United States

The recent convergence of chiral molecules with metal halide perovskite frameworks gives rise to an interesting family of chiral systems: two-dimensional, chiral hybrid organic-inorganic perovskites (2D-chiral-HOIPs). While possessing photovoltaic properties of traditional HOIPs, this class of materials is endowed with chirality through its organic ligands in which the degeneracy of the electron spin in charge transport is broken, i.e., the Chirality-Induced Spin Selectivity (CISS) effect manifests, making it a promising platform to bridge opto-spintronic studies and the CISS effect. In this work, 2D-chiral-HOIP/NiFe heterostructures are studied by means of the magneto-optical Kerr effect using a Sagnac interferometer. Upon illumination of the HOIPs, the Kerr signal at the HOIP / NiFe interface changes, and a linear dependence of the response on the magnetic field is observed. The sign of the slope was found to depend on the chirality of the HOIPs. The results demonstrate the utility of chiral HOIP materials for chiral Opto-spintronic applications. D.S. acknowledges the support from National Science Foundation ECCS 1936527. D.H.W. acknowledges the support of the Department of Energy (Grant No. ER46430). Work at Utah is supported by US Department of Energy (Grant No. DE-FG02-04ER46148). F.L. acknowledges NERSC and CHPC at the University of Utah for providing the computing resources.

MMM 2020

Magneto-Optical Detection of Photoinduced Magnetism via Chirality Induced Spin Selectivity in 2D Chiral Hybrid Organic-Inorganic Perovskites

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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technical paper

Permanent magnet dipole can generate magnetic field higher than the Br of the material with structures such as the Halbach array [1] and Stelter array [2]. These arrays usually generate magnetic field inside a cavity. And the magnetic field is perpendicular to the axis of the cavity. For applications such as Faraday Effect and erasing hard disks perpendicular media, a strong in-plane magnetic field is required. In the case of erasing hard disks, the disk needs to rotate in the air gap with magnetic field perpendicular to the surface of the disks. This requires that the dipole must open along the field direction. And it is often difficult to generate strong magnetic field when magnets can only be placed around the path of the magnetic flux. This paper presents a permanent magnet dipole design for strong in-plane magnetic field. The design starts with a Halbach structure that generates strong magnetic field in the cavity. To create access to the cavity from the in-plane field direction, a pair of magnets with orientation opposite to the magnetic field of the Halbach dipole are substituted along the field direction. The magnetic field from the substitute magnets is analyzed using current sheet equivalent method. This field is then subtracted from the Halbach magnet to generate the magnetic field from the new structure. We then used Bound Analysis software to simulate the structure. For the prototype, we choose an 8-segment Halbach magnet as the basic structure and used triangular magnets so that the magnets fit inside a rectangular steel housing. The magnetic material is NdFeB 48 MGOe with Br of 13600 G. Fig 1 shows the prototype dipole and the magnetic field in the air gap. In conclusion, this paper proposes a permanent magnet dipole with strong in-plane magnetic field. The prototype dipole generates strong in-plane magnetic field that is in line with the predictions from the analytical equation as well as our BEA analysis. This strong dipole can be used in many applications such as Faraday Effect and hard disk erase. The design is granted with US patent 10529362 [3]References: [1] K.Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt materials”, Nucl. Instru. and Methods, Vol.169, pp 1-10 (1980) [2] Richard E. Stelter, US patent 5635889, 1997 [3] Chun Li, US patent 10,529,362, 2020

