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Stripe domains in thin films form through a complex competition of perpendicular anisotropy and demagnetizing energy and are still lacking a complete micromagnetic description, despite being investigated since 50 years. This work elucidates the formation of stripe domains with a special focus on the dependence of stripe domain width on film thickness with varying ratio of the two major energy contributions. An overview and review of the most established analytical models for the calculation of this dependency is given with respect to existing experimental data as well as to new experimental data on epitaxial Fe-Co-C films with perpendicular anisotropy. Since the analytical models are limited in their predictive power and compatibility in terms of the suitable material parameter range, an effcient but rigorous micromagnetic simulation method was developed, which proved to be comparable or better than previous models in describing experimental finding, especially for films with strongly dominating demagnetizing. Comprehensive simulations where performed to determine thickness dependent stripe width for various material parameters which can serve as a benchmark for analytical theories or can be used directly for comparison with experimental results. At a given combination of exchange constant and saturation polarization there exists a specifc thickness at which the stripe width is independent of the uniaxial anisotropy.

Epitaxial Mn–Ge films with varying composition have been prepared on Ir/Cr buffered MgO(0 0 1) substrates using DC magnetron sputtering. The effect of composition on phase formation, texture and magnetic properties of Mn–Ge films has been investigated. These films grow epitaxially on Ir/Cr buffered MgO(0 0 1) with a tetragonal D022 type structure. From the pole figure analysis the epitaxial relationship is determined to be: D022 Mn–Ge [1 0 0] (0 0 1)||Ir [1 0 0] (0 0 1)||Cr [1 1 0] (0 0 1)||MgO [1 0 0] (0 0 1). Mn–Ge films close to stoichiometric composition (Mn77.5Ge22.5) exhibit perpendicular magnetic anisotropy with crystallographic c-axis being the easy axis of magnetization. The room temperature measured values of coercivity µ 0 H c, saturation magnetization M S and anisotropy field µ 0 H A for Mn77.5Ge22.5 are 2.86 T, 90 kA m−1 and 9.6 T, respectively. Mn–Ge films with low and high Mn concentration possess high coercivity but extremely low magnetization and that can be ascribed to the presence of secondary phases of non-magnetic/low magnetic nature, possible interdiffusion, and especially partial substitution of excess Mn atoms into the Ge site.

Relations between discrete quantities such as people, genes, or streets can be described by networks, which consist of nodes that are connected by edges. Network analysis aims to identify important nodes in a network and to uncover structural properties of a network. A network is said to be bipartite if its nodes can be subdivided into two nonempty sets such that there are no edges between nodes in the same set. It is a difficult task to determine the closest bipartite network to a given network. This paper describes how a given network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems. The algorithm also produces a node permutation which makes the possible bipartite nature of the initial adjacency matrix evident, and identifies the two sets of nodes. We finally show how the same procedure can be used to detect the presence of a large anti-community in a network and to identify it., Comment: 30 pages

Tetragonally strained interstitial Fe-Co-B alloys were synthesized as epitaxial films grown on a 20 nm thick Au0.55Cu0.45 buffer layer. Different ratios of the perpendicular to in-plane lattice constant c/a = 1.013, 1.034 and 1.02 were stabilized by adding interstitial boron with different concentrations 0, 4, and 10 at.%, respectively. Using ferromagnetic resonance (FMR) and x-ray magnetic circular dichroism (XMCD) we found that the total orbital magnetic moment significantly increases with increasing c/a ratio, indicating that reduced crystal symmetry and interstitial B leads to a noticeable enhancement of the effect of spin-orbit coupling (SOC) in the Fe-Co-B alloys. First-principles calculations reveal that the increase in orbital magnetic moment mainly originates from B impurities in octahedral position and the reduced symmetry around B atoms. These findings offer the possibility to enhance SOC phenomena - namely the magnetocrystalline anisotropy and the orbital moment - by stabilizing anisotropic strain by doping 4 at.% B. Results on the influence of B doping on the Fe-Co film microstructure, their coercive field and magnetic relaxation are also presented.

