A Physics-Informed Recurrent Neural Network for RRAM Modeling

Abstract

Extracting behavioral models of RRAM devices is challenging due to their unique “memory” behaviors and rapid developments, for which well-established modeling frameworks and systematic parameter extraction processes are not available. In this work, we propose a physics-informed recurrent neural network (PiRNN) methodology to generate behavioral models of RRAM devices from practical measurement/simulation data. The proposed framework can faithfully capture the evolution of internal state and its impacts on the output. A series of modifications informed by the RRAM device physics are proposed to enhance the modeling capabilities. The integration strategy of Verilog-A equivalent circuits, is also developed for compatibility with existing general-purpose circuit simulators. The Verilog-A model can be easily adopted into the SPICE-type simulator for the circuit design with a variable step that differs from the training process. Numerical experiments with real RRAM devices data demonstrate the feasibility and advantages of the proposed methodology.

1. Introduction

Resistive random access memory (RRAM) has attracted much attention due to its advantages, including high density, low power consumption, long endurance, simple metal–insulator–metal construction, and good compatibility with CMOS technology since it was predicted by Leon O. Chua in 1971 [1] and first manufactured by HP Labs in 2008 [2]. Recent work has been dedicated to verifying the application of RRAM in neuromorphic computing, next-generation non-volatile memory technology and in-memory acceleration [3,4,5,6,7].

High-quality compact models of RRAM are a key for RRAM-based circuit design and optimization [8,9,10,11]. Existing modeling methods can be roughly divided into physics-based and empirical categories. Physical modeling provides excellent accuracy, but often at the cost of high-consumption computing power for its use in circuit simulation. Empirical modeling is more efficient but usually suffers from a poor generalization capacity.

To balance accuracy and efficiency, aphysics-informed compact modeling strategy is often applied, which incorporates device physics into the development of empirical-like compact models. However, developing high-quality physical-informed models is a time-consuming process, especially for emerging RRAM devices due to its much more complex conduction mechanism. Modern RRAM technologies use a wide variety of materials as dielectric layers to enable memristic properties. The switching of high- and low-resistance states in RRAM also depends heavily on the electrode materials. Therefore, it is increasingly challenging for physics-based model development to catch up with the rapid innovations in materials and structures in RRAM and other mem-device areas.

An effective strategy to combat these modeling challenges is to adopt neural network (NN) techniques since for emerging devices, high-quality measurement data are often more readily available than comprehensive physical understanding. In several application domains, machine learning has already demonstrated some promise [10,11,12,13,14]. Neural networks using back propagation can model the relationships between different variables with certain generalization capabilities. A recent methodological investigation indicates that using artificial neural networks (ANNs) for advanced transistors effectively reduces the demand of expertise and turn-around-time, and facilitates automatic parameter extraction [15]. Ref. [16] developed the NN-assisted compact model to capture cycle-to-cycle variations to evaluate the effectiveness of RRAM using advanced materials. To model the I-V curves of RRAM, Ref. [17] employs convolutional neural networks (CNNs) using I-V loop photos to extract the fitting parameters of the dynamic memdiode model. Although a purely data-driven NN model usually can fit the measurement data well, it still lacks a physical foundation, leading to poor generalization or prediction in practical use. Meanwhile, a purely data-driven neural network model would need a large amount of data, negating its benefit of rapid modeling. Therefore, neural networks with prior knowledge derived from physical or empirical understanding are proposed to enhance network performance. Physics-informed neural networks shows advantages of good generalizability on limited training datasets. Since physics is incorporated, the network model is effectively limited to a low-dimensional manifold and can be trained with a small amount of data [18]. Hence, the design of NNs should be informed or guided by an appropriate extent of device physics to avoid the low generalizability that a purely data-driven model might suffer from. A preliminary effort of using NNs to model thin-TFETs was reported [19], in which knowledge of different $V_{gs}$ and $V_{ds}$ characteristics guides the choices of excitation functions of the NN. The proposed model was proven to be accurate and predictive.

