It has been shown previously that spin-Hall oscillators based on current-driven bi-layered film structures containing an antiferromagnet (AFM) and a normal metal can generate ultra-short ( $\u223c$2 ps) “spike-like” pulses in response to an external current stimulus of a sufficient amplitude, thus operating as ultra-fast artificial “neurons.” Here, we report the results of numerical simulations demonstrating that a single AFM “neuron” can perform the logic functions of or, and, majority, or q-gates, while a circuit consisting of a small number $n<5$ of AFM “neurons” can function as a full-adder or as a dynamic memory loop with variable clock frequency. The clock frequencies of such AFM-based logic devices could reach tens of GHz, which make them promising as base elements of future ultra-fast high-efficiency neuromorphic computing.

## I. INTRODUCTION

The operation of the human brain, the most complex organ of a living being, which is by far the least studied object in biology, medicine, and physics, differs substantially from the operation of conventional information and signal processing devices (like conventional semiconductor processors in computers based on Boolean logic).^{1}

The brain is a huge neural network consisting of an enormous number of neurons coupled via synapses, where each neuron behaves as a nonlinear oscillator generating spike-like pulses in reaction to an external stimulus. These pulses can act on other neurons, and these neurons can respond to the pulses generated by other neurons.^{1–3} The intelligent behavior provided by the brain emerges from the cooperative and competitive interactions between multiple parts of the brain’s neural network and its synaptic environment.^{1}

Although the brain’s neural network is not very efficient for performing complex and/or routine numerical calculations, its performance is extremely efficient for image and pattern recognition, classification of multi-dimensional signals, prediction, synthesis, solution of control problems, etc.^{4} Also, in contrast to classical semiconductor processors, the performance of which is limited by their substantial power consumption,^{5} neuromorphic systems can operate at much smaller power levels and, therefore, could form a basis for novel energy-efficient signal processing and computing systems. Consequently, the “brain-inspired” computing and signal processing devices have the potential to revolutionize the computer industry, which made them a subject of intensive study and development during the last two decades.^{6–8}

Note that emulation of neural networks using the software running on traditional semiconductor computers has not been very successful, as most machine-learning algorithms suffer heavily from the, so-called, Von Neumann architecture bottleneck.^{5} Usually, such bio-inspired algorithms lose their most precious qualities—speed, error tolerance, and low energy consumption—when running on conventional processors with no devoted hardware support for neuromorphic computing.^{6} The development of a neural network specific hardware could make bio-inspired calculations more efficient than software-based neural networks.^{6} Since the beginning of the neural network theory development, artificial neurons based on magnetic materials have been considered as one of the promising basic elements suitable for the creation of an “electronic brain.”^{6}

The development of magnetization dynamics and spintronics has resulted in the appearance of numerous concepts and proposals for neuromorphic computing hardware. A list of magnetic phenomena that can be used for this purpose includes the dual mode spin wave excitation,^{9,10} the spin transfer torque,^{11–13} the giant or tunneling magnetoresistance,^{14,15} the spin pumping,^{16,17} and the spin Hall effects.^{18,19} It has been realized recently that spintronic microwave nano-oscillators utilizing ferromagnetic materials^{5–7} are of particular interest for this purpose. These nonlinear ferromagnetic (FM) oscillators driven by spin-polarized current could generate spin wave signals at frequencies up to 10–30 GHz, and the frequency could be easily controlled externally,^{11,12,20–25} thus providing a possibility to design artificial FM “neurons.” The advantages of the spin wave-based “neurons” are their small sizes ( $\u223c$10–100 nm), the absence of an electric charge flow, and the relatively low exciting current ( $\u223c\mu $A).^{6,7} However, the frequency of operation of FM-based artificial “neurons” is limited mainly by the magnitude of the experimentally achievable bias magnetic field.

