A theoretical study is given for the self-oscillation excited in a spin torque oscillator (STO) consisting of an in-plane magnetized free layer and a perpendicularly magnetized pinned layer in the presence of a perpendicular magnetic field. This type of STO is a potential candidate for a microwave source of microwave assisted magnetization reversal (MAMR). It is, however, found that the self-oscillation applicable to MAMR disappears when the perpendicular field is larger than a critical value, which is much smaller than a demagnetization field. This result provides a condition that the reversal field of a magnetic recording bit by MAMR in nanopillar structure should be smaller than the critical value. The analytical formulas of currents determining the critical field are obtained, which indicate that a material with a small damping is not preferable to acheive a wide range of the self-oscillation applicable to MAMR, although such a material is preferable from the view point of the reduction of the power consumption.
An excitation of a large-amplitude out-of-plane magnetization oscillation in a ferromagnet by the spin Hall effect is of great interest for practical applications such as microwave generator and neuromorphic computing. However, both experimental and theoretical works have revealed that only small-amplitude oscillation around an in-plane easy axis can be excited via the spin Hall effect. Here, we propose that an out-of-plane oscillation can be excited due to an assistance of field-like torque. We focus on an in-plane magnetized ferromagnet with an easy axis parallel to current direction. We not…
We study the influence of the spin torque, which depends on the space and time derivative of magnetization, on magnetization reversal time in a ferromagnetic fine particle. The spin torque operates to dissipate the angular momentum of the magnetization precession, and the torque increases in a spin vortex structure. We calculate the magnetization reversal time under a DC magnetic field using the Landau-Lifshitz-Gilbert equation containing a spin torque term. We found that the spin torque changes the magnetization switching speed significantly during the reversal process by maintaining a spin v…
Control of magnetization dynamics is one of the primary goals in spintronics. It has been demonstrated using spin Hall effect i.e charge current to spin current conversion in non-magnetic metal which has large spin-orbit coupling such as Pt, W etc. Recently different groups have shown generation of spin current in Pt, W while thermal gradient is created by virtue of spin Nernst effect. In this work we show the evidence of magnetization control by spin Nernst torque in Pt/Py bi-layer. We compared relative strength of spin Nernst Torque and spin Hall torque by measuring the systematic variation …
The spin-torque ferromagnetic resonance (ST-FMR) in a bilayer system consisting of a magnetic insulator such as Y3Fe5O12 and a normal metal with spin-orbit interaction such as Pt is addressed theoretically. We model the ST-FMR for all magnetization directions and in the presence of field-like spin-orbit torques based on the drift-diffusion spin model and quantum mechanical boundary conditions. ST-FMR experiments may expose crucial information about the spin-orbit coupling between currents and magnetization in the bilayers.
We study the role of thermal fluctuations on the spin dynamics of a thin permalloy film with a focus on the behavior of spin torque and find that the thermally assisted spin torque results in new aspects of the magnetization dynamics. In particular, we uncover the formation of a finite, spin torque-induced, in-plane magnetization component. The orientation of the in-plane magnetization vector depends on the temperature and the spin-torque coupling. We investigate and illustrate that the variation of the temperature leads to a thermally-induced rotation of the in-plane magnetization.
Spin transfer torque from spin-polarized electrical current can excite large-amplitude magnetization dynamics in metallic ferromagnets of nanoscale dimensions. Since magnetic anisotropy energies of nanomagnets are comparable to the thermal energy scale, temperature can have a profound effect on the dynamics of a nanomagnet driven by spin transfer torque. Here we report the observation of unusual types of microwave-frequency nonlinear magnetization dynamics co-excited by alternating spin transfer torque and thermal fluctuations. In these dynamics, temperature amplifies the amplitude of GHz-rang…
Measurements of spin-orbit torques in a ferromagnetic/nonmagnetic multilayer are typically based on an assumption that the entire ferromagnetic layer uniformly responds to the spin-orbit torque. This assumption breaks down when the thickness of the ferromagnetic layer is comparable to the dynamic exchange coupling length, which can be as short as a few nanometers in certain measurement geometries. The nonuniform magnetization reorientation coupled with nonuniform contribution of each magnetic sublayer to the magnetoresistance or the Kerr effect may impact the accuracy in the extrapolation of s…
In a nanopillar with dipolarly coupled vortices, we present an experimental and simulation study to understand how the interplay between the bias field and spin transfer torque impacts reversal of the vortex cores. We find that, depending on the current values, vortex cores might experience different physical mechanisms for their reversal, namely a static or a dynamic switching. We believe that our results might be useful in the context of vortex based non volatile memories, as a current controlled selective core switching is proposed.
