paper

1. 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, Germany
b
Walter Schottky Institut, Technische Universität München, Am Coulombwall 4, 85748 Garching, Germany
c
Physik-Department, Technische Universität München, 85748 Garching, Germany
a r t i c l e i n f o
Article history:
Received 12 July 2013
Accepted 20 July 2013
by G.E.W. Bauer
Available online 26 July 2013
Keywords:
A. Multiferroic hybrids
D. Magnetostriction
E. Ferromagnetic resonance
E. SQUID magnetometry
a b s t r a c t
We investigate different approaches towards a nonvolatile switching of the remanent magnetization
in single-crystalline ferromagnets at room temperature via elastic strain using ferromagnetic thin
film/piezoelectric actuator hybrids. The piezoelectric actuator induces a voltage-controllable strain
along different crystalline directions of the ferromagnetic thin film, resulting in modifications of its
magnetization by converse magnetoelastic effects. We quantify the magnetization changes in the hybrids
via 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. To
overcome these obstacles, we address practicable engineering concepts and use a model to demonstrate
that a strain-controlled, nonvolatile magnetization switching should be possible in appropriately
engineered ferromagnetic/piezoelectric actuator hybrids.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
In magnetoelectric multiferroics, where the ferromagnetic and
ferroelectric order parameters are coupled, an electric-field control
of the magnetic properties becomes possible [1–3]. This opens the
way for appealing novel magnetization control schemes in future
spintronic devices [4]. Unfortunately, single-phase multiferroics
with strong magnetoelectric coupling remain rare [5,2]. Attractive
alternatives are composite material systems made from ferro-
electric and ferromagnetic compounds [6–9]. In such systems, an
electric-field control of magnetism can be realized using electric
field effects in carrier-mediated ferromagnets [10,11], or exchange
coupling 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 depositing
ferromagnetic thin films on ferroelectric substrates [14–25].
Another approach to realize a strain-mediated control of the
magnetization is to fabricate ferromagnetic thin film/piezoelectric
actuator hybrids by either depositing or cementing ferromagnetic
thin films onto commercially available PbðZrxTi1ÀxÞO3 (PZT) multi-
layer piezoelectric actuator stacks [cf. Fig. 1(a)] [26–31]. In these
hybrids, the application of a voltage to the piezoelectric actuator
results in a deformation, which is transferred to the overlaying
ferromagnetic thin film, changing its magnetic anisotropy due to
the converse magnetoelastic effect.
In this paper, we report on two different experimental approaches
towards a strain-mediated, nonvolatile, voltage-controlled magnetiza-
tion switching in the complete absence of magnetic fields. They
are based on ferromagnetic thin film/piezoelectric actuator hybrids
using Fe3O4 as the ferromagnet. Our experiments show that a
significant modification of the magnetic anisotropy is possible via
voltage-controlled strain. This work extends our previous studies on
ferromagnetic/ferroelectric hybrids [26,27,30,20], where we achieved a
reversible reorientation of the magnetization by up to 901 in Ni based
hybrids. However, a true switching of the magnetization between two
(or more) remanent states solely by means of an electric field induced
strain has not been realized experimentally up to now [32–37].
2. The spin-mechanics concept
The orientation of a well-defined homogeneous magnetization
in a ferromagnet depends on external mechanical stress due
to magnetostriction [38,39]. We exploit this so-called spin-
mechanics scheme to control the magnetic anisotropy in Fe3O4
thin films cemented on PbðZrxTi1ÀxÞO3 (PZT) multilayer piezo-
electric actuator stacks [cf. Fig. 1(a)]. In particular, we compare
different hybrids fabricated by cementing Fe3O4 thin films
with different angles α between the crystallographic axes fx; yg
of the film and the principal elongation axes fx′; y′g of the actuator
with z∥z′.
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Solid State Communications
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ssc.2013.07.019
n
Corresponding author. Tel.: +49 892 891 4255.
E-mail address: stephan.gepraegs@wmi.badw.de (S. Geprägs).
Solid State Communications 198 (2014) 7–12

2. The application of a voltage Vp 40 ðVp o0Þ to the piezoelectric
actuator causes an elongation ϵ′
2 40 (contraction ϵ′
2 o0) along the
actuator's dominant elongation axis y′, which is due to elasticity
accompanied by a contraction (elongation) along the two ortho-
gonal directions x′ and z′ [cf. Fig. 1(a)]. This leads to a change of
the strain state ϵ of the Fe3O4 thin film elastically clamped onto the
piezoelectric actuator. This causes a modification of the magnetic
anisotropy, and thus alters the direction of the magnetization M.
