Understanding memristive switching via in situ characterization and device modeling

Owing to their attractive application potentials in both non-volatile memory and unconventional computing, memristive devices have drawn substantial research attention in the last decade. However, major roadblocks still remain in device performance, especially concerning relatively large parameter variability and limited cycling endurance. The response of the active region in the device within and between switching cycles plays the dominating role, yet the microscopic details remain elusive. This Review summarizes recent progress in scientific understanding of the physical origins of the non-idealities and propose a synergistic approach based on in situ characterization and device modeling to investigate switching mechanism. At last, the Review offers an outlook for commercialization viability of memristive technology.

M emristive devices have been studied intensively since the link between memristor theory and physical resistive switching devices was established in 2008 1 , which was initially driven by the need for high-performance non-volatile memory and has more recently been fueled by energy-efficient unconventional computing 2 . Postulated as the fourth fundamental passive circuit element in addition to resistor, capacitor, and inductor, the memristor can store information in a form of resistance, which can be modulated by the history of its external stimuli 3 . The typical structure of memristors is a two-terminal three-layered stack, consisting of a switching layer sandwiched between two metallic electrodes. The switching layer ranges from semiconducting to insulating inorganic or organic materials. Through materials engineering, memristive devices can be tailored to provide non-volatile or volatile memory. Non-volatile memristor maintains its resistance state after the removal of the applied switching voltage or current. The stable resistance state is used to represent stored information, making memristive devices suitable for data storage applications. Integrated memristive crossbar arrays are considered promising candidates for future application in mainstream non-volatile memory. This is because memristive devices can store information at the sub-2-nm scale 4 and possess many other desired properties, including high speed 5 , low energy consumption, three-dimensional integration capability, and compatibility with complementary metal oxide semiconductor (CMOS) technologies 6 . Furthermore, in-memory analog computing is being developed to process information where it is stored 7 . Such in-memory computing is expected to offer an efficient and reconfigurable solution to process analog information in artificial intelligence (AI) applications 8 . On the other hand, the programmed resistance state of a volatile memristor gradually relaxes toward a thermodynamically stable state upon the removal of the programming signal, offering desirable dynamics for emulating biological synapses and neurons 9 . Memristors at the individual device level have shown short-and long-term plasticity similar to that of biological synapses 9,10 . Neural networks with memristors at the array level have been used to demonstrate brain-inspired functions 11 .
Consequently, memristive devices have attracted significant attention in the past decade as a key enabler of new computing paradigms to overcome the limitations of the conventional von Neumann computing architecture. However, although significant research efforts have been directed toward memristive devices, a large-scale commercialization of these devices has not yet been achieved. In addition to challenges at the circuit, algorithm and architecture levels, issues at the device level are likely still the primary reason. Two of the major remaining challenges at the device level are relatively large parameter variability and poor cycling endurance (Fig. 1a). To improve the cycling endurance and reduce the parameter variability to a level sufficient for large-scale commercialization, it is necessary to acquire an in-depth understanding of ion migration and its coupling with electron transport-the dominating dynamics in memristive mechanism-during the switching process. To reveal the dynamic process of switching, in situ characterization techniques are necessary; then, device modeling is needed to thoroughly explain the phenomena observed in situ. As such, in situ characterization combined with device modeling is the most efficient approach to achieve a complete and in-depth understanding of the switching mechanism. In this Review, we summarize state-of-the-art understanding of memristive switching mechanism and discuss future research directions. We primarily focus on in situ characterization techniques and device modeling methods, aiming to monitor and analyze the switching behavior of memristors at the single-atom level.
