Magnetic Switching of a Stoner-Wohlfarth Particle Subjected to a Perpendicular Bias Field

electronics

Article Magnetic Switching of a Stoner-Wohlfarth Particle Subjected to a Perpendicular Bias Field

Dong Xue 1,* and Weiguang Ma 2 1 Department of Physics and Astronomy, Texas Tech University, Lubbock, TX 79409-1051, USA 2 Department of Physics, Umeå University, 90187 Umeå, Sweden; [email protected] * Correspondence: [email protected]; Tel.: +1-806-834-4563

Received: 28 February 2019; Accepted: 21 March 2019; Published: 26 March 2019

Abstract: Characterized by uniaxial magnetic anisotropy, the Stoner-Wohlfarth particle experiences a change in magnetization leading to a switch in behavior when tuned by an externally applied field, which relates to the perpendicular bias component (hperp) that remains substantially small in comparison with the constant switching field (h0). The dynamics of the magnetic moment that governs the magnetic switching is studied numerically by solving the Landau-Lifshitz-Gilbert (LLG) equation using the Mathematica code without any physical approximations; the results are compared with the switching time obtained from the analytic method that intricately treats the non-trivial bias field as a perturbation. A good agreement regarding the magnetic switching time (ts) between the numerical calculation and the analytic results is found over a wide initial angle range (0.01 < θ0 < 0.3), as h0 and hperp are 1.5 × K and 0.02 × K, where K represents the anisotropy constant. However, the quality of the analytic approximation starts to deteriorate slightly in contrast to the numerical approach when computing ts in terms of the field that satisfies hperp > 0.15 × K and h0 = 1.5 × K. Additionally, existence of a comparably small perpendicular bias field (hperp << h0) causes ts to decrease in a roughly exponential manner when hperp increases.

Keywords: Stoner-Wohlfarth particle; Landau-Lifshitz-Gilbert equation; anisotropy; bias ﬁeld

1. Introduction Growing research attention has been paid to controlling the magnetization reversal in ferromagnetic multilayer configurations in the past two decades [1–6]. To primitively describe the magnetic dynamics of single-domain ferromagnets [7], it appears necessary to revisit the Stoner-Wohlfarth (SW) model that was originally proposed by E. C. Stoner and E. P. Wohlfarth in 1948 [8]. Based on their early work, many additional extensions [9–11] of the simplified SW model have undergone intensive investigation motivated by the current progress in computational technology applied to solid-state physics. The concept of “magnetic switching” plays a key role in the reading and writing of information and monitoring the magnetization in magnetic material. In our study, the time-dependent magnetization vector of a particle satisfying the SW model requirement is examined and associated with the applied torque (due to externally applied magnetic fields) that causes magnetic moment reversal to occur. For the crucial importance of investigating the dynamical behavior of magnetization in a SW particle modulated by a significantly small bias field, one notices that discerning how to efficiently manipulate the magnetic state of a primitive single-domain ferromagnet is key to the development of ferromagnetic spintronics. Specifically, there exists two major aspects that motivate our current study. First, the local modulation of magnetization being done by tuning the magnitude of the bias magnetic field should be completed in a more efficient manner than the traditional method which uses electric fields to store information in the computer memory named “dynamic random-access memory” (DRAM), because information storage in such a conventional method is transient and easily lost when turning

Electronics 2019, 8, 366; doi:10.3390/electronics8030366 www.mdpi.com/journal/electronics Electronics 2019, 8, 366 2 of 12 off the computer. Therefore, newer way of memorizing information such as “magnetic random-access memory” (MRAM), which uses long-lasting ferromagnetism and ferroelectricity, is strongly desired. Second, increasing the magnetic switching rates is required by the disk speed and recorded density with respect to the physical equipment in computer memory part. Accordingly, ferromagnetic materials have become increasingly essential and demonstrate many promising practical applications including nano-scale memory devices and magnetic field probes [12–14]. At present, our work discusses the switching behavior of a Stoner-Wohlfarth particle (sometimes called single-domain magnetic grain) for the instance when the external fields that involve the driving field (h0) along with the small bias field (hperp) are simultaneously present. Among the results, the time it takes for the magnetization vector to experience an angle change, which is a bit smaller than π/2, and how the bias field strength (or initial angle parameter) effects the switching time are extensively explored in this paper. There exist many computational techniques that can be used to reveal the magnetic dynamics governed by the Landau-Lifshitz-Gilbert (LLG) equation [15], which was developed after the Landau-Lifshitz (LL) equation [16,17]. In general, many codes utilizing Monte Carlo methods, fast Fourier transformations, etc., have been utilized to obtain a numerical simulation of the magnetization occurring in actual magnetic grains, as well as the time evolution of other physical quantities, namely, the vortex profile, dipole–dipole interaction, etc. [18–20]. In addition to these programs, which aim to acquire the time evolution of magnetic quantities without any physical approximations, one learns that an analytic solution within the framework of the LLG equation provides another valid and effective method to interpret the actual magnetic behavior of various systems [21–23], including our study, which uses appropriate analytic approximations, such as the perturbation theory. Specifically, for our work, we compare the magnetic switching time gained using two different computational methods (numerical and analytical approaches) and examine the apparent agreement (or slight discrepancy) between these approaches. This paper is structured as follows. We briefly introduce the fundamental model (LLG equation) used to explain the time evolution of the magnetic moment in the context of the Bloch equation along with the LL equation, with the goal of establishing the analytic approach (involving appropriate approximations) that applies to the switching time computation. Then, the interplay of the magnetic driving field and the perpendicular bias field occurring in a Stoner-Wohlfarth particle is reviewed and contrasted with our calculated results, which suggest that magnetic switching can be effectively tuned using a significantly small bias field. In the discussion section, the magnetic switching time in the case of axial symmetry (no bias field), as well as in the case of a small perpendicular bias field, is analytically examined, exhibiting fair agreement with the numerically obtained data.

