Dynamics of Bound Monopoles in Artificial Spin Ice: How to Store Energy in Dirac Strings.

Dirac strings in spin ices are lines of reversed dipoles joining two quasiparticle excitations. These excitations behave as unbound emergent monopoles if the tension of Dirac strings vanishes. In this Letter, analytical and numerical analysis are used to study the dynamics of two-dimensional dipolar spin ices, artificially created analogs of bulk spin ice, in the regime of bound monopoles. It is shown that, in this regime, strings, rather than monopoles, are effective degrees of freedom explaining the finite-width band of Pauling states. A measurable prediction of path-time dependence of endpoints of a stretched and, then, released Dirac string is made and verified via simulations. It is shown that string dynamics is defined by the characteristic tension-to-mass ratio, which is determined by the fine structure constant and lattice dependent parameter. It is proposed to use string tension to achieve spontaneous magnetic currents. A concept of an energy storing device on the basis of this principle is proposed and illustrated by an experimental demonstration. A scheme of independent measurement at the nanoscale is proposed.

Stability of single skyrmionic bits.

The switching between topologically distinct skyrmionic and ferromagnetic states has been proposed as a bit operation for information storage. While long lifetimes of the bits are required for data storage devices, the lifetimes of skyrmions have not been addressed so far. Here we show by means of atomistic Monte Carlo simulations that the field-dependent mean lifetimes of the skyrmionic and ferromagnetic states have a high asymmetry with respect to the critical magnetic field, at which these lifetimes are identical. According to our calculations, the main reason for the enhanced stability of skyrmions is a different field dependence of skyrmionic and ferromagnetic activation energies and a lower attempt frequency of skyrmions rather than the height of energy barriers. We use this knowledge to propose a procedure for the determination of effective material parameters and the quantification of the Monte Carlo timescale from the comparison of theoretical and experimental data.

Topologically protected magnetic helix for all-spin-based applications.

The recent years have witnessed an emergence of the field of all-spin-based devices without any flow of charge. An ultimate goal of this scientific direction is the realization of the full spectrum of spin-based networks as in modern electronics. The concept of energy-storing elements, indispensable for those networks, are so far lacking. Analyzing analytically the size dependent properties of magnetic chains that are coupled via either exchange or long-range dipolar or Ruderman-Kittel-Kasuya-Yosida interactions, we discover a particularly simple law: magnetic configurations corresponding to helices with integer number of twists, which are commensurate with the chain's length, are energetically stable. This finding, supported by simulations and an experimentally benchmarked model, agrees with the study [R. Skomski et al., J. Appl. Phys. 111, 07E116 (2012)] showing that boundaries can topologically stabilize structures that are not stable otherwise. On that basis, an energy-storing element that uses spin at every stage of its operation is proposed.

Domain wall manipulation with a magnetic tip.

A theoretical concept of local manipulation of magnetic domain walls is introduced. In the proposed procedure, a domain wall is driven by a spin-polarized current induced by a magnetic tip, as used in a scanning tunneling microscope, placed above a magnetic nanostripe and then moved along its long axis with a current flowing through the vacuum barrier. The angular momentum from the spin-polarized current exerts a torque on the magnetic moments underneath the tip and leads to a displacement of the domain wall. Particularly, the manipulation of a ferromagnetic 180° transverse domain wall has been studied by means of Landau-Lifshitz-Gilbert dynamics and Monte Carlo simulations. Different relative orientations of the tip and the sample magnetization have been considered.

Indirect control of antiferromagnetic domain walls with spin current.

The indirect controlled displacement of an antiferromagnetic domain wall by a spin current is studied by Landau-Lifshitz-Gilbert spin dynamics. The antiferromagnetic domain wall can be shifted both by a spin-polarized tunnel current of a scanning tunneling microscope or by a current driven ferromagnetic domain wall in an exchange coupled antiferromagnetic-ferromagnetic layer system. The indirect control of antiferromagnetic domain walls opens up a new and promising direction for future spin device applications based on antiferromagnetic materials.

Atomic-level control of the domain wall velocity in ultrathin magnets by tuning of exchange interactions.

