Magnetic Domain Transition of Adjacent Narrow Thin Film Strips with Inclined Uniaxial Magnetic Anisotropy

micromachines

Article Magnetic Domain Transition of Adjacent Narrow Thin Film Strips with Inclined Uniaxial Magnetic Anisotropy

Tomoo Nakai

Industrial Technology Institute, Miyagi Prefectural Government, 2-2 Akedori, Aoba-ku, Sendai-city, Miyagi 981-3206, Japan; [email protected]; Tel.: +81-22-377-8700

Received: 13 January 2020; Accepted: 5 March 2020; Published: 8 March 2020

Abstract: This study deals a phenomenon of magnetic domain transition for the stepped magneto-impedance element. Our previous research shows that an element with 70◦ inclined easy axis has a typical characteristic of the domain transition, and the transition can be controlled by the normal magnetic ﬁeld. In this paper, we apply this phenomenon and controlling method to the line arrangement adjacent to many body elements, in which mutual magnetic interaction exists. The result shows that the hidden inclined Landau–Lifshitz domain appears by applying a distributed normal ﬁeld the same as an individual element.

Keywords: stepped magneto-impedance element; magnetic domain; domain transition; distributed field; normal field; many-body elements; thin film

1. Introduction The observation of a magnetic domain of a thin film magnetic element is carried out using measuring methods such as electron beam [1], Kerr magneto-optic effect [2], scanning tunneling microscope, and magnetic force microscopy. Recent topics of investigations on the domain structure range from the nanoscale to the submillimeter-scale, such as a formation of vortices [3], an effect caused by ion irradiated sheet [4] and a property improvement of the silicon steel magnetic core [5]. The structure of the magnetic domain formed in a thin film element having a dimension from several to hundreds of micrometers was previously reported, showing that it forms several typical domain patterns, the Landau-Lifshitz domain [6], for example, and the variations as a function of the strength of the external field [7]. For the research field of soft magnetic thin films, the domain structure has been studied in accordance with the study of magneto-impedance magnetic field sensors. The recent performance improvement of an apparatus of the magneto-optic Kerr effect (MOKE) has contributed to the investigation of the magneto-impedance (MI) sensor from the viewpoint of the dependence of high-frequency impedance on the magnetic domain structure of the element. The beginning of it seemed to be a physical investigation of the MI sensor for clarifying the sensing mechanisms [8–10]. Thereafter, the study of the magnetic domain for thin film MI sensors has been developing continuously for the purpose of improving the properties of the MI sensor based on the control of the magnetic domain. The following are the typical example of the study: A layered and laminated structure of thin film element [11–13], a miniaturization of the element [14], a consideration of magnetostriction [15], an effect of high-temperature annealing [16], an effect of element dimension for a magnetic property such as coercivity Hc [17], direct current (DC) biasing [18], and a biasing caused by the exchange force [19]. A domain structure simulation was originally developed for the methods which were suitable for the actual size of the MI element which is unable to apply the conventional micro-magnetic simulations due to larger dimensions of the element [20,21].

Micromachines 2020, 11, 279; doi:10.3390/mi11030279 www.mdpi.com/journal/micromachines Micromachines 2020, 11, 279 2 of 17

The stepped-MI element has the unique property of a step-like magneto-impedance change, in the case where the sensor has an in-plane uniaxial inclined easy axis [22,23]. A domain observation shows that the step-like change is due to a magnetization transition within three states, including the longitudinal single domain with parallel state, with an anti-parallel state and the inclined Landau–Lifshitz domain (ILLD). This phenomenon is expected to create a sensor with a memory function [24]. In a condition where the sensor has an easy axis of 70◦ relative to the short-side axis of the rectangular element, the transition is limited between the parallel and anti-parallel states despite the existence of a stable ILLD [25]. We call this stable ILLD state as a hidden ILLD state. The hidden ILLD state is able to be appear by applying a normal magnetic field with a distributed inclined angle. The normal direction is defined as a surface normal direction to the substrate plane, and the inclination angle of the field is defined as an angle between the normal direction and the direction of the magnetic field having a certain inclination toward the length direction of the element. The distribution of the inclination means a variation of the inclination angle as a function of the length position of the element. The appearance of the hidden ILLD state was estimated by numerical analysis [21,26,27] and experimentally confirmed [28,29] using a single element. The artificial transition from the single domain state toward the hidden ILLD state was also realized experimentally [30]. It is expected to apply to an unerased memory and its reset procedure. In this study, an extension of the application of this phenomenon to clustered many-body elements was experimentally investigated. The individual element has the same width and thickness, and also has the direction of magnetic anisotropy the same as an element having the stepped-MI property with a hidden ILLD state. A certain number of the elements were arranged in a planar cluster which has a configuration of a line arrangement adjacent to many-body elements. This trial is important for realizing a high-density device using this phenomenon. The magnetic hysteresis loop (MH-loop) of the planar clustered element was measured in accordance with a domain observation. An effect of the application of the distributed normal field was also investigated.

