# Metastable Memristive Lines for Signal Transmission and Information Processing Applications

###### Abstract

Traditional studies of memristive devices have mainly focused on their applications in non-volatile information storage and information processing. Here, we demonstrate that the third fundamental component of information technologies – the transfer of information – can also be employed with memristive devices. For this purpose, we introduce a metastable memristive circuit. Combining metastable memristive circuits into a line, one obtains an architecture capable of transferring a signal edge from one space location to another. We emphasize that the suggested transmission lines employ only resistive components. Moreover, their networks (for example, Y-connected lines) have an information processing capability.

Currently, the term memristive device Chua and Kang (1976) (memristor) is primarily used to denote resistive switching memories that have been considered the most promising candidates for replacing the state-of-the-art memory technology. Moreover, it has been established that memristive networks (the networks of memristive devices) are useful to implement neuromorphic Pershin and Di Ventra (2010); Prezioso et al. (2015), digital Borghetti et al. (2010); Pershin et al. (2016a) and some unconventional computing architectures Pershin and Di Ventra (2011a). The main advantages of computing with memristors (as well as with memcapacitors Di Ventra et al. (2009); Traversa et al. (2014); Pershin et al. (2015)) are related, in particular, to their ability to store and process information on the same physical platform, massively-parallel dynamics of memristors in networks Pershin and Di Ventra (2011a), sub-nanosecond computing times Torrezan et al. (2011) and low power consumption. Computing with memory circuit elements Di Ventra et al. (2009) (memcomputing Di Ventra and Pershin (2013)) is thus a promising alternative to the conventional von Neumann computing Backus (1978).

The information transfer is another important aspect of modern information technologies. Typically, the signal transmission is considered in the framework of transmission line models having a wide applicability range Goleniewski and Jarrett (2006); Martín (2015); Christophe Caloz (2005); Wallraff et al. (2004). The conventional transmission line models employ reactive components – capacitors and inductors – for signal transmision. The transmission line losses are taken into account by resistors. Recently, reconfigurable transmission lines utilizing memcapacitors Di Ventra et al. (2009) instead of capacitors were suggested Pershin et al. (2016b). Transmission characteristics of such lines and thus their functionality can be pre-programmed on demand Pershin et al. (2016b).

The present Letter introduces a different approach to signal transmission uniquely based on the resistive devices. Fig. 1(a) presents its basic block – a metastable memristive circuit – combining a resistor R and memristor M. This circuit employs the most common type of memristors characterized by the bipolar threshold-type switching Pershin and Di Ventra (2011b). According to the selected connection polarity of M in Fig. 1(a), increases at positive voltages across M, . Here, is the memristance (memory resistance) of M changing between and (the low- and high-resistance states of memristor), and is the threshold voltage. Moreover, (the resistance of R in Fig. 1(a) circuit) is selected so that at , is slightly below (see Fig. 1(b)). This circuit configuration can be referred to as a metastable state. The circuit can spend an extended time in this state prior being driven out by an input signal or its fluctuation triggering an abrupt (accelerated Pershin et al. (2013)) switching of M. The final state of the circuit (see Fig. 1(c)) is perfectly stable and thus can be referred to as the ground state.

Next, let us consider a chain (line) of metastable memristive circuits presented in Fig. 2. We argue that under appropriate conditions, this line can transfer a signal edge from one space location to another. Indeed, as it is shown below, there is a certain parameter space where the switching of M (from to ) initiates the switching of M, the switching of M initiates the switching of M, and so forth. The applied signal (see Fig. 2), thus, can set off a chain of switching events propagating along the line. We note that since our approach relies on metastable states of memristive devices, such states should be prepared in advance and periodically refreshed (similarly to the laser pumping Svelto (2009) in the area of lasers). Below, we investigate numerically the dynamics of pulse edge propagation along the line and develop a theory of this phenomenon. In particular, we formulate a time-nonlocal equation describing an infinite line and find its solution in a certain limit.

In our calculations, we use the following model of first-order Chua and Kang (1976) bipolar memristive system with threshold Pershin et al. (2009)

(1) | |||

(2) | |||

where is the current, is the voltage, is the memristance changing continuously between and , is the constant defining the rate of change of , and is the sign of the argument. According to the above equations, the memristance changes only when , and the direction of change is defined by the device connection and applied voltage polarities.

