This thesis presents the theoretical studies of chiral magnetic structures, which exist or are affected by antisymmetric Dzyaloshinskii-Moriya interactions. The theoretical approach is based on the phenomenological model of ferromagnetic materials lacking inversion symmetry. Equilibrium magnetic states are described as static structures in the micromagnetic low temperature limit with a fixed magnitude of the magnetization. The studies are focused on two cases: (i) magnetization structures that are affected by chiral exchange so that a particular chirality of these structures is selected, and (ii) novel solitonic states that are called chiral Skyrmions and only exist because of the chiral exchange. Vortex states in magnetic nanodisks provide the simplest example of a handed magnetization structure, where effects of the chiral couplings may become noticeable. A chiral exchange here favours one chirality of such a vortex state over the other. This effect can stem from surface-induced or other defect-related chiral Dzyaloshinskii-Moriya exchange. The different chiral versions of the vortex states are shown to display strong dependencies on the materials properties of such nanodisks. Within a micromagnetic model for these effects, numerical calculations of the shape, size, and stability of the vortices in equilibrium as functions of magnetic field and the material and geometrical parameters provide a general analysis of the influence of the broken mirror symmetry caused by the surface/interfaces or structural defect on their properties. The Dzyaloshinskii-Moriya interactions impose differences in the energies and sizes of vortices with different chirality: these couplings can considerably increase sizes of vortices with one sense of rotation and suppress vortices with opposite sense of rotation. Torsions related to lattice defects can cause similar to the surface-induced chiral couplings. A general phenomenological magneto-elastic formulation for this torsional chirality selection is given. It is applied to calculate similar effects on vortex states in magnetic disks with a screw dislocation at their center. In systems with strong chiral exchange the magnetic equilibrium states themselves become chiral twisted structures. The most interesting structures in this context are the two-dimensional solitonic states that are now known as chiral Skyrmions. The properties and stability of multiply twisted states composed of these particle-like units are the subject of the second part of this thesis. These states compete with the well known onedimensionally modulated helical states in non-centrosymmetric magnetic systems. Studies of modulated states in cubic helimagnets have shown, that in absence of additional effects, the only thermodynamically stable state is a cone helix. Uniaxial distortions, that can be caused by uniaxial stresses in the bulk samples or arise due to surface effects in thin films, suppress the helical states and stabilize Skyrmion lattices in a broad range of thermodynamical parameters. Using the phenomenological theory for modulated and localized states in chiral magnets, the equilibrium parameters of the Skyrmion and helical states have been derived as functions of applied magnetic field and induced uniaxial anisotropy. These results show that due to a combined effect of induced uniaxial anisotropy and an applied magnetic field, Skyrmion lattices can be formed as thermodynamically stable states. The theoretical results provide a comprehensive description of the evolution of modulated states in an applied magnetic field depending on type of anisotropy. The cases of a uniaxial anisotropy of easy axis and easy plane type with fields applied along its axis are investigated in detail. Existence of Skyrmion-lattice states in the easy axis case as thermodynamic field-induced phase is demonstrated. The results explain recent observation of Skyrmion lattices by magnetic Lorentz microscopy in thin foils of cubic chiral magnets. In systems with easy plane type of anisotropy, Skyrmion states do not form thermodynamic phases in applied fields along the axis. However, distorted Skyrmion phases can exist in fields applied perpendicularly to the axis. In this configuration of anisotropy axis and fields, both the helical states and the Skyrmions display elliptical distortions. The investigated micromagnetic model maps out the basic helical and Skyrmionic states expected to exist in cubic and nearly cubic chiral magnets.

Increased complexity of novel magnetic materials in the last decade has placed high demands on the manufacturing process as well as on the characterization. One of the possibilities for characterization of magnetic samples is to exploit the magneto-optical effects. The presented work uses the magneto-optical Kerr effect as a major characterization technique to probe the magnetic properties of samples. We have developed a mathematical model describing the effect of the magnetization on the polarized light and present an apparatus capable of measuring the response given by the light-matter interaction. The experimental results show the performance of the apparatus on the various magnetic systems including meta-stable iron layers, Stoner-Wohlfarth particles and magnetic vortices. The scanning vectorial Kerr magnetometer allowed us to probe the vector of magnetization with diffraction limited resolution below 500 nm.

