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A demonstration of parametric oscillation of magnetization in nanostructured ferromagnets via voltage-controlled magnetic anisotropy (VCMA) effect provided an alternative approach to spintronic oscillator applications with low-energy consumption. However, the phase of this voltage-controlled parametric oscillator was unable to be locked uniquely by microwave VCMA effect. The oscillation phase is locked in one of two possible states, which originates from the fact that the frequency of the microwave voltage is twice that of the magnetization oscillation. In this work, we investigate the phase locking by additional microwave voltage through analytical and numerical studies of the Landau-Lifshitz-Gilbert equation. An analytical study suggests that the additional voltage makes one of two phases more stable than the other by having asymmetric potential for the phase. The simulation results indicate a trigonometric-function-like dependence of the locked rate on the phase of the additional voltage, which qualitatively agrees with the analytical theory and also suggests a possibility to manipulate the phase by the additional voltage.

Recent observations of chaos in nanomagnet suggest a possibility of new spintronics applications such as random-number generator and neuromorphic computing. However, large amount of electric current and/or magnetic field are necessary for the excitation of chaos, which are unsuitable for energy-saving applications. Here, we propose an excitation of chaos in three-terminal spin-torque oscillator (STO). The driving force of the chaos is voltage-controlled magnetic anisotropy (VCMA) effect, which enables us to manipulate magnetization dynamics without spending electric current or magnetic field, and thus, energy efficient. In particular, we focus on the VCMA effect generated by feedback signal from the STO since feedback effect is known to be effective in exciting chaos in dynamical system. Solving the Landau-Lifshitz-Gilbert (LLG) equation numerically and applying temporal and statistical analyses to its solution, the existence of the chaotic and transient-chaotic magnetization dynamics driven by the feedback VCMA effect is identified., Comment: 6 pages, 4 figures

An excitation of highly nonlinear, complex magnetization dynamics in a ferromagnet, for example chaos, is a new research target in spintronics. This technology is applied to practical applications such as random number generator and information processing systems. One way to induce complex dynamics is applying feedback effect to the ferromagnet. The role of the feedback electric current on the magnetization dynamics was studied in the past. However, there is another way to apply feedback effect to the ferromagnet, namely feedback magnetic field. In this paper, we developed both numerical and theoretical analyses on the role of the feedback magnetic field causing complex magnetization dynamics. The numerical simulation indicates the change of the dynamical behavior from a simple oscillation with a unique frequency to complex dynamics such as amplitude modulation and chaos. The theoretical analyses on the equation of motion qualitatively explain several features found in the numerical simulations, exemplified as an appearance of multipeak structure in the Fourier spectra. The difference of the role of the feedback electric current and magnetic field is also revealed from the theoretical analyses., Comment: 9 pages, 5 figures

Antiferromagnetic magnons possess a distinctive feature absent in their ferromagnetic counterparts: the presence of two distinct handedness modes, the right-handed (RH) and left-handed (LH) precession modes. The magnon handedness determines the sign of spin polarization carried by the propagating magnon, which is indispensable for harnessing the diverse functionalities. However, the control of coherently propagating magnon handedness in antiferromagnets has remained elusive so far. Here we demonstrate the manipulation and electrical readout of propagating magnon handedness in perpendicularly magnetized synthetic antiferromagnets (SAF). We find that the antiferromagnetic magnon handedness can be directly identified by measuring the inverse spin Hall effect (ISHE) voltage, which arises from the spin pumping effect caused by the propagating antiferromagnetic magnons in the SAF structure. The RH and LH modes of the magnon can be distinguishable particularly when the SAF structure is sandwiched by heavy metals with the same sign of spin Hall angle. Moreover, we succeed in controlling the handedness of propagating antiferromagnetic magnons by tuning the excitation microwave frequency. This work unveils promising avenues for harnessing magnon unique properties in antiferromagnet-based magnonic applications.

