Spin wave modes of two magnetostatic coupled spin transfer torque nano-oscillators

During the last two decades, the study of spin transfer torque nano-oscillators (STNOs) has received quite a bit of attention due to their potential applications in wireless telecommunication technology, in the implementation of magnetic field sensors, and in several other radio frequency (RF) nano-scale devices.^1–4 A frequently addressed STNO is a nanopillar, typically composed of two ferromagnetic layers separated by a non-magnetic spacer. The ferromagnetic layers are known as the fixed and the free layer, since their magnetizations are mainly fixed or free to move in response to external stimulus. When a current is injected into the system, the electron spins get polarized in the direction of the magnetization of the fixed layer and then exert a torque on the magnetization of the free layer. This spin transfer torque has an anti-damping effect that can generate an auto-oscillatory regime in the normal modes of the free layer.^5,6 The minimum dc current density necessary to observe these oscillations is called the critical current density.

Due to their properties, such structures are proposed as a model-system to emulate neurons and synapses through their nonlinear dynamics.^7–9 Currently, coupled oscillator networks are candidates for the realization of associative memories, in which a recognition is obtained when this network converges to a synchronized state.^10–12 This synchronization may be obtained through the interaction between the STNOs, as through electrical connections,^13–15 emission of spin waves,^16,17 or by magnetostatic interactions.^18–21 Some studies have addressed the magnetostatic coupling between nano-oscillators; however, most of them consider a vortex state as the equilibrium magnetization configuration^19,22,23 instead of a uniform state of the magnetization.^21,24 In particular, Chen et al.²¹ developed an interesting work determining the critical current needed for synchronizing two nano-oscillators considering that the same or different currents are applied to both oscillators. A similar study was done by Zhang et al.,²⁴ who determined the different degrees of synchronization that two nano-oscillators show in the presence of different external magnetic fields and injected currents to each STNO.

Despite the advances made by these authors, an open question is the role played by the STNO’s relative locations when the equilibrium magnetizations are fixed by an external magnetic field applied along a particular direction. In these systems, the magnetostatic interaction plays a significant role because the relative position of the nano-oscillators can favor the parallel or antiparallel magnetization alignment. An example of this can be seen in the studies of the interaction between two magnetic nanotubes²⁵ or two rough slabs,²⁶ in which the magnetic interaction energy changes sign depending on the relative positions between the samples.

Thus, in this work, we study the uniform normal modes of two STNOs located in the same plane, coupled by the magnetostatic interaction, as a function of their relative position. Using a standard Hamiltonian formalism,^27–29 we determined the spin wave modes of the coupled disks in the macrospin approximation. We found that there are two uniform modes, with the magnetic moments oscillating in-phase or anti-phase, respectively. The frequencies of the modes depend on the relative positions of the two nanopillars and the applied magnetic field. We also obtained the critical current density needed to induce self-oscillations for identical and different STNOs.

This paper is organized as follows: in Sec. II, we describe the model and formalism used to determine the spin wave modes and the critical current densities; in Sec. III, we present and discuss our results; and finally, in Sec. IV, we present our conclusions.

Our model system consists of two nanopillar devices of circular cross sections, with radii $R1$ and $R2$⁠. Each nanopillar is composed of two ferromagnetic layers separated by a non-magnetic metallic spacer. We define $S$ as the shortest distance between the surfaces of the nanopillars, i.e., they have a center to center distance $D=R1+R2+S$⁠. The unit vector along the direction that connects nanopillars $1$ and $2$ is $n^$⁠, as illustrated in Fig. 1. There is an external magnetic field applied to the system, $H→=Hxx^$⁠. We consider that the free layers, located in the same plane, are disks of soft ferromagnets with a saturation magnetization $Ms$ and a very small thickness $L$⁠, as shown in Fig. 1. Both free layers are magnetized in plane along the direction of the applied field, i.e., the x-direction. In addition, the same current density $J→=Jz^$ is injected to each nanopillar. Both nanopillars are coupled through magnetostatic interactions (see  Appendix A for details). In this work, we focus our attention on the magnetostatic interaction between the free layers, and we vary the angle $θ$ between $n^$ and $x^$⁠, i.e., $cos⁡(θ)=n^⋅x^$⁠. Also, the magnetostatic interaction between the fixed and free layers slightly modifies the eigenfrequencies and critical current of the system, as is discussed in Appendix A 3.

