1. Introduction
The magnetic domain is a region within a magnetic material [
1,
2]. The magnetization within each domain points in a uniform direction, but the magnetization of different domains may point in a different direction. The regions separating magnetic domains are called domain walls. The magnetic domains and the domain walls are generated in order to reduce the overall free energy, mainly the magneto-static energy, in magnetic systems. Thus, without the application of external magnetic fields, the magnetic domain and domain walls are unavoidable in patterned soft-magnetic material systems. As they can have different shapes and widths, the magnetic domains are an exciting playground for fundamental research and have become in recent years the subject of intense studies [
3]; these studies have mainly been focused on controlling, manipulating, and moving their internal magnetic configuration; their usage in information storage [
4,
5,
6]; their performance in logic operations [
7,
8,
9]; high frequency inductors [
10]; and sensor applications.
The domain structure has been studied in the research field of magnetic inductors and sensors using soft magnetic materials. The relationship between the magnetic domain and permeability has been extensively studied for ferromagnetic materials. The basic magnetization process consists of domain wall movement, magnetization rotation, and magnetization reversal. They are intricately intertwined in a manner that depends on the inner structure, such as crystal structure, anisotropy property, magnetostriction, defects, and grain size [
11].
For the study of the inductor, the relationship between permeability and the magnetic domain was recognized as indispensable. It was initiated in the study of silicon iron [
12,
13,
14]. The improvement of magnetization properties was studied based on the structural and dimensional change in the magnetic materials, such as in the study of amorphous powder, the size dependence of spherical ferromagnetic particles, and the soft magnetic multilayered films [
15,
16,
17]. Some recent studies on modeling and simulation had advantageous results, such as the improvement of the performance of magnetic shielding [
18]. In order to clarify the magnetization process for the actual application, the domain wall propagation is essential; therefore, some attempts were made to achieve it, such as those in the study on domain boundary propagation velocity, the domain wall transverse caused by fast-rising current pulses, and the effect of plastic deformation on the permeability [
19,
20,
21].
Many studies have also been conducted on magneto-impedance (MI) sensors. The recent performance improvement of the magnetic domain observation has contributed to the investigation of the sensor from the viewpoint of the dependence of high-frequency impedance on the magnetic domain structure of the thin-film element. This electrical response against the alternating current which depends on the magnetic domain is closely related to this study; an application to the MI sensor is one of the expected candidates. The physical fundamentals of the MI sensor have been studied to clarify the sensing mechanisms [
22,
23,
24]. Thereafter, the study of the thin-film MI sensors was carried out for the purpose of improving the properties of the MI sensor based on the control of the magnetic domains. The study subjects ranged from sensor structures, materials, and treatment methods to biasing [
25,
26,
27,
28,
29,
30,
31,
32,
33,
34]. Regarding the magnetic domain simulation, an original method was developed which was adaptive for the actual size of the MI element, which are difficult to apply using the conventional micro-magnetic simulations due to the larger dimensions of the element [
35,
36].
This paper considers a narrow rectangular element made of amorphous soft magnetic thin film with an inclined in-plane magnetization easy axis. It is well known that a sputter-deposited amorphous thin film is able to induce a uniaxial magnetization easy axis using magnetic field annealing [
11]. Our previous study shows that a narrow rectangular strip of amorphous thin film with an in-plane inclined easy axis has two different magnetic domain transition properties. One is a multi-domain with inclined striped domain configuration, in which the width of the stripe gradually changes as a function of the external magnetic field. The other is a single domain with a magnetic moment along the length direction of the strip, in which the magnetization reversal appears as a function of the external field. A typical characteristic of magnetic domain transition was reported when the easy axis lay at around 70° [
37], which is a transition condition of the easy axis angle between these different magnetic domain variations. This study aims to apply this phenomenon to a functional device with a switching as well as a memorizing property. In this transitional condition, the magnetic domain transition between the single domain and the multi-domain is artificially controlled by applying a surface normal magnetic field with a certain distribution [
38,
39]. The details are explained in the next paragraph.