2. Model

2.1. Magnetic Moment Dynamics For the purpose of characterizing the magnetic dynamics that govern the time evolution of the magnetization vector M, the Bloch equation [24] was initially proposed via relation dM/dt = γ0 × M × H, where γ0 described the gyromagnetic ratio while H represented the external ﬁeld. Then, Landau and Lifshitz extended the Bloch equation to the parameter M subjected to the effective ﬁeld (Heff), which arose from the negative functional magnetization derivative with respect to the magnetic energy (−δEm/δM), as well as the dissipation terms, which were related to the loss and relaxation processes in ferromagnetic materials. Consequently, the derived Landau-Lifshitz (LL) equation is as follows:

dM/dt = −γ0 × M × Heff + λ × γ0/|M| × M × (M × Heff) (1) where Landau and Lifshitz assumed that the dissipation was described by a nonlinear term (second term in Equation (1)) of the form restricted by the same effective ﬁeld, while introducing the coefﬁcient Electronics 2019, 8, 366 3 of 12

λElectronicsand taking 2019, 8 into, x FOR account PEER REVIEW the gyromagnetic ratio γ0. The first term on the right-hand side (RHS)3 of of12 Equation (1) describes the dissipationless motion of the moment. Notice that the LL equation conserves theconserves magnetic the moment magnetic | Mmoment|= (M·× |MM|=)1/2 (Mand·× M takes)1/2 and the takes scalar the product scalar of productM and allof M terms, and exceptall terms, for theexcept nonlinear for the termnonlinear appearing term appearin in Equationg in (1), Equation which leads(1), which to M ·leads(dM/d tot )M =·(d 0.M Therefore,/dt) = 0. Therefore, one finds thatone |findsM| =thatMs |=M a| constant,= Ms = a constant, where M swhererepresents Ms represents the magnitude the magnitude of the saturation of the saturation magnetization magnetization under the circumstanceunder the circumstance of a negligible of a negligible second term second in the term RHS in of the Equation RHS of Equation (1). This can(1). beThis illustrated can be illustrated with the examplewith the ofexample a uniaxial of a anisotropicuniaxial anisotropic magnet placed magnet into placed an external into an field external (Heff )field directed (Heff) alongdirected the along same axis.the same The axis. left panel The left of thepanel figure of the below figure shows below that sh withoutows that dissipation, without dissipation, the moment the performsmoment performs a circular motiona circular along motion the parallelsalong the of parallels a unit sphere, of a unit so thatsphere, |M |so = that a constant |M| = a is constant maintained. is maintained. In the absence In the of dissipation,absence of dissipation, the precession the precession of the moment of the persists moment for persists an infinitely for an longinfinitely time andlong is time shown and inis theshown left panelin the ofleft Figure panel1 .of Figure 1.

Figure 1. Dynamics of the magnetic moment M in the absence of dissipation losses (a) and with dampingFigure 1. actionDynamics (b) withinof the themagnetic scheme moment of the Landau-Lifshitz M in the absence equation. of dissipation The blue losses arrow (a notifies) and with the directiondamping ofaction moment (b) within trajectory. the scheme of the Landau-Lifshitz equation. The blue arrow notifies the direction of moment trajectory. In real experiments, dissipation losses play a crucial role in regulating the dynamics of the moment. EvenIn a smallreal experiments, variation in the dissipation dissipation losses torque play can a causecrucial the role moment in regulating to change the significantly. dynamics Asof the rightmoment. panel Even of Figure a small1 shows, variation the dissipation in the dissipation torque is the torque only torque can cause that actually the moment pushes toM towardchange thesignificantly. minimum As magnetic the right energy, panel resulting of Figure in a1 magnetizationshows, the dissipation trajectory torque that is onis the the sphericalonly torque surface that dueactually to its pushes constant M toward module the |M minimum|, and eventually magnetic precessesenergy, resu tolting the direction in a magnetization which is almost trajectory aligned that alongis on the the spherical−Heff field surface direction. due to its constant module |M|, and eventually precesses to the direction whichThe is almost divergence aligned problem along arises the − whenHeff field significantly direction. large dissipation losses (λ >> 1) are available in the LLThe equation. divergence To conquer problem this arises trouble, when Gilbert significantly modified large the dissipation dissipation losses term by(λ >> taking 1) are a dampingavailable −1 quantityin the LL of equation. the form αTo× Mconquers × M this× (dtrouble,M/dt) intoGilbert account. modifiedα acts the as thedissipation Gilbert damping term by coefficient. taking a Consequently,damping quantity the motionof the form of a magneticα × Ms−1 × moment M × (dM in/d thet) into context account. of a redefined α acts as damping the Gilbert factor damping along withcoefficient. effective Consequently, field was mathematically the motion of given a magnet by theic so-called moment Landau-Lifshitz-Gilbert in the context of a redefined (LLG) damping equation: factor along with effective field was mathematically given by the so-called Landau-Lifshitz-Gilbert M t − × M × H × M −1 × M × M t (LLG) equation: d /d = γ eff + α s (d /d ) (2) where γ is another gyromagneticdM/d ratio,t = −γ and × Mα ×turns Heff + out α × to M bes−1 the× M Gilbert × (dM/d dampingt) parameter. It appears(2) straightforward to verify that we can retrieve the LL equation format from the LLG counterpart. Taking where γ is another gyromagnetic ratio, and α turns out to be the Gilbert damping parameter. It the cross product of the left-hand side (LHS) of Equation (2) with M, and using M norm conservation appears straightforward to verify that we can retrieve the LL equation format from the LLG (M·× (dM/dt) = 0), one can obtain the following: counterpart. Taking the cross product of the left-hand side (LHS) of Equation (2) with M, and using M norm conservation (M·× (dM/dt) = 0), one can obtain the following: M × dM/dt = −γ × M × (M × Heff) + α × Ms × (dM/dt) (3) M × dM/dt = −γ × M × (M × Heff) + α × Ms × (dM/dt) (3) Notice that Ms represents the magnitude of the moment vector M. Inserting M × (dM/dt), which Notice that Ms represents the magnitude of the moment vector M. Inserting M × (dM/dt), which roots from the LHS of Equation (3) into Equation (2), one eventually obtains the following: roots from the LHS of Equation (3) into Equation (2), one eventually obtains the following:

dM/dt = −γ × (1 + α2)−1 × (M × Heff) – α × γ × Ms−1 × (1+α2)−1 × M × (M × Heff) (4)

Here, one may identify γ0 as γ × (1 + α2)−1 and λγ0/|M| as α × γ × Ms−1 × (1+α2)−1, then both equations (LL and LLG) become identical.

