Spin-current driven spontaneous coupling of ferromagnets
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.
pacs:85.75.-d, 75.78.-n, 05.45.Xt, 72.25.-b
Current driven magnetization dynamics in ferromagnets has been attracting much attention in the field of spintronics [Slonczewski, 1996; Berger, 1996; Kiselev et al., 2003; Rippard et al., 2004; Houssameddine et al., 2007]. In particular, an excitation of the coupled or forced dynamics of the magnetizations, such as synchronization of self-oscillations in spin torque oscillators (STOs), is currently an exciting topic. This is because it has a possibility in enhancing emission power of practical devices, such as microwave generator and magnetic sensors, and applicability to new devices such as phased array and brain-inspired computing [Locatelli et al., 2014; Grollier et al., 2016; Kudo and Morie, 2017; Torrejon et al., 2017]. The coupled dynamics is excited as a result of electric and/or magnetic interaction between the ferromagnets. Several mechanisms of coupling have been proposed theoretically and/or demonstrated experimentally, such as spin wave propagation [Kaka et al., 2005; Mancoff et al., 2005; Houshang et al., 2016; Urazhdin et al., 2016; Awad et al., 2017], electric current injection and/or feedback [Rippard et al., 2005; Kudo et al., 2006; Zhou and Akerman, 2009; Khalsa et al., 2015; Tsunegi et al., 2016; Bhuktare et al., 2017], microwave field [Urazhdin et al., 2010], stochastic noise in current [Nakada et al., 2012], and dipole interaction [Locatelli et al., 2015; Chen et al., 2016]. Each coupling mechanism has interesting characteristics. As an example, the dipole interaction can excite spontaneous synchronization without adding interconnections between STOs, while the number of STOs to be synchronized is restricted due to the spatial decay of the interaction. On the other hand, the electric current injection can lead to a long-range coupling owing to the conservation law of the current, but makes the circuit complicated. Moreover, every mechanism has the possibility leading to different types of synchronizations, depending on the experimental setup. For example, the electric coupling results in either in-phase or antiphase synchronization, depending on the method connecting the oscillators [Taniguchi et al., 2018]. Therefore, the investigation of a new coupling mechanism between STOs and clarification of its role on the synchronization provide rich physical insights for both magnetism and nonlinear science.
An excitation of spin-current driven coupled motion of magnetizations is a relatively new and interesting research target. The coupled magnetization dynamics through spin current has been investigated mainly in ferromagnetic resonances (FMR) [Tserkovnyak et al., 2003a, b; Heinrich et al., 2003; Takahashi, 2014; Skarsvag et al., 2014; Chiba et al., 2015]. However, a harmonic oscillation excited by an oscillating force, as in the case of FMR, should be distinguished by a self-oscillation excited by a direct force [Pikovsky et al., 2003]. A synchronization of the self-oscillation between STOs through spin current has not been fully investigated yet. The spin current decays within a characteristic length scale called spin diffusion length. Thus, the STOs should be connected with each other within a distance shorter than the spin diffusion length. It has been experimentally confirmed that several nonmagnetic metals, such as Cu and Al, have the spin diffusion length longer than one hundred nanometers [Bass and W. P. Pratt, 2007]. Connecting STOs by such materials within a distance shorter than the spin diffusion length is expected. Therefore, it is of great interest to develop a model of coupling between ferromagnets through spin current and investigate synchronized magnetization dynamics.
In this work, we propose a theoretical model embodying spin-current driven synchronization between STOs in spin Hall geometry. The coupling mechanism in this work comes purely from spin degree of freedom in the system without using electric and/or magnetic interactions. We consider two STOs placed onto different nonmagnetic heavy metals. The spin currents generated by the spin Hall effects in the bottom electrodes excite the self-oscillations in the STOs through the spin-transfer effect. At this stage, two STOs oscillate independently. Here, the spin currents simultaneously create spin accumulation in the ferromagnets. Next, the top surfaces of the STOs are connected by another nonmagnetic metal having a long spin diffusion length. Then, additional spin currents flow through the top connector according to the gradient of the spin accumulation and due to the spin pumping mechanism. These spin currents excite additional spin torques resulting in a coupled motion of magnetizations. As a result, a phase synchronization is then spontaneously excited between the STOs. The phase difference between the STOs in the synchronized state is determined according to the competition between two coupling mechanisms; one originates from the spin current generated by the gradient of the spin accumulation whereas the other comes from the spin pumping effect. We find these results by developing the theoretical model of the additional spin torque from the diffusive spin transport theory and solving the equation of motion of the magnetizations both numerically and analytically.
This paper is organized as follows. In Sec. II, we present a description of the system in this study. We first derive the distribution of the spin accumulation in a single STO in the presence of the spin Hall effect. Secondly, we consider connecting the top surfaces of two independent STOs, and investigate the coupling due to spin current flowing caused by the gradient of the spin accumulation and spin pumping in the connector. In Sec. III, we study the coupled motion of the magnetizations by solving the Landau-Lifshitz-Gilbert (LLG) equation. In Sec. IV, the role of the spin current generated by the gradient of the spin accumulation on the coupled motion of the magnetization is studied both numerically and analytically. Section V shows the conclusions of this work.