Figure 1 shows the schematics of the domain transition of this case and the effect of the applied distributed normal field. This figure shows the cases both with and without the application of the distributed normal field. When the normal field is not applied, the domain apparently switches between the single domain + direction and the − direction (
Figure 1a). There is a non-apparent multi-domain state, which is shown by a short line in the middle of this figure. This state has lower energy compared with the single-domain one; thus, this state is a stable one [
40].
Figure 1b shows a case when a distributed normal field is applied. In this case, the multi-domain state appears and the transition from the single domain to the multi-domain is available. This was confirmed experimentally [
38], and a sensor with a memory function or a three-state memory as an application was proposed.
We also discovered that this phenomenon appears for a clustered many-body element, with an almost 70° easy axis, even in the conditions of the existing magnetic mutual interaction between the multiple elements [
41]. It was also reported that the magnetic permeability for the alternating current (AC) magnetic field changes drastically in the frequency range from 10 kHz to 10 MHz as a function of the strength of the distributed magnetic field [
42,
43]. It was also noticed in a newly reported typical phenomenon, which has recently been reported [
44], that there was an appearance of the multi-domain state when a small magnetic particle was placed just above the strip to control the applied surface normal magnetic field. These reports provide hope for the achievement of an integrated device applying this phenomenon. Based on the background, this article’s study aims to clarify the basis of the permeability caused by the magnetic domain transition of this phenomenon.
Figure 1.
Magnetization diagram with a hidden multi-domain state. (
a) Without surface normal magnetic field. (
b) With a distributed normal magnetic field. The numbers in the figure indicate as follows: 1—longitudinal single domain (parallel direction); 2—longitudinal single domain (anti-parallel direction); 3—inclined Landau–Lifshitz domain [
45,
46] (hidden stable state); 3’—inclined Landau–Lifshitz domain (state of transition available); 4—canted normal field with distribution, where
Bz = const. and Δ
Bx/Δ
x = const.; 5—sensor element. The “Memorized state” and the “Reset state” indicates a function of three-state memory.
Figure 1.
Magnetization diagram with a hidden multi-domain state. (
a) Without surface normal magnetic field. (
b) With a distributed normal magnetic field. The numbers in the figure indicate as follows: 1—longitudinal single domain (parallel direction); 2—longitudinal single domain (anti-parallel direction); 3—inclined Landau–Lifshitz domain [
45,
46] (hidden stable state); 3’—inclined Landau–Lifshitz domain (state of transition available); 4—canted normal field with distribution, where
Bz = const. and Δ
Bx/Δ
x = const.; 5—sensor element. The “Memorized state” and the “Reset state” indicates a function of three-state memory.
In this paper, the correspondence between the AC permeability and the structure of the magnetic domain as a function of the intensity of the distributed field is investigated. It was confirmed that the extension of the area of the Landau–Lifshitz-like multi-domain (LLD) [
45,
46] on the clustered narrow strips was observed as a function of the intensity of the distributed magnetic field. This domain extension was compared with the permeability variation, and it was revealed that they match with each other. The effect of the distributed field was compared with an element which has the easy magnetization axis along the length direction (90°), and it was revealed that the 70° easy axis has the clear transition to the LLD domain structure and also that it needed a lower distributed field to change the permeability. The result leads to the application of this phenomenon to a tunable inductor, electromagnetic shielding, or a sensor for detecting and memorizing the existence of distributed magnetic field generated by a magnetic nanoparticle in the vicinity of the sensor.
