A Collective Study on Modeling and Simulation of Resistive Random Access Memory
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Abstract
In this work, we provide a comprehensive discussion on the various models proposed for the design and description of resistive random access memory (RRAM), being a nascent technology is heavily reliant on accurate models to develop efficient working designs and standardize its implementation across devices. This review provides detailed information regarding the various physical methodologies considered for developing models for RRAM devices. It covers all the important models reported till now and elucidates their features and limitations. Various additional effects and anomalies arising from memristive system have been addressed, and the solutions provided by the models to these problems have been shown as well. All the fundamental concepts of RRAM model development such as device operation, switching dynamics, and currentvoltage relationships are covered in detail in this work. Popular models proposed by Chua, HP Labs, Yakopcic, TEAM, Stanford/ASU, Ielmini, BercoTseng, and many others have been compared and analyzed extensively on various parameters. The working and implementations of the window functions like Joglekar, Biolek, Prodromakis, etc. has been presented and compared as well. New welldefined modeling concepts have been discussed which increase the applicability and accuracy of the models. The use of these concepts brings forth several improvements in the existing models, which have been enumerated in this work. Following the template presented, highly accurate models would be developed which will vastly help future model developers and the modeling community.
Background
This new age of computing requires a technology being equally capable to match its growth. The new technology should be able to meet the demands of improved performance and scalable to cater to the future devices. Memristors, postulated in 1971 [1] by Leon O. Chua seems to fulfill these requirements and laid the foundation for new classes of devices. Memristors, short for “memoryresistors,” are basic twoterminal devices which remember their internal resistance state depending on the history of the input stimulus provided. Chua devised that the memristors are characterized by a relationship between flux and charge, which are the time integrals of current and voltage, respectively.
Later in 1976, Chua and Kang [2] generalized the memristors to include in a new class of dynamical systems called memristive systems. In the end of twentieth century, the interest in these devices had waned despite its many benefits. This was partly because of the advances in silicon integrated circuit technology. But with the aging on silicon technologies and their incapability to support scaling down, the search for alternative switching devices gained attraction in the early twentyfirst century. It was equally aided by the advances in the growth and characterization of nanoscale materials. This invariably leads to significant progress in understanding microscopic memristive switching.
Memristor technology got a major breakthrough in the year 2008 when Strukov et al. [3] established a link between the theory and experiment for their TiO_{ x }based devices. Also, they obtained a pinched hysteresis in the currentvoltage relationship, which is one of the identifiable features of memristive systems [4, 5]. This opened up the memristor technology to a wide array of devices following the footprints of the metal/oxide film/metal structure. Some of the similar types of popular devices were Oxygen RRAM (OxRRAM) [6, 7, 8, 9, 10] and Conductive Bridge RAM (CBRAM) [11, 12, 13] among many others. These devices are generally classified on the basis of their switching mechanism.
Resistive Random Access Memory (RRAM)
Research interest into these emerging devices heightened because the nonvolatile memristive behavior demonstrated could be harnessed into nonvolatile memory. They are being seen as potential alternatives of the flash memory technology. With present age computing being more and more data driven, there has been demands for a memory technology which is more intune with the present and future requirements. Compared to the several emerging devices, RRAM devices are more scalable [14, 15, 16, 17, 18], have high density [19, 20, 21, 22, 23, 24], consume low power [25, 26, 27, 28, 29], are faster [30, 31, 32, 33], have higher endurance and retention [34, 35, 36, 37] and highly CMOS compatible [38, 39, 40, 41, 42]. RRAM devices are one of the most popular nonvolatile memory technologies with extensive study being undertaken to understand their mechanism and develop models to realize the device operation and design accurate and simple device structure. The devices are simple twoterminal metalinsulatormetal (MIM) structure and switch between two resistance states lowresistance state (LRS) and highresistance state (HRS). A LRS suggests the device is in the SET or ON state. A contrasting HRS means the device is in the RESET or OFF state. Through this switching of resistance states in the device, the data bit is stored [43, 44, 45]. RRAM devices can be classified into bipolar and unipolar devices, depending on the polarity of switching. In unipolar switching, the devices switch in the same polarity bias, whereas in bipolar switching, bias of both the polarities is required.
Several approaches have been proposed to explain the switching mechanism of RRAM devices, but the most popular and widely accepted, for binary oxidebased RRAM devices, is the formation and ruptured of localized conductive filaments (CF) by the drift of oxygen ions/ vacancies [9, 16, 46, 47, 48, 49]. The SET/RESET occurs as a result of the combination/regeneration of the oxygen ions/vacancies [50, 51, 52]. It has been demonstrated that the performance of the RRAM devices is strongly affected by the choice of the active oxide layer [53, 54, 55]. A variety of oxide systems such as HfO_{ x }, TiO_{ x }, NiO_{ x }, TaO_{ x }, ZnO_{ x }, etc. [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66] have been used to demonstrate resistive switching behavior. There have been some controversies whether RRAM devices are actually memristive devices. To make the position of RRAM devices clear, Chua provided clarifications that they are indeed memristive devices [67].
Importance of RRAM Modeling
A very important aspect of developing electronic devices based on new semiconductor technologies is the role of modeling. An accurate and comprehensive model is of paramount importance in understanding the device operation, designing it for optimum performance, and verifying that it matches the required specifications. A number of models have been proposed with varying degrees of accuracy, different features, and mixed results. So, any developer aiming to design a robust and flexible model for RRAM devices should have information about the methods tried before and the constraints faced.
In this work, we have discussed in detail all the features and characteristics of the various RRAM models. General memristor models are also considered to explain RRAM devices [67]. Starting from the Chua model [1] which provides the basics of memristors, we discuss the fundamental definition of memristors. The breakthrough for memristors and RRAM devices provided by the HP model [3] is discussed in detail. Linear ion drift effects, which form the basics of the mechanism of these devices, along with the nonlinear effects [46, 68, 69], are considered. The PickettAbdalla model [70, 71, 72] which laid the foundation for SPICE compatible physicsbased models is covered indepth. Its various features which have been adopted and refined by the Yakopcic model [73, 74] are also covered.
Comparative analysis of the models
Model  Device type  State variable  Control mechanism  Threshold exists  Supports boundary effects  Simulation compatible 

