Variations in the cellular magnetic domain structure in Fe–Ga alloys

Alloys of iron and gallium, known also as Galfenol, have attracted much attention recently for their unique combination of high magnetostriction of up to 400 ppm^1,2 with good ductility and machinability.^3–5 Other well-known functional materials like magnetostrictive Terfenol-D⁶ (an alloy of iron, terbium, and dysprosium) and piezoelectric PZT⁷ (lead zirconate titanate) exhibit high strains, but without the ductility offered by Galfenol. Galfenol is used in actuator and sensor applications where its machinability and ability to bear tensile loads open up new possibilities.

The large magnetostriction in Galfenol is composition dependent. Pure Fe has a saturation magnetostriction $(32λ100)$ of only 40 ppm, but as was shown by Clark et al.,¹ Ga additions cause increases in λ₁₀₀. There is a complex double peak dependency of λ₁₀₀ on Ga concentration, with maxima at 17 at. % Ga and 26 at. % Ga. Annealing treatments followed by rapid quenching increase the magnetostriction even further.

The increased interest in Galfenol has also resulted in studies on the magnetic domain structures that underlie the properties of this material. Since magnetostriction is strongly dependent on the composition and magnetostriction influences the domain structure, it should be no surprise that a variety of different domain structures have been observed at different compositions. Of special interest is the unusual cellular structure observed by Chopra and Wuttig.⁸ This structure was first found on the (100) surface of an annealed and quenched single crystal of Fe–26.1 at. % Ga. It consisted of regularly spaced chains of “cells,” each rectangular with zigzag boundaries and a zigzag line at its center, connected to the outer corners by four straight diagonal lines. In their interpretation, the zigzag-shaped domain boundaries are 180° walls and each cell forms a closure loop, as recreated here in Fig. 1(a). At smaller scales, the magnetization was assumed to wave back and forth staying parallel to the zigzagged wall. They suggested that this peculiar structure is caused by charge density waves and is responsible for the unique magnetoelastic properties of Galfenol. Chopra et al. reported on further observations of this structure in single crystals of annealed and quenched Fe–17.1 at. % Ga and slow-cooled Fe–26.1 at. % Ga.⁹ Tianen et al. also found cellular domains on an annealed and quenched single crystal of Fe–18.07 at. % Ga and in an arc melted polycrystal of Fe–19 at. % Ga.¹⁰ In that work, the cellular domains were explained as assemblages of V-lines, which are formed by the meeting of two subsurface 90° domain walls. That V-line interpretation of the cellular domains is consistent with the work of Stephan, who observed a similar cellular domain structure in the 1950s on annealed and quenched single crystals of Fe–Si.^11,12 The presence of this structure in a material other than Galfenol is intriguing. Carey also found cellular domains in cube-textured polycrystalline Fe–Si.¹³ The interpretation of Stephan,^11,12 and later Tianen et al.,¹⁰ of the zigzag V-lines is well founded in the classical domain theory. Zigzag-shaped V-lines were observed and analyzed by Chikazumi and Suzuki in Fe–Si crystals¹⁴ and are understood to arise from considerations of the energy of 90° walls in cubic magnetic materials of positive anisotropy without the need to invoke charge density waves.¹⁵ Stephan's interpretation of the structure was not a closure loop on the surface but instead had V-lines with head-to-head and tail-to-tail magnetization configurations as illustrated in Fig. 1(b). It should be noted that this configuration does not produce charged domain walls because they are comprised of two uncharged 90° walls that form closure domains beneath the surface, as will be explained in detail later (see Figs. 5 and 6).

It is worth noting that highly periodic cellular domains have been observed so far in Fe–Ga alloys of relatively high Ga concentrations of 17.01%–26.01 at. % Ga.^8–10 Tianen et al. recently observed stripe domains consisting of parallel zigzag boundaries on a slow-cooled single crystal of Fe–8.5 at. % Ga.¹⁶ They argued that the zigzag boundaries in the stripe domains and the cellular domains are indeed the same type of V-lines and provided experimental evidence by comparing their responses to perpendicular magnetic fields. Those observations of zigzag boundaries forming the cellular domain pattern at a higher Ga concentration while forming the stripe domain pattern at a lower Ga concentration justify an investigation of the magnetic domains in Fe–Ga alloys of intermediate concentrations. These would hopefully feature a transition between the two domain patterns. Indeed, on an annealed and quenched Fe–15 at. % Ga single crystal, a variety of domains are observed spanning the range from plain zigzag lines to periodic cellular domain structures as will be presented later. Similar variations in the cellular structure were also observed in Fe–Si alloys,¹² confirming that those variations are a generic domain phenomenon in cubic magnetostrictive materials. This work analyzes the variations in the cellular structure, explains the underlying energetics mechanisms for the cellular domains and variations, clarifies the cause and identity of the cellular domains, and discusses the composition dependency of the magnetic domains in Fe–Ga alloys.

