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This work introduces an approach to compute periodic phase diagram of micromagnetic systems by solving a periodic linearized Landau-Lifshitz-Gilbert (LLG) equation using an eigenvalue solver with the Finite Element Method formalism. The linear operator in the eigenvalue problem is defined as a function of the periodic phase shift wave vector. The dispersion diagrams are obtained by solving the eigenvalue problem for complex eigen frequencies and corresponding eigen states for a range of prescribed wave vectors. The presented approach incorporates a calculation of the periodic effective field, including the exchange and magnetostatic field components. The approach is general in that it allows handling 3D problems with any 1D, 2D, and 3D periodicities. The ability to calculated periodic diagrams provides insights into the spin wave propagation and localized resonances in complex micromagnetic structures.

Periodic micromagnetic finite element method (PM-FEM) is introduced to solve periodic unit cell problems using the Landau-Lifshitz-Gilbert equation. PM-FEM is applicable to general problems with 1D, 2D, and 3D periodicities. PM-FEM is based on a non-periodic FEM-based micromagnetic solver and extends it in several aspects to account for periodicities, including the computation of exchange and magnetostatic fields. For the exchange field, PM-FEM modifies the sparse matrix construction for computing the Laplace operator to include additional elements arising due to the periodicities. For the magnetostatic field, the periodic extensions include modifications in the local operators, such as gradient, divergence, and surface magnetic charges as well as the long-range superposition operator for computing the periodic scalar potential. The local operators are extended to account for the periodicities similar to handling the Laplace operator. For the long-range superposition operator, PM-FEM utilizes a periodic Green's function (PGF) and fast spatial convolutions. The PGF is computed rapidly via exponentially rapidly convergent sums. The spatial convolutions are accomplished via a modified fast Fourier transform based adaptive integral method that allows calculating spatial convolutions with non-uniform meshes in $O(N\log N)$ numerical operations. PM-FEM is implemented on CPU and GPU based computer architectures. PM-FEM allows efficiently handling cases of structures contained withing the periodic unit cell touching or not touching its boundaries as well as structures that protrude beyond the unit cell boundaries. PM-FEM is demonstrated to have about the same or even higher performance than its parent non-periodic code. The demonstrated numerical examples show the efficiency of PM-FEM for highly complex structures with 1D, 2D, and 3D periodicities.

Synchronization of coupled oscillators is a fundamental phenomenon that has been observed in many natural and engineered systems, ranging from biological cells to electronic circuits. In recent years, spin torque nano-oscillators (STNOs) have emerged as a promising platform for a variety of applications in nanoscale systems such as neuromorphic computing, due to their compact size, low power consumption, and tunable properties. However, achieving synchronization in a large array of STNOs remains a challenge, for the complexity of the interactions between the oscillators. To address this challenge, we report on the achievement of synchronization across a 2D array of STNOs. We designed a honeycomb model that created a network of interactions among the oscillators that only involved nearest neighbors. Using micromagnetic simulations, we demonstrated the effectiveness of our design and observed that global synchronization across the whole array. Such global synchronization can also be tuned by enable/disenable next-nearest neighbor interactions. We also found that our design enables synchronization at room temperature, which is crucial for practical applications. Our work contributes to the growing body of research on the synchronization of coupled oscillators in spintronics and could pave the way for the development of new types of spintronic devices and potential applications.

Coherent X-ray Diffraction Imaging (CXDI) technique offers unique insights into the nanoscale world, enabling the reconstruction of 3D structures with a nanoscale resolution achieved through computational phase reconstruction from measured scattered intensity maps. Computational demands of 3D CXDI, however, limit its real-time application in experimental settings. This work presents a carousel phase retrieval algorithm (CPRA) that enables the real-time, high-resolution reconstruction of computationally complex 3D objects. CPRA is based on representing the 3D reconstruction problem as a set of 2D reconstructions of projected images corresponding to different experimentally collected angles via the Fourier slice theorem. Consistency between the 2D reconstructed images is based on an iterative procedure, in which each 2D reconstruction accounts for the adjacent 2D reconstructed images in a periodic (carousel) manner. Demonstrations on complex systems, including a lithium-rich layered oxide particle and a Staphylococcus aureus biological cell, demonstrate that CPRA significantly enhances the reconstruction quality and enables the reconstruction process to be completed in real time during experiment.