Memristive switching mechanism Based on the active switching region and geometry, memristors can be classified as filament type, interface type, and bulk type. Among them, the filament type memristors have been attracted more research attentions due to their overall superior performance as well as greater challenges in mechanism studies. Thus, the Review will discuss more on filament type memristors than other types. On the other hand, according to the type of mobile species and migration behavior, memristors can be classified as cation devices, anion devices, and dual ionic devices (Box 1). To understand the underlying memristive switching mechanisms, the current research has been focused on the following five aspects: chemical composition of materials in the active region; driving forces for ion migration; filament morphology; electron conduction mechanisms; and switching dynamics ( Table 1). It is notable that a complete mechanistic understanding of resistive switching still remains largely ambiguous at this stage due to limited assessable experimental data, especially those concerning switching dynamics, such as the evolution of filament morphology and transport mechanism during repetitive cycling.

Cation memristors
Among the active electrode metals in cation devices, Ag has been studied intensively by in situ transmission electron microscopy (TEM) for understanding the switching mechanism of memristive devices. During electroforming, the growth of conduction filaments has been observed in the form of a chain of Ag nanoparticles 12 or an Ag crystalline phase 13 . The growth direction and switching dynamics of conduction filaments are governed by kinetic factors, such as the ion mobility and the redox rate 14,15 . Under the driving force of electric field, the migration of Ag nanoparticles follows a step-by-step mass transport process to form a complete conduction filament 16 .Under the reverse voltage, the conduction filaments are ruptured and a gap region is formed. In volatile cation memristor, which is also called diffusion memristor, the Ag filament relaxed into a round shape to minimize the interfacial energy after the removal of the bias 9 . Ag or Cu can function as the active electrode for both non-volatile and volatile memristors. Whether or not they maintain the conduction filaments after set operations without external bias depends on the parameters defining the system, such as the diameter of the conduction channel, the interfacial energy between the channel and dielectric materials, the heat dissipation, and local temperature, and others 17 .

Anion memristors
The oxygen migration in anion devices is important because it is involved in the switching mechanism of various materials systems. In fact, the conduction path in anion device is a chemically reduced metal with a lower valence state, which can be regarded as being composed of oxygen vacancies as n-type dopants. In memristors made up of complex oxides with a singlecrystalline structure, such as Pr 0.7 Ca 0.3 MnO 3 18 , BiFeO 3 19 , and La 2/3 Sr 1/3 MnO 3 20 , the dynamic behavior of oxygen ions/vacancies has been clarified with conclusive experimental evidence because oxygen vacancies can be observed directly. In these single-crystalline materials, the oxygen vacancies originate from the crystalline switching layer 19    Non-italicized indicates the conclusive findings, and italicized represents arguments not fully conclusive yet the motion of both cations and anions has been experimentally observed. In these dual ionic devices, the motion of cations or/ and anions can play the dominating role in the switching process, depending on the migration energy barrier of mobile species in the switching layer 31 , such as oxygen vacancies in Ni/HfO 2 /SiO 2 / Si 32 , Ag ion in Ag/Ta 2 O 5 /Pt 33 , and simultaneously migration of Ta ions and oxygen vacancies in Ta/HfO 2 /Pt 31 . In addition, noble metal ions can also act as migrating cations, such as the Pd ions in a SiO 2 switching layer 34 . As a result of the cation movement towards the cathode and the anion movement towards the anode, the conduction filaments can be identified as metal atoms 31,35 or a combination of metal atoms and oxygen vacancies 33 . However, the dynamic evolution of cations and anions needs further explorations to reveal the details.