Different from conventional semiconductor devices, such as transistors, the output of RRAM, e.g., currents, are determined not only by the current inputs, but also by an internal hidden state that reflects the influence of previous inputs. In other words, the currents can be different even for the same voltage at different points of the hysteresis loop. As a consequence, conventional ANNs, in which the output depends solely on the current inputs, are not suitable for RRAM modeling. NNs with certain “memory” effects are needed to capture the unique physics of RRAM as well as other types of mem-devices.

In this paper, we investigate the possibility of using a recurrent neural network (RNN) with the embedded devices physics to model the steady-state I-V behavior of the metal-oxide-based RRAM devices. The RNN consists of a feedback loop to update the internal state to account for the input history, and the output depends on both the current input and the hidden state. The entire network structure is carefully crafted according to the RRAM device physics, rendering a physics-informed RNN (PiRNN) model that can mimic the physical behavior of a RRAM device. The training and testing procedures are then detailed, followed by integration with SPICE-type simulators. The Python model is converted into a Verilog-A equivalent circuit to employ a different step size in the simulation from that used in the training process or a variable step size. The feasibility and accuracy of the proposed PiRNN model are finally verified by experimental measurement data from realistic RRAM devices.

2. Fundamentals and Existing Models for RRAM

The conductive filament (CF) model is the mainstream theory for metal-oxide-based RRAM devices. The low-resistance state (LRS) and high-resistance state (HRS) of RRAM are associated with the formation and rupture of the CF controlled by the applied voltage waveforms [20,21,22,23,24]. As shown in Figure 1, the generation and recombination of oxygen vacancies and oxygen ions lead to the growth and rupture of CF, i.e., changes of the tunneling gap distance g. The internal state variable g of this theory can be viewed as the average distance from the tip of the CF to the top electrode (TE). And g controls the resistance of the RRAM through the electron tunneling conduction mechanism.

The conductance and current of RRAM are strongly correlated with the CF geometry evolution, which depends on the history of the applied voltages. During the SET process, oxygen vacancies and ions are generated with the applied bias. The continuous drift of the oxygen ions towards the top electrode produces a conductive filament linking the top electrode to the bottom electrode, eventually leading to the conversion of the HRS to the LRS. During the RESET process, the recombination of oxygen ions and oxygen vacancies leads to the continuous dissolution of CF and ultimately causes the transformation of LRS into HRS.

The relationship of the I-V can be expressed as [22]

$$I=I_{0}exp\left(-\frac{g}{g_0}\right)sinh\left(\frac{V}{V_0}\right)$$

(1)

where $I_0$, $g_0$ and $V_0$ are fitting parameters and the current is strongly correlated with the gap distance and the applied voltage. The evolution of g can be estimated as the following functions [22]:

$$\begin{array}{c}\hfill \frac{dg}{dt}=-v_0\left[exp\left(-\frac{q{E}_{ag}}{kT}\right)exp\left(\frac{\gamma a_0}{L}\frac{qV}{kT}\right)-exp\left(-\frac{q{E}_{ar}}{kT}\right)exp\left(-\frac{\gamma a_0}{L}\frac{qV}{kT}\right)\right]\end{array}$$

(2)

$$\begin{array}{c}\hfill \gamma ={\gamma }_0-\beta {\left(\frac{g}{g_1}\right)}^3\end{array}$$

(3)

where $dg/dt$ represents the growth/dissolution velocity of the CF and the RHS is the difference between oxygen vacancy generation and recombination rates. Equation (2) describes the temporal evolution of g, where $E_{ag}$ and $E_{ar}$ are the activation energies for the generation and recombination of oxygen ions and oxygen vacancies, respectively. g contains $g_{max}$ and $g_{min}$. During the SET process, the generation of the vacancies contributes to the growth of CF, and g gradually decreases to $g_{min}$. During the RESET process, the recombination of the vacancies leads to the rupture of CF, and g gradually increases to $g_{max}$. $\gamma$ is a g-dependent local field acceleration parameter which considers the strong polarizability in high-k dielectrics [23], and L is the thickness of the oxide. $v_0$, $a_0$, ${\gamma }_0$ and $g_1$ are fitting parameters.

