Unveiling the Fundamental Role of Temperature in RRAM Switching

Feb 1, 2018 - The circuit is solved by means of Kirchhoff's laws and linear algebra operations with an external applied voltage that corresponds to ou...
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Research Article Cite This: ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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Unveiling the Fundamental Role of Temperature in RRAM Switching Mechanism by Multiscale Simulations Federico Raffone* and Giancarlo Cicero Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy S Supporting Information *

ABSTRACT: Even though resistance switching memories (RRAMs) can be potentially employed in a broad variety of fields, such as electronics and brain science, they are still affected by issues that prevent their application in circuitry. These problems are a consequence of the lack of detailed knowledge about the physical processes occurring in the device. In this work, we propose multiscale simulations, combining kinetic Monte Carlo and finite difference methods, to shed light on the yet-unclear switching process occurring in the valence change RRAMs, which are believed to work as a consequence of the drift and diffusion of crystalline defects that act as dopants. Results show that the height of the defect diffusion barrier influences the switching process, the retention, and the switching time. In particular, nonvolatile switching can be achieved only by means of the fundamental role of temperature variations induced by Joule heating if the diffusion barriers of the defects are larger than ∼1 eV. High barriers prevent defects from hopping when no voltage is applied. During the transition from the high-resistance to the low-resistance state of the device, a heating stage of the material precedes the defect drift because the applied electric field by itself is not enough to lead to a drift velocity such that switching is achieved within microseconds. The temperature increase has, therefore, the double effect of activating the motion of the defects and enhancing their drift velocity. The switching process can occur only if a sufficiently high temperature is reached thanks to the Joule effect. On the basis of these findings, the RRAM design could aim at a better temperature management to achieve at the same time reproducibility and reliability. KEYWORDS: RRAM, memristor, TiO2, thin film, KMC

1. INTRODUCTION Memristors are devices that are able to switch among different resistive states, typically a high-resistance state (HRS) or the OFF state and a low-resistance state (LRS) or the ON state by means of an applied external bias. The devices conserve the last resistance state in which they were left even if the external voltage is removed. The transition from the HRS to the LRS is called the SET process whereas the opposite is called RESET. The existence of the memristor was theorized by Chua in 1971,1 but only in 2008 a first oxide-based prototype was realized,2,3 later addressed as a redox-based resistance switching memory (RRAM). Since its discovery, scientists have been working to understand the mechanism behind the device switching and to eliminate the intrinsic device failures and poor reproducibility4−6 that prevent its practical applications in commercial circuitry. The first RRAM was made by sandwiching a TiO2 thin film between two platinum contacts. These materials are nowadays among the most widely employed in the fabrication of RRAM.7 For this class of devices, also known as valence change memories (VCMs), the inner oxide is said to be divided into two parts.3 One is stoichiometric and therefore is electrically insulating. The other one has a high density of oxygen vacancies. These defects are © XXXX American Chemical Society

known to act as donor impurities; thus, they locally enhance the conductivity of the oxide.8 It is argued that the resistance switching occurs because of the motion of the crystallographic defects when the voltage overcomes a threshold (VSET). Once the understoichiometric part extends over the defect-free region or once a conductive filament of stacked defects connects the two electrodes, a low-resistance path is created for electrons and an abrupt change in current is measured externally: the device switches to the LRS.9,10 So far in the literature, the main drive for the motion of vacancies has been identified in the electric field generated by the high voltage, in particular because of the nonlinearity of the relationship between the defect drift velocity and the electric field.11,12 Nevertheless, from the atomistic point of view, oxygen vacancies in TiO2 possess high diffusion barriers,13 which can be hardly overcome by applying the typical operating voltages to RRAM. The temperature generated from self-heating11 could potentially help solve such an inconsistency arising from the lack of a link between the atomistic nature of the switching and measurable external Received: January 9, 2018 Accepted: February 1, 2018 Published: February 1, 2018 A

