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Ferromagnetic materials are used in a wide range of electromagnetic applications (energy converters, sensors, inductances …). In this domain, we observed a growing interest in the development of simulation tools reducing the experimental campaigns and improving the knowledge and the performances. Accurate simulation results can only be obtained by coupling precise electromagnetic equations to the exact material laws (hysteresis, saturation, frequency dependence) [1]-[3]. Under the influence of an external magnetic excitation, the local magnetic state through the ferromagnetic specimen is ruled by the combination of both the magnetic domain kinetics and the external magnetic field diffusion. The usual methods for the simulation of the magnetic behavior are all based on the separation of the magnetic contributions, where the microscopic Eddy currents due to the domain wall motions and the macroscopic ones due to the extern magnetic field variations are considered separately [4]. This separation remains artificial, since practically both losses mechanisms occurs simultaneously and interact on each other [5]. In this study, an alternative solution is proposed through the resolution of an anomalous fractional magnetic field diffusion (1, 2 or 3D depending on the experimental situation). The fractional order constitutes an additional degree of freedom in the simulation scheme [6]-[8]. It is identified through comparisons to experimental results. By adjusting precisely this order, very accurate local and global simulation results can be obtained on a very broad frequency bandwidth. It allows to predict precisely the dynamic magnetic behavior of classic ferromagnetic components.References: [1] B. Zhang et al., IEEE Transaction on Magnetics, vol. 54, iss. 11 (2018) [2] B. Zhang et al., IEEE Transaction on Magnetics, Vol. 54, iss. 3, (2017). [3] B. Gupta et al., IEEE Transaction on Magnetics, Vol. 54, Iss. 3, (2018). [4] G. Bertotti, IEEE Transaction on Magnetics, vol. 24, iss. 1, pp. 621-630, (1980). [5] M. A. Raulet et al., IEEE Transactions on Magnetics, Vol. 40, iss 2, pp. 872-875, (2004). [6] B. Ducharne et al., International journal of dynamics and control, vol. 6, pp. 89-96, (2018). [7] B. Ducharne et al., Journal of Applied Physics, vol. 117, iss. 24, (2015). [8] B. Ducharne et al., The European Physical Journal Plus, 135:325, (2020).

MMM 2020

Fractional equation for the anomalous magnetic diffusion in ferromagnetic materials

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In the present investigation, the magnetic and electromagnetic character of cobalt substituted M-type barium hexaferrite with composition BaCoxFe12-xO19 (x = 0.0 to1.0 in the steps of 0.2) by co-precipitation method has been studied. After the appropriate heat treatments, the structural, magnetic and electromagnetic properties at X band ranging from 8-12 GHz are thoroughly examined. XRD confirmed the formation of M phase and revealed that with an increase in the content of cobalt ions the average crystallite size is found to be increased from 28 to 45 nm whereas the morphology examined from FESEM and TEM confirmed the formation of high quality and densely packed nano ferrites. From all the six samples the composition BaCo0.8Fe11.2O19 exhibit remarkable magnetic and electromagnetic properties and found to be one of the best compositions. For this particular composition very higher value of coercivity of the order of 3789 Oe along with good saturation magnetisation value of 65 emu/g is observed. Similar kinds of results are observed form electromagnetic properties of the same composition. Again the composition BaCo0.8Fe11.2O19 showed some excellent and sustainable values of complex permittivity and tangent losses i.e 11, 0.024 and 0.035. So based on the magnetic and electromagnetic performances, the composition BaCo0.8Fe11.2O19 may emerge as an ideal candidate for high-speed recording, permanent magnet applications, microwave absorption at high-frequency applications ranges from 8-12 GHz and many more.References: Robert C Puller, Progress in Material Science., 57, 1191-1334 (2012). K. Kumar and D. Pandey, Physical Review B., 96, 024102 (2017). H. Lou, S. Sun, X. Lu, C. He, Y. Pan, Q. Song, W. Xuejin, W. Jianjiang, J Mater Sci: Mater Electron., 27, 11231–11240 (2016). A. Baykal, I. A. Auwal, S. Guner, H. Sozeri,J. Magn. Magn Mater. 430, 29-35 (2017). S Kumar, M K Manglam, S Supriya and H Kumar et.al., J Magn. Magn. Mater., 473, 312-319 (2019).

Investigation of Magnetic and Electromagnetic Properties of Cobalt Substituted M-type Barium Hexaferrite in X-band.