Generic  Flux or charge  Current  NA  NA  NA  
Linear ion drift [3]  Bipolar  0 ≤ w ≤ D Doped region physical width  Current  No  External window functions  Possible with SPICE 
Bipolar  0 ≤ w ≤ 1 Doped region normalized width  Voltage  No  External window functions  No  
Exponential [69]  Bipolar  Switching speed  Voltage  No  Yes  No 
Bipolar  a_{off} ≤ w ≤ a_{on} Undoped region width  Current  No  No  SPICE  
Bipolar  0 ≤ w ≤ 1 Not explained physically  Voltage  Yes  External window functions  SPICE/Verilog/MAPP  
Bipolar  x_{on} ≤ x ≤ x_{off} Undoped region width  Current  Current  Implicit window functions  SPICE/Verilog/MAPP  
VTEAM [77]  Bipolar  x_{on} ≤ x ≤ x_{off} Undoped region width  Voltage  Voltage  Implicit window functions  SPICE/Verilog/MAPP 
Bipolar  Filament gap (g)  Voltage  Temperature  No  SPICE/Verilog/MAPP  
Unipolar  Concentration of ions  Voltage  Temperature  No  COMSOL  
Physical electro thermal [87]  Bipolar  Concentration of ions  Voltage  Temperature  Practically yes  COMSOL 
Bocquet unipolar [90]  Unipolar  Concentration of ions  Voltage  Temperature  Yes  COMSOL/SPICE 
Bipolar  CF radius  Voltage  Temperature  Yes  SPICE  
GonzalezCordero [93]  Bipolar  CF radius (top and bottom)  Voltage  Temperature  Yes  SPICE 
Various models based on window function implementations such as Joglekar [94], Biolek [95], BenderliWey [96], Shin [97], Prodromakis [98, 99], etc. have also been accounted for the limitations and constraints in the various models, and the methods used by subsequent models to overcome them have been presented in a comprehensive manner. Significant work done by Wang and Roychowdhury [100] to improve RRAM modeling has also been reviewed in depth as it is a considerable push in the right direction for the whole RRAM modeling community. Along with those examples, covering simulation and verification studies of the devices in different platforms are discussed. This is the most comprehensive review relating to RRAM and memristor models at present stage. The description of the models has been divided into those that describe bipolar devices and unipolar devices. Window function implementation models are described in a separate section.
Earlier, there have been multiple reviews on RRAM device mechanisms [46, 101, 102, 103, 104, 105], fabrication technology [106, 107, 108, 109], material stacks [110, 111, 112, 113], and a concise discussion on some of the models present at that time [114]. Very recently Villena et al. [115] combined the theory of all RRAM modeling and proposed an optimize model. In this study, we focused more on the various modeling techniques along with the solutions provided to various drawbacks. A comprehensive discussion on boundary condition models which can be classified as pseudocompact models have also been discussed. Some critical modeling techniques have been investigated in this work which can significantly help model developers. Also, a discussion on various simulation techniques and platforms for RRAM models such as SPICE [116, 117] has been included which is highly essential. Our work aims to fill a significant gap in the RRAM modeling community.
RRAM Models for Bipolar Devices
Chua Model
Leon O. Chua in 1971 put forward the idea of memristor [1] that it was indeed the fourth basic element alongside the resistor, capacitor, and inductor. The basic characteristics of a memristor are believed to be flux controlled (φ) or charge controlled (q) and are defined by a relation of the type g (φ,q) = 0.
The value of incremental memristance (memductance) at a time instant t_{0} depends on the time integration of the complete memristor current (voltage) from t = − t to t = t_{0}. So, this translates to the fact that while a memristor acts as a normal resistor at any instant of time t_{0}, but its resistance (conductance) values depend on the complete past history of the device current (voltage), hence the justification of the name memory resistor.
Interestingly, at the time of specified memristor voltage v(t) or current i(t), the memristor behaves as a linear timevarying resistor. But in the case when the φq curve is a straight line, i.e., M(q) = R or W(φ) = G, the memristor acts like a linear timeinvariant resistor. So, a memristor device cannot be used in linear network theory but can be used to define circuits where the present state of the parameters is dependent on the past states.
The memristor equations were used reasonably to define the variable state of a threshold switch by Chua [1], which are the first instance of using memristors in device modeling. Formulation of the memristor by Chua rightfully laid the foundation for a new class of devices and varied applications which use a basic circuit element to store data. This basic concept of memristors led to the design of new architectures for future nonvolatile memory applications of which RRAM is a promising candidate. There has been significant amount of theories explaining the working of RRAM devices and models defining them, which are fundamentally based on the memristor model.
Here, ϕ_{ r } and ϕ_{ s } are the RESET and SET flux, respectively. These equations can be implemented into a SPICEcompatible circuit comprising of a network of capacitors. The SPICE implementation results were found to be closely following the experimental results with the model able to reproduce almost identical memristor characteristics. It validates the use of the Chua fluxcharge model [1] to be used for modeling unipolar devices as well.
Linear Ion Drift Model
With a considerable gap in the consequent decades after the formulation of the memristor by Chua, researchers at HP Labs [3] in 2008 made an exciting find regarding memristor devices. Although Chua had formulated the presence of an element such as a memristor, there had not been a realizable circuit or model developed after that although several efforts were reported to fabricate RRAM devices in the very beginning of twentyfirst century. The team at HP Labs led by Strukov et al. [3] realized a functional nanoscale memristive system where memristance occurs naturally, where solidstate electronic and ionic transport are coupled together under an external voltage bias. Those systems show a hysteretic relation between the current and voltage characteristics similar to other nanoscale electronic devices, thus leading to a fundamental understanding of memristive systems and the design of similar systems.
A simple twoterminal device was reported, where an oxide (TiO_{2}) of thickness D was sandwiched in between two Pt electrodes. Hysteresis IV switching curves have been compared with the simulated curve. Although the exact mechanism of these devices was not completely understood at that time, it was one of the first instances where resistive switching memories were classified into memristive systems.
In the above Eq. (15), the qdependent term is the primary contribution to memristance. An interesting analysis provided as to why this particular phenomenon was hidden for so long is due to that magnetic field did not play an explicit role in the mechanism. For a memristor to be realized in simple terms, there should exist a nonlinear relationship between the integrals of voltage and current.
The Eqs. (13)–(15) also incorporate the fundamentals of bipolar switching, that is the device switches from one state to another by the application of voltage of two polarities. As a result, devices showing bipolar hysteretic IV relationships are capable of being modeled by these equations, and hence leading to the classification of such devices as memristive systems. Such behavior is observed in many material systems such as organic films [119, 120, 121, 122, 123], chalcogenides [124, 125, 126], metal oxides [127, 128, 129], dielectric oxides [130, 131, 132], perovskites [133, 134, 135, 136], etc. The HP team themselves used a TiO_{2} [3] system and observed similar bipolar switching characteristics, with the dopant or impurity motion through the active region as the reason for such dramatic changes in the resistance. This is shown in Fig. 2b, c with the current showing drastic drop and rapid rise with the change in voltage.
This model could be attributed to laying the foundations for future RRAM models. It can also be used for two terminal semiconductor devices having bipolar hysteretic IV relationships. Taking the mechanism of a memristor as the reference, numerous future models for RRAM devices have been developed.
Nonlinear Ion Drift Model
Linear ion drift model developed by HP [3] primarily demonstrated linear drift effects in the bulk region of the memristor device. They observed some nonlinear effects at the boundaries but did not define it comprehensively. Nonlinear dependence of the dopant drift on applied voltage was observed and formulated by Yang et al. [46] in 2008. They proposed a currentvoltage relationship accounted for the nonlinear effects accurately. It was later improved and added upon by Eero Lehtonen and Mika Laiho [68].
Conduction in memristive devices is controlled by a spatially heterogeneous metal/oxide electronic barrier was reported by Yang et al. [46]. The switching is caused by the drift of positively charged oxygen vacancies acting as native dopants to form or dissolute conductive channels through this electronic barrier. The concentration of vacancies is higher at the boundaries or metal/oxide interfaces. The ON and OFF switching took place at the top interface only, which indicates that top electrode acts as the active electrode.
Here, β, γ, n, and χ are fitting constants. In the above equation, the first term βsinh(αv) approximates [1] the ON state of the memristor where the electrons tunnel through the thin residual electronic barrier. w is defined as the state variable of the device in the range of 0 (OFF) and 1 (ON). Second part of the equation approximates the OFF state of the device with the other parameters acting as fitting constants. Parameter n here acts as the free parameter used to modify the switching between the states. During the adjustment of n, the nonlinear effects come into picture. IV curve from the fabricated device is modeled using the Eq. (16). The best fitting is obtained at 14 ≤ n ≤ 22. This can be interpreted as evidence that the effective vacancy drift velocity depends in a very highly nonlinear way with the applied voltage to the device. So, the majority of the dopant drift effects at the boundaries/interfaces could then be understood as nonlinear in nature.
Here, the exponent q is used to mimic the rapid switching process. Transition between ON and OFF state in a memristor generally takes place very fast. An input voltage with a very high sweep rate is used to obtain such behavior. This is the first implementation of memristor models in the SPICE platform [116, 117].The major advantage of SPICE implementation is the ability of the model to be used in analog circuits and simulations and can be verified as fit to be circuit implementable or not. Although many improvements were made in subsequent models, this model lays the foundation for the rest of the RRAM models by accurately taking into consideration and explaining the nonlinear dopant drift effects [3, 46].
Exponential Ion Drift Model
In practice, resistance switching characteristics are nonlinear in nature. To analyze such exponential characteristics, Strukov et al. [69] proposed exponential ion drift model in 2009. This nonlinearity caused a significant variation in retention time and write speed. Due to the exponential dependence of the switching rate for high electric field, the exponential ion drift model is generalized to explain the phenomenon by the nonlinear microscopic drift of charged species in the dielectric at high field and temperature.
Here, ν is the drift velocity, f_{e} the frequency of escape attempts, T the device temperature, a_{p} the periodicity, E_{a} the activation energy, and E the applied electric field.
Simmons Tunneling Barrier Model
Though Lehtonen and Laiho [68] first proposed SPICEbased simulations model for nonlinear ion drift model as mentioned in the “Nonlinear Ion Drift Model” section, but this modeling is not suitable for use in an electricalbased time domain simulation, due to the lack of proper definition of simulation parameters and equations. This situation changed with the PickettAdballa et al. [70, 71, 72] model where a new class of model based on the device physics was demonstrated, which is capable of being explained and compatible with SPICE. The equations were modified to fit the requirements for SPICE implementation.
The parameters have been adjusted here such that the barrier height φ_{ b } is in volts (not in electron volts), and the timevarying tunnel barrier width w is in nanometers. In the equations above, A is the channel area of the memristor, e is the electron charge, h is the Planck’s constant, ε is the dielectric constant, m is the mass of electron, φ_{0} is a standard barrier height taken from reference [70], and v is the voltage across the tunnel barrier. B is a fitting constant. In lieu of the analytical form of the equations, they can be conveniently described and implemented in SPICE, or it can be implemented with the any SPICE compatible electrical simulator.
Here, f_{1,}i_{1}, a_{1}, b, w_{c}, f_{2}, i_{2}, and a_{2} are fitting parameters. The abovementioned equations are used to model the memristor on the circuit level considering the electron tunnel barrier as a voltagedependent current source, and the conducting channel (TiO_{2}) is modeled as a series resistance. The voltage drops across the tunnel barrier and the series resistance make up the complete voltage drop across the circuit.
