Empirical Characterization of ReRAM Devices Using Memory Maps and a Dynamic Route Map

Abstract

Memristors were proposed in the early 1970s by Leon Chua as a new electrical element linking charge to flux. Since that first introduction, these devices have positioned themselves to be considered as possible fundamental ones for the generations of electronic devices to come. In this paper, we propose a new way to investigate the effects of the electrical variables on the memristance of a device, and we successfully apply this technique to model the behavior of a TiN/Ti/HfO${}_2$/W ReRAM structure. To do so, we initially apply the Dynamic Route Map technique in the general case to obtain an approximation to the differential equation that determines the behaviour of the device. This is performed by choosing a variable of interest and observing the evolution of its own temporal derivative versus both its value and the applied voltage. Then, according to this technique, it is possible to obtain an approach to the governing equations with no need to make any assumption about the underlying physical mechanisms, by fitting a function to this. We have used a polynomial function, which allows accurate reproduction of the observed electrical behavior of the measured devices, by integrating the resulting differential equation system.

1. Introduction

At the beginning of the 1970s, Leon Chua noticed a symmetry in the pattern formed by the equations relating current i, voltage v, charge q, and flux $\phi$, within the frame of the electric theory [1]. As a consequence of this symmetry, the existence of a fourth element emerged, next to the three already known passive elements, namely capacitor, inductor, and resistor. Each one of the latter three elements relates a pair of electrical variables: charge with voltage, flux with current, and current with voltage, respectively. Then, Chua suggested the existence of “memristor” as the new fourth electrical element, and this way the missing relation between charge and flux could be fulfilled, by attributing to the introduced element (nonlinear) dynamics that allowed it to behave as a resistor with a memory.

In fact, the first experimental description of such elements had already been performed long time ago [2], but they did not manage to become part of the circuit theory mainstream. The theoretical introduction of these devices paved the way to develop a generalized description of a class of devices and systems: the memristive devices or systems. These devices (or systems) are intrinsically nonlinear and defined by a set of differential equations relating the electrical magnitudes to a set of internal state variables [3].

Currently, this novel kind of device is considered as one of the most promising for the next generation of integrated circuits, those beyond Moore [4]. For instance, these devices are expected to be a path for alleviating the classical problem of the Von Neumman architecture, caused by the need to transferr data between memories and processors, enabling in-memory computing [5]. As a result, an increasing number of applications using memristors have been been proposed, such as innovative memories (ReRAMs, MRAM, etc.) [6,7,8], new sensor devices [9,10,11,12], elementary blocks for artificial neural networks (ANNs), and other bioinspired systems [13,14,15], to mention a few.

Related to fabrication technologies, memristive devices are currently being implemented in many different flavors, ranging from spintronics [16] to many different oxides [17,18,19,20], organic materials [21,22], and even emulators [23,24,25,26,27]. However, due to the relative immaturity of the current technology, only a handful of foundries are providing memristors in their libraries as standard blocks. As an example, we mention that as far as we know, RRAMs are offered by TSMC, Intel, and Towerjazz (where this last one seems to be in the process of being acquired by Intel), but they use old nodes. This implies that they can be implemented in cheaper designs, allowing their use in low-cost edge IoT platforms [28]. This low level of technological maturity also leads to the same problems that were also encountered in other technologies, such as reliability or forms of variability: the usual device–to–device, plus a cycle to cycle variability [29,30].

Simulating these devices is also a hot topic inside the modeling community [31,32], and there are two main approaches to this task. The first approach uses current and voltage [33,34,35,36,37] as the electrical variables of interest. The second approach, in line with Chua’s original proposal, uses charge and flux instead [38,39,40,41,42,43,44,45,46,47].

An interesting way to examine a model is using a tool named the Dynamic Route Map (DRM). This technique helps to represent how the system’s governing variable evolves versus its rate of change (the temporal derivative). As discussed in the corresponding section, this representation can provide a significant insight into how state variables evolve and project their relations as well. The DRM method can also be understood within the formalism proposed by Corinto [48], as the equation relating the internal variables with their derivatives.

The DRM technique has already been used to represent experimental ReRAM device measurements [49] and PCM devices [50]. Furthermore, in [51], it was shown how memristive devices with a well-known behavior (i.e., thermistors) can be discussed and represented using DRM. In this paper, we move this work a step forward and show how DRM can be used with experimental measurements to obtain an approximation to the differential equations defining the behavior of the device.

