Session S19: Magnetic Nanoparticles: Dynamics, Domains and Vortices

Mechanics of magnetic solitons

The dynamics of magnetization in a ferromagnet is governed by the Landau-Lifshitz equation, a nonlinear PDE that is rarely solvable exactly. To approximate soliton dynamics, we introduced a method of collective coordinates [1] generalizing Thiele's rigid-translation approximation [2] by including arbitrary deformations. The method has been applied to a variety of solitons (domain walls, skyrmions, vortices), extended to include the effects of spin-transfer torque, and adopted to antiferromagnetic solitons [3]. I will describe our recent applications of the method to several new problems: an extended domain wall in a thin ferromagnetic film behaves as a nonreciprocal string with transverse waves propagating on it with unequal speeds; annihilation of a vortex-antivortex pair in a thin film; and the propulsion of antiferromagnetic solitons under combined magnetic field and spin-polarized current.

[1] O. A. Tretiakov et al., Phys. Rev. Lett. 100, 127204 (2008).
[2] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).
[3] E. G. Tveten et al., Phys. Rev. Lett. 110, 127208 (2013).   

Annihilation of Vortex-Antivortex Pairs in Thin Ferromagnetic Films

We study theoretically the annihilation of a vortex-antivortex pair in a two-dimensional easy-plane ferromagnet. The motion is governed by the interplay of three types of forces: attraction between the vortices mediated by exchange interaction, a gyroscopic force proportional to the vortex's velocity and orthogonal to it, and viscous forces. For opposite out-of-plane orientations of the vortex cores, the two solitons orbit each other. In the presence of finite damping, the radius of the orbit gradually decreases until the cores overlap. We show that the viscous forces involve several components. In addition to the expected viscous force proportional to the vortex's velocity and opposing it, there is mutual drag exerted by vortices on each other. In addition, image vortices, created by the sample edge, also generate appreciable drag on the real ones. We compare predictions of analytical theory to results of micromagnetic simulations.
  

Ferromagnetic domain wall as a nonreciprocal string

The collective coordinate method has been a great success in analyzing the dynamics of point-like topological defects, such as vortices in two dimensions and a domain wall in one dimension. To study a domain wall in a two-dimensional ferromagnetic film, we generalize the discrete collective coordinates into a continuous field describing the displacement of the wall. The domain wall moves like a string; however, its dynamics differs significantly from the familiar case of a taut string thanks to an additional Berry-phase term reflecting the precessional nature of the spins. The Berry phase gives rise to nonreciprocal dynamics, where the waves traveling left and right have different speeds. Given an initial deformation, the nonreciprocal string with free ends exhibits uniform translational motion in addition to oscillations, as if it is self-driven, which is allowed by the broken time-reversal symmetry. This can be understood from the view of the conserved transverse momentum derived from Noether’s theorem. It is then evident how the domain wall evolves in the presence of an external magnetic field and dissipation.   

Spin-transfer torque, stress-energy tensor, and Magnus force

In micromagnetics, equations of motion for the magnetic moments m(r,t) are given by the Landau Lifshitz Gilbert equation, which can be equivalently derived from a Lagrangian. The Lagrangian for ferro and antiferromagnets contains terms with a spatial derivative of the order parameter in the presence of a spin carrying electric current, (u●▽)n. Here u is the electron drift velocity proportional to the current density j. In terms of collective coordinates (q) this term appears as a vector potential coupling to u, A(q)●u. It is free of time derivatives and thus appears to be a form of potential energy. But it is explicitly gauge dependent and should not directly represent a physical quantity like a potential energy density. Treating it like one yields incorrect generalized forces and a gauge dependent stress energy tensor. Motivated by this anomaly we outline the procedure to correctly define the energy density and energy flow caused by the electric current interacting with the soliton and provide a conjectural form of a gauge independent stress energy tensor. We illustrate with examples of a 1D ferromagnetic wire and an antiferromagnetic vortex.   

Influence of Interplay of Volume and Interface Spin Transfer Torques Affecting Current Induced Domain Wall Transport in Atomically Engineered Multilayered Racetracks

Advances in our understanding of the current controlled motion of domain walls in perpendicularly magnetized multilayer stacks has led to very high domain wall velocities. The motion is controlled by several spin orbit derived phenomena including, notably, the Spin Hall Effect, and the Dzyaloshinskii-Moriya interaction. Here we study atomically engineered multilayered racetracks in which interface and volume derived spin transfer torques compete with each other. Of especial significance is the dependence of the domain wall motion on the magnitude and direction of in plane magnetic fields that compete with the Dzyaloshinskii-Moriya exchange fields. The various contributions to the interface and volume torques are independently varied to allow their respective contributions to be separated. Insight is provided by 1-D analytical modeling of the domain wall motion to extract relevant parameters.   