2. Experimental Procedure

The element was fabricated by a thin-film process. An amorphous Co85Nb12Zr3 film was RF-sputter deposited onto a soda glass substrate and then micro-fabricated into rectangular elements by a lift-off process. The element was thousands of µm in length, 20 µm wide, and 2.1 µm thick. A uniaxial magnetic anisotropy was induced by magnetic field annealing, 240 kA/m at 673 K for 1 h. The easy axis of the magnetic anisotropy is made along the processing magnetic field. The annealing apparatus used in this study had an accuracy in the angle position of 0.5◦. In this study it was induced in several different conditions for a comparison of their properties. We made two different layouts of the elements. One was the configuration of the line arrangement adjacent many-body elements having a mutual magnetic interaction with each other, and the other layout were dispersed individual elements. These two types of different layouts were made for a comparison between the elements having the mutual force and those without it. The prior many-body element was prepared in three different conditions for the purpose of confirming the effect of the direction of easy axis with a focus on the element having the hidden domain state, which is around θ = 70◦. The element dimensions were as follows: In the case of the adjacent many-body elements, the length was 3000 µm, the width was 20 µm, and the thickness was 2.1 µm. The element length was determined by a knowledge obtained by the previous study that the element would have a residual domain at the both end of element. In our previous report, the element length was set as 2000 µm [29]. This element was assembled to form a parallel line arrangement configuration with the line and space (L/S) as 20 µm and 20 µm. The two dimensional area of this assembled elements was 3000 µm 3020 µm, which was suitable for a measurement of magnetization loop using a vibrating × sample magnetometer (VSM). The three different directions of easy axis were made in θ = 61◦, θ = 71◦, and θ = 90◦. Our previous study showed that the element of θ = 61◦ has a changing property from single domain, , to the ILLD and then to single domain, +, with the increasing magnetic field. The − Micromachines 2020, 11, 279 3 of 17 Micromachines 2020, 11, x 3 of 17 Micromachines 2020, 11, x 3 of 17 single domain − and +. These are the neighboring conditions in the direction of easy axis in θ = 71˚ , elementsingle domain of θ = − 90and◦ has +. These a changing are the propertyneighboring between conditions the single in the domain direction ofand easy+. axis These in θ are = 71 the˚ , which is the focusing condition in this study. The element layout on the glass− substrate is shown in neighboringwhich is the conditionsfocusing condition in the direction in this study. of easy The axis element in θ = 71 layout◦, which on isthe the glass focusing substrate condition is shown in this in study.FigureFigure 1. The1. AA schematic elementschematic layout explanationexplanation on the of glassof thethe substratedirectiondirection of isof shownthethe easyeasy in axisaxis Figure andand1 alsoalso. A schematicanan enlargedenlarged explanation viewview ofof aa partpart of theofof aa direction fabricatedfabricated of theelementelement easy axisisis shownshown and also inin anFigureFigure enlarged 2.2. OnOn view thethe of otherother a part hand,hand, of a fabricated thethe latterlatter element layout,layout, is whichwhich shown hashas in Figuredisperseddispersed2. On individualindividual the other elements,elements, hand, the consisted latterconsisted layout, ofof 5454 which elelementsements has dispersed onon aa 2626 mmmm individual ×× 2626 mmmm elements, glassglass substrate.substrate. consisted TheThe of 54distancedistance elements ofof eacheach on a elementelement 26 mm waswas26 mmsetset asas glass follows:follows: substrate. AA longitlongit Theudinaludinal distance distancedistance of each ofof 20002000 element µmµm andwasand a seta laterallateral as follows: distancedistance A × longitudinalofof 25002500 µm.µm. The distanceThe dimensionsdimensions of 2000 µofofm eacheach and elementelement a lateral was distancewas asas follows:follows: of 2500 AAµ m.lengthlength The of dimensionsof 20002000 µm,µm, ofaa widthwidth each element 2020 µm,µm, ˚ wasandand asaa thickness follows:thickness A 2.12.1 length µm.µm. of TheThe 2000 angleangleµm, of aof width thethe magneticmagnetic 20 µm, and easyeasy a thickness axisaxis waswas 2.1 directeddirectedµm. The inin angleθθ == 6767 of˚ the againstagainst magnetic thethe width direction of the element strips. This layout on the glass substrate is shown in Figure 3. The easywidth axis direction was directed of the inelementθ = 67 ◦strips.against This the layout width directionon the glass of the substrate element is strips. shown This in Figure layout on3. The the element with θ = 67˚ was expected to have a property of the three stable states [24] with a narrow glasselement substrate with θ is = shown67˚ was in Figureexpected3. The to have element a property with θ =of67 th◦ewas three expected stable states to have [24] a propertywith a narrow of the threemulti-domainmulti-domain stable states range.range. [24 ] with a narrow multi-domain range.

Figure 1. Layout of the many-body element on a glass substrate. Figure 1. Layout of the many-body elementelement on a glass substrate.

FigureFigure 2.2. Direction of easy axisaxis θθ andand anan enlargedenlarged picturepicture ofof thethe fabricatedfabricated element.element. Figure 2. Direction of easy axis θ and an enlarged picture of the fabricated element.

Figure 3. Layout of dispersed individual elements on a glass substrate. Figure 3. Layout of dispersed individual elements on a glass substrate.

The measurements carried out here were a MH-loop measurement and a magnetic TheThe measurementsmeasurements carriedcarried outout herehere werewere aa MH-loopMH-loop measurementmeasurement andand aa magneticmagnetic domaindomain domain observation. The MH-loop was measured by a vibrating sample magnetometer (VSM) observation.observation. TheThe MH-loopMH-loop waswas measuredmeasured byby aa vibratingvibrating samplesample magnetometermagnetometer (VSM)(VSM) (TM-VSM211483-HGC,(TM-VSM211483-HGC, TAMAGAWATAMAGAWA Co.,Co., Ltd.,Ltd., Sendai,Sendai, Japan).Japan). TheThe magneticmagnetic domaindomain waswas observedobserved byby aa KerrKerr microscopemicroscope (BH-762PI-MAE,(BH-762PI-MAE, NEOARKNEOARK Corporation,Corporation, Tokyo,Tokyo, Japan)Japan) byby applyingapplying aa

