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.

Due to the opportunities of low energy consumption,1 strong non-linearity,2,3 and spatial miniaturization4–6 provided by spin waves, magnonics7,8 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.

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 = ± . 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 Hbias 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 Hbias cycling and the difference in the magnetic reversal field between the YIG film and the CoFeB stripe.

FIG. 1.

Schematic illustration of (a) the SOT ECMR design and (b) the direct waveguide SOT pumping set up. In these 2D models of (a) and (b), the structures are supposed to extend to y = ± . Dimension parameters details: d = 20 nm, w = 50 nm, h = 2.5 nm, s = 12.5 nm, s1 = 5 nm, g = 100 nm, and L = 2400 nm.

FIG. 1.

Schematic illustration of (a) the SOT ECMR design and (b) the direct waveguide SOT pumping set up. In these 2D models of (a) and (b), the structures are supposed to extend to y = ± . Dimension parameters details: d = 20 nm, w = 50 nm, h = 2.5 nm, s = 12.5 nm, s1 = 5 nm, g = 100 nm, and L = 2400 nm.

Close modal

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 Hbias 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.

Micromagnetic evolution of the magnetic moment m in the model described in Figs. 1(a) and 1(b) is governed by the Landau–Lifshitz–Gilbert equation,
m ˙ = μ 0 γ m × H + α m × m ˙ ,
(1)
where μ 0 is the permeability constant, γ is the gyromagnetic ratio (see Table I), α is the Gilbert damping factor, and
H = H eff + H Oe + H FL p FL + H DL m × p DL .
(2)
TABLE I.

Material parameters used in the SOT ECMR model [Fig. 1(a)] and the direct waveguide pumping model [Fig. 1(b)]. The source in literature studies for the parameter values used are enlisted beside the values themselves.

ParametersSymbolsValuesUnits
Gyromagnetic ratio γ 176 rad GHz T−1 
Saturation magnetization of YIG Ms,YIG 140 × 10320  Am−1 
Exchange constant of YIG Aex, YIG 3.6 × 10−1220  Jm−1 
Saturation magnetization of CoFeB Ms,CoFeB 1194 × 10321  Am−1 
Exchange constant of CoFeB Aex, CoFeB 2.8 × 10−1121  Jm−1 
Surface anisotropy of CoFeB in z direction Ks 1.3921  mJ m−2 
Gilbert damping factor of CoFeB αCoFeB 0.0122  … 
Gilbert damping factor of YIG αYIG 2 × 10−420  … 
Damp-like SOT efficiency for the Pt\CoFeB interface θDL 0.07423  … 
Field-like SOT efficiency for the Pt\CoFeB interface θFL −0.01223  … 
Damp-like SOT efficiency for the YIG\Pt interface θDL 0.02624  … 
Field-like SOT efficiency for the YIG\Pt interface θFL 0.001524  … 
Electrical resistivity of CoFeB ρCoFeB 170025  Ω nm 
Electrical resistivity of Pt ρPt 40026  Ω nm 
ParametersSymbolsValuesUnits
Gyromagnetic ratio γ 176 rad GHz T−1 
Saturation magnetization of YIG Ms,YIG 140 × 10320  Am−1 
Exchange constant of YIG Aex, YIG 3.6 × 10−1220  Jm−1 
Saturation magnetization of CoFeB Ms,CoFeB 1194 × 10321  Am−1 
Exchange constant of CoFeB Aex, CoFeB 2.8 × 10−1121  Jm−1 
Surface anisotropy of CoFeB in z direction Ks 1.3921  mJ m−2 
Gilbert damping factor of CoFeB αCoFeB 0.0122  … 
Gilbert damping factor of YIG αYIG 2 × 10−420  … 
Damp-like SOT efficiency for the Pt\CoFeB interface θDL 0.07423  … 
Field-like SOT efficiency for the Pt\CoFeB interface θFL −0.01223  … 
Damp-like SOT efficiency for the YIG\Pt interface θDL 0.02624  … 
Field-like SOT efficiency for the YIG\Pt interface θFL 0.001524  … 
Electrical resistivity of CoFeB ρCoFeB 170025  Ω nm 
Electrical resistivity of Pt ρPt 40026  Ω nm 
In Eq. (2), H eff includes the global bias field (Hbias), exchange interaction field (Hexc), and dipolar interaction field (Hdip) 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 by26,
μ 0 H D L ( F L ) = | θ D L ( F L ) J 2 e M s t m | ,
(3)
where is the Planck constant, 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). θ D L and θ 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 by27 
p D L ( F L ) = sign ( θ D L ( F L ) ) j × n ,
(4)
where n is the normal unit vector at the Pt/CoFeB interface [Fig. 1(a)] and the YIG/Pt interface [Fig. 1(b)]. Also, j is the unit vector for the current density flow vector (i.e., J = J j), set to be in the −x direction for both the SOT ECMR and the direct waveguide pumping model [Figs. 1(a) and 1(b)]. Since θ D L is positive at both the Pt/CoFeB and YIG/Pt interface (see Table I), the damp-like field p D L will point toward +y and −y in the SOT ECMR and the direct waveguide model, respectively, and both will point opposite to the local magnetic moment m, a prerequisite for spin wave amplification.
Before we can analyze the spin wave amplification behavior of the SOT ECMR, one has to deal with the shift of the ECMR resonant frequency due to the Oersted field and the field-like SOT effective field. Figure 2(a) displays the results of the transmission spectrum for the case of J = 0 (black dotted line), which displays an attenuation maximum at around 12.83 GHz corresponding to the resonator “dark mode” frequency.19 This dark mode attenuation maximum is downshifted to 12.76 GHz after J is increased to 9 × 10 11 A m 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 × 10 11 A m 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 f0,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 f0,J and J is established as
f 0 , J = C J f J + f 0 ,
(5)
where C J f = 3.6666 × 10 5 Hz / ( A m 2 ) and f 0 = Ω 0 / ( 2 π ) = 12.833 × 10 9 Hz, plotted as the yellow solid straight line in Fig. 2(b). Here, we define the effective anti-damping factor for the SOT ECMR as
α J = γ μ 0 H D L Ω 0 , J ,
(6)
where Ω 0 , J = 2 π f 0 , J. For the direct waveguide pumping model, resonant frequency in the YIG film depends on the wavevector of the spin wave excited, i.e., there is no particular preferred resonant frequency from the point of view of the YIG film. The effective anti-damping factor in this case is subsequently defined as
α J = γ μ 0 H D L ω ,
(7)
where ω is the frequency of the spin wave in the waveguide. In Fig. 2(c), we plot the dependence of α J on J according to Eqs. (6) and (7), where the ω in Eq. (7) is set to be 2 π × 12.804 rad GHz, which is the resonant angular frequency of the ECMR at J = 8 × 10 11 A m 2 according to Eq. (5).
FIG. 2.