This thesis presents the theoretical studies of chiral magnetic structures, which exist or are affected by antisymmetric Dzyaloshinskii-Moriya interactions. The theoretical approach is based on the phenomenological model of ferromagnetic materials lacking inversion symmetry. Equilibrium magnetic states are described as static structures in the micromagnetic low temperature limit with a fixed magnitude of the magnetization. The studies are focused on two cases: (i) magnetization structures that are affected by chiral exchange so that a particular chirality of these structures is selected, and (ii) novel solitonic states that are called chiral Skyrmions and only exist because of the chiral exchange. Vortex states in magnetic nanodisks provide the simplest example of a handed magnetization structure, where effects of the chiral couplings may become noticeable. A chiral exchange here favours one chirality of such a vortex state over the other. This effect can stem from surface-induced or other defect-related chiral Dzyaloshinskii-Moriya exchange. The different chiral versions of the vortex states are shown to display strong dependencies on the materials properties of such nanodisks. Within a micromagnetic model for these effects, numerical calculations of the shape, size, and stability of the vortices in equilibrium as functions of magnetic field and the material and geometrical parameters provide a general analysis of the influence of the broken mirror symmetry caused by the surface/interfaces or structural defect on their properties. The Dzyaloshinskii-Moriya interactions impose differences in the energies and sizes of vortices with different chirality: these couplings can considerably increase sizes of vortices with one sense of rotation and suppress vortices with opposite sense of rotation. Torsions related to lattice defects can cause similar to the surface-induced chiral couplings. A general phenomenological magneto-elastic formulation for this torsional chirality selection is given. It is applied to calculate similar effects on vortex states in magnetic disks with a screw dislocation at their center. In systems with strong chiral exchange the magnetic equilibrium states themselves become chiral twisted structures. The most interesting structures in this context are the two-dimensional solitonic states that are now known as chiral Skyrmions. The properties and stability of multiply twisted states composed of these particle-like units are the subject of the second part of this thesis. These states compete with the well known onedimensionally modulated helical states in non-centrosymmetric magnetic systems. Studies of modulated states in cubic helimagnets have shown, that in absence of additional effects, the only thermodynamically stable state is a cone helix. Uniaxial distortions, that can be caused by uniaxial stresses in the bulk samples or arise due to surface effects in thin films, suppress the helical states and stabilize Skyrmion lattices in a broad range of thermodynamical parameters. Using the phenomenological theory for modulated and localized states in chiral magnets, the equilibrium parameters of the Skyrmion and helical states have been derived as functions of applied magnetic field and induced uniaxial anisotropy. These results show that due to a combined effect of induced uniaxial anisotropy and an applied magnetic field, Skyrmion lattices can be formed as thermodynamically stable states. The theoretical results provide a comprehensive description of the evolution of modulated states in an applied magnetic field depending on type of anisotropy. The cases of a uniaxial anisotropy of easy axis and easy plane type with fields applied along its axis are investigated in detail. Existence of Skyrmion-lattice states in the easy axis case as thermodynamic field-induced phase is demonstrated. The results explain recent observation of Skyrmion lattices by magnetic Lorentz microscopy in thin foils of cubic chiral magnets. In systems with easy plane type of anisotropy, Skyrmion states do not form thermodynamic phases in applied fields along the axis. However, distorted Skyrmion phases can exist in fields applied perpendicularly to the axis. In this configuration of anisotropy axis and fields, both the helical states and the Skyrmions display elliptical distortions. The investigated micromagnetic model maps out the basic helical and Skyrmionic states expected to exist in cubic and nearly cubic chiral magnets.

By allowing to measure the magnetic field distribution inside a material, muon spin rotation experiments have the potential to provide valuable information about microscopic properties of high-temperature superconductors. Nevertheless, information about the intrinsic superconducting properties of the material is masked by random thermal and static fluctuations of the magnetic field which penetrates the material in the form of vortices of quantized magnetic flux. A good understanding of the fluctuations of those vortices is needed for the correct determination of intrinsic properties, notably the coherence length &xi;, and the field penetration depth &lambda;. We develop a simulation based on the Metropolis algorithm in order to understand the effect, on the magnetic field distribution, of disorder- and thermally-induced fluctuations of the vortex lattice inside a layered superconductor. <br /><br /> Our model correctly predicts the melting temperatures of the YBa₂Cu₃O_{6. 95} (YBCO) superconductor but largely underestimates the observed entropy jump. Also we failed to simulate the high field disordered phase, possibly because of a finite size limitation. In addition, we found our model unable to describe the first-order transition observed in the highly anisotropic Bi₂Sr₂CaCu₂O_8+<em>y</em>. <br /><br /> Our model predicts that for YBCO, the effect of thermal fluctuations on the field distribution is indistinguishable from a change in &xi;. It also confirms the usual assumption that the effect of static fluctuations at low temperature can be efficiently modeled by convolution of the field distribution with a Gaussian function. However the extraction of &xi; at low fields requires a very high resolution of the field distribution because of the low vortex density.

