# Micromagnetism as a Prototype for Complexity

Authored by: Anthony S. Arrott

# Spintronics Handbook: Spin Transport and Magnetism, Second Edition

Print publication date:  May  2019
Online publication date:  May  2019

Print ISBN: 9781498769525
eBook ISBN: 9780429423079

10.1201/9780429423079-3

#### Abstract

Magnetostatic dipole-dipole interactions play an important role in determining the properties of nanomagnetic systems along with the exchange coupling and magnetocrystalline anisotropy, which were discussed in Chapter 2. The integral-differential equations of micromagnetics lead to complexity. These equations are non-linear because the magnetization is a vector of constant magnitude, M = m M s, where m is a unit vector in the direction of the magnetization. The nature of the dipole-dipole interactions favor patterns that are divergence free in volume and put magnetic charges on the surfaces that create demagnetizing fields as opposed to applied fields. In spintronics with its spin-polarized currents in semiconductors, the magnetic moment of the electron is just the tail of the dog. Much of the work on Giant Magnetoresistance (see Chapter 4, Volume 1), Tunneling Magnetoresistance (see Chapters 11, 12, and 13, Volume 1), and Magnetic Random Access Memories (see Chapter 13, Volume 3) comes from clever ways of avoiding the complexities of dipole-dipole interactions. The use of ultrathin films reduces the role of dipole-dipole interactions in the switching of the direction of magnetization, while it eliminates the variation of magnetization in the z-direction, perpendicular to the plane of the ultrathin film. By 1890, J. A. Ewing already understood the importance of the dipole-dipole interaction in ferromagnetism. Ewing coined the term “hysteresis”.

#### 3.1  Introduction

Magnetostatic dipole-dipole interactions play an important role in determining the properties of nanomagnetic systems along with the exchange coupling and magnetocrystalline anisotropy, which were discussed in Chapter 2. The integral-differential equations of micromagnetics lead to complexity. These equations are non-linear because the magnetization is a vector of constant magnitude, M = m M s, where m is a unit vector in the direction of the magnetization. The nature of the dipole-dipole interactions favor patterns that are divergence free in volume and put magnetic charges on the surfaces that create demagnetizing fields as opposed to applied fields. In spintronics with its spin-polarized currents in semiconductors, the magnetic moment of the electron is just the tail of the dog. Much of the work on Giant Magnetoresistance (see Chapter 4, Volume 1), Tunneling Magnetoresistance (see Chapters 11, 12, and 13, Volume 1), and Magnetic Random Access Memories (see Chapter 13, Volume 3) comes from clever ways of avoiding the complexities of dipole-dipole interactions. The use of ultrathin films reduces the role of dipole-dipole interactions in the switching of the direction of magnetization, while it eliminates the variation of magnetization in the z-direction, perpendicular to the plane of the ultrathin film. By 1890, J. A. Ewing already understood the importance of the dipole-dipole interaction in ferromagnetism. Ewing coined the term “hysteresis”.

The present author addressed the “Past, present and future of soft magnetic materials” at the turn of the century [1]. The “past” reintroduced Ewing to audiences that had never known the importance of his work. 100 years later, Ewing would have understood much of what was presented at present-day conferences. The “present” was a tribute to Alex Hubert who was the major figure in the theory of Magnetic Domains. The book of that name [2], which he coauthored with Rudolf Schäfer, is the most comprehensive work examining the many aspects of domains in magnetic materials. Hubert was a collector of seashells. The book is organized as a biological treatise. It remains a challenge for micromagnetism to account for the over 1000 beautiful illustrations of magnetic configurations. The study of three-dimensional magnetization configurations was envisioned as the “future.” It is the focus of this chapter.

In 1935, Landau provided the first three-dimensional model of a structure that minimized the effects of the dipole-dipole interaction; see Figure 3.1. In the 1970s a beautiful new technology was created using magnetic bubbles for which understanding of the dipole-dipole interactions was critical. This chapter is about three-dimensional magnetic structures where some of the simplifications from ultrathin films no longer apply. In this second edition, note is taken of the understanding of these structures that results from advances in computing power. What is certainly more prevalent than at the time of the first edition is the wide use of parallel processing to treat bigger problems faster. Part of this comes from gamers who have motivated the development of the GPU with thousands of CPUs for the cost of, at the most, a few dollars per CPU, using CUDA programming. Random Access Memory has become cheaper and cheaper. This and the use of the 64-bit word facilitate attacks on problems too big for CUDA.

Figure 3.1   The Landau structure as presented in domain theory is shown in the top panel with the four domains separated by four 90º walls and one 180º Bloch wall. This is a vortex structure with winding number +1. It does not have inversion symmetry. The structure with a diamond in the center has inversion symmetry and a winding number of +2. The structure in the bottom panel is an antivortex with winding number −1, where on following the circumference clockwise the magnetization rotates counterclockwise by one cycle.

The work of Abert, in his thesis and as the author of programs for both finite element and finite grid using CUDA, is recommended as an up-to-date starting source for potential users of micromagnetics [3]. Commercially available micromagnetic codes are listed there. Perhaps the most used code is OOMMF. This a public domain finite grid code from NIST with more than 20000 hits on Google. It is described as a “code with a steep learning curve.”

This chapter is about the three-dimensional patterns of magnetization in (generalized) cylinders of iron of circular, hexagonal, and rectangular cross-section. The nanobrick presented here in detail is a parallelepiped with typical dimensions X = 130 nm, Y = 80 nm, and Z = 50 nm, typically using 2 × 2 × 2 nm3 voxels. Currently, problems with 1000 times more voxels are carried out on a desktop computer. The iron nanobrick is compared with an ellipsoid of comparable dimensions to show that the main actors in this drama of complexity are tube structur es capable of playing many roles. Tube structures terminate at surfaces in patterns like the center-back of a head where the hair tries to lie in the surface yet twists as it goes away from the center. Hubert has named these as swirls [4] in a remarkable paper that remains of interest after almost 30 years.

The swirls are a three-dimensional effect requiring a ∂m z/∂z to account for their structure. The center of the swirl need not correspond to the geometric center of a cylinder. Consider a cylinder of length L and radius R with swirls on the end surfaces, z = ±L/2. If the swirls have the opposite handedness, this is inversion symmetry, favored by the Landau-Lifshitz formulation of micromagnetics. In three-dimensional magnetic patterns, inversion symmetry is broken, generally. This is already the case when the swirls have the same handedness. There is a contour line, m z = 1, that goes from one swirl to the other. In most cases that line moves from off the z-axis to on the z-axis as one proceeds in from an end surface. In some cases that line forms a helix from the top to the bottom surface.

To describe such patterns, one can look at surfaces of constant mz. Such isosurfaces surround the path of the line m z = 1, but the curl of the magnetization need not be centered on that line. An isosurface of the maximum of | curl m | would be more appropriate, but not easily programmed. Expediency leads to the use of isosurfaces of constant component of m z or a constant value of a scalar such as an energy density or magnetic charge. In the first edition of this chapter, those isosurfaces were referred to as vortices, sometimes correctly and sometimes not. In this edition, they are referred to as tubes.

In the detailed example of the nanobrick, the tube starts out as a vortex at high, but not too high, magnetic field H, and then in lower applied fields morphs into the famous Landau structure of Figure 3.1. Each tube has a life of its own. It is shown here that a tube can take a helical pattern as a means of reducing the magnetostatic energy, which comes from the magnetic charge that results from −M s div m . Because div B = 0, div H = −f div M . It is seen that – div M is a source term for the field H . The f is a factor (4π or 1) that depends on the units that may be chosen to discuss magnetism (units are treated in detail in Section 3.2.). A field that arises from – div M is called a demagnetizing field. The self-energy of a local magnetic charge density is positive. In magnetically soft materials, that self-energy is minimized by creating divergence-free patterns with magnetic charge appearing only on surfaces, as like charges repel.

In a soft magnetic material, the surface charges create fields that are approximately equal and opposite to any externally applied fields. The boundary conditions created by the magnetostatics require non-uniform magnetization patterns that decrease the parallel alignment of the electron spins, which are favored by the exchange interaction responsible for ferromagnetism. That decrease in the degree of local alignment of the spins is called an increase in the exchange energy as discussed below. The boundary conditions increase the exchange energy and lead to div M in the bulk.

These magnetic charges are paid for by lowering the increase in the exchange of energy with respect to the divergence-free patterns that satisfy the same boundary conditions. Once those magnetic charges exist in the bulk, they interact with unlike charges attracting, but not annihilating, one another because that would increase the local exchange energy. The attractive magnetostatic interaction accounts for the movement of the swirls off center at the ends of the cylinder, and also for what happens when the conditions of geometry and applied magnetic fields are right for the onset of helical magnetic patterns.

When a swirl moves away from a symmetry axis, it develops the beginning of the helical tube pattern to spread out the magnetic charge and at the same time to put the positive magnetic charge closer to the negative magnetic charge. Under recently determined conditions, the helix can propagate from one swirl to the other, providing, at last, an explanation for the striped domain patterns on hexagonal iron whiskers with [111] orientation along the long axis, discovered by Hanham 40 years ago [5]. The basic mechanism is with the arrangement of patches of magnetic charge into the formation of magnetic extended quadrupoles when the charge density is mostly confined to a plane, as is the case on a surface. There are four helical tubes of magnetic charge density that form one tube of quadrupole-like magnetic charge density.

If the diameter of the tube of a quadrupole-like magnetic charge measured in the z-direction is h, then the pitch of the helix makes the spacing between the centers of adjacent turns of the helix ~2.5 h for a helix with a period of 750 nm in a 320 nm diameter cylinder of iron; see Section 3.7.

The tube configurations discussed here have minimal problems from surfaces and can be manipulated by sufficiently small fields and currents to be attractive for nanoscale devices.

The external magnetic fields from a vortex in an ultrathin film are small because the magnetic charge on one surface cancels the effects of the charge on the other surface. In a nanobrick, the swirls on the two ends of the tube can be moved far apart so that the external fields can be greater than one-quarter of the saturation induction of iron; see Figure 3.2.

Figure 3.2   External fields for a 156 × 96 × 60 nm3 nanobrick using 1.25-nm cubes for calculation. The scan is on a line of constant y through the center of the swirl at positions 1, 10, and 20 nm above the centers of the uppermost calculation cells. The large peak in the H z field is from one swirl on the top surface. The swirl on the bottom surface is 60 nm away and creates a negative contribution to H z that is negligible on the top surface. There are also large H z fields at the four vertical corners of the nanobrick.

The swirls can be moved easily and rapidly over large distances. They become highly non-linear oscillators that serve as prototypes for the complexity of bifurcations and chaos.

The calculations for the nanobrick serve also as a primer on the effective use of modern codes for micromagnetism using fields from magnets and fields from electrical currents to provide symmetry breaking.

#### 3.2  Symmetry Breaking in a Cycle of Magnetization

For an ellipsoid in a mathematical model with continuous variation of magnetization, the application of a large applied magnetic field H along the i -direction parallel to the long x-axis can result in a uniform magnetization. For an iron nanobrick in a high field, this is not the case. If the magnetization is along i at the x-axis it will deviate from i for finite y and z. At high fields, that deviation is outward at the positive x-surface and inward at the –x-surface. This is called the flower state. It applies to fields along any of the three principal axes of the nanobrick. In this discussion of magnetization processes, the first example is field H z applied perpendicular to the largest face of the nanobrick. The flower state is discussed in the caption of Figure 3.3.

Figure 3.3   An iron nanobrick with dimensions of 156 × 96 × 60 nm3 in the flower state at a field of μ0 H z = 2.2 T, showing the directions of the magnetization as hollow cones on three sides of the brick. Each cone represents the average components of the magnetization in a volume of 72 nm3 combining 36 computational cells in a plane. The computational cells are cubes with 2-nm edges. The magnetic surface charge density on the top surface is σm ≈ M s, where M s is the saturation magnetization of iron at room temperature, taken here to be 1714 emu or μ0 M s = 2.154 T. This charge produces a magnetic field that is in the plane of the top surface and points outward. The charge density on the bottom surface is σm ≈ −M s. The field from this is inward in the plane of the bottom surface. There is also magnetic charge density on the side surfaces that varies almost proportional to z, measured from the center of the brick. The flower state appears in high fields for nanobricks of all dimensions, including a cluster of nine iron atoms at T = 0 K in a cube with one of the atoms in the body-centered position. The flower state has inversion symmetry at equilibrium. Changing the uniformly applied field will cause the magnetic moments to precess, yet the inversion symmetry is maintained. Changing the field non-uniformly by passing a current along the z-axis will break the inversion symmetry.

Discussion of the visual patterns and details of the calculations are found in the figure captions throughout this chapter.

As the field is lowered, the magnetization pattern changes continuously, starting at a critical field, from the flower state to the curling state in which the magnetization has components that circulate around the central axis; see Figure 3.4. The sense of the circulation, cw or ccw, is determined while breaking the reflection symmetry of the flower state. This is as a result of the dipole-dipole interaction using the concept of splay saving, which reduces the magnetic charge density by having opposite signs for ∂m z/∂z and (1/ρ) ∂(ρm ρ)/∂ρ.

Figure 3.4   An iron nanobrick with dimensions of 156 × 96 × 60 nm3 in the curling state at a field of μ0 H z = 0.8 T. The contour lines are for direction cosines m z = 0.95 and m z = 0.8. The area between these is shaded gray. The cones on the left, a, are the average of the magnetization for nine cells of 4-nm cubes in the topmost plane centered at z = 28 nm. In the blow up on the right, b, the cones are the magnetization in each cell. As the field is lowered below μ0 H z = 0.9 T, each cone on the top surface rotates counterclockwise from its position in the flower state as it would if a current were passed up the z-axis. This configuration at and near the surface is a surface soliton called a swirl. The position where the component along each edge changes sign is the center of an edge soliton. Each cone is treated as if it were centered on a position one-third of the distance from the apex, denoted, in detail at c in the lower right, as a sphere superimposed on the cone that points out of the plane of the page. The cones are hollow, so that when viewed from below, light is usually not reflected. The hollow cone at d points into the page. Using cones to indicate the direction of magnetization creates artificial waves in a visual pattern when the base of the cone moves from one side of a grid line to the other, for example, the line of cones at x = 0.