3. Results
In this section, the results of the magnetic domain variation for two different directions of the magnetization easy axis are shown. One is the case for θ = 71°, and the other is in the direction of almost θ = 90°. The element structures were the same as the structure of the adjacent narrow strips of a 3000 μm length, with L/S = 20 μm/20 μm, which were formed as a 3000 μm × 3000 μm square element. The quasi-steady magnetization curve, which was the MH loop measured by VSM, indicated, along the longitudinal direction, their soft magnetic property, which was reported in our previous papers [
41,
42]. According to the papers, the saturation magnetization of 0.96T appeared at 3.5 Oe of the applied magnetic field. The coercivity had almost the same value, 0.45 Oe, for both easy axis cases; then, they had a soft magnetic property. Under these conditions, the magnetic domain of the individual strips in the clustered element forms a single domain with the magnetic moment along the bi-directional length direction. The number of strips having the same direction as the direction of the external magnetic field increases as a function of the strength of the field. Under the condition of a zero external field, both directional strips are mixed and form the equilibrium state. As shown in the introduction of this paper, the variation in the magnetic domain throughout the whole strip length was investigated, when a distributed surface normal field was applied; the correlation of the permeability of the alternating current magnetic field was discussed in the following study of this paper. Firstly, the condition of θ = 71° is dealt with based on the previously shown typical AC permeability of the element [
43]. Secondly, a newly reported magnetic domain of almost θ = 90° and its AC permeability are reported and discussed, depending on the distributed surface normal field.
The magnetic domain was observed from the left end to the right end of the 10 strips placed in the middle of the width direction of the clustered many-body elements. The observation division and their corresponding index symbols are schematically shown in
Figure 10. The observed domain photos are continuously connected from one end to the other end. The 3000 μm length elements were divided into six areas of photographs: L1, L2, and L3 from the left end and R1, R2, and R3 from the right end. Due to the observation procedure of the Kerr microscope, in which the magnetically saturated reference image must be obtained in advance of the observation of the domain in a certain external magnetic field, it should be noticed and explained that the structure of the magnetic domain of the same element in the neighboring photos is not always smoothly connected. In the following results, both the actual domain photographs and the schematic illustrations of the tendency of the domain variation are shown, depending on the strength of the distributed field.
3.1. Magnetic Domain Variation for θ = 71° Element
3.1.1. Domain Observation
Figure 11 shows a magnetic domain variation in the case of θ = 71°, as a function of the applied intensity of the distribution of the surface normal magnetic field. The whole lengths of the 10 strips were observed in the neighboring six photographs. The observation layout and the index symbols are shown in
Figure 10.
Figure 11a indicates a case when the intensity of the distributed field was 0.89 G/mm. This condition was obtained when the current of the field generating coil was set at zero, due to the existing residual magnetization of the steel core.
Figure 11a consists of contiguous bright and dark elements, which indicate an opposite directional single domain. There were some residual LLD stripe domains in the edge area of each strip. There are several photos which have an LLD domain around the middle area of the length of the strip, which is observed in the bottom strip of the photo. The observation magnetic field includes a slightly distributed field, due to the residual magnetization of the core; thus, it is assumed that this is the reason for the appearance of the LLD domain in the middle area of the length of the strip.
Figure 11c indicates a case when the intensity of the distributed field was 5.39 G/mm. In this case, the LLD stripe domain area extends toward the middle area of the strips, and it appears in every other strip in the observation area. L2 and R2 are the typical cases of this. In the neighborhood of the LLD element, a single-domain strip exists; the single domain of the bright ones appear in the L area, which is in the left moment direction, and the dark ones appear in the R area, which is in right moment direction.
Figure 11d indicates a case when the intensity of the distributed field was 9.89 G/mm. It is a relatively large intensity for the element. There is a single-domain area in the outermost edge areas, L1 and R1, and also around the middle position of the next edge areas, L2 and R2. The LLD stripe domain area shrinks and is limited to the vicinity of the edge area, the residual edge domain, and the middle area of strip.
In
Figure 12, a schematic of the magnetic domain variation as a tendency of the whole clustered element, which corresponds to
Figure 11, is shown.
Figure 12a shows a case which corresponds to
Figure 11a. It is a domain structure without a distributed surface normal field for the magnetization easy axis of θ = 71°.
Figure 12b indicates a case which corresponds to
Figure 11c, in which the LLD extends toward the middle area of the element strips. In this case, the single strip, which consists of the clustered element, is divided into two areas; one is the LLD stripe domain and the other is the single domain with a magnetic moment in the longitudinal direction. When the left side of the strip has the stripe domain, the right side has a single domain towards right-end direction. In the case that the right side of the strip has the stripe domain, the left side has a single domain towards the left-end direction.
