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Alterations in the ground state of thin (001) films of a BiFeO3 (BFO)-type multiferroic in a magnetic field are studied theoretically, with changes in the energy of induced anisotropy taken into account. The anisotropy vs. field phase diagrams identifying the stability regions of homogeneous antiferromagnetic states and the regions of emergence of spatially modulated antiferromagnetic states are constructed for three mutually orthogonal orientations of applied magnetic field. We show that, as the magnetic field decreases, the transformation of a homogeneous phase into a spatially modulated state occurs at the instability point of the homogeneous state via gradual emergence of the conical phase that transforms into a planar cycloid with the decreasing magnetic field. A multiferroic film grown on a (001) substrate develops considerable anisotropy of the energy of spatially modulated state, depending on the modulation orientation. Meanwhile, cycloids with different orientations undergo the transitions from incommensurate phase into the homogeneous state differently: either the conical cycloid is formed followed by its collapse into the homogeneous state or an unlimitedly growing domain of the homogeneous phase is formed within the flat cycloid. Examples of field-induced changes in magnetization, with changes in spin states taken into account, are provided. These results are of value in practical applications of multiferroic strain engineering.

The use of the spin-torque diode effect, which is generated upon the current-induced transfer of spin angular momentum in magnetic tunnel junctions, opens the prospect of considerably improving microwave sensitivity in the gigahertz frequency band over that of semiconductor Schottky diodes. In this work, the spin-diode effect of microwave signal rectification in a magnetic tunnel junction upon the resonant excitation of spin waves in the free magnetic layer as a result of the current-induced spin-transfer torque effect is analyzed theoretically. Frequency characteristics of the resonant response of a spin-torque diode to a microwave signal are calculated in the linear macrospin approximation as functions of the direction and value of the applied magnetic field and bias current. It is shown that with zero bias current, the maximum rectified voltage across a junction is obtained in the geometry of mutually perpendicular magnetizations of magnetic layers; as the resonant frequency of oscillations in a magnetic field increases, this voltage drops at the retained equilibrium orientation of spins in the layers. After the bias current is switched on, the resonance amplitude of forced oscillations of the free layer spins upon microwave excitation grows sharply at the critical point of the spin-torque diode’s loss of equilibrium state stability. The linewidth at this point is limited only by nonlinear effects. Improving spin-torque diode sensitivity is important for its use in microwave holographic vision systems.

The microwave sensitivity of a spin diode consisting of a tunnel junction with two magnetic electrodes is investigated. The specific feature of the magnetic tunnel structure under consideration is a skew of the electrode magnetizations in the plane of the layers, arising due to exchange interaction with the adjacent antiferromagnetic layers with different superparamagnetic blocking temperatures. Within the framework of the macrospin model the stability analysis of the stationary magnetization states is carried out as a function of the electric current and the skew angle between the magnetizations of the magnetic layers on the angle-current plane, taking into account the effect of spin torque transfer. For the obtained stationary states, variations of the resonant response to an ac current of microwave range and volt-watt sensitivity of the spin diode far away from and near the critical bias-current lines of the transition to self-oscillations are determined. It is shown that with increasing the bias curre...

Spintronics is a new trend in the development of nanoelectronics and, therefore, requires the development of new software tools for modeling aimed at the development and further miniaturization of spintronic devices. In this paper, we present a new software package for the technology computer aided design modeling of spintronic devices based on magnetic tunnel junctions. The theoretical models and scenarios of the software package are described. Examples of application of the software package to solving the main problems arising in the design of magnetoresistive memory elements and consistent miniaturization of these devices are considered.

At present, there is a growing interest in the field of spin caloritronics, which demonstrates the advantages of using thermoelectric effects when considering spin-transport phenomena. One of its most promising directions are the thermal transfer of the spin-transfer torque and magneto-tunnel Seebeck effect in magnetic tunnel junctions in the presence of a non-equilibrium temperature gradient across the tunnel barrier, which, in combination with the voltage rectification effect driven by the alternating current, can serve as an additional source of microwave energy harvesting and has not been considered before. In this work, we estimated the bolometric properties of a spin-torque diode based on the magnetic tunnel junction under the microwave heating, and also compared the thermal dependences of its sensitivity in comparison with the Schottky semiconductor diode. The thermal contribution to the voltage rectification can be significantly increased in some cases, which plays an important role in the development of new types of microwave energy harvesters operating on the thermoelectric effects.

