We propose theoretically an active material for electromagnetic radiation with frequency of GHz by use of spin-torque oscillators. The origin of the amplification is the energy supplied to the magnetization by the injected current. We show that close to a resonance with current-driven magnetization, the imaginary part of magnetic permeability becomes indeed negative for either of the two circular polarizations, resulting in negative imaginary part of refractive index. Besides, the real part of the refractive index is also manipulated by the current. Our system thus realizes an active filter to obtain circular polarized radiation and/or an electromagnetic metamaterial having negative refractive index, both controlled electrically.
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.
The self-oscillation of the magnetization in a spin torque oscillator (STO) with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer in the absence of an applied magnetic field was studied by numerically solving the Landau-Lifshitz-Gilbert equation. It was pointed out that field-like torque was necessary to realize stable self-oscillation in this type of STO at zero field. The numerical simulation at finite temperature showed that the presence of the field-like torque led to a high power with a relatively high oscillation frequency.
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…
Based on theoretical models, the dynamics of spin-torque nano-oscillators can be substantially modified by re-injecting the emitted signal to the input of the oscillator after some delay. Numerical simulations for vortex magnetic tunnel junctions show that with reasonable parameters this approach can decrease critical currents as much as 25 % and linewidths by a factor of 4. Analytical calculations, which agree well with simulations, demonstrate that these results can be generalized to any kind of spin-torque oscillator.
Skyrmion-based spin torque nano-oscillators are potential next-generation microwave signal generators. However, ferromagnetic skyrmion-based spin torque nano-oscillators cannot reach high oscillation frequencies. In this work, we propose to use the circular motion of an antiferromagnetic skyrmion to create the oscillation signal in order to overcome this obstacle. Micromagnetic simulations demonstrate that the antiferromagnetic skyrmion-based spin torque nano-oscillators can produce high frequencies (tens of GHz). Furthermore, the speed of the circular motion for an antiferromagnetic skyrmion …
In this paper, we propose to control the strength of phase-locking between two dipolarly coupled vortex based spin-torque nano-oscillators by placing an intermediate oscillator between them. We show through micromagnetic simulations that the strength of phase-locking can be largely tuned by a slight variation of current in the intermediate oscillator. We develop simplified numerical simulations based on analytical expressions of the vortex core trajectories that will be useful for investigating large arrays of densely packed spin-torque oscillators interacting through their stray fields.
The self-synchronization of spin torque oscillators is investigated experimentally by re-injecting its radiofrequency (rf) current after a certain delay time. We demonstrate that the emission power and the spectral linewidth are improved for optimal delay times. Moreover by varying the phase difference between the emitted power and the re-injected one, we find a clear oscillatory dependence with a 2π periodicity of the frequency of the oscillator as well as its power and linewidth. Such periodical behavior within the self-injection regime is well described by the general model of nonlinear aut…
We show that the phase of a spin-torque oscillator generically acquires a geometric contribution upon slow and cyclic variation of the parameters that govern its dynamics. As an example, we compute the geometric phase that results from a cyclic excursion of the magnitude of the external field and the current. We consider thermal phase fluctuations and conclude that the geometric phase should be experimentally observable at room temperature with current setups. We briefly comment on arrays of spin-torque oscillators and possible applications.
We present a combined analytical and numerical study on double-vortex spin-torque nano-oscillators and describe a mechanism that suppresses the windmill modes. The magnetization dynamics is dominated by the gyrotropic precession of the vortex in one of the ferromagnetic layers. In the other layer the vortex gyration is strongly damped. The dominating layer for the magnetization dynamics is determined by the current polarity. Measurements on Fe/Ag/Fe nano-pillars support these findings. The results open up a new perspective for building high quality-factor spin-torque oscillators operating at s…
The oscillation properties of a spin torque oscillator consisting of a perpendicularly magnetized free layer and an in-plane magnetized pinned layer are studied based on an analysis of the energy balance between spin torque and damping. The critical value of an external magnetic field applied normal to the film plane is found, below which the controllable range of the oscillation frequency by the current is suppressed. The value of the critical field depends on the magnetic anisotropy, the saturation magnetization, and the spin torque parameter.