In a macrospin model, the magnetization M of the Fe3O4 thin film
described by MðMs; Θ; ΦÞ ¼ MsmðΘ; ΦÞ aligns in such a way that
the 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 a
phenomenological thermodynamic model based on the free energy
density
F ¼ FZeeman þ F001
u;eff þ Fmc þ Fel þ Fme ð1Þ
with the Zeeman energy density FZeeman ¼ Àμ0MsmðΘ; ΦÞHhðθ; ϕÞ,
the effective uniaxial anisotropy contribution F001
u;eff ¼ 1
2 μ0M2
s m2
z þ
K001
u m2
z , which comprises the demagnetization contribution and the
uniaxial contribution K001
u resulting from the pseudomorphic
growth of the ferromagnetic thin film, the first-order magnetocrys-
talline anisotropy contribution Fmc ¼ Kcðm2
x m2
y þ m2
ym2
z þ m2
z m2
x Þ
with the cubic anisotropy constant Kc, the elastic energy density
[40] Fel ¼ 1
2 c11 ϵ2
1 þ ϵ2
2 þ ϵ2
3
À Á
þ c12 ϵ1ϵ2 þ ϵ2ϵ3 þ ϵ1ϵ3ð Þ þ 1
2 c44 ϵ2
4þ
À
ϵ2
5 þ ϵ2
6Þ, and the magnetoelastic contribution
Fme ¼ B1 ϵ1 m2
x À
1
3
þ ϵ2 m2
yÀ
1
3
þ ϵ3 m2
z À
1
3
!
þB2ðϵ4mymz þ ϵ5mxmz þ ϵ6mxmyÞ: ð2Þ
The magnetoelastic coupling coefficients B1 and B2 can be written as
a function of the magnetostrictive constants λ100 and λ111, which
yields B1 ¼ À 3
2 λ100 c11Àc12ð Þ and B2 ¼ À3λ111c44, respectively. Here
we use bulk values for the magnetostrictive constants (λ100 ¼
À19:5 Â 10À6
and λ111 ¼ þ 77:6 Â 10À6
) as well as for the elastic
stiffness constants cij (c11 ¼ 27:2 Â 1010
N=m2
, c12 ¼ 17:8 Â 1010
N=m2
, and c44 ¼ 6:1 Â 1010
N=m2
) [41–43].
To determine the modification of the magnetic anisotropy
caused by strain effects induced by the piezoelectric actuator, we
first derive the strain tensor ϵ of the Fe3O4 thin film. In the
fx′; 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 thin
film is clamped to the piezoelectric actuator, the in-plane strains ϵ′
1
and ϵ′
2 are not independent. Due to the actuator's elastic proper-
ties, these strain components are related via the Poisson ratio
ν ¼ 0:45 according to ϵ′
1 ¼ Àνϵ′
2. To obtain the strain tensor ϵ in the
coordinate system of the Fe3O4 thin film fx; y; zg, we apply a tensor
transformation as described in detail in Refs. [38,44]. The strain
components ϵi ði ¼ 3; 4; 5Þ can then be deduced according to the
mechanical equilibrium condition si ¼ ∂F=∂ϵi ¼ 0 (i¼3, 4, 5). With
this relation, we finally obtain ϵ as a function of ϵ′
2, neglecting
comparably small magnetoelastic terms
ϵ ¼
À
1
2
À1 þ ν þ 1 þ νð Þ cos 2αð Þ½ Šϵ′
2
1
2
1Àν þ 1 þ νð Þ cos 2αð Þ½ Šϵ′
2
À
c12
c11
1Àνð Þϵ′
2
0
0
ð1 þ νÞ sin ð2αÞϵ′
2
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: ð3Þ
Now we are in a position to derive the magnetization orienta-
tion mðΘ; ΦÞ by tracing the minimum of the total free energy
density F as a function of ϵ′
2, which can be controlled by Vp. The
corresponding evolution is calculated by minimizing Eq. (1) with
respect to the orientation of the magnetization Θ. Since the strain
induced in the ferromagnetic thin film is of the order of 10À3
in
our 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 magnetic
field, which results in Φ ¼ 901.