To find the normal modes and critical current densities of our system as a function of the magnetostatic interaction between the free layers, we analyze the magnetization dynamics of them by means of the Landau-Lifshitz-Slonczewski (LLS) equation⁴ using the Hamiltonian formalism.^27–29 We use the macrospin approximation,^30,31 that is, we assume that the magnetizations of the free layers are uniform.

The magnetizations of the free layers are free to move in response to the external current applied to the system, and their dynamics can be obtained through the LLS equation⁴ given by

where $m→k=M→k/Ms$ is the normalized magnetization of the $k$ free layer, with $k={1,2}$ and $Ms$ the saturation magnetization. Time is normalized through $τ=|γ|4πMst$⁠, where $|γ|=1.76⋅107Oe−1s−1$ is the absolute value of the gyromagnetic ratio. The normalized effective field is $h→eff,k=H→eff,k/(4πMs)$⁠, $α$ is a dimensionless quantity and corresponds to the damping parameter, $J$ represents the current density applied on each free layer, $p^$ the spin polarization direction, and $β=2πℏϵ/[(4πMs)2eL]$ is the spin transfer torque parameter,²⁷ with $ϵ$ the polarization factor that varies between 0 and 1 and $e$ the magnitude of the electron charge.

To study the dynamics of our system, we introduce new variables $ak$ and $ak∗$ through a classical Holstein-Primakoff transformation,^32,33 which are related with the magnetization components through

where super-indices $x$⁠, $y$⁠, and $z$ represent the components of the magnetization in the corresponding directions and $k$ identifies free layer 1 or 2. The variable $ak∗$ is the complex conjugate of $ak$⁠. These new variables represent perturbations around the equilibrium magnetization, which is in plane. Through this change of variables, the LLS equations for the magnetizations $m→k$ are transformed, to linear order, into the following equations^29,34,35

where $Vk$ is the volume of the $k$ free layer and $Utot=Etot/4πMs2$ is the normalized total energy of the system defined as

In this expression, $UZ$⁠, $UD$⁠, and $UI$ are the normalized Zeeman, self demagnetizing, and magnetostatic interaction energies, respectively. The expressions for these energies, as a function of $ak$ and $ak∗$⁠, are found in  Appendix B (see Sec. B 1 for $UZ$⁠, Sec. B 2 for $UD$⁠, and Sec. B 3 for $UI$⁠). We do not include an exchange term because we assume that the magnetizations are uniform.

In this subsection, we develop the methodology to obtain the normal mode frequencies and the critical current densities of our system. We start with a well known result for an isolated nanopillar. After that, we introduce the magnetostatic coupling between the free layers.

In an isolated nanopillar with free layer magnetized in plane and from Eq. (3), we obtain

where $Ak=hx+hkd/2$ and $Bk=hkd/2$⁠. The expressions $hx=Hx/(4πMs)$ and $hkd$ are the magnitudes of the normalized external magnetic field and the normalized demagnetizing factor in the z direction, respectively [see Eq. (B7) for $hkd]$⁠. The normalized demagnetizing field points along the z-axis with $h→kd=−hkdmkzz^$⁠, where the demagnetization factor $hkd$ depends on the geometrical parameters of the disk, i.e., the thickness $L$ and the radius $Rk$⁠. To obtain the normal modes of an isolated nanopillar, we diagonalize the set of Eq. (5) through the change of variables³² 

with $λk=(Ak+ωk)/(2ωk)$⁠, $μk=(Ak−ωk)/(2ωk)$⁠, and $ωk$ the normalized angular frequency. Finally, Eq. (5) and its conjugate are equivalent to

and its complex conjugate equation. The solution of Eq. (7) corresponds to a damped oscillator, $bk(τ)=bk0e(−iωk+γk)τ$⁠, in which^34,36

The constant $bk0$ represents the initial condition at $τ=0$ and $−γk$ corresponds to the effective dissipation constant. In this expression, $γk$ depends on the current density, $J$⁠. To obtain the critical current density necessary to induce auto-oscillation of the magnetization, the effective dissipation must be zero. With this condition, the critical current density of the isolated macrospin mode is^34,36

If we now include the magnetostatic interaction between the free layers, Eq. (3) for $ak$ and $ak∗$ and their conjugates become

where $a→=(a1,a1∗,a2,a2∗)$ and $M′=[(1−iαI′)M+iβJI]$⁠. The elements of the $I′$ matrix are $Iij′=δij(−1)i+1$⁠, with $I$ the identity matrix and

