Simulation of coupled spin torque oscillators for pattern recognition

Over the last few decades, the downscaling of CMOS integrated circuits (ICs) has followed the empirical law stated by Gordon Moore¹ reaching features size in the few-10 nm range.² Additional benefits of scaling are achievable high operating frequencies in the GHz range, supply voltages below one volt, and low production cost—just to name a few.³ However, the phenomenal increase in the performance of silicon ICs has not solved all challenging computational problems. There are in fact still some special purpose computational tasks where the standard von Neumann machine based logical units are highly inefficient. Associative memories and pattern recognition⁴ are just two special classes of computational problems where digital number crunching is widely believed to be inefficient.

One alternative approach that was suggested and shows promising results is neuromorphic computing.⁵ This non-Boolean computing strategy is inspired by the human brain for tasks like complex pattern recognition. Various physical implementations of neuromorphic computing have been demonstrated: artificial neural networks (Hopfield networks),⁶ cellular neural networks (CNN),⁷ or oscillatory neural networks (ONN),^8–11 just to mention a few.

In this work, we will focus on the last one, the oscillatory neural network, using coupled spin torque oscillators (STOs) as building blocks. In these networks, the information is stored in the frequency and/or phase of the oscillations rather than in analog signal levels, as in classical neural networks, and the onset of synchronization (i.e., frequency/phase locking) indicates a pattern match. In particular, we will design the circuit architecture studied in Ref. 11, but using STOs interconnected by CMOS circuit elements.

Locking phenomena in coupled oscillators have been described as early as the 1920s by Vincent¹² and Appleton,¹³ and the first analytical model for locking of electrical and mechanical oscillators was delivered by Adler.¹⁴ 

Several studies have investigated the synchronization of coupled spin torque oscillators in the previous years. Rippard et al.¹⁵ demonstrated that STOs can lock to a microwave signal hundreds of MHz shifted from its natural frequency. Slavini and Kabos¹⁶ have extensively studied the locking phenomena in STOs. Mutual coupling of two STOs by means of spin wave interactions in the free ferromagnetic layer has been presented by Kaka et al.¹⁷ and Mancoff et al.¹⁸ Locking of electrically connected STOs has been analytically studied by Georges et al.,¹⁹ and some basic conditions for the onset of synchronization for series and parallel arrays are formulated. Very recently, STO-based neuromorphic computing primitives have been demonstrated,^27,28 pointing toward promising applications.

Csaba and Porod²⁰ presented a full micromagnetic model that can reproduce the magnetization dynamics of realistic STO structures. Csaba et al. also developed a circuit model for the STOs, based on a single domain approximation (a macrospin model);^21,22 a variant of this model will be used throughout this paper. The circuit model presented in Refs. 21–24 can be implemented in commercial SPICE simulators allowing the simulation of large STO arrays, which are combined with standard circuit elements like biasing networks and sensing amplifiers. The technology of STOs is compatible with IC manufacturing processes, and an integrated STO-CMOS array would be a practically realizable special-purpose computing circuit.

In fact, when it comes to physical implementations of ONNs, there are very few studies—most published works assume mathematical oscillator models.^11,22–24 Maffezzoni et al.¹¹ provide a very detailed simulation study of an array of LC oscillators coupled via transconductance elements and also derive a useful non-linear phase-domain model for array design. In their implementation, the information of the associative memory is stored in the relative phases of the oscillators at synchronization, a so-called Phase Shift Keying (PSK). Csaba et al. perform with the already mentioned compact model of the STO a SPICE simulation of an associative memory,^21,24 employing a Frequency Shift Keying (FSK). This means that the information is encoded in the relative frequencies of the different spin torque oscillators, and a pattern matching condition can be derived from the intensity of the output signal, which has its maximum at full synchronization.

In the present study, we use the array topology suggested by Maffezzoni et al.,¹¹ but substituting the LC resonant oscillator block with the compact model of the STOs. This way, we provide a case study about integrating nanoelements in a complementary metal–oxide–semiconductor (CMOS) neuromorphic circuit in order to understand the benefits and drawbacks of an ONN that uses an emerging nanoscale oscillator at its heart.