Permanent Magnet Dipole Design with Strong In-plane Magnetic Field

I Introduction Accurate modeling of the hysteresis characteristics of magnetic materials is of great significance to analysis the electromagnetic performance of electrical equipment[1]. The Preisach model is widely used in the simulation of the magnetic characteristic of transformers through the FEM calculation, while its parameter identification problem is still a complicated task[2]. This paper proposed a new parameter identification method based on an intelligent optimization algorithm with the explicit-form Everett function, which can speed the identification process. II Method and Discussion Firstly, the hysteresis loops of the B30P105 steel under quasi-static conditions are measured according to the IEC60404-2; Secondly, the Preisach model is built through the explicit-form Everett function. The explicit-form Everett function transformed the integral on the Preisach plane into the sum of a series of exponential functions which gives an explicit expression of the Everett function. To simulate the hysteresis loops of the material by the proposed method, the parameters identification problems of the explicit function need to be fully investigated. The optimization process of calculating the Everett function is shown in Figure 1 (where a, b, c, γ, Hs, Ms are the parameters of the proposed model); Finally, a Velocity-Controlled Particle Swarm (VCPSO) algorithm, which only needs the limiting hysteresis loop of the experimental data is developed and implemented to find the optimal parameters of the model. The simulation results are in good agreement with the experimental data as shown in Figure 2. III Conclusion A new parameter identification method of the Preisach model with explicit Everett function is proposed. By using the VCPSO algorithm, a fast parameter identification process, which only needs the limiting loop of the experimental data is developed. The proposed method are proved to be very efficient and accurate to construct the Preisach model, which has great significance on fast simulating magnetic performance of the transformer when considering hysteresis properties.References: [1] X. Zhao, X. Liu, Z. Zhao, X. Zou, Y. Xiao, and G. Li, "Measurement and modeling of hysteresis characteristics in ferromagnetic materials under DC magnetizations," Aip Advances, vol. 9, no. 2, Feb 2019, Art no. 025111, doi: 10.1063/1.5054948. [2] S. Hussain and D. A. Lowther, "An Efficient Implementation of the Classical Preisach Model," IEEE Trans. Magn., vol. 54, no. 3, Mar 2018, Art no. 7300204, doi: 10.1109/tmag.2017.2748100.

Parameter Identification of Preisach Model Based on Velocity-Controlled Particle Swarm Optimization Method

Millimeter-wave electronics are crucial for next-generation (5G) electronic communication systems. However, the development of desirable millimeter (mm) wave absorbers remains an ongoing challenge due to the difficulties of fabricating suitable magnetic absorbers in mm regimes. For ideal mm wave magnetic absorbers, materials should have 1) large anisotropy field (HA) and 2) saturation magnetization (Ms). The resonance frequency is proportional to HA and thus materials must have a large HA are essential for their resonance frequency to be located in mm wave range (20-100 GHz)1. However, the permeability decreases with increasing resonance frequency according to Snoek’s limit2. Thus, materials with larger Ms, are also required to mitigate this challenge. Combing both properties will enable us to achieve enhanced absorption intensity and broaden the bandwidth at mm waves. Rather than focusing on optimization of a single material, here we employ exchange coupled core/shell strategies to generate composite structures with both large Ms and HA. Here, we have fabricated novel magnetic core/shell nanostructures with the goal to realize a highly efficient mm wave absorber. Specifically, we have successfully synthesized ε-Fe2O3 core particles and ε-Fe2O3/Co core/shell nanoparticles using wet-chemistry methods specifically via sol-gel and thermal decomposition of metal precursors. The structural analysis of core and core/shell structures was performed by TEM, HRTEM and EDX, revealing well-defined interfaces and good crystal structure. Furthermore, the coercivity of resulting ε-Fe2O3 core particles is in the range of 2 – 18 kOe, which is significantly affected by the synthetic conditions. Details of the magnetic and the frequency response from the mm wave range of ε-Fe2O3 core and ε-Fe2O3/Co core/shell nanostructures performed by DC-magnetometry, X-ray magnetic circular dichroism and custom built vector network analyzer system will be discussed in the presentation. Prepared by LLNL under Contract DE-AC52-07NA27344.References: 1. S. Ohkoshi, S. Kuroki, S. Sakurai, K. Masumoto, K. Sato, S. Sasaki, Angew. Chem. 119, 8544-8547 (2007). 2. J. V. I. Timonen, R. H. A. Ras, O. Ikkala, M. Oksanen, E. Seppälä, K. Chalapat, J. Li, G.S. Poraoanu, Magnetic Nanocomposites at Microwave Frequencies. In Trends in Nanophysics: Theory, Experiment and Technology; Bârsan, V., Aldea, A., Eds.; Springer Berlin Heidelberg: Berlin, Heidelberg, pp 257-285 (2010).

Development of magnetic core/shell nanostructures for 5G mm wave absorbers.