We propose a Fast Fourier Transform based Periodic Interpolation Method (FFT-PIM), a flexible and computationally efficient approach for computing the scalar potential given by a superposition sum in a unit cell of an infinitely periodic array. Under the same umbrella, FFT-PIM allows computing the potential for 1D, 2D, and 3D periodicities for dynamic and static problems, including problems with and without a periodic phase shift. The computational complexity of the FFT-PIM is of $O(N \log N)$ for $N$ spatially coinciding sources and observer points. The FFT-PIM uses rapidly converging series representations of the Green's function serving as a kernel in the superposition sum. Based on these representations, the FFT-PIM splits the potential into its near-zone component, which includes a small number of images surrounding the unit cell of interest, and far-zone component, which includes the rest of an infinite number of images. The far-zone component is evaluated by projecting the non-uniform sources onto a sparse uniform grid, performing superposition sums on this sparse grid, and interpolating the potential from the uniform grid to the non-uniform observation points. The near-zone component is evaluated using an FFT-based method, which is adapted to efficiently handle non-uniform source-observer distributions within the periodic unit cell. The FFT-PIM can be used for a broad range of applications, such as periodic problems involving integral equations in computational electromagnetic and acoustic, micromagnetic solvers, and density functional theory solvers.

We present a general finite element linearized Landau-Lifshitz-Gilbert equation (LLGE) solver for magnetic systems under weak time-harmonic excitation field. The linearized LLGE is obtained by assuming a small deviation around the equilibrium state of the magnetic system. Inserting such expansion into LLGE and keeping only first order terms gives the linearized LLGE, which gives a frequency domain solution for the complex magnetization amplitudes under an external time-harmonic applied field of a given frequency. We solve the linear system with an iterative solver using generalized minimal residual method. We construct a preconditioner matrix to effectively solve the linear system. The validity, effectiveness, speed, and scalability of the linear solver are demonstrated via numerical examples.

The transport response of dipole skyrmions in amorphous centrosymmetric Fe/Gd multilayer is investigated by temperature and field-dependent resistivity measurements collected in three current and magnetic field configurations. It is shown that a dipole skyrmion lattice phase may form at certain temperatures leading to a unique signature of the polar longitudinal resistivity. This signature differs from the conventional field-varying parabolic response associated with stripe phases, which transition to a disordered skyrmion phase under applied fields. Transport measurements under different field history protocols reveal that the anomaly in the polar longitudinal resistivity appears under specific field history reversal processes. Our experimental results are reproduced using micromagnetic simulations that show the anomaly in the polar longitudinal resistivity is related to a domain wall reconfiguration that occurs as the domain morphology transitions from disordered stripes to a skyrmion lattice under applied perpendicular fields.

Neuromorphic computing approaches become increasingly important as we address future needs for efficiently processing massive amounts of data. The unique attributes of quantum materials can help address these needs by enabling new energy-efficient device concepts that implement neuromorphic ideas at the hardware level. In particular, strong correlations give rise to highly non-linear responses, such as conductive phase transitions that can be harnessed for short and long-term plasticity. Similarly, magnetization dynamics are strongly non-linear and can be utilized for data classification. This paper discusses select examples of these approaches, and provides a perspective for the current opportunities and challenges for assembling quantum-material-based devices for neuromorphic functionalities into larger emergent complex network systems.

We present an eigenvalue-based approach for studying the magnetization dynamics in magnetic nanostructures driven by spintronic excitations, such as spin transfer torque and spin orbit torque. The approach represents the system dynamics in terms of normal oscillation modes (eigenstates) with corresponding complex eigenfrequencies. The dynamics is driven by a small number of active eigenstates and often considering just a single eigenstate is sufficient. We develop a perturbation theory that provides semi-analytical dynamic solutions by using eigenstates for the case in the absence of damping and spintronic excitations as a basis. The approach provides important insights into dynamics in such systems and allows solving several difficulties in their modeling, such as extracting the switching current in magnetic random access memories (MRAM) and understanding switching mechanisms. We show that the presented approach directly predicts the critical switching current, i.e., switching current for an infinite time. The approach also provides solutions for the switching dynamics allowing obtaining the switching current for a finite switching time, provided that the system symmetry is broken, e.g., by tilting the polarizer, so that switching by a finite pulse is possible.

