bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

Domain theory

The discovery of enhanced magnetoresistance and oscillatory interlayer exchange coupling in transition metal multilayers just over a decade ago has enabled the development of new classes of magnetically engineered magnetic thin-film materials suitable for advanced magnetic sensors and magnetic random access memories. Magnetic sensors based on spin-valve giant magnetoresistive (GMR) sandwiches with artificial antiferromagnetic reference layers have resulted in enormous increases in the storage capacity of magnetic hard disk drives. The unique properties of magnetic tunnel junction (MTJ) devices has led to the development of an advanced high performance nonvolatile magnet random access memory with density approaching that of dynamic random-access memory (RAM) and read-write speeds comparable to static RAM. Both GMR and MTJ devices are examples of spintronic materials in which the flow of spin-polarized electrons is manipulated by controlling, via magnetic fields, the orientation of magnetic moments in inhomogeneous magnetic thin film systems. More complex devices, including three-terminal hot electron magnetic tunnel transistors, suggest that there are many other applications of spintronic materials.

Magnetically engineered spintronic sensors and memory

The surface magneto-optic Kerr effect (SMOKE) has significantly impacted research on magnetic thin films. This is due to its sensitivity, local probing nature, and experimental simplicity. The polar and longitudinal Kerr effects are characterized by a complex rotation of the plane of polarization of linearly polarized incident light upon reflection from the surface of a ferromagnetic material. The rotation is directly related to the magnetization of the material within the probing region of the light. Light penetrates into metals >20 nm deep, but the SMOKE technique derives its surface sensitivity from the limited thickness of the deposited magnetic film, which can be as thin as one atomic layer. Basic principles, experimental arrangements, and applications of SMOKE are reviewed in order to acquaint the nonspecialist with the technique and place it into perspective.

Surface magneto-optic Kerr effect

(1993). Ultrathin metallic magnetic films: magnetic anisotropies and exchange interactions. Advances in Physics: Vol. 42, No. 5, pp. 523-639.

Ultrathin metallic magnetic films: magnetic anisotropies and exchange interactions

A theoretical description of magneto-optical effects in metallic transition metal systems on the basis of an itinerant description for the underlying electronic structure is presented and applications to the magneto-optical Kerr effect and the circular dichroism in x-ray absorption are discussed Simple arguments based on symmetry considerations as well as electronic excitation schemes are given to show that magneto-optical effects are caused by the interplay of magnetic ordering and spin - orbit coupling The various band structure techniques developed to deal with this situation are reviewed with an emphasize on fully relativistic methods based on the Dirac equation The theoretical framework to calculate magneto-optical spectra on the basis of the underlying band structure is outlined in some detail In addition, a number of sum rules derived by various authors are presented that allow one to estimate spin and orbital magnetic moments of an absorbing atom from the magnetic x-ray dichroism spectra For the magneto-optical Kerr effect, as well as the magnetic x-ray dichroism, application of the theoretical formalism is presented for a great variety of systems, ie pure elements, compounds, alloys and multilayer systems For both kinds of spectroscopies it is demonstrated that model calculations which allow one to manipulate the exchange splitting and the spin - orbit coupling strength give valuable information for a better understanding of the rather complex spectra Concerning the sum rules, this technique together with the direct calculation of the dichroism spectra allow for a stringent test of the various assumptions on which these rules are based

https://iopscience.iop.org/article/10.1088/0034-4885/59/12/003/pdf

Magneto-optical effects in transition metal systems

The recently discovered oscillating exchange effect in iron–chromium multilayers can best be studied on an epitaxial iron sandwich with a wedge shaped chromium interlayer. Domain patterns are analyzed as well as magnetization curves on such samples using magnetooptical techniques. Unusual domains are found in zones around the transitions between ferromagnetic and antiferromagnetic exchange. Also the magnetization curves for larger chromium thickness show some peculiar features. The interpretation of these patterns invokes coupling between the iron layers which favours a right angle between the magnetization directions in both layers. Such a non-collinear interaction can be derived phenomenologi-cally from a biquadratic coupling as introduced earlier in the analysis of spin structures in oxides. A biquadratic coupling can be understood either as a second order Heisenberg or a second order Dzyaloshinskii interaction. In the course of these investigations domain observations proved to be a useful and sensitive method to detect locally the nature of interactions between ferromagnetic layers.