To illustrate the concept of a strain-induced, nonvolatile mag-
netization switching in zero magnetic and electric fields, Fig. 2
exemplary shows free energy density FðΘ; ϵ′
2Þ contours within the
film plane for α ¼ 01 [Fig. 2(b)–(d)] and α ¼ 451 [Fig. 2(f)–(h)]. In
both cases, the induced uniaxial strain is symmetric with respect
to the crystallographic axes of the cubic ferromagnetic thin film.
This results in two energetically equivalent minima in the free
energy density F, which forces domain formation. To lift this
degeneracy a small uniaxial magnetic anisotropy contribution
in the film plane is introduced in the simulations given by
Fip
u ¼ Kip
u ðmx sin Θu þ my cos ΘuÞ2
with the uniaxial anisotropy
constant Kip
u . For illustration purposes, we here use Θu ¼ 101 and
Kip
u 40 with jKip
u =Kcj ¼ 1=15. To meet the experimental conditions
of Fe3O4 thin films, we choose K001
u 40 and Kc o0.
In case of α ¼ 01, the ferromagnetic thin film is elongated and
contracted along the cubic axes (x′∥x and y′∥y) [cf. Fig. 2(a)] and
thus no shear strains appear ðϵ6 ¼ 0Þ [cf. Eq. (3)]. Starting at
ϵ′
2 ¼ 0 ðVp ¼ 0 VÞ [cf. black line in Fig. 2(b)], the magnetization
orientation Θ is aligned along a magnetically easy axis, e.g., at
Θ ¼ 471 (point A). Upon increasing ϵ′
2 ðVp 40 VÞ, the magnetically
easy axis and thus the magnetization orientation Θ continu-
ously rotates towards Θ ¼ 981 (point B). The corresponding free
energy 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 to
ϵmax ¼ 1 Â 10À3
in case of Fe3O4 thin films. By decreasing ϵ′
2 back
to 0, the magnetization orientation continuously rotates to the
energetically stable direction Θ ¼ 1331 (point C) at ϵ′
2 ¼ 0. This
demonstrates that a reorientation of the magnetization by about
861 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 from
Θ ¼ 1331 (point C) to Θ ¼ 1651 (point D) by inducing ϵ′
2 ¼ Àϵmax.
However, upon reducing ϵ′
2 back to 0, the easy axis rotates back
to Θ ¼ 1331 (point C). Thus, the magnetization remains in point C
x
y
z
z
[001]
x
[100]
y
[010]
x
[100]
y
[010]
z
[001]
M
H
Vp
dominant
elongation
axis
Fig. 1. (Colour online) (a) Schematic illustration of a Fe3O4 thin film/piezoelectric
actuator hybrid. The coordinate system of the thin film and the actuator enclosing
an angle α are denoted by fx; y; zg and fx′; y′; z′g, respectively. (b) Orientation of the
magnetic field HðH; θ; ϕÞ and the magnetization MðMs; Θ; ΦÞ with respect to the
crystallographic axes 〈100〉 of the Fe3O4 thin film.
S. Geprägs et al. / Solid State Communications 198 (2014) 7–128

3. and a further strain-induced switching process is not possible.
Consequently, the configuration α ¼ 01 allows for a single, irrever-
sible, and nonvolatile magnetization switching. The whole reor-
ientation process of the magnetization for α ¼ 01 is shown in Fig. 2
(d), which displays the free energy density surface FðΘ; ϵ′
2Þ. The full
yellow line traces the minimum of F. By applying the sequence
0- þ ϵmax-0-Àϵmax-0 for ϵ′
2, the remanent magnetization
aligns along A-B-C-D-C.