An experimental demonstration of a parametric oscillation of a magnetization in a ferromagnet was performed recently by applying a microwave voltage, indicating the potential to be applied in a switching method in non-volatile memories. In the previous works, the modulation of a perpendicular magnetic anisotropy field produced by the microwave voltage was small compared with an external magnetic field pointing in an in-plane direction. A recent trend is, however, opposite, where an efficiency of the voltage controlled magnetic anisotropy (VCMA) effect is increased significantly by material research and thus, the modulated magnetic anisotropy field can be larger than the external magnetic field. Here, we solved the Landau-Lifshitz-Gilbert equation numerically and investigated the magnetization dynamics driven under a wide range of the microwave VCMA effect. We evaluated bifurcation diagrams, which summarize local maxima of the magnetization dynamics. For low modulation amplitudes, the local maximum is a single point because the dynamics is the periodic parametric oscillation. The bifurcation diagrams show distributions of the local maxima when the microwave magnetic anisotropy field becomes larger than the external magnetic field. The appearance of this broadened distribution indicates complex dynamics such as chaotic and transient-chaotic behaviors, which were confirmed from an analysis of temporal dynamics.

Recently, an associative memory operation by a virtual oscillator network, consisting of a single spintronic oscillator, was examined to solve issues in conventional, real oscillators-based neural networks such as inhomogeneities between the oscillators. However, the spintronic oscillator still carries issues dissipating large amount of energy because it is driven by electric current. Here, we propose to use a single ferromagnet manipulated by voltage-controlled magnetic anisotropy (VCMA) effect as a fundamental element in a virtual neural network, which will contribute to significantly reducing the Joule heating caused by electric current. Instead of the oscillation in oscillator networks, magnetization relaxation dynamics were used for the associative memory operation. The associative memory operation for alphabet patterns is successfully demonstrated by giving correspondences between the colors in a pattern recognition task and the sign of a perpendicular magnetic anisotropy coefficient, which could be either positive or negative via the VCMA effect., Comment: 18 pages, 5 figures, publication information was added

We provide high-speed implementations for simulating reservoirs described by $N$-coupled spin-torque oscillators. Here $N$ also corresponds to the number of reservoir nodes. We benchmark a variety of implementations based on CPU and GPU. Our new methods are at least 2.6 times quicker than the baseline for $N$ in range $1$ to $10^4$. More specifically, over all implementations the best factor is 78.9 for $N=1$ which decreases to 2.6 for $N=10^3$ and finally increases to 23.8 for $N=10^4$. GPU outperforms CPU significantly at $N=2500$. Our results show that GPU implementations should be tested for reservoir simulations. The implementations considered here can be used for any reservoir with evolution that can be approximated using an explicit method.

We develop a high-throughput method for measuring the composition dependence of magnetoresistance (MR) and spin-transfer-torque (STT) effects in current-perpendicular-to-plane giant magnetoresistance (CPP-GMR) devices and report its application to the CoFe system. The method is based on the use of composition-gradient films deposited by combinatorial sputtering. This structure allows the fabrication of devices with different compositions on a single substrate, drastically enhancing the throughput in investigating composition dependence. We fabricated CPP-GMR devices on a single GMR film consisting of a Co$_{x}$Fe$_{1-x}$ (0 $\le$ $\textit{x}$ $\le$ 1) composition-gradient layer, a Cu spacer layer, and a NiFe layer. The MR ratio obtained from resistance-field measurements exhibited the maximum in the broad Co concentration range of 0.3 $\le$ $\textit{x}$ $\le$ 0.65. In addition, the STT efficiency was estimated from the current to induce magnetization reversal of the NiFe layer by spin injection from the Co$_{x}$Fe$_{1-x}$ layer. The STT efficiency was also the highest around the same Co concentration range as for the MR ratio, and this correlation was theoretically explained by the change in the spin polarization of the Co$_{x}$Fe$_{1-x}$ layer. The results revealed the Co$_{x}$Fe$_{1-x}$ composition range suitable for spintronic applications, demonstrating the advantages of the developed method., Comment: 19 pages, 5 figures