Micromachines 2020, 11, x 4 of 17 Micromachines 2020, 11, 279 4 of 17 Micromachines 2020, 11, x 4 of 17 magnetic field in a certain direction and also at a certain controlled strength. These applied magnetic Micromachines 2020, 11, x 4 of 17 (TM-VSM211483-HGC,fieldsmagnetic were field generated in a certain by TAMAGAWA directionthe following and Co., alsotwo Ltd., at appa a Sendai,certainratuses. Japan).controlled The Thedistributed strength. magnetic normalThese domain applied field was was magnetic observed made fields were generated by the following two apparatuses. The distributed normal field was made usingbymagnetic a Kerr a ring-shaped microscopefield in a certain (BH-762PI-MAE,magnet direction which and NEOARKwas also positionedat a certain Corporation, belowcontrolled Tokyo,the strength.observation Japan) These by applyingstage applied of a magneticthe Kerr using a ring-shaped magnet which was positioned below the observation stage of the Kerr microscope.fieldfields in were a certain generatedThe magnetic direction by the andfield following alsoalong at the a certaintwo element’s appa controlledratuses. longitudinal strength.The distributed in-plane These direction appliednormal field magneticwas wascontrolled fieldsmade microscope. The magnetic field along the element’s longitudinal in-plane direction was controlled wereusing generatedaa Helmholtzring-shaped by thecoil magnet following(Custom-made which two apparatuses.was one, positionedRyowa TheElectronics below distributed the Co., observation normalLtd., Sendai, field stage was Japan) madeof whilethe using Kerr the a using a Helmholtz coil (Custom-made one, Ryowa Electronics Co., Ltd., Sendai, Japan) while the observationring-shapedmicroscope. of magnetThe the magnetic magnetic which field wasdomain positionedalong was the carried element’s below out. the longitudinalFigure observation 4 and stagein-plane Figure of 5 the directionshow Kerr the microscope. wasschematic controlled and The observation of the magnetic domain was carried out. Figure 4 and Figure 5 show the schematic and photomagneticusing ofa Helmholtzthe field measurement along coil the (Custom-made element’s apparatus longitudinal of theone, magnetic Ryowa in-plane Electronicsdomain direction used Co., was in Ltd.,this controlled study. Sendai, using Japan) a Helmholtz while the photo of the measurement apparatus of the magnetic domain used in this study. coilobservation (Custom-madeThe definition of the magnetic of one, the Ryowadistributed domain Electronics wasnormal carried Co.,field out. Ltd.,is shown Figure Sendai, in 4 Figureand Japan) Figure 6, while which 5 show the is the observationthe same schematic definition of and the The definition of the distributed normal field is shown in Figure 6, which is the same definition asmagneticphoto the previousof the domain measurement article was carried[29]. apparatus out. Figures of the4 andmagnetic5 show domain the schematic used in and this photo study. of the measurement as the previous article [29]. apparatusThe definition of the magnetic of the distributed domain used normal in this field study. is shown in Figure 6, which is the same definition as the previous article [29].

Figure 4. Schematic of the measurement apparatus. Figure 4. SchematicSchematic of the measurement apparatus.

Figure 4. Schematic of the measurement apparatus.

Figure 5. Figure 5. Photo of the measurement apparatus. Figure 5. Photo of the measurement apparatus. The definition of the distributed normal field is shown in Figure6, which is the same definition as the previous article [29]. Figure 5. Photo of the measurement apparatus.

Figure 6. Definition of the distributed normal field. Figure 6. Definition of the distributed normal field.

Figure 6. DefinitionDeﬁnition of the dist distributedributed normal ﬁeld.field.

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3. Results A photo of the fabricated many-body elements on a glass substrate are shown inin FigureFigure7 7.. EachEach element waswas divideddivided individually individually by by a dicinga dicing apparatus apparatus and and then then used used for the for measurements. the measurements. The front The threefront elementsthree elements of the photo of the were photo the measuredwere the meas samplesured of samples the L/S = 20of /20theµ mL/S layout. = 20/20 The µm measurements layout. The inmeasurements this study were in carriedthis study out usingwere thiscarried element. out us Theing results this element. are shown The in results the following are shown subsections. in the following subsections.

Figure 7. View of the fabricated many-body elements. 3.1. Magnetization Process of the Many-Body Elements 3.1. Magnetization Process of the Many-Body Elements Firstly, the MH-loop measurement for the many-body elements with different directions of the Firstly, the MH-loop measurement for the many-body elements with different directions of the easy axis was carried out. The length direction of the element line was set in the measurement direction easy axis was carried out. The length direction of the element line was set in the measurement of the VSM apparatus. Measured MH-loops are shown in Figure8. Figure8a is a case for the easy direction of the VSM apparatus. Measured MH-loops are shown in Figure 8. Figure 8a is a case for axis in θ = 61 , Figure8b for θ = 71 , and Figure8c for θ = 90 . The measurement sweep speed the easy axis in◦ θ = 61°, Figure 8b for◦ θ = 71°, and Figure 8c for◦ θ = 90°. The measurement sweep was 0.1 Oe/s (7.96 (A/m)/s) and the measurement time constant was 100 ms. The MH-loop in each speed was 0.1 Oe/s (7.96 (A/m)/s) and the measurement time constant was 100 ms. The MH-loop in condition resembles each other. The coercivity slightly increased with the increase in the θ value, and each condition resembles each other. The coercivity slightly increased with the increase in the θ the linearly-inclined part of the curve changes to a slightly bending configuration in the vicinity of value, and the linearly-inclined part of the curve changes to a slightly bending configuration in the the zero field with the increase in the θ value. In these figures, the value of Meff (T) is estimated from vicinity of the zero field with the increase in the θ value. In these figures, the value of Meff (T) is the volume 3000 µm 3020 µm 2.1 µm. This is an effective value, because it is estimated as if the estimated from the volume× 3000× μm × 3020 μm × 2.1 μm. This is an effective value, because it is element has a shape of a single square sheet which corresponds to the volume of many-body elements estimated as if the element has a shape of a single square sheet which corresponds to the volume of as a whole. many-body elements as a whole.

Micromachines 2020, 11, 279 6 of 17 Micromachines 2020, 11, x 6 of 17

(a)

(b)

(c)

° FigureFigure 8. MH-loop 8. MH-loop of theof the fabricated fabricated elements. elements. ((aa)) TheThe case case for for easy easy axis axis θ θ= 61= 61. (b◦.() Theb) Thecase casefor easy for easy axis θ = 71°. (c) The case for easy axis θ = 90°. axis θ = 71◦.(c) The case for easy axis θ = 90◦.