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2 −1 −1 2 −1 dM/dt = −γ × (1 + α ) × (M × Heff) − α × γ × Ms × (1+α ) × M × (M × Heff) (4)

2 −1 −1 2 −1 Here, one may identify γ0 as γ × (1 + α ) and λγ0/|M| as α × γ × Ms × (1+α ) , then both equations (LL and LLG) become identical. The next step was to transform Equation (4) into the dimensionless form. Dividing both sides of Electronics 2019, 8, x FOR PEER REVIEW 4 of 12 Equation (2) by γ × Ms gave rise to the below expression:

The next step was−1 to transform Equation−1 (4) into the dimensionless−1 − form.2 Dividing both sides of (γ × Ms) × dM/dt = −Ms × M × Heff + (α × γ × Ms ) × M × (dM/dt) (5) Equation (2) by γ × Ms gave rise to the below expression:

with the following(γ deﬁnitions:× Ms)−1 × dMn/d=t M= /−Ms−1and × Mh ×eff H=effγ +× (αH ×eff, γ−1one × M cans−2) rearrange× M × (dM Equation/dt) (5) to obtain(5) the dimensionless equation: dn/dt = −n × heff + α × n × dn/dt. Then, taking the cross product of n With the following definitions: n = M/Ms and heff = γ × Heff, one can rearrange Equation (5) to obtain and the expression involving dn/dt, one can ﬁnally determine the following: the dimensionless equation: dn/dt = −n × heff + α × n × dn/dt. Then, taking the cross product of n and the expression involving dn/dt, one can2 finally determine the following: (1 + α ) × dn/dt = −n × heff + α × heff (6) (1 + α2) × dn/dt = −n × heff + α × heff (6) 2.2. Stoner-Wohlfarth Model

2.2. Stoner-WohlfarthTo understand Model how the presence of magnetic anisotropy leads to the existence of preferred directions for moment M, one realizes that the origin of anisotropy energy EA stems from two factors: To understand how the presence of magnetic anisotropy leads to the existence of preferred The crystalline anisotropy ultimately caused by the spin–orbit interactions in the material [25] and directions for moment M, one realizes that the origin of anisotropy energy EA stems from two factors: the shape anisotropy caused by the energy of the ﬁelds induced by the magnetic moment [26]. The crystalline anisotropy ultimately caused by the spin–orbit interactions in the material [25] and The properties of these two types of energy remain complex in multi-domain samples, while the the shape anisotropy caused by the energy of the fields induced by the magnetic moment [26]. The anisotropy energy EA(n) of a single-domain case has the uniaxial form: properties of these two types of energy remain complex in multi-domain samples, while the anisotropy energy EA(n) of a single-domain case has the uniaxial2 form: EA = −1/2 × K × (n × z) (7)

EA = −1/2 × K × (n × z)2 (7) where K is the anisotropy constant that stays positive, and z denotes a unit vector along the z-axis. whereThe minimum K is the anisotropy of the anisotropy constant is achieved that stays for positive,n = +z or andn = −z denotesz in the context a unit vector of a positive along Kthe, while z-axis. the Thez-axis minimum here is defined of the anisotropy as the easy is axis achieved of the magneticfor n = +z material.or n = −z Consideringin the context a single-domainof a positive K, magnet while thethat z-axis is placed here intois defined an external as the magnetic easy axis field of theH, onemagnetic notices material. that an Considering additional contribution a single-domain to the magnetenergy that is equivalentplaced into toan (– externalM·× H) magnetic arises. Hence, field theH, one total notices energy that is consequently an additional defined contribution by the tofollowing: the energy that is equivalent to (–M·× H) arises. Hence, the total energy is consequently defined by the following: 2 EA = −1/2 × K × (n × z) − M·× H (8) 2 In terms of spherical angles,EAθ =and −1/2ϕ × areK × ( commonlyn × z) − M·× used H to parameterize the unit vector(8) n = (sinIn termsθcosϕ ,of sin sphericalθsinϕ, cos angles,θ), and theθ and total φ energyare commonlyEA is rewritten used to based parameterize on the re-expressed the unit vector component n = (sin(nxθ, ncosy, φnz, )sin asθ thesinφ following:, cosθ), and the total energy EA is rewritten based on the re-expressed component (nx, ny, nz) as the following: 2 EA = −1/2 × K × cos θ − |M| × |H| × cosθ (9) EA = −1/2 × K × cos2θ − |M| × |H| × cosθ (9) Such energy, which depends solely on angle θ, can now be plotted as a series of E(θ) graphs in the Such energy, which depends solely on angle θ, can now be plotted as a series of E(θ) graphs in following (see Figure2). The qualitative shape of each diagram appears to be different relying on the the following (see Figure 2). The qualitative shape of each diagram appears to be different relying on value of |H| = Hz/cosθ. the value of |H| = Hz/cosθ.

Figure 2. Calculated magnetic energy E as a function of angle θ with respect to the applied ﬁeld H Figure 2. Calculated magnetic energy E as a function of angle θ with respect to the applied field H (−1.4K/|M| ≤ |H| ≤ 1.4K/|M|) conﬁned along the z-axis. (−1.4K/|M| ≤ |H| ≤ 1.4K/|M|) confined along the z-axis.