For this work, domains were observed on the (100) surface of an annealed and quenched Fe–15 at. % Ga single crystal, which was prepared at the Ames Laboratory. The surface was prepared by mechanical polishing on a Struers Labopol-1 autopolisher using the method described by Tianen et al. previously.¹⁶ To image the domains, Ferrotec EMG707 ferrofluid was diluted 30:1 with distilled water. A single drop was added to the surface and covered with a glass coverslip. To enhance domain contrast, they were imaged using Nomarski differential interference contrast (DIC) microscopy. When needed, perpendicular magnetic fields were applied using an electromagnet coil placed below the sample.

Cellular magnetic domains were seen on the Fe–15 at. % Ga sample, as shown in Fig. 2. The structure is similar to previously published cellular domains by both Stephan¹¹ and Chopra and Wuttig,⁸ with four zigzag outer lines enclosing each rectangular cell and a single zigzag running lengthwise down its center.

This form of the cellular domains with rectangular cells is consistent with the literature, but it is not the only form that they can take. The cellular domain structure can have a variety of different but related shapes. Several of these varieties are shown in Fig. 3. There are cells with rounded edges which taper to a point like in Fig. 3(a), relatively rectangular cells (but less square than in Fig. 2) like in Fig. 3(b), chains which branched to form smaller chains like in Fig. 3(c), and nested structures with cells growing from the zigzag lines of other cells like in Fig. 3(d). Similar variations, including the complex nested structures, were also observed by Stephan in Fe–Si single crystals.¹² 

The chains of cellular domains can turn at 90°, with cells aligned along the [100] or [010] direction. Some clues to the magnetization state can be found by applying a small magnetic field perpendicular to the surface, as in Fig. 4. When this field was applied, many of the zigzag lines disappeared. In some chains, all of the outer lines of the cell disappeared, no longer collecting any ferrofluid, while the centerline remained visible. In adjacent chains, all of the centerlines disappeared, but the outer lines forming the border of the cells remained visible. This appearance and disappearance of zigzag boundaries alternated from one chain to the next.

The alternating disappearance of zigzag V-lines was observed previously in Galfenol by Tianen et al. for regular arrangements of parallel zigzag lines.¹⁶ As discussed in that work, the disappearance is dependent upon the orientation of the adjacent domains. If the adjacent domains are magnetized to point inward toward the zigzag V-line, the line will disappear in an outward pointing field. If they point away from the line, the line will disappear in an inward pointing field. This behavior can, therefore, be used to infer the magnetization state of the cellular domains. Since all of the outer boundaries of the cells disappear together on the same chain, it can be inferred that they are magnetized similarly, i.e., they all point inward or all point outward. The state of the centerline is then opposite of the outer boundaries. This is consistent with Stephan's interpretation of the cellular domains, as shown in Fig. 1(b). The cellular domains in Fig. 4 are of two different types. There are rounded cells that taper to a point like in Fig. 3(a) and more rectangular cells like in Fig. 3(b). The behavior of these different forms of cells under an applied field is the same, with the same pattern of lines disappearing in both types. This uniformity of behavior demonstrates that the different forms of the cellular domains share similar micromagnetic states.

This behavior, with all four of the domains within the cell magnetized inward toward the centerline or outward away from it, implies the presence of charged 90° domain walls in the rectangular cells. The four 90° walls that divide the cell, as shown in Fig. 1(b), must be charged for this interpretation to hold. As will be shown, these charged 90° domain walls are small vertical areas of triangular shape that extend only a small depth beneath the surface (see Fig. 7). It is noted that, using the Bitter method, these charged 90° domain walls are invisible under zero field while they can become visible as straight line segments under a perpendicular magnetic field, as demonstrated in Ref. 10. The formation of these walls is important to understand the transition between different forms of cellular domains. To understand why such charged domain walls might form, the three-dimensional (3D) domain structure must be considered. Here, we will consider the 3D structure of the ideal rectangular domains shown in Fig. 2 for simplicity, but the same principles, like the V-line structure of the zigzag lines, apply also to other forms of cellular domains. Figures 5(a) and 5(b) show the domain structure with magnetization vectors, with transverse and longitudinal cross sections, respectively, through the cellular chains. The cellular domains shown on the top surfaces of Fig. 5 represent a small section of a surface like the one shown in Fig. 2. The magnetization arrangement on the surface is the same as shown in Figs. 4(a) and 1(b). The front surface in Fig. 5(a) shows a transverse cross section through the cellular domains, revealing the domains beneath the surface. This shows how the zigzag V-lines are formed by the meeting of two subsurface 90° domain walls.

