Antiferromagnets (AFM) have two magnetic sublattices orientated in opposing directions. The direction of AFM magnetic sublattices relative to the crystal lattice can be manipulated using spin currents. Usually, this is achieved in spin Hall geometry, in which a layer of heavy metal covers an AFM element. When a DC electric current flows in the heavy metal layer, it induces a perpendicular spin current that penetrates into the AFM.

The most interesting effect of spin current on the AFM dynamics happens when the spin polarization of the spin current is perpendicular to the easy plane of the AFM. In this case, spin-transfer torque induced by a sufficiently large spin current causes the AFM sublattices to rotate in the easy plane [22]. For AFM materials with bi-axial anisotropy, the rotation of the sublattices is not uniform with time. This results in a sequence of short spin-pumping spikes at a frequency that can reach the THz range. The threshold current needed to achieve this auto-oscillating regime depends on the easy-plane anisotropy of the AFM material and is of the order of \({10}^{8} \text{A}/{\text{c}\text{m}}^{2}\) for NiO AFM [22].

If the driving current is below the generation threshold, the AFM oscillator will not have enough energy to overcome the anisotropy, but the equilibrium orientation of the AFM sublattices will be moved towards the hard direction in the easy plane. With an additional impulse of current, the AFM magnetizations will surpass the anisotropy energy barrier and perform a single half-turn in the easy plane, which will cause a single spike of the spin-pumping voltage. This response of a sub-threshold AFM spin Hall oscillator is similar to the reaction of a biological neuron to an external stimulus [18]. The AFM neurons and their networks also have other properties that resemble biological neural systems, such as refraction and delayed response [19].

As it was shown in Ref. [22], the dynamics of an AFM neuron can be described by the in-plane angle \(\varphi\) that the Neel vector of the AFM makes with the easy axis of the AFM. Under rather general assumptions, the angle \(\varphi\) obeys the second-order dynamical equation,

$$\frac{1}{{\omega }_{ex}}\ddot{\varphi }+\alpha \dot{\varphi }+\frac{{\omega }_{e}}{2}\text{sin}2\varphi =\sigma I$$

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where \({{\omega }_{ex}=2\pi f}_{ex}\) is the exchange frequency of the AFM, \(\alpha\) is the effective Gilbert damping constant, \({\omega }_{e}=2\pi {f}_{e}\) is the easy axis anisotropy frequency, \(\sigma\) is the spin-torque efficiency defined by Eq. (3) in Ref. [22], \(I\) is the driving electric current. Further details about the derivation of Eq. (1) can be found in [18], [22]. Note that the spin-pumping signal produced by the AFM is proportional to the angular velocity of the sublattice rotation \(\dot{\varphi }\). Namely, the inverse spin Hall voltage produced by the AFM neuron can be found as

$$V=\beta \dot{\varphi }$$

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where the efficiency \(\beta =0.11x{10}^{-15} V\bullet s\) is defined by Eq. (2) in Ref. [19].

In this work, we study the dynamics of a network of interconnected AFM neurons. Each neuron is described by its own phase \({\varphi }_{i}\) and obeys an equation similar to Eq. (1) with additional terms describing synaptic connections between the neurons:

$$\frac{1}{{\omega }_{ex}}{\ddot{\varphi }}_{i}+\alpha {\dot{\varphi }}_{i}+\frac{{\omega }_{e}}{2}\text{sin}2{\varphi }_{i}=\sigma I+\sum _{i\ne k}{\kappa }_{ik}{\dot{\varphi }}_{k}$$

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Here, \(i\) and \(k\) are indices that represent the \(i\)-th and \(k\)-th neurons, and \({\kappa }_{ik}\) represents a matrix of coupling coefficients. Note that the coupling signal produced by the \(k\)-th neuron is proportional to \({\dot{\varphi }}_{k}\), in agreement with Eq. (2).

The coupling coefficients that constitute \({\kappa }_{ik}\) can behave as the synaptic weights in a machine learning system. To a large extent, the challenge of building a fast and efficient neuromorphic computing system depends on the efficient implementation of variable spintronic synapses capable of changing inter-neuron connectivity. The need for variable synapses dramatically increases the complexity of a neuromorphic neural network. This problem is even more serious for AFM neurons since, to fully employ ultra-fast AFM dynamics in neuromorphic hardware, the reaction times of artificial synapses should be on the timescale of AFM neuron dynamics. As AFM neurons spike with a duration that can be less than 5 ps, it should be noted that traditional CMOS technology would severely limit the capabilities of an AFM neural network. To our knowledge, no variable weight synapses have been developed that are suitable to work in conjunction with AFM neurons. As no CMOS or spintronic hardware is capable of being used as variable synapses for AFM neurons, circuit simulations such as SPICE simulations cannot be done. Therefore, in this paper, which primarily focuses on investigating the dynamics of AFM neurons, we did not assume any particular physical model of a synapse. Instead, the simulated synapses are considered to be “ideal” such that they can be adjusted instantaneously and to any value.

