Introduction

Oxide-based resistance-change memories (RRAMs) have been studied intensively during the past decade and have attracted increasing interest as potential next-generation nonvolatile memories1. To explain their resistive switching behaviors, researchers have proposed various mechanisms, among which the concept of formation/rupture of conductive filaments (CFs) has been widely accepted2. The transition from a high-resistance state (HRS) to a low-resistance state (LRS) is called the set process, which is interpreted as a defect-induced soft breakdown associated with the migration of non-lattice oxygen ions toward the electrode3. In contrast, the transition from the LRS to the HRS is called the reset process, in which electric-field-enhanced oxygen ion-hopping ruptures the CFs4. In the case of bipolar resistive switching, the voltages for the set and reset processes have the opposite polarities, thus revealing the role of voltage-driven migration of oxygen ions5,6,7,8. Moreover, recent studies have pointed out the crucial role of temperature in the voltage-driven migration process at the moment of set or reset transition9,10.

On the basis of comprehensive and intensive studies, the nature of the filament during the set and reset processes is now comprehensible. The presence of one or more localized CFs has also been confirmed by direct microscopic methods11,12 and indirect electrical measurements. In addition, several phenomenological or physical models have predicted the consistent resistance-change behavior (i.e., current-voltage (I–V) characteristics) of RRAMs5,8,9. However, existing models have apparently overlooked an important fact. As mentioned earlier, the thermal effect is an essential factor of oxygen ion migration during the set and reset processes. Therefore, to predict the temperature inside the CF accurately, the thermal properties of both the electrodes and the oxide layer should be included in a model. However, most previous models have ignored the effect of electrodes and assumed either the CF/electrode interface to be at room temperature5,9 or the temperature inside the CF to be constant13,14. Although Menzel et al. have recently proposed an advanced electro-thermal model that considers both the top and the bottom electrodes to calculate the temperature of the CF precisely10, a physical model that predicts the I–V characteristics of an RRAM accurately in a complete bi-layered RRAM structure has not yet been demonstrated.

In this work, a numerical model, which is based on temperature- and field-accelerated migration of oxygen vacancies, is presented for bipolar RRAMs. A 2-D axis-symmetric finite element simulation model, which allows a quantitative analysis of field and temperature contributions, accounts accurately for the set/reset characteristics of both the DC and the AC responses. The proposed model also provides a microscopic interpretation of intermediate set/reset states based on differences in the CF morphology.

Results

The tantalum-oxide-based bi-layered RRAM consisted of a high resistive Ta2O5 layer on top of a less resistive TaOx layer (see Methods for details), which was sandwiched by a top electrode (TE) of Pt and a bottom electrode (BE) of W, as shown in Fig. 1(a). When a large negative voltage is applied to the pristine device (a process called “forming” shown in Fig. 1(b)), positively charged oxygen vacancies are attracted from the TaOx layer to the TE, forming a CF15,16. This CF is described as a doped region where oxygen vacancies (Vo) act as dopants and thus contribute to local electrical and thermal conductivity values17. The simulation, which is started immediately after the forming process, features a continuous CF that connects the top and bottom electrodes (Fig. 1(c)). After the forming process, the set and reset processes are described through Vo migration induced by the local electric field and temperature due to Joule heating5,17. Thus, simulation of the set/reset transitions requires the self-consistent solution of three partial differential equations (PDEs): (1) a drift/diffusion continuity equation for Vo transport, (2) a current continuity equation for electrical conduction and (3) a Fourier equation for Joule heating, as shown in Fig. 2.

Figure 1
figure 1

Schematic diagram and simulated geometry.

(a) Schematic and cross-sectional TEM images of the Pt/Ta2O5/TaOx/W bi-layered RRAM device. (b) Typical I–V characteristic of Pt/Ta2O5/TaOx/W device. (c) Simulated geometry used in calculation. The axisymmetric geometry reduces the problem from one of three dimensions to one of two dimensions.

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Figure 2
figure 2

The equations and parameters in the proposed model.

Three PDEs are self-consistently solved within a numerical solver.