Ii Spin torque formula from spin transport theory
In this section, we derive the spin torque formula in the spin Hall geometry from the diffusive spin transport theory.
ii.1 System description
Let us first consider the injection of spin current into a single ferromagnet by the spin Hall effect. The system we consider is schematically shown in Fig. 1(a). The electric current flowing in the nonmagnet N along the direction is converted to a spin current flowing into the direction by the spin Hall effect, where is the conductivity of the nonmagnet N and is the electric field. The spin current creates the spin accumulation in the metallic ferromagnet F. Here, we define the spin accumulation in the ferromagnet as , where () is the electrochemical potential of spin- electrons, and is the unit vector pointing in the magnetization direction of the ferromagnet. The spin accumulation in the ferromagnet obeys the diffusion equation [Valet and Fert, 1993; Takahashi and Maekawa, 2008]
where is the spin diffusion length in the ferromagnet.
The solution of the spin accumulation is determined by identifying the spin currents at the boundaries. Let us denote that spin current density at the F/N interface () flowing in the positive direction, i.e., from the N to F layer, as . In both the ferromagnet and the nonmagnet, the spin currents are zero at the outer boundaries, and , where and are the thicknesses of the bottom nonmagnet and ferromagnet, respectively. Note that the spin currents in the ferromagnet and nonmagnet are given by
where is the conductivity of the ferromagnet, and is its spin polarization. The spin Hall angle in the bottom nonmagnet is . The suffixes and represent the spatial direction in the real space, whereas represents the direction of the spin polarization. Note that the suffix is not added to the spin current in the ferromagnet because the spin polarization in the ferromagnet is assumed to be parallel to the magnetization. The Levi-Civita asymmetric tensor is with . The electrochemical potential is denoted as . Then, the solutions of the spin accumulation in the ferromagnet and nonmagnet are given by
The spin current at the F/N boundary is given by the circuit theory [Brataas et al., 2001] as
where is the cross section area of the F/N boundary. The interface conductance with its spin polarization is related to the interface resistance via . The real and imaginary parts of the mixing conductance are denoted as and , respectively. Substituting Eqs. (4) and (5) into Eq. (6), we find that Eq. (6) can be rewritten as [Y.-T.Chen et al., 2013; Taniguchi, 2016]
where we introduce the following notations,
For typical ferromagnetic/nonmagnetic metallic interface, [Zwierzycki et al., 2005]. Therefore, in the following, we neglect the terms related to .
The absorption of the transverse spin current at the F/N interface leads to an excitation of spin torque acting on the magnetization in the ferromagnet. We denote the spin current at the F/N interface flowing from the ferromagnet to the nonmagnet as . By using Eq. (7), the conventional spin Hall torque generated from the spin current injected from the bottom nonmagnet is given by
We note that the spin torque in Eq. (12) is expressed in terms of the current density, . Throughout this paper, we use the current density , and not a current which is a product of and the cross-section area of the nonmagnet in the plane. This is, from the viewpoint of simplicity, it is preferable to avoid to including geometrical factor [Fert and Jaffrés, 2001] related to the different size between the ferromagnet and nonmagnet, although the experimental results are often expressed in terms of current.
In the next section, we consider connecting the ferromagnets by adding another electrode on their top surfaces. In this case, it is necessary to evaluate the spin accumulation at the top surface of the ferromagnet to understand the role of the connection on the magnetization dynamics. Substituting Eq. (7) into Eq. (4), the solution of the spin accumulation in the ferromagnet is given by
where is defined as
As schematically shown in Fig. 1(a) and described by Eq. (13), the spin accumulation in the ferromagnet is maximized at the F/N interface, and decreases exponentially from the interface. We note that the spin accumulation at the outer boundary, , is finite. It should also be emphasized that the magnitude and polarized-direction of the spin accumulation depend on the magnetization direction through the term in Eq. (13).
ii.2 Spin current in nonmagnetic connector
Now let us consider two STOs and assume placing another nonmagnet N onto the top surface of the ferromagnets, as shown in Fig. 1(b). In the following, we add the suffix to the quantities related to the F layer to distinguish the ferromagnets. We assume for simplicity that the cross section area of the ferromagnet in the plane and that of the nonmagnetic connector in the plane are the same.
When two STOs are connected by the nonmagnet N, spin currents are driven in the nonmagnet N according to the drop of the spin accumulation at the F/N interface and the gradient of the spin accumulation inside the connector. The spin current at the interface is described by
where is the spin accumulation in the connector N. We note that Eq. (15) is basically identical to Eq. (6) except for the fact that Eq. (15) is defined at the F/N interface whereas Eq. (6) is defined at the F/N interface. The sign difference on the right hand sides of Eqs. (6) and (15) is due to the fact that the spin current defined by Eq. (6) flows from the nonmagnet N to the ferromagnet F, whereas that defined by Eq. (15) flows from the ferromagnet F to the nonmagnetic connector N. In Eq. (15), we add the prime symbols to the quantities related to the F/N interface to distinguish them from those defined at the F/N interface in Eq. (6). We assume that the quantities related to the interface resistance at F/N are identical to those at F/N interface.
is a source term of this additional spin torque. There are several approaches to derive the spin torque formula from Eq. (16).