DOI: 10.1021/acsami.8b00443 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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of the highly resistive layer.26 The device area in the x-y plane is 102 nm × 2 nm. The TiO2 is in the rutile phase, with the [001] direction corresponding to the direction of the electric field.27 The reason for the choice of the rutile phase lies in the different diffusivity of oxygen vacancies (VO) with respect to titanium interstitials (Tii). In fact, not only oxygen vacancies but also Tii were reported to actively take part in the switching process by altering the local conductance.28 Oxygen vacancies can be considered as slow-moving defects as their limiting diffusion barrier in the direction of the electric field is above 1 eV.13 Titanium interstitials are fast-moving defects whose limiting diffusion barrier in the direction of the electric field is around 0.4 eV.13 This difference in mobility allows for the analysis of the effect of the barrier height on the switching mechanism. In anatase, the diffusion barriers of both oxygen vacancies and titanium interstitials are above 1 eV in the [001] direction according to ref 29. Despite the arbitrary defects typology chosen in our simulation, the defect content of the understoichiometric layer can be experimentally tuned during the growth process favoring the formation of one kind of defects over the other.13,28,30,31 To provide a comprehensive description of an RRAM, it is necessary to account for both the atomistic and macroscopic aspects of the device. To this end, we proposed a multiscale simulation approach that is made up of three different subunits as shown in Figure 2. Each subunit deals with a different domain. In the first one (labeled A in Figure 2), the defect drift and diffusion inside the oxide are described thanks to KMC simulations in the variable step-size method.32 The first subunit treats only the crystalline TiO2 layer as a domain. To achieve an off-stoichiometry of TiO1.9 in the bottom half of the device, either 900 titanium interstitials or 1800 oxygen vacancies were randomly generated in the oxide. The system was assumed closed, and so no generation or annihilation of defects was considered. Also, no explicit interaction among defects was accounted for. Periodic boundary conditions were applied in the directions parallel to the TiO2/Pt contact plane. Given the crystalline nature of the oxide, defects can move only on a grid of fixed positions representing the loci of the defect potential energy minima inside the TiO2 lattice. Free diffusion is prevented by energy barriers so that the motion occurs by thermally activated hopping events between adjacent minima positions. The oxide is then simplified as a grid of all possible positions that a defect can occupy. According to the KMC methodology, in every Monte Carlo step, a defect moves. This represents the biggest advantage of KMC simulations compared to classical molecular dynamics as all the time between two events is skipped. Thus, it is possible to achieve larger timescales.33 The defect and its hop direction are randomly selected on the basis of the defect transition rates Γi. The rate of each hop event can be obtained treating the defect diffusion in terms of a thermally activated process within the transition-state theory.34 The rate is determined as follows

quantities such as total current. Many factors are reported to influence the device performance in terms of endurance, retention, and switching speed. However, the source of these effects is still unclear.14,15 These phenomena are not yet well related to the material characteristics. In particular, the impact of atomic-level properties has not been so far taken into consideration. In this regard, multilevel simulations could constitute a means to link these properties with the device performance and to predict quantities such as the SET voltage or the internal temperature required to avoid possible device breakdowns or unwanted structural changes. We herein propose a mixed continuum and kinetic Monte Carlo (KMC) approach to take into account both atomic-level physical quantities such as defect diffusion barriers and macroscopic quantities such as internal temperature and overall device current to predict the conditions under which the device switches. Thanks to this combination of computational methods, in our approach, the temperature can dynamically change during the course of the simulation. We highlight that in previous RRAM atomistic simulations, the temperature has often been kept constant during switching,16−18although it is known that Joule heating derived from the current flowing into the oxide leads to temperature variations. Both previous experiments and finite element modeling simulations suggest that the temperature rises even before switching on the device.19−21 For example, during LRS, temperatures up to 1800 K were predicted theoretically.22 These results seem to indicate that thermal variations cannot be ignored as temperature alters the defect motion, increasing the diffusivity23 or leading to phase transitions.24−26 In this paper, we show how temperature has a fundamental role in triggering the switching of a VCM and how the oxide characteristics, such as the defect diffusion barriers, determine the switching mechanism.

2. METHOD In our analysis, we focused on a simple model device to unveil the basic, yet unclear, principles of switching and to ensure a broad applicability of the model. The prototype device that we focused on (Figure 1) is made of a top and a bottom electrode in platinum, both 10 nm thick. Between the electrodes, a thin layer (3 nm of thickness) of crystalline and stoichiometric TiO2 is stacked on 3 nm of defective TiO1.9 that acts as a defect reservoir in accordance with literature suggestions, which describe the switching as a small modulation of the thickness

⎛ U ⎞ Γi = ν0 exp⎜ − A ⎟ ⎝ kBT ⎠

(1)

where the escape-attempts frequency ν0 was set to 1013 Hz as experimentally established in accordance with previous studies,11,35 kB is the Boltzmann constant, and T denotes the local temperature at a given step. The activation energy UA corresponds to the defect diffusion barrier of TiO2. For our study, we used the barriers calculated by means of density