We demonstrate that the propagation velocity of field driven magnetic domain walls in ultrathin Au/Co/Au films with perpendicular anisotropy on vicinal substrates is anisotropic and strongly depends on the step density of the substrate. The velocity of walls oriented perpendicular to the steps drastically increases with increasing local step density while being unchanged or only weakly decreased for the walls oriented parallel to the steps. We develop an analytical model revealing the step-modified exchange interactions as the main driving force for this anisotropic behavior. The enhancement of the domain wall velocity at low magnetic fields far below the Walker instability threshold makes this phenomenon interesting for magnetic nanodevices.

Magnetic ground state of single and coupled permalloy rectangles.

We have studied the magnetic domain structure in Permalloy rectangles that reveal flux-closure domain configurations. Arrays with varying spacing between the rectangles are investigated by scanning electron microscopy with polarization analysis as well as by micromagnetic simulation. In contrast to general expectation, rectangles in the flux-closure Landau state show significant coupling and form a magnetic pattern of common chirality. The coupling is due to the stray field that originates from small changes of the magnetization alignment, which is sensitive to the exact shape and the separation of the rectangles.

Real-space observation of a right-rotating inhomogeneous cycloidal spin spiral by spin-polarized scanning tunneling microscopy in a triple axes vector magnet.

Using spin-polarized scanning tunneling microscopy performed in a triple axes vector magnet, we show that the magnetic structure of the Fe double layer on W(110) is an inhomogeneous right-rotating cycloidal spin spiral. The magnitude of the Dzyaloshinskii-Moriya vector is extracted from the experimental data using micromagnetic calculations. The result is confirmed by comparison of the measured saturation field along the easy axis to the respective value as obtained from Monte Carlo simulations. We find that the Dzyaloshinskii-Moriya interaction is too weak to destabilize the single domain state. However, it can define the sense of rotation and the cycloidal spiral type once the single domain state is destabilized by dipolar interaction.

Magnetization reversal of nanoscale islands: how size and shape affect the arrhenius prefactor.

The thermal switching behavior of individual in-plane magnetized Fe/W(110) nanoislands is investigated by a combined study of variable-temperature spin-polarized scanning tunneling microscopy and Monte Carlo simulations. Even for islands consisting of less than 100 atoms the magnetization reversal takes place via nucleation and propagation. The Arrhenius prefactor is found to strongly depend on the individual island size and shape, and based on the experimental results a simple model is developed to describe the magnetization reversal in terms of metastable states. Complementary Monte Carlo simulations confirm the model and provide new insight into the microscopic processes involved in magnetization reversal of smallest nanomagnets.

Quantized spin waves in antiferromagnetic Heisenberg chains.

The quantized stationary spin wave modes in one-dimensional antiferromagnetic spin chains with easy axis on-site anisotropy have been studied by means of Landau-Lifshitz-Gilbert spin dynamics. We demonstrate that the confined antiferromagnetic chains show a unique behavior having no equivalent, neither in ferromagnetism nor in acoustics. The discrete energy dispersion is split into two interpenetrating n and n' levels caused by the existence of two sublattices. The oscillations of individual sublattices as well as the standing wave pattern strongly depend on the boundary conditions. Particularly, acoustical and optical antiferromagnetic spin waves in chains with boundaries fixed (pinned) on different sublattices can be found, while an asymmetry of oscillations appears if the two pinned ends belong to the same sublattice.

Atomic-scale spin spiral with a unique rotational sense: Mn monolayer on W(001).

Using spin-polarized scanning tunneling microscopy we show that the magnetic order of 1 monolayer Mn on W(001) is a spin spiral propagating along 110 crystallographic directions. The spiral arises on the atomic scale with a period of about 2.2 nm, equivalent to only 10 atomic rows. Ab initio calculations identify the spin spiral as a left-handed cycloid stabilized by the Dzyaloshinskii-Moriya interaction, imposed by spin-orbit coupling, in the presence of softened ferromagnetic exchange coupling. Monte Carlo simulations explain the formation of a nanoscale labyrinth pattern, originating from the coexistence of the two possible rotational domains, that is intrinsic to the system.

Atomic spin structure of antiferromagnetic domain walls.

