Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)
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9.3 DOMAIN WALL OBSERVATION |
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Fig. 9.12 Principle of domain observation by the Kerr effect.
Kerr Effect. This effect is a rotation of the plane of polarization of a light beam during reflection from a magnetized specimen. The amount of rotation is small, generally much less than one degree, and depends the material and on the direction and magnitude of the magnetization relative to the plane of incidence of the light beam. Specifically, the degree of rotation depends on the component of magnetization parallel to the direction of propagation of the light beam. Figure 9.12 shows the experimental arrangement. Light from a source passed through a polarizer which transmits only plane polarized light, or naturally polarized light from a laser, is incident on the specimen. For simplicity the specimen is assumed to contain only two domains, magnetized antiparallel to each other as indicated by the arrows. During reflection the plane of polarization of beam 1 is rotated one way and that of beam 2 the other way, because they have encountered oppositely magnetized domains. The light then passes through an analyzer and into a low-power microscope. The analyzer is now rotated until it is “crossed” with respect to reflected beam 2; this beam is therefore extinguished and the lower domain appears dark. However, the analyzer in this position is not crossed with respect to beam 1, because the plane of polarization of beam 1 has been rotated with respect to that of beam 2. Therefore beam 1 is not extinguished, and the upper domain appears light. Figure 9.13 shows domains in a thin film revealed in this way; the light and dark bands are domains magnetized in opposite directions.
Because of the small angle of rotation of the plane of polarization, the contrast between adjoining domains tends to be low, so all the optical elements must be of high quality and well adjusted. However, the Kerr method is ideal for observation of domain walls in motion and has supplanted the Bitter method for such studies. It has no limitations with respect to specimen temperature beyond the usual ones of thermal insulation and protection against oxidation. It can be applied both to bulk specimens and thin films. However, a component of the magnetization vector must be parallel to the direction of propagation of the light, which means for most materials the light beam strikes the sample surface at a fairly small angle. This limits the area that can be observed, especially at high magnification.
Note that the term “Kerr effect” is also applied to an electro-optic effect. If certain organic liquids are placed in a transparent container, called a Kerr cell, and subjected to an electric field, plane polarized light passing through the cell will be rotated by an amount depending on the applied voltage.
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Fig. 9.13 Domains in a film of 81 permalloy observed using the Kerr effect.
Faraday Effect. This effect is a rotation of the plane of polarization of a light beam as it is transmitted through a magnetized specimen. The optical system is the same as the Kerreffect system, except that source, polarizer, specimen, analyzer, and microscope are all in line. The method is, of course, limited to specimens thin enough, or transparent enough, to transmit light; it is applied most often to thin sections of ferrimagnetic oxides, up to
˚
about 0.1 mm in thickness, although metallic films less than 400 A thick have also been examined.
For thin sections of oxides, the amount of the Faraday rotation is a few degrees. This results in high contrast between adjoining domains and yields photographs of remarkable clarity. Like the Kerr method, the Faraday method is unrestricted as to temperature and is excellent for wall motion studies.
9.3.4Scanning Probe; Magnetic Force Microscope
The scanning tunneling microscope (STM) was introduced in 1986. It makes use of the fact that the dimensions of a piezoelectric crystal can be controlled by the application of electric fields down to displacements less than the diameter of an atom. A very sharp-pointed tip can be scanned across a small sample area with a set of xy piezoelectric crystals while its distance above the surface z is controlled by a third piezoelectric crystal, as shown in Fig. 9.14. The original version of the scanning probe microscope applies a voltage between the tip and the sample, and brings the tip close enough to the surface so that a tunneling current flows. Then the tunneling current is measured as the tip is scanned over the sample surface, or (usually) the tip-to-sample distance is varied to keep the tunneling current constant. The variation of tunneling current or tip spacing with position then maps out the geometry of the surface at atomic-scale resolution.
A number of variations on this basic idea quickly followed. One of these uses a tip coated with a thin layer of magnetic material, which experiences a measurable force when it enters the field gradient where a domain wall meets the sample surface. This is the magnetic force microscope, or MFM. The force acting on the tip can be measured optically, by the deflection of the cantilever holding the tip, or from the change in amplitude of
9.3 DOMAIN WALL OBSERVATION |
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Fig. 9.14 Scanning probe microscope.
