The recently proposed concept of electric chiral magnonic resonator (ECMR) has been extended to include usage of spin–orbit torques (SOT). Unlike the original version of ECMR which was based on voltage controlled magnetic anisotropy (VCMA), the spin wave amplification power by this new version of ECMR (pumped by SOT) no longer depends on the phase of the incident wave, which is highly desirable from an application point of view. The performance of the SOT pumped ECMR has been compared with the case of amplification by applying SOT pumping directly to a waveguide (without any ECMR involved). It is argued that at the expense of narrowing the bandwidth (i.e., slower amplifier response), the advantage of the former configuration (amplification by a SOT pumped ECMR) over the latter (amplification by direct SOT pumping the waveguide) is to offer gain, while at the same time, maintaining system stability (avoidance of auto-oscillations). Non-linear behavior of the SOT pumped ECMR has been analyzed. It is demonstrated that by cascading a SOT ECMR operating in an off-resonance mode together with a VCMA biased passive ECMR, it is possible to produce a magnonic neuron with a transmitted signal magnitude larger than the input in the firing state.

## I. INTRODUCTION

Due to the opportunities of low energy consumption,^{1} strong non-linearity,^{2,3} and spatial miniaturization^{4–6} provided by spin waves, magnonics^{7,8} has been regarded as one of the top candidates in providing a platform for constructing future non-von Neumann architectures computing system. However, development of magnonics over the years was largely hindered by the lack of efficient means to amplify spin waves. Studies in early years were concentrated on parametric amplification of magnetostatic spin waves in permalloy,^{9–12} where parametric pumping is made possible through modulation of spin wave precession ellipticity by a rf oersted field delivered through electric current leads in proximity with the permalloy waveguide. However, due to the demand for device miniaturization and the desire for low spin wave propagation loss, interest has been shifted recently to the study of sub-100 nm wavelength exchange spin waves in nanometers thick yttrium iron garnet (YIG) films.^{4–6} Unfortunately, these exchange spin waves in YIG are highly circular polarized and their precession ellipticity is relatively insensitive to bias field strength. This renders the former mentioned parametric amplification through precession ellipticity modulation by a rf magnetic field unsuitable to amplify such a spin wave. Therefore, recently, attention has migrated to another mechanism: the spin–orbit torque (SOT).^{13,14} The spin pumping nature of the SOT makes it more versatile to deal with different kinds of spin wave excitations including the aforementioned short wavelength spin wave in YIG films. Historically, this SOT strategy has been plagued by the occurrence of auto-oscillations, which scatter non-linearly with the input spin wave signal and suppresses its amplification.^{15} Recently, after introducing a perpendicular anisotropy in the YIG film, true amplification of the spin wave has been claimed to be observed.^{16} However, even in this improved circumstance, spin wave amplification has to be conducted under pulse mode since prolonged SOT pumping would still finally lead to the occurrence of uncontrolled auto-oscillations and ruin the system.

Recently, to circumvent the inefficiency of direct parametric pumping of the YIG based exchange spin wave, it has been proposed instead to parametrically pump a metallic stripe coupled to the YIG film through dipolar interaction. A spin wave propagating in the YIG film could be amplified indirectly through “chiral resonant scattering (CRS)”^{17} with the metallic stripe. The metallic stripe is electrically modulated through a phenomenon named “voltage controlled magnetic anisotropy” (VCMA),^{18} which turns the stripe into a so-called “electric chiral magnonic resonator” (ECMR).^{19} However, due to the small spatial dimension of the proposed VCMA based ECMR, the amplification power of the device depends strongly on the oscillation phase of the incident spin wave, which is highly undesirable. In fact, the device may turn from a spin wave amplifier into an attenuator by simply misplacing the device for a distance equal to ¼ of the spin wave wavelength in the wave propagation path. This brings a very stringent requirement in the positional accuracy of devices in magnonic circuitry fabrication. In this article, inspired by the recent success of the aforementioned spin wave amplification by direct SOT pumping of a YIG waveguide,^{16} it is proposed to apply the same SOT pumping mechanism to the ECMR. The primary advantage of replacing the original VCMA pumping with SOT is that since in the latter case, the electrical input is a dc current (instead of a rf voltage as in the former case), the spin wave amplification no longer depends on the incident spin wave phase, which is highly favorable because of the earlier mentioned reasons. In this article, it is shown that when compared with the case of direct SOT pumping in the waveguide, the SOT pumped ECMR suffers substantial bandwidth loss, but obtains system stability while maintaining amplifier gain at the same time. The mean to maximize the ECMR bandwidth and the trade off with power consumption is discussed. Last, it is demonstrated by connecting a SOT ECMR with a VCMA ECMR operating in the passive mode, a magnonic neuron could be produced which is largely lossless in its “on” state. Such neuron forms the basis for neuromorphic computation where successive layers of neurons are cascaded together while the spin wave amplitude is maintained largely constant throughout all the layers.

## II. DEVICE DESIGN

Figure 1 illustrates the proposed SOT ECMR device. In our simplified two dimensional (2D) treatment, the entire structure in the x–z plane is supposed to extend to $ y=\xb1\u221e$. In the same manner as the VCMA ECMR,^{19} a spin wave incident from the left through the YIG film waveguide chiral resonant scatter with the CoFeB stripe through dipolar interaction. Frequency of the incident spin wave is selected to excite the “dark” mode in the CoFeB stripe where reflection of the spin wave back to the left in the YIG by the CoFeB resonator is virtually zero. After scattering, the spin wave continues its travel toward right in the YIG film, which will either be attenuated or amplified, depending on the current density **J** (toward negative x) in the Pt layer formed beneath and is in direct contact with the CoFeB resonator. The two ends of the Pt layer are deposited with Cr/Au electrodes to provide the necessary voltage bias for the in-plane current inside Pt underneath the CoFeB resonator. The MgO layer above the CoFeB resonator is to provide out-of-plane (z direction) magnetic anisotropy for the CoFeB material, which serves to enhance dipolar interaction with the YIG film and also to suppress the resonant frequency values of various excitation modes (including the aforementioned dark mode) in the CoFeB stripe. The Ta layer atop MgO is simply to provide physical protection. The global magnetic bias field **H**_{bias} is applied in the +y direction. Magnetic moment (**m**) in CoFeB and YIG film is aligned toward the −y and +y direction, respectively. This magnetic moment alignment configuration could be achieved by proper history of H_{bias} cycling and the difference in the magnetic reversal field between the YIG film and the CoFeB stripe.

