Memory computing based on thermal memory elements

. The article analyses the possibility of using elements of thermal memory to create a system that allows to perform calculations in memory. Such a computing system is built on devices that are used simultaneously for storing input data, performing a logical operation, and storing the output result. The authors conclude that it is possible to emulate this behaviour by using thermal memory elements with dielectric (SiO 2 ) by a layer of thermal insulation. Special attention is paid to the logic gates of computing systems and their realisation on the basis of thermal memory elements. Simulation modelling of the work of such elements is carried out on the ANSYS Workbanch platform using the Transient Thermal module for non-stationary thermal calculations. On the basis of the modelling, the possibility of creating two basic logic gates "AND" and "OR" is established. The results can be used to create more integrated structures, such as artificial neural networks.


Introduction
Artificial intelligence tasks such as image and speech recognition, prediction, decision making, etc. every year require increasing the performance of computing systems for more efficient solution of such tasks.At the same time, systems based on traditional von Neumann architecture have almost reached the maximum of their efficiency [1].To further increase the computational power of traditional systems, it is necessary to increase the number of their main constituent elements, transistors, without increasing the size of the computing systems themselves.This is possible only by reducing the topological norms of transistors, the sizes of which have already approached their physical limit [1].Therefore, today's research is aimed at the development of a new architecture of computingcomputing in memory -similar to the cognitive processes of the human brain.In-memory computing is a promising approach in which analogue memory elements, organised in a computational memory block, are used for both data processing and storage [2].
Resistive analogue memory elements, so-called memristors, have gained popularity in research in recent years [3][4][5][6][7][8][9][10][11][12].These elements are realised on materials with phase transition under the influence of electric, magnetic and thermal fields.Such materials include chalcogenides, segnetoelectrics (polar dielectrics) and some types of dielectrics in which the phase transition occurs under the influence of high temperatures.
In [3], Julien Borghetti and colleagues showed that a NAND logic operation could be performed using three memristive devices connected to a load resistor, implying an extension to all Boolean logic.It was later shown that such logic could be realised in a memristive die with two memristors on the same bit line but in different word lines [4].Additional NOR-based memristive logic jumper architectures have been demonstrated [5][6][7].Other adaptations of memristive device physics to in-memory logic operations include the large pattern dynamics demonstrated by switching of segmentoelectric domains [8], crystallisation [9] and melting [10] physics of phase transition materials.The accumulation property of phase transition materials has been used to perform basic arithmetic processes of addition, multiplication, division and subtraction while simultaneously storing the result [11].
However, the limited number of non-volatile conduction states offered by modern memristors is a challenge for their use in memristor computing.Moreover, the inability to precisely replicate the conductance suggests the presence of defective and volatile devices [12].Therefore, further research and discussions on new physical models of computation are required to develop a computing device based on interacting memristor elements to solve complex computational problems.
In this paper, we consider the application of thermal memory elements, which we introduced in [13][14][15], as a computational memory unit.Thermal memory has a dynamic range of heating degree, non-linearity and asymmetry of the heating response and device variability [13], i.e., properties similar to those of memristive devices.The similarity in the operation of these analogue memory elements discovered by the authors allowed us to propose the use of thermal memory elements to implement a new model of computation.The property of variability of the device, which appears due to the thermodynamic influence of elements on each other is accepted to be levelled with the help of a layer of dielectric thermal insulation.
First, we will discuss basic Boolean logic operations that can be realised on the basis of thermal memory elements.We will also briefly review the logic of operation of the thermal memory element and present a geometrical model of the device obtained as a result of forming a dielectric layer of thermal insulation.And at the end of the article, the results of calculation of the three-dimensional model of the designed logic gates will be presented.