The paper analyzes microwave resonant response of the spin-torque diode. The considered spin-torque diode is a magnetic tunnel junction with a nano-pillar structure. The magnetization of the free layer has a tilt caused by an action of the inclined magnetic field in the plane of the structure. Taking into account the effect of spin torque transfer we define stability regions of stationary states of magnetization in the free layer as a function of the azimuth angle of the magnetic field and bias DC current. Microwave volt-watt sensitivity of the spin diode for the obtained stationary states is calculated. It is shown that non-linear frequency shift of the resonance line width limits maximum sensitivity of the spin diode near the critical bias current corresponding to the point of the transition to the self-oscillating mode. Besides that, overlapping of frequency branches occurs in the resonant response as the critical point approach, which is different from the foldover effect in the nonlinear ferromagnetic resonance induced by an alternating magnetic field.

Thermally assisted magnetic random access memory (TAS-MRAM) nanodevices are promising candidates for future memory applications, because they provide non-volatile data storage at fast operation and low energy consumption. In this work, we analyze operating regimes of these devices using macrospin and micromagnetic modeling. It is shown that, for the successful performance of write and read operations in a TAS-MRAM device, the intrinsic magnetic anisotropy of ferromagnetic layers must not exceed some critical value. The relationship between the device parameters and the value of critical magnetic anisotropy is explained using analytical model and proved by the results of numerical simulation. This analysis is important for the future downscaling of TAS-MRAM nanodevices.

The field dependence of critical voltages of switching of magnetic states of a synthetic multiferroic structure is studied based on a bifurcation analysis of Landau–Lifshitz–Gilbert equations with the torques caused by a tunneling spin transport, taking into account the voltage dependence of transferred spin momenta at the variations in the magnitude and direction of electric polarization. The voltage dependences of transferred spin momenta are determined based on the free electron model, taking into account exchange splitting of electron energy subbands in magnetic beaches and the effect of changing the tunnel barrier height at the variations in the polarization magnitude and state.

The specific features of spin-transfer torque in vacuum tunnel structures with magnetic electrodes are investigated using the quasi-classical Sommerfeld model of electron conductivity, which takes into account the exchange splitting of the spin energy subbands of free electrons. Using the calculated voltage dependences of the transferred torques for a tunnel structure with cobalt electrodes and noncollinear magnetic moments in the electrodes, diagrams of stable spin states on the current–field parameter plane in the in-plane geometry of the initial magnetization are obtained.

The output characteristics of a membrane-based heat-resistant calorimetric gas flow sensor are analyzed. The dependence of the output characteristics on the design parameters of the flow sensor is investigated. The change in heat distribution along the membrane depending on the flow rate is demonstrated and the effect of the distance between thermal resistors, as well as the thickness and material of the membrane, on the characteristics of the flow sensor is investigated. The simulation results are compared with the experimental data. The dependences found can be used for optimizing the design of flow sensors in order to increase their sensitivity and expand the dynamic range of a thermal mass flow meter.

The Landau-Lifshitz equations are generalized to the case of spin accumulation and dynamics of localized spins in the conductive magnet, in terms of the Rashba spin-orbit scattering mechanism. The effects of the spin-orbit transfer torque caused by longitudinal current flowing through the conducting heterostructure with a thin magnetic layer, on the spin state of the structure, are analyzed in the macrospin approximation on the basis of modified Landau-Lifshitz equations. It is found that the possibility of realizing the spin switch modes in the magnetic layer of the heterostructure with an easy-plane magnetic layer exists, and depends on the geometry of the magnetization, which is also the case for transverse spin-transfer torque in spin-valve structures. We built examples of diagrams of spin states on the current-field plane, that show a satisfactory agreement between the results of the experimental measurements of the critical fields and spin switching currents in the magnetic layers of asymmetric hete...

The influence of an electric field on the structural changes and related transformation of the magnetic subsystem in a BiFeO3 multiferroic material has been investigated in the framework of the phenomenological model based on the Ginzburg-Landau theory. The hysteresis dependences of the electric polarization, the counter-rotation of oxygen octahedra, the magnetization, and the antiferromagnetic vector have been obtained as a function of the electric field applied along the crystallographic directions [110] and [001] of the pseudocubic structure. The model parameters are consistent with the results of the ab initio calculations of the structural and magnetic transitions in this material. It has been shown that, in the region of the existence of a spatially modulated spin state, an abrupt change in the polarization vector leads to a stepwise transformation of the antiferromagnetic structure. Within the proposed approach, the electric-field-controlled change in the spin structure of the multiferroic material is adequately described in the range of weak and strong magnetic fields.

A study of the initial stages of crystallization in RF magnetron-sputtered ferrite garnet films is reported, in which a series of ultrathin Bi2Dy1Fe4Ga1O12 layers is fabricated and characterized. The spectral and temperature dependencies of magnetic circular dichroism (MCD) of these films are studied in the temperature range from 300 K down to 8 K. Measured magneto-optical properties are reported in the spectral range between 300 and 600 nm. In ultrathin garnets at temperatures below 160 K, we found that between 360 and 520 nm, the spectral MCD dependencies were typical of bismuth-substituted garnets with high levels of gallium dilution in the tetrahedral sublattice. The MCD signal strength measured at its 440 nm peak grows linearly with reducing temperature between 160 K and 8 K. This observed temperature dependency of MCD differed dramatically from these measured in thicker (3.7 nm) nanocrystalline garnet films. The peak MCD signal at 440 nm in these 3.7 nm-thick samples grows linearly from 215 K down to 100 K, resembling the same dependency seen in 1.7 nm films. In thinnest layers of thickness 0.6 nm, no MCD signals were observed at any temperature in the range between 8 and 300 K.

We investigate transformations of space modulated structure in BiFeO3 films induced by change of magnetic anisotropy, strain, or temperature and applied magnetic field. We show that critical field of the transition between incommensurate and commensurate structures is lower in films as compared with single crystals. There are two incommensurate phases at H∥[111]: plane and conical cycloidal phases with the second order transition between them. An important feature of this transition is softening of spin wave activation mode localized on spin cycloid. Cycloidal spin wave spectrum contains a countable number of branches, thus BiFeO3 can be considered as a magnonic crystal.

Spin oscillations and their phase synchronization are discovered in a spin 1D waveguide (nanowire) after the local injection of a spin-polarized current in the vertical (perpendicular) geometry of magnetization. The mode composition of nonlinear spin self-oscillations is analyzed for a single nano-oscillator using the effect of spin transfer torque with regard to spin wave runaway over the 1D waveguide and synchronized spin self-oscillations at current pumping by two nano-oscillators. It is shown specifically that, along with simple (“nontopological”) oscillation modes, in which singular points in the oscillation amplitude spatial distribution are absent, the 1D waveguide may support modes with pole-type singular points inside the current pumping domain, which are characteristic of the geometry of a precessing 2π-domain boundary. A diagram for frequency- and current-detuning-synchronized spin self-oscillations that are excited by two spin nano-oscillators in the 1D spin waveguide is constructed.

The influence of variations of magnetic parameters on the transformations of antiferromagnetic space – modulated structure in BiFeO3– like multiferroics is investigated. The specificity of variations of magnetic parameters of different origin namely the magnetic anisotropy, the stiffness, the lattice mismatch between multiferroics film and oriented substrate which can change the conditions of space-modulated structures emergence and the character of space – modulated structures transformations in bulk crystals and thin films is discussed. Threshold parameters of phase transitions between cycloidal structures and homogeneous magnetic states have been found out. It has been shown that plane cycloid can be transformed into conical state during the transition from incommensurate phase into homogeneous phase.

The spin wave spectra of antiferromagnetic BiFeO3-type multiferronics are analyzed theoretically. The presence of a spatially modulated cycloidal antiferromagnetic structure leads to a countable number of frequency branches of two oscillatory modes (Goldstone and activation) for spin waves propagating along a cycloid. When there is no magnetic field and anisotropy, the magnon spectrum is characterized by the absence of frequency gaps. The spectral features of the spin oscillations with changing anisotropy and application of a magnetic field are identified and the limits on the existence of an antiferromagnetic cycloid are established up to its transformation into a conical structure. In the transverse direction the spin oscillations have a mixed character which indicates that the cycloid is stable with respect to bending throughout its domain of existence.

Comprehensively studied have been structural elements of ecosystems of Siberian natural plague foci, as well as levels of integration among epizootic process components, and ways of their functional interaction. Application of the complex approach to the surveillance over structural-functional elements of the parasitic system along with investigations of epizootic process dynamics has provided for identification of peculiarities as regards epizootics development, transformation and evolution of population and carrier/vector coenosis structure in time and space. Revealed is the genetic diversity of plague microbe circulating within the bounds of separate foci and zones of focality. Determined is a long-lasting anti-epidemic effect (more than 20 years) of the field desinsection in the Saglinsk meso-focus of the Tuva natural plague focus.

The magnetic-field-dependent spin ordering of strained BiFeO3 films is determined using nuclear resonant scattering and Raman spectroscopy. The critical field required to destroy the cycloidal modulation of the Fe spins is found to be significantly lower than in the bulk, with appealing implications for field-controlled spintronic and magnonic devices., Work supported by ERC Consolidator grant MINT (Contract No. 615759)

The phenomena of spin tunneling and spin torque transfer between magnetic layers of a tunnel spin-valve setup under weak and strong field emissions of spin-polarized electrons are considered. Bifurcational features of changes in the macrospin states under the impact of a tunnel current are discussed for varying directions of the spin-polarization vector.

The magnetic-field-dependent spin ordering of strained BiFeO

Spin current manipulation of magnetic tunnel junction (MTJ) opens the attractive way for the development of low-power and high-speed spintronic memory devices (SMD) based on this effect for data storage, high-performance logic systems and mobile telecommunication [1]. The influence of the bias-voltage dependence of in-plane and out-of-plane spin-transfer torque (STT) on the switching thresholds plays an important role in the optimization of SMD during the process of technological design [2]. In this work we present a compact model of MTJ that can be integrated in HSPICE electrical simulator using the standard BSIM4 (Berkeley Short-channel IGFET model) model of the nMOS transistor, where the macrospin magnetization dynamics driven by the applied bias voltage is self-consistent with the direct STT calculation in MTJ from the quantum-mechanical tunneling problem in the same electronic circuit. A comparison of current-voltage characteristics of the nMOS (metal-oxide-semiconductor) transistor in the SPICE simulation within BSIM4 model and TCAD (Technology Computer Aided Design) simulation is performed for the standard 130 nm CMOS (complementary-MOS) technology. The results obtained in the simulation of read and write operations in 1T(transistor)-1MTJ magnetoresistive cell based on our HSPICE compact model of MTJ voltage-driven by STT are in good agreement with the phase diagrams of the stability of spin states experimentally measured in the in-plane magnetized MTJs [3].

In this article the mechanism of the linear magnetoelectric (ME) effect in the rhombohedral multiferroic BiFeO$_3$ is considered. The study is based on the symmetry approach of the GinzburgLandau type, in which polarization, antiferrodistortion, and antiferromagnetic momentum vectors are viewed as ordering parameters. We demonstrate that the linear ME effect in BFO is caused by reorientation of the antiferrodistortion vector in either electric or magnetic field. The numerical estimations, which show quantitative agreement with the results of the recent measurements in film samples, have been performed. A possibility of significant enhancement of the magnetoelectric effect by applying an external static electric field has been investigated. The considered approach is promising for explaining the high values of the ME effect in composite films and heterostructures with BFO., 12 pages, 2 figures, submitted to Physical Review Letters

The phenomena of spin transport and spin-transfer torque between layers of a magnetic hetero-structure with conducting and tunneling interlayers, which can lead to switching of micromagnetic states in the structure’s layers and microwave generation, are considered. The features of spin transport in the field-emission mode for magnetic tunnel structures and problems of the current generation of spin waves in the magnetic film during local spin pumping are discussed. The possibility of the phase locking of the system of nanogenerators on a common magnetic platform and prospects for using nanogenerators based on the effect of spin transport in microwave communication engineering are analyzed.

Possible types of spatially modulated periodic antiferromagnetic structures in a uniaxial rhombohedral multiferroic with BiFeO3 crystal symmetry are studied depending on the ratio of the parameters of uniaxial and basal anisotropy and on magnetoelectric coupling.

We report on possible types of spatially modulated antiferromagnetic structures in the biaxial multiferroic and their stability depending on the ratio of the magnetic anisotropy and flexomagnetoelectric interaction parameters.

In this work it is shown that in magnetic heterostructure with the structural inversion symmetry due to the presence of the Rashba e ective eld responsible for the e ects of spin orbit scattering, the spin dynamics in the magnetic layer is described by a modi ed system of coupled equations for the exchange-coupled localized spins and spin accumulation. As a result, this system is reduced to an e ective Landau Lifshitz equation for the total magnetization with the additional torques which act similarly to the spin transfer torque in magnetic tunnel junction. Based on these equations, we calculated numerically the eld dependences of the switching current density for the di erent magnetization geometries, in particular for that was used in work of Miron et al., which are in a good agreement with the experimental results for the reasonable tting parameters.

Possible types of spatially modulated periodic antiferromagnetic structures in a uniaxial rhombohedral multiferroic with BiFeO3 crystal symmetry have been studied depending on the ratio of the uniaxial anisotropy and magnetoelectric interaction parameters. It has been shown that, along with symmetric cycloid antiferromagnetic structures with zero transverse component of the antiferromagnetism vector, there are changes in the antiferromagnetism vector direction with both right and left nonzero components of the antiferromagnetic moment, which are branched from the high-symmetry spatially modulated distribution. These solutions degenerate into a homogeneous state at a critical value of the normalized easy-plane anisotropy parameter. The existence of the found spatially inhomogeneous antiferromagnetic states with an incommensurate period can lead to additional features in magnetoelectric properties in multiferroics of the type under consideration near magnetic phase transitions in electric and magnetic fields.

We consider a vortex spin-torque diode represented by a magnetic tunnel junction with tilted magnetization in the fixed layer and vortex magnetization distribution in the free layer. Vortex dynamics and the corresponding spin-torque diode effect are analyzed using micromagnetic modelling for different input RF signal powers and different DC biases. The analytical model of the vortex spin-torque diode based on the Thiele equation is developed. We demonstrate that the considered spin-torque diode has sub-gigahertz resonant frequency, which can help to expand the frequency range of spintronic rectification. Relatively high sensitivity is observed in the case of ultra-low power even without the external magnetic field.

The work is devoted to the discussion of magnetic dynamics features in mesoscopic magnetic structures under the action of spin transport and spin-torque transfer in the nanopillar structures. Tunneling magnetic structure consisting of autoemission cathode and patterned magnetic anode is considered. Main bifurcation changes in the states of a dynamical system, that models mesoscopic structure, are discussed for varied spin polarized current and an external magnetic field.

The effect of the geometric shape of elements of a spin-dependent tunneling gate on their magnetic characteristic is studied theoretically and experimentally. The hysteresis dependences of the magnetization and magnetoresistance of rectangular, hexagonal, and ringed elements on the magnetic field are calculated for different characteristically geometric parameters of the elements. In the studies of the magneto-optical Kerr effect, magnetic hysteresis loops are observed. The calculated and experimental magnetization curves are compared, and qualitative agreement between the curves is gained.

The static and dynamic equilibrium states of spins in a thin layer of a conducting nanosized column containing two magnetic layers are analyzed theoretically. The magnetization of one of the layers is assumed to be fixed. The analysis is performed in terms of a macrospin model with allowance for the Slonczewski-Berger torque transfer. Bifurcation diagrams are constructed describing the change of spin states in the current-field plane. The relation of the specific features of varying magnetization and the spin precession frequency to bifurcations in the dynamic system under study is discussed. It is shown that the soft creation of cycles with a zero amplitude is accompanied by precession at a finite frequency and that the precession frequency becomes zero when a cycle with a finite amplitude disappears or arises in a jump. Comparative analysis is performed for two orientations of a magnetic field (parallel and perpendicular to the easy magnetization axis in the layer plane) in the presence of a current with a given spin orientation.

Reduced equations are derived for the slow evolution of the phase and amplitude of spin self-oscillations arising in a free magnetic layer of a nanosized microwave oscillator under the action of an over-threshold spin-polarized current. These equations are used to calculate the spectral intensity of the fundamental-harmonic signal of a spin self-oscillator allowing for thermal noise in the spin subsystem. It is shown that the line width as a function of the current is determined by a bifurcation-type variation in the spin state near the self-oscillation threshold. The self-oscillation nonisochronism makes the spectral line profile asymmetric. The line-profile asymmetry can change as the cycle-appearance point is approached.