Temperature Dependent Magnetization Reversal Process on a Single Magnetic Vortex

Magnetic vortices can be the ground state of certain ferromagnetic systems. They have been subject of study for the past decade because they are good candidates for a variety of applications. Magnetic vortices can be nucleated in micron and sub-micron sized soft ferromagnetic disks because geometry constraints. Previous studies have shown the important effect that pinning sites can have in the magnetization reversal process of an ensemble of disks and in the dynamics of single disks. Moreover, low temperature measurements of single disks showed thermal dependence of the nucleation and annihilation fields. However, a detailed study of domain wall motion at low temperatures in a single disk is missing.

In this work, we used electrical contacts on top of a disk to measure the change in resistance when an in-plane magnetic field is applied. We find that the nucleation and annihilation fields depend on temperature, with different thermal energy barriers for nucleation and annihilation, pinning and depinning affect the reversal process. Micromagnetic simulations are also performed to confirm the results.   

Subnanosecond magnetization reversal of magnetic nanoparticle driven by chirp microwave field pulse

We investigate the magnetization reversal of single-domain magnetic nanoparticle driven by linear down-chirp microwave magnetic field pulse. Numerical simulations based on the Landau-Lifshitz-Gilbert equation reveal that solely down-chirp pulse is capable of inducing subnanosecond magnetization reversal. With certain range of initial frequency and chirp rate, the field amplitude required is much smaller than that of constant frequency microwave field. The fast reversal is because the down-chirp microwave field acts as energy source and sink for the magnetic particle before and after crossing over the energy barrier respectively. Applying a spin-polarized current additively to the system further reduces the microwave field amplitude. Our findings provide a new way to realize low-cost and fast magnetization reversal.   

Chiral Edge Mode in the Coupled Dynamics of Magnetic Solitons in a Honeycomb Lattice

Motivated by a recent experimental demonstration of a chiral edge mode in an array of spinning gyroscopes, we theoretically study the coupled gyration modes of topological magnetic solitons, vortices and magnetic bubbles, arranged as a honeycomb lattice [1]. The soliton lattice under suitable conditions is shown to support a chiral edge mode like its mechanical analogue, the existence of which can be understood by mapping the system to the Haldane model for an electronic system. The direction of the chiral edge mode is associated with the topological charge of the constituent solitons, which can be manipulated by an external field or by an electric-current pulse. The direction can also be controlled by distorting the honeycomb lattice. Our results indicate that the lattices of magnetic solitons can serve as reprogrammable topological metamaterials.

[1] S. K. Kim and Y. Tserkovnyak, “Chiral edge mode in the coupled dynamics of magnetic solitons in a honeycomb lattice,” Phys. Rev. Lett. 119, 077204 (2017)   

Micromagnetic simulations for modulated permalloy cylindrical nanowires

Modulated magnetic nanowires attract the attention of both experimental and theoretical
scientists due to the possibility of tuning their properties by simply controlling
a few geometrical parameters. In the present work we consider a cylindrical rod of
length H, where a disk-like portion of height h<<H on the same axis as the rod is defined.
The diameter of the disk is varied from wider than the cylinder to
narrower than the cylinder: vortex and ferromagnetic behavior are found respectively.
Then the wider disk is moved along the axis of the cylinder finding a nontrivial
reversion behavior, namely, the system is magnetically softer when the wide disk portion is at a critical
distance from the ends. Then 2, 3 and 4 wide disk portions are defined along the cylinder's axis
discussing their different properties. Results are obtained by Micromagnetic simulations.
The presentation includes variation of the external magnetic field as to go over the
complete hysteresis loops; coercive fields are studied as functions of the geometry;
polarization and vortex orientations are discussed.
Movies are includes to illustrate reversion mechanisms. Conditions for harder and
softer magnetic configurations are given.   