The search for uncompensated magnetic moments on antiferromagnetic surfaces is of great technological importance as they are responsible for the exchange-bias effect that is widely used in state-of-the-art magnetic storage devices. We have studied the atomic spin structure of phase domain walls in the antiferromagnetic Fe monolayer on W(001) by means of spin-polarized scanning tunnelling microscopy and Monte Carlo simulations. The domain wall width only amounts to 6-8 atomic rows. Although walls oriented along <100> directions are found to be fully compensated, detailed analysis of <110>-oriented walls reveals an uncompensated perpendicular magnetic moment. Our result represents a major advance in the field of antiferromagnetism, and may lead to a better understanding of the magnetic interaction between ferromagnetic and antiferromagnetic materials.

Observation of a complex nanoscale magnetic structure in a hexagonal Fe monolayer.

We have observed a novel magnetic structure in the pseudomorphic Fe monolayer on Ir(111). Using spin-polarized scanning tunneling microscopy we find a nanometer-sized two-dimensional magnetic unit cell. A collinear magnetic structure is proposed consisting of 15 Fe atoms per unit cell with 7 magnetic moments pointing in one and 8 moments in the opposite direction. First-principles calculations verify that such an unusual magnetic state is indeed lower in energy than all solutions of the classical Heisenberg model. We demonstrate that the complex magnetic structure is induced by the strong Fe-Ir hybridization.

Multipolar ordering and magnetization reversal in two-dimensional nanomagnet arrays.

The low-temperature stable states and the magnetization reversal of realistic two-dimensional nanoarrays with higher-order magnetostatic interactions are studied theoretically. For a general calculus of the multipole-multipole interaction energy we introduce a Hamiltonian in spherical coordinates into the Monte Carlo scheme. We demonstrate that higher-order interactions considerably change the dipolar ground states of in-plane magnetized arrays favoring collinear configurations. The multipolar interactions lead to enhancement or decrease of the coercivity in arrays with in-plane or out-of-plane magnetization.

Noncollinear magnetic order in quasicrystals.

Based on Monte Carlo simulations, the stable magnetization configurations of an antiferromagnet on a quasiperiodic tiling are derived theoretically. The exchange coupling is assumed to decrease exponentially with the distance between magnetic moments. It is demonstrated that the superposition of geometric frustration with the quasiperiodic ordering leads to a three-dimensional noncollinear antiferromagnetic spin structure. The structure can be divided into several ordered interpenetrating magnetic supertilings of different energy and characteristic wave vector. The number and the symmetry of subtilings depend on the quasiperiodic ordering of atoms.

Domain wall orientation in magnetic nanowires.

Scanning tunneling microscopy reveals that domain walls in ultrathin Fe nanowires are oriented along a certain crystallographic direction, regardless of the orientation of the wires. Monte Carlo simulations on a discrete lattice are in accordance with the experiment if the film relaxation is taken into account. We demonstrate that the wall orientation is determined by the atomic lattice and the resulting strength of an effective exchange interaction. The magnetic anisotropy and the magnetostatic energy play a minor role for the wall orientation in that system.

Decagonal quasiferromagnetic microstructure on the penrose tiling.

The stable magnetization configurations of a ferromagnet on a quasiperiodic tiling have been derived theoretically. The magnetization configuration is investigated as a function of the ratio of the exchange to the dipolar energy. The exchange coupling is assumed to decrease exponentially with the distance between magnetic moments. It is demonstrated that for a weak exchange interaction the new structure, the quasiferromagnetic decagonal configuration, corresponds to the minimum of the free energy. The decagonal state represents a new class of frustrated systems where the degenerated ground state is aperiodic and consists of two parts: ordered decagon rings and disordered spin-glass-like phase inside the decagons.

Magnetic microstructure of the spin reorientation transition: a computer experiment.

The scenario of the spin reorientation in two-dimensional films within first-order anisotropy approximation is theoretically studied by means of Monte Carlo simulations. The magnetic microstructure is investigated as a function of the ratio of the perpendicular anisotropy energy to the dipolar one. If the anisotropy dominates, out-of-plane domains will be found while in-plane vortices appear for a vanishing anisotropy. In the range of comparable anisotropy and dipolar energies a complex domain pattern evolves yielding a continuous transition between the two structures. The structure with equally distributed magnetic moment orientations is stable at the point where anisotropy and dipolar energies cancel each other.