Existing RRAM models, e.g., [25,26], consist of two modules: one to describe the evolution of the internal state due to the applied bias and device parameters, often in the form of a differential equation or a dynamic sub-circuit, and the other to link the output current to the applied bias and the internal state of the device as shown in Figure 2. These internal state modules often demand substantial developmental efforts and complicate the parameter extraction process.

The existence of internal states complicates the development and deployment of RRAM compact models. First, the physical mechanisms governing internal states evolution remains less understood compared to other full-fledged devices, rendering the development of physics-based models a challenging and time-consuming task. Rapid innovations in RRAM technologies dictate frequent reworking of the models. Second, fully automatic parameter extraction is cumbersome for physics-based models. Third, interfacing with existing circuit simulators is not convenient. Some models even require modifications at the simulator level to handle the internal state variable. All these challenges hamper the quick turnaround in RRAM model development, undesirable for advanced design-technology co-optimization (DTCO) or path-finding activities.

Alternatively, one may choose to apply data-driven approaches to model RRAM devices, motivated by the fact that high-quality measurement data are often available earlier than comprehensive physical insights. However, purely data-driven models, such as look-up tables or simple NNs, usually do not provide adequate accuracy and generalizability for compact modeling [20]. In addition, the unique flux-dependent feature of RRAM renders the conventional ANNs without memory effect not applicable. New types of NNs are needed to account for the RRAM physics.

3. Physics-Informed RNN for RRAM Modeling

3.1. Basic Structure of PiRNN

We propose a physics-informed recurrent neural network (PiRNN) to model the steady-state behaviors of RRAM devices. Figure 3 depicts the structure of the proposed PiRNN model. The entire PiRNN takes in two consecutive voltages and the device width, then outputs the current flowing through the device.

Inspired by the two-module framework of existing RRAM models as shown in Figure 2, the PiRNN consists of an RNN module to model the internal state evolution associated to a given input voltage sequence, and an MLP module to generate the output current based on the internal state and the inputs. The RNN module, corresponding to the state module, takes the current voltage V, the previous voltage $V_{pre}$[27], and the device width W as the primary inputs and outputs the hidden state h, which is an m-bit vector. The hidden state is then fed back to form another input to the RNN, which is a critical operation that handles time-series data or the memory effect of RRAM devices in our application scenario. Given a sequence of input voltages, the goal of the RNN model is to determine a unique trajectory of the internal states of the RRAM devices.

The MLP module, on the other hand, is for determining the final output current, with a series of specialized designs guided by RRAM device physics. The MLP module is equivalent to the conduction module in Figure 2. By obtaining the internal state of the conductive filament, the current voltage, and the geometric parameters, it determines the current flowing through the device. In this work, we mainly follow the physical model reported in [28], which relates the output current to the input voltages and the internal state via Equation (1). Clearly, Equation (1) indicates that the current I depends on g and V in different (nonlinear) manners, representing different physical mechanisms. Informed by these physical insights, we choose to apply additionally an exponential activation function to the internal state h and a $sinh$ function to the input voltage V before sending them to the MLP. The width W is also squared to produce the conduction area input to the MLP based on the physical analysis in [29].

We adopt the following specific network structures for the PiRNN in this work. The RNN module has one layer with m $tanh$ neurons whose outputs form the m-bit hidden state h in Figure 3. The MLP module consists of three hidden layers, with $n_1$, $n_2$, and $n_3$ neurons, respectively. The above structure results in $m^2+9m$ parameters for the RNN and $m{n}_1+n_1{n}_2+n_2{n}_3+3n_1+n_2+2n_3+1$ parameters for the MLP.

3.2. RNN Cells

Figure 4 describes the structure of an RNN module in the proposed model and unrolled hidden layers through time. Each hidden layer in an RNN network can have an RNN cell with an n-bit hidden state. The input sequence updates the hidden state $h_t$ continually. Similar to an ANN, the RNN generates the hidden state for the current time step with the current input and the previous hidden state at each time step. The length of the input sequence determines the size of the unfolded RNN.

