domain wall structures in single-crystal magnetite

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. B12, PAGES 30,551-30,560, DECEMBER 10, 1998

Domain wall structures in single-crystal magnetite investigated by magnetic force microscopy

Sheryl Foss, 1'2 Bruce M. Moskowitz, 3 Roger Proksch, •'4'5 and E. Dan Dahlberg •

Abstract. Domain walls in bulk single-crystal magnetite were studied using a variable magnetic field magnetic force microscope (MFM). Classical configurations of 180 ø, 109 ø, and 71 o walls were observed on (110) surfaces. Magnetostatic effects on these different walls were compared. Profiles of the MFM response above the walls were measured with the MFM tip magnetized in different directions. The contribution to the profiles from the z component of the sample field was distinguished from the in-plane components. An asymmetry of the z component of the response profiles for all wall types was observed, consistent with the existence of N6el caps which terminate the interior Bloch walls near the surface. The wall profiles of the non- 180 ø walls were more asymmetric than that of the 180 ø walls. The 180 ø walls were observed to be subdivided into alternating polarity segments of average length 15 gm. These walls formed a characteristic zig-zag structure in which the Bloch lines separating segments were located at the corners of the zig-zag. Only unusually long 109 ø walls were observed to contain a single Bloch line, and the 71 o walls, although the longest, were never observed to be subdivided. An applied field perpendicular to the sample plane moved the Bloch lines within the walls without translating the walls themselves. Multipolar walls were converted to unipolar in perpendicular applied fields from 0 to 100 mT. Profiles of opposite polarity segments of a subdivided wall indicated that the N6el cap formation does not alternate sides of the wall from segment to segment. Alignment of opposite polarity segments of parallel subdivided walls provided an example of long range magnetostatic interactions between walls and possibly their N6el caps.

1. Introduction

1.1. Background

The understanding of the magnetic domain and domain wall fine structures in magnetite and related phases as well as the behavior of these micromagnetic structures in variable, external magnetic fields are essential for understanding the macro- scopic, magnetic properties of rocks used to construct records of geomagnetic field behavior, lithospheric plate motions, or paleoclimatic change. Micromagnetic model simulations of experimentally observable magnetic structures in magnetite and other magnetic minerals have provided a theoretical framework for thermoremanent magnetization and its grain size dependence in terms of uniformly magnetized single domain (SD) and nonuniformly magnetized multidomain (MD) structures. MD structures, or simply structures containing uniformly magnetized regions called domains separated by narrow transition regions with rapidly varying spin direction called domain walls, can exhibit what has been termed pseudo-single domain (PSD)behavior [Stacey, 1963; Stacey and Banerjee,

•Magnetic Microscopy Center, School of Physics and Astronomy, University of Minnesota, Minneapolis.

2Now at Imation Corporation, St. Paul, Minnesota. 31nstitute for Rock Magnetism, Department of Geology and

Geophysics, University of Minnesota, Minneapolis. nAlso at Department of Physics, St. Olaf College, Northfield,

Minnesota.

5Now at Digital Instruments, Santa Barbara, California.

Copyright 1998 by the American Geophysical Union.

Paper number 98JB00152. 0148-0227/98/98JB-00152509.00

1974]. The domain walls play central roles in models of PSD behavior; these models include surface pinning of domains postulated by Stacey and Banerjee [1974] and extended by Banerjee [ 1977], domain wall pinning by crystal defects [Xu and Merrill, 1989, 1990; Moskowitz, 1993]; transdomain processes involving nucleation and denucleation of domain walls [Halgedahl and Fuller, 1983; Moon and Merrill, 198 5; Halgedahl, 1991, 1995; McClelland and Shcherbakov, 1995]; and intrinsic SD-like moments associated with the internal

structures of a domain wall [Dunlop, 1977]. Hence experimental studies of domain wall structures can provide essential tests of existing micromagnetic models as well as the basis for the formulation of new ones that can lead to better under-

standing of the stability of natural remanent magnetization over geologic time.

Experimental observations of surface micromagnetic structures in fine particles (< 50 gm) of magnetite and other magnetic minerals using the Bitter or magneto-optical Kerr effect methods have yielded much useful data; such data helped to establish domain wall nucleation as a possible mechanism for PSD behavior [Halgedahl and Fuller, 1983; Halgedahl, 1991, 1995; Heider and Hoffmann, 1992; Geiss et al., 1996]. Similarly, observations using these techniques on large, MD, single crystals with crystallographically oriented surfaces have provided information on the fundamental types of domain walls and closure domain structures in magnetite [Bogdanov and Vlasov, 1965; Ozdernir and Dunlop, 1993; Ozdernir et al., 1995].