In order to assess the performance of the SOT ECMR, we also consider a controlled model of direct SOT pumping in the YIG waveguide, illustrated as in Fig. 1(b). In this case, the Pt layer is deposited directly on the top of the YIG film. Electrodes (Cr/Au) formed on the two ends of the Pt film provide the in-plane current density **J** toward negative x. Both the **H**_{bias} field and the YIG magnetic moment (**m**) are aligned toward the +y direction. The spin wave propagating toward right in the YIG film will be amplified and the amplification value depends on **J**.

**m**in the model described in Figs. 1(a) and 1(b) is governed by the Landau–Lifshitz–Gilbert equation,

Parameters . | Symbols . | Values . | Units . |
---|---|---|---|

Gyromagnetic ratio | γ | 176 | rad GHz T^{−1} |

Saturation magnetization of YIG | M_{s,YIG} | 140 × 10^{3}^{20} | Am^{−1} |

Exchange constant of YIG | A_{ex, YIG} | 3.6 × 10^{−12}^{20} | Jm^{−1} |

Saturation magnetization of CoFeB | M_{s,CoFeB} | 1194 × 10^{3}^{21} | Am^{−1} |

Exchange constant of CoFeB | A_{ex, CoFeB} | 2.8 × 10^{−11}^{21} | Jm^{−1} |

Surface anisotropy of CoFeB in z direction | K_{s} | 1.39^{21} | mJ m^{−2} |

Gilbert damping factor of CoFeB | α_{CoFeB} | 0.01^{22} | … |

Gilbert damping factor of YIG | α_{YIG} | 2 × 10^{−4}^{20} | … |

Damp-like SOT efficiency for the Pt\CoFeB interface | θ_{DL} | 0.074^{23} | … |

Field-like SOT efficiency for the Pt\CoFeB interface | θ_{FL} | −0.012^{23} | … |

Damp-like SOT efficiency for the YIG\Pt interface | θ_{DL} | 0.026^{24} | … |

Field-like SOT efficiency for the YIG\Pt interface | θ_{FL} | 0.0015^{24} | … |

Electrical resistivity of CoFeB | ρ_{CoFeB} | 1700^{25} | Ω nm |

Electrical resistivity of Pt | ρ_{Pt} | 400^{26} | Ω nm |

Parameters . | Symbols . | Values . | Units . |
---|---|---|---|

Gyromagnetic ratio | γ | 176 | rad GHz T^{−1} |

Saturation magnetization of YIG | M_{s,YIG} | 140 × 10^{3}^{20} | Am^{−1} |

Exchange constant of YIG | A_{ex, YIG} | 3.6 × 10^{−12}^{20} | Jm^{−1} |

Saturation magnetization of CoFeB | M_{s,CoFeB} | 1194 × 10^{3}^{21} | Am^{−1} |

Exchange constant of CoFeB | A_{ex, CoFeB} | 2.8 × 10^{−11}^{21} | Jm^{−1} |

Surface anisotropy of CoFeB in z direction | K_{s} | 1.39^{21} | mJ m^{−2} |

Gilbert damping factor of CoFeB | α_{CoFeB} | 0.01^{22} | … |

Gilbert damping factor of YIG | α_{YIG} | 2 × 10^{−4}^{20} | … |

Damp-like SOT efficiency for the Pt\CoFeB interface | θ_{DL} | 0.074^{23} | … |

Field-like SOT efficiency for the Pt\CoFeB interface | θ_{FL} | −0.012^{23} | … |

Damp-like SOT efficiency for the YIG\Pt interface | θ_{DL} | 0.026^{24} | … |

Field-like SOT efficiency for the YIG\Pt interface | θ_{FL} | 0.0015^{24} | … |

Electrical resistivity of CoFeB | ρ_{CoFeB} | 1700^{25} | Ω nm |

Electrical resistivity of Pt | ρ_{Pt} | 400^{26} | Ω nm |

**H**

_{bias}), exchange interaction field (

**H**

_{exc}), and dipolar interaction field (

**H**

_{dip}) which are routine in micromagnetic simulation. On the other hand, $ H Oe$ is the Oersted field that arises due to the in-plane electric current in the Pt layer and its computation method is detailed in Appendix A. The last two terms in Eq. (2) ( $ H FL p FL$ and $ H DL p DL$) are the field-like and damp-like SOT effective field, respectively. They exist only inside the CoFeB resonator in the SOT ECMR model [Fig. 1(a)] and in the YIG film underneath Pt with in-plane current nearby in the direct waveguide pumping model [Fig. 1(b)]. Their magnitude is given by

^{26}

^{,}

*e*is the electron charge magnitude, $ M s$ is the saturation magnetization of the SOT pumped material (i.e., CoFeB in the SOT ECMR model and YIG in the direct waveguide pumping model), and J is the magnitude of the current density, while $ t m$ is the magnetic material thickness (equal

*h*in the SOT ECMR model and

*d*in the direct waveguide pumping model). $ \theta D L$ and $ \theta F L$ are the damp-like and field-like SOT efficiency given in Table I. On the other hand, the unit vector for the damp-like and field-like SOT fields is given by

^{27}

**m**, a prerequisite for spin wave amplification.

^{19}This dark mode attenuation maximum is downshifted to 12.76 GHz after J is increased to $9\xd7 10 11 A m \u2212 2$ with only the Oersted field included in the calculation (blue solid curve). If, instead, only the field-like SOT field is concerned, the dark mode attenuation maximum upshifted to 12.87 GHz (brown solid curve). The combined consequence at $ J=9\xd7 10 11 A m \u2212 2$ of the Oersted field and the field-like SOT field is displayed as the yellow solid curve, in which the dark mode attenuation maximum is compromised at 12.80 GHz. Summary of the attenuation maximum frequency (i.e., dark mode resonant frequency f

_{0,J}of the resonator) against J for these various situations of field inclusion is displayed in Fig. 2(b). For the combined effect of the Oersted field and the field-like SOT field, a linear relationship between f

_{0,J}and J is established as

After establishing a shift in the resonant frequency of the ECMR by the Oersted field and the field-like SOT field, we now include contribution from the damp-like SOT field to investigate the associated amplification effect. The transmission spectrum of the SOT ECMR for an increasing value of J is plotted in Fig. 2(d). The attenuation maximum at J = 0 Am^{−2} is located at 12.83 GHz, while the amplification maximum at $ J=8.5\xd7 10 11 A m \u2212 2$ is located at 12.80 GHz. This downshift behavior of the resonant frequency in the simulation agrees with the prediction of Eq. (5).