We have investigated the control of effective magnetic anisotropy in Permalloy (Ni80Fe20) linear chain arrays by tuning the symmetry arrangements of the ellipsoidal nanomagnets and the film thickness. The coupled ellipsoidal nanomagnets are arranged in two distinct configurations (Fig. 1 (a) and (b)), namely: type-I, where elements are arranged along the major axis of the ellipsoid, and type-II, in which the neighboring elements are coupled along the minor axis of the ellipsoid [1, 2]. Hence, the shape anisotropy of each nanomagnet and the configurational anisotropy due to the lattice arrangements favor the same direction of magnetization for type-I arrays while competition exists between the shape and configurational anisotropy for type-II arrays. The hysteresis loops depicted in Fig. 2 (a) and (c) clearly show the variation of the effective anisotropy with thickness which is also supported by the ferromagnetic resonance (FMR) spectra, shown in Fig. 2 (b) and (d) for type-I and type-II arrays respectively. A clear transition from a single domain states to a combination of complex flux closure states such as vortex, double vortices are observed at different applied field angles (φ) when the film thickness is varied in the range from 20 nm to 100 nm. Tunable microwave absorption spectra established the complex interplay between the shape anisotropy and magnetostatic interactions at different film thicknesses and applied field angles. Micromagnetic simulations are in good agreement with the experimental results. Our results demonstrate the possible ways of manipulating the effective magnetic anisotropy in the arrays of nanomagnets for magnonics and microwave applications.References: [1] S. Jain, A. O. Adeyeye, N. Singh, Nanotechnology 2010, 21 (28), 285702. [2] A. Talapatra, A. O. Adeyeye, Linear Chains of Nanomagnets: Engineering the Effective Magnetic Anisotropy (submitted).

Linear Chains of Nanomagnets: Engineering the Effective Magnetic Anisotropy

For next generation wireless systems requirements for multi-gigabit speed communication, researchers are focusing on millimeter wave frequency band up to 100 GHz for potential solution. Meanwhile hexagonal barium and strontium ferrite material are attracting increasing attention because its Ferromagnetic Resonance (FMR) frequency is at the vicinity of aforementioned frequency band of interest. Other features such as large saturation magnetization (4πMs), high anisotropy field (HA), low dielectric loss and most importantly, its potential of achieving self-bias, paves the way for fabrication of integrated, planar ferrite devices that can play a crucial role in next generation wireless system. BaFe12O19 (BaM) and SrFe12O19 (SrM) are M-type (magnetoplumbite structure) hexaferrites, anisotropic hard magnetic oxides which have great technological importance as low-cost permanent magnets and maintain great scientific interest due to potential use in magnetic recording and millimeter wave (MMW) filters and absorbers. Many electromagnetic properties of materials change as particle sizes are reduced to the nanoscale. Due to high magneto crystalline anisotropy, both BaM and SrM hexaferrites are expected to have narrow (20 Oe) spin resonances in millimeter-wave frequency range, while YIG and nickel ferrites have rather wide (300 Oe and wider) ferromagnetic resonance characteristics. Measurements conducted confirm the idea of using BaM and SrM ferrite materials as sharp, narrow ferromagnetic resonance-based devices, operating in millimeter waves. Our recent study reveal that the grain size of commercially available nano ferrite powders must be greater than single domain dimension (about 80-100 nanometers in diameter) to have any sharp ferromagnetic resonance to appear in millimeter waves. These broadband MMW transmittance measurements were performed using a quasi-optical MMW spectrometer and a series of backwards wave oscillators encompassing the 30–120 GHz range. BaM and SrM powders can be further investigated by DC magnetization, electron microscopy, x-ray diffraction, and light scattering to assess magnetic behavior, size, size distribution, and structure.References: Millimeter wave hexagonal nano-ferrite circulator on silicon CMOS substrate Liu Chao; Hassan Oukacha; Enjin Fu; Valencia Joyner Koomson; Mohammed N. Afsar 2014 IEEE MTT-S International Microwave Symposium (IMS2014) Year: 2014 Pages: 1 - 4, DOI: 10.1109/MWSYM.2014.6848467 Effects of particle size on magnetic and dielectric properties of ferrites at microwave and millimeterwave frequencies Anjali Sharma and Mohammed N. Afsar 2013 European Microwave Conference Year: 2013 Pages: 786 - 789

Self-Biased Hexagonal Ferrite Thin Film on GaN Substrate for Millimeter-Wave Devices