The objective of the rare-earth free permanent magnets (REFREEPM) project is to develop a new generation of high-performance permanent magnets (PMs) without rare earths. Our approach is based on modeling using a combinatorial approach together with micromagnetic modeling and the realization of the modeled systems (I) by using a novel production of high-aspect-ratio (> 5) nanostructrures (nanowires, nanorods, and nanoflakes) by exploiting the magnetic shape anisotropy of the constituents that can be produced via chemical nanosynthesis polyol process or electrodeposition, which can be consolidated with novel processes for a new generation of rare-earth free PMs with energy product in the range of 60 kJ/m(3) < (BH)max < 160 kJ/m(3) at room temperature, and (II) by using a high-throughput thin-film synthesis and high-throughput characterization approach to identify promising candidate materials that can be stabilized in a tetragonal or hexagonal structure by epitaxial growth on selected substrates, under various conditions of pressure, stoichiometry, and temperature. In this article, we report the progress so far in selected phases.

In order to convert the well-known Fe-Co-B alloy from a soft to a hard magnet, we propose tetragonal strain by interstitial boron. Density functional theory reveals that when B atoms occupy octahedral interstitial sites, the bcc Fe-Co lattice is strained spontaneously. Such highly distorted Fe-Co is predicted to reach a strong magnetocrystalline anisotropy which may compete with shape anisotropy. To probe this theoretical suggestion experimentally, epitaxial films are examined. A spontaneous strain up to 5% lattice distortion is obtained for B content up to 4 at%, which leads to uniaxial anisotropy constants exceeding 0.5 MJ m(-3). However, a further addition of B results in a partial amorphisation, which degrades both anisotropy and magnetisation.

We describe a novel method for reducing a pair of large matrices {A;B} to a pair of small matrices {H;K}. The method is an extension of Golub-Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix A represents a discretization of a compact integral operator and B is a regularization matrix. Keywords: Generalized Golub-Kahan bidiagonalization, eneralized Lanczos bidiagonalization, generalized Krylov method, matrix pair decomposition, ill-posed problem, Tikhonov regularization, multi-parameter regularization.

The straightforward solution of discrete ill-posed linear systems of equations or least-squares problems with error contaminated data does not, in general, give meaningful results, because the propagated error destroys the computed solution. The problems have to be modified to reduce their sensitivity to the error in the data. The amount of modification is determined by a regularization parameter. It can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available. This paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation. It is based on comparing computed solutions determined by Tikhonov regularization and truncated singular value decomposition. Analogous comparisons are proposed for large-scale problems. The technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data. Keywords: Ill-posed problem; Regularization; Noise level estimation; TSVD; Tikhonov regularization; Heuristic parameter choice rule

The straightforward solution of discrete ill-posed linear systems of equations or least-squares problems with error contaminated data does not, in general, give meaningful results, because the propagated error destroys the computed solution. The problems have to be modified to reduce their sensitivity to the error in the data. The amount of modification is determined by a regularization parameter. It can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available. This paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation. It is based on comparing computed solutions determined by Tikhonov regularization and truncated singular value decomposition. Analogous comparisons are proposed for large-scale problems. The technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data. Keywords: Ill-posed problem; Regularization; Noise level estimation; TSVD; Tikhonov regularization; Heuristic parameter choice rule

We describe a novel method for reducing a pair of large matrices {A;B} to a pair of small matrices {H;K}. The method is an extension of Golub-Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix A represents a discretization of a compact integral operator and B is a regularization matrix. Keywords: Generalized Golub-Kahan bidiagonalization, eneralized Lanczos bidiagonalization, generalized Krylov method, matrix pair decomposition, ill-posed problem, Tikhonov regularization, multi-parameter regularization.

The objective of the rare-earth free permanent magnets (REFREEPM) project is to develop a new generation of high-performance permanent magnets (PMs) without rare earths. Our approach is based on modeling using a combinatorial approach together with micromagnetic modeling and the realization of the modeled systems (I) by using a novel production of high-aspect-ratio (>5) nanostructrures (nanowires, nanorods, and nanoflakes) by exploiting the magnetic shape anisotropy of the constituents that can be produced via chemical nanosynthesis polyol process or electrodeposition, which can be consolidated with novel processes for a new generation of rare-earth free PMs with energy product in the range of 60 kJ/m3 < (BH)max < 160 kJ/m3 at room temperature, and (II) by using a high-throughput thin-film synthesis and high-throughput characterization approach to identify promising candidate materials that can be stabilized in a tetragonal or hexagonal structure by epitaxial growth on selected substrates, under various conditions of pressure, stoichiometry, and temperature. In this article, we report the progress so far in selected phases.

