Boundary Surface of 5-Valued Memory

The subject of research in this paper is multiple-valued (MV) memory cell—particularly the morphology of boundary surface of five-valued memory. By accepting the values of parasitic accumulation elements on the chip, very complicated morphology of the boundary surfaces occurs, which separates various attractors from each other. This is due to the occurrence of undesirable oscillations—a stable limit cycles, which makes it impossible to control memory. These dynamic attractors are so dominant that their regions of attraction even surround regions of attraction of static attractors—required logic levels of memory.Therefore, in the realization of the MV memory on the chip is necessary to know the values of the parasitic elements, because their presence may cause amalfunction of thememory. In this case, only calculation and displaying the boundary surface provides exact answers related to operation of the MV memory.


Introduction
Multiple-valued logic (MVL) compared to the binary logic has the advantage that in circuits with MVL, according to [1], are reduced the circuits providing transfer into higher orders.Thus, integration density increases and amount of arithmetic operations reduces.This advantage used the Intel corporation for development of MV ROM memory exploited in the coprocessor, but also companies such as NEC, Motorola, General Instruments, and Hitachi used up to 16-value memory [2].
While in the 80s of the last century MVL memories were designed based on MOS transistors [3][4][5], in the 90s of the last century they have already dominant resonant tunneling diodes (RTDs) [6][7][8][9].To create elementary MVL memory, only two RTDs are sufficient instead of 6 transistors.This was also one of reasons why RTD broke into the MVL memory design.It should be noted that the use of transistors in the MVL is now up to date [10] thanks to a new carbon nanotube FETs [11][12][13].Other advantages of using RTDs are the ability to work in the GHz region, occupying a tiny area on the chip, the design of multipeak RTD being easy, and having a timeindependent I-V characteristic.Change of I-V characteristic of GaAs tunnel diodes due to ageing in the 60s of the last century resulted in malfunction of computers based on these semiconductor devices [14].Although at 90s of the last century the most publication activity of using RTD in MVL memories was noticed, US patent [15] proves continued recency of the problems MVL memories based on RTDs.The advantage of using RTDs lies also in the fact that if RTD is characterised by -peaks and both elements are identical, it is possible to create a memory that will be characterised up to 2 + 1 stable states-logical levels.So, if  = 4, using two RTDs only, it is possible to design such memory which has up to 9 stable states.If we had used MOS transistors, we would have needed greater than 6 MOS transistors.
In order to control any memory, not only just MVL memory, exact investigation of dynamic properties of the elementary memory is necessary.It is not possible without the knowledge of the morphological characteristics of the boundary surface (BS) that separates the regions of attraction of particular attractors.BS is used not only in memories [16][17][18][19][20], but also in chaos generating circuits [21][22][23][24].Therefore, this paper presents morphology of BS in the form of 2D crosssections for two cases of five-valued elementary memory.The first ternary memories consisting of RTDs were analyzed in [25].For  = 10 mH,  1 =  2 = 4 F ternary memory was characterized by three stable states, and memory control by current source Δ was trouble-free.Work [26] showed that if ,  1 , and  2 are comparable with size of parasitic accumulation elements on the chip, a problem may occur.Ternary memory will be characterised not only by three, but four attractors!Newly created attractor is a stable limit cycle (SLC), which disables the memory.Later, two SLCs were found or even three (!) SLCs [27].Circuit was then characterized by five or even six (!) steady states.Figure 2 illustrates the distribution of singularities in the projection to the  2 ,  plane and Figure 3 Monge projection of 2D crosssections of BS through the singularity N1.Symbols S1, S2, and S3 in Figure 2 correspond to stable singularities (attractors) and symbols N1 and N2 to unstable singularities.The same marking is used also in Figure 3.
For a color distinction of particular regions of attraction in Figure 3 the following applies: (i) the green, gray, and red colors represent regions of attraction for stable singularities S1, S2, and S3; (ii) yellow, purple, and blue colors represent regions of attraction of undesired SLCs-L1, L2, and L3.
Marks on SLCs as a cross and dot in a circle represent the intersection of L1, L2, and L3 through corresponding current (plane  2 ,  1 ) or voltage plane (plane  2 , ) [27].The arrows on SLC illustrate the direction of representative point movement in the state space.Time marks in Figure 3 (projection into  2 ,  1 plane) illustrate the time interval of representative point movement on SLCs.L1, L2, and L3 are special because they are not tied to an unstable limit cycle as in the cases described in [14,22,28,29].The symbol L  denotes absolutely unstable limit cycle.It is absolutely unstable because it lies on the boundary of the four (!) regions of attraction and can be calculated only by backward integration [30].
The line denoted that EG1 graphically proves if BS is calculated correctly, because element of BS passing through unstable singularity N1 is tangent plane of green and gray regions of attraction just in N1.More information on the elements of singularities can be found in [16,27,31].The presence of the three SLCs in ternary memory is interesting in terms of the theory of nonlinear circuits, but in terms of implementation of the memory, SLCs represent a warning.But it is possible to find such parameters ,  1 , and  2 , when SLCs are not present and memory can be realised [19].The author of this paper is interested in BS morphology for five-valued elementary memory.A sure thing should be five attractors-logic states of elementary memory.However, the question is whether they will again present undesirable SLCs also in structures described in the next section, where ,  1 , and  2 will be comparable with parasitic values on the chip.