In systems with small spin-orbit coupling, current-induced torques on the magnetization require inhomogeneous magnetization textures. For large spin-orbit coupling, such torques exist even without gradients in the magnetization direction. Here, we consider current-induced torques in ferromagnetic metals with both Rashba spin-orbit coupling and inhomogeneous magnetization. We first phenomenologically construct all torques that are allowed by the symmetries of the system, to first order in magnetization-direction gradients and electric field. Second, we use a Boltzmann approach to calculate the …
We proposed and demonstrated a simple method for detection of in-plane magnetization switching by spin-orbit torque (SOT) in bilayers of non-magnetic / magnetic materials. In our method, SOT is used not only for magnetization switching but also for detection. Our method can detect arbitrary Mx and My component without an external magnetic field, which is useful for fast characterization of type-X, type-Y, and type-XY SOT magnetization switching. Our SOT detection scheme can be utilized not only for fast characterization of SOT switching in bilayers, but also for electrical detection of in-plan…
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 t…
Oscillation frequency of spin torque oscillator with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer is theoretically investigated by taking into account the field-like torque. It is shown that the field-like torque plays an important role in finding the balance between the energy supplied by the spin torque and the dissipation due to the damping, which results in a steady precession. The validity of the developed theory is confirmed by performing numerical simulations based on the Landau-Lifshitz-Gilbert equation.
Parametric mechanism of the magnetization reversal as a low-power recording mechanism for MRAM. Measurement of spin-accumulation-induced in-plane magnetic field in a FeB nanomagnet
The optimum condition of the magnetization alignment to maximize the spin torque diode voltage is derived by solving the Landau-Lifshitz-Gilbert equation. We show that the optimized diode voltage can be one order of magnitude larger than that of the conventional alignment where the easy axes of the free and the pinned layers are parallel. These analytical predictions are confirmed by numerical simulations.
Spin torque majority gates are modeled and several regimes of magnetization switching (some leading to failure) are discovered. The switching speed and noise margins are determined for STMGs and an adder based on it. With switching time of 3ns at current of 80uA, the adder computational throughput is comparable to that of a CMOS adder.
The dynamics of magnetization in the presence of spin-transfer torque was studied. We derived the equation for the motion of magnetization in the presence of a spin current by using the local equilibrium assumption in non-equilibrium thermodynamics. We show that, in the resultant equation, the ratio of the Gilbert damping constant, $α$, and the coefficient, $β$, of the current-induced torque, called non-adiabatic torque, depends on the relaxation time of the fluctuating field $τ_{c}$. The equality $α=β$ holds when $τ_c$ is very short compared to the time scale of magnetization dynamics. We app…
We simulate the spin torque-induced reversal of the magnetization in thin disks with perpendicular anisotropy at zero temperature. Disks typically smaller than 20 nm in diameter exhibit coherent reversal. A domain wall is involved in larger disks. We derive the critical diameter of this transition. Using a proper definition of the critical voltage, a macrospin model can account perfectly for the reversal dynamics when the reversal is coherent. The same critical voltage appears to match with the micromagnetics switching voltage regardless of the switching path.