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As shown in the figure above, for −K/|M| < |H| < K/|M|, there are two minima of energy in the angle interval (0, π), while for |H| > K/|M| (or |H| < −K/|M|) there exists only one minimum within the same parameter range. If one initially sets the field to a positive value, for instance, |H| Electronics> K/|M 2019|, the, 8, x available FOR PEER energyREVIEW minimum can only be acquired at θ = 0, which corresponds5 to ofthe 12 moment pointing in the +z direction. By decreasing the field until it eventually approaches its negative fromElectronicscounterpart up 2019(cos, (from8θ, x= FOR 1)| toPEERH |down = REVIEW 1.4 (cos× Kθ/| =M −1).| to The |H switching| = −1.4 × behaviorK/|M| indeed in Figure occurs2), one when can switchthe energy5 of the 12 minimummoment from in the up curves (cosθ at= 1)θ = to 0 downdisappears. (cosθ = −1). The switching behavior indeed occurs when the from up (cosθ = 1) to down (cosθ = −1). The switching behavior indeed occurs when the energy energyIt is minimum apparent in to the deduce curves from at θ =Equation 0 disappears. (9) that this critical phenomenon can be numerically minimum in the curves at θ = 0 disappears. capturedIt is at apparent the field to value deduce |H| from= −K/| EquationM|. Continuing (9) that to this tune critical the field phenomenon toward −1.4 can × K/| beM numerically|, one finds It is apparent to deduce from Equation (9) that this critical phenomenon can be numerically thatcaptured the moment at the field abruptly value jumps |H| = into−K /|a stateM|. with Continuing θ = π, (i.e., to tune being the directed field toward along− –z1.4). In× otherK/|M words,|, one captured at the field value |H| = −K/|M|. Continuing to tune the field toward −1.4 × K/|M|, one finds thefinds moment that the stays moment in the abruptly downstate jumps situation. into astate If the with fieldθ is= πincreased, (i.e., being up from directed a negative along − valuez). In other(e.g., that the moment abruptly jumps into a state with θ = π, (i.e., being directed along –z). In other words, |words,H| < −K the/|M moment|), the moment stays in theremains downstate in the situation.minimum If state the field(θ = π is) increaseduntil the field up from reaches a negative the value value of the moment stays in the downstate situation. If the field is increased up from a negative value (e.g., |(e.g.,H| = | +HK|/|

Figure 3. Calculated magnetic energy E as a function of angle θ for the special case where |H| =

±K/|M|. Figure 3. Calculated magnetic energy E as a function of angle θ for the special case where |H| = ±K/|M|. Figure 3. Calculated magnetic energy E as a function of angle θ for the special case where |H| = ±K/|M|.

Figure 4. Magnetic hysteresis describing the moment’s |M| dependence on the applied field strength Figure 4. Magnetic hysteresis describing the moment’s |M| dependence on the applied ﬁeld strength ||HH|| is is shown shown on on the the basis basis of of the the anisot anisotropicropic energy energy guided guided by by Equation Equation (9). (9). Figure 4. Magnetic hysteresis describing the moment’s |M| dependence on the applied field strength It|ItH is is| isobserved shown on that that the basisthis this magnetic magneticof the anisot hysteresis hysteresisropic energy ha has sguided a a rectangular rectangular by Equation hysteresis hysteresis (9). loop, loop, and and its its width width (or (or coercivecoercive ﬁeld)field) isis KK/|/|MM|. A A more more complicated situationsituation wherewhereH Hhas has two two nonzero nonzero components components given given by It is observed that this magnetic hysteresis has a rectangular hysteresis loop, and its width (or by H = (Hx, 0, Hz) is of great significance and is analogous to a Stoner-Wohlfarth particle. It is assumed coercive field) is K/|M|. A more complicated situation where H has two nonzero components given that the longitudinal component Hz stays extremely larger than the perpendicular component Hx whichby H = is(H freex, 0, toHz be) is tunedof great and significance is defined andas a isbias analogous field for tolater a Stoner-Wohlfarth discussion. Currently, particle. the It study is assumed of the switchingthat the longitudinal behavior of componenta uniaxial magnet Hz stays in extremelythe presence larger of a thanbias field,the perpendicular in addition to componenta longitudinal Hx field,which has is free been to attracting be tuned andincreasing is defined attention as a bias [27,28]. field Forfor laterthe purpose discussion. of seeking Currently, the thecritical study value of the of switching behavior of a uniaxial magnet in the presence of a bias field, in addition to a longitudinal Hx at which the switching occurs supposing that Hz is fixed, the functional dependence Hz(Hx) field, has been attracting increasing attention [27,28]. For the purpose of seeking the critical value of Hx at which the switching occurs supposing that Hz is fixed, the functional dependence Hz(Hx)

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H = (Hx, 0, Hz) is of great significance and is analogous to a Stoner-Wohlfarth particle. It is assumed that the longitudinal component Hz stays extremely larger than the perpendicular component Hx which is free to be tuned and is defined as a bias field for later discussion. Currently, the study of the switching behavior of a uniaxial magnet in the presence of a bias field, in addition to a longitudinal field, has

Electronicsbeen attracting 2019, 8, x increasing FOR PEER REVIEW attention [27,28]. For the purpose of seeking the critical value of Hx at6 which of 12 the switching occurs supposing that Hz is fixed, the functional dependence Hz(Hx) describing interplay describingof these two interplay critical fields of these is plotted two incritical the following fields is figure plotted and givenin the mathematically following figure as [11and,29 –given31]: mathematically as [11,29–31]: 2/3 2/3 2/3 Hz + Hx = (K/|M|) (10) Hz2/3 + Hx2/3 = (K/|M|)2/3 (10) Depicting the above relation between the bias field field and the fixed fixed field field leads to the resulting diagram, which is referred to as the asteroid curve, in which the dashed line in Figure 55 showsshows howhow the total field H changes when H varies itself at a fixed value of H . the total field H changes when Hz varies itself at a fixed value of Hx.