## III. COMPARISON OF SOT ECMR AND DIRECT WAVEGUIDE PUMPING

^{19}[Fig. 2(d)]. In this section, we investigate the subject using the cw excitation method (see Appendix A), which is numerically more brutal and trustworthy. In Fig. 3(a), the time evolutions of m

_{z}(x

_{1},t) and m

_{z}(x

_{2},t) that resulted from cw excitation at $f=\omega /( 2 \pi )=12.804 GHz$ at $ J=8\xd7 10 11 A m \u2212 2$ are displayed. The simulation is repeated for various values of J. The steady state of m

_{z}(x

_{2},t) attained slightly before t = 30 ns can be used to compute the magnitude and phase of S

_{21}against $ \alpha J$ [associated to J by Eq. (6)] and the results are displayed in Fig. 3(b) as blue and brown circles, respectively. An analytical model for the passive (unpumped) case of the ECMR has been developed in Ref. 32 and reiterated in Ref. 19. The analytical transmission coefficient for our SOT pumped ECMR is, therefore, given by

The values of the fitted parameters $ \Gamma R$, $ \Gamma 0$, $ \Omega 0$, and $ \Phi R$ (see below) are enlisted in Table II. According to Ref. 19, the magnitude and phase of the micromagnetic simulated transmission is related to the analytical one by $| S 21|=| \tau R , J|$ and $\u2220 S 21=\u2212( \u2220 \tau R , J + \Phi R)$. The resultant predictions of $| S 21|$ and $\u2220 S 21$ dependence on $ \alpha J$ through Eqs. (6) and (8) are displayed in Fig. 3(b) as blue and brown solid curves, which agree fairly well with the micromagnetic simulation data.

Parameters . | Unit . | SOT ECMR . | VCMA ECMR . |
---|---|---|---|

K_{s} | mJ m^{−2} | 1.390 | 1.414 |

Ω_{0} | GHz rad | 12.833 × 2π | 12.630 × 2π |

Γ_{0} | Ω_{0} | 0.0102 | 0.0101 |

Γ_{R} | Ω_{0} | 0.0082 | 0.0086 |

Γ_{L} | Ω_{0} | 0 | 0 |

$\u2220 \Delta R$ | Degree | $\u2212 91 \xb0$ | $\u2212 92 \xb0$ |

$\u2220 \Delta L$ | Degree | N/A | N/A |

Φ_{R} | Degree | $ 2 \xb0$ | $ 2 \xb0$ |

Parameters . | Unit . | SOT ECMR . | VCMA ECMR . |
---|---|---|---|

K_{s} | mJ m^{−2} | 1.390 | 1.414 |

Ω_{0} | GHz rad | 12.833 × 2π | 12.630 × 2π |

Γ_{0} | Ω_{0} | 0.0102 | 0.0101 |

Γ_{R} | Ω_{0} | 0.0082 | 0.0086 |

Γ_{L} | Ω_{0} | 0 | 0 |

$\u2220 \Delta R$ | Degree | $\u2212 91 \xb0$ | $\u2212 92 \xb0$ |

$\u2220 \Delta L$ | Degree | N/A | N/A |

Φ_{R} | Degree | $ 2 \xb0$ | $ 2 \xb0$ |

_{z}(x

_{1},t) and m

_{z}(x

_{2},t) under an aforementioned pulsed sinusoidal spin wave input (i.e., at the same excitation frequency $f=\omega /( 2 \pi )=12.804 GHz$) are displayed in Fig. 3(c). Unlike the case that utilizes a resonator that requires a transient period of nearly 20 ns to reach the steady state [Fig. 3(a), brown data from t = 10 to 30 ns], amplifying response at the output m

_{z}(x

_{2},t) by directly pumping the waveguide is instantaneous with respect to the input m

_{z}(x

_{1},t) [Fig. 3(c) sharp rise of brown data signal amplitude at around t = 4 ns], indicating that direct SOT pumping in the waveguide provides a much more broadband amplification ability than the SOT ECMR. Figure 3(d) summarizes the resultant magnitude and phase of S

_{21}calculated using time evolution data similar to those provided in Fig. 3(c) for various values of pumping strength $ \alpha J$, defined according to Eq. (7). The time delay of spin wave arrival between x

_{2}and x

_{1}(namely, Δt

_{21}) is 1.676 ns [see Fig. 3(c); time delay of the initial rise of signal amplitude between blue and brown data]. Theoretically, the magnitude of S

_{21}could be predicted using the following formula:

_{YIG}given in Table I). The phase of S

_{21}is fitted according to

_{21}against $ \alpha J$ through Eqs. (7) and (12)–(14) are plotted as blue and brown solid curves in Fig. 3(d), where the agreement with the micromagnetic simulations data (blue and brown circles) is satisfactory.