Cellular domains occur where the magnetization vectors of internal domains are perpendicular to the surface. The cells themselves are only surface domains, not extending deep into the bulk (the depth is only a fraction of the cell size). They provide flux closure for internal domains. Figure 5(b) shows a longitudinal cross section of the cellular structure, representing a cut through the dashed red line in Fig. 5(a), which was then rotated 90°. This shows that the closure domains are present in both cross sections. An internal domain magnetized perpendicular to the surface connects with a cell from beneath and flux closure is provided by all four domains of the cell along four different directions (see Fig. 7).

The magnetization flux closure provided near a surface by V-lines is well documented. Simple stripe domains separated by an array of parallel V-lines were widely observed in Fe–Si¹⁵ and were first analyzed by Chikazumi and Suzuki in Fe–Si.¹⁴ Stripe domains were also recently observed by Tianen et al. in Galfenol.¹⁶ The cellular domains represent a more complex two-dimensional array of closure domains. This more complex geometry is a consequence of influences from elastic energy resulting from magnetostriction. Where the wedge-shaped surface domains meet the internal ±[001] domains, a strain is generated, as illustrated in Fig. 6. Figure 6(a) shows the closure magnetization configuration at a cross section through two parallel V-lines, representing just one small region on the front face of Fig. 5(a). Figure 6(b) shows the magnetostrictive strain associated with that closure domain structure. Each domain is elongated along the magnetization direction and contracted in its perpendicular directions. As a result, the wedge domains at the surface elongate while the internal domains shrink in the horizontal direction, creating incompatible shapes. Fitting the surface domain into the space below results in elastic deformation, increasing elastic energy. This elastic energy is a volume energy, which increases with the size of the surface domains.

The strain energy due to the surface domains can be reduced by forming the cellular domain pattern. As shown in Fig. 5, the surface domains have magnetization along one of four easy directions on the surface: [100], $[1¯00]$⁠, [010], or $[01¯0]$⁠. To aid in visualization, Fig. 7 shows a 3D exploded view of the cellular domains. The internal domains are shown at the bottom in Fig. 7(b). The surface domains have been removed and shown separately at the top in Fig. 7(a). These surface domains fit into the cavities in the lower part of the figure. The magnetization of the surface domains is represented by arrows on the surface. Due to the geometric complexity, the magnetization of the internal domains is shown by dashed lines on the surface. Each dashed line represents one of the zigzag-shaped V-lines that were seen on the top surfaces in Fig. 5. As shown on the frontal cross section surfaces of Fig. 5, each V-line has an internal domain region beneath it. The magnetization of these internal regions is represented in Fig. 7 by the color of the dashed line. The white dashed lines mean that the domain below the line is magnetized outward in the [001] direction with a tail-to-tail magnetization configuration on the surface, while the green dashed lines mean it is magnetized inward in the $[001¯]$ direction with a head-to-head magnetization configuration on the surface. The color of the domains represents the strain state, so each color represents an elastic domain. The internal domains are magnetized along ±[001] directions, so they are all elongated along the [001] axis, represented by their purple color. Therefore, these magnetic domains form one uniform elastic domain with the magnetostrictive strain ɛ(m = ±[001]),

The surface domains, shown separately in Fig. 7(a), are magnetized along one of four easy directions on the surface: [100], $[1¯00]$⁠, $[01¯0]$⁠, or $[01¯0]$⁠. These are associated with two elastic domains. Elongation along the [100] axis is represented by orange color in Fig. 7, while elongation along the [010] axis is represented by blue color. The magnetostrictive strains of these domains are ɛ(m = ±[100]) and ɛ(m = ±[010]), respectively,

These two strain states, and their associated misfits with the internal domains, are complementary. A mixture of complementary strain states lowers the overall strain and, as a result, the elastic energy. In the case of Fig. 7, the mixture of the different strain states within each individual cell reduces the strain energy resulting from the misfit with the internal domains below. The cellular domains thus reduce the elastic energy by periodically alternating between [100] and [010] domains.

The above energetic analysis has considered only the rectangular cellular domains of the type seen in Fig. 2, but these principles can be applied to explain the full range of cellular domain structures. As was shown in Fig. 3, the cellular domains can exhibit a variety of forms. These forms can all be described by the same general principles as a combination of V-lines and charged 90° walls, whose shape can vary as a result of the competition between elastic energy and domain wall energy.