Figure 12c indicates a case which corresponds to
Figure 11d. In this case, the intensity of the distribution of the applied magnetic field has a certain large value, then the end-side area of the strip has single domain toward the outer direction, due to the magnetic force toward the outer direction. Even in this case, the edge residual stripe domain was still observed.
To summarize the magnetic domain variation in this sub-section, which is when the element has the magnetization easy axis direction of θ = 71°, the distributed field has the effect of changing the sectional ratio of the LLD stripe domain versus the single domain.
Figure 13 shows a schematic explanation of this effect. As the intensity of the distributed field increases, the single-domain area gradually decreases, and the border of the single domain and the multi-domain moves toward the middle of the narrow strip. The red arrow in this figure indicates the variation direction of this phenomenon.
Figure 14 shows a variation in an area ratio of the LLD stripe domain as a function of the intensity of the distributed magnetic field. As the intensity increases, the area of the LLD stripe domain increases until the horizontal value of the intensity is up to 5.4 G/mm. The intensity of the distributed magnetic field, 5.4 G/mm, corresponds to
Figure 11c and
Figure 12b. When the intensity of the distributed field increases more than 5.4 G/mm, both of the end areas of the narrow strip change to a single domain with a moment direction toward the outer direction in the length of each strip, as shown in
Figure 11d and
Figure 12c.
3.1.2. Comparison of Alternating Current Permeability and Magnetic Domain
Figure 15 indicates our previously reported alternating current (AC) permeability of the clustered element with the magnetization easy axis of θ = 71° [
43]. It indicates a measured complex permeability as a parameter of the intensity of the distributed field.
Figure 15a shows a real part of the permeability, and
Figure 15b shows an imaginary part of the permeability. Both of them are indicated as a function of the frequency of the AC magnetic field.
Here, a discussion of the effect of the distributed magnetic field is provided, based on the comparison of the permeability and the magnetic domain variation.
As reported in the previous paper [
43], which is reposted as
Figure 15 in this paper, the real part of the permeability gradually increases its value and then saturates around the relative permeability of 370. This variation in the increment and saturation of the permeability appears initially at a low frequency; then, it extends its frequency range up to 1 MHz. The highest permeability and the wider frequency range were obtained when the applied intensity of the distributed field had the value of 5.4 Oe/mm. The value of the 5.4 Oe/mm distribution intensity was a value in which a wide multi-domain extension was observed experimentally in this paper, as shown in the previous subsection. It is assumed that the higher AC permeability in the frequency range of our experiment comes from the domain wall movement; thus, the extension of the multi-domain area is a reason for this phenomenon. The imaginary part of the AC permeability has a typical tendency, in which the peak frequency shifts from low frequency to high frequency. This phenomenon appeared in accordance with the variation in the distributed fields. The observation shows that there is almost a single domain in the element under the condition of the weak intensity of the distributed field, and the area ratio of the multi-domain increases and extends more widely when the intensity is 5.4 Oe/mm. It is well known that the imaginary permeability means a lost parameter. Then, the phenomenon would be assumed to come from a change in the resonant condition depending on the change in the domain state. A comparison between the AC permeability and the magnetic domain variation shows that it has a certain linear relation, which is assumed to come from a tendency of the proportional increment of the LLD stripe domain sectional ratio, as shown in
Figure 14.
3.2. Magnetic Domain Variation for θ = 90° Element
In this subsection, an observation of the magnetic domain and a discussion on the comparison of the permeability, as a function of the intensity of the distributed magnetic field, is given for the element which has the magnetization easy axis of θ = 90°. The experimental methods and the development of the argument are the same as those of the previous subsection for the different easy axis angle, θ = 71°. In our previous study [
41], the clustered element with a θ = 90° easy axis has a directional switching property in the magnetic domain. When the distributed magnetic field does not apply to the element, the direction of the magnetic moment in each magnetic narrow strip in the clustered element has the single domain along the length direction and its direction is switched by applying a certain value of the in-plane uniform external magnetic field. In this subsection, the effect of the distributed field for the element of θ = 90° is investigated.
