Cellular magnetic domains in Fe–Ga alloys

Binary Fe_100-xGa_x alloys (Galfenol) with compositions of 10 < x < 30 have recently attracted great attention for their large magnetostriction up to 400 ppm.^1,2 Fe–Ga alloys possess good ductility, machinability, and high tensile strength, exhibit significant magnetostriction under low magnetic field, show small hysteresis, perform under both compression and tension, and operate in a wide temperature range.^3–9 The unique combination of these good properties gives Galfenol many advantages over other currently leading functional materials (e.g., magnetostrictive Terfenol-D,¹⁰ piezoelectric PZT,¹¹ and magnetic shape memory Ni–Mn–Ga¹²); thus, it has stimulated strong interest in developing Fe–Ga alloys in bulk, film, ribbon, wire, and composite forms for sensor, actuator, transducer, vibrator, damping, and energy harvesting applications. In spite of the intense interest in Fe–Ga alloys, the magnetic domain processes underlying their magnetoelastic behaviors are yet to be understood.

Bai et al.^13,14 observed maze-like domains in Fe–Ga alloys of compositions 12%–25%Ga and two different thermal histories (furnace cooled and post annealed), where they inferred that nanoscale heterogeneities cause an out-of-plane magnetic anisotropy. While more observations of similar maze-like domains were reported,^15,16 the findings of the maze-like domains in Fe–Ga alloys were overturned by Mudivarthi et al.,¹⁷ who decisively demonstrated that the maze-like domains are merely a result of the damaged surface layers caused by standard mechanical polishing. Removal of the damaged surface layer by an additional polishing step using colloidal silica reveals true domain patterns of in-plane magnetization with straight 90° and 180° domain walls. These domain structures consisting of straight 90° and 180° walls are expected in cubic Fe–Ga alloys according to the classical domain theory and have been observed on the (001) surface in Fe-Ga single crystals of compositions 15.8%–19%Ga.^17–20 A significant development in Galfenol magnetic domain research was recently reported in a Nature paper by Chopra and Wuttig,²¹ where a peculiar magnetic domain pattern, the so-called cellular domain structure, was observed on the (001) surface in a quenched single crystal of 26.1%Ga. The authors did not explain the cellular domains by the classical magnetic domain theory but instead proposed a micromagnetic model (hereafter Chopra–Wuttig model for brevity) by resorting to hypothetical charge density waves and claimed that the charge density waves and resultant cellular domains are the mechanisms responsible for the long-sought hysteresis-free and isotropic magnetization behaviors of the material. Subsequently, Chopra et al.²² observed similar cellular domains also in a slow-cooled single crystal of 26.1%Ga and quenched single crystal of 17.1%Ga, emphasizing the generic aspects of the cellular domain structure over a wide composition range and further interpreting them as manifestations of long-wavelength elastic waves that are ultimately caused by charge density waves.

So far, the cellular domain phenomenon has been observed only in Fe–Ga single crystals with special thermal history (annealing followed by slow cooling in furnace or water quenching).^21,22 In this Letter, we report the observation of cellular domain structure in as-cast crystalline Fe–Ga alloys which are simply synthesized by arc-melting without special thermal treatment. For comparison, the cellular domains observed in an Fe–Ga single crystal specimen of rectangular prism shape are also presented. It confirms the generic aspects of the cellular domains in Fe–Ga alloys regardless of specific thermal history or specimen shape. More importantly, in contrast to the Chopra–Wuttig model that resorts to hypothetical charge density waves, this Letter presents a simple while convincing explanation of the cellular structure in Fe–Ga alloys within the framework of the classical domain theory. Our findings unveil the mystery of the cellular structure and shed light on the domain phenomena in Fe–Ga alloys, helping the development of Galfenol and similar magnetostrictive alloys for sensor and actuator applications.