(a) Frequency spectra of transmission coefficient magnitude obtained from the broadband pulse excitation method19 for the case of J = 0 Am2 (black dotted curve), J = 9 × 10 11 A m 2 with only the Oersted field included (blue solid curve), J = 9 × 10 11 A m 2 with only the field-like SOT effective field included (brown solid curve) and, J = 9 × 10 11 A m 2 with both field-like SOT field and Oersted field included but not the damp-like SOT field (yellow solid curve). (b) Frequency of the attenuation maximum for various situations in (a) against current density J. The yellow solid straight line represents the best linear fit [Eq. (5)] to the case of the combined effect from the fields like SOT and the Oersted field. (c) Effective anti-damping factor αJ against current density J according to the definition in Eqs. (6) and (7) where for the case of direct waveguide pumping [Eq. (7)], spin wave frequency ω / ( 2 π ) is set at 12.804 GHz. (d) Magnitude of the transmission coefficient against frequency of the SOT ECMR calculated by the broadband pulse excitation method for J = 0 Am2 (blue solid curve), J = 3 × 10 11 A m 2 (brown solid curve), J = 6 × 10 11 A m 2 (yellow solid curve), and J = 8.5 × 10 11 A m 2 (purple solid curve). Other than the Oersted field and the field-like SOT field, contribution from the damp-like SOT field is also included.

FIG. 2.

(a) Frequency spectra of transmission coefficient magnitude obtained from the broadband pulse excitation method19 for the case of J = 0 Am2 (black dotted curve), J = 9 × 10 11 A m 2 with only the Oersted field included (blue solid curve), J = 9 × 10 11 A m 2 with only the field-like SOT effective field included (brown solid curve) and, J = 9 × 10 11 A m 2 with both field-like SOT field and Oersted field included but not the damp-like SOT field (yellow solid curve). (b) Frequency of the attenuation maximum for various situations in (a) against current density J. The yellow solid straight line represents the best linear fit [Eq. (5)] to the case of the combined effect from the fields like SOT and the Oersted field. (c) Effective anti-damping factor αJ against current density J according to the definition in Eqs. (6) and (7) where for the case of direct waveguide pumping [Eq. (7)], spin wave frequency ω / ( 2 π ) is set at 12.804 GHz. (d) Magnitude of the transmission coefficient against frequency of the SOT ECMR calculated by the broadband pulse excitation method for J = 0 Am2 (blue solid curve), J = 3 × 10 11 A m 2 (brown solid curve), J = 6 × 10 11 A m 2 (yellow solid curve), and J = 8.5 × 10 11 A m 2 (purple solid curve). Other than the Oersted field and the field-like SOT field, contribution from the damp-like SOT field is also included.

Close modal

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 × 10 11 A m 2 is located at 12.80 GHz. This downshift behavior of the resonant frequency in the simulation agrees with the prediction of Eq. (5).

In the last section, we have a glimpse of the amplification behavior of the SOT ECMR calculated by the broadband pulse excitation method19 [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 mz(x1,t) and mz(x2,t) that resulted from cw excitation at f = ω / ( 2 π ) = 12.804 GHz at J = 8 × 10 11 A m 2 are displayed. The simulation is repeated for various values of J. The steady state of mz(x2,t) attained slightly before t = 30 ns can be used to compute the magnitude and phase of S21 against α 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
τ R , J ( ω ) = 1 2 i Γ R , J ω Ω 0 , J + i Γ tot , J ,
(8)
which bear the same form as in Refs. 19 and 32, but with Ω 0, Γ 0, Γ R, and Γ tot replaced by Ω 0 , J, Γ 0 , J, Γ R , J, and Γ tot , J, where
Γ tot , J = Γ R , J + Γ 0 , J α J Ω 0 , J ,
(9)
Γ R , J = ( Γ R / Ω 0 ) Ω 0 , J ,
(10)
Γ 0 , J = ( Γ 0 / Ω 0 ) Ω 0 , J .
(11)
FIG. 3.