It must be noted that the pattern matching scheme we describe here is only one possible way to do oscillatory neural networks. One could possibly exploit pattern formation in a partially synchronized, AC pumped oscillator network (as described in Refs. 25 and 26 and experimentally demonstrated in Ref. 28). Another option is to implement reservoir computing, possibly with a single STO and appropriate interface circuitry around it.²⁸ Our paper represents a feasibility study for implementing a different type of neuromorphic computing protocol, a pattern matching network (or autoassociative memory). The way of doing computing with the oscillator network follows closely the work of Maffezzoni et al.¹¹ and prior work referenced there, essentially implementing a Hopfield-type neural network with the oscillator phases being the state variables of the network. Interconnected STOs may synchronize in-phase and out of phase to each other depending on the couplings between them. A network of such oscillators will converge to a phase pattern representing a black and white image. If multiple patterns are encoded in the interconnection network, then the phase pattern will converge to the one resembling most to the initial (starting) phase pattern of the oscillators, hence implementing associative memory functionality. Hopfield-type networks tend to converge to a stationary pattern, and throughout the paper, we will assume that the STO-based network will inherit this property.

This article is organized as follows: Sec. II gives a brief description of the STO model, including some details about the materials and geometry that we assumed for the simulations.

In Sec. III, we turn our attention to the sensitivity of basic STO coupling schemes to parameter variations. One of the main challenges with nanoscale building blocks is that they show rather large variations in their parameters. This motivates us to perform a sensitivity analysis for various STO topologies versus process variations in radius, shape, or thickness of the STOs. Many of the simplest coupling schemes turn out to be extremely sensitive to STO process and geometry variations and therefore are not feasible to use. After demonstrating the robustness of the cross coupled coupling configuration, we perform some pattern recognition tests in Sec. IV. For the examples, we consider a simple array consisting of 9 STOs, for processing a 3 × 3 image. In Sec V, we extract the impulse sensitivity function (ISF) of the STO and adapt the phase domain model shown in Ref. 11 for this oscillator. We show that it is a valid approximation, and the results from the phase domain model match the numerical simulation reasonably well. Finally, in the last section of our study, the conclusions of our study are drawn, some improvements of the simulation model are suggested, and an outlook of future theoretical investigations is given.

In Fig. 1(a), a schematic layout for a typical metallic, giant magnetoresistance (GMR) based STO is shown. Current flowing through this metallic stack will initiate precession of the magnetic moments in the magnetic free layer (typically in the GHz frequency range), and these magnetization oscillations will modulate the resistance of the STO stack. From the circuit point of view, the STO can be viewed as a current controlled oscillator.

The equation of motion describing the dynamics of magnetic moments in ferromagnetic materials is the Landau-Lifshitz (LL) equation. In the presence of a spin-polarized current, a so-called spin transfer process occurs. In the STOs we study, the spin-transfer torque coming from the current flow can be modeled by adding the Slonczewski term²⁹ to the LL equation. In our study, we consider STOs where the magnetization of the free layer remains uniform during precession—this is usually the case if the STOs are sufficiently small (i.e., their diameters are under 30 nm as shown in Ref. 20). In this case, one can approximate the magnetization distribution by a single vector. This so-called macrospin model leads to a system of ordinary differential equations, and it is no longer necessary to solve the partial differential equations of a full micromagnetic model. The equations governing STO dynamics are

Here $M$ represents the magnetization vector, which describes the internal degree of freedom for the STO; the S vector points out of plane describing the fixed layer spin polarization; $α$⁠, $γ$⁠, and $Ms$ are constants. $J(t)$ is the potentially time-dependent driving current flowing through the free layer of $tfree$ thickness. The magnetic moments in the free layer experience a $Heff$ effective field, which depends on the shape of the magnetic layer and its magnetic anisotropy—this requires additional equations to describe. Some of the more relevant simulation parameters are as follows: radius and free layer thickness are 15 nm and 5 nm, respectively; the saturation magnetization of the free layer is M_s = 8.6 × 10⁵ A/m and the out of plane anisotropy is K = 5 × 10⁵ J/m³. The applied magnetic field in the z-direction measures 0.5 T, and finally the spin polarization and spin asymmetry parameters are P = 0.5 and Λ = 1.5. The M vector modulates the STO resistance—the net resistance is R = 10 Ω and a magnetoresistance ratio (the amplitude of the change of R with magnetization) is ΔR = 0.1—these parameters link the free layer magnetization to the electrical resistance of the STO. The simulation parameters we used are similar to the ones encountered in “classic” GMR STO with a Cobalt Platinum (CoPt) out of plane free layer^20–22 and copper spacer. Current flows across the fixed magnetic layer (orange in Fig. 1) and becomes spin polarized along the x-axis. This current then passes the free layer (in gray) where the external applied magnetic field shows in the z-direction (B_z), and due to the spin transfer effect, the magnetization of the free layer starts to rotate around the z-axis—as sketched in Fig. 1(b).

