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

The observation of magnetic domains by magneto-optical microscopy, based on the Kerr and the Faraday effect, is one of the most prominent techniques for the visualization of distributions of magnetization within magnetic materials. The method has gained increased attention due to the possibility to visualize field and current induced phenomena in nanostructured magnetic materials on fast time-scales. Fundamental concepts and recent advances in methodology are discussed in order to provide guidance on the usage of wide-field magneto-optical microscopy in applied magnetism. Recent applications of magneto-optical microscopy in bulk and thin film materials are reviewed at the end.

https://iopscience.iop.org/article/10.1088/0022-3727/48/33/333001/pdf

Progress in magnetic domain observation by advanced magneto-optical microscopy

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

By means of light scattering from spin waves and magneto-optical Kerr measurements we investigated the exchange interaction of two Fe films across a Cr film as a function of the Cr thickness dCr. Using good-quality epitaxial samples with a special wedge geometry of the interspacer we observed up to four full periods of the long-range oscillations of the exchange, including its ferromagnetic parts. The decay of the oscillations does not follow the dCr-2-dependence predicted by classical two-dimensional RKKY-type theory. A fine structure in the first antiferromagnetic region is also observed.

Evidence for Oscillations in the Interlayer Coupling of Fe Films Across Cr Films from Spin Waves and M(H) Curves

Since the discovery of antiferromagnetic (AF) type interlayer coupling in 1986 numerous systems have been investigated. Generally, it is found that the coupling of ferromagnetic films separated by an interlayer oscillates as a function of the interlayer thickness between ferro (F)- and AF-type and the transition zones are characterized by 90°-type coupling. The oscillation periods are in agreement with a theory that attributes the microscopic origin of the coupling to an indirect exchange of the ferromagnetic layers via the electrons of the interlayer. For the 90°-type coupling various mechanisms have been proposed but the situation is still unclear. We would like to present here a condensed overview of the field and illustrate the development with some own results.

J. A. Wolf

Papers

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

Evidence for Oscillations in the Interlayer Coupling of Fe Films Across Cr Films from Spin Waves and M(H) Curves

Long-range exchange interactions in epitaxial layered magnetic structures