One is to solve the diffusion equation of . The spin accumulation in the nonmagnetic connector obeys the diffusion equation with the spin diffusion length , and is given by
where is the conductivity of the nonmagnetic connector. We assume that F and F layers locate at and , respectively. Substituting Eq. (18) into Eq. (16), the spin current at the F/N interface, , will be expressed as a function of the source term. This approach has been used in, for example, Refs. [Chiba et al., 2015; Y.-T.Chen et al., 2013] for a situation, where a ferromagnet is an insulator, and therefore, . The coupling spin torque derived by this approach will explicitly depend on the length of the nonmagnetic connector , similar to Eq. (37).
Another approach is to assume a ballistic transport in the nonmagnetic connector N. In this case, the spin current is conserved in the nonmagnetic connector, i.e., . Then again, the spin current at the F/N interface will be expressed as a function of the source term. The spin torque formula derived by this approach, however, does not include the length of the nonmagnetic connector due to the assumption of the conservation law of the spin current. The derivations of the spin torque formulas based on this approach were developed in, for example, Refs. [Tserkovnyak et al., 2003a; Taniguchi et al., 2015a], however, in different systems compared with ours.
There are even many other methods deriving the spin torque formula in the presence of the interface scattering [Slonczewski, 1996; Kovalev et al., 2002; Stiles and Zangwill, 2002; Barnás et al., 2005; Theodonis et al., 2006]. A spin torque formula including interface effect is often complex, as can be seen in the references mentioned above. In this work, we use the first approach mentioned above to calculate the coupling spin torque because it provides more accurate evaluation of the coupling spin torque. Before further discussing the spin torque formula, however, let us consider an approach to include the spin pumping in the present formalism.
ii.3 Spin pumping
The discussion in Sec. II.2 is valid when . On the other hand, when the magnetization dynamics is excited by the spin torque given by Eq. (12), the spin pumping becomes another and unavoidable source of the spin current flowing in the nonmagnetic connector. In this section, including the effect of the spin pumping in the above formulas is discussed.
The spin pumping is a phenomenon where a pure spin current is emitted from a ferromagnet to an adjacent metal as a result of the magnetization dynamics. The pumped spin current density at the F/N is given by [Tserkovnyak et al., 2002a]
In the presence of the spin pumping, the total spin current at the F/N interface becomes , i.e.,
Accordingly, Eq. (18) is modified as
i.e., in Eq. (18) is replaced by the total spin current density including the spin pumping effect. Substituting Eq. (21) into Eq. (20), the spin current at the F/N interface, as well as the spin torque excited at the interface, can be calculated; see Sec. II.4.
Before ending this section, we note that the spin pumping from the ferromagnet emits spin current not only to the nonmagnetic connector N but also to the bottom nonmagnet N. The spin torque due to the spin pumping into the nonmagnetic connector also excites a spin torque given by [Tserkovnyak et al., 2002b]
ii.4 Definition of coupling spin torque
where on the right hand side is the source term in Eq. (20), i.e.,
The ( or ,,) components of matrices, and , in Eq. (24) are given by
where . By solving Eq. (24), the total spin current density at the F/N is obtained. The spin torque excited at the F/N interface is then calculated as
As implied from Eq. (24), the spin torque given by Eq. (28) generally depends on the magnetizations of two STOs, and . In other words, the spin torque given by Eq. (28) acting on depends on the magnetization in the other STO, ( and ). As a result, a coupled motion of the magnetizations is excited in the STOs. In Sec. III, we investigate such coupled dynamics by solving the LLG equation with Eq. (28).
At the end of this section, let us give some comments on Eq. (28). According to the existence of two source terms of the spin current in the nonmagnetic connector shown in Eq. (25), the torque given by Eq. (28) can be decomposed into two contributions as
Here, the first term, , originates from the source term given by Eq. (17), and depends on the magnetizations directions and . On the other hand, the second term, , originates from the spin pumping given by Eq. (19). We emphasize here that the torque is newly found in this work. On the other hand, the role of the spin pumping on coupled STOs was studied in Ref. [Taniguchi, 2018] for a current-perpendicular-to-plane (CPP) structure. We should, however, note that the formalism developed in Ref. [Taniguchi, 2018] is slightly different from the present work because Ref. [Taniguchi, 2018] uses the second approach mentioned in Sec. II.2 to calculate the spin torque, i.e., the conservation of the spin current in the nonmagnetic connector is assumed.