Die kiurzlich entdeckte oszillierende Wechselwirkung in Eisen–Chrom-Vielfachschichten kann am besten an einer epitaxialen Schichtanordnung mit einer keilformigen Chromzwischenschicht studiert werden. Die auf solchen Proben beobachtbaren Domanenstrukturen sowie die Magnetisierungskurven werden mit magnetooptischen Verfahren untersucht. Im Bereich der Ubergange zwischen ferroma-gnetischer und antiferromagnetischer Kopplung werden Zonen ungewohnlicher Muster entdeckt. Auch die Magnetisierungskurven fur grosere Chrom-Schichtdicken zeigten eigentumliche Zuge. Die Beobach-tungen lassen sich erklaren, wenn eine zusatzliche Kopplung der Schichten angenommen wird, die einen rechten Winkel zwischen den Magnetisierungsrichtungen in beiden Schichten begunstigt. Phanomenologisch ergibt sich eine solche nicht-kollineare Kopplung aus der Annahme einer biquadrati-schen Wechselwirkung, wie sie fruher in der Analyse von Spinstrukturen magnetischer Oxide eingefuhrt wurde. Eine biquadratische Kopplung Iast sich entweder als Heisenberg- oder Dzyaioshinskii-Wechselwirkung zweiter Ordnung interpretieren. Domanenbeobachtungen erwiesen sich im Laufe dieser Untersuchung als empfindliche Methode zur Untersuchung des Charakters der Kopplung in Vielfachschichten.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssa.2211250225

Domain Observations on FeCrFe Layered Structnres. Evidence for a Biquadratic Coupling Effect

The stray field, magnetic microstructure, and switching behavior of high‐resolution electron beam fabricated thin film tips for magnetic force microscopy (MFM) are investigated with different imaging modes in a transmission electron microscope (TEM). As the tiny smooth carbon needles covered with a thermally evaporated magnetic thin film are transparent to the electron energies used in these TEMs it is possible to observe both the external stray field emanating from the tips as well as their internal domain structure. The experiments confirm the basic features of electron beam fabricated thin film tips concluded from various MFM observations using these tips. Only a weak but highly concentrated stray field is observed emanating from the immediate apex region of the tip, consistent with their capability for high resolution. It also supports the negligible perturbation of the magnetization sample due to the tip stray field observed in MFM experiments. Investigation of the magnetization distributions within the tips, as well as preliminary magnetizing experiments, confirm a preferred single domain state of the high aspect ratio tips. To exclude artefacts of the observation techniques both nonmagnetic tips and those supporting different magnetization states are used for comparison.

/pdf/electron-beam-fabrication-and-characterization-of-high-4sqiciwur1.pdf

Electron beam fabrication and characterization of high- resolution magnetic force microscopy tips

Wall Transitions and Coupling of Magnetization in NiFe/C/NiFe Double Films

We optimized the losses in iron-rich metallic glasses by surface crystallization and by annealing in a field perpendicular to the ribbon surface. Both methods lead to a weak perpendicular anisotropy and rotational magnetization processes. The origin of surface crystallization could be traced to the effects of oxygen during quenching and annealing. While in surface crystallized samples the ratio between measured losses and classical eddy current losses was at best 6 at 2000 Hz (0.2 T), a much better behaviour could be achieved by field annealing. With careful sample preparation and annealing, a Ni-content of 2.5% proved to be sufficient to obtain low losses between 2-3 times the classical eddy current losses in a large range of magnetization amplitudes and frequencies at permeabilities in excess of 10000. Domain observations show uniform magnetization processes.

Loss optimization for iron-rich metallic glasses

Domain wall configurations in soft magnetic materials are invisible in the regular polar magnetooptical Kerr effect at perpendicular incidence. Earlier we reported about two other contrast mechanisms under these conditions: the Voigt effect and the magnetooptical gradient effect. The net contrast from both effects vanishes for certain wall orientations relative to the polarization axis of the light; the case of a 180°-wall oriented parallel to the E-vector of the light is an example. The weak[residual]contrasts observable under such circumstances are the subject of this paper. The residual contrasts have a dipolar character. They can be consistently analysed in terms of magnetooptical gradient and Voigt effect contributions due to the surface wall structure itself. The two contributions interfere constructively or destructively depending in a predictable fashion on domain magnetization.



Domanenwande in weichmagnetischen Materialien sind im normalen polaren magnetooptischen Kerreffekt (bei senkrechter Inzidenz) unsichtbar. Fruher hatten wir uber zwei Effekte berichtet, die unter diesen Bedingungen doch zu einem Kontrast fuhren: den Voigteffkt und den magnetooptischen Gradienteneffekt. Fur bestimmte Orientierungen der Wand relativ zur Polarisationsrichtung des Lichts verschwinden aber die resultierenden Beitrage beider Effekte; eine 180°-Wand, die paralle zum E-Vektor des Lichts ausgerichtet ist, ist ein Beispiel. Die dann noch sichtbaren[Restkontraste]sind das Thema dieser Arbeit. Der Restkontrast hat einen dipolaren Charakter. Er kann auf den Voigteffekt und den Gradienteneffekt der Wand selber zuruckgefuhrt werden. Beide Beitrage uberlagern sich und interferieren konstruktiv oder destruktiv je nach Domanenmagnetisierung.

M. Rührig

Papers

Domain Observations on FeCrFe Layered Structnres. Evidence for a Biquadratic Coupling Effect

Electron beam fabrication and characterization of high- resolution magnetic force microscopy tips

Wall Transitions and Coupling of Magnetization in NiFe/C/NiFe Double Films

Loss optimization for iron-rich metallic glasses

Magnetooptical Domain Wall Observation at Perpendicular Incidence