Figure 5. Functional dependence of Hz on Hx simulating the Stoner-Wohlfarth model. Figure 5. Functional dependence of Hz on Hx simulating the Stoner-Wohlfarth model. It is assumed that the longitudinal field Hz is abruptly tuned from the field point A to the one correspondingIt is assumed to the that field the at longitudinal point B, as a functionfield Hz is of abruptly a constant tunedHx. This from is the due field to the point fact that,A to tangentsthe one correspondingto the asteroid giveto the rise field to at magnetization point B, as a function directions of witha constant external Hx. energy. This is due For ato system the fact with that, a tangents uniaxial toanisotropy, the asteroid the give tangents rise to that magnet are closerization to directions the easy axis with can external result inen stableergy. For solutions a system (minimum with a uniaxial energy anisotropy,configuration). the tangents Specifically, that inare the closer initial to the state easy A, axis the magneticcan result momentin stable issolutions pointing (minimum mostly in energy the +z configuration).direction because Specifically, of a slight deviationin the initial from state +z due A, the to the magnetic presence moment of a nonzero, is pointing yet very mostly small, inH thex ( H+zx <

3. Results and Discussion

3.1. Switching Behavior under Axial Symmetry Before performing practical calculations to determine the magnetic switching time of an SW particle subjected to an additional bias field, it was necessary to convert the derived vector Equation (6) within the small damping approximation (α << 1) to a system of two scalar equations on (θ, φ). Figure 6 introduces the two vectors eθ and eφ, both of which are tangent to the unit sphere.

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3. Results and Discussion

3.1. Switching Behavior under Axial Symmetry Before performing practical calculations to determine the magnetic switching time of an SW particle subjected to an additional bias ﬁeld, it was necessary to convert the derived vector Equation (6) within the small damping approximation (α << 1) to a system of two scalar equations on (θ, ϕ). Figure6 Electronicsintroduces 2019 the, 8, x two FOR vectors PEER REVIEWeθ and eϕ, both of which are tangent to the unit sphere. 7 of 12 Electronics 2019, 8, x FOR PEER REVIEW 7 of 12

Figure 6. Vector analysis ofof angleangle quantitiesquantities ofof eeθ, eφ andand the the unit unit moment moment parameter parameter nn.. Figure 6. Vector analysis of angle quantitiesθ ofϕ eθ, eφ and the unit moment parameter n.

By differentiating the expressions n = (sin (sinθθcosφϕ, sin θsinϕφ,, coscosθ),), eeθθ = (cos (cosθθcosφϕ, cosθsinϕφ,, −sinsinθθ)) By differentiating the expressions n = (sinθcosφ, sinθsinφ, cosθ), eθ = (cosθcosφ, cosθsinφ, −sinθ) and eφϕ == ( (−−sinsinφϕ, cos, cosφϕ, 0), 0) with with respect respect to to time time t twhilewhile making making use use of of the the right-hand right-hand coordinate coordinate system, system, and eφ = (−sinφ, cosφ, 0) with respect to time t while making use of the right-hand coordinate system, including a standard unit basis (e.g., eθ ×× eφe ϕ= =n),n ),one one can can transform transform Equation Equation (6) (6) into into the the form: form: including a standard unit basis (e.g., eθ × eφ = n), one can transform Equation (6) into the form: (dθ/dt) × eθ + (dφ/dt) × sinθeφ = (əESW/əθ) × eφ − sin−1θ × (əESW/əφ) × eθ + α × ((dθ/dt) −1 −1 (dθ(d/dθt/d) ×t) ×eθ eθ+ + (d (dϕφ/d/dtt)) ×× sinsinθθeφϕ == (( əE ESWSW/əθθ)) ×× eφe ϕ− −sinsinθ × θ(ə×ESW( /EəφSW) /× eϕθ )+× α e×θ ((d+ θ/dt)(11) × eφ − (dφ/dt) × sinθeθ) (11)(11) α × ((dθ/dt)×× eφe −ϕ (d−φ(d/dϕt/d) × tsin) ×θsineθ) θeθ) where ESW is identified as the energy pertaining to an SW particle of interest, and the LHS of Equation (6) where ESW is identified as the energy pertaining to an SW particle of interest, and the LHS of Equation (6) iswhere approximatedESW is identified as dn/d ast. Next, the energy by equating pertaining the tocoefficients an SW particle pertinent of interest,to both andtangent the LHSvectors of is approximated as dn/dt. Next, by equating the coefficients pertinent to both tangent vectors appearingEquation (6) in isthe approximated LHS of Equation as d (11)n/d andt. Next, rearranging by equating the appropriate the coefficients time pertinent derivatives to bothwith tangentrespect appearing in the LHS of Equation (11) and rearranging the appropriate time derivatives with respect tovectors the angles, appearing one finally in the LHSfinds of the Equation following: (11) and rearranging the appropriate time derivatives with respectto the to theangles, angles, one onefinally finally finds finds the thefollowing: following: dθ/dt = −sin−1θ × (əESW/əφ) – α × (əESW/əθ) dθ/dt = −sin−1θ × (əESW/əφ) – α × (əESW/əθ) (12) −1 (12) dθ/dt = −sin θ × ( ESW/ ϕ) − α × ( ESW/ θ) (12) dϕ/dt = sin−1θ × (əESW/əθ) – α × sin−2θ × (əESW/əφ) dϕ/dt = sin−1θ × (əESW/əθ) – α × sin−2θ × (əESW/əφ) (13) −1 −2 (13) dφ/dt = sin θ × ( ESW/ θ) − α × sin θ × ( ESW/ ϕ) (13) We now investigated the switching behavior in the case of axial symmetry (Hx = 0) in the context We now investigated the switching behavior in the case of axial symmetry (Hx = 0) in the context whereWe the now magnetic investigated energy the was switching assumed behavior to depend in thesolely case on of the axial θ parameter symmetry ( (EH(θx)= = 0)−1/2 in the× K context× cos2θ where the magnetic energy was assumed to depend solely on the θ parameter (E(θ) = −1/2 × K ×2 cos2θ −where |M| the× |H magnetic| × cosθ energy). Therefore, was assumed two independent to depend solelyequations on the associatedθ parameter with (E the(θ) =angle−1/2 variables× K × cos areθ − |M| × |H| × cosθ). Therefore, two independent equations associated with the angle variables are given:− |M | × |H| × cosθ). Therefore, two independent equations associated with the angle variables are given:given: dθ/dt = −α × sinθ × (K × cosθ + hz) (14) dθ/dtd=θ−/dαt ×= −sinα ×θ sin× θ(K ×× (Kcos × cosθ +θh z+) hz) (14)(14) Here, conducting an integration of Equation (14) over θ and t subjected to a provided initial Here,Here, conducting conducting an integration an integration of Equation of Equation (14) over (14)θ andovert θsubjected and t subjected to a provided to a provided initial angle initial angle θ0 ≡ θ(t = 0) was apparently responsible for resolving the magnetic switching time. The θ0 ≡angleθ(t = θ 0)0 was≡ θ( apparentlyt = 0) was responsibleapparently forresponsible resolving for the resolving magnetic the switching magnetic time. switching The switching time. The switching field hz is denoted as Hz × |M|. Suppose one initiates with a magnetic moment positioned fieldswitchinghz is denoted field ashz Hisz denoted× |M|. as Suppose Hz × |M one|. Suppose initiates one with initiates a magnetic with momenta magnetic positioned moment close positioned to close to the +z direction and applies a negative magnetic field –H0 (aligned along −z) sufficient to the +closez direction to the and+z direction applies aand negative applies magnetic a negative field magnetic−H (aligned field – alongH0 (aligned−z) sufficient along − toz) inducesufficient a to induce a switch. A representation of the dependence of θ on0 t (in the unit of α × K) according to the switch.induce Arepresentation a switch. A representation of the dependence of the dependence of θ on t (in of the θ uniton t of(inα the× Kunit) according of α × K) toaccording the model to the model described mathematically in Equation (14) is shown in Figure 7. As the switching process describedmodel mathematicallydescribed mathematically in Equation in (14) Equation is shown (14) in Figure is shown7. As in the Figure switching 7. As process the switching progressed, process progressed, different curves reflecting angle θ (given in the unit of radian) versus time, arose due to differentprogressed, curves different reflecting curves angle reflectingθ (given in angle the unit θ (given of radian) in the versus unit of time, radian) arose versus due time, to the arose distinct due to the distinct values of θ0. By the end of the switching process, the moment had approached the –z valuesthe ofdistinctθ . By thevalues end of of θ the0. By switching the end process,of the switching the moment process, had approached the moment the had− zapproacheddirection, and the –z direction, and0 the angle θ approached π. the angledirection,θ approached and the angleπ. θ approached π.