3.2.1. Domain Observation
Figure 16 shows a magnetic domain variation in the case of θ = 90°, as a function of the applied intensity of the distributed magnetic field.
Figure 16a indicates a case when the intensity of the distributed field was 0.89 G/mm, which is the value of the coil current at zero.
Figure 16a consists of contiguous bright and dark elements, which indicate a single domain with an opposite direction. There were typical residual magnetic domains in the edge area of each strip, which consisted of long triangular domain lengthening along the length direction of the strip, as if the domain wall divided the magnetic strip into two along the length direction.
Figure 16c indicates a case when the intensity of the distributed field was 5.39 G/mm. In this case, the multi-domain area extends towards the inner area of the strip. There are several patterns of multi-domain in this area, which are different from the previous θ = 71° case.
Figure 17 shows the schematics of the typical multi-domain structure, which is observed in
Figure 16c. These are typical domain patterns for the multi-domain in the case of θ = 90° with the application of the distributed field. The observed index number of the photo and the line number are indicated below each schematic. The multi-domain appeared in the border area between the single domain+ and the single domain−. The multi-domain structure is not clear compared with the previous case of θ = 71°; consequently, a quantitative discussion of the area ratio is difficult to carry out.
Figure 16d indicates a case when the intensity of the distributed field was 9.89 G/mm. It is a relatively large intensity for the element. There is a whole single domain in the near area of the edge of L1 and R1. In this case, the multi-domain area seems to shrink and is limited in the vicinity of the middle of the strip.
In
Figure 18, a schematic of the magnetic domain variation as a tendency of the whole clustered element, corresponding to
Figure 16, is shown.
Figure 18a shows a case which corresponds to
Figure 16a. It is a domain structure without a distributed surface normal field for the magnetization easy axis of θ = 90°.
Figure 18b indicates a case which corresponds to
Figure 16c, in which the multi-domain area extends toward the middle area of the element strips. In this case, the single strip, which consists of the clustered element, is divided into three areas; the first one is the single domain with a magnetic moment in the longitudinal direction +; the second one is the single domain in the longitudinal – direction; and the third one is their border area with the abovementioned typical multi-domain structure in
Figure 17. When the left side area of the strip has the single domain −, the right side has single domain + towards the right end direction. The multi-domain area appears between them. In
Figure 18, the multi-domain area is indicated as the bottom pattern of
Figure 17, the triangular multi-domain, to simplify the expression.
Figure 18c indicates a case which corresponds to
Figure 16d. In this case, the intensity of the distribution of the applied magnetic field has a certain large value; then, the end-side area of the strip has a single domain toward the outer direction, due to the magnetic force toward the outer direction. Even in this case, a small edge residual triangular domain was observed, and the multi-domain area also exists between the opposite single-domain areas.
To summarize the magnetic domain variation in this subsection, which concerns the element with an easy axis of θ = 90°, the distributed field has an effect of generating the multi-domain area between the single-domain + and − areas.
Figure 19 shows a schematic explanation of this effect. As the intensity of the distributed field increases, the multi-domain area appears around the border area of the single-domain + and − areas. The multi-domain border area moves toward the middle of the narrow strip, as the intensity of the distributed surface normal magnetic field increases. The red arrow in this figure indicates the variation direction of this phenomenon. The multi-domain area seems to widen; then, the area ratio would increase when the intention of the distributed field was around 5.4 G/mm; however, a quantitative discussion is difficult due to the uncertainty of the discrimination of the multi-domain existence.
3.2.2. Comparison of Alternating Current Permeability and Magnetic Domain
Figure 20 indicates an AC permeability of the clustered element with the magnetization easy axis of almost θ = 90°. It indicates a measured complex permeability as a parameter of the intensity of the distributed field; the real part is in
Figure 20a, and the imaginary part is in
Figure 20b. Both of them are shown as a function of the frequency of the AC magnetic field.