3.3. Training and Testing Procedures

Figure 5 illustrates the training process of the proposed model. The proposed PiRNN is trained with multiple sequences of steady-state I-V data for different device widths W acquired from experimental measurements or physics-based simulation. At a particular index of the sequence (called time t hereafter), the current voltage $V_t$, the previous voltage $V_{t-1}$ and the width W are fed to the model. These inputs, together with the previous state $h_{t-1}$ and the previous output current $I_{t-1}$, determine the current hidden state $h_t$. Then $h_t$, $V_t$ and W, after the corresponding physics-informed transforms, are input into the MLP to generate the output current $I_t$. The network equations read as

$$\begin{array}{cc}\hfill h_t& \hfill =f_{RNN}(V_t,V_{t-1},W,h_{t-1},I_{t-1})\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill I_t& \hfill =g_{MLP}(exp\left(h_t\right),sinh\left(V_t\right),W^2)\hfill \end{array}$$

(5)

where $f_{RNN}$ and $g_{MLP}$ refer to the manipulation of the inputs by the RNN module and the MLP module, respectively. In Equation (4), $V_{t-1}$ stands for the previous input $V_{pre}$, which is the input of the RNN module.

Specifically, before each run, the hidden state and the output current $(h_0,I_0)$ are initialized to zeros. All the weights and biases are randomly initialized within the range $(-\sqrt{1/m},\sqrt{1/m})$, where m is the hidden state size. The first two voltages $V_0$ and $V_1$, and W, are input to the RNN to generate the new state $h_1$. Then, $h_1$, $V_1$ and W are used to generate $I_1$ by the MLP. At the next time point, the pair $(h_1,I_1)$ is fed back to the input of the RNN, together with $V_1$ and $V_2$, to update the internal state to $h_2$. The process repeats for all the data points in the training sequence. If there are multiple training data sequences for, e.g., different widths, they will be applied sequentially to train the model.

For validation and testing, one can send the corresponding voltage sequences into the trained model and evaluate the error between the generated currents and the true currents. The step size, i.e., the interval between two adjacent voltage points in the sequence, should be similar to the one used in the training process. The entire training and testing process can be fully automated utilizing existing machine learning software and hardware infrastructures.

In applications, the trained PiRNN model can be implemented in Verilog-A for integration with SPICE-type simulators [15,30], or used directly as a behavioral model in high-level memristor simulation platforms such as MNSim [31].

3.4. Verilog-A Implement

The RNN finite difference equations are transformed into differential equations in order to utilize a different step size in the simulation from that used in the training procedure or a variable step size. The equivalent model for these differential equations is ultimately implemented by Verilog-A [32].

At the given moment $T=T_{t-1}+\alpha h$, a first-order expansion of continuous-time input voltages and hidden states is performed. $0<\alpha <1$ is the model parameter, and s is the time step of the training data. The input and hidden states of a discrete-time RNN can thus be expanded as follows:

$$\begin{array}{cc}\hfill V_t& \hfill =V+(1-\alpha )s\dot{V}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill V_{t-1}& \hfill =V-\alpha s\dot{V}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill h_t& \hfill =h+(1-\alpha )s\dot{h}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill h_{t-1}& \hfill =h-\alpha s\dot{h}\hfill \end{array}$$

(9)

During the implementation, the physical dimension width and the output current are treated as constants and thus involved in the calculation. Substituting Equations (6)–(9) into the hidden state Equation (4) yields the following. The physical dimension width and the output current are considered constants throughout implementation and are incorporated in the calculation. Equations (6)–(9) are substituted into the hidden state Equation (4) and the results are as follows:

$$\begin{array}{c}\hfill h+(1-\alpha )s\dot{h}=f_{RNN}\left(V+(1-\alpha )s\dot{V},V-\alpha s\dot{V},h-\alpha s\dot{h},W,I\right)\end{array}$$

(10)

The output is given by

$$\begin{array}{c}\hfill I=g_{MLP}\left(exp(h+(1-\alpha )s\dot{h}),sinh(V+(1-\alpha )s\dot{V}),W^2\right)\end{array}$$

(11)

