Manuscript-Mechanical Engineering

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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

[email protected]

Charles Hansen Prof.Dr.

Technical University of Denmark

[email protected]

Jrg Schrder Prof.Dr.

Institute of Mechanics, Universitt [email protected]

Min-Zhong Wang Prof.Dr.

Department of Mechanics and Aerospace Engineering, Peking University

[email protected]

Ralf Mller Prof.Dr.

Institute of Applied Mechanics, Technische Universitt Kaiserslautern

[email protected]

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

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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 C44 75.5 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

[1] K. Kirk, Nanomagnets for sensors and data storage, Contemporary Physics 41 (2) (2000) 6178. doi:10.1080/

001075100181187.

[2] E. A. Dobisz, Z. Z. Bandic, T.-W. Wu, T. Albrecht, Patterned media: nanofabrication challenges of future disk drives,

Proceedings of the IEEE 96 (11) (2008) 18361846. doi:10.1109/JPROC.2008.2007600.

16
http://dx.doi.org/10.1080/001075100181187http://dx.doi.org/10.1080/001075100181187http://dx.doi.org/10.1080/001075100181187http://dx.doi.org/10.1109/JPROC.2008.2007600http://dx.doi.org/10.1109/JPROC.2008.2007600http://dx.doi.org/10.1109/JPROC.2008.2007600http://dx.doi.org/10.1080/001075100181187http://dx.doi.org/10.1080/001075100181187


21/23

[3] N. Thiyagarajah, M. Asbahi, R. T. J. Wong, K. W. M. Low, N. L. Yakovlev, J. K. W. Yang, V. Ng, A facile approach

for screening isolated nanomagnetic behavior for bit-patterned media, Nanotechnology 25 (22) (2014) 225203. doi:

10.1088/0957-4484/25/22/225203 .[4] S. Piramanayagam, Perpendicular recording media for hard disk drives, Journal of Applied Physics 102 (1) (2007) 011301.

doi:10.1063/1.2750414.

[5] 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, et al.,

Electric-field control of local ferromagnetism using a magnetoelectric multiferroic, Nature Materials 7 (6) (2008) 478482.

doi:10.1038/nmat2184.

[6] M. Fechner, P. Zahn, S. Ostanin, M. Bibes, I. Mertig, Switching magnetization by 180 with an electric field, Physical

Review Letters 108 (19) (2012) 197206. doi:10.1103/PhysRevLett.108.197206.

[7] T. Wu, A. Bur, K. Wong, P. Zhao, C. S. Lynch, P. K. Amiri, K. L. Wang, G. P. Carman, Electrical control of reversible

and permanent magnetization reorientation for magnetoelectric memory devices, Applied Physics Letters 98 (26) (2011)

262504. doi:10.1063/1.3605571.

[8] C. A. Vaz, Electric field control of magnetism in multiferroic heterostructures, Journal of Physics: Condensed Matter

24 (33) (2012) 333201. doi:10.1088/0953-8984/24/33/333201.

[9] P. Shabadi, A. Khitun, K. Wong, P. K. Amiri, K. L. Wang, C. A. Moritz, Spin wave functions nanofabric update, in:

Proceedings of 2011 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH), 2011, pp. 107113.

doi:10.1109/NANOARCH.2011.5941491.

[10] K. Wang, J. Alzate, P. K. Amiri, Low-power non-volatile spintronic memory: Stt-ram and beyond, Journal of Physics D:

Applied Physics 46 (7) (2013) 074003. doi:10.1088/0022-3727/46/7/074003.

[11] S. Mangin, Y. Henry, D. Ravelosona, J. Katine, E. E. Fullerton, Reducing the critical current for spin-transfer switching

of perpendicularly magnetized nanomagnets, Applied Physics Letters 94 (1) (2009) 012502. doi:10.1063/1.3058680.

[12] E. Myers, D. Ralph, J. Katine, R. Louie, R. Buhrman, Current-induced switching of domains in magnetic multilayer

devices, Science 285 (5429) (1999) 867870. doi:10.1126/science.285.5429.867.[13] A. Khan, D. E. Nikonov, S. Manipatruni, T. Ghani, I. A. Young, Voltage induced magnetostrictive switching of nano-

magnets: Strain assisted strain transfer torque random access memory, Applied Physics Letters 104 (26) (2014) 262407.

doi:10.1063/1.4884419.

[14] J. Wang, G.-P. Li, T. Shimada, H. Fang, T. Kitamura, Control of the polarity of magnetization vortex by torsion, Applied

Physics Letters 103 (24) (2013) 242413. doi:10.1063/1.4847375.

[15] K. Roy, S. Bandyopadhyay, J. Atulasimha, Switching dynamics of a magnetostrictive single-domain nanomagnet subjected

to stress, Physical Review B 83 (22) (2011) 224412. doi:10.1103/PhysRevB.83.224412.

[16] K. Roy, S. Bandyopadhyay, J. Atulasimha, Energy dissipation and switching delay in stress-induced switching of

multiferroic nanomagnets in the presence of thermal fluctuations, Journal of Applied Physics 112 (2) (2012) 023914.

doi:10.1063/1.4737792.