Figure 1. Structure of the simulated TiO2-based RRAM device. B

DOI: 10.1021/acsami.8b00443 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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Figure 4. Sketch of the TiO2 unit cell containing a titanium interstitial. The diffusion paths are indicated by the blue and yellow arrows. The number next to each arrow expresses the diffusion barrier in eV.

by Iddir et al. in ref 13. In the case of VO, the direct diffusion along the direction [001] (the electric field direction) is highly unlikely because of the 1.77 eV barrier. An indirect path, which involves two hopping processes represented in Figure 3 with a barrier of 1.10 eV,13 is instead more probable. Titanium interstitials diffuse thanks to two different mechanisms. In one case, the defect hops along the channel formed by the nearby titanium atoms (blue arrow of Figure 4) and involves a barrier of 0.37 eV.13 In the other, the motion occurs by displacing from its position a titanium atom belonging to the TiO2 crystal (yellow arrows). During this process, having a 0.23 eV barrier,13 the titanium interstitial takes the place of an atom in a crystal lattice position while simultaneously creating another interstitial. The defects travel around the crystal according to such allowed hops. An external electric field is applied to the TiO2 layer. The effect of the electric field is to reduce the UA by a factor of −qaE/2, where q is the defect charge (+4 for Tii and +2 for VO), E is the electric field intensity, and a is the displacement in the direction of the electric field.12 Because of the flatness of the TiO2−x/TiO2 and TiO2/Pt interfaces, the electric field was calculated as the ratio between the applied voltage and the total oxide thickness.15 The temporal gap between hopping events varies according to a uniform random variable r and the sum of the rates of all possible moves 1 ln r Δt = − ∑i Γi (2)

Figure 2. Flowchart representing the subunit division of the multiscale algorithm of the mixed KMC/continuum approach. On the left side, the different algorithms employed in the simulation steps are shown and on the right side is the related domain involved in the calculation. The KMC domain A is made by a grid of locations where the defects (blue balls) can move. The TiO2 structure, constituted by red (oxygens atoms) and gray (titanium atoms) sticks, is sketched only for reference. The dark blue dashed lines show the division of the oxide into bins. In the subunit B, the circuit made by resistors among the bins is connected to an external voltage. In system C, the heat equation is solved using the finite difference method over the domain formed by the TiO2 and the platinum contacts. At the top of the device, the cooling effect of the air is accounted, whereas at the bottom insulating boundary conditions are assumed. The information passed between systems is indicated by light blue labels.

functional theory (DFT) simulations in ref 13. Although ν0 depends on temperature36 and the diffusion barrier height should be better expressed as enthalpy difference rather than internal energy difference, both effects were found to be negligible and therefore excluded from the computation.37 In Figures 3 and 4, the rutile defect diffusion barriers and directions are sketched, respectively, for oxygen vacancies and titanium interstitials, calculated by means of DFT simulations

The time gap and the hopping rates are, thus, a function of the electric field and temperature, which is provided by the other algorithm subunit. After every dynamic step, the standard KMC algorithm jumps forward in time in a discrete way. In the KMC methodology, during this time gap, no other defect is assumed to hop because the temperature is kept constant. However, in our case, the temperature is allowed to rise in the meanwhile due to Joule heating. Although T variations can be of the order of 1 K/ns and at room temperature the time between two hopping events can be as high as 105 s, updating T so rarely would give rise to abrupt discontinuities. To cope with this issue, the next hop is rejected if the hopping process occurs in more than a threshold time tth. The threshold time is defined as the time it takes to dissipate a power of P = V2/RHRS causing a temperature increase ΔTth = (αPt)/mcp, where m is the mass of the oxide, cp is the TiO2-specific heat capacity at fixed pressure, V is the current input voltage, RHRS denotes the resistance in the HRS, and α is a factor that accounts for heat dispersion at the metal contacts and in air and that depends on the materials, the geometry, and the thermal boundary

Figure 3. Sketch of the TiO2 unit cell containing an oxygen vacancy. The diffusion paths are indicated by the blue arrows. The number next to each arrow expresses the diffusion barrier in eV. C