Micromagnetics simulation based on the classical Landau-Lifshitz-Gilbert (LLG) equation has long been a powerful method for modeling magnetization dynamics and reversal of three-dimensional (3D) magnets. For two-dimensional (2D) magnets, the magnetization reversal always accompanies the collapse of the magnetization even at low temperatures due to intrinsic strong spin fluctuation. We propose a micromagnetic theory that explicitly takes into account the rapid demagnetization and remagnetization dynamics of 2D magnets during magnetization reversal. We apply the theory to a single-domain magnet …
In ferromagnetic trilayers, a spin-orbit-induced spin current can have a spin polarization of which direction is deviated from that for the spin Hall effect. Recently, magnetization switching in ferromagnetic trilayers has been proposed and confirmed by the experiments. In this work, we theoretically and numerically investigate the switching current required for perpendicular magnetization switching in ferromagnetic trilayers. We confirm that the tilted spin polarization enables field-free deterministic switching at a lower current than conventional spin-orbit torque or spin-transfer torque sw…
Spin torque from spin current applied to a nanoscale region of a ferromagnet can act as negative magnetic damping and thereby excite self-oscillations of its magnetization. In contrast, spin torque uniformly applied to the magnetization of an extended ferromagnetic film does not generate self-oscillatory magnetic dynamics but leads to reduction of the saturation magnetization. Here we report studies of the effect of spin torque on a system of intermediate dimensionality - a ferromagnetic nanowire. We observe coherent self-oscillations of magnetization in a ferromagnetic nanowire serving as the…
A mechanism of current-induced magnetization reversal based on the parametric resonance is described. The source of the magnetization reversal is a current-induced magnetic field, which is applied perpendicularly to the easy axis of magnetic anisotropy of a ferromagnetic nanomagnet. The current-induced magnetic field was measured in a FeCoB nanomagnet to be 60 Gauss at a current density of 65 mA/um2. Two mechanisms of the magnetization reversal are described and calculated. The first mechanism is the reversal by a RF electrical current, which modulated at a frequency close to the precession fr…
We show that ideal memristors - devices whose resistance is proportional to the charge that flows through them - can be realized using spin torque-driven viscous magnetization dynamics. The latter can be accomplished in the spin liquid state of thin-film heterostructures with frustrated exchange, where memristive response is tunable by proximity to the glass transition, while current-induced Joule heating facilitates nonvolatile operation and second-order memristive functionality beneficial for neuromorphic applications. Ideal memristive behaviors can be achieved in other systems characterized…
The length scale of the magnetization gradients in chiral magnets is determined by the relativistic Dzyaloshinskii-Moriya interaction. Thus, even conventional spin-transfer torques are controlled by the relativistic spin-orbit coupling in these systems, and additional relativistic corrections to the current-induced torques and magnetization damping become important for a complete understanding of the current-driven magnetization dynamics. We theoretically study the effects of reactive and dissipative homogeneous spin-orbit torques and anisotropic damping on the current-driven skyrmion dynamics…
Spin valves incorporating perpendicularly magnetized materials are promising structures for memory elements and high-frequency generators. We report the angular dependence of the spin-transfer torque in spin valves with perpendicular equilibrium magnetization computed by first-principles circuit theory and compare results with experiments by W.H. Rippard c.s. [Phys. Rev. B 81, 014426 (2010)] on the CoFe|Cu|CoNi system. Furthermore, we predict a non-monotonous ("wavy") spin-transfer torque when the Cu spacer is replaced by a Ru layer.
A crossover between two mechanisms destabilizing the magnetization in equilibrium by the spin transfer effect is found in a ferromagnetic multilayer consisting of an in-plane magnetized free layer and a perpendicularly magnetized pinned layer, where an in-plane magnetic field is applied, and electric current flows from the pinned to the free layer. A fast transition from the in-plane to the out-of-plane state occurs in the low-field region, whereas a slow transition with small-amplitude oscillation becomes dominant in the high-field region. On the other hand, only the fast transition mechanism…
We demonstrate (using full-scale micromagnetic simulations) that the spin injection driven steady-state precession of a thin magnetic nanoelement exhibit a complicate transition from the quasi-macrospin to the chaotic behaviour with the increasing element size. For nanoelement parameters typical for those used experimentally we have found that the macrospin approximation becomes invalid already for very small nanoelement sizes (~ 30 nm), in contrast to the previously reported results (Li and Zhang, Phys. Rev. B, vol. B68, 024404-1 (2003))
The asymptotic behavior of switching time as a function of current for a uniaxial macrospin under the effects of both spin-torque and thermal noise is explored analytically by focusing on its diffusive energy space dynamics. The scaling dependence ($I\rightarrow 0$, $<τ\propto\exp(-ξ(1-I)^2)$) is shown to confirm recent literature results. The analysis shows the mean switching time to be functionally independent of the angle between the spin current and magnet's uniaxial axes. These results have important implications for modeling the energetics of thermally assisted magnetization reversal of …