The energy contours present a three-tined fork [6]. The paths to both the right and the left become lower in energy than the path along the central tine. It is useful to provide a bias field so that the magnetization does not stay on the higher energy path. If one relies on the round-off error of a computation to provide the bias, the computation time can be much too long. An electric current I z is passed, briefly, up the z-axis to choose the path and save computation time.

#### 3.2.1  Breaking Inversion Symmetry and Choosing a Handedness

The inversion symmetry of the flower state may or may not be maintained in the curling state. Note that inversion symmetry for a magnetic dipole is the opposite to that of an electric dipole. Consider the moments at two points equidistant from a center of inversion symmetry on opposite ends of a line through that center. For electric dipoles, the moments point in opposite directions. For magnetic dipoles, they point in the same direction. If the inversion symmetry were maintained on the transition to the curling state, the circulation at the bottom surface would be opposite to that on the top surface; see Figure 3.5, where the “top” and “bottom” are along the x-axis. This leads to complex magnetization patterns as the field is lowered [1]. Away from the axis, the in-plane magnetization reverses on going from top to bottom, but the component along the axis remains m z = 1.

Figure 3.5   Curling with inversion symmetry for an iron nanobrick with dimensions of 112 × 70 × 42 nm3. The moments near the front surface at x = X/2 show a cw circulation. The moments near the back surface at x = –X/2 show a ccw circulation. The line of moments joining the front and back surfaces shows a Néel wall in the form of an end-over-end helix. Inversion symmetry is maintained. Note that the two cones labeled A are connected by a line through the center of the brick and have the same orientation. The rendering of the magnetization as cones shows an object that does not have inversion symmetry (a line joining the tips of the cones doe s not go through the center of the brick). It is the magnetization at the grid points that has the inversion symmetry. The field along the x-axis where the two surfaces are separated by 112 nm was used for this illustration of inversion symmetry. When the field is along the z-axis, the two surfaces are separated by 42 nm. Then the exchange energy presents a sufficiently large barrier that the cw/ccw structure is suppressed; see Figure 3.6. The distinction in the meaning of inversion symmetry between a polar vector and an axial vector is illustrated in the lower left corner using the polar vector i for the current in a ring to produce an axial vector m. The two current vectors, equidistant from the center of inversion symmetry, point in opposite directions, while the magnetizations they produce point in the same direction.

There are many ways to break the inversion symmetry to arrive at the Landau configuration. Here the discussion is focused on achieving that configuration by passing a small current along the z-axis while lowering H z from saturation and then passing a small current along the y-axis to further break the symmetry. That is just one of many paths from saturation to the Landau configuration. A more complicated path is discussed in Section 3. 7

#### 3.2.2  Response of Isosurfaces to Changing Fields

In the presence of the field from a small current I z, the flower pattern develops a slight swirl even at the highest fields. This is analogous to the magnetization of a ferromagnetic material in a small field above its Curie temperature. When the temperature is lowered below T c, the response to the small field increases rapidly as the spontaneous magnetization develops. In the nanobrick, below a critical field H zc, there is a spontaneous contribution to the angle of rotation of the moments away from their direction in the flower state. The swirls are centered on the z-axis; see Figure 3.6. As the swirls develop with decreasing H z, the action is first concentrated at the core of the swirl. As the field is decreased to zero, the moments in the corners are the last to fully participate in the curling pattern, turning from pointing out, along a line at 45º, to pointing perpendicular to that line. The places where the magnetization with respect to an edge changes sign is the center of an edge soliton. In Figure 3.6, the solitons move from the center of each of the top edges to the corners of the nanobrick.

Figure 3.6   The curling pattern that develops on reducing μ0 H z from 0.9 T for the nanobrick in Figure 3.4 is called theIvortex state. The circulation is cw (or ccw) in all planes of constant z. The I-vortex state persists in zero field for the dimensions of 102 × 70 × 42 nm3, but, if Z > 42 nm, the I-vortex state becomes unstable at a low field and moves off center. When there is a computational cell centered at the origin, the cells along the z-axis all have m z = 1. Here the center of the vortex is midway between two cells on the y-axis displaced in x by 1 nm on either side of the vortex center. When the cells are grouped in bundles of 49 cells in a plane to produce the cones shown here, the center of the bundles nearest the core of the swirl are displaced by 7 nm from the axis. This is far enough from the center of the swirl that the magnetization lies close to the surface when H goes to zero. The development of the I-vortex state is followed here with the 42-nm-high brick in the presence of a small current I z. Note the progress of the corner moments. These change little at high fields. At lower fields, they rotate so they are parallel to one of the two corner edges. In the lowest fields, they lose their radial component and at the same time turn down in response to the dipole field from the central core of the vortex structure. Note that the hollow cone in the lower left corner of the large panel for 0.00 T appears darker at the bottom of the cone where the light enters. This cone points into the iron nanobrick as is the case for all corners. It is more evident in the lower left.

In general, reversal of the magnetization of the edges takes place by the propagation of edge solitons, either from one end or the other or from both at the same time. It can also happen through the formation of a pair of solitons somewhere along an edge, with the two solitons moving to opposite ends to reverse the edge. The propagation of edge solitons was important in the switching of the original thin film MRAM bits that had lateral dimensions in the order of a micron. In 1998, most of the work in micromagnetics concentrated on calculating hysteresis loops, without much attention paid to the visualization of the reversal processes. Solitons were overlooked even when the visualizations were presented that showed them. They had to be pointed out to the presenters. (Solitons along the axis of the nanobrick are seen in Figure 3.13 and discussed in Section 3.7.) The swirls are surface solitons.

Figure 3.6 shows the curling pattern that develops on reducing Hz. It is seen that the sense of the circulating component is forced to be the same at all z by the small current along z . The radial component is inward for z < 0 and outward for z > 0 as a result of the dipole-dipole interaction. For the exchange interaction, the energy does not depend on the angle of the in-plane magnetization in cylindrical symmetry. The m z-isosurfaces define the tube structure. The role of these tubes in magnetism has often been overlooked.

The direction cosines of the magnetization M are denoted by m x, m y, and m z. Starting from 1, m z decreases with distance from the z-axis. Tubes of constant m z, called m z-isosurfaces, connect the top and bottom surfaces; see Figure 3.7. The cross-sections of the tubes in planes of constant z are elliptical (reflecting the geometry of the brick) and centered on the z-axis. The ellipses are largest in the midplane z = 0. The magnetization does not lie in the m z-isosurfaces except in the midplane. Elsewhere it has a radial component as well as a circulating component.

Figure 3.7   The S-tube state develops from the I-vortex state with decreasing H z for the iron brick with dimensions of 156 × 100 × 60 nm3. Each cone represents 144 cubes with 4-nm edges in a plane of constant y. The coarse grid is used in this calculation for visual purposes, even though it is too crude for accurate assessment of critical fields. The width of the iron nanobrick was increased by one computational cell compared to Figures 3.3 and 3.4 in order to show the central cross-section, y = 0, of the I -vortex state. The formation of the Stube state is followed using m z-isosurfaces that connect the top and bottom swirls of the Ivortex state. The intersections with the midplane at y = 0 for μ0 H z > 0.4 T or at y = 4 nm for μ0 H z < 0.4 T are shown for the contour lines in that plane for m z = 0.95 and m z = 0.80 with the central white regions corresponding to m z > 0.95 and the gray regions to 0.8 < m z < 0.95. A small bias current in the x-direction creates a field, H y, that is positive at the top and negative at the bottom, tilting the magnetization in the +x-direction at the top and in the −x-direction at the bottom. To make this clearer, a contour for m y = 0 is shown in the middle of the central white region. The tilt of the m y = 0 contour in the second panel at 0.75 T is the result of the bias current. The larger displacement of the swirls in the panel at 0.50 T is almost all the result of a spontaneous displacement of the swirls in opposite directions to form the S-tube state. In zero field, the swirls reach their maximum displacement and sit near x = (X–Y)/2, y = y 1, z = Z/2 and x = –(X–Y)/2, y = y 1, z = –Z/2, where y 1 is a small displacement; see Figure 3.9. At Z = 60 nm, the I-state vortex can persist as a structure in an unstable equilibrium all the way to H z = 0. The width of the contours at z = 0 is essentially the same for the unstable I-state vortex and the S-state tube.

#### 3.2.3  Displacing the Swirls

Bias fields perpendicular to the z-axis can move the vortex structure off center. The vortex moves perpendicular to the bias field with the swirls remaining above one another, but the ellipsoidal cross-sections distort and need not be centered with respect to the swirls. That is for equilibrium configurations. In dynamic responses, the swirls move with respect to each other. The m z-isosurface takes on a life of its own. The tubes can bulge, twist, bend, and even take helical patterns. If the bias fields are derived from a current i x along the x-axis in the +x-direction, the upper swirl moves in the +x-direction and the lower swirl moves in the –x-direction as a result of fields in the +y-direction at the top and in the –y-direction at the bottom. The H y fields from i x are largest at the top and bottom surfaces. When a uniform H y field is superimposed on the field from i x, one can independently manipulate the two swirls in any H z.

#### 3.2.4  Onset of the Landau Structure

If i x is maintained while H z is lowered, there is a critical H z, below which the displacements of the swirls in opposite directions increase rapidly with decreasing H z. The development of the S-tube state from the I-vortex state is shown in Figure 3.7. If the dimensions are at the threshold for the instability of the I-vortex state, the correspondence of the S-tube state with the well-known Landau configuration is not obvious. When viewed in the central cross-section, z = 0, the pattern more closely reflects the Landau configuration; see Figure 3.8. When the dimensions X and Y are much larger than the values used here, the swirls sit at the ends of the 180º Bloch wall of the Landau configuration at x = (X–Y)/2, y = y 1, z = Z/2 and x = (X–Y)/2, y = y 1, z = –Z/2, where y 1 is a small displacement; see Figure 3.9 for a portrait of the S-tube that is the heart of the Landau configuration. If the current i x were in the –x-direction while H z was lowered through the critical H z, the swirls would sit at x = (X–Y)/2, y = y1, z = Z/2, and x = (X–Y)/2, y = y1, z = –Z/2. The statics and dynamics of switching between these two configurations, called the S+ tube and the S sup− tube, are emphasized in this chapter.

Figure 3.8   Central cross-section for z = 0 for an iron nanobrick with dimensions of 156 × 100 × 60 nm3. The core of the S-tube and center of the 180º Bloch wall is the central white region where mz > 0.95. In the surrounding gray region, 0.95 > m x > 0.8. In the dark gray region, y < 0, –0.4 < m z < –0.2. The open contour around the central region marks m z = 0. In the corners, m z ≈ 0.8. A negative H z is required to turn the corners down for the Z = 60 nm structure, while for Z = 42 nm in Figure 3.6, the corners are already down at H z = 0. The slightly wavy horizontal line is the contour for m x = 0. The vertical thin line is the contour for m y = 0. The gray lines from the four corners show the positions of walls in the divergence-free van den Berg construction [2].

Figure 3.9   Portrait of an S-tube state core showing contours of m z = 0.95 in successive planes in y or z spaced by 1.5 nm. The S-tube state is a 3D structure that is not fully characterized by a single slice through the m z-isosurfaces. The upper panel shows that the swirl is offset from the core of the 180˚ wall, which itself is offset from the plane y = 0. The silhouette of the structure in the l ower panel has contributions from several slices in y. The lines in the upper panel show the positions of the 180° Bloch wall and the four 90° walls delineating the two end-closure domains in the classic structure first postulated by Landau in 1935. These lines are also the positions of the walls obtained by van den Berg in his solution to the problem of ideally soft magnetic materials in the limit of ultrathin films.

#### 3.2.5  Landau’s Structure

Landau’s structure, as shown in Figure 3.1, was postulated for a large iron brick where the anisotropy causes a clearer distinction between the domains and the domain walls. The Landau structure does not have inversion symmetry, but the structure in Figure 3.1 with a diamond in the center, often observed in the 1950s by researchers at General Electric and General Motors, does have inversion symmetry. Landau put the walls at 90º to avoid a discontinuity in the component of m perpendicular to the walls of the end-closure domains. The magnetization was assumed to be in the z-plane everywhere except in the 180º Bloch wall. The Bloch wall creates surface charges, σ = n · m , which Néel eliminated by having the magnetization lie on the surface as it turned through 180º. These are the Néel caps on the Bloch wall first calculated by LaBonte in the 1960s in his treatment of a never-ending Bloch wall; see Figure 3.10. Consequently, it was pointed out that the Néel cap was an extension of one or the other of the end-closure domains with that extension terminating in a swirl at the opposite end [7]. The swirls satisfy the topological necessity of the magnetization pointing out of the surface in at least two places. One of the first SEMPA (scanning electron microscopy with polarization analysis) experiments was to show that this is the case in iron whiskers [8]. Structures, where both swirls are on the same surface, are possible [9] but not considered here. Which end of the Bloch wall claims the swirl depends on the sense of rotation in the Néel cap, which itself depends on the sense of rotation of the magnetization around the y-axis on transversing the Bloch wall in the y-direction. The diamond structure, also shown in Figure 3.1, requires additional tubes as there are now two swirls and two half anti-swirls on the top and bottom surfaces. But that discussion will be left for another time.

Figure 3.10   Cross-section in the plane x = 0 for the 156 × 100 × 60 nm3 iron nanobrick. Each cone represents a 4-nm cubic computational cell. The almost vertical bowed line in the center is the contour for m x = 0. It is centered along the top and bottom surfaces but bows to the right by 4 nm at the center of the nanobrick. In the white region about y = 4 nm and z = 0, m z > 0.95. This is the core of the S-tube and the center of the 180° Bloch wall of the Landau structure. The light line at z = 0 is also the contour for m y = 0. Along that line, the cones rotate through an angle greater than 180º from y = –Y/2 to y = Y/2. This is the Bloch wall separating the two principal domains of the Landau structure, where in the white region on the far left, m x < –0.95 and in the white region on the far right, m x > 0.95. In the gray region on the left, –0.95 < m x < –0.8. In the gray region on the right, 0.95 > m x > 0.8. In the gray region in the center, 0.95 > m z > 0.8. This figure is in one to one correspondence with LaBonte’s calculation of the cross-section of a neverending Bloch wall from the 1960s, which clearly showed the existence of the Néel caps to the left of the center on both surfaces. The magnetization in the Néel caps and the core of the S-tube circulate around the white bulge on the left, which forms the core of a partial vortex in the x-direction. This partial vortex accounts for the component <m x> that accompanies the transformation of the I-vortex to the S-tube as shown in Figure 3.14.