Despite these successes, the experiments mentioned above were limited by the spatial resolution of the techniques em- ployed. Owing to computational limitations, the upper spatial limit of three-dimensional micromagnetic simulations is a few microns which barely exceeds the resolution of the Bitter and

30,551

30,552 FOSS ET AL.: MFM STUDY OF DOMAIN WALLS IN MAGNETITE

Kerr optical domain imaging methods [Halgedahl, 19 87, 1995; Hoffmann et al., 1987; Heider and Hoffmann, 1992]. In addition, standard Bitter method images of some micromagnetic structures may not give enough information to effectively test numerical predictions [Williams et al., 1992a; Newell et al., 1993]. Significant advances in domain imaging methods have come with the introduction of such magnetic probes as magnetic force microscopy (MFM) and scanning electron microscopy with polarization analysis (SEMPA) which provide submicron resolution and higher sensitivity than the earlier optical techniques [Martin and Wickramasinghe, 1987; Celotta and Pierce, 1986]. These relatively new techniques have been used in several earlier studies dealing with geophysically important minerals including magnetite, titanomagnetite, and titanohematite [Williams et al., 1992b; Haag et al., 1993; Haag and Allenspach, 1993; Proksch et al., 1994, 1995; Moloni et al., 1996; Pokhil and Moskowitz, 1996]. The MFM has been used for this work to provide images of domain wall structures in magnetite with resolution approaching a few tens of nanometers. Although quantitative interpretation of such MFM data is very difficult [Griitter and Allenspach, 1994; Proksch et al., 1995; Foss et al., 1996], using careful experimental procedures, valuable information on appropriate length scales can be obtained about the micromagnetic structure of a sample.

1.2. Micromagnetics and Magnetostatic Effects

The domain and domain wall structures of a magnetic system are governed by several different contributions to the total free energy. The state of lowest energy is the most probable to be achieved. The energy terms contributing to the total energy include magnetocrystalline anisotropy energy, magnetoelastic energy, exchange energy, Zeeman energy (of the external magnetic field), and magnetostatic (demagnetizing) energy. Among these, the exchange and anisotropy energies are localized interactions, making them easy to manage in theoretical simulations. The magnetoelastic energy is often small enough to neglect. Of all of these terms, the most difficult to handle is the magnetostatic energy. Originating from dipole-dipole interactions, this term is long range and nonlinear such that analytic solutions are often intractable and simulations on even the smallest systems require large numbers of iterations.

At surfaces, distributions of free magnetic poles result from the component of magnetization perpendicular to the surface. The near-surface spins respond to reduce the resultant magnetostatic energy by forming flux closure configurations. The formation of closure domains at crystal surfaces and edges has been predicted by classical domain theory and confirmed by observations near both internal and external boundaries [e.g., Landau and Lifshitz, 1935; Williams et al., 1949; Ozdemir et al., 1995].

Micromagnetic models have also shown that particle surfaces profoundly affect the micromagnetic structure of domain walls. At the intersection of a Bloch wall with a crystal surface, the self-magnetostatic energy of the wall will produce wall closure features. One example is the division of the domain wall into a periodic configuration of domain wall segments which alternate in polarity. Opposite polarity segments are separated by narrow transitional regions in which the magnetization rotates from one domain wall polarity to the other along the domain wall [Shtrikman and Treves, 1960a; Dunlop, 1977]. A schematic diagram of one of these spin transitions is shown in Figure la. These transition "lines" are Bloch-like if considered relative to the opposite polarity domain wall segments they separate or N6el-like relative to the body domains that the domain wall separates. These different viewpoints resulted in two names for the same structure: Bloch lines and N6el lines. Here, we refer to them as Bloch lines. The change in domain wall polarity across a Bloch line reduces the magnetostatic energy of the domain wall at the surface in much the same way as main body domains reduce the magnetostatic energy of the whole crystal. The number of wall segments is governed by the additional energy of the Bloch lines.

Subdivided walls form characteristic zig-zag configurations about the average wall direction near the sample surface; the corners of the zig-zag are coincident with Bloch line positions [Shtrikman and Treves, 1960b; Proksch et al., 1994]. The equilibrium zig-zag structure and its depth of penetration in the sample result from the interplay between surface and volume magnetostatic effects. Surface charges tend to increase the angles between segments and the average wall direction to bring opposite poles of neighboring segments closer together. However, as the angle between the segments and the average wall direction increases, the component of magnetization perpendicular to the wall segments increases, causing volume charges on the sides of wall segments to accumulate.