The increasing demand of permanent magnets (PM) based technology going along with the limitation of global supply of rare-earths (RE) elements drives the development of RE-free PM [1]. Based on the material requirements regarding magnetic properties, costs and criticality, Mn-Al(-C) alloys have been pointed as candidates for PM applications. The hard magnetic properties originate from the metastable τ-phase which is usually formed during annealing from the parent high temperature ε-phase [2,3]. However, the annealing parameters have to be optimized in order to avoid the decomposition of the ferromagnetic phase into the nonmagnetic stable phases in the Mn-Al system. In this context, the Spark Plasma Sintering (SPS) can be a useful technique to optimize this process by combining uniaxial pressure and fast heating/cooling rates due to efficient Joule heating, caused by direct current pulses, and possible side effects related to electric field/current [4]. In this study, we investigate the transformation from ε- to τ-Mn54Al44C2 using the SPS and compared the results with conventional annealing. The ε- to τ-phase transformation has been done by SPS at different temperatures (400, 450, 500 and 550°C) and different times (5 and 15 min). A pure τ-MnAlC is obtained at 500°C/5min (Figure 1 (a)). Below this temperature there is still presence of ε-MnAlC and above we observe the formation of undesired β-phase. On the other hand, conventional heat treatment shows a full transformation at 500°C only after 30 min. Magnetic properties are dependent on the phase formation (Figure 1 (b)), but similar coercivity and saturation magnetization are obtained for the pure τ-MnAlC formed either by SPS or by conventional annealing. However, the SPS method allows to significantly shorter the time necessary to complete the transformation and limits the grain growth.References: [1] K.P. Skokov, O. Gutfleisch, Scripta Mater., 154 289 (2018) [2] V. Øygarden et al., J. Alloys Compd. 779 776 (2019) [3] H. Jian et al., J. Alloys Compd. 622 524 (2015) [4] O. Guillon et al., Adv. Eng. Mater., 16 830 (2014)

Formation and Magnetic Properties of τ-MnAlC Synthesized by Spark Plasma Sintering Process

Hard magnetic ferrites have broad applications ranging from permanent magnets and data storage media to high-frequency devices. The only known ferrite material exhibiting giant coercivity over 20 kOe at room temperatures is epsilon-iron oxide (ε-Fe2O3). In addition, it displays sub-THz natural ferromagnetic resonance (NFMR) frequency of 182 GHz. The Rh-doping of the material leads to an increase in both the NFMR up to 209 GHz and the coercivity up to 27 kOe [1]. However, since ε-Fe2O3 is a metastable polymorph and its production as a pure phase is quite complex, it still has not been applied in industry. Recently we have reported [2] an effective method for production of single-domain M-type hexaferrite particles with composition of Sr0.67Ca0.33Fe8Al4O19, which demonstrated coercivity of 21.3 kOe and saturation magnetization of 15 emu/g, which is close to ε-Fe2O3. The proposed preparation technique is based on citrate-nitrate auto-combustion method; therefore, it is simple, economical, and readily scalable, and it can be efficiently integrated into modern ferrite technology. Herein, we show that further increase of Al substitution degree leads to significant improvement of coercivity (up to 36 kOe) and NFMR frequency (up to 250 GHz), which are record-high values for ferrites to date [3]. Furthermore, we discuss temperature dependence of the magnetic and microwave properties, crystal and magnetic structures of the ultra-hard magnetic hexaferrites Sr1−x/12Cax/12Fe12−xAlxO19 (x = 3 – 5.5) in a range of 4.2 – 700 K. Temperature dependencies of both NFMR and coercivity have a maximum shifted to low temperatures with increase of aluminium content. For x = 5.5 the compound coercivity is raised from 36 kOe at 300 K up to 41 kOe at 180 K. Furthermore, we developed a technique for preparation of dense ceramics with single-domain grains and studied their room temperature magnetic properties and NFMR. The obtained ceramic materials possess coercivity higher than 18 kOe at specific density of 67 - 95%. NFMR frequency of the ceramics is 200 GHz, which is 20% blue-shifted compared to that of a powder sample. The ferrite ceramics with a coercivity of more than 10 kOe and sub-terahertz absorption were obtained for the first time. The work was supported by RFBR grant No. 20-02-00887.References: [1] A. Namai, et al, Nature Communications, Vol. 3, p. 1035, (2012). [2] L.A. Trusov, E.A. Gorbachev, et al, Chemical Communications, Vol. 54, p. 479-482 (2018). [3] E.A. Gorbachev, L.A. Trusov, et al, Materials Today, Vol. 32, p. 13-18 (2020).