The Bloch wall with the Néel caps is seen in Figure 3.10 for an S-tube in the iron nanobrick for a cross-section in the plane x = 0. This structure is essentially the same as that calculated by LaBonte in the 1960s for a never-ending 180° Bloch wall [10]. That there is a regular vortex structure around a line in the x-direction has long been noted without recognizing that the vertical section of that circulation is actually the core of an S-tube connecting the upper and lower surfaces. The swirls do not appear in LaBonte’s calculation for they are displaced to infinity in the never-ending Bloch wall. They do not appear in Figure 3.10 because there the cross-section is midway between the two swirls. But the displacement of the core of the S-tube does appear in Figures 3.8 and 3.9 and in LaBonte’s calculation. The reason for the displacement has long been understood. It is to make room for the Néel caps that not only remove magnetic charge from the surfaces but also minimize volume charge by curling about the line parallel to the x-axis to form part of the x-axis vortex structure.

#### 3.2.6  The Nano-Ellipsoid

The S-tube state for a nano-ellipsoid is shown in Figure 3.11. The nano-ellipsoid avoids the discussion of what happens along all 12 edges in the nanobrick. The central core of the S-tube in the nano-ellipsoid is the Bloch wall terminating in the displaced swirls. The S-tube is illustrated by an m z-isosurface for m z = 0.8 which corresponds to the 3-4-5 triangle with acos (m z) ≈ 37º. (The m z-isosurfaces in Figure 3.7 are also for m z = 0.8. In Figure 3.9, for the portrait of the S-tube, the m z-isosurfaces are for m z = 0.95.) The Néel caps and the closure domains are represented by the m y-isosurfaces for m y = ±0.95 in Figure 3.11. The contours lines on the circular slices at various values of x, show the m x components. The principal domains of the Landau structure in the ellipsoid are suggested by contours with m x > 0.8 and m x < –0.8.

Figure 3.11   The Landau structure in an ellipsoid (260 × 161 × 100 nm3) calculated using R. Hertel’s finite element program TetraMag. This program results in “slices” which allow viewing of the internal structure. The isosurfaces at my = +0.95 and my = –0.95 delineate the combined closure domains and Néel walls at each end of the core of the Bloch wall, outlined by the isosurface mz = 0.8. The core of the Bloch wall terminates at the surfaces in swirls. The Bloch and Néel walls separate the regions of high magnetization in the +x and –x directions, indicated by the darker regions of the five planes perpendicular to the x-axis. The same sequence of fields used in Figure 3.7 to produce the Landau structure starting from the high field flower state. But, the flower state exists only to the extent that the finite elements are not sufficiently effective in producing an exact ellipsoid. The ellipsoid is simpler than the brick because there are no corners or edges. For the ellipsoid, this configuration is one of the eight ground states which differ in the choice of polarization with respect to the z-axis, the handedness of the circulation about the z-axis and whether the upper swirl is on the left in an S − state as shown here or on the right in an S + state. The two swirls and the core of the Bloch wall are displaced from the y = Y/2 plane just as shown for the brick in Figure 3.9. The swirls can be displaced in opposite directions by the field from a current in the x-direction as shown in Figure 3.7 for the brick. A uniform H y will move both the swirls toward one end or the other of the ellipsoid. A large H y will drive an S + pattern into an I* pattern (see Figure 3.16) near the end of the ellipsoid. Then, on decreasing Hy, the I* pattern becomes an S + pattern again if biased by a field in H x. If the bias field is in –H x the I* pattern becomes an S − pattern. In a high enough H y field, the swirls move together out one end of the ellipsoid.

#### 3.2.7  Manipulating the Landau Structure

Near the critical field, it is easy to drive the swirls back and forth between their small offset positions. At H z = 0 it is harder. But it can still be done. It is easier to do this dynamically with the right time sequence of the fields. The moving swirls form a highly non-linear oscillator. They can move over distances of 50 nm in 200 ps. The swirls carry with them localized external magnetic fields with μ0 H z ≈ 0.5 T from the surface magnetic charges. At resonance, they will oscillate as long as the energy is supplied to compensate for the damping losses. As the swirls oscillate back and forth in the x-direction they also make excursions in the y-direction as they follow paths of almost constant energy. This is in contrast with the motion of the swirls in slowly varying fields where the paths are perpendicular to the paths of constant energy; it is easier to go around a barrier than to climb over it. The ability to easily and quickly move well-localized sources of large external magnetic fields is a phenomenon waiting to be exploited in the world of nanoscience and nanotechnology.

#### 3.2.8  Core Reversal by Pair Creation

When H z is increased in the – k -direction, the swirls move back toward the z-axis reaching the axis at a critical field that is lower in magnitude than the critical field for forming the displaced swirls on the decreasing field in the k -direction. During all these changes the m z = 1 line still goes from the bottom surface to the top surface. But at a high enough field in the – k -direction, a pair of solitons are created that propagate from near the midplane outward toward the surfaces, reversing the core of the vortex as they propagate; see Figure 3.12.

Figure 3.12   Pair creation in a reversed applied magnetic field μ0 H z = –0.8 T. The centers of the propagating singularities are in the middle of the four central cones in planes of constant z in each of the panels A, B, C, D on the right. The panel on the left is a cut in a plane of constant y through the two cones just above that center. By symmetry, the two central columns of cones on the left are two views of each cone of the four central cones. In panels A, B, C, D the four central cones all rotate in unison as one views this as a sequence of layers one above the other. It is also the time sequence for any given layer as the singularity passes up the axis. For the upper singularity, ∂m z/∂z is positive and in the lower singularity, it is negative. For splay canceling, a negative m r is required for the first case and a positive m r for the lower case, as ∇·m ≈ m r/r + ∂m z/∂z. As m r is positive in both regions, there is splay saving in the case of the lower singularity. The lower one propagates more quickly as the splay saving lowers the energy barrier.

Each soliton contains a singularity in a continuum model. On a finite grid, the core of the vortex centers itself on a position between the grid points, so that the singularity itself is a mathematical point between the grid points. This should be the case for a real lattice where the center of the vortex would lie between the atoms. But that is a classical description for which there is no quantum mechanical calculation to support the concept of the atoms maintaining a rigid magnetic moment in the core of the soliton. The magnetic moment density can vary in direction as well as magnitude across an atom [11].

#### 3.3  The LLG Equations and Their Numerical Solutions

The above description is for a vortex that intersects the top and bottom surfaces. Starting with fields in the x- or y-directions, vortices can form with swirls on the end and/or side surfaces. In either case, the removal or reversal of these fields returns the system to the state where the swirls are on the top and bottom surfaces. This process can be quite complex with the swirls moving from one surface to another or by the vortices leaving the brick and then re-entering. Starting with the Landau structure in zero field, there is a rich landscape of responses, steady and dynamic, to fields and field gradients applied in the x-y plane.

Solving the Landau-Lifshitz-Gilbert (LLG) equations of motion explains most of the above. A modern review of micromagnetics is given in Volume 2 of The Handbook of Magnetism where 33 authors over 500 pages provide a modern review of micromagnetics. The chapters on numerical methods are quite detailed with hundreds of equations [12].

There are only a few cases where these equations can be treated by analytical techniques. The LLG equations are simultaneous integral-differential equations for the direction cosines of the components of the magnetization. The differential part comes from the exchange energy responsible in the first place for ferromagnetism. Any variation of the moments from parallel alignment locally changes the negative exchange energy in the positive direction. The integral part comes from the dipole-dipole interactions in which, if there are N moments, there are N(N1)/2 pairs to consider.

#### 3.3.1  Zeeman Energy

The independent variable is the applied magnetic field H ext. The Zeeman energy is E ext = –μ· B for a magnetic moment μ in an externally applied magnetic induction, B , defined by the force on a moving charge. In free space, B = μ0 H ext, where μ0 ≡ 4π 10−7 H/m, is an arbitrary constant called the permeability of free space in SI units, which were enacted by a one-vote margin at a conference in 1931 dominated by electrical engineers with little appreciation of magnetism. Magneticians have resisted the adoption of these units in favor of Gaussian units for the reasons of good physical insight. In Gaussian units μ0 = 1. The magnetic moment of a nanobrick is μ = < M> V, where < M> is the average magnetization in the volume V. In SI units, the magnetization M is replaced by the magnetic polarization J , where μ = < J> V/μ0. In terms of the unit vector m representing the direction of magnetization or magnetic polarization, the Zeeman energy in SI units is E ext = –J s m · H ext V = –M s m ·(μ0 H ext) V, where J s and M s are the spontaneous polarization and the spontaneous magnetization, respectively, of the iron nanobrick. For iron, J s = 4 π 1714/104 T = 2.154 T, where 1714 comes from M s for iron in Gaussian units and the 104 comes from the conversion of Gauss to Tesla. In Gaussian units the polarization has the same units as the magnetization but differs by the factor 4π; that is, J s = 4πM s. In SI units J s = μ0 M s. This means that one cannot convert from SI to Gaussian just by setting μ0 = 1. The magnetic susceptibility, which is dimensionless in both systems, differs by that factor of 4π.

The engineers removed the 4π’s in Maxwell’s equations from where they properly belong, in front of the source terms. In Maxwell’s equations, M appears as a current source given by j m = ∇× M. That 4π will come up again when the dipole-dipole interaction is considered. The anisotropy and the exchange energies depend only on the directions of magnetization, so the question of units should not affect their energy expressions.

#### 3.3.2  Anisotropy

When a magnetic moment is rotated with respect to the atomic lattice, there is a change in energy because of spin-orbit coupling to the lattice. The vector derivatives of the anisotropy energy density with respect to the magnetization components produce effective fields acting on the magnetization. These are not real fields and do not necessarily obey the Rabi-Schwinger theorem [13], that a magnetic moment in a real field can be treated classically, but they are treated as real fields in the LLG equations. As the anisotropy fields play very minor roles in the behavior of an iron nanobrick, quantum mechanical subtleties will be ignored here.

The magnetization patterns for an iron nanobrick and those for a permalloy nanobrick are indistinguishable in a zero field at the level of visually comparing graphs of lines of constant components of the reduced magnetization. The anisotropy becomes a player in the magnetization patterns when the dimensions are in microns rather than nanometers. Even then the anisotropy in single crystal iron is much more noticeable than in polycrystalline iron when the grain size is less than the dimensions of the element being considered.

Magnetostatics act as a constraint on the magnetization of adjacent grains forcing them to have close to the same normal component to their mutual boundary. The anisotropy rotates the magnetization away by a small amount from the pattern that would be there in the absence of anisotropy. That small amount accounts for the phenomena called buckling patterns or ripple structures where there are Néel-like walls perpendicular to the direction of the magnetization. The helical pattern in hexagonal iron whiskers occurs whether the anisotropy is considered or not for dimensions less than 1 µm. Anisotropy requires large distances to be effective in iron. To see where anisotropy is important in spintronics visit Chapter 2 in this volume.

#### 3.3.3  Exchange Interaction

The effective field from the exchange interaction can be written using the Laplacian of the direction cosines of the magnetization m , leading to nine terms in the torque equations representing curvatures in each of the components in each of the three directions. By definition, the exchange energy density e ex is A times the sum of the squares of the nine first derivatives of the three components of the magnetization direction, with respect to the three axes. The coefficient A is called the exchange stiffness constant. For the unit vector m , | m |2 = 1 and hence m · ∂ m /x = 0; e ex can be written as

3.1 $e ex = A ( ∇ m ) 2 = − A m ⋅ ∇ 2 m ,$

where the Laplacian can be written also as

3.2 $∇ 2 m = ∇ ( ∇ ⋅ m ) − ∇ × ( ∇ × m ) .$

That an expression with only first derivatives can be replaced by an expression with only second derivatives is the property of a unit-vector field. To see this, put the unit-vector field along the y-direction and then rotate one moment into the x-direction, with respect to its neighbors, by a small angle. Then (dm x/dx)2 = –d2 m y/dx2. The minus sign appears in front of the Laplacian.

Equation 3.2 can be considered as the definition of the vector Laplacian. When expressed in Cartesian coordinates there are nine terms in ∇ (∇· m ) and 12 terms in × ( × m ), but the six terms in the first cancel the six terms in the second. The vector Laplacian in Cartesian coordinates has nine second derivatives, whereas the starting expression has the squares of the nine first derivatives. The vector Laplacian has 15 components in cylindrical coordinates and 18 in spherical coordinates.

The appeal of Equation 3.2 is that the curl can be envisioned by the Stokes theorem for the line integral around an area and the divergence by the Gauss law for a pillbox. Despite all its terms, a simple insight can be seen by looking at the vector Laplacian in spherical coordinates. For the spherical hedgehog, the magnetization is along the radial vector everywhere, $m r = r ^$ (The spherical hedgehog has no swirl, a real hedgehog does.) The curvature is 1/r in each of the directions orthogonal to the radial vector $r ^$ . One of the 18 terms is $2 m r / r 2 r ^$ , with no derivatives. The exchange energy density is 2A/r 2. If this is integrated for a sphere of radius R, the total energy is 8πAR. The energy density is all from one term in ∇ (∇· m ). There is no curl in the spherical hedgehog. This hedgehog has a point singularity at its center. Next, take the hedgehog pattern and rotate every spin by π around the z-axis. Now in spherical coordinates, the directions of the spins are given by m = –cos(2θ) R + sin(2θ) Θ. This is a splay-saving pattern that decreases ∇ (∇· m ). It will increase | × ( × m )|.

An easy way to visualize the component of the vector Laplacian in the direction of the magnetization is to put the unit-vector magnetization in the x-direction for one voxel, add the x-components of the six nearest neighbor voxels, and subtract the number six, then divide by the square of the voxel side. This measure of curvature will be a negative number and that is why there is a minus sign in the expression for the exchange energy with the vector Laplacian.

As an exercise in vector analysis, the exchange energy was evaluated for that splay-saving pattern using: (1) the liquid crystal expression A·{(∇· m )2 + (× m )2}, (2) the first derivative expression A ( m )2, and (3) the second derivative vector Laplacian –A m ·{∇ (∇· m ) − × (× m )} for each of the three choices of the coordinate system. There are so many terms that it is easy to make a mistake, so it helped to know the answer before starting. If one looks at A ( m )2, the exchange energy as originally defined, there are nine derivatives each of which is squared. Rotating the spin by 180º about the z-axis changes the signs of the derivatives but does not change the squares of the derivatives. The energy density must be the same as for the spherical hedgehog. The splay energy is reduced, but the total energy remains the same. The energy for the sphere with the splay-saving pattern is also 8πAR.