Basic Boolean logic operations
Computation in a computer processor occurs by sequentially performing elementary operations.These include: set, shift, receive, transform, add, and some others.To perform each of these operations, electronic nodes are constructed -registers, counters, adders, code converters, etc.These nodes are used to build integrated circuits of a very high level: microprocessors, RAM modules, controllers of external devices, etc.These nodes themselves are assembled from the basic elements of Boolean logic -both the simplest ones, realising logical functions AND, OR, NOT, NAND, NOR and similar, and more complex ones, such as triggers.
To implement Boolean logic gates on the basis of thermal memory elements, let us turn to the operation of logic gates.Each element of Boolean logic is considered as an operation over logical statements and has its own designation.We shall give two logic gates: -AND -the operation expressed by the conjunction "and" is called conjunction (Latin: Conjunctio) or logical multiplication and is denoted by the dot " ." (may also be denoted by &).The statement A .B is true if and only if both statements A and B are true.
-OR -the operation expressed by the conjunction "or" (in the non-exclusive sense of the word) is called disjunction (Latin disjunctio -separation) or logical addition and is denoted by the sign v (or plus).The statement A v B is false if and only if both statements A and B are false.
For the logic gate "AND", the output Q will contain logical 1 only if both inputs ("A" and "B") are signalled with logical 1 (Fig. 1).

Fig. 1. Table of the logic gate "AND" validity
The output Q, of the "OR" element, will have a logical 1 if a logical 1 is applied to either or both inputs at once. (Fig. 2).

Fig. 2. Table of the logic gate "OR" validity
Let us proceed to the consideration of the functioning of the thermal memory element and present a geometrical model of the device obtained as a result of the formation of a dielectric layer of thermal insulation.

Thermal memory element
The thermal memory element is a structure based on a metallisation system -a thin film of metal (Al) deposited on the surface of a semiconductor (Si).The logic of operation of the thermal memory element is based on the dynamic temperature change of the conductive track of aluminium (Al) deposited on silicon (Si).Heat dissipation is mainly through contact with the silicon wafer.The thermal field propagates not only deep into the silicon wafer, but also across its width around the heat source.That leads to influence on neighbouring memory elements and change of their thermodynamic state.To solve this problem, we propose a method of creating dielectric (SiO2) insulation inside the silicon in the form of pockets.
To investigate the influence of thermal insulation pockets on the thermal field propagation around the thermal memory element, the modelling of the thermal memory elements isolated in this way has been carried out on the ANSYS Workbach platform with calculation in the Steady-State Thermal module.
The geometrical model (1 in Fig. 3) with thermal memory elements isolated from each other by dielectric (SiO2) pockets is a silicon wafer of 2 mm height with inclusions of 4 dielectric layers (3 in Fig. 3).Each dielectric layer is inserted 30 µm deep into the silicon wafer and has a thickness of 1 µm, which corresponds to the dimensions of the insulating silicon dioxide film that can be obtained from the EPIC process [16].The dimensions of the thermal cells correspond to the experimental data (2 in Fig. 3) [13,15].The length of the insulating layer is equal to the length of the conductive track of the thermal cell and corresponds to 4 mm (Fig. 3).The insulating films in pairs form two insulated pockets with thermal cells arranged in their boundary.The geometrical model includes two thermal memory cells without insulated heat sink, located at a distance of 3.8 mm from each other (Fig. 3).
As a result of the modelling, the regions of thermal field propagation are obtained, which show the limitation of thermal field propagation when an insulated pocket is placed in the silicon wafer.Temperature values at equidistant areas from the boundaries of memory elements isolated and non-isolated were found.The calculations showed that when the distance from the boundary of the conductive track to the wall of the insulating pocket is reduced by 20 times, the depth of the thermal field propagation decreases by four times.The results of the calculations were saved as images, Fig. 4.  The presented calculations clearly show the possibility of controlling the propagation of the thermal field from a heated memory element in a silicon substrate through the inclusion of silicon dioxide layers.The use of a dielectric layer limiting the propagation of the thermal field made it possible to design circuits of two logic gates "AND" and "OR" on the basis of thermal memory elements.Simulation modelling of the operation of such elements was carried out on the ANSYS Workbanch platform using the Transient Thermal module for unsteady thermal calculations.