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Figure 5. SET process of a TiO2-based memristor device containing only oxygen vacancies. (Panel a) The dynamics of defects after the application of an external voltage of a 5 V pulse. In the color panels from I to V, the local off-stoichiometry x of each bin comprising the titanium oxide is represented by a color map. Letters h and l indicate the height and the length of the oxide layer. The metallic contacts, located on the top and on the bottom of the oxide, are not shown. (Panel b) The resulting evolution of the current and the average temperature in time. The roman numeral labels in the plot indicate the corresponding structures of panel a. In the inset, a zoom of the SET stage is shown.

domain is assumed as two-dimensional. Each bin has four resistors linking the surrounding bins so that a square network is then built. The resistance is taken as the mean between the resistance of the two linked bins as suggested in ref 40. The circuit is solved by means of Kirchhoff’s laws and linear algebra operations with an external applied voltage that corresponds to our input voltage. The current passing into the external loop and in all of the interconnecting resistors is calculated so that the dissipated power in each bin is then found. This information is passed to the last algorithm subunit (labeled as C in Figure 2). As suggested by Padgett et al.,40,41 the bin power dissipation is included in the heat equation in the form of Joule heating. The heat equation is then solved in the finite difference scheme. In this case, the domain of the system is extended so as to include the platinum contacts. Periodic boundary conditions were applied in the plane parallel to the device area. At the bottom of the device, a thermal insulation was assumed because typically the bottom contact lies on a highly insulating silicon oxide layer used to avoid any sneak path current. The top electrode is in contact with air through convective boundary conditions, with a heat transfer coefficient of 20 W/m2·K.42 For the Pt/TiO2 boundary, the temperature and flux continuity are assumed within the matching conductivity condition.43 From the heat equation, it is possible to establish the temperature in each bin, which is then sent back to the KMC subunit where the defect hopping rates are updated on the basis of the local temperature. With such a multiscale approach, it is then possible to monitor how defect positions, local temperature, and total current simultaneously evolve in time so as to provide an overall description of the switching process.

conditions. In the case of 10 nm-thick platinum contacts and convective heat transfer on top of the device, the value of α was found to be 0.178 by running a test simulation in the HRS without defect dynamics and calculating the slope of the temperature−time curve. ΔTth is the maximum increase in the temperature that is allowed to occur between two Monte Carlo steps. We found that a value of ΔTth = 1K is a good trade-off to avoid dynamics perturbation and maintain computational efficiency. By combining the two formulas, one obtains t th =

ΔTthmcpRHRS αV 2

(3)

The time and all other quantities, such as device temperature, are updated according to tth,T = tth/2 (i.e., corresponding to a ΔTth of 0.5 K) to have a frequent check of the defect motion. The simulation space constituted by the TiO2 is then divided into bins having 1× 2 ×1 nm size each. After each dynamic step, the defects are allocated in the corresponding bin on the basis of their updated position. By knowing the number of defects per bin, we can extract the local off-stoichiometry of TiO2−x. The off-stoichiometry x is related to the electrical resistivity ρ of the bin as defects (both interstitials and vacancies) add electrons to the conduction band. The relation between x and electrical resistivity ρ was derived from the work of Bak et al.,30,38 where the partial oxygen pressure is related to both defect content and electrical conductivity. The effect of temperature on conductivity was also accounted in an Arrhenius-like fashion with an activation energy of 0.028 eV39 (see Supporting Information for further details on the x−ρ relation). The information on the local bin resistivity and the time elapsed during dynamics are then passed to the second algorithm subunit (labeled B in Figure 2), which treats the device from the circuit point of view. Adjacent bins are electrically interconnected through resistors. Because of the small dimension of the device along the y axis (2 nm), the

3. RESULTS AND DISCUSSION We first discuss the result of a simulation of a TiO2-based device containing the most common kind of defects, that is, oxygen vacancies (see Figure 5). As said before, these defects D