A B

Figure 1. Schematic diagrams of micromagnetic wall structures which form to reduce the magnetostatic energy of surface pole densities. (a) A schematic diagram of a portion of a 180 ø Bloch domain wall (surface, viewed from above) that has divided into opposite polarity segments. Two opposite polarity segments are separated by a transition region referred to as a Bloch line. Such walls typically form a zig-zag structure near the surface in which the corners of the zig-zag are the locations of Bloch lines. (b) A vertical cross section of a 180 ø domain wall that is Bloch-like in the interior of the sample, but near the surface, the spins of the wall rotate into the plane of the surface and form a N6el-like wall portion referred to as a N6el cap.

FOSS ET AL.: MFM STUDY OF DOMAIN WALLS IN MAGNETITE 30,553

Another magnetostatically induced wall feature occurs in Bloch walls at sample surfaces. The vertical magnetization of the Bloch wall near the surface rotates 90 ø to one side of the

wall and becomes parallel to the surface forming a N6el-like wall portion referred to as a N6el cap [LaBonte, 1969]. This wall feature, shown schematically in Figure lb, forms for reasons analogous to those causing the formation of closure domains at crystal edges. The N6el cap is predicted to penetrate approximately one Bloch wall width below the surface [Scheinfein et al., 1989, 1991; Xu and Dunlop, 1996].

In this paper, we describe measurements of domain wall structures on {110} surfaces of single-crystal magnetite made using variable applied field MFM. Although millimeter-sized crystals of magnetite are magnetically soft and do not con- tribute to stable remanence in rocks, they can be oriented and sectioned along particular crystal planes. Domain structures are relatively easy to interpret when the viewing surface contains one or more magnetic easy axes [e.g., Chikazumi, 1964].

In magnetite with cubic magnetocrystalline anisotropy (K i<0) and <111> magnetic easy directions, domain boundaries can be of three fundamental types in which the direction of magnetization within a domain wall changes by 180 ø, 109 ø, or 71 ø. Therefore a { 110} surface which contains two <111 > directions is well-suited for domain and domain wall observations. Arrays of the three types of walls can be observed and should be representative of the internal domain structure when the viewing surface is { 110}; otherwise, complex, flux closure patterns will decorate the surface if it is not perfectly planed [e.g., Halgedahl, 1987; Ozdemir et al., 1995]. In this work, the magnetostatic effects described above have been considered on all three domain wall types, significantly extending the results of previous MFM experiments on single-crystal magnetite [Williams et al., 1992b; Proksch et al., 1994; Moloni et al., 1996] and complementary to MFM results for PSD-sized magnetite particles [Pokhil and Moskowitz, 1996].

2. Experimental Methods

The magnetite sample studied in this work was a synthetic single crystal grown along the [110] direction by the floating- zone technique using an image furnace [Wanamaker and Moskowitz, 1994]. The crystal slab of the order of 1 mm thick was cut parallel to a { 110} plane, and the orientation was confirmed using the Laue backreflection method. The surface was mechanically polished with diamond compounds followed by a final polish using an amorphous silica solution to achieve a smooth surface and to reduce the strained surface layer produced during the initial mechanical polishing [Hoffmann et al., 1987]. Hysteresis loops of the magnetization were consistent with typical MD behavior. Prior to imaging with the MFM, the sample was AF demagnetized in a peak field of 100 mT.

A Nanoscope TM III in conjunction with a Multimode TM scanning probe microscope from Digital Instruments was used to obtain MFM images of the magnetite single crystal. The technique which interleaves TappingMode TM and LiftMode TM allowed separate magnetic and topographic images of the same area to be measured so that correlations between surface defects

such as scratches and magnetic structures could be observed [Babcock et al., 1995]. The probes used were batch-fabricated, single-crystal silicon cantilevers with integrated tips that had been sputter-coated with a magnetic thin film [Griitter et al., 1990; Babcock et al., 1994]. In LiftMode, the phase or amplitude of the cantilever oscillation gives a measure of the magnetic force gradient acting on the tip [Martin et al., 1987]. For

this work, the phase was measured at constant tip-sample separations ranging from 5 nm to several hundred nanometers.