Thermal fluctuations of nanomagnets driven by spin-polarized currents are treated via the Landau-Lifshitz-Gilbert equation generalized to include both the random thermal noise field and the Slonczewski spin-transfer torque term. By averaging this stochastic (Langevin) equation over its realizations, the explicit infinite hierarchy of differential-recurrence relations for statistical moments (averaged spherical harmonics) is derived for arbitrary demagnetizing factors and magnetocrystalline anisotropy for the generic nanopillar model of a spin-torque device comprising two ferromagnetic strata r…
An effective method for ultrafast magnetization reversal of nanoclusters is suggested. The method is based on coupling a nanocluster to a resonant electric circuit. This coupling causes the appearance of a magnetic feedback field acting on the cluster, which drastically shortens the magnetization reversal time. The influence of the resonator properties, nanocluster parameters, and external fields on the magnetization dynamics and reversal time is analyzed. The magnetization reversal time can be made many orders shorter than the natural relaxation time. The reversal is studied for both the case…
Magnetization reversal of interconnected Kagome artificial spin ice was studied by the first-order reversal curve (FORC) technique based on the magneto-optical Kerr effect and magnetoresistance measurements. The magnetization reversal exhibits a distinct six-fold symmetry with the external field orientation. When the field is parallel to one of the nano-bar branches, the domain nucleation/propagation and annihilation processes sensitively depend on the field cycling history and the maximum field applied. When the field is nearly perpendicular to one of the branches, the FORC measurement reveal…
We apply a recently developed quantum theory of spin transfer torque to study the effect of the quantum noise in spin transfer process on the magnetization switching in spin-torque-driven devices. The quantum noise induces considerable fluctuation of the switching time at zero temperature. By including the thermal noise, the temperature dependence of the expectation value and standard deviation of the switching time are obtained in the Monte Carlo simulations, and the results are fitted to an effective first passage model. We expect that the predictions here are observable in the single-shot m…
We demonstrate a quasi ballistic switching of the magnetization in a microscopic mag-neto resistive memory cell. By means of time resolved magneto transport we follow the large angle precession of the free layer magnetization of a spin valve cell upon applica-tion of transverse magnetic field pulses. Stopping the field pulse after a 180 degree precession rotation leads to magnetization reversal with reversal times as short as 165 ps. This switching mode represents the fundamental ultra fast limit of field induced magnetization reversal.
Local magnetic moments can be created in graphene by incorporating different defects. The possibility of regulating dynamics of magnetization in graphene, by employing the Purcell effect, is analyzed. The role of the system parameters in magnetization reversal is studied. The characteristics of such a reversal can be varied in a wide range, which can be used for various applications in spintronics. It is shown that fast magnetization reversal generates coherent radiation.
The critical current of the spin transfer torque-driven magnetization dynamics was studied by taking into account both spin pumping and the finite penetration depth of the transverse spin current. We successfully reproduced the recent experimental results obtained by Chen et al. [Phys. Rev. B {\bf 74}, 144408 (2006)] and found that the critical current remains finite even in the zero-thickness limit of the free layer. We showed that the remaining value of the critical current is determined mainly by spin pumping.
We demonstrate magnetization switching in out-of-plane magnetized Ta\CoFeB\MgO nanowires by current pulse injection along the nanowires, both with and without a constant and uniform magnetic field collinear to the current direction. We deduce that an effective torque arising from spin-orbit effects in the multilayer drives the switching mechanism. While the generation of a component of the magnetization along the current direction is crucial for the switching to occur, we observe that even without a longitudinal field thermally generated magnetization fluctuations can lead to switching. Analys…
Using an optimally coupled nanometer-scale superconducting quantum interference device, we measure the magnetic flux originating from an individual ferromagnetic Ni nanotube attached to a Si cantilever. At the same time, we detect the nanotube's volume magnetization using torque magnetometry. We observe both the predicted reversible and irreversible reversal processes. A detailed comparison with micromagnetic simulations suggests that vortex-like states are formed in different segments of the individual nanotube. Such stray-field free states are interesting for memory applications and non-inva…
We study current-induced switching in magnetic tunnel junctions (MTJs) in the presence of a field-like spin-transfer torque and titled pinned-layer magnetization in the high current limit at finite temperature. We consider both the Slonczewski and field-like torques with coefficients $a_J$ and $b_J $, respectively. At finite temperatures, $σ= b_J/a_J = \pm1$ leads to a smaller mean switching time compared that with $σ= 0$. The reduction of switching time in the presence of the field-like term is due to the alignment effect (for $σ> 0$) and the initial torque effect.