2. Memristor Modelling Framework

As mentioned in the introduction, a framework for memristor modeling was proposed by Corinto and Chua in [48], as an expansion of the previous work in [38]. In that work, the authors proposed a consolidated theoretical framework that allows a unified approach to modeling, and they also examined the possible advantages of the flux–charge ($\phi$-q) variables used in this context. In this proposal, they defined the different memristors presented in the taxonomy by [52] (the ideal, the generic, and the extended memristor) as being different approximations in the equations. The most general type in this classification is the extended memristor, which is a memristor described by additional state variables $\mathbf{X}=(X_1,X_2,\dots )$ (other than just the electrical $\phi$ and q, or v and i). These state variables can be, for instance, the physical dimensions of a conductive filament, the temperature of the device, or anything else that may be needed for a complete description of the device. The most general expression relating all these variables can be expressed as follows:

$$\begin{array}{c}\hfill Q=f(\phi ,v,\mathbf{X})\end{array}$$

(1)

where the flux ($\phi$) and the charge (Q) are linked to the voltage (v) and the current (i), respectively, as follows

$$\begin{array}{c}\hfill \frac{d\phi }{dt}=v\end{array}$$

(2)

$$\begin{array}{c}\hfill \frac{dQ}{dt}=i\end{array}$$

(3)

In addition, the evolution of the internal variables $\mathbf{X}$, is described by a differential equation:

$$\begin{array}{c}\hfill \frac{d\mathbf{X}}{dt}={\mathbf{g}}_{\phi }(\phi ,v,\mathbf{X})\end{array}$$

(4)

In order to obtain an equivalent expression to Ohm’s law, Equation (1) can be derived with respect to time:

$$i=\frac{dQ}{dt}=\frac{\partial f(\phi ,v,\mathbf{X})}{\partial \phi }\frac{d\phi }{dt}+\frac{\partial f(\phi ,v,\mathbf{X})}{\partial v}\frac{dv}{dt}+\sum _j\frac{\partial f(\phi ,v,\mathbf{X})}{\partial X_j}\frac{d{X}_j}{dt}$$

(5)

The above equation can be further simplified using the Lagrangian L and the Jacobian $\mathbf{J}$, as expressed in Equations (6) and (7), respectively:

$$\begin{array}{c}\hfill L(\phi ,v,\mathbf{x})=\frac{\partial g_{\phi }(\phi ,v,\mathbf{x})}{\partial v}\end{array}$$

(6)

$$\begin{array}{c}\hfill \mathbf{J}(\phi ,v,\mathbf{X})=\left(\frac{\partial g_{\phi }(\phi ,v,\mathbf{X})}{\partial X_1},\dots ,\frac{\partial g_{\phi }(\phi ,v,\mathbf{X})}{\partial X_n}\right)\end{array}$$

(7)

$$i=\frac{\partial f(\phi ,v,\mathbf{X})}{\partial \phi }v+L(\phi ,v,\mathbf{x})\frac{dv}{dt}+\mathbf{J}(\phi ,v,\mathbf{x})·\frac{d\mathbf{X}}{dt}$$

(8)

It is interesting to point out that the two last right terms in Equation (8) have a clear physical meaning, being identified as parasitic elements [48]. Iin particular, the part including the term $dv/dt$ can be connected to an inductive element, while the term $d{X}_j/dt$ can be identified as a current source. In the case where parasitic elements are present, the most general (i.e., extended) memristors may be reduced to generic memristors (plainly, just memristors). This is caused by function $g_{\phi }$ being dependent only on flux $\phi$ and the state variables $\mathbf{x}$; thus, $L=0$. Moreover, the original memristors, as defined in [1] are generic memristors that do not depend on any internal state variable (i.e., $J=0$). It has also to be noticed that it has been demonstrated in [48] that the following relation has to hold:

$$L(\phi ,v,\mathbf{x})\frac{dv}{dt}+\mathbf{J}(\phi ,v,\mathbf{x})·\frac{d\mathbf{X}}{dt}=0$$

(9)

Therefore, we can determine the memristance M to be as follows

$$M(\phi ,v,\mathbf{X})=\frac{\partial f(\phi ,v,\mathbf{X})}{\partial \phi }=f_M(\phi ,v,\mathbf{X})$$

(10)

The above expressions can also be formulated in terms of charge and current as in [51]. In this case, the parasitic elements would have been inductors and voltage sources [48].

