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Non-linear parametric excitation phenomena of magnons, with their interesting properties, can be utilized to realize magnonic devices for futuristic computation [1,2] and processing of microwave signals [3]. However, for robust device-design, much more remains to be understood regarding the effects of noise on magnons. The modification of the parametric resonance effects, in the presence of an additional signal, is a problem of both theoretical as well as practical interest. We use micromagnetics to model an yttrium iron garnet film of dimensions 5.1 μm x 100 μm x 1250 μm, employing 128x128x1 cells. A parallel-pumping configuration that scatters magnons largely in a plane perpendicular to the bias-field is used, allowing the use of a 2D simulation paradigm [3]. Saturation magnetization of 145 emu/cm3, exchange constant 3.77e-7 ergs/cm, and intrinsic-damping 0.0007 along with 2D periodic boundary conditions and thermal-fields (300 K) are used. A bias field of 200 Oe and a primary frequency of 6 GHz are used. The secondary signal is polarized parallel to the bias-field. Introduction of a secondary signal is found to limit the non-linearity by increasing the threshold-field. This effect, as shown in Fig. 1, is found to dominate at a frequency that corresponds to the relaxation rate of the longitudinal magnetization. Oscillatory behavior of the threshold with secondary frequency is noticeable especially for higher frequencies. These features have been explained analytically and are physically brought about by the modification of the relative phase of the magnons with respect to the microwave photons. The magnon modes that are dominant differ from the original 3 GHz modes by integer multiples of half the secondary frequency. The variation in the number of such modes w.r.t secondary frequency is also studied. We also examine phase noise and find that it affects the magnon instability by reducing the non-linearity and growth rate of magnons as shown in Fig. 2.References: 1. A. Chumak, A. Serga, B. Hillebrands, Nat. Comm. 5, 4700, 2014. 2. A. Khitun, M. Bao, K. L. Wang, J. Phys. D: Appl. Phys. 43, 264005, 2010. 3. A. Venugopal, T. Qu and R. H. Victora, IEEE Trans. Microw. Theory and Techn., vol. 68, no. 2, pp. 602-610, Feb. 2020.

MMM 2020

Modification of Magnon Threshold and Characteristics Using a Secondary Signal and Phase Noise

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The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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The perpendicular magnetic media has the potential to improve data storage beyond 1 Tb/in2. Improving data density is a major challenge in data recording industries due to its sustainability and performance issues [1]. In the present study, by employing Ti as under-layer (UL) and spacer-layer (SL) the bulk perpendicular magnetic anisotropy (PMA) of Tb-Fe thin films is found to be improved considerably. 100 nm thick Tb-Fe film devoid of UL exhibited a maze-like domain pattern with strong out-of-plane (OOP) contrast. With 10 nm thin Ti UL, the domain width, and the OOP contrast of Tb-Fe (100 nm) film were found to improve considerably (fig. 1). Accordingly, the coercive field for Tb-Fe film with Ti UL was doubled when compared with the bare Tb-Fe film. The possible reason for the enhancement in OOP properties could be due to the lateral extent of d-d hybridization that exists at the interface. Similarly, PMA and domain areal density also increased by placing a 2 nm thin Ti SL: Tb-Fe (100 nm) / Ti (2 nm) / Tb-Fe (100 nm) (fig. 1). Here, we observed a large dipolar coupling that exists between the top and bottom Tb-Fe layers. The Hc of about 0.5 T observed for bare Tb-Fe (200 nm), was found to increase to about 1 T for the Tb-Fe films with Ti SL. Interestingly, Tb-Fe film with SL exhibited a hybrid domain pattern probably due to the bottom Tb-Fe layer. Thus, Ti when used either as UL or SL improved the OOP magnetic properties of Tb-Fe thin films considerably. The fundamental observation and understanding will have a significant impact on perpendicular media which may also useful in other areas of spintronic applications.References: 1. K. Tang, et. al, IEEE transactions on magnetics, Vol. 45 p. 786-792 (2009).