One could notice that after the main spin wave excitation terminates at t ≈ 12 ns, direct SOT pumped waveguide magnetization moments do not relax back to their ground state [Fig. 3(c)]. Instead, the main excitation left a tail of auto-oscillations that grows beyond the magnitude of the main excitation after t ≈ 30 ns. To generally address the behavior of these auto-oscillations, spin wave excitation at the transducer is turned off (i.e., set b_{rf} to 0 mT) and simulations similar to those displayed in Figs. 3(a) and 3(c) are repeated. The results for the SOT ECMR and the direct waveguide pumping model are displayed in Figs. 4(a) and 4(b), respectively. It is noticed that even without any input spin wave excitation from the transducer, there is a spontaneous occurrence of oscillation in the system that grow exponentially with time. If we plot the data in a logarithmic scale as in Figs. 4(c) and 4(d), it results in straight lines with tilted slopes, which enables one to extract their exponentials. The resultant extracted $| \Gamma tot , J|$ against $ \alpha J$ are plotted in Fig. 4(e) as blue circles and brown triangles. For the SOT ECMR, the simulation data (blue circles) agree pretty well with the analytical prediction from Eq. (9) (blue solid line). On the other hand, for the direct waveguide pumping model, it was found that α_{wg} has to be slightly upshifted from $ \alpha YIG(=2\xd7 10 \u2212 4)$ to 0.001 to get the best fit of analytical prediction from Eq. (13) (brown solid line) with the simulation data (brown triangles).

_{2}− x

_{1}in Fig. 7(b)] in the direct waveguide approach [Fig. 1(b)]. The huge reduction of length required to be pumped in the ECMR approach compared to the direct waveguide approach not only translates to smaller spatial footprint of the device but also huge reduction of power consumption by ohmic loss of the electric current that provides SOT pumping. However, this comes with the price of a reduced bandwidth, i.e., prolonged transient time of the ECMR method [around 20 ns, see Fig. 3(a)] compared to the direct waveguide pumping approach [less than 1 ns, see Fig. 3(c)]. But the ultimate advantage of the ECMR approach lies in the avoidance of auto-oscillations while obtaining gain at the same time. From Fig. 3(b), one can notice that for the ECMR, micromagnetic simulation states that gain start to occur at $ \alpha J>0.01$. Assume that $| \omega \u2212 \Omega 0 , J|\u226a \Gamma tot , J$; then Eq. (8) reduces to

Model . | SOT ECMR . | SOT ECMR . | Direct waveguide pumping . |
---|---|---|---|

Excitation method | Broadband pulse | CW | CW |

x_{0} | 20 μm | 1100 nm | N/A |

x_{1} | 15 μm | 700 nm | 700 nm |

x_{2} | 25 μm | 1500 nm | 3100 nm |

x_{3} | 10 μm | 400 nm | 400 nm |

x_{4} | 30 μm | 1800 nm | 3400 nm |

x_{total} | 40 μm | 2200 nm | 3800 nm |

x_{tdr} | 12.5 μm | 500 nm | 500 nm |

w_{tdr} | 5 μm | 30 nm | 30 nm |

d | 20 nm | 20 nm | 20 nm |

s | 12.5 nm | 12.5 nm | N/A |

h | 2.5 nm | 2.5 nm | N/A |

w | 50 nm | 50 nm | N/A |

s_{1} | 5 nm | 5 nm | 5 nm |

g | 100 nm | 100 nm | N/A |

Model . | SOT ECMR . | SOT ECMR . | Direct waveguide pumping . |
---|---|---|---|

Excitation method | Broadband pulse | CW | CW |

x_{0} | 20 μm | 1100 nm | N/A |

x_{1} | 15 μm | 700 nm | 700 nm |

x_{2} | 25 μm | 1500 nm | 3100 nm |

x_{3} | 10 μm | 400 nm | 400 nm |

x_{4} | 30 μm | 1800 nm | 3400 nm |

x_{total} | 40 μm | 2200 nm | 3800 nm |

x_{tdr} | 12.5 μm | 500 nm | 500 nm |

w_{tdr} | 5 μm | 30 nm | 30 nm |

d | 20 nm | 20 nm | 20 nm |

s | 12.5 nm | 12.5 nm | N/A |

h | 2.5 nm | 2.5 nm | N/A |

w | 50 nm | 50 nm | N/A |

s_{1} | 5 nm | 5 nm | 5 nm |

g | 100 nm | 100 nm | N/A |

One could observe from Eq. (15) that $| \tau R 0 , J|$ starts to exceed unity when $ \Gamma tot , J$ drops to below $ \Gamma R , J$ according to Eq. (9) as J increases, i.e., $ \alpha J \Omega 0 , J$ surpasses $ \Gamma 0 , J( = 0.0102 \Omega 0 , J)$, which agrees very well with our previous simulation results that gain start to occur when $ \alpha J>0.01$. On the other hand, auto-oscillations do not occur in the ECMR system until $ \alpha J>0.019$ as displayed in Fig. 4(e) (blue circles). At the same time, Fourier transform of Eq. (8) into time domain would lead to a transient term with an exponential factor in the form of $ exp( \u2212 \Gamma tot , J t)$. Therefore, the theory predicts the system to become unstable only when $ \Gamma tot , J$ according to Eq. (9) turns negative, i.e., $ \alpha J \Omega 0 , J> \Gamma R , J+ \Gamma 0 , J=0.0184 \Omega 0 , J$, which agrees pretty well with the micromagnetic simulated values of $ \alpha J>0.019$ mentioned above.

Now, let us shift our attention to the direct waveguide pumping approach. Non-zero $| \Gamma tot , J|$ data points occur almost immediately after $ \alpha J$ deviates from zero [Fig. 4(e), brown triangles] which agrees fairly well with the prediction from Eq. (13) [Fig. 4(e) , brown straight line]. If one compares these auto-oscillations data with |S_{21}| against $ \alpha J$ in Fig. 3(d), one could infer that the occurrence of gain in the direct waveguide pumping approach is almost always accompanied with system instability. Therefore, the amplification process becomes a competition between gain and auto-oscillations. The pumping process has to be performed in the pulse mode,^{16} i.e., turning off the pump power as long as the signal gets enough amplification to prevent auto-oscillation outgrow the signal. It is also noted that the signal itself sow seeds for further auto-oscillations [see Fig. 3(c)] which can accumulate over time and ultimately contaminate the magnonic circuitry system entirely. Here, we see that the primary advantage of the ECMR approach over the direct waveguide pumping approach is to provide a range of $ \alpha J \Omega 0 , J$ between $ \Gamma 0 , J$ and $ \Gamma R , J+ \Gamma 0 , J$, where gain exists but auto-oscillations are avoided.