We simulate the stochastic Landau-Lifshitz-Gilbert (LLG) dynamics of a uniaxial nanomagnet out to sub-millisecond timescales using a graphical processing unit based micromagnetic code and determine the effect of geometrical tilts between the spin-current and uniaxial anisotropy axes on the thermally assisted reversal dynamics. The asymptotic behavior of the switching time ($I\rightarrow 0$, $<τ>\propto\exp(-ξ(1-I)^2)$) is approached gradually, indicating a broad crossover regime between ballistic and thermally assisted spin transfer reversal. Interestingly, the mean switching time is shown to …
Spin-orbit torques are studied in Ta/TbFeCo patterned structures with a bulk perpendicular magnetic anisotropy (bulk-PMA) for the first time. The current-induced magnetization switching is investigated in the presence of a perpendicular, longitudinal, or transverse field. In order to rule out Joule heating effect, switching of the magnetization is also demonstrated using current pulses. It is found that the anti-damping torque correlated with spin Hall effect is very strong, and a spin Hall angle of about 0.12 is obtained. The field-like torque related with Rashba effect is negligible in this …
We investigate the domain wall dynamics of a ferromagnetic wire under the combined influence of a spin-polarized current and magnonic spin-transfer torque generated by an external field, taking also into account Rashba spin-orbit coupling interactions. It is demonstrated that current-induced motion of the domain wall may be completely reversed in an oscillatory fashion by applying a magnonic spin-transfer torque as long as the spin-wave velocity is sufficiently high. Moreover, we show that the motion of the domain wall may be fully terminated by means of the generation of spin-waves, suggestin…
We demonstrate efficient current induced spin orbit torque switching in perpendicularly magnetized epitaxial MgO(001)//Pd/Co2FeAl/MgO heterostructures grown by magnetron sputtering. The advantage of such heterostructures for spin orbit torque MRAM devices is that they allow a unique combination of critical ingredients: i) low resistivity required for reduced electric energy consumption during writing, ii) high TMR ratio required for efficient reading and iii) strong perpendicular magnetic anisotropy for increased thermal stability.
In this article we analyze by modeling two possible mechanisms for magnetization switching using spin orbit torques, which have been reported to cause field-free deterministic switching in experiments. Here we compare the field-free magnetization switching due to a tilt of the anisotropy direction against the use of an antiferromagnetic bias field. Simple results obtained analytically show that a bias field not only causes the magnetization reversal but also reduces the corresponding energy barrier. The critical current required for magnetization switching is analyzed on the basis of a macrosp…
We show that asymmetrical mesoscopic superconductors bring new insight into vortex physics where we found the remarkable coexistence of long and short vortices. We study an asymmetrical mesoscopic sphere, that lacks one of its quadrants, and obtain its three-dimensional vortex patterns by solving the Ginzburg-Landau theory. We find that the vortex patterns are asymmetric whose effects are clearly visible and detectable in the transverse magnetization and torque.