Effect of Ti as an Under and Spacer-layer on the Perpendicular Magnetic Anisotropy of Tb-Fe Thin Film

Atomistic spin models [1] provide a natural bridge between electronic and micromagnetic length scales, where the magnetic properties are described by localised atomic spins within the Heisenberg model. The model parameters can be calculated ab-initio but the macroscopic magnetization dynamics and properties can also be simulated. While powerful, atomistic models are computationally expensive. A particular difficulty is the calculation of long-range dipole-dipole interactions, arising from the stray magnetic fields of localised atomic moments and causes complex magnetisation structures, such as magnetic vortices or Skyrmions. The dipole -dipole interactions are typically solved using a Fast-Fourier-Transform (FFT) [2] or the Fast Multipole Method (FMM) [3]. However, for atomistic simulations the continuum approximation does not accurately reflect the local dipole-dipole interactions at the atomic level and leads to systematic errors in the calculation of the dipole field. A brute force atomistic dipole-dipole approach is infeasible for the time (nanosecond+) and length (100 nm+) scales needed to extract useful data. Here we present modifications to the VAMPIRE software package [1] to include accurate and scalable calculations of dipole-dipole interactions for improved performance and scalability on ARCHER, the UK national supercomputer. We have developed a space-time hierarchical calculation of the dipole field calculation with order NlogN scalability with problem size, a comparison to the brute force tensor approach [4] is shown in Fig. 1. Our novel approach combines exact near-field computation of the dipole field with an M-Level hierarchical representation of the far-field dipole field [5]. The method is hierarchical in time and space, since the far-field time-evolution of the magnetization is much slower than the near-field which changes on the femtosecond timescale. Our improved algorithm enables the inclusion of accurate dipole-dipole interactions in atomistic simulations for large system sizes in the micrometre range for the first time.References: [1] R. F. L. Evans et al, J. Phys.: Condens. Matter 26, 103202 (2014) [2] S. Yuan and H. Bertram, IEEE Trans. Mag. 28, 2031 (1992) [3] G. Brown et al, IEEE Trans. Mag. 40 2146 (2004) [4] G.J. Bowden et al., J. Phys. Condens. Matter 28, 066001 (2016) [5] J.J. Miles and B.K. Middleton, J. Magn. Magn. Mater. 95, 99 (1991)

Scalable space and time hierarchical dipole-dipole interactions in the VAMPIRE code

With the development of electronic low-dimension systems, numerical modeling has become a powerful method for studying and predicting the physical properties of materials. To model dynamic magnetic systems the micromagnetic approach [1] and the methods of GPU acceleration of calculations on the video core of the system [2] are successfully applied. In our work, this approach is used to simulate the effect of nonlinear autoresonant excitation [3, 4] of nanoscale films of yttrium iron garnet (YIG) by a weak external magnetic field. The proposed analytical equations for autoresonance do not take into account a number of important real conditions, for example: attenuation, three-dimensional geometry of the sample, dipole – dipole interaction of spins, surface anisotropy, etc., which does not allow one to directly verify the theoretical model in practice. The aim of the work was to find the conditions for the excitation of autoresonance in a film with some real physical parameters using micromagnetic simulation in the MUMAX3 software. As a planar model, a YIG film with a size of 1600 × 1600 × 160 nm was chosen. The film thickness was taken to satisfy the conditions of a single-mode spin wave excitation and proximity to real planar samples. During simulations the sample was separated into rectangular elements in the form of cells, the interactions between them were calculated by using the Landau-Lifshitz equation. Since the YIG has the minimum level of attenuation of the magnetic subsystem, the damping parameter was excluded at this stage. Then the sample saturated along the axis of easy magnetization by H = 20 kOe was excited by a weak perpendicular pump field h with an amplitude of about 1 Oe. As a result, the stable nonlinear regime of excitation of magnetic oscillations of high magnitude in the film by a chirp rate of the pump field frequency was obtained in in the MUMAX3 simulations for the first time - the autoresonance effect. For the modelled YIG film the optimal chirp rate was a 4.3×10^16 s^(– 2). Finally, the full conditions for the autoresonance excitation of magnetization in the quasi-three-dimensional magnetic film by numerical methods were defined. Supported by the RFBR grant No. 19-32-90014.References: 1 A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez // AIP Advances. 2014. Vol. 4. P. 107133. 2 V. S. Teplov, V. D. Bessonov. Resource and Mechanics of materials and structures. 2018. V. 6.P. 222—228. 3. S.V. Batalov, A. G. Shagalov. Physics of Metals and Metallography, 2010. Vol 109, N1, P. 3-8 . 4 S,V. Batalov, A. G. Shagalov.. Physics of Metals and Metallography, 2013, Vol 114, N 2, P. 115–119