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Electronics 2019, 8, 366 8 of 12

FigureFigure 7. θ 7.angle θ angle of the momentof the moment is analytically is analytically determined determined in terms of parameterin terms toffor parameter three characteristic t for three characteristic initial angles θ0 of approximately◦ 0.03 (~1.72°),◦ 0.17 (~9.74°), ◦and 0.32 (~18.33°). A solid initial angles θ0 of approximately 0.03 (~1.72 ), 0.17 (~9.74 ), and 0.32 (~18.33 ). A solid straight line representingstraight lineθ = representingπ/2 acts as a θ guide = π/2 for acts reference. as a guide for reference.

TwoTwo features features of thisof this graph graph are are important important for for us tous revisit.to revisit. First, First, the the switching switching behavior behavior occurred occurred muchmuch faster faster for for larger larger values values of θof0. θ Second,0. Second, each each trajectory trajectory path path describing describing the the time-dependent time-dependent angle angle approachedapproached a final a final value value that that was was sufficiently sufficiently close close to πtoin π anin an exponential exponential form. form. That That is, theis, the curve curve formallyformally took took a comparably a comparably long long time time to achieveto achieve the the entire entire variation variation process process (from (fromθ < θ0.5 < 0.5 to toθ = θπ =). π). ToTo efficiently efficiently define define a finite a finite switching switching time, time, it is it customary is customary to selectto select a cutoff a cutoff angle angleθs andθs and to calculateto calculate thethe corresponding corresponding time timets itts takesit takes to reachto reach that that angle. angle. Figure Figure7 shows 7 shows that that the the switching switching behavior behavior confinedconfined in thein the angle angle range range of 0of to 0 πtoexists π exists over over a relatively a relatively short short time time interval. interval. Prior Prior to thisto this interval, interval, thethe angles angles inside inside different different curves curves were were close close to θto0, θ and0, and beyond beyond this this interval, interval, the the angles angles remained remained close close to πto. Theπ. The midpoint midpoint for for the the time time of thisof this interval interval was was roughly roughly located located at θ sat= θπs /2.= π/2. This This explains explains why why it it mademade sense sense to selectto selectπ/2 π/2 as as a cutoffa cutoff angle. angle. Such Such a criticala critical angle angle value value has has been been frequently frequently used used in thein the literatureliterature [32 ,[32,33].33]. Additionally, Additionally, it is it noted is noted that that at the at the same same time time (t = (tts =), t thes), the projection projection moment momentMz Mhadz had reachedreached zero. zero. ContinuingContinuing with with the exactthe exact analytic analytic expression expression of Equations of Equations (12) and (12) (14), and we resolved(14), we the resolved analytic the solutionanalytic to solution obtain the to obtain switching the switching time of the time magnetic of the magnetic particle particle in the context in the context of axial of symmetry axial symmetry by meansby means of integration of integration using separation using separation of variables. of Byvariables. integrating By Equationintegrating (14) Equation over time (14) parameter over timet as wellparameter as angle t asθ wellwhile as imposing angle θ while the condition imposingθ( thet = 0)condition = θ0, the θ analytic(t = 0) = solutionθ0, the analytic finally solution becomes finally the following:becomes the following:

