle of magneti-zation reorientation is not large enough to induce a magnetizationswitching in the former, and magnetic domain formation impedesa coherent magnetization switching in the latter approach.This shows that inefficient strain transfer and magnetic domainformation are major obstacles towards a non-volatile strain-con-trolled magnetization switching. Using the experimental freeenergy and strain transfer parameters, we find in simulations thatskillful alignment of the strain within the films should reduce thestrain values required to switch the magnetization and impededomain formation. More explicitly, our experiments suggest thatan all-voltage-controlled, nonvolatile magnetization switching atroom temperature and zero magnetic field should be possible.

Acknowledgments

Financial support via DFG Project no. GO 944/3-1 and theGerman Excellence Initiative via the “Nanosystems InitiativeMunich (NIM)” are gratefully acknowledged.

References

[1] M. Fiebig, J. Phys. D: Appl. Phys. 38 (2005) R123.[2] W. Eerenstein, N.D. Mathur, J.F. Scott, Nature 442 (2006) 759.[3] R. Ramesh, N.A. Spaldin, Nat. Mater. 6 (2007) 21.[4] T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A. Doran, M.P. Cruz, Y.H. Chu,

C. Ederer, N.A. Spaldin, R.R. Das, D.M. Kim, S.H. Baek, C.B. Eom, R. Ramesh, Nat.Mater. 5 (2006) 823.

[5] N.A. Hill, J. Phys. Chem. B 104 (2000) 6694–6709.[6] S. Geprägs, M. Opel, S.T.B. Goennenwein, R. Gross, Philos. Mag. Lett. 87 (2007)

141.[7] C.-W. Nan, M.I. Bichurin, S. Dong, D. Viehland, G. Srinivasan, J. Appl. Phys. 103

(2008) 031101.[8] J. Ma, J. Hu, Z. Li, C.-W. Nan, Adv. Mater. 23 (2011) 1062.[9] C.A.F. Vaz, J. Phys.: Condens. Matter 24 (2012) 333201.[10] I. Stolichnov, S.W.E. Riester, H.J. Trodahl, N. Setter, A.W. Rushforth, K.W. Edmonds,

R.P. Campion, C.T. Foxon, B.L. Gallagher, T. Jungwirth, Nat. Mater. 7 (2008) 464.[11] C.A.F. Vaz, J. Hoffman, C.H. Ahn, R. Ramesh, Adv. Mater. 22 (2010) 2900–2918.[12] P. Borisov, A. Hochstrat, X. Chen, W. Kleemann, C. Binek, Phys. Rev. Lett. 94

(2005) 117203.[13] Y.-H. Chu, L.W. Martin, M.B. Holcomb, M. Gajek, S.-J. Han, Q. He, N. Balke,

C.-H. Yang, D. Lee, W. Hu, Q. Zhan, P.-L. Yang, A. Fraile-Rodriguez, A. Scholl,S.X. Wang, R. Ramesh, Nat. Mater. 7 (2008) 478.

[14] J.-W. Lee, S.-C. Shin, S.-K. Kim, Appl. Phys. Lett. 82 (2003) 2458.

[15] W. Eerenstein, M. Wiora, J.L. Prieto, J.F. Scott, N.D. Mathur, Nat. Mater. 6 (2007)348.

[16] S. Sahoo, S. Polisetty, C.-G. Duan, S.S. Jaswal, E.Y. Tsymbal, C. Binek, Phys. Rev.B 76 (2007) 092108.

[17] C. Thiele, K. Dörr, O. Bilani, J. Rödel, L. Schultz, Phys. Rev. B 75 (2007) 054408.[18] C. Israel, S. Kar-Narayan, N.D. Mathur, Appl. Phys. Lett. 93 (2008) 173501.[19] M. Liu, O. Obi, J. Lou, Y. Chen, Z. Cai, S. Stoute, M. Espanol, M. Lew, X. Situ,

K.S. Ziemer, V.G. Harris, N.X. Sun, Adv. Funct. Mater. 19 (2009) 1826.[20] S. Geprägs, A. Brandlmaier, M. Opel, R. Gross, S.T.B. Goennenwein, Appl. Phys.