Splay saving is very important as it reduces the magnetic charge density, but not because it lowers the exchange energy as it appears to do in the liquid crystal formulation of the strain energy.

The hedgehog shows the scale of the exchange energy. It increases in proportion to a length. For a 100 nm sphere in the splay-saving configuration, 8πAR is ~500 picoergs for iron. The numerical solution using the LLG equations for a 100 nm iron sphere removes most of the enormous magnetostatic energy that a hedgehog would have. The pattern is a vortex with much of the exchange energy, ~400 picoergs, in the two swirls at the north and south poles. When the exchange energy of a sphere of iron is calculated using the full power of micromagnetics, the result shows that, as the sphere gets larger, the total exchange energy approaches that of the hedgehog, within one part in 100 by a radius of 150 nm.

Another configuration where it is simple to determine the exchange energy is when the magnetization circulates around the axis making an angle χ with the radial vector from the z-axis. It doesn’t matter for the exchange whether that angle is 90º for which the div m = 0 (the pattern of a z-axis vortex) or 0º for wh ich the curl m = 0 (the pattern of a cylindrical hedgehog). If m ρ = 1 in cylindrical coordinates, ρ, ϕ, and z, the vector Laplacian is −1/ρ2 in the ρ-direction. If m ϕ = 1, the Laplacian is unchanged, and the energy density is also A/ρ2. The first case is pure splay and the second case is pure curl. The total energy for a cylinder of length 2 L and radius R is 4πA log (R/R 0)L, where R 0 is the radius of the hole that Néel first put down the center to avoid the logarithmic singularity in estimating the critical radius for the onset of curling in an infinitely long cylinder.

The toroid is a simple problem. There is no divergence, so it is only necessary to use Stokes theorem twice to find curl(curl ( m )) for a toroidal shell magnetized with components circulating about either the minor radius or the major radius.

The energy for producing a vortex-antivortex pair in an ultrathin film (no dependence on z) is 8πAh where h is the height of the film.

In a divergence-free pattern of magnetization, the only contributions to the exchange field come from × m . In cylindrical coordinates, there are 15 terms in the Laplacian. Even so, it is easier to think about divergence-free patterns using the Laplacian rather than ( m )2. In the simplest magnetization pattern for a vortex: m ρ = 0, m ϕ = tanh (aρ) and m z = sech (aρ), where 1/a is a constant called the effective exchange length, as it measures the region where the exchange energy is more important than the dipole-dipole energy that would come from the z-component of the magnetization at a surface. There are no derivatives with respect to ϕ or z, there is no m r component, and ∇·m = 0. In this case, there are still five terms in the Laplacian. When the dot product of the Laplacian and the magnetization is considered, two of the terms cancel and two of the terms combine, leaving the local exchange energy density e ex(r) as

3.3 $e ex ( ρ ) = A ( ( sec h 2 ( a ρ ) a 2 + tan h 2 ( a ρ ) ρ 2 ) .$

Even after these simplifications, the integration of Equation 3.3 requires approximations [14]. It would be helpful to have the total exchange energy in terms of <m z>, but that requires the evaluation of $∫ ρ sec h ( a ρ ) d ρ$ , which does not have an analytic expression. Progress in micromagnetics is difficult without numerical methods.

The exchange fields calculated analytically, in the simple case with m z = sech(aρ) and m ρ= tanh(aρ), do not quite point in the same direction as the magnetization, indicating that these are not self-consistent solutions of the torque equations. They are, however, quite good approximations to the computed magnetization pattern around a vortex in an ultra-thin circular disk (for which there is no significant dependence on z) using the full micromagnetic equations including all the dipole-dipole interactions [14].

#### 3.3.4  Dipole-Dipole Interactions

The treatment of the dipole-dipole interaction is different in the finite element program TetraMag by Hertel [15] and the finite grid program LLG Micromagnetic Simulator by Scheinfein [16]. In TetraMag the moments in each element are used to solve Poisson’s equation for the magnetic potential from which the fields are derived. In the LLG Micromagnetic Simulator, the dipoles on a uniform grid are summed using fast Fourier transforms, which require that every grid point is treated the same, so that the interactions depend only on the vector connecting any two grid points. Using fast Fourier transforms reduces the calculation from the order of N 2 to the order of N ln(N). Fortunately, the equations of micromagnetics have attracted mathematicians who have brought sophisticated methods to bear on the problem of treating the dipole-dipole problem, sufficiently sophisticated to be beyond the intent of this chapter.

There is a problem with the finite grid approach in treating boundaries that are not aligned with the grid. This is avoided here by choosing the parallelepiped as the object of interest. One solution to the jagged-boundary problem is to make boundary corrections by treating local regions at the order of n(n–1)/2 and farther regions at the level of N ln(N) for all N points, adding Nn(n–1)/2 calculations, where n is the number of points in the local region. It takes but a small n for n(n–1)/2 to be bigger than ln(N).

In Section 3.7, for circular geometry, the jagged boundaries appear in the graphics for the charge density. The LLG equations are solved with boundary corrections. but the graphics using the correct magnetization do not correct for the jagged edges in plotting div m .

The dipole-dipole energy is written using the demagnetizing field. This follows from fact that the sources of ∇× B are currents and there are no sources for ∇· B . The vector H is a mixed vector combining the sou rce vector M with the field vector B . In Gaussian units the combination that defines H is H B − 4π M , where the 4π belongs in front of the source vector. In SI units H is defined as μ0 H B J. The sources of ∇× H are real currents. The sources of ∇· H are magnetic charges given by −∇· M . This part of H that derives from those charges is called the demagnetizing field H D. In SI units, ∇· H D = (Js0) ∇· m . In Gaussian units, ∇· H D = –4πMs ∇· m . The demagnetizing field energy is the integral over the nanobrick to get $E dem = − ( 1 / 2 ) ∫ H D ⋅ J d V$ in SI units and $E dem = − ( 1 / 2 ) ∫ H D ⋅ M d V$ in Gaussian units. The factor (1/2) comes from these being self-energies.

For a sphere uniformly magnetized in the z-direction, in Gaussian units H D = 4π(1/3) M s k and M = M s k for which $E dem = − ( 1 / 2 ) 4 π ( 1 / 3 ) M s 2$ . In SI units, J = J s k and H D = (1/3)J s0 k for which $E dem = − ( 1 / 2 ) ( 1 / 3 ) J s 2 V / μ 0$ or $E dem = − ( 1 / 2 ) ( 1 / 3 ) μ 0 M s 2 V$ . In all cases, the (1/3) is the demagnetizing coefficient N for a sphere. In the general ellipse, N x + N y + N z = 1. In Gaussian units, 4πN is called the demagnetizing factor.

#### 3.3.5  The Torque Equation

The basic equation of micromagnetics is the torque equation for the precession with time t of an electron in a magnetic induction B . The electron has a magnetic moment μ = –gμB S and an angular momentum , where ħ is the reduced Planck’s constant, g is very close to 2 and S = 1/2. The minus sign appears because the spin and the moment are in opposite directions for a negatively charged electron. The ratio of the angular momentum to the magnetic moment of the electron is γe = –gμB/ħ. These are used in a classical equation of motion, where the angular momentum of the magnetic moment is L μ = μe and the torque acting on that moment is μ × B ; that is,

3.4 $( 1 γ e ) d μ d t = μ × B .$

A mystery of this equation lies in questions about nutation. Nevertheless, it works. One can replace B by μ0 H in the torque. Dividing bo th sides by a small volume and letting μ stand for the moment in that volume, μ can be replaced by M to give

3.5 $( 1 γ e ) d M d t = μ 0 M × H ,$

which is in SI units, but produces the torque equation in Gaussian units by replacing μ0 = 4π 107 H/m by a dimensionless 1. It is common practice to absorb the μ0 into the gyromagnetic ratio to write

3.6 $( 1 γ 0 ) d M d t = M × H ,$

where γ0 ≡ μ0 γe. As M , the magnetization, appears on both sides of the equation, it can be replaced by J , the magnetic polarization ( J = μ0 M ), to obtain

3.7 $( 1 γ 0 ) d J d t = J × H { − ∇ j ( e ext ) } .$

The H in Equations 3.5 to 3.7 is treated as an effective field by taking minus the vector gradient of the energy density terms with respect to the magnetization or the polarization as in Equation 3.8,

3.8 $H eff = − ∇ J ( e tot ) = − ∇ m ( e tot ) J s = − ∇ m ( e tot ) ( μ 0 M s )$

The price for using the magnetic polarization as the variable in the torque equations is to put some μ0’s into expressions for the anisotropy and exchange energies where there is no physical reason for them being there. The effective field includes the applied field, the demagnetizing field from all the other magnetic moments, the exchange field from the variation of the moment direction with position, the anisotropy field, and the damping field.

The damping field was formulated by Gilbert to be proportional to the rate of change of the components of the magnetization [17]. The damping in iron comes from the repopulation of the Fermi surfaces of spin-up and spin-down electrons as the magnetization direction is rotated locally. These spin currents dissipate energy. Spin currents can be created externally and used to cause the magnetization to rotate by forced repopulation of the Fermi surfaces. The two processes differ in the sign of the coefficient of the contribution of ∂ m /∂t to the effective field. There is much more about the subject of damping in Chapter 2 in this book. Spin currents are not discussed in this chapter because the author has not yet applied them to the nanobrick.

#### 3.4  Applying the Micromagnetic Equations of Motion

This chapter describes in some detail the results of calculations for the iron nanobrick. The parameters are those of iron except for the calculation of the equilibrium configurations where the damping coefficient α is greatly increased from the low value of iron to the value that gives critical damping. The dynamic calculations use α = 0.02 and the equilibrium calculations use α = 1. For α = 0.02, the time constant for the approach to equilibrium is typically 10’s of ns. For α = 1, the time constants can be less than 1 ns, unless the system is near a critical point for which the torques vanish. Then the time can be too long to compute and the usual approach to equilibrium as exponential changes to a 1/t approach. As critical points are of interest in describing magnetic configurations, it is necessary to have techniques to obtain the answers more quickly.

The time steps used in micromagnetic calculations are small compared to the time resolution adequate to describe the fastest of dynamic responses. The torque equations have mathematical instabilities that often require that the time steps be as small as a fraction of a femtosecond. The time step must be smaller when the grid size is smaller.

The grid size itself should be smaller, by at least a factor of two, than the exchange length λ given by λ2 = A/K, where A is exchange stiffness constant of Equation 3.1 and K is a magnetostatic energy density; $K = μ 0 M s 2 / 2$ .

In classical micromagnetics [18], the language is that of Gaussian units, where M, H, and B all have the same dimensions. Then, μ0 is replaced by 4π in the magnetostatic energy used in the definition of the exchange length. The use of a grid spacing that is too large can lead to results that are completely misleading. In a treatment of magnetization processes in a nanobox with square sides, X = Y > Z, it was shown that the moments in the I z vortex have m z = 0, even at the axis, when the grid is greater than 3λ [19]. Critical fields are sensitive to the grid size even for grid size < λ/10.

#### 3.4.1  Bias Fields

The fields from currents used to break the symmetry at critical points also provide a means of avoiding prohibitively long computation times. Such bias fields are called anticipatory fields [20] as they select among three prongs of a three-tined fork. Without a proper bias field, one can stay on the central tine even though the other tines lead to lower energies. If an anticipatory field is not used, the middle tine will be abandoned after numerical round-off errors propagate exponentially with time for more than ten-time constants.

Away from critical points, the total energy comes to equilibrium exponentially, while the components of the magnetization and individual terms in the energy come to equilibrium as damped oscillators. The convergence of a calculation is achieved when the extrapolation of the exponential or the damped oscillators to infinite time no longer changes significantly with time. From the magnetization at any three points equally spaced in time, it is simple to extrapolate to infinite time to get the apparent equilibrium magnetization for a simple exponential approach. For example, if the points are ma, mb, and mc, the extrapolated mf = (mb 2 − ma mc)/(2mb − ma − mc) provides criteria for terminating the calculation.

At a critical field, it is convenient to use a bias field that takes the magnetization from one configuration to another quickly and then analyze the results of that calculation to obtain what would have happened in the absence of that bias field. This is called the path method [21].

#### 3.4.2  Internal Energy

The path method relies on the insensitivity of the internal energy E int to the values of the external fields needed to reach configurations that have the same values of the average components of the magnetization <m x>, <m y>, <mz>. E int is the sum of all the energy terms excluding the Zeeman term. The energy, divided by the volume V, along an equilibrium path can be written as

3.9 $E V = − ⟨ m * ⟩ ⋅ ( J s H − ∇ ⟨ m ⟩ ( E int ) | ⟨ m ⟩ = ⟨ m * ⟩ ) ,$

where < m* > is < m > at some point along the path. Knowing the gradient of the internal energy E int, with respect to the components of the average magnetization as a function of < m> along an equilibrium path, one can calculate the field necessary to reach equilibrium (not necessarily stable) for a given < m *> along that path. The path method assumes that knowledge of E int along a path that is close to a given equilibrium path will give the same result; that is, ∇ <m >(Eint)|< m >=< m *>, which is assumed to be insensitive to small bias fields.

In the path method the calculations of the magnetic response to applied fields are carried out for a sufficient number of fields to determine the functional form of E int, but no more than necessary. Once one has an analytic expression for E int(< m >) of a particular type of pattern, one can produce the entire dependence of the magnetization on the field. When this works, it can save many orders of magnitude in computation time. To determine whether it works can take time, but that need only be done once for a given type of system. The path method does not work when there are two or more independent responses to the applied field.

#### 3.4.3  Using Anticipatory Fields

An example of the use of the path method and anticipatory fields is the calculation of the field for a vortex parallel to the z-axis to re-enter an iron nanobox [19] after being driven out through the y = Y/2 surface by a field H x. After the vortex leaves, the magnetization is in a C-state which can be viewed as a virtual I z-vortex just outside the brick. When H x is reduced sufficiently, the vortex should re-enter the surface through which it exited. The three-tined fork, in this case, is the fact that the virtual vortex must choose the direction of magnetization for the core in order to re-enter. The virtual vortex does not have polarization before it enters unless the C-state itself has a bias in the direction of the vortex that exited, which can happen for particular geometries. If a bias field along the z-axis is not applied, the system stays on the central tine and the vortex does not re-enter until long after a round-off e rror provides an initial bias. If a bias field is applied along the z-axis, it slightly changes the H x at which the vortex enters, but it changes by many orders of magnitude how long one would have to wait for that to happen.