A number of recent experimental works have shown that the dynamics of a single spin torque oscillator can exhibit complex behavior that stems from interactions between two or more modes of the oscillator. Examples are observed mode-hopping or mode coexistence. There has been some intial work indicating how the theory for a single-mode (macro-spin) spin torque oscillator should be generalized to include several modes and the interactions between them. In the present work, we derive such a theory starting with the Landau-Lifshitz-Gilbert equation for magnetization dynamics. We compare our result…
Spin-torque nano-oscillators (STNO) are microwave auto-oscillators based on magnetic resonances having a nonlinear response with the oscillating amplitude, which provides them with a large frequency tunability including the possibility of mutual synchronization. The magnetization dynamics in STNO are induced by spin transfer torque (STT) from spin currents and can be detected by changes in electrical resistance due to giant magnetoresistance or tunneling magnetoresistance. The STT effect is usually treated as a damping-like term that reduces magnetic dissipation and promotes excitation of magn…
The magnetization dynamics of spin torque oscillator (STO) consisting of a perpendicularly magnetized free layer and an in-plane magnetized pinned layer was studied by solving the Landau-Lifshitz-Gilbert equation. We derived the analytical formula of the relation between the current and the oscillation frequency of the STO by analyzing the energy balance between the work done by the spin torque and the energy dissipation due to the damping. We also found that the field-like torque breaks the energy balance, and change the oscillation frequency.
Phase coupling between auto-oscillators is central for achieving coherent responses such as synchronization. Here we present an experimental approach to probe it in the case of two dipolarly coupled spin-torque vortex nano-oscillators using an external microwave field. By phase-locking one oscillator to the microwave field, we observe frequency pulling on the second oscillator. From coupled Thiele equations we show analytically that this frequency pulling results from concerted actions of dipolar and microwave couplings. The analysis allows us to determine the amplitude of dipolar coupling and…
Manipulating operation states of coupled spin-torque nano-oscillators (STNOs), including their synchronization, is essential for applications such as complex oscillator networks. In this work we experimentally demonstrate selective control of two coupled vortex STNOs through microwave-assisted switching of their vortex core polarities. First, the two oscillators are shown to synchronize due to dipolar interaction in a broad frequency range tuned by external biasing field. Coherent output is demonstrated along with strong linewidth reduction. Then, we show individual vortex polarity control of …
We investigate how the ability of the vortex oscillation mode of a spin-torque nano-oscillator to lock to an external microwave signal is modified when it is coupled to another oscillator. We show experimentally that mutual electrical coupling can lead to locking range enhancements of a factor 1.64. Furthermore, we analyze the evolution of the locking range as a function of the coupling strength through experiments and numerical simulations. By uncovering the mechanisms at stake in the locking range enhancement, our results will be useful for designing spin-torque nano-oscillators arrays with …
The mutual interaction between the different eigenmodes of a spin-torque oscillator can lead to a large variety of physical mechanisms from mode hopping to multi-mode generation, that usually reduce their performances as radio-frequency devices. To tackle this issue for the future applications, we investigate the properties of a model spin-torque oscillator that is composed of two coupled vortices with one vortex in each of the two magnetic layers of the oscillator. In such double-vortex system, the remarkable properties of energy transfer between the coupled modes, one being excited by spin t…
The correlation of phase fluctuations in any type of oscillator fundamentally defines its spectral shape. However, in nonlinear oscillators, such as spin torque nano oscillators, the frequency spectrum can become particularly complex. This is specifically true when not only considering thermal but also colored $1/f$ flicker noise processes, which are crucial in the context of the oscillator's long term stability. In this study, we address the frequency spectrum of spin torque oscillators in the regime of large-amplitude steady oscillations experimentally and as well theoretically. We particula…
This paper reviews the state of the art in spin-torque and spin Hall effect driven nano-oscillators. After a brief introduction to the underlying physics, the authors discuss different implementations of these oscillators, their functional properties in terms of frequency range, output power, phase noise, and modulation rates, and their inherent propensity for mutual synchronization. Finally, the potential for these oscillators in a wide range of applications, from microwave signal sources and detectors to neuromorphic computation elements, is discussed together with the specific electronic ci…