The magnetic force gradient images were essentially proportional to o•2Hs/ogz 2, where Hs is the sample field [Rugar et al., 1990]. Depending on the direction of magnetization of the MFM tip, different components of the sample field can be sensed. For instance, if the tip is magnetized entirely in the • direction (which is defined to be perpendicular to the sample surface), the magnetic force gradient measurement is sensitive to the z component, o•2Hsz/ogz 2 only. For conventional imaging, the MFM tip magnetization is aligned primarily in the .• direction; however, as will be seen later, it is useful to measure the same sample feature various times with the MFM tip magnetized in different directions each time. In addition, it will be seen that it is critical for the interpretation of the MFM response to domain wall structures to consider measurements of mixtures of sample field components, since it is experimentally difficult to measure a single field component.

The conventional model of MFM response described above is based on the assumption that the MFM response is linear in the sample field, i.e., that no alteration of the sample or tip magnetizations occurred while imaging [Hartmann, 1988]. The standard MFM tips used for this work were coated with a CoCr alloy thin film which was magnetically hard, making it reasonable to assume in this case that the magnetic properties of the tip remained fixed while imaging [Babcock et al., 1996]. However, in some instances, the sample micromagnetic structure was observed to be affected by the field of the tip. The effects of the tip field on domain wall structures could essentially be extracted from the data using an experimental technique described elsewhere [Foss et al., 1996]. In addition, other tips with lower stray fields were used to verify the results obtained with standard tips [Riihrig et al., 1996].

3. Results

3.1. Domain Wall Types and N6el Caps

The domain wall configurations observed on the magnetite single-crystal sample confirmed the {110} surface plane as well as the orientation of the crystallographic axes relative to the sample geometry. Most of the sample surface was divided by classical arrangements of domain walls like those in the images of Figure 2 [Bogdanov and Vlasov, 1965; Ozdemir et al., 1995]. Typically, long 71 ø walls directed along the <001> axis (upper right to lower left) were connected by 180 ø wall segments which alternated between the two easy axes (one of them vertical). The 71 ø walls switch polarity in crossing the intersection with the 180 ø walls. In some instances, the 71 ø walls were offset along the <110> axis by relatively short 109 ø walls as in the lower wall intersection. The angles between the walls were as expected for 180 ø, 71 ø, and 109 ø walls based on the assumption that a domain wall bisects the angle between the magnetization directions in adjacent domains to ensure zero divergence of the magnetization normal to the domain wall [e.g., Chikazumi, 1964].

As mentioned previously, insight about the micromagnetic structure of a domain wall can only be gained if the MFM tip magnetization is well understood. For example, Figures 2a-2d were all measured above the same area of the sample but with the same MFM tip magnetized in a different direction for each scan. The tip magnetization was defined relative to the sample origin' • perpendicular to the sample, .• in the plane of the sample and perpendicular to W1, and •9 in the sample plane


Figure 2. Four MFM images of approximately the same area of a magnetite single crystal measured at a height of 50 nm above the sample surface. The same MFM tip magnetized in a different direction relative to the sample origin was used for each. Domain walls of the three possible types can be seen in these images. The central, vertical wall (W1) is a 180 ø domain wall running parallel to one of the <111> easy axes. The lower right domain wall and the upper left domain wall are also 180 ø walls. Four 71 ø domain walls (including W2) lie along the hard <001> axis and one short 109 ø domain wall along the intermediate <110> axis, perpendicular to the hard axis, offsets the two lower 71 ø domain walls at the junction. For Figures 2a-2d, the cantilever length was parallel to W1. For Figures 2a and 2b the tip was magnetized in the +2 direction, perpendicular to the sample surface. For Figures 2c and 2d the tip was magnetized in the -2 direction. For Figures 2b and 2d the sample was rotated 180 ø to effectively reverse any x component of the tip magnetization relative to the sample without changing the z component of the tip magnetization. Reasons for the different greyscale shades in the various domains of these images have been suggested; however an anomalous feature across the lower 71ø wall on the right remains unidentified. It is likely the result of a surface imperfection, and is not pertinent to the overall domain structure. The scale bar in Figure 2a represents a length of 20 gm. The arrows in Figure 2c indicate the direction of magnetization in each domain.

and parallel to W1. For all of the images, the tip was magnetized perpendicular to the cantilever plane prior to mounting in the MFM. However, the cantilever was always slightly in- clined relative to the sample surface after mounting which results in an additional tip magnetization component in the sample plane as well. Differences in MFM signal contrasts above the various domain walls in an image are due in part to the varying ratio of perpendicular and in-plane field components being sensed. This ratio depends on the orientation of the cantilever relative to the length of a wall [Foss et al., 19951.