It may be useful for readers to show the explicit form of the coupling spin torque to apprehend its physical insight. The exact solution of is, however, complex; see Appendix A. Therefore, in the numerical simulation of the LLG equation discussed in Sec. III, the coupling spin torque is calculated numerically. In Sec. IV, on the other hand, we derive an approximated analytical expression of to clarify the role of the torque on the coupled dynamics of the magnetization.
ii.5 Relation to previous works
Before proceeding to further calculations, let us discuss the relation between the present and previous works. In 2017, Kudo and Morie proposed an array of STOs for practical application of pattern recognition based on spintronics technology [Kudo and Morie, 2017]. Each STO has a bottom electrode consisting of a nonmagnetic heavy metal driving the spin Hall effect. The system also has a common top electrode. Therefore, the structure is similar to the present model. They considered, however, supplying electric power from the top electrode, which injects electric currents into the STOs. There are two kinds of spin torques excited in their geometry; one arises from the spin Hall effect in the bottom nonmagnet, and the other originates from the electric current directly injected from the top electrode. The crucial point is that the total resistance of the system between the top and bottom electrodes depend on the magnetization directions of STOs due to the tunnel magnetoresistance effect. Therefore, the magnitude of the electric current injected from the top electrode to one STO depends on the magnetization directions of the other STOs. As a result, the magnetization dynamics in STOs are coupled [Kudo and Morie, 2017]. On the other hand, the present model demonstrates that a coupling is spontaneously induced without applying the electric current from the top electrode because spin currents naturally flow according to the gradient of the spin accumulations in the top electrode and due to the spin pumping. In this respect, this work can be partially regarded as a theoretical study elaborating the hidden mechanism of coupling in previously proposed STO arrays.
The mutual and self-synchronization of the STOs with the spin Hall effect have been demonstrated both experimentally and theoretically. The coupling is driven by, for example, the spin wave [Awad et al., 2017] or the feedback of the oscillating electric current [Bhuktare et al., 2017]. A mutual synchronization by the electric current generated by the spin Hall magnetoresistance [Nakayama et al., 2013; Althammer et al., 2013; Y.-T.Chen et al., 2013] was also proposed in our previous work [Taniguchi, 2017]. The coupling mechanism in Ref. [Taniguchi, 2017] is a long-range interaction due to the conservation law of the electric current. The strength of the coupling is, however, weak because it is proportional to the third order of the spin Hall angle. On the other hand, the coupling mechanism proposed here comes purely from the spin degree of freedom, where the spin currents driven by the gradient of the spin accumulation and spin pumping lead to the synchronized dynamics of the magnetizations. The coupling strength of is proportional to the first order of the spin Hall angle, and therefore, is relatively strong, although the interaction length is restricted by the spin diffusion length.
The present system is suitable to study the role of the coupling by spin current on the magnetization dynamics. Since the ferromagnets are connected by a nonmagnet, we can exclude a possibility of the coupling through spin wave [Kaka et al., 2005; Mancoff et al., 2005]. Also, since the dipole interaction decays according to the inverse cube detection law, we expect that the dipole coupling can be excluded by setting a sufficiently long distance, but shorter than the spin diffusion length, between STOs. One might also consider a structure where two STOs are placed onto a single nonmagnet, instead of the model shown in Fig. 1(b). The same coupling mechanism appears even in such geometry. At the same time, however, a different mechanism of the coupling due to the spin Hall magnetoresistance [Taniguchi, 2017] will also appear when the STOs have a common bottom nonmagnet. The purpose of this work is to clarify the role of the coupling spin torque generated by spin current on the magnetization dynamics, and the structure having two STOs and one common bottom nonmagnet is therefore excluded. We also note that, although we consider the spin Hall geometry in this work as an example, a coupled dynamics of the magnetizations driven by a spin current is expected even in a CPP structure.
It might be currently difficult to connect STOs within a short distance. An integration of STOs will be, however, an inevitable topic in the field of spintronics. In the integrated system, the STOs will be assembled in a distance shorter than the spin diffusion length. The coupling mechanism proposed in this work hence will be of great interest in such situation.
Iii Coupled magnetization dynamics
In this section, we study the magnetization dynamics in the presence of the coupling torque by solving the LLG equation numerically.