Here,

The quantities $gx$⁠, $gy$⁠, and $gz$ are related to the magnetostatic interaction between the free layers and are derived in  Appendix B [see Eqs. (B13a)–(B13d)]. To obtain the normal modes and the critical current densities of the coupled nanopillars, we need to diagonalize the $M′$ matrix. In this step, we write $M′=PDP−1$⁠, where the $P$ matrix is related to the eigenstates of the system and the $D$ diagonal matrix is related to the eigenvalues of the system. Then, we introduce a new change of variables $a→=Pb→$⁠, with $b→=(bip,bip∗,bap,bap∗)$⁠, where the subscripts ip and ap represent in-phase and anti-phase, respectively. In this way, we obtain

where the diagonal elements of $D$ are $D11=ωip+iγip$⁠, $D22=−D11∗$⁠, $D33=ωap+iγap$⁠, and $D44=−D33∗$⁠. The quantities $ωip$⁠, $ωap$⁠, $γip$⁠, and $γap$ are real numbers. $ωip$ and $ωap$ represent the normalized oscillation frequencies (angular), and $−γip$ and $−γap$ represent the effective damping, as we can see from the solutions of Eq. (12):

Initial conditions at $τ=0$ are given by $bip0$ and $bap0$⁠. The above solutions show that the normal modes oscillate in-phase or anti-phase, with the eigenvectors given by $P$⁠. The in-phase mode is characterized by the $y^$ and $z^$ components of the magnetization having the same sign in both nanopillars. On the other hand, the anti-phase mode is characterized by opposite signs for these components of the magnetization, i.e., $sgn(m1y)=−sgn(m2y)$ and $sgn(m1z)=−sgn(m2z)$⁠, as shown in Fig. 2. In both modes, we have $sgn(m1x)=sgn(m2x)$ when the amplitudes of the spin waves are small. These modes are also called binding and anti-binding modes³⁷ or optical and acoustic modes³⁸ and have been observed by ferromagnetic resonance force microscopy.³⁹ 

From Eq. (10), it is possible to obtain analytical expressions for the eigenfrequencies, by diagonalizing the matrix $M′$ when the system is free of dissipation and spin transfer torque, i.e., $α=0$ and $J=0$ or $M′=M$⁠. In this case, the angular eigenfrequencies are

where

It is important to note that the frequencies $ω±$ are functions of $θ$⁠. It is possible to identify a particular angle, $θ1$⁠, such that for $θ<θ1$⁠, $ω+=ωip$ and $ω−=ωap$⁠, and for $θ>θ1$⁠, $ω+=ωap$ and $ω−=ωip$⁠. To determine $θ1$⁠, it is necessary to interpret the eigenvectors that can be numerically obtained.

Finally, to obtain the critical current densities, we solved numerically Eq. (10) including dissipation and spin transfer torque. Then, the critical current densities, $JipC$ and $JapC$⁠, can be obtained imposing $γip=0$ and $γap=0$⁠, respectively.

We consider two permalloy free layers defined by magnetic parameters $Ms=860$ kA/m and $A=1.3×10−11$ J/m. To solve the LLS equation, we used a damping parameter $α=0.01$ and a polarization factor $ϵ≈0.11$⁠. In all the cases, we considered an applied magnetic field with magnitude $|H→|=1$ kOe. With these parameters, we addressed two cases, identical and non-identical nano-oscillators, where the shortest distance between them is $S=10$ nm (see  Appendix C for other values of $S$⁠). Both free layers have the same thickness, $L=5$ nm.

In this section, we find the frequencies and critical current densities for two identical free layers of radius $R=50$ nm, with demagnetizing factors $h1d=h2d=hd$⁠, and normalized applied magnetic field $hx$⁠. In the absence of dissipation and spin transfer torque, we obtained an analytical expression for the angular eigenfrequencies from Eq. (14):

where $gx$⁠, $gy$⁠, and $gz$ were already introduced after Eq. (11) [see Eqs. (B13a)–(B14c)]. According to Eq. (8a), the eigenfrequency for an isolated nanopillar is $ωk=hx(hx+hkd)$⁠. By comparing this with Eqs. (17a) and (17b), we identify effective applied fields $hx±=hx+(gx±gy)$ and effective demagnetizing factors $hd±=hd±(gz−gy)$⁠. In this way, we write Eqs. (17a) and (17b) as