Rare earth free alloys are in focus of permanent magnet research since the accessibility of the elements needed for nowadays conventional magnets is limited. Tetragonally strained iron-cobalt (Fe-Co) has attracted large interest as promising candidate due to theoretical calculations. In experiments, however, the applied strain quickly relaxes with increasing film thickness and hampers stabilization of a strong magnetocrys- talline anisotropy. In our study we show that already 2 at% of carbon substantially reduce the lattice relaxation leading to the formation of a spontaneously strained phase with 3% tetragonal distortion. In these strained (Fe0.4Co0.6)0.98C0.02 films, a magnetocrystalline anisotropy above 0.4 MJ/m3 is observed while the large polarization of 2.1 T is maintained. Compared to binary Fe-Co this is a remarkable improve- ment of the intrinsic magnetic properties. In this paper, we relate our experimental work to theoretical studies of strained Fe-Co-C and find a very good agreement.

Straightforward solution of discrete ill-posed linear systems of equations or least-squares problems with error contaminated data does not, in general, give meaningful results, because the propagated error destroys the computed solution. The problems have to be modified to reduce their sensitivity to the error in the data. The amount of modification is determined by a regularization parameter. It can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available. This paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation. It is based on comparing computed solutions determined by Tikhonov regularization and truncated singular value decomposition. Analogous comparisons are proposed for large-scale problems. The technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data. Keywords: Ill-posed problem; Regularization; Noise level estimation; TSVD; Tikhonov regularization; Heuristic parameter choice rule

Smallest variations of the lattice parameter result in significant changes in material properties. Whereas in bulk, lattice parameters can only be changed by composition or temperature, coherent epitaxial growth of thin films on single crystals allows adjusting the lattice parameters independently. Up to now only discrete values were accessible by using different buffer or substrate materials. We realize a lateral variation of in-plane lattice parameters using combinatorial film deposition of epitaxial Cu-Au on a 4-in. Si wafer. This template gives the possibility to adjust the in-plane lattice parameter over a wide range from 0.365 nm up to 0.382 nm.

Straightforward solution of discrete ill-posed linear systems of equations or least-squares problems with error contaminated data does not, in general, give meaningful results, because the propagated error destroys the computed solution. The problems have to be modified to reduce their sensitivity to the error in the data. The amount of modification is determined by a regularization parameter. It can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available. This paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation. It is based on comparing computed solutions determined by Tikhonov regularization and truncated singular value decomposition. Analogous comparisons are proposed for large-scale problems. The technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data. Keywords: Ill-posed problem; Regularization; Noise level estimation; TSVD; Tikhonov regularization; Heuristic parameter choice rule

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices A^TA (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small and large-scale problems are considered. Keywords: Ill-posed problem; fractional Tikhonov regularization; fractional Lavrentiev regularization; fractional power regularization matrix

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices A^TA (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small and large-scale problems are considered. Keywords: Ill-posed problem; fractional Tikhonov regularization; fractional Lavrentiev regularization; fractional power regularization matrix

Straightforward solution of discrete ill-posed least-squares problems with error-contaminated data does not, in general, give meaningful results, because propagated error destroys the computed solution. Error propagation can be reduced by imposing constraints on the computed solution. A commonly used constraint is the discrepancy principle, which bounds the norm of the computed solution when applied in conjunction with Tikhonov regularization. Another approach, which recently has received considerable attention, is to explicitly impose a constraint on the norm of the computed solution. For instance, the computed solution may be required to have the same Euclidean norm as the unknown solution of the error-free least-squares problem. We compare these approaches and discuss numerical methods for their implementation, among them a new implementation of the Arnoldi-Tikhonov method. Also solution methods which use both the discrepancy principle and a solution norm constraint are considered. Key words: Ill-posed problem, regularization, solution norm constraint, Arnoldi-Tikhonov, discrepancy principle

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of the Moore-Penrose pseudoinverse of AAT as weighting matrix. Properties of this regularization method are discussed. Numerical examples illustrate that the proposed scheme for a suitable fractional power may give approximate solutions of higher quality than standard Tikhonov regularization.

