Ferromagnetic Resonance Frequency Research Articles

Ga-, Y-, and Sc-substituted M-type ferrites for self-biasing circulators in LTCC microwave modules

The partial substitution of Fe3+ by diamagnetic Me3+ in BaMexFe12-xO19 with Me = Ga and Y (0 ≤ x ≤ 2) was studied. Samples were synthesized by the mixed-oxide route and their properties were compared to BaScxFe12-xO19 ferrites. It was shown by XRD that the hexagonal lattice parameters with x for Me = Ga and Sc vary continuously. For Me = Y no single-phase ferrites were obtained indicating that the solubility limit for Y in Ba ferrite is very limited. For Me = Ga and Sc, the saturation magnetization at room temperature is reduced with x. The coercivity as well as the ferromagnetic resonance frequency remain almost unchanged for Ga-substituted ferrites, whereas for Sc-substituted ferrites the coercivity and ferromagnetic resonance frequency decrease with x. Thick films of Sc-ferrite (x = 0.5) were screen-printed onto LTCC tape substrates and post-fired at 900°C. Application of a magnetic field during drying allows fabrication of textured ferrite layers. Alternatively, ferrite tapes and LTCC tapes were successfully co-fired at 900°C.

Nano-Size Hexagonal Ferrites for Microwave and Millimeter-Wave Devices

Complex dielectric and magnetic properties of micro- and nano-size hexagonal ferrites have been studied. Two different measurement techniques have been applied to characterize the samples in the broad frequency range from 1.7 to 120 GHz. It was observed that the constitutive material properties, namely permittivity and permeability, as well as the ferromagnetic resonance frequency of the samples vary with the change in particle dimensions. Based on the results of these measurements, a model for calculating the ferromagnetic resonance frequency of ferrite powders has been derived, which takes into account the size and shape of the particles in the sample. It can be concluded from the size-dependent absorption properties observed in this study that these materials are considered to be tunable millimeter-wave absorbers.

Ferromagnetic resonance frequency and spin wave mode of asymmetric strip nanomagnet

Recently, the operating frequency of nanomagnetic logic device has reached the spin wave frequency of nanomagnets. Therefore, the dynamic magnetic properties of nanomagnets, which are excited by microwave magnetic field, have been explored by many researchers. In this paper, the micro-magnetic model of asymmetric strip nanomagnets under microwave excitation is established. By using the anisotropic stress field (along the <i>x</i>-axis direction) that is generated by a constant voltage and the SINC function microwave magnetic field (along the <i>y</i>-axis direction) to excite the nanomagnets at the same time, the effects of tilt angle and defect angle on the ferromagnetic resonance (FMR) spectrum and spin wave mode of the asymmetric strip nanomagnets are studied. Spectral analysis is performed on the micromagnetic simulation data. Simulation results show that as the tilt angle of the asymmetric strip nanomagnet increases, the ferromagnetic resonance frequency increases. What is more, this phenomenon is independent of the defect angle of the nanomagnet. When the tilt angle is constant, there exists a monotonically increasing relation between the ferromagnetic resonance frequency of the asymmetric strip nanomagnet and the defect angle. The spin wave modes of the nanomagnets differ a lot as defect angle changes. The asymmetric strip nanomagnet is compared with the rectangle nanomagnet, and the spin wave mode of the asymmetric strip nanomagnet is localized. Specifically, the spin wave mode of the asymmetric strip nanomagnets is asymmetric and the high precession region exists at the edge, which is termed asymmetric edge mode. The changes of the tilt angle lead to the changes in the demagnetizing field inside the nanomagnet, which gives rise to the movement of the edge mode. However, the center mode is not sensitive to the change of tilt angle. Finally, the magnetic loss of the model under the excitation of high frequency microwave magnetic field is analyzed and the reliability of the model is verified. These findings indicate that the defect angle and tilt angle can be used to tune the spin wave mode and the ferromagnetic resonance frequency of nanomagnets, and thus providing an important theoretical basis for designing the tunable microwave nanomagnetic devices.