Figures 2a and 2b were measured with a positive z component of the tip magnetization, and Figures 2c and 2d were measured with a negative z component. The central 180 ø domain wall (W 1) in Figures 2a and 2b is white and in Figures 2c and 2d, the same domain wall is black. In all images shown in this paper, white (black) represents a repulsive (attractive) interaction. Repulsive (attractive) essentially refers to the magnetostatic interaction in each case. In Figure 2, the cantilever length was parallel to W1 so that the major in-plane component of the tip magnetization induced by the cantilever tilt is in the .9 direction parallel to this wall. The differences

FOSS ET AL.: MFM STUDY OF DOMAIN WALLS IN MAGNETITE ,

30,555

i i i i

I I I I

1000 2000 3000 4000

Lateral Distance (nm)

Figure 3. Average profiles of the 180 ø domain wall, W1, in Figure 2. Each profile is the average of 20 successive line scans measured perpendicular to the domain wall at a height of 30 nm. The C and D refer to the experimental conditions of Figures 2c and 2d. Although the x component of the tip magnetization was reversed between measurement of these images, the two profiles are essentially identical. From this, it is assumed that the MFM signal was due primarily to the z component of the tip magnetization.

between Figures 2a and 2b or between Figures 2c and 2d were due to the reversal of only the x component of the tip magnetization obtained by rotating the sample 180 ø underneath the cantilever. Examination of the wall profiles measured under these various conditions allowed determination of intrinsic

wall features.

Profiles of the MFM response across 180 ø domain walls were consistent with the existence of N6el caps on these walls. The MFM profiles in Figure 3 are the average of 20 scans along the same line above W1 in Figure 2. They were measured with the tip magnetized as in Figures 2c and 2d, respectively. Even though the x component of the tip magnetization for Figure 2c was opposite that for Figure 2d, the profiles were identical, indicating that mainly the z component of the sample field was sensed. In this case, the profiles indicate an asymmetry intrinsic to the wall spin structure. Since •2Hsz/oaz2 of a Bloch wall is symmetric and 32Hsz/O3Z 2 of a Ndel wall is antisymmetric, the asymmetry of the profiles in Figure 3 supports a wall structure which combines an interior Bloch wall with a N6el cap. This is consistent with theoretical predictions for 180 ø wall structures in magnetite [Xu and Dunlop, 1996].

Measurements of 71 ø and 109 ø domain walls supported the existence of N6el caps on these as well. Because the magnetization (along <111> easy axes) forms a nonzero angle with these domain walls, the component normal to the surface which varies across the wall is reduced relative to that of the

180 ø walls. Hence the 71 ø and 109 ø Bloch walls have lower

surface pole densities and less need for N6el caps than the 180 ø wall. In comparing MFM profiles of the different wall types, those of the non-180 ø walls were more asymmetric, consistent with a larger contribution to the MFM signal from the N6el cap relative to the contribution from the underlying Bloch wall. For example, the symmetries of 71 ø and 180 ø wall profiles can be compared in Figure 4b. The upper and lower profiles in

Figure 4b were measured above a 71 o wall and a 180 ø wall in the image in Figure 4a, respectively. The reduced contribution from the underlying Bloch wall makes the N6el cap contribution more pronounced.

Signs of long range flux closure via N6el caps were observed in regions where domain walls closely neighbored other domain walls, for example, near domain wall triple junctions where the three fundamental wall types intersect. As in Figure 4a, the MFM signal is larger closer to the intersection of the domain walls. The N6el caps on the walls in this region support a vortex structure around the intersection as depicted by the small arrows perpendicular to the walls in the image. The MFM signal increases in amplitude as the intersection is approached because the divergence of the near-surface magnetization increases. This behavior results in different MFM

signal contrasts above the various domains as can be seen in the image in Figure 4a. This divergence of in-plane magnetization produces spatial variation of the vertical (z) component of the sample field (sensed by the MFM) because of the finite size of the domains, i.e., because of the presence of the domain boundaries. Although the effect is relatively small, it was observed to be very long range. Differences in the MFM s ig-

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nals above neighboring domains were measured in scanning several hundreds of nanometers above the surface. The long range of this behavior is correlated with the large sizes of the domains.