The observation of the magnetic domain was also carried out and shown in Figure9. The MH-loop and corresponding photo of the magnetic domain at a certain magnetic ﬁeld is shown in the ﬁgure. Micromachines 2020, 11, 279 7 of 17

Note that the magnetic domain observation at different strengths of the magnetic field was carried out individually, not sequentially. The MH-loop measurement was a sequential measurement, on the other hand. Figure9a is the case for the easy axis at θ = 61◦, Figure9b for θ = 71◦, and Figure9c for θ = 90◦. The elements were magnetically saturated in the value around 300–400 A/m. Based on our previous study the variation of the magnetic domain for the individual single element, having an easy axis at θ = 61 , has a property of domain variation which changes from a single domain in the direction to ◦ − ILLD around zero magnetic field, and then a single domain in the + direction, with the increase in the magnetic field from minus value to the plus value. In the case of θ = 71◦, it is the element having the hidden-stable state. This means that the element has an energetically stable ILLD state, however, it does not appear using a method of application of a uniform external field. Apparently the domain switched between the single domain and the + single domain for the θ = 71 element. In the case − ◦ of θ = 90 , the domain simply switched between the single domain and the + single domain. The ◦ − measured result of this study, as shown in the Figure9, has a changing property of each element but not all at once. In our fabrication process, the magnetic property of the thin film on a glass substrate was a uniform one, therefore, the domain transition should occur simultaneously. However, the result shows that it does not occur all at once. It is expected that it comes from the magnetically mutual interaction between the accumulated many-body elements. Each element in a many-body element has a certain magnetization, therefore, each element has a certain value of the magnetic pole at the end of the narrow strip. The magnetic pole of an element makes an inverse magnetic field to the surrounding elements. According to the applied external field the number of transited elements changes, as shown in Figure9. A magnetic field which was felt in a certain element was a sum of the field caused by other elements in a many-body and the external field. Therefore, it has a certain inverse effect of the magnetic field as if there exists a well-known “demagnetizing field”. Based on these three experimental results the magnetic domain transition of different easy axis conditions is explained as follows:

(a) θ = 61 : The initial single domain in the direction of whole elements as shown in the photo ◦ − view change to the ILLD gradually and individually with the increase in the applied field. After the domain of all elements changes to the ILLD, then the width of the area of the striped domain gradually changes and, finally, the stripe of the ILLD is erased and all elements change to the single domain in the + direction. When the elements are all in the state of a single domain, +, the many-body element is getting magnetically saturated. (b) θ = 71 : The initial single domain in the direction of whole elements as shown in the photo ◦ − view change to the single domain in + gradually and individually with the increase in the applied field. When the elements are all in the state of single domain +, the many-body element is getting magnetically saturated.

Micromachines 2020, 11, x 8 of 17

(a)

(b)

Figure 9. Cont.

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(c)

FigureFigure 9.9. MH-loopMH-loop andand corresponding corresponding photo photo of of the the magnetic magnetic domain. domain. (a) ( Thea) The case case for easyfor easy axis θaxis= 61θ ◦=. ° ° ° (61b) The. (b) caseThe case for easy for easy axis θaxis= 71 θ ◦=.( 71c). The (c) The case case for easyfor easy axis axisθ = 90θ =◦ .90 .

FromFrom thethe viewview point point of of the the magnetic magnetic domain, domain, both both Figures Figure9b 9b and and9c had,Figure qualitatively, 9c had, qualitatively, the same profilethe same of domainprofile of transition. domain transition. This is explained This is explained based on thebased characteristic on the characteristic of the domain of the transition domain oftransition individual of elements.individual The elements. individual The domain individual transition domain was transition the switched was property the switched between property+ and betweensingle domains.+ and − single The domains. hidden ILLD The inhidden Figure ILLD9b did in Figure not appear 9b did even not inappear the case even of in many-body the case of − configuration.many-body configurat It is the sameion. asIt theis the case same of the as individual the case element. of the Theindividual domain element. transition The in Figure domain9a istransition clearly di infferent Figure from 9a the isother clearly two different conditions. from In thisthe caseother it istwo characterized conditions. by In the this appearance case it ofis thecharacterized ILLD around by zerothe appearance external field. of the ILLD around zero external field. FigureFigure 1010 showsshows aa variationvariation ofof elementelement numbernumber ofof thethe appearanceappearance ofof thethe domaindomain transitiontransition asas aa functionfunction ofof thethe appliedapplied magneticmagnetic field.field. ItIt isis plottedplotted basedbased onon thethe numbernumber ofof transitedtransited elementselements inin thethe middlemiddle areaarea of of the the many-body many-body rectangular rectangular plane. plane. Figure Figure 10 a10a shows shows a variation a variation for thefor casethe case of easy of easy axis inaxisθ =in71 θ◦ =, and71°, Figureand Figure 10b shows10b shows the case the forcaseθ for= 90 θ◦ =. The90°. measuredThe measured MH-loop MH-loop in Figure in Figure8 must 8 must be a wholebe a whole sum ofsum the of magnetization the magnetization of individual of individual elements elements in the many-bodyin the many-body element. element. Then the Then plot the of Figureplot of 10Figure would 10 expectwould toexpect match to withmatch the with MH-loop the MH in-loop Figure in 8Figure. Actually, 8. Actually, both profiles both profiles are in goodare in agreement,good agreement, especially especially the appearance the appearance of bending of be propertynding property of the magnetization of the magnetization curve in the curve vicinity in the of thevicinity zero of field. the zero field.

Micromachines 2020, 11, x 10 of 17

Micromachines 2020, 11, 279 10 of 17 Micromachines 2020, 11, x 10 of 17 (a) (b)

Figure 10. Number of domain transitions as a function of applied magnetic field. (a) The case for easy axis θ = 71°. (b) The case for easy axis θ = 90°.