Electromagnetic Properties of Hexaferrite Polymer Composite Materials

In this paper the results of studies of the electromagnetic characteristics of composite radio materials based on Z-, W-, M-type barium ferrites and epoxy resin are presented. The frequency dependences of the reflection, transmission, and absorption coefficients are shown. The obtained results of the electromagnetic response are in good agreement with the published data on the frequencies of natural ferromagnetic resonance for the types of ferrites considered. Based on the obtained data, the spectra of complex dielectric and magnetic permeabilities were calculated. It has been suggested that the above changes in the complex magnetic permeability are associated with the presence of domain wall resonance in the region of 1 GHz and natural ferromagnetic resonance at higher frequencies.

Экспериментальное получение поликомпонентных кристаллов со структурой гексаферрита М-типа

The experimental production of multicomponent crystals with M-type hexaferrite structure, the qualitative and quantitative composition of which reflects the formula AB 12 O 19 , where A is Ba, Pb, Sr, Ca, Zn, and B is Fe, Mn, Ni, Ti, Al, Cu, W is carried out. The resulting material of this composition in the future can provide the ability to smoothly change the frequency of ferromagnetic resonance and throughput. In this way, the properties required by electronic equipment manufacturers can be obtained. In the course of the research, have been studied the possibilities of using different methods for synthesizing experimental samples – solid-phase sintering, melting in a platinum crucible, and melting in a stainless steel crucible. According to SEM and EDX results of the obtained samples, two main types of multicomponent crystalline phases were found: hexagonal crystals having the structure of hexaferrite M-type and octahedral crystals having the structure of spinel AB 2 O 4

Achieving low frequency electromagnetic wave absorption by gyromagnetic ferrite

Under the action of static bias magnetic field, the magnetized ferrite has a permeability tensor which can be adjusted by the applied magnetic field. In this paper, the absorption properties of bulk gyromagnetic ferrites under different magnetized conditions are studied and the great potential of gyromagnetic ferrite in achieving low frequency electromagnetic wave absorption is demonstrated. Full wave electromagnetic simulations are performed based on the finite element method (FEM). A floquet port is adopted at the top boundary of the unit cell to simulate a normally incident plane wave. The unit cell boundary conditions are used in the <i>x</i>-<i>y</i> plane to simulate a periodic structure. Orthogonality magnetization in plane is utilized to solve the polarization selectivity in the condition of transverse magnetic field. The influence on absorption capacity of discrete ferrite array structure and the coupling effect of ferrite elements with different sizes are also studied in consideration of the size effect. The simulation results show that a thin bulk gyromagnetic ferrite layer whose thickness is only 4 mm can possess frequency as low as 0.48 GHz and reflectivity below –10 dB. Gyromagnetic ferrite presents different absorption properties under longitudinal magnetization and transversal magnetization, and different polarization directions in transversal magnetization as well. When longitudinal bias magnetic field <i>H</i><sub>0</sub> = 200 Oe, the bandwidth of the reflectivity below –10 dB ranges from 0.48 to 1.84 GHz. The resonant absorption frequency can be regulated by adjusting bias magnetic field and the size of ferrite element. In general, a large bias magnetic field leads to a high resonant frequency due to the ferromagnetic resonance frequency positively associated with the applied magnetic field, but a ferrite array consisting of larger size elements provides a lower resonant frequency for the size resonance negatively associated with the size. By introducing the coupling between elements with different sizes, the reflection bandwidth below –10 dB can be effectively extended to above 80% of the sum of the bandwidth possessed by single unit cell, especially 105.7% under transversal bias magnetic field 700 Oe. And the broadening effect is effective in both longitudinal and transverse magnetized state but it will be weaker when the two absorption peaks are closer. To further understand the absorption mechanism of the two-element absorber, the distribution of the electric field, magnetic field and power loss density are examined. The results prove that the two peaks at the lower frequency exactly originates from <inline-formula><tex-math id="M1">\begin{document}$ \Delta R = 0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191229_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191229_M1.png"/></alternatives></inline-formula> and the higher frequency originates from <inline-formula><tex-math id="M2">\begin{document}$ \Delta R = 4$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191229_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191229_M2.png"/></alternatives></inline-formula>, and therefore the widened absorption is contributed by the coupled multiple resonances provided by the elements with different sizes.