The dynamic behavior of the device is visibly complex as it is physicsbased modeling approach and has been articulated as such by the Eqs. (27) and (28). The rate of switching possibly has contributions from the nonlinear drift at high electric fields and local Joule heating of the junction speeding up the thermally activated drift of oxygen vacancies [16, 46, 82, 83]. This can be clearly seen in the case of Fig. 6a, c [70] where the nature of the curves at high electric fields is quite different to those in low fields. The switching in the device is directly affected by the width of the gap. Application of a positive bias on the top electrode increases the state variable w resulting in an exponential increase in the resistance of the device as illustrated in Fig. 6b, d [70]. An opposite phenomenon occurs when negative bias is applied on the top electrode. This signifies the bipolar nature of the switching characteristics and their dependence on the dynamic state variable w.
Yakopcic Model
Although not validated specifically for RRAM devices at the time of development, the Yakopcic model [73, 74] closely resembled a variety of RRAM devices. The model was initially tested for TiO_{2} systems [73], and these systems are indeed one of the most popular ones along with HfO2based RRAM devices.
This model was based on the PickettAdballa model [70, 71, 72] using a similar state variable, but it was modified to include neuromorphic systems as well. It was one of the first models to consider the functioning of synapses into their equations. This model was verified for the device used by the HP lab team to explain the working of memristive systems.
A_{ p } and A_{ n } indicate the rate of the change of state once the voltage threshold is crossed. It can be understood as the dissolution or the rupture of the filament in terms of RRAM devices. There is inbuilt support for threshold values in the model, which enhances its applicability.
Here, f_{ p }(w,w_{ p }) is a window function which limits the value of f(w) to 0 when x(t) = 1 and v(t) > 0. f_{ n }(w,w_{ n }) is a similar window function which does not allow the value of w(t) to become less than zero when the current flow is reversed.
Owing to the analytical nature of the coupled equations, they can be solved using a mathematical solver such as MATLAB [138, 139]. The differential equation can also be solved in MATLAB using the inbuilt solvers idt() and ddt() functions, which employ the time step integration method. This particular model was simulated using the characterization data of the TiO_{2} memristor from HP Labs [3], and the fitting obtained was pretty good when the fitting parameters are properly calibrated.
A separate SPICE implementation of the same model was reported by Yakopcic et al. [74] which were fitted and characterized for a multitude of devices for both sinusoidal and repeated sweep inputs. The SPICE implementation revealed a good accuracy and applicability of the model at the circuit level. The model was correlated with a variety of experimental data, and low error rates of about 6% were obtained. It was one of the first SPICE implementation where the model was tested under sinusoidal as well as repetitive sweeping inputs. This helps in determining the AC behavior of the device. Along with that, very important device variability analysis is performed which defines the error tolerance in the device. Variability is an important issue, when the RRAM device is used in large systems, such as arrays. The variability analysis performed is essential in knowing until which point the system can tolerate the variability. After reaching the critical point, there is possibility of errors in device read/write.
The model was also tested for read/write operations using 256 devices, which helps determine its usability in crossbar arrays. Similarly, it can be used for neuromorphic read/write operations to test the model applicability in that system. Device variability in the model is defined with change in the device parameters. So, changing the device parameters leads to a change in the simulated device IV which is very useful in fitting the model with the experimental data. The values of the device parameters used can help define the accepted values of the particular parameters in the real case scenario. No convergence errors were found in the 256 array system, but with new RRAM array systems reaching higher density, applicability of the model there remains a question. Higher density array systems generally pose a convergence problem in SPICE simulations, but with proper parameter definition, it can be avoided. This model can be considered a new paradigm when it comes to circuit level SPICE simulations, variability analysis, and read/write operation simulations for RRAM devices.
TEAM/VTEAM Model
Threshold Adaptive Memristor (TEAM) model [75, 76] builds based on the Simmons Tunneling Barrier model [70, 71, 72] (discussed in the “Simmons Tunneling Barrier Model” section) and delivers a much simpler physicsbased modeling approach for memristive systems. IV relationship in this case is not fixed and can be chosen to fit any device which provides some amount of flexibility in the model. TEAM model arose from the need of simpler analytical equations which describe the mechanism of memristive systems accurately and which take less computation time.
The window functions describe the dependence of the derivative in the state variable x. They work well within the described boundaries, but the problem arises when the device goes beyond the boundaries. There are no limiting parameters here, and the window function only describes the state variable inside a particular limit. If the device goes beyond the boundaries, it can cause convergence issues with the simulator and it does not make sense for good modeling practice in case of analog devices.
Here, λ is a fitting parameter and R_{ON} the equivalent effective resistance at the bounds.
IV relationship for this model can be seen in Fig. 8a [76]. Although there is a presence of a pinched hysteresis, the form and structure of the curve are not welldefined. The model is driven with a sinusoidal input of 1 V. The verification done for this model is different from the tunneling model [70, 71, 72] in terms of the platform used to simulate it. The latter model uses a SPICE macro model [72] to describe the equations, but SPICE takes up a significant amount of computation time. Modeling in VerilogA [140, 141, 142, 143] is much more efficient, and the TEAM model [75] utilizes this functionality to model the equations presented by them.
A slightly modified version of the TEAM model with the introduction of voltage threshold levels was reported by the same group, called Voltage Threshold Adaptive Memristor model (VTEAM) [77]. Discussed TEAM model was based on threshold currents, whereas VTEAM is based on threshold voltages. The major advantages cited for using threshold voltages is that comparison with current causes performance and reliability issues if the condition is not satisfied, i.e., a lowcurrent threshold will automatically have a lowvoltage threshold as well. This might affect the overall performance of the device. Also with a threshold voltage, there is no risk with going overboard with high power and voltage destroying the device as the values are automatically controlled.
Stanford/ASU Model
A physicsbased model which has become very popular is the one developed by Guan et al. and Chen et al. of Stanford University and ASU, known as Stanford/ASU model [78, 79, 80]. This model is exclusively developed for RRAM devices, rather than a generalized one for memristive systems which was fitted for those particular devices. It included the effect of critical phenomenon of switching such as Joule heating and temperature change, which had been neglected before. The developed model was applied in the IV switching characteristics of HfO_{2}RRAM [144]. Along with it, VerilogA [79] and SPICE [81] implementations of the model are also presented.
The parameter E_{ a } is the activation energy for vacancy generation and oxygen vacancy migration in the SET and RESET processes, respectively. v is the applied voltage across the device, ν_{0} the velocity containing the attempttoescape frequency, L the switching material thickness and a_{ h }, the hopping site distance.
Here, T_{crit} is defined as a threshold temperature beyond which there is a significant change in the gap size. This can be understood as the point where the device undergoes a physical transformation such as transitioning into a SET or RESET state. In this case, threshold is considered in terms of temperature, rather than voltage or current, whatever employed in the previous models [75, 76, 77]. So, the equation basically depicts the resistance fluctuation that occurs when the CF temperature is increased beyond the room temperature.
A major feature of this model is implemented in the neuromorphic systems and RRAM synaptic device design [147]. This model has been tested against a HfO_{ x }/TiO_{ x } multistack RRAM system [148] which is implemented in a neuromorphic system. This gives the model great flexibility and wide applications as there are only a few models that are actually applicable for neuromorphic systems. Also, the model defined for these systems has been deemed tolerant to training error caused by device variation [149]. The gradual resistance modulation which is critical to the learning process in a synaptic device can be quantified in the model [150] which marks a significant development in using RRAM synaptic stacks in neuromorphic computing systems.
Physical ElectroThermal Model
This model is an extensive physical model which describes the bipolar operation in RRAM devices using equations closely resembling the physical mechanisms. This model was reported by Kim et al. [87], and it was verified with a tantalum pentoxide (Ta_{2}O_{5})based bilayered RRAM structure [15, 151, 152]. It makes use of the finite element solving method employed in the previous model to solve the differential equations. The major value addition by this model over the model proposed by Larentis et al. [86] was the proper description provided for the SET state in the bipolar RRAM device. The previous model was inadequate in accommodating the complete transition and explaining it properly but this model makes up for that. Also, it improved upon a physical electrothermal model reported by Menzel et al. [153] which attempts at calculating the CF temperature precisely.
It also uses the electrothermal physics phenomenon approach for modeling which we have seen in the previous model [86]. The major advantage with models based on this concept is their ease of use owing to the simple fundamental equations and the flexibility to employ a proper finite element method (FEM) solver to simulate the system very accurately. But a major disadvantage is that the model becomes very difficult to implement in circuit solvers based on SPICE and providing an equivalent implementation in Verilog. This is because of the lack of support in SPICE and Verilog for properly defining partial differential equations which make up for the vastness of the model. Normal ordinary differential equations and the ones which are in analytical form can be solved in circuit solvers but partial differential equations (PDE) cannot be solved.
Electrothermal models are equally important as compared to the other physicsbased models discussed before because temperature is an important factor governing the set and reset processes. Ion and vacancy migration plays a dominant role for switching mechanism [16, 46], although the governing factors are behind this process and the exact type of ions is still up for debate. So, the fact that temperature is a governing factor in this process makes these models attention worthy. Also, experiments [85, 154] in this regard suggest that there is significant change in the temperature in the CF during the switching process. Some of the previous models discussed above have neglected this effect by considering conducting filamentoxide interface to be at room temperature or by taking constant conducting filament temperature [39, 86, 88, 89, 144].
The major difference between this model and the previously discussed electrothermal model is in the expressions used to describe the driftdiffusion process. CF is described as a doped region where the oxygen vacancies act as dopants, and the CF runs from the top electrode to the bottom electrode. This is an assumption that many models take that the CF runs from one end of the electrode to the other when the state variable is considered as the length of CF. A few models discussed previously [78, 80] have used the filament gap to the top electrode as state variable. So, the assumptions generally vary from system to system and are dependent on what mechanism is employed to describe the device.
Here, l_{m} is the mesh size. So, using the Eqs. (48)–(50), the oxygen vacancy transport given in Eq. (47) can be defined which contains all the factors of driftdiffusion as well as the vacancy regeneration. These equations govern the CF growth and rupture which defines the physical transformation of the device during the SET and RESET transition of the device. So, it basically acts as a dynamic internal state variable which controls the switching rate of the device.
Equations (95) and (98)mentioned further are also used in the model to describe the current conduction and the temperature change due to Joule heating in the device. The equations are simultaneously solved in COMSOL to generate the required simulated profiles. The obtained simulated profiles are compared and verified against a TaO_{x} bilayered RRAM system [87]. In addition to the DC IV characteristics the model was also used to generate timedependent reset characteristics by investigating its response to square pulses.
Huang’s Physical Model