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To describe the drift/diffusion of Vo migration, a simple 1D rigid point ion model by Mott and Gurney is employed18. Although the ion-hopping model by Mott and Gurney is applicable oxygen ions rather than oxygen vacancies, here, we believe that both oxygen vacancies and ions can be described by the same equations except for the detailed physical parameters. Diffusion is given by D=1/2 a2 f exp(−Ea/kT) and the drift velocity is given by v=a f exp(−Ea/kT) sinh(qaE/kT), where f is the attempt-to-escape frequency (1013 Hz)18, a is the effective hopping distance (0.05 nm) and Ea is the activation energy of migration. In fact, the effective hopping distance (a) has a range of a few nanometers (~1 nm)18. However, the value of 0.05 nm was used in this work for the purpose of best fitting the experimental data. This inconsistency is presumably due to differences in the physical nature of oxygen vacancies and oxygen ions. Considering the drift and diffusion, the time-dependent evolution of Vo concentration (nD) can be expressed by the continuity equation, as follows:

Here, G is the generation term to describe the set process. The kinetics of the set process can be interpreted as a dielectric soft breakdown associated with the migration or generation of Vo through thermally activated hopping. The CF growth rate is assumed to be proportional to the ion migration rate, which is given by G=A exp(−(Eb − qV)/kT)=A exp(−(Eb − qβE)/kT)19,20, where A is a pre-exponential constant, Eb is the energy barrier for ion hopping (1 eV) and β is the mesh size in the simulation (0.5 nm). Term qβE describes the lowering of the barrier attributed to the applied field. This lowering forces the Vo to move along the direction of the electric field. To express the set process that occurs only at a negative bias, the generation term is also applied exclusively at a negative bias. Equation (1) can be solved with the current continuity equation for electrical conduction, as follows:

where σ is the electrical conductivity and with the steady-state Fourier equation for Joule heating

where kth is the thermal conductivity. The transient contribution to the Fourier equation was disregarded, given the relatively fast thermal response of approximately 1 ps (see Supplementary Information)5,9.

The three PDEs in equations (1)–(3) are self-consistently solved by a numerical solver (COMSOL) to calculate nD, ψ (potential) and T (temperature). To solve equation (3), models for electrical conductivity (σ) and thermal conductivity (kth) in the oxide and in the CF are required. To this end, both the electrical and the thermal conductivities are assumed to depend on nD, as shown in Fig. 35,9. The electrical conductivity is given by the Arrhenius equation21, σ=σ0 exp(−EAC/kT), where σ0 is a pre-exponential factor and EAC is the activation energy for conduction. As shown in Figure 3(a), σ0 is assumed to linearly increase from 5 to 450 Ω−1cm−1 with an increase in nD; these values correspond to the measured values of resistivity at both the LRS and the HRS, respectively. In addition, Figure 3(b) shows the conduction activation energy EAC used in the calculations. The activation energy is 0.016 eV for high nD and linearly increases to 0.052 eV with a decrease in nD, consistent with the measured data shown in Figure 3(c). Moreover, a linear dependence of kth on nD is assumed on the basis of the Wiedemann-Franz law, as shown in Figure 3(d)5. The minimum value for nD=0 refers to the thermal conductivity of the insulating Ta2O5, kTa2O5=0.12 Wm−1K−1 for T=300 K22. The maximum kth value at high nD corresponds to that of the metallic CF, i.e., thermal conductivity of tantalum kTa=57.5 Wm−1K−1. Here, a maximum Vo density of 1 × 1022 cm−3 is chosen approximately where it corresponds to a relative atomic concentration of 20%. Although the particular choice of the maximum Vo value and the linear approximations of σ0, EAC and kth appear to have a weak physical background, the calculation results based on these assumptions show good consistency with the experimental data; therefore, these assumptions do not limit the validity of the calculations.

Figure 3
figure 3

Parameters from measurements and assumptions.

(a) Electrical conductivity pre-exponential factor σ0, (b) assumed activation energy for conduction EAC, (c) measured EAC at both HRS and LRS and (c) assumed thermal conductivity kth as a function of local Vo density nD.

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Figure 4(a) shows the simulated region of the tantalum-oxide-based bi-layered RRAM. In the actual calculations, the axisymmetric geometry allowed the 3-D problem to be reduced to a 2-D solution with a radial coordinate and a vertical coordinate. The active oxide materials (Ta2O5 and TaOx layers) are contacted by two electrodes and all layers including the two electrodes are considered in the calculations (Pt TE, σPt=105 Ω−1cm−1, kPt=71.6 Wm−1K−1; W BE, σW=2 × 105 Ω−1cm−1, kW=173 Wm−1K−1). The boundary conditions for equation (2) are ψ=0 and ψ=V at the bottom and top electrodes, respectively. The outermost boundaries of the two electrodes are defined as ideal heat sinks with boundary conditions T=300 K; this simplifying assumption is reasonable because the electrode area is generally large with respect to the CF. For Vo drift/diffusion, no Vo flux was assumed at the TE/Ta2O5 and TaOx/BE interfaces. In addition, only diffusion was considered in the TaOx layer (i.e., the drift term (v) of equation (1) is zero) because this layer only serves the role of an oxygen vacancy reservoir15,16. A uniform doping concentration of nD=1 × 1022 cm−3 was defined within the CF and TaOx layer as the initial state (i.e., the forming state). The CF size was set to a diameter of 10 nm, in agreement with direct evaluations by using conductive atomic force microscopy23.