We show how a charge current through a single antiferromagnetic layer can excite and control self-oscillations. Sustained oscillations with tunable amplitudes and frequencies are possible in a variety of geometries using certain classes of non-centrosymmetric materials that exhibit finite dissipative spin-orbit torque. We compute the steady-state phase diagram as a function of the current and spin-orbit torque magnitude. The anisotropic magnetoresistance causes the conversion of the resulting AF oscillations to a terahertz AC output voltage. These findings provide an attractive and novel route…
We report an approach to improving the performance of spin torque nano-oscillators (STNOs) that utilizes power-dependent negative feedback to achieve a significantly enhanced dynamic damping. In combination with a sufficiently slow variation of frequency with power this can result in a quasi-linear STNO, with very weak non-linear coupling of power and phase fluctuations over a range of bias current and field. An implementation of this approach that utilizes a non-uniform spin-torque demonstrates that highly coherent room temperature STNOs can be achieved while retaining a significant tunabilit…
We have proposed a method to synchronize multiple spin-transfer torque oscillators based on spin pumping, inverse spin Hall, and spin Hall effects. The proposed oscillator system consists of a series of nano-magnets in junction with a normal metal with high spin-orbit coupling, and an accumulative feedback loop. We conduct simulations to demonstrate the effect of modulated charge currents in the normal metal due to spin pumping from each nano-magnet. We show that the interplay between the spin Hall effect and inverse spin Hall effect results in synchronization of the nano-magnets.
The concept of spin torque driven high frequency magnetization dynamics has opened up the field of spintronics to non-linear physics, potentially in complex networks of dynamical systems. In the scarce demonstrations of synchronized spin-torque oscillators, the local nature of the magnetic coupling that is used have largely hampered a good understanding and thus the control of the synchronization process. Here we take advantage of the non-local nature of an electrical coupling to mutually synchronize spin-torque oscillators through their self-emitted microwave currents. The control of the sync…
Stochastic dynamics of spin torque oscillators (STOs) can be described in terms of magnetization drift and diffusion over a current-dependent effective energy surface given by the Fokker-Planck equation. Here we present a method that directly probes this effective energy surface via time-resolved measurements of the microwave voltage generated by a STO. We show that the effective energy approach provides a simple recipe for predicting spectral line widths and line shapes near the generation threshold. Our time domain technique also accurately measures the field-like component of spin torque in…
The dynamics of vortex based spin-torque nano-oscillators is investigated theoretically. Starting from a fully analytical model based on the Thiele equation approach, fine-tuned data-driven corrections are carried out to the gyrotropic and damping terms. These adjustments, based on micromagnetic simulation results, allow to quantitatively model the response of such oscillators to any dc current within the range of the vortex stability. Both, the transient and the steady-state regimes are accurately predicted under the proposed data-driven Thiele equation approach. Furthermore, the computation …
Two-mode model of spin-torque nano-oscillator (STNO) under the action of thermal noise is considered. Langevin equations for mode amplitudes were derived starting from general nonlinear oscillator model. Stationary probability distribution function describing mean mode generation powers was obtained using Fokker-Planck equation. It was shown that thermal noise can lead to two-mode generation in STNO. An increase of thermal noise power leads to excitation of the second mode in a system and to a two-mode generation regime through intermediate state when two modes coexist only in some range of th…
A model for a spin-torque nano-oscillator based on the self-sustained oscillation of a magnetic skyrmion is presented. The system involves a circular nanopillar geometry comprising an ultrathin film free magnetic layer with a strong Dzyaloshinkii-Moriya interaction and a polariser layer with a vortex-like spin configuration. It is shown that spin-transfer torques due to current flow perpendicular to the film plane leads to skyrmion gyration that arises from a competition between geometric confinement due to boundary edges and the vortex-like polarisation of the spin torques. A phenomenology fo…
Experiments on spin torque oscillators commonly observe multi-mode signals. Recent theoretical works have ascribed the multi-mode signal generation to coupling between energy-separated spin wave modes. Here, we analyze in detail the dynamics generated by such mode coupling. We show analytically that the mode-hopping dynamics broaden the generation linewidth and makes it generally well described by a Voigt lineshape. Furthermore, we show that the mode-hopping contribution to the linewidth can dominate in which case it provides a direct measure of the mode-hopping rate. Due to the thermal drive …
We measure oscillator phase from the zero crossings of the voltage vs. time waveform of a spin torque nanocontact oscillating in a vortex mode. The power spectrum of the phase noise varies with Fourier frequency $f$ as $1/f^2$, consistent with frequency fluctuations driven by a thermal source. The linewidth implied by phase noise alone is about 70 % of that measured using a spectrum analyzer. A phase-locked loop reduces the phase noise for frequencies within its 3 MHz bandwidth.