DOI: 10.1021/acsami.8b00443 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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ACS Applied Materials & Interfaces are characterized by a diffusion barrier of about 1 eV. In Figure 5a, the oxide area is subdivided in many bins whose color indicates the local off-stoichiometry. Defects are initially located at the lowest half of the oxide (panel I). The perfectly stoichiometric top half of the oxide ensures that the device is initially in the HRS. The Pt contacts, not represented in the picture, are located below and above the oxide film. A pulse of 5 V is applied to the system. The voltage induces an electric field in the direction of the upper contact. From the results of our simulation, we see that even after 1600 ns no defect moved toward the top high-resistive area although lateral motion took place (panel II). In the meanwhile, the temperature steadily increased reaching more than 700 K in the OFF state due to the Joule heating. The first hop in the direction of the electric field is seen after 1620 ns. Afterward, a progressively increasing number of vacancies drift into the upper direction (panel III). This transition is remarked by the change in slope of the total current in the device (Figure 5b) because the stoichiometric region becomes thinner as vacancies start moving. Eventually, a defect establishes a highly conductive path between electrodes and the device switches to the ON state (panel IV). The ON state current is capped by a compliance current (ICC) 4×102 times larger than the OFF state current as usually done in the experiments. We will return later in the text to the effect of the ICC on the physical behavior of the device and in particular on the defect dynamics. As soon as the device is shorted, the mean temperature sharply rises. In the early stage of the switch, the ON state is not stable (see the inset of Figure 5b). The current abruptly switches between the LRS and HRS because the vacancies jump back and forth breaking and reforming the conductive filament. After 50 ns, a stable ON state is achieved. As we keep applying the voltage, the filament enlarges (we consider part of a single filament all the bins that connect the two electrodes) because more defects have moved closer to the top electrode and the temperature slope reduces until a plateau is reached (panel V). We now analyze the behavior of the device during the first linear part of the IV characteristics reported in Figure 5b by studying more in detail what happens in the oxide under a microscopic point of view. In particular, to better understand why the vacancies do not move before 1600 ns from the application of the voltage and why a temperature of at least 700 K is needed for the switching to occur, we calculated the average hopping time of VO in the [001] direction as function of the temperature (T) and of the electric field (here expressed in the form of voltage applied to a 10 nm-thick oxide: V@10 nm). This quantity is reported in Figure 6a. It is apparent that the electric field has no significant effect on the reduction of the average hopping time. Therefore, the application of a voltage that leaves the temperature of the device unaltered is not enough to produce a resistance switching. By contrast, the temperature has a tremendous impact: as the temperature passes from 300 to 900 K, the average hopping time is reduced by 12 orders of magnitude. Therefore, to activate the motion of vacancies and achieve reasonable hopping times, a heating stage is needed. Additionally, such a T requirement ensures the required nonvolatility for memory applications. We highlight that the activation of the motion of defects by the temperature does not lead necessarily to a LRS because of the nondirectionality of the hopping that it induces. To better describe the tendency of hopping toward the direction of the electric field, we calculated the drift velocity, defined as

Figure 6. Average hopping time (panel a) and drift velocity (panel b) along the direction of the electric field for the oxygen vacancies as a function of the temperature and the electric field (shown as voltage applied to a 10 nm-thick oxide). In panel b, the blue line represents the average velocity at which a vacancy must have to switch the device whereas the dashed black lines indicate its standard deviation.

⎛1 1 ⎞⎟ vd = a⎜⎜ − tap ⎟⎠ ⎝ tp

(4)

where a represents the distance between two minimum energy hopping sites along the [001] direction, whereas tp and tap correspond to the average hopping time in the direction parallel (shown in Figure 6a) and antiparallel to the electric field, respectively. This quantity gives an estimation of the flux of defects that drift to eventually form a filament and it is dependent on the temperature and the applied voltage as reported in Figure 6b. From an analysis of Figure 6b, we understand how decisive is the role of temperature, which exponentially increases the likelihood of jumping parallel rather than antiparallel to the applied electric field. Thus, the temperature has a double effect: it activates the motion of the vacancies, as said previously, and it increases their likeliness to drift in the direction of the electric field. In the case of very high temperatures (above the typical T reached during the switching), the latter effect saturates. The mere application of an electric field is not enough to significantly enhance the flux. For instance, if we apply 10 V to a 10 nm-thick oxide at 300 K statistically, a vacancy will jump forward every 1012 ns. Clearly, at this rate no switching takes place. A threshold value that distinguishes the “frozen” regime where defects are immobile from the “unfrozen” regime can be inferred from our simulations. In Figure 5b, we showed how the OFF current changes slope at around 1620 ns. We have already remarked how this is due to the motion of defects. Before the slope change, the defect distribution is similar to the initial condition (panel I and II of Figure 5a), but after the slope change, the defects start moving upward (panel III of Figure 5a). As a result, the high-resistive region becomes thinner and the overall E