3.2. Comparison with Individual Element For the purpose of discovering the effect of magnetic mutual interaction, a comparison of the magnetization process between the adjacent many-body element and the dispersed individual elements was carried out based on the variation of the domain transition number as a function of the applied field, as plotted in Figure 11. This transition was counted based on an observed domain transition of the individual element of the fabricated elements of the layout as shown in Figure 3. Figure 12 shows a detailed domain transition for an individual element as a function of applied field. The applied field was in the in-plane length direction. This θ = 67° element has a splitting appearance of the(a) ILLD as a function of the applied field. When the(b fi) eld is increasing the − single domain switched to the ILLD in the + region of the applied field, and it switches within the − Figure 10.10. NumberNumber ofof domain domain transitions transitions as as a functiona function of appliedof applied magnetic magnetic field. field. (a) The(a) caseThe forcase easy for region when the field is decreasing. This profile was already reported and explained previously [24]. axiseasyθ axis= 71 θ =.( 71b)°. The (b) caseThe case for easy for easy axis θaxis= 90 θ =. 90°. In Figure 11 the◦ number of transitions was ◦counted when the domain changed from the single domain3.2.3.2. Comparison to the ILLD with in Individual both directions Element of the changing field. Based on Figure 11; Figure 12, the domain transition of the individual elements, which was For the purpose of discovering the effect of magnetic mutual interaction, a comparison of the fabricatedFor the on purposethe same of substrate, discovering occur the at effect almost of thmagne sameetic valuemutual of interaction,the magnetic a field.comparison It is derived of the magnetization process between the adjacent many-body element and the dispersed individual elements frommagnetization the experimental process phenomenonbetween the thatadjacent the transiti many-bodyon occurs element sharply and in the the dispersed vicinity of individual the zero was carried out based on the variation of the domain transition number as a function of the applied field,elements as shown was carried in Figure out based 11. With on the the variation comparis ofon the of domain this result transition and Figure number 10a, as athe function gradual of field, as plotted in Figure 11. This transition was counted based on an observed domain transition of transitionthe applied of field,the domain as plotted is explained in Figure as 11. a resultThis tr ofansition the magnetic was counted mutual based interaction. on an observed domain thetransition individual of the element individual of the element fabricated of the elements fabricat ofed the elements layout asof the shown layout in Figure as shown3. in Figure 3. Figure 12 shows a detailed domain transition for an individual element as a function of applied field. The applied field was in the in-plane length direction. This θ = 67° element has a splitting appearance of the ILLD as a function of the applied field. When the field is increasing the − single domain switched to the ILLD in the + region of the applied field, and it switches within the − region when the field is decreasing. This profile was already reported and explained previously [24]. In Figure 11 the number of transitions was counted when the domain changed from the single domain to the ILLD in both directions of the changing field. Based on Figure 11; Figure 12, the domain transition of the individual elements, which was fabricated on the same substrate, occur at almost the same value of the magnetic field. It is derived from the experimental phenomenon that the transition occurs sharply in the vicinity of the zero field, as shown in Figure 11. With the comparison of this result and Figure 10a, the gradual transition of the domain is explained as a result of the magnetic mutual interaction. FigureFigure 11. 11. NumberNumber of domain of domain transitions transitions as a function as a function of the applied of the magnetic applied field magnetic for individual field for elements.individual elements. Figure 12 shows a detailed domain transition for an individual element as a function of applied field. The applied field was in the in-plane length direction. This θ = 67◦ element has a splitting appearance of the ILLD as a function of the applied field. When the field is increasing the single − domain switched to the ILLD in the + region of the applied field, and it switches within the region − when the field is decreasing. This profile was already reported and explained previously [24]. In Figure 11 the number of transitions was counted when the domain changed from the single domain to the ILLD in both directions of the changing field. Based on Figures 11 and 12, the domain transition of the individual elements, which was fabricated on the same substrate, occur at almost the same value of the magnetic field. It is derived from the experimental phenomenon that the transition occurs sharply in the vicinity of the zero field, as shown in Figure 11. With the comparison of this result and Figure 10a, the gradual transition of the domain is Figure 11. Number of domain transitions as a function of the applied magnetic field for individual explainedelements. as a result of the magnetic mutual interaction.