Figure 4
figure 4

Simulated boundary conditions and calculated results.

(a) Cross section of the simulated cell. A uniform doping concentration of nD=1 × 1022 cm−3 was defined within the CF and TaOx layer as the initial forming state. (b) Measured and calculated I–V characteristics.

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Discussion

Figure 4(b) shows the measured and calculated I–V characteristics during the set and reset processes after the forming process. The reset transition starts at 1.85 V and the resistance gradually increases, finally achieving a difference of roughly one decade after VRESET=3.0 V. Similarly, the set transition starting at −0.5 V also shows good consistency with the measured data. Figure 5 shows calculated (a) 2-D maps of nD as well as 1-D profiles of (b) nD, (c) T and (d) ψ at bias points A, B and C. As the voltage increases during the reset sweep, the temperature increases as a consequence of Joule heating, as shown in Fig. 5(c). Vo migration is localized at the point of maximum temperature (i.e., near the TE) owing to the activation of drift velocity by exponential T. Consequently, Vo migrates toward the BE, resulting in the opening of a depleted gap. The opposite process occurs in the set transition. Figure 6 shows the calculated (a) 2-D maps of nD and the 1-D profiles of (b) nD, (c) E (electric field) and (d) G (generation rate of Vo) at the bias points D, E and F. Given that the electric field at the depleted gap increases with an increase in the negative bias, the generation rate of Vo increases, as shown in Figs. 6(c) and 6(d). Consequently, the depleted gap at the CF is re-filled by Vo, after which the high electric field is no longer applied to the CF. This suppresses further Vo generation or migration and is responsible for a self-limiting set transition.

Figure 5
figure 5

Simulation results for the reset transition.

(a) Calculated map of Vo density (nD). Calculated profiles of (b) nD, (c) T and (d) ψ for states A (1.0 V), B (1.7 V) and C (3.0 V) as depicted in Fig. 4(b). The position of z=15 nm indicates the Ta2O5/TaOx interface. The depleted gap, defined for nD < 5 × 1021 cm−3, is marked by the shaded area.

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Figure 6
figure 6

Simulation results for the set transition.

(a) Calculated map of Vo density (nD). Calculated profiles of (b) nD, (c) E and (d) G for states D (−0.25 V), E (−1.5 V) and F (−0.25 V), as depicted in Fig. 4(b).

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By applying different boundary conditions of temperature, the thermal effect on the I–V characteristics can be predicted by calculations. Figure 7(a) shows the calculated I–V curves under different temperature conditions. The reset transition voltage (i.e., the moment at which the resistance starts to decrease) noticeably decreases with an increase in the device temperature. In addition, the resistance at the HRS increases with an increase in the device temperature. The reset transition occurs at a lower bias voltage because Vo migration (drift and diffusion) is more favorable at higher device temperatures. Moreover, the difference of conduction curve at the HRS is due to the temperature-dependent conductivity enhancement of σ through the Arrhenius equation. These two distinctive features agree well with the experimental data shown in Figs. 7(b) and 7(c), where the measurements are carried out in the range of −20°C–85°C; accordingly, the simulation accurately reflects the thermal effect for the set and reset transitions. To further verify the simulated results, Figures 8(a) and 8(b) show the measured and calculated I–V curves, including reset sweeps stopped at various values of VRESET. The simulation accurately reproduces the different features of resistive switching with different values of VRESET and predicts a gradual reset transition behavior.

Figure 7
figure 7

Temperature dependency.

(a) Calculated I–V characteristics at different device temperatures. Measured and calculated data for the (b) reset transition voltage and (c) resistance value of HRS.

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Figure 8
figure 8

Gradual reset process.

(a) Measured and (b) calculated I–V curves, showing reset transition, obtained by varying VRESET (1.8 V to 3 V).

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The application of the proposed model is not limited to reproducing the DC I–V characteristics. Time-dependent reset characteristics were further explored by investigating the response to square pulses. Time-resolved experiments were carried out in which a series of set/reset pulses was applied to the RRAM device with a serially connected resistor and the response to an AC pulse was simultaneously detected by an oscilloscope. Figures 9(a) and 9(b) show the calculated and measured response profiles to the applied reset pulse as a function of time. When a single square pulse for the reset transition is applied to the device, the response current increases following a voltage increment because the initial state of the device is in the LRS. Over time, the response current rapidly decreases, which indicates the moment of reset transition. The measured time to the reset transition (tRESET) is 1.3 μs for the input pulse shown in Fig. 9(b), which is comparable to the calculated value of tRESET=3.5 μs. Consequently, the simulation can describe the AC resistance-change response reliably in addition to the DC I–V characteristics.