Fig. 9.15 SEMPA analysis of domain wall structure. This is a scan across a 1808 domain wall in iron. Solid bars show vertical polarization of electrons vs. position; open circles show horizontal polarization. Note that there is no visible separation of the 1808 wall into two 908 walls. [Data from H. P. Oepen and J. Kirschner, Phys. Rev. Lett., 62 (1989) p. 819.]
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the cantilever motion when set into resonant oscillation. MFM requires minimal surface preparation, and works on both conducting and insulating materials. The same instrument can determine the surface contours of the sample, and so can largely separate surface effects from magnetic effects.
The MFM offers the prospect of observing magnetic structures at extremely high resolution, and it has been used successfully in some investigations. However, there are problems in interpreting the results. The magnetic material on the tip is in the form of a hollow cone or a portion of a hollow cone, and it is typically subjected to a highly nonuniform field above the surface of the sample. It is then not clear how to relate the magnetic field above the sample, which is the desired information, to the force on the tip, which is the measured quantity. There is also the possibility that if the domain walls in the sample are highly mobile, they may be displaced by the interaction with the magnetic material of the tip. The tip coating is usually a soft magnetic material, but it may also be a permanent magnet material, in which case the tip magnetization may be regarded as fixed, but the chance of the tip causing motion of the domain walls is increased.
9.3.5Scanning Electron Microscopy with Polarization Analysis
This is another powerful technique for domain observation. A tightly focused electron beam is scanned over the sample surface, and the emitted secondary electrons are analyzed to determine their magnetic polarization, which depends on the direction of magnetization of the region from which they were emitted. The experiment requires special apparatus carefully adjusted, but can give very detailed results, including the variation in magnetization direction across an individual domain wall, as in Fig. 9.15.
9.4MAGNETOSTATIC ENERGY AND DOMAIN STRUCTURE
We turn now from the observation of domains to an examination of the reasons for their formation and their relative arrangement in any given specimen. We will find that magnetostatic energy plays a primary role.
9.4.1Uniaxial Crystals
Consider a large single crystal of a material with uniaxial anisotropy. Suppose it is entirely one domain, spontaneously magnetized parallel to the easy axis, as in Fig. 9.16a. Then free poles form on the ends, and these poles are the source of a large H field. The magnetostatic energy of this crystal is 1/8p Ð H2 dv (cgs) or 1/2 Ð H2 dv (SI), evaluated over all space. This considerable energy can be reduced by almost a factor of 2 if the crystal splits into two domains magnetized in opposite directions as in Fig. 9.16b, because this brings north and south poles closer to one another, thus decreasing the spatial extent of the H field. If the crystal splits into four domains as in Fig. 9.16c, the magnetostatic energy again decreases, to about one-fourth of its original value, and so on. But this division into smaller and smaller domains cannot continue indefinitely, because each wall formed in the crystal has a wall energy per unit area, which adds energy to the system. Eventually an equilibrium domain size will be reached.
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Fig. 9.16 Division of a crystal into domains. Only external H fields are shown.
The magnetostatic energy of the single-domain crystal is, from Equation 7.59,
Ems ¼ |
1 |
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2NdMs2 |
(9:26) |
per unit volume, where Nd is the demagnetizing factor. The value of Nd for a cube, in a direction parallel to an edge, is 4p/3 (cgs) or 1/3 (SI). If we take this value as applying approximately to the crystal of Fig. 9.16, the magnetostatic energy of the crystal per unit area of its top surface is
Ems ¼ |
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pMs2L (cgs) or |
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Ms2L (SI), |
(9:27) |
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where L is the thickness.
