France



We use micromagnetic simulations based on the stochastic Landau-Lifshitz-Gilbert (sLLG) equation to calculate dynamic magnetic hysteresis loops relevant to magnetic nanoparticle hyperthermia.&lt;br/&gt;&lt;br/&gt;Our simulation model is inspired by a preclinical study reported by Dennis et al. [1] using iron oxide nanoparticles for cancer treatment in mice. The nanoparticles consist of nanorods nominally 6.7 nm x 20 nm x 47 nm in size. Instead of simulating atomic spins, micromagnetics suggests assuming uniformly magnetized cells with dimensions of the cubic unit cell (&lt;i&gt;a&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;=0.839 nm), resulting in a rod composed of 10752 cells.&lt;br/&gt;&lt;br/&gt;The approximation that all spins within a cell are uniformly aligned via a relatively strong exchange interaction is valid at zero temperature for cell sizes smaller than the exchange length. At finite temperature, spin decorrelation must be taken into account, and a coarse-graining procedure is required to determine how micromagnetic parameters vary with cell size and temperature.&lt;br/&gt;&lt;br/&gt;To avoid unwanted heating of surrounding healthy tissues due to Eddy currents, restrictions on the applied field frequency and amplitude limit the field sweep rate to SR=4&lt;i&gt;H&lt;/i&gt;&lt;sub&gt;max&lt;/sub&gt; &lt;i&gt;f&lt;/i&gt; &amp;lt; 0.25 Oe/ns [2, 3]. Simulating a rod made of cells with size &lt;i&gt;a&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt; takes months, so using fewer but larger cells is computationally attractive. Our study models the rods using larger cells (of length &lt;i&gt;a&lt;/i&gt;&lt;sub&gt;b&lt;/sub&gt; = b &lt;i&gt;a&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt; for b &amp;gt; 1) to be able to simulate complex nanoparticles in a reasonable time (see Fig. 1).&lt;br/&gt;&lt;br/&gt;As the cell size affects the system’s simulated properties, magnetic parameters need to be scaled to keep the model’s magnetic response, i.e., &lt;i&gt;m&lt;/i&gt;-&lt;i&gt;H&lt;/i&gt; hysteresis loops, invariant with the simulation cell size. Scaling also appears within the realm of phase transitions and critical phenomena, driven by thermodynamic considerations, resulting in the powerful renormalization group (RG) approach based on the correlation length. In the present study, we use the RG approach of Grinstein and Koch [4] to determine how the field H(b), anisotropy constant K(b), and exchange constant A(b) should scale with b (see Fig. 1) at finite temperature. We modify their approach to describe the scaling of the magnetization m(b) by introducing a phenomenological parameter δ that accounts for the nonlinear magnetization temperature dependence [5]. We also extend their approach to include magnetostatic interactions that were not considered in Grinstein and Koch’s work. The scaling algorithm produces nearly identical loops over several decades in the model grain volume, as shown in the top four curves of Fig. 2; for details, see Ref. [6]. Larger cell sizes also allow using a larger Δt when solving the equations of motion. Using the scaling approach decreases the rod’s simulation time from months to days.&lt;br/&gt;&lt;br/&gt;At the heart of the LLG equation is α, responsible for describing energy loss from the macroscopic motions of the local magnetization to microscopic thermal motion [7]. A larger α signifies a faster loss of energy and a shorter relaxation time required to align the magnetic moments to the field and results in a smaller hysteresis loop. Likewise, a slower SR is equivalent to a longer measurement time and consequently yields a smaller loop. Considering Sharrock&#39;s work [8], which relates the coercive field to the relaxation time and attempt frequency, we argue that, for the small α, so long as SR/α = constant, the coercive field should remain the same. In Fig. 2, we plot the &lt;i&gt;m&lt;/i&gt;-&lt;i&gt;H&lt;/i&gt; hysteresis loops for two sets of values for SR and α, such that SR/α =2.5, and interestingly even for large values of α, the loops overlap. Using this approach reduces the simulation time from days to hours, for details see Ref. [5].&lt;br/&gt;&lt;br/&gt;To summarize, we show here that the RG approach of Grinstein and Koch yields a scaling of magnetostatic, exchange and anisotropy parameters that, with some modification for the rescaling of the resulting magnetization, yields finite temperature nanorod hysteresis loops that are, to approximately 10%, invariant with cell size. Scaling results hold even to the point where the nanorod is represented by a single magnetization vector that experiences only anisotropy and self-demagnetization. In addition to the computational speedup resulting from the use of fewer micromagnetic cells, the invariance of loops when SR/α is fixed provides another avenue for computational efficiency by allowing one to use a larger SR than the target value. This paves the way for efficient simulations of nanoparticles (composed of nanorods) and their response to AC fields [6], in support of experimental studies of hyperthermia.

**References:**

[1] C. L. Dennis, et al. &lt;i&gt;Nanotechnology, 20(39),&lt;/i&gt; 395103 (2009).&lt;br/&gt;&lt;br/&gt;[2] S. Dutz, and R. Hergt, &lt;i&gt;Int. J. Hyperth., 29(8), &lt;/i&gt;790-800 (2013).&lt;br/&gt;&lt;br/&gt;[3] R. Hergt, and S. Dutz, &lt;i&gt;J. Magn. Magn. Mater., 311(1),&lt;/i&gt; 187-192 (2007).&lt;br/&gt;&lt;br/&gt;[4] G. Grinstein, and R. H. Koch, &lt;i&gt;Phys. Rev. Lett., 90(20)&lt;/i&gt;, 207201 (2003).&lt;br/&gt;&lt;br/&gt;[5] R. Behbahani, M. L. Plumer and I. Saika-Voivod, &lt;i&gt;J. Phys.: Condens. Matter,&lt;/i&gt; 32, 35LT01 6pp (2020).&lt;br/&gt;&lt;br/&gt;[6] R. Behbahani, M. L. Plumer and I. Saika-Voivod, arXiv:2010.08848 (2020).&lt;br/&gt;&lt;br/&gt;[7] T. L. Gilbert, &lt;i&gt;IEEE Trans. Magn., 40(6)&lt;/i&gt;, 3443-3449 (2004).&lt;br/&gt;&lt;br/&gt;[8] M. Sharrock, and J. McKinney, &lt;i&gt;IEEE Trans. Magn., 17(6),&lt;/i&gt; 3020-3022 (1981).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/53d389db281adb19e35bb9b508f104a5.jpg)
Figure 1. Coarse-grained modelling of a magnetite nanorod. The smallest micromagnetic cell models eight atomic spins within a cubic unit cell of length &lt;i&gt;a&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt; = 0.839 nm with a single magnetic moment. Ultimately, the nanorod is modelled as a single block, corresponding to b = 22. Spherical magnetic nanoparticles studied in Ref. 2 are made up of magnetite nanorods.