For the above mentioned parameters, the macrospin model shows oscillatory behavior of the STOs, comparable to those of real fabricated devices, as for example in Ref. 30. Extensive details about the macrospin model derivation, its validity range, and the exact meaning of all the above mentioned simulation parameters can be found in Refs. 20, 21, and 32.

Next, this macrospin model can be implemented into a SPICE circuit model enabling the simulation of hybrid CMOS/STO circuits. In this study, we make use of the SPICE circuit model^21,24 and the free LTSPICE software in order to simulate different coupling schemes for pattern recognition.

In the first arrangement, we simulate four STOs which are coupled in series with a voltage source. Two resistors provide the bias current of around 3 mA needed for an oscillation. In such a topology, the phase shift between the individual STOs should be in an ideal case $Δφn=360/n$⁠; so for $n=4$⁠, $Δφn=90$⁠. We observe exactly this behavior in Fig. 2 where the voltage drops over the four oscillators are plotted together versus time. Identical voltage amplitudes with a phase shift of 90°, as expected, and an oscillation frequency of around 13.53 GHz can be observed for the four almost sinusoidal waveforms.

For the active coupling, we use the cross coupled scheme normally employed for negative g_m oscillators,¹¹ but replacing the resonant tank with a STO as shown in Fig. 3(a). The additional n-type transistor pair T₁ and T₂ together with the current source I₁ serves as a variable transconductance coupling as proposed in Ref. 11. It is important to note that in this circuit the STO is replacing a passive RC element of the original circuit,¹¹ and therefore the STO is not necessarily operating in an auto-oscillatory mode. The transistors are largely responsible for oscillator feedback.

For simplicity, we first use only 2 STO blocks in this section to illustrate the coupling scheme and then step on to simulating 4-STO systems for studying parameter variations. There are two possible ways of interconnecting the two blocks shown in Fig. 3(a): either the gates $G1+$ and $G1−$ of the coupling transistors T₁ and T₂ are connected “straight” to the terminals $R2+$ and $R2−$ of STO2 or they can be “twisted” $G1+$ to $R2−$ and $G1−$ connected to $R2+$⁠. The first type of connection results in an anti-phase synchronization as can be seen in Fig. 3(b), whereas the second type of connection leads to in-phase synchronization as shown in Fig. 3(c), where the two waveforms overlap. Since the gates of the coupling transistors T₁ and T₂ are connected to the STO2 outputs, the injected current I_D1 which is the sum of the drain current through T₁ and T₂ will be a differential current and proportional to the oscillating voltage at the terminals of STO2. We can write³¹ 

where g₁₂ can be simply set via the current source I₁. A positive coupling parameter leads to anti-phase synchronization while a negative g₁₂ translates into in-phase synchronization.³³ In all simulations, we use symmetric coupling, i.e., g₁₂ = g₂₁, which means that also STO1 injects a current in the STO2 block via the coupling transistors T₅ and T₆. The arrangement shown in Fig. 3(a) can be extended to any N number of STO blocks where each STO injects a current to each of the other $N−1$ diff-pairs and also gets injected by a current controlled by all the other $N−1$ STOs (The all-to-all coupling scheme severely restricts the number of STOs that can be practical in this computing scheme—see Conclusion). For this coupling, a simulated oscillation frequency of 17.03 GHz has been obtained, slightly higher than the one obtained in the series coupling mainly due to the larger RC constant given by the series resistors in the latter case.

It should be noted that this coupling circuitry represents just one possibility for bringing STOs into interconnection—besides using physical coupling mechanisms,^17,34 a passive (but robust) electrical coupling scheme was recently demonstrated experimentally.³⁵ Also, the cross coupled circuit we study here is not a weak coupling scheme that is studied for most oscillator-based computing work. The transistors in the embedding circuit significantly influence the STO dynamics—the STOs act as the tank circuit of the oscillators and are not (or not necessarily) in the auto oscillation mode.

Previous studies have shown that the oscillation frequency of STOs is highly sensitive to the material parameters of the free magnetic layer.¹⁵ This is a serious drawback of any array architecture involving a large number of STOs. Minor process variations will inevitably lead to different free layer geometries and mismatches between the STOs—geometries are very hard to control in the few-10 nm range. It would be therefore desirable to find and exploit a coupling scheme that is invariant or robust enough to handle mismatches induced by process variations.