It is mentioned that the variation tendency is the same as that of the θ = 71° element, although it has a different domain transition property, especially for the structure of the multi-domain. The point of difference between them is the existence of the LLD stripe domain for θ = 71°, but the point of these phenomena is assumed to be the appearance of the multi-domain area and the enhancement of the effect of the domain wall movement. A comprehensive and detailed discussion is provided in the following discussion section.
4. Discussion
In this study, the effect of the distributed field on the magnetic domain variation for the clustered soft magnetic strips with a certain inclined magnetization easy axis was investigated. The observed variation in the magnetic domain was compared with the permeability variation, in order to reveal the effect of the distributed field on the AC permeability. The element, which has the in-plane inclined magnetization easy axis of θ = 71° and θ = 90°, and also has the width of the single strip of the clustered element as 20 μm, was experimentally confirmed as having low AC permeability in the frequency range from 10 kHz to 10 MHz, when it has a single domain structure in each strip. In the case of the distributed magnetic field when a suitable value is applied to the element, the original single domain with the magnetic momentum along the length direction changes to the multi-domain state. The sectional ratio of the multi-domain area changes as a function of the intensity of the distributed field, which was clearly observed in the case of θ = 71°. For the different easy axis cases, which were between θ = 71° and θ = 90°, the experimentally observed domain structures of the multi-domain were quite different, whereas the tendency of the permeability variation was almost the same under these two conditions. There is a possibility of the existence of an effect of the enhancement of the domain wall movement caused by the distributed field and the inclined easy axis in a certain domain structure.
Additionally, It should be mentioned that a typical domain transition was experimentally confirmed for the θ = 71° clustered element, which would seem to be the effect of the existence of the hidden stable multi-domain state. It was recently reported [
44] that a small soft magnetic particle placed on the element was able to induce a domain transition from a single domain to a multi-domain. It was explained that a distributed magnetic field generated by the dipole magnetization of the particle enhanced the domain transition.
In this discussion, a comprehensive overview of the phenomenon concerning the element with the hidden multi-domain state and its neighboring condition is summarized. The future investigation of these phenomena is also discussed.
Firstly, the phenomenon of low AC permeability is discussed for the clustered element with the inclined easy axis. As explained in references [
42,
43], the low permeability typically appeared for the narrow strips, which have a width of 20 μm and the easy axis direction of more than θ = 70°. In the previous study [
42], the parameters of the experiment ranged from strip widths of 20 μm to 100 μm and from θ = 61° to θ = 90° in the easy axis direction. The single domain in the strips, which causes the low AC permeability, was obtained when the easy axis was larger than θ = 71° and also when the width was 20 μm. The results are summarized in
Figure 21. The Figure shows a matrix expression for the element width W and the easy axis direction θ. The “
M” indicates a multi-domain state, and the “
S” indicates a single-domain state. The hatched area indicates the low AC permeability conditions. Despite the single domain, the θ = 90° and the W = 50 μm element has a high-permeability property. The triangular-shaped residual magnetic domain in the edge area of the strip appeared to be the same as that of the 50 μm width and the 20 μm width; therefore, the element width would be expected to be the key parameter for the low AC permeability. The low permeability was not restricted by the existence of the hidden multi-domain state. The element with the hidden state is included in the condition range of the single domain because the magnetic domain of the condition apparently forms the single domain, and the strip width is narrow enough.
In the case of the narrow single domain, with a low AC permeability, it is possible to control the permeability by controlling the intensity of the distributed magnetic field within the surface normal magnetic field. It would be possible to control the permeability of the easy axis ranging from θ = 71° to θ = 90° without depending on the re-constructed multi-domain structure, which was shown in our study, including that of this paper. As a result of this study, the enhancement of permeability with application of the distributed magnetic field is caused by the formation of a multi-domain structure; this is assumed to be the effect of the enhancement of the domain wall movement of these domains.