![](https://assets.underline.io/uploads/markdown_image/1/image/6ef098f20be94e527108ac4581410602.jpg)
Figure 2: Hysteresis loop invariance.  The top four loops show invariance with cell size when magnetic parameters are scaled according to a modified RG prescription in addition to the magnetostatic interactions scaling. The last two loops show the invariance of MH loops when the ratio SR/α is constant.

INTERMAG 2021

Coarse-graining in micromagnetic simulations of dynamic hysteresis loops

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We use micromagnetic simulations based on the stochastic Landau-Lifshitz-Gilbert (sLLG) equation to calculate dynamic magnetic hysteresis loops relevant to magnetic nanoparticle hyperthermia. Our simulation model is inspired by a preclinical study reported by Dennis et al. [1] using iron oxide nanoparticles for cancer treatment in mice. The nanoparticles consist of nanorods nominally 6.7 nm x 20 nm x 47 nm in size. Instead of simulating atomic spins, micromagnetics suggests assuming uniformly magnetized cells with dimensions of the cubic unit cell (a0=0.839 nm), resulting in a rod composed of 10752 cells. The approximation that all spins within a cell are uniformly aligned via a relatively strong exchange interaction is valid at zero temperature for cell sizes smaller than the exchange length. At finite temperature, spin decorrelation must be taken into account, and a coarse-graining procedure is required to determine how micromagnetic parameters vary with cell size and temperature. To avoid unwanted heating of surrounding healthy tissues due to Eddy currents, restrictions on the applied field frequency and amplitude limit the field sweep rate to SR=4Hmax f &lt; 0.25 Oe/ns [2, 3]. Simulating a rod made of cells with size a0 takes months, so using fewer but larger cells is computationally attractive. Our study models the rods using larger cells (of length ab = b a0 for b &gt; 1) to be able to simulate complex nanoparticles in a reasonable time (see Fig. 1). As the cell size affects the system’s simulated properties, magnetic parameters need to be scaled to keep the model’s magnetic response, i.e., m-H hysteresis loops, invariant with the simulation cell size. Scaling also appears within the realm of phase transitions and critical phenomena, driven by thermodynamic considerations, resulting in the powerful renormalization group (RG) approach based on the correlation length. In the present study, we use the RG approach of Grinstein and Koch [4] to determine how the field H(b), anisotropy constant K(b), and exchange constant A(b) should scale with b (see Fig. 1) at finite temperature. We modify their approach to describe the scaling of the magnetization m(b) by introducing a phenomenological parameter δ that accounts for the nonlinear magnetization temperature dependence [5]. We also extend their approach to include magnetostatic interactions that were not considered in Grinstein and Koch’s work. The scaling algorithm produces nearly identical loops over several decades in the model grain volume, as shown in the top four curves of Fig. 2; for details, see Ref. [6]. Larger cell sizes also allow using a larger Δt when solving the equations of motion. Using the scaling approach decreases the rod’s simulation time from months to days. At the heart of the LLG equation is α, responsible for describing energy loss from the macroscopic motions of the local magnetization to microscopic thermal motion [7]. A larger α signifies a faster loss of energy and a shorter relaxation time required to align the magnetic moments to the field and results in a smaller hysteresis loop. Likewise, a slower SR is equivalent to a longer measurement time and consequently yields a smaller loop. Considering Sharrock's work [8], which relates the coercive field to the relaxation time and attempt frequency, we argue that, for the small α, so long as SR/α = constant, the coercive field should remain the same. In Fig. 2, we plot the m-H hysteresis loops for two sets of values for SR and α, such that SR/α =2.5, and interestingly even for large values of α, the loops overlap. Using this approach reduces the simulation time from days to hours, for details see Ref. [5]. To summarize, we show here that the RG approach of Grinstein and Koch yields a scaling of magnetostatic, exchange and anisotropy parameters that, with some modification for the rescaling of the resulting magnetization, yields finite temperature nanorod hysteresis loops that are, to approximately 10%, invariant with cell size. Scaling results hold even to the point where the nanorod is represented by a single magnetization vector that experiences only anisotropy and self-demagnetization. In addition to the computational speedup resulting from the use of fewer micromagnetic cells, the invariance of loops when SR/α is fixed provides another avenue for computational efficiency by allowing one to use a larger SR than the target value. This paves the way for efficient simulations of nanoparticles (composed of nanorods) and their response to AC fields [6], in support of experimental studies of hyperthermia.

**References:**

[1] C. L. Dennis, et al. Nanotechnology, 20(39), 395103 (2009). [2] S. Dutz, and R. Hergt, Int. J. Hyperth., 29(8), 790-800 (2013). [3] R. Hergt, and S. Dutz, J. Magn. Magn. Mater., 311(1), 187-192 (2007). [4] G. Grinstein, and R. H. Koch, Phys. Rev. Lett., 90(20), 207201 (2003). [5] R. Behbahani, M. L. Plumer and I. Saika-Voivod, J. Phys.: Condens. Matter, 32, 35LT01 6pp (2020). [6] R. Behbahani, M. L. Plumer and I. Saika-Voivod, arXiv:2010.08848 (2020). [7] T. L. Gilbert, IEEE Trans. Magn., 40(6), 3443-3449 (2004). [8] M. Sharrock, and J. McKinney, IEEE Trans. Magn., 17(6), 3020-3022 (1981).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/53d389db281adb19e35bb9b508f104a5.jpg)
Figure 1. Coarse-grained modelling of a magnetite nanorod. The smallest micromagnetic cell models eight atomic spins within a cubic unit cell of length a0 = 0.839 nm with a single magnetic moment. Ultimately, the nanorod is modelled as a single block, corresponding to b = 22. Spherical magnetic nanoparticles studied in Ref. 2 are made up of magnetite nanorods.