In contrast to α ¼ 01, the configuration with α ¼ 451 leads to
a finite shear strain component ϵ6≠0, since the piezoelectric
actuator exerts stress along the in-plane 〈110〉 directions of the
ferromagnetic thin film [cf. Fig. 2(e)]. For simplicity, we assume
jB1ϵmax=Kcj ¼ jB2ϵmax=Kcj. At the beginning ðϵ′
2 ¼ 0Þ [cf. black line
in 2(f)], the magnetization orientation Θ is aligned along 1331
(point A). While increasing ϵ′
2, the easy axis basically retains its
initial orientation. However, the free energy density minimum
gradually transforms into a maximum. Upon a certain critical
induced strain ϵsw
, the easy axis changes discontinuously to
Θ ¼ 461 (point B), indicating an abrupt magnetization switching
[cf. green arrow in 2(f)]. The orientation of the easy axis essentially
stays along Θ ¼ 461 while reducing ϵ′
2 back to 0 (point C). Sub-
sequently, we continuously increase the inverted induced strain
ϵ′
2 o0 [Fig. 2(g)]. Starting from point C the easy axis abruptly
rotates to Θ ¼ 1331 (point D). This magnetization orientation
remains unchanged, while increasing ϵ′
2 back to zero again. Thus,
in case of α ¼ 451, a reorientation of the magnetization back to the
initial state is possible, which demonstrates that a reversible,
nonvolatile magnetization switching in the absence of a magnetic
field is possible. The switching of the magnetization from point A
to point C and back to point A upon applying the strain sequence
0- þ ϵmax-0-Àϵmax-0 is further illustrated in Fig. 2(h).
3. Towards a nonvolatile magnetization switching via strain
in 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 discuss
two hybrids corresponding to the configurations discussed in
Section 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 two
pieces, which were cemented onto the piezoelectric actuators in
such a way that stress is either exerted along the 〈100〉 crystal axes
ðα ¼ 01Þ or along 〈110〉 ðα ¼ 451Þ. The fabrication process of the thin
film/piezoelectric actuator hybrid structure is described in detail in
Ref. [26]. The samples thus obtained are referred to as hybrid 〈100〉
and hybrid 〈110〉, respectively. The magnetic anisotropy of the
Fe3O4 thin film was determined by angular-dependent ferromag-
netic resonance (FMR) spectroscopy at constant actuator voltages
Vp with the magnetic field applied in the film plane hðθ; ϕ ¼ 901Þ
at room temperature [26].
For the hybrid 〈100〉 ðα ¼ 01Þ, the evolution of the obtained FMR
fields μ0HresðθÞ as a function of the external magnetic field
orientation θ reveals a superposition of a cubic magnetic aniso-
tropy with a uniaxial one [cf. Fig. 3(a)] [26]. For a quantitative
simulation of the experimental data, the FMR angular dependence
is simulated according to Eq. (1) [45,26]. The best agreement
between the FMR fields for Vp¼0 V observed in experiment [cf.
black symbols in Fig. 3(a)] and simulation [cf. black line in Fig. 3
M orientation
0
0
0
- max
max
0° 90° 180°0° 90° 180°
M orientation
0
0
0
- max
max
F(arb.units)
0° 90° 180°
F(arb.units)
0° 90° 180°
F(arb.units)
M orientation M orientation
F(arb.units)
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
A
C
D
CA
B
C
D
2
=+ max
2
=+ max
/2
2
=0
2
=- max
/2
2
=- max
2
=0
C
A
B
D
C
A
B
D
A
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-plane
crystallographic 100h i axes ðα ¼ 01Þ. (b)–(d) Corresponding free energy density
contours FðΘ; ϵ′
2Þ, with capital letters indicating the equilibrium magnetization
orientations. The full yellow line in (d) traces the minimum of F. Forward switching
occurs ðA-CÞ, while a back switching ðC-AÞ is not possible. (e)–(h) Deformation of
a cubic crystal along the in-plane 〈110〉 axes ðα ¼ 451Þ. Both a forward and a back
switching is feasible. The green arrows illustrate the discontinuous change of the
magnetization orientation.
250
300
H orientation θ
250
300
μ0Hres(mT)μ0Hres(mT)
0
4
F/Ms(mT)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
-30V
0V
+90V
+60V
+150V
-30V
0V
Fig. 3. (Colour online) FMR fields μ0Hres for a rotation of the magnetic field in the
film plane hðθ; ϕ ¼ 901Þ as a function of Vp (symbols) for the hybrid 〈100〉 (a) and the
hybrid 〈110〉 (c). The lines depict the simulated FMR fields. (b), (d) Calculated free
energy density contours as a function of the magnetization orientation Θ in the film
plane at zero external magnetic field.
