## Modeling of computing cell

The behavior of the described structures under the influence of electric pulses was modelled.

For the modelling of the described system some well-known material parameters were used. These parameters are listed in Table 1.

Table 1

– Main parameters of materials which used in modelling

Resistivity of crystalline GST (Crystalline/amorphous) ρcr/ ρam | 10− 5 Ohm*m / 1 Ohm*m |

Resistivity of the material between nanoparticles (Ge) | 1 Ohm*m |

GST phase transition temperature (crystallization/melting) Tcr/Tmelt | 393 K / 1000 K |

Specific heat capacity of GST(Crystalline/amorphous) | 200 J/(kg·K) |

Density of GST(Crystalline/amorphous) | 5000 kg/m3 |

In the initial state all PCM NP are amorphous. The space between the particles is filled with a material whose resistivity is not less than that of the amorphous GST. Therefore, the directions of currents in such a system can be modelled using a graph. Such a graph can be constructed using the Delaunay triangulation method. Then the system will have the form of an electric circuit presented in Fig. 2.

The edges of this graph are resistors whose resistance is determined by the material between NP. The nanoparticles are also resistors and the resistance values of the entire edge is defined as R + R1 + R2. Where the resistance R is determined by the length of the fill material from one nanoparticle to another. R1 and R2 are the resistances of the nanoparticle hemispheres connected by the edge.

The state change of the nanoparticles after the electric pulse depends on their heating temperature. This leads to a change in their resistivity relative to the current ρncur (n is the number of electric pulses in the system) according to the rule:

$${\rho }_{cur}^{n+1}=\left\{\begin{array}{c}{\rho }_{cur}^{n}, T<{T}_{cr}\\ {\rho }_{cr}, { T}_{cr}<T<{ T}_{melt}\\ {\rho }_{am}, T>{ T}_{melt}\end{array}\right.$$

Such a simple approximation is not applicable to thin films because the cooling rate of the material must be taken into account for GST phase transitions. However, for nanoparticles it can be assumed that the electrical pulse time is much longer than the cooling time due to their small size and large surface area. The particle temperature after the pulse is determined in the described model by the heat released in the nanoparticle by the current according to the Joule-Lenz law.

The behavior of the system described has been modelled for the effect of a sequence of short pulses. It can be assumed that during a pulse the voltage between the contacts of the system is constant. The amplitude of the voltage increases with each pulse. This approach makes it possible to find out how smoothly the state of the cell can be tuned by controlling only the amplitude of the electrical influence. The amplitude of the first pulse is 50 V and increases by 5 V with each subsequent pulse. The pulse duration is 100 ns. The particle sizes are randomly distributed in the range 50 to 100 nm. The radius of the circle in which the NPs are located is 1 µm. The pulse voltage source is connected to pin 3 (diametrically opposite to ground). Possible connections of the electrical pulse source to the cell are shown in Fig. 3.

The result of modelling the effect of the described pulses on the NP array is shown in Fig. 4.

Particles in the low-resistance crystalline state are indicated by red circles in the figures. Colors of the edges indicate the currents flowing through these edges. It should be noted that the maximum current flowing in such a circuit does not exceed 1 µA. It is easy to see that the phase transition after the first pulses occurs close to the contacts. Furthermore, when the number of pulses increases, the phase inhomogeneity of the cell also increases. As can be clearly seen in Fig. 4, at the 10th pulse the crystalline nanoparticles formed a segment of low resistance. The current density is the highest in the vicinity of this segment. A similar pattern can be obtained for any cell size with any number of nanoparticles. An example for 450 nanoparticles ranging in size from 10 to 30 nm is shown in Fig. 5.

The same sequence of pulses was used to construct Fig. 6. The peculiarities of the evolution of the phase structure of the cell remain the same when the geometry of the arrangement and the number of nanoparticles are changed. However, the chaotic arrangement of nanoparticles makes such a system poorly predictable when using a small number of pulses.

The above simulation examples were verified for simple cases using COMSOL simulation software.

The examples described show the general character of the dynamics of the systems described under the action of sequences of electrical impulses. In order to assess the limits of application of such systems in real electronics, it is necessary to carry out statistical studies of the evolution of their electrical properties.

The resistivity dynamics of the NP array were investigated. These dynamics are slightly different for each cell realization due to the stochasticity of the NP array. Therefore, the graphs were plotted by averaging the results of 100 different cells of the same size and number of NP.

Figure 6 shows the dependence of the cell resistance on the number of pulses. The resistance is measured between the same contacts to which the pulse voltage source is connected. The initial value of the pulse voltage amplitude was 50 V. This value increased by 1 V every pulse.

The graph shows the dynamics of the resistivity contrast. This contrast is the ratio of the current resistance to the initial resistance when all particles are amorphous. It is easy to see that the resistance of the cell changes smoothly with the crystallization of different sites. If 5V is added to the voltage amplitude with each pulse, the non-monotonic resistive dynamics shown in Fig. 7 can be obtained.

It is clearly seen that with increasing power of electric pulses the resistance of the cell starts to increase smoothly up to its initial value. It should be noted that it is not possible to obtain smooth switching of the electrical resistance by using pulse sequences whose amplitude does not change from pulse to pulse in any configuration of the cell. In this regime stationary state is reached much faster. In addition, a bifurcation is clearly visible. This is due to the fact that the system can switch to a regime where some groups of nanoparticles change their states equally with each pulse. Such particles switch to a different state with each pulse.

In addition, the described system has an important property for neuromorphic computing. A change in any of the resistances Rn−0, where n = 1...5, is accompanied by a change in all other resistances. These resistances can be thought of as synaptic weights. Then all weights in such a neuron are not independent. An example of the effect of excitation of one contact on the others is shown in Fig. 8.

It is clearly seen that different resistances (horizontal connections of neuron weights) are not independent. This property is useful in neuromorphic systems. For example, when the physical location of inputs carries information. Many modern neural network architectures have horizontal connections between neurons. In such a system, the properties of such horizontal connections can be controlled by changing the shape of the cell.

The strength of horizontal bonds in the described design can be adjusted by changing the shape of the cell with nanoparticles. Horizontal connections can be made very strong or completely eliminated.

As the modelling of the described cell shows it can be used on the neural network element. The described cell is a convolutional neuron with dependent weights of inputs (horizontal links). Such architecture can be used both for realization of impulse data-driven neural networks and for physical realization of classical neural networks. In the first case, the system has a threshold activation function associated with a phase transition [20]. In the second case, the cell needs to be supplemented with a nonlinear function, which is easy to realize by circuit engineering [21]. At the same time, it should be noted that the character of signal processing by the cell can be applied for pulse signals and for computing architectures built on the non-Von Neumann principle, but far from neuromorphic.

It is important to note that the modelling did not take into account the hopping conductivity, which can occur when electrons overcome the potential barrier when nanoparticles are close to each other. This effect will not significantly change the described properties of the system, but will add an additional nonlinear effect.

The cell described and the methods of controlling its state have a number of disadvantages. Firstly, the modelled architecture has a high resistance between the contacts (up to tens of megohms) and a complex amplitude-frequency characteristic that changes during operation. This can lead to difficulties in tightening the fronts of electrical control pulses. And the low contrast between resistances in different states can make it difficult to read the state of the system without error. These problems can be overcome by using an NP array consisting of several vertical layers, by increasing the NP density and by reducing the NP size in conjunction with a reduction in the overall cell geometry.