List of Figures

• 2.1. Kuhn triangulation of a cube into six tetrahedral finite elements (exploded view) [16].
• 2.2. Hat function (linear basis function) for a triangulation in 2D.
• 3.1. Local coordinate system and various vectors required for the discretization of the boundary integral Eq. (3.47) [43].
• 5.1. Structure of the PETSc library.
• 5.2. Structure of additional libraries.
• 5.3. Coarse flow chart of the program.
• 5.4. Flow chart of the single processor initialization section.
• 5.5. Mesh partitioning of a soft magnetic nanodot for two, three, four, and ten processors, respectively. Different colors correspond to parts which are assigned to different processors.
• 5.6. Flow chart of the parallel initialization section.
• 5.7. Matrix-vector multiplication with matrix and vector elements distributed over four processors.
• 5.8. Sparsity pattern of the stiffness matrix of the nanodot model for a single processor (left) and distributed over two processors (right).
• 5.9. Solid angle of a trihedral angle made up by three faces of a tetrahedron.
• 5.10. Flow chart of the solution loop.
• 5.11. Flow chart of the final clean up section.
• 6.1. Global refinement of a ``parent'' tetrahedral finite element (thick lines) into eight ``children'' [16].
• 6.2. Barycentric coordinates in a regular tetrahedron.
• 6.3. Sparsity pattern of the interpolation matrix for graphics output.
• 6.4. RGB color intensities for data encoding and resulting color map.
• 6.5. Speedup of initialization, solution, and total execution time of the parallel energy minimization algorithm (TAO) on an AlphaServer.
• 6.6. Speedup of initialization, solution, and total execution time of the parallel time integration (PVODE) on an AlphaServer.
• 6.7. Speedup of initialization, solution, and total execution time of the parallel time integration (PVODE) on a Beowulf type AMD cluster.
• 7.1. TEM micrograph of the microstructure of a Sm(Co,Fe,Cu,Zr)$_z$ precipitation hardened magnet. The lamella phase perpendicular to the $[0001]$ direction of the cell matrix phase gives strong contrast in this image [89].
• 7.2. Bright field TEM micrograph of the cellular precipitation structure of a Sm(Co,Fe,Cu,Zr)$_z$ magnet. The Sm$_2$(Co,Fe)$_{17}$ cells are surrounded by Sm(Co,Cu)$_5$ cell boundaries [89].
• 7.3. Lorentz electron micrograph of a Sm(Co,Fe,Cu,Zr)$_z$ magnet. The magnetic domain wall between two domains with opposite magnetization (bright and dark cells) is pinned on the precipitation structure [87].
• 7.4. Model geometry for domain wall pinning on a perfectly planar interface with parallel anisotropy axes. The chain of arrows indicates the magnetization distribution of a pinned domain wall.
• . Dependence of the pinning field $H_\mathrm{pin}$ (in units of the anisotropy field of material II) on the ratio of exchange ($\varepsilon _A$) and anisotropy constants ($\varepsilon _K$) for given $A^\mathrm{II}$ and $K^\mathrm{II}_1$ according to a 1D analytical model [91]. The thick curve represents $\varepsilon _A \cdot \varepsilon _K = 1$, where the two materials have equal domain wall energy and $H_\mathrm{pin}=0$.
• 7.6. Model geometry for domain wall pinning on an intercellular phase with parallel anisotropy axes. The chain of arrows indicates the magnetization distribution of a pinned domain wall.
• 7.7. Pinning field for attractive and repulsive pinning as a function of the thickness $t$ of the intercellular phase. The thickness is given in units of the exchange length of the intercellular phase. The pinning field is given in units of the pinning field for infinite $t$.
• 7.8. Model geometry for domain wall pinning on a coherent precipitation structure (with parallel anisotropy axes). The shaded areas indicate the faces of the cells, where the magnetic domain wall gets pinned.
• 7.9. Pinning field for attractive and repulsive pinning of a magnetic domain wall on the cell structure as a function of the relative thickness $t/T$. The data marked ``area x2'' and ``area x3'' have been obtained with a model scaled to twice and three times the initial size. The dashed line is just a guide to the eye.
• 7.10. Domain wall energy as a function of the average edge length of the finite element model. The average edge length is given in units of the exchange length. The solid line indicates the analytical Bloch wall energy $4\sqrt {A K_1}$.
• 7.11. Demagnetization curves for different average edge length of the finite element model. The analytical pinning field has been calculated with the model presented in Sec. 7.3.1.
• 7.12. Pinning fields for different average edge length of the finite element model. As the mesh size increases the domain wall gets artificially pinned on the finite element mesh.
• 7.13. Finite element model of the rhomboidal cell structure of precipitation hardened Sm(Co,Fe,Cu,Zr)$_z$ magnets.
• 7.14. Demagnetization curves for reduced magnetocrystalline anisotropy $K_1$ of the cell boundary phase (values in the legend in MJ/m$^3$) - attractive pinning.
• 7.15. Demagnetization curves for enhanced magnetocrystalline anisotropy $K_1$ of the cell boundary phase (values in the legend in MJ/m$^3$) - repulsive pinning.