DOI: 10.1021/acsami.8b00443 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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device. It is then possible to work in the direction of neuronlike memristor reproducing features like integrate-and-fire, as proven for phase-change devices.44 Provided that the input pulse raises the temperature above the transition region, its duration and intensity are a means to directly control the status of the device in the OFF state. The pulses must be carefully engineered to last enough to raise the temperature so as to unfreeze the defects but not much to lead to the switching. We finally discuss the influence on the device switching of an important parameter that is fixed during the experiments: the compliance current. We have seen in Figure 5b how this quantity limits the current intensity during the LRS. The main purpose of the ICC in real devices is to preserve the RRAM from breaking down. In fact, the cap to the total current that passes through the RRAM is used to avoid those extremely high temperatures that are generated inside the oxide. However, it could also have a substantial effect on the defect dynamics. In many current control circuits, the RRAM is placed in series with either a resistor or a transistor, which determines the magnitude of the compliance current.45,46 Whenever the RRAM switches to LRS, the external voltage drops mainly on the additional component rather than on the RRAM because of the higher resistance of the former. Consequently, the internal electric field in the oxide abruptly changes before and after the switch. The intensity of the effect depends on the ratio between the ON current and the compliance current according to the relation Er/E = ICC/ILRS, where Er is the reduced electric field after switching, E is the electric field before switching, ICC is the compliance current, and ILRS is the uncapped LRS state current. In the case of a high Er/E ratio as for our simulations, once the device turns into the LRS, the defect motion stops because the drift velocity is also a function of the electric field and the filament morphology remains unaltered. For instance, in the simulation shown in Figure 5, the drift velocity passes from 9.7×10−4 nm/ns before switching to 3.0×10−7 nm/ns afterward. The compliance current is an important self-limiting process that preserves the status of the device. In fact, if no compliance current were applied, the defects, because of the increased temperature of the ON state, would suddenly accumulate in the top part of the device, leaving a perfectly stoichiometric layer in the bottom (see Supporting Information), possibly leading to the phase transition of part of the oxide. The compliance current is then another powerful tool that, coupled with the temperature, allows for the control of the defect distribution. Right after the switch ON, vacancies stop moving as a consequence of the ICC. A sudden increase in temperature is reported in the following nanoseconds. The enhanced temperature copes with the compliance current reduction in the electric field intensity, and the defects gradually return to drift. More bins then connect the two electrodes so the current flows through a larger area (panel V of Figure 5a). As a result, the temperature stops increasing so sharply, and ultimately it reaches a plateau. We remark that in our simulations, the plateau temperature is enough to cause a small flux of defects into the electric field direction (otherwise we would not have any filament enlargement); therefore, the internal morphology of a device that is pulsed for a few microseconds will be completely different from the one of a very slowly swept device. So also the method of switching influences the final state. For comparison, we simulated a device with TiO2 containing only titanium interstitials, which, as said, have a diffusion barrier of 0.37 eV. The results are completely different with respect to

current increases. In other words, the slope change marks the threshold between the frozen regime and the unfrozen regime. We run up to 15 calculations with three different voltage pulse intensities (3, 5, and 8 V) checking at which temperature the slope change occurred to mark the regime transition. The average velocity required to switch the device was then generalized for every T and V (see the solid blue line in Figure 6b). The position of this line in the figure is only indicative because the change between the two regimes is not sharp and can be thought more of as a transition range. In Figure 6b, the two dashed black lines, which define the transition region, represent the standard deviation resulting from the set of calculations we performed. Within this transition range, the switching time greatly varies with a small change in T and V. The whole switching process can be described in terms of motion on the drift velocity surface of Figure 6b. During the switching, T and V evolve affecting consequently the flux of defects, so that a “trajectory” on the surface can be traced. Only the “trajectories” that overcome the transition region will lead to a consistent switching whereas for all the others paths altering the resistance state will be either unlikely, if the “trajectory” ends within the transition region, or impossible, if it stays far below. When designing a device, it has to be ensured that the device is provided with a voltage that generates the right temperature to create a sufficient flux of defects. On the basis of the drift velocity analysis, we can identify the right switching SET voltage. Let us now consider more explicitly the fact that V and T are not two independent variables and discuss the consequences of the device performance. Indeed, the stationary temperature, that is, the maximum temperature that is reached after a transient, is a function of the applied voltage because the current is the only heating source. Therefore, the voltage alone controls the drift velocity as it governs the stationary temperature in the device. The switching VSET is the one that heats the material to the temperature necessary for the defects to start moving in the direction of the electric field and that permits to accomplish the SET within a time span comparable with RRAM applications. Ideally, one wants to measure the device resistance without perturbing its state. A pulse that heats the oxide below the transition range of the defect motion statistically does not cause any significant change in the defect rearrangement along the electric field direction. For example, in our system, if we apply 0.05 V, the stationary temperature is around 360 K so no defect motion can take place in the amount of time of a measure according to Figure 6. The correct identification of the VSET that leads to the unfrozen regime (and in general of all the voltage-dependent stationary temperatures of a device) has paramount importance in the RRAM endurance. We have, indeed, already showed that internal temperature can reach up to 900 K during the switching (Figure 5b). It is then important not to overheat the oxide to avoid potential phase transitions or any unwanted temperatureinduced effect by the application of the proper voltage. Because temperature, voltage, and defect location are all related variables, it is not enough to rely on the simple relation in Figure 6b but it is necessary to run dynamical simulations like the one we proposed in this paper. The prediction of the stationary temperature, however, is not an easy task because it might require the knowledge of the internal morphology. Such an understanding will be completely achieved only after a more accurate description of the forming stage. Governing the temperature is a means to control the defect location inside the F