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Figure 12. Detail of the domain transition for an element of easy axis θ = 67◦. Figure 12. Detail of the domain transition for an element of easy axis θ = 67°. 3.3. Effect of Distributed Normal Field 3.3. Effect of Distributed Normal Field From here an examination of the reconstruction of the hidden ILLD is shown. The method used here wasFrom previously here an examination developed by of us the [29 ]reconstruction for the case of a of single the element.hidden ILLD It is the is methodshown. ofThe application method usedof a distributed here was previously normal field. developed In this paper by us a trial [29] forfor the the many-body case of a single element elemen is investigatedt. It is the formethod the first of applicationtime. Prior toof this a distributed paper the many-body normal field. element In this having paper the a easy trial axis for at theθ = many-body71◦ was experimentally element is investigatedshown to behave for the as iffirst it istime. the individualPrior to this element paper havingthe many-body the same element easy axis having direction. the easy The domainaxis at θ of = 71the° was consisted experimentally each element shown switched to behave as expected as if it is from the individual the behavior element of the having individual the same single easy element axis withdirection. hidden The ILLD domain state, of whereasthe consisted the domain each elemen switchingt switched does notas expected occur simultaneously from the behavior within of the individualmany-body single element. element This dispersedwith hidden behavior ILLD instat thee, whereas switched the field domain would switching be expected does to come not occur from simultaneouslythe mutual magnetic within interaction. the many-body In this section,element. an Th applicationis dispersed of thebehavior distributed in the normal switched field field was wouldtried for be the expectedθ = 71◦ tomany-body come from element. the mutual magnetic interaction. In this section, an application of the distributedFigure 13 shows normal a vectorfield was distribution tried for the of magnetic θ = 71° many-body flux density element. generated by a ring-shape magnet. ThisFigure magnet 13 was shows used a forvector generating distribution the distributedof magnetic normal flux density field in generated the apparatus by a ofring-shape domain magnet.observation, This Kerr magnet microscope. was used The for dimensions generating of thethe NdFeBdistributed ring-shape normal magnet field werein the as apparatus follows: The of domainouter diameter observation, was 30 Kerr mm, micros the innercope. diameter The dimensions was 15 mm, of the the NdFe thicknessB ring-shape was 2 mm. magnet The magnetic were as polesfollows: were The placed outer bothdiameter on the was upper 30 mm, and athe bottom inner surfacesdiameter of was the ring-disc.15 mm, the The thickness vector distributionwas 2 mm. Theshows magnetic that there poles are were two phases,placed oneboth is on an the area u inpper which and thea bottom magnetic surfaces flux forms of the a closedring-disc. circular The vectorconfiguration distribution with shows a small that diameter there are and two the phases, other is one an areais an in area which in which the vectors the magnetic bound forflux or forms from aoutside closed havingcircular almost configuration parallel configuration.with a small diameter The prior and was the placed other in is the an near area of in the which magnet the andvectors the boundlatter was for relativelyor from outside far from having the magnet. almost Inparallel this study, configuration. the many-body The prior element was was placed placed in the in thenear area of theof almost magnet parallel and the vector latter which was relatively was relatively far from far fromthe magnet. the magnet. In this study, the many-body element was placedFigure in14 athe shows area of a variationalmost parallel of Bx as ve actor function which of was the relatively X-position far at from different the magnet. distances from the magnet,Figured = 14a8.5 mmshows and a dvariation= 10 mm. of “Bdx” as was a function defined inof Figurethe X-position4. The position at differentx = 0 distances was placed from on the magnet,center axis d = of 8.5 the mm ring-shaped and d = 10 magnet, mm. “d and” was the defined measured in Figure element 4. The was position placed on x the= 0 was X–Y placed plane at on a thecertain center Z-position, axis of the which ring-shaped means di ffmagnet,erent d-positions and the measured in this measurement. element was The placed variations on the in X–Y the figureplane atshow a certain linear configurationsZ-position, which having means diff erentdifferent inclination, d-positions one isin Bthisx (mT) measurement.= 0.54 x (mm) The atvariationsd = 10 mm in × theand figure the other show is Blinearx (mT) configurations= 1.22 x (mm) having at d = different8.5 mm. inclination, The normal one field isB Bz xas (mT) a function = 0.54 × of x X-position(mm) at d × =is 10 shown mm and in Figure the other 13b. is The Bx normal(mT) = 1.22 field × at x x(mm)= 0 was at d 11.5 = 8.5 mT mm. at d The= 10 normal mm and field 8.7 B mTz asat a dfunction= 8.5 mm. of TheX-position normal is field shown in the in element’sFigure 13b. rectangular The normal area field varies at x at = most0 was 8%, 11.5 therefore, mT at d it= 10 has mm a slight and di8.7ff erencemT at dfrom = 8.5 the mm. ideal The variation normal (seefield Figure in the6 ).element’s rectangular area varies at most 8%, therefore, it has a slightThe difference result of from the applicationthe ideal variation of the distributed (see Figure normal6). field to the element having the easy axis The result of the application of the distributed normal field to the element having the easy axis in θ = 71◦ is shown as follows: Figure 15 shows a magnetic domain at Bx = 0, without applying a indistributed θ = 71° field. is shown Figure as 16follows:a shows Figure a magnetic 15 shows domain a magnetic when the domaind = 10 at mm Bx in= 0, which without the distributedapplying a distributed field. Figure 16a shows a magnetic domain when the d = 10 mm in which the distributed parameter ∆Bx/∆x = 0.54 (T/m), and Figure 16b shows a magnetic domain when d = 8.5 mm in which

Micromachines 2020, 11, 279 12 of 17

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Micromachinesparameter ∆ 2020Bx/∆, 11x, =x 0.54 (T/m), and Figure 16b shows a magnetic domain when d = 8.5 mm in12 which of 17 the distributed parameter parameter ∆∆BBx/x∆/∆xx == 1.221.22 (T/m). (T/m). These These results results show show that that the the ILLD ILLD state state appears appears in the in central area of the many-body element when the distributed normal field is applied. These ILLD the distributed central area parameter of the many-body ∆Bx/∆x = element 1.22 (T/m). when These the distributedresults show normal that the field ILLD is applied. state appears These in ILLD the centralhave aa certainareacertain of periodical theperiodical many-body pattern pattern element in thein the element when element linesthe dilines instributed spite in ofspite thenormal randomly-switchedof the field randomly-switched is applied. single These domain singleILLD haveappearancedomain a certainappearance at the periodical zero at field the pattern withoutzero field in the withoutthe distributed element the normal distributedlines in field. spite normal This of periodicalthe field. randomly-switched This pattern periodical has a variation pattern single domainashas a a function variation appearance of as the a distributedfunction at the zero of parameter.the field distributed without parameter.the distributed normal field. This periodical pattern has a variation as a function of the distributed parameter.

Figure 13. The vector of the magnetic ﬂux density generated by a ring-shaped magnet. Figure 13. The vector of the magnetic flux density generated by a ring-shaped magnet. Figure 13. The vector of the magnetic flux density generated by a ring-shaped magnet.

(a) (b)

Figure 14. The distribution(a) of the magnetic field generated by a ring-shaped(b) magnet. (a) Variation of Bx as a function of x. (b) Variation of Bz as a function of x. Figure 14. TheThe distribution distribution of of the the magnetic field ﬁeld generated by a ring-shaped magnet. ( (aa)) Variation Variation of Bx asas a a function function of of x.x. ((bb)) Variation Variation of of Bzz asas a a function function of of xx..

Figure 15. Magnetic domain without applying distributed field (θ = 71°). Figure 15. Magnetic domain without applying distributed ﬁeldfield (θ = 71°◦).).

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(a) (b)

FigureFigure 16. 16.Magnetic Magnetic domain domain with with applying applying distributed distributed ﬁeld field (θ (θ= =71 71◦).°). ( a()a) Distributed Distributed parameter parameter ∆∆BxB/x∆/∆xx= =0.54 0.54 (T (T/m)./m). ( b(b)) Distributed Distributed parameter parameter∆ ∆BBx/x∆/∆xx= =1.22 1.22 (T(T/m)./m).