The importance of understanding switching mechanism The fundamental reasons for the non-ideal performance of memristive devices, including the large parameter variability, cycling endurance degradation, and so on, are attributed to uncontrolled ion migration. For example, in the filament type memristor, the ion migration dominates the evolution of conduction filaments. The initial formation of conduction filaments decides the device-to-device variability, while the repeated rupture and re-formation of the filaments affect the cycle-to-cycle variability and cycling endurance (Fig. 1a, b). Parameter variability and cycling endurance issues are not concerns for metal oxide semiconductor (MOS) transistors because they rely on the transport and storage of a large number of carriers (electrons or holes). In contrast, the resistive switching in a memristor is realized via a limited number of ions migrating through a switching layer or at the electrode/switching layer interface (Box 1), so it is more challenging to quantify and control the ions engaged in the switching process. Meanwhile, the location and morphology of the ion migration region, e.g., conduction filament region, appear to be largely random, governed by a number of currently unknown factors, which in turn exacerbate the parameter variability issue 21 . The situation can be further complicated due to the expansion of active areas by ion migration to new locations

Box 1 | Classification of memristive switching mechanisms
During memristive switching, the atomic motion induces microscopic changes in memristors following three types of reconfiguration. The first type is filamentary switching, the mechanism of which is correlated with the growth of single or multiple localized conduction filaments through the switching layer 80 . This type is very common, and the active switching region is concentrated at the tip of a filament typically close to the interface between the switching layer and an electrode. The second type is non-filamentary interfacial switching, in which resistive switching is resulted from the modulation of the carrier transport barrier at the electrode/switching layer interface induced by ion migration 81,82 . This type of switching is typically slow, likely because of the lack of effective Joule heating associated with a localized conduction filament to induce a local temperature increase and to enable a fast ion motion at elevated temperature during switching. The third type of resistive switching occurs inside the bulk of the entire or a part of the switching layer, which is typically triggered by a phase change, and it involves short-term atomic rearrangement. The phase change devices include an amorphous-crystalline transition as in the Ge x Sb y Te z memristors (also called phase change memory, PCM) 83 , and topological transition or lattice structure transition as in VO 2 and NbO 2 memristors (also called Mott memristors) 27 .
According to the types of mobile species responsible for the resistive switching, memristors can be classified into three categories, namely cation devices (also called electrochemical metallization memory, ECM), anion devices (also called valence charge memory, VCM), and dual ionic devices. A typical cation device consists of an electrochemically active metal/dielectric/inert metal structure. When the scale is reduced to nano-size, the dielectric layer can be converted into solid electrolytes 84 . The mobile ions are usually the cations of the active metal, such as Ag, Cu, Ni, or an alloy of these metals 14,15 . In anion devices, the switching layers are made of compound or organic materials, including binary metal oxides, complex oxides, carbon-based oxides, nitrides, chalcogenides, etc 2 . Among them, metal oxides are the most common material system, in which the mobile ions are normally oxygen ions, or equivalently, oxygen vacancies 6 . Recently, a multi-switching mechanism involving the motion of both cations and anions has been established by a number of reports 30,33 , and thus, these memristors can be defined as dual ionic devices. Schematics of switching mechanisms using different classification criteria: during cycling, leading to the irreversible damage to devices 36 . All of the above-mentioned uncertainties and complications can cause the eventual failure of memristive switching. Therefore, from a materials science perspective, understanding ion migration in active materials during resistive switching is crucial for building robust memristors, which is the main focus of this Review (Fig. 1c).

A synergistic characterization approach
To characterize the localized ion motion in memristors, especially the dynamics during real-time cycling operations, it is essential to combine in situ characterization techniques with device modeling -a synergistic approach-as outlined in Fig. 1c. Using a variety of in situ techniques, the dynamic evolution of electrochemical and structural changes inside the functioning device during resistive switching can be visualized to a certain degree, and some of the dominant dynamic factors can be captured via the direct observations. However, experimental characterization techniques always have certain limitations in observing the switching process due to finite spatiotemporal resolution, which can be complemented and completed by modeling studies. Based on the input of the experimental results, equations describing the dynamic switching process can be developed using appropriate physical modeling, and the driving forces, electron conduction, as well as ion migrations can be simulated at the atomic level. With the combination of in situ techniques and device modeling, a holistic picture of switching mechanism can be eventually established.