Transmission of Yersinia pestis to the long-tailed suslik (Citellus undulatus) by fleas (Citellophilus tesquorum) in the Tuva natural plague focus in different seasons (spring, summer, and autumn) was studied experimentally. Between feeding periods, insects were kept in an artificial nest under temperature and humidity closely corresponding to seasonal ones. The character of the agent transmission was estimated according to the fraction of fleas with the agent in the aggregated state (bacterial lumps, partial blocks of proventriculus), the fraction of blocked individuals, and the fraction of infected susliks and of those with the generalized form of infection. Seasonal dynamics of epizootic process of the Y. pestis transmission corresponded to the results obtained in the epizootic examination of the Tuva natural plague focus and reflected the dynamics of the epizootic process (increase-peak-decline). The activity of the formation of a proventriculus block in C. t. altaicus, the infection ability of the fleas, and the sensitivity of long-tailed Siberian susliks to Y. pestis were the highest in mid-summer (July-first ten days of August), during the period of epizooty activation in the focus. The maximal number of C. t. altaicus with the plague agent at the aggregated state was observed in the cold period, before wintering of insects and after their hibernation.

A simplified model of the Jahn-Teller band effect in a conducting ferromagnet with two degenerate subbands with the peak density of states of itinerant electrons is considered. It is found that the martensite transition temperature in a narrow-band conductor as a function of the position of the Fermi level near the peak of the energy density of states varies nonmonotonically in the narrow spin electron subband. The magnetic field dependence of the martensite-austenite structural phase transition temperature in the ferromagnet is analyzed. The developed theory and calculated data for the electron density of states in Ni{sub 2}MnGa are used as the basis for estimating the variation of the martensite transition temperature with the magnetic field ({partial_derivative}T{sub m}/{partial_derivative}H), which demonstrates a satisfactory agreement with experimental data for the Heusler alloy Ni{sub 2+x}Mn{sub 1-x}Fe{sub y}Ga{sub 1-y}.

The room temperature multiferroic ${\mathrm{BiFeO}}_{3}$, by far the most studied experimentally, exhibits outstanding ferroelectric properties with a cycloidal magnetic order in the bulk and many unexpected advantages for possible applications in spintronics, sensor techniques, and photovoltaics. To consider ferroelectric and magnetic phase transitions in multiferroic ${\mathrm{BiFeO}}_{3}$ under electric field, we suggest the Ginsburg-Landau-like approach based on the symmetry and $\mathbf{P}\ensuremath{-}\ensuremath{\omega}\ensuremath{-}\mathbf{L}$ coupling, where the order parameters are: $\mathbf{P}$ is the electric polarization, $\ensuremath{\omega}$ is the axial vector of antidistorsion (describing a rotation of the oxygen octahedrons), and $\mathbf{L}$ is the antiferromagnetic vector. The theoretical model is consistent with experiment and ab initio calculations data. We give the complete set of numerical coefficients of the model and explore the behavior of $\mathbf{P}$ and $\ensuremath{\omega}$ vectors in strong electric field. The proposed approach is particularly promising for the analysis of magnetoelectric phenomena whose length scale is significantly larger than the length of the cell used in ab initio calculations. The considered cycloid problem is the clear example of such a system. Electric field-induced transformations of cycloid are exemplified on an epitaxial ${\mathrm{BiFeO}}_{3}$ film grown on the (001)-oriented substrate. We show that the jump of vectors $\mathbf{P}$ and $\ensuremath{\omega}$ in the field $E=6$ MV/m is accompanied by a jump of a cycloid spin rotation plane. This effect is of particular interest for spintronics and nanoelectronics.

The subject of discussion is calibration of the tip of a magnetic force microscope using the field of a ring-shaped current loop. To calculate the calibration parameters, the magnetic contribution from the extended tip of the probe in the field of the current loop to the rigidity of the cantilever is approximated by the contribution from a point magnetic dipole and magnetic “charge” in terms of the theoretical model adopted. Three simplified models of the conic tip (with a sharpened, blunted, and rounded top) are considered. The calculated dependences of the effective calibration parameters on the radius of the current loop are compared with experimental data. It is found that the model of a uniformly magnetized tip in the form of a blunted cone provides the best fit to the experiment. The calculation results may be helpful in simulating images obtained with a magnetic force microscope and numerically testing magnetic objects.