Figure 6 shows the Verilog-A implementation of Equations (10) and (11). In the Verilog-A model, the hidden states are represented by node voltages of the controlled current source, and n equals 20. Each current source is connected to the ground and implements Equation (12), which is the difference of the LHS and RHS of Equation (11). The current flowing through each current source equals zero. The result of the KCL equation for each node is the solution of Equation (11). The $ddt$ function of Verilog-A is applied to generate $\dot{V}$ and $\dot{h}$. Thus, the entire Verilog-A model takes the voltage and width as input since the previous voltage $V_{i-1}$ is implemented as Equation (7), and generates the current flowing through the device:

$$\begin{array}{c}\hfill I[1:n]=h+(1-\alpha )s\dot{h}-f_{RNN}\left(V+(1-\alpha )s\dot{V},V-\alpha s\dot{V},h-\alpha s\dot{h},W,I\right)\end{array}$$

(12)

4. Numerical Results

4.1. Model Configuration and Data Acquisition

The PiRNN model used in our numerical experiments consists of an RNN module with 1 hidden layer and $m=20$ hidden state neurons, and a 3-layer MLP module with 256, 64, and 8 neurons in each layer, resulting in 28,565 parameters in total. The model is implemented by Python with the open-source deep learning framework PyTorch [33], with some network and training parameters reported in

Table 1. Details of the proposed model.

Parameters	Quantity
Model configuration	(20, 256, 64, 8)
RNN parameters	580
MLP parameters	$27,985$
Learning rate	$0.001$
Training Epochs	less than 5000

.

To collect data, we fabricated several ${\mathrm{HfO}}_2$ RRAM devices with different widths, and measured their steady-state I-V data with the probe station as illustrated in Figure 7. The width, or side lengths, of the RRAM devices are respectively $3\mathsf{\mu }\mathrm{m}$, $5\mathsf{\mu }\mathrm{m}$, $10\mathsf{\mu }\mathrm{m}$, and $15\mathsf{\mu }\mathrm{m}$. A 501-point sequence is produced by the voltage sweeping from $-2$ V to 3 V with a uniform step size of $0.02$ V. The sequence is generated by sweeping the voltage along $0\to$ positive limit $\to 0\to$ negative limit $\to 0$. The compliance current is set to $1\mathrm{mA}$ in the measurement of the I-V data. All the I-V curves are plotted in Figure 8.

To improve the numerical stability, the values of I in the sequences are converted to the logarithmic scale in the training process. The mean squared error (MSE) loss function with respect to the output current is used

$$MSE=\frac{1}{N}\sum _{i=1}^N{\left(I_i-{\widehat{I}}_i\right)}^2$$

(13)

where ${\widehat{I}}_i$ is the prediction of PiRNN, $I_i$ is the measured data, and N is the length of the current sequence. The adaptive moment estimation (Adam) [34] is applied.

4.2. Result Analysis

We first confirm the accuracy of the proposed PiRNN for devices with a single width. We used two I-V data sequences collected with $W=5\mathsf{\mu }$m and $W=10\mathsf{\mu }$m to train two PiRNN models. For each data set, we pick the 251 odd data points for training and the 252 even data points for testing (with overlapping start and end points), respectively. As shown in Figure 9, the test loss drops rapidly in the first 200 epochs and converges at about 1500 epochs. The final training and test losses are $0.000226$ and $0.000435$, respectively. The whole training process takes about 6 min. Figure 10 demonstrates good agreement between the predicted and measured currents in both linear and logarithmic scales for the two cases. The unique hysteresis loops of RRAM devices are well captured by the PiRNN model, owing to its modeling capability of internal state evolution.

Next, we show that the physics-informed modifications introduced in Section 3 indeed improve the model quality. Figure 10e,f show the IV curves from the PiRNN with the $exp\left(h\right)$ and the $sinh\left(V\right)$ functions removed, respectively, for the $W=10\mathsf{\mu }$m case. In comparison to the full PiRNN in Figure 10d at the same training epochs, one can see that removing these physics-informed modifications results in accuracy degradation, which in turn demonstrates the importance of including the relevant physical insights.