The switching mechanism needs to be studied over a broad spatiotemporal range, which requires a closely collaborative application of in situ techniques and device modeling ( Fig. 2 and Table 2). The time scale of ion migration can vary over a range from picoseconds to seconds depending on the condition of the driving force. Due to the fast speed of ion migration under high electric field and/or elevated temperature, resistive switching may occur as rapidly as a sub-nanosecond 37 . Typically, the switching speed is exponentially dependent on the applied voltage 38 . The requirement for data retention in commercialization is normally ten years at ambient temperature. On the other hand, the parameter variability of memristors results from the unstable behavior of the conduction filaments, as well as non-uniformity during fabrications at the device, array, chip, and wafer levels. The diameters of the conduction filaments range from angstroms to nanometers 9,13 . In Fig. 2, the temporal ranges of the physical properties and the electrical behaviors are positioned along the top horizontal axis, and the spatial ranges from atom level to wafer level along the right vertical axis. In principle the combination of in situ techniques and device modeling is capable of covering a broad spatiotemporal range and thus providing a comprehensive understanding of device behavior. The in situ techniques can typically work in a temporal range from microsecond scale to several days and in a spatial range from sub-angstrom level to several micrometers. Device modeling further expand this temporal-spatial region. The first principles (FP), molecular dynamics (MD), and kinetic Monte Carlo (KMC) methods can be used to investigate the resistive switching of memristors down to less than a picosecond, whilst KMC and compact models cover long-time domains up to years.
The strategy of combining in situ characterization with device modeling has shown a great potential in mechanism understanding and performance optimization of memristors. For instance, Li et al. have used in situ holography to characterize the evolution of oxygen vacancies distribution in a HfO 2 -based memristor under electrical biases 24 . The dynamics of the oxygen vacancies were also modeled by KMC simulation, revealing that the rupture of conduction filaments occurred in the region near the top electrode. Using scalpel scanning probe microscopy (SPM), Celano et al. differentiated the oxygen exchange layer (OEL) from the conduction filament in Hf/HfO 2 and Ta/Ta 2 O 5 systems with a nanometer lateral resolution 39 . First principles (FP) calculation indicated that the diffusion of oxygen at the OEL/oxide interface can induce variations in the density of states near the Fermi level in the oxide. The minimum thickness of the OEL/oxide interface has to be~3 nm to achieve the optimized performance. For a Ag/ZrO 2   Memristive switching studied by in situ techniques From a microscopic view, the in situ characterization of switching dynamics can be divided into four different physical stages (Fig. 3a). In stage I, the initial step is the generation of mobile ions in the switching layer or at the switching layer/electrode interface, followed by the drift of ions under an electric field. The redox reactions in memristors are essentially electrochemical reactions, which are closely related to the interfaces and moisture concentration in the ambient atmosphere 41 . Although diffusion coefficients have been determined by experimental methods 41 , this dynamic process, from the redox reactions to the nucleation of mobile atoms in the switching layer, has not been directly characterized using existing in situ techniques, partially because of the high temporal resolution required. During stage II, after the aggregation of mobile ions develops into nanoparticles for cation devices or the concentration of oxygen ions reaches a detectable level for anion devices, in situ TEM is the most suitable technique for visualizing the dynamic evolution of morphology at nanoscale and the moving paths of the nanoparticles. Once the formation of conduction region (filamentary or non-filamentary) is completed, the memristor is switched from the high resistance state (HRS) to the low resistance state (LRS) in stage III. The morphological evolution of the ion migration region and the resistive switching of the device can be characterized simultaneously by most of in situ techniques. The correlation between the morphology change and the electric behavior provides clues to parameter variability of memristors. During stage IV, the memristor is switched back and forth between the HRS and LRS by applying switching voltages. To characterize this repetitive cycling, in situ scanning tunneling microscopy (STM) and in situ conductive atomic force microscopy (CAFM) can be employed to monitor local morphological changes in the conduction channel and the resistive switching of the device 30,42 . Therefore, in situ STM and in situ CAFM are key techniques for studying device failure during cycling endurance.