![](https://assets.underline.io/uploads/markdown_image/1/image/6ef098f20be94e527108ac4581410602.jpg)
Figure 2: Hysteresis loop invariance. The top four loops show invariance with cell size when magnetic parameters are scaled according to a modified RG prescription in addition to the magnetostatic interactions scaling. The last two loops show the invariance of MH loops when the ratio SR/α is constant.

technical paper

Modern MRAM technology relies on the magnetic tunnel junction (MTJ) consisting of the state-of-the-art CoFeB/MgO heterostructure. Although current studies support the importance of this heterostructure to the technology, the concern of MgO dielectric breakdown limits MRAM lifetime. We found that inserting an ultra-thin Mg (0.4nm) layer between CoFeB and MgO could promote lifetime and change polarity dependence on applied voltage (Fig.1). However, it traded off the perpendicular magnetic anisotropy and tunneling magneto-resistance. In this work we visualized the modified CoFeB/MgO with Mg insertion against dielectric breakdown by using an in-situ x-ray absorption spectroscopy technique. We find this technique very helpful to understand the origin of interface reconstruction in association with the changes of energy barrier-height and electronic occupation, which are invisible but essential to the MTJ’s performance. This is due to the uniqueness of the element-/orbital specificity and the real-time design of the setup. We established a bridge correlating the heterostructure’ microscopic and macroscopic properties, by cross-utilizing the x-ray technique with spin transport measurements and atomistic magnetic simulation (Vampire). This thus offers us comprehensive insights which may improve contact engineering of the MTJ technology.

**References:**

[1] Van Beek, Simon, et al. "Impact of processing and stack optimization on the reliability of perpendicular STT-MRAM." 2017 IEEE International Reliability Physics Symposium (IRPS). IEEE (2017). [2] Van Beek, Simon, et al. "Understanding and empirical fitting the breakdown of MgO in end-of-line annealed MTJs." 2020 IEEE International Reliability Physics Symposium (IRPS). IEEE (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/706c071b51b55198e8156662bc6a6fd7.jpg)
Fig. 1: Welbull distribution versus breakdown voltage for the pMTJ stack consisting of (a) SAF/CoFeB(8Å)/MgO(10Å)/CoFeB(11.5Å)/Ta(3Å) (referred to w/o Mg sample), (b) SAF/CoFeB(8Å)/MgO(10Å)/Mg(4Å)/CoFeB(11.5Å)/Ta(3Å) (referred to with Mg sample). Blue and red data correspond to positive and negative voltages applied to the MTJ, respectively. SAF refers to synthetic antiferromagnetic layer which consists of [Co(4Å)/Pt(3Å)]7/Co(6Å)/Ru(8Å)/[Co(4Å)/Pt(3Å)]3/Co(4Å). β and η refer to slope parameter and extrapolated breakdown voltage, respectively.

Virtualizing CoFeB/MgO Reconstruction Effects on the STT-pMTJ's Performance

We present a simple device that measures the optical transmission of a suspension of magneto-tactic bacteria under application of a magnetic field. The system consists of a measurement head with a RGB LED, a photodetector and three sets of orthogonal coils that generate a field up to 3.5 mT. In addition we realized a module based on an ESP32 microprocessor that sets the coil currents, performs data collection, and features a touchscreen and Wifi access. Source files such as source code for the software as well as links to the designs for hardware (3D prints, circuit boards) are available online at https://github.com/LeonAbelmann/MagOD, so that anyone with average electronic skills (for instance EE students) can construct such a magnetic Optical Density meter (MagOD) for under 1000 Eu. We demonstrate to application of the MagOD system on two types of magneto-tactic bacteria. Magnetospirillum bacteria (MSR-1) are elongated and have very different scattering properties depending on the direction of the incident light. Figure 1 shows that there is a clear change in transmitted light when the magnetic field is rotated from parallel to perpendicular to the light path. The signal difference (red lines) is a measure for the fraction of magnetic bacteria in the culture. Since the MagOD meter measures at data acquisition rates up to 800 Hz, we can also time-resolve the rotation of the bacteria (blue lines). The time constant for rotation is a measure for the ratio between bacteria length and their magnetic moment. Unlike MSR-1, which swim along the magnetic field lines but reverse frequently in direction, Magnetococcus bacteria (MC-1) swim in one direction only. This allows us to perform marathon experiments, as shown in figure 2. First, the bacteria are collected at the bottom of the container by a vertical magnetic field. After reversal of the field, the entire groups swim up and passes the photodetector after some time. From the reduction in transmission, we can obtain the distribution in swimming velocity. These are just two examples of the wealth of information one can obtain with such an inexpensive system. Not all possibilities have been explored yet. Next to magneto-tactic bacteria, also suspensions of magnetic nanoparticles can be analyzed with respect to sedimentation and clustering. Since the LED used for the light beam has three different wavelengths, which can be individually tuned in intensity, simple spectral analysis methods can be applied. Moreover next to transmitted light also the scattered light can be detected. Since the housing is 3D printed, other geometries can be easily realized. By making the source code available, we hope that researchers in other fields will make use of and expand on our efforts.

**References:**



**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/34803d5b19fbdfb0bfe99155bb581fe1.jpg)
Fig 1. The scattering properties of magnetospirillum bacteria is dependent on their orientation with respect to the incident light beam. By rotating the magnetic field, we can obtain information on the amount of magnetic bacteria in the suspension from the amplitude of the signal (Cmag) and information on the magnetic moment from the time response (gamma).

![](https://assets.underline.io/uploads/markdown_image/1/image/7ceb7ab967a7f5ec83236c2905f337fb.jpg)
Fig. 2. Magnetococcus bacteria can be sent to the bottom of the container by a vertical magnetic field. When the field is reversed, the entire group swims up (like in a marathon). From the resulting dip in transmission, we can determine the velocity distribution of the bacteria.

