Reduction of energy cost of magnetization switching in a biaxial nanoparticle by use of internal dynamics Mohammad H.A. Badarneh1Grzegorz J. Kwiatkowski1and Pavel F. Bessarab1 2

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Reduction of energy cost of magnetization switching in a biaxial nanoparticle by use

of internal dynamics

Mohammad H.A. Badarneh,1, ∗Grzegorz J. Kwiatkowski,1and Pavel F. Bessarab1, 2

1Science Institute of the University of Iceland, 107 Reykjavík, Iceland

2Department of Physics and Electrical Engineering, Linnaeus University, SE-39231 Kalmar, Sweden

(Dated: June 21, 2023)

A solution to energy-eﬃcient magnetization switching in a nanoparticle with biaxial anisotropy is

presented. Optimal control paths minimizing the energy cost of magnetization reversal are calculated

numerically as functions of the switching time and materials properties, and used to derive energy-

eﬃcient switching pulses of external magnetic ﬁeld. Hard-axis anisotropy reduces the minimum

energy cost of magnetization switching due to the internal torque in the desired switching direction.

Analytical estimates quantifying this eﬀect are obtained based on the perturbation theory. The

optimal switching time providing a tradeoﬀ between fast switching and energy eﬃciency is obtained.

The energy cost of switching and the energy barrier between the stable states can be controlled

independently in a biaxial nanomagnet. This provides a solution to the dilemma between energy-

eﬃcient writability and good thermal stability of magnetic memory elements.

I. INTRODUCTION

Identiﬁcation of energy limits for the control of mag-

netization is an important fundamental problem of con-

densed matter physics. It is also a prerequisite for the de-

velopment of energy-eﬃcient technologies based on mag-

netic materials. An important application is magnetic

memory where writing of data is realized via selective

magnetization reversals in nanoelements. Magnetization

reversal can be achieved by various means, including op-

tical pulses [1–3], spin-polarized electric current [4,5],

external magnetic [6–9] and electric ﬁeld [10], microwave-

assisted reversal switching [11–13], stress [14], tempera-

ture gradient [15,16], etc. The challenge is to minimize

the energy cost of the control stimulus generation.

In conventional bit recording, magnetization reversal

in a memory element is achieved by applying a static ex-

ternal magnetic ﬁeld in an opposite direction to the initial

magnetization. This results in a relatively slow reversal

process governed by damping as long as the magnitude of

the external ﬁeld exceeds the coercive ﬁeld [17,18]. The

coercive ﬁeld and, thereby, the energy cost of switching

can be reduced by decreasing the magnetic anisotropy,

but this may lead to unwanted reversals induced by

thermal ﬂuctuations due to decrease in the energy bar-

rier separating the stable states. One solution to this

dilemma between good thermal stability and energy-

eﬃcient writability of magnetic elements for memory ap-

plications is use of exchange spring magnets [19], where

the energy barrier and the coercive ﬁeld can be tuned

independently.

Decrease in the switching time and/or the switching

ﬁeld can also be achieved via realization of special rever-

sal protocols such as precessional magnetization switch-

ing [20]. Precessional switching is typically induced by

applying a magnetic ﬁeld pulse transverse to the initial

∗Corresponding author: mha5@hi.is

magnetization, but the pulse duration must be chosen

accurately so as to avoid back switching [21]. Addition-

ally, precessional switching is prone to instabilities due to

the magnetization ringing eﬀect [22] unless the switching

pulse is properly shaped [22–24]. In microwave-assisted

reversals, the switching ﬁeld can also be reduced thanks

to resonant energy pumping [11–13,25].

Clearly, the possibility to achieve the reversal by sev-

eral diﬀerent methods implies the existence of an optimal

protocol, but its deﬁnite identiﬁcation is a challenging

problem. Barros et al. employed the optimal control

theory (OCT) [26] to establish a formal approach to the

magnetization switching optimization [27,28]. Within

the approach, the optimal switching pulse is found as a

result of a direct minimization of the switching cost func-

tional under the constraint deﬁned by a system-speciﬁc

magnetization dynamics. In our previous article, we re-

visited the OCT due to Barros et al. using unconstrained

minimization, which helped us ﬁnd a complete analytical

solution to the energy-eﬃcient reversal of a nanomagnet

with uniaxial anisotropy [29].

We also reported decrease in the switching cost for sys-

tems with biaxial anisotropy, the result of the internal

torque produced by the hard axis [29]. That the internal

torque can assist magnetization reversal was recognized

earlier for several systems, for example for Co ﬁlms [30]

and Co nanoclusters [31]. The aim of the present study

is to explore this eﬀect quantitatively. We focus on nano-

magnets with biaxial anisotropy, which can arise due to

the demagnetizing ﬁeld [32]. This scenario is realized in

ﬂat elongated nanoelements; see Fig. 1. Such systems are

used, e.g., as single bits in in-plane memory [33], or as

elements of artiﬁcial spin ice arrays [34,35].

We investigate by means of the OCT to what extent

the energy cost of magnetization switching can be min-

imized by pulse shaping and how this depends on the

parameters of the biaxial system and the switching time.