4. Discussion 4. Discussion AA consideration consideration concerningconcerning thethe magneticmagnetic mutual interaction within within the the many-body many-body element element is isdiscussed discussed here. here. A Aqualitative qualitative estimation estimation of the of the interaction interaction is carried is carried out outbased based on a on magnetic a magnetic field fielddistribution distribution originating originating from from one one narrow narrow magnetic magnetic strip strip which which is is a a construction elementelement ofof the the many-bodyMicromachinesmany-body 2020 element. element., 11, x 14 of 17 FigureFigure 17 17 shows shows a vector a vector diagram diagram of magnetic of magnetic flux density flux density arising arising from one from narrow one strip narrow element. strip monotonically decreases as a function of De, such as Bx = 2.7 × 10−6 T (27 mG) at De = 20 µm and Bx = Weelement. call the We magnetic call the fluxmagnetic density flux the density “magnetic the “mag field”,netic the field”, same asthe previous same as section. previous The section. analysis The 1.0 × 10−6 T (10 mG) at De = 1.5 mm. This magnetic field is much smaller than the maximum value of isanalysis done using is done a 3D using FEM a simulation3D FEM simulation of the static of magneticthe static magnetic field. In thisfield. analysis In this theanalysis single the narrow single the one in Figure 18, whereas when we consider a cumulative effect of the many-body element elementnarrow having element a magnetichaving a singlemagnetic domain single in domain the longitudinal in the longitudinal direction was direction modeled was as onemodeled which as was one which has 76 adjacent lines in an element, the sum of the magnetic fields would be expectedM to magneticallywhich was magnetically saturated along saturated the X+ directionalong the with X+ direction a value of with saturation a value magnetization of saturation asmagnetizations = 0.93 T. −4 ThisexceedM sthevalue transition was the field measured 500 mOe value (0.5 of× 10 Co85 T).Nb 12Zr3 amorphous thin film. Figure 17 is the vector as Ms = 0.93 T. This Ms value was the measured value of Co85Nb12Zr3 amorphous thin film. Figure 17 According to the experimental observation in this study, the blocking effect of the domain distributionis the vector on thedistribution substrate on plane the which substrate comes pl fromane which the saturated comes narrowfrom the strip saturated element. narrow This figure strip transition in the neighboring element of a transited single narrow element was observed, as can be indicateselement. a This simple figure directional indicates distribution a simple directional diagram, therefore,distribution the diagram, length of ther theefore, arrow the does length not showof the seen in Figure 9. The measured MH-loop of the many-body element, as shown in Figure 8, also thearrow strength does ofnot the show magnetic the strength field. It of is easilythe magnetic presumed field. that It theis easily distribution presumed forms that a the magnetic distribution field suffered an effect as if it suffered a demagnetizing field. These effects would be considered to come arisingforms froma magnetic one source field and arising one sink from configuration. one source and In this one case sink both configuration. the source and In the this sink case are both placed the from both the short-range effect of magnetic field generated at the edge of a single narrow element, atsource the end and of the narrowsink are element. placed at The the simulation end of the layout narrow was element. set as the The element simulation axis was layout placed was along set as and also the cumulative effect based on the accumulation of the widely spread weak field within thethe X element axis and axis the centerwas placed position along of thethe element X axis and was th sete center at the origin.position A of distance the element from the was element set at the is shownthe whole as D areae. of adjacent many-body elements. origin. A distance from the element is shown as De. Figure 18 shows the magnetic field distributions along the axis of elements in the neighboring position of the line arrangement many-body element. The L/S = 20/20 μm configuration made the distance of axes to be the multiples of 40 µm. In this figure, variations of magnetic field Bx, which is the field in the direction of longitudinal axis is shown as a parameter of element distance De. The horizontal axis of the figure shows a longitudinal position x on the axis of the neighboring virtual element. The direction of magnetic field Bx has a negative value, due to the magnetized direction of the existing element being positive, as shown in Figure 17. For the reason of easy understanding the vertical value was made as an absolute value of Bx. The element distance De ranges from 40 µm to 200 µm, which corresponds to the position from the just neighboring to the 5th neighboring element. Figure 18 shows that the |Bx| has a maximum value in the vicinity of the end of the element, the range was almost 0.3 mm from the end, and the maximum value rapidly decreased with the increase in the De value. From Figure 12 the magnetic domain transition appears around 500 mOe (0.5 × 10−4 T). The magnetic field which exceeds this value would expect to have a promoting effect of domain transition. The inverse field generated by a transited element have a blocking effect for the neighboring elements. The effect is expected to be strongly affected to the 1st, 2nd, and 3rd neighboring elements. Figure 17. VectorVector diagram of the magnetic flux flux density arising from one narrow strip element. The magnetic field distribution in Figure 17 also shows that a certain weak field widely spread around the single element. The field ranges more than 1.5 mm. Figure 19 shows the variation of magnetic field Bx on the Y-axis of Figure 17 as a function of the distance from the element De. The Bx

Figure 18. Magnetic field distributions along the axis of elements in the neighboring element position.

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monotonically decreases as a function of De, such as Bx = 2.7 × 10−6 T (27 mG) at De = 20 µm and Bx = 1.0 × 10−6 T (10 mG) at De = 1.5 mm. This magnetic field is much smaller than the maximum value of the one in Figure 18, whereas when we consider a cumulative effect of the many-body element which has 76 adjacent lines in an element, the sum of the magnetic fields would be expected to exceed the transition field 500 mOe (0.5 × 10−4 T). According to the experimental observation in this study, the blocking effect of the domain transition in the neighboring element of a transited single narrow element was observed, as can be seen in Figure 9. The measured MH-loop of the many-body element, as shown in Figure 8, also suffered an effect as if it suffered a demagnetizing field. These effects would be considered to come from both the short-range effect of magnetic field generated at the edge of a single narrow element, and also the cumulative effect based on the accumulation of the widely spread weak field within the whole area of adjacent many-body elements.