We simulate the effect of geometrical variations on the STO array behavior. In a first simulation, we varied the radius of the STOs. We change the STO radius from 15 nm to 13 nm in a four-STO cluster, using both of the above-described coupling schemes. As can be seen in Fig. 4(a) for the cross coupled case, this radius misfit has no impact on the voltages over the four STOs; they remain basically unchanged when compared to Figs. 3(b) and 3(c). On the other hand, in Fig. 4(b), one can notice a large impact of the radius variation on the series resistive connection. Additional simulations confirm that geometry changes on the order of 1 nm are sufficient to destroy any synchronization phenomena. STOs in the cross coupled scheme remain perfectly synchronized.

Before we move on to study the transistor-based coupling scheme, it is worthwhile to understand where the limits of this particular passive series coupling come from. From the definition of the negative g_m oscillator, we know that a stable oscillation is possible if and only if the negative transconductance g_m seen at the nodes R⁺–R⁻ is larger than or equal the resistance Rs_to of the resonant tank—STO in our case:³⁶ 

From the equation above, we see that with an appropriate choice of transistors in the differential pair (i.e., transistors with sufficiently large transconductance), small variations in the STO geometry, which have a direct impact on the electrical resistivity in the magnetic layer, can be successfully compensated—and also the coupling can be made sufficiently strong to overcome the variations in the natural frequencies of STOs.

In this section, we demonstrate an associative memory function consisting of 9 STO blocks like those depicted in Fig. 2 and with the three patterns to be recognized shown in Fig. 5. We follow the procedure of Ref. 11 to construct the associative memory.

Since from the discussion in Sec. II B we saw that there are only two ways the STOs can synchronize, either in phase and their output waveforms overlap or out of phase with a 180° phase offset (the antiphase synchronization), we can use these two phase states to encode a binary pattern. In-phase or anti-phase couplings with different strengths can realize positive or negative coupling weights between the oscillators, which then can act as the neurons of a Hopfield network, with phase representing the binary bits. A black and white image (binary pattern) may be hard-wired in the oscillator interconnections in such a way that oscillators belonging to like-colored pixels are positively interconnected (pull each other toward the same phase), and negative weights (anti-phase synchronization) belong to differently colored pixels. The oscillatory ground states will correspond to the stored patterns (images), and an initial phase pattern of the oscillators will converge to one of the resembling stored patterns. This is the way how the autoassociative memory function is verified in Ref. 11 and we also follow this way.

If we take the phase of the first STO as a reference and define all other phase shifts relative to it, we have $Δφn=φn−φ1$ that can be only 0 or 180° and $Δφ1=0$ by definition since $φ1$ is our reference phase. If we construct our information bits as $ξn=cos(Δφn)$⁠, they can take two values: 1 if $Δφn=0$ or −1 if $Δφn=180°$⁠. Furthermore, we define a pattern to be stored as the vector $ξ→$ consisting of all $ξn$ and each array $ξ→k$ can store $k=1…p$ patterns. The binary patterns can be viewed as images, as it is shown in Fig. 5(a) where 1 is represented by white and 0 by black color, respectively.

The pattern identification process consists of two sequences: First the array has to be initialized with the pattern to be recognized (by setting the initial phases of the STOs), and the actual recognition phase takes place in the next phase, when the network finds its oscillatory ground state and the phases of the STOs represent the recognized pattern.

For the first pattern, we can define the so-called connection coefficient $snj$ as¹¹ 

where $snj$ is the connection coefficient between the n-th and j-th STO and 0 denotes index of the pattern.

In order to recognize a pattern, one needs to modify the connection coefficients, from their initial state into a new state, using the well-established “Hebbian rule”:³⁷ 

where $snj$ is the new connection coefficient between the n-th and j-th STO in our associative memory and p is the number of stored patterns—in our case 3. In the table shown in Fig. 5(b), the connection coefficients for the 3 stored patterns are calculated according to the above formula. Only the upper half of the table is filled because as we already mentioned the coupling between STOs is symmetric: $snj=sjn$⁠. These coupling coefficients, however, can take only values of +1 or −1 which means that they simply indicate the type of coupling: positive or negative, if we recall the discussion in Sec. III B. In our case, the coupling is achieved through the differential pairs with variable transconductance that injects currents in the different STOs. Maffezzoni et al. derive a simple formula linking the connection coefficients with the real valued transconductance parameter in¹¹ 

where $σnj$ is a phenomenological parameter characterizing the coupling strength. We recall that the real $gnj$ is set by the variable current sources. A weak coupling is desirable as the injected differential currents should not modify the amplitude of the oscillation only they should shift its phase, where the information is actually stored. We chose for the differential currents a value of 450 μA which is 10% of the current flowing in the STOs (4.5 mA), value that translates into a g_m of 9 mS.