Micromagnetic modeling magnetization excitation by a resonance

Spherical nanoparticles with hollow structure have received much attention because of their unique magnetic features reflecting their large surface area [1,2]. Recent works suggested the formation of a spin vortex for hollow spherical submicron Fe3O4 particles, whose stability is enhanced with increasing the diameter [3]. In this work, the formation mechanism of spin vortexes have been studied for a hollow submicron sphere using micromagnetic simulations, where possible realization of a double vortex reflecting the hollow structure was revealed. The magnetization reversal and magnetic configurations under magnetic fields along the [100] or [111] axis were studied using OOMMF for a single hollow Fe3O4 spherical particle with the varying diameter (d = 300 - 700 nm) and inner/outer diameter ratio (γ = 0 - 0.75). Electron holography was also performed for 400- nm hollow spheres to directly characterize a magnetic state. Figure 1 shows a typical example of the M-H curves for a hollow sphere (d = 300 nm, γ = 0.5). Before entering a single vortex state (phase IV) typically observed in a solid sphere, two types of vortex structures are formed for a hollow sphere, associated with multi-step magnetization changes. We find that as the magnetic field is reduced from high fields, a clockwise and anti-clockwise vortices are formed at the ends of the sphere (phase II) and then transform into a single vortex with cores displaced by changing a magnetic field (phase III). The formation of such double vortices can be due to reduced Zeeman energy by spins at and around the equatorial plane of the sphere. With increasing d or γ, a field range, where the phases II and III are stabilized, expands for both [100] and [111] field orientations. Magnetic phase images obtained by electron holography revealed no significant stray field around the hollow sphere in zero fields, being consistent with the formation of a spin vortex.References: 1] H. Khurashid et al., Scientific Reports 5, 15054 (2015) 2] Y. Liang et al., Colloids and Surfaces A 530 191-199 (2017) 3] M. Chiba et al., Journal of Magnetism and Magnetic Materials 512 167012 (2020)

Possible realization of double spin vortex state in hollow submicron particles

The ground state of nanoscale circular magnetic disks of certain geometric aspect ratios is a spontaneously forming stable vortex configuration with circulating in-plane magnetization and a vortex core pointing out-of-plane. Resonantly exciting the vortex core via either an rf magnetic field or an rf spin-polarized current yields a gyrotropic motion around its equilibrium position, characterised by a specific eigen-frequency, which depends on the material parameters and the disk geometry [1]. Such oscillations, which can be read out via periodic magnetoresistance changes, generate rf signals with high quality factors (>10000) in the sub-GHz bandwidth [2,3]. While all these features make vortex-based nano-oscillators interesting as nanoscale rf sources, the major drawback remains their low frequency tunability associated with the linear characteristics of the gyrotropic mode. Here, we investigate, by micromagnetic simulations, the role of magnetostriction in improving the tunability of vortex nano-oscillators. Specifically, we consider a double-disk structure comprising magnetostrictive (CoFe) and non-magnetostrictive (Py) layers separated vertically by a non-magnetic (NM) spacer. We show that, when the two vortices have different eigen-frequencies and the magnetostatic coupling between them is sufficiently strong, the stress-induced magnetoelastic anisotropy can lead to the synchronized gyration of the two vortex cores (Fig. 1). The stress-induced transition from double-frequency to single-frequency dynamics is mostly controlled by the polarization of the vortices and the magnetostatic coupling strength (i.e. spacer thickness). These findings offer a frequency tunability of vortex-based oscillators via mechanical stress, which can be generated and controlled electrically, for example, using piezoelectric substrates [4]. Funding from the EU Horizon 2020 project No. 737038 (TRANSPIRE) is acknowledged.References: [1] K. Yu. Guslienko, et al, J. Appl. Phys. 91, 8037 (2002). [2] A. Dussaux, et al, Nat. Commun. 1, 1 (2010). [3] N. Locatelli, et al, Appl. Phys. Lett. 98, 062501 (2011). [4] M. Filianina, et al, Appl. Phys. Lett. 115, 062404 (2019).