Figure 9
figure 9

Transient response.

(a) Measured and (b) calculated AC response as a function of time.

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Although the morphology of the CF has been observed by direct imaging with transmission electron microscopy (TEM), the effort required was extensive and the extracted results were insufficiently informative for comprehensively understanding the mechanism of resistive switching11. In contrast, a merit of the simulation is that it provides a microscopic interpretation of intermediate set/reset states based on differences in the CF morphology. As mentioned above, the thermal properties of both the electrodes and the oxide layer should be included in the model to accurately describe the resistive switching phenomenon. Previously reported models predicted that the depleted gap at the reset transition should be positioned in the middle of the CF when temperature-driven process is dominant4,9,17. On the other hand, several papers have verified by electrical measurements that the depleted gap is located near the electrode/oxide interface. To overcome this disagreement between the model and the measurement result, modified models have been explained that the depleted gap can be located near the electrode/oxide interface, particularly when the shape of the CF is conical24,25. However, the analyses based on these modified pre-existing models are also inaccurate because they did not consider the thermal property of the electrodes. Figure 10 shows the I–V characteristics and 2-D maps of nD corresponding to filaments differing in their initial shape: (a) rectangular, (b) trapezoidal, or (c) inverted trapezoidal. It should be noted that the location of the depleted gap in the CF is independent of the shape. In particular, the filament with an inverted trapezoidal shape changes to a rectangular one after consecutive switching cycles. These results imply either that the depleted gap is not always located in the middle of the CF or that its location depends on the shape of the filament. Indeed, a crucial factor for the reset transition is the thermal conductivity ratio between the TE and the oxide. Figure 11(a) shows the calculated I–V characteristics based on different TE materials and Figure 11(b) shows the corresponding 2-D maps of nD and the temperature profiles. The position of the depleted gap in the CF is determined by the thermal conductivity of the TE and the DC I–V characteristics are also distinct. (The BE material has a negligible effect owing to the presence of a relatively thick TaOx layer. In the case of a single-layered RRAM, the effect of the BE is also significant). As the thermal conductivity of the TE increases, the generated heat by Joule heating is confined within the oxide layer; consequently, the depletion gap of the CF is positioned away from the TE/oxide interface. Therefore, the effect of the electrodes should be considered in an analytical model to predict the nature of resistive switching accurately. The model proposed here provides an accurate interpretation of intermediate set/reset states based on differences in the CF morphology.

Figure 10
figure 10

Calculated I–V curves for filaments of different shapes.

(a) rectangular, (b) trapezoidal and (c) inverted trapezoidal. Despite the different shapes, the position of the depleted gap is almost identical. In addition, the I–V characteristics are also independent of the filament shape.

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Figure 11
figure 11

Effect of different TE materials.

(a) Calculated I–V curves for different TE materials. (b) 2-D maps of nD corresponding to TE materials and temperature profiles in the CF and TaOx layer.

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In summary, a physical and numerical model has been proposed on the basis of temperature/field-driven migration of oxygen vacancies in bi-layered bipolar RRAM devices. The proposed model allows the calculation of resistance-change characteristics for both the DC and the AC input signals. By considering the top and bottom electrodes, a more accurate temperature profile than that of the pre-existing models is obtained, which in turn leads to more accurate simulation results. The proposed model is potentially useful for gaining physical insights into the morphology of the CF during set and reset transitions, which is valuable for exploring the scaling and multilevel capabilities of RRAM devices.

Methods

All measured samples (Pt/Ta2O5/TaOx/W devices) had a 1 μm × 1 μm crossbar structure. A 50-nm-thick W layer was deposited by sputtering and then patterned by dry etching to form the bottom electrode (BE) on a 3-nm-thick adhesion Ti/500-nm-thick SiO2/Si substrate. Then, a 20-nm-thick conducting TaOx layer was deposited on the BE by DC magnetron sputtering in a reactive O2 + Ar ambient, with Ta as the target. Subsequently, an insulating Ta2O5 layer was formed by a pulsed O2 plasma oxidation process that involved a 2-s plasma-on step and a 4-s plasma-off (purge) step in a single cycle. The total oxidation time was controlled by the number of cycles, which was 80 in the present study. Finally, a 20-nm-thick Pt layer was evaporated and patterned by a lift-off process.

The DC electrical characterizations were carried out by an Agilent B1500A semiconductor analyzer in the voltage-sweep mode. During AC pulse measurements, a series resistor (1 kΩ) was serially connected to the RRAM device. Input pulses were generated by a B1500A and the response was detected by an oscilloscope (Tektronix DPO 70604).