In brief, the relevant relations for a flux-controlled memristor can be described using Equations (3), (4), and (11):

$$\begin{array}{c}\hfill v=M(Q,i,\mathbf{X})·i\end{array}$$

(11)

The latter, Equation (11), shows the dependence of memristance in the most general case (the extended memristor). In this equation, the voltage across the terminals is named v, while i is the current and Q the charge (the first momentum of the current, as in [48]). The internal variables are included in the form of a vector $\mathbf{x}$, and their dynamics are defined by ${\mathbf{g}}_{\mathbf{Q}}$ and Equation (4).

The framework described above has already been employed to successfully model various sorts of memristive systems, additionally enhancing the generalized framework for compact modeling in the flux–charge space [53]. Some interesting works utilising this approach may be found in the literature, as in [54] where a memristor model was developed using a charge-dependent mobility. A model for PCM was proposed in [39] and in [40] or [43], where the authors proposed semi-empirical models for ReRAMs, considered to behave as memristors, and in [55,56], where a Monte Carlo model for ReRAMs was presented. A delay model for memristive crossbars was derived in [57] utilizing a flux–charge definition. Finally, in [58], light bulbs based on incandescent filaments were shown to be generic memristors, while in [59,60], an experimental characterization of a memristive system was performed by using the flux–charge notation, and they present a discussion of the influence of the waveform’s frequency and shape.

3. The Dynamic Route Map

The use of state space to study dynamics was initially introduced by J. Liouville [61] and further utilized by H. Poincaré to study the three-body problem [62]. Since then, the study of nonlinear systems and the relevant methods in nonlinear dynamics have been developed within the phase space approach. This was due to the fact that important features and attributed of nonlinear systems are clearly emerging within the state space. It should be noted that the case of the two-dimensional phase space, also mentioned as phase portrait (x vs. $dx/dt$), had been utilized as early as the beginning of the 20th century by the Ehrenfests [63].

On the other hand, the study of memristors is conducted within the previously described framework (Section 2), where the dynamics of an extended memristor that is flux-controlled is presented. It is apparent that due to the principle of duality, the corresponding charge-controlled extended memristor could be easily described within the same frame. Clearly, memristors are nonlinear elements, and that is due to their hysteretic behaviour, which is their main fingerprint [64]. As a result, phase space emerges as the proper space for studying memristor dynamics, and this way useful information on their dynamical properties can be provided. Toward this, a new tool in the phase space, the recently introduced novel technique of the Dynamic Route Map, has been proposed for studying memristor devices. This technique allows for acquiring information regarding their switching properties and features in general [65].

The dynamic route concept within the phase space, applied in the case of memristors, considers the definition of the trajectory of one of the system’s state variables as the voltage or the current through the device (according to whether the memristor is flux or charge controlled), which obtains a specific value.

This procedure can be considered to be equivalent to plotting Equation (4). Expanding this concept from a specific trajectory to a family of trajectories, the Dynamic Route Map is defined as a a parametric collection of a theoretically infinite number of dynamic routes [65]. An illustration of a typical DRM is presented in Figure 1, where three representative cases of DRM are presented. The one in the middle is a characteristic case that includes one stable and two unstable equillibria. It is apparent that the DRM is infinitely dense and that for proper visualization usually only a finite number is presented.

A comprehensive apposition of the features demonstrated by the DRM technique appears in [65]. For the reader’s convenience, we present the most important here:

• Any point belonging to the positive half-plane, i.e., $dx/dt>0$, moves to the right of its dynamic route, thus increasing the value of variable x.
• any point belonging to the negative half-plane, i.e., $dx/dt<0$, moves to the left of its dynamic route, thus decreasing the value of variable x.
• The higher a dynamic route is located, the faster a point will move along it, as long as this route belongs to the upper half-plane (thus, its points move to the right).
• The lower a dynamic route is, the faster a point will travel along it, as long as this route belongs to the lower half-plane (thus, its points move to the left).
• All the points with null velocity $dx/dt=0$, i.e., those found on the horizontal axis, define equilibrium states of the system and are called equilibrium points.
• Equilibrium points may be stable or unstable. This means that the dynamic routes that lead to them may converge towards them or diverge from them, correspondingly.