A very comprehensive physical model of RRAM devices is developed by Huang et al. [88, 89]. Its major feature is its consideration of the multitude of factors affecting the CF dynamics in the RRAM device. This model is comprehensive in the sense that it considers both the width of CF as well as filament gap to the electrode as factors affecting the state variable dynamics. The model was validated in a TiO_{2} based device and also applied in a 2 × 2 RRAM array cell [88].
Covering bipolar devices primarily, it also accounts for the temperature distribution in the device with multiple heating sources. SET/RESET process is considered to be caused due to generation/recombination process of the oxygen ions (O^{2−}) and oxygen vacancies (V_{o}). Top electrode (TE) is the active electrode and acts as an oxygen reservoir for the release or absorption of oxygen ions [88]. The CF evolution during the SET process is modeled based on the width of the CF. Growth of the CF is thought to start from the tip of the active electrode. With an increase of voltage the CF enlarges along the radius resulting in a final width of the CF as w. So, the value of w is critical to determine the LRS resistance in the SET process. Huang et al. [88] assumed that the CF grows in a symmetrical cylindrical shape which is simplifying at best. While the cylinder has been the most popular to describe the shape of the CF, it might not be the most accurate.
Rupture of the CF during the reset process is considered to start from the TE first. CF disconnects from the starting point and then dissolves internally with increase in the voltage. Distance between the tip of the CF and the active electrode layer is defined as the filament gap distance (x). The value of x determines the resistance of HRS during the RESET process. x and dx/dt are thus critical in defining the RESET process. A very important feature of the model is that there are two parameters defining the state of the system, in place of one parameter. The parameter w acts as the state variable for the SET process and x for the RESET process. So dx/dt and dw/dt define the dynamics of the device during the SET/RESET transition. Analytical model for a RRAM cell presented by Huang et al. [88] is developed by modeling the parameters x, w and their evolving speeds.
This model also presents one of the most detailed descriptions for the processes involved behind the RESET process. The rate of the CF shortening is affected by three processes, (a) O^{2−} release by the electrode, (b) O^{2−} hopping in the oxide layer, and (c) recombination between O^{2−} and V_{o}. Slowest process among the three dominates the CF reduction process which is defined by the parameter x. Speed of the processes is affected by the specific device characteristics and the oxide used.
Temperature effects in the model are considered from the Filament Dissolution model [82, 83] discussed further in the “Filament Dissolution Model” section. Validation of the model is performed in HfO_{x}/TiO_{x} system [88, 89]. Transient results obtained from simulating the model are compared against the data from the device, which shows a good match as demonstrated by Huang et al. [88]. The model is also validated against devices fabricated by other groups [144, 159] and the parameters are adjusted accordingly. A pretty accurate match between the simulation and the experimental results suggests a good level of flexibility with the model. The model also demonstrates that the switching speed of the device is highly dependent on the input voltage sweep rate.
Although the model is very comprehensive and takes into account a variety of detailed processes affecting the RRAM operation; it has some critical shortcomings. A major one is the noncompatibility with SPICE or VerilogA. Implementations in any of the circuit simulators based on these platforms has not been demonstrated which raises a question on its readiness for simulations. Also, boundary conditions and nonlinear effects have not been applied in the model which leaves it open to unphysical solutions. There has been no attempt to fit a window function with the model to account for this effect. These shortcomings make the model difficult for application for simulations, but its physics give a lot of insights into the functioning of RRAM devices.
Bocquet Bipolar Model
A very interesting and unique model from Bocquet et al. [90, 92] which utilizes a physics based modeling approach to describe bipolar oxide based resistive switching memories. This was a model developed exclusively for the RRAM devices. Although a point of speculation still exists, it has been more or less accepted that the bipolar resistive switching mechanism is governed by the valence change mechanism which occurs in specific transition metal oxides and the fieldassisted motion of oxygen ions O^{2−} [160].
This is also one of the few models that can describe electroforming process. This process basically initiates the CF growth for the first time when the device is in a pristine state. It requires significantly higher voltage as compared to the set or reset voltage because the CF formation requires an electric breakdown of the oxide and this requires higher voltage and energy. However, forming free RRAM devices have been reported [85] by adjusting the oxygen stoichiometry of the active layer. Removal of the forming process will reduce the voltage requirement of the device and make it more energy efficient.
Bocquet bipolar model uses some concepts from the Bocquet unipolar model [90] and modifies it significantly according to the bipolar switching characteristics. Major features of the model are its intrinsic simplicity in the model equations, full compatibility with SPICE based electric simulators and inclusion of voltage and time dependencies of the device. Internal state variable here is the radius of the CF which governs the switching rate. Radius of the CF varies with growth/rupture mechanism of the CF which is explained in the model with the help of local electrochemical redox processes [82, 83, 105, 161] which are dependent on the applied bias polarity. A single master equation in which both the SET and RESET processes are accounted for simultaneously is controlled by the CF radius which thus gives the switching rate of the device.
On the face of it, the equations seem pretty complex to evaluate. But in reality, they are analytical in nature which makes them easily solvable in a numeric solver and can be implemented in an electric simulator. This is a major advantage of this model. Almost all of the models which employ the concept of temperature change in the filament follow the basic principles of the filament dissolution model [82, 83] discussed further in the “Filament Dissolution Model” section. During set operation, the temperature rises due to the increase in the CF radius, while it falls due to a decrease in the CF radius during the reset operation. This creates a positive feedback loop between the two processes leading to a selfaccelerated reaction. This forms the basis of the filament dissolution model and all models incorporating the temperature effects in the device converge on this phenomenon [82, 83, 86, 87, 88, 89, 92].
Another important feature of OxRAM that has been highlighted in the model is the softreset [168]. It mainly induces the dependence between resistance in HRS and the stop voltage during the preceding reset operation. This phenomenon is basically due to the incomplete destruction of the CF during the reset process. So, the CF radius and temperature decrease during this process, leading to a decrease in the reaction rate. This means a selflimited reaction rate thus getting the name softReset. This model can account for the device to device variability very efficiently [169, 170]. The standard deviation obtained for the important parameters such as the length of the oxide (L_{x}) is well within the accepted range, thus accounting for the variations when the materials change in different devices [167, 168].
A shortcoming in the model which can be highlighted is the lack of a voltage or current threshold. Also, it works on the simplifying assumption that the CF radius grows from one end of the top electrode to the other end of the bottom electrode. This makes the model immune to significant fluctuations if the growth of the CF is not complete, thus leaving a filament gap. There is no provision to account for the effect of the filament gap if it occurs.
BercoTseng Model
The proposed model and simulation approach [171, 172, 173, 174, 175] by BercoTseng for RRAM devices is based on describing the CF growth process. The Gibbs free energy criteria [174, 175] is used as an indicator to model the growth dynamics of the CF. Simulation approach for the forming, set and reset process in the model is based on the Metropolis Monte Carlo algorithm [174]. This approach importantly does not rely on time evolution of the CF, thus increasing the efficiency of comparison of the relative retention properties of MIM structures.
The model is quite comprehensive in terms of describing the underlying physical parameters which affect the CF kinetics in the resistive switching layer. It also introduces the concept of “hotspots” [172, 173, 174] which are random localized initial clustering of oxygen vacancies which facilitate the formation of the CF. The major parameter governing the Gibbs free energy is the enthalpy of formation of an oxygen vacancy [174] is used to define the CF growth dynamics in the switching layer and integrate it into the Monte Carlo simulator. As a result, all the CF processes, namely forming, set and reset can be effectively simulated.
Typical boundary condition, such as V_{dd} at top electrode and ground at bottom electrode is applied to the device. For modeling the CF accurately, it is divided into a grid structure to discretize it, which is in line with the finite element analysis (FEA) method. The various parameters defining each grid site are its spatial coordinates (x, y), local potential φ, temperature T, N_{ o }, N_{ ov }, trap occupancy c, electrical conductivity σ and thermal conductivity k_{ t }h. The various processes associated with the evolution of CF within the oxide layer involves generation, recombination and hopping of oxygen (O), oxygen vacancies (OV) and electrons.
Here, E is the local electric field, C_{n} represents the ratio of N_{ov} (density of oxygen vacancies) in the low state to the maximal one at site n (n^{th} grid site). E_{ a } and E_{ h } are the activation energies for oxygen species generation and hopping respectively. Similarly, a_{ a } and a_{ h } are the field lowering factor for O generation and hopping.
Here, d_{mn} is the distance between m and n, α is the typical attenuation length of the electron wave function in the trap and c being the trap occupancy.
GonzalezCordero et al. Bipolar Model
It is a compact physical model proposed by GonzalezCordero et al. [93] describing the working of bipolar RRAM systems. The model is unique because it considers the CF as a truncated cone, which is a significant departure from previous models considering the CF shape generally as a cylinder. Also, the model is validated by implementing it in VerilogA which gives us a closer look into the description and simulation of RRAM devices on the circuit level using VerilogA. The proposed model builds upon the concepts introduced in the previous Bocquet bipolar model [91, 92] and modifies it accordingly to suit the new CF shape proposed.
One of the important aspects about the model is the consideration of a truncated cone shaped CF [176, 177, 178, 179]. Majority of the models we have encountered till now consider the CF as a symmetrical cylinder which is more of a simplifying assumption [91, 92]. This is because it has been shown that the CF can grow [39, 51, 55] from one end of either electrode to the other depending on the active electrode. So, it is quite possible that the CF in this case might not be a perfect cylinder. So, a truncated cone is equipped to account for any variability and fluctuations arising due to the shape of the CF. Shapes other than simplified geometrical shapes are not considered in the models because of algebraic complexities. In previous models [91, 92], we noticed that device to device and cycle to cycle variability’s have a significant effect on the application of particular models to devices. So, by taking a truncated cone as the CF shape provides this model more flexibility than the others.
Another significant feature is the role of temperature in the CF and the reset process. Majority of the models which describe the CF rupture due to the selfaccelerated dissolution, consider that the process takes place at the CF narrowing point and temperature increases at that particular point [82, 83, 84, 91, 92]. This point is generally in the middle of the cylindrical CF due to its symmetry. So, when we look at it from a physical standing point, the temperature at each of the points in all the RRAMs stacked together in a circuit has to be evaluated. Realizing this from the circuital standing point and simulating thousands of devices in the circuits is a very timeconsuming process and slows down the simulation. This problem can be circumvented by considering two temperatures in the CF instead of the general single temperature; this approach also keeps the simplicity of the model intact. Two temperatures represent the main CF body that is not destroyed during the reset operation and the CF narrowing. This has been implemented in this model by considering the two temperatures as the wide region and the narrow region of the truncated cone, respectively.