Micromachines 2020, 11, 279 14 of 17

Figure 18 shows the magnetic field distributions along the axis of elements in the neighboring position of the line arrangement many-body element. The L/S = 20/20 µm configuration made the distance of axes to be the multiples of 40 µm. In this figure, variations of magnetic field Bx, which is the field in the direction of longitudinal axis is shown as a parameter of element distance De. The horizontal axis of the figure shows a longitudinal position x on the axis of the neighboring virtual element. The direction of magnetic field Bx has a negative value, due to the magnetized direction of the existing element being positive, as shown in Figure 17. For the reason of easy understanding the vertical value was made as an absolute value of Bx. The element distance De ranges from 40 µm to 200 µm, which corresponds to the position from the just neighboring to the 5th neighboring element. Figure 18 shows that the |Bx| has a maximum value in the vicinity of the end of the element, the range was almost 0.3 mm from the end, and the maximum value rapidly decreased with the increase in the 4 De value. From Figure 12 the magnetic domain transition appears around 500 mOe (0.5 10 T). The × − magnetic field which exceeds this value would expect to have a promoting effect of domain transition. The inverseFigure field 17. generated Vector diagram by a transitedof the magnetic element flux have density a blocking arising from effect one for narrow the neighboring strip element. elements. The effect is expected to be strongly affected to the 1st, 2nd, and 3rd neighboring elements.

Figure 18. Magnetic field distributions along the axis of elements in the neighboring element position. Figure 18. Magnetic field distributions along the axis of elements in the neighboring element Theposition. magnetic field distribution in Figure 17 also shows that a certain weak field widely spread around the single element. The field ranges more than 1.5 mm. Figure 19 shows the variation of magnetic field Bx on the Y-axis of Figure 17 as a function of the distance from the element De. The 6 Bx monotonically decreases as a function of De, such as Bx = 2.7 10− T (27 mG) at De = 20 µm and 6 × Bx = 1.0 10 T (10 mG) at De = 1.5 mm. This magnetic field is much smaller than the maximum × − value of the one in Figure 18, whereas when we consider a cumulative effect of the many-body element which has 76 adjacent lines in an element, the sum of the magnetic fields would be expected to exceed the transition field 500 mOe (0.5 10 4 T). × − According to the experimental observation in this study, the blocking effect of the domain transition in the neighboring element of a transited single narrow element was observed, as can be seen in Figure9. The measured MH-loop of the many-body element, as shown in Figure8, also su ffered an effect as if it suffered a demagnetizing field. These effects would be considered to come from both the short-range effect of magnetic field generated at the edge of a single narrow element, and also the cumulative effect based on the accumulation of the widely spread weak field within the whole area of adjacent many-body elements. Micromachines 2020, 11, x 15 of 17 Micromachines 2020, 11, 279 15 of 17

Figure 19. Variation of magnetic field Bx as a function of the distance from the element De. Figure 19. Variation of magnetic field Bx as a function of the distance from the element De. 5. Summary 5. Summary An investigation of the stepped-MI phenomenon for the clustered many-body elements was experimentallyAn investigation carried of out. the stepped-MI The 76 elements phenomen wereon arranged for the clustered in a planar many-body cluster with elements the line was/ spaceexperimentally= 20 µm/20 carriedµm having out. aThe configuration 76 elements of were the line arranged arrangement in a planar adjacent cluster to many-body with the line/space elements. = The20 µm/20 individual µm having element a in configuration it has the same of widththe line and arrangement thickness, andadjacent also hasto many-body the direction elements. of magnetic The anisotropyindividual the element same asin anit elementhas the same having width the stepped-MI and thickness, property and withalso has a hidden the direction ILLD state, of magnetic which is θanisotropy= 71◦. The the MH-loop same as of an the element planar having clustered the elementstepped-MI was property measured with in accordancea hidden ILLD with state, a domain which observation.is θ = 71°. The The MH-loop MH-loop of of the the planar clustered clustered many-body element element was measured which has in the accordance easy axis inwithθ = a71 domain◦ was almostobservation. the same The as MH-loop the one having of the the clustered easy axis ma inny-bodyθ = 61◦ and elementθ = 90 which◦. The has each the element easy axis has suinff θered = 71 a° mutualwas almost magnetic the same interaction as the between one having the consistingthe easy axis element in θ strips,= 61° and as if θ it = su 90ffered°. The the each demagnetizing element has force.suffered The a variation mutual ofmagnetic magnetic interaction domain for between the θ = the71◦ consistingmany-body element element strips, was a as gradually if it suffered changed the fromdemagnetizing the number force. of switched The variation elements of having magnetic the singledomain domain, for the and θ = which 71° many-body is the same aselement the domain was a variationgradually of changedθ = 90◦. from An eff theect ofnumber the application of switched of theelements distributed having normal the single field wasdomain, also investigated.and which is Thethe ILLDsame appearedas the domain by applying variation the of distributed θ = 90°. An normal effect field. of the A periodicityapplication of of the the domain distributed distribution normal formedfield was by thealso consisting investigated. elements The was ILLD observed, appeared and by the applying periodical the pattern distributed changed normal as a function field. ofA theperiodicity distributed of parameterthe domain∆ Bdistributionx/∆x. formed by the consisting elements was observed, and the periodical pattern changed as a function of the distributed parameter ∆Bx/∆x. Funding: This research received no external funding. Funding: This research received no external funding. Acknowledgments: The author’s special thanks are due to Hiroyuki Abe, Industrial Technology Institute, Miyagi PrefecturalAcknowledgments: Government, The for author’s support special of micro-fabrication thanks are due process. to Hiro Theyuki author Abe, also Industrial gratefully Technology acknowledges Institute, the ITIM sta for kind support of this research activity. Miyagi ffPrefectural Government, for support of micro-fabrication process. The author also gratefully Conflictsacknowledges of Interest: the ITIMThe staff author for declareskind support no conflict of this of research interest. activity.

Conflicts of Interest: The author declares no conflict of interest. References

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