It is well known that the magnetization process of soft magnetic materials consists of both the domain wall movement and the magnetization rotation. The high-permeability property of soft magnetic material is mainly caused by the domain wall movement. When the frequency of the external magnetic field increases, the domain wall movement is restricted by the generation of a microscopic eddy current, which appears in the vicinity of the domain wall. In this case, the permeability is dominated by the magnetization rotation [
11]. The study in this paper probably includes a transition frequency range of these phenomena. At 10 kHz, which is the lowest frequency of this study, the domain wall movement would be dominant, due to the observation of the sensitivity of the permeability to the domain existence in the element. At around 10 MHz, which is the highest frequency of this study, a typical resonant profile of the complex permeability was observed [
42], and the permeability decreased suddenly in the frequency occurring here. In the case of several hundred MHz, the domain walls are stuck in a certain position, which is the position formed initially in the stable energy state. In this case, the domain structure is observable even in the case that a high-frequency current is induced in the magnetic element [
47]. In this study, the magnetization process is dominated by the domain wall movement; therefore, the single-domain structure has low permeability and the multi-domain structure has high permeability; this comes from the existence of movable domain walls in the element.
The difference in the effect of the easy axis direction for the permeability variation generated by the intensity of the distributed field mainly appeared in an intensity value which increased the real permeability from a low to a high value. In the case of 71°, this is the increase from 0.14 to 0.27 Oe/mm given in
Figure 15a. In the case of 90°, this is the increase from 0.27 to 0.54 Oe/mm given in
Figure 20a. The inclined easy axis direction, which would have hidden the stable state, has a lower value, almost half of the intensity of the distributed field for making the transient phenomenon of permeability. It is assumed that it comes from the easiness of generating the multi-domain structure for the 71° easy axis rather than the 90° one. It is experimentally observed in the domain photo as the portion of the multi-domain area, between
Figure 11b and
Figure 16b, which is the condition of the intensity just after the permeability increase.
Concerning the formation and transition of the magnetic domain that appeared in the soft magnetic narrow strip and the magnetic energy estimation of these different states, the consideration of the threshold barrier that needs to be overcome to make the change in the domain state is important. It was discussed in my previous paper [
48] with regard to the case of a single narrow strip. Based on the previous knowledge, the effect of the distributed field should be discussed. In this paper, the formation of the multi-domain for a 90° easy axis indicates that there is another multi-domain stable state existing in the distributed field, other than the ordinally longitudinal strip domain, such as that in
Figure 17. Because of this, the energy difference between the initial domain state and the final state would be changed by applying the distributed field. It would be the reason for the different rising point of the permeability for the different easy axis directions. In order to make a strict discussion, a magnetic energy simulation is needed; this would be the subject of a future investigation.
For a brief comment about the effect of distance between the strips, a consideration was made in my previous paper based on magnetic field simulation [
41]. It was mentioned that “There are two types of mutual interaction. One is a strong short-range effect of magnetic field generated at the edge of a single narrow strip, and the other is a cumulative effect of the widely spread weak field which come from the other adjacent many-body elements”. In order to investigate the distance effect, we have to consider both the short-range field and the cumulative widely spread field, based on the previous consideration.
It is noticed that a newly reported typical phenomenon, which appears in the element with the hidden multi-domain state of θ = 71° has been recently reported [
44]. It is an appearance of the LLD multi-domain state, which happens when a small magnetic particle is placed just above the strip and controls the applied surface normal magnetic field. The parallel surface normal field makes the particle magnetize, and it generates a magnetic dipole field. The magnetic dipole of the magnetized particle generates a distributed magnetic field on the magnetic strip of the element; then, it has a possibility to enhance the re-construction of the LLD stripe domain. This phenomenon is assumed to be a typical one for the element with the hidden stable state. The appearance of the transited domain of a certain strip would possibly be detected by an electrical parameter such as impedance or inductance. The detection and discrimination of the appearance of the domain transition are two of the objectives of this paper for the sensor application of this phenomenon.
For further understanding of this physical phenomenon, a more detailed study should be conducted based on the knowledge of this paper and our related papers.