Stress-Induced Modification of Gyration Eigen-Frequencies in Stacked Double-Vortex Structures

Recent improvements in nanoparticle synthesis have allowed the application of magnetic nanoparticles (MNPs) in many fields where particle size control is key, such as data storage, spintronics or biomedicine, in which drug delivery and cancer treatment using hyperthermia stand out. In MNP systems, shape can trigger magnetostatic anisotropy effects induced by the demagnetizing factor. This feature is most notorious in highly degenerated systems on which shape anisotropy can be induced to alter properties like coercive field and remanence of non-interacting MNPs. In addition, magnetostatic dipolar interactions between MNPs can be tuned in disordered and crystalline structures by adding non-magnetic coatings, which can also control the lattice parameter, in a core-shell scheme, thus affecting the effective anisotropy of the system. This work focuses on the angular study of magnetization reversal processes in quasi-spherical (ε~1) single MNPs with different shapes and magnetic properties. Simulations show different hysteresis loops for different sphericalnesses of the MNPs, which can be related to the presence of different magnetization pinning events at the sharp edges of less spherical MNPs[Fig. 1]. Moreover, the effect of dipolar interactions has been tested out with simulations of the remanent state in MNP superlattices with simple, body-centered and face-centered cubic arrangements[Fig. 2].References: D.L. Huber, small., Vol. 1(5), p.482-501 (2005) D. Kim et al., Journal of Magnetism and Magnetic Materials., Vol. 320(19), p.2390-2396 (2008) Q. Li et al., Scientific Reports., Vol. 7, p.9894 (2017) D. Peddis et al., Chemistry of Materials., Vol. 25(10), p.2005-2013 (2013) A. Vansteenkiste et al., AIP Advances, Vol. 4, p.107133 (2014)

Shape-related Effects on Ferromagnetic Nanoparticles: A Micromagnetic Approach for Highly Diluted and Highly Ordered Systems

Inhomogeneous magnetic phases like magnetic vortex are expected to open up a wide range of opportunities both from a fundamental and technological perspective. Asymmetric nanostructures offer alternative data storage possibilities by suggesting a control over the vortex formation and thereby chirality [1-3]. Asymmetry introduced in the geometry also has a strong impact on magnetization reversal. Increased coercivity, remanence, and switching field distribution with the degree of asymmetry establishes the existence of an induced anisotropy, which in turn could even destabilize the magnetic vortex depending on the degree of asymmetry in a particular geometry [4]. Following these ideas, in this work, we investigate how the asymmetry modifies the magnetization reversal in hemispherical shells. We address this issue by breaking the symmetry of non-interacting hemispherical cap structures through a thickness modulation inclined at an angle with the symmetry plane and by analyzing the magnetization reversal mechanism for varying thicknesses. Micromagnetic simulations indicate that the asymmetric nanocaps reverse through the propagation of a curling state, while in the symmetric caps, magnetization reversal happens via coherent rotation. With an increase in thickness, the field at which curling state nucleates, shifts towards a higher applied field. The non-uniform magnetization reversal is apparently due to the altered energy contributions resulting from the asymmetry introduced in the nanocaps.References: [1] M. Schneider, H. Hoffmann, and J. Zweck, Appl. Phys. Lett., Vol. 79, p.3113 (2001) [2] D. B. Gopman, Y. P. Kabanov, J. Cui, et al., Appl. Phys. Lett., Vol. 109, p.082407 (2016) [3] J. Rothman, M. Kläui, L. Lopez-Diaz, et al., Phys. Rev. Lett., Vol. 86, p.1098 (2001) [4] K. M. Wu, J. F. Wang, Y. H. Wu, et al., J. Appl. Phys., Vol. 103, p.07F314 (2008)

Asymmetry Induced Inhomogeneous Magnetization Reversal in Hemispherical Shells

Achieving a strong perpendicular magnetic anisotropy is of great importance for materials intended for use in spintronics devices. Perpendicular anisotropy in magnetic multilayers arises from the broken symmetry at the interfaces due to the enhanced orbital and spin moments relative to the bulk system1. In order to accurately describe the quasi-static properties of these materials using a macrospin approach, the inclusion of both second-order (K2) fourth-order (K4) uniaxial anisotropies is essential. Beik Mohammadi et al2 have shown that lateral inhomogeneities of the second-order uniaxial anisotropy in thin films with perpendicular uniaxial anisotropy can give rise to higher-order anisotropy contributions. Here, we report on the study of the layer to layer variation of the second-order perpendicular anisotropy in magnetic bilayers and its influence on the quasi-static properties. We analyze the results of micromagnetic simulations3 in magnetic bilayers using a macrospin model. We show that in order to describe the micromagnetic simulation data using macrospin model, one requires a fourth-order uniaxial anisotropy contribution although the underlaying micromagnetic model does not include such term. Quasi-static calculations show that the fourth-order uniaxial anisotropy is caused by the deviation of the magnetization from the applied field direction.References: 1. J. M. Shaw, H. T. Nembach and T. J. Silva, Physical Review B 87 (5), 054416 (2013). 2. J. Mohammadi, K. Cole, T. Mewes and C. Mewes, Physical Review B 97 (1), 014434 (2018). 3. T. Mewes, C. Mewes, MATLAB based micromagnetics code M3, (2018). URL http://magneticslab.ua.edu/micromagnetics-code.html