• 7.16. Pinning field vs. difference in anisotropy constant between the cells and the cell boundary phase in a 3D model of the rhomboidal SmCo microstructure and comparison with the analytical 1D model of Kronmüller, Goll [91].
• 7.17. Finite element model with $3 \times 3 \times 3$ cells.
• 7.18. Domain wall bending of a magnetic domain wall in the (softer) intercellular phase (attractive pinning).
• 7.19. Demagnetization curves for varying thickness $t$ (values in the legend in nm) of the intercellular phase around large cells with $D=250 \mathrm{nm}$.
• . Magnetization distribution for $D=250 \mathrm{nm}$ and $t=10 \mathrm{nm}$. The green surface indicates the domain wall, which separates the two domains (red and blue areas) with antiparallel magnetization.
• . Magnetization distribution for $D=250 \mathrm{nm}$ and $t=20 \mathrm{nm}$. The green surface indicates the domain wall, which separates the two domains (red and blue areas) with antiparallel magnetization.
• 8.1. TEM image and material properties of FePt [100]
• 8.2. FePt nanoparticle with different easy axes [101] and FE model
• 8.3. Nucleation field of an FePt nanoparticle with uniaxial magnetocrystalline anisotropy as a function of the angle of the applied field with respect to the easy axis. The finite element simulation gives the correct result of a Stoner-Wohlfarth particle. If the demagnetizing field is taken into account, the nucleation field is reduced by less than 5 % due to the dominating high magnetocrystalline anisotropy.
• 8.4. The $z$-component of the demagnetizing field has been measured along the $x$-axis through the center of the nanoparticle as shown in the left image. The result is shown in the graph on the right.
• 8.5. The $z$-component of the demagnetizing field has been measured along the $z$-axis through the center of the nanoparticle as shown in the left image. The result is shown in the graph on the right.
• 8.6. The $z$-component of the demagnetizing field has been measured along a line parallel to the $z$-axis close to an edge of the nanoparticle as shown in the left image. The result is shown in the graph on the right.
• 8.7. Coercivity as a function of the easy axis distribution and edge length of the nanoparticle. The easy axis distribution is shown in the top left figure for the ``3 easy axes'' (where two neighboring parts of the model have the same easy axis) and on the top right for the ``6 easy axes'' (where all neighboring pairs have perpendicular easy axes) distribution, respectively.
• 9.1. MFM image of nanodots with 50 nm thickness and different diameters (0.3 to 1 $\mu$m) [107]. The dark spots in the center of the dots indicate the magnetic vortex core, where the MFM detects the stray field caused by the perpendicular magnetization.
• 9.2. Geometry of a flat cylindrical nanodot.
• 9.3. Magnetization distribution of the vortex state on the coarse mesh. The vortex core cannot be resolved.
• 9.4. Magnetization distribution of the vortex state on the fine mesh. The vortex core is properly resolved.
• 9.5. Magnetization distribution of the vortex state on the adapted mesh. The vortex core is nicely resolved, but the total number of elements and vertices is similar to that of the coarse mesh.
• 9.6. Profile of $M_z$ along the $x$-axis through the center of the nanodot for different meshes.
• 9.7. Isovolume ($M_z < 0$) plot of the vortex core.
• 9.8. Profiles of $M_z$ along the $x$-axis through the center of the nanodot for the analytical model, its finite element approximation, and the relaxed magnetization in equilibrium.
• . Hysteresis curve of a nanodot with $L/R=20 \mathrm{nm}/100 \mathrm{nm}=0.2$ for an in-plane external field. The circles mark the position on the hysteresis curve at which the snapshots in Fig. 9.10 have been taken.
• 9.10. Typical magnetization distributions along the hysteresis loop. The snapshots have been taken at the corresponding position on the hysteresis curve indicated in Fig. 9.9.
• 9.11. Total energy as a function of the external field for both branches (solid line for decreasing field - dashed line for increasing field) of the hysteresis loop.
• 9.12. Exchange and magnetostatic energy and their sum as a function of the external field (for decreasing external field).
• 9.13. Hysteresis curve of a nanodot with a radius of 200 nm and a thickness of 40 nm.
• 9.14. Profiles of $M_z$ along the $y$-axis through the center of the nanodot for a vortex moving in $-y$ direction due to an external field increasing in $x$ direction.
• . Comparison of $\langle M_x \rangle$ as a function of the vortex displacement between the FE simulation and the rigid vortex model. $y=0$ corresponds to a centered vortex, $y=-0.78$ is the maximum shift before vortex annihilation occurs in the FE simulation.
• 9.16. Contour plot of the surface charge density on the nanodot in zero field. Green indicates no surface charges, blue indicates negative surface charges.
• 9.17. Contour plot of the surface charge density on the circumference of the nanodot in zero field.
• 9.18. Contour plot of the surface charge density on the nanodot in an external field. Blue indicates negative charges, green zero charges, and red positive surface charges.
• 9.19. Contour plot of the surface charge density on the circumference of the nanodot in an external field.
• 9.20. Surface charge density on the circumference of the nanodot for different applied fields.