Figure 3 illustrates the frequencies $fip=2|γ|Msωip$ and $fap=2|γ|Msωap$⁠, obtained from Eqs. (18a) and (18b) for different $θ$ values, for $R=50$ nm and $D=110$ nm. In addition, the frequency of an isolated disk, $f0=2|γ|Msω0$⁠, is also depicted. We observe that for $θ=0$⁠, the lower frequency mode is the anti-phase one, while for $θ=π/2$⁠, the lower frequency mode is the in-phase one. Also, there is a degeneracy in the frequency at $θ1$⁠, i.e., $fap=fip$⁠. This angle can be obtained through the condition

and for the parameters previously mentioned $θ1≈0.6$ rad. This behavior can be understood through the analysis of the surface magnetic charges of the free layers. In Fig. 4, we show a schematic representation of these magnetic charges. Figures 4(a) and 4(c) represent the in-phase modes at $θ=0$ and $θ=π/2$⁠, respectively. Figures 4(b) and 4(d) represent the anti-phase modes at $θ=0$ and $θ=π/2$⁠, respectively. At $θ=0$⁠, we observe that the distance between the positive charge of the free layer 1 and the negative charge of the free layer 2 is smaller in the anti-phase mode. For this reason, when $θ=0$⁠, we may argue that the lower frequency mode is the anti-phase one. On the contrary, when $θ=π/2$⁠, we observe that this distance is smaller in the in-phase mode, and then we may argue that the lower frequency mode is the in-phase one.

In addition, from Fig. 3, we observe that the changes in frequency with angle for the in-phase mode is larger than the equivalent variation for the anti-phase mode. To explain qualitatively this behavior, first, we calculate the difference of the expected value of the total energy between $θ=0°$ and $θ=90°$⁠, for the in-phase and anti-phase modes, in the absence of dissipation and spin transfer torque (⁠$γip=γap=0$⁠). Then, we compare these differences between the two modes. To calculate the difference between the expected value of the total energy between $θ=0°$ and $θ=90°$ for one mode, it is necessary to calculate only the expected value of the magnetostatic interaction energy. It is because the Zeeman and the self-magnetostatic energies are the same if we consider the same oscillation amplitude for both angles. The magnetostatic interaction energy [see Eq. (B15)], normalized by the volume $V=πR2L$ of a disk is

(we neglected terms that couple $mx$ and $my$ that produce a small effect on the linear spin wave modes). We write the magnetizations of the free layers as a function of the normal modes, and we calculate the time average of the normalized magnetostatic energy over a period of oscillation, $uIip(θ)$ and $uIap(θ)$⁠, for the in-phase and anti-phase modes, respectively:

In these calculations, we only consider a time average over the interaction energy of the perturbations around the equilibrium magnetization of the system (there is a time invariant interaction energy associated with the equilibrium magnetizations that depends on the relative positions but that is independent of the modes involved). Then, we consider the interaction energy to second order, written with respect to $bip$ and $bip∗$⁠, or to $bap$ and $bap∗$⁠. In this context, the results obtained from Eqs. (21) and (22) are proportional to $|bip0|2$ or $|bap0|2$⁠, and they are time invariant in an oscillation period. For the in-phase mode, we obtain $|uIip(π/2)−uIip(0)|/|bip0|2≈0.021$ and for the anti-phase mode $|uIap(π/2)−uIap(0)|/|bap0|2≈0.001$⁠. Since it is possible to relate the variations of the mode interaction energies with frequency variations, one may conclude that the anti-phase frequency exhibits smaller changes than the in-phase frequency as a function of $θ$⁠, which corresponds to what is illustrated in Fig. 3.

Finally, we analyze the critical current densities of both normal modes. According to Eq. (9), the critical current density of an isolated free layer is $JkC=α(hx+hkd/2)/β$⁠. Using this expression and replacing the applied field by the effective applied fields, $hx±=hx+(gx±gy)$⁠, and the demagnetizing factor by the effective demagnetizing factors $hd±=hd±(gz−gy)$⁠, the critical current densities obtained for the two modes when the free layers are coupled through the magnetostatic interaction are $J±C=α(hx±+hd±/2)/β$⁠, i.e., the critical current densities for the in-phase and the anti-phase modes would be