3.2. Domain Wall Types and Bloch Lines

In addition to N6el caps, domain walls were also observed to be subdivided into alternating polarity segments separated by Bloch lines, confirming previous MFM results by Williams et al. [1992b]. The minimum length of a domain wall necessary for it to subdivide into opposite polarity segments depends on the domain wall type. As the surface pole density of the wall increases, the minimum wall length needed for subdivision decreases. Our experimental findings were consistent with this expectation. The image in Figure 5 displays typical behavior; the long 180 ø wall in the image is subdivided by two Bloch lines, while the other domain walls in this area do not contain

Bloch lines. The average segment length of 180 ø domain walls was observed to be 15 gm. The typical 109 ø walls were 2-3 gm in length and contained no Bloch lines. Only two unusually long 109 ø walls (-• 100 gm) were observed to be divided with a single Bloch line, but no statistically significant segment length could be determined. Although the 71 ø walls were the longest of the three wall types, typically reaching 100 gm in length, no subdivided 71 ø walls were ever observed. The magnetostatic energy of a 71 ø Bloch wall is proportional to Ms2sin20 with 0--35 ø, which is approximately one third that of the 180 ø wall.

The Bloch lines in subdivided 180 ø walls often, but not

always, resided at positions where the wall intersected scratches. A typical image is shown in Figure 6. The two Bloch lines in the portion of the 180 ø wall seen in the magnetic image (Figure 6a)running from upper right to lower left are situated at intersections of the wall and scratches. The stray fields from several scratches can also be seen in this image. The directions of the fields from •cratches which intersect the domain wall switch across the wall with the reversal of the

magnetization. In addition to the reversal of scratch stray fields, domain walls in images could be distinguished from scratches using small applied magnetic fields to move the walls. In order to make the scratches more visible in the topographic image of Figure 6b, the image was high-pass filtered. The intersection of the largest scratch (running from upper left to lower right in the image) with the wall resulted in a kink in the wall at this point, but not in a Bloch line. Bloch lines were also observed at positions corresponding to no visible defects. Nevertheless, correlations between Bloch lines and

scratches were observed, consistent with previous observations in iron whiskers which showed that defects, being sources of high local demagnetizing fields, provide sites of easy Bloch line nucleation [Hartmann and Mende, 1986].

Using simple energy considerations, Shtrikman and Treves [1960a] predicted the ratio of the domain wall segment period to the sample thickness. Their prediction agreed fairly well with experimental results for the shortest wall segments observed in iron. For example, a 0.5 gm thick iron film supported subdivided walls with segments of average length 0.56 gm which is only about 4 times longer than predicted by the Shtrikman and Treves theory [Proksch et al., 1994]. This theory does not take into account the energy required to nucleate a Bloch line. Hence it is reasonable, in this case, that

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Figure 5. (top) A 41 x 45 gm MFM image of a typical domain wall group including all three types: 180 ø , 109 ø , and 71 ø. Two Bloch lines (BL) can be seen in the long 180 ø wall, whereas the other walls contain no Bloch lines in this area. It

should also be noted that the polarities of the observed spike domains (which formed at surface defects) provide an indica- tion of the magnetization direction in the surrounding domains. (bottom) A schematic representation identifying important features of the image.

magnetite single crystal studied in this work, the observed average segment length was 15 gm, -40 times shorter than predicted by the Shtrikman and Treves model. This comparison to the model assumes the subdivided walls penetrate through the entire thickness of the crystal (-1 mm). This assumption may be false; it is possible that the walls do not penetrate

FUSS ET AL.: MFM STUDY OF DOMAIN WALLS IN MAGNETITE 30,557

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Figure 6. A 12 gm square magnetic and topographic image pair. (a) Magnetic image in which a subdivided 180 ø domain wall runs from upper right to lower left. Two Bloch lines in this section of the wall are situated at intersections of the wall

and scratches. (b) Topographic image which has been high pass filtered to make the scratches more clearly visible.

directly through to the opposite surface of the crystal, but even more likely, it is possible that the subdivision of the walls does not penetrate the sample entirely. It is reasonable to suggest that wall segments of a subdivided wall gradually become longer as the distance from the surface increases such that deep in the sample, the wall has a single polarity. The average segment length at the surface of 15 gm provides a reasonable estimate of the depth of the penetration of the subdivision. This estimate is consistent with an estimate obtained

by scaling to the results for iron described above. Shtrikman and Treves [1960b] also stated that the segments

of subdivided walls are expected to zig-zag about the easy axis near the surface to further reduce the magnetostatic energy, but this was not taken into account in the model described in the

above discussion. In the magnetite studied in this work, the

subdivided 180 ø wall segments formed a zig-zag configuration as expected in which the corners of the zig-zag were the locations of Bloch lines. The segments formed an average angle of 4 ø with the easy axis. A relationship between the segment length and the angle it made with the easy axis was also observed; as the segment length increased, the angle it made with the easy axis decreased; that is, the amplitude of the zig- zag was approximately constant along the wall. This behavior, which was also observed in iron [Proksch, 1993], represents an equilibrium configuration minimizing surface and volume magnetostatic effects. Opposite surface polarity wall segments attract each other which promotes the zig-zag configuration. However, the additional volume magnetic charges that accumulate on the sides of the wall when the wall makes an angle with the adjacent domain magnetization result in the attenuation of the zig-zag amplitude below the surface. If the depth of the zig-zag below the surface, that is, the spatial rate of the attenuation of the zig-zag were known, more accurate analytical predictions of the segment periodicity could be made. More sophisticated 3-D micromagnetic simulations are needed to obtain this information.