The Fe-19%Ga polycrystal investigated in this study was prepared by arc-melting 99.99% Fe and Ga in argon gas atmosphere. The Fe-19%Ga ingot was melted and flipped three times to ensure homogeneity. The ingot was mounted in epoxy, and the flatter bottom surface was polished using a Struers LaboPol-1 autopolisher through the following sequential steps: 400, 600, 800, and 1200 Grit SiC sandpapers, 1 μm diamond compound, and 0.04 μm colloidal silica. Domain observation was performed by the bitter method using a Ferrotec EMG 707 water-based ferrofluid diluted 1:30 with distilled water. A drop of the diluted ferrofluid was applied to the polished surface and covered with a glass coverslip. The surface domain patterns were then observed under an optical microscope in Nomarski differential interference contrast (DIC) mode.

Figure 1 shows the observed magnetic domains in the arc-melted Fe-19%Ga polycrystal. The average grain size is about a few hundred micrometers, and the domain pattern appears to be discontinuous across the grain boundaries as expected. In particular, one of the grains exhibits the cellular domains that are clearly visible in the zoomed-in image. Since the cellular domains are unique to the {001} crystallographic surfaces, the ⟨001⟩ direction of this grain must by chance be oriented perpendicular to the polished surface. The cellular domains form long chains that run parallel along another ⟨001⟩ direction of the grain. The typical size of the cells in the chains (the length in the longitudinal direction) varies, increasing from ∼8 μm in the left region to ∼16 μm in the right region. To compare the cellular domains observed in the polycrystal with that observed in single crystals of a similar composition, Fig. 2 presents two domain observation images taken on the (001) surface of quenched Fe–Ga single crystals: (a) Fe-17.1%Ga in disk shape [Reproduced with permission from Chopra et al., Phys. Status Solidi B 255, 1800214 (2018). Copyright 2018 John Wiley and Sons] and (b) Fe-18.07%Ga in rectangular prism shape (prepared at Ames Lab). The thicknesses of the Fe–Ga samples in Figs. 1 and 2(b) are, respectively, ∼5 mm and 3 mm. The typical size of the cells from Ref. 22 is unknown due to the lack of the scale bar. The typical cell size is ∼10–17 μm in Fig. 2(b), which is consistent with the previously mentioned 8–16 μm in the Fe-19%Ga polycrystal shown in the right region of the zoomed-in area in Fig. 1, and also the reported value of 12–14 μm in quenched Fe-26.1%Ga single crystal from Ref. 21 and ∼9–16 μm in slow-cooled Fe-26.1%Ga single crystal measured from Ref. 22. Therefore, the typical size of the cells is qualitatively consistent in the polycrystal of arc-melted Fe-19%Ga and the single crystals of quenched Fe-18.07%Ga, quenched Fe-26.1%Ga, and slow-cooled Fe-26.1%Ga. It is concluded that the cellular domain structure consisting of parallel chains of cells with typical size on the order of ∼10 μm generally forms on {001} surfaces of Fe-Ga alloys of varied compositions (at least in the range of 17–26%Ga) and thermal histories. Therefore, the cellular domain structure as a generic magnetic domain phenomenon in Fe–Ga alloys deserves further investigation. First, the magnetic structure of the cellular domains must be determined.

Figure 3 shows schematically the magnetization configuration of the cellular domains on the (001) surface. Fe–Ga alloys are cubic crystal systems with ⟨100⟩ easy axes; thus, on the (001) surface, four surface domains are expected with the magnetization, respectively, along $[100]$⁠, $[1¯00]$⁠, [$010],$ and $[01¯0]$ directions. Figure 3(a) shows two types of cells repeating in chains aligned along the [100] direction. Each cell consists of the four ⟨100⟩ magnetic domains with magnetization in each domain indicated by the red arrows. The two types of cells in the respective chains differ only in the opposite magnetization direction of corresponding domains. Rotating the cellular domains by 90° results in the cells aligned in the chains along the [010] direction. The cellular domains shown in Figs. 1 and 2 are just colonies of such cells forming clusters of parallel chains, where the types of the cells alternate between the chains.