By allowing to measure the magnetic field distribution inside a material, muon spin rotation experiments have the potential to provide valuable information about microscopic properties of high-temperature superconductors. Nevertheless, information about the intrinsic superconducting properties of the material is masked by random thermal and static fluctuations of the magnetic field which penetrates the material in the form of vortices of quantized magnetic flux. A good understanding of the fluctuations of those vortices is needed for the correct determination of intrinsic properties, notably the coherence length &xi;, and the field penetration depth &lambda;. We develop a simulation based on the Metropolis algorithm in order to understand the effect, on the magnetic field distribution, of disorder- and thermally-induced fluctuations of the vortex lattice inside a layered superconductor. <br /><br /> Our model correctly predicts the melting temperatures of the YBa₂Cu₃O_{6. 95} (YBCO) superconductor but largely underestimates the observed entropy jump. Also we failed to simulate the high field disordered phase, possibly because of a finite size limitation. In addition, we found our model unable to describe the first-order transition observed in the highly anisotropic Bi₂Sr₂CaCu₂O_8+<em>y</em>. <br /><br /> Our model predicts that for YBCO, the effect of thermal fluctuations on the field distribution is indistinguishable from a change in &xi;. It also confirms the usual assumption that the effect of static fluctuations at low temperature can be efficiently modeled by convolution of the field distribution with a Gaussian function. However the extraction of &xi; at low fields requires a very high resolution of the field distribution because of the low vortex density.

Zur Aufklärung der lokalen Magnetisierungs- und magnetischen Streufeldstruktur in ferromagnetischen und supraleitenden Materialien wurden magnetkraftmikroskopische (Magnetkraftmikroskopie-MFM) Untersuchungen durchgeführt und quantitativ ausgewertet. Für eine solch quantitative Auswertung muss der Einfluß der verwendeten MFM-Spitzen auf das MFM-Bild bestimmt und in geeigneter Weise subtrahiert werden. Hierzu wurden Spitzenkalibrierungsroutinen und ein Verfahren zur Entfaltung der gemessenen MFM-Daten implementiert, das auf der Wiener Dekonvolution basiert. Mit Hilfe dieser Prozedur können sowohl die räumliche Ausdehnung als auch die Größe der Streufelder direkt aus gemessenen MFM-Bildern bestimmt werden. Gezeigt wurde diese Anwendung für die Durchmesserbestimmung von Blasendomänen in einer (Co/Pd)-Multilage und für die Bestimmung der temperaturabhängigen magnetischen Eindringtiefe in einem supraleitendem BaFe2(As0.24P0.76)2 Einkristall. Desweiteren konnte durch die Kombination von mikromagnetischen Rechnungen und der quantitativen MFM-Datenanalyse die Existenz einer dreidimensionalen Vortex-Struktur am Ende von Co48Fe52-Nanodrähten nachgewiesen werden. Damit ist es gelungen die Tiefensensitivität der Magnetkraftmikroskopie erfolgreich in die Rekonstruktion der vermessenen Magnetisierungsstruktur einzubeziehen.