(a) Reduced z component magnetization dynamics mz(x1,t) and mz(x2,t) against time t as blue and brown data, respectively, for the SOT ECMR model under cw excitation with tstart = 10 ns and tend = 30 ns. Current density J is set at 8 × 10 11 A m 2. The excitation frequency fcw is fixed at 12.804 GHz, which coincides with the dark mode frequency in the CoFeB resonator at J = 8 × 10 11 A m 2 according to Eq. (5). The excitation field magnitude brf equals 0.1 mT. (b) Summary of the magnitude and phase of S21 of the SOT ECMR under cw excitation (blue and brown circles) against α J. Blue and brown solid curves represent the prediction of S21 magnitude and phase through Eq. (8). (c) Same as in (a), but for the direct waveguide pumping model with tstart = 2.5 ns and tend = 10 ns. In this case, J is set at 6 × 10 11 A m 2. (d) Same as in (b), but for the direct waveguide pumping model. Blue and brown solid curves represent predictions from Eqs. (12) and (14).

FIG. 3.

(a) Reduced z component magnetization dynamics mz(x1,t) and mz(x2,t) against time t as blue and brown data, respectively, for the SOT ECMR model under cw excitation with tstart = 10 ns and tend = 30 ns. Current density J is set at 8 × 10 11 A m 2. The excitation frequency fcw is fixed at 12.804 GHz, which coincides with the dark mode frequency in the CoFeB resonator at J = 8 × 10 11 A m 2 according to Eq. (5). The excitation field magnitude brf equals 0.1 mT. (b) Summary of the magnitude and phase of S21 of the SOT ECMR under cw excitation (blue and brown circles) against α J. Blue and brown solid curves represent the prediction of S21 magnitude and phase through Eq. (8). (c) Same as in (a), but for the direct waveguide pumping model with tstart = 2.5 ns and tend = 10 ns. In this case, J is set at 6 × 10 11 A m 2. (d) Same as in (b), but for the direct waveguide pumping model. Blue and brown solid curves represent predictions from Eqs. (12) and (14).

Close modal

The values of the fitted parameters Γ R, Γ 0, Ω 0, and Φ 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 | = | τ R , J | and S 21 = ( τ R , J + Φ R ). The resultant predictions of | S 21 | and S 21 dependence on α 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.

TABLE II.

Material parameters and fitted analytical parameters for the two different ECMRs described in this article. The methodology to obtain the fitted parameters is detailed in Ref. 19.

ParametersUnitSOT ECMRVCMA ECMR
Ks mJ m−2 1.390 1.414 
Ω0 GHz rad 12.833 × 2π 12.630 × 2π 
Γ0 Ω0 0.0102 0.0101 
ΓR Ω0 0.0082 0.0086 
ΓL Ω0 
Δ R Degree  91 °  92 ° 
Δ L Degree N/A N/A 
ΦR Degree  2 °  2 ° 
ParametersUnitSOT ECMRVCMA ECMR
Ks mJ m−2 1.390 1.414 
Ω0 GHz rad 12.833 × 2π 12.630 × 2π 
Γ0 Ω0 0.0102 0.0101 
ΓR Ω0 0.0082 0.0086 
ΓL Ω0 
Δ R Degree  91 °  92 ° 
Δ L Degree N/A N/A 
ΦR Degree  2 °  2 ° 
Now, let us turn to the direct waveguide pumping model. The time evolutions of mz(x1,t) and mz(x2,t) under an aforementioned pulsed sinusoidal spin wave input (i.e., at the same excitation frequency f = ω / ( 2 π ) = 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 mz(x2,t) by directly pumping the waveguide is instantaneous with respect to the input mz(x1,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 S21 calculated using time evolution data similar to those provided in Fig. 3(c) for various values of pumping strength α J, defined according to Eq. (7). The time delay of spin wave arrival between x2 and x1 (namely, Δt21) 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 S21 could be predicted using the following formula:
| S 21 | = exp ( Γ tot , J Δ t 21 ) ,
(12)
with
Γ tot , J = ( α wg α J ) ω ,
(13)
where α wg is the Gilbert damping factor in the waveguide (i.e., αYIG given in Table I). The phase of S21 is fitted according to
S 21 = S 21 , J = 0 + γ μ 0 ( H O e H F L ) Δ t 21 ,
(14)
where S 21 , J = 0 is obtained from the simulation data as 153 °, Oersted field H O e = J s 1 / 2, and H F L is given by Eq. (3). The predicted magnitude and phase of S21 against α 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 brf 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 | Γ tot , J | against α 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 αwg has to be slightly upshifted from α YIG ( = 2 × 10 4 ) to 0.001 to get the best fit of analytical prediction from Eq. (13) (brown solid line) with the simulation data (brown triangles).

FIG. 4.

(a) Reduced z component magnetization dynamics mz(x2,t) against time t (brown data) for the SOT ECMR with brf = 0 mT and J = 1.25 × 10 12 A m 2. (b) Same as in (a), but for the direct waveguide pumping model with J = 7.5 × 10 11 A m 2. (c) Logarithmic plot of oscillation root mean square magnitude (averages are taken over five cycles of oscillation) against time for simulation similar to those data presented in (a) (i.e., from SOT ECMR) for various values of current density J. (d) Same as those in (c) but for data presented in (b) (i.e., from direct waveguide pumping model). (e) Assuming spin wave magnitude |Mz/Ms| in (c) and (d) to grow as exp ( | Γ tot , J | t ), the extracted growth factor | Γ tot , J | against α J for the data in (c) (i.e., from SOT ECMR model, blue circles) and the data in (d) (i.e., from direct waveguide pumping model, brown triangles). Blue and brown solid lines represent predictions from Eqs. (9) and (13) with αwg = 0.001.

FIG. 4.