This model extends the previously discussed Bocquet bipolar model [91, 92] in the “Bocquet Bipolar Model” section. In the previous model, the equations were defined keeping a cylindrical CF in mind, so the equations here have been modified to account for the change in the CF shape. The truncated cone CF is described by two different radii. CF is considered to grow from the top electrode to the bottom; the interface radius with the top electrode (TE) is r_{CFT} which is always greater than the radius of the interface with the bottom electrode (BE) r_{CFB}. This adheres to the structure of the truncated cone. An assumption is made here that during CF rupture, height of the cone is not affected; this makes the model open to fluctuations if there is any filament gap produced due to premature growth of CF. Although a forming process is considered for the device, it is not included in the model making the model not suitable for application to devices where forming is a significant factor. A possible explanation for leaving out the forming process is to avoid adding more complexity to the model because the forming parameters have to be included in the set/reset equations as well.
Here r_{CF,TM}/r_{CF,BM} is the maximum top/bottom radius that can be achieved and r_{CF,Tm} the minimum value of the top radius. This equation indicates CF geometry following a truncated cone structure and the top radius is greater than the bottom radius at all times. The model employs a numeric solving method similar to the one used in the previous model to find the discrete solutions for the master differential equations. But the solution for this model is not found, which means it is difficult to validate the reliability of the equations.
A very interesting point here is that a separate local diffusion process is not added in this model to describe the reset process in addition to the oxidation/reduction process. Many of the previously discussed models [91, 92] have a separate equation for the diffusion. But in this model diffusion has been integrated into the Eqs. (80) and (81) for redox reactions by considering different activation energies for the reduction and oxidation rates. This has been deliberately done considering the fact that the equation used in previous models to describe the diffusion velocity is similar in structure to the redox equations. As a result, the activation energies for both are combined together to consider a single activation energy which includes diffusion as well.
However, this approach is not the best one for truncated cones. It leads to increased complexity and improper calculation of temperature. So, GonzelezCordero et al. [93] proposed a different approach where they have developed simplified analytical equations which are suitable for simulation. In the previous Bocquet model [92] they have assumed a cylindrical uniform geometry where the calculated unique temperature is uniform throughout the CF. But to consider a more detailed physical description in this model; they have considered two temperatures in the CF. One temperature is at the hottest CF point where the reset process ruptures the CF and the other temperature is considering the main CF volume. This is a more reasonable model for a truncated cone structure as the two radii grow independently of each other depending on the oxidation/reduction processes [171, 181].
Here, α_{ T } is the conductivity temperature coefficient. The model is simulated and compared with the results from the previous Bocquet bipolar model [93] on which it is based. The results compare the findings from the model considering a cylindrical CF to the one considering the truncated cone CF. There is some evidence [93] presenting a better fit with the experimental data for this particular model as compared to the previous models where cylindrical CFs are considered and also results pertaining to the cases where multiple CFs are also presented; this shows the model’s flexibility in accommodating devices where multiple CFs is existent.
Primary aim of the model was to be simplistic enough to be implemented in electric circuit simulators. Analytical equations are properly laid out to be used in SPICE simulations and it has been represented through a 1T/1R circuit. The model is also represented through a VerilogA representation [93] which shows its applicability in digital circuits as well.
RRAM Models Based on Unipolar Devices
Random Circuit Breaker Network Model
In 2008, Noh’s et al. [182] proposed a random circuit breaker model to explain the switching in unipolar resistive switching devices. This model evolved to clear the considerable debate regarding the switching mechanism in unipolar devices, at the early stage of the study mechanism of resistive switching memory. Some reported that switching is the result of homogeneous/nonhomogeneous transition of current distribution, while some other says due to the formation and rupture of conducting filaments. A new percolation model was reported by Noh’s group [182] in this regard which was based on a network of circuit breakers with two switchable metastable states. The device used for the study is a polycrystalline TiO_{2} RRAM device. It shows wide distributions of SET and RESET voltage with uniform resistance change at the particular transition voltage. Conductive atomic force microscopy (CAFM) tip was used as a top electrode, and external voltage was applied through it for the resistive switching operations.
This model basically laid the foundation for future percolation models used to describe RRAM switching behavior, since it is dealt with the stochastic reversible dynamic processes. Most percolation models either investigate static cluster topology problems or dynamic percolation problems. A combination of reversible and dynamic processes is quite interesting. This also enabled future model developers to account for stochastic switching in the physical equations describing unipolar RRAM devices.
Filament Dissolution Model
Filament dissolution model was proposed by Russo et al. [82, 83, 84] exclusively for unipolar RRAM devices, although later revisions by the same group [85, 86] made this model suitable for bipolar devices as well. This model is based on the fundamental concept of Joule heating and filament temperature change. The model primarily focuses on the RESET transition of the devices, i.e., the transition from LRS to HRS. This is because of the high resistance associated with the RESET transition and this is where the major physical transformation in the device takes place. The model is based on the concept of conductive filament ruptured or dissolute under the effect of significant temperature change [84]. This temperature change in the filament is caused due to Joule heating. The proposed filament dissolution model has been deemed as selfaccelerated due to the process of the rupture of filament accelerates by itself under suitable conditions.
Major advantage of this model is that it makes use of simple wellknown coupled partial differential equations which describe the various effects in the device. The model is applied on a NiO based unipolar system [82, 83] where the oxide layer is sandwiched between two Pt electrodes and the filament is considered to grow from one end of the electrode to the other. Temperature profile in the oxide layer across its geometry is considered as parabolic; meaning that the temperature in the filament is minimum at the electrodes and maximum at the middle.
The mechanism behind the filament dissolution can be explained by the basic concept of Joule heating and dissolution which acts as an activator for the CF rupture. With the application of bias across the top electrode of the device, heat is produced in the filament due to the current flowing. The temperature steadily rises with an increase in the bias and when the bias reaches a significant level called reset voltage, the temperature rises above a value called the critical temperature. At this point, the dissolution of the filament is activated and the filament gets ruptured at a very fast rate leading to the device reaching a HRS.
Here, ρ is the resistivity of the oxide and v the electric potential developed in the device due to the application of an external bias voltage v_{term}. The voltage bias is applied at one of the electrodes, while the other electrode is connected to ground which act as boundary conditions for the device. NiO is the switching oxide, and the CF is formed as a sublayer comprising primarily of metal ions and oxygen vacancies. The CF is considered to have a diameter of φ_{ d }.
Where k is the thermal conductivity of the oxide layer, T is the device temperature and J is current density. Thermal conductivity and electrical resistivity values are dependent on the position they are applied in. So, ρ = ρ_{CF} inside the CF while it is equal to ρ_{ OX } in the oxide layer. The same analogy applies for the values of thermal conductivity as well. The temperature is considered to be equal to room temperature T_{0} at the electrodes, i.e., they act as heat sinks.
Where, E_{ a } is the activation energy, k_{ B } is the Boltzmann constant, v_{DISF} is a fitting parameter and v_{DIS} is the velocity of the CF boundary toward the symmetry axis.
where c is the experimentally calculated temperature coefficient of resistivity and ρ_{CFRT} the standard CF resistivity at room temperature.
Filament dissolution model discussed here has been modified and presented for bipolar RRAM devices by Larentis et al. [85, 86]. It is based on the same temperature and field accelerated ion migration. The set and reset processes in the device are defined by the mechanisms of drift migration induced by local electric field, ionic/electronic conduction and Joule heating. This is a point of departure from the model for the unipolar devices where the switching mechanisms for the set state were not properly defined and understood.
The current conduction defined in Eq. (95) and Joule heating described in Eq. (96) are used in the model for the bipolar devices. This suggests an assumption that the temperature profile for both types of devices follows a similar pattern. Along with it, the current conduction mechanism is also assumed to be similar. This in a sense might be an oversimplified assumption because many of the models described for bipolar resistive switching devices have been unable to be used for unipolar RRAM devices [90, 91, 92, 186] due to a marked mismatch in the conduction and switching mechanisms.
Bocquet Unipolar Model
This model was developed by Bocquet et al. [90] for describing both the set and reset processes in unipolar RRAM devices. It is basically a modified extension of the model proposed by Russo et al. [82, 83] in the sense that it can model both the transitions of the RRAM device while the former only considers reset transition. For set process, a local electrochemical reduction of the oxide is considered to be responsible for formation of conductive filaments. However, the reset mechanism follows the tried and tested formula for unipolar devices which considers thermally assisted destruction of the formed metallic filaments by Joule heating as the primary mechanism. Also, it has to be mentioned that the model proposes equations which are analytical in nature and can be conveniently solved in an electric circuit solver.
As the dissolution velocity is exponentially dependent on temperature, it gets activated only when there is significant amount of temperature. Temperature value is high only when there is a comparable amount of voltage, i.e., the reset voltage has to cross a critical value during CF dissolution. This acts as means of a selfactivation voltage threshold where the voltage controls the CF dissolution.
The coupled equations in this model have to be solved simultaneously and continuously due to the fact that the model relies on selfconsistent kinetic equations accounting for both CF growth and destruction mechanisms. This is a key feature which has to be implemented when using simulation tools to attain numerical accuracy.
Numerical values obtained from simulation in general profoundly interlinked between the set and reset transitions. This is because from a practical stand point the CF profile obtained after the set operation is used as the initial state to simulate the subsequent reset operation. Also, the reset current and LRS resistance depends significantly on the maximum current reached during the previous set operation [187, 188, 189]. This is basically due to the minimization of the CF radius which subsequently increases the resistance of the device [190].
The Bocquet unipolar model is compared against a NiO system similar to the one used in the previous Filament dissolution model [82, 83]. It is to be noted that the model is applicable for a unipolar device only. But the comparison with the NiO system is limited to a single system using a numerical solver. This is a major shortcoming in this particular model regarding the nonavailability of exact experimental characteristics data comparison from other sources to calibrate the model. It means that the fitting parameters have not been tested for a variety of characterization data or other models as well. So, it is difficult to judge the accuracy and viability of the model even though it uses some interesting concepts to explain the switching process in unipolar devices.
Window Function Models
Window functions are introduced in the “Linear Ion Drift Model” section [3]. These functions are generally required to limit the values that the internal state variable can reach. The dynamics of the state variable governs the switching property of the device. So, the state variable has to be set bounds within which it can grow so that the device always remains in the permissible state and does not go out of bounds. For example, if the growth/rupture of CF is being modeled, the CF physically can only grow from one electrode end to the other. If the model growth crossed that limit, it suggests a mismatch between the physical phenomenon and the model. As a result, certain window functions [94, 95, 96, 97, 98, 99] which acts as limiting functions is introduced into the model to set bounds for the device.