(0) −1 −1 −1 −1 (0) ts ≡ t s(θ0) = −(1/2)− ×1 α × ((K + hz−) 1 × In(hz × (1 − cosθ0) × (K × cosθ0 + hz) ) + (K− 1− hz) ts ≡ t (θ ) = −(1/2) × α × ((K + h ) × In(h × (1 − cosθ ) × (K × cosθ + h ) ) + (15) s 0 z z 0 −1 0 z −1 × In(hz × (1 + cosθ0) × (K × cosθ0 + hz) ))− 1 (15) (K − hz) × In(hz × (1 + cosθ0) × (K × cosθ0 + hz) )) One crucial thing about the understanding of the initial angle θ0 requires emphasis. The switchingOne crucial time thing t(0)s about(θ0) ultimately the understanding approached of the zero initial once angle θ0 θwas0 requires close emphasis.to π/2. Additionally, The switching this (0) timeconclusiont s(θ0) ultimately was certainly approached consistent zero with once commonθ0 was sense. close A to similarπ/2. Additionally, expression concerning this conclusion Equation was (15) certainlywas derived consistent in another with common report, in sense. particular A similar for the expression case where concerning K = 0, and Equation for an arbitrary (15) was K derived [34]. in another report, in particular for the case where K = 0, and for an arbitrary K [34]. 3.2. Switching Behavior of an SW particle in a Small Perpendicular Bias Field 3.2. Switching Behavior of an SW particle in a Small Perpendicular Bias Field In the presence of the switching field H0 pointing in the negative z-axis and the bias field Hperp In the presence of the switching field H pointing in the negative z-axis and the bias field H pointing in the positive x-axis, the energy0 density of a Stoner-Wohlfarth particle is expressed perpas the pointing in the positive x-axis, the energy density of a Stoner-Wohlfarth particle is expressed as following: the following: Esw = −1/2 × K × cos2θ − |M| × |Hperp| × sinθ × cosϕ − |M| × |H0| × cosθ (16) 2 Esw = −1/2 × K × cos θ − |M| × |Hperp| × sinθ × cosφ − |M| × |H0| × cosθ (16) Here, the switching field and the bias field were also recognized as hperp = Hperp × |M| and hz = H0 × |M|. If the bias field Hperp was significantly small (hperp = |hperp| << hz = |hz|), one can hope to find Here, the switching field and the bias field were also recognized as hperp = Hperp × |M| and an analytic approximation to obtain the switching time by taking such a bias field into account as a hz = H0 × |M|. If the bias field Hperp was significantly small (hperp = |hperp| << hz = |hz|), one can

Electronics 2019, 8, 366 9 of 12 hope to find an analytic approximation to obtain the switching time by taking such a bias field into account as a small perturbation. Similar reports addressing the precessional magnetization switching under a biased perpendicular anisotropy can be found in [35,36]. The result can be formulated as follows. Combining Equation (12) and (15), one notes that the magnetic switching time was identified from the following expression after integration over angle θ and time t, respectively: Electronics 2019, 8, x FOR PEER REVIEW 9 of 12

dθ/dt = −α × sinθ × (K × cosθ + hz) − hperp × sinφ (17) small perturbation. Similar reports addressing the precessional magnetization switching under a biased perpendicular anisotropy can be found in [35,36]. The result can be formulated as follows. Treating the −hperp × sinφ term as a perturbation and depending on the previously obtained Combining Equation (12) and (15), one notes that the magnetic switching time was identified from switching time appearing in Equation (15), the determination of t(0) (θ ) in the case of a perpendicular the following expression after integration over angle θ and time t, respectively:s 0 bias field was found by substituting an “effective initial angle” θin for the initial angle θ0. The initial dθ/dt = −α × sinθ × (K × cosθ + hz) − hperp × sinϕ (17) unit vector n(t = 0) was designated by the angles (θ0, ϕ0). Accordingly, this effective initial angle was definedTreating as the the angle −hperp between × sinϕ term the initialas a perturbation unit vector andn( tdepending= 0) and theon the position previously of the obtained total energy switching time appearing in Equation (15), the determination of t(0)s(θ0) in the case of a perpendicular −1 maximum, which was characterized by the spherical angles θmax = θ* = |Hperp| × (|H0| − K/|M|) , bias field was found by substituting an “effective initial angle” θin for the initial angle θ0. The initial φmax = π. In a word, the following approximation was effective as a method of computing the magnetic unit vector n(t = 0) was designated by the angles (θ0, φ0). Accordingly, this effective initial angle was switching time as long as the angle θ stayed typically small (e.g., θ < 0.5), while the calculational defined as the angle between the initial* unit vector n(t = 0) and the position* of the total energy maximum, accuracy turned out to be quadratic in terms in θ*: which was characterized by the spherical angles θmax = θ* = |Hperp| × (|H0| − K/|M|)−1, ϕmax = π. In a word, the following approximation was effective as a method of computing the magnetic switching time as t = t(0) (θ ) = t(0) (θ ) + O(θ 2) (18) long as the angle θ* stayed typicallys smalls (e.g.,0 θ* < s0.5),in while the* calculational accuracy turned out to be quadratic in terms in θ*: Furthermore, relying on the primitive LLG model, we numerically solved the magnetic switching ts = t(0)s(θ0) = t(0)s(θin) + O(θ*2) (18) time of an SW particle in the case of a small perpendicular bias field via the computational package MathematicaFurthermore, [37] while relying fulfilling on the the primitive requirement LLG thatmodel,θ( tswe) = numericallyπ/2. Performing solved the this magnetic calculational processswitching for different time of initial an SW directions particle in the governed case of a bysmalln(t perpendicular= 0), we determined bias field undervia the whatcomputational circumstance package Mathematica [37] while fulfilling the requirement that θ(ts) = π/2. Performing this the approximate expression (18) may lose its validity. First, we tested the case in which the field Hperp calculational process for different initial directions governed by n(t = 0), we determined under what was assumed to be constant, and the moment was set up at the initial position stated by (θ , 0), while circumstance the approximate expression (18) may lose its validity. First, we tested the case in which0 the switching field satisfied h0 = hz = 1.5 × K. Using the analytic solution derived from Equation (18) the field Hperp was assumed to be constant, and the moment was set up at the initial position stated alongby with (θ0, the 0), while numerically the switching solved field LLG satisfied equation, h0 = h wez = 1.5 then × K plotted. Using the analyticts(θ0) relations solution inderived Figure from8, where time parameterEquation (18) was along shown with inthe the numerically unit of α solved× K, andLLG theequation, magnitude we then of plottedHperp thewas ts( alternativelyθ0) relations in given by hperpFigure= 0.02 8, where× K. Thetime initial parameter angle wasθ0 isshown expressed in the inunit the of unit α × ofK, radian.and the magnitude of Hperp was alternatively given by hperp = 0.02 × K. The initial angle θ0 is expressed in the unit of radian.