Micromagnetic Simulations of the Quasi-static Properties of Dissimilar Bilayers

Two-dimensional (2D) magnetic materials, such as chromium trihalides and manganese dichalcogenides, have recently drawn immense attention of both theoretical and experimental research, due to both fundamental significance and promising technological applications. Particularly, some chromium trihalides have been shown able to host terahertz spin-waves (SW) [1] and are promising candidates for ultra-fast information transport and processing based on magnons. In this work we investigated the spin-wave propagation in the presence of strategically-placed defects (halide vacancies) in the honeycomb structure of chromium trihalides. We performed Density Functional Theory (DFT) calculations to obtain the magnetic parameters of the considered structures, followed by spin-dynamics simulations of the SW propagation. We reveal that the lattice defects host strong Dzyaloshinskii–Moriya interaction (DMI), so that a designed pattern of defects can serve as a spin-wave guide. We show the spectra of spin-waves propagating across periodic defect lines that prove such structures can work as a magnonic crystal [2] for terahertz SWs, exhibiting key features such as band gaps where SWs are not allowed to propagate. Such broad degree of manipulations available in 2D materials therefore suggests these systems as a front-runner for terahertz magnonics, applicable in cutting-edge devices.References: [1] W. Jin, H. Kim, Z. Ye, et al. Nature communications, 9, 5122 (2018). [2] A. V. Chumak, A. A. Serga, and B. Hillebrands. Journal of Physics D: Applied Physics, 50, 244001 (2017).

Terahertz Magnonics in 2D Chromium Trihalides with Atomically-Engineered Dzyaloshinskii–Moriya Interaction

Square Artificial Spin Ices (S-ASI)1-3 are the 2-dimensional realization of the standard 3-dimensional spin ice material of the type Dy2Ti2O7 (pyrochlore compound), with the alluring possibility of studying geometric frustration in a controlled way. In addition to the frustration, these kinds of structures enable the investigation of a variety of fascinating phenomena such as frustration, emergent magnetic monopoles, collective magnetization dynamics of interacting spins, and phase transition, etc. Recently, there is a focus on the understanding of dynamical behavior of spins or spin wave (sw) modes in such dipolar coupled systems. Detailed understanding of such behavior may be of importance to the applications in magnonics4,5. Here, we have investigated the sw modes of strongly dipolar coupled, highly anisotropic nanoislands forming S-ASI system using micro-magnetic simulation in MUMAX3 in combination with MATLAB coding. For the same, we have systematically scrutinized the sw spectra of the building block of S-ASI which is a square ring-like structure of highly shape anisotropic nanoislands of permalloy6,7. The evolution of the spectra as a function of no. of rings of up to four rings forming a S-ASI with closed edges exhibit a direct relation between the sw modes and the micro-states of the system. The study reveals that a single ring-type structure can alone be adequately used to understand the sw modes of S-ASI as well as it’s behavior in the presence of an external bias field. All the ring-type structures studied show ubiquitous switching identifier mode at approximately 14 GHz. Detailed analysis of the spin wave propagation modes in the building block as well as individual vertices of S-ASI structure will be discussed in this work.References: [1] P.W. Anderson, Phys. Rev., Vol. 102, p.1008 (1956). [2] C. Castelnovo et al., Nature, Vol. 451, p.42–45 (2008). [3] L. Balents, Nature Vol. 464, p.199–208 (2010). [4] E. Iacocca et al., Phys. Rev. B, Vol. 93, p.134420 (2016). [5] S. Mamica et al., Adv. Cond. Matter Phys., Vol. 2012 (2012). [6] N. Keswani and P. Das, AIP Adv., Vol. 8, p.101501 (2018). [7] N. Keswani et al., Appl. Phys. Lett., Vol 116, p. 102401 (2020).

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Effect of Ti as an Under and Spacer-layer on the Perpendicular Magnetic Anisotropy of Tb-Fe Thin Film

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