Different device structures have been in situ characterized, including vertical, lateral, or tip-based structures. The vertical structure is a thin slice of the stacked device (Fig. 3b) 43 or a stack of a three-layer device on a supporting membrane 12 . Such vertical structure more closely represents the actual devices in applications 44 . With the lateral structure (Fig. 3c) 45 , the memristive device is usually fabricated directly on a specialized chip designed for in situ characterization. Since the gap between the two electrodes is wider than the vertical structure, the dynamics of ion migration can be easily observed. The tip-based structure utilizes a metallic tip on the instrument as the opposite electrode to form a memristive device with the oxide layer and metal substrate (Fig. 3d) 46 . In this structure, ion migrations happen around the tip electrode, which is easier for device preparation and provides a better control for the resistive switching during dynamic observations.
Currently, several in situ techniques have been employed individually to study the switching mechanisms of memristive switching. Table 2 lists the details of their operational features. In situ TEM plays an important role in elucidating the underlying microscopic mechanisms of switching dynamics due to its unprecedented spatial resolution 47,48 . For the cation devices, the formation of a conductive bridge composed of Ag nanoparticles under an applied bias was observed by high-resolution brightfield imaging, under the driving force of electric field and Joule heating (Fig. 4a) 9 . The dynamic relaxation process of the conduction filament was recorded after the removal of biasing. Furthermore, the combined application of in situ TEM and atom probe tomography (APT) has revealed the elemental distribution of conduction filaments in Ag/TiO 2 /Pt devices in threedimensional space 13 . The results showed that the conduction filament was Ag-doped TiO 2 , which means that the switching dynamic was accompanied by doping/de-doping with mobile ions rather than the connection/disconnection of the entirely metallic conduction filament. For anion devices, the evolution of oxygen ions in a single crystal during resistive switching was identified by the combination of TEM and electron energy loss spectroscopy (EELS) 49 . It was found that the change in the overall oxygen vacancy concentration within the device is associated with the electrocatalytic release and reincorporation of oxygen at the electrode/oxide interface. Although it is challenging to characterize the oxygen ions/vacancies in amorphous materials, the combination of in situ TEM and electro-holography provides a workable solution 24 . The experimental results demonstrated that the conduction filaments constituted by oxygen vacancies were initiated in the bulk of the HfO 2 layer and ruptured at the interface of the HfO 2 /TiN top electrode 24 .
Besides the significant findings via in situ TEM, switching mechanisms were also partially elucidated via other in situ techniques. In situ scanning transmission X-ray microscopy (STXM) enables nondestructive studies of a vertical specimen to reveal chemical composition distribution 50 . The formation of conduction filaments in Pt/HfO 2 /Hf/Pt devices was captured under biasing, finding a ring-like feature in Fig. 4b 51 . The local distribution of an electronic structure in TiO 2 -based memristors was measured with in situ electron beam-induced current (EBIC), as shown in Fig. 4c, providing a clear image of the underlying switching region, and reveals propagating polarization domains of symmetrical device structures 52 . Electron transport and electrochemical reactions of HfO 2 -based memristors were investigated with in situ CAFM. As shown in Fig. 4d, increasing the applied voltage generated an oxygen-deficient region, and oxygen vacancies accumulated in the location with higher applied bias, leading to the formation of conduction filaments with diameters of~15 nm 53 . As shown in Fig. 4e, the highly conductive regions in TaO x -based memristors were visualized by in situ STM 30 . The measured corresponding I-V curve indicated that a purely metallic or mixed metallic-semiconductor structure was formed during resistive switching, suggesting that both cations and anions participate in the switching process in oxide-based memristors 30 . Valov et al. have developed an approach involving the STM technique to visualize the electrochemical formation and dissolution of Ag clusters 54 . Through further material design, a single ion/atom transfer can be manipulated in an atomic layer-by-layer manner, enabling a discrete control of the electrical properties 55 . Combined with in situ TEM, in situ STM-TEM system have been applied successfully to reveal the details of the electrochemical dynamics in cation memristive devices, including the migration and redox reactions of Cu + , and the nucleation and growth of Cu protrusion. This finding convincingly demonstrates that the chemical potential difference is a key driving force for the oxidation of Cu, besides the external electric field. The dynamics of the redox reaction for a lateral graphene oxide device was visualized by in situ optical microscopy (Fig. 4f) 56 , presenting effective information of filament growth kinetics. Redox reaction at the interface of the active electrode and the electrolyte was studied with in situ cyclic voltammetry 57,58 . The activation barrier for the reduction (Ag + to Ag, Ag 2+ to Ag + ) and oxidation (Ag to Ag + , Ag + to Ag 2+ ) reactions experienced by Ag atoms/ions at the cathode and anode can be measured (Fig. 4g) 57 , and a second-order oxidation of Ag at the interface was observed.