In order to substantially increase the operational frequency of artificial “neurons,” it is possible to use antiferromagnetic (AFM) materials, having extremely high internal magnetic fields ( $ H e x \u223c1000$ T) caused by the uniform exchange interaction, for the development of ultra-fast artificial “neurons” operating at the frequencies close to the typical frequencies of the AFM resonance $\u223c$0.1–1 THz.^{26–29} We note that the AFM spintronic systems have recently attracted strong interest from researchers and became a subject of intensive theoretical and experimental studies.^{19,26–32}

In this work, we demonstrate by numerical simulations that a circuit based on a small number ( $n<5$) of AFM-based artificial “neurons” can efficiently perform logic operations at rather high clock frequencies ( $\u223c$50–100 GHz). The AFM artificial “neurons” are based on AFM spin-Hall oscillators (SHOs)^{33,34} working in a sub-critical regime and are capable of generating ultra-short ( $\u223c$2 ps) spike-like pulses in response to an external stimulus of a sufficient amplitude. The excitation mechanism in such devices uses the spin Hall effect (SHE), while the inverse spin Hall effect (ISHE) is used for the extraction of the output pulse generated by the AFM artificial “neuron.”^{33,34} The generation of the spike-like pulses in the above described AFM pulse generator is a threshold process, possible only when the amplitude of the external current stimulus exceeds a certain threshold defined by the material’s anisotropy and dissipation, which allows one to consider such a pulse generator as an ultra-fast analog of a biological neuron.^{34}

Below, we show that an artificial AFM “neuron” can have the functionality of a basic logic element and can perform the functions of an or, and, majority, or q-gate, while a simple circuit consisting of only three ( $n=3$) such AFM “neurons” can be used as a full-adder. Also, a concept of a dynamic spintronic memory can be realized using several AFM artificial “neurons.”

The paper is organized as follows. Following the introduction, the basic physical principles governing the operation of an artificial AFM “neuron” (spintronic AFM pulse generator) are presented in Sec. II. The description of the performance of a chain consisting of several coupled AFM “neurons” is described hereafter in Sec. II. Implementation of different logic operations using one or several AFM “neurons” is proposed in Sec. III. Next, the possibility of the development of a dynamic memory cell based on AFM “neurons” is discussed in Sec. IV. Finally, the obtained results are summarized in Sec. V.

## II. AFM SHO AS AN ARTIFICIAL NEURON

The proposed AFM-based pulse generator (AFM artificial “neuron”) is based on a bi-layered film structure, where the current-driven layer is made of a normal metal (NM) with a strong spin-orbit coupling (usually, platinum or tungsten), and the other layer is a thin film of an AFM material; see Fig. 1. An electric current, flowing in the NM layer, generates a spin current via the spin-Hall effect, which, in turn, creates a torque, which tilts the sublattice magnetizations $ M 1 $ and $ M 2 $ of the adjacent AFM layer out of the AFM “easy plane,” exposing them to the torque created by the huge internal exchange field $ H e x $. This exchange-related torque tends to push the tilted magnetizations of the AFM sublattices into rotation.^{26,27}

Here, we consider a bi-anisotropic AFM material with a strong anisotropy keeping the sublattice magnetizations inside the “easy plane,” and with an additional weaker anisotropy in the plane perpendicular to the direction of polarization $ p $ of the incoming spin current ( $ H e \u22a5 p $, where $ H e $ is the effective anisotropy field). This anisotropy field creates a potential barrier for the continuous rotation of the tilted magnetic sublattices, so for an insufficient driving current, the oscillator behavior is *sub-critical *with no AFM dynamics. However, when an additional current stimulus is applied at a certain moment, and the *total current overcomes the anisotropy-related threshold*, the orientation of $ M 1 $ and $ M 2 $ is rapidly switched, causing the generation of a short pulse of a spin current.^{34} This spin current pulse, flowing back into the NM layer, is converted into a short electric pulse via the inverse spin-Hall effect. Such a behavior of an AFM oscillator is similar to the behavior of a biological neuron,^{34} thus making current-driven AFM-based oscillators to work as artificial AFM “neurons” and become promising candidates for the basic elements of the next-generation computing devices.

The dynamics of the sublattices’ magnetizations $ M 1 $ and $ M 2 $ in an AFM layer of the above described oscillator is described by two coupled Landau-Lifshitz-Gilbert-Slonczewski equations, which can be reduced to a single scalar equation for the azimuthal angle $\varphi $ of the Neel vector $ l =( M 1 \u2212 M 2 )/(2 M s )$:^{33}

which is valid in the low-frequency limit (i.e., when $ \varphi \u02d9 \u2261\u2202\varphi /\u2202t\u226a \omega e x $, where $ \omega e x $ is an exchange frequency).