➢Consider and take into account the boundary conditions at the top and bottom electrodes of the device.

➢Be capable of imposing nonlinear drift over the entire active core of the device.

➢Provide linkage between the linear and nonlinear dopant drift models.

➢Be scalable, meaning a range of fmax(x) can be obtained such that 0 ≤ fmax(x) ≤ 1.

➢Utilize an inbuilt control parameter for adjusting the model.
Comparison of the window function implementations
Window Function  Symmetrical  Solves all boundary conditions  Accounts for nonlinear effects  Scaling possible  Fits which model  DC response 

Linear Ion Drift [3]  Yes  No  No  No  Linear ion drift  No 
Joglekar [94]  Yes  No  Partially  No  Linear/nonlinear/TEAM  No 
Biolek [95]  Yes  Partially  Partially  No  Linear/nonlinear/TEAM  No 
BenderliWey [96]  NA  No  Partially  No  Linear  No 
Shin [97]  NA  Almost all  Yes  Yes  Chua/linear drift  No 
Prodromakis [98]  No  Almost all  Yes  Yes  Linear/nonlinear/TEAM  No 
BCM [99]  No  Partially  No  No  Linear drift  No 
No  Partially  Practically yes  No  TEAM/TEAM for Simmons tunneling barrier  No 
Joglekar Window
One of the very first window functions proposed by Yogesh Joglekar and Stephen Wolf [94] is based on the linear ion drift model [3]. It was developed when memristors were still in its early stages of development after the breakthrough by the HP team proposed linear model. Window functions are aim to generalize the behavior of the model around the device boundary.
This characteristic also acts as a significant limitation for the model. On the one hand, at low values of p, the window function does not perform as per expectation. But at significantly high values of p where the nonlinear effects are taken into account, the difference between the linear drift and nonlinear drift in the model disappears. This means that there is no proper way to account for both the linear and nonlinear drift effects at the same time, while implementing this window function. Also, the mobility at the boundaries was suppressed down to zero, which made the function to be stuck at the 0 value at the terminal states. Shortcomings of this function and the improvements made over it by the Biolek window function [95] are discussed further.
Window function implemented here should be understood as extensions to the physics based models. General limitation with the physics based models is that the models do not account for the effects at the physical boundaries of the device. Memristive devices have been found to behave differently in the bulk and the boundary of the device. So, these window functions can help to overcome these limitations in the device by setting a boundary for the model and properly accounting the boundary effects.
Joglekar et al. [94] also investigated memristive device implementation in standard fundamental circuits along with the other basic elements R, L and C. Combining these four basic elements (memristor, inductor, resistor, and capacitor), the functioning of standard circuits such as MC, MLC, etc. were studied. They came to a similar conclusion as the HP team [3] that the primary property of a memristor is the memory of the charge that has passed through it. The memristor was dimensionally characterized as a magnetic flux D^{2}/μ_{D} where D is the memristor size and μ is the mobility.
Biolek Window Function
Biolek window function [95] developed by Zdenek and Dalibor Biolek in 2009 was modeled on the proposed memristor equations by Strukov et al. [3] Its primary aim was to provide a marked improvement over the previous Joglekar model [94] and also provided a SPICE implementation of the Linear drift model [3]. They proposed changes to the way the window functions are defined so that a closer approximation between the model and the real circuit element can be achieved. They also reported the SPICE implementation of the linear drift model [3], which opened up the model to a wide range of circuit applications at that time.
Strukov et al. [3] first pinpointed the pertinent problems in the Joglekar function while implementing in SPICE and then proposed improvements over it. First major problem with Joglekar function is its way of setting up of the terminal state R_{ON} and R_{OFF}. State equation and the window function defined, respectively, in Eqs. (113) and (114) bound the value of the variable to 0 at the boundary and it is forced to hold that value. This state cannot even be changed by an external stimulus. This happens due to the HP memristor remembering the xcoordinate of the boundary between two layers and not the amount of electric charge passed through it. As a result, when a new set or reset transition is to be started from a terminal value, the device has to start from 0 and not the actual value it had in its previous state.
Second problem of the window function is noticed when the model is implemented as an actual circuit element. The circuit component exactly remembers the entire charge which is passing through it. So, in case of the Joglekar window function, to transpose the memristor from a state x_{0} to x_{1}, a certain amount of charge q is required. Now the same amount of charge but in the opposite polarity, i.e., –q is required to bring the memristor back from x_{1} to x_{0}. Thus, when a memristor is being driven by a constant current with a time interval, say t, the same time t is also required for restoring the device back to its original state.
This occurs regardless of the fact that the device could be in its terminal state all the while when the current flows. This leads to significant operating delays as documented by the SPICE simulation presented by Biolek et al. [95]. Also, when the current direction is reversed, the boundaries start to move in an opposite direction regardless of the past state, thus the state is lost along another curve.
If the function increases the width of the doped layer, or x → 1, the current is positive. The function value is 0 at the boundaries. When increasing the value of p, the function yields a flat window with steep troughs to zeros at x = 0 and x = 1.
BenderliWay Window Function
Another window function based on the basic HP model was proposed by Benderli and Wey [96] in 2009. The end result they set out to get was similar to the Biolek function [95]. The developers wanted to develop a SPICE compatible macro model based on the HP memristor which would be suitable for applications in circuit simulations. They proposed a clipping circuit which will bind it within the constraints of the length of the device (D).
The proposed clipping circuit was comprised of four comparators whose job was to ensure that the state variable function w(t) did not go beyond its limits. The comparators clipped w(t) at its top and bottom boundaries. It basically acts as a switch in which if the comparators detect a certain value in the device they activate a switch and set the device at a particular voltage. So, when w(t) reached the upper boundary of the device, the device is connected to a voltage source of value D, which effectively clips w(t) at D. This operation occurs when the voltage bias is positive.
Also by increasing the capacitance near the boundaries, the nonlinear effects could be accounted for it in the circuit. The major shortcomings of this function are its simplifying approximations and the lack of a clear description of how the linear and nonlinear drift can be modeled in the circuit. Although it manages to obtain a hysteresis relation for the device, it suffers from similar limitations as the Joglekar model [94]. Additionally, lack of clear information regarding the nonlinear effects was an equal deterrent to the application of the function in circuits.
Shin Window Function
Shin, Kim, and Kang [97] in 2010 tried to circumvent the issue of window functions by proposing a constitutive relationship derived from the basics of the memristors developed by Chua [1]. This is different from the previously reported window functions in the sense that they tried to model the memristors perfectly by relating charge and flux together. This was the fundamental essence of the Chua model and is a stark contrast to the linear ion drift mechanics proposed by Strukov et al. [3].
Here R_{ M }(q) is the memristance defined by a derivative of the chargeflux relationship with respect to the charge. Thus, R_{ M }(q) = df (q)/dq defines it as a current controlled memristor.
So, on similar terms, G_{M}(φ) is a voltagecontrolled memductance whose values can be calculated by measuring slopes of chargeflux relationship g(φ).
The above written Eqs. (118) and (119) can be used in compact models for circuit implementations. But they are inadequate when it is needed to define it within a bounded resistance range. This is where window functions are written to modify the circuit parameters so that the model operates within the resistance range of R_{MIN} and R_{MAX}. Thus, in mathematical terms it means that R_{M}∈ [R_{MIN,}R_{MAX}]. Memristor needs to be confined within the available range of resistance so as to adhere to design requirements. When the device reaches one of its boundary values, it has to stay in that state after any excess charge or flux is applied to the device. This has to be ensured so that the device does not violate its boundary conditions under hard switching conditions.
Above equations disregard any excess input current or voltage in the model space. The boundary state of the device is held until the polarity of the input source is reversed. When the polarity is reversed, it indicates the start of a new transition; the function will force the memristor to move back into memristive region. The operation is similar to the clipping circuit proposed in the BenderliWay function [96] discussed previously, but here, there is no requirement of a complicated comparator circuit. The major purpose of using this approach to model the devices was to remove the usage of a special window function as done previously but still be able to adhere to the boundary conditions implicitly.
Prodromakis Window Function
Prodromakis window function was proposed in 2011 by Themis Prodromakis et al. [98] of Imperial College, London, which aimed for a simple and efficient function modeled the memristor device characteristics [191] effectively. Some of the limitations and constraints of the previous models were alleviated which made the function easy and accurate to use.
Control parameter is critical in this function as it helps to remove many of the constraints and limitations of the previous functions. The function can scale upwards due to the control parameter, which suggests that f_{max}(x) can take any value between 0 and 1 inclusive. Also a very large value of p provides a linkage with the linear dopant drift effects. A serious limitation with the previous Joglekar [94] and Biolek [95] models was that the control parameter was allowed to take only integer values. But here, p could have real values as well which added more flexibility to the model.
Hysteresis loop obtained using the window function is asymmetrical which has been explained by Prodromakis et al. [98] as a result of the different switching rates of the ON and OFF rates which is quite reasonable. The hysteresis also suggests there is no terminal state problem as highlighted in the Joglekar function. Width of the doped region does not go higher than D, and the memristance is correctly limited. In the case of reverse polarity, it does not get stuck at a zero value and does not take any error states as highlighted by the results.
Boundary ConditionBased Model (BCM) Function
This window function developed in 2012 by Fernando Corinto and Alan Ascoli [99] was aimed at improving the models proposed by Joglekar [94] and Biolek [95]. They identified possible limitations with the previous functions with respect to their exhibition of singlevaluedness and multivaluedness, respectively. Also tuning the range and the boundary conditions were not possible with the Joglekar [94] and Biolek [95] functions. This was handled by the BCM window function by deriving novel methods to propose closedform solutions for memristor devices. Along with that they also added tuning parameters to increase the flexibility of the boundary conditions used in the models.
Design of the Joglekar function limits it to a single value of memductanceflux characteristics at all input values. Similarly, the input dependent Biolek function limits it to only multivalues of the function under sign varying input. But the BCM function allows for both singlevalued and multivalued memductanceflux characteristics under a single sign varying input. The function assumes a linear dopant drift effect, which simplifies the analytic integration as well as makes it suitable for closed form solutions under any initial condition state. But this invariably neglects the nonlinear boundary effects in the device. So, on the one hand, the BCM function proposes a very simplified expression for defining the boundaries of the device but it misses out on accounting for the nonlinear effects due to the simplifying assumptions.
Here, η is a linear control parameter and ∃ is a quantifier denoting “there exists” which signifies that for x(t) there exists exactly one solution. Values of the nonnegative parameters v_{th,1} and v_{th,0} determine the occurrence of such transitions. The conditions in the first Eq. (125) are established when x(t) obeys the boundary conditions x = 1 and x = 0. But the conditions in the second Eq. (126) is established when the function x(t) no longer obeys these boundaries.
The window function qualitatively works similar to other functions. At the boundary conditions, the vertical transition from 0 to 1 or vice versa occurs depending on the polarity of the input stimulus. Thus, the input used here is sign varying in nature.
Model Verification
Several ways are there to verify the working of the presented models, in this work. Some of the implementations and verification have been included with the description of the model. Models which have been described quantitatively using mathematical equations can be verified by solving the equations in a simulator. Generally, IV characteristics of the simulated model are compared with the corresponding experimental data from the device. This gives a fair idea on the reliability of the model. Physical models described by mathematical equations can be solved by a multitude of solvers such as MATLAB, Mathematica, COMSOL, etc. Compact models which have been translated to work in the circuit level are generally simulated using the SPICE framework. There are a variety of SPICE based simulators in the market such as HSPICE, Ngspice, etc. which could be utilized. Corresponding output characteristics can be matched with the experimental results.
Physically verifying the switching mechanism in a model is trickier. It generally involves insitu observation of the switching process [192] which requires a lot of precision and highend equipment. However, it is very solid evidence regarding the viability of the switching mechanism presented in the model. There have been some novel methods reported to observe the growth of the conductive filaments during the switching process. Conductive atomic force microscopy (CAFM) [182] has been used to visualize the formation and rupture of conducting filaments. IV switching curve is shown in Fig. 17, which is clearly shows HRS/LRS states and the corresponding state of the filaments. Electrostatic force microscopy (EFM) [193] can also be used to visualize the migration and accumulation of oxygen ions by calculating the electrostatic force between the probe and the sample. It is one kind of insitu TEM, where the focus is primarily on the charge of the carrier. Formation of conductive channels can be observed by high resolution energydispersive Xray spectroscopy (EDS) which can provide accurate detailing of the composition in the filaments. These two methods have been proven to be quite effective in verifying the physical switching mechanism and the visualization of conductive filaments in RRAM devices.
WellPosed Memristive System Definitions
An excellent work published in 2015 by Wang and Chowdhury [100] of UC Berkeley set a new paradigm for memristor and RRAM modeling. It was a push in the right direction for the whole memristor modeling community. The major features of the work were the significant improvements made on the preexisting models. Some tweaks were proposed in the fundamental understanding of memristor models which contribute to eradicating some of the longstanding issues which has plagued the memristor models. Also, they demonstrated implementation of the models in SPICE, VerilogA as well as their own prototyping platform based on MATLAB called as MAPP [191].
The root cause of many of the issues affecting memristor models were improper mathematical implementation. As a result, it limited their application in a variety of simulation and design scenarios. Simulation of the models is a very critical factor in determining the applicability of the model; however, the existing models were unable to be applied in a variety of simulation studies such as DC, transient etc. This work aimed at modifying the models into a form where simulating them would be a simple task.
The common illposed or erroneous definitions that many of the previous models suffer are not being properly defined at all biases, outputs not being unique or continuity problems. These basic problems are to be avoided in the models for wide application. All the various problems that the authors have encountered and the improvements they have presented are discussed.
A very valid point highlighted by them is that a wellwritten model for a particular circuit should be able to replicate its characteristics or be valid beyond its actual boundaries as well. Even if getting outputs beyond the applicable range might not be physically possible, but in simulation environments like SPICE it is imperative that the circuits work at all level of biases. This will lead to efficient simulation and produce smooth varying outputs at all biases. Many of the models we have discussed previously have sought to ignore the operation of the model beyond the available range, leading to their incompatibility for use in circuit simulators. This requirement can also be understood by the underlying algorithms of circuit simulators such as the NewtonRalphson (NR) algorithm [194] which is commonly used for solving nonlinear equations.
The NR algorithm [194] works on the principle of applying a sequence of biases to devices so that convergence can be reached for a valid solution of the circuit. So even if the bias is physically possible or not, for the NR algorithm to find a solution the model must evaluate at all bias. This invariably means that if the NR finds a solution to a welldesigned model, the input bias will be physically reasonable. In the cases where the converged solution is physically not possible, it provides insights into the problematic areas of the models and is critical to troubleshooting. Therefore, in order for the NR algorithm to work correctly and find a proper solution, models should be designed to be evaluated at all biases.
Another fundamental problem is the dividebyzero error. Many of the models have terms such as 1/(x − a) which cause these errors. It leads to the solutions getting unbounded and causing discrepancies. Along with that, some expressions use square root (√) with negative arguments which give rise to complex arguments. Nonreal numbers are not valid arguments for models and can cause nonconvergence to a valid solution in simulators.
Almost all of the models we have discussed earlier do not account for a very important aspect of device model simulations. Any mathematically viable input must produce a mathematically viable output, and the most basic among this is the DC analysis. It is commonly the crux and starting point of any analysis and a proper model should produce an accurate DC solution. A proper welldesigned model should work consistently with all kind of analysis. But almost all of the models suffer from significant DC response problems. So, this is another area that needs to be improved. Wang et al. [100] also addresses the problems of the models generally faced when being defined in circuit level languages such as VerilogA.
The crux of the improvement to the models Wang et al. [100] have proposed revolve around the correct way of modeling hysteresis itself, i.e., using internal unknowns and implicit equations. This is because the dynamics of the filament in a RRAM closely follows a hysteresis characteristic. Also, the improvements make the models simulation ready with all the major analyses like DC, AC, and transient providing acceptable results. Various techniques are also proposed to aid convergence in electric simulators including a proposed new limiting function which replaces the functionality of window functions and overcomes all their limitations.
Accurate Description of Hysteresis
The above equations serve as the model template for modeling hysteresis. Value of the s(t) at a particular time instant t is governed by the history or the state of v(t). Thus, the device is considered to be having an internal memory of the input voltage. Functions f_{1} and f_{2} are chosen accordingly to define the characteristics of the device. Choice of these functions could be termed as critical in defining the dynamic of the device.
Proper Definition of Internal Unknown Variables in VerilogA
It has been discussed several times during review of the various models, implementation of the models accurately in SPICE [116, 117] and VerilogA [142] is critical for their acceptability. This is because SPICE is the most commonly used circuit simulation platform, and VerilogA is the most widely used hardware description language. So, simulating in these platforms is as close as it gets to the real physical devices. A major shortcoming of the previous models is the way that internal unknown variables were handled in VerilogA.