Five-Valued Memory
Circuit in Figure 1, as for ternary memory even for five-valued memory, is described by system The expressions ( 2) and (3) from work [32] have been modified to express the PWL  2 ( 2 ) and  1 ( 1 ) characteristics, to form where    are conductances and    are the break points I-V characteristics.If  = 1, it concerned load, if  = 2, it concerned active device.Expression (2) and parameters (4) or expression (3) and parameters ( 5) correspond to such I-V characteristics in order to achieve five stable singularities S1, S2, S3, S4, and S5.As is clear from Figure 4(a), both I-V characteristics are identical-defined by the relation (2).Since one RTD is active device and the second is load, it is possible to make 5 stable singularities.A similar comment applies to Figure 5(a), with the difference that the active and the load device are defined by the relation (3).To parameters (4) corresponds Figure 4 (5) When Δ = 0, the number of singularities of the system (1) and their coordinates is given by the system of algebraic equations As in the case of ternary memory [27], control crosssections of BS were made through unstable singularities by  This image is a color key to Figure 6.Since  = 0, projection of intersections I-V surfaces with the plane  2 ,  1 is line.The symbol (•) on the line indicates stable singularities S1-S5 and symbol (+) unstable singularities N1-N4.The fact, that N1-N4 lie exactly on the boundary of color regions proves that the cross-section of BS is calculated correctly.Designation S1-S5 is located in the color region corresponding to the region of attraction of incident attractor.Surprising finding was that the memory was characterized by another 10 (!) SLCs.
They are labeled as L1-L10, and colored regions are symmetrically distributed around the linear projection of I-V surfaces.For example, the region of the attraction of SLC L1 is marked by yellow.In one of the two yellow regions, you can find label L1.SLC L2 is marked by light green color labeled L2 (dark green color corresponds to the region of attraction for attractor S2).Last SLC-L10 in Figure 4(b) is marked by white color.Because of the symmetry of I-V characteristics and uniform spacing of singularities, BS crosssection in Figure 4(b) evokes a chessboard.This implies relatively the same size of the regions of attraction as for S1-S5, as well as for L1-L10.Since the presence of up to 10 SLCs was surprising for the author, he proposed hypothetical parameters of RTD (5).These parameters would be difficult to realize, but for verification, BS morphology of other fivevalued memory is sufficient.Because the distribution of singularities is not as uniform as in the previous case, Figure 5 6 and 7, respectively.The number placed at the top of the figure indicates the current in mA.Labels N2, N3, or N1, N4 instead of number labels in Figure 7 indicates, that the cross-sections of BS correspond to current levels of unstable singularities N2, N3, or N1, N4.These are listed in the figure caption.If we compare Figures 6 and 7, we can conclude that the BS morphology is quite complicated.At low or high current levels are dominant regions of attraction for SLC.Regions of attraction for S1-S5 are actually "encapsulated" by regions of attraction for SLC and in 3D state space form objects like "cones."

Conclusion
Since the change of ,  1 , and  2 can significantly affect the functionality of the newly designed prospective memory, it is almost a necessity to know the morphology of BS.Without its knowledge, it is very difficult to find the optimal parameters of the control pulse.Otherwise, the designer of memory is forced to use inefficient method of "trial-error" that does not explain the failure of memory, for example, due to the presence of SLCs.The possibility of practical utilization of SLCs, occurring in memories, is unknown to the author.Therefore, he evaluates negatively their presence in memory structures.Given that in the ternary memory such parameters ,  1 , and  2 were found, leading to the extinction of SLCs, it can be assumed that even in fivevalued memory the same analogy would be possible.Then, BS morphology would be much easier.Presence of virtual saddle singularity, or cases of very complex BS morphology can even happen [19].Mentioned cases make control memory impossible, despite the absence of SLC.Even under these notes, the knowledge of BS morphology is important part in the analysis of MV memories.Other activities for examining the points mentioned earlier will continue if author will meet the interest in science public.

Figure 2 :
Figure 2: I-V characteristics of RTDs and singular points projected into the plane ,  2 .

Figure 6 :
Figure 6: The projections of cross-sections into  2 ,  1 plane for the parameters (4) and current denoted in the figure (in mA).
(b) is not characterised by approximately equal regions of attraction as shown in the Figure 4(b).Cross-section of BS was realised through current level  = 0.94 mA, which is current level corresponding to unstable singularities N2 and N3.Symbols of labelling S1-S5 and N1-N4 and principles of SLC marking L1-L8 are the same as in the commentary to Figure 4(b).Although this structure contains 8 SLCs, the presence of even a single SLC memory makes this MV memory unusable.