A theoretical formula of the linewidth caused by the thermal activation in a spin torque oscillator with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer was developed by solving the stochastic Landau-Lifshitz-Gilbert equation in the energy-phase representation. It is shown that the linewidth can be suppressed down to 0.1 MHz by applying a large current (10 mA for typical material parameters). A quality factor larger than 10^{4} is predicted in the large current limit, which is two orders of magnitude larger than the recently observed experimental value.
Phononic metamaterials have attracted extensive attention since they are exibly adjustable to control the phonon transmission. In this work, we study a one-dimensional phononic metamaterial, made of mechanical resonant oscillators and chiral couplings. We show that by design, the oscillator mass and inter-oscillator coupling, although both are positive naturally, can be either single negative or double negative effectively within a certain frequency range. At the frequency where the effective mass and coupling are both infinite, a flat band emerges that will induce an extremely high density of…
We are reporting a new type of synchronization, termed dancing synchronization, between two spin-torque nano-oscillators (STNOs) coupled through spin waves. Different from the known synchronizations in which two STNOs are locked with various fixed relative phases, in this new synchronized state two STNOs have the same frequency, but their relative phase varies periodically within the common period, resulting in a dynamic waving pattern. The amplitude of the oscillating relative phase depends on the coupling strength of two STNOs, as well as the driven currents. The dancing synchronization turn…
This Habilitation Thesis written in 2013 reviews my research work on spin torque nano-oscillators (from zero-field oscillations, to synchronization and vortex oscillators) and memristive devices (spin Torque and ferroelectric memristors).
A theoretical study on how synchronization and resonance-like phenomena in superparamagnetic tunnel junctions can be driven by spin-transfer torques is presented. We examine the magnetization of a superparamagnetic free layer that reverses randomly between two well-defined orientations due to thermal fluctuations, acting as a stochastic oscillator. When subject to an external ac forcing this system can present stochastic resonance and noise-enhanced synchronization. We focus on the roles of the mutually perpendicular damping-like and field-like torques, showing that the response of the system …
We investigate analytically and numerically the synchronization dynamics of dipolarly coupled vortex based Spin-Torque Nano Oscillators (STNO) with different pillar diameters. We identify the critical interpillar distances on which synchronization occurs as a function of their diameter mismatch. We obtain numerically a phase diagram showing the transition between unsynchronized and synchronized states and compare it to analytical predictions we make using Thiele approach. Our study demonstrates that for relatively small diameters differences the synchronization dynamics can be described qualit…
Spin torque nano-oscillator has been extensively studied both theoretically and experimentally in recent decades due to its potential applications in future microwave communication technology and neuromorphic computing. In this work, we present a skyrmion-based spin torque nano-oscillator driven by a spatially uniform direct current, where the skyrmion is confined by two pinning sites. Different from other skyrmion-based oscillators that arise from the circular motion or the breathing mode of a skyrmion, the steady-state oscillatory motions are produced by the periodic deformation of an elonga…
Theoretical conditions to excite self-oscillation in a spin torque oscillator consisting of a perpendicularly magnetized free layer and an in-plane magnetized pinned layer are investigated by analytically solving the Landau-Lifshitz-Gilbert equation. The analytical relation between the current and oscillation frequency is derived. It is found that a large amplitude oscillation can be excited by applying a small field pointing to the direction anti-parallel to the magnetization of the pinned layer. The validity of the analytical results is confirmed by comparing with numerical simulation, showi…
Magnetic systems have been extensively studied both from a fundamental physics perspective and as building blocks for a variety of applications. Their topological properties, in particular those of excitations, remain relatively unexplored due to their inherently dissipative nature. The recent introduction of non-Hermitian topological classifications opens up new opportunities for engineering topological phases in dissipative systems. Here, we propose a magnonic realization of a non-Hermitian topological system. A crucial ingredient of our proposal is the injection of spin current into the mag…