iii.1 Equation of motion
The spin torques derived in Sec. II lead to the excitation of the magnetization dynamics in the ferromagnets. It was experimentally shown that the spin torque , given by Eq. (12), originated from the spin current injected from the bottom nonmagnet by the spin Hall effect can excite an self-oscillation of an in-plane magnetized ferromagnet [Liu et al., 2012]. Therefore, we assume that the magnetic field consists of an in-plane anisotropy field and the demagnetization field in the perpendicular direction as
In addition, in the present system, the spin current flowing in the top electrode also provides the spin torque given by Eq. (28). Therefore, the LLG equation describing the magnetization dynamics in the F layer is given by
where and are the gyromagnetic ratio and the intrinsic Gilbert damping constant, respectively, whereas is given by Eq. (23). The intrinsic damping constant is the damping constant in the absence of the spin pumping, and is on the order of [Oogane et al., 2006]. As mentioned with regard to the explanation of Eq. (29), the torque is decomposed into two contributions. As a result, the LLG equation given by Eq. (41) can be rewritten in the form of
where the effect of the torque due to the spin pumping into the connector is included in a matrix . The explicit forms of and and their values are summarized in Appendix A. In the following sections, we will show the solutions of by numerically calculating Eq. (32). The material parameters used in the following discussion are derived from recent experiments on the spin Hall magnetoresistance in W/CoFeB metallic bilayer [Kim et al., 2016], where knm, , nm, knm, nm, and , whereas nm and nm. We assume that knm, , and nm. Also, we use emu/c.c. [Kim et al., 2016], Oe, and rad/(Oe s). The value of the intrinsic damping constant is mentioned below. The detail of the numerical methods to calculate the LLG equation is summarized in Appendix B.
The magnetization initially stays near the stable state of . We give different initial conditions to the magnetizations and . Therefore, in the absence of the coupling due to the spin currents in the connector, two magnetizations oscillate independently with different phases. In the presence of the coupling spin torque, on the other hand, the dynamics of the magnetizations interact each other. As a result, the phase difference between the magnetizations will be stabilized and attains a certain value. The purpose of this section is to investigate such phase synchronization.
iii.2 Coupled dynamics in STOs
In this section, we study the coupled dynamics of the magnetizations by solving Eq. (32) numerically.
First, let us assume that the damping constant is relatively small, . In this case, we find that the magnetizations stay in the stable state near the initial state for MA/cm and switch their directions to the other stable state for MA/cm. Therefore, the oscillations of the magnetizations are excited when MA/cm. Figure 2(a) shows the oscillations of (red solid line) and (blue dashed line) in a steady state near the critical point MA/cm. As shown, an in-phase synchronization of the magnetizations is excited. However, when the current magnitude is slightly increased to MA/cm, the in-phase synchronization disappears. Instead, an oscillation accompanying a long-period beat appears, as can be seen in Fig. 2(b), where we note that the time range of Fig. 2(b) in the horizontal axis is different from that in Fig. 2(a). In this case, a steady sate depends on the initial state of the magnetizations, and accordingly, the phase difference in this region is not well-defined.
Next, let us consider a relatively large damping case, . In this case, we observe the oscillations of the magnetizations in the current range of MA/cm. Figure 3(a) shows the oscillations of and near the critical point MA/cm. Similar to Fig. 2(b), the oscillation accompanying beat having a long period appears. When the current magnitude is increased to MA/cm, however, the beat virtually disappears, and antiphase synchronization between STOs is excited, as can be seen in Fig. 3(b).
The in-phase synchronization is useful to enhance the emission power generated from an array of STOs, and thus, is useful for practical applications. On the other hand, the antiphase synchronization, or more generally an out-of-phase synchronization, has been found in several physical system such as Huygens pendulum clock [Pikovsky et al., 2003; Balanov et al., 2009], and therefore, is of great interest from the viewpoint of fundamental physics. One may consider that the antiphase synchronization is not preferable for practical applications. This is because the oscillating signals become totally zero or at least attenuate, and therefore, is not beneficial in terms of the enhancement of emission power from the devices. The out-of-phase synchronization, however, becomes of interest not only from the perspective of nonlinear science but also from for practical application viewpoint such as a phased array [Sun et al., 2013] and brain-inspired computing [Kudo and Morie, 2017; Vassilieva et al., 2011].
iii.3 Summary of this section and question to be addressed
The numerical simulations shown in Figs. 2 and 3 indicate that the steady state of the coupled magnetization dynamics depends on the damping constant and current magnitude. For small damping and small current region, the in-phase synchronization is stabilized. On the other hand, for a large damping case, the antiphase synchronization appears. In the intermediate region, the oscillation with a long-period beat is observed.
It has been revealed theoretically in a CPP structure that the coupling between in-plane magnetized STO via spin pumping results in an in-phase synchronization [Taniguchi, 2018]. Therefore, one might consider that the in-phase synchronization shown in Fig. 2(a) is caused by the spin pumping. However, the origin of the antiphase synchronization in Fig. 3(b), as well as that of the beat shown in Figs. 2(b) and 3(a), is unclear. We emphasize that there are two origins of the spin currents leading to the coupled motion of the magnetizations given by Eqs. (17) and (19). Therefore, to fully understand the results shown in Figs. 2 and 3, it is necessary to reveal the role of the coupling spin torque originated from the source term of Eq. (17). This analysis is developed in the next section.