Nevertheless, it is important to consider how the latency, or synaptic delay, would impact our model. In a previous work [19], copper bridges with fixed dimensions were used to provide constant weight synaptic coupling or fixed connections \({\kappa }_{ik}\) between AFM neurons. Copper is capable of carrying spin current from one neuron to the next, allowing the output of one neuron to act as the input for a second neuron. The synaptic delay of copper bridges can be found by solving the diffusion equation for spin accumulation in copper. By using standard diffusion parameters for copper [23] and AFM neuron dimensions found in Ref. [19], the synaptic delay for a copper bridge with a length of 100 nm can be found to be about 1.5 ps. We consider this delay to be short enough to have a negligible impact on our system.

There is a remarkable similarity between the equation describing the AFM neuron and that describing the dynamics of a physical pendulum; therefore, each term in Eq. (3) can be characterized by its mechanical analog. As a result, the coefficient of the first term on the left-hand side of Eq. (3) defines an effective mass, indicating that the AFM neuron possesses an effective inertia due to AFM exchange. This inertia results in a delay between a neuron receiving an input and the resulting output, an effect not found in conventional artificial neurons. When AFM neurons are linked together, such that the output of one neuron acts as the input of the next, the delay is dependent on the coupling strength \({\kappa }_{ik}\) between the neurons. The delay caused by inertia decreases as the strength between neurons increases. Thus, the firing time of the neuron can be easily controlled. This means that the AFM neurons are well-suited for neuromorphic algorithms in which time encoding of neuron spikes is used.

One such time-encoding approach, namely, spike pattern association neuron (SPAN) [20], is studied in this paper. The architecture of an AFM neural network realizing the SPAN algorithm is shown in Fig. 1(a). It consists of one output “SPAN” neuron connected to many neurons of the input layer. In our simulations, the input layer consisted of 25 neurons and encoding input symbols drawn in a \(5\times 5\) binary grid. We used several shapes of the input symbols shown in Fig. 1(b). A blackened pixel in the input symbol causes a spike in the corresponding input neuron, while a white pixel will have no spike. The SPAN neuron is trained to output a spike at a certain prescribed time if the input symbol matches the pattern to be recognized. To achieve this, synaptic connections between the input layer and the SPAN neuron are adjusted during the training, as explained below.

We used parallel encoding of the input layer; namely, the input symbol triggers the input neurons to fire simultaneously. If the combined weights connected to the SPAN are strong enough, there will be an output spike. The goal of training is to move this output spike to the desired time for a chosen symbol. If the spike is produced earlier (later) than the target time, the weights connected to the SPAN should reduce (increase).

In more detail, the SPAN training algorithm is based on the Widrow-Hoff rule, where the difference between the desired spike time \({t}_{d}\) and the actual spike time \({t}_{a}\) is used to update the synaptic weights. After some manipulation, shown in Ref. [24], the Widrow-Hoff rule is transformed to describe the change in weights during training:

$${\Delta }\kappa = \lambda {\left(\frac{e}{2}\right)}^{2}\left[\left({t}_{d}-{t}_{i}+{\tau }\right){e}^{-\left({t}_{d}-{t}_{i}\right)/\tau }-\left({t}_{a}-{t}_{i}+{\tau }\right){e}^{-({t}_{a}-{t}_{i})/\tau }\right]$$

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where \(\lambda\) is a positive and constant learning rate, \({t}_{i}\) is the timing of the input spike, \({t}_{d}\) is the desired timing of the output spike, \({t}_{a}\) is the actual timing of the output spike, and \(\tau\) is a time constant corresponding to the width of a spike. Due to the simplicity of the SPAN algorithm, it is only capable of training a neuron to a single symbol. Upon training, a SPAN should output its spike at the target time \({t}_{d}\) for the correct symbol and spike away from the target time for any other symbol.

A library of 20 symbols is used to train the neural network. These symbols are all variations of the correct symbol chosen from one of the symbols shown in Fig. 1(b). Variations include symbols with multiple additional or missing pixels. Initially initialized with random synaptic weights \kappa_{ik}, the neural network receives each symbol as an input during one training epoch. A symbol is associated with a target time corresponding to the image's difference from the correct symbol. Using this time and the actual timing of the output neuron, the SPAN algorithm determines how the weights should be changed in accordance with Eq. (4). The algorithm is modified to ensure that the weights cannot become negative, ensuring a more straightforward implementation in hardware. When all images have been processed, the weight changes resulting from each symbol are averaged, the neural network is updated, and the next epoch begins.

Figure 2 shows the output spikes of a SPAN neural network after training. When the correct symbol is supplied as input, the SPAN spikes within a 10 ps time window of the target time; this implies that the neural network has recognized chosen symbol. Any other symbol should cause a spike outside the target time window, indicating that a different symbol was used as input.