The discovery of new mechanisms of controlling magnetic properties by electric fields or currents furthers the fundamental understanding of magnetism and has important implications for practical use. Here, we present a novel approach of utilizing resistive switching to control magnetic anisotropy. We study a ferromagnetic oxide that exhibits an electrically triggered metal-to-insulator phase transition producing a volatile resistive switching. This switching occurs in a characteristic spatial pattern: the formation of a transverse insulating barrier inside a metallic matrix resulting in an unusual ferromagnetic/paramagnetic/ferromagnetic configuration. We found that the formation of this voltage-driven paramagnetic insulating barrier is accompanied by the emergence of a strong uniaxial magnetic anisotropy that overpowers the intrinsic material anisotropy. Our results demonstrate that resistive switching is an effective tool for manipulating magnetic properties. Because resistive switching can be induced in a very broad range of materials, our findings could enable a new class of voltage-controlled magnetism systems.

We present a micromagnetic formulation for modeling the magnetization dynamics and thermal equilibrium in ferrimagnetic materials at low and elevated temperatures. The formulation is based on a mean field approximation (MFA). In this formulation, the ferrimagnet is described micromagnetically by two coupled sublattices with corresponding interactions, including inter- and intra-sublattice micromagnetic exchange as well as four-spin interactions described as an inter-sublattice molecular field with a cubic dependence of the magnetization. The MFA is used to derive a Landau Lifshitz Bloch type equation for ferrimagnetic material, including cases with a ferromagnetic - antiferromagnetic phase transitions. For validation, the results obtained via the presented model are compared with recent experimental data for phase transitions in FeRh., Comment: 20 pages, 6 figures, 3 tables

All-optical switching of the magnetization in magnetic nanostructures by femtosecond circularly polarized laser pulses has been demonstrated in several systems. We present a Landau-Lifshitz-Lambda (LLL) model which describes the magnetization dynamics using three density states: two ferromagnetic grounds states and an excited optical state. One of the ferromagnetic ground states is optically excited by circularly polarized light to a spin reversed state, which is then "Coulomb collapsed" to the magnetization reversed ground state. The time evolution of the optically excited states is described by a Lindblad master equation, in which the optical excitation is introduced via the Hamiltonian. Dissipation terms are introduced via Lindblad operators. The LLL model combines the precessional motion of the magnetization described by the Landau-Lifshitz theory, with the response of the three level Lambda system. The optical excitation lasts for the duration of the laser pulse and the system relaxes at a fast rate due to the electron-electron interaction. We study the solution of the eigenvalues problem of the optical equation of motion for the magnetization and identify a coherent and incoherent regimes and derive an LLL model that can be integrated with existing micromagnetic codes to describe optical excitation of magnetic materials., Comment: 22 pages, 9 figures

The field of magnonics, which aims at using spin waves as carriers in data processing devices, has attracted increasing interest in recent years. We present and study micromagnetically a nonlinear nanoscale magnonic ring resonator device for enabling implementations of magnonic logic gates and neuromorphic magnonic circuits. In the linear regime, this device efficiently suppresses spin-wave transmission using the phenomenon of critical resonant coupling, thus exhibiting the behavior of a notch filter. By increasing the spin-wave input power, the resonance frequency is shifted leading to transmission curves, depending on the frequency, reminiscent of the activation functions of neurons or showing the characteristics of a power limiter. An analytical theory is developed to describe the transmission curve of magnonic ring resonators in the linear and nonlinear regimes and validated by a comprehensive micromagnetic study. The proposed magnonic ring resonator provides a multi-functional nonlinear building block for unconventional magnonic circuits., Comment: 21 pages, 6 figures

A formulation for thermal noise in the stochastic form of the Landau Lifshitz Bloch equation used for modeling the magnetization dynamics at elevated temperatures is presented. The diffusion coefficients for thermal fluctuations are obtained via the Fokker-Plank equation using the mean field approximation of the field in defining the free energy. The presented model leads to a mean magnetization consistent with the equilibrium magnetization for small and large particles. The distribution of the magnetization magnitude is of a Poisson-like type rather than the classical Boltzmann distribution. The presented model was tested by studying the equilibrium magnetization in macrospin particles at high temperatures. The model is appealing for multi-scale modeling, such as modeling heat assisted magnetic recording systems and all-optical magnetization reversal, Comment: 16 pages, 4 figures, 1 appendix

We describe the inclusion of electrodynamic fields in Time-Dependent Density Functional Theory (TDDFT) by incorporating both the induced scalar and vector potentials within the time-dependent Kohn-Sham equation. The Hamiltonian is described in both the Coulomb and Lorenz gauges, and the advantages of the latter are outlined. Integral expressions are defined for the retarded potentials of each gauge and a methodological approach to evaluate these nontrivial expressions with low computational cost is adopted. Various molecular structures of relatively small sizes are studied, including water, benzene, and conductive carbon chains. Absorption cross sections resulting from both pulse and boost excitations suggest a preserved gauge-invariance of the proposed formal approach to TDDFT in the weak magnetic field limit.