Proof of Eq. (17) is referred to Appendix B. To repeat the message: given that ECMR suffers huge loss in response speed when compared with the direct waveguide pumping approach, the only way to remedy this problem while maintaining gain at the same level is to increase waveguide-resonator coupling strength and SOT pumping strength. This would be a complicate engineer trade-off between device speed and power consumption. In this manner, the ECMR amplifier here and even the field of magnonics share the same fate with the whole spintronics community in material research for the most power efficient method of electric current to spin conversion,^{28} which will be further discussed in Sec. V.

## IV. NON-LINEAR BEHAVIOR

_{rf}) are displayed in Fig. 5(a). The observed frequency downshift of the attenuation maximum against increasing b

_{rf}is consistent with the previous reports.

^{29}The data should be fitted analytically with Eq. (8), where $ \Omega 0 , J$ is replaced by $( 1 \u2212 \xi ) \Omega 0 , J$, i.e.,

The origin of Eq. (19) is referred to in Appendix C. In Eq. (19), the YIG waveguide spin wave phase velocity *v* is set to 0.96 *μ*m/ns (determined from micromagnetic simulation). In Ref. 19, $\eta $ is described as a length dummy variable. However, logically, it is the dimension of the scatterer along the spin wave propagation path, and, therefore, it is natural to equate it with the width of the CoFeB stripe w (i.e., $\eta =w=50 nm$). In Ref. 19, $| \psi 0|$ is the amplitude of the incident spin wave in the analytical model; therefore, it is equivalent to |M_{z}/M_{s}| in the micromagnetic simulation. Finally, $ \lambda J$ is a fitting parameter, which according to analytical derivation (see Appendix C) should equal 0.5 under the assumption that magnetization precession of the resonator is completely circular polarized. It is found that best fits to micromagnetic simulations data in Fig. 5(a) are obtained by setting $ \lambda J=0.55$, which is not far from the naïve theoretical value of 0.5. It should be emphasized that in case of the existence of multiple roots of $\xi $ in Eq. (19), the root with the smallest value has always been chosen as the accepted root value.

After inspecting the ECMR non-linear behavior in its passive mode, we now turn on SOT pumping by setting $ J=8\xd7 10 11 A m \u2212 2$. The resultant data for various spin wave excitation strengths b_{rf} are shown in Fig. 5(b). It is noticed that since in the pumped case, the resonator oscillation amplitude is amplified, non-linearity occurs much more severe than in the passive case [Fig. 5(a)], manifested in the form of a larger frequency downshift of the resonance peak for the same b_{rf} [Fig. 5(b)]. This also comes with the suppression of the amplifying power of peak value at near to 3.0 for b_{rf} = 0.1 mT [Fig. 5(b), blue circles and solid curve] to 2.0 for b_{rf} = 15 mT [Fig. 5(b), brown triangles and solid curve], and further to 1.6 and 1.5 for b_{rf} = 25 mT [Fig. 5(b), yellow inverted triangles and solid curve] and b_{rf} = 35 mT [Fig. 5(b), purple squares and solid curve]. It should be emphasized that the solid curves are predictions from Eqs. (18) and (19) which work almost equally well in the pumped case as in the passive case, simply by relying on the $ \Gamma tot , J$ dependence on $ \alpha J$ according to Eq. (9), although the theory predicts a slightly earlier transition to low transmission state than the simulation data as frequency decrease in the case of b_{rf} = 25 and 35 mT [Fig. 5(b), solid yellow and purple curves vs yellow inverted triangles and purple squares].

Here, we see again the puzzle that has already been mentioned in Ref. 19. The “neuron” presented in Refs. 3 and 29 is lossy in the “on” state (transmission $\u22450.5$). The mission of the ECMR amplifier is to construct a lossless neuron with transmission > 1 when firing. Suppose we try to accomplish this task by putting a pumped ECMR in series with a passive ECMR. Suppose, we are operating them on resonance (f_{cw} = 12.804 GHz). By the time when non-linearity kicks in which bring the transmission of the passive ECMR to 0.5 at f_{cw} = 12.804 GHz [Fig. 5(a), purple squares and solid curve], which corresponds to b_{rf} = 35 mT, then the amplifying power of the pumped ECMR would be greatly compromised. [The data for b_{rf} = 15, 25 and 35 mT in Fig. 5(b) at f_{cw} = 12.804 GHz are all below 1.5, reduced from the linear case of 3.0 at b_{rf} = 0.1 mT in the same figure.] The amplifying power has to be greater than 2.0 to compensate the attenuation of the passive ECMR with transmission equal 0.5. This presents a great obstacle to construct our lossless neuron.

Here, we present a solution by operating the pumped ECMR in an off-resonance manner. In Fig. 5(c), the |S_{21}| against spin wave amplitude |M_{z}/M_{s}| for the pumped ECMR with f_{cw} = 12.804 GHz (on resonance) are presented as blue circles and a solid curve. The pumping current density J is slightly increased to $9\xd7 10 11 A m \u2212 2$. The amplifier power deteriorates quickly with increasing |M_{z}/M_{s}|, same as in Ref. 19. Now, we reduce the excitation frequency f_{cw} to 12.630 GHz and repeat the calculations. The results are presented as brown triangles and a solid curve in Fig. 5(c). At this frequency, |S_{21}| starts off small (at 1.5) at low values of |M_{z}/M_{s}|, but jump abruptly to 3.0 at |M_{z}/M_{s}| = 0.013, and then slowly deteriorates with further increase in |M_{z}/M_{s}|. Since the device natural resonance frequency is at $ \Omega 0 , J/( 2 \pi )=12.800 GHz$, according to Eq. (5), the device is considered as operating at off resonance. In the same figure, we present the transmission data of a passive ECMR at f_{cw} = 12.630 GHz (yellow inverted triangles and a solid curve) with the natural resonance frequency of the ECMR $ \Omega 0/( 2 \pi )$ also reduced to 12.630 GHz (which could be done easily through the VCMA phenomenon^{19}). Now, since the device corresponds to the brown triangles and the yellow inverted triangles operate at the same f_{cw} ( = 12.630 GHz), it is legitimate to place them in series [as in the inset of Fig. 5(d), namely, a “ECMR neuron”] and look at their combined transmission. The results are presented in Fig. 5(d). |S_{21}| starts off at a low value around 0.15 and remains roughly at this same level until |M_{z}/M_{s}| gets near to 0.013. At |M_{z}/M_{s}| = 0.013, |S_{21}| jumps abruptly to 1.5 and gradually falls back to 1.0 at higher values of |M_{z}/M_{s}|. The observed dependence of |S_{21}| on |M_{z}/M_{s}| is highly desirable for a neuron as (1) it has a very high on-off ratio (jumping from lower than 0.2 in the “off” state to higher than 1.0 in the “on” state), (2) when |M_{z}/M_{s}| increases, the transition from the “off” state to the “on” state is abrupt (satisfying the desired threshold firing behavior of a neuron), and (3) the transmission in the “on” state is lossless and even with a small gain at some |M_{z}/M_{s}| values.