Different Verilog compilers handle variables declared using “real” differently. Then, this will lead to very inconsistent results.

Differential equations should not be defined by using the inbuilt idt() function. Because this function has very inconsistent support in the compilers and causes many limitations [140, 142].

Time integration to obtain analytical solutions should not be coded inside the model. The process is pretty simple but has serious pitfalls as given below.

➢This method makes use of “abstime” function. To define the starting point of the integration it also uses “initial_step.” These have been termed as bad modeling practices in analog simulation [140, 143].

➢The internal unknowns are defined as a memory state in this method, which can create problems for periodic steady state (PSS) analysis.

➢This method bypasses many of the simulators built in facilities such as the convergence aids, time step control etc.

➢It can cause serious convergence issues for stiff systems due to its dependence on the Forward Euler (FE) algorithm [195].
These problems are generally a combination of bad modeling practices and the incapability of VerilogA to handle internal unknowns efficiently. As a result, declaring s as a voltage has been demonstrated as an effective way of getting around the problem.
Developing Generic RRAM Models
Taking the previously discussed hysteresis equations as a template, Wang et al. [100] presented a generic way of developing compact models for RRAM devices. To demonstrate the RRAM model, ASU/Stanford model [78, 80] is considered.
The γ here is the local field enhancement factor [196] which contributes to the abrupt SET (filament growth) and gradual RESET (filament dissolution). A common property among most of the RRAM models is the fact that the sign of f_{2} is same as that of –sinh (vtb). This means in terms of the gap that it starts to decrease whenever vtb is positive and vice versa. But this growth or dissolution cannot be indefinite for numerical simulation to work. For the simulations to work in reality, they have to be bounded which has been discussed in depth in the next section.
Various methods have been proposed to account the boundary effects of the devices. It will come up short with the methods having some serious limitations. Some of the models have implemented direct “ifthenelse” statements in the Verilog code [80, 81]. But the problem is that the use of “ifelse” statements removes the model from the differential equations framework which is not acceptable. It also introduces hard discontinuities in the model whereas we need smooth continuous curves.
As is seen in Fig. 24b, the f_{2} = 0 curve has three lines, V = 0, maxGap, and minGap. Beyond the values of maxGap and minGap, there will not be any stable DC solutions so they will not show up in transient analysis. Therefore, when sweeping between those values, the transient solution will work fine with the window function multiplied to f_{2}. But with the other analyses there are several problems. In DC operating point analysis, unphysical solutions can show up owing to the fact that all the lines consisting of the f_{2} = 0 curves are valid. So, when the voltage is zero, any arbitrary value can be the value of gap and it no longer follows the boundaries. Hence, DC analysis is a major limitation for the window functions.
The functions safeexp() and smoothstep() are smoother versions of the normal variants of the exp() and step() function. They have been developed by Wang et al. [100] in their MAPP [191] platform and is available to use within the platform. The clipping functions here closely mimic the actual physical effects occurring in the device. It can be termed as a huge force which keeps the state variable gap within its acceptable physical limits.
The templates provided by Wang and Chowdhury [100] for RRAM modeling is capable of widespread applicability and can be used as an ideal platform to develop other models. It consists of accurate modeling of hysteresis, includes proper handling of internal unknowns in VerilogA and does not need to use incompatible functions like “idt()” and “initial_step” in the differential equation framework. They also circumvent the various limitations of the window functions by the use of mathematically accurate clipping functions. The model templates support a variety of analyses such as DC, AC, transient, PSS, and homotopy [197] in VerilogA, SPICE, and MAPP. This is a very exhaustive list of advantages which should be used for development of future models by the RRAM community.
Improving Solution Convergence
Obtaining solution convergence is one of the most important features, every RRAM model ought to possess. The convergence of the solution points to the fact that it is valid and acceptable. This has been a problem for many of the compact models describing RRAM devices. Several techniques have been proposed [100] to aid convergence of solutions in these models. The use of limiting functions compatible with SPICE is very important so that it limits the solutions whenever they cross the acceptable range. A very simple way to make sure the solutions converge is to properly scale the unknowns and variables. Proper scaling makes sure that any results obtained are defined relatively accurate to the inputs.
The major feature here is that the limiting function does not use very large values of x_{ new }, instead it increases the value in iterations. This function is fully compatible with SPICE and can be implemented in any SPICE compatible circuit simulator. This marks a new addition to the number of limiting functions available for use in circuit simulators apart from the ones developed decades ago.
Improving Existing Models
On the basis of the accurate generic model templates discussed in the previous section, Wang and Chowdhury [100] have proposed some improvements in some of the existing popular models. We discuss them here and present it in a concise form so that it becomes easy to understand the changes and implement it forward in future models.
The improvements have been proposed on the linear ion drift [3], nonlinear ion drift [46], Yakopcic [73, 74], TEAM/VTEAM [75, 76, 77], and ASU [78] models. Many of the models have IV relationships in common with other models. Therefore, the most common and important among those are considered and termed as f_{1} functions as discussed earlier. The state variable equations of those models are termed as f_{2} functions are corresponding improvements are proposed.
Both the f_{1} and f_{2} functions are generally nonlinear and asymmetric. And the reported models use very discontinuous and fastgrowing terms like exponential, sinh, and power (pow) functions which results in difficulties during the convergence of the solutions. So, these can be overcome by using “smooth” and “safe” functions as proposed by Wang et al. [100]. The smooth functions are used in place of discontinuous functions. Major design criteria in the smooth functions used is a common smoothing factor which combines common elementary functions to approximate the original nonsmooth functions. Smoothing factor controls the tradeoff between better approximation and more smoothness. The safe functions are versions of the fastgrowing functions which limit the maximum slope the functions can attain, and then linearize it to keep the slopes constant beyond it. For some functions like sqrt, log, etc., the “safe” versions clip the inputs using smoothclip so that nonvalid outputs can be avoided.
In the particular f_{1} and f_{2} functions used in the models, the ifelse statements are replaced by smoothswitch which removes the discontinuity of the former. The exp and sinh functions are correspondingly replaced by safeexp and safesinh. The authors have demonstrated the definition of the functions in MAPP and VerilogA which makes it easy for future model developers to integrate it into their system. A very common problem with the f_{2} functions, i.e., the state variable dynamics is the uncertainty over the range of the internal unknown. The previous models counter this by either introducing window functions that bound it within a range or do not account for the effects at all. This has been very efficiently handled by introducing selfmodeled clipping functions which define the acceptable range of the internal unknown.
Improved IV relationships of the various models
Model  Original IV relationship (f_{1})  Improved IV using concepts from Wang and Roychowdhury^{100} 