In summary, most in situ characterization results have been focusing on the formation of conduction filaments (stage III) and the corresponding resistive switching, while the initial ion migration (stage I), nanoparticle movement (stage II), and cycling between LRS and HRS (stage IV) have been less often revealed using in situ techniques. However, it has become more and more clear that the parameter variability of memristive devices is closely related to the non-uniformity of the conduction path during set process and the variation of the gap length after the rupture of conduction path during reset process. Unfortunately, the  corresponding details remain poorly understood to date, while the factors determining cycling endurance haven't yet been well investigated either. Therefore, an immediate attention to the switching mechanism in stages I, II, and IV via in situ studies is urgently demanded.
Device modeling based on in situ characterizations Device modeling and in situ techniques complement each other to improve our understanding of resistive switching. According to different application requirements, device modeling can be classified into physical modeling and compact modeling (Box 2). One target of physical modeling is to reconstruct the ion migration during the resistive switching process. For example, while in situ techniques are capable of characterizing the morphology of conduction filaments as shown in Fig. 1c, physical modeling can be further used to shed light on the intermediate filament growing/rupturing processes at high spatial and temporal resolutions beyond those of experiments [59][60][61] . Modeling results can also clarify the driving force by simulating the distribution of electric field, temperature, and chemical potential in the active layers [61][62][63] . Several physical modeling techniques have been explored for this purpose. First Principles (FP) (also called ab initio) calculations are widely used to obtain the conduction property of a stable state and the transition energy between different stable states 63 . As illustrated in Fig. 5a, FP calculations were used to construct a supercell with tens to hundreds of atoms. With this technique, there is no need to introduce any model parameters. A typical task with the FP approach is to calculate the activation barrier of ion migration. This calculation is very useful for evaluating whether a specific doping configuration can help improve the uniformity or reliability of a memristor 64 . FP calculations have also been used to investigate some simple dynamics, such as single oxygen migration events, charging/decharging events 63,65 , or resistance volatility 66 . However, existing FP approaches still face the challenge of how to accurately model the polycrystalline, amorphous, or complex stack structures in real memristive devices. In addition, it is also challenging to simulate long-term and multievent dynamic processes owing to its complex computation procedure. Dynamic simulation techniques, such as MD 67 (Fig. 5b) and KMC 59 (Fig. 5c), are powerful tools for studying the ion migration process in memristor and related variabilities. A reactive MD with the charge equilibration method was proposed to investigate the formation of metastable atom-chain and stable filaments in the cation memristor 67 . KMC is essentially a stochastic and event-driven method. It does not need to perform calculations for every time period; instead, it counts only the duration between two events. Thus, it is easy to cover a broad time scale ranging from resistive switching over nanoseconds to long-term retention over years 59,68 . To model the ion distribution accurately, the KMC method is applied in combination with the finite element (FE) method (Fig. 5d). Whilst, the FE method is sometimes used separately to simulate the resistive switching without considering the parameter variability 69 . In this case, only the concentration of oxygen vacancies is calculated, as illustrated in Fig. 5d, thus simplifying the simulation. The disadvantage of the KMC and FE methods is that they require many model parameters and assumptions.