Linear discrete ill-posed problems of small to medium size are commonly solved by first computing the singular value decomposition of the matrix and then determining an approximate solution by one of several available numerical methods, such as the truncated singular value decomposition or Tikhonov regularization. The determination of an approximate solution is relatively inexpensive once the singular value decomposition is available. This paper proposes to compute several approximate solutions by standard methods and then extract a new candidate solution from the linear subspace spanned by the available approximate solutions. We also describe how the method may be used for large-scale problems. Key words: Ill-posed problem, linear combination, solution norm constraint, TSVD, Tikhonov regularization, discrepancy principle.

Straightforward solution of discrete ill-posed least-squares problems with error-contaminated data does not, in general, give meaningful results, because propagated error destroys the computed solution. Error propagation can be reduced by imposing constraints on the computed solution. A commonly used constraint is the discrepancy principle, which bounds the norm of the computed solution when applied in conjunction with Tikhonov regularization. Another approach, which recently has received considerable attention, is to explicitly impose a constraint on the norm of the computed solution. For instance, the computed solution may be required to have the same Euclidean norm as the unknown solution of the error-free least-squares problem. We compare these approaches and discuss numerical methods for their implementation, among them a new implementation of the Arnoldi-Tikhonov method. Also solution methods which use both the discrepancy principle and a solution norm constraint are considered. Key words: Ill-posed problem, regularization, solution norm constraint, Arnoldi-Tikhonov, discrepancy principle

Linear discrete ill-posed problems of small to medium size are commonly solved by first computing the singular value decomposition of the matrix and then determining an approximate solution by one of several available numerical methods, such as the truncated singular value decomposition or Tikhonov regularization. The determination of an approximate solution is relatively inexpensive once the singular value decomposition is available. This paper proposes to compute several approximate solutions by standard methods and then extract a new candidate solution from the linear subspace spanned by the available approximate solutions. We also describe how the method may be used for large-scale problems. Key words: Ill-posed problem, linear combination, solution norm constraint, TSVD, Tikhonov regularization, discrepancy principle.

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of the Moore-Penrose pseudoinverse of AAT as weighting matrix. Properties of this regularization method are discussed. Numerical examples illustrate that the proposed scheme for a suitable fractional power may give approximate solutions of higher quality than standard Tikhonov regularization.

The truncated singular value decomposition is a popular solution method for linear discrete ill-posed problems. These problems are numerically underdetermined. Therefore, it can be beneficial to incorporate information about the desired solution into the solution process. This paper describes a modification of the singular value decomposition that permits a specified linear subspace to be contained in the solution subspace for all truncations. Modifications that allow the range to contain a specified subspace, or that allow both the solution subspace and the range to contain specified subspaces also are described.

Tikhonov regularization is one of the most popular approaches to solve discrete ill-posed problems with error-contaminated data. A regularization operator and a suitable value of a regularization parameter have to be chosen. This paper describes an iterative method, based on Golub-Kahan bidiagonalization, for solving large-scale Tikhonov minimization problems with a linear regularization operator of general form. The regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method. Key words. Discrete ill-posed problem, iterative method, Tikhonov regularization, general linear regularization operator, discrepancy principle.

The truncated singular value decomposition is a popular solution method for linear discrete ill-posed problems. These problems are numerically underdetermined. Therefore, it can be beneficial to incorporate information about the desired solution into the solution process. This paper describes a modification of the singular value decomposition that permits a specified linear subspace to be contained in the solution subspace for all truncations. Modifications that allow the range to contain a specified subspace, or that allow both the solution subspace and the range to contain specified subspaces also are described.

Tikhonov regularization is one of the most popular approaches to solve discrete ill-posed problems with error-contaminated data. A regularization operator and a suitable value of a regularization parameter have to be chosen. This paper describes an iterative method, based on Golub-Kahan bidiagonalization, for solving large-scale Tikhonov minimization problems with a linear regularization operator of general form. The regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method. Key words. Discrete ill-posed problem, iterative method, Tikhonov regularization, general linear regularization operator, discrepancy principle.