Analysis of magneto-tactic bacteria using an open-source optical density meter

Electronic structure and magnetism of complex materials have been a field for decades extensively investigated in materials science. Although the scientific community has in hands outstanding tools to probe such properties in a microscopic level, resonant x-ray techniques are still a unique instrument for probing electronic and magnetic properties with chemical and site selectivity. In particular, x-ray techniques are ideal for providing additional information for AC/DC susceptibility, electrical resistivity, neutron scattering, etc. In order to grant the community with a powerful instrument for investigating 4d/5d transition metals, lanthanide and actinide complex magnetic materials, beamline P09 at Petra III/DESY has employed diverse x-ray scattering, diffraction and absorption techniques. [1] Particular attention has been drawn to x-ray methods that employ circular radiation and able to probe interfacial magnetism. A few recent examples include measurements carried out by means of x-ray magnetic circular dichroism (XMCD)[2-6] and x-ray resonant magnetic reflectivity (XRMR)[6], in which a wide variety of samples, such as powder, single crystals, thin films can be investigated. Using an in vacuum double phase retarder system, P09 is able to provide full polarization control (e.g. linear or circular polarization) between 2.7 and 15 keV [7]. Dichroic magnetic x-ray data combined with magneto-optical sum rules [8] are able to individually provide spin and angular momentum of the probed ions. Hereby I will present a few experimental set-ups currently available at P09 along with some of novel scientific results obtained mainly by means of XMCD and XRMR methods.

**References:**

[1] J. Strempfer, S. Francoual, D. Reuther, J. Synchrotron Rad. 20, 541 (2013) [2] J. Mardegan, S. Francoual, G. Fabbris, Phys. Rev. B 93, 024421 (2016) [3] A. Khanas, S. Zarubin, A. Dmitriyeva, Adv. Mater. Interfaces, 2000411, 1 (2020) [4] Y. Liu, A. Luchin, S. M. Sanchez, ACS Appl. Mater. Interfaces 12, 8780 (2020) [5] Z. Liu, X. Wang, X. Ye, accepted Phys. Rev. B (2021) [6] D. Graulich, J. Krieft, A. Moskaltsova, Appl. Phys. Lett. 118, 012407 (2021) [7] S. Francoual, J. Strempfer, D. Reuther J. Phys.: Conf. Ser. 425 132010 (2013) [8] B. T. Thole, P. Carra, F. Sette, Phys. Rev. Lett. 68, 1943 (1992)

**Images:**

Exploring magnetic materials by means of XMCD and XRMR methods at beamline P09 at Petra III/DESY

Magneto-rheological materials are composites consisting of hard or soft magnetic particles embedded in a cross-linked elastomer with various filling ratios. They belong to the class of smart materials whose stiffness and damping can be controlled by an external stimuli as magnetic field. Due to their great flexibility and potential strain amplitude under a various magnetic field, they can find an interest in the field of mechatronic systems such as, soft-robotic or medical technologies. Magneto-mechanical coupling of flexible magnet needs to be clearly understood for development of applications in which they can be submitted to magnetic and mechanical loads. Some researchers have already worked on a full magneto-mechanical experimental characterization for soft magneto-rheological elastomer in case of large deformation [1-3]. However, due to high deformation range, the distances between magnetic particles increase. Thus, the magneto-rheological effect, which is defined by an increase of the stiffness under various magnetic field, is lowered. In the first part of this work we present the recent development of a magneto-mechanical bench for magneto-rheological elastomer filled with hard or soft ferromagnetic particles (figure 1). Concerning the mechanical part, the setup is composed by a system which consist of a displacement loading with a stress controller. By monitoring the displacement, we are able to work up to 1% of its deformation. This deformation mode allows to avoid non-linearities in case of large deformation range. The magnetic loading consists of a reluctant magnetic circuit which is able to produce homogeneous and large magnetic field (up to 40kA/m) in a variable air gap. Mechanical stress and magnetic loading are oriented toward the same direction. A 3D simulation analysis has been performed with a finite element simulation software (Altair Flux ©) for sizing and optimization of the magnetic loading such as intensity and homogeneity. The goal of this study is to expand as large as possible the magnetic field range to which our material would be submitted. For the numerical study, we have chosen a permanent magnet made of NdFeB (Br = 1.3T) and the magnetic flux is driven through a pure iron core. Various magnetic strength field can be reached by adjusting the air-gap thickness of the device. In case of hard magneto-rheological elastomer, composed by silicon-based elastomer filled with 36% vol of NdFeB particles, we are able to explore 10% of the demagnetization curve. The second part of this work focuses on the experimental characterization of magneto-mechanical coupling based on the setup developed. Thanks to this characterization tool, we are able to perform mechanical loading and unloading cycles with and without magnetic field. We observe, without any magnetic field applied, a hysteretic behavior and a remanent deformation specific to elastomeric material (figure 2) [4]. From this unloading stress state, a magnetic field strength of 40kA/m is applied. As it can be seen (figure 2), a tensile stress around 6.5kPa is induced due to magnetic particles interaction in response of the magnetic field applied. These first results are very promising and confirm the ability of the setup to characterize the magneto-mechanical coupling of hard flexible magnet. Further tests will be performed on composites with different elastomeric matrices with various Young’s moduli, different filling ratios and polarization orientation.

**References:**

[1] C.bellan, & Bossis, Georges. (2012). Field dependence of viscoelastic properties of MR Elastomers. Int J Mod Phys B;16(17n18):2447-2453 [2] Schubert, Gerlind & Harrison, Philip. (2015). Large-strain behaviour of Magneto-Rheological Elastomers tested under uniaxial compression and tension, and pure shear deformations. Polym Test; 42:122-134 [3] Pössinger, Tobias. (2015). Experimental Characterization, Modeling and Simulation of Magneto-Rheological Elastomers. PhD THESIS. Ecole Polytechnique, France. [4] Chagnon, Gregory. (2003). Modélisation de l'effet Mullins dans les élastomères. PhD THESIS. Ecole Centrale de Nantes, France.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d30aa3cd104288673cbffe08ee05f8b5.jpg)
Figure 1: Illustration of the magneto-mechanical bench

![](https://assets.underline.io/uploads/markdown_image/1/image/254d7b2a27c44c2635c7592b269ecf07.jpg)
Figure 2: Experimental characterization of the mechanical behavior of a hard magneto-rheological elastomer

Development of a magneto-mechanical bench and experimental characterization for magneto-rheological elastomers