Another goal of physical modeling is to connect the microscopic ion distribution with the macroscopic electrical performance. In some cases, the electron transport mechanisms of memristors can directly fit in the macroscopic model (Table 1). However, with complex structural and material components, memristors may exhibit unique I-V curves that cannot be fit well with existing models. In this case, the modeling of I-V characteristics should start from the microscopic mechanism correlated with a specific distribution of ions. A simplified way to model I-V characteristics is to use a resistor network. The resistors can vary according to the corresponding oxygen vacancy distribution 62 . A more general way is to solve the Poisson equation and current continuity equation. Figure 5e summarizes a more complete flow to model and simulate resistive switching processes based on the approaches reported previously [59][60][61][62][63] . First, decisions should be made on which physical mechanisms are taken into account and how to quantify these physics processes with appropriate equations. Typically, most Box 2 | Classification of memristive device models Based on different application requirements, device modeling of memristor can be classified into different levels. Physical modeling starts from microscopic physical processes of resistive switching, using numerical simulation approach to accurately reproduce to device behaviors 59,60 . Physical modeling aims to clarify switching mechanisms in complementary with experimental characterizations and provide guidelines for device performance optimization 61 . Compact modeling introduces some empirical assumption without concerning the underlying physics. Compared to physical modeling, compact modeling provides the possibility of rapidly reproducing the phenomenological electrical behavior of memristors with a low computation cost. Some compact models are established by mathematically fitting the measured electrical behaviors of memristors 85,86 , which are also called behavioral models. The behavioral models are highly simplified and easy to extend to different types of memristors; however, they cannot capture the variabilities in a real circuit. For example, in a crossbar array, the voltage drop on each memristor cell is different because of the interconnected resistance 70,87 , therefore, the resistive switching behaviors are different for each cell, which is almost impossible to be captured by the behavioral model. Therefore, the behavioral models are mainly suitable for system-level simulations, such as the benchmarking of a memristor-based neuromorphic network 71 . On the other hand, a physics-based compact model is more useful for circuit-level simulation. Most physics-based compact models of memristors are derived using the FE method by quantifying the morphological or compositional changes with simplified physical equations 88,89 . The simulation program with integrated circuit emphasis (SPICE) model is a special type of compact model. It should be analytical so that it can be easily embedded into the commercialized technology computer-aided design tools. A good SPICE model for a real memristive device should include the parasitic circuit elements, so that it can be accurate for the simulation of a complex mixed memristor/CMOS circuit 87,90 . A physics-based model can help to understand the device mechanism while a purely behavior model does not. Schematic for the device modeling of memristors: a Physical modeling. b Memristive device design with a physical model. c Compact modeling. d Crossbar array simulation using a compact model. e SPICE modeling. f Memristor/CMOS mixed circuit simulation using a SPICE model. Reproduced from ref. 91 , Nature Publishing Group (b); ref. 72 , IEEE (c); ref. 87  models start from ion migration or oxygen vacancy generation based on the defect theory. The parameters should be input based on the experimental extraction or FP calculations, and the boundary conditions should be fixed. Subsequently, the distributions of various driving forces are calculated, and a dynamic method is used to describe the ion migration. With an updated ion distribution, the dynamic switching processes including microscopic morphology evolutions and macroscopic electrical characteristics are finally simulated. Compact modeling also plays a crucial role for memristor research, especially in circuit analysis. Unlike CMOS transistors, the compact model of memristors should not only focus on the basic electrical characteristics but also capture the parameter variability and cycling endurance degradation characteristics. Typically, to construct a compact model of memristor, the basic model framework is firstly developed using a simplified physical image and analytical equations. Then, the degradation parameters and randomness parameters are introduced in the framework to describe the cycling endurance and parameter variability characteristics [70][71][72] . After that, the model parameters are extracted based on the statistical measurements from amount of memristive devices and repetitive cycling tests. Finally, a SPICE model (simulation program with integrated circuit emphasis model) can be constructed by further extracted the parameters of parasitic circuit elements. To ensure the accuracy of circuit simulation, the compact model of memristors also calls for a complete picture of switching mechanism, since most of the compact models are physics-based.