(a) Reduced z component magnetization dynamics mz(x2,t) against time t (brown data) for the SOT ECMR with brf = 0 mT and J = 1.25 × 10 12 A m 2. (b) Same as in (a), but for the direct waveguide pumping model with J = 7.5 × 10 11 A m 2. (c) Logarithmic plot of oscillation root mean square magnitude (averages are taken over five cycles of oscillation) against time for simulation similar to those data presented in (a) (i.e., from SOT ECMR) for various values of current density J. (d) Same as those in (c) but for data presented in (b) (i.e., from direct waveguide pumping model). (e) Assuming spin wave magnitude |Mz/Ms| in (c) and (d) to grow as exp ( | Γ tot , J | t ), the extracted growth factor | Γ tot , J | against α J for the data in (c) (i.e., from SOT ECMR model, blue circles) and the data in (d) (i.e., from direct waveguide pumping model, brown triangles). Blue and brown solid lines represent predictions from Eqs. (9) and (13) with αwg = 0.001.

Close modal
Here, we are ready to evaluate the pros and cons of the SOT ECMR approach against the direct waveguide pumping approach. First, one can notice from Figs. 3(b) and 3(d) that both approaches attain similar amplification power under similar values of pumping strength α J. However, the length of region that is required to be SOT pumped is w = 50 nm in the ECMR approach [Fig. 1(a)] and L = 2400 nm [ = x2 − x1 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 α J > 0.01. Assume that | ω Ω 0 , J | Γ tot , J; then Eq. (8) reduces to
τ R 0 , J = 1 2 Γ R , J Γ tot , J .
(15)
TABLE V.

Dimension parameters depicted in Fig. 7.

ModelSOT ECMRSOT ECMRDirect waveguide pumping
Excitation method Broadband pulse CW CW 
x0 20 μ1100 nm N/A 
x1 15 μ700 nm 700 nm 
x2 25 μ1500 nm 3100 nm 
x3 10 μ400 nm 400 nm 
x4 30 μ1800 nm 3400 nm 
xtotal 40 μ2200 nm 3800 nm 
xtdr 12.5 μ500 nm 500 nm 
wtdr 5 μ30 nm 30 nm 
20 nm 20 nm 20 nm 
12.5 nm 12.5 nm N/A 
2.5 nm 2.5 nm N/A 
50 nm 50 nm N/A 
s1 5 nm 5 nm 5 nm 
100 nm 100 nm N/A 
ModelSOT ECMRSOT ECMRDirect waveguide pumping
Excitation method Broadband pulse CW CW 
x0 20 μ1100 nm N/A 
x1 15 μ700 nm 700 nm 
x2 25 μ1500 nm 3100 nm 
x3 10 μ400 nm 400 nm 
x4 30 μ1800 nm 3400 nm 
xtotal 40 μ2200 nm 3800 nm 
xtdr 12.5 μ500 nm 500 nm 
wtdr 5 μ30 nm 30 nm 
20 nm 20 nm 20 nm 
12.5 nm 12.5 nm N/A 
2.5 nm 2.5 nm N/A 
50 nm 50 nm N/A 
s1 5 nm 5 nm 5 nm 
100 nm 100 nm N/A 

One could observe from Eq. (15) that | τ R 0 , J | starts to exceed unity when Γ tot , J drops to below Γ R , J according to Eq. (9) as J increases, i.e., α J Ω 0 , J surpasses Γ 0 , J ( = 0.0102 Ω 0 , J ), which agrees very well with our previous simulation results that gain start to occur when α J > 0.01. On the other hand, auto-oscillations do not occur in the ECMR system until α 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 ( Γ tot , J t ). Therefore, the theory predicts the system to become unstable only when Γ tot , J according to Eq. (9) turns negative, i.e., α J Ω 0 , J > Γ R , J + Γ 0 , J = 0.0184 Ω 0 , J, which agrees pretty well with the micromagnetic simulated values of α J > 0.019 mentioned above.

Now, let us shift our attention to the direct waveguide pumping approach. Non-zero | Γ tot , J | data points occur almost immediately after α 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 |S21| against α 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 α J Ω 0 , J between Γ 0 , J and Γ R , J + Γ 0 , J, where gain exists but auto-oscillations are avoided.

As aforementioned, the primary sacrifice of replacing the direct waveguide pumping system with ECMR is the loss of bandwidth. Write the bandwidth (full width half maximum) of the transmission coefficient square modulus as 2 δ ω 1 / 2, then δ ω 1 / 2 is related to the on resonance transmission coefficient (i.e., gain) τ R 0 , J and the radiative linewidth Γ R , J by
δ ω 1 / 2 = | τ R 0 , J | τ R 0 , J 2 2 ( 2 Γ R , J 1 + | τ R 0 , J | ) .
(16)
Proof of Eq. (16) is given in  Appendix B. One can see from Eq. (16) that for a fixed gain ( | τ R 0 , J | ), the only way to increase the bandwidth is to increase the radiative linewidth Γ R , J [or Γ R more precisely according to Eq. (10), which could be seen as the coupling strength between the resonator and the waveguide]. From Ref. 19, we know that by decreasing the waveguide-resonator spacing s from the currently 12.5 nm to 5 nm, Γ R can increase three times. Therefore, we know the bandwidth (transient response time) of the ECMR could at least be increased three times (shorten to one-third, or around 6 ns) as compared to now. However, for a fixed gain, a larger bandwidth also requires a larger pumping strength to maintain. They are related to each other as follows:
α J Ω 0 = Γ 0 , J + τ R 0 , J 2 2 | τ R 0 , J | ( 1 + | τ R 0 , J | 2 ) δ ω 1 / 2 .
(17)

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.

