Analytical modeling electrical conduction in resistive-switching memory through current-limiting-friendly combination frameworks

The resistive-switching memory (RSM) device is one of the most promising candidates used for next-generation edge computing, such as in-memory computation and neuromorphic computing, due to its simple structure, low operating power, high device endurance, and high switching speed.^1–3 Various dielectric materials, such as HfO₂,⁴ MgO,³ AlN_x,⁵ and SiO₂,⁶ are widely used in RSM devices. Driven by an electric field produced by a bias voltage applied to the top electrode, oxygen or nitrogen vacancies can be generated to form a conduction path. Correspondingly, the device can switch from the high-resistance state (HRS) to low-resistance state (LRS), and this process is defined as the set process.⁷ Conversely, when an opposite bias voltage is applied, trapped oxygen or nitrogen ions tend to be released from the bottom electrode and they can recombine with the oxygen/nitrogen vacancies in a conduction path. Such a conduction path has a tendency to be ruptured, and the device can switch from LRS to HRS, which is known as the reset process.⁷ 

Numerous theoretical models have been proposed to explain the resistive-switching process in RSM devices, and its compact modeling mostly consists of three related stages: (i) evolution of the conduction filament (CF), (ii) conduction mechanism, and (iii) dynamic temperature evolution.^8–10 The evolution of the CF is generally attributed to the effects of thermal, electrical, or ion migration, but such assignments lack direct evidence. Therefore, singling out the dominant conduction mechanism of RSM devices from several possibilities is a challenge.^8,11 Recently, several attempts have been proposed to explain the conduction mechanism of RSM devices. Typical conduction behaviors observed can be space-charge-limited current (SCLC),^12–14 trap-assisted tunneling (TAT),^15–17 Poole–Frenkel emission,¹⁸ Schottky emission,¹⁹ and so on. However, a complete physical model that can describe the conduction process is still missing. To improve device performance and predict the conduction process, it is essential to obtain a complete physical model of conduction mechanism, especially as the device footprint is reduced to a few nanometer scale. An analysis of the potential change in temperature during the conduction process is important because it is possible that unexpected heat accumulation occurs in both set^12,20–22 and reset^23–26 processes. The change in temperature might yield changes in electronic properties, such as the mobility of electrons^27,28 and depth of traps,²⁹ and influence the conduction behavior of RSM devices.^30,31 However, there is still a lack of complete physical model to investigate the significance of the change in the mobility of electrons in the set process and change in the depth of traps in the reset process, which inhibits the development of RSM devices. Here, we present a complete physical model that can accurately describe the rich conduction behavior by harnessing a combination of electronic and thermal considerations via electron mobility and trap depth. By considering the model of conduction behaviors and the assumptions related to the mobility of electrons and depth of traps, we can precisely predict the conduction behavior in an iridium/aluminum-nitride/titanium RSM device. More importantly, both the conduction and switching behaviors can be described by just a single set of expressions. Our electrothermal simulations elucidate the role of thermal effects in the set and reset processes. Furthermore, the rich conduction and switching behaviors can be fully accounted for by describing device characteristics obtained by a completely new set of general current-limiting parameters.

Figure 1(a) displays the schematic of the device structure of a RSM device. A heavily doped n-type silicon substrate was used as the starting material, on which a 45-nm-thick titanium bottom electrode was deposited. Then, a 12-nm-thick aluminum-nitride thin film was deposited to form the switching layer. Finally, the 50-nm-thick iridium top electrode and 300-nm-thick aluminum capping layer were deposited to complete the structure. The iridium and aluminum metal layers were patterned by different shadow masks to form device pillars with various diameters from 50 µm to 200 µm. To analyze the resistive-switching characteristics, all samples were tested using a standard electrical-characterization system (Keithley 4200 SCS). The heavily doped n-type silicon substrate was grounded, and direct-current (DC) voltage was applied to the top electrode.

The typical bipolar resistive-switching behavior is shown in Fig. 1(b). The compliance current (CC) value was kept constant at about 1 mA, and bias voltage was applied from −2 V to 2 V. During the set process, when the DC voltage reaches the threshold voltage, the resistance of the device changes from HRS to LRS due to the formation of CFs. The CC value limits the maximum current passing through the device for damage reduction.^32,33 The fittings of the expressions of SCLC and TAT models of conduction mechanisms to experimental data are shown in Figs. 1(c) and 1(d), respectively. The SCLC model is best fitted to the experimental data of the device in the HRS at the low voltage region from 0.01 V to 0.30 V and the intermediate voltage region from 0.3 V to 1.2 V by simple expressions of the linear Ohmic conduction (I ∝ V) and power-law dependence (I ∝ V²), respectively.^8,34 Additionally, the TAT model is well fitted to the experimental data of the device in the LRS by the expression of V ∝ ln(J) [Fig. 1(d)].^8,16 Moreover, R-squared values of the fittings to experimental data were close to 1, indicating that the fittings were excellent.