In one of the previous studies of magnetite with MFM, Williams et al. [ 1992b] considered the effects of N•el caps and zig-zagging separately in trying to explain their experimental data. In our work, behavior resulting from the interplay of these two effects was observed. First, as in Figure 7, profiles of opposite polarity segments in a subdivided wall were observed to have the same asymmetry. In Figure 7, the attractive (black) profile has been inverted for comparison. These profiles were measured with a tip with a lower moment and narrower geometry than standard tips. These qualities allow for a tip with a lower field and hence a measurement less invasive of the sample magnetic structure. These measurements, which show similar asymmetries to the same side of the wall, indicate that the N•el cap does not alternate sides of the wall from segment to segment (as indicated by the arrows in Figure 7a). This would allow for vortex-like closure of N•el caps of adjacent wall segments in the vicinity of a Bloch line to one side of the wall.

Another surprisingly long range effect was observed in parallel subdivided walls like those shown in Figure 8. In this 50 gm square image, two parallel, subdivided 180 ø walls run diagonally along one of the easy axes. All other features in this image are due to the stray fields above surface defects and scratches. It can be seen that segments of one polarity aligned with segments of the opposite polarity perpendicular to the walls, indicative of a long range interaction between wall segments and possibly their N6el caps.

3.3. Magnetic Field Dependence of Bloch Lines

The Bloch lines in subdivided walls were movable with the

stray field of the MFM tip. This is demonstrated in Figure 9. These MFM images were measured over the same area with the same tip, but the tip magnetization for Figure 9a was opposite that for Figure 9b, perpendicular to the sample surface. A 180 ø wall runs along one of the easy axes from upper right to lower left in the images. Two Bloch lines divide this portion of the wall into alternating polarity segments. All other features in the images are due to the stray fields above scratches. If the field from the MFM tip had no effect on the micromagnetic structure of the sample, these two images would have been


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0 0.5 I 1.5 '•

Lateral Position (gm) Figure 7. (a) A 5 gm square MFM image of a subdivided 180 ø domain wall. One Bloch line can be seen in this portion of the wall. The arrows indicate possible N6el caps on these wall segments to the same side of the wall. (b) Profiles of the two opposite polarity segments taken from Figure 7a as indicated by the dashed lines. The attractive (black) profile has been inverted and offset in order to compare the symmetries of the profiles.

essentially the same, but opposite in greyscale. This, however, was not the case. Upon reversal of the tip magnetization, the new attractive (black) segments grew. In Figure 9a, the central (white) segment is --24 gm long whereas in Figure 9b this same segment (now black) is --28 gm and the Bloch lines reside at the intersections of the wall and scratches. This area

was scanned repeatedly, reversing the tip magnetization each time. The positions of the Bloch-lifi•s alternated between their positions in Figure 9a and those in Figure 9b. Quantitative electron holography measurements of the magnetic field of MFM tips like those used for this work show that the field is -5 mT at a distance of a few tens of nanometers from the tip [Streblechenko et al., 1996]. These measurements combined with our observations put an upper bound on the pinning field of a Bloch line at a defect.

Although the motion of Bloch lines within their host walls during motion of the host wall has been described theoretically and observed using MFM in an in-plane applied field [Malozemoff and Slonczewski, 1979; T. G. Pokhil and B. M. Moskowitz, unpublished data, 1996], it is interesting to

observe Bloch line behavior independently of their host walls. With a magnetic field applied perpendicular to the sample plane, it was possible to observe Bloch line translation within subdivided 180 ø walls without translation of the walls them-

selves. Multipolar walls were converted to unipolar walls through the process of Bloch line translation in perpendicular fields from 0 to 10 mT. This experiment was similar to that done by Hartmann and Mende [ 1986] on subdivided walls in iron whiskers and reminiscent of the domain wall inversion

process proposed by Dunlop [1977]. Upon decreasing the applied field, a saturated (i.e., single polarity) wall segment again subdivided as Bloch lines gradually renucleated.