Finally, one has to examine the non-linear behavior of the ECMR amplifier at a large spin wave input amplitude since this will be the regime that useful neuromorphic computing would be operating at. First, the micromagnetic simulated transmission frequency spectra of the ECMR device with pumping shut off (J = 0) for various excitation field strengths (brf) are displayed in Fig. 5(a). The observed frequency downshift of the attenuation maximum against increasing brf is consistent with the previous reports.29 The data should be fitted analytically with Eq. (8), where Ω 0 , J is replaced by ( 1 ξ ) Ω 0 , J, i.e.,
τ R , J ( ω ) = 1 2 i Γ R , J ω ( 1 ξ ) Ω 0 , J + i Γ tot , J ,
(18)
where ξ is a frequency reduction factor found by solving the following cubic equation:
ξ 3 + 2 ( ω Ω 0 , J ) Ω 0 , J ξ 2 + [ ( ω Ω 0 , J ) 2 + Γ tot , J 2 ] Ω 0 , J 2 ξ 2 v λ J Γ R , J η Ω 0 , J 2 | ψ 0 | 2 = 0.
(19)
FIG. 5.

(a) Micromagnetic simulated |S21| of the SOT ECMR in the passive mode (J = 0) against transducer excitation frequency fcw for excitation field brf = 0.1 mT (blue circles), 15 mT (brown triangles), 25 mT (yellow inverted triangles), and 35 mT (purple squares). According to a separated calibration (not shown), brf can be converted into a normalized spin wave amplitude (|Mz/Ms|) in the YIG waveguide incident onto the ECMR by a conversion factor equal 1.4 T1 (for example, for brf = 15 mT, | M z / M s | = 15 × 10 3 × 1.4 = 0.021). The blue, brown, yellow, and purple solid curves represent predictions from Eqs. (18) and (19) for brf values of micromagnetic data points that share the same color with the curves. Best agreement between analytical model and micromagnetic simulation were obtained by setting λ J = 0.55. (b) Same as in (a), but with J of the ECMR changed to 8 × 10 11 A m 2. (c) Micromagnetic simulated |S21| of the SOT pumped ECMR with J = 9 × 10 11 A m 2 against the incident spin wave amplitude in the YIG waveguide |Mz/Ms| with excitation frequency operating almost on resonance with respect to the ECMR [ f c w = 12.804 GHz Ω 0 , J / ( 2 π ), blue circles] and off resonance [ f c w = 12.630 GHz Ω 0 , J / ( 2 π ), brown triangles]. Yellow inverted triangles correspond to data with a VCMA biased ECMR (with no SOT current pumping) in which the resonant frequency of the ECMR is reduced to Ω 0 / ( 2 π ) = 12.630 GHz by an applied voltage. Excited spin wave frequency fcw is also set at 12.630 GHz (i.e., the VCMA device is operating on resonance). Geometrical parameters of the VCMA device are identical to the SOT pumped device (with same d, s, h, and w given in Table V). Solid curves represent predictions from Eqs. (18) and (19) of the main text for simulation data point where color matches with the curves. One should notice that the VCMA biased device has a different natural resonance frequency from the SOT pumped device; therefore, it requires a separate set of intrinsic parameters for the analytical model [as calculated by Eqs. (18) and (19)] as enlisted in Table II. (d) Micromagnetic simulated magnitude and phase of S21 against incident spin wave amplitude |Mz/Ms| (blue and brown circles) for an “ECMR neuron” (see the inset) formed by placing a SOT pumped ECMR operating in the off-resonance mode in series with a VCMA biased ECMR [both as described in (c)]. Incident spin waves are excited at fcw = 12.630 GHz. The edge-to-edge distance of the SOT device to the VCMA device in the x direction is set at 350 nm to reduce direct magnetic dipolar interaction between the two resonators to a negligible level. The blue and brown solid curves represent predictions of |S21| magnitude and phase by the analytical formulas (18) and (19) with parameters described in Table II.

FIG. 5.

(a) Micromagnetic simulated |S21| of the SOT ECMR in the passive mode (J = 0) against transducer excitation frequency fcw for excitation field brf = 0.1 mT (blue circles), 15 mT (brown triangles), 25 mT (yellow inverted triangles), and 35 mT (purple squares). According to a separated calibration (not shown), brf can be converted into a normalized spin wave amplitude (|Mz/Ms|) in the YIG waveguide incident onto the ECMR by a conversion factor equal 1.4 T1 (for example, for brf = 15 mT, | M z / M s | = 15 × 10 3 × 1.4 = 0.021). The blue, brown, yellow, and purple solid curves represent predictions from Eqs. (18) and (19) for brf values of micromagnetic data points that share the same color with the curves. Best agreement between analytical model and micromagnetic simulation were obtained by setting λ J = 0.55. (b) Same as in (a), but with J of the ECMR changed to 8 × 10 11 A m 2. (c) Micromagnetic simulated |S21| of the SOT pumped ECMR with J = 9 × 10 11 A m 2 against the incident spin wave amplitude in the YIG waveguide |Mz/Ms| with excitation frequency operating almost on resonance with respect to the ECMR [ f c w = 12.804 GHz Ω 0 , J / ( 2 π ), blue circles] and off resonance [ f c w = 12.630 GHz Ω 0 , J / ( 2 π ), brown triangles]. Yellow inverted triangles correspond to data with a VCMA biased ECMR (with no SOT current pumping) in which the resonant frequency of the ECMR is reduced to Ω 0 / ( 2 π ) = 12.630 GHz by an applied voltage. Excited spin wave frequency fcw is also set at 12.630 GHz (i.e., the VCMA device is operating on resonance). Geometrical parameters of the VCMA device are identical to the SOT pumped device (with same d, s, h, and w given in Table V). Solid curves represent predictions from Eqs. (18) and (19) of the main text for simulation data point where color matches with the curves. One should notice that the VCMA biased device has a different natural resonance frequency from the SOT pumped device; therefore, it requires a separate set of intrinsic parameters for the analytical model [as calculated by Eqs. (18) and (19)] as enlisted in Table II. (d) Micromagnetic simulated magnitude and phase of S21 against incident spin wave amplitude |Mz/Ms| (blue and brown circles) for an “ECMR neuron” (see the inset) formed by placing a SOT pumped ECMR operating in the off-resonance mode in series with a VCMA biased ECMR [both as described in (c)]. Incident spin waves are excited at fcw = 12.630 GHz. The edge-to-edge distance of the SOT device to the VCMA device in the x direction is set at 350 nm to reduce direct magnetic dipolar interaction between the two resonators to a negligible level. The blue and brown solid curves represent predictions of |S21| magnitude and phase by the analytical formulas (18) and (19) with parameters described in Table II.