Iv Role of coupling spin torque
As shown in Sec. III, the STOs show several kinds of the coupled dynamics, depending on the damping constant and current density. The purpose of this section is to clarify the physical insight of these results. Our recent work [Taniguchi, 2018] already revealed that the coupling due to the spin pumping results in an in-phase synchronization between in-plane magnetized STOs, where the effect of the spin pumping in this study is described by the torque . On the other hand, the coupling spin torque is newly proposed in this study, and therefore, its role on the magnetization dynamics is not revealed yet. Therefore, in this section, we will concentrate on the magnetization dynamics in the presence of but neglecting the spin pumping effect. This approach clarifies the fact that the coupling spin torque prefers the antiphase synchronization between STOs, and provides a physical picture which is useful to understand the results shown in Figs. 2 and 3.
iv.1 Approximated formula of coupling torque
We remind the readers that the explicit form of the coupling spin torque is given as Eq. (51). However, the formula is complex and is not useful in the following analysis. Therefore, let us first derive another expression of . Here, we use an approximation that the F/N interface is transparent. In this case, the spin accumulation is continuous at the interface, contrary to the assumption used in Eq. (15), where the spin accumulation at the interface is discontinuous. When the interface is transparent, the solution of the spin accumulation in the N layer is given by
The spin current inside the N layer is given by
When the spin diffusion length of the N layer is much longer than its length, we notice that
Therefore, in the limit of , the spin current density flowing in the N layer, from the F to F layer [ or (2,1)], can be approximated as,
The emission of the spin current at the top interface leads to the spin torque given by
where is defined as
The value of the coupling constant is for the present parameter. Equation (37) is the approximated formula of given by Eq. (51). We emphasize that using Eq. (37), instead of Eq. (51), does not change the qualitative picture of magnetization dynamics; see also Appendix A. It should also be noted that Eq. (37) is useful to develop an analytical theory of the phase synchronization because of its simplified form; see Sec. IV.4 below.
iv.2 Scaling currents
In the absence of the coupling spin torque, the spin Hall effect excites the self-oscillation of the magnetization around the in-plane easy () axis when the magnitude of the current density is in the range of , where
The current , called critical current density [Grollier et al., 2003], is the minimum current necessary to destabilize the magnetization staying near the easy axis and excites self-oscillation. On the other hand, the current is the switching current density to reverse the magnetization direction between two stable states, . The magnetization oscillates around the positive (negative) direction of the axis when is positive (negative). We note that the currents and determining the oscillation region of the magnetization are slightly affected by the coupling spin torque, as will be mentioned in the next section.
iv.3 Numerical simulation
In this section, we solve the LLG equation in the presence of Eq. (37). Since the purpose of this section is to clarify the role of Eq. (37), we have neglected the spin pumping effect in this section. Therefore, the LLG equation is given by
For the damping constant, we use . The values of and are and MA/cm, respectively.
Figures 4(a), 4(b), and 4(c) respectively show the examples of the oscillations of the , , and components of (red solid line) and (blue dotted line) for MA/cm. The results indicate that an antiphase synchronization of two magnetizations is excited by the coupling torque given by Eq. (37). Notice that the oscillation trajectory is suppressed along the direction due to the large demagnetization field. The oscillation trajectory is well described by the elliptic functions, as discussed in Appendix C. We will revisit this point in the next section to investigate the stable phase-difference by an analytical calculation of the LLG equation. Although the results shown in Figs. 4(a)-4(c) are examples for one set of parameters and initial conditions, we confirmed that the antiphase synchronization of two magnetizations is achieved for a wide range of the coupling constant , current density , and initial conditions.
Figure 4(d) compares the oscillations of for MA/cm for the coupled and uncoupled (free-running) STOs by the red solid and black dashed lines, respectively. We notice that the oscillation frequency of the coupled oscillator, 2.4 GHz, is lower than the free-running frequency, 4.7 GHz. Figure 5 summarizes the dependences of the oscillation frequencies of the coupled and free-running STOs on a wide range of . The frequency dependence on current shifts to the low current region due to the coupling. This result can be explained as follows. The current range for the self-oscillation is determined by two characteristic current densities and given by Eqs. (39) and (40), as mentioned earlier. We find that and for coupled two identical STOs should instead be substituted to (see also Appendix D)
Since the factors and are positive, the values of and for two coupled STOs are smaller than those without coupling. The values of and for two coupled STOs are and MA/cm, respectively. Therefore, the curve that represents the relation between the current density and frequency shifts to a low current region due to the coupling.
iv.4 Analytical theory
The numerical simulation indicates that the coupling spin torque given by Eq. (37) prefers antiphase coupling of the magnetizations. This result implies that the coupling torque in the present model acts as a repulsive force of the phase in the oscillators. We notice that this conclusion can also be explained analytically.
A standard approach in the field of nonlinear science to clarify the stable phase difference between the coupled oscillators is to reduce the equation of motion to the Kuramoto model [Kuramoto, 2003; Strogatz, 2001; Pikovsky et al., 2003; Stankovski et al., 2017]. The Kuramoto model argues that the oscillation properties of any kind of oscillator are characterized by the phase defined from the oscillation period. This is based on the assumption that any kind of the oscillation trajectory can be transformed into the trigonometric functions after a proper transformation of the coordinate called phase reduction [Stankovski et al., 2017]. The phase in the proper coordinate always satisfy the relation , where is a constant oscillation frequency. Solving the equation of motion of the phase difference near the fixed points, the stability of the phase difference can be investigated. Therefore, we are naturally motivated to derive the Kuramoto model in the present system.