The calculation, which is not easy, of the magnetostatic energy of the multi-domain crystal of Fig. 9.16c has been given by S. Chikazumi [Physics of Ferromagnetism, 2nd ed. Oxford (1997), pp. 434–435]. This energy, per unit area of the top surface, is
Ems ¼ 0:85 Ms2D, |
(9:28) |
where D is the thickness of the slab-like domains, provided that D is small compared to L. The total energy is the sum of the magnetostatic and wall energies:
E ¼ Ems þ Ewall, |
(9:29) |
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E ¼ 0:85 Ms2D þ s |
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where s is the domain wall energy per unit area of wall and L/D is the wall area per unit area of the top surface of the crystal. The minimum energy occurs when
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For cobalt, taking s ¼ 7.6 erg/cm2 and L ¼ 1 cm, we find
s
D ¼ (7:6)(1) 2 10 3 cm ¼ 20 mm (0:85)(1422)2
(9:30)
(9:31)
which means there will be about 1/(2 1023) ¼ 500 domains in a 1 cm cube crystal. The ratio of total energy before and after division into domains is
E (single-domain) |
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where the values appropriate to cobalt have been inserted. Thus the creation of domains has lowered the energy by a factor of 500.
A still larger reduction in magnetostatic energy will result if the unlike poles on each end of the crystal are “mixed” more intimately. This can be done if the domain walls become curved rather than flat, although still parallel to the easy axis, as in Fig. 9.17a. A section of such a crystal parallel to the easy axis will show straight lines separating the domains, and a section normal to the easy axis will show curved lines. Curvature of the walls increases the wall area, and this type of domain structure is therefore found mainly in very thin crystals. In thick crystals, wall curvature involves too much extra wall energy,
Fig. 9.17 Curved domain walls and surface spike domains.
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and another method of reducing magnetostatic energy is favored, as in Fig. 9.17b. Here spike-shaped domains of reversed magnetization are formed at the surface. This has the desired effect of producing a fine mixture of opposite poles on the end surfaces without adding too much wall energy, because the spike domains are short. However, there is a discontinuity in the normal component of Ms on the walls of the spike domains, and free north poles must form there in accordance with Equation 2.29. These interior poles are the source of an H field and therefore contribute to the magnetostatic energy. The number and size of the spike domains will be such as to balance the reduction in main magnetostatic energy due to the surface poles against the increase in wall energy and in magnetostatic energy due to interior poles, assuming that there is no barrier to the formation of additional domains.
9.4.2Cubic Crystals
The domain structure of cubic crystals tends to be more complicated, because there are now three or four easy axes, depending on the sign of the anisotropy constant K1. Furthermore, it is now possible for the flux to follow a closed path within the specimen so that no surface or interior poles are formed, and the magnetostatic energy is reduced to zero. Figure 9.18a shows how this is done. Triangular domains are formed at the ends and, because they are paths by which the flux can close on itself, they are called closure domains. One might think that the domains in such a crystal could be very large, since the only obvious source of energy is wall energy. However, there is also magnetoelastic energy. If l100 is positive, as it is in iron, then the [100] closure domain would strain magnetostrictively as shown by the dotted lines in Fig. 9.18b, if not restrained by the main [010] and
¯
[010] domains on either side. The closure domains are therefore strained, and the magnetoelastic energy stored in them is proportional to their volume. The total closure-domain volume can be reduced by decreasing the width D of the main domains. The crystal will
¯
therefore split into more and more [010] and [010] domains, until the sum of the magnetoelastic and domain wall energies becomes a minimum. Closure domains of the type shown in Fig. 9.18 are often been seen at the edge of cubic crystal. Note that a closure-domain structure will ideally have Ms parallel to the surface at all free surfaces.
The avoidance of free poles is also the guiding principle controlling the orientation of domain walls. For example, a 1808 wall must be parallel to the Ms vectors in the adjacent domains; if not, as in the spike domains of Fig. 9.17b, free poles will form on the wall, creating magnetostatic energy. For the same reason, a 908 wall, such as those bounding the closure domains of Fig. 9.18, must lie at 458 to the adjoining Ms vectors.
Fig. 9.18 Closure domains in a cubic crystal with k100l easy axes.
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The principle of free-pole avoidance does not entirely fix the orientation of a domain wall. For example, suppose a 1808 wall, in a material with k100l easy directions, separates
¯
two domains magnetized along [100] and [100]. Then the wall could tilt about [100] as an axis through an infinity of orientations and still remain parallel to the Ms vectors on either side of it, thus remaining free of poles. Calculations show that the energy s per unit area of wall varies with the tilt angle and goes through a maximum for some orientations and a minimum for others. Similar conclusions apply to 908 walls. But the ratio of maximum to minimum values of s is less than 2, and this effect is small compared to the creation of magnetostatic energy by walls so oriented that they have free poles.