An artificial spin ice consisting of numerous ferromagnets has attracted attention because of its applicability to practical devices. The ferromagnets interact through their stray magnetic field and show various functionality. The ferromagnetic element in the spin ice was recently made in elliptical-shape or stadium-shape. The former has a narrow edge, expecting to generate a large stray magnetic field. The latter has a large volume and is also expected to generate a large stray magnetic field. Here, we estimate the stray magnetic field by numerically integrating the solution of the Poisson equation. When magnetization is parallel to an easy axis, the elliptical-shaped ferromagnet generates a larger stray magnetic field than the stadium-shaped ferromagnet. The stray magnetic fields from both ferromagnets for arbitrary magnetization directions are also investigated., Comment: The publication information is added

A coupled oscillator network may be able to perform an energy-efficient associative memory operation. However, its realization has been difficult because inhomogeneities unavoidably arise among the oscillators during fabrication and lead to an unreliable operation. This issue could be resolved if the oscillator network were able to be formed from a single oscillator. Here, we performed numerical simulations and theoretical analyses on an associative memory operation that uses a virtual oscillator network based on a spin-torque oscillator. The virtual network combines the concept of coupled oscillators with that of feedforward neural networks. Numerical experiments demonstrate successful associations of $60$-pixel patterns with various memorized patterns. Moreover, the origin of the associative memory is shown to be forced synchronization driven by feedforward input, where phase differences among oscillators are fixed and correspond to the colors of the pixels in the pattern., Comment: 15 pages, 4 figures (main text), Supplemental Information was added. Trigonometric functions in Eqs. (6), (7), (9), (10), (11), (12), (14), and (15), as well as related terms in main text are corrected, but they do not affect the conclusions

A new research topic in spintronics relating to the operation principles of brain-inspired computing is input-driven magnetization dynamics in nanomagnet. In this paper, the magnetization dynamics in a vortex spin-torque oscillator (STO) driven by a series of random magnetic field are studied through a numerical simulation of the Thiele equation. It is found that input-driven synchronization occurs in the weak perturbation limit, as found recently. As well, chaotic behavior is newly found to occur in the vortex core dynamics for a wide range of parameters, where synchronized behavior is disrupted by an intermittency. Ordered and chaotic dynamical phases are examined by evaluating the Lyapunov exponent. The relation between the dynamical phase and the computational capability of physical reservoir computing is also studied., Comment: 11 pages, 4 figures

Recent studies have shown that nonlinear magnetization dynamics excited in nanostructured ferromagnets are applicable to brain-inspired computing such as physical reservoir computing. The previous works have utilized the magnetization dynamics driven by electric current and/or magnetic field. This work proposes a method to apply the magnetization dynamics driven by voltage control of magnetic anisotropy to physical reservoir computing, which will be preferable from the viewpoint of low-power consumption. The computational capabilities of benchmark tasks in single MTJ are evaluated by numerical simulation of the magnetization dynamics and found to be comparable to those of echo-state networks with more than 10 nodes., Comment: 13 pages, 5 figures

Input-driven dynamical systems have attracted attention because their dynamics can be used as resources for brain-inspired computing. The recent achievement of human-voice recognition by spintronic oscillator also utilizes an input-driven magnetization dynamics. Here, we investigate an excitation of input-driven chaos in magnetization dynamics by voltage controlled magnetic anisotropy effect. The study focuses on the parametric magnetization oscillation induced by a microwave voltage and investigates the effect of random-pulse input on the oscillation behavior. Solving the Landau-Lifshitz-Gilbert equation, temporal dynamics of the magnetization and its statistical character are evaluated. In a weak perturbation limit, the temporal dynamics of the magnetization are mainly determined by the input signal, which is classified as input-driven synchronization. In a large perturbation limit, on the other hand, chaotic dynamics are observed, where the dynamical response is sensitive to the initial state. The existence of chaos is also identified by the evaluation of the Lyapunov exponent., Comment: 21 pages, 6 figures