In Fig. 3(a), the green arrows show the magnetization directions in the subsurface domains, the red solid lines represent surface domain boundaries, and the green dashed lines represent subsurface 180° domain boundaries that separate [001] and [00$1¯$] domains. To visualize the 3D subsurface domain structure of the cellular domains, a folded paper model was constructed, and the surface domain boundaries are highlighted in red lines on its photo shown in Fig. 3(b). The 3D model shows that the depth of the subsurface domain structure (not the bulk domain structure further beneath the subsurface domains) is a fraction of the cell size. As shown in Figs. 3(a) and 3(b), there are two types of surface cellular domain boundaries, namely, zigzag segments along [100] and [010] directions and straight segments along [110] and $[11¯0]$ directions. It is worth noting that the zigzag boundaries feature head-on or tail-on magnetization configurations on the (001) surface and, indeed, are not ordinary domain walls. As shown in Fig. 3(c), the zigzag boundaries are actually lines on the (001) surface where two subsurface 90° domain walls meet, capping the [001] and $[001¯]$ magnetic domains below. These zigzag boundaries are the well-known V-lines that are commonly observed in iron and iron-silicon and were well investigated in 1950s.^23–25 The V-line is named after its V-shaped internal structure,²⁴ and the zigzag feature arises from the zigzag folding of the subsurface 90° domain walls. The energetics of the zigzag folding of 90° domain walls in cubic magnetic materials has been analyzed.^23,25 The straight boundaries are head-to-head or tail-to-tail 90° geometrically necessary domain walls (GNDWs) that only extend a small distance beneath the surface: they are the small-area vertical 90° domain walls of triangular shape under the straight boundary segments shown in Fig. 3(b), which are elastically compatible but magnetically charged. Reducing the energy of the charged GNDWs can be achieved by shrinking the size of the square boundaries formed by the four straight segments, leading to deformation of the long zigzag boundaries and necking of the chains at the cell joints, as observed in Fig. 2(b). The rectangular and square features of the cellular domains illustrated in Fig. 3(a) are for the purpose of simplicity, while their deformation should be kept in mind. Therefore, the cellular domains are surface domains where the [100] and [010] zigzag boundaries are V-lines formed on the (001) surface, while the [110] and $[11¯0]$ straight boundaries are domain walls extending a small depth (a fraction of the cell size) beneath the surface.

Our interpretation of the cellular domains is in a stark contrast to the Chopra–Wuttig model shown in Fig. 4(a).²¹ While the cellular domains in our interpretation are surface domains that are formed only in the surface layer to cap the subsurface domains (shown in Fig. 3), the cellular domains in the Chopra–Wuttig model are bulk domains in the crystal and extend to the surface. The most prominent difference is in the interpretation of the zigzag boundaries. While in our model shown in Fig. 3, the zigzag boundaries are simply the well-known V-lines (for more examples of domain observations involving V-lines, see Hubert and Schafer²⁵), the zigzag boundaries in the Chopra–Wuttig model are interpreted as a consequence of the schematically illustrated non-uniform (zigzagged) magnetization vectors in Fig. 4(a), which are attributed to the hypothesized lattice modulations caused by charge density waves. Instead of interpreting the cellular domain phenomenon by the classical domain theory, a theory based on the hypothesized charge density waves is introduced in the Chopra–Wuttig model. However, it cannot explain the existence of regular magnetic domains with straight 90° and 180° domain walls observed in Fe–Ga alloys with a wide range of compositions 15.8%–19%Ga, thermal history, and specimen shape^17–20 (these samples have similar composition and thermal history, and some of them were prepared in the same Ames Lab as the samples that exhibit cellular domains and zigzag boundaries^21,22). Moreover, the theory introduces a hypothesis of charge density waves and the consequential lattice modulation, wavy magnetization, and zigzag domain walls, which have not been confirmed. In contrast, the classical domain theory can explain the cellular domains with the zigzag boundaries, where the latter are the V-lines widely observed and well-studied in 1960s.^23–25 Therefore, we conclude that the theory in the Chopra-Wuttig model can be rejected by Occam's razor.