Here, $ \omega e =\gamma H e $ is the frequency determined by the anisotropy field $ H e $ ( $ H e \u22a5 p $), $\gamma $ is the gyromagnetic ratio, $\alpha $ is the effective Gilbert damping parameter of the AFM material, $\sigma $ is the torque–current density proportionality coefficient determined by the degree of the current spin polarization, and $ M s =| M 1 |=| M 2 |$ is the static AFM sublattice magnetization. It should be noted that the anisotropy field $ H e $ in the considered AFM thin films can have both surface and intrinsic bulk origin.

Equation (1) is mathematically analogous to the equation describing the dynamics of a pendulum under the action of an external torque. The external torque, that is constant in time (i.e., created by the driving direct current $ j d c $), tilts the pendulum to a certain angle $ \varphi 0 $. The maximum angle $\varphi $, at which the pendulum remains stationary, is equal to $\pi /4$, which takes place at the value of the driving force equal to $\sigma j d c t h = \omega e /2$. If the supplied current density $j(t)$ exceeds the threshold value $ j d c t h $, the Neel vector $ l $ starts to rotate infinitely. This rotation of the Neel vector of the AFM creates a spin pumping into the adjacent layer of the NM, resulting in the ISHE-caused output current density:

where $ g r $ is the spin-mixing conductance and $\u210f$ is the reduced Planck constant. Since the spin-mixing conductance heavily depends on the properties of the AFM/NM interface, it is convenient to characterize the output signal of the AFM oscillator by the value $ \varphi \u02d9 $, which we will do in the following text.

As one can see from (2), the uniform rotation of the Neel vector (when $ \varphi \u02d9 =0$) will give zero output ac signal. Thus, the AFM layer of a considered oscillator should be made of a anisotropic AFM material, such as nickel oxide (NiO), with the anisotropy field $ H e $ causing a temporal non-uniformity in the rotation of the vector $ l $. Therefore, in our numerical simulations of the dynamics of an artificial AFM “neuron,” we used the following parameters, typical for NiO: $ \omega e x =2\pi \u22c527.5 THz $, $ \omega e =2\pi \u22c51.76 GHz $, and $\alpha =0.01$ since we assume a thin AFM layer.

To describe the current-induced pulse generation in the artificial AFM “neuron,” we are interested in the subcritical case, when the DC is close, but still below the threshold value, $ j d c < j d c t h $. Let us assume that at a certain moment of time, we supply an additional current stimulus, so that the total current density overcomes the threshold for only a short period of time, tilting the pendulum to the position $ \varphi 0 =\pi /4+\u03f5$, $0<\u03f5\u226a1$.

In such a case, the Neel vector starts to rotate to the new stationary point $ \varphi 0 +n\pi $ ( $n$ is an integer number) where it should be stopped by damping. Note that in the above described AFM oscillator, the effective damping parameter $\alpha $ could be substantial, because it contains the contribution from the spin-pumping losses in addition to the contribution from the intrinsic Gilbert damping. Therefore, for the qualitative analysis, one can neglect the first term in Eq. (1) and get the following simple solution for the rotation angle:

and a corresponding solution for the output signal of the AFM pulse generator:

It is clear that the second term in Eq. (3), which depends on the initial conditions, only shifts the solution in time but does not affect its shape or duration. Note also that the shape and duration of the output signal do not depend on the parameters of the supplied current stimulus and are determined by the intrinsic properties ( $\alpha $ and $ \omega e $) of the AFM oscillator.

Now, we can treat the output signal [Eq. (4)] of one of the AFM artificial “neurons,” attenuated by a factor $\kappa $, as an input signal for another similar AFM “neuron” subjected to the same direct current $ j d c $. Thus, the parameter $\kappa $ defines the unidirectional coupling between the two “neurons.” We analyze the response of the second “neuron” on the pulse produced by the first one, by solving numerically Eq. (1).

Our simulations show that three different regimes are possible: absence of generation, generation of a single peak, and multi-peak (double peaks, triple peaks, and so on) generation, also known as the regime of “bursting” for the biological neurons.

The calculated state diagram for an artificial AFM “neuron” illustrating the possible pulse generation regimes is presented in Fig. 2. As one can see, by choosing the initial state of the AFM “neuron” (driving current density $ j d c $) and controlling the coupling (coefficient $\kappa $) between the “neurons,” one can easily change the “neuron’s” response on some input pulsed signal.

Note that the shape of the output pulse generated by an AFM “neuron” and described by Eq. (4) is approximate. Moreover, a signal pulse propagating in a neural network can pass through a long chain of neurons, which raise a question about the stability of a neuron response with respect to the shape deviations of the input signal.