We report on a theoretical study about the magneto-dipolar coupling and synchronization between two vortex-based spin-torque nano-oscillators. In this work we study the dependence of the coupling efficiency on the relative magnetization parameters of the vortices in the system. For that purpose, we combine micromagnetic simulations, Thiele equation approach, and analytical macro-dipole approximation model to identify the optimized configuration for achieving phase-locking between neighboring oscillators. Notably, we compare vortices configurations with parallel (P) polarities and with opposite…
Spin torque and spin Hall effect nanooscillators generate high intensity spin wave auto oscillations on the nanoscale enabling novel microwave applications in spintronics, magnonics, and neuromorphic computing. For their operation, these devices require externally generated spin currents either from an additional ferromagnetic layer or a material with a high spin Hall angle. Here we demonstrate highly coherent field and current tunable microwave signals from nanoconstrictions in single 15 and 20 nm thick permalloy layers. Using a combination of spin torque ferromagnetic resonance measurements,…
We study the magnetization dynamics of spin valve structures with a free composite synthetic ferromagnet (SyF) that consists of two ferromagnetic layers coupled through a normal metal spacer. A ferromagnetically coupled SyF can be excited into dynamical precessional states by an applied current without external magnetic fields. We analytically determine the stability of these states in the space spanned by the current density and SyF interlayer exchange coupling. Numerical simulations confirm our analytical results.
The influence of dynamic interlayer interactions on the spin torque driven and damped excitations are illustrated for a three layer macrospin model system that corresponds to a standard spin-torque oscillator. The free layer and a synthetic antiferromagnetic (SyF) pinned layer of the spin-torque oscillator are in-plane magnetized. In order to understand experimental results, numerical simulations have been performed considering three types of interlayer interactions: exchange interaction between the two magnetic layers of the SyF, mutual spin torque between the top layer of the SyF and the fre…
We consider theoretically the impact of Rashba spin-orbit coupling on spin torque oscillators (STOs) in synthetic ferromagnets and antiferromagnets that have either a bulk multilayer or a thin film structure. The synthetic magnets consist of a fixed polarizing layer and two free magnetic layers that interact through the Ruderman-Kittel-Kasuya-Yosida interaction. We determine analytically which collinear states along the easy axis that are stable, and establish numerically the phase diagram for when the system is in the STO mode and when collinear configurations are stable, respectively. It is …
Spin torque nano oscillators (STNO) are nano-scale devices that can convert a direct current into short wave-length spin-wave excitations in a ferromagnetic layer. We show that arrays of STNO can be used to create directional spin-wave radiation similar to electromagnetic antennas. Combining STNO excitations with planar spin waves also creates interference patterns. We show that these interference patterns are static and have information on the wavelength and phase of the spin waves emitted from the STNO.We describe means of actively controlling spin-wave radiation patterns with the direct cur…
Spin transfer torque nano-oscillators are potential candidates for replacing the traditional inductor based voltage controlled oscillators in modern communication devices. Typical oscillator designs are based on trilayer magnetic tunnel junctions which are disadvantaged by low power outputs and poor conversion efficiencies. In this letter, we theoretically propose to use resonant spin filtering in pentalayer magnetic tunnel junctions as a possible route to alleviate these issues and present device designs geared toward a high microwave output power and an efficient conversion of the d.c. input…
We report on a time-domain study of parametric synchronization in a magnetic tunnel junction based spin torque nano-oscillator (STNO). Time-domain measurements of the instantaneous frequency ($f_{i}$) of a parametrically synchronized STNO show random short-term unlocking of the STNO signal for low injected radio-frequency (RF) power, which cannot be revealed in time-averaged frequency domain measurements. Macrospin simulations reproduce the experimental results and reveal that the random unlocking during synchronization is driven by thermal fluctuations. We show that by using a high injected R…