Injection Locking at Fractional Frequencies of Magnetic-Tunnel-Junction-Based Read Sensors’ Ferromagnetic Resonance Modes

Being nonlinear dynamic systems, magnetic read sensors should respond to an excitation signal of a frequency considerably different from their natural ferromagnetic resonance (FMR) frequencies. Because of the magnetization dynamics' inherent nonlinear nature, the sensors' response should be measured at the DC, excitation frequency, and its multiples (harmonics). In this paper, we present results of such measurements, accomplished using a one-port nonlinear vector network analyzer (NVNA), which show distinct resonances at fractional frequencies of the free layer (FL) FMR mode. Identification of these resonances, resulting from the nonlinear nature of the spin-torque (ST)-induced magnetization dynamics, was performed using micromagnetic modeling. In particular, we show that the measured DC response at the above-mentioned fractional frequencies can be explained by a low-order nonlinearity and strong magnetodipolar feedback between magnetic layers adjacent to an MgO barrier. Additionally, we determined that the simulated harmonic response is strongly enhanced by the mutual ST effect between these layers. Finally, we demonstrate that the read sensors' nonlinear magnetization dynamics and, by extension, their harmonic response are highly sensitive to various magnetic and ST parameters. Thus, this study shows that using NVNA measurements in conjunction with micromagnetic modeling can clarify the uncertainty in the definition of these parameters.

High rectification sensitivity of radiofrequency signal through adiabatic stochastic resonance in nanoscale magnetic tunnel junctions

Rectification is an important stage in electronic circuits for any wireless radio frequency power transfer application. Currently, Schottky diodes are widely used as rectifiers; however, they are inefficient at low power levels of microwatts or less (providing maximum sensitivities around 4 mV/μW). Nanoscale magnetic tunnel junctions can serve as alternative rectifiers by utilizing the so-called spin-torque diode effect, demonstrating a much higher rectification sensitivity (200 mV/μW) compared to Schottky diodes. However, for this mechanism to work, the signal frequency must match the ferromagnetic resonance frequency, which typically lies in the gigahertz range. For signals in the megahertz range or lower, Schottky diodes remain the only option for rectification. Here, we demonstrate a mechanism based on thermally activated adiabatic stochastic resonance in magnetic tunnel junctions to produce low frequency (up to tens of megahertz) signal rectification at low input power (submicrowatt), with a sensitivity of up to 35 mV/μW—higher than state-of-the-art Schottky diode rectifiers at this frequency and power range. These findings suggest magnetic tunnel junctions as potential alternatives to Schottky diodes for low frequency and low power applications.

Frequency Dependency of the Delta-E Effect and the Sensitivity of Delta-E Effect Magnetic Field Sensors.

In recent years the delta-E effect has been used for detecting low frequency and low amplitude magnetic fields. Delta-E effect sensors utilize a forced mechanical resonator that is detuned by the delta-E effect upon application of a magnetic field. Typical frequencies of operation are from several kHz to the upper MHz regime. Different models have been used to describe the delta-E effect in those devices, but the frequency dependency has mainly been neglected. With this work we present a simple description of the delta-E effect as a function of the differential magnetic susceptibility of the magnetic material. We derive an analytical expression for that permits describing the frequency dependency of the delta-E effect of the Young’s modulus and the magnetic sensitivity. Calculations are compared with measurements on soft-magnetic thin films. We show that the frequency of operation can have a strong influence on the delta-E effect and the magnetic sensitivity of delta-E effect sensors. Overall, the delta-E effect reduces with increasing frequency and results in a stiffening of the Young’s modulus above the ferromagnetic resonance frequency. The details depend on the Gilbert damping. Whereas for large Gilbert damping the sensitivity continuously decreases with frequency, typical damping values result in an amplification close to the ferromagnetic resonance frequency.

Fundamental Limits on the Repetition Rate of Photomagnetic Recording

In the last decade it was demonstrated that the fastest way to write information employs ultrashort laser pulses. Naturally such experiments raise questions about the ultimate limit of repetition rate at which light can switch a medium between stable bit states. Here the authors demonstrate that with femtosecond pulses it is possible to write and rewrite magnetic bits in iron garnet with a frequency of up to 20 GHz, with the maximum repetition rate being defined by the frequency of ferromagnetic resonance in the field of photoinduced magnetic anisotropy. This finding reveals the principles to be employed in achieving magnetic recording at frequencies far beyond today's state of the art.