Thanks to the internal torque generated by the hard-axis

anisotropy, the energy cost can be reduced below the free-

arXiv:2210.13514v2 [physics.comp-ph] 20 Jun 2023

Figure 1. Optimal switching of a ﬂat elongated nanomagnet

representing a biaxial anisotropy system. The direction of

the normalized magnetic moment ⃗s is shown with the blue

arrow. Orientations of ⃗s that correspond to the minima and

the saddle points on the energy surface are marked with the

green and magenta crosses, respectively. The calculated opti-

mal control paths between the energy minima are shown with

the solid and the dashed green lines. The damping factor αis

0.1, the switching time Tis 8τ0, and the hard-axis anisotropy

constant is twice as large as the easy-axis anisotropy constant.

The green arrows along the reversal paths show the velocity

of the system at t=T/6,t=T/3,t=T/2,t= 2T/3, and

t= 5T/6, with the size of the arrowheads being proportional

to the magnitude of the velocity. The contours of constant

azimuthal angle φ(meridians) and polar angle θ(parallels)

are shown with thin black lines.

macrospin level. Based on the perturbation theory, we

show some analytical estimates of the energy cost reduc-

tion. We show that in a biaxial system the energy barrier

separating the stable states and energy cost of switching

between them can be tuned independently, which pro-

vides a solution to the magnetic recording dilemma.

The article is organized as follows. Sec. II provides

a theoretical framework for energy-eﬃcient control of

magnetization by means of external magnetic ﬁeld: In

Sec. II A, the OCT for magnetic systems is presented

and the corresponding Euler-Lagrange equation for the

optimal control path (OCP), a dynamical trajectory min-

imizing the energy cost of magnetization switching, is

derived; In Sec. II B, the numerical method for ﬁnding

OCPs and corresponding energy-eﬃcient control pulses

via direct minimization of the cost functional is pre-

sented; In Sec. II C, a method for ﬁnding an approxi-

mate solution for the minimum energy cost is worked out

based on the perturbation theory. The application of the

methodology to a biaxial anisotropy system is presented

in Sec. III. Conclusions and discussion are presented in

Sec. IV.

II. METHODOLOGY

A. Optimal control theory

We deﬁne the cost of the magnetization switching as

the amount of energy used to generate the control pulse

that produces the desired change in the magnetic struc-

ture of the system. Assuming the control to be an exter-

nal magnetic ﬁeld generated by an electric circuit, the en-

ergy cost is mostly deﬁned by Joule heating due to the re-

sistance of the circuit. This is proportional to the square

of the electric current integrated over the switching time.

Taking into account the linear relationship between the

current magnitude and the strength of the generated

ﬁeld, the cost functional can be written as [27,29,36]

Φ = ZT

|⃗

B(t)|2dt, (1)

where Tis the switching time and ⃗

B(t)is the generated

external magnetic ﬁeld at time t. The aim of the OCT is

to identify the optimal pulse ⃗

Bm(t)that brings the sys-

tem to the desired ﬁnal state such that Φis minimized.

Whenever thermal ﬂuctuations are negligible, the sys-

tem dynamics can be described by the Landau-Lifshitz-

Gilbert (LLG) equation [37]:

1 + α2˙

⃗s =−γ⃗s ×⃗

b+⃗

B−αγ⃗s ×h⃗s ×⃗

b+⃗

Bi,(2)

where ⃗s is the normalized magnetic moment vector, γ

is the gyromagnetic ratio, αis the damping factor, and

⃗

bis the internal magnetic ﬁeld deﬁned by the magnetic

conﬁguration through the following equation:

⃗

b=⃗

b(⃗s) = −1

∂E

∂⃗s (3)

with µbeing the magnetic moment length and Ethe

internal energy of the system.

Both ⃗

B(t)and ⃗s(t)can be treated as independent vari-

ables, and Φminimized subject to the constraint deﬁned

by Eq. (2) [27,28]. Alternatively, the optimal pulse

⃗

Bm(t)can be calculated via unconstrained minimization

of Φ. For this, Eq. (2) is used to express the external

magnetic ﬁeld in terms of the dynamical trajectory and

the internal magnetic ﬁeld [29]:

⃗

B(⃗s, ˙

⃗s) = α

γ˙

⃗s +1

γh⃗s ×˙

⃗si−⃗

b⊥,(4)

with ⃗

b⊥=⃗

b−⃗s⃗

b·⃗sbeing the transverse component of

the internal ﬁeld, and the result substituted into Eq. (1).

Subsequently, the energy cost Φbecomes a functional of

the switching trajectory ⃗s(t):

Φ = Φ[⃗s(t)] = ZT

A(⃗s, ˙

⃗s)dt, (5)

where A(⃗s, ˙

⃗s)is given by

A(⃗s, ˙

⃗s) = α2+ 1

γ2|˙

⃗s|2−2α

γ˙

⃗s ·⃗

b⊥−2

γ⃗s ×˙

⃗s·⃗

b⊥+|⃗

b⊥|2.