Logic gates AND and OR
The geometry of the structure (Fig. 5) of the logic gates includes three thermal memory elements, two of which are used to store input data and one of which is designed to perform the calculation operation and save the result, let us call it the weight element by analogy with the weight of the synapse in neuromorphic devices [17].The memory elements receiving the input data are arranged parallel to each other and separated by a dielectric layer to limit heat propagation and reduce the probability of changing each other's thermodynamic state.A third memory element is located below this dielectric layer.Fig. 5 shows the geometry of the structure, which is bordered on three sides by the dielectric layer.The absence of a separating dielectric layer on the side of the weight element allows its connection to other logic gates to be realised.Input data can take the value of logical "0" or "1".The logic of operation of the logic gate "OR" here manifests itself as an increase in the temperature value of the weight element to the level of logic "1" due to the propagation of the thermal field from the heated elements of the thermal memory with input data.And the power of propagation of the thermal field should be sufficient to heat the weight element of the thermal memory even in the case of receipt of logic "1" only on one element with input data.It is known that the power of the thermal field decreases with distance from the hot body.Therefore, the logic of operation of the logic gate "OR" on the basis of thermal memory elements is realised at the expense of the minimum distance between the weight element of thermal memory and elements with input data.In Fig. 1, this distance is marked in the range from 0.1 mm to 0.5 mm.The distance of 0.1 mm corresponds to the logic gate "OR", and the distance of 0.5 mm corresponds to the logic gate "AND".
Results of calculation of the three-dimensional model.Carrying out simulation modelling allowed us to see the propagation of thermal fields and different degree of heating of the weight element depending on the distance between the elements.The study of thermal field propagation was carried out at the same values of heat release power, which is set distributed over the volume of material of elements with input data.Calculations have shown that at a distance to the weight element 0.1 mm its temperature will reach the value of logical "1" in the area close to the memory elements with input data.Which is sufficient to correctly estimate the logic state of the thermal memory element.The result is shown in Fig. 6.It shows the propagation of the thermal field around one memory element heated to a temperature whose maximum value on the scale is 25.767 ℃.At the same time, the weight memory element is heated to 23.65 ℃, which will give a temperature increase of two degrees when rounded off.Such an increase in temperature in accordance with the material of the previously published work of the authors [13] leads to the recording of the information state of logical "1" on the memory element.The obtained result characterises the functioning of the logic gate "OR", the value of logical "1" on any of the inputs will give a logical one on the output.As well as logical "1" on the output will appear in the case of two input signals equal to logical "1".Such an algorithm of operation is shown in Fig. 7. Heating the two input memory elements, increases the value of the maximum temperature to 28.308 degrees Celsius and obviously increases the value of the temperature on the weight memory element significantly.The calculations showed that at a distance to the weight element of 0.5 mm, its temperature will reach the value of logic "1" only when two input memory elements are heated simultaneously.The result is presented in Fig. 8.It shows the propagation of the thermal field around two simultaneously heated input memory elements to a temperature whose maximum value on the scale is 28.291 ℃.At the same time, the weight memory element is heated to 24.593 ℃, which corresponds to an increase in the temperature of the memory material by two degrees, which also results in the information state of logical "1" being written to the memory element.This behaviour characterises the functioning of the logic gate "AND", the output will be the value of logic "1" only in case of arrival of logic "1" on both input memory elements.Insignificant change in the temperature of the weight memory element in the case of receipt of logic "1" only on one element with input data is shown in Fig. 9.This confirms the possibility of implementing the "AND" logic gate based on thermal memory elements with an increased distance between the elements that store the input data and the weight element.

Results
Simulation modelling of thermal fields propagation from hot bodies demonstrates the influence of these thermal fields on other solid bodies.The experiments performed on the constructed models of calculating Boolean logic operations on the basis of thermal memory elements proved the possibility of using these elements to create a system of calculations in memory.We managed to model the logical operations AND and OR.
As a result of this research, we determined the number of thermal memory elements in the device that allows us to perform calculations in memory.The device should consist of three stipulated elements with spacing between elements from 0.1 mm to 0.5 mm in the given model.This range can be varied proportionally by changing the geometry of the elements themselves.And it is also necessary to introduce a dielectric layer of thermal insulation according to the geometry given in Fig. 5 to levelling the thermodynamic effects of the elements on each other.
The constructed models of devices for calculating two operations of Boolean logic can be applied in the development of experimental samples of such devices and for modelling the architecture of neuromorphic computing system.