The HRS of the iridium/aluminum-nitride/titanium-based RSM device can be dominated by SCLC. To describe the resistive-switching behavior, the SCLC model was extended from the low/intermediate voltage region (from 0.01 V to 1.2 V) to high voltage region (>1.2 V) where the switching occurs. The current and voltage values of the SCLC model for the set process were calculated by^8,35

where μ is the electron mobility; ε is the permittivity of nitride; ε₀ is the dielectric constant and dependent on the material; V is the applied voltage, which is considered as the variable; θ₀ is the ratio of free and shallow trapped charge; and L is the thickness of the dielectric material. The key parameter values are shown in supplementary material, Table SI.

The current and voltage values calculated by the general expression of the SCLC model^8,35 (named unoptimized SCLC model) for the set process are plotted in Fig. 2(a) (green line). Both experimental data and calculated values show an increase in current values with an increase in negative-bias-voltage values for the set process, but a large difference between the current values obtained from experiments (pink line) and those from calculations (green line) before the onset of the resistive switching is observed. Such a difference could be due to an increase in temperature,^12,20–22 and this increase in temperature could bring forth change in the mobility of electrons at the HRS.^27,28 Thus, it is necessary to investigate the role of temperature in the set process. It is likely that an increase in negative-bias voltage might produce heat in the aluminum-nitride switching layer, and nitrogen ions might be released from equilibrium positions, which results in the generation of nitrogen vacancies. This process tends to be very similar to the creation of oxygen vacancies.^5,7,20 When an increased negative-bias voltage is applied, the current increases slowly due to the ionic motion that reduces the resistance level of a device.²⁰ 

The simulated thermal distribution of the device with different bias voltages is shown in Figs. 3(a) and 3(b) and supplementary material, Fig. S2. The device showed an increased peak temperature value (from 300 K to 420 K) with an increase in the negative-bias-voltage value (from 0 to −0.8 V) [Fig. 3(c)]. To investigate the temperature-dependent mobility of electrons in an aluminum-nitride switching layer, the variational-principle method is proposed. The drift mobility of electrons μ is expressed as^27,36

where z₁ is the reduced phonon energy, m* is the effective mass, F_0.5(η) is the Fermi–Dirac integral, η is the reduced Fermi energy, δ is the constant that depends on the ionic species and impurity degree, and D(x) is the determinants that contains variational integrals.^27,36–38 In an aluminum-nitride switching layer, only phonon-limited-drift mobility is considered to be influenced by temperature because of its deep donors²⁷ and high compensation³⁹ level. The calculated values for the temperature-dependent mobility of electrons are shown in Fig. 3(d). The value of the drift mobility of electrons for the aluminum-nitride switching layer decreases from ∼300 cm²/V s to 140 cm²/V s with an increase in temperature from 300 K to 420 K.

During the set process, a conduction path grows slowly and subsequently at rapid rates due to increased generation of heat with an increase in the negative-bias-voltage value.⁴⁰ Self-heating of the device during the set process might not be high enough for filament rupture. However, it can accelerate the formation of the filament and reduce the mobility of vacancies in a dielectric layer. To obtain a relationship between J and V with modification of the Frenkel effect, a modified expression of the SCLC model that considers the Simmons equation, Simpson rule, and iterative solution^21,22,40,41 is given by

where the relationship between the mobility μ and temperature T is shown as

with Θ(V) being the Heaviside step function, Θ(V) = 0 for V < 0 and Θ(V) = 1 for V ≥ 0, A is the compliance-current factor, μ_o is the mobility of electron at 300 K, and δ is the deteriorate parameter, which is dependent on the material property.^27,36,37 We used Eq. (3) to describe an optimized model, which included the mobility parameter in Eq. (4) for the set process, and the calculated current values are shown in Fig. 2(a) (brown lines). Notably, the optimized model can show around 50 times lower error-from-experimental-data value for the set process (∼6.7 × 10⁻¹³) compared to that of the traditional unoptimized model (∼3.3 × 10⁻¹¹) [see supplementary material, Fig. S3(a)].