RF voltage-controlled magnetization switching in a nano-disk

Nanomagnetic oscillators are key components for radio-frequency (RF) signal generation in nanoscale devices. However, these oscillators are primarily electric current-based, which is energy inefficient at the nanoscale due to ohmic losses. In this study, we present an actuation mechanism for magnetization switching using a multiferroic structure that relies on an RF voltage input instead of electrical current. An AC voltage with a DC bias is applied to the piezoelectric substrate and the magnetic nanodisk with perpendicular magnetic anisotropy that is attached onto the substrate, which can achieve steady magnetic oscillation when the driven voltage is at ferromagnetic resonance (FMR) of the nanodisk. Changing the DC bias changes the magnetic anisotropy of the magnetoelastic nanodisk, hence changes the FMR and oscillation frequency. The frequency modulation is quantified using the Kittel equation. Parametric studies are conducted to investigate the influence of voltage amplitude, frequency, waveform, and the thickness of the magnetoelastic nanodisk. This multiferroic approach opens possibilities for designing energy efficient nanomagnetic oscillators that have both large amplitude and broad frequency range.

The temperature dependence of magnetic losses in CoO-doped Mn-Zn ferrites

CoO-doping is known to stabilize the temperature dependence of initial permeability and magnetic losses in Mn-Zn ferrites, besides providing, with appropriate dopant contents, good soft magnetic response at and around room temperature. These effects, thought to derive from the mechanism of anisotropy compensation, are, however, poorly assessed from a quantitative viewpoint. In this work, we overcome such limitations by providing, besides extensive experimental investigation vs frequency (DC–1 GHz), CoO content (0 ≤ CoO ≤ 6000 ppm), and temperature (−20 °C ≤ T ≤ 130 °C) of permeability and losses of sintered Mn-Zn ferrites, a comprehensive theoretical framework. This relies on the separate identification of domain wall motion and moment rotations and on a generalized approach to magnetic loss decomposition. The average effective anisotropy constant ⟨Keff⟩ is obtained and found to monotonically decrease with temperature, depending on the CoO content. The quasistatic energy loss Wh is then predicted to pass through a deep minimum for CoO = 3000–4000 ppm at and below the room temperature, while becoming weakly dependent on CoO under increasing T. The rotational loss Wrot(f) is calculated via the complex permeability, as obtained from the Landau-Lifshitz equation for distributed values of the local effective anisotropy field Hk,eff (i.e., ferromagnetic resonance frequency). Finally, the excess loss Wexc(f) is derived and found to comply with suitable analytical formulation. It is concluded that, by achieving, via the rotational permeability, value and behavior of the magnetic anisotropy constant, we can predict the ensuing properties of hysteresis, excess, and rotational losses.

Ultra-wideband 1–20 GHz Non-contact FMR Test System for TMR HGA

The read sensor of hard disk drive abnormality at the bar level is detected by a ferromagnetic resonant analyzer (FMRA) system requiring a galvanic contact to the sensor. After the read head is assembled to the slider, a quasistatic tester is used to check the slider functionality. At this stage, the FMR information of the read sensor cannot be easily accessed since there is no suitable measurement technique available. In this research, an innovative non-contact ferromagnetic resonant (FMR) measurement for the read head sensor quality control at slider level is reported for the first time. The proposed test system is designed for detecting the FMR from 1 to 20 GHz. The FMR is interpreted from the peak of read sensor resistance over the frequency span. This new measurement technique has been applied to a model of tunneling magnetoresistance (TMR) sensor and compared with the FMRA. The FMR frequency at 5.4 GHz can be detected by the proposed measurement system with only 1 MHz difference from the value obtained from conventional FMRA. The higher effective stiffness field of the read sensor is also observed from the lower FMR resistance in the shorter stripe height head.