The prospect of thermal imaging and control in magnetic particle imaging (MPI) is an important advancement for non-invasive medical diagnostics and therapeutics. Current MPI systems focus on imaging only, from measurement of magnetic nanoparticle concentrations, while leaving out the important in vivo diagnostic of temperature. Here, we report on our progress toward accurate thermal imaging with high resolution MPI and discuss our achievement of 410 mK total temperature uncertainty at 100 ms integration time near room temperature using a high-precision magnetic field-stabilized magnetic particle spectrometer. The magnetic field is stabilized to ~ 14 nTRMS uncertainty, which allows for precise and accurate determination of magnetic nanoparticle temperature by measurement of magnetization under AC driving fields. Figure 1 shows the obtained temperature Allan deviation plot extracted from the measurement of magnetization in 10 nm ferrite magnetic nanoparticles driven by a 2.5 kHz magnetic field [1]. It has been reported that the imaging resolution of MPI is critically impacted by nanoparticle relaxation dynamics [2]; specifically, spin relaxation under AC drive fields causes a lagged response that results in image blurring. The dynamic response of magnetic nanoparticles in magnetic fields is intricate, depending strongly on the particles’ inherent magnetic and structural properties, inter-particle interactions, as well as the local environment in which they are suspended. Thus, comprehensive understanding of these effects requires high sensitivity over a broad parameter space. We have recently developed an arbitrary-wave magnetic particle spectrometer for characterizing relaxation dynamics for particles ranging in sizes from 10 – 70 nm. The combined advantage of arbitrary drive waveforms (sine, pulse, chirp, composite, etc.) and high drive amplitudes in our system allows for facile characterization over a large parameter space, using magnetic field frequencies from DC to 50 MHz at amplitudes 0-10 mTRMS. This capability to drive and observe magnetization dynamics occurring on the order 20 ns is required for isolating independent Brownian and Néel relaxation processes, as well as their dependence on particle environment and magnetic field amplitude. We have characterized the dynamic AC susceptibility of magnetic nanoparticles ranging in diameter from 10 nm to 70 nm and in composition (ferrite, cobalt-doped ferrite, zinc-doped ferrite, core-shell, etc.), and we have observed peaks in the imaginary part of the AC susceptibility (χ’’) at frequencies from as low as 50 Hz to &gt; 50 MHz, depending on particle size and composition. As an example, Figure 2 show the χ’’ peak frequency for 15 nm (blue arrow) and 20 nm (green arrow) Fe3O4 nanoparticles suspended in water. This peak is an indication of the effective relaxation timescale (τ ~ 16 ms for 20 nm, and τ ~ 160 ns for 15 nm), which will impact the spatial resolution of MPI, depending on the excitation frequency used. This knowledge will eventually inform strategies for design and synthesis of magnetic nanoparticles with properties targeted for accurate and sensitive thermal imaging using MPI.

**References:**

[1] Thinh Q. Bui, Weston L. Tew, Solomon I. Woods. AC magnetometry with active stabilization and harmonic suppression for magnetic nanoparticle spectroscopy and thermometry. J. Appl. Phys. 128, 224901 (2020). [2] Zhi Wei Tay, Daniel W Hensley, Erika C Vreeland, Bo Zheng, Steven M Conolly. The Relaxation Wall: Experimental Limits to Improving MPI Spatial Resolution by Increasing Nanoparticle Core size. Biomed Phys. Eng. Express, 3 (3), 035003 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/99ecda540b1e19510881aae0f2006451.jpg)
Figure 1: Temperature Allan deviation for 10 nm ferrite nanoparticles at 2.5 kHz and 2.5 mTRMS

![](https://assets.underline.io/uploads/markdown_image/1/image/778f9916b80bb72d4d72ad20e1dbea97.jpg)
Figure 2: AC susceptibility for 15 and 20 nm ferrite nanoparticles. χ’ and χ” are the real and imaginary components, respectively.

Magnetic nanoparticle for thermal and magnetic particle imaging

Thorough characterization of magnetic nanoparticles (MNP) is a demanding task. Still, due to growing usability and demand, there is a constant pressure to develop better means of MNP characterization, especially in a solution, which is a typical environment for majority of MNP applications. Techniques well developed so far are well suited to probe nanopowders or thin films, but are scarce in case of magnetic nanoparticles embedded in a matrix, due to short probing distance, weak selectivity, or incompatibility of method’s characteristic time and MNP’ relaxation time. It has been shown that RIXS-MCD (Magnetic Circular Dichroism in Resonant Inelastic X-ray Scattering) is a sensitive, element selective probe of structure, environment and magnetic moment of transition metals in solids, nanomaterials and frozen solutions of MNP [1-3]. 1s2p RIXS-MCD is a high-resolution second-order optical transition spectroscopy (K absorption followed by Kα relaxation) resulting in dichroism effect comparable to L-edge XMCD. Any material embedded in a hard X-ray transparent matrix can be studied volumetrically with weak self-absorption. By the use of bend crystal analyzers Ka emission can be recorded free of background, omitting constituents not containing studied element, what results in a high sensitivity often required for nanoparticles in solutions. Well-resolved spectral features in the K pre-edge energy range holds information on ion coordination and MCD signal in that region can be used to quantitatively determine magnetic moment projection onto the beam/magnetic field direction. MNP solution magnetization can be measured free of diamagnetic or paramagnetic contributions from carrier or chemical additives in the solution, what is especially important for the evaluation of the smallest nanoparticles that are very hard to magnetize and easy to disregard as background (fig. 1) [4]. We demonstrate applicability of RIXS-MCD to study structural and magnetic state of ferrites nanoparticles being synthesized via decomposition reaction starting from pure precursors. Growth of MNP by means of thermal decomposition of iron/cobalt acetylacetonates syntheses were studied on-the-fly both on iron and cobalt edges using a special cell designed for in-situ synchrotron experiments on solutions (fig.2) [5]. Recorded spectra are used to discern main components taking part in the MNP growth process and to follow evolution of local atomic coordination and valence of transition metal ions. Development of magnetization in iron and cobalt ferrites MNP are linked to changes in atomic structure and chemical state. RIXS-MCD supported by ex-situ characterization (TEM, SAXS, Mössbauer spectroscopy) provides a comprehensive characterization of magnetic nanoparticles in solution. It opens a new pathway for controlled synthesis of MNP with defined magnetic response.