To confirm the effects of temperature on the device, iridium/aluminum-nitride/titanium-based RSM devices with different values of a general device parameter, i.e., CC, were analyzed by using the same calculation methods. The compliance current value typically controls the strength of the conduction filament formed in a device.⁴² Interestingly, the difference between the current values of experimental data and those of unoptimized models before the onset of the resistive switching for the set process tends to be larger for a RSM device with 1.0 mA CC than that of a RSM device with 0.5 mA CC [Figs. 2(a) and 2(b)]. A possible explanation for this difference for various CC conditions might be that when an increased CC value is used, the temperature of the local filament also increases. Therefore, the variability in temperature-dependent electron mobility and trap depth increases, which can produce the difference between the current values of experimental data and those of unoptimized models for different CC conditions (consideration of temperature is absent). Additionally, the excellent agreement between the current values of the optimized model and those of experiment data for different CC conditions for the set process [Figs. 2(a) and 2(b)] confirms the importance of the analysis of temperature in the iridium/aluminum-nitride/titanium-based RSM device under different CC conditions.

To further understand the influence of heat generation on the reset process of iridium/aluminum-nitride/titanium-based RSM devices, the temperature-dependent depth of traps of a device is discussed.^29,43 Due to high concentration of traps in an insulator layer, it is possible that a TAT model consists of several processes: (1) tunneling from electrodes to traps, (2) emission and tunneling from traps to the conduction band, and (3) trap-to-trap tunneling in the form of hopping from traps to electrodes.^8,15,17 The prototypical expression for the TAT model (unoptimized TAT model) is given by⁸ 

where ϕ_T is the energy of electron traps with respect to the conduction band edge, A is the constant, h is the Planck constant, and E is the electric field of the dielectric layer.

The RSM device in the reset process shows a similar increase in current values with an increase in the positive-bias-voltage values, but there exists a large difference between the current values of experiments and those of calculations [Fig. 2(a)]. In the reset process, Joule heating occurs when a positive-bias voltage is applied and it contributes to an increase in the resistance level before switching occurs.^20,40 Supplementary material, Fig. S4 shows thermal distributions of a RSM device. The voltage-dependent-temperature of the RSM device calculated by electrothermal simulations is shown in Fig. S4(e). The device exhibits an increased temperature value (from ∼300 K to 670 K) with an increase in positive-bias voltage (from ∼0.1 V to 1.1 V) due to Joule heating,^25,26,44,45 which also contributes to the reduction in the density of defects. By using the expression proposed by Chen⁴⁶ and the truncated Boltzmann (quasi-thermal) distribution for trap-depth calculation, the expression of the trap depth in eV is shown as

where k is the Boltzmann constant, T is the peak temperature, w is the full-width at half-maximum (FWHM) of the glow curve, and μ_g is the geometrical-shape-factor.^15,29,47,48

The temperature-dependent trap depth is shown in supplementary material, Fig. S4. The plot is obtained by using Eq. (6). The depth of traps increases (from 0.5 eV to 1.8 eV) with an increase in temperature (from 300 K to 670 K). The calculated values of trap depth were verified by variable heating-rate methods.²⁹ Interestingly, the optimized TAT model, which further considered the change in trap depth [Eqs. (5) and (6)], can show around 700 times lower error-from-experimental-data value for the reset process (∼7.7 × 10⁻⁹) compared to that of the traditional unoptimized model (∼5.6 × 10⁻⁶) [see supplementary material, Fig. S3(a)]. Furthermore, the same trend is observed for different CC conditions [Fig. S3(b)], indicating the importance of the change in trap depth in the reset process.

By considering the temperature, mobility of electrons, and depth of traps, we developed a complete and accurate physical model that is able to explain the conduction process. The proposed model allows accurate prediction of conduction behavior via harnessing a combination of electronic and thermal considerations via electron mobility and trap depth and is able to reproduce device characteristics obtained by an entirely new set of general current-limiting parameters. Both the conduction and switching behaviors can be described using only a single set of expressions. Our model reveals that the set and reset processes involve a thermal-driven conduction process. We believe that such an analysis of conduction behavior not only provides a reliable and accurate physical picture of the conduction process but also produces much needed guidelines for continued design and optimization of this important class of memory applications.

See the supplementary material for further details on our simulation of the RSM device, calculations for the errors of models compared to the experimental data of the RSM device, and the list of the values of the parameters used in equations.