Simple domain arrangements, giving rise to a set of parallel 1808 walls, as in the central portion of the crystal of Fig. 9.18, are seen only when the surface of the crystal is accurately parallel to an easy direction. If the surface deviates only a few degrees from a f100g plane, the complex “fir tree” pattern of Fig. 9.9 is formed. It has “branches” jutting out from the main 1808 walls at an angle of about 458, as shown in detail in Fig. 9.19. Here the (100) planes make an angle u with the crystal surface, and so do the Ms vectors in the two main domains. North and south poles are therefore formed on the top surfaces of these domains. To decrease the resulting magnetostatic energy, the branch domains form. These carry flux parallel to the crystal surface, in the easy y or [010] direction, and therefore have no poles on their top surface. These branch domains are shallow and are bounded on the bottom by curved, nearly 908 walls which have some free poles distributed on them. Nevertheless, the total energy is reduced by the formation of the branch domains.
The complex tree pattern is instructive, when one considers that well below the surface the domain structure of this crystal is extremely simple. This observation is generally true. Many of the complex domain arrangements seen on some crystal surfaces would not exist if that surface had not been exposed by cutting; they are closure domains which form when the cut is made, in order to reduce magnetostatic energy. Surface domain structures can therefore be very different from the basic domain structure of the interior. On the other hand, domains in very thin specimens, such as films and foils, normally extend completely through the specimen thickness. The domain structure revealed by surface examination, by the Bitter or Kerr techniques, is therefore the same as the interior structure in this case.
Fig. 9.19 Interpretation of the fir tree pattern. Lines on the sides of the crystal are parallel to f100g planes.
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The simplest kinds of domain structure are seen in properly prepared single crystals, and the most complex in polycrystals. The great accomplishment of Williams, Bozorth, and Shockley lies precisely in the fact that they established this conclusion. They succeeded in showing that the complex pattern of Fig. 9.9, for example, was nothing but a surface artifact, caused by the fact that the specimen surface was not accurately parallel to a f100g plane.
The most nearly perfect metal single crystals available are whiskers. These are fine filaments, generally a few millimeters in length and several tens of micrometers thick, grown by the reduction of metal bromide vapor by hydrogen at about 8008C. Whiskers first attracted scientific interest because of their extremely high mechanical strength. Later it was realized that they offered excellent opportunities for magnetic domain studies. Iron whiskers, for example, often grow with a k100l axis and with sides parallel to f100g planes. Moreover, these sides are optically flat, which means that no specimen preparation is needed before the magnetic suspension is added to form a Bitter pattern. Figure 9.20 shows the domain structure observed on a f100g k100l iron whisker and the wall motion that occurs in an applied field. The sketch at the top shows the domain structure through the volume of the whisker, and the Bitter patterns are of the top face. The arrows below
Fig. 9.20 Reversible wall motion in an iron whisker. Bitter patterns under dark-field illumination.
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Fig. 9.20 Continued.
each pattern indicate the direction and magnitude of the applied field, which had a maximum value of about 10 Oe. The changes shown are entirely reversible; that is, the configuration at top could be regained at any stage of the magnetization process by reducing the field to zero.
Closely analogous to whiskers are metal “platelets,” which are grown by the same methods and are also single crystals. Here the growth habit is edgewise rather than axial; the result is a platelet hundreds of micrometers in its lateral dimensions and from less
m ˚ m
than 0.1 m (1000 A) to over 10 m in thickness. The platelet surfaces and edges are crystal planes and directions of low indices, such as f100g and k100l. These platelets are structurally more nearly perfect than the usual thin films formed by evaporation of a metal in vacuum onto a substrate. Figure 9.21 shows an exceptionally simple domain structure in a demagnetized Ni–Co alloy platelet. The crystal structure is face-centered cubic, with k100l easy directions (positive K1). When a field of 3.6 Oe is applied to the right, the domain walls move reversibly to the positions shown in Fig. 9.21b. Here the corners of the platelet have pinned the ends of the 908 walls, forcing the latter to bend. The normal components of Ms are no longer continuous across these walls, so they are now the sites of free poles. With further increase in field, to 4.1 Oe, the domain structure