**References:**

1. A. Juhin, A. Lopez-Ortega, M. Sikora, et al., Nanoscale, Vol. 6, 11911 (2014) 2. N. Daffé, M. Sikora, M. Rovezzi, et al., Advanced Materials Interfaces, Vol. 4, 1700599 (2017) 3. N. Daffé, J. Zečević, A. Juhin, et al., Nanoscale, Vol. 12, 11222 (2020) 4. J. Kuciakowski, M. Sikora, D. Koziej, et al., Nanoscale, Vol. 12, 12640 (2020) 5. M. Staniuk, O. Hirsch, N. Kränzlin, et al., Chemistry of Materials, Vol. 26, p. 2086-2094 (2014)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9eebf768e8201652248821acd3ad6a19.jpg)
Fig. 1. Magnetization profiles of ZnxFe3-xO4 MNP in solution measured using VSM (black squares) and RIXS-MCD (orange circles). Lines correspond to VSM signal corrected for diamagnetic background using different approaches [4].

![](https://assets.underline.io/uploads/markdown_image/1/image/4c8a57800b76886a7ecb28cd4308d35c.jpg)
Fig. 2. Sketch of RIXS-MCD experimental setup with reaction cell for in-situ experiments on solutions [5].

RIXS-MCD as selective in-situ probe of structure and magnetization of nanoparticles throughout synthesis

Polarized neutron diffraction (PND) is a powerful method for investigating magnetic structures. It gives unique access to contributions from nuclear and magnetic scattering, their interference terms, magnetic chirality, and permits to distinct between them. In contrast to non-polarized neutron diffraction, where the scattered intensity depends on the square of the magnetic structure factor, PND has a linear nuclear–magnetic interference term as part of the scattered intensity. This increases the precision in the determination of the ordered magnetic moment by at least one order of magnitude. In the classical PND the sample is situated in a strong magnetic field. For each Bragg reflection, two scattered cross sections are measured for the two antiparallel oriented directions of the incoming neutron spin, and the ratio between them is built. PND measurements are used for the refinement of magnetization density distribution maps [1,2] and the determination of local anisotropy in the magnetic susceptibility at the unit-cell level [3]. It is also used for the high-quality determination of magnetic form factors [4,5], to untangle complex (e.g. chiral) magnetic structures and to follow the movement of magnetic domains [6]. Recently, a first PND setup using a compact high Tc superconducting magnet and a 3He spin filter polarizer has been successfully implemented [7,8] on the hot neutrons single crystal diffractometer POLI at Maier-Leibnitz Zentrum (MLZ) in Garching near Munich, Germany. Although this setup performs well and first scientific output from it starts to appear [1], it suffers from the relatively low maximal available field of only 2.2 T. For studying many modern-topic (e.g quantum, topologic-order, complex frustrated, etc.) magnetic materials with small ordered magnetic moment, the field-limit of about 2 T is insufficient to produce significant measurable signal, even for PND. To overcome this limitation, a new 8 T split-coil superconducting magnet has been procured and implemented for measurements on POLI. It provides a large angular acceptance for the scattering beam of 300° in the horizontal and 30° (-5°/+25°) in the vertical plane respectively, permitting both a right- and left-handed scattering geometry, the registering of the out-of-plane reflections and the usage of large 2D detectors. The field polarity can be swiped between positive and negative. The vertical asymmetric field configuration is realized as the standard operation mode to shift the zero-field node out of the beam path to avoid a neutron depolarization. The magnet is equipped with a re-condensing liquid He cryostat using a Joule-Thomson cold head on top to liquefy the exhaust He gas back into the cryogen bath. Under a constant temperature condition, a zero He boil-off rate could be maintained over a long time (weeks or even month). This significantly reduces the operation costs and maintenance effort for the regular refilling during the neutron experiments. Using two sample rods with dedicated thermometry, sample temperatures between 1.7 and 800 K are available. An extension to the mK range is planned. The sample rod is equipped with an axial rotation and a vertical translation for precise sample centering and diffraction measurements. The spacious sample tube (Ø=34 mm) and large coil-split space of 53 mm facilitates multiparameter studies by fitting e.g. pressure cells inside [9]. The installation of high-voltage lines for an additional electric field at the sample position is ongoing. Although this new magnet is actively shielded, reducing the stray field by an order of magnitude compared to the classical design (50 Oe at 1 m radial distance from the magnets center), its fringe fields are still too large to be used with the sensitive 3He polarizer of the previous setup. To overcome this issue, a new large-beam-cross-section solid-state supermirror bender polarizer (BP) has been developed for POLI. It was realized by a Fe/Si multilayer coating on both sides of the thin Si wafers (m=3) by the NOB company. An additional Gd oxide layer is deposited on the convex side of each wafer to absorb the neutrons with the wrong polarization. An existing shielded Mezei-type flipper is used between the magnet and BP. A dedicated guide field construction was numerically simulated, optimized and built to link the magnetic field of the BP to the flipper and to the stray field of the magnet. The neutron beam path between the monochromator-drum to the sample magnet is shown in the Fig. 1. The new setup was successfully tested using a 3He spin filter as analyzer at low fields (&lt; 1 T), and a high quality Cu2MnAl single-crystal situated in the magnet as analyzer at high fields (&gt; 2T). An almost loss-free spin transport within the instrument for the complete field range of the new magnet was achieved. A high polarization efficiency of above 99% even for short wavelength neutrons could be experimentally reached using the new BP (Fig.2). The new high–field PND setup is now available for precise magnetic structure investigations on POLI for the internal and external user communities.

**References:**

[1] J. Jeong, B. Lenz, A. Gukasov et al., Phys. Rev. Lett. 125, 097202 (2020). [2] M. Deutsch, B. Gillon, N. Claiser et al., IUCrJ 1, 194 (2014). [3] A. Gukasov, P. J. Brown, J. Phys. Condens. Matter 14, 8831 (2002). [4] B. Lebech, B. D. Rainford, P. J. Brown et al., J. Magn. Magn. Mater. 14, 298 (1979). [5] C. Wilkinson, D. A. Keen, P. J. Brown et al., J. Phys. Condens. Matter 1, 3833 (1989). [6] R. Nathans, S. J. Pickart, H. A. Alperin et al., Phys. Rev. 136, A1641 (1964). [7] H. Thoma, H. Deng, G. Roth et al., J. Phys.: Conf. Ser. 1316, 012016 (2019). [8] H. Thoma, W. Luberstetter, J. Peters et al., J. Appl. Cryst. 51, 17 (2018). [9] A. Eich, M. Hölzle, Y. Su et al., High Pressure Research (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fbd43fea7ad0ff16112e879b64c38c02.jpg)
Fig.1: Components of the PND setup on POLI.