[17] S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, V. Preobrazhensky, Thermal effects in magnetoelectric memories with

stress-mediated switching, Journal of Physics D: Applied Physics 46 (32) (2013) 325002. doi:10.1088/0022-3727/46/32/

325002.

[18] Y. Shu, M. Lin, K. Wu, Micromagnetic modeling of magnetostrictive materials under intrinsic stress, Mechanics of Mate-

rials 36 (10) (2004) 975997. doi:10.1016/j.mechmat.2003.04.004.

[19] K. Singh, C. Tipton, E. Han, T. Mullin, Magneto-elastic buckling of an euler b eam, Proceedings of the Royal Society A:

Mathematical, Physical and Engineering Science 469 (2155) (2013) 20130111. doi:10.1098/rspa.2013.0111.

[20] J. Li, Y. Ma, Magnetoelastic modeling of magnetization rotation and variant rearrangement in ferromagnetic shape

memory alloys, Mechanics of Materials 40 (12) (2008) 10221036. doi:10.1016/j.mechmat.2008.06.003.

17
http://dx.doi.org/10.1088/0957-4484/25/22/225203http://dx.doi.org/10.1088/0957-4484/25/22/225203http://dx.doi.org/10.1088/0957-4484/25/22/225203http://dx.doi.org/10.1063/1.2750414http://dx.doi.org/10.1063/1.2750414http://dx.doi.org/10.1038/nmat2184http://dx.doi.org/10.1038/nmat2184http://dx.doi.org/10.1103/PhysRevLett.108.197206http://dx.doi.org/10.1103/PhysRevLett.108.197206http://dx.doi.org/10.1063/1.3605571http://dx.doi.org/10.1063/1.3605571http://dx.doi.org/10.1088/0953-8984/24/33/333201http://dx.doi.org/10.1088/0953-8984/24/33/333201http://dx.doi.org/10.1109/NANOARCH.2011.5941491http://dx.doi.org/10.1109/NANOARCH.2011.5941491http://dx.doi.org/10.1088/0022-3727/46/7/074003http://dx.doi.org/10.1088/0022-3727/46/7/074003http://dx.doi.org/10.1063/1.3058680http://dx.doi.org/10.1063/1.3058680http://dx.doi.org/10.1126/science.285.5429.867http://dx.doi.org/10.1126/science.285.5429.867http://dx.doi.org/10.1063/1.4884419http://dx.doi.org/10.1063/1.4884419http://dx.doi.org/10.1063/1.4847375http://dx.doi.org/10.1063/1.4847375http://dx.doi.org/10.1103/PhysRevB.83.224412http://dx.doi.org/10.1103/PhysRevB.83.224412http://dx.doi.org/10.1063/1.4737792http://dx.doi.org/10.1063/1.4737792http://dx.doi.org/10.1088/0022-3727/46/32/325002http://dx.doi.org/10.1088/0022-3727/46/32/325002http://dx.doi.org/10.1088/0022-3727/46/32/325002http://dx.doi.org/10.1016/j.mechmat.2003.04.004http://dx.doi.org/10.1016/j.mechmat.2003.04.004http://dx.doi.org/10.1098/rspa.2013.0111http://dx.doi.org/10.1098/rspa.2013.0111http://dx.doi.org/10.1016/j.mechmat.2008.06.003http://dx.doi.org/10.1016/j.mechmat.2008.06.003http://dx.doi.org/10.1016/j.mechmat.2008.06.003http://dx.doi.org/10.1098/rspa.2013.0111http://dx.doi.org/10.1016/j.mechmat.2003.04.004http://dx.doi.org/10.1088/0022-3727/46/32/325002http://dx.doi.org/10.1088/0022-3727/46/32/325002http://dx.doi.org/10.1063/1.4737792http://dx.doi.org/10.1103/PhysRevB.83.224412http://dx.doi.org/10.1063/1.4847375http://dx.doi.org/10.1063/1.4884419http://dx.doi.org/10.1126/science.285.5429.867http://dx.doi.org/10.1063/1.3058680http://dx.doi.org/10.1088/0022-3727/46/7/074003http://dx.doi.org/10.1109/NANOARCH.2011.5941491http://dx.doi.org/10.1088/0953-8984/24/33/333201http://dx.doi.org/10.1063/1.3605571http://dx.doi.org/10.1103/PhysRevLett.108.197206http://dx.doi.org/10.1038/nmat2184http://dx.doi.org/10.1063/1.2750414http://dx.doi.org/10.1088/0957-4484/25/22/225203http://dx.doi.org/10.1088/0957-4484/25/22/225203


22/23

[21] E. Galipeau, P. P. Castaneda, Giant field-induced strains in magnetoactive elastomer composites, Proceedings of the Royal

Society A: Mathematical, Physical and Engineering Science 469 (2158) (2013) 20130385. doi:10.1098/rspa.2013.0385.

[22] A. K. Biswas, S. Bandyopadhyay, J. Atulasimha, Complete magnetization reversal in a magnetostrictive nanomagnet withvoltage-generated stress: A reliable energy-efficient non-volatile magneto-elastic memory, Applied Physics Letters 105 (7)

(2014) 072408. doi:10.1063/1.4893617.