On the other hand, in the experiments using STOs, the word ”phase” has often been used to describe the oscillation of the electric power generated from the STO [Tsunegi et al., 2016], which directly reflects the magnetization oscillation in the real space. When the oscillation of the magnetization is described by the trigonometric functions with the frequency , the phase measured in the experiments is simply given by . In general, however, the magnetization oscillation cannot be described exactly by the trigonometric function. For example, the trajectory of the self-oscillation for the in-plane magnetized STO is described by the elliptic function, as discussed in the last section and Appendix C. The oscillation trajectories in the other types of an STO are, except special cases [Silva and Keller, 2010], also not described by the trigonometric function due to, for example, symmetry breaking by an external magnetic field [Taniguchi et al., 2015b] or angular dependence of spin torque [Taniguchi et al., 2017]. Breaking a symmetry is often necessary in STO devices for an electrical detection of the oscillating signal because, in a highly symmetrical system, no electrical signal is obtained through giant magnetoresistance or tunnel magnetoresistance effect, even if a self-oscillation of the magnetization is excited. In these cases, the phase defined from the experiments is, strictly speaking, different from the phase in the Kuramoto model. It is difficult and/or complicated to apply the phase reduction or similar approximations to the LLG equation analytically [Slavin and Tiberkevich, 2008; Kudo et al., 2010; Li et al., 2010, 2011]. Moreover, it has been shown that an extension of the Kuramoto model to the nonlinear region is necessary to describe the self-oscillation in STOs [Slavin and Tiberkevich, 2009], where the nonlinearity means that the oscillation frequency, as well as the phase, strongly couples to the oscillation amplitude.
We can, nevertheless, show that the coupling spin torque in the present model acts as repulsive force between the STOs by deriving the Kuramoto model from the LLG equation. This approach will also be useful to discuss the frequency locking in the next section. When the amplitude of the self-oscillation is small, we can apply the linear approximation to the LLG equation, where the oscillation trajectory is well approximated by the trigonometric function. We note that a fixed point for the in-plane magnetized STO is . In this case, it is convenient to introduce the zenith and azimuth angles in a spherical coordinate as , although this definition is different from the conventional definitions of the zenith and azimuth angles. The reason why we use this modified relation between and is as follows. The assumption of the small amplitude oscillation means that , and therefore, . In this limit, the oscillation of the magnetization around the axis is approximately described by the trigonometric function, and can be directly regarded as the phase of the Kuramoto model; see Appendix C. As a result, we can determine the stable phase difference between STOs by using the Kuramoto model [Strogatz, 2001]. In the limit of , the LLG equation for becomes
where we introduce the notation
The double sign means the upper for and the lower for . We neglect the spin torque and the damping torque because these torques cancel each other to sustain the self-oscillation. We note that in Eq. (44) should be replaced by when we focus on the small amplitude self-oscillation around the positive (negative) axis. As mentioned earlier in this study, the positive (negative) spin current excites the self-oscillation around the positive (negative) direction of the axis. In addition, the parameter has the same sign with . Therefore, we can replace in Eq. (44) with in the present approximation.
Let us define the phase difference between two STOs as . According to Eq. (44), the phase difference near the in-phase () or antiphase () state obeys
Let us imagine that the phase difference slightly shifts from a fixed point corresponding to the in-phase state as (). The shift obeys
The solution of this equation is given by , where is the integral constant. This solution means that the shift from the in-phase state increases with time, indicating that the in-phase state is an unstable fixed point. On the other hand, a shift of the phase difference from the other fixed point corresponding to the antiphase state, defined as , obeys
The solution decreases with time, indicating that the antiphase state is the stable fixed point. Therefore, the coupling spin torque given by Eq. (37) leads to the antiphase synchronization of the magnetizations. As mentioned above, the calculation in this section is valid only for a small amplitude oscillation. The result of the numerical simulation, however, indicates that the analysis explained so far can also be applied to large amplitude oscillation.
iv.5 Summary of this section and answer to the question in the last section
The results shown in Secs. IV.3 and IV.4 indicate that the coupling spin torque given by Eq. (37) results in the antiphase synchronization between STOs. This conclusion explains the results shown in Figs. 2 and 3 in Sec. III.2 as follows.