DOI: 10.1021/acsami.8b00443 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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the system containing only vacancies. In fact, for interstitials, the electric field is as important as the temperature; therefore immediately after the voltage is applied, the defects move toward the upper contact so that the LRS is achieved within subnanoseconds (Figure S4 of the Supporting Information). During the entire process, for low diffusion barriers the mean temperature inside the oxide remains around 300 K, in accordance with literature simulations.47 Nevertheless, the fast switching has a downside: at 300 K when no voltage is applied, the average hopping time is 100 ns. The hops occur so frequently that the ON state is quickly destroyed. Therefore, more generally, if the diffusion barriers are around 0.40 eV or less, the resistance state is volatile. In conclusion, we proposed a mixed continuum and KMC approach to take into account both atomic-level properties such as defect diffusivity and macroscopic quantities such as internal temperature and overall device current. Thanks to this combination of computational methods, temperature can dynamically change during the course of the simulation. We have shown how atomic-level quantities such as defect diffusion barriers in TiO2 have a major effect on device-level characteristics such as switching time and internal temperature. When the barriers are low (∼0.4 eV), the defects move as soon as the voltage is applied and correspondingly the device switching is fast; however, at room temperature, the resistance state is volatile. By contrast, if the barriers are relatively high (∼1 eV), the oxide is first to heat up in order for the defects to have enough energy to overcome the barriers. The temperature is then a fundamental quantity to control the defect distribution. Additionally, the current compliance was found to deeply affect the defect dynamics in the ON state. A proper characterization of T and ICC could be the key to control the switching at the microscopic level. Advanced techniques of voltage pulse engineering or technological improvements of the design of the RRAM, aimed at confining the temperature increase to a small region, could be the means to experimentally enhancing the reproducibility of the device.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.8b00443. TiO2 resistivity as a function of the defect concentration, defect dynamics in the absence of compliance current, and titanium interstitial simulation (PDF)

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Research Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: federico.raff[email protected] ORCID

Federico Raffone: 0000-0001-5045-7533 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the CINECA award under the ISCRA initiative and HPC@POLITO for the availability of highperformance computing resources and support. The authors declare no competing financial interests. G

DOI: 10.1021/acsami.8b00443 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