Close modal

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 μm/ns (determined from micromagnetic simulation). In Ref. 19, η 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., η = w = 50 nm). In Ref. 19, | ψ 0 | is the amplitude of the incident spin wave in the analytical model; therefore, it is equivalent to |Mz/Ms| in the micromagnetic simulation. Finally, λ 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 λ 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 ξ 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 × 10 11 A m 2. The resultant data for various spin wave excitation strengths brf 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 brf [Fig. 5(b)]. This also comes with the suppression of the amplifying power of peak value at near to 3.0 for brf = 0.1 mT [Fig. 5(b), blue circles and solid curve] to 2.0 for brf = 15 mT [Fig. 5(b), brown triangles and solid curve], and further to 1.6 and 1.5 for brf = 25 mT [Fig. 5(b), yellow inverted triangles and solid curve] and brf = 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 Γ tot , J dependence on α 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 brf = 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 0.5). 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 (fcw = 12.804 GHz). By the time when non-linearity kicks in which bring the transmission of the passive ECMR to 0.5 at fcw = 12.804 GHz [Fig. 5(a), purple squares and solid curve], which corresponds to brf  = 35 mT, then the amplifying power of the pumped ECMR would be greatly compromised. [The data for brf = 15, 25 and 35 mT in Fig. 5(b) at fcw = 12.804 GHz are all below 1.5, reduced from the linear case of 3.0 at brf = 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 |S21| against spin wave amplitude |Mz/Ms| for the pumped ECMR with fcw = 12.804 GHz (on resonance) are presented as blue circles and a solid curve. The pumping current density J is slightly increased to 9 × 10 11 A m 2. The amplifier power deteriorates quickly with increasing |Mz/Ms|, same as in Ref. 19. Now, we reduce the excitation frequency fcw 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, |S21| starts off small (at 1.5) at low values of |Mz/Ms|, but jump abruptly to 3.0 at |Mz/Ms| = 0.013, and then slowly deteriorates with further increase in |Mz/Ms|. Since the device natural resonance frequency is at Ω 0 , J / ( 2 π ) = 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 fcw = 12.630 GHz (yellow inverted triangles and a solid curve) with the natural resonance frequency of the ECMR Ω 0 / ( 2 π ) also reduced to 12.630 GHz (which could be done easily through the VCMA phenomenon19). Now, since the device corresponds to the brown triangles and the yellow inverted triangles operate at the same fcw ( = 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). |S21| starts off at a low value around 0.15 and remains roughly at this same level until |Mz/Ms| gets near to 0.013. At |Mz/Ms| = 0.013, |S21| jumps abruptly to 1.5 and gradually falls back to 1.0 at higher values of |Mz/Ms|. The observed dependence of |S21| on |Mz/Ms| 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 |Mz/Ms| 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 |Mz/Ms| values.

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 wwg = 500 nm in the y direction (see Fig. 6 inset) instead of extending to y = ± . The Pt and CoFeB blocks in the Comsol model are treated as heat sources with power volume density equal to ρ local J local 2, where the local resistivity ρ local and current density Jlocal 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 ° C at J = 9 × 10 11 A m 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.

FIG. 6.

Temperature measured at the black dot (see the inset) against current density J in the Pt layer. The black dot is aligned laterally (in the x–y direction) with the center of the CoFeB stripe and is s1/2 below the Pt\CoFeB interface in the z direction. Width of the YIG waveguide in the y direction (wwg) is set to be 500 nm. Material parameters adopted in the Comsol heat transfer model are displayed in Table III.

FIG. 6.

Temperature measured at the black dot (see the inset) against current density J in the Pt layer. The black dot is aligned laterally (in the x–y direction) with the center of the CoFeB stripe and is s1/2 below the Pt\CoFeB interface in the z direction. Width of the YIG waveguide in the y direction (wwg) is set to be 500 nm. Material parameters adopted in the Comsol heat transfer model are displayed in Table III.

Close modal
FIG. 7.

Set up of the micromagnetic simulation for (a) SOT pumped ECMR coupled to a waveguide and (b) direct SOT pumping in the waveguide without ECMR. Dimension parameters in the figure are enlisted in Table V. The “SO layer” stands for the “spin–orbit interaction layer,” corresponding to the Pt layer in Figs. 1(a) and 1(b). The “resonator” corresponds to the CoFeB stripe in Fig. 1(a), while the “waveguide” represents the YIG film in Figs. 1(a) and 1(b).

