A convergent finite element approximation for Landau–Lifschitz–Gilbert equation

Shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy (XMCD-PEEM) is a recent technique, in which the photon intensity in the shadow of an object lying on a surface, may be used to gather information about the three-dimensional magnetization texture inside the object. Our purpose here is to lay the basis of a quantitative analysis of this technique. We first discuss the principle and implementation of a method to simulate the contrast expected from an arbitrary micromagnetic state. Text book examples and successful comparison with experiments are then given. Instrumental settings are finally discussed, having an impact on the contrast and spatial resolution : photon energy, microscope extraction voltage and plane of focus, microscope background level, electric-field related distortion of three-dimensional objects, Fresnel diffraction or photon scattering.Progress is continuous in the decreasing size and increasing complexity of nanosized magnetic systems being designed for either fundamental science or devices. Magnetic microscopies are crucial tools to monitor and understand the properties of such systems. Various types of information are desirable to gather, leading to multiple criteria to classify microscopies: spatial and time resolution, compatibility with environmental parameters such as variable temperature and applied magnetic field, requirements on the sample preparation and compatibility for ex-situ processing such as lithography, correlation with structural information, elemental sensitivity, quantity measured (magnetization, induction, stray field etc.), sensitivity. The most common magnetic microscopies offering spatial resolution below 50 nm and direct sensitivity to magnetization are X-ray Magnetic Circular Dichroism PhotoEmission Electron Microscopy (XMCD-PEEM) [ [7][8][9].Yet another criterion is the volume of the sample probed. This criterium is gaining in importance in the context of the emergence of three-dimensional (3D) magnetic objects and textures. The distribution of magnetization may be truly 3D if along the three directions in space the size of a system lies above magnetic characteristic length scales, such as the dipolar exchange length ∆ d for soft magnetic materials, or the anisotropy exchange length ∆ u for a hard magnetic material [10]. While this is obviously fulfilled in macroscopic materials, the complexity of magnetic textures is such that it cannot be measured in detail, and besides it cannot be controlled to achieve specific functions. The progress in nanofabrication techniques now allows to design suitable systems, both with top-down and bottom-up approaches. Let us give some examples. Flat magnetic elements are basic building blocks in spintronics, being patterned with great 2D versatility with thin film and lithography technologies. Large lateral dimensions may give rise to 2D magnetic textures, such as the so-called vortex state in disks [11]. Such textures are being investigated to design RF oscillator components, relying on the gyrotropic motion ...

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“…For the micromagnetic configurations, we use the home-made code FeeLLGood [39]. FeeLLGood is based on the finite element method.…”

Section: B Numerical Implementationmentioning

confidence: 99%

Quantitative analysis of shadow x-ray magnetic circular dichroism photoemission electron microscopy

et al. 2015

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“…Micromagnetic simulations were performed using feellgood, a home-built code based on the temporal integration of the Landau-Lifshitz-Gilbert equation in a finite element scheme, i.e. using tetrahedra to discretize the nanowires [21]. Only exchange and magnetostatic interactions were taken into account, to deal with the present case of magnetically-soft wires.…”

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confidence: 99%

Observation of Bloch-point domain walls in cylindrical magnetic nanowires

Col

Jamet²,

Rougemaille³

et al. 2014

Phys. Rev. B

168

151

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Topological protection is an elegant way of warranting the integrity of quantum and nanosized systems. In magnetism one example is the Bloch-point, a peculiar object implying the local vanishing of magnetization within a ferromagnet. Its existence had been postulated and described theoretically since several decades, however it has never been observed. We confirm experimentally the existence of Bloch points, imaged within domain walls in cylindrical magnetic nanowires, combining surface and transmission XMCD-PEEM magnetic microscopy. This opens the way to the experimental search for peculiar phenomena predicted during the motion of Bloch-point-based domain walls.There is a rising interest for physical systems providing topological protection. The interest is both fundamental to elucidate the underlying physical phenomena, and applied as a mean to provide robustness to a state against external perturbations and decoherence pathways. For example, the peculiar topology of the band structure of carbon nanotubes and graphene forbids backscattering of charge carriers [1], an effect which is often invoked to explain the high mobilities up to room temperature [2]. A similar effect occurs at the surface of so-called topological insulators, together with a locking of the spin of charge carriers; this provides spin currents protected against depolarization [3]. A photonic analogue was also designed by combining helical wave guides on a lattice with a graphene-like honeycomb topology, removing time-reversal symmetry and thereby preventing backscattering of light [4].In systems displaying a directional order parameter such as liquid crystals and ferromagnets, interesting phenomena are associated with the slowly-varying texture of the order field (magnetization for a ferromagnet). The requirement of local continuity of a vector field with fixed magnitude provides a topological protection against the transformation of the texture. A prototypical case in magnetism is skyrmions, which are essentially local two-dimensional chiral spin textures stabilized by the Dzyaloshinskii-Moriya interaction, embedded in an otherwise uniformly-magnetized surrounding. Despite these surroundings skyrmions cannot unwind continuously as explained by the above continuity constraints of the magnetization field, explaining their topological protection. Skyrmions have first been predicted theoretically [5], then confirmed experimentally in both bulk [6] and thin film forms [7].Bloch points are yet another type of topologicallyprotected magnetic texture which cannot be unwound, however of a three-dimensional nature. Bloch points are such that given the distribution of magnetization set on a closed surface like a sphere, the enclosed volume cannot be mapped with a continuous magnetization field of finite magnitude. This occurs e.g. for hedge-hog configurations, or more generally whenever all directions of magnetization are mapped on the closed surface (Fig. 1a-b). Such boundary conditions imply the local cancellation of the modulus of magnetization on a...