Coercive field enhancement in single-domain to vortex switching in magnetic Co nanodisks

The impressive magnetic properties at the nanoscale have given special attention to the magnetic nanodisks (ND). Among the most prominent applications stand data storage or magneto plasmonic devices. For the present work, we have focused in Cobalt (HCP), using the colloidal lithography technique we have obtained circular ND varying the Cobalt layer from 5-15nm and diameters of 170nm. Here we have studied experimentally, by vibrating sample magnetometry (VSM). Theoretically studies of nanodisks by micromagnetic simulations (OOMMF) have been performed in a wide range of thicknesses between 5-15 nm and diameters between 100-500 nm. From the hysteresis loop measurements, the coercive field as a function of thickness shows an unexpected peak at a critical thickness. Simulations predict the appearance of such peak and indicate that is strongly dependent on the dimensions of the Co layer. The fabrication and subsequent magnetic measurements confirm those results, such as the presence of magnetic single domain and vortex states in the ND. Finally the critical thickness where the hysteresis loop shows the highest coercive field turns out to be the value of thickness which separates the single domain state from the vortex state, due to dipolar exchange energies, and shape anisotropy.   

Magnetization switching by manipulating the thickness of FM&HM layers

Magnetization switching with spin–orbit torque has attracted lots of attention due to its fast switching characteristics and low-power consumption. Increasing the conversion efficiency from charge current to spin current is one of hot topics in this area. We examine a method to find the optimum layer thickness of the ferromagnet and of the normal metal for efficient magnetization switching. By controlling the thickness of the ferromagnet and of the normal metal, we can change the sign and magnitude of the overall field-like torque, which enables faster switching. Our results suggest a method for fast switching in spin-orbit torque-induced cases, where the film thickness of the ferromagnet and the normal metal are controllable parameters to provide an efficient way of implementing spin-orbitronics devices.
  

YIG/Permalloy spin torque oscillator

We grow 20 nm thick epitaxial YIG films on GGG substrates and deposit 5 nm Py capping layers, using sputter deposition. Nanowires of 280 nm width and 2.2 μm length are fabricated using single-step e-beam lithography and ion-milling. We investigate the spin-wave spectrum of the nanowires using spin-torque ferromagnetic resonance and micromagnetic simulations. By applying a direct bias current to the nanowire, magnetic damping can be tuned by injecting spin current from the Py layer. We discover that with increasing bias current, the damping can be fully compensated in this system. Above a critical current, the nanowire enters the auto-oscillatory regime and emits a microwave signal. We find that using Py as a spin injector allows us to achieve microwave emission power levels (pW-scale) that significantly exceed those observed in the YIG/Pt systems (fW-scale). We discuss the physics of current-driven auto-oscillatory dynamics, spin-orbit and thermal spin torques in nanodevices based on bi-layer ferromagnets.   

Magnonic Architectures for Quantum Circuits and Measurements.

The development of 2- and 3-dimensional circuit architectures for experiments at millikelvin temperatures aimed at exploring and exploiting magnonic systems at the quantum level is fast becoming an established area of magnetics research. Magnons are readily excited over the frequency range used by established solid-state quantum circuit technology and couple readily to EM fields. These facts, in combination with the field-tunable dispersion of the excitations make them an interesting proposition in the context of quantum device design. In this talk, we discuss recent progress made in the combination of planar superconducting circuit technology with magnon systems. We discuss the technical and materials requirements of successful magnonic experiments at millikelvin temperatures. We go on to describe experimental investigations including the study of spin-wave propagation in magnetic waveguides at the single magnon level [1, 2], the investigation of magnon modes in spherical magnetic resonators [3], the measurement of magnonic crystals, and the development of systems incorporating Josephson junction based qubits.

[1] A. Karenowska et al., arXiv:1502.06263 (2014). [2] A. van Loo et al., arXiv:1610.08402 (2016). [3] R. Morris et al., Scientific Reports, 7, 11511 (2017).   

Relation between switching time distribution and damping constant

Magnetization dynamics is strongly influenced by the damping process, which is energy dissipation of the magnetization. Especially for spin-transfer torque magnetic random-access memories (STT-MRAMs) applications, the critical current density for magnetization switching is strongly affected by the damping constant because the spin-transfer torque must overcome the damping torque for the switching. However, it is not easy to determine the damping constant of nano-scale patterned samples.
In this talk, we show the relation between the damping constant and switching time distribution in case of nano-scale patterned samples by solving Fokker-Planck equation (FPE). We derive a FPE for the probability distribution of the magnetization angle as a function of time in case of perpendicular anisotropy samples. The approximated analytic expression of FWHM of switching time distribution is obtained, and we found that the inverse of the FWHM is proportional to the applied field and the slope is related with damping constant. The analytic relation shows a good agreement with full numerical results in the limit of high energy barrier. We believe our work is helpful to estimate the damping constant of the nano-scale patterned samples.