We study the theoretical conditions to excite a stable self-oscillation in a spin torque oscillator with an in-plane magnetized free layer and a perpendicularly magnetized pinned layer in the presence of magnetic field pointing in an arbitrary direction. The linearized Landau-Lifshitz-Gilbert (LLG) equation is found to be inapplicable to evaluate the threshold between the stable and self-oscillation states because the critical current density estimated from the linearized equation is considerably larger than that found in the numerical simulation. We derive a theoretical formula of the thresho…
We investigate the spectral characteristics of spin torque oscillator (STO) excited by the spin Hall-induced spin current. We observe that the modest spin current injection triggers the conventional single peak oscillating behavior of STO. As the spin current is further increased to enter the non-linear regime, we find the transition of the spectrum from a single peak to multipeak structure whose frequency spacing is constant. This behavior can be primarily explained by the extremely broadened peak of the STO, which is accompanied by the frequency-dependent filtering by the transmission line. …
The dynamics of a spin-torque-oscillator (STO) coupled with a nano-magnet through dipole-dipole interaction was studied numerically by using the macrospin model for the application of the STO as a read head sensor of hard disk drives. We found that the current, which is required to induce the oscillation of the free-layer (FL) of the STO, depends strongly on the distance between the FL and the nano-magnet as well as on the relative orientation of the magnetizations between them. To determine the dynamics of the STO it is indispensable to consider the dynamics of the dipole-coupled nano-magnet.…
Using micromagnetic simulation, we investigate the self-sustained oscillation of magnetic skyrmion in a ferromagnetic circular nanodot, driven by spin-torque which is generated from a reference layer of a circular nanopillar device. We demonstrate, by lowering the value of uniaxial anisotropy constant ($K_u$), the velocity of the skyrmion can be increased and using this property, gyration frequency of the skyrmion oscillator can be enhanced. Annihilation of the skyrmion at higher current densities, limit the gyration frequency of the oscillator, whereas by modifying the $K_u$ value at the edge…
Magnetization dynamics in nanomagnets has attracted broad interest since it was predicted that a dc-current flowing through a thin magnetic layer can create spin-wave excitations. These excitations are due to spin-momentum transfer, a transfer of spin angular momentum between conduction electrons and the background magnetization, that enables new types of information processing. Here we show how arrays of spin-torque nano-oscillators (STNO) can create propagating spin-wave interference patterns of use for memory and computation. Memristic transponders distributed on the thin film respond to th…
We consider a biaxial macrospin with an easy and hard axis, and study its dynamical evolution under the combined effects of thermal noise and spin transfer torque. The spin-torque is associated with both a perpendicularly magnetized polarizer and an in-plane magnetized reference layer, leading to an effective tilt between the easy and spin polarization axes. Using techniques based on energy averaging over the relevant dynamical trajectories, we analyze the effects of tilt on the dynamics and derive the conditions for the occurrence of stable out-of-plane precessionary states. The presence of t…
We consider a synthetic antiferromagnetic spin-torque oscillator with anisotropic interlayer exchange coupling. This system exhibits exceptional points in its linearized dynamics. We find the non-linear dynamics and the dynamical phase diagram of the system both analytically and numerically. Moreover, we show that, near one of the exceptional points, the power of the oscillator depends extremely sensitively on the injected spin current. Our findings may be useful for designing sensitive magnetometers and for other applications of spin-torque oscillators.
We demonstrate how magnetic droplet soliton pairs, nucleated by two separated nano-contact (NC) spin torque oscillators, can merge into a single droplet soliton. A detailed description of the magnetization dynamics of this merger process is obtained by micromagnetic simulations: A droplet pair with a steady-state in-phase spin precession is generated through the spin-transfer torque effect underneath two separate NCs, followed by a gradual expansion of the droplets volume and the out phase of magnetization on the inner side of the two droplets, resulting in the droplets merging into a larger d…
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 s…