#### 3.4.4  Using the Path Method

A hypothetical example of the path method is given in Figure 3.13. The left panel in this figure shows the dependence of the internal energy E int upon <M x> as the solid curve for an arbitrarily constructed Eint given by

Figure 3.13   Illustration of the path method. The dependence of the internal energy E int upon <M x> is shown in the left panel as the solid curve for an arbitrarily constructed E int = E 2<M x>2 + E 4<M x>4 + E 6<M x>6)/(1–<M x>12) to produce four inflection points and also mimic the approach to saturation. The system is in an unstable equilibrium for the region between the inflection points at a and b. The dashed line shows dE int/d<M x>, which is shown again in the right panel as the independent variable (~H x) to produce a magnetization curve with hysteresis.

3.10 $E int = ( E 2 ⟨ m x ⟩ 2 + E 4 ⟨ m x ⟩ 4 + E 6 ⟨ m x ⟩ 6 ) ( 1 − ⟨ m x ⟩ 12 )$

to produce four inflection points and also mimic the approach to saturation. The system is in unstable equilibrium for the region between the inflection points at a and b. The dashed line shows dE int/d<m x>, which is shown again in the right panel as the independent variable for producing a magnetization curve with hysteresis. For equilibrium, the applied field must be equal and opposite to the internal field given by –dE int/d<m x>, which explains why there are no minus signs in this illustration of the path method. E int can be constructed by analysis of the calculations in regions of stable equilibrium to interpolate the regions of unstable equilibrium. E int can also be calculated using the regions of unstable equilibrium if the timescale of changes is appropriate for obtaining close to equilibrium configurations while the configuration is moving in time as a whole.

The path method works only for large damping. For small damping, the magnetization moves on a path of almost constant energy, while the path method has the magnetization moving in the direction of the maximum gradient of the energy.

#### 3.5  Around the <M z>–H z Hysteresis Loops

The computations are designed to obtain E int as a function of < M > and < M > as a function of H z for each configuration using appropriate bias fields when necessary. Hysteresis loops are shown for <m z> and <m x> in Figure 3.14, with and without a small bias field from currents i x = 0.1 mA. The iron brick has dimensions of 50 × 80 × 130 nm3. The sequence of configurations starting from high H z includes:

Figure 3.14   Hysteresis loops for an iron nanobrick with dimensions of 130 × 80 × 50 nm3. The first quadrant of the major hysteresis loop is shown for both <m z> and <m x>, the latter with much magnification (its ordinate is on the right). The gray curves for <m z> and <m x> were calculated with a bias current i x = 0.1 mA to anticipate the transitions to and from the I state. The bias current has little effect on <m z> but facilitates following the transitions from the effects it has on <m x>. The gray curve for <m x> measures the degree of curling between points a and b and measures the susceptibility of the I state for forming the S state in the field region between 0.3 and 0.4 T. The two black curves for <m x> show the sharp transition between the S and I state near 0.3 T that occur in the absence of the bias current. The curve for <m x> labeled A is for the vortex core m z in the –z-direction after coming from saturation at a high negative field. The curve for <m x> labeled B is for the vortex core m z in the +z-direction with the transition occurring at a slightly higher field, d compared to h. All the hysteresis at low fields is from the switching of the magnetization directions in the four corners of the nanobrick. For <m z> the black curves, one of which hides the gray curve, are for no bias current. The curve for <m z> labeled A is for the core of the vortex in the –z-direction with the transition for the state with the core m z in the +z-direction taking place by pair creation and propagation at j. There are two different states within the +z-direction and the other has the core in the –z-direction. This would give different values for the magnetization in zero field if it were not for the almost complete compensation of the net magnetization by the moments in the four corners, which are opposed, to the core magnetization in both cases.

#### 3.5.1  The Flower State

The magnetization splays out from the center on the top surface and inward toward the axis on the bottom surface, with most of the magnetization along the +z-axis; see Figure 3.3 and the section of Figure 3.14 labeled c, μ0 H z ~ 1.

#### 3.5.2  The Curling State with Inversion Symmetry

The circulation is cw on one-half of the nanobrick and ccw on the other; see Figure 3.5. The magnetization dependence on H is not shown in Figure 3.14. (The reader will be spared the complexities of magnetization processes proceeding from this state.)

#### 3.5.3  The I z-Vortex State

The same handedness throughout the nanobrick is achieved by applying a bias field from a current along the z-axis, which can be removed once the handedness is chosen.

The gray curves in Figure 3.14 were obtained in the presences of a bias field from a current along the x-axis, i x = 0.1 mA, which anticipates the transformation from the I z-vortex state to the S z-tube state as H z is lowered.

The degree of the displacement of the swirls is tracked by <m x>. The inflection point on the gray curve at μ0 H z = 0.32 T corresponds to the field at which the transition takes place in the absence of the bias field, labeled d. The bias current, i x, does not produce any <m x> in the flower state but does in the curling state, so that <m x> tracks the onset of curling at b and the approach to saturation of that effect at a on the gray curves.

#### 3.5.4  The Landau-Type Curling State

The I z vortex distorts spontaneously into either the $S z +$ -tube or the $S z −$ -tube depending on the bias field applied from a current along the x-axis, see Figures 3.6 to 3.10. The presence of the S z -tube is signaled by the presence of <m x> in the absence of bias fields. The presence of the S z -tube has a minor effect on <m z> that can be noticed after subtracting the demagnetizing field, the dominant effect of the magnetostatic energy.

#### 3.5.5  States of the Vertical Edges

The Landau-type state has four choices of polarization, plus (p) or minus (m) for the virtual vortices along the four vertical edges. Without additional bias fields, the four vertical edges for the $S z +$ -tube state are either pppp, ppmm, or mmmm, labeled cw from the corner (–X/2, Y/2). For the $S z −$ -tube state, the sequence is pppp, mmpp, mmmm. Each of these states has its range of stability with hysteresis in the minor loops for the switching of these edges by soliton propagation; see 3.7, where edge solitons play a major role in achieving the Landau structure. The minor hysteresis loops at low fields are shown in Figure 3.14 and in more detail in Figure 3.15, where the mean effect of the demagnetizing field has been subtracted. All of this is avoided in the ellipsoid, which has no edges.

Figure 3.15   The hysteresis loops for the difference ∆<m z> = <m z> – <m z>D found by subtracting <m z>D = H z/H s from each <m z> in Figure 3.14. The hysteresis in low fields accompanies the reversal of the magnetization in the four vertical edges. All four vertical edges flipped on decreasing the field in a single large step in the field on decreasing the field, but on increasing the field in smaller steps, two flip back first and then after a second step in the field the other two flip back. When resolved in time, the four edges flip independently from one another. The onset of the S-tube is signaled by ∆<m z> beginning to decrease below d. The lower curve is for the magnetization of the core opposite to the applied field. The S-tube goes back to the I-vortex at h. The higher slope for the reversed field reflects the smaller demagnetizing field for the trapped vortex. Point j is where the core reverses by pair creation and propagation.

#### 3.5.6  The S-Tube in Reversed Field

The extension of the S-tube in the x-direction is maximum for H z = 0. For H z < 0, the field is opposite to the magnetization of the core of the S-tube, but most of the magnetization outside the tube follows H z. When H z is sufficiently negative, the S-tube returns to the z-axis becoming a very compact version of the I z vortex. The <m x> component goes to zero for a smaller magnitude of H z when the fields are in the direction opposite to the core magnetization; compare points d and h in Figure 3.14. The curves labeled A are calculated for negative fields and then replotted in the first quadrant for the direct comparison with the positive fields. Those curves would be obtained directly in the first quadrant if the magnetization process started with large negative fields and then proceeded to positive fields.

#### 3.5.7  The Trapped I z-Vortex State

When the field is large in the reversed direction, the magnetization is negative everywhere except in the core of the vortex. The demagnetizing field becomes smaller for the trapped state leading to a higher slope of m z vs. H z in A compared to B.

#### 3.5.8  The Transient Core Reversal State

Starting with the trapped + I z vortex, as the field becomes more negative, near −μ0 H z= 0.8 T, a pair of point singularities is created near the midplane. These propagate up and down the z-axis reversing the magnetization of the core to produce the −I z vortex with the same circulation as maintained in all these processes; see Figure 3.12 and point j in Figure 3.14. The derivative of m z with respect to z changes sign at each singularity but the derivatives of m ρ with respect to ρ do not, providing a clear picture of the role of splay saving in Figure 3.12.

#### 3.5.9  Saturation

At a high negative H z, the –I z-vortex state goes to the flower state with the moments splaying outward from the center of the bottom surface and inward toward the center of the top surface, with most of the magnetization along the –z-axis

#### 3.6  Discussion of Magnetization Processes

The programs for micromagnetic calculations keep track of the components of the magnetization for each grid point for every-so-many iterations as well as the net magnetizations and each term in the energy of the nanobrick as a whole. These are analyzed to gain insight into the competition among the energy terms and how they lead to the various magnetization patterns. The leading energy term is the Zeeman energy. Even when the external fields are all zero, the configuration in a real nanobrick is one that reflects the past history of the magnetization. On the computer one can arbitrarily assign a configuration A and calculate the configuration B that minimizes the energy starting from A. Then, if H z = 0, the dominant term in the energy of the iron nanobrick is the dipole-dipole energy. At all other fields, the competition is between the Zeeman energy and the dipole-dipole energy with the exchange playing a supporting role and the anisotropy almost no role at all.

#### 3.6.1  Ideally Soft Magnetic Materials

For a magnetic system to act as a conductor, the magnetization pattern has to adjust itself to produce the required surface charges while remaining divergence free within the volume. It can do this if there is no anisotropy and the system is large enough that the increase in exchange energy required by the divergence-free pattern is very small. A large enough ferromagnetic body without anisotropy behaves as a magnetic conductor with no net field inside the body, except that in a singly connected body, the magnetic conductor cannot topologically escape the need for two swirls. For a large enough body the magnetization pattern is divergence free everywhere except in the vicinity of the swirls. This is called the ideally soft magnetic material. This is realized experimentally in iron whiskers with X = Y ~ 0.1 mm and Z ~ 10 mm just below the Curie temperature where the magnetic anisotropy goes to zero much faster than the spontaneous magnetization [22]. Even though this work was inspired by measurements at high temperature, the calculations are all for low temperature where thermal agitation is completely neglected, except for its effect on the material constants that are those of ambient temperature.

The electrical charge on the surface of an electrical conductor is a very small fraction of the charge on a surface atom. The magnetic charge on the surface of a magnetic conductor is limited by the finite moment of the surface atom. As the charge necessary to cancel an applied field at a corner of a brick goes to infinity, the corners become saturated (in the direction of the net field) as the external field penetrates the surface.

In high fields in the curling state, the iron nanobrick also mimics a magnetically soft material as long as the high field is not so high as to force the flower state.

#### 3.6.2  Demagnetizing Factor

To a good approximation, the dipole-dipole energy is quadratic in <m z> for all the states of the nanobrick described above. The Zeeman energy is, of course, linear in <m z>. The magnetization curve is then approximately linear in H z until the flower state is approached at high fields. The magnetization is given to a good approximation by H z <m z> H s =0, where <m z>H s is called the effective demagnetizing field and H s is a fitting parameter. To emphasize the role of the supporting actors in this drama and to show the non-quadratic terms in Edem, a magnetization <m z>DH z/H s is subtracted from each point in Figure 3.14 to produce Figure 3.15, where the value of H s has been chosen to make the difference ∆<m z> = <m z> <m z>D independent of H z over much of the range of H z.

#### 3.6.3  Vortex Statics and Dynamics

To some extent, the iron nanobrick behaves like an ideally soft magnetic material, but the S-tube has its own life. The I-vortex builds up during the transition from the flower state to the curling state. It costs exchange energy to do this. During the buildup, the exchange energy is proportional to the deviation of <m z> from unity. The derivative of the exchange energy with respect to <m z> is constant during this process, giving rise to a constant effective exchange field that aids the Zeeman field in maintaining the magnetization.

Once the I state breaks free of its confinement to the symmetry axis and becomes the S-tu be, it is like a moving domain wall in which changes in exchange energy are slight and compensated by changes in magnetostatic self-energy. In the dynamic response in the constant applied field, the tube can move back and forth between the S +- and the S -tube states along paths of constant internal energy, along which there are oscillations in the exchange energy and the magnetostatic energy that are equal and opposite to one another. When the S-tube is free to move, the exchange energy has little effect on the magnetization loop.

#### 3.6.3.1  Vortex Contributions to the Magnetization

The core of the tube does have an effect on the magnetization loop because the core is magnetized. It is a separate permanent magnet that has its own magnetization loop, reversing only in fields of the order of μ0 H z = 1 T. The volume of that permanent magnet changes somewhat with the field, shrinking as the field is lowered from saturation and continuing to shrink as the field is increased in the reverse direction.

#### 3.6.3.2  The I z-Vortex State

In the nanobrick, the curling pattern with the swirls centered on the z-axis is called the I z-vortex state. The m z-isosurfaces are elliptical in cross-section. The central bulge along the x-axis corresponds to the 180º wall of the Landau structure. As field H z is lowered, the bulge extends toward the positions x = ±(X/2–Y/2). When the I z-vortex state is maintained down to H z = 0, for |x| > (X/2–Y/2) and y = 0, magnetization lies almost in the midplane with |m y| ~ 1 corresponding to the closure-domain pattern of the Landau structure.

For Z above a critical thickness that depends somewhat on X and Y, the I z-vortex is not stable for H z = 0, but it is always an equilibrium state that persists as long as there is no symmetry-breaking field or the inevitable effect of computational round-off error has not yet developed. Once one sees the correspondence between the Landau structure and the S-tube state, one can view the I z-vortex state as the Landau structure with its two swirls centered on the z-axis. Or one can view the Landau structure as an S-tube with its two swirls moved off the z-axis. The two swirls can be manipulated to move along the top and bottom surfaces distorting the 180º Bloch wall as they move.