S. Geprägs et al. / Solid State Communications 198 (2014) 7–12 9

4. (a)] was obtained by using the voltage-independent anisotropy
fields K001
u;eff =Ms ¼ 80:2 mT, Kc=Ms ¼ À14:9 mT, and Kip
u =Ms ¼ 3:2
mT. An additional uniaxial contribution in the film plane Fip
u with
θu ¼ 01, which is not observed in the as-grown Fe3O4 thin film and
caused by an anisotropic thermal expansion during the curing
process, has to be included in the free energy density F. For
Vp≠0 V, i.e., ϵ′
2≠0, the FMR fields μ0Hres are modeled by using ϵ′
2 as
fit parameter [cf. red and blue lines in Fig. 3(a)]. The derived strain
Δϵ′
2 ¼ ϵ′
2ðþ90 VÞÀϵ′
2ðÀ30 VÞ ¼ 0:23  10À3
induced in the ferro-
magnetic thin film amounts to only about 27% of the nominal
stroke of Δϵideal
2 ¼ 0:87 Â 10À3
of the piezoelectric actuator [46].
This is most likely caused by an imperfect strain transmission
between the piezoelectric actuator and the Fe3O4 thin film, which
can be described by Δϵ′
2 ¼ χ100
Δϵideal
2 with χ100
¼ 0:27.
The corresponding free energy contours within the film plane
F=MsðΘ; Φ ¼ 901Þ in the absence of an external magnetic field are
shown in Fig. 3(b). In agreement with Fig. 2(b)–(d), the contour for
Vp
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 01
in the experiment. Upon the application of Vp≠0 we observe a
change of the relative strength of the magnetic hard axes, as
evident from the different magnitudes of the maxima of the free
energy, while they retain their orientation. The easy axes—i.e., the
free energy density minima—almost retain their strength, but the
orientation Θ of the easy axes clearly is dependent on Vp. This is
the basis for the continuous and reversible rotation of M in the
spin-mechanics scheme and confirms the simulations of Fig. 2.
However, the free energy density contours in Fig. 3(a) reveal a
rotation of the easy axes by ΔΘ ¼ 61 for À30 V ≤Vp ≤ þ 90 V at
room temperature. Hence, a continuous and reversible voltage
control of magnetization orientation is possible, but a voltage-
controlled magnetization switching is out of reach in the present
hybrid, since the induced strain ϵ′
2 is much lower than the nominal
strain of the piezoelectric actuator.
In case of α ¼ 451, the experimentally obtained and simulated
FMR fields μ0HresðθÞ are shown in Fig. 3(c). In analogy to the
configuration α ¼ 01, the total free energy density F for the present
sample is composed of Eq. (1), with the additional, thermally
induced uniaxial anisotropy Fip
u in the film plane along Θu ¼ 51. The
solid lines in Fig. 3(b) represent the numerically simulated FMR
fields using the voltage-independent anisotropy fields K001
u;eff =Ms ¼
75:3 mT, Kc=Ms ¼ À14:5 mT, Kip
u =Ms ¼ 1:1 mT, and the strain ϵ′
2 as
fit parameter. From these values a non-ideal strain transfer of
χ110
¼ 0:09 can be inferred. The corresponding calculated free
energy density curves in the film plane F=MsðΘ; Φ ¼ 901Þ are
depicted in Fig. 3(d). According to Section 2, upon the application
of a voltage Vp, the energy minima mainly retain their orientation.
More importantly, the relative strengths of the magnetic easy axes
considerably change, as illustrated for the energy density mini-
mum at Θ ¼ 1331, which remarkably loses depth for Vp ¼ þ 150 V
and approaches transforming into a maximum. Due to the low χ110
value the strain-induced anisotropy is unfortunately not large
enough to cause an abrupt magnetization switching as shown in
Fig. 2(f) and (g). However, by optimizing the strain transmission
efficiency, magnetization switching should be possible for α ¼ 451.
Fig. 3 demonstrates that angular-dependent FMR measure-
ments allow to quantitatively determine the contributions to the
total free energy density F. However, it does not directly measure
the remanent magnetization orientation. Moreover, Eq. (1) is
applicable only to homogeneously magnetized samples. This is
valid for FMR measurements, since the applied external field
suffices to fully saturate the magnetization for the present hybrids.
As we are particularly aiming at a magnetization switching at
vanishing external magnetic field, magnetic domain formation
might be important. Therefore, in the following, we utilize super-
conducting quantum interference device (SQUID) magnetometry
measurements as a function of the in-plane magnetic field
orientation θ to directly measure the remanent magnetization
as a function of Vp. For these angular-dependent magnetization
measurements, we magnetized the hybrid along a magnetically
easy 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 voltage
Vp. After the preparation of the magnetization, we recorded the
projection of the magnetization on the magnetic field direction as
a 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 À30 V with the external magnetic
field oriented along θprep ¼ 501, which corresponds to a minimum
of the free energy density F [cf. Fig. 3(b)]. The results obtained by
carrying out angular-dependent magnetometry measurements are
shown in Fig. 4(a). Since in the absence of an external magnetic
field the magnetization preferably aligns along a magnetic easy
axis, maxima in the MðθÞ curves correspond to minima in the FðΘÞ
contours. The respective maxima of the MðθÞ curves are evaluated
in Fig. 4(b), regarding their orientation θmax (black squares) and
magnitude Mmax (green circles) as a function of Vp. Mmax changes
by only 1% in the voltage range À30 V ≤Vp ≤ þ 90 V and thus is
almost independent of Vp. This demonstrates that domain forma-
tion plays only a negligible role in case of α ¼ 01. However, θmax
changes by about 91. This proves that the macroscopic, homo-
geneous remanent magnetization M rotates by about 91 in the film
plane for À30 V ≤Vp ≤ þ 90 V, which confirms the results obtained
by 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)]. As
the free energy density at this orientation continuously evolves
from 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 local
minimum in the following. The angle-dependent SQUID measure-
ments reveal a qualitatively different behavior compared to the
measurements on the hybrid 〈100〉. In case of the hybrid 〈110〉,
both the magnitude Mmax as well as the orientation θmax of the
maximum significantly change as a function of Vp [cf. Fig. 4(d)].
Upon increasing Vp from À30 V to +150 V, Mmax decreases by 49%
of its initial value, while the orientation of Mmax rotates by 201
towards the free energy density minimum at 471 [cf. Fig. 2(f)].
The reduction of Mmax elucidates magnetic domain formation with
increasing Vp. After the magnetic preparation at Vp ¼ À30 V,
the Fe3O4 thin film exhibits a single-domain state with the
-200
0
200
M(kA/m)
50°
60°
1
2
θmax
Mmax(102kA/m)Mmax(102kA/m)
0° 180°
-200
0
200
M(kA/m)
H orientation θ
0 100
120°
130°
1
2
θmax
Vp (V)
360°
hybrid 100
hybrid 110
-30V
+150V
+60V
0V
-30V
+90V
0V
Fig. 4. (Colour online) SQUID magnetometry measurements as a function of θ with
ϕ ¼ 901 at different voltages Vp using the hybrid 〈100〉 (a), (b) and the hybrid 〈110〉
(c), (d). All measurements were carried out at μ0H ¼ 0 mT. The symbols represent
the 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 a
function of Vp.
S. Geprägs et al. / Solid State Communications 198 (2014) 7–1210

5. homogeneous magnetization oriented along Θ ¼ 1331. As the
applied voltage Vp increases, the local energy density minimum
at Θ ¼ 1331 looses depth and thus magnetically hardens, while the
global minimum of the energy density at Θ ¼ 471 magnetically
softens, favoring domain formation [cf. Fig. 3(c)]. Thus, the
angular-dependent magnetization measurements shown in Fig. 4
(c) and (d) are not consistent with the single-domain free energy
density approach used to calculated the energy density contours in
Fig. 3(d).
In a second set of experiments, we repeated the SQUID
measurements with the magnetic preparation field Hprep applied
along θprep ¼ 431, close to the global minimum of the free energy
density at Θ ¼ 471. The experimental data coincide in good appro-
ximation for different applied voltages Vp [not shown here]. The
magnetization M retains its orientation at Θ ¼ 471 independent
of Vp, while the magnitude of the magnetization at this orientation
Mmax changes by only 5% within the full voltage range. Considering
the free energy density contours [cf. Fig. 3(d)], the energy barrier
for domain formation is much larger in this case, such that the
Fe3O4 thin film remains in a magnetically single domain state and
can be described by Eq. (1).
4. Impact of different strain orientations
The experimental results discussed above show that an align-
ment of the strain axes along crystallographic axes might favor
magnetic 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 α ¼ 201 in Fig. 5(a), (b),
and (c).
Starting at ϵ′
2 ¼ 0, we assume an initial magnetization orienta-
tion along Θ ¼ 1351 [point A in Fig. 5(a)]. Upon increasing ϵ′
2 40 in
the thin film, the easy axis continuously rotates, until it switches
discontinuously to point B at a certain critical strain ϵ′;crit
2 ¼ ϵsw
ðαÞ
[green arrow in Fig. 5(a)]. When the strain ϵ′
2 is reduced back to 0,
the easy axis rotates to point C at Θ ¼ 451. Upon subsequently
increasing the strain ϵ′
2 o0 with opposite sign [Fig. 5(b)], the
situation appears qualitatively different from the situation illu-
strated in Fig. 2(g), as we do not observe a back switching to the
initial orientation (point A), but a further switching process along
the original direction of rotation via point D to point E at Θ ¼ À451
[Fig. 5(b)]. Hence, iteratively applying ϵsw
with alternating sign
provides a concept to discontinuously rotate the equilibrium
magnetization 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 a
very 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 be
derived using the free energy density F given in Eq. (1) with
a cubic anisotropy field Kc=Ms ¼ À14:7 mT, which is the averaged
value 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 magnetization
switching process significantly decreases with increasing angle α,
exhibits a minimum at α ¼ 331, and finally slightly increases with α
approaching 451. Overall, ϵsw
has comparatively moderate values
lower than 10À3
, which are experimentally achievable using the
concept described in Fig. 1(a). These values correspond to switch-
ing voltages Vsw
p ðαÞo150 V in our hybrid concept, which are
experimentally accessible [cf. Fig. 5(d)].
To furthermore lower the switching strain ϵsw
, the properties
of the ferromagnetic film itself must be fine-tuned, as ϵsw
linearly
depends on the cubic anisotropy constant Kc and inversely depends
on the magnetostriction constants λ100 and λ111. Most promising
candidates regarding the realization of a magnetization switching
therefore evidently are materials with a small cubic anisotropy and
high 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 room
temperature. The possibility to induce strain along different
directions in the film plane with respect to the crystallographic
axes depending on the cementing procedure, allows to investigate
the switching behavior and particularly to take advantage of the
magnetostriction constants λ along different crystalline orienta-
tions. We have discussed the qualitatively different switching
behavior for two different configurations, namely strain exerted
along the in-plane crystalline Fe3O4 〈100〉 and along the in-plane
〈110〉 directions. The free energy density of the ferromagnetic thin
films was determined by FMR spectroscopy, which allows to infer
the equilibrium magnetization orientation in a Stoner–Wohlfarth
model. The results show a rotation of the easy axes by a few
degrees and a significant modification of the relative strength of
the easy axes, respectively. However, in combination with SQUID
0 5 10 15 20 25 30 35 40 45
0.00
0.02
0.04
0.06
0.08
0.10
εsw(%)
α (°)
0
20
40
60
80
100
120
140
V(V)
sw
p
F(arb.units)
0° 90°
F(arb.units)
M orientation Θ
-90° 0° 90° 180° 270°
M orientation Θ
χ=0.27
χ=1.0
A
E
B
D
C
α=20°
C
A
B
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
/2
ε 2
=-εmax
ε 2
=0
α=20°
α=20°
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 α ¼ 201.
The full yellow line in (c) traces the minimum of F. Discontinuous switching
processes of the magnetization by 901 from A-C-E while applying ϵsw
with
alternating sign are visible. (d) Calculated critical strain ϵsw
ðαÞ values and the
corresponding switching voltages Vsw
p ðαÞ at which a discontinuous switching of the
magnetization, which is illustrated by the green arrows in (a) and (b), occurs. The
blue curve depicts perfect strain transmission ðχ ¼ 1:0Þ and the black curve
represents the experimentally realized strain in the hybrid 〈100〉ð χ ¼ 0:27Þ.
S. Geprägs et al. / Solid State Communications 198 (2014) 7–12 11

6. magnetometry measurements we find that the angle of magneti-
zation reorientation is not large enough to induce a magnetization
switching in the former, and magnetic domain formation impedes
a coherent magnetization switching in the latter approach.
This shows that inefficient strain transfer and magnetic domain
formation are major obstacles towards a non-volatile strain-con-
trolled magnetization switching. Using the experimental free
energy and strain transfer parameters, we find in simulations that
skillful alignment of the strain within the films should reduce the
strain values required to switch the magnetization and impede
domain formation. More explicitly, our experiments suggest that
an all-voltage-controlled, nonvolatile magnetization switching at
room temperature and zero magnetic field should be possible.
Acknowledgments
Financial support via DFG Project no. GO 944/3-1 and the
German Excellence Initiative via the “Nanosystems Initiative
Munich (NIM)” are gratefully acknowledged.
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