It is very important to mention that the dynamic route corresponding to a null parameter value (i.e., for memristors that would be a voltage v = 0 or a current i = 0) defines the so-called Power-Off Plot (POP), which provides the stability points [49]. Specifically, memristors having one stable and two unstable equilibria appear to be volatile, while memristors with one unstable and two stable equilibrium points appear to be nonvolatile.

As a final point, we would like to mention that the DRM can be used to study the dynamics of the switching process in a memristive device (i.e., the setting and resetting procedures). These dynamics can be visualized on the DRM, since these two operations are achieved when the operation point is obliged to shift its dynamic route, during some time; this is something that happens when positive or negative voltage pulses are applied, further resulting in switching between two equilibria.

4. Experimental Results

4.1. Description of the Used Devices

The devices used in this paper were TiN/Ti/HfO${}_2$/W metal–insulator–metal (MIM) stacks [66]. They were made on a 100 mm-diameter Si-n${}^{++}$ wafer by first creating a deposition of a 200 nm-thick Ti adherence layer, which was then followed by a deposition of a 50 nm-thick W layer. Both of those layers were grown using Magnetron sputtering. A plasma-enhanced chemical vapor deposition (PECVD) using silane (SiH${}_4$) as the precursor process was used to create a 100 nm SiO${}_2$ isolation oxide. This was in turn patterned using photolithography and dry etching. The apertures of the SiO${}_2$ layer define the active area of the MIM devices. After that, the 10 nm HfO${}_2$ film was deposited by atomic layer deposition (ALD) at 225 °C using Tetrakis(dimethylamino)hafnium (TDMAH) and H₂O as precursors and N${}_2$ as the carrier and purge gas. Magnetron sputtering was used again to define the top electrode as a metal layer of 10 nm-thick Ti and a 200 nm-thick TiN. The electrode was patterned by a lift-off process.

During the last step, the magnetron sputtering was used to deposit a 500 nm Al layer on the back of the wafer for electrically contacting the bottom electrode through the Si-n${}^{++}$ substrate. The final TiN/Ti/HfO${}_2$/W devices presented a square shape, and the fabricated areas ranged between 2 × 2 $\mathsf{\mu }$m${}^2$ and 120 × 120 $\mathsf{\mu }$m${}^2$.

A typical curve representing the evolution of the conductance G for a set of different amplitudes of a triangular input waveform is presented in Figure 2.

4.2. Model Generation

Following the work published in [51], we propose an approach that utilizes experimental approximation to the variation of the memristance:

$$\begin{array}{c}\hfill i=\frac{1}{M}vs.=Gv\end{array}$$

(12)

where the dynamics of the memconductance G are determined by a differential equation:

$$\begin{array}{c}\hfill \frac{dG}{dt}=g_G(G,v)\end{array}$$

(13)

It is clear that Equation (13) corresponds to a possible DRM. Due to our election of G as the variable of interest, it is possible to calculate an approximation for $dG/dt$ from the experimental data, plotting it as a function of G and v.

As a first step, we have approximated the temporal variation of G by a discrete approximation:

$$\begin{array}{c}\hfill \frac{dG}{dt}\approx \frac{G\left(t\right)-G(t-dt)}{dt}\end{array}$$

(14)

where $dt$ is the time interval between two samples. Once this is calculated, we can plot the DRM for this data in the form of a 3D picture, as shown in Figure 3. This figure shows a subset of the data, marked as the lines with an ‘x’ symbol. This subset of data has been then used to obtain an approximation to Equation (13) with a polynomial function (using MATLAB polyfit55).

$$g_G(G,v)\approx \sum _{i=0..5,j=0..3}{p}_{i,j}{v}^i{G}^j$$

(15)

The value of each coefficient $p_{i,j}$ is provided in

Table 1. Coefficients for the approximation of $g_G$ in Equation (15). The units for coeffient $p_{ij}$ are $s^{-1}{V}^{-i}{\mathsf{\Omega }}^{j-1}$.