Zur Aufklärung der lokalen Magnetisierungs- und magnetischen Streufeldstruktur in ferromagnetischen und supraleitenden Materialien wurden magnetkraftmikroskopische (Magnetkraftmikroskopie-MFM) Untersuchungen durchgeführt und quantitativ ausgewertet. Für eine solch quantitative Auswertung muss der Einfluß der verwendeten MFM-Spitzen auf das MFM-Bild bestimmt und in geeigneter Weise subtrahiert werden. Hierzu wurden Spitzenkalibrierungsroutinen und ein Verfahren zur Entfaltung der gemessenen MFM-Daten implementiert, das auf der Wiener Dekonvolution basiert. Mit Hilfe dieser Prozedur können sowohl die räumliche Ausdehnung als auch die Größe der Streufelder direkt aus gemessenen MFM-Bildern bestimmt werden. Gezeigt wurde diese Anwendung für die Durchmesserbestimmung von Blasendomänen in einer (Co/Pd)-Multilage und für die Bestimmung der temperaturabhängigen magnetischen Eindringtiefe in einem supraleitendem BaFe2(As0.24P0.76)2 Einkristall. Desweiteren konnte durch die Kombination von mikromagnetischen Rechnungen und der quantitativen MFM-Datenanalyse die Existenz einer dreidimensionalen Vortex-Struktur am Ende von Co48Fe52-Nanodrähten nachgewiesen werden. Damit ist es gelungen die Tiefensensitivität der Magnetkraftmikroskopie erfolgreich in die Rekonstruktion der vermessenen Magnetisierungsstruktur einzubeziehen.:Introduction 6 1 Contrast formation in Magnetic Force Microscopy (MFM) 9 1.1 Type of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Relevant interaction forces . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Magnetic interaction mechanisms . . . . . . . . . . . . . . . . . . . 11 1.2 Basic magnetostatics of the tip-sample system . . . . . . . . . . . . . . . . 12 1.2.1 General magnetostatic expressions . . . . . . . . . . . . . . . . . . . 12 1.2.2 Description of the tip sample system . . . . . . . . . . . . . . . . . 14 1.2.3 Magnetostatics in Fourier space . . . . . . . . . . . . . . . . . . . . 15 2 Instrumentation 20 2.1 Scanning Force Microscopy (SFM) . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.1 Measurement principle and operation modes . . . . . . . . . . . . . 20 2.1.2 Dynamic mode SFM . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Lift mode MFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Non-contact MFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Vibrating Sample Magnetometry . . . . . . . . . . . . . . . . . . . . . . . 26 3 Quantitative Magnetic Force Microscopy 28 3.1 The challenge of MFM image inversion . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Description of the problem and state of the art . . . . . . . . . . . 28 3.1.2 The point probe approximations . . . . . . . . . . . . . . . . . . . . 31 3.1.3 The transfer function approach . . . . . . . . . . . . . . . . . . . . 33 3.2 Tip calibration: Adapted Wiener deconvolution . . . . . . . . . . . . . . . 39 3.2.1 Details of the procedure . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Evaluation of possible errors . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Noise measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 MFM probes and their specific characteristics . . . . . . . . . . . . . . . . 49 3.5 Calibration samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.6 Detection of tip-sample modification . . . . . . . . . . . . . . . . . . . . . 55 4 Quantitative MFM with iron filled carbon nanotube sensors (Fe-CNT) 56 4.1 The monopole character of Fe-CNT sensors . . . . . . . . . . . . . . . . . . 57 4.1.1 Calibration within the point probe approximation . . . . . . . . . . 57 4.1.2 Calibration results and discussion . . . . . . . . . . . . . . . . . . . 59 4.1.3 Quantitative MFM on a [Co/Pt]/Co/Ru multilayer . . . . . . . . . 62 4.2 Inplane sensitive MFM with Fe-CNT sensors . . . . . . . . . . . . . . . . . 63 4.2.1 Bimodal MFM technique . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2.2 Comparison between calculated and measured in-plane contrast . . 66 5 Quantification of magnetic nanoobjects in MFM measurements 70 5.1 Bubble domains in a [Co/Pd]80 multilayer . . . . . . . . . . . . . . . . . . 71 5.1.1 Micromagnetic model . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.2 MFM image simulation . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Quantitative assessment of the magnetic penetration depth in superconductors 78 5.2.1 Comparison of methods . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2.2 Experimental determination of the temperature dependent penetration depth in a BaFe2(As0:24P0:76)2 single crystal . . . . . . . . . . . 83 6 Magnetization studies of CoFe nanowire arrays on a local and global scale 87 6.1 Revisiting the estimation of demagnetizing fields in magnetic nanowire arrays 88 6.1.1 Available approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.1.2 Calculation of demagnetizing fields in nanowire arrays . . . . . . . . 91 6.2 Micromagnetic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3 Combination of demagnetizing field calculations and micromagnetic simulation100 6.4 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.5 Global hysteresis measurements of CoFe nanowire arrays with varying length 104 6.6 Local magnetic characterization of a CoFe nanowire array by quantitative MFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.6.1 Magnetic structure of individual nanowires . . . . . . . . . . . . . . 107 6.6.2 Magnetization reversal of the nanowire array . . . . . . . . . . . . . 110 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Conclusions and Outlook 119 Bibliography 121 Acknowledgements 135