![](https://assets.underline.io/uploads/markdown_image/1/image/a314b63d747f0ae70b578b751d0280e1.jpg)
Fig.2: Experimentally measured polarization efficiency of the new PND setup on POLI. Above 1 T, the spin filter cell (SFC) gets magnetized and the Heusler crystal is not yet fully saturated up to 2 T. Thus, both analyzers are not perfectly suitable for the calibration in this field portion. Nonetheless, further measurements with magnetic samples showed a continuous high polarization efficiency in this field region as well.

New polarized neutron diffraction setup for precise high-field investigations of magnetic structures up to 8 T at the MLZ

Insomnia is a common sleep disorder that affects a great percentage of the population. Magnetic stimulation technology is a new non-invasive and painless treatment for brain disorders. However, the effectiveness of magnetic stimulation technology in the treatment of insomnia has not been fully studied. In this paper, magnetic stimulation was used to stimulate acupuncture points (acupoints). 64-channel electroencephalography (EEG) signals under different states (healthy subjects' resting state, sub-healthy insomnia subjects' resting state before and after magnetic stimulation) were recorded. The brain functional networks based on EEG were constructed and analyzed to investigate the effects of magnetic stimulation on insomnia. The results showed that after magnetic stimulation, the connection and characteristic parameters of brain function network in EEG beta band of insomniacs were improved. Keywords: Magnetic stimulation; Electroencephalography (EEG); Brain functional network; Insomnia I. Introduction Insomnia is a functional disorder caused by the imbalance of brain function. Magnetic stimulation technology has the advantage of safe and painless, and it could stimulate deep tissues and nerves [1]. The synchronization of EEG signals could reflect the correlation between various brain regions [2]. II. Materials and methods Eighteen (nine insomnia and nine healthy) right-handed subjects were chosen in this experiment. According to traditional Chinese medicine, Neiguan (PC6), Shenmen (HT7) and Sanyinjiao (SP6) acupoints were selected in this experiment. Stimulation frequency was 1Hz. Intensity was 1.76 T (80% of the maximal). Each subject was stimulated for three days. 64-channel EEG signals were recorded before and after magnetic stimulation. Pearson correlation coefficient of EEG beta band was calculated. III. Results For insomnia subjects, after magnetic stimulation, the connections and the characteristic parameters of brain in EEG beta band were tending to those of healthy subjects, as shown in Fig.1 and Table 1. IV. Conclusion The results showed that after magnetic stimulation, insomnia subjects’ connections and characteristic parameters of brain functional network were improved. It suggested that magnetic stimulation of acupoints might be an effective treatment for insomnia.

**References:**

[1] Penghui S, Hua L, Siran L, Sleep Med., vol.56, pp.157-163 (2019) [2] Yunna K, Chorom B, Sunhye C, Int J Psychophysiol.,vol.124, pp.26-32 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a6ef9c96488e69c84623f29c30b4f744.jpg)
Fig.1 Node degree of brain information maps of EEG beta band under different sates

![](https://assets.underline.io/uploads/markdown_image/1/image/f13a86538b506441b80d8949c8f8b44c.jpg)
Table 1 Characteristic parameters of brain functional networks of EEG beta band under different states

Magnetic Stimulation Reduced Electroencephalography Beta Band Excitability in Insomniacs

The first-order reversal curve (FORC) method is one of the most appealing magnetic characterization methods that has been used over decades for detailed analyses of nanoscaled magnetic systems. These detailed analyses are accompanied by numerous magnetic features in FORC heat-maps that usually cannot be fully explained1. In this direction, several theoretical studies have been done to provide a coherent framework to demonstrate those features1,2. However, due to computational limitations, the theoretical models consider mostly perfect arrangements of ideal nanomagnets, such as magnetic nanowires (MNWs), that do not match the real experiments3,4. Here, the FORC heat-maps of several MNW arrays were modeled and discussed. First, several MNW arrays with different arrangements (e.g. hexagonal, square, and random) are modeled and the effect of arrangement on the interaction fields (Hu) are explored. Next, the FORC heat-maps of these MNW arrays are calculated to visualize the importance of the arrangement on the magnetic characteristics of these systems. Then, geometrical imperfections (e.g. variations in shape and size) of MNWs were modeled and randomly assigned to investigate their effects on the FORC heat-maps. Figure 1 shows simulation results for different MNWs arrangements and geometrical imperfections that dictate the Hu among MNWs and the FORC heat-maps, respectively. It was found that the average interaction field (Huave) in hexagonal arrangements is higher than those in square arrangements regardless of the filling factor (defined as the area of MNWs to the total area of the simulation box). That is because the hexagonal arrangement has an odd number of MNWs in which the middle one is always in a frustrated state. The MNWs in random arrangements have significantly higher Hu variations (defined as the difference between the interaction field maximum (Humax) and minimum (Humin)) because the interwire distance in this arrangement is not constant, as opposed to hexagonal and square arrangements. Thus, the Hu is less uniform, and the mean-field effect theory ceased to be a reasonable approximation for simulating these randomly arranged MNWs. Moreover, modeling the geometrical imperfections as variations in coercivity (Hc) will result in presence of the well-known “Wish-bone” and “T-shape” features in the FORC heat-maps. Interestingly, we show that the presence of these features not only depends on the variation of the Hc but also depends on the relative strength of both Hc and Hu.

**References:**

1 C.-I. Dobrotă and A. Stancu, J. Appl. Phys., 2013, 113, 043928. 2 D. A. Gilbert, G. T. Zimanyi, R. K. Dumas, M. Winklhofer, A. Gomez, N. Eibagi, J. L. Vicent and K. Liu, Sci. Rep., 2014, 4, 1–5. 3 M. R. Zamani Kouhpanji, A. Ghoreyshi, P. B. Visscher and B. J. H. Stadler, Sci. Rep., 2020, 10, 15482. 4 M. R. Zamani Kouhpanji and B. J. H. Stadler, Nano Express, 2020, 1, 010017.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/18107c60ea58df5b3d2fffb6001ab6fa.jpg)
Figure 1: Subfigures (a) and (b) show the average interaction field (Hu) and the difference between the maximum and minimum Hu for various MNW arrangements. Subfigures (c) and (d) show the FORC heat-maps for the hexagonal arrangement, where the filling factor and average coercivity (Hc) are 5% and 100 Oe, respectively, for both cases. The Hc standard deviation is 10 Oe and 50 Oe for (c) and (d), respectively. As the ratio of Hu standard deviation (which is constant here because both cases have the same filling factor) to Hc standard deviation decreases from 10 to 2, the FORC heat-map of MNWs changes from “T-shape”, (c), to “Wish-bone”, (d), respectively.