#### 3.6.3.3  The S-tube States

The swirls of the I z-vortex state can be manipulated. In response to a current i x in the +x-direction, the I z-vortex state takes the $S z +$ -tube configuration. For the dimensions of the iron nanobricks chosen for this chapter, the $S z +$ -tube configuration appears spontaneously below a critical magnitude of H z. In the absence of bias fields in the x- or y-directions, the two swirls of the spontaneous $S z +$ - and $S z −$ -tube states have coordinates (x s, y s, Z/2) and (–x s, y s, –Z/2), respectively, where x s and y s increase with decreasing H z, reaching a maximum at H z = 0. As x s increases it is accompanied by an increase in <m x> as shown in Figure 3.14. This occurs because there is a displacement of the core in the –y-direction increasing the volume in which the magnetization in the +x-direction is dominant. In the midplane where x = 0, the core of the vortex with m z = 1 and the two Néel caps with m y = 1 at the top and m y = 1 at the bottom form a circulating magnetization pattern on one side of the S-tube. The circulation is around an x-axis displaced from the midplane in the –y direction, as originally calculated by LaBonte for a never-ending Bloch wall; see Figure 3.10.

The lowest Z for the spontaneous appearance of an S z-tube structure is Z crit = 25 nm for Y = 35 nm with X varying from 120 to 126 nm. For X = 119 nm the S z-tube configuration goes to the I z-vortex state. For X = 127 nm, an S z-tube structure is unstable with respect to the formation of an I x-vortex along the x-axis. For X = 130 nm and Y = 80 nm, the spontaneous S z-tube structure occurs for Z > 42 nm.

There are also limits on the sizes of the nanobrick for which the I z-vortex state is stable in the absence of a magnetic field. If the nanobrick is too small, the I z-vortex state moves away from the axis and disappears out the nearby Y face as H z is reduced. The range of dimensions (X, Y, Z) and of applied fields (H x, H y, or H z) for which the $S z +$ - and $S z −$ -tube states and the I z-vortex states are stable has been studied by T. L. Templeton [23] who includes the effects of the configurations in the four vertical edges in his elaborate phase diagrams.

#### 3.6.3.4  The I z * -Vortex State

The spontaneous $S z +$ -tube is distorted by applying a uniform field H y. The swirl on the left moves toward the swirl on the right, which moves only slightly to the right. There is a critical field H y0 H y ~ 0.1 T) at which the left swirl catches up to the right swirl. The $I z *$ -vortex state is the Landau structure with both swirls on the same end of the 180º Bloch wall. The centers of the two swirls are at the same x-position, but the small displacements of the swirls in the y-direction are in opposite y-directions. The m z-isosurfaces are far from symmetric. The 180º Bloch wall is attached to one side of the $I z *$ -vortex. The central cross-section of the m z-isosurfaces bulges to include the Bloch wall, see Figure 3.16. On lowering H y there is a critical field for re-nucleating the S z-tube that restores the Landau state with the swirls on opposite ends as H y goes to zero. The transitions from S z to $I z *$ in H y are not quite continuous and hysteresis occurs.

Figure 3.16   The I*-vortex formed by displacing both swirls to the same end of the Bloch wall, using a field H y. The two swirls have the same x-coordinate but are displaced in y by –4 nm on the bottom surface and +4 nm on the top. The regions in light gray (m z > 0.8) and dark gray (0.8 > m z > 0.5) are for m z-isosurfaces intersecting the y = 0 plane. The slightly curved, almost vertical line is the contour for m y = 0 in that plane. The I*-vortex maintains a memory of the Bloch wall as it bulges to the right. The area in white labeled 0.95 is for the plane y = –4 nm and the dotted contour is in the plane y = 4 nm. The I*-vortex has two-fold symmetry on rotation about the x-axis.

The transitions from $I z *$ to S z depends on a bias to select between $S z +$ and $S z −$ . From the view of the swirls, one of the swirls traveled further to form the $I z *$ -vortex state in the large H y. The choice of which swirl travels back along the nanobrick when H y is reduced is, again, a three-tined fork. If no decision is made, the $I z *$ -vortex state persists for a long time in a region where an S z-tube would be more stable. A small bias field in H x would make the selection. For a finite step in H y, the dynamics make the selection. In experimental studies at Grenoble of faceted nanogems of iron, it was the facets on the top surface that favored the upper swirl as the one that determined the handedness on reducing H y [24].

The transition to the $I z *$ -vortex state from the spontaneous S z-tube state can also be made by applying a large current, i y (~20 mA). The latter produces a large gradient field in the z-direction (μ0 H z = 0.06 T at the end surface) which stabilizes the $I z *$ -vortex state in a first-order jump, with the complication of turning the magnetization in the corners into the direction of the gradient field, that is mmmm goes to mppm for the four corners ordered clockwise from (–X/2,Y/2). Removing the large i y does not restore the corners to their original configurations.

#### 3.6.3.5  Switching

A dramatic result of the attack on the micromagnetics of the nanobrick was the discovery by computation that the two swirls of the Landau structure, when viewed as the ends of the S-tube, could be switched back and forth, using modest driving fields, over long distances in short times carrying with them large external fields [25]. The switching of the S-tube had already been observed experimentally in 2004 but not specifically identified with the reversal of the positions of the two swirls [26]. The switching between two stable states is discussed here in terms of the energy landscape correlated with the positions of the two swirls. This is a gross simplification, but in equilibrium and for heavily damped dynamics there is some usefulness in thinking about the internal energy along and near the equilibrium path.

The combination of the field from a current i x, producing a field that is +H y at the top surface and –H y at the bottom, and a uniform H y permits the independent manipulation of the positions of the two swirls in any given H z. The internal energy at equilibrium in the combined fields changes with the positions of the two swirls. At constant H z, one has an energy landscape with minima at the symmetric positions that the two swirls for S + or S - take in the absence of bias fields.

If the motion of the two swirls between S + and S is determined by a slowly varying current oscillating between +i x and –i x, the internal energy along the path goes through the minima when the current goes through zero. The displacement in x from that equilibrium has the separation between the two swirls first increase reaching a maximum for the highest current, returning to equilibrium as the current goes through zero, and then having the displacement in x go toward zero as the swirls approach each other. But before they reach each other, the swirls have reached an energy position where it is all downhill toward a stable position that lies beyond the far equilibrium position in a zero current. That position will be reached if the current is maintained at the critical current for switching, i x = 1.2 mA, which produces a maximum field μ0 H z = 0.06 T at the surfaces for the nanobrick 156 × 96 × 60 nm3.

#### 3.6.3.6  Double-Well Potential

Along the path to the critical current for switching from the S + to S configuration, the internal energy contour is one-half of a double-well potential. That potential is completely determined at each current up to the critical current. The inflection point on that curve is reached at the critical current. If the motion is calculated using large damping, a path beyond the inflection point can be determined from the damped dynamic response and the full double-well potential can be determined. This path leads through the position where the separation in x of the swirls goes through zero, but when the separation in x is zero, the separation in y is not zero. The paths of the two swirls on the opposite faces of the nanobrick are narrow ellipses tilted in the x-y planes. The double-well potential is defined along the ellipses. The swirls do not pass over the maximum in the potential where the displacements in x and y are both zero.

#### 3.6.3.7  Forced Oscillations

An example of the non-linear oscillations of the S-tube with large amplitude is found using a 1.28 GHz driving current i x = 0.6 mA with a period of 780 ps. This current is one-half of that necessary to reverse the S-tube with slowly varying currents. The swirls move in elliptical paths on the top and bottom surfaces that avoid the region of the local maximum in the internal energy at the center of the faces, as they follow contours of almost constant energy near the saddle point on the +y and –y sides of the origin. The path on the top surface is shown in Figure 3.17 for one period of oscillation. The path on the bottom surface is the mirror image in either the x = 0 or y = 0 plane with the two ends of the m z-isosurface moving counterclockwise along the paths. When the swirls pass one another at x = 0, the two ends are displaced in y = ±8 nm. At their greatest separation x = ±36 nm at y = 16 nm.

Figure 3.17   The dynamic response of a nanobrick with dimensions of 156 × 96 × 60 nm3 to an ac driving current i x = 0.6 mA with a period τ = 780 ps. The path of the swirl on the bottom surface is followed in the upper panel by tracing the contours m z = 0.95 in steps of 12.8 ps. The calculation was carried out with a grid of 4-nm cubes, which exaggerates the interaction of the swirl with the grid resulting in the steps of 4 nm in both the x- and y-directions. A much smoother ellipse is obtained for a grid of 1.25 nm At no time does the central core of the S-tube lie in a plane, let alone in the y = 0 plane, but the contours with m y = 0 in the plane y = 0 are used in the bottom panel to reflect the distortions of the m z-isosurfaces, which to be fully appreciated require 3D movies that someday should be on youtube.

#### 3.6.3.8  Mired at the Central Maximum

In the dynamic response with low damping, the path of either swirl can go through the origin but generally not at the same time. But in one dynamical calculation, the two swirls came through the origin at the same time during one cycle of a damped oscillation. The two swirls then stayed there for a time equal to the period of the non-linear oscillator. During this time a higher harmonic of the dynamic response, corresponding to a wave propagating up and down the z-axis, provided the driving force to allow the two vortices to move away from the central energy maximum.

#### 3.6.3.9  Vertical Edges as Partial Antivortices

The vertical edges of the nanobrick are each one-quarter of a virtual vortex that is just outside the corners. These partial-virtual vortices are antivortices with a winding number of 1. An example of an antivortex is given in Figure 3.1, where the Bloch wall is the core of the antivortex. Each of the corner partial-virtual antivortices can be described using Preisach diagrams (with sloping sides). When a corner reverses there is a change in the exchange energy in the region between the corners and the core of the S or I-vortex in the center. The change in exchange energy with <m z> during the reversal of a corner is an exchange field that adds to or subtracts from the applied field at the same time that the magnetization of the antivortex is changing its contribution to the magnetization. The flipping of an antivortex shifts the sloping line of m z vs. H z both sideways from the exchange field effect and up or down from the change in a magnetic moment.

The composite picture is then of five Preisach diagrams added to the ideal soft magnet plus some exchange energy to be provided for the buildup of the five Preisach regions.

#### 3.6.4  Ellipsoids

From the beginning with Ewing, the ellipsoid has been useful in understanding magnetism. With the work of Stoner and Wohlfarth, it became the model for understanding the origin of permanent magnets and the applications to magnetic recording. The earliest micromagnetic calculations addressed the particles in magnetic recording media. It was concluded that magnetic recording depended so much on particle-particle interactions that understanding the micromagnetics of a single ellipsoid was not sufficient for understanding magnetic tapes. Yet the properties of a single ellipsoid led to a fuller appreciation of the Landau structure.

#### 3.6.4.1  Absence of the Flower State in Ellipsoids

When the calculations are carried out for an ellipsoid, the flower state does not occur. Splay saving works right up to saturation. The swirl in the ellipsoid is not at a flat surface. It is the curvature that forestalls the breakdown of splay saving. The ellipsoid goes directly from saturation to the swirl with the handedness chosen by a symmetry-breaking field. As the swirl forms, the exchange energy increases in proportion to (M s <M z>). The linear increase in exchange energy with decreasing <M z> is a constant exchange field that adds to the applied field. When H z is increasing below saturation, the exchange field brings the ellipsoid to saturation at a lower H z than one would obtain for a paramagnet with infinite susceptibility to reach M s. The magnetization is linear in the applied field with the slope determined completely by the demagnetizing field. The demagnetizing field line is offset by the constant exchange field.

For the ellipsoid, this is true for a field along any of the three principal axes. The slope is different for each axis because of the change in the demagnetizing factor. The constant offset is different because the curvatures of the surface change the contribution of the exchange energy to the energy of formation of the swirl.

#### 3.6.4.2  Curling in an Ellipsoid

To predict the straight line of <M z> versus H z for an ellipsoid at high fields, all one needs is Osborn’s formulas [27] for the demagnetizing factors of the ellipsoid and a single number for ∂E ex/∂<M z> for the chosen axis. The latter can be obtained from a micromagnetic calculation of <M z> at a single field below saturation. Precise agreement with the analytic formulas has been found using TetraMag [9] to calculate the properties of the mathematical ellipsoid with a triangular mesh on the boundaries; see Figure 3.11. A full micromagnetic calculation of <M z> versus H z for the approach to saturation for an ellipsoid would require very long computational times because the torques become very small as the swirl saturates.

Modeling the results of the calculations can lead to insights that greatly shorten the computational time for a given problem. The ellipsoid at high fields is an extreme example in which one calculation in a single field, where the torques are large and the relaxation time short, produces the entire magnetization “curve” in the region where swirl remains along the axis and ∂E ex/∂<M z> remains constant with change in H z. In this case, the field for saturation is determined precisely. The instability field at which the I z vortex moves off axis cannot be determined without calculating the pattern changes when the swirls move off axis.

#### 3.6.4.3  Reversal of a “Stoner-Wolhfarth Particle”

Below a second critical field, the central position of the swirls on the ends of the principal axis of the ellipsoid becomes an energy maximum and the swirls move off center if the dimensions of the ellipsoid are sufficiently large. At small enough dimensions, the ellipsoid remains “uniformly magnetized” in all fields. The magnetization process is limited to rotations in the Stoner-Wohlfarth model. It is assumed that the exchange energy does not change in the process. That model has served as the starting point for 60 years for understanding magnetization processes as a competition between the Zeeman energy and the anisotropy energy, where the anisotropy energy includes the dipole-dipole interactions and the crystalline anisotropy. Variations of the exchange energy in “uniform” rotation would also appear as an addition to the anisotropy. Even an ellipsoid in the Stoner-Wohlfarth model requires bias fields for the uniform rotation of the magnetization away from a principal axis.

#### 3.6.4.4  Bias fields in the Stoner-Wolhfarth Model

The importance of bias fields for the reversal of the magnetization was first pointed out by D.O. Smith at the 2nd MMM conference in Boston in 1956. The subject of the session was the failure of experiments to show switching with the time constant predicted by the Landau-Lifshitz equations. The experimentalists were asking what was wrong with the Landau-Lifshitz equations. Smith showed that if the experiments and the theory are done using bias fields they agree.