Linear ion drift [3]  f_{1} = (R_{ON} × s + R_{off} × (1 − s))^{−1} × vpn  Can have division by zero error when s = R_{off}/(R_{on}/R_{off}). Modified equation: y = smoothclip(s − R_{off}/(R_{on} − R_{off}), smoothing) + R_{off}/(R_{on} − R_{off}) Then, f_{1} = (R_{on} × y + R_{off} × (1 − y))^{−1} × vpn 
I = s^{ n }β sinh(α × vpn) + χ(exp(γ × vpn) − 1)  sinh can be changed to safesinh(), exponential function to safeexp()  
\( I(t)=\left\{\begin{array}{c}{A}_1\times s\times \sinh (Bvpn),\kern0.5em vpn\ge 0\\ {}{A}_2\times s\times \sinh (Bvpn),\kern2.25em vpn<0\end{array}\right. \)  sinh is changed to safesinh(). The function is then smoothed. f_{1p} = A_{1} × s × safesinh(B × vpn, maxslope) f_{1n} = A_{2} × s × safesinh(B × vpn, maxslope) f_{1} = smoothswitch(f_{1n}, f_{1p}, vpn, smoothing)  
\( v(t)={R}_{\mathrm{ON}}{e}^{\left(\lambda /{x}_{\mathrm{off}}{x}_{\mathrm{on}}\right)\left(x{x}_{\mathrm{on}}\right)}\times i(t) \)  The exponential function is changed to safeexp()  
\( I\left(g,V\right)={I}_0\exp \left(\frac{g}{g_0}\right)\sinh \left(\frac{V}{V_0}\right) \)  The gap is expressed using s: gap = s × min gap + (1 − s) × maxgap Then sinh is changed to safesinh(), exponential function to safeexp() 
The state variable equations presented in an improved form
Model  Original state variable dynamics (f_{2})  Modified using concepts from Wang and Roychowdhury^{100} 

Linear ion drift [3]  f_{2} = μ_{ v } × R_{ on } × f_{1}(vpn, s)  DC hysteresis not present. Clipping technique is used to set bounds for s, so that 0 ≤ s ≤ 1 
f_{2} = a × vpn^{ m }  DC hysteresis not present. Clipping technique is used to set bounds for s, so that 0 ≤ s ≤ 1  
\( {f}_2=\left\{\begin{array}{c}{c}_{\mathrm{off}}\times \sinh \left(\frac{i}{i_{\mathrm{off}}}\right)\times \exp \left(\exp \left(\frac{s{a}_{\mathrm{off}}}{w_c}\frac{i}{b}\right)\frac{s}{w_c}\right),\mathrm{if}\ i\ge 0\\ {}{c}_{\mathrm{on}}\times \sinh \left(\frac{i}{i_{\mathrm{on}}}\right)\times \exp \left(\exp \left(\frac{a_{\mathrm{on}}s}{w_c}+\frac{i}{b}\right)\frac{s}{w_c}\right),\mathrm{otherwise}\end{array}\right. \) where i = f_{ 1 } (vpn, s)  No DC hysteresis present. Consists of fast growing functions. sinh is changed to safesinh(), exp to safeexp(). Smoothing is performed and bounds for s, so that 0 ≤ s ≤ 1  
\( {f}_2=\left\{\begin{array}{c}{k}_{\mathrm{off}}\times {\left(\frac{vpn}{v_{\mathrm{off}}}1\right)}^{a_{\mathrm{off}}},\mathrm{if}\ vpn>{v}_{\mathrm{off}}\\ {}{k}_{\mathrm{on}}\times {\left(\frac{vpn}{v_{on}}1\right)}^{a_{\mathrm{on}}},\mathrm{if}\ vpn<{v}_{\mathrm{on}}\\ {}0,\kern10.25em \mathrm{otherwise}\end{array}\right. \)  The equation is redesigned as: \( {f}_2=\left\{\begin{array}{c}{k}_{\mathrm{off}}\times {\left(\frac{vpn{v}^{\ast }}{v_{\mathrm{off}}}\right)}^{a_{\mathrm{off}}}, if\ vpn>{v}_{\mathrm{off}}\\ {}{k}_{\mathrm{on}}\times {\left(\frac{vpn{v}^{\ast }}{v_{\mathrm{on}}}\right)}^{a_{\mathrm{on}}},\kern2.75em \mathrm{otherwise}\end{array}\right. \) where v^{∗} = (1 − s) × v_{off} + s × v_{on}, Such that when s = 1 and s = 0, it is equivalent to the VTEAM equation in the vpn > v_{off} and vpn < v_{on} regions, respectively. The functions are also smoothened by: \( {f}_{2p}={k}_{\mathrm{off}}.{\left( vpn{v}^{\ast }/{v}_{\mathrm{off}}\right)}^{\alpha_{\mathrm{off}}} \), \( {f}_{2n}={k}_{\mathrm{on}}.{\left( vpn{v}^{\ast }/{v}_{\mathrm{on}}\right)}^{\alpha_{\mathrm{on}}} \), f_{2} = smoothswitch(f_{2n}, f_{2p}, vpn − v^{∗}, smoothing) The bounds for s are set using clipping techniques.  
f_{2} = g(vpn) × f(s), where \( g(vpn)=\left\{\begin{array}{c}{A}_p\times \left(\exp (vpn)\exp \left({V}_p\right)\right),\kern0.75em if\ vpn>{V}_p\\ {}{A}_n\times \left(\exp \left( vpn\right)\exp \left({V}_n\right)\right),\kern0.75em if\ vpn<{V}_n\\ {}0,\kern13.00em \mathrm{otherwise},\end{array}\right. \) and \( f(s)=\left\{\begin{array}{c}\exp \left({\alpha}_p\times \left(s{x}_p\right)\right),\kern0.75em if\ s\ge {x}_p\\ {}\exp \left({\alpha}_n\times \left(s1+{x}_n\right)\right),\kern0.75em if\ s\le 1{x}_n\\ {}1,\kern2.5em \mathrm{otherwise}\ \end{array}\right. \)  The equations are designed to get proper DC hysteresis: \( g(vpn)=\left\{\begin{array}{c}{A}_p\times \left(\exp (vpn)\exp \left({v}^{\ast}\right)\right),\kern0.75em if\ vpn>{v}^{\ast}\\ {}{A}_n\times \left(\exp \left( vpn\right)\exp \left({V}_n\right)\right),\kern0.75em otherwise\end{array}\right. \) where v^{∗} = − V_{ n } × s + V_{ p } × (1 − s) Also exponential function is changed to safeexp(). The whole function is made smooth. Clipping is used to set bounds for s.  
\( {f}_2={v}_0\times \exp \left(\frac{q\times {E}_a}{k\times T}\right)\times \sinh \left(\frac{vpn\times \gamma \times {a}_0\times q}{k\times T\times {t}_{ox}}\right) \) where γ = γ_{0} − β_{0} × Gap^{3}  The d/dt Gap is converted to d/dt s: \( {f}_2=\left( maxGap minGap\right)\times {v}_0\times \exp \left(\frac{q\times {E}_a}{k\times T}\right)\times \sinh \left(\frac{vpn\times \gamma \times {a}_0\times q}{k\times T\times {t}_{ox}}\right) \) Also exp is changed to safeexp() and sinh tosafesinh (). Clipping is used to set bounds for s. 
Novel RRAM Applications
There have been several new breakthroughs with RRAM architectures and applications. Among them noteworthy in case of architectures is the use of materials such as graphene, amorphous carbon films, transition metal dichalcogenides (TMDs), black phosphorous in a RRAM device. Neuromorphic computing is a novel application scheme for RRAM devices which utilize the memory retention property to use them as synapse devices.
RRAM devices based on graphene and related materials [198] have showcased performance similar to conventional metal oxide devices. These devices are different due to their unique lattice structure and belong to a completely different family of materials. So, investigation on modeling of such devices is highly necessary. Whether conventional modeling techniques such as the ones presented in this work can be used for these devices depends on the physical phenomena governing them and the corresponding IV charcateristics. The hypotheses presented to explain the switching in graphene oxide (GO) based devices [198] are consistent with standard bipolar RRAM switching mechanism. The absorption and creation of the conductive filaments are thought to be a result of diffusion of metallic ions from the electrode to the switching layer or transport of oxygen related carriers in the switching media. RRAM devices based on amorphous carbon [198] as the switching media are thought to operate under a similar mechanism. Owing to the similarity in the nature of the physical transport mechanisms, existing physical models can be used to explain the novel GO and amorphous carbon based devices.
Neuromorphic computing [199] is a novel architecture scheme which employs RRAM as synapse devices as its fundamental building block. It is believed that these RRAM based neuromorphic systems can replicate how our brain functions, harnessing the ability of memristors to remember their state. This enables the system to be trained for specific applications just like the human brain. With RRAM forming the crux of these systems, it is critical that device characteristics for the RRAM devices are well modeled. But modeling of RRAM based synapse devices is challenging owing to the fact that the RRAM devices used need welldefined analog behavior, which is the precondition for brain like functions. Device performance under AC stress and cycle to cycle variability are factors which affect the potentiation and depression of the synapse device. Standard models reported in this work can predict the digital behavior of the RRAM, but one may implement them for the analog behavior. Though a few models are reported [200, 201] to describe the switching mechanism in analog RRAM, but it has been difficult to mathematically define them and translate it into a compact form. However, significant research is ongoing to quantitatively describe these characteristics and translate it into a compact form to be used on the circuit level.
Conclusions
In summary, the important features of all widely accepted RRAM models have been discussed. This work fulfills the requirement of the modeling community for a unified discussion on the various RRAM models. Many of the recent models, such as Stanford/ ASU model, GonzelezCordero et al. model, Prodromakis model, have provided apt explanations for RRAM processes based on the early models. Implementations of different window functions like Joglekar, Biolek, and Prodromakis have been presented and compared. Various unexplained phenomena occurring in the devices are numerically validated in the models. No one model can be deemed as the perfect one, owing to the variety of materials, fabrication processes and device operations exist in the RRAM devices. Each model has been tuned accordingly to fit the device used. Researchers are still some time away from developing a generic RRAM model owing to these factors and also due to the deficiencies in the modeling techniques. Accurate and welldefined modeling techniques have been discussed in the “WellPosed Memristive System Definitions” section, which should act as a competent template for future model development. Combined with the detailed analysis provided for past RRAM models, this review work can potentially act as a focal point for RRAM model developers.
Notes
Acknowledgements
This work was supported in part by the Ministry of Science and Technology, Taiwan, under Project MOST 1052221E009134MY3.
Authors’ contributions
DP collected important papers and made a roadmap for the manuscript. PPS collected all papers and made first draft of the article with the guidance of DP. DP modified, revised and make shaped for publication. TYT participated in the discussion, and modification. DP and PPS modified the manuscript after revision. All authors read and approved the final manuscript.
Competing Interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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