Layer-selective detection of magnetization directions from two layers of antiferromagnetically-coupled magnetizations by ferromagnetic resonance using a spin-torque oscillator

We use micromagnetic simulation to demonstrate layer-selective detection of magnetization directions from magnetic dots having two recording layers by using a spin-torque oscillator (STO) as a read device. This method is based on ferromagnetic resonance (FMR) excitation of recording-layer magnetizations by the microwave field from the STO. The FMR excitation affects the oscillation of the STO, which is utilized to sense the magnetization states in a recording layer. The recording layers are designed to have different FMR frequencies so that the FMR excitation is selectively induced by tuning the oscillation frequency of the STO. Since all magnetic layers interact with each other through dipolar fields, unnecessary interlayer interferences can occur, which are suppressed by designing magnetic properties of the layers. We move the STO over the magnetic dots, which models a read head moving over recording media, and show that changes in the STO oscillation occur on the one-nanosecond timescale.

Higher ferromagnetic resonance frequency in NiFe/FeMn film obtained by flash annealing in reversing field

The unsaturated magnetization reversal by pulsed current annealing was investigated in exchange biased NiFe(15 nm)/FeMn(10 nm) bilayer. The magnetization reversal was conducted by energizing and thus heating the film under an external applied magnetic field ( $$H_{\text{a}}$$ ) with the direction opposite to the deposition field ( $$H_{\text{d}}$$ ). The static and dynamic magnetic properties of the films before and after rapid annealing were characterized using vibrating sample magnetometer, anisotropic magnetoresistance (AMR) and vector network analyzer measurements. When the samples were annealed at 40 and 45 V condenser voltages, greater values of rotational anisotropy $$(H_{\text{rot}} )$$ , ferromagnetic resonance frequency ( $$f_{\text{r}}$$ ) and smaller values of exchange bias field $$(H_{\text{e}} )$$ , exchange bias field in AMR measurement ( $$H_{\text{e}}^{\text{MR}}$$ ), and magnetoresistance in the samples were obtained, which is attributed to the unsaturated reversed magnetization in reversing magnetic field. A suggested model for which the scattered magnetization of magnetic moment results in additional unstable antiferromagnetic spins in the film was proposed to explain this interesting phenomenon.

Optimization of acoustically-driven ferromagnetic resonance devices

Acoustically-driven ferromagnetic resonance (ADFMR) has recently emerged as a powerful scientific test-bed toward understanding complex interactions between phonons and magnons. In this technique, a traditional surface acoustic wave (SAW) delay-line filter interfaces with a ferromagnetic thin-film which can be driven into precession at the ferromagnetic resonance (FMR) frequency by the SAWs. SAW filters are used extensively in industry, but in the context of ADFMR, their design considerations are largely absent from the literature. We produced a variety of ADFMR devices by systematically changing parameters including the material and the number of pairs of interdigital transducers, the ferromagnetic thin-film growth technique, and the presence or the absence of a capping layer on the ferromagnetic thin-film. We quantitatively compare results by adapting traditional ferromagnetic resonance techniques. This work describes the parameters relevant to the development and optimization of SAW-driven FMR.

In Operando Study of the Hydrogen-Induced Switching of Magnetic Anisotropy at the Co/Pd Interface for Magnetic Hydrogen Gas Sensing.

Heterostructures exhibiting perpendicular magnetic anisotropy (PMA) have traditionally served the magnetic recording industry. However, an opportunity exists to expand the applications of PMA heterostructures into the realm of hydrogen sensing using ferromagnetic resonance (FMR) by exploiting the hydrogen-induced modifications to PMA that occur at the interface between Pd and a ferromagnet. Here, we present the first in operando depth-resolved study of the in-plane interfacial magnetization of a Co/Pd film which features tailorable PMA in the presence of hydrogen gas. We combine polarized neutron reflectometry with in situ FMR to explore how the absorption of hydrogen at the Co/Pd interface affects the heterostructures spin-resonance condition during hydrogen cycling. Experimental data and modeling reveal that the Pd layer expands when exposed to hydrogen gas, while the in-plane magnetic moment of the Co/Pd film increases as the interfacial PMA is reduced to affect the FMR frequency. This work highlights a potential route for magnetic hydrogen gas sensing.

Highly Efficient Wideband Microwave Absorbers Based on Zero-Valent Fe@γ-Fe2O3 and Fe/Co/Ni Carbon-Protected Alloy Nanoparticles Supported on Reduced Graphene Oxide.

Electronic systems and telecommunication devices based on low-power microwaves, ranging from 2 to 40 GHz, have massively developed in the last decades. Their extensive use has contributed to the emergence of diverse electromagnetic interference (EMI) phenomena. Consequently, EMI shielding has become a ubiquitous necessity and, in certain countries, a legal requirement. Broadband absorption is considered the only convincing EMI shielding solution when the complete disappearance of the unwanted microwave is required. In this study, a new type of microwave absorber materials (MAMs) based on reduced graphene oxide (rGO) decorated with zero-valent Fe@γ-Fe2O3 and Fe/Co/Ni carbon-protected alloy nanoparticles (NPs) were synthesized using the Pechini sol-gel method. Synthetic parameters were varied to determine their influence on the deposited NPs size and spatial distribution. The deposited superparamagnetic nanoparticles were found to induce a ferromagnetic resonance (FMR) absorption process in all cases. Furthermore, a direct relationship between the nanocomposites’ natural FMR frequency and their composition-dependent saturation magnetization (Ms) was established. Finally, the microwave absorption efficiency (0.4 MHz to 20 GHz) of these new materials was found to range from 60% to 100%, depending on the nature of the metallic particles grafted onto rGO.

MECHANOSYNTHESIS, CRYSTAL STRUCTURE, MAGNETIC AND ABSORPTION PROPERTIES OF Al SUBSTITUTED BaFe12O19

The Aluminums (Al) substituted on M-type barium ferrite (BaFe12O19) is one of the magnetic materials, which can be applied to the microwave band working at high frequencies. The purpose of this paper is to investigate the effect of Al substitution for Fe3+ ions on the structure, magnetic and absorption behavior of M-type barium ferrite. The sample was prepared by mechano-synthesis using high-energy milling (HEM). In this research, Fe ion in BaFe12O19 was substituted by Al ion to form BaFe12-xAlxO19 for x = 0.0,BaFe10Al2O19 for x = 2.0 and BaFe8Al4O19 for x = 4.0 which is called as BAl-0, BAl-2 and BAl-4 sample, respectively. The stainless steel balls were used for the milling with a ball-to-powder sample ratio of 5. The mixing and milling for each the sample was conducted for 5 hours in ethanol medium and dried at 100oC in oven for 24 hours and then followed by heat treatment at 1100°C during 1h in the atmosphere media. The sample was characterized using X-rays diffractometer (XRD), Scanning Electron Microscope (SEM), Vibrating Sample Magnetometer (VSM) and Vector Network Analyzer (VNA). The result indicates that the addition of Al ion lead to the change cell parameters, volume, and the particles size. The magnetic behavior such as magnetic coercivity (Hc), magnetization saturation (Ms) and remanent (Mr) changed significantly with the substitution of Al ions. The optimum reflection loss (RL) is found to be –35dB at 14GHz for BaFe10Al2O19 (BAl-2) sample. It is shown that Al substitutions change the particle size, ferromagnetic resonant frequency, and structural and magnetic behavior of M-type barium ferrite.

Tunable ferromagnetic resonance in coupled trilayers with crossed in-plane and perpendicular magnetic anisotropies

An original approach to tune the ferromagnetic resonance frequency of a soft magnetic Ni80Fe20 (Permalloy = Py) film with in-plane magnetic anisotropy (IMA) based on the controlled coupling to a hard magnetic NdCox film with perpendicular magnetic anisotropy (PMA) through a nonmagnetic Al spacer is studied. Using the transverse magneto-optical Kerr effect (T-MOKE), alternating gradient magnetometry (AGM), and vector network analyzer ferromagnetic resonance (VNA-FMR) spectroscopy, the influence of both the Co concentration and the Al spacer thickness on the static and dynamic magnetic properties of the coupled IMA/PMA system is investigated. Compared to a single Py film, two striking effects of the coupling between IMA and PMA layers can be observed in their FMR spectra. First, there is a significant increase in the zero-field resonance frequency from 2.0 GHz up to 6.4 GHz, and second, an additional frequency hysteresis occurs at low magnetic fields applied along the hard axis. The maximum frequency difference between the frequency branches for increasing and decreasing magnetic fields is as high as 1 GHz, corresponding to a tunability of about 20% at external fields of typically less than ±70 mT. The origin of the observed features in the FMR spectra is discussed by means of magnetization reversal curves.