While we strive to justify our model based on the classical domain theory (shown in Fig. 3) over the Chopra–Wuttig model based on a theory involving hypothetical charge density waves shown in Fig. 4(a), we surprisingly came across Stephan's work published over 60 years ago²⁶ (Stephan model hereafter). Stephan not only observed similar cellular domains (though not so-called) in Fe–Si alloys but also discussed the zigzag boundaries on the (001) surface. Stephan interpreted the cellular domains using the micromagnetic model shown in Fig. 4(b), where the cells also feature narrower ends (i.e., necking) as previously discussed. The prior observation of the cellular domain structure in Fe–Si alloys and its interpretation by the Stephan model further confirm that the cellular domain phenomenon in Fe–Ga is not unusual and the classical domain theory does provide a convincing explanation.

Finally, an experiment on the response of the cellular domains to a magnetic field perpendicular to the (001) surface further supports our model and disproves the Chopra–Wuttig model. As shown in Fig. 3(a), each cell consists of four outer zigzag V-line segments and one inner zigzag V-line segment; and the perpendicular magnetizations in the subsurface domains beneath the outer and inner zigzag V-line segments are of opposite directions (i.e., [001] vs [00$1¯$]), as shown in Fig. 3(c). Such an asymmetry between the structures of the outer and inner zigzag V-line segments is expected to result in their different responses to a magnetic field perpendicular to the (001) surface, depending on whether the subsurface magnetization is parallel or antiparallel to the magnetic field. Such a prediction is confirmed by the experimental observations shown in Fig. 5: one type of the zigzag V-line segments is enhanced while the other type is faded, and these opposite responses are reversed when the magnetic field direction is switched. Moreover, since all magnetization vectors switch directions for the cells in the neighboring chains as shown in Fig. 3(a), we anticipate opposite responses of the outer and inner zigzag V-line segments in the neighboring chains, which is confirmed in Fig. 5. As a result, the cellular domains lose either the outer or inner zigzag V-line segments, which alternates between the chains. The experimental results in Fig. 5 agree with our model in Fig. 3. In contrast, the inner and outer zigzag segments in the Chopra–Wuttig model shown in Fig. 4(a) are equivalent with respect to [001] magnetic field; thus, it cannot explain their opposite responses to the field as demonstrated in Fig. 5. It is worth noting that Stephan²⁶ also demonstrated the opposite responses of the inner and outer zigzag V-line segments of the cells to a perpendicular field,²⁶ the same phenomenon as shown in Fig. 5, which further proves that the cellular domains in Fe–Ga and Fe–Si alloys have the same structures (i.e., Stephan model). It is also worth noting that a Kerr image can, in principle, help determine the surface magnetization vector directions in the cellular domains; however, to date, neither the cellular domains nor the zigzag V-lines have been directly observed by Kerr microscopy.¹⁹ Further study of the magnetic domains in Fe–Ga alloys using Kerr microscopy is highly desirable.

In summary, cellular domains observed in some Fe–Ga single crystals of specific compositions and thermal histories have been recently postulated to manifest an unusual crystal lattice modulation phenomenon caused by hypothetical charge density waves and to be responsible for special magnetic properties. This Letter reports an observation of the cellular domains in an Fe–Ga polycrystalline sample synthesized by arc-melting without special thermal history. It is found that similar cellular domains were also observed in Fe–Si alloys over 60 years ago and are not a new magnetic domain phenomenon. The cellular domains are interpreted using the classical magnetic domain theory in agreement with the Stephan model, which are a regular surface domain phenomenon that does not require a new mechanism or theory to explain. Being surface domains, the cellular domains form only within a surface layer and are different from the bulk domains beneath the surface. Our findings unveil the mystery of the cellular structure and shed light on the domain phenomena in Fe–Ga alloys, aiding the development of Galfenol and similar magnetostrictive alloys for sensor and actuator applications.

Support from NSF under Grant No. DMR-1409317 is acknowledged.