To describe the line dynamics, we set up a system of equations based on Kirchoff’s rules supplemented by Eqs. (1) for the evolution of memristive components. The voltages across memristive systems M are chosen as unknown variables. The equation for -th metastable circuit () reads

(3) |

The boundary conditions for the -st and -th circuits are selected as follows: is the input voltage voltage (see Fig. 2), and . Generally, equations (1), (3) for variables , supplemented with initial conditions (specifically, the initial memristances) fully define the transmission line dynamics.

In what follows we consider a homogeneous metastable memristive line with , , and for . Let us take a closer look at the memristive line dynamics triggered by a rectangular voltage pulse shown in Fig. 3(b). Fig. 3 presents a numerical solution of the line equations found with a set of parameters specified in the figure caption. In particular, Fig. 3(a) demonstrates that the switchings of memristors occur sequentially with almost the same time interval between adjacent switchings. The time dependencies of voltages (see Fig. 3(b)) are similarly shifted with respect to each other. Their waveforms (neglecting the boundary effects noticeable in and lines) are essentially the same. Moreover, taking a closer look at any of these voltages, say , one can notice that a slow increase of changes to a fast increase followed by a slow increase. These stages of voltage growth are mainly associated with the switchings of , and memristors, respectively.

It is amply clear that the dynamics in the central part of the line is determined solely by the line properties but not by the boundary conditions (for example, the input pulse waveform or coupling to the external circuit). Next, we consider the dynamics of pulse edge propagation in the limit of an infinite line as we are not interested in the boundary effects. It is evident that we can safely assume that and are simply time-shifted with respect to each other, namely, , where is the pulse edge propagation time per metastable circuit (time interval between adjacent switchings). Then, the system of coupled equations (3) reduces to a single time-nonlocal equation of the form

(4) |

Eq. (4) together with Eq. (1) thus describe the pulse edge propagation in an infinite line. Unfortunately, even the set of Eqs. (1), (4) is rather complicated one, not only because Eq. (4) includes retarded and advanced times, but also because it is nonlinear. Even for one of the simplest possible given by Eq. (2) we cannot solve the system (1), (4) analytically.

In fact, an approximate solution to the problem can be obtained in the limit of independent dynamics. In this limit, we assume that at each instant of time only one memristor is changing its state (switching). To proceed, let’s focus on Eqs. (3) for , -th and metastable circuits:

(5) | |||||

(6) | |||||

(7) |

Next, the following approximations are made: (i) the voltages at and nodes are replaced by some constant values, , , and (ii) it is assumed that and memristors are in the and states, respectively, that is and . Here, and are given by

(8) |

representing the voltages in the line with all memristors in either or state. The above approximations make possible to truncate the system of equations (3).

Now taking into account Eq. (9) and using the initial condition we integrate Eq. (1) and obtain implicitly the time dependence of :

(16) |

We note that Eq. (16) can be used to find the switching time of every memristor in the line. For this purpose, one should just substitute in the right-hand side of Eq. (16). We have found a very good agreement between the results of exact numerical calculations and analytical solution (9),(16). Fig. 4 presents and found using the analytical model. Eqs. (9) and (16) determine the time dependence of voltage across the switching memristor. The pairs of Eqs. (10) , (16) and Eqs. (11) , (16) can be used to obtain the voltage across -th memristor in the situation when memristor is switching, and, correspondingly, when memristor is switching.

In the above consideration, the -th memristor starts switching at . Therefore, the time it takes for the switching edge to move from th to metastable circuit along the line can be found as the time when a suitable condition to start the switching of memristor is established, namely, when

(17) |

By using Eqs. (17), (10), and (9) we determine the value of such that the condition (17) is satisfied:

(18) |

Plugging this value of into the right-hand side of Eq. (16) we get . Note that the switching time is always longer than the pulse edge propagation time per metastable circuit, .

In conclusion, in this Letter we introduced metastable memristive states, circuits and lines. The signal transmission through metastable transmission lines was investigated using both numerical and analytical approaches. An approximate analytical solution was found in the framework of a single memristor switching approximation. Thus, we have established an innovative approach to signal transmission, which is unique in being based on only resistive components. Moreover, one can also envisage purely capacitive transmission lines, where the capacitive components replace the corresponding resistive ones. However, this idea needs further investigation.