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“…For discretization of the LLG equation, we follow the approach of Alouges (2008), which has been generalized in Alouges et al (2011), Goldenits et al (2011), Bruckner et al (2012 and Goldenits (2012). The main idea is to introduce a new free variable v ≈ m t and to interpret LLG as a linear equation in v. This ansatz exploits the formulation…”

Section: Algorithmmentioning

confidence: 99%

“…For a review of the analysis of LLG, we refer to Kružík & Prohl (2006), García-Cervera (2007) and Cimrak (2008) or the monographs (Hubert & Schäfer, 1998;Prohl, 2001) and the references therein. As far as the numerical analysis is concerned, mathematically reliable and convergent LLG integrators are found in Bartels & Prohl (2006), Alouges (2008), Baňas et al (2008), Bartels et al (2008), Alouges et al (2011), Goldenits et al (2011), Bruckner et al (2012), Goldenits et al (2012), Le & Tran (2012), Rochat (2012) and Baňas et al (2013). Of particular interest are unconditionally convergent integrators, which do not impose a coupling of spatial mesh size h and timestep size k to ensure stability of the numerical scheme.…”

Section: Introductionmentioning

confidence: 99%

“…Of particular interest are unconditionally convergent integrators, which do not impose a coupling of spatial mesh size h and timestep size k to ensure stability of the numerical scheme. Those integrators are split into two groups; first, midpoint-scheme-based integrators (Baňas et al, 2008;Rochat, 2012), which rely on the seminal work (Bartels & Prohl, 2006); secondly, projection-based first-order integrators (Alouges et al, 2011;Goldenits et al, 2011;Bruckner et al, 2012;Goldenits et al, 2012;Le & Tran, 2012;Baňas et al, 2013), which build on the seminal work (Alouges, 2008) and are the basis for our scheme. All of the above integrators have in common that they allow for constructive existence proofs of the corresponding problems.…”

Section: Introductionmentioning

confidence: 99%

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A decoupled and unconditionally convergent linear FEM integrator for the Landau-Lifshitz-Gilbert equation with magnetostriction

Baňas

Page

Praetorius

et al. 2013

IMA Journal of Numerical Analysis

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To describe and simulate dynamic micromagnetic phenomena, we consider a coupled system of the nonlinear Landau-Lifshitz-Gilbert equation and the conservation of momentum equation. This coupling allows one to include magnetostrictive effects into the simulations. Existence of weak solutions has recently been shown in Carbou et al. (2011) (Global weak solutions for the Landau-Lifschitz equation with magnetostriction. Math. Meth. Appl. Sci., 34, 1274-1288. In our contribution, we give an alternate proof which additionally provides an effective numerical integrator. The latter is based on linear finite elements (FEs) in space and a linear-implicit Euler time-stepping. Despite the nonlinearity, only two linear systems have to be solved per timestep, and the integrator fully decouples both equations. Finally, we prove unconditional convergence-at least of a subsequence-towards, and hence existence of, a weak solution of the coupled system, as timestep size and spatial mesh size tend to zero. We conclude the work with numerical experiments, which study the discrete blow-up of the LLG equation as well as the influence of the magnetostrictive term on the discrete blow-up.

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A convergent finite element approximation for Landau–Lifschitz–Gilbert equation

Abstract: In this paper, we rigorously study an order 2 scheme that was previously proposed by some of the authors. A slight modification is proposed that enables us to prove the convergence of the scheme while simplifying in the same time the inner iteration.

Cited by 39 publications

References 13 publications

Quantitative analysis of shadow x-ray magnetic circular dichroism photoemission electron microscopy

Quantitative analysis of shadow x-ray magnetic circular dichroism photoemission electron microscopy

Observation of Bloch-point domain walls in cylindrical magnetic nanowires

A decoupled and unconditionally convergent linear FEM integrator for the Landau-Lifshitz-Gilbert equation with magnetostriction

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