• . Surface charge distribution as a function of the polar angle on the circumference of the nanodot for different vortex shifts (indicated by different $\langle M_x \rangle$).
• 9.22. Surface charge distribution on the circumference of the nanodot as a function of the polar angle. An in-plane external field shifts the vortex (cf. Fig. 9.23) and leads to surface charges on the circumference. The rigid vortex model (gray lines) overestimates the charge density as compared to the FE simulation (black lines).
• . Contour plot of $d=\vert\boldsymbol{M}_\mathrm{FE}-\boldsymbol{M}_\mathrm{rv}\vert$. Blue areas indicate good agreement of the magnetization distribution between the rigid vortex model (black cones) and the FE model (gray cones), red areas indicate larger differences.
• . $H^\mathrm{dem}_z$ and $H^\mathrm{dem}_r$ across the nanodot.
• 9.25. Phase diagram of magnetic ground states of magnetic nanodots.
• 9.26. Equilibrium magnetization distributions in zero field inside the nanodots (cut along the cylinder axis) for dots with aspect ratios $L/R=1$.
• 9.27. Magnetization distributions for dots with aspect ratios $L/R=2$.
• . Oscillation of $\langle M_x \rangle$ as the vortex core precesses towards equilibrium.
• . Oscillation of $\langle M_y \rangle$ as the vortex core precesses towards equilibrium.
• 9.30. Oscillation of the magnetostatic energy.
• . Simulation results for vortex precession in a nanodot with $L/R=20 \mathrm{nm}/100 \mathrm{nm}=0.2$ under applied in-plane field $\mu_0 H_x=0.01 \mathrm{T}$.
• 9.32. Translation mode eigenfrequencies versus aspect ratio $L/R$ for nanodots with $R=100 \mathrm{nm}$.
• 9.33. Energy over time for a dot with $L/R=0.1$ and $\mu_0 H_x=0.01 \mathrm{T}$.
• 9.34. Snapshots of $M_z$ as a function of the radius at different times during one oscillation period. The magnetization fluctuation outside the vortex core is very uniform, while the vortex core remains undisturbed (``rigid'').
• 9.35. $M_z$ as a function of time at different radii. The magnetization is perfectly in phase in all spots, which emphasizes the uniformity of the excitation mode.
• 9.36. Fourier spectrum of $M_z$ (of Fig. 9.35) at different radii.
• 9.37. $M_z$ as a function of time at $(R\vert)$ and $(0\vert R)$.
• 9.38. Fourier spectrum of $M_z$ (of Fig. 9.37) at $(R\vert)$ and $(0\vert R)$.
• 9.39. $M_r$ as a function of the radius at different $z$ positions.
• 9.40. $M_r$ as a function of the radius at different times.
• 9.41. $M_r$ and $M_z$ as a function of time at different radii.
• 9.42. Fourier spectrum of $M_r$ and $M_z$ (of Fig. 9.41) at different radii.
• 9.43. $M_r$ as a function of time at $(R\vert)$ and $(0\vert R)$.
• 9.44. Fourier spectrum of $M_r$ (of Fig. 9.43) at $(R\vert)$ and $(0\vert R)$.
• 9.45. $M_\varphi$ as a function of the radius at different times.
• 9.46. $M_\varphi$ as a function of time at different radii.
• 9.47. Simulation results for a nanodot with $L/R=0.2$ under applied out-of-plane field $\mu_0 H_z=0.005 \mathrm{T}$.
• . Simulation results for a nanodot with $L/R=40 \mathrm{nm}/200 \mathrm{nm}=0.2$.
• 10.1. Shape and finite element mesh of an elliptical nanoparticle.
• 10.2. Snapshot of the magnetization reversal process of a single particle.
• 10.3. Snapshot of the magnetization reversal process of a chain of six elliptical particles.
• 10.4. Demagnetization curves for a chain of elliptical and rectangular particles with parallel and antiparallel initial magnetization.
• 10.5. Snapshot of the magnetization reversal process of a chain of six rectangular particles.
• 10.6. Chain of particles with the initial magnetization parallel to the chain axis.
• 10.7. Snapshot of the relaxation process of a chain of six elliptical and rectangular particles with their initial magnetization parallel to the chain axis.
• 10.8. Equilibrium magnetization distribution in zero field of a chain of six elliptical and rectangular particles with their initial magnetization parallel to the chain axis.
• 10.9. Demagnetization curves for a chain of elliptical particles with (``touch.'' - $50 \times 10$ nm) and without (``isol.'') contact faces with parallel and antiparallel initial magnetization.
• 10.10. Equilibrium magnetization distribution in zero field of a chain of 6 elliptical particles with a contact area of $50 \times 10$ nm.
• 10.11. Demagnetization curves for a chain of elliptical particles with (``touch.'' - $10 \times 10$ nm) and without (``isol.'') contact faces with parallel and antiparallel initial magnetization.
• 10.12. Equilibrium magnetization distribution in zero field of a chain of six elliptical particles with a contact area of $10 \times 10$ nm.
• 10.13. Demagnetization curves for a chain of elliptical particles with $10 \times 10$ nm and $50 \times 10$ nm contact faces with parallel and antiparallel initial magnetization.