Although topology was recognized by Gauss and Maxwell to play a pivotal role in the formulation of electromagnetic boundary value problems, it is a largely unexploited tool for field computation. The development of algebraic topology since Maxwell provides a framework for linking data structures, algorithms, and computation to topological aspects of three-dimensional electromagnetic boundary value problems. This book, first published in 2004, attempts to expose the link between Maxwell and a modern approach to algorithms. The first chapters lay out the relevant facts about homology and cohomology, stressing their interpretations in electromagnetism. These topological structures are subsequently tied to variational formulations in electromagnetics, the finite element method, algorithms, and certain aspects of numerical linear algebra. A recurring theme is the formulation of and algorithms for the problem of making branch cuts for computing magnetic scalar potentials and eddy currents.
Direct integration of the Landau–Lifshitz–Gilbert equation in a three-dimensional rectangular lattice shows that fabrication asymmetry of tapered ends affects the magnetic switching behavior of the magnetic layers of a pseudospin valve (PSV) memory cell in word disturb condition. When a 10 nm asymmetry is introduced with the longer sides of the tapers on the same side of the PSV memory cell, a “360°” wall in the storage layer is formed when a 40 Oe uniform word disturb field is applied opposite to the magnetization direction in the storage layer. The initial magnetization state is recovered when the field is turned off. When the longer sides of the tapers are on opposite sides of the cell, complete magnetization reversal occurs in the storage layer. The sense layer and the storage layer relax to an antiparallel configuration magnetization that is opposite to the initial configuration in the memory cell when the field is turned off. The information is lost. Successful operation of the memory depends upon a fabrication accuracy that is better than 10 nm.
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