A recent experimental demonstration of a parametric magnetization oscillation excited by applying a microwave voltage to a ferromagnetic metal will be applicable not only to a new magnetization switching method but also to bio-inspired computing. It should be, however, noted that a phase of the parametric magnetization oscillation is not uniquely locked, related to the fact that a frequency of the microwave voltage is twice the value of the magnetization oscillation. There are two possible phases in the parametric oscillation state, and which of the two is realized depends on the initial condition of the magnetization. Here, we examine two approaches to lock the phase uniquely. One is to suppress the distribution of the initial state by enhancing the perpendicular magnetic anisotropy before applying microwave voltage, and the other is to use a sweeping frequency. Through numerical simulation of the Landau-Lifshitz-Gilbert equation and quantification of locked rate, we find that the sweeping frequency is more effective to lock the phase of the parametric magnetization oscillation., Comment: 22 pages, 5 figures

A numerical analysis on the computational capability of physical reservoir computing utilizing a spin-torque oscillator with two free layers is reported. Conventional spintronics devices usually consist of two ferromagnets, where the direction of magnetization in one layer, called the free layer, can move while that of the other, the reference layer, is fixed. Recently, however, devices with two free layers, where the reference layer is replaced by another free layer, have been developed for various practical applications. Adding another free layer drastically changes the dynamical response of the device through the couplings via the spin-transfer effect and the dipole magnetic field. A numerical simulation of the Landau-Lifshitz-Gilbert equation and a statistical analyses of the Lyapunov exponent and the synchronization index reveal the appearance of an amplitude-modulated oscillation and chaos in the oscillators with two free layers. Such complex dynamics qualitatively change the computational capability of physical reservoir computing because the computational resource is dynamics of the physical system. An evaluation of the short-term memory capacity clarifies that oscillators with two free layers have a larger capacity than those of conventional oscillators. An enhancement in capacity near the edge of echo state property, i.e., the boundary between zero and finite synchronization index, is also found., Comment: 16 pages, 11 figures

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 notice that the field-like torque with an appropriate sign provides an additional field modifying the dynamic trajectory of the magnetization and drives the auto-oscillation. The condition on the sign of the field-like torque is satisfied for typical nonmagnet used in spin Hall devices such as tungsten., Comment: 5 pages, 4 figures

Physical reservoir computing is a type of recurrent neural network that applies the dynamical response from physical systems to information processing. However, the relation between computation performance and physical parameters/phenomena still remains unclear. This study reports our progress regarding the role of current-dependent magnetic damping in the computational performance of reservoir computing. The current-dependent relaxation dynamics of a magnetic vortex core results in an asymmetric memory function with respect to binary inputs. A fast relaxation caused by a large input leads to a fast fading of the input memory, whereas a slow relaxation by a small input enables the reservoir to keep the input memory for a relatively long time. As a result, a step-like dependence is found for the short-term memory and parity-check capacities on the pulse width of input data, where the capacities remain at 1.5 for a certain range of the pulse width, and drop to 1.0 for a long pulse-width limit. Both analytical and numerical analyses clarify that the step-like behavior can be attributed to the current-dependent relaxation time of the vortex core to a limit-cycle state. }, Comment: 9 pages, 5 figures

Recent studies on magnetization dynamics induced by spin-orbit torque have revealed a weak dependence of the critical current for magnetization switching on the damping constant of a ferromagnetic free layer. This study, however, reveals that the damping constant nevertheless plays a key role in magnetization switching induced by spin-orbit torque. An undesirable switching, returning to an initial state, named as back switching, occurs in a ferromagnet with an easy axis parallel to the current direction. Numerical and theoretical analyses reveal that back switching is strongly suppressed when the damping constant of the ferromagnet is large., Comment: 5 pages, 4 figures