FIG. 7.

Set up of the micromagnetic simulation for (a) SOT pumped ECMR coupled to a waveguide and (b) direct SOT pumping in the waveguide without ECMR. Dimension parameters in the figure are enlisted in Table V. The “SO layer” stands for the “spin–orbit interaction layer,” corresponding to the Pt layer in Figs. 1(a) and 1(b). The “resonator” corresponds to the CoFeB stripe in Fig. 1(a), while the “waveguide” represents the YIG film in Figs. 1(a) and 1(b).

Close modal
TABLE III.

Material parameters adopted by the Comsol heat transfer model.

MaterialThermal 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 
SiO2 1.3 2650 680 
Pt 71.6 21450 125 
CoFeB 69 7800 418 
MaterialThermal 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 
SiO2 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 = M B. The spin wave energy density with bias field By along the y direction is, therefore, U sw = M y B y ( M s B y ) = M s B y 1 ( M z / M s ) 2 ( M s B y ) = ( 1 / 2 ) M s B y ( M z / M s ) 2 = ( 1 / 2 ) M s ( ω / γ ) ( M z / M s ) 2. The power flux through a YIG waveguide with width wwg and thickness d would, therefore, be given as P sw = U sw w wg v sw d, where v sw is the spin wave group velocity. Because of the ECMR bandwidth limitation, the duration of spin wave pulse Δ 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 Δ t sw. Plugging in the following values: M s = 140 × 10 3 A m 1, ω = 2 π × 12.630 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 μ m / ns, and d = 20 nm, one can determine E s w = 6.1 × 10 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 = 25) 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 × 10 18 × 5 = 1.76 × 10 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.

TABLE IV.

Summary of energy and time duration required to process an equivalent of five bits of data for various neuromorphic building blocks described in Sec. V. The thermal fluctuation energy at room temperature is also enlisted for comparison.

Building blockEnergy to process equivalent of five bits of digital data (J)Time duration to process five bits of data
Thermal fluctuation at room temperature (kBT) 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 blockEnergy to process equivalent of five bits of digital data (J)Time duration to process five bits of data
Thermal fluctuation at room temperature (kBT) 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 Δ t sw, where P SOT = ρ Pt J 2 w wg w s 1. Plug in the values: ρ Pt = 400 Ω nm, J = 9 × 10 11 A m 2, w wg = 100 nm, w = 50 nm, and s 1 = 5 nm, one would obtain that E SOT = 8.1 × 10 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 Δ t sw with P direct = ρ Pt J 2 w wg L s 1. Because direct waveguide pumping is broadband in frequency, wave pulse duration Δ t sw could be shortened to 5 ns. Plugging in J = 1.5 × 10 12 A m 2, L = 2.4 μ m, one would obtain E direct = 5.4 × 10 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 ΔKs equal to 0.04 mJ m−2. The VCMA amplifier in Ref. 19 that requires ΔKs = 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 s2; then, capacitance of the VCMA amplifier is given as C = ϵ r ϵ 0 w w wg / s 2, where ϵ 0 is the dielectric constant and ϵ r is the relative permittivity of the spacer. Peak current of the device is then given by I = ω C V 0, where V 0 is the peak applied voltage. Therefore, the peak current density is J = I / w wg / ( h + s 1 ) = ω / ( h + s 1 ) × ( V 0 / s 2 ) × ϵ r ϵ 0 w. Plugging in ω = 2 π × 12.83 rad GHz, h + s 1 = 10 nm, V 0 / s 2 = 0.750 V / nm, ϵ r = 10 and w = 50 nm, one would find J = 2.68 × 10 10 A m 2, which is much lower than the SOT ECMR ( 9 × 10 11 A m 2 ). Subsequently, the energy for the VCMA amplifier to process a 100 ns spin wave pulse would be E VCMA = P VCMA Δ t sw, where P VCMA = ρ Ta J 2 w wg w s 1. Plugging in values: ρ Ta = 1900 Ω nm and s 1 = 7.5 nm give E VCMA = 5.1 × 10 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 ( 10 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 ( 10 15 J, see Table IV). Lowering the energy consumption for another order of magnitude would make the ECMR based magnonics competitive with CMOS ( 10 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.

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.

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.

The authors have no conflicts to disclose.

Yat-Yin Au: Conceptualization (lead).

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

Mumax3 was deployed to perform micromagnetic simulation of two separated models illustrated in Fig. 7 with their dimensional parameters displayed in Table V. The cell size is set to be 5 × 5 × 2.5 nm3 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 Hbias = 50 Oe (i.e., Bbias = 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 Ks = 1.39 mJ m−2 in the z direction has been introduced. The Gilbert damping factor in the waveguide is set to 2 × 10 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 = μ 0 H D L m × 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 = μ 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 −Jx. 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(ρPtCoFeB)x and −J[1 + (hρPt/s1CoFeB)]x, respectively (ρPt and ρCoFeB are the electrical resistivity given in Table I). It is derived that a sheet of electric current with density Jxx which begins at x = xstart, ends at x = xend, extends to y = ± , and centered at z with thickness Δz would produce a magnetic field at position (x′, y′, and z′) given as
B Oe ( x , y , z ) = μ 0 2 π J x Δ z y [ tan 1 ( x x start z z ) tan 1 ( x x end z z ) ] ,
(A1)
where 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 S21. For the cw excitation method, the excitation field takes a temporal pulsed sinusoidal form b rf sin ( 2 π f c w t ) rect ( ( t 0.5 ( t end + t start ) ) / ( t end t start ) ) and is spatially uniform in the transducer region. The time (t) dependence of the z component of reduced magnetization (Mz/Ms) dynamics at x = x1 and x2 are recorded and averaged across the thickness of the YIG film and labeled as mz(x1,t) and mz(x2,t), respectively. Reference magnetization dynamics mz,ref(x1,t) and mz,ref(x2,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 S 21 = phase { m z ( x 2 , t ) } 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, S11 is not considered (formulas for S11 in the general case where reflection is non-zero can be found in Ref. 19).