Furthermore, we prove that PiRNN can capture the dependence of the output current on device geometric parameters such as the width. We use the data sets of $3\mathsf{\mu }\mathrm{m}$, $10\mathsf{\mu }\mathrm{m}$, $15\mathsf{\mu }\mathrm{m}$ for training, and $5\mathsf{\mu }\mathrm{m}$) for testing. After applying the three training data sets sequentially, we achieve the final training and test losses of $0.187$ and $0.196$, respectively. For the proposed model, the R-squared value is $0.98$. The whole training process takes about 20 min. Figure 11 compares the predicted and the measured currents at the testing data ($W=5\mathsf{\mu }$m), which demonstrates reasonable accuracy.

4.3. Verilog-A Model

Finally, we implement the proposed PiRNN model in Verilog-A, build a simple test circuit with the Verilog-A model, and simulate it by Cadence Spectre. To validate the model, a piecewise-linear voltage source is applied in Figure 12 with $\alpha$ and s set to $0.1$ and $0.005$, respectively. Figure 13 shows that the Verilog-A model correctly reproduces the results of the Python model.

5. Discussion

This paper presents a novel approach for modeling the steady-state behaviors of RRAM devices, utilizing a physics-informed recurrent neural network. The proposed model consists of two sub-modules, namely the MLP module and the RNN module. The RNN module, acting as the state module, captures the RRAM history-dependent features by leveraging its “memory” capacity to represent the internal state quantity g. On the other hand, the MLP module, analogous to the conduction module, generates the current output by incorporating the current internal state, voltage, and geometric parameters with physics-informed modifications. The accuracy of the proposed model is initially showcased for a device of single width in Figure 10a–d, and its capacity to apprehend the correlation between the output current and the geometrical parameters of the devices with a reasonable degree of precision is depicted in Figure 11. Figure 10e,f serve to demonstrate that the model’s quality is indeed enhanced through the incorporation of physics-informed adjustments. The model’s development and implementation are facile, and its accuracy is validated against measurement data obtained from practical RRAM devices. The PiRNN is implemented in Verilog-A for integration with SPICE-type simulators and thus allows a different step size in the simulation from that used in the training procedure or a variable step size.

Previous works report several physics-based compact models for RRAM, which commonly involve a dynamic module or sub-circuit to model the temporal evolution of the internal state (such as the conductive filament length) and the device responses (such as the current). These internal state modules often demand substantial developmental efforts and complicate the parameter extraction process. The proposed model distinguishes itself from prior models by integrating physics-informed neural networks (NNs) into the modeling process, thereby addressing the conventional challenge of developing high-quality physics-informed compact models in a time-efficient manner. Specifically, the model employs a specific neural network with fundamental physics embedded and reliable measurement data to rapidly model RRAM, obviating the need for crafting differential equations or dynamic circuits based on physical principles. Additionally, despite the limited training data, the model exhibits strong extrapolation capabilities, owing to the incorporation of recurrent neural networks (RNNs) and fundamental physics. Hence, the proposed PiRNN presents an effective method to cope with the swift progressions in RRAM technology, as well as the prompt development of RRAM models and the co-optimization of advanced design-technology co-optimization (DTCO) or path finding activities. Furthermore, the proposed approach is versatile and can be applied to other emerging devices that demonstrate history-dependent behavior.

This paper presents a proposed I-V model that incorporates the device geometry W. Nonetheless, the temperature during the switching process exerts a noteworthy influence on the growth and dissolution of CF. Consequently, temperature is an indispensable factor that warrants consideration in forthcoming research. Additionally, to meet the demands of circuit design, a charge-voltage, Q-V, model is formulated, utilizing a neural network and the C-V data obtained from the RRAM devices.

6. Conclusions

Motivated by the demand of rapid yet quality model development of emerging devices, we proposed a PiRNN model combining an RNN module and an MLP module for the behavioral modeling of RRAM devices. The model exploits the ”memory” effects of RNN to mimic the internal state evaluation of RRAMs and apply an array of physical insights to guide the model design for better modeling accuracy and capability. The model is validated against measurement data from practical RRAM devices and is also implemented in Verilog-A for SPICE integration. All the results confirm the feasibility and practicability of PiRNN. More thorough studies of the temperature and charge-voltage model will be conducted in the future.