## V. DISCUSSIONS

To address the possible temperature rise of the SOT ECMR device, a 3D Comsol heat transfer model has been developed. While the x–z cross section of the device remains the same as in Fig. 1(a), the YIG waveguide with the ECMR structure on the top is truncated to have width equal to w_{wg} = 500 nm in the y direction (see Fig. 6 inset) instead of extending to $ y=\xb1\u221e$. The Pt and CoFeB blocks in the Comsol model are treated as heat sources with power volume density equal to $ \rho local J local 2$, where the local resistivity $ \rho local$ and current density J_{local} are detailed in Appendix A. The model is assumed to be at room temperature at J = 0. The simulation results state that the device measured at the body center of the Pt layer (Fig. 6 inset, black dot) rise to slightly above $50 \xb0 C$ at $ J=9\xd7 10 11 A m \u2212 2$. This indicates that at the highest current density utilized by the SOT ECMR in this article, the device will get warm, but will not heat up to a temperature that leads to structural damage or micromagnetic failure.

Material . | Thermal conductivity (W m^{−1} K^{−1})
. | Mass density (kg m^{−3})
. | Heat capacity (J kg^{−1} K^{−1})
. |
---|---|---|---|

GGG | 7.4 | 7100 | 390 |

YIG | 7.4 | 5110 | 590 |

SiO_{2} | 1.3 | 2650 | 680 |

Pt | 71.6 | 21450 | 125 |

CoFeB | 69 | 7800 | 418 |

Material . | Thermal conductivity (W m^{−1} K^{−1})
. | Mass density (kg m^{−3})
. | Heat capacity (J kg^{−1} K^{−1})
. |
---|---|---|---|

GGG | 7.4 | 7100 | 390 |

YIG | 7.4 | 5110 | 590 |

SiO_{2} | 1.3 | 2650 | 680 |

Pt | 71.6 | 21450 | 125 |

CoFeB | 69 | 7800 | 418 |

In order to analyze the energy economy of the SOT ECMR, one has to start with the Zeeman energy density $U=\u2212 M\u22c5 B$. The spin wave energy density with bias field B_{y} along the y direction is, therefore, $ U sw=\u2212 M y B y\u2212( \u2212 M s B y)=\u2212 M s B y 1 \u2212 ( M z / M s ) 2$ $\u2212( \u2212 M s B y)=( 1 / 2) M s B y ( M z / M s ) 2=( 1 / 2) M s( \omega / \gamma ) ( M z / M s ) 2$. The power flux through a YIG waveguide with width *w _{wg}* and thickness

*d*would, therefore, be given as $ P sw= U sw w wg v swd$, where $ v sw$ is the spin wave group velocity. Because of the ECMR bandwidth limitation, the duration of spin wave pulse $ \Delta t sw$ has to be as long as 100 ns.

^{19}The energy of a single spin wave pulse operating in a SOT ECMR system is, therefore, given as $ E s w= P sw \Delta t sw$. Plugging in the following values: $ M s=140\xd7 10 3 A m \u2212 1$, $\omega =2\pi \xd712.630 rad GHz$, $| M z / M s|=0.01$ [assuming the spin wave to be operating at an amplitude slightly below the firing threshold of the ECMR neuron according to Fig. 5(d)], $ w wg=100 nm$, $ v sw=0.96\mu m / ns$, and d = 20 nm, one can determine $ E s w=6.1\xd7 10 \u2212 19 J$. Assuming that using amplitude modulation and the noise floor allows the amplitude of this spin wave to be discretized to represent 32 distinctive levels. This would be equivalent to five (32 = 2

^{5}) bits of information in a CMOS system. According to Ref. 30, energy consumption of the CMOS system with 7 nm technology is 35.3 aJ per bit. Therefore, the energy to represent five bits of information in CMOS would be $ E CMOS=35.3\xd7 10 \u2212 18\xd75=1.76\xd7 10 \u2212 16 J$, which is almost 300 folds the value of $ E s w$. Here, one would appreciate that the potential of magnonics (assuming adoption of the ECMR technology) is to reduce the energy consumption of computation by two orders of magnitude compared with conventional computers. The obtained values of $ E s w$ and $ E CMOS$ are now summarized in Table IV.

Building block . | Energy to process equivalent of five bits of digital data (J) . | Time duration to process five bits of data . |
---|---|---|

Thermal fluctuation at room temperature (k_{B}T) | 4.1 × 10^{−21} | N/A |

CMOS | 1.76 × 10^{−16} | 0.33 ns |

SOT ECMR (spin wave only) | 6.1 × 10^{−19} | 100 ns |

SOT ECMR (Ohmic loss included) | 8.1 × 10^{−13} | 100 ns |

Direct waveguide pumping | 5.4 × 10^{−12} | 5 ns |

VCMA ECMR | 5.1 × 10^{−15} | 100 ns |

Building block . | Energy to process equivalent of five bits of digital data (J) . | Time duration to process five bits of data . |
---|---|---|