(6)

The optimal reversal mechanism can be found by mini-

mizing Φwith respect to path connecting the initial and

the ﬁnal state in the conﬁguration space. Correspond-

ing OCP ⃗sm(t)can be identiﬁed by solving the Euler-

Lagrange equation:

⃗s ·⃗

bˆ

I−1

µˆ

H1

γ⃗s ×˙

⃗s −⃗

b⊥

µ⃗s ·ˆ

H1

γ⃗s ×˙

⃗s −⃗

b⊥⃗s

−1 + α2

γ2h¨

⃗s −⃗s ·¨

⃗s⃗si+1

µγ ⃗s ×ˆ

H˙

⃗s = 0

(7)

supplemented by the boundary conditions deﬁned by the

initial and the ﬁnal orientation of the magnetic moment.

Here, ˆ

Iis a 3×3identity matrix and ˆ

His the matrix

of second derivatives of the energy Ewith respect to

components of the magnetic moment sx,sy,sz. Note

that Eq. (7) is derived under the constraint |⃗s|= 1. The

optimal switching pulse is found upon substituting the

OCP into Eq. (4).

It is not possible to ﬁnd a general analytical solution

to the Euler-Lagrange equation except for special cases

where the symmetries of the system make it possible to

simplify the problem. For example, for a free magnetic

moment (E= 0) Eq. (7) simpliﬁes to:

⃗s −⃗s ·¨

⃗s⃗s = 0,(8)

and the solution is a constant-speed rotation over the

shortest distance between the initial and ﬁnal states.

The corresponding energy cost Φffor reversing of a free

macrospin reads

Φf=π2(1 + α2)/(γ2T).(9)

Another case with a fully analytical solution is the re-

versal of a macrospin with uniaxial anisotropy [29]. Be-

cause of the rotational symmetry of the problem, the

separation of variables in the spherical coordinate sys-

tem is possible if the z-direction is chosen to be along the

anisotropy axis. This leads to a well-known Sine-Gordon

equation for the polar angle θof the magnetic moment

and makes the azimuthal angle φcompletely deﬁned by

θ(see Fig. 1for the deﬁnition of θand φ):

τ2

0¨

θ=α2

4(1 + α2)2sin 4θ, τ0˙φ=cos θ

1 + α2,(10)

Figure 2. Illustration of the midpoint scheme used in the

numerical method for ﬁnding OCPs. Two images ⃗spand ⃗sp+1

are connected by a geodesic path in the conﬁguration space.

The position ⃗sp+1

2and the velocity ˙

⃗sp+1

2at the midpoint of

the path are deﬁned by ⃗spand ⃗sp+1, and the angle δpbetween

them.

where τ0=µ/(2γK)deﬁnes the timescale, and Kis the

anisotropy constant. Solution of Eq. (10) is explicitly ex-

pressed in terms of the Jacobi amplitude [29]; it describes

a superposition of the steady rotation of the moment be-

tween the energy minima and its precession around the

anisotropy axis, where the precession direction reverses

when the system reaches the top of the energy barrier.

The corresponding optimal switching ﬁeld rotates syn-

chronously with the magnetic moment in such a way that

it generates the torque only in the direction of increasing

θ[29]. The amplitude of the optimal switching ﬁeld re-

mains constant over time when α= 0, but it exhibits a

maximum (minimum) before (after) crossing the energy

barrier for α > 0[29]. The optimal switching ﬁeld is al-

ways perpendicular to the magnetic moment, see Eq. (4).

Nevertheless, most cases are impossible to solve ana-

lytically, and numerical methods for ﬁnding OCPs are

required. One such method is presented in the following.

B. Numerical calculation of optimal control paths

We ﬁnd OCPs numerically via the direct minimization

of the cost functional. For this, we discretize Φusing the

midpoint rule [36]:

Φ[⃗s(t)] ≈Φ[s] =

p=0

|⃗

Bp+1

2|2(tp+1 −tp),(11)

where {tp}is a partition of the time interval [0, T ]such

that 0 = t0< t1< . . . < tQ+1 =T. Here, the partition

has a regular spacing, i.e. tp+1 −tp= ∆t=T/(Q+1),p=

0, . . . , Q. A switching trajectory ⃗s(t)is represented by a

polygeodesic line connecting Q+ 2 points, referred to as

‘images’: ⃗s(t)→ {⃗s0, ⃗s1, ..., ⃗sQ+1}, with ⃗sp=⃗s(tp). The

ﬁrst image ⃗s0and the last image ⃗sQ+1 correspond to the

initial and the ﬁnal orientation of the magnetic moment,

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ReductionofenergycostofmagnetizationswitchinginabiaxialnanoparticlebyuseofinternaldynamicsMohammadH.A.Badarneh,1,∗GrzegorzJ.Kwiatkowski,1andPavelF.Bessarab1,21ScienceInstituteoftheUniversityofIceland,107Reykjavík,Iceland2DepartmentofPhysicsandElectricalEngineering,LinnaeusUniversity,SE-39231Kalmar,S...

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Reduction of energy cost of magnetization switching in a biaxial nanoparticle by use of internal dynamics Mohammad H.A. Badarneh1Grzegorz J. Kwiatkowski1and Pavel F. Bessarab1 2

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