[23] M. S. Fashami, K. Roy, J. Atulasimha, S. Bandyopadhyay, Magnetization dynamics, b ennett clocking and associated

energy dissipation in multiferroic logic, Nanotechnology 22 (15) (2011) 155201. doi:10.1088/0957-4484/22/15/155201.

[24] M. Yi, B.-X. Xu, A constraint-free phase field model for ferromagnetic domain evolution, Proceedings of the Royal Society

A: Mathematical, Physical and Engineering Science 470 (2171) (2014) 20140517. doi:10.1098/rspa.2014.0517.

[25] D. Sander, The correlation between mechanical stress and magnetic anisotropy in ultrathin films, Reports on Progress in

Physics 62 (5) (1999) 809. doi:10.1088/0034-4885/62/5/204.

[26] J. Rothman, M. Klaui, L. Lopez-Diaz, C. Vaz, A. Bleloch, J. Bland, Z. Cui, R. Speaks, Observation of a bi-domain state

and nucleation free switching in mesoscopic ring magnets, Physical Review Letters 86 (6) (2001) 1098. doi:10.1103/

PhysRevLett.86.1098.

[27] H. Szambolics, L. Buda-Prejbeanu, J.-C. Toussaint, O. Fruchart, A constrained finite element formulation for the landau

lifshitzgilbert equations, Computational Materials Science 44 (2) (2008) 253258. doi:10.1016/j.commatsci.2008.03.

019.

[28] G. S. Abo, Y.-K. Hong, J.-H. Park, J.-J. Lee, W. Lee, B.-C. Choi, Definition of magnetic exchange length, IEEE Trans-

actions on Magnetics 49 (8) (2013) 49374939. doi:10.1109/TMAG.2013.2258028.

[29] P. Krone, D. Makarov, M. Albrecht, T. Schrefl, D. Suess, Magnetization reversal processes of single nanomagnets and their

energy barrier, Journal of Magnetism and Magnetic Materials 322 (23) (2010) 37713776. doi:10.1016/j.jmmm.2010.07.

041.

[30] A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, R. Wiesendanger, Direct observation of internal spin

structure of magnetic vortex cores, Science 298 (5593) (2002) 577580. doi:10.1126/science.1075302.[31] Y. Wang, R. Ott, T. Van Buuren, T. Willey, M. Biener, A. Hamza, Controlling factors in tensile deformation of nanocrys-

talline cobalt and nickel, Physical Review B 85 (1) (2012) 014101. doi:10.1103/PhysRevB.85.014101.

18
http://dx.doi.org/10.1098/rspa.2013.0385http://dx.doi.org/10.1098/rspa.2013.0385http://dx.doi.org/10.1063/1.4893617http://dx.doi.org/10.1063/1.4893617http://dx.doi.org/10.1088/0957-4484/22/15/155201http://dx.doi.org/10.1088/0957-4484/22/15/155201http://dx.doi.org/10.1098/rspa.2014.0517http://dx.doi.org/10.1098/rspa.2014.0517http://dx.doi.org/10.1088/0034-4885/62/5/204http://dx.doi.org/10.1088/0034-4885/62/5/204http://dx.doi.org/10.1103/PhysRevLett.86.1098http://dx.doi.org/10.1103/PhysRevLett.86.1098http://dx.doi.org/10.1103/PhysRevLett.86.1098http://dx.doi.org/10.1016/j.commatsci.2008.03.019http://dx.doi.org/10.1016/j.commatsci.2008.03.019http://dx.doi.org/10.1016/j.commatsci.2008.03.019http://dx.doi.org/10.1109/TMAG.2013.2258028http://dx.doi.org/10.1109/TMAG.2013.2258028http://dx.doi.org/10.1016/j.jmmm.2010.07.041http://dx.doi.org/10.1016/j.jmmm.2010.07.041http://dx.doi.org/10.1016/j.jmmm.2010.07.041http://dx.doi.org/10.1126/science.1075302http://dx.doi.org/10.1126/science.1075302http://dx.doi.org/10.1103/PhysRevB.85.014101http://dx.doi.org/10.1103/PhysRevB.85.014101http://dx.doi.org/10.1103/PhysRevB.85.014101http://dx.doi.org/10.1126/science.1075302http://dx.doi.org/10.1016/j.jmmm.2010.07.041http://dx.doi.org/10.1016/j.jmmm.2010.07.041http://dx.doi.org/10.1109/TMAG.2013.2258028http://dx.doi.org/10.1016/j.commatsci.2008.03.019http://dx.doi.org/10.1016/j.commatsci.2008.03.019http://dx.doi.org/10.1103/PhysRevLett.86.1098http://dx.doi.org/10.1103/PhysRevLett.86.1098http://dx.doi.org/10.1088/0034-4885/62/5/204http://dx.doi.org/10.1098/rspa.2014.0517http://dx.doi.org/10.1088/0957-4484/22/15/155201http://dx.doi.org/10.1063/1.4893617http://dx.doi.org/10.1098/rspa.2013.0385


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