Coefficient	Value	95% Confidence
p${}_{00}$	−3.19 × 10${}^{-4}$	(−5.93 × 10${}^{-4}$, −4.56 × 10${}^{-5}$)
p${}_{10}$	3.77 × 10${}^{-4}$	(−1.03 × 10${}^{-4}$, 8.57 × 10${}^{-4}$)
p${}_{01}$	3.42 × 10${}^{-1}$	(−7.88 × 10${}^{-2}$, 7.63 × 10${}^{-1}$)
p${}_{20}$	1.96 × 10${}^{-3}$	(1.04 × 10${}^{-3}$, 2.88 × 10${}^{-3}$)
p${}_{11}$	−4.70 × 10${}^{-1}$	(−9.22 × 10${}^{-1}$, −1.84 × 10${}^{-2}$)
p${}_{02}$	−1.24 × 10${}^2$	(−3.43 × 10${}^2$, 9.56 × 10${}^1$)
p${}_{30}$	3.52 × 10${}^{-3}$	(2.18 × 10${}^{-3}$, 4.87 × 10${}^{-3}$)
p${}_{21}$	2.68 × 10${}^{-1}$	(−2.57 × 10${}^{-1}$, 7.93 × 10${}^{-1}$)
p${}_{12}$	2.28 × 10${}^2$	(8.40 × 10${}^1$, 3.72 × 10${}^2$)
p${}_{03}$	1.33 × 10${}^4$	(−3.44 × 10${}^4$, 6.10 × 10${}^4$)
p${}_{40}$	2.26 × 10${}^{-3}$	(1.37 × 10${}^{-3}$, 3.14 × 10${}^{-3}$)
p${}_{31}$	−1.41 × 10${}^{-1}$	(−5.16 × 10${}^{-1}$, 2.34 × 10${}^{-1}$)
p${}_{22}$	−1.62 × 10${}^2$	(−2.46 × 10${}^2$, −7.74 × 10${}^1$)
p${}_{13}$	−3.65 × 10${}^4$	(−5.37 × 10${}^4$, −1.93 × 10${}^4$)
p${}_{04}$	−4.13 × 10${}^4$	(−4.57 × 10${}^6$, 4.49 × 10${}^6$)
p${}_{50}$	4.40 × 10${}^{-4}$	(2.31 × 10${}^{-4}$, 6.48 × 10${}^{-4}$)
p${}_{41}$	−2.46 × 10${}^{-1}$	(−3.50 × 10${}^{-1}$, −1.42 × 10${}^{-1}$)
p${}_{32}$	−2.41 × 10${}^1$	(−4.71 × 10${}^1$, −9.93 × 10${}^{-1}$)
p${}_{23}$	1.15 × 10${}^4$	(7.47 × 10${}^3$, 1.55 × 10${}^4$)
p${}_{14}$	1.80 × 10${}^6$	(1.11 × 10${}^6$, 2.49 × 10${}^6$)
p${}_{05}$	−3.56 × 10${}^7$	(−1.92 × 10${}^8$, 1.20 × 10${}^8$)

, together with the 95% confidence interval, as provided by MATLAB. Notice that this polynomial is actually an approximation to $g_G$. Figure 4 shows the surface defined by this approximation and the original data.

The validity of the proposed method has been tested using the fitted polynomial approximation to predict the response of the system to a specific input. This has been performed by integrating the differential equation system formed by Equations (12) and (13), where $g_M$ is the fitted approximation using the coefficients from Table 1, and the initial condition was chosen as the experimental initial value of G. The results of this integration are depicted in Figure 5, where the thick lines are the model and the thin lines featuring symbols correspond to the experimental data.

5. Discussion

The goal of this paper was to show how the DRM technique can be used to provide insight into the governing equations of a memristor or a memristive device. To do so, we first established a working framework that can be naturally linked to DRM [48]. Using this framework, we defined a variable of interest (in this case, this was conductance), and we represented the experimental measurements in the phase space. As a result, we plotted $dG/dt$ as a function of both G and v. Notice that this plot provided us with the required information to approximate the differential equation that included the underlying physical and chemical mechanisms that determine the behavior of the device. Approximating this plot by the polynomial function defined by Equation (15) and the corresponding coefficients in Table 1, we generated a semi-empirical model of the device. It is noted that this was due to information provided by the DRM technique.

Then the model was used to reproduce the observed electrical curves. To do so, we integrated the resulting set of differential equations, using a time-dependent voltage-input, defined as a ramp, just as in the real measurements. Notice that this is required, since the use of a different input signal would provide a different curve [59,67]. Comparison between the model and the experimental data showed a fairly good agreement, justifying the use of the DRM approach. The modeling could be improved by using a different approximation to the DRM instead of a polynomial, using for instance the so-called hysterons proposed in [68].

Further work related to this should be performed to check this conclusion. In particular, the effect of the input signal velocity has to be tested, as well as the variability in the cycle to cycle switching characteristics.