#### 3.6.4.5  Propagating Singularities in Particles

In a micromagnetic calculation, one can eliminate all the geometrical biases and cause the small ellipsoid to reverse its magnetization through a non-uniform distortion in which a singularity propagates along the principal axis starting at the swirl. The field must be applied fast enough that the round-off errors in the numerical calculation do not have sufficient time to nucleate the uniform rotation by the displacement of the swirls. Even then one needs a symmetry-breaking field to choose the handedness of the swirls. Here again, the numerical round-off error can provide the handedness. The field must be larger than necessary and applied fast enough that the round-off error favors the reversal through singularity propagation rather than through uniform rotation.

The reversal can also take place with the creation of a vortex-antivortex pair away from the ends where the swirls reside.

#### 3.6.5  Pair creation and Propagation in the I z Vortex

A rule for the formation of a pair of singularities in micromagnetics has been given by Sebastian Gliga who worked in high-energy physics before coming to magnetism [28]. If enough energy is provided to create each of the singularities in a given region, the program for solving the micromagnetic equations will find the solution in which the pair is created. The energy for the creation of a vortex-antivortex p air is of the order of 8πAλ, where λ is the exchange length and A is the exchange constant of Equation 3.1.

As the field becomes more negative for a + I z-vortex state, the magnetization turns to the –z direction everywhere except in the immediate vicinity of the axis of the vortex. There is a wall in which exchange energy becomes higher as the field in the –z-direction becomes more negative. In a bcc lattice, the singularity appears at the center of a tetrahedron of iron atoms. The four moments can no longer sustain the wall when their m z decreases to a critical value. A pair of solitons is created which then propagate to reverse the core of the vortex.

The creation and annihilation of vortices was envisioned in the original discussion of point singularities in the micromagnetics of a cylinder [7] in the 1970s but the terminology of skyrmion was not in the literature of magnetism until Tretiakov and Tchernyshyov [29] recognized “that a peculiar annihilation of a vortex-antivortex pair observed numerically by Hertel and Schneider [97] represents the formation and a subsequent decay of a skyrmion.” These had been studied for many years, but this gave them a new name and popularity.

#### 3.7  Recent Studies of Whiskers and Cylindrical Toroids on the Micron Scale

In the original manuscript for this chapter, Section 3.7 was to obtain the Landau structure starting at high fields along the long axis and arriving by a path that did not require fields from currents. This would match the experimental approach, which generally required cycling between positive and negative fields. With 40 years of experiments on the magnetization processes in measuring the susceptibility of iron whiskers, it seemed that there were too many effects to be explained with the computational tools available at that time, particularly because dimensions of the whiskers were so many orders of magnitude larger (1016) than an exchange-length voxel.

The new age of computing, with desktop computers that would have been called super computers not too long ago, has shown calculations with dimensions still very small compared to the experimental whiskers are able to provide insights into their magnetization processes. Already, from computations with the smallest dimension near 300 nm, the behavior of systems with smallest dimensions of 100 µm is understandable. These systems include:

1. a.The formation of the Landau structure starting from saturation along the long axis in real iron whiskers with the [100] orientation as studied by Heinrich and coworkers in the 1970’s using ac susceptibility [22].
2. b.The formation of striped domain patterns on hexagonal iron whiskers with the hard [111] axis of magnetization along the long axis as seen by Hanham [5].
3. c.The role of exchange springs in the interaction of domain walls with surfaces.
4. d.The formation of single domain walls in toroidal geometry.
5. e.The suppression by magnetostatics of the effects of anisotropy on the approach to saturation in polycrystalline materials permitting the use of the full saturation magnetization of iron in a transformer.

#### 3.7.1  Barber-Pole Configurations in Iron Whiskers

The whisker is considered as having one central tube of magnetization around the line with m x = 1 that goes from the center of the swirl on one end to the center of the swirl on the other end for the case that the two end swirls have the same handedness. It is now seen how a single handedness can occur without using a current along the whisker. Once a single handedness is established, it takes a field of the order of µ0 H = 1 T to return the whisker to the inversion symmetry state preferred by the inversion symmetry of the LLG equations. The whiskers were not subjected to such high fields. The whiskers are below the Curie temperature when they form.

The are several possible states that can occur when starting at moderate fields with the same handedness at each end. To achieve the Landau state in a [100] whisker, it is necessary to reverse the magnetization on two adjacent side edges. It is less complicated if the cross-section of the whisker is rectangular, breaking the degeneracy of the square cross-section.

A Landau-like structure can appear on adjacent faces at opposite ends of the square cross-section whisker. This effect was seen by Graham in the 1950s. The author published a cut out drawing of the three-dimensional transition domain between the two Landau structures [30] in the 1990s after Graham told him that the original wooden model was lost.

The reversal of the magnetization of the edges takes place through the propagation of edge solitons, either from one end or the other or from both at the same time. It can also happen via the creation of a pair of solitons somewhere along an edge, with the two solitons moving to opposite ends to reverse the edge.

As there are four side edges, it matters whether one, two, three, or four edges reverse. There are stable states in which edges are partially reversed with one, two, or more solitons. The Landau structure in a rectangular cross-section whisker has two closely adjacent edges magnetized opposite to the central core. The two reversed edges can also be on two nonadjacent edges, accounting for a state seen in the ac susceptibility studies where the losses are strongly field dependent.

When the swirls are on the two ends and the four sides are magnetized in the same direction, there are two states, one of which has the central Ix-vortex with its core magnetized in the same direction as the four edges, while the other state has the four edges opposite to the core. In either case, in addition to the central vortex, there are four edge vortices, which are quarter-vortices with their core trapped at the edge or they can be virtual vortices with their cores lying just outside of the whisker.

When the field is reduced, the end swirls move off center and the central vortex becomes a tube of magnetization around the line m x = 1 that goes from one end swirl to the other. When the swirl approaches one of the four end edges, it is repulsed from that edge. That the swirl moves off center when the field is lowered is discussed below. The moving off center was an important unanswered question in the original manuscript of this chapter.

The swirl cannot cross an end edge without first having the direction of the magnetization on that edge reverse by soliton motion. Again, that can take place when a soliton breaks away from a corner or by the creation of a pair of solitons along that edge, instigated by the approaching swirl.

Before the end edge reverses by soliton motion, a second swirl appears on the side surface. Both swirls are connected to the central vortex in complicated patterns that are more readily grasped by three-dimensional plots of tubes of the energy density. The energy-density isosurfaces have arms going from the central vortex to each of the swirls (two on each end). Sometimes even a third swirl appears near the end. Once the edge reverses, the end swirl merges with the side swirl to produce the Landau structure via the propagation of solitons along adjacent edges on the smaller of the side surfaces . This is where it is better to have a rectangular rather than a square cross-section to avoid the Graham configuration. If the solitons propagate from both ends but not on the same two edges, the structure is complicated. This is one reason why it was often necessary to cycle the magnetization to achieve the field dependence of the ac susceptibility assigned to the presence of the Landau structure.

The side surface becomes magnetized in the direction opposite to the central vortex, which now should be called a tube because the core of the x-vortex does not come to the surface at the center of a swirl. The tube in m z, described in Section 3.4, coexists with the vortex in m x. The tube around m x interacts with the two long edge quarter-vortices that are magnetized in the same x-direction. The combined structure of the central tube and the two like-directed quarter-vortices is called the Bloch wall with Néel caps as first calculated by LaBonte. The contour of that tube, where m x = 0, is the domain wall separating the regions of reversed magnetization as shown in Figure 3.10.

From this picture, it is clear that it should make a difference whether the Néel caps of LaBonte’s wall are pointing toward or away from the surface through which the domain wall moves with fields that are large enough in either direction to saturate the central cross-section of the whisker. The Néel caps act as springs as the wall approaches either side of the whisker. That spring is stronger if the Néel caps are displaced from the center of the Bloch wall in the direction of the approaching side surface. That this effect was never observed in the measurements of the ac susceptibility reflects the structure of a real wall. The real wall is not translationally invariant as calculated by LaBonte. It has reversals of the direction of the Néel caps from place to place along the wall as seen in all studies of domain walls intersecting surfaces and thoroughly discussed in Magnetic Domains. The net strength of the exchange spring is then the same approaching either side surface.

That the tube of m x has a life of its own is seen in the calculations related to the barber-pole effect, first seen experimentally by Hanham [5] and recently in the calculations of Templeton [23] for the hexagonal [111] whisker. The barber-pole effect is the tube of m x taking on a helical pattern as one proceeds from the swirl on one end to the swirl on the other. The beginning of the helix has been present in calculations of swirls for the last two decades. It usually goes smoothly into the central vortex near the end surfaces.

The propagation all the way from one end of the swirl to the other first appeared in a hysteresis loop in a calculation for a 150 nm-diameter hexagonal iron whisker 1500 nm high. It gave an account of the reversal of the six edges by both the creation of soliton pairs along the edges and the breaking away of a soliton from either of the corners where the side edge ends. The helical pattern of the m x-tube formed sequentially in a critical negative field. It was stable for more negative fields until all six edges reversed relative to the core. At an even more negative field, the core reversed and aligned with the edges. If the field was increased (less negative) after the barber pole formed, it was stable until, at a still negative field, it switched back by soliton propagations to the state where the six edges and the central vortex were aligned in the original direction of the core. The formation of a stable helix requires a diameter greater than 99 nm.

The barber-pole pattern observed on the surface of the hexagonal whisker is an outward manifestation of the helix when the radius of the helix brings the center of the m z-tube close enough to the surface to change the magnetization direction in patches along the six edges.

It was then calculated that the barber-pole effect still occurred on the removal of the anisotropy of Fe, which is a small effect at 150 nm. It was next calculated that the replacement of the hexagonal cross-section by a cylindrical cross-section lowered the range of stability somewhat, but the barber-pole effect was still there on the cylindrical surface. After searching, conditions were found for the helical periodic structure to appear in a whisker with close to square cross-section; see Figure 3.18.

Figure 3.18   (See color insert.) The appearance of a 0.6 µm periodic structure in a [100] oriented iron whisker with dimensions of 1.54 × 0.20 × 0.24 µm3. The four long sides are shown as if they were unwrapped, with a repetition of the front surface to see the configuration on each of the four long edges. The color code is red to the right, green to the left. Both blue and yellow are downward on the page. The core of the helical structure comes close enough to the side surfaces every 0.6 µm to turn the magnetization perpendicular to the x-axis and slightly reversed. On two of the edges on the backside, two edge solitons have reversed part of the edges and part of the top, bottom, and backside surface, showing green to the left.

Unstable helices are found by looking at the patterns of magnetization reversal calculated previously, once one knows where to look.

During the course of these discoveries, it was paramount to find the driving forces for the formation of the swirl and its preference for moving away from the center of symmetry while connecting to the vortex down the centerline by a path that traced out a helix of diminishing radius starting at the surface. And then to explain why sometimes it propagates all the way to the far swirl.

All of these searches fixed upon the idea of using splay saving as the mechanism.

The discussion of the hedgehog in Section 3.3.3 explains why this was the wrong approach. Changing the hedgehog into splay-saving configurations does not change the exchange energy. It does in liquid crystal theory applied to magnetism, but Hubert in his treatise on Magnetic Domains showed two decades ago why one cannot use (div m)2 + (curl m)2 for the energy in micromagnetic calculations. Yet, thinking in terms of divergences and curls was sufficiently useful in that the author brought them back into magnetism in Equation 3.2.

In writing the revision of this chapter, the author took the time to calculate the splay and curl terms in Equation 3.2 and found that the exchange energy of the splay-saving configuration was the same as the hedgehog. It appeared to be perfectly obvious for the energy to be written as the sum of the squares of the first derivatives. All the energy to be that was saved in the divergence term appears in the curl term using Equation 3.2. The helix could not be explained from splay saving.

The answer is that the splay is created by the exchange energy in trying to meet the magnetostatic boundary conditions at the surfaces and, in particular, at the positions of the swirls. Even though magnetic configurations can be created that are divergence free except at the surface, these are not torque-free solutions of the LLG equations. Those torques create divergences in the bulk. In the swirl, they are near to the surface. In the helical pattern, they are throughout the bulk. The divergences are both positive and negative. The self-magnetos tatic energy of the regions of positive or negative divergence is paid for by the minimizing of the exchange energy taking into account the magnetostatics of the boundaries and the applied field. Once these regions of charge density exist, it pays to move them to lower the magnetostatic energy.

In ultrathin film memory elements, a rectangle can take either the S or the C configuration. In the S configuration, the magnetic charges of opposite sign are on the diagonally opposite far corners. In the C configuration, they are on corners along the same longer edge but not on the same shorter edge because that would cost too much exchange energy. The C-state is lower energy for a rectangle that is not too close to a square and not too small.

Given four patches of charge, two of each sign, the lowest energy configuration is a square with charges arranged with the four nearest neighbors of opposite sign and two pairs of diagonal neighbors of the same sign. This is an extended quadrupole. The extended quadrupole appears in the barber pole. It also has been noted in the chemistry and biology of the arrangement of charges on surfaces. The helical configuration is analyzed in Figures 3.19 and 3.20 to make the case that the helix takes advantage of the energy to be saved from the magnetostatic interactions of unlike charges. The usefulness of thinking about magnetic charges has been presented in an editorial titled “Visualization and Interpretation of Magnetic Configurations Using Magnetic Charge” printed in IEEE Magnetic Letters [31] and illustrated in “Using magnetic charge to understand soft-magnetic materials” [32].

Figure 3.19   (See color insert.) Magnetization A and demagnetizing fields B for one pitch of 750 nm of the helical magnetization patterns in an iron cylinder of radius 160 nm. The central figures are crosssections at constant z. The patterns rotate as z is changed. The panels on the sides are meridian cuts. The positions of the colors move up and down as the angle of the meridian cut is changed. The demagnetizing fields in a slice at constant z point along the sides of a rectangle with green, blue, red, and yellow in the middle of the sides. The green region points left, the blue region points up, the red region points right, and the yellow region points down indicating the presence of four regions of charge density on the four corners of the rectangle. The charge is positive in the corner between the yellow and the green and in the opposite corner between the blue and the red. The charge is negative at the other two corners. There are no charges along the radial line from the center as ∂m z/∂z (1/ρ) cancels ∂(ρmρ)/∂ρ. This is seen by the direct calculation of div m as shown in Figure 3.20. To reconcile the two figures, it is necessary to remember that the magnetic charge is –div m.