In the first simulation of our associative memory, we test its pattern recognition capabilities. We initialize the coupling coefficients with the values corresponding to pattern 1 from Fig. 5(a), and we let the oscillators achieve steady state; after this has occurred, we change the coupling with the values from Fig. 5(b). In other words, we modify the values of the injected differential currents (in some cases also their sign), and we let the oscillators achieve steady state. The simulated transient waveforms for the 9 STOs can be seen in Fig. 6(a) together with a color-coded representation of the recognized pattern 1 in Fig. 6(b). When we compare Fig. 6(b) with the pattern 1 in Fig. 5(a), we see that the recognition process was successful with the exception of “bits” 2 and 8 that are represented by light gray shading. In Fig. 6(a), we can see this more accurately, the transient signals for STO2 and STO8 do not overlap with STO1-STO3-STO7-STO9 as they should, but neither with STO4-STO5-STO6; they lock exactly in between the two main 180° phase shifted wave-forms. We repeat the experiment initializing our associative memory with pattern 2 this time. In Figs. 7(a) and 7(b), the simulation results are shown; also for this pattern, we have successful pattern recognition with the exception of two “bits”—4 and 6—marked also with gray shading. From the transient outputs in Fig. 7(a), we observe that also in this case the two signals lock in between the two nominal waveforms. A third simulation was carried out with a distorted input pattern. We initialized the array with a modified pattern 2 where bit 4 has the wrong (white) value. We see in Fig. 8 that the pattern is recognized even with faulty initialization, albeit the input had one erroneous bit.

In the final part of our study, we investigate if the analytical phase domain modeling presented in Ref. 11 is applicable also in the case of the STOs—similarly as it was done in Ref. 39 for a different STO-based scheme. The applicability of the phase-domain model is not at all trivial, due to the highly nonlinear magnetization dynamics of these devices. For completeness, we write here the two equations describing the phase model:

where $αn(t)$ is the time dependent phase shift of the n-th STO, $Γn$ is the impulse sensitivity function, and $IDn$ is the differential current injected in the n-th STO by all the other j STO blocks. The $Γn$-function describes the phase sensitivity of an oscillator to a perturbation current. In fact, as can be seen, Eq. (6) states that the transient phase shift of an STO depends on the sensitivity function and the amount of injected current. Eq. (7) states that each differential current is composed of several components, which are the product of the coupling transconductance and the control voltage. From Eqs. (6) and (7), we have a system of coupled ordinary differential equations that is time dependent and can be solved for the unknown $αn(t)$⁠. Once $αn(t)$ is known, $φn(t)$ and $Δφn$ are straightforwardly computed as

with $ω0$ being the constant oscillation frequency. The system of partial differential equations (PDEs) can be trivially solved using numerical methods. But first one has to determine the impulse sensitivity function, and from there we have all the information needed to find all $αn(t)$⁠. For the extraction of $Γn$⁠, we use the standard method, extensively described in literature, i.e., by Pepe et al.³⁸ In brief, we run a simulation of the STO and chose a reference time t_ref where we annotate the value of the output voltage—for simplicity, we can take t_ref at the first zero-crossing of the oscillator’s voltage. Next, we are perturbing the oscillator with a charge pulse at a time t_ref + τ and observe how the voltage reference (zero-crossing) has changed. If the zero-crossing has shifted on the time axis, we annotate the new value t₁ and the phase shift is calculated from this $Δt$⁠. We repeat this for several charge pulses at different times until the zero-crossing matches again our reference value t_ref. Since the output voltage is periodic with a period T, it has to reach its initial value after a maximum time of T seconds. By choosing an appropriate small τ, we can determine the number of sampling points where we perturb the STO as $T/τ$ and get a highly resolved $Γn$⁠; in our analysis, we used 100 points per period. Finally, we plot the phase shift (in rad or deg) versus time (normed at one or two periods) and we have the $Γn$⁠. More details about the extraction procedure can be found in Ref. 38.