Due to its low coercivity and negligible magnetostriction, permalloy (Ni80Fe20) is one of the most used materials in thin film and micro/nano magnetism. By means of ion irradiation the magnetic properties can be modified [1], and in combination with a lithographically defined mask or the use of a focused ion beam magnetic patterning can be achieved [2]. The changes of the magnetic properties due to ion-solid- interaction must be related to different origins, e.g. direct implantation, surface sputtering and interfacial mixing. Their respective influence depends strongly on the chosen multilayer system as well as on the implantation conditions. Here we present a systematic study of irradiation of permalloy with common ion species. Special emphasis is put on the separation of the effect of direct implantation from mixing and sputtering. By transferring this knowledge to laterally resolved irradiation, direct exchange coupled magnetic stripes of submicron width are created. The magnetization reversal process of this lateral exchange spring structures depends on the interaction between adjacent soft and hard magnetic stripes. By scaling the stripe sizes down, fundamental questions regarding the maximum domain wall density and the domain–wall interaction may be addressed. References [1] J. Fassbender et al., Physical Review B 73, 184410 (2006). [2] J. McCord et al., Advanced Materials 20, 2090 (2008).

Due to its low coercivity and negligible magnetostriction, permalloy (Ni80Fe20) is one of the most used materials in thin film and micro/nano magnetism. By means of ion irradiation the magnetic properties can be modified [1], and in combination with a lithographically defined mask or the use of a focused ion beam magnetic patterning can be achieved [2]. The changes of the magnetic properties due to ion-solid- interaction must be related to different origins, e.g. direct implantation, surface sputtering and interfacial mixing. Their respective influence depends strongly on the chosen multilayer system as well as on the implantation conditions. Here we present a systematic study of irradiation of permalloy with common ion species. Special emphasis is put on the separation of the effect of direct implantation from mixing and sputtering. By transferring this knowledge to laterally resolved irradiation, direct exchange coupled magnetic stripes of submicron width are created. The magnetization reversal process of this lateral exchange spring structures depends on the interaction between adjacent soft and hard magnetic stripes. By scaling the stripe sizes down, fundamental questions regarding the maximum domain wall density and the domain–wall interaction may be addressed. References [1] J. Fassbender et al., Physical Review B 73, 184410 (2006). [2] J. McCord et al., Advanced Materials 20, 2090 (2008).

We present FORTRAN subroutines that update the QR decomposition in a numerically stable manner when A is modified by a matrix of rank one, or when a row or a column is inserted or deleted. These subroutines are modifications of the Algol procedures in Daniel et al. 5. We also present a subroutine that the elements in the lower right corner of R will generally be small if the columns of A are nearly linearly dependent. This subroutine is an implementation of the rank revealing QR decomposition scheme recently proposed by Chan (3). The subroutines have been written to perform well on a vector computer. Algorithms Additional Key Words and Phrases: QR decomposition, updating, subset selection. Computer programs.

Let H epsilon C be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrix H is split into two smaller unitary right Hessenberg matrices H1 and H2 by a rank-one modification of H. The eigenproblems for H1 and H2 can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues of H. The eigenvectors of H can be determined from the eigenvalues of H and the eigenvectors of H1 and H2. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer. When H epsilon R sub nxn is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paper.

The approximation of analytic or harmonic functions on plane finitely connected domains with smooth boundary curves is considered, and several easily computable families of near-orthonormal functions are described. The functions can be used in fast Galerkin schemes for numerical approximation. Several numerical examples are presented. (Author)

A collocation method for evaluating integrals of rapidly oscillating functions is analyzed. The method is based on the approximation of the antiderivative of the integrand as the solution of an ordinary differential equation via polynomial collocation. Numerical examples are presented. (Author)

The aim of this paper is to describe a computational method for determining traveling edge waves along a straight infinite beach with a cylindrical, not necessarily plan bottom. The determination of a periodic edge waves along a straight infinite beach can be formulated as a 2-dimensional eigenvalue problem for a generalized wedge. A least squares boundary collocation method for computing the lower discrete eigenvalues is described, with an error estimate for the least eigenvalue. Numerical examples are presented.