Furthermore, we note that metastable memristive lines can also find applications in the area of information processing. For example, the time delays introduced by these lines could be of use in the development of race logic architectures Madhavan et al. (2015). Moreover, capacitively Y-connected lines (see Fig. 5 for an example) are capable to implement some boolean logic operations, such as AND and OR. Some additional information regarding this idea is provided in Fig. 5 caption.

This work has been partially supported by the Russian Scientific Foundation grant No. 15-13-20021. VAS acknowledges the support by the Erasmus Mundus Action 2 ACTIVE programme (Agreement No. 2013-2523/001-001 EMA2).

## References

- Chua and Kang (1976) L. O. Chua and S. M. Kang, Proceedings of IEEE 64, 209 (1976).
- Pershin and Di Ventra (2010) Y. V. Pershin and M. Di Ventra, Neural Networks 23, 881 (2010).
- Prezioso et al. (2015) M. Prezioso, F. Merrikh-Bayat, B. D. Hoskins, G. C. Adam, K. K. Likharev, and D. B. Strukov, Nature 521, 61 (2015).
- Borghetti et al. (2010) J. Borghetti, G. S. Snider, P. J. Kuekes, J. J. Yang, D. R. Stewart, and R. S. Williams, Nature 464, 873 (2010).
- Pershin et al. (2016a) Y. V. Pershin, L. K. Castelano, F.Hartmann, V. Lopez-Richard, and M. D. Ventra, IEEE Trans. Circ. Syst. II 63, 558 (2016a).
- Pershin and Di Ventra (2011a) Y. V. Pershin and M. Di Ventra, Phys. Rev. E 84, 046703 (2011a).
- Di Ventra et al. (2009) M. Di Ventra, Y. V. Pershin, and L. O. Chua, Proc. IEEE 97, 1717 (2009).
- Traversa et al. (2014) F. L. Traversa, F. Bonani, Y. V. Pershin, and M. D. Ventra, Nanotechnology 25, 285201 (2014).
- Pershin et al. (2015) Y. V. Pershin, F. L. Traversa, and M. D. Ventra, Nanotechnology 26, 225201 (2015).
- Torrezan et al. (2011) A. C. Torrezan, J. P. Strachan, G. Medeiros-Ribeiro, and R. S. Williams, Nanotechnology 22, 485203 (2011).
- Di Ventra and Pershin (2013) M. Di Ventra and Y. V. Pershin, Nature Physics 9, 200 (2013).
- Backus (1978) J. Backus, Comm. Assoc. Comp. Machin. 21, 613 (1978).
- Goleniewski and Jarrett (2006) L. Goleniewski and K. W. Jarrett, Telecommunications Essentials, Second Edition: The Complete Global Source (2Nd Edition) (Addison-Wesley Professional, 2006).
- Martín (2015) F. Martín, Artificial Transmission Lines for RF and Microwave Applications, Wiley Series in Microwave and Optical Engineering (Wiley, 2015).
- Christophe Caloz (2005) T. I. Christophe Caloz, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley-IEEE Press, 2005), 1st ed.
- Wallraff et al. (2004) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004).
- Pershin et al. (2016b) Y. V. Pershin, V. A. Slipko, and M. D. Ventra, Appl. Phys. Lett. 107, 253101 (2016b).
- Pershin and Di Ventra (2011b) Y. V. Pershin and M. Di Ventra, Advances in Physics 60, 145 (2011b).
- Pershin et al. (2013) Y. V. Pershin, V. A. Slipko, and M. Di Ventra, Phys. Rev. E 87, 022116 (2013).
- Svelto (2009) O. Svelto, Principles of Lasers (2009), 4th ed., ISBN 0306457482.
- Pershin et al. (2009) Y. V. Pershin, S. La Fontaine, and M. Di Ventra, Phys. Rev. E 80, 021926 (2009).
- Madhavan et al. (2015) A. Madhavan, T. Sherwood, and D. Strukov, IEEE Micro 35, 48 (2015).