In addition to this analysis, we have computed the critical current densities numerically by diagonalizing $M′$ in Eq. (10). We verified that both procedures deliver the same results. The critical current densities are plotted in Fig. 5 as a function of $θ$⁠. In this figure, it is shown an angle, $θ2$⁠, at which both critical current densities are the same, i.e., $JipC=JapC$⁠. Then, by using Eqs. (23a) and (23b), we have the condition $gy(θ2)+gz=0$⁠. For our parameters, $θ2≈1.03$ rad. For $θ<θ2$⁠, the lower critical current density corresponds to the anti-phase mode, while for $θ>θ2$⁠, it corresponds to the in-phase mode. Therefore, there is a region between $θ1$ and $θ2$ in which the lowest critical current density corresponds to the normal mode of the highest frequency, i.e., the anti-phase mode. In the same way, for this region, the highest critical current density corresponds to the normal mode of the lowest frequency, i.e., the in-phase mode. This behavior occurs because the effective dissipation for the in-phase mode, $−γip=α(hx−+hd−/2)−βJ$⁠, and the anti-phase mode, $−γap=α(hx++hd+/2)−βJ$⁠, is not proportional to the frequencies, as shown from Eqs. (18a) and (18b).

It is interesting to note that there is a region in which the critical current densities of both modes are lower than the critical current density of an isolated free layer represented by $J0C$⁠. This region, shown in Fig. 5, corresponds to $θ≳θ2$⁠.

In this section, we find the frequencies and critical current densities for two non-identical oscillators with $R2≠R1$⁠, and therefore, $h1d≠h2d$⁠. We considered an applied magnetic field $Hx$ that normalized by $4πMs$ defines $hx$⁠. In the absence of dissipation and spin transfer torque, the frequencies are given by Eq. (14). Figure 6 illustrates these frequencies as a function of $θ$⁠. We consider two cases: for the first one, we used $R1=50$ nm and $R2=48$ nm [see Fig. 6(a)]. For the second case, we used $R1=50$ nm and $R2=40$ nm [see Fig. 6(b)].

In Figs. 6(a) and 6(b), the horizontal lines illustrate the frequencies of the isolated free layers, denoted $f1$ and $f2$⁠, and the vertical line shows the transition between the different modes at the angle $θ1$⁠. Below $θ1$⁠, the frequency of the in-phase mode is higher than the frequency of the anti-phase mode, but above $θ1$⁠, the opposite occurs. This behavior can be understood from Fig. 4, as occurs for identical oscillators. A difference between identical and non-identical oscillators is the split of frequencies, $δf=|fip−fap|$ for non-identical oscillators at $θ=θ1$⁠. This split of frequencies increases as the difference between $R1$ and $R2$ increases. For example, we have $δf≈0.01$ GHz for $R2=48$ nm, see Fig. 6(c), while $δf≈0.06$ GHz for $R2=40$ nm, see Fig. 6(b). This split of frequencies can be understood from Eq. (14). When we have identical oscillators, the expression inside the root $(δ2−Δ+Δ−)$ in Eq. (14) is zero at $θ=θ1$⁠, then $ω+=ω−$⁠. On the contrary, $(δ2−Δ+Δ−)$ is different from zero when we have non-identical oscillators (different demagnetization and magnetostatic fields); therefore, there is a split in the frequencies at $θ=θ1$⁠. To understand the transition between the two modes shown in Figs. 6(a) and 6(b), we run simulations for the magnetization given by Eq. (1) for different $θ$⁠. We consider two non-identical nano-oscillators with $R1=50$ nm, $R2=40$ nm, and $S=10$ nm. The objective was to compare these results with our analytical and numerical results and to confirm the validity of our model.