Magnetization switching in a fine-structured ferromagnet of nanoscale by the spin-transfer torque excited via the spin Hall effect has attracted much attention because it enables us to manipulate the magnetization without directly applying current to the ferromagnet. However, the switching mechanism is still unclear in regard to the ferromagnet having an in-plane easy axis parallel to the current. Here, we develop an analytical theory of the magnetization switching in this type of ferromagnet, and reveal the threshold current formulas for a deterministic switching. It is clarified that the current should be in between a certain range determined by two threshold currents because the spin-transfer torque due to a large current outside the range brings the magnetization in an energetically unstable state, and causes magnetization precession around the hard axis., Comment: 13 pages, 7 figures

Spin-orbit torque (SOT) magnetoresistance (MR) devices have attracted attention for use in next-generation MR devices. The SOT devices are known to exhibit different write properties based on the relative angle between the magnetization direction of the free layer and the write-current direction. However, few studies that compare the write properties of each type have been reported. In this study, we measured the external perpendicular-magnetic field dependence of the threshold write current density and the write current switching probability using two types of in-plane magnetization SOT-MR devices., Comment: 17 pages, 5 figures

The role of the feedback effect on physical reservoir computing is studied theoretically by solving the vortex-core dynamics in a nanostructured ferromagnet. Although the spin-transfer torque due to the feedback current makes the vortex dynamics complex, it is clarified that the feedback effect does not always contribute to the enhancement of the memory function in a physical reservoir. The memory function, characterized by the correlation coefficient between the input data and the dynamical response of the vortex core, becomes large when the delay time of the feedback current is not an integral multiple of the pulse width. On the other hand, the memory function remains small when the delay time is an integral multiple of the pulse width. As a result, a periodic behavior for the short-term memory capacity is observed with respect to the delay time, the phenomenon of which can be attributed to correlations between the virtual neurons via the feedback current., Comment: 10 pages, 9 figures

A theoretical analysis is developed on spin-torque diode effect in nonlinear region. An analytical solution of the diode voltage generated from spin-torque oscillator by the rectification of an alternating current is derived. The diode voltage is revealed to depend nonlinearly on the phase difference between the oscillator and the alternating current. The validity of the analytical prediction is confirmed by numerical simulation of the Landau-Lifshitz-Gilbert equation. The results indicate that the spin-torque diode effect is useful to evaluate the phase of a spin-torque oscillator in forced synchronization state., Comment: 5 pages, 3 figures

We have studied the spin orbit torque (SOT) in Pt/Co/Ir multilayers with 3 repeats of the unit structure. As the system exhibits oscillatory interlayer exchange coupling (IEC) with varying Ir layer thickness, we compare the SOT of films when the Co layers are coupled ferromagnetically and antiferromagnetically. SOT is evaluated using current induced shift of the anomalous Hall resistance hysteresis loops. A relatively thick Pt layer, serving as a seed layer to the multilayer, is used to generate spin current via the spin Hall effect. In the absence of antiferromagnetic coupling, the SOT is constant against the applied current density and the corresponding spin torque efficiency (i.e. the effective spin Hall angle) is $\sim$0.09, in agreement with previous reports. In contrast, for films with antiferromagnetic coupling, the SOT increases with the applied current density and eventually saturates. The SOT at saturation is a factor of $\sim$15 larger than that without the antiferromagnetic coupling. The spin torque efficiency is $\sim$5 times larger if we assume the net total magnetization is reduced by a factor of 3 due to the antiferromagnetic coupling. Model calculations based on the Landau Lifshitz Gilbert equation show that the presence of antiferromagnetic coupling can increase the SOT but the degree of enhancement is limited, in this case, to a factor of 1.2-1.4. We thus consider there are other sources of SOT, possibly at the interfaces, which may account for the highly efficient SOT in the uncompensated synthetic anti-ferromagnet (SAF) multilayers.

Synchronization and chaos caused by alternating current and microwave field in a spin torque oscillator consisting of a perpendicularly magnetized free layer and an in-plane magnetized reference layer is comprehensively studied theoretically. A forced synchronization by the alternating current is observed in numerical simulation over wide ranges of its amplitude and frequency. An analytical theory clarifies that the nonlinear frequency shift, as well as the spin-transfer torque asymmetry, plays a key role in determining locking range and phase difference between the oscillator and current. Chaos caused by the alternating current is identified for a region of large alternating current by evaluating the Lyapunov exponent. Similar results are also obtained for microwave field, although the parameter regions causing chaos are narrower than those by the alternating current.