Atomically thin, magnetic materials have gained a lot of attention since 2017, when the first 2D ferromagnet was reported [1]. This breakthrough has triggered research on 2D magnetic materials [2]. They are not only important from a fundamental point of view -to understand the theory of magnetism in reduced dimensions-, but also for technological applications. However, probing the magnetic order of the 2D systems by conventional magnetic experimental setups is very challenging. On the other hand, it is well known, that even in the single layer limit, semiconducting two-dimensional materials strongly absorb light. Therefore, optical spectroscopy is a good method for their characterization. In order to shed light on the intriguing phenomena of 2D magnetism, we present comprehensive theoretical investigations of the optical properties of the van der Waals layered material MnPS3, which is one important example from the large family of transition metal phosphorus trisulfide (MPS3) [3]. Our study reveals, that the interband absorption spectrum, which is proportional to the imaginary part of the dielectric function, is very similar for different possible magnetic states of MnPS3. On the other hand, the calculated effective masses of electrons and holes exhibit an anisotropic behaviour and they strongly depend on the magnetic order. Aforementioned properties are reflected in the binding energy of excitons in the studied systems. The impact of the magnetic order on electronic band gap is also revealed, and indirect character of electronic band gap is confirmed as the groundstate of bulk system. Our results demonstrate that the magnetic order of 2D materials can be indirectly inferred via optical measurements. The study was accomplished thanks to the funds allotted by the National Science Centre, Poland within the framework of the research project ‘SONATA12’ no. UMO-2016/23/D/ST3/03446. Access to computing facilities of TU Dresden ZIH for the project ”TransPheMat”, PL-Grid Polish Infrastructure for Supporting Computational Science in the European Research Space, and of the Interdisciplinary Center of Modeling (ICM), University of Warsaw are gratefully acknowledgedReferences: [1] B. Huan et al., Nature 546, p. 270-273 (2017) [2] M. Gibertini et al., Nat. Nanot. vol. 14, p. 408 (2019). [3] R. Brec, Solid State Ionics vol. 22, p. 3-30 (1986).

Controlling magnetic properties of the van der Waals crystal MnPS3 via the absorption spectrum.

Magneto-optical (MO) devices based on the Faraday effect are widely used for optical communication and optical sensing. Due to their weak MO effect in the optical frequency range, the size of MO devices is typically in the range of 10~1000 wavelengths, limiting their application in integrated photonic systems. Currently, most studies for enhancing the Faraday effect are based on surface plasmon resonance.1, 2 However, due to the Ohmic loss of metals, these structures are optically lossy. Therefore, all dielectric nanostructures provide an alternative route to achieve high Faraday rotation and high transparency simultaneously.3,4 Here, we report theoretical and experimental study on enhanced Faraday effect in all dielectric MO metasurfaces based on Si/Ce:YIG nanostructures. The device is composed of a 2D periodic array of amorphous Si nanodisks patterned on Ce:YIG/YIG oxide thin film stacks grown on silica. By adjusting the geometric parameters, magnetic dipole and electric dipole Mie resonance modes can be excited in the Si disks. Thanks to the high refractive index of Ce:YIG/YIG, the modes are also strongly confined in the Ce:YIG/YIG layers. The transmission spectrum is simulated using the finite element time-domain difference method, which agrees excellently with experimental results. The Faraday rotation spectrum of the sample was measured using a home-made microscale Faraday rotation characterization set-up, which shows obvious enhancement peaks in the vicinity of each resonance. The MO response of the Ce:YIG thin film deposited on the same silica substrate is also presented as a reference. With the transmission close to 40%, we achieved 2.3-fold Faraday rotation enhancement compared to bare Ce:YIG thin films. Our device shows effective MO enhancement in the sub-wavelength scale, which is promising for applications such as free space non-reciprocal photonic devices and MO imaging.References: 1. Liu, M. and Zhang, X., Nature Photonics., Vol. 7, p.430-431 (2013) 2. Armelles, G. and Cebollada, A., Advanced Optical Materials., Vol. 1, p.10-35 (2013) 3. Christofi, A., Kawaguchi, Y. and Alu, A., Optics Letters, Vol. 43, p.1838-1841 (2018) 4. Royer, F., Varghese, B. and Gamet, E., ACS omega, Vol. 5, p.2886-2892 (2020)