Figure 8. The magnetic switching time ts calculated using a numerical approach (points) and an Figure 8. The magnetic switching time ts calculated using a numerical approach (points) and an analytical method (solid line) is shown as a function of the initial angle θ0 in the context of the analytical method (solid line) is shown as a function of the initial angle θ0 in the context of the switching ﬁeld (h0 = 1.5 × K) supplied along the −z axis, in addition to the bias ﬁeld (hperp = 0.02 × K) switching field (h0 = 1.5 × K) supplied along the −z axis, in addition to the bias field (hperp = 0.02 × K) given in the (π/2, 0) direction. given in the (π/2, 0) direction.

In general, the analytical results obtained on the basis of Equation (12), (16) and (18) qualitatively agreed well with the numerical solutions (solid line) considering that the perturbation parameter θ*

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Electronics 2019, 8, x FOR PEER REVIEW 10 of 12 In general, the anasof Equations (12), (16) and (18) qualitatively agreed well with the numerical wassolutions kept small (solid throughout line) considering the entire that simulation the pserturbation process. Moreover, parameter weθ* observedwas kept from small the throughout graph that consistentthe entire simulationand accurate process. correspondence Moreover, we was observed found fromamong the graphthese thattwo consistentdifferent andcalculational accurate approachescorrespondence for nearly was found all of amongthe values these of two θ0. differentAdditionally, calculational the existence approaches of the typically for nearly small all of bias the × × fieldvalues (h ofperpθ 0=. Additionally,0.02 × K << theh0 = existence 1.5 × K) of somehow the typically significantly small bias ﬁelddecreased (hperp =the 0.02 switchingK << h 0time= 1.5 fromK) × × ≈ approximatelysomehow signiﬁcantly 600 × α decreased× K at θ0 ≈ the 0.01 switching to approximately time from 320 approximately × α × K at θ 6000 = 0.15α (correspondingK at θ0 0.01 to ◦ approximately 8.6°). 320 × α × K at θ0 = 0.15 (corresponding to approximately 8.6 ). Second, we considered a situation where the vector n(t = 0) 0) started started its its motion motion from from the the energy energy minimum statestate correspondingcorresponding to to the the applied applied field fieldH 0H, which0, which was was aligned aligned along along the positivethe positivez-axis. z-axis. This −1 Thistypical typical starting starting point point (initial (initial unit unit vector) vector) was was parameterized parameterized by the by anglesthe anglesθ0 = θh0perp = hperp× (×h (0h+0 +K K))−1 and ϕφ0 == 0.0.Figure Figure9 illustrates 9 illustrates how analytichow analytic approximation approximation coincides coincides with the numericalwith the resultsnumerical represented results representedby discrete points. by discrete points.

Figure 9. Magnetic switching time ts calculatedcalculated using using the the numerical numerical approach approach (points) (points) and and analytical

method (solid line) is shown asas aa functionfunction ofof thethe biasbiasﬁeld fieldh hperpperp applied along thethe ((ππ/2, 0) 0) direction direction in × − thethe context context of of the the switching switching field ﬁeld (h (0h =0 1.5= 1.5 × K) suppliedK) supplied along along the − thez axisz inaxis addition in addition to the toinitial the initialangle −1 ×−1 determinedangle determined to be h toperp be × h(hperp0 + K) (.h 0 + K) .

Compared to Figure 88,, thethe correspondencecorrespondence amongamong thethe twotwo differentdifferent calculationalcalculational approachesapproaches became slightly worse as the bias field hperp was increased beyond approximately 0.15 × K, as seen became slightly worse as the bias field hperp was increased beyond approximately 0.15 × K, as seen in in Figure9. On the other hand, with the increase of the bias field hperp lying in the range of 0.01 × K Figure 9. On the other hand, with the increase of the bias field hperp lying in the range of 0.01 × K and × 0.15and 0.15× K, theK ,determined the determined switching switching time timearising arising from fromthe analytic the analytic approximation approximation decreased decreased and and agreed fairly well with the numerical data points, indicating that the remainder term (O(θ 2)) agreed fairly well with the numerical data points, indicating that the remainder term (O(θ**2)) occurring in Equation (18) became non-trivial. Al Althoughthough the the quality of the approximation begins to deteriorate slightly when hperp > 0.15 × K, the analytical approach in studying the magnetic switching deteriorate slightly when hperp > 0.15 × K, the analytical approach in studying the magnetic switching isis still still believed believed to to be be acceptable acceptable because because the the actual actual bias bias field field is normally normally set to to be be a a few few percent percent of of the the anisotropy field field K. 4. Conclusions 4. Conclusions The SW model appears to be quite an effective simulation of magnetic systems and exhibits The SW model appears to be quite an effective simulation of magnetic systems and exhibits extremely rich physical properties from, for instance, a dynamic point of view. The combined use of extremely rich physical properties from, for instance, a dynamic point of view. The combined use of a uniaxial anisotropy and an externally applied field provided a thorough insight into the magnetic a uniaxial anisotropy and an externally applied field provided a thorough insight into the magnetic switching behavior tuned by a weak bias component hperp (or small initial angle θ0). The validity of switching behavior tuned by a weak bias component hperp (or small initial angle θ0). The validity of the analytic approximation that treats a comparably small hperp (<

Electronics 2019, 8, 366 11 of 12 consistently found that the calculated magnetic switching time during which the moment vector changed its angle by π/2 − θ0 demonstrated decreasing behavior in a somewhat exponential manner when hperp was increased from 0.01 × K to 0.15 × K (or θ0 was increased from 0.01 to 0.3). Furthermore, a slight discrepancy existed between the numerical data points and the analytic results, given that hperp was tuned beyond 0.15 × K when ts was shown as a function of the perpendicular bias field. It is expected that the derived analytic expression, which treats hperp as small perturbation, can be extended to simplify the determinations for magnetic switching in granular magnetic media; however, the bias field can originate from the induced field of other grains or from the misalignment of the individual grain’s axes with the applied field.

Author Contributions: D.X., the first author, undertook all of the (sometimes computationally intense) analytic calculations that are reported in the paper. He also wrote the majority of the first draft and participated in the editing of that draft which produced the final, submitted version. W.M., the second author, is a visiting Professor of Physics at Umeå University with beneficial suggestions and comments. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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