In Fig. 9, one can see the periodic impulse sensitivity function obtained by injecting a small current in the drain of the transistor T4 for a configuration very similar to that in Fig. 2. The circuit studied consists of a single STO block and therefore without a coupling differential transistor pair. From the shape of the function, we see that only for some small intervals of the total period T there is an actual phase shift produced by the injected drain current. Those time intervals correspond to the periods when a positive bias is applied on the gate of T4 so that the transistor channel is open. Obviously, for the times when the channel is closed, the perturbation will not have any effect on the oscillator. Since the oscillator topology is a differential one, there is also a sensitivity function when injecting a pulse into the drain of T3; the complete $Γn$ is the superposition of the two and is simply the function in Fig. 9 mirrored to the y-axis.

With this function, we can solve Eqs. (6) and (7) and plot the differential phase shift for the different patterns to be recognized. For the initialization, we set the coupling coefficients according to our pattern to be recognized, and we integrate Eq. (6) in time until synchronization is achieved. Then, we simply change the coefficients according to the prescription in Eq. (4), and we integrate again until steady state is reached. In Fig. 8(a), the procedure has been applied for pattern 1 but using a normalized time axis due to numerical accuracy problems for the real time scale of nanoseconds. This means just the normalization in Eqs. (8) and (9) with a different $ω0$⁠, but it has no impact on the differential phase shift which is of importance to us. We change the coupling coefficients between initialization and recognition set at 25 s, and one can see how the various phase shifts reach synchronization. What is remarkable if we analyze the recognized pattern is that all the phases are in the correct position except for $φ2$ and $φ8$⁠, the same erroneous “bits” as in the circuit simulation earlier presented in Fig. 10. We see that the phase shift is not truly 1 rad (3.14), but this can have many reasons: the accuracy of the extracted $Γn$⁠, the renormalized time axis, etc., nonetheless we have a good qualitative agreement with the SPICE simulations. Also for pattern 2, we observe a good agreement; in fact for this case, we have no erroneous bit at all since after 25 s no phase switches to a wrong state.

In summary, the analysis based on the phase-sensitivity function, gives a qualitatively correct result, just as it was found in Ref. 11. On the one hand, this result is remarkable, considering the highly nonlinear nature of STO dynamics. On the other hand, the oscillator dynamics is largely determined by the CMOS embedding circuitry, not the auto-oscillatory dynamics of the STOs, so one expects a result similar to what was found in Ref. 11. It is likely, that the phase domain model will not be generally applicable to other STO-based oscillatory circuits.

In this work, we presented a case study of 9-STO associative memory. To drive and couple the STOs, we have chosen a transistor coupled scheme, similar to the one described in Ref. 11. We have shown that this scheme is reasonably robust against unavoidable STO parameter variations. We employed a SPICE macro-model for the STOs to perform a full, realistic circuit simulation of the STO/CMOS hybrid circuit. Simulations verified that the circuit works as an oscillatory associative memory. In some cases, erroneous pattern bits are obtained after the recognition step, but such spurious patterns are well known in the literature of Hopfield (like) networks.

We have demonstrated that an analytical phase model can also deliver qualitatively correct results for the array behavior. Some key parameters for the simulation can be extracted fast, like the coupling strength for example. They can be used then in the complete SPICE simulation and provide accurate, realistic output patterns.

Benchmarking our devices against established CMOS-based hardware accelerators is beyond the scope of this paper. But the benchmarks should be similar to other STO-based neuromorphic computing concepts (such as Ref. 28) and should promise potentially significant payoffs.

A significant limitation to the proposed network is that as any all-to-all connected network, it will allow very limited scalability. Still, the associative memory function can ubiquitously be used in neuromorphic computing pipelines, and the network does not have to be scalable to large sizes in order to be useful. For example, most image processing algorithms require operations on scaled images or image patches, which are often just a few pixels in size.

To our knowledge, this is the first work to perform circuit-level simulations of STO-based all-to-all coupled Hopfield networks, using realistic models for the circuit elements, the STOs, and considering parameter variations. It serves as an example of CMOS/STO co-design, even if majority of the circuit functions are done by the CMOS part. We hope that our work adds a potentially useful component to the toolbox of neuromorphic magnetoelectronic computing devices.

The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) (Grant LU 1198/18-1) for financial support. In addition, partial support came from the TUM Graduate School (TUM GS), the International Graduate School for Science and Engineering (IGGSE), and the Institute of Advanced Study (IAS). G.C. and W.P. were supported, in part, by the Defense Advanced Research Projects Agency (DARPA) UPSIDE program.