Figure 7 illustrates the y-component of the magnetization dynamics of the two oscillators, $m1y$ and $m2y$⁠, as a function of time, where the time $T$ is much larger than the transient time. The applied current densities are higher than the minimum critical current density, and then, we observe oscillations of the magnetization associated with the mode of lowest frequency. In Fig. 7(a), for $θ=0°$ and $J=5.61×107$ A/cm$2$⁠, the oscillations correspond to the anti-phase mode. In Fig. 7(b), for $θ=37°$ and $J=5.56×107$ A/cm$2$⁠, the oscillations still correspond to the anti-phase mode. In Fig. 7(c), for $θ=39°$ and $J=5.56×107$ A/cm$2$⁠, the oscillations now represent an in-phase mode. Finally, in Fig. 7(d), for $θ=90°$ and $J=5.48×107$ A/cm$2$⁠, the oscillations describe an in-phase mode. We observe that the amplitude of the oscillator 2 is always bigger than the oscillation amplitude of the oscillator 1, but the difference of amplitudes increases for angles near to $37°$ and $39°$⁠, where the amplitude of the oscillator 1 tends to zero. We conclude that there is a continuous transition from the anti-phase mode to the in-phase mode for an angle between $37°$ and $39°$⁠. At the specific angle of transition, $θ1$⁠, the amplitude of the oscillator 1 is zero when the applied current density is equal to the critical current density. This behavior is qualitatively consistent with our previous results; however, there is a quantitative difference in the angle of transition, given that our analytical calculations predict $θ1≈34.55°$⁠. The difference appears because in our analytical calculations, for simplicity, we consider that the equilibrium magnetization is in the x-direction, i.e., we consider that $gxy=0$ [see  Appendix B, specifically Sec. 3]. In addition, we analyze the Fourier spectrum of the signal $m1y$⁠. We observe that the peaks of the signal are: $f≈8.63$ GHz for $θ=0°$⁠, $f≈8.51$ GHz for $θ=37°$⁠, $f≈8.50$ GHz for $θ=39°$⁠, and $f≈7.88$ GHz for $θ=90°$⁠. This behavior is consistent with Fig. 6(b), where the frequency decreases as $θ$ increases.

Finally, Fig. 8 illustrates the critical current densities for the in-phase and anti-phase modes as a function of $θ$⁠. These current densities are obtained from numerical calculations. The behavior illustrated in Fig. 8(a) is similar to the one exhibit for identical oscillators shown in Fig. 5, but in the region closer to $θ1$⁠, the critical current densities change abruptly their slopes. At $θ1$⁠, one of the amplitudes of the magnetization oscillation is zero, that is, there is a steady state with a saturated magnetization in one direction. This magnetization (from free layer $i$⁠) exerts a magnetic field in the x-direction on the other free layer (free layer $j$⁠) equal to $(Ri/Rj)gx$ [see Eq. (B12)], and then

The difference between both critical currents at $θ1$ is

This quantity increases as the difference between the radius of the free layers increases.

In summary, by means of analytical and numerical calculations, we studied the normal modes of two free layers coupled through magnetostatic interactions. Two cases were considered, identical and non-identical free layers. We focused on the eigenfrequencies of the in-phase and anti-phase modes as well as on the critical current densities necessary to induce self-oscillations of the magnetization. Both the eigenfrequencies and the critical current densities were studied as a function of the relative position between the nanopillars.

While the frequency of the magnetization oscillation of the anti-phase mode is almost constant as a function of the relative position of the nanopillars, in the in-phase mode, such frequency changes with this position. This behavior is explained analyzing the time average of the normalized magnetostatic energy over a period of oscillation for the in-phase and anti-phase modes. In addition, for identical oscillators, both modes are degenerate at a particular position, while for non-identical nanopillars, the difference in the geometry induces a split in the frequencies that removes the degeneracy. This split increases with the difference between the radius of both STNOs.

We also observe that the coupling of two identical nanopillars generates a decrease of the critical current densities of both modes for some positions, as compared to an isolated nanopillar. For non-identical oscillators, there exists a particular relative position at which the critical current densities of both modes behave quite sensitively. This behavior is explained through an analysis of the amplitude of the oscillation of the magnetization in both free layers. Therefore, by changing the relative position of two STNOs, it is possible to control their normal modes and critical current. In consequence, relative position is a powerful element and can be used for the design and control of STNOs for associative memory.

The authors thank A. O. León for fruitful discussions and acknowledge financial support in Chile from FONDECYT (Nos. 1161018, 1160198, and 1170781), Financiamiento Basal para Centros Científicos y Tecnológicos de Excelencia (No. FB 0807), and AFOSR Neuromorphics Inspired Science (No. FA9550-16-1-0384). D.M.-A. acknowledges financial support in Chile from CONICYT, Postdoctorado FONDECYT 2018, folio 3180416.