To address this issue, the signal of Eq. (4) was applied to the input of a serial chain consisting of 6 artificial AFM “neurons” with the attenuation $\kappa $ after each of them. The output signals of each of the “neurons” connected in the chain are shown in Fig. 3. The calculated shape of the signal propagating in the chain is changing and is, obviously, different from the trial analytical input at the beginning of the chain. This shape, however, becomes almost constant and does not substantially change in the course of the further propagation in the chain. Thus, a neuromorphic network of artificial AFM “neurons” demonstrates a high stability with respect to the shape deviations of a transmitted signal.

Figure 3 also shows that the response time of the artificial AFM “neuron” strongly depends on the coefficient $\kappa $ characterizing the coupling between the “neurons.” At the large values of $\kappa $, one can reach the response time smaller than the width of the generated spike-like pulse, and, therefore, the strong coupling is preferable for the ultra-fast computing circuits having a large clock frequency. Moreover, one could control the required phase shifts and delays between the different “neurons” by varying the corresponding coupling parameters.

## III. LOGIC OPERATIONS PERFORMED BY ARTIFICIAL AFM “NEURONS”

The biological neurons in a human brain form large coupled arrays for their joint performance. The artificial AFM “neurons” could also be coupled in a network for the realization of complex signal processing tasks, so that an output pulse from one “neuron” could be used as an input signal for the other AFM “neurons.” The coupling between the AFM “neurons,” characterized by the coefficient $\kappa $, can be varied externally by creating a particular circuit of the coupled AFM “neurons.”

Then, the equation for the variable $ \varphi i $, describing the state and the dynamics of the $i\u2212$th “neuron” in a particular circuit coupling it to all the other “neurons,” can be written as

Solving this system of equation for the known shapes and amplitudes of the external input spike-like signals and for the known coupling coefficients $ \kappa i k $, one can determine all the output signals generated by coupled AFM “neurons.”

If we now consider a circuit consisting of several AFM “neurons” connected in a certain particular manner (the simplest cases are $ \kappa i k =0$ or $ \kappa i k \u22600$), we can obtain an output signal from a set of AFM SHOs, which depends on the architecture of such a neural network and, consequently, is determined by the logic operations performed over the external input signal by all the AFM “neurons” of the circuit. Below, we assume a unidirectional coupling, i.e., $ \kappa i k \u22600$ and $ \kappa k i =0$, the value of which is controlled by the coupling circuit (“axons”).

Below, we demonstrate that a small array of coupled AFM “neurons” could successfully perform the Boolean logic operations. For all the examples presented below, we used the cycle duration of $40 ps $ (clock frequency of 25 GHz) and the coupling coefficient of $\kappa =1.5\u22c5 10 \u2212 3 $.

First of all, let us consider a single AFM “neuron” with two independent input lines (two “synapses” attached to a “neuron”). Depending on the initial state of the AFM “neuron” (i.e., the value of the driving current density $ j dc $—see Fig. 2), either any one of the input spike signals (logical operation or) or only both of them (logical operation and) can “push” the AFM “neuron” over the threshold causing the generation of the output spike signal similar to the input one. This behavior of a single AFM “neuron” is demonstrated in Figs. 4(a) and 4(b), for the cases of or and and operations, respectively. Note that to clearly demonstrate both input signals, we intentionally created a nonzero phase shift between them (blue and green peaks in Fig. 4).

The delay time between the output signal (shown by solid line in Fig. 4) and the input signal (shown by dashed lines) is the time which the AFM “neuron” needs to respond to the input signal. This time is comparable to the duration of a generated spike-like pulse and is about 10 ps. This delay, however, can be significantly decreased by the increase of the coupling coefficient $\kappa $.

Also, note that if in the or regime the phase shift between the input signals is larger than the generated peak duration, we will not receive a correct result, so in such systems, coupled AFM “neurons” will necessarily use some control schemes for proper signal synchronization. The sign of the coupling coefficient $\kappa $ in the and and or regimes is the same. Thus, one can switch the operational regime from or to and simply by changing the $ j dc $ value: in the and regime, it should be lower than the or regime.