Figure 3.20   (See color insert.) Four helical tubes of magnetic charge density shown as a cut in the z-plane in the central panel and as meridian cuts in x and y on the left and right, respectively. The pattern is the same in every cross-section but rotating with z. The charge density along the line labeled E on the right is the same as along the line labeled D in the center. As E is moved down the page, the pattern of the cut in z rotates counterclockwise changing the charge along line D. The gray ellipse in the left side meridian cut represents the charges that are not there directly across from all the charges that are there. The charge density in red sees the charge density in blue across and downward as well as directly above. The length of the gray ellipse is 3/2 of the length of the red or blue regions of the charge distribution. This accounts for the observation that the outer red moves to where the outer blue is presently on a rotation of 2π/7. The outer charges are at the same distance from the inner charges as the inner charges are from the cylinder axis.

#### 3.7.2  Microtransformers

The simplest of all magnetic configurations is the cylindrical toroid. A single LaBonte wall would divide the cylindrical toroid into two domains, one clockwise and one counterclockwise. The energy of the LaBonte wall would increase linearly with the radius of the wall making the wall unstable. The wall can be stabilized by a magnetic field from a current in a wire through the hole in the doughnut. It could be further stabilized by passing a second current anti-parallel to the first through the doughnut itself, producing a field that changes sign between the inner and outer radius. Yet it is not necessary to use currents to produce a stable LaBonte wall in the toroid.

If the Néel caps are directed toward the inner radius, they provide strong exchange springs that keep the wall from leaving. The direction of the Néel caps will be directed inward if the domain wall is nucleated at the outer radius. This is accomplished using a current through the doughnut itself, as the field will then be at maximum at the outer radius. When this is modeled using micromagnetics there are several interesting results.

The first result relates to the process of magnetization before the current is sufficient to nucleate a domain wall. The others are about what happens after nucleation. Three different metallurgical preparations were considered. A single grain of iron, in which the direction of the anisotropy axes could be chosen in any orientation, was considered first. The other two preparations were polycrystalline iron. For one of these the grain size was smaller than the smallest dimension of the toroidal cylinder. The other had a grain size so small (less than 8 nm) that the exchange energy averaged out the effects of crystalline anisotropy. The latter is the ideally soft magnetic material.

Ultrafine grains have been the subject of research in magnetism for over 25 years. No one has yet to produce a commercial magnetic material with higher saturation than grain-oriented Fe(Si). Material with nanocrystalline grains produced by mechanical milling proved too mechanically hard to compact into useful material because mechanical strength is inversely proportional to grain size.

Cases, where grain size was smaller than the smallest dimension, showed that it was not necessary to produce nanocrystalline grains. With grain sizes in the range of 100 nm, the material still averages over the effects of the anisotropy, but because of magnetostatic interactions rather than exchange. When the magnetization in the toroid tries to rotate in the direction preferred in each of the grains, the magnetic charge develops at the grain boundaries. The fields from those charges keep the magnetization directions very close to the mean direction of the magnetization determined by the circular boundaries of the toroid. For this to happen, it is necessary that the height of the toroid be less than 300 nm.

If the height is greater than that, the magnetization in adjacent grains can each rotate toward the z-axis, one with +m z and one with –m z. When the height is less than 300 nm each grain produces enough magnetic charge on the top and bottom surface to supp ress those rotations. Toroids 300 nm high can be stacked to any height if the separation between the toroids is 4 nm and the magnetization still does not rotate into the z-axis; see Figure 3.20.

The magnetization in the polycrystalline toroid is close to saturation in the direction of the circulating magnetization, but it is not actually saturated because a ripple structure develops wherein the magnetization alternately rotates inward and outward in the plane of the toroidal cylinder.

When a current is used to try to reverse the magnetization, the ripple structure increases as a means to lower the Zeeman energy without producing much charge on the surfaces. It is not sensitive to the crystalline microstructure.

At a critical current, surface charges appear as full reversal nucleates at many places on the outer or inner cylindrical surface depending on whether the current is in a wire through the hole in the doughnut or through the doughnut itself. Despite multiple nucleations, the magnetization in the doughnut re-organizes itself to form one LaBonte wall that separates the volume into two countercirculating domains; see Figure 3.21.

Figure 3.21   (See color insert.) Effect of anisotropy on the nucleation of a domain wall at the outer radius in a cylindrical toroid with radii R o = 1000 nm and R i = 400 nm with height L = 80 nm for a differing microstructure. For a to d, the configuration in a single crystal is changing with time. In a, nucleation is just starting. In e there are 80 grains just after nucleation as is the case for b. In f there are 800 and in g and h there are 8000 grains. From b to d, the vortices and antivortices are annihilating in pairs. The last four pairs shown in d are about to annihilate. The wall is stable in h, which also shows the ripple structure in the almost completely magnetized domains. The grain size does not significantly change the structure of the nucleation process.

As nucleated, the Néel caps are not all in the same direction and the wall is decorated by vortices and antivortices. With time, these annihilate in pairs until there is a single direction for the caps all the way around; see Figure 3.21. But if any of the vortices have their cores in opposite directions, they do not annihilate and the direction of the Néel caps changes from place to place along the circular wall. This effect is eliminated by carrying out the nucleation of the wall in a vertical field of µ0 H z = 0.4 T. The nucleation of the single domain wall is insensitive to the grain size.

The calculations have been carried out for infinitely high stacks using periodic boundary conditions for toroids with outer radii up to 5 µm, which is almost big enough to use to produce transformers with close to the full saturation magnetization of iron.

Calculations of single crystal picture frames at the level of five micron dimensions should be applicable to large sizes if magnetostriction is taken into account. In going to large sizes for polycrystalline toroids, it is not obvious at what dimensions the averaging over anisotropy by magnetic charges at the grain boundary will no longer be effective. Fortunately, the sizes of the calculated toroids have reached the range where they can be used to interpret the surface observations of magneto-optics on bigger systems.

In over 1,000 pictures in their treatise Magnetic Domains, Rudi Schäfer and the late Alex Hubert have shown how complex and beautiful the patterns of magnetization can be. It now seems possible that many of those 1,000 pictures could be reproduced using micromagnetics on smaller systems (Figure 3.22 ) The studies of toroidal geometry have now been adapted to propose the use of patterned polycrystalline iron in motors, generators and transformers [33]. Substantial gains in performance are to be achieved by operating close to the full saturation inductance of iron.

Figure 3.22   (See color insert.) Continuous stacking of cylindrical toroids with an 8-nm gap to suppress rotation of the magnetization into the z-direction (4-nm is sufficient to do this). Each toroid has an outer radius of 1000 nm, an inner radius of 400 nm and a height of 240 nm. A vertical cross-section through the stack is shown in all panels. In a, three consecutive toroids are shown to illustrate that the magnetization lies close to the planes where the toroids are separated. It takes very little magnetic charge to suppress the rotations into the z-direction. In b, an enlargement of the cross-sections on the left show the partial vortex that circulates around the z-axis. The Néel caps are red, pointing toward the z-axis and green pointing outward. The blue is the center of the Bloch wall. Black is either into the paper as it is on the far left or out of the paper as it is in the vortex and on the right. The panels labeled c through h are contours for the three components of the magnetization, m x in the lower panels, m y in the middle panels, and m z in the upper panels. More contours are shown in the panels to the right, to illustrate the compressing of the exchange springs as the wall approaches the inner radius. Panel d shows the two domains. Panel c shows the Bloch wall. Panel e shows the Néel caps displaced to the right of the Bloch wall.

#### Acknowledgments

Bretislav Heinrich and the author started their studies of the magnetic response of iron whiskers in 1969. Dan S. Bloomberg, Murray J. Press, Scott D. Hanham, Amikam Aharoni, T. L. Templeton, and J.-G. Lee have all contributed to the partial understanding of a long list of phenomena observed in whiskers. The micromagnetics of a nanobrick is a continuation of that work made possible by Riccardo Hertel, Attila Kakay, and Mike Scheinfein, who have provided tools for, and participated in, this attack on complexity and now have added parallel processing using graphic processing units to speed up the calculations. This work has been supported by the Natural Sciences and Engineering Research Council of Canada from 1968 to 1998. Since then Dr. Harold Weinstock of the Air Force Office of Scientific Research sponsored the work on ultrafine grain sizes that led to the study of doughnuts at Morgan State University with Professor Conrad Williams and Dr. Ezana Negusse and at Simon Fraser University with Professor Karen Kavanagh. Templeton, Hanham, and Scheinfein each contributed to the development of this manuscript.

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12
H. Kronmüller and S. Parkin , Handbook of Magnetism and Advanced Magnetic Materials, Vol. 2, Micromagnetics, John Wiley & Sons, Chichester, 2007. See in particular: J. E. Miltat and J. Donahoe , Numerical Micromagnetics: Finite Difference Methods, John Wiley & Sons, New York, 2007; T. Schrefl et al., Numerical Methods in Micromagnetics (Finite Element Method), John Wiley & Sons, New York, 2007.
13
I. Rabi , N. E. Ramsey , and J. Schwinger , Use of rotating coordinates in magnetic resonance problems, Rev Mod Phys. 26, 167–171 (1954).
14
A. S. Arrott . Introduction to micromagnetics, in Ultrathin Magnetic Structures IV, B. Heinrich and J. A. C. Bland (Eds.), Springer-Verlag, Berlin, 2005. See p. 131 for the approximation to the integral of tanh2 (ρ/λ)/ρ. This reference follows micromagnetics from a single iron atom imbedded in a lattice to a 4-nm thick circular disk of 96 nm diameter. The present chapter continues that development of thicknesses where the patterns of magnetization are fully 3D.
15
A concise summary of TetraMag is found in TetraMag - A general-purpose finite element micromagnetic simulation package and high resolution large-scale micromagnetic simulations with hierarchical matrices, both by Riccardo Hertel and Attila Kakay. These are available by inserting TetraMag Hertel into the Google search box.
16
A full description of Michael R. Scheinfein’s LLG Micromagnetic Simulator is available by inserting “Scheinfein micromagnetic simulator” into the Google search box. The integrated graphics are native to the Microsoft operating system, but that is one of the strengths of the LLG Micromagnetic Simulator. Visualization is very important in micromagnetics.
17
The story of Gilbert damping starts with an abstract by T. Gilbert in 1955. Gilbert’s idea was formalized by R. Kikuchi . On the minimum of magnetization reversal time. J Appl Phys. 27, 1352–1357 (1956). Fifty years later, Gilbert commented on his work in 2004. IEEE Trans. Magn. 40, 3443. The basis for Gilbert’s phenomenology is firmly established in experiments by R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 2173 (2001). The theoretical foundation is of current interest as described in Chapter 2 by Bret Heinrich and in the interesting exchange in Physical Review Letters that can be accessed by entering Hickey Replies in the Google search box.
18
Classical micromagnetics refers to two monographs by William Fuller Brown, Jr. published in 1962 and 1963 by Interscience, New York: Magnetostatic Principles in Ferromagnetism and Micromagnetism. These are still worth reading today, not only for the physics but also for the style. Brown was also an English teacher. A detailed discussion of the conversion between Gaussian units and SI units is given in the introduction of Ultrathin Magnetic Structures I, B. Heinrich and J. A. C. Bland (Eds.), Springer-Verlag, Berlin, 1994.
19
D. Dotze and A. S. Arrott , Micromagnetic studies of vortices leaving and entering square nanoboxes, J Appl Phys. 97, 10E307 (2005).
20 A. S. Arrott and R. Hertel , Mode anticipation fields for symmetry breaking, IEEE Trans Magn. 43, 2911 (2007).
21
M. R. Scheinfein and A. S. Arrott , Increased efficiency and accuracy in micromagnetic calculations of switching asteroids, J Appl Phys. 93, 6802 (2003).
22
For the years 1971–1975, the Proceedings of the MMM Conference were published in books by the American Institute of Physics. These are known in magnetism as the “lost years” because the work was so infrequently cited and are still not referenced in the Web of Science (The AIP now has its conference proceedings online at \$28 per paper.) In those years Heinrich and Arrott studied iron whiskers at the Curie temperature. The conjectured curling pattern just below Tc appeared in the seldom-cited and hard-to-find paper by B. Heinrich and A. S. Arrott, in Proceedings of the International Conference of Magnetism ICM-73, Vol. IV, Publishing House NAUKA, Moscow, Russia, pp. 556–561, 1974.
23
T. L. Templeton , S. D. Hanham, A. S. Arrott, Helical patterns of magnetization and magnetic charge density in iron whiskers, AIP ADVANCES 8, 056022 (2018).
24
F. Cheynis , A. Masseboeuf , O. Fruchart et al., Controlled switching of Néel caps in flux-closure dots, Phys Rev Lett. 102, 107201 (2009).
25
A. S. Arrott and R. Hertel , Large amplitude oscillations (switching) of bi-stable vortex structures in zero field, J Magn Magn Mater. 322, 1389–1391 (2010).
26
S. B. Choe , Y. Acremann , A. Scholl et al., Vortex core driven magnetic dynamics, Science 304, 420–422 (2004). Their Figure 1 panel IV shows the oscillation of a swirl in the Landau structure. See also the discussion by Xiaowei Yu. 2009. in Time-resolved x-ray imaging of spin-torque-induced magnetic vortex oscillation. Ph.D. Thesis, Applied Physics, Stanford.
27
J. A. Osborn , Demagnetizing factors of the general ellipsoid, Phys Rev. 67, 351 (1945).
28
S. Gliga , Energy thresholds in the magnetic vortex core reversal, J Phys: Conf Ser. 303, 012005 (2011).
29
O. A. Tretiakov and O. Tchernyshyov , Vortices in thin ferromagnetic films and the skyrmion number, Phys. Rev. B 75, 012408 (2007).
30
J. G. Lee , S. A. Govorkov , and A.S. Arrott , Iron whisker domain patterns imaged by garnet films J Appl Phys. 79, 6051 (1996).
31.A. S. Arrott, Visualization and interpretation of magnetic configurations using magnetic charge, IEEE Magnetic Letters 7, 1108505 (2016).
32.A. S. Arrott and T. L. Templeton, Using magnetic charge to understand soft-magnetic materials, AIP ADVANCES 8, 047301 (2018).
33.A. S. Arrott, C. M. Williams, E. Negusse, Magnetic charges suppress effects of anisotropy in polycrystalline soft ferromagnetic materials, AIP ADVANCES 8, 056114 (2018).

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