The range of observed domain structures is presented in Fig. 8, along with schematics of their magnetization state inferred from the zigzag V-line model. On one extreme end of the spectrum is a structure containing plain, parallel V-lines like the one in Fig. 8(a). The cellular domains form as chains along these zigzag V-lines. There are sometimes transitions between plain V-lines and cellular structure like in Fig. 3(a). Lens-shaped cells with tapered ends are chained along the V-line, as illustrated in Fig. 8(b). This type of cell has no internal 90° domain walls, consisting simply of three V-lines, which meet at the ends. While the centerline of these tapered cells is not clearly visible under zero magnetic field, the application of perpendicular fields makes them visible as shown in Fig. 4. When these cells are numerous enough to form a continuous chain, they form rounded cells like in Fig. 8(c). At the contact points, 90° walls form producing the additional domain regions at the end of the cell, which are magnetized perpendicular to the other domains within the surface. The square-cornered domains in Fig. 8(d) are functionally the same as those in Fig. 8(c). The domains at the end of the cell are just larger and the cell shape is more rectangular. These are the type of domain seen by Chopra and Wuttig.⁸ The sizes of these cells of varied forms are similar, about 10–20 μm in length and below 10 μm in width. Finally, the cells may also form complex nested structures like in Fig. 8(e) [full image in Fig. 3(d)]. Therein, the cells are nested within larger cells ∼50 μm in size, which will be referred to as macrocells. It is a hierarchical structure where the cells are grown on the zigzag border V-lines and center V-lines of the macrocells.

The range of structures shown in Fig. 8 is indicative of an energy competition between domain wall energy and elastic energy. Consider a domain structure with parallel V-lines. As was shown in Figs. 6 and 7, the surface domains are wedge-shaped structures, which experience shape misfit strains, and the elastic energy produced is dependent upon the volume of those domains. Forming chains of lens-shaped cells like in Fig. 8(b) reduces the wedge-shaped surface domains, which reduces the elastic energy while increasing the surface energy as a result of the associated additional domain walls. The formation of the domains at the ends of the cells like in Fig. 8(c) causes a further reduction of the elastic energy by creating a mixture of complementary elastic domains, but at the cost of producing additional charged 90° domain walls. The energy cost incurred by these charged walls is minimized by their shallow depth (a fraction of the cell size). The necking at the meeting of the cells also further reduces the surface area of charged walls. Increasing the size of these complementary domains further and straightening outer border lines by forming rectangular cells like in Fig. 8(d) lowers the elastic energy further, again at the cost of additional charged 90° domain wall area. In principle, when the elastic energy dominates (e.g., because the magnetostriction is stronger), the material system would choose elastically favorable structures more like Fig. 8(c) and 8(d) [forming periodic cellular domains like in Figs. 2 and 3(b)], and when the domain wall energy dominates (e.g., because the magnetostriction is weaker), the material system would choose a domain structure with plain V-lines with large spacing between them (forming simple stripe domains). It is not difficult to predict that when the elastic energy and surface energy are comparable, a range of structures can be equally favorable, including the above discussed cellular domain variations that bridge between the highly periodic rectangular cellular domains and the simple stripe domains without cells. It is important to point out that magnetic domains now become sensitive to underlying material inhomogeneities (e.g., composition, surface finish, and residual stress) since the small energy perturbations caused by them can shift the domain structure from one to another. In the annealed and quenched Fe–15% Ga single crystal sample considered in this work, while the heterogeneity in terms of composition is unlikely, heterogeneous residual stress and surface finish condition likely resulted from quenching and polishing of the sample. This may explain our observations of the variety of domain structures appearing as patches over the surface area that are not in an ordered manner. Nevertheless, we did find qualitatively that structures of plain zigzag lines with no cells are more common near the edges of the rectangular surface area. This appears to agree with the above argument on the role of the stress due to misfit strain (of the wedge-shaped surface domains) in driving the cell structure formation; the presence of another free surface at the edge provides stress relaxation near edges suppressing cell formation. The correlation of the cell structure variations and distributions with material heterogeneities require further investigation.

The above argument based on the V-line model appears to be backed up by magnetic domain studies. As mentioned previously, Clark et al.¹ measured the magnetostriction in Galfenol alloys systematically and found a strong composition dependence. Those magnetostriction values are used here for comparison. Highly periodic cellular domains have been observed in Fe–Ga alloys of higher Ga concentration, namely, 17.01%–26.1% Ga^8–10 with higher magnetostriction of $λ100≳200ppm$⁠. Meanwhile, at a low Ga concentration, namely, slow-cooled Fe–8.5 at. % Ga with λ₁₀₀ ∼ 75 ppm, only the stripe domains consisting of parallel V-lines without cells have been observed.¹⁶ For an intermediate value of λ₁₀₀ ∼ 170 ppm, the quenched Fe–15 at. % Ga exhibits the range of variations in the cellular structure, as demonstrated in this paper. This is the lowest Ga concentration so far shown to contain cellular domains.