The magnetostatic field produced by the $i$ fixed layer can be calculated through the magnetostatic potential $Φifix$⁠, and then the demagnetizing field can be obtained through the relation $H→ifix=−∇→Φifix$⁠. The magnetostatic potential is given by

where $σifix=n^i⋅M→(x→i)=ρ^i⋅Msx^=Mscos⁡(ϕi)$ is the effective surface magnetic charge density on the surface of the disk $i$⁠. We consider that fixed and free layers are made of the same material and that they are magnetized along the x-direction. The fixed layer $i$ has a thickness $Lfix$ and a radius $Ri$⁠. The demagnetizing potential can be written as

where $Jm(kρ)$ are the Bessel functions of the first kind. Calculating the integrals, the magnetostatic potential of the $i$ fixed layer, for $z>Lfix$⁠, is

Finally, the magnetostatic field can be written as

In this section, we consider the magnetostatic interaction between the fixed and free layers of the same nanopillar. The average of the magnetostatic field produced by the fixed layer in the volume of the $i$ free layer is

where $Ls$ is the thickness of the non-magnetic spacer. When integrating in the $ϕ$ coordinate, only the x-component of the field remains. This field, normalized by $4πMs$⁠, is

where $g1(q,Ri)=(1−e−qLfix/Ri)(1−e−qL/Ri)e−qLs/Ri$⁠. In particular, if we consider $R1=R2=50$ nm, $Lfix=5$ nm, $L=5$ nm, and $Ls=10$ nm, we obtain $⟨h→iifix⟩≈−0.02x^$⁠, which is approximately 20% of the applied field. The effect of this interaction is to decrease both the eigenfrequencies and the critical current densities.³⁵ 

In this section, we consider the magnetostatic interaction between a fixed layer and a free layer of a different nanopillar. Coordinates $ρ$ and $ϕ$ in Eq. (A3) are related to the center of the $i$ fixed layer. In order to relate these coordinates to the polar coordinates of the $j$ free layer, we write

where the new coordinates $ρ′$ and $ϕ′$ are related to the center of the $j$ free layer. The next step is to average the magnetostatic field produced by the fixed layer in the volume of the $j$ free layer, i.e.,

where $Ls$ is the thickness of the non-magnetic spacer. The normalized magnetostatic field can then be written as

with

where $g2(q)=(1−e−qLfix/D)(1−e−qL/D)e−qLs/D$⁠. In particular, if we consider $R1=R2=50$ nm, $Lfix=5$ nm, $L=5$ nm, $Ls=10$ nm, and $D=110$ nm, we obtain $k1≈0.0060$⁠, $k2≈−0.0027$⁠, and $k3≈0.0028$⁠. With these parameters, the maximum values of the field for each axis are in the x-axis $⟨hijfix⟩V≈0.0060$⁠, in the y-axis $⟨hijfix⟩V≈0.0044$⁠, and in the z-axis $⟨hijfix⟩V≈0.0028$⁠. These values represent less than 6% of the applied field, and then in a first approximation, we neglect this interaction.

In this section, we discuss the effect of the magnetostatic interaction between the free and the fixed layers in the eigenfrequencies of the coupled free layers. In summary, the fixed layers exert an external field on the free layers. These fields have a component in all the directions. The components of the fields in the $y^$ and $z^$ directions just change the direction of the equilibrium magnetization, but it is negligible for the range of parameters we have considered here. In relation to the fields in the $x^$ direction, they change the eigenfrequencies and the critical current densities of the spin waves. We can identified an effective applied field in every disk as

where $⟨hiifix⟩V$ is the magnetic field of the fixed layer $i$ in the free layer $i$ and $⟨hijfix⟩V$ is the magnetic field of the fixed layer $i$ in the free layer $j$⁠. With these new fields, we observe that some of the elements of the matrix $M$ in Eq. (11) change. Then, $M11=h1x+h1d/2+(R2/R1)gx(θ)$ and $M21=h2x+h2d/2+(R1/R2)gx(θ)$⁠. In consequence, the eigenfrequencies and critical current densities are modified.

In this section, we obtain the expressions for the total normalized free energy in the macrospin approximation given by

where $WZ$ and $WD$ represent the Zeeman and the self-magnetostatic energy densities of the two free layers, respectively, and $WI$ is the magnetostatic interaction energy density between the free layers. Next, we show the expressions for the different contributions to the energy.

The Zeeman energy of the two nanopillars $UZ$⁠, normalized by $4πMs2$⁠, is given by

where we have neglected the constant terms. $hx=Hx/(4πMs)$ is the normalized applied magnetic field, and $V1$ and $V2$ are the volumes of disks 1 and 2, respectively.