Research Article

ACS Applied Materials & Interfaces Switching of TiO2Nanocrossbar Devices. IEEE Electron Device Lett. 2011, 32, 1116. (23) Strukov, D. B.; Alibart, F.; Williams, R. S. Thermophoresis/ Diffusion as a Plausible Mechanism for Unipolar Resistive Switching in Metal-Oxide-Metal Memristors. Appl. Phys. A 2012, 107, 509. (24) Kamaladasa, R. J.; Sharma, A. A.; Lai, Y.-T.; Chen, W.; Salvador, P. A.; Bain, J. A.; Skowronski, M.; Picard, Y. N. In Situ TEM Imaging of Defect Dynamics Under Electrical Bias in Resistive Switching Rutile-TiO2. Microsc. Microanal. 2015, 21, 140. (25) Kwon, D.-H.; Kim, K. M.; Jang, J. H.; Jeon, J. M.; Lee, M. H.; Kim, G. H.; Li, X.-S.; Park, G.-S.; Lee, B.; Han, S.; Kim, M.; Hwang, C. S. Atomic Structure of Conducting Nanofilaments in TiO2 Resistive Switching Memory. Nat. Nanotechnol. 2010, 5, 148. (26) Strachan, J. P.; Pickett, M. D.; Yang, J. J.; Aloni, S.; Kilcoyne, A. L. D.; Medeiros-Ribeiro, G.; Williams, R. S. Direct Identification of the Conducting Channels in a Functioning Memristive Device. Adv. Mater. 2010, 22, 3573. (27) The rutile unit cell has the following lattice parameters: a = 4.739 Å, c = 2.862 Å and u = 0.3050 Å. The parameters were obtained by means of a classical potential.48 (28) Moballegh, A.; Dickey, E. C. Electric-Field-Induced Point Defect Redistribution in Single-Crystal TiO2−x and Effects on Electrical Transport. Acta Mater. 2015, 86, 352. (29) Uberuaga, B. P.; Bai, X.-M. Defects in Rutile and Anatase Polymorphs of TiO2: Kinetics and Thermodynamics Near Grain Boundaries. J. Phys.: Condens. Matter 2011, 23, 435004. (30) Bak, T.; Nowotny, J.; Rekas, M.; Sorrell, C. C. Defect Chemistry and Semiconducting Properties of Titanium Dioxide: II. Defect Diagrams. J. Phys. Chem. Solids 2003, 64, 1057. (31) Nowotny, M. K.; Bak, T.; Nowotny, J. Electrical Properties and Defect Chemistry of TiO2 Single Crystal. I. Electrical Conductivity. J. Phys. Chem. B 2006, 110, 16270. (32) Jansen, A. P. J. An Introduction to Kinetic Monte Carlo Simulations of Surface Reactions; Springer, 2012. (33) Although it is not possible to straightforwardly account for phase transitions in KMC, one can get information on the highest temperature achieved in the simulation to understand if a phase change can potentially occur. (34) Machlin, E. S.An Introduction to Aspects of Thermodynamics and Kinetics Relevant to Material Science; Elsevier, 1991; p 207. (35) Spitzer, W. G.; Miller, R. C.; Kleinman, D. A.; Howarth, L. E. Far Infrared Dielectric Dispersion in BaTiO2, SrTiO2, and TiO2. Phys. Rev. 1962, 126, 1710. (36) Reuter, K.; Scheffler, M. First-Principles Kinetic Monte Carlo Simulations for Heterogeneous Catalysis: Application to the CO Oxidation at RuO2(110). Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 045433. (37) He, J. First Principles Calculations of Intrinsic Defects and Extrinsic Impurities in Rutile Titanium Dioxide. Doctoral Dissertation, 2006. (38) Bak, T.; Nowotny, J.; Rekas, M.; Sorrell, C. C. Defect Chemistry and Semiconducting Properties of Titanium Dioxide: III. Mobility of Electronic Charge Carriers. J. Phys. Chem. Solids 2003, 64, 1069. (39) Iguchi, E.; Yajima, K.; Asahina, T.; Kanamori, Y. Resistivities of Reduced Rutile (TiO2) from 300 K to Exhaustion Range. J. Phys. Chem. Solids 1974, 35, 597. (40) Padgett, C. W.; Brenner, D. W. A Continuum-Atomistic Method for Incorporating Joule Heating Into Classical Molecular Dynamics Simulations. Mol. Simul. 2005, 31, 749. (41) Schall, J. D.; Padgett, C. W.; Brenner, D. W. Ad Hoc Continuum-Atomistic Thermostat for Modeling Heat Flow in Molecular Dynamics Simulations. Mol. Simul. 2005, 31, 283. (42) Bergman, T. L.; Lavine, A. S.; Incropera, F. P.; Dewitt, D. P. Fundamentals of Heat and Mass Transfer; Wiley, 2011. (43) Hickson, R. I.; Barry, S. I.; Mercer, G. N.; Sidhu, H. S. Finite Difference Schemes for Multilayer Diffusion. Math. Comput. Model. 2011, 54, 210. (44) Tuma, T.; Pantazi, A.; Le Gallo, M.; Sebastian, A.; Eleftheriou, E. Stochastic Phase-Change Neurons. Nat. Nanotechnol. 2016, 11, 693.

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DOI: 10.1021/acsami.8b00443 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX