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Elsevier Editorial System(tm) for Mechanics of Materials
Manuscript Draft
Manuscript Number: MOM-14-406
Title: Mechanically induced deterministic 180^o switching in nanomagnets
Article Type: Research Paper
Keywords: mechanical-magnetic coupling; 180^o switching; nanomagnets; ferromagnetic materials
Corresponding Author: Dr. Min Yi,
Corresponding Author's Institution: Technische Universitt Darmstadt
First Author: Min Yi
Order of Authors: Min Yi; Bai-Xiang Xu, Prof.Dr.; Dietmar Gross, Prof.Dr.
Manuscript Region of Origin: GERMANY
Suggested Reviewers: George Weng Prof.Dr.
Department of Mechanical & Aerospace Engineering, Rutgers University
Charles Hansen Prof.Dr.
Technical University of Denmark
Jrg Schrder Prof.Dr.
Institute of Mechanics, Universitt [email protected]
Min-Zhong Wang Prof.Dr.
Department of Mechanics and Aerospace Engineering, Peking University
Ralf Mller Prof.Dr.
Institute of Applied Mechanics, Technische Universitt Kaiserslautern
Opposed Reviewers:
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Author Statement
For the submitted original manuscript, entitled Mechanically induced deterministic 180o
switching in nanomagnets,
(a) it is notconcurrently submitted for publication elsewhere;
(b) the paper, in its entirety, in part, or in a modified version, has notbeen published elsewhere;
(c) the paper has notpreviously been submitted for possible publication elsewhere.
Authors: Min Yi, Bai-Xiang Xu, Dietmar Gross
thor Statement
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phical Abstract (for review)
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A constraint-free phase model is used for magneto-mechanically coupled
nanomagnets.
Mechanically induced magnetization switching dynamics is readily captured.
Size dependence of mechanically induced switching mode is identified.
Mechanically driven/assisted 180oswitching of nanomagnets is indeed feasible.
180oswitching under combined mechanical loading and magnetic field is explored.
hlights (for review)
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Mechanically induced deterministic 180 switching in nanomagnets
Min Yia,b,
, Bai-Xiang Xub,
, Dietmar Grossc
aSchool of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Xueyuan Road 37,
Beijing 100191, ChinabMechanics of Functional Materials Division, Institute of Material Science, Technische Universitat Darmstadt,
Jovanka-Bontschits-Strasse 2, Darmstadt 64287, GermanycDivision of Solid Mechanics, Technische Universitat Darmstadt, Franziska-Braun-Strasse 7, Darmstadt 64287, Germany
Abstract
The mechanically induced magnetization switching in nanomagnets is studied by a constraint-
free phase field model, which permits exactly constant magnetization magnitude and explicit
magneto-mechanical coupling. Depending on the geometry of the nanomagnets, there exist
two distinct switching modes: one is the coherent mode where the magnetization vector
remains homogeneous during the switching, and the other is the incoherent mode where
heterogeneous magnetization distribution occurs. For the application of nanomagnets-based
logic and memory devices, the coherent mode is of great interest. Results show that a deter-
ministic 180 switching can happen if mechanical loading is removed once the magnetization
rotates to the largest switching angle. The switching time decreases with the magnitude of
the applied strain. In addition, the 180 switching under a combination of magnetic field and
mechanical strain is also investigated. Simulations demonstrate that an optimum additional
strain to reduce the switching time is around 0.2%. This work provides a foundation for the
study of mechanically driven/assisted nanomagnets-based logic and memory devices.
Keywords: phase field model, mechanical-magnetic coupling, 180 switching, nanomagnets, ferromagnetic
materials
1. Introduction
Nanomagnets in the form of individual bits have shown huge potential for the bit pat-
terned media (BPM) concept in the magnetic data storage[13]. In the BPM concept, each
Corresponding authorEmail addresses: [email protected] (Min Yi), [email protected] (Bai-Xiang Xu),
[email protected] (Dietmar Gross)
Preprint submitted to Mechanics of Materials October 16, 2014
nuscript
ck here to view linked References
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magnetic entity can save one single bit of information [2]. The technology based on this
concept is believed to be able to overcome the areal density limitation of the current hard
disk drives which are constituted by perpendicular granular media [4]. Within the BPM
concept, the nanomagnets-based logic and memory applications require a 180 switching of
the magnetization, i.e. the nanomagnet changes its bit state from 0 to 1, or vice versa.
Thus, in order to achieve reliable and stable performance, a deterministic 180 switching is
indispensable [5,6].
A few approaches have been used to switch the magnetization in nanomagnets. As the
simplest way, a magnetic field with sufficient intensity can induce a deterministic 180 switch-
ing. By using the coupling between the electric field and the magnetic order in multiferroic
heterostructures, a deterministic 90 switching can be induced by an electric field [68]. But
experiments showed that 180 switching occurs only when the electric field is applied in
the plane of the multiferroic layers [9]. Furthermore, through the spin-transfer torque, a
spin-polarized current can also result in a deterministic 180 switching [1013]. However,
the critical current density for the switching is as high as 106 A/cm2, which leads tosignificant energy dissipation [13].
A more promising alternative is to switch nanomagnets by mechanical loading[1417],
thanks to the magneto-mechanical coupling [1821]. It has been demonstrated that switch-
ing a nanomagnet by mechanical loading dissipates much less energy than by a magnetic
field or a spin-polarized current[15,22,23]. Thus, mechanically mediated switching shows
potential applications in low-energy nanomagnets-based logic and memory devices. How-
ever, the mechanism of a deterministic 180 switching by mechanical loading requires further
study. Take a nanomagnet with uniaxial anisotropy as an example. Assume that the initial
magnetization is parallel to the easy axis. It is known that mechanical loading, compression
or tension along the easy axis, can switch the magnetization by 90. After removal of the
mechanical loading, the magnetization rotates further and becomes either parallel or an-
tiparallel to the easy axis, with equal probability in the both cases. Thus, a deterministic
180 switching cannot be achieved.
In order to obtain a deterministic 180 switching, Khan et al [13] have tried recently
the combination of mechanical loading and spin-polarized current. Royet al. [15] used an
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ellipsoid as a special example and showed that a 180 switching can happen after the re-
moval of the mechanical loading. It should be noted that results in both work were based on
micromagnetics calculations, where only the anisotropy energy and the magneto-mechanical
coupling energy were considered, and the exchange energy, the elastic energy, and the mag-
netostatic energy were ignored. However, the exchange energy and the magnetostatic energy
play important roles in determining the magnetization state in nanomagnets. Furthermore,
due to the neglect of elastic energy, the micromagnetics calculations cannot take into ac-
count the mechanical boundary condition on nanomagnets, which can be influential in the
switching mechanism.
In this paper, we use a fully coupled constraint-free phase field model to study the me-
chanically induced switching dynamics in nanomagnets. The phase field model allows ex-
plicit magneto-mechanical coupling, mechanical boundary conditions, and a straightforward
realization of the constraint on a constant magnetization magnitude. In the analysis, the
exchange energy, the anisotropy energy, the elastic energy, the magneto-mechanical energy,
and the magnetostatic energy are all incorporated. The magnetization switching mecha-
nism in hexagonal close-packed (hcp) Cobalt nanomagnets of regular cuboidal shape with
different aspect ratios are investigated. The transition between the coherent and incoherent
switching modes is identified with respect to the geometry of the cuboid. It is found that in
the incoherent regime, mechanical loading and unloading cannot result in 90 switching of
all the magnetization vectors. But in the coherent regime, once the switching angle reaches
the possible largest value during the mechanical loading, mechanical unloading can result
in a deterministic 180 switching. The 180 switching by mechanical loading and unloading
combined with a magnetic field is also studied. The time of 180 switching and its optimum
value are explored. The study should provide a solid basis for designing nanomagnets with
mechanically driven deterministic 180 switching.
2. Constraint-free phase field model
The constraint-free phase field model, which was developed for the study of ferromagnetic
domain structure in cubic crystal nanomagnets [24], is extended in the present work for the
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investigation of a hexagonal crystal nanomagnet. As an illustrative example, the hcp crystal
Cobalt nanomagnets will be considered.
It is known that when the temperature is far below the Curie point, the magnitude
of the magnetization vector M should be constant, i.e. M = Msm with Ms being the
constant saturation magnetization and mthe magnetization unit vector. In contrast to the
conventional micromagnetic or phase field models, the constraint-free phase field model takes
the polar and azimuthal angles (1, 2), instead of the Cartesian components (m1, m2,m3) of
the magnetization unit vector, as the order parameters. Please see the illustration in Figure
1. In this way, the constraint on the magnitude of the magnetization vector is fulfilled
automatically, and thus no additional numerical treatment on the constraint is required.
Furthermore, the degrees of freedom per node in the finite element implementation are also
reduced by one. For more details, the readers are referred to the previous work [ 24].
The important point of a fully coupled phase field model is to construct the magnetic
enthalpy for a hexagonal crystal. The total magnetic enthalpy of the ferromagnetic material
consists of the pure mechanical contributionHmech, the magneto-elastic coupling contribu-tion
Hmag-ela, the magnetocrystalline anisotropy contribution
Hani, the exchange contribution
Hexc, and the magnetostatic contributionHmag, i.e.
H = Hmech +Hmag-ela +Hani +Hexc +Hmag (1)
For a hcp Cobalt crystal, the pure mechanical contribution can be given as [ 25]
Hmech =12C11(
211+
222)+
1
2C33
233+C121122+C13(1133+2233)+2C44(
223+
231)+(C11C12)212
(2)
whereC11, C12, C33, C13, and C44 are the elastic constants for the hcp Cobalt crystal, andij =
12(ui,j+ uj,i) are the strain components with ui being the mechanical displacement
components. Hereafter, the Latin indices i and j run over the range of 1-3. A comma in a
subscript denotes partial differentiation, for example, ui,j =ui/xj in which xj is the j -th
Cartesian coordinate direction. By using the order parameters (1, 2), the magneto-elastic
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coupling energy can be written as[25]
Hmag-ela
=B1(sin
2
1cos
2
211+ 2 sin
2
1sin 2212+ sin
2
1sin
2
222) + B2sin
2
133
+ B3sin2 1(11+ 22) + B4(sin21sin 223+ sin 21cos 223)
(3)
whereB1, B2, B3, and B4 are the magneto-elastic coupling coefficients for the hcp Cobalt
crystal. The easy axis of the hcp Cobalt crystal is assumed to be along the x3 direction,
and thus the magnetocrystalline anisotropy energy can be given as
Hani =Kusin2 1 (4)
whereKuis the anisotropy constant. The exchange energy can be expressed as the gradient
of magnetization, i.e.
Hexc =Ae(1,j1,j+ sin2 12,j2,j) (5)
whereAe is the exchange stiffness constant. The magnetostatic energy has the form
Hmag = 120HjHj 0Ms(H1sin1cos 2+ H2sin 1sin2+ H3cos 1) (6)
in which0= 4107 H/m is the vacuum permeability and Hj =,j is the magnetic fieldwith being the magnetic scalar potential.
With the above specified magnetic enthalpy, a combination of the configurational force
balance and the second law of thermodynamics leads to a generalized evolution equation for
the order parameters 1 and 2, which takes the form [24]
1
Ms
H,j
,j
1Ms
H
+ ex = 1
0L
t
(7)
in which
L=
sin1
sin1 sin2
1
, (8)
ex is the external field, is the damping coefficient, and 0 = 1.76 1011/(Ts) is thegyromagnetic ratio. The Greek indices and run over the range of 1-2.
In addition to the evolution equation (7), the following mechanical equilibrium equation
and the Maxwells equation which governs the magnetic part are also incorporated in the
model: Hij
,j
= 0 (9)
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and
HHj
,j
= 0 (10)
With the six degrees of freedom as [u1, u2, u3, ,1, 2]T, a 3D nonlinear finite element
implementation is performed to solve equations (7), (9), and (10). It should be noted that
the degrees of freedom are reduced by one when compared to the conventional phase field
model. For more details about the model and its implementation, one is referred to the
previous work[24].
The prism shape of the hcp Cobalt nanomagnet is schematized in Figure 1,with a height
ofh and a square base plane with width ofw. The material parameters are listed in Table1[25, 26]. The choice of the finite element mesh size is based on a physical criterion: the
maximum mesh size must be smaller than the minimum of the exchange length and the
Bloch length[27]. Using the parameters in Table 1, the Bloch length is much larger than
the exchange length which can be estimated as lex=
2Ae/(0M2s ) 5.1 nm[28]. Basedon this, the mesh size in all the simulations is taken as 2.5 nm. Cuboid nanomagnets with
different w and h are simulated under the mechanical loading along the x3 direction. The
mechanical loading is represented by strain in the x3 direction, denoted by for simplicity,as shown in Figure1. It should be noted that when the magnetization is exactly along the
x3direction, the torque induced by mechanical loading along thex3direction will vanish and
no magnetization rotation will occur numerically. In order to avoid this numerical problem,
the initial magnetization is chosen to be almostalong the positive x3 direction with a small
Fig. 1. Schematic of the cuboid hcp Cobalt model under study, with x3 as the easy axis of magnetization.
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angle of1 = 0.01. Simulations are performed to investigate whether the mechanical loading
can switch the magnetization from the initial positive x3 direction to the final negative x3
direction.
Table 1. Material parameters of the hcp Cobalt nanomagnet
Parameter Value
Elastic constant C11 307 GPa
Elastic constant C12 165 GPa
Elastic constant C44 75.5 GPa
Elastic constant C13 103 GPa
Elastic constant C33 358 GPaMagneto-elastic coupling constant B1 -8.1 MPa
Magneto-elastic coupling constant B2 -29 MPa
Magneto-elastic coupling constant B3 28.2 MPa
Magneto-elastic coupling constant B4 29.4 MPa
Anisotropy constant Ku 6.5104 J/m3
Saturation magnetization magnitude Ms 1.424106 A/m
Exchange stiffness constantAe 3.31011 J/m
Damping coefficient 0.01
3. Results and discussions
3.1. Mechanically induced switching mode
The size dependence of the switching dynamics in cuboidal hcp Cobalt nanomagnets
under mechanical loading is investigated. Due to the negative magnetostriction effect,
compressive loading cannot switch magnetization from the above-prescribed initial state.
Therefore, a tensile loading of = 1.0% is used. It is found that when the nanomagnets
are small, coherent switching occurs and all the magnetization vectors rotate to the x1-x2
plane. While in the case of large nanomagnets, incoherent switching occurs and the obtained
stable magnetization distribution is spatially inhomogeneous. As an example, Figure2gives
the switching process of two nanomagnet samples. For the small sample with w= h = 15
nm, all the magnetization vectors rotate coherently, as shown in Figure 2(a). Whereas, for
the relatively large sample with w= 15 nm and h= 30 nm, coherent rotation only exists
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Fig. 2. Temporal evolution of the magnetization vectors under the mechanical loading = 1.0%. (a)
Coherent switching process in the nanomagnet with w=h=15 nm. (b) Incoherent switching process in the
nanomagnet with w=15 nm andh=30 nm.
in the first 10 ns. Afterwards incoherent rotation predominates and the exchange energy
remarkably increases, as shown in Figures2(b) and3(c). Figures3(a) and (b) compare the
temporal evolution curves of magnetization components in these two cases. The plotted
curves for incoherent switching are averaged over the whole sample. It can be seen that
due to the small damping in hcp Cobalt, the magnetization components m1 and m2 oscil-
late violently. In the small nanomagnet, coherent rotation under mechanical loading leads
to a stable homogeneous magnetization state with 1= 90 and 2= 135
, which implies
m3= 0 and m1= m2= 2/2 (Figure 3(a)). On the contrary, in the large nanomagnet,the incoherent rotation leads to a totally different magnetization state, which is featured
by inhomogeneous magnetization and the net magnetization along the x2 direction (Figure
3(b)). This is also different from the magnetic field induced switching, in which the final
net magnetization is parallel to the applied magnetic field in spite of the incoherent rotation
[29].
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Fig. 3. Temporal evolution of the magnetization components in nanomagnets with the geometry of (a)
w=h=15 nm and (b) w=15 nm, h=30 nm. (c) Temporal evolution of the exchange energy. (d) Switching
mode as a function ofw and h.
Furthermore, we performed simulations on a series of nanomagnets with different w and
h to identify the transition between these two distinctive switching modes. Figure 3(d)
depicts the switching modes of the nanomagnets with respect to geometry parameters h
and w, with a transition boundary identified. It can be found that when w 12.5 nm,coherent switching happens for any values ofh, whereas incoherent switching happens when
w>15 nm. For w= 15 nm, h= 22.5 nm becomes the boundary for coherent and incoherent
switching. It seems that the transition boundary is at wtb= 12.5-15 nm, which is nearly
3 times of the exchange length, i.e. wtb 3lex. It should be noted that the experimentalwork has shown that the width of magnetic vortex in a nanostructured film is also around
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3lex [30]. This implies that the coherent switching is preferable when w is smaller than the
vortex width. The coherent switching should be free from the vortex formation, while the
incoherent switching suffers from the tendency to form vortex. Indeed, as shown in Figure
2(b) the exchange energy distribution in the incoherent switching is really accompanied with
the tendency of vortex formation; because the vortex core often initializes at the site whose
exchange energy is high.
Fig. 4. (a) Trajectories of the magnetization vectors under different tensile strain . The inset cubes
accompanied by arrows illustrate the initial and final state of the magnetization. (b) Evolution of the angle1 as a function of time under different . The simulated nanomagnet has the geometry ofw=h=15 nm.
3.2. Mechanically induced deterministic 90 switching
As shown above, in the case of incoherent switching mode, although the net magnetization
component m3 of the nanomagnet as a whole can reach zero, vortex tends to form and only
a portion of magnetization vectors undergo 90 switching. Therefore, we will focus on the
deterministic 90 switching in the case of coherent switching mode. The nanomagnet with
w = h = 15 nm is chosen for the following simulations. The final stable magnetization
state under the mechanical loading is highly dependent on the tensile strain , but this
dependence seems to follow no definite laws, as shown in Figure4(a). For different , the
evolution trajectories of the magnetization which is initially nearly parallel to the x3direction
(i.e. 1 0) are almost the same before it reaches the x1-x2 plane (i.e. 1 = 90). Whenthe magnetization vector approaches the x1-x2 plane, it will rotate and finally stochastically
arrive at four equilibrium states denoted by (1, 2): (90, 45), (90, 135), (90, 225), and
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(90, 315). Figure4(b) shows the temporal evolution of1 as a function of tensile strain
. Results indicate that there exists a critical value c, and strains above this value can
lead to a deterministic 90 switching. From Figure4(b),c can be estimated as 0.16-0.17%.
= 0.16% can only result in a state with 1 45, while = 0.17% can lead to a stable197. However, when
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the switching process is displayed in Figure 6(b). After the mechanical loading is removed,
the nanomagnet is stress-free and the pure mechanical contributionH
mech reduce to zero
abruptly (not plotted here). The deterministic 180 switching is a result of the evolution of
magneto-elastic energy and anisotropy energy. In the stress-free stage, the magneto-elastic
energy increases while the anisotropy energy decreases. Therefore, during the switching
from 90 to 180, the anisotropy energy is the driving force.
Fig. 6. (a) Mechanical loading history and temporal evolution of magnetization components in the nano-
magnet (w=h=15 nm) subjected to the tensile strain = 1.0%. (b) Temporal evolution of different energy
terms related to magnetization in the nanomagnet (w= h = 15 nm) subjected to loading and subsequent
unloading. The pure mechanical contributionHmech is not included inH
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Fig. 7. (a) Temporal evolution of1 in the nanomagnet (w=h=15 nm) subjected to different mechanical
loading and subsequent unloading. Once 1 reaches its maximum value during the loading, the load is
removed and the nanomagnet is stress-free. (b) The time for 180 switching as a function of strain. The
insets illustrate the switching process.
As shown in Figure 7(a), this 180 switching is also dependent on the applied strain.
When 0.2%, the time for switching from 90 to 180 is even much smaller than thatfor the former 90 switching. While in the case of > 0.2%, the former switching from 0
to 90 is much faster. In Figure7(b), the total time for the deterministic 180 switching
is plotted as a function of the applied strain. It can be seen that in spite of the different
intermediate states achieved in 90 switching, the switching time decreases with the applied
strain, from 620 ns at= 0.17% to 49 ns at= 3.0%. But the decreasing tendency becomes
less notable as the strain increases, and a saturation seems to appear. Although the hcp
Cobalt single crystal can undergo strain more than 3% [31], this mechanically dependent
behavior indicates that a moderate mechanical loading should be sufficient.
3.4. Deterministic 180 switching under mechanical loading combined with a magnetic field
Deterministic 180 switching under the combination of mechanical loading and a magnetic
field is also investigated. The magnetic field is antiparallel to the initial direction of the
magnetization. The magnetic field are applied through the whole time regime, while the
strain is applied merely till the maximum 1 is reached, as it is discussed in the subsection
3.3. Figure 8(a) demonstrates the evolution of magnetization components for two cases:
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one is solely under a magnetic field of 105 A/m, and the other is under the same magnetic
field but with an additional tensile strain of 0.2%. It is apparent that the switching time is
remarkably reduced by the additional strain.
Fig. 8. (a) Temporal evolution of magnetization components in the nanomagnet (w=h=15 nm) subjected
to mechanical loading and magnetic field. (b) The time for 180 switching as s function of mechanical strain
under different magnetic field in the nanomagnet ofw=h=15 nm.
The results which capture the relationship among the switching time, strain, and mag-netic field are summarized in Figure 8(b). In particular, a magnetic field of 0.5105 A/malone cannot switch the magnetization by 180. But the addition of a strain of 0.05% can
accomplish the 180 switching. A compressive strain or a large tensile strain can always
lead to an increase in the switching time. But low tensile strain can shorten the switching
time. The optimum minimum switching time lies around a tensile strain of 0.2%, almost
independent on the magnitude of the magnetic field. This additional strain of 0.2% can
result in a decrement of around 71 ns in the switching time under the magnetic field of 10 5
A/m. This effect becomes less notable as the magnetic field increases. For instance, the
strain reduces the switching time only by 9 ns under a magnetic field of 2105 A/m and by1 ns under a field of 5105 A/m. Hence, application of an additional tensile strain is morefavourable when the magnetic field is relatively low.
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4. Conclusions
The mechanically induced deterministic 180
switching in hcp Cobalt nanomagnets isstudied by a constraint-free phase field model, in which the constraint on the magnetization
magnitude can be strictly satisfied automatically and the magneto-elastic coupling can be
explicitly included. It is found that there exist two switching modes, coherent switching
and incoherent switching. The switching mode is demonstrated to be controlled by the
magnetostatic energy and thus the geometry of the nanomagnet. The transition between
these two switching modes is identified. In the case of coherent switching mode, a deter-
ministic 90
switching is possible when the strain exceeds a critical value. The removal ofthe mechanical loading once the magnetization rotates to the maximum angle can lead to
a deterministic 180 switching. The 180 switching is driven firstly by the magneto-elastic
coupling and then by the magnetocrystalline anisotropy. The time for 180 switching de-
creases with the applied strain and a saturation appears at high strain. The magnetization
switching by the magnetic field combined with the mechanical loading and unloading is also
explored. It is shown that an optimum switching time is obtained when the mechanical
strain is around 0.2%. These results indicate that mechanically driven/assisted 180
switch-ing of nanomagnets is feasible. It should stimulate the study on mechanically switchable
nanomagnets-based logic and memory devices.
Acknowledgmets
The support from the LOEWE research cluster RESPONSE (Hessen, Germany), the
China Scholarship Council , and the Innovation Foundation of BUAA for PhD Graduates
(YWF-14-YJSY-052) is acknowledged.
References
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