Thermal fluctuation at room temperature (k_{B}T) | 4.1 × 10^{−21} | N/A |

CMOS | 1.76 × 10^{−16} | 0.33 ns |

SOT ECMR (spin wave only) | 6.1 × 10^{−19} | 100 ns |

SOT ECMR (Ohmic loss included) | 8.1 × 10^{−13} | 100 ns |

Direct waveguide pumping | 5.4 × 10^{−12} | 5 ns |

VCMA ECMR | 5.1 × 10^{−15} | 100 ns |

Of course, the naïve conclusion arrived above has ignored Ohmic loss in the SOT ECMR. The energy consumption for the SOT ECMR to process the aforementioned 100 ns pulse would be given by $ E SOT= P SOT \Delta t sw$, where $ P SOT= \rho Pt J 2 w wgw s 1$. Plug in the values: $ \rho Pt=400\Omega nm$, $ J=9\xd7 10 11 A m \u2212 2$, $ w wg=100 nm$, $w=50 nm$, and $ s 1=5 nm$, one would obtain that $ E SOT=8.1\xd7 10 \u2212 13 J$. This is six orders of magnitude higher than $ E s w$. The situation of the direct waveguide pumping approach would be even worse, where the energy consumption is $ E direct= P direct \Delta t sw$ with $ P direct= \rho Pt J 2 w wgL s 1$. Because direct waveguide pumping is broadband in frequency, wave pulse duration $ \Delta t sw$ could be shortened to 5 ns. Plugging in $ J=1.5\xd7 10 12 A m \u2212 2$, $L=2.4\mu m$, one would obtain $ E direct=5.4\xd7 10 \u2212 12 J$. We also analyze the VCMA amplifier in Ref. 19 as follows: according to Ref. 21, an electric field change equal 0.667 V/nm leads to a change in the surface anisotropy ΔK_{s} equal to 0.04 mJ m^{−2}. The VCMA amplifier in Ref. 19 that requires ΔK_{s} = 0.045 mJ m^{−2} to operate would, therefore, require an ac electric field with a peak amplitude that equals 0.750 V/nm. Assume the thickness of the dielectric spacer between the CoFeB layer and the top electrode to be s_{2}; then, capacitance of the VCMA amplifier is given as $C= \u03f5 r \u03f5 0w w wg/ s 2$, where $ \u03f5 0$ is the dielectric constant and $ \u03f5 r$ is the relative permittivity of the spacer. Peak current of the device is then given by $I=\omega C V 0$, where $ V 0$ is the peak applied voltage. Therefore, the peak current density is $ J=I/ w wg/( h + s 1)=\omega /( h + s 1)\xd7( V 0 / s 2)\xd7 \u03f5 r \u03f5 0w$. Plugging in $\omega =2\pi \xd712.83 rad GHz$, $h+ s 1=10 nm$, $ V 0/ s 2=0.750 V / nm$, $ \u03f5 r=10$ and $w=50 nm$, one would find $ J=2.68\xd7 10 10 A m \u2212 2$, which is much lower than the SOT ECMR $(9\xd7 10 11 A m \u2212 2)$. Subsequently, the energy for the VCMA amplifier to process a 100 ns spin wave pulse would be $ E VCMA= P VCMA \Delta t sw$, where $ P VCMA= \rho Ta J 2 w wgw s 1$. Plugging in values: $ \rho Ta=1900 \Omega nm$ and $ s 1=7.5 nm$ give $ E VCMA=5.1\xd7 10 \u2212 15 J$. The obtained values of $ E SOT$, $ E direct$, and $ E VCMA$ are now enlisted in Table IV.

The high value of $ E SOT(\u223c 10 \u2212 13 J )$ is the result of choice of Pt as the SOT interaction material, which is pretty much at the bottom of material hierarchy in terms of electron to spin conversion efficiency. Reference 28 gives a very good review on the recent advance in search of alternative materials that would lower energy consumption of a SOT device. In particular, by replacing the heavy metal (i.e., Pt in this article) with a topological insulator, power consumption could be lowered for two orders of magnitude.^{31} This would bring SOT ECMR energy economy to be in par with the VCMA version ( $\u223c 10 \u2212 15 J$, see Table IV). Lowering the energy consumption for another order of magnitude would make the ECMR based magnonics competitive with CMOS $(\u223c 10 \u2212 16 J )$. This has not included yet the benefit of architecture simplification in implementing neuromorphic computation by magnonics for being capable to deviate away from the conventional von Neumann architecture because of the wave nature of the spin wave. As explained above, the theoretical lower limit of ECMR magnonics energy consumption could be 300 times smaller than CMOS, therefore there is still a long way to tap into the full potential of spin wave technology.

## VI. CONCLUSIONS

In conclusion, we have presented a new version of ECMR amplifier that relies on SOT rather than VCMA parametric pumping. Unlike the VCMA version, new SOT based amplifier transmission is independent of the incident spin wave oscillation phase, which is highly desirable. It has been explained that the advantage of applying SOT pumping to an ECMR over direct pumping the waveguide is the avoidance of auto-oscillations while maintaining gain. This comes with the price of reduced bandwidth (prolonged transient time). The relation of bandwidth, gain, resonator-waveguide coupling strength, and pumping strength has been clarified. The currently presented ECMR design has potential to improve the bandwidth three times (reducing transient time to 6 ns) by reducing the waveguide-resonator spacing s and increase the pumping strength correspondingly. At last, by placing a SOT pumped ECMR operating in the off-resonance mode in series with a passive ECMR under the influence of VCMA, one would be able to construct an “ECMR neuron” with ideal neuronic behavior: a high on-off ratio, true threshold behavior, and lossless transmission in the firing mode. The proposed ECMR neuron, together with the VCMA synapse presented in Ref. 19 will form the basis of a neural network composed of ECMRs on top of YIG waveguides.