In the SOT ECMR model, the way we define S21 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 = x1 and x2, the S21 definition is modified as | S 21 | = mag { m z , ( x 2 , t ) } / mag { m z , ( x 1 , t ) } and S 21 = phase { m z ( x 2 , t ) } phase { m z , ( x 1 , t ) }.

Since 2 δ ω 1 / 2 is defined as the full width half maximum of the transmission coefficient square modulus, according to Eq. (8) in the main text, we have
| 1 2 i Γ R , J δ ω 1 / 2 + i Γ tot , J | 2 = 1 2 | 1 2 i Γ R , J i Γ tot , J | 2 .
(B1)
Expand (B1) and replace Γ R , J with τ R 0 , J using Eq. (15) in the main text, and one would obtain
δ ω 1 / 2 2 + τ R 0 , J 2 Γ tot , J 2 = 1 2 τ R 0 , J 2 δ ω 1 / 2 2 + 1 2 τ R 0 , J 2 Γ tot , J 2 ,
which could be rearranged to become
δ ω 1 / 2 = | τ R 0 , J | τ R 0 , J 2 2 Γ tot , J .
(B2)
On the other hand, given that τ R 0 , J < 1 when there is gain, Eq. (15) of the main text could be rearranged to become
Γ tot , J = 2 Γ R , J 1 + | τ R 0 , J | .
(B3)
Substitute (B3) into (B2), one would obtain Eq. (16) in the main text. On the other hand, substitute Eq. (9) in the main text into (B2), one would obtain
( τ R 0 , J 2 2 | τ R 0 , J | ) δ ω 1 / 2 = Γ R , J + Γ 0 , J α J Ω 0 , J .
(B4)
Also, Eq. (16) in the main text could be rewritten as
τ R 0 , J 2 2 | τ R 0 , J | ( 1 + | τ R 0 , J | 2 ) δ ω 1 / 2 = Γ R , J .
(B5)

Eliminate Γ R , J by substituting (B5) into (B4), one would be able to recover Eq. (17) in the main text.

Write the magnetization of the resonator in the analytical model as m = φ x x + 1 φ x 2 φ z 2 y + φ z z and adopt the simple scalarization transformation,
{ φ x = 1 2 ( φ + φ ) , φ z = 1 2 i ( φ φ ) .
(C1)
Using the identity φ x 2 + φ z 2 = | φ | 2, m can be rewritten as m = φ x x + 1 | φ | 2 y + φ z z. With magnetic field H = N x M s φ x x + H 0 , J y N z M s φ z z, where N x, N z are the demagnetization factors in x and z directions. Also, H 0 , J is the bias magnetic field in the y direction, where the suffix J remind us that H 0 , J could be affected by a nearby electric current (i.e., in the Pt layer). Subsequently, the x and z components of the Landau–Lifshitz equation m ˙ = μ 0 γ m × H can be written as
{ φ ˙ x = Ω z , nl φ z , φ ˙ z = Ω x , nl φ x ,
(C2)
where Ω σ , nl = γ μ 0 [ H 0 , J + N σ M s ( 1 ( 1 / 2 ) | φ | 2 ) ] for σ = x , z. On the other hand, plugging (C1) into (C2) would lead to a single complex scalar equation given by
i φ ˙ = ( Ω x , nl + Ω z , nl 2 ) φ + ( Ω x , nl Ω z , nl 2 ) φ .
(C3)
If we set N x = N z = N, one would find Ω x , nl Ω z , nl = 0 and ( Ω x , nl + Ω z , nl ) / 2 = γ μ 0 [ H 0 , J + N M s ( 1 ( 1 / 2 ) | φ | 2 ) ]. Define Ω 0 , J = γ μ 0 ( H 0 , J + N M s ) and λ J = ( 1 / 2 ) γ μ 0 N M s / Ω 0 , J, then (C3) becomes
i φ ˙ = Ω 0 , J ( 1 λ J | φ | 2 ) φ .
(C4)

If one compare (C4) with Eq. (2) in Ref. 19 with ϵ = 0, one would conclude that the effect of non-linearity is simply to replace Ω 0 with Ω 0 , J ( 1 λ J | φ | 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 λ J to be near to 0.5. In practice, we found that λ J = 0.55 gives best fit to the simulation data.

In analogy to Eq. (14) of Ref. 19 with δ Ω 0 = 0, the state equation that needs to be solved should read as
( i ω + Γ tot , J + i Ω 0 , J ( 1 λ J | φ 0 | 2 ) ) φ 0 = i I 0 .
(C5)
Taking the absolute square of both sides of equation (C5) would give
[ ( ω Ω 0 , J + λ J Ω 0 , J | φ 0 | 2 ) 2 + Γ tot , J 2 ] | φ 0 | 2 = | I 0 | 2 .
(C6)

According to Ref. 19, | I 0 | 2 = | Δ R | 2 | ψ 0 | 2 = ( 2 v Γ R / η ) | ψ 0 | 2. Substitute this into (C6) and define a new parameter ξ = λ J | φ 0 | 2; then, one would easily arrive at Eq. (19) of the main text.

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