Outlook and perspectives
Despite recent progress in the in situ characterization and device modeling of resistive switching, there are still many aspects not fully understood yet. The currently available data from in situ measurements has only uncovered a small portion of the entire picture of memristive switching, which in turn limits the application of device models to further understand it. This situation thus calls for in situ characterizations of memristors at high resolution in three dimensions (3D) and physical modeling with high degree of accuracy capable of providing practical guidelines for device optimization in the future. To this end, here we propose several research directions that are crucial to tackle the challenges.
First, we need to continuously improve the observation resolution in order to monitor the dynamics of ion migration in a region of atomic distances and sub-ns intervals. Currently the biggest obstacle to increase spatial resolution is the mobility of anions migrating through amorphous oxide layers, making it extremely challenging to identify their positions. One possible solution is the use of isotope elements to trace the movement of ions 73 . The switching dynamics occurring in a short time range of nanoseconds to milliseconds can be investigated via ultrafast transmission electron microscopy (UTEM) 74 . Another plausible approach is to reduce the speed of resistive switching using low temperatures and to characterize the dynamic evolution using cryo-TEM techniques 53 .
Secondly, the characterization of switching dynamics needs to be conducted at higher dimensions. The 3D characterization of conduction filaments is expected to provide a complete representation of their morphology 13,42 . A 4D evolutionary representation that demonstrates the 3D switching dynamics over time can provide more direct information for understanding the switching mechanisms. In situ techniques that can be applied in this aspect include electron tomography 75 , 3D STEM tomography 76 , and 4D UEM 74 . Studying the influence of environmental conditions, such as different ambient atmosphere and thermal field, can not only provide additional evidence of switching mechanism, but also help to evaluate the potential of memristors for some special applications, like automotive electronics. Environmental TEM 77   Then and distribution of driving force and ion motion in the switching layer are calculated. These calculations are repeated for many times as the whole switching process evolves multiple ion motion events. Finally, the electrical behaviors (e.g. variability and switching) of the memristor is output based on the microscopic simulation and electron transport modeling simulated environment with a controlled humidity, gas composition, liquid mixture, and temperature. Finally, we should continuously use device modeling to further understand the switching mechanism, especially beyond the resolutions of the characterization techniques, and thus possibly use the insights obtained from modeling to guide device optimization and design. With the help of predictive models, we could be able to understand how to best combine materials or design device architecture to improve device uniformity and reliability. We can also use the modeling to investigate the theoretical limit of the minimal cycle-to-cycle and device-to-device variation in addition to the maximum cycling endurance, based on which we decide on how to cope with this non-ideal performance at a circuit level. The accuracy of memristor models needs to be continuously improved. It is essential to develop realistic physical models for the amorphous and polycrystalline states of active materials to accurately reproduce that in testing devices. The combination of percolation theory and Monte Carlo simulation provides a possible solution to model the ion migration and electron transport in an amorphous system. To capture the variability and endurance degradation over trillions of switching cycles, it is recommended to utilize machine learning in the simulation of memristors 78,79 . After training a neural network with the pre-calculated electrical data in a few switching cycles via device modeling, a neural network can fast predict the data during the following cycles, no need to repeat the time-cost atomic-scale simulation.
In view of all of above-mentioned opportunities, it is our hope that large parameter variability and poor cycling endurance issues that we are currently encountering in memristor research can be eventually solved based on a comprehensive understanding of the resistive switching mechanism. To this end, one possible approach is to design material stacks with predictable ion migrating paths by confining the formation/rupture of conduction filaments to local regions using nano-fabrication technologies, such as deliberated doping, multilayer and side-wall protected structure, capping layer modulation, etc.
In conclusion, we have reviewed the current research status of the switching mechanism of memristors, outlined a synergistic strategy of combining in situ techniques and device modeling to uncover the detailed switching mechanism and examined its potential. We advocate immediate action to further investigate memristive switching dynamics at higher resolutions and higher dimensions in experiments, and a larger degree of accuracy in modeling. Continued research efforts are required to solve the practical issues associated with parameter variability and cycling endurance of memristive devices before they could be possibly transferred to commercialization for future memory and computing technologies.