## ACKNOWLEDGMENTS

The research leading to these results has received funding from the EPSRC of the UK (Project Nos. EP/L019876/1 and EP/T016574/1) and UK Research and Innovation (UKRI) under the UK government's Horizon Europe funding guarantee (Grant No. 10039217) as part of the Horizon Europe (HORIZON-CL4-2021-DIGITAL-EMERGING-01) under Grant Agreement No. 101070347.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Yat-Yin Au:** Conceptualization (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: MICROMAGNETIC SIMULATION DETAILS

^{3}in the x, y, and z directions, respectively. Both models are set to be one-cell wide in the y direction, with periodic boundary conditions employed in both the x and y directions. A global static bias magnetic field is applied in the +y direction with H

_{bias}= 50 Oe (i.e., B

_{bias}= 5 mT). The initial magnetizations in the YIG film waveguide [in the model of both Figs. 7(a) and 7(b)] and in the CoFeB stripe resonator [in Fig. 7(a)'s model only] are oriented toward the +y and −y direction, respectively, and the system is relaxed for 50 ns to obtain the ground state. Subsequent dynamic simulations involving the scattering process are done with the obtained ground state as the starting magnetization configuration. The material parameters and device dimensions are given in Tables I and V. The CoFeB film is supposed to be in contact with a MgO layer such that a natural surface anisotropy K

_{s}= 1.39 mJ m

^{−2}in the z direction has been introduced. The Gilbert damping factor in the waveguide is set to $2\xd7 10 \u2212 4$. To prevent spin wave reflection, ramped damping regions at the two ends of the waveguide are set up. Details of the variation of the Gilbert damping factor against x in the damped region can be found in Ref. 19. SOT pumping is applied to the resonator only in the model of Fig. 7(a) and directly to the waveguide in the model of Fig. 7(b). SOT pumping by the damp-like term is represented by introducing an additional magnetic field term into the Mumax3 software given by $ B D L= \mu 0 H D L m\xd7 p D L$, where $ m$ is the local magnetic moment unit vector, $ H D L$ is the damp-like SOT effective field magnitude [given by Eq. (3)], and $ p DL$ is the injected damp-like electron spin polarization unit vector [given by Eq. (4)]. The SOT field-like term is represented by introducing an additional magnetic field term into the Mumax3 software given by $ B F L= \mu 0 H F L p F L$. Again, $ H F L$ and $ p F L$ are given by Eqs. (3) and (4), respectively. To compute the Oersted field associated with the in-plane electric current, it is assumed that current density in the Pt layer [the “SO layer” region in Figs. 7(a) and 7(b)] is spatially uniformly given by −J

**x**. Subsequently, in the SOT ECMR model [Fig. 7(a)], the current density in the CoFeB resonator and the Pt layer lead region are given as −J(ρ

_{Pt}/ρ

_{CoFeB})

**x**and −J[1 + (hρ

_{Pt}/s

_{1}/ρ

_{CoFeB})]

**x**, respectively (ρ

_{Pt}and ρ

_{CoFeB}are the electrical resistivity given in Table I). It is derived that a sheet of electric current with density J

_{x}

**x**which begins at x = x

_{start}, ends at x = x

_{end}, extends to $ y=\xb1\u221e$

_{,}and centered at z with thickness Δz would produce a magnetic field at position (x′, y′, and z′) given as

**y**is the unit vector in the y direction. Finally, the total Oersted field at any position in the simulation could be computed by summing up the field contribution from all the current sheet layers existing in the model with aforementioned current density.

To launch spin waves, an excitation magnetic field pointing toward the z direction confined in the transducer region inside the YIG film is deployed. For the broadband pulse excitation method, the spatial and temporal dependence of the excitation field inside the transducer region is referred to Ref. 19. The same reference also details the formula to obtain broadband excitation generated frequency spectrum of the scattering parameter S_{21}. For the cw excitation method, the excitation field takes a temporal pulsed sinusoidal form $ b rf sin( 2 \pi f c w t) rect( ( t \u2212 0.5 ( t end + t start ) ) / ( t end \u2212 t start ))$ and is spatially uniform in the transducer region. The time (*t*) dependence of the z component of reduced magnetization (M_{z}/M_{s}) dynamics at x = x_{1} and x_{2} are recorded and averaged across the thickness of the YIG film and labeled as m_{z}(x_{1},t) and m_{z}(x_{2},t), respectively. Reference magnetization dynamics m_{z,ref}(x_{1},t) and m_{z,ref}(x_{2},t) are produced similarly from the direct waveguide pumping model with the current turned off (i.e., J = 0). Regarding the SOT ECMR model, magnitude and phase of the spin wave transmission can be calculated by $| S 21|= mag{ m z , ( x 2 , t )}/ mag{ m z , ref ( x 2 , t )}$ and $\u2220 S 21= phase{ m z ( x 2 , t )}\u2212 phase{ m z , ref ( x 2 , t )}$ after $ m z ,( x 2 , t)$ and $ m z , ref( x 2 , t)$ have attained the steady state, where mag{} and phase{} denote amplitude and phase values of a temporal sinusoidal fit to magnetization time dependence. In this article, CRS with the “dark mode” excited in the resonator is reflectionless;^{19,32} therefore, S_{11} is not considered (formulas for S_{11} in the general case where reflection is non-zero can be found in Ref. 19).

In the SOT ECMR model, the way we define S_{21} in cw excitation is designed to eliminate any contribution from the waveguide and retain contribution only from the resonator. On the other hand, there is no resonator in the direct waveguide pumping model and amplification of the spin wave is solely contributed by the waveguide, where there is an ambiguity regarding which particular section of the waveguide we are concerning with. In order to specify that we are interested in the waveguide section between x = x_{1} and x_{2}, the S_{21} definition is modified as $| S 21|$ $= mag{ m z , ( x 2 , t )}/ mag{ m z , ( x 1 , t )}$ and $\u2220 S 21= phase{ m z ( x 2 , t )}\u2212 phase{ m z , ( x 1 , t )}$.

### APPENDIX B: RELATION OF THE BANDWIDTH TO RADIATIVE LINEWIDTH, GAIN, AND PUMPING STRENGTH OF ECMR

### APPENDIX C: NON-LINEAR ANALYTICAL MODEL FOR THE ECMR

If one compare (C4) with Eq. (2) in Ref. 19 with $\u03f5=0$, one would conclude that the effect of non-linearity is simply to replace $ \Omega 0$ with $ \Omega 0 , J( 1 \u2212 \lambda J | \phi | 2)$. In this article, $ H 0 , J$ includes contribution from $ H bias$, $ H Oe$, and $ H FL$, where their magnitudes are much smaller than $N M s$. Therefore, one would expect $ \lambda J$ to be near to 0.5. In practice, we found that $ \lambda J=0.55$ gives best fit to the simulation data.

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