There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

Skyrmions represent topologically stable field configurations with particle-like properties. We used neutron scattering to observe the spontaneous formation of a two-dimensional lattice of skyrmion lines, a type of magnetic vortex, in the chiral itinerant-electron magnet MnSi. The skyrmion lattice stabilizes at the border between paramagnetism and long-range helimagnetic order perpendicular to a small applied magnetic field regardless of the direction of the magnetic field relative to the atomic lattice. Our study experimentally establishes magnetic materials lacking inversion symmetry as an arena for new forms of crystalline order composed of topologically stable spin states.

Skyrmion Lattice in a Chiral Magnet

Magnetic skyrmions are particle-like nanometre-sized spin textures of topological origin found in several magnetic materials, and are characterized by a long lifetime. Skyrmions have been observed both by means of neutron scattering in momentum space and microscopy techniques in real space, and their properties include novel Hall effects, current-driven motion with ultralow current density and multiferroic behaviour. These properties can be understood from a unified viewpoint, namely the emergent electromagnetism associated with the non-coplanar spin structure of skyrmions. From this description, potential applications of skyrmions as information carriers in magnetic information storage and processing devices are envisaged.

Topological properties and dynamics of magnetic skyrmions

We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE), focusing on recent developments that have provided a more complete framework for understanding this subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergy between experimental and theoretical work, both playing a crucial role, has been at the heart of these advances. On the theoretical front, the adoption of Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents which originate from spin-orbit coupling. On the experimental front, new experimental studies of the AHE in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors, have more clearly established systematic trends. These two developments in concert with first-principles electronic structure calculations, strongly favor the dominance of an intrinsic Berry-phase-related AHE mechanism in metallic ferromagnets with moderate conductivity. The intrinsic AHE can be expressed in terms of Berry-phase curvatures and it is therefore an intrinsic quantum mechanical property of a perfect cyrstal. An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highly conductive ferromagnets. We review the full modern semiclassical treatment of the AHE together with the more rigorous quantum-mechanical treatments based on the Kubo and Keldysh formalisms, taking into account multiband effects, and demonstrate the equivalence of all three linear response theories in the metallic regime. Finally we discuss outstanding issues and avenues for future investigation.

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Anomalous hall effect

Recent small angle neutron scattering suggests that the spin structure in the A phase of MnSi is a so-called triple-Q state, i.e., a superposition of three helices under 120 degrees. Model calculations indicate that this structure in fact is a lattice of so-called Skyrmions, i.e., a lattice of topologically stable knots in the spin structure. We report a distinct additional contribution to the Hall effect in the temperature and magnetic field range of the proposed Skyrmion lattice, where such a contribution is neither seen nor expected for a normal helical state. Our Hall effect measurements constitute a direct observation of a topologically quantized Berry phase that identifies the spin structure seen in neutron scattering as the proposed Skyrmion lattice.

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.102.186602

Topological Hall Effect in the A Phase of MnSi

Chiral magnets like MnSi form lattices of skyrmions, i.e., magnetic whirls, which react sensitively to small electric currents $j$ above a critical current density ${j}_{c}$. The interplay of these currents with tiny gradients of either the magnetic field or the temperature can induce a rotation of the magnetic pattern for $jg{j}_{c}$. Either a rotation by a finite angle of up to ${15}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{\circ}}$ or---for larger gradients---a continuous rotation with a finite angular velocity is induced. We use Landau-Lifshitz-Gilbert equations extended by extra damping terms in combination with a phenomenological treatment of pinning forces to develop a theory of the relevant rotational torques. Experimental neutron scattering data on the angular distribution of skyrmion lattices suggest that continuously rotating domains are easy to obtain in the presence of remarkably small currents and temperature gradients.

Rotating skyrmion lattices by spin torques and field or temperature gradients

MnSi is an itinerant magnet which at low temperatures develops a helical spin-density wave. Under pressure it undergoes a transition into an unusual partially ordered state whose nature is debated. Here we propose that the helical spin crystal (the magnetic analog of a solid) is a useful starting point to understand partial order in MnSi. We consider different helical spin crystals and determine conditions under which they may be energetically favored. The most promising candidate has bcc structure and is reminiscent of the blue phase of liquid crystals in that it has line nodes of magnetization protected by symmetry. We introduce a Landau theory to study the properties of these states, in particular, the effect of crystal anisotropy, magnetic field, and disorder. These results compare favorably with existing data on MnSi from neutron scattering and magnetic field studies. Future experiments to test this scenario are also proposed.

Theory of the helical spin crystal: a candidate for the partially ordered state of MnSi.

High pressure studies in MnSi suggest the existence of a non-Fermi liquid state without quantum criticality. The observation of partial magnetic order in a small pocket of the pressure versus temperature phase diagram of MnSi has additionally inspired several proposals of complex spin textures in chiral magnets. We used neutron scattering to observe the formation of a two-dimensional lattice of skyrmion lines, a type of magnetic vortices, under applied magnetic fields in metallic and semiconducting B20 compounds. In strongly disordered systems the skyrmion lattice is hysteretic and extends over a large temperature range. Our study experimentally establishes magnetic materials lacking inversion symmetry as an arena for new forms of spin order composed of topologically stable spin textures.

https://iopscience.iop.org/article/10.1088/0953-8984/22/16/164207/pdf

B. Binz

Papers

Skyrmion Lattice in a Chiral Magnet

Topological Hall Effect in the A Phase of MnSi

Rotating skyrmion lattices by spin torques and field or temperature gradients

Theory of the helical spin crystal: a candidate for the partially ordered state of MnSi.

Skyrmion lattices in metallic and semiconducting B20 transition metal compounds.