In contrast to the V-line model, the charge density wave model can explain neither the responses of the cellular domains to a perpendicular magnetic field nor the variations in the cellular domains. In the V-line model, the perpendicular magnetizations beneath the outer border and inner center zigzag V-lines are of opposite directions (see Fig. 7), which results in their different responses to a perpendicular magnetic field in terms of the selective disappearance of zigzag boundaries shown in Fig. 4. In the charge density wave model shown in Fig. 1(a), the outer and inner zigzag segments are equivalent with respect to the perpendicular magnetic field; thus, it cannot explain their opposite responses exemplified in Fig. 4 (see also Ref. 10). The observed variations in the cellular domains, which are explained by the V-line model from the energetics point of view, cannot be explained by the charge density model. The hypothetical charge density waves were proposed to be the origin for the zigzagging of the domain boundaries in highly periodic cellular domains, and it is incomprehensible why such periodic charge density waves would result in irregular and variable cellular structures. Moreover, the V-line model for the cellular domain structure is rooted in the classical domain theory, according to which other typical domain structures consisting of in-plane magnetizations with straight 90° and 180° walls are expected on the (001) surface in Fe–Ga alloys. Indeed, these other domain structures typical for cubic magnetic materials were widely observed in Fe–Ga alloys of compositions 15.8–19 at. % Ga.^17–20 The existence of the straight 180° domain walls in Fe–Ga alloys are contradicted by the charge density waves that are claimed to account for the zigzagging of the 180° boundaries as shown in Fig. 1(a).⁸ It is insightful to relate the magnetic domain phenomena in Fe–Ga alloys recently to that seen in Fe–Si alloys more than 60 years ago.^11,12 The two alloy systems were shown to exhibit similar cellular domain behaviors including the variations in the cellular domains and the selective disappearance of zigzag boundaries in response to an external perpendicular field. Against the new charge density wave model⁸ proposed recently in 2015, the V-line model proposed by Stephan over 60 years ago offers convincing explanations for the cellular domains in both Fe–Si and Fe–Ga, which are both ironlike cubic magnetic materials. Finally, it is worth noting that a micromagnetic simulation study can, in principle, provide the quantitative descriptions of the cellular domain structure and zigzag boundaries. Further study of magnetic domains in Fe–Ga alloys using micromagnetic simulations is highly desirable.

The periodic cellular magnetic domains observed in Fe–Ga alloys have emerged as an important issue in recent magnetostrictive materials research, but their cause and identity remain controversial. We report here the presence of various deviations in Fe–Ga alloys in the cellular domains from their previously known ideal periodic structure. The variations include differences in cell shape and spacing, as well as cell branching and nesting. These variations are observed in Fe–Ga alloys of an intermediate composition of 15 at. % Ga, which are shown to bridge the stripe domains observed at a lower Ga composition of 8.5 at. % and the highly periodic cellular domains at higher Ga compositions above 17 at. %. With similar cellular domain variations being previously observed in Fe–Si alloys as well, it is apparent that these variations are a general magnetic domain phenomenon with broad implications in ironlike cubic magnetostrictive materials.

Between the two existing viewpoints for the cause of cellular domains, one based on the novel charge density waves and the other based on the traditional V-lines from the classical domain theory, the latter is shown to agree with the experimental observations. In the V-line model, the cellular domains exist only on the surface and are distinct from the magnetic domains in the bulk. The cellular domains and their variations are explained in light of the energetics, highlighting the important roles of the elastic energy associated with the magnetostriction-induced strain misfits in magnetic domain structures. The 3D micromagnetic identity of the cellular domains given by the V-line model helps elucidate the observed domain responses to a perpendicular magnetic field, which feature patterned disappearance of zigzag boundaries. In comparison, in the charge density wave model, the cellular domains are bulk domains that are extended to the surface. The disagreements of the charge density wave model with the experimental observations are discussed. The findings help unravel the true nature of the cellular domains and advance our understanding of the magnetic domain phenomena in ironlike cubic magnetic materials, assisting in the development of Fe–Ga and other magnetostrictive alloys.

Support from the National Science Foundation (NSF) under Grant No. DMR-1409317 is acknowledged.

The data that support the findings of this study are available from the corresponding author upon reasonable request.