Enhanced Magneto-Optical Effect in Ce:YIG Based All Dielectric Metasurfaces

Observation of All-Optical Switching (AOS) of magnetization in ferrimagnetic GdFeCo alloys, using ultrashort circularly polarized laser pulses [1], triggered a renewed interest in ferrimagnetic rare Earth-transition metal (RE-TM) alloys [1-5]. Tb-containing ferrimagnetic alloys exhibit all-optical helicity-dependent switching (AO-HDS), and have strong perpendicular magnetic anisotropy, a result of a high spin-orbit coupling in Tb [6]. In this work we study the magnetic properties and AOS performance of amorphous TbCo alloys, prepared by a DC magnetron sputtering (for more details see [7]). We measured μ0Hc using polar magneto-optical Kerr effect (MOKE), and investigated the AOS behaviour using a pump-probe based static magneto-optical imaging setup. We show that AO-HDS is possible in TbxCo100−x for certain compositions (x=24-30 at.%) and thicknesses (Fig. 1), if the laser pulse parameters such as pump fluence, are optimized, and the number of pulses in one burst is at least 50-100. Samples with the compositions out of this range show only helicity-independent thermal demagnetization. We also performed ab initio calculations of the inverse Faraday effect to explain the cumulative nature of the multi-shot AO-HDS switching. It is triggered by circularly polarized laser light, inducing a helicity dependent magnetic moment in the Co atoms of the alloys (Fig. 2). This study shows that ferrimagnetic TbCo films can exhibit AO-HDS in a certain compositional range and laser pulse parameters, with a minimal amount of shots.References: 1. C. D. Stanciu et al Phys. Rev. Lett. 99, 047601 (2007) 2. B. Hebler et al, Frontiers in Materials 3, 8 (2016). 3. S. Alebrand et al Applied Physics Letters101,162408 (2012). 4. S. Mangin et al, Nature Materials 13, 286 (2014). 5. C. H. Lambert et al, Science 345, 1337 (2014) 6. R. Moreno et al, Phys. Rev. B 96, 014409 (2017). 7. A. Ciuciulkaite et al. arXiv:2002.07544

Design of TbxCo100-x for All Optical Switching of Magnetization

On-chip optical isolators are key components in photonic integrated circuits (PICs). In recent years, passive magneto-optical (MO) isolators based on nonreciprocal phase shift (NRPS) have been widely studied [1]. Micro-ring and Mach-Zehnder interferometer (MZI) isolators have achieved considerable progress on silicon on insulator (SOI) platforms [2, 3]. As a different material platform, silicon nitride (SiN) based PICs show much lower propagation loss, which are highly promising for telecommunication, data communication and sensing applications [4, 5]. However, high performance MO isolators have not been demonstrated on SiN, which is fundamentally limited by the low Faraday rotation and weak NRPS for MO thin films grown on SiN waveguides. Here, we report high-performance integrated MO isolators on SiN platforms for both TM and TE polarizations using cerium doped yttrium iron garnet (Ce: YIG) thin films deposited by pulsed laser deposition (PLD). By increasing the Ce solubility up to 40% [6], Ce:YIG thin films grown on SiN show high Faraday rotation up to -5500 deg/cm at 1550 nm wavelength, which is 83% higher than previously reported Ce1Y2Fe5O12 thin films [7, 8]. Based on these materials, broadband SiN MZI isolators are designed and fabricated. The TM isolators show 24 dB isolation and 4 dB insertion loss around 1567 nm wavelength, with a footprint of 0.9×0.25 mm2. For TE isolators, a maximum isolation ratio of 28 dB with insertion loss of 7.5 dB is observed at 1565 nm wavelength. These devices have exceeded the performance of their counterparts on SOI [7]. Our work demonstrates that high performance MO thin films and optical isolators can be monolithically integrated on SiN PICs.References: [1] Bi L, H.J. and Jiang P, Nature Photonics., Vol. 5, p.758 (2011) [2] Shoji Y, F.A. and M. T, IEEE Journal of Selected Topics in Quantum Electronics., Vol. 22, p.264-270 (2016) [3] Pintus P, H.D. and Morton P A, Journal of Lightwave Technology., Vol. 37, p.1463-1473 (2019) [4] Baets R, S.A.Z and Clemmen S, Optical Fiber Communication Conference., p. Th3J-1 (2016) [5] Gaeta A L, Lipson M and Kippenberg T J, Nature Photonics., Vol. 13, p.158-169 (2019) [6] Zhang Y, Wang C T and Liang X, Journal of Alloys and Compounds., Vol. 703, p.591-599 (2017) [7] Zhang Y, Du Q and Wang C, Optica., Vol. 6, pp.473-478 (2019) [8] Du Q, Wang C and Zhang Y, ACS Photonics., Vol. 5, p.5010-5016 (2018)