Let us now consider a more interesting case when the AFM artificial “neuron” operates with three input signals. In such a case, a single AFM “neuron” can operate as a majority gate (for comparison, the realization of a majority gate using conventional semiconductor transistors needs at least five elements). The functionality of a majority gate on a single AFM “neuron,” obtained from a numerical solution of Eq. (5), is demonstrated in Fig. 5. Note that the level of the dc driving current density $ j dc $ in this regime of the AFM “neuron” operation is close to its value for the or regime.

Another important and interesting logic element that can be realized on a single AFM “neuron” is the element performing the following logic function:

Below, we will call this logic element a *Q-gate*. Similar to the majority gate, it has three independent inputs, but one of the input signals is inverted in sign. The logic operations of a q-gate can be interpreted as follows. The output signal will be “true” (“1”) in the case when the signal 3 is “false” (“0”) and one of the signals 1 or 2 is “1,” or if the signal 3 is “1” and both signals 1 and 2 are equal to“1.” Physically, it is realized that signals 1 and 2 increase current density and signal 3 has the opposite sign of the current. In practice, this can be done when the coupling coefficients $\kappa $ at one of the inputs in Eq. (5) is negative. The logic “truth” table for the q-gate is presented in Table I, and the input and output signals of this logic circuit are shown in Fig. 6.

$1 st $ input . | $2 nd $ input . | $3 rd $ input . | Output . |
---|---|---|---|

(Blue) . | (Green) . | (Red) . | (Black) . |

0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 |

0 | 1 | 0 | 1 |

0 | 0 | 1 | 0 |

1 | 0 | 1 | 0 |

0 | 1 | 1 | 0 |

1 | 1 | 0 | 1 |

1 | 1 | 1 | 1 |

$1 st $ input . | $2 nd $ input . | $3 rd $ input . | Output . |
---|---|---|---|

(Blue) . | (Green) . | (Red) . | (Black) . |

0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 |

0 | 1 | 0 | 1 |

0 | 0 | 1 | 0 |

1 | 0 | 1 | 0 |

0 | 1 | 1 | 0 |

1 | 1 | 0 | 1 |

1 | 1 | 1 | 1 |

The combined use of both the majority gate (m-gate) and the q-gate provides a possibility to create a simple full-adder based on three AFM “neurons.” The scheme of such a logic circuit is shown in Fig. 7. Note that the number of common logical elements (transistors) necessary for a conventional full-adder is at least five. The logic “truth” table for the AFM full-adder comprising three AFM “neurons” is presented in Table II. The input (red, blue, and green) and output (dashed blue for the “sum” bit and black for “carry-out” bit) signals for this logic circuit are shown in Fig. 8.

$1 st $ input . | $2 nd $ input . | Carry-in . | Sum . | Carry out . |
---|---|---|---|---|

(Blue) . | (Green) . | (Red) . | (Dashed-blue) . | (Black) . |

0 | 0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 | 0 |

0 | 1 | 0 | 1 | 0 |

0 | 0 | 1 | 1 | 0 |

0 | 1 | 1 | 0 | 1 |

1 | 0 | 1 | 0 | 1 |

1 | 1 | 0 | 0 | 1 |

1 | 1 | 1 | 1 | 1 |

$1 st $ input . | $2 nd $ input . | Carry-in . | Sum . | Carry out . |
---|---|---|---|---|

(Blue) . | (Green) . | (Red) . | (Dashed-blue) . | (Black) . |

0 | 0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 | 0 |

0 | 1 | 0 | 1 | 0 |

0 | 0 | 1 | 1 | 0 |

0 | 1 | 1 | 0 | 1 |

1 | 0 | 1 | 0 | 1 |

1 | 1 | 0 | 0 | 1 |

1 | 1 | 1 | 1 | 1 |

## IV. DYNAMIC MEMORY BASED ON AFM “NEURONS”

The other important component necessary for the development of any neuromorphic or classical logic systems is memory. It is convenient to relate the possible information (“0” or “1” signals) stored in the memory cell to the state of an AFM “neuron” in the following way. Let us assume that the signal “1” means that the AFM “neuron” is turned on (generates pulses), while the signal “0” means the absence of pulse generation. Then, it is simple to implement the “writing” and “erasing” signals which will change the state of the AFM “neuron” and realize a dynamic memory based on several AFM “neurons” connected in a loop.