4. Conclusions

The MFM was used to study intrinsic domain wall structures on a (110) surface of single crystal magnetite. This surface contains two magnetic easy axes and allowed micromagnetic features associated with the three fundamental wall types (180 ø, 109 ø, and 71 o) for magnetite to be investigated. Our results demonstrate that the magnetostatic energy resulting from the surface termination of domain walls has significant effects on the wall structures, emphasizing the need for more sophisticated 3-D micromagnetic wall models for magnetite. Our results also provide an experimental approach which can be used when studying wall features in PSD-sized (1-50 gm) magnetite particles and can help in recognizing how intrinsic wall structures may be altered by the finite particle size. The major conclusions of our study are the following:

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Figure 8. A 50 gm square MFM image showing two subdivided, 180 ø domain walls running parallel to each other along one of the easy axes. The large arrows indicate the direction of magnetization in the bulk of each domain. Opposite polarity segments of the two parallel walls appear to be aligned with each other possibly through the extension of the walls' N6el caps in order to promote flux closure.

FOSS ET AL.: MFM STUDY OF DOMAIN WALLS IN MAGNETITE 30,559

A

43 Figure 9. Two 43 I. tm square MFM images measured over the same area; the only difference in the experimental conditions for these images was the tip magnetization for Figure 9a was opposite that for Figure 9b. A subdivided 180 ø domain wall runs from the upper right to the lower left in the images. All other features in this image are due to the stray fields above scratches in the crystal surface.

1. MFM Profiles of all three wall types suggested that the walls have surface N6el caps terminating their interior Bloch walls. The results for the 180 ø walls were consistent with

recent micromagnetic models for the wall structure in magnetite [Xu and Dunlop, 1996]. Signs of the N6el caps on the 71 ø and 109 ø walls were more pronounced than those of the 180 ø walls; possibly merely the result of the reduced contributions to the MFM measurements from the underlying Bloch walls of these walls. Additional modeling is needed to more critically compare the N6el caps on the various wall types.

2. The subdivision of the different types of walls into alternating polarity segments separated by Bloch lines displayed trends which are consistent with magnetostatic energy con-

siderations. The 180 ø domain walls, the most magnetostatically favorable for subdivision, were observed to be subdivided when longer than 15 I. tm on average. Because the 109 ø domain walls were observed to be very short compared to the other two types of walls (typically 2-3 I. tm), a measure of the necessary length for subdivision was difficult to approximate. Only anomalously long 109 ø wall segments were observed to contain a single Bloch line. Although the 71 ø walls were the longest of the three types, no Bloch lines were ever observed in these walls, and hence no minimum length was determined in this case either.

3. Subdivided 180 ø walls displayed a characteristic zig-zag structure about the <111> easy axis with an average zig-zag angle of 4 ø .

4. The N6el caps were also observed to be correlated around triple junctions where the three wall types intersect such that the caps form a long-range, vortex-like structure.

5. Bloch line segments were correlated across adjacent 180 ø walls such that segments of one polarity were aligned with segments of the opposite polarity in the neighboring wall. This correlation could be the result of flux closure via the N6el

caps on these walls or long-range magnetostatic interaction between wall segments.

6. Profiles of adjacent, opposite polarity wall segments were consistent with N6el caps extending to the same side of the wall. In the vicinity of Bloch lines, this allowed N6el caps to form vortex-like structures around the Bloch lines.

7. Bloch lines moved independently of their host walls when a field was applied normal to the domain wall. The con- version of multipolar walls to unipolar walls occurred in perpendicular fields of ~ 0 to 10 mT. This shows that walls can become saturated by Bloch line translation before domain wall displacement occurs and is reminiscent of the domain wall inversion process proposed by Dunlop [1977].

Acknowledgments. We thank Sanghamitra Sahu for assistance in preparing the magnetite samples, Matt Dugas of Advanced Research Corporation for sputter coating the MFM tips used for this work, Song Xu and David Dunlop for sharing micromagnetic simulation results prior to publication, and Taras Pokhil for useful discussions. We gratefully acknowledge the support from the Office of Naval Research grant N00014-94-1-0123 (S.F., R.P., E.D.D.). In addition, B.M. was supported by NSF grants EAR-9304520 and EAR-9526812. This is contribution 9708 of the Institute for Rock Magnetism (IRM). The IRM is supported by grants from the Keck Foundation and the NSF.

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(Received May 22, 1997; revised December 8, 1997; accepted January 7, 1998.)

domain wall structures in single-crystal magnetite

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