Novel ferrite materials possessing record magnetic hardness

Alnico permanent magnets contain a number of major alloying additions such as Fe, Co, and Ni, along with some other minor elemental additions; this alloying convention is similar to the high entropy alloys (HEAs) concept [1]. In this paper, we investigate the hard magnetic properties of FeCoNiAl-based HEAs with additions of Cu/Ti. Additions of both Cu and Ti to an equimolar FeCoNiAl alloy are more effective at enhancing coercivity compared to Cu only, but they also decrease the saturation magnetization and remanence (Figure 1). By changing the Fe/Co amounts, however, we can regain some of these properties. Fe2CoNiAlCu0.4Ti0.4 shows promising hard magnetic properties as an isotropic, cast magnet, with a coercivity (HC) of 1.1 kOe and maximum energy product, (BH)max, of 2.06 MGOe, which is slightly better than the performance of isotropic, cast Alnico 2 magnets [2]. The thermal stability is also sufficient to be used at temperatures over 200 °C. The drastic change in HC might be strongly associated with spinodal decomposition at annealing temperatures of ~650-700 °C for all samples (Figure 2).References: [1] D. B. Miracle and O. N. Senkov, Acta Mater. Vol. 122, p.448 (2017). [2] https://en.wikipedia.org/wiki/Alnico.

Hard magnetic properties of FeCoNiAlCuxTix based high entropy alloys

The coherent transfer of information between different waveform is an important ingredient of information technology as it allows transport of quantum states without loss of frequency or phase. An efficient hybridization, though, requires to reach the strong coupling regime, where the interaction rate between two collective states becomes larger than their relaxation rates. This has revived interest for insulating magnetic garnets with low magnetic and acoustic dampings. It was recently shown that the circularly polarized phonons in dielectric crystals can transfer angular momentum over millimeters due to the strong magnon phonon coupling in a dielectric spin valve system, which is composed of two yittrium iron garnet (YIG) layers on both sides of gadolium gallium garnet substrate [1]. The evidence of this coupling is the emergence of a bright and a dark collective states that can be revealed through an interference pattern in the ferromagnetic resonance signal. In this work, we will focus on the collective behavior when the splitting between two Kittel modes is within the coupling strength. We tune the splitting either i) by varying the polar angle of an external magnetic field or ii) by applying a temperature gradient along the stack direction. We describe their mutual coupling as provided by a dissipative cavity, which represents the nearly degenerate standing acoustic shear wave mode that resonates across the whole crystal thickness. The reactive part of this mutual coupling is responsible for the usual level repulsion between the two modes. The contrast between the bright and dark states enhances as the splitting between two Kittel modes reduces. However, when the zero detuning becomes degenerate with an acoustic resonance, the dissipative part dominates over the reactive part and it leads to partial level attraction between the two collective modes. From the experimentally measured voltage contrast between two frequency tones, we extract a qualitative behavior of the coupling strength as a function of detuning, which reproduces the reactive and dissipative part of the phonon mediated coupling. We show that our results can be explained by three coupled oscillator model.References: [1] K. An, A. N. Litvinenko, R. Kohno et al., Physical Review B 101.6 060407 (2020)

Resonant Coupling between Magnetic Layers by Acoustic Phonons

Non-linear interactions related to magnons are expected to be applied to advanced computing devices [1,2]. Synthetic antiferromagnets (SAFs) are candidate materials to observe non-linear magnon interactions because they have two fundamental magnon modes, that are the acoustic (AC) and optical (OP) modes [3]. However, there are no reports whether the AC and OP mode can interact with each other. Here, we report parametric amplification of the AC mode by the OP mode in SAFs using time-resolved magneto-optical Kerr effect (TRMOKE) technique [4]. SAF films, Si/SiO2 (subs.)/Ta(3)/Co20Fe60B20(3)/Ru(0.4)/Co20Fe60B20(3)/Ta(3) (thickness is in nm), was prepared by a sputtering. Fig. 1(a) shows the TRMOKE data with various pump laser power Ppump at the critical magnetic field angle θH,c . At this angle, 2fAC = fOP is satisfied, where fAC and fOP are the frequency of the AC and OP mode, respectively. ΔθK and Δt are the change in the Kerr rotation angle and the pump-probe delay time, respectively. In case of Ppump > 5 mW, the temporal amplification of the magnetization precession was observed (dashed curves). Fig. 1(b) shows the θH dependence of the peak intensity of the FFT spectrum at the Ppump = 10 mW for the AC and OP mode, respectively. At around θH = 45°~ θH,c , the intensity of the AC mode rapidly enhanced. This indicates the non-linear interaction between the AC and OP modes is allowed for SAFs. The threshold of the OP mode to the amplification will be also discussed. This work was partially supported by KAKENHI (Nos. 26103004 and 19K15430). A.K. thanks to GP-Spin at Tohoku University.References: [1] S. Manipatruni et al., Nat. Phys. 14, 338 (2018). [2] J. Torrejon et al., Nature 547, 428 (2017). [3] A. Kamimaki et al., Appl. Phys. Lett. 115, 132402 (2019). [4] A. Kamimaki et al., Phys. Rev. Appl. 13, 044036 (2020).

Parametric Amplification of Magnons in Synthetic Antiferromagnets

The physics of nonlinear magnon scattering has been exploited in nonlinear microwave devices for wireless and satellite communication applications. The latent state, which is the time delay in the magnon response to a microwave pulse, is required to be reduced for rapid information transfer. We have studied the latent state in time-resolved nonlinear magnon scattering in thin films, in both theory and experiment. The experiment is performed using phase-sensitive time-resolved heterodyne ferromagnetic resonance. The theory is a hybrid time-resolved model which uses an analytical equation of motion based on the Holstein-Primakoff transformation, and realistic micromagnetic simulation. From the experiment in Fig. 1(a), we find the latent state is extraordinarily sensitive to the power. When the power is near the threshold, the lifetime t0 of the latent state is of order ten microseconds. At high powers, t0 decreases sharply to a hundred nanosecond. Our simulation shows consistent behavior of the latent state in the same dynamic power range in Fig. 1(b). From capturing the time-resolved magnon number of all the existing modes, we find that the origin of the latent state is the magnon redistribution: the k=0 mode takes time to scatter into non-uniform magnum modes k≠0. The power controls the scattering rate from k=0 to k≠0, where the rate is time dependent on the magnon number. We thank DARPA, XSEDE, and the SMART center (funded by SRC and NIST) for support of this project.

Latent state in time-resolved nonlinear magnon scattering in thin films

The aim of ab-initio spin dynamics is to go beyond the limitations of standard atomistic spin dynamics, which is based on a mapping of the original electronic Hamiltonian to an effective spin Hamiltonian. This spin Hamiltonian assumes constant magnetic moment lengths and constant interaction parameters and is therefore only valid in the magnetic ground-state configuration. Recent studies [1] have shown that non-collinearity in the magnetic state reduces the spin-spin interaction and gives improved accuracy compared to experiment. However, the proposed method in Ref. [1] is not applicable for the general case and reveals limitations of the above mentioned mapping [2]. In ab-initio spin dynamics the magnetic moment lengths and the quasi-classical torques acting on the atomic magnetic moments are calculated from a quasi-equilibrium electronic ground state within the adiabatic approximation [3], where constraining fields are required to stabilize non-collinear magnetic moment configurations [4]. The constraining field has therefore to cancel the internal effective magnetic field, which is expected to be minus the gradient of the energy with respect to the magnetization. In practical calculations the constraints on the moment directions are often only implemented approximately without constraining fields, which can introduce serious errors [5]. We present surprising numerical results from constrained density functional (cDFT) calculations which demonstrate that the constraining field and the effective field obtained from the energy gradient are not exactly opposite, as would be expected [4]. Furthermore, we provide a theoretical explanation of this discrepancy based on a theorem that relates the constraining field and the energy gradient: the constraining field theorem. We argue that in the case of cDFT the correct effective magnetic field is actually given by the energy gradient and not by the constraining field.References: [1] Szilva et al., Phys. Rev. Lett. 111, 127204 (2013). [2] Szilva et al., Phys. Rev. B 96, 144413 (2017). [3] Antropov et al., Phys. Rev. Lett. 75, 729 (1995). [4] Stocks et al., Philos. Mag. B 78, 665 (1998). [5] Singer et al., Phys. Rev. B 71, 214435 (2005).

Discrepancy Between Constraining and Effective Fields in Ab-Initio Spin Dynamics

There exists a wide array of uses for ferromagnetic materials with easy-axis magnetic anisotropy, including motors, magnetic imaging devices, and acoustic devices. As most of the highest quality permanent magnets are financially and environmentally costly, the search for alternative magnetic materials is ongoing and paramount. Magnetic elements are rare, and the best way to predict novel magnetic materials is to constrain a search to analyzing the magnetic properties of compounds containing iron, cobalt, and manganese. However, while many databases exist for the purposes of analyzing structural properties of compounds, any such database of magnetic materials that exists is either outdated or incomplete. The magnetic properties of 10,760 compounds containing the element cobalt are being gathered through a literature search, with the goal to eventually complete the entire list, as well as those of iron and manganese compounds. As of now, a total of 1,282 ferromagnetic compounds have been found and catalogued. Structure types, lattice parameters, cobalt percentage, and lattice angles of each compound were also collected; the shortest cobalt-cobalt bond lengths within the crystal lattice were calculated; and data trends were analyzed to predict novel ferromagnetic and anisotropic compounds. From this data, a previously unstudied member of the Cu2MnAl family was confirmed to be ferromagnetic with a Curie temperature above room temperature. Currently, compounds in the MgZn2, La6Co11Ga3, and Mg3Cd families are being studied to check for ferromagnetism and uniaxial anisotropy. The research was supported by the Critical Materials Institute, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office.

Novel Database Driven Method for Discovering New Co-based Ferromagnets INVITED

First-order reversal-curves (FORCs) are a powerful tool, which is increasingly used in material science and nano-magnetism as well as ferroelectricity, geology, archeology, and for spin-transition materials [1-4]. Ideally, it can access microscopic distributions of interaction and coercive fields without the need for lateral resolution [5]. Unfortunately, the reliable data analysis and interpretation poses a major challenge. This is why FORC is often seen as a magnetic fingerprint instead of a measurement method yielding quantitative information. To push past these limitations, we present a fast and user-independent analysis algorithm and possibilities to interpret the resulting FORC diagrams beyond the Preisach model. We present a new evaluation approach which exploits the diversity of Fourier space to not only speed up the calculation by a factor of 1000 but also move away from the conventional smoothing factor towards real field resolution. By comparing the baseline resolution of the new and the old algorithm we are able to deduce an analytical equation which converts the smoothing factor into field resolution making the old and new algorithm comparable. We find excellent agreement not only for various systems of increasing complexity but also over a large range of smoothing factors. The achieved speed up enables us to calculate a large number of first-order reversal-curve diagrams with increasing smoothing factor allowing for an autocorrelation approach to find a hard criterion for the optimum smoothing factor. This previously computational prohibitive evaluation of first-order reversal-curves solves the problem of over- and undersmoothing by increasing general readability and preventing information destruction. However, these measured FORC densities are not always straightforward to interpret, especially if the system is interaction dominated and the Preisach-like interpretation of the FORC density breaks down. To understand additional features arising from the interactions in the system, we purposely designed permalloy microstructures which violate the Mayergoyz criteria [7]. These artificial systems allow us to isolate the origin of an additional interaction peak in the FORC density. Modeling the system as a superposition of dipoles allows us to extract interaction strength parameters from this static simulation. Additionally, we suggest a linear relation between integrated interaction peak volume and interaction strength within the system. The presented correlation could be used to investigate the interaction behavior of samples as a function of structural parameters within a series of FORC measurements. This is an important step towards a more quantitative understanding of FORCs which violate the Mayergoyz criteria and away from a fingerprint interpretation.References: [1] J. Gräfe, Phys. Rev. B, 93 (2016) 014406 [2] F. Groß, Phys. Rev. B, 99 (2019) 064401 [3] C.-I. Dobrota, J. Appl. Phys., 113 (2013) 043928 [4] C. R. Pike, Phys. Rev. B, 71 (2005) 134407 [5] J. Gräfe, Rev. Sci. Instr., 85 (2014) 023901 [6] D. Cimpoesu, J. Appl. Phys., 125 (2019) 023906 [7] I. D. Mayergoyz, Phys. Rev. Lett., 56 (1986) 1518

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Investigation of Magnetic and Electromagnetic Properties of Cobalt Substituted M-type Barium Hexaferrite in X-band.

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