Waveguide Integrated Broadband Magneto-Optical Isolators on Silicon Nitride Platforms

Ten years ago, it was shown that ultrafast laser-induced heating of a thin film of amorphous Gd(FeCo)3 can magnetically switch the irradiated area [1-2]. A train of light pulses toggles the magnetization between two equivalent directions perpendicular to the film. Since then an intense research effort, both experimental and theoretical, has sought to unravel the underlying mechanisms of this single-pulse, all-optical switching (SP-AOS). Empirically it was established that the material needs to be ferrimagnetic with two magnetic lattices that demagnetize at different rates. As the initial, purely thermal demagnetisation rate is proportional to the local atomic moments on the sublattices, efforts were therefore focused on the gadolinium/transition metal alloys where the moments differs by a factor of almost four. It is therefore surprising that a second material, where we recently discovered SP-AOS [3], was the XA-ordered ferrimagnetic Heusler alloy Mn2RuxGa (MRG) where both magnetic sublattices are composed of manganese, with similar atomic moments. In this talk, we discuss how ~200 femtosecond light pulses toggle the magnetization of MRG.[3]. MRG is a half-metallic ferrimagnet with two antiparallel sublattices made up of manganese occupying inequivalent 4a and 4c positions in the F-43m space group[4]. On account of its unusual electronic structure, 4c electrons at the Fermi level dominate both magneto-optic and magneto-transport properties [5-6]. The sublattice moments compensate at a temperature Tcomp that depends on x and the thin film preparation conditions. Usually, at low temperatures the moment of the 4c sublattice is bigger than that of the 4a sublattice, but as the temperature rises, it falls off faster than 4a, and the two cross at Tcomp where they are equal but opposite. Unlike Gd(FeCo)3, which exhibits toggle switching for a range of initial temperatures ranging from 100 K below to 100 K above its compensation temperature, because the difference in atomic moments ensures the TM sublattice is demagnetised before the RE one, MRG will only switch when the initial temperature lies below Tcomp. This suggests that exchange relaxation plays a much more important role in the switching of MRG. MRG is a versatile material magnetic material. Its compensation temperature can easily be tuned and it is impervious to high magnetic fields in the vicinity of Tcomp, where the coercive field rises to 10 T. Furthermore, it possesses a large anomalous Hall effect and a very high intrinsic spin-orbit torque that may be useful for spintronic applications [7-8]. The observation of SP-AOS in a material with these properties opens both new technological prospects, and new challenges for understanding the fundamental non-equilibrium electronic processes involved.References: 1. Radu et al., Nature, Vol. 472, p. 205 (2011) 2. Ostler at al., Nature Communications, Vol. 3, p. 666 (2020) 3. C. Banerjee et al., arXiv, Vol. 1909, p. 09085 (2020) 4. H. Kurt et al., Physical Review Letters, Vol. 112, p. 027201 (2014) 5. K. Fleisher et al., Physical Review B, Vol. 98, p. 134445 (2018) 6. K. E. Siewierska et al., IEEE Transactions on Magnetics, Vol. 55, p. 2 (2019) 7. N. Thiyagarajah et al., Applied Physics Letters, Vol. 106, p. 122402 (2015) 8. S. Lenne et al., arXiv, Vol 1903, p. 04432 (2019)

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Magneto-Optical Detection of Photoinduced Magnetism via Chirality Induced Spin Selectivity in 2D Chiral Hybrid Organic-Inorganic Perovskites

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Permanent Magnet Dipole Design with Strong In-plane Magnetic Field

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Linear Chains of Nanomagnets: Engineering the Effective Magnetic Anisotropy

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