We report on experimental and computational investigations of the domain structure of ~0.2 x 0.2 x 8 {\mu}m single-crystal Ni nanowires (NWs). The Ni NWs were grown by a thermal chemical vapor deposition technique that results in highly-oriented single-crystal structures on amorphous SiOx coated Si substrates. Magnetoresistance measurements of the Ni NWs suggest the average magnetization points largely off the NW long axis at zero field. X-ray photoemission electron microscopy images show a well-defined periodic magnetization pattern along the surface of the nanowires with a period of {\lambda} = 250 nm. Finite element micromagnetic simulations reveal that an oscillatory magnetization configuration with a period closely matching experimental observation ({\lambda} = 240 nm) is obtainable at remanence. This magnetization configuration involves a periodic array of alternating chirality vortex domains distributed along the length of the NW. Vortex formation is attributable to the cubic anisotropy of the single crystal Ni NW system and its reduced structural dimensions. The periodic alternating chirality vortex state is a topologically protected metastable state, analogous to an array of 360{\deg} domain walls in a thin strip. Simulations show that other remanent states are also possible, depending on the field history. Effects of material properties and strain on the vortex pattern are investigated. It is shown that at reduced cubic anisotropy vortices are no longer stable, while negative uniaxial anisotropy and magnetoelastic effects in the presence of compressive biaxial strain contribute to vortex formation., Comment: 15 pages, 11 figures

Recently, Lorentz transmission electron microscopy (LTEM) has helped researchers advance the emerging field of magnetic skyrmions. These magnetic quasi-particles, composed of topologically non-trivial magnetization textures, have a large potential for application as information carriers in low-power memory and logic devices. LTEM is one of a very few techniques for direct real space imaging of magnetic features at the nanoscale. For Fresnel-contrast LTEM, the transport of intensity equation (TIE) is the tool of choice for quantitative reconstruction of the local magnetic induction through the sample thickness. Typically this analysis requires collection of at least three images.Here we show that for uniform thin magnetic films which includes many skyrmionic samples, the magnetic induction can be quantitatively determined from a single defocused image using a simplified TIE approach.

We introduce a new fundamental property of waveguides induced by the forces of the guided light, namely, the ability to self align or be in instability. A nanoscale waveguide broken by an offset and a gap may tend to self align to form a continuous waveguide. Conversely, depending on the geometry and light polarization, the two parts of the waveguide may be deflected away from each other, thus being in an unstable state. These effects are unique as they rely on the presence of both the guided mode and the scattered light. Strong self alignment forces may be facilitated by near field interaction with polarization surface charges.

Hybrid spectral-spatial representations are introduced to rapidly calculate periodic scalar and dyadic Green's functions of the Helmholtz equation for 2D and 3D configurations with a 1D (linear) periodicity. The presented schemes work seamlessly for any observation location near the array and for any practical array periodicities, including electrically small and large periodicities. The representations are based on the expansion of the periodic Green's functions in terms of the continuous spectral integrals over the transverse (to the array) spectral parameters. To achieve high convergence and numerical efficiency, the introduced integral representations are cast in a hybrid form in terms of (i) a small number of contributions due to sources located around the unit cell of interest, (ii) a small number of symmetric combinations of the Floquet modes, and (iii) an integral evaluated along the steepest descent path (SDP). The SDP integral is regularized by extracting the singular behavior near the saddle point of the integrand and integrating the extracted components in closed form. Efficient quadrature rules are established to evaluate this integral using a small number of quadrature nodes with arbitrary small error for a wide range of structure parameters. Strengths of the introduced approach are demonstrated via extensive numerical examples., Comment: 28 pages, 8 figures, accepted for publication in IEEE Transactions on Antennas and Propagation