Lett. 96 (2010) 142509.[21] M. Liu, O. Obi, Z. Cai, J. Lou, G. Yang, K.S. Ziemer, N.X. Sun, J. Appl. Phys. 107

(2010) 073916.[22] S. Geprägs, M. Opel, S.T.B. Goennenwein, R. Gross, Phys. Rev. B 86 (2012)

134432.[23] T.H.E. Lahtinen, K.J.A. Franke, S. van Dijken, Sci. Rep. 2 (2012) 258.[24] R. Streubel, D. Köhler, R. Schäfer, L.M. Eng, Phys. Rev. B 87 (2013) 054410.[25] M. Ghidini, R. Pellicelli, J.L. Prieto, X. Moya, J. Soussi, J. Briscoe, S. Dunn,

N.D. Mathur, Nat. Commun. 4 (2013) 1453.[26] A. Brandlmaier, S. Geprägs, M. Weiler, A. Boger, M. Opel, H. Huebl, C. Bihler,

M.S. Brandt, B. Botters, D. Grundler, R. Gross, S.T.B. Goennenwein, Phys. Rev. B77 (2008) 104445.

[27] C. Bihler, M. Althammer, A. Brandlmaier, S. Geprägs, M. Weiler, M. Opel,W. Schoch, W. Limmer, R. Gross, M.S. Brandt, S.T.B. Goennenwein, Phys. Rev. B78 (2008) 045203.

[28] M. Overby, A. Chernyshov, L.P. Rokhinson, X. Liu, J.K. Furdyna, Appl. Phys. Lett.92 (2008) 192501.

[29] A.W. Rushforth, E.D. Ranieri, J. Zemen, J. Wunderlich, K.W. Edmonds, C.S. King,E. Ahmad, R.P. Campion, C.T. Foxon, B.L. Gallagher, K. Výborný, J. Kučera,T. Jungwirth, Phys. Rev. B 78 (2008) 085314.

[30] M. Weiler, A. Brandlmaier, S. Geprägs, M. Althammer, M. Opel, C. Bihler,H. Huebl, M.S. Brandt, R. Gross, S.T.B. Goennenwein, New J. Phys. 11 (2009)013021.

[31] A. Brandlmaier, S. Geprägs, G. Woltersdorf, R. Gross, S.T.B. Goennenwein,J. Appl. Phys. 110 (2011) 043913.

[32] N.A. Pertsev, Phys. Rev. B 78 (2008) 212102.[33] N.A. Pertsev, H. Kohlstedt, Appl. Phys. Lett. 95 (2009) 163503.[34] J.-M. Hu, C.W. Nan, Phys. Rev. B 80 (2009) 224416.[35] J.-M. Hu, Z. Li, J. Wang, C.W. Nan, J. Appl. Phys. 107 (2010) 093912.[36] J.-M. Hu, G. Sheng, J.X. Zhang, C.W. Nan, L.Q. Chen, J. Appl. Phys. 109 (2011)

123917.[37] Y. Iwasaki, J. Magn. Magn. Mater. 240 (2002) 395.[38] D. Sander, Rep. Prog. Phys. 62 (1999) 809.[39] S. Chikazumi, Physics of Ferromagnetism, 2nd edition, Oxford University Press,

New York, 1997.[40] The matrix notation (also called Voigt notation) is used throughout the paper.[41] E.W. Gorter, Proc. IRE 43 (1955) 1945.[42] L.R. Bickford, J. Pappis, J.L. Stull, Phys. Rev. 99 (1955) 1210.[43] H. Schwenk, S. Bareiter, C. Hinkel, B. Lüthi, Z. Kakol, A. Koslowski, J.M. Honig,

Eur. Phys. J. B 13 (2000) 491.[44] J.F. Nye, Physical Properties of Crystals, Oxford University Press, 1985.[45] M. Farle, Rep. Prog. Phys. 61 (1998) 755.[46] Low voltage co-fired multilayer stacks, rings and chips for actuation, Piezo-

mechanik GmbH, Germany, 2010.