At first, we should remind the readers that the LLG equation used in Sec. III.2 includes two coupling mechanisms between STOs, where one originates from the spin accumulation whereas the other comes from the spin pumping. As implied in Eqs. (17) and (19), the strength of the coupling spin torque due to the spin accumulation is proportional to the current density whereas that of the spin pumping is proportional to the oscillation frequency of the magnetization, . In addition, the oscillation frequency of the in-plane magnetized ferromagnet decreases with increasing the current density, as shown in Fig. 5. These facts indicate that the coupling due to the spin pumping is dominated in a relatively low current region, whereas that originated from the spin accumulation becomes large in a relatively high current region. As clarified in Ref. [Taniguchi, 2018], the spin pumping prefers the in-phase synchronization between the in-plane magnetized STOs. Therefore, the in-phase synchronization is found in Fig. 2(a). When the current magnitude is increased, however, the spin torque due to the spin accumulation becomes a dominant contribution to the coupled motion of the magnetizations. Since this coupling torque prefers the antiphase synchronization, as clarified in Secs. IV.3 and IV.4, a beat of the oscillations appears in the intermediate current region shown in Figs. 2(b) and 3(a) as a result of the competition between two coupling mechanisms. When the current becomes sufficiently large, the antiphase synchronization is stabilized by the coupling torque originated from the spin accumulation, as can be seen in Fig. 3(c).
iv.6 Frequency locking
In the above sections, we assume that two STOs have identical parameters. An interesting characteristic of synchronization is, on the other hand, the frequency locking. A typical value of the locking range of the frequency between STOs in the previous works is on the order of 10-100 MHz [Kaka et al., 2005; Mancoff et al., 2005; Rippard et al., 2005]. At the end of this section, let us briefly study the possibility of a frequency locking for two STOs having different free-running frequencies by the coupling torque given by Eq. (37) because it provides another viewpoint to catch the strength of the coupling.
The range of the frequency locking is determined by the strength of the coupling spin torque. Although it is difficult to derive an exact analytical expression of the locking range, Eq. (46) implies that the locking range is roughly given by
In the following, the possibility of the frequency locking is confirmed using numerical simulations. The value of in the F layer is changed to 192 Oe to make the difference of the oscillation frequencies of STOs to about 100 MHz, which is the maximum value of the locking range we obtained from the numerical simulation. In Fig. 6(a), we show the free-running oscillations of and by red solid and blue dotted lines, respectively, where the coupling spin torque is set to zero. As mentioned, the frequency difference between and is about 100 MHz, as can be seen from their Fourier transformations in Fig. 6(b). On the other hand, in the presence of the coupling spin torque, the frequency locking is observed, as schematically shown in Figs. 6(c) and 6(d), where the oscillations of () and their Fourier transformations are shown. Figure 7 shows the current dependences of the oscillation frequencies (red square) and phase difference (blue circle) of the coupled STOs. The in-phase and antiphase synchronizations correspond to the phase difference of 0 and 0.5, respectively (see also Appendix B). The frequency locking occurs for the current of MA/cm. We note that the locked frequency is different from the free-running frequencies of the STOs or their average, and the phase difference is different from the antiphase due to the frequency difference, as expected for the nonlinear oscillator [Slavin and Tiberkevich, 2009]. The numerical simulation indicates that the phase difference in the locked state is almost constant.
In conclusion, a theoretical framework for the spin-current driven synchronization in spin torque oscillators in the spin Hall geometry was proposed. The spin current generated from the spin Hall effect excites the self-oscillation of the magnetization and simultaneously creates the spin accumulation in the oscillator. The magnitude and direction of the spin accumulation in the ferromagnet depend on the magnetization direction of the oscillator. Then, by connecting the top surfaces of the oscillators with a nonmagnet having a long spin diffusion length, a spin current spontaneously flows between the oscillators, according to the gradient of the spin accumulation. The spin current excites an additional spin torque acting on the magnetization. As a result, the self-oscillations in the oscillators are naturally coupled. The coupling mechanism comes purely from the spin degree of freedom, contrary to the previous proposals based on the electric and/or magnetic interactions. Both the numerical simulation and analytical theory show that the coupling torque acts as a repulsive force, and therefore, the antiphase synchronization of the self-oscillation is preferred by this coupling mechanism. These conclusions are obtained by deriving the theoretical formulas of the coupling spin torque from the spin transport theory, and by solving the equation of motion of the magnetizations with the coupling spin torque both numerically and analytically. In the self-oscillations state, however, the spin pumping becomes another source of the coupling because the spin current generated by the spin pumping also flows in the nonmagnetic connector. When the spin pumping is taken into account, a competition between the coupling spin torque and spin pumping appears because the spin pumping prefers an in-phase synchronization. As a result, the in-phase synchronization appears in a relatively low current region whereas the antiphase synchronization appears in a relatively high current region.
The author is thankful to Yoji Kawamura, Kiwamu Kudo, Hitoshi Kubota, Sumito Tsunegi, Takehiko Yorozu, Hidekazu Saito, Takahiro Chiba, Yasuhiro Utsumi, and Masamitsu Hayashi for valuable discussions. The author is also grateful to Satoshi Iba, Aurelie Spiesser, Hiroki Maehara, and Ai Emura for their support and encouragement. This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 16K17486.
Appendix A Explicit forms of and and validity of
a.1 Definitions of and
To make the notation simple, we define a matrix from Eq. (24) as
The coupling spin torque, , is defined as [see also Eq. (37)],
where the ( or ,,) component of is given by