A theoretical formula is derived for the threshold current to switch a perpendicular magnetization in a ferromagnet by the spin Hall effect. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates that magnetization switching is achieved when the steady-state solution of the magnetization in the presence of the current is outside an energetically unstable region. Based on the numerical result, an analytical theory deriving the threshold current is developed by focusing on the first-order perturbation to the unstable state. The analytical formula clarifies that the magnitude of the magnetic field applied to the current direction should be larger than 15\% of the perpendicular magnetic anisotropy field, and the current is less than the derived threshold value., Comment: 8 pages, 5 figures

We have studied the magnetic properties of multilayers composed of ferromagnetic metal Co and heavy metals with strong spin orbit coupling (Pt and Ir). Multilayers with symmetric (ABA stacking) and asymmetric (ABC stacking) structures are grown to study the effect of broken structural inversion symmetry. We compare the perpendicular magnetic anisotropy (PMA) energy of symmetric Pt/Co/Pt, Ir/Co/Ir multilayers and asymmetric Pt/Co/Ir, Ir/Co/Pt multilayers. First, the interface contribution to the PMA is studied using the Co layer thickness dependence of the effective PMA energy. Comparison of the interfacial PMA between the Ir/Co/Pt, Pt/Co/Ir asymmetric structures and Pt/Co/Pt, Ir/Co/Ir symmetric structures indicate that the broken structural inversion symmetry induced PMA is small compared to the overall interfacial PMA. Second, we find the magnetic anisotropy field is significantly increased in multilayers when the ferromagnetic layers are antiferromagnetically coupled via interlayer exchange coupling (IEC). Macrospin model calculations can qualitatively account for the relation between the anisotropy field and the IEC. Among the structures studied, IEC is the largest for the asymmetric Ir/Co/Pt multilayers: the exchange coupling field exceeds 3 T and consequently, the anisotropy field approaches 10 T. Third, comparing the asymmetric Ir/Co/Pt and Pt/Co/Ir structures, we find the IEC and, to some extent, the interface PMA are stronger for the former than the latter. X-ray magnetic circular dichroism studies suggest that the proximity induced magnetization in Pt is larger for the Ir/Co/Pt multilayers than the inverted structure, which may partly account for the difference in the magnetic properties. These results show the intricate relation between PMA, IEC and the proximity induced magnetization that can be exploited to design artificial structures with unique magnetic characteristics.

Nonlinear magnetization dynamics excited by spin-transfer effect with feedback current is studied both numerically and analytically. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates the positive Lyapunov exponent for a certain range of the feedback rate, which identifies the existence of chaos in a nanostructured ferromagnet. Transient behavior from chaotic to steady oscillation is also observed in another range of the feedback parameter. An analytical theory is also developed, which indicates the appearance of multiple attractors in a phase space due to the feedback current. An instantaneous imbalance between the spin-transfer torque and damping torque causes a transition between the attractors, and results in the complex dynamics.

We quantitatively evaluate a spin anomalous Hall effect (SAHE), generating spin angular momentum flow (spin current, $J_{\rm s}$), in an L1$\rm_{0}$-FePt ferromagnet by exploiting giant magnetoresistance devices with L1$\rm_{0}$-FePt / Cu / Ni$\rm_{81}$Fe$\rm_{19}$ . From the ferromagnetic resonance linewidth modulated by the charge current ($J_{\rm c}$) injection, the spin anomalous Hall angle ($ \alpha_{\rm SAH} $) is obtained to be 0.25 $ \pm $ 0.03. The evaluation of $ \alpha_{\rm SAH} $ at different configurations between $J_{\rm c}$ and magnetization enables us to discuss the symmetry of SAHE and gives the unambiguous evidence that SAHE is the source of $J_{\rm s}$. Thanks to the large $ \alpha_{\rm SAH} $, we demonstrate the SAHE-induced magnetization switching., Comment: 16 pages, 4 figures