When we have a loop of $n$ oscillator with coupling realized in such a way, that a single starting spike-like signal exceeds the generation threshold, this signal will continue to circulate in a loop from one “neuron” to another. Obviously, this signal could be erased if a spike of the opposite sign enters the loop. It is possible to control the time interval between the pulses circulating in the loop by changing the number $n$ of AFM “neurons” inside the loop (see Fig. 9).

It is known that in a human nervous system, the information-carrying signals are encoded by the clock frequency of propagating pulses, rather than by the pulse amplitude. A similar scheme can be realized in a neural network based on AFM “neurons,” where the loops containing different numbers of AFM “neurons” will provide different frequencies if the pulse sequences circulate in these memory dynamic memory loops (see Fig. 9). The other opportunity is to tune the coupling strength $\kappa $ and, as a result, vary the delay time of the AFM “neuron” response to the input stimulus, as was discussed above (see Fig. 3).

## V. DISCUSSION

One can notice from Fig. 2 that the coupling $\kappa $, required for the operation, decreases with the increase of $ j d c $. However, at this point, the effect of thermal noise comes to the fore. Since thermal fluctuations are not limited in amplitude, they can switch the direction of magnetic sublattices $ M 1 $ and $ M 2 $ without any external stimulus and, therefore, create a so-called “soft-error” of the operation. The probability of this process is $P\u2243\Delta E/ k B T$, where $\Delta E$ defines the potential barrier between two states (i.e., $\varphi = \varphi 0 $ and $\varphi =\pi + \varphi 0 $), $ k B $ is a Boltzmann constant, and $T$ is a temperature. In the absence of the bias current, the barrier is defined by the energy of the anisotropy, which reads as $ W a =V M s H e cos\u20612\varphi $, where $V$ is the volume of the AFM material in the oscillator. The dc decreases the potential barrier as

where $\xi = j d c / j d c t h $. Thus, now we can calculate the required in-plane size $a$ of an oscillator, shown in Fig. 10, by assuming the square shape, the thickness of the AFM film $d=5 nm $, $ M s =350 kA / m $, and desirable $\Delta E/ k B T=10$. As expected, the side length rapidly increases, when $ j d c \u2192 j d c t h $, and for the bias current $ j d c / j d c t h =0.95$, used above in this paper, one gets $a>135 nm $ or $a>200 nm $ for $\Delta E/ k B T=20$. Since the direction of the Neel vector rotation (clockwise or counterclockwise) depends only on the polarization of the input spin current, the circuits are robust against the domain structure of the AFM, in a sense that each auto-oscillator can be placed in the different domain of the AFM thin film. However, as we assume uniform spin dynamics inside each oscillator, it should be placed within the single domain, which can also restrict the maximum size of the oscillator.

We also want to admit that both the efficiency of the input spin current and the output amplitude decrease with the increase of angle between the direction of the spin-current polarization $ p $ and the AFM hard axis $ n h $ as $ p \u22c5 n h $.^{33} Particularly, for the NiO thin film, grown in [001] direction, and $ p $ is along [110], the relative angle with the hard [111] axis is equal to $ 35.3 \u2218 $, which means the increasing of the threshold and decreasing of the output currents in 1.22 times.

In summary, it is demonstrated by numerical simulation that AFM-based spin-Hall oscillators operating in a sub-critical regime could perform the functions of artificial “neurons,” working with the spike-like pulses of the duration below 4 ps at clock frequencies of tens of GHz with a response time below 2 ps. Simple circuits based on a small number ( $n<5$) of these AFM “neurons” could perform the logic operations of and, or, or majority gates, and more complex functions of a q-gate, full-adder, or a dynamic memory loop with a variable clock frequency of the pulsed signals circulating in the loop. The obtained simulation results show that the proposed AFM artificial “neurons” could become standard base elements of future ultra-fast neuromorphic computers and signal processing systems.

## Acknowledgments

This work was supported by the Knut and Alice Wallenberg Foundation (KAW), by Grant Nos. EFMA-1641989 and ECCS-1708982 from the National Science Foundation (NSF) of the USA, and by the Defense Advanced Research Projects Agency (DARPA) M3IC Grant under Contract No. W911-17-C-0031. This work was also supported by the President’s of Ukraine grant for competitive projects (F 78) and by the grant F 76 from the State Fund for Fundamental Research of Ukraine. This work was also supported in part by Grant Nos. 16BF052–01 and 18BF052–01M from the Taras Shevchenko National University of Kyiv and the grant 7F from the National Academy of Sciences of Ukraine.

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