We report quantum and semi-classical calculations of spin current and spin-transfer torque in a free-electron Stoner model for systems where the magnetization varies continuously in one dimension.Analytic results are obtained for an infinite spin spiral and numerical results are obtained for realistic domain wall profiles. The adiabatic limit describes conduction electron spins that follow the sum of the exchange field and an effective, velocity-dependent field produced by the gradient of the magnetization in the wall. Non-adiabatic effects arise for short domain walls but their magnitude decr…
The Casimir-Lifshitz torque between two biaxially polarizable anisotropic planar slabs is shown to exhibit a non-trivial sign-reversal in its rotational sense. The critical distance $a_c$ between the slabs that marks this reversal is characterized by the frequency $ω_c\!\sim \!c/2a_c$ at which the in-planar polarizabilities along the two principal axes are equal. The two materials seek to align their principal axes of polarizabilities in one direction below $a_c$, while above $a_c$ their axes try to align rotated perpendicular relative to their previous minimum energy orientation. The sign-rev…
A common ploy to reduce the switching current and energy dissipation in spin-transfer-torque driven magnetization switching of shape-anisotropic single-domain nanomagnets is to employ magnets with low saturation magnetization $M_s$ and high shape-anisotropy. The high shape-anisotropy compensates for low $M_s$ to keep the static switching error rate constant. However, this ploy increases the switching delay, its variance in the presence of thermal noise, and the dynamic switching error rate. Using the stochastic Landau-Lifshitz-Gilbert equation with a random torque emulating thermal noise, we s…
We investigate magnetic anisotropy and magnetization reversal mechanism in chromium telluride thin films grown by molecular beam epitaxy. We report existence of strong perpendicular anisotropy in these thin films, along with a relatively strong second order anisotropy contribution. The angular variation of the switching field observed from the magnetoresistance measurement is explained quantitatively using a one-dimensional defect model. The model reveals the relative roles of nucleation and pinning in the magnetization reversal, depending on the applied field orientation. Micromagnetic simula…
The spin-orbit torque switching of ferrimagnetic Gd$_x$(Fe$_{90}$Co$_{10}$)$_{100-x}$ films was studied for both transition metal (TM)-rich and rare earth (RE)-rich configurations. The spin-orbit torque driven magnetization switching follows the same handedness in TM-rich and RE-rich samples with respect to the total magnetization, but the handedness of the switching is reversed with respect to the TM magnetization. This indicates that the sign of the spin-orbit-torque-driven magnetic switching follows the total magnetization, although transport based techniques such as anomalous Hall effect a…
The switching probability of a single-domain ferromagnet under spin-current excitation is evaluated using the Fokker-Planck equation(FPE). In the case of uniaxial anisotropy, the FPE reduces to an ordinary differential equation in which the lowest eigenvalue $λ_1$ determines the slowest switching events. We have calculated $λ_1$ by using both analytical and numerical methods. It is found that the previous model based on thermally distributed initial magnetization states \cite{Sun1} can be accurately justified in some useful limiting conditions.
Magnetization reversal mechanisms and impact of magnetization direction are studied in square arrays of interconnected circular permalloy nanorings using MOKE, local imaging, numerical simulations and transport techniques.
We report on the current-induced magnetization switching of a three-terminal perpendicular magnetic tunnel junction by spin-orbit torque and the read-out using the tunnelling magnetoresistance (TMR) effect. The device is composed of a perpendicular Ta/FeCoB/MgO/FeCoB stack on top of a Ta current line. The magnetization of the bottom FeCoB layer can be switched reproducibly by the injection of current pulses with density $5\times10^{11}$ A/m$^2$ in the Ta layer in the presence of an in-plane bias magnetic field, leading to the full-scale change of the TMR signal. Our work demonstrates the proof…
We show that the spin-orbit torque induced magnetization switching in nanomagnets presenting Dzyaloshinskii-Moriya (DMI) interaction is governed by a chiral domain nucleation at the edges. The nucleation is induced by the DMI and the applied in-plane magnetic field followed by domain wall propagation. Our micromagnetic simulations show that the DC switching current can be defined as the edge nucleation current, which decreases strongly with increasing amplitude of the DMI. This description allows us to build a simple analytical model to quantitatively predict the switching current. We find tha…
The reversal of perpendicular magnetization (PM) by electric control is crucial for high-density integration of low-power magnetic random-access memory (MRAM). Although the spin-transfer torque (STT) and spin-orbit torque (SOT) technologies have been used to switch the magnetization of a free layer with perpendicular magnetic anisotropy, the former has limited endurance because of the high current density directly through the junction, while the latter requires an external magnetic field or unconventional configuration to break the symmetry. Here we propose and realize the orbit-transfer torqu…
The dependence of the critical current of spin transfer torque-driven magnetization dynamics on the free-layer thickness was studied by taking into account both the finite penetration depth of the transverse spin current and spin pumping. We showed that the critical current remains finite in the zero-thickness limit of the free layer for both parallel and anti-parallel alignments. We also showed that the remaining value of the critical current of parallel to anti-parallel switching is larger than that of anti-parallel to parallel switching.