Exploring effects of magnetic nanowire arrangements and imperfections on first-order reversal curve heat-maps

For more than 30 years, the properties of Co/Pt multilayers have intrigued magneticians for their high coercivity and well-developed perpendicular magnetic anisotropy (PMA) [1], with a view to applications in magneto-optical recording and perpendicular magnetic recording media and MRAM, to name but a few. The magnetic and structural properties of this system usually rely on the induction of &lt;111&gt; crystallographic texture and do vary substantially with thickness of the component layers within and number of repetitions of the Co/Pt bi-layer building block [2]. The quality of PMA typically deteriorates beyond 10-20 repetitions and mechanisms of magnetisation reversal complicate as a result. Here we describe the measurement of dynamic hysteresis loops from essentially DC to about 1 kHz in this type of multilayer using an in-house developed MOKE dynamic hysteresiograph and their interpretation within a model of viscous dynamics, accounting for both hysteretic switching and magnetisation rotation. The samples are prepared by DC magnetron sputtering in a Shamrock multi-chamber deposition tool and have the following structure SiO2/Ta(5)/Pt(2)/[Co(0.4)/Pt(0.8)]10/Co(0.4)/Pt(2), where the dimensions in brackets are given in nanometre. The measurement setup, shown on fig. 2, consists of an optical system utilizing a crossed polarizer/analyser close-to-polar geometry, with a CW He-Ne laser light source (focused down to about 30 μm) and high-speed biased (&gt; 1 MHz bandwidth) photo-diode detector. The magnetic field excitation is provided by an air-gap resonant coil, connected in series to a variable capacitance bank and a high-bandwidth linear operational power amplifier. The system is able to swing over 60 V and at least 5 A of current and deliver fields &gt; 250 mT at low frequency and 150 mT at 1 kHz. Fundamental frequency current resonance is maintained by adjusting the magnitude of the series capacitance, for each magnetic field ramp rate of interest. Two typical hysteresis loops are shown on fig. 1, demonstrating two distinct regions – one of relatively abrupt magnetisation reversal, followed by another, consisting primarily of magnetisation rotation. The behaviour of the normalized differential susceptibility is illustrated on fig. 2, for a number of frequencies in the region 5 – 810 Hz (linear ramp rates of up to 100 T/s). The same is computed using a statistical β-spline (of 3rd order) interpolation of the normalized hysteresis curves, in time domain. The peak in the differential susceptibility is becoming lower and broader as the frequency increases, while the plateau (associated with rotation) remains constant in amplitude, while shifting towards larger fields. To account for this behaviour a model is constructed, comprised of a statistical ‘normal’ hysteronic distribution and constant ‘rectangular profile’ plateau, time-convolved with a function (convolution kernel), which is a decaying exponent for positive times and zero for negative times, the time-constant of which is related to both the magnetic viscosity of the system and the magnetic field ramp rate. The resulting integral model yields excellent non-linear least squares fits to the experimental data, as demonstrated of fig. 2, and allows for the quantification of the quasi-static coercive field Hc, its distribution ΔHc, the rotational torque amplitude ΔHr and the magnetic viscosity parameter η, at different magnetic field ramp rates. The scaling of the amplitude of the model is essentially constant in the frequency range DC-1 kHz, while the extracted velues of the average coercive field, the width of the hysteronic distribution and effective viscosity parameter, all scale linearly, with factors of 25 %, 50 % and 500 %, respectively. As the rotational plateau of the normalized differential susceptibility is essentially unchanged up to field ramp rate of ~100 T/s, it can be infurred that both the magnetisation reversal and rotational dynamics (ferromagnetic resonance frequencies are expected to range betweeen 0.5 and 3 GHz, in the field region of interest) is governed by timescales much shorter than 1 ms, and the probing spot size chosen, is sufficiently small, to not impact significantly the results even for domain wall viscous propagation velocities as low as 0.1 m/s. It is hoped that the here-illustrated simple dynamic model of the dynamic hysteresis in perpendicular anisotropy thin films will also help with the parametrisation, interpretation and analysis (in broad regions of field, frequency and temperature) of experimental data on systems of great practical interest, other than Co/Pt, such as L10 FePt, FeNi and CoPd, to name just a few.

**References:**

[1] C.-J.Lin, G.L.Gorman, C.H.Lee, R.F.C.Farrow, E.E.Marinero, H.V.Do, H.Notarys and C.J.Chien “Magnetic and structural properties of Co/Pt multilayers”, Journal of Magnetism and Magnetic Materials 93 194-206(1991). [2] D. M. Donnet, V. G. Lewis, J. N. Chapman, K. O'Grady and H. W. van Kesteren “Microstructure and hysteresis in Co/Pt multilayers”, Journal of Applied Physics 26 (1993).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ed09a37c8867302bb2be444eb0563174.jpg)
(left) Illustrative dynamic hysteresis loops of a Co/Pt multilayer, measured via MOKE, at two different frequencies of the triangular profile drive field (5 Hz and 810 Hz). (right) Views of the various components of the experimental setup and, including the optical and magnetic field sub-systems and the sample mounting. Also shown is a real-time oscillogram of the field and MOKE sample signal, for sinusoidal excitation.

![](https://assets.underline.io/uploads/markdown_image/1/image/059da582912ae82a6d2610b805cecf02.jpg)
(left) Normallized differential susceptibility of the Co/Pt multilayer described in the text, at several different frequencies in the range (5 - 810 Hz). (right) A non-linear least squares fit to the data obtained at 440 Hz, using the integral model described in the text and the corresponding fittting parameters, as an inset.

A dynamic model for the hysteresis of CoPt multilayers and its use for the interpretation of MOKE hysteresis loops aquired at different field ramp rates

Coarse-graining in micromagnetic simulations of dynamic hysteresis loops

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