Discussion
Our results show that the thermodynamic interaction of thermal memory elements with each other enables the construction of computational logic gates that perform memory and logic functions.With smoothly adjustable temperature under current pulses, thermal memory elements are exemplary neurosynaptic bionic components providing a potential solution for brain-inspired computing chips.Alongside the well-known memristive devices, there is the prospect of developing cross-bar arrays of thermal memory elements to create more integrated structures such as artificial neural networks.
Memory computing or mem-computing is now a hot topic of research and development for many scientists.In the work of scientists from the Berlin Institute of Technology Andreas Ney and Paul Drud [18], combined elements that process and store data in one, built on the basis of a magnetic memory cell.In contrast to our research, the scientists were able to design a cell that can perform any of the five basic logical operations: AND, OR, NOT, NAND and NOR.Using two input channels, the cell (depending on the direction of current in the output channel) behaves like an AND or OR element.And by adding a third input channel, the researchers obtained an inverter.In this case, the cell can still be used as a non-volatile memory element.
Realisation of NAND and NOR logic gates on the basis of selected elements of thermal memory is possible at creation of inverting logic of work of these elements.Such a logic of operation can be obtained by using the Peltier element.
According to the materials of the Chinese Academy of Sciences [19], perovskite is a promising material for designing the next generation of neuromorphic memristors (CaTiO3).In view of the discovered new optoelectronic properties such as mixed ionelectronic conductivity switched by the concentration of basic charge carriers and slow photocurrent decay.The performance of the perovskite-based memristor remains robust under harsh conditions and withstands irradiation for 60 s with total radiation dose 5*10 5 rad, but there is a disadvantage, such as incompatibility with CMOS processes, which limits their practical application.
Radiation exposure does not interfere with the operation of thermal memory elements [20].Devices based on thermal memory elements can be compatible with CMOS processes.

Conclusions
In this study, Boolean logic gates based on thermal memory elements have been modelled on ANSYS Workbanch platform in the Transient Thermal module for unsteady thermal calculations.The geometry of the logic gates structure includes three thermal memory elements, two of which are used to store input data and one of which is designed to carry out the calculation operation and save the result.For levelling the thermodynamic influence of elements on each other and organisation of directional propagation of a thermal field, the dielectric layer (SiO2) of thermal insulation in the form of pockets was introduced.
The logic of operation of the logic gate "OR" on the basis of elements of thermal memory is realised due to the distance of 0.1 mm between the weight element of thermal memory and elements with input data.The distance of 0.5 mm corresponds to the logic gate "AND".Carrying out simulation modelling allowed us to see the distribution of thermal fields and different degrees of heating of the weight element depending on the distance between the elements.The calculations performed showed that at a distance to the weight element of 0.5 mm its temperature will reach the value of logic "1" only when heating two input memory elements simultaneously.
Simulation modelling of thermal fields propagation allows to estimate expediency of implementation of calculations in memory on the basis of thermal memory elements.Thermal calculations have shown the performance of our proposed models of logic gates.Further research will allow to realise inverted logic gates and to create artificial neural networks on the basis of thermal memory elements.

Fig. 4 .
Fig. 4. Thermal field propagation: a -around a cell in the boundary of a wide pocket and a cell without thermal insulation, b -around a cell in the boundary of a narrow pocket.

Fig. 6 .
Fig. 6.Propagation of the thermal field around one heated memory element to a temperature of 25.767 ℃

Fig. 7 .
Fig. 7. Heating of two memory input elements with increasing maximum temperature value up to 28.308 ℃

Fig. 8 .
Fig. 8. Thermal field propagation around two simultaneously heated memory elements with input data up to temperature 28.291 ℃

Fig. 9 .
Fig. 9. Weighing element heating to a temperature insufficient for switching its information state