A phase diagram of the magnetization dynamics is studied by numerically solving the Landau-Lifshitz-Gilbert (LLG) equation in a spin torque oscillator consisting of asymmetric two free layers that are magnetized in in-plane direction. We calculated the dynamics for a wide range of current density for both low and high field cases, and found many dynamical phases such as synchronization, auto-oscillation with different frequencies, and chaotic dynamics. The observation of the synchronization indicates the presence of a dynamical phase which has not been found experimentally by using the conventional electrical detection method. The auto-oscillations with different frequencies lead to an oscillation of magnetoresistance with a high frequency, which can be measured experimentally. The chaotic and/or periodic behavior of magnetoresistance in a high current region, on the other hand, leads to a discontinuous change of the peak frequency in Fourier spectrum.

An array of spin torque oscillators (STOs) for practical applications such as pattern recognition was recently proposed, where several STOs are connected by a common nonmagnet. In this structure, in addition to the electric and/or magnetic interactions proposed in previous works, the STOs are spontaneously coupled to each other through the nonmagnetic connector, due to the injection of spin current. Solving the Landau-Lifshitz-Gilbert equation numerically for such system consisting of three STOs driven by the spin Hall effect, it is found that both in-phase and antiphase synchronization of the STOs can be achieved by adjusting the current density and appropriate distance between the oscillators., Comment: 4 pages, 2 figures

The theoretical investigation on magnetization dynamics excited by the spin Hall effect in metallic multilayers having two ferromagnets is discussed. The relaxation of the transverse spin in one ferromagnet enables us to manipulate the direction of the spin-transfer torque excited in another ferromagnet, although the spin-polarization originally generated by the spin Hall effect is geometrically fixed. Solving the Landau-Lifshitz-Gilbert-Slonczewski equation, the possibility to excite an out-of-plane auto-oscillation of an in-plane magnetized ferromagnet is presented. An application to magnetic recording using microwave-assisted magnetization reversal is also discussed., Comment: 5 pages, 3 figures

A theoretical framework is proposed for the spin-current driven synchronized self-oscillations in ferromagnets in the spin Hall geometry. The spin current generated by the spin Hall effect in a bottom nonmagnetic heavy metal excites a self-oscillation of the magnetization in an attached ferromagnet through spin-transfer effect. The spin current simultaneously creates spin accumulation inside the ferromagnet. Therefore, when the top surfaces of two ferromagnets are connected by a nonmagnetic material having a long spin diffusion length, another spin current flows according to the gradient of the spin accumulations between the ferromagnets. This additional spin current excites an additional spin torque leading to a coupled motion of the magnetizations. This coupling mechanism comes purely from spin degree of freedom in the system without using electric and/or magnetic interactions. The additional spin torque acts as a repulsive force between the magnetizations, and prefers an antiphase synchronization between the oscillators. The phase difference in a synchronized state is determined by the competition between this additional spin torque and spin pumping. Eventually, either an in-phase or antiphase synchronization is spontaneously excited in the individual ferromagnets, depending on the current magnitude. These conclusions are obtained by deriving the theoretical formula of the additional spin torque from the diffusive spin transport theory and solving the equation of motion of the magnetizations both numerically and analytically., Comment: 19 pages, 8 figures

A theoretical formalism is developed to simultaneously solve equation of motion of the magnetizations in two ferromagnets and the spin-pumping induced spin transport equation. Based on the formalism, a coupled motion of the magnetizations in a self-oscillation state is studied. The spin pumping is found to induce an in-phase synchronization of the magnetizations for the oscillation around the easy axis. For an out-of-plane self-oscillation around the hard axis, on the other hand, the spin pumping leads to an in-phase synchronization in a small current region, whereas an antiphase synchronization is excited in a large current region. An analytical theory based on the phase equation reveals that the phase difference between the magnetizations in a steady state depends on the oscillation direction, clockwise or counterclockwise, of the magnetizations., Comment: 11 pages, 7 figures, published

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., Comment: 16 pages, 8 figures, published

An analytical formulation to compute a magnetic field generated from an uniformly magnetized cylinder ferromagnet is developed. Exact solutions of the magnetic field generated from the magnetization pointing in an arbitrary direction are derived, which are applicable both inside and outside the ferromagnet. The validities of the present formulas are confirmed by comparing them with demagnetization coefficients estimated in earlier works. The results will be useful for designing practical applications, such as high-density magnetic recording and microwave generators, where nanostructured ferromagnets are coupled to each other through the dipole interactions and show cooperative phenomena such as synchronization. As an example, the magnetic field generated from a spin torque oscillator for magnetic recording based on microwave assisted magnetization reversal is studied., Comment: published

Transverse magnetoresistance in a ferromagnetic/nonmagnetic/ferromagnetic trilayer originated from charge-spin conversion by anomalous Hall effect is investigated theoretically. Solving the spin diffusion equation in bulk and using the spin-dependent Landauer formula at the ferromagnetic/nonmagnetic interface, an analytical formula of the transverse resistivity is obtained. The charge-spin conversion by the anomalous Hall effect contributes to the magnetoresistance in a manner proportional to the square of the spin anomalous Hall angle. The angular dependence of the magnetoresistance is basically identical to that of planar Hall effect, but has an additional term which depends on the relative angle of the magnetizations in two ferromagnets., Comment: 10 pages, 1 figure, conference proceeding of MMM 2017

A mutual synchronization of spin-torque oscillators coupled through current injection is studied theoretically. Models of electrical coupling in parallel and series circuits are proposed. Solving the Landau-Lifshitz-Gilbert equation, excitation of in-phase or antiphase synchronization, depending on the ways the oscillators are connected, is found. It is also found from both analytical and numerical calculations that the current-frequency relations for both parallel and series circuits are the same as that for a single spin-torque oscillator., Comment: 5 pages, 4 figures

A possibility to excite a stable self-oscillation in a perpendicularly magnetized ferromagnet by the spin Hall effect is investigated theoretically. It had been shown that such self-oscillation cannot be stabilized solely by the direct spin torque by the spin Hall effect. Here, we consider adding another ferromagnet, referred to as pinned layer, on the free layer. The pinned layer provides another spin torque through the reflection of the spin current. The study shows that the stable self-oscillation is excited by the additional spin torque when the magnetization in the pinned layer is tilted from the film plane., Comment: 6 pages, 3 figures

The relaxation phenomena of spin-torque oscillators consisting of nanostructured ferromagnets are interesting research targets in magnetism. A theoretical study on the relaxation time of a spin-torque oscillator from one self-oscillation state to another is investigated. By solving the Landau-Lifshitz-Gilbert equation both analytically and numerically, it is shown that the oscillator relaxes to the self-oscillation state exponentially within a few nanoseconds, except when magnetization is close to a critical point. The relaxation rate, which is an inverse of relaxation time, is proportional to the current. On the other hand, a critical slowing down appears near the critical point, where relaxation is inversely proportional to time, and the relaxation time becomes on the order of hundreds of nanoseconds. These conclusions are primarily obtained for a spin-torque oscillator consisting of a perpendicularly magnetized free layer and an in-plane magnetized pinned layer, and are further developed for application to arbitrary types of spin-torque oscillators., Comment: 13 pages, 7 figures

Spin torque oscillators placed onto a nonmagnetic heavy metal show synchronized auto-oscillations due to the coupling originating from spin Hall magnetoresistance effect. Here, we study a system having two spin torque oscillators under the effect of the spin Hall torque, and show that switching the external current direction enables us to control the phase difference of the synchronization between in-phase and antiphase., Comment: 7 pages, 5 figures, published as a conference proceeding of Intermag 2017