Nonlinear ion drift-diffusion memristance description of TiO2 RRAM devices

The nature and direction of the hysteresis in memristive devices is critical to device operation and performance and the ability to realise their potential in neuromorphic applications. TiO2 is a prototypical memristive device material and is known to show hysteresis loops with both clockwise switching and counter-clockwise switching and in many instances evidence of negative differential resistance (NDR) behaviour. Here we study the electrical response of a device composed of a single nanowire channel Au–Ti/TiO2/Ti–Au both in air and under vacuum and simulate the I–V characteristics in each case using the Schottky barrier and an ohmic-like transport memristive model which capture nonlinear diffusion and migration of ions within the wire. The dynamics of this complex charge conduction phenomenon is obtained by fitting the nonlinear ion-drift equations with the experimental data. Our experimental results support a nonlinear drift of oxygen vacancies acting as shallow donors under vacuum conditions. Simulations show that dopant diffusion under bias creates a depletion region along the channel which results in NDR behaviour, but it is overcome at higher applied bias due to oxygen vacancy generation at the anode. The model allows the motion of the charged dopants to be visualised during device operation in air and under vacuum and predicts the elimination of the NDR under low bias operation, in agreement with experiments.


Fitting parameters for nonlinear ion drift-diffusion models
Tables below present the values of the optimized parameters resulted from the numerical fittings of Figure 5 in the main manuscript. Our selected fitting shown in Figure 5(a) in the main manuscript (MM1) was obtained using the functional form for the voltage-controlled element in the drift equation [1] as and Joglekar window function [2]. The function is added to account for the dual CW (…) orientation of the I-V curve. The initial guess and optimized values for this model are shown on Table  S1. The determination of the parameters initial guess was performed using a systematic "pre-search" technique in which a subset of initial parameters is kept fixed while others are allowed to fluctuate under certain constraints. Table S1: Initial guess and optimized parameters for the MM1 discussed in the main text using equations (1+S1) to describe the nonlinear ion-drift dynamics. The resulting fitting is depicted on Figure 5(a) in the main manuscript which shows the I-V characteristics of the Au-Ti/TiO 2 /Ti-Au device in vacuum and at room temperature. The initial condition for the internal state variable was set at . ( = 0) = 0.9 Our selected fitting shown in Figure 5(b) in the main manuscript (MM1+ ) was obtained using the functional form for the voltage-controlled element in the drift equation [3] as

Fitting Parameters
Electronic Supplementary Material (ESI) for Nanoscale Advances. This journal is © The Royal Society of Chemistry 2020 and Joglekar window function [2]. This functional form is flexible in terms of establishing the correct  voltage thresholds for the model and accounts for the dual CW orientation of the I-V curve. The initial  guess and optimized values for this model are shown on Table S2. Table S2: Initial guess and optimized parameters for the MM1+ discussed in the main text using equations (1+S2) to describe the nonlinear ion-drift-diffusion dynamics. The resulting fitting is depicted on Figure 5(b) in the main manuscript which shows the I-V characteristics of the Au-Ti/TiO 2 /Ti-Au device in vacuum and at room temperature. The initial condition for the internal state variable was set at .    Table S2. Their arithmetical addition results in the I-V curve of Figure 5(b) in the main manuscript.
Our selected fitting shown in Figure 5(c) in the main manuscript (MM2) was obtained using the functional form for the voltage-controlled element in the drift equation [3] as expressed in equation (S2) and for [4] as The initial guess and optimized values for this model are shown on Table S3. The optimized solution for is depicted in Figure S2. When increases (decreases) points to a decrease (increase) in the ( ) diffusion effects within the device. Its dynamical behaviour agrees with our transport picture schematized in Figure 3 in the main manuscript. Table S3: Initial guess and optimized parameters for the MM2 discussed in the main text using equations (1+S2+S3) to describe the nonlinear ion-drift-diffusion dynamics. The resulting fitting is depicted on Figure  5(c) in the main manuscript which shows the I-V characteristics of the Au-Ti/TiO 2 /Ti-Au device in vacuum and at room temperature. The initial conditions for the dynamical quantities were set at for the ( = 0) = 0.9 internal state variable and a.u. for the diffusion rate.

Fitting Parameters
Initial   Table S3. This ( ) solution was obtained from fitting MM2 onto the I-V characteristics of Au-Ti/TiO 2 /Ti-Au devices in vacuum and at room temperature (cf. Figure 5(c) in the main manuscript).
Our selected fitting shown in Figure 5(d) in the main manuscript (MM3) was obtained using the functional form for the voltage-controlled element in the drift equation [3] expressed as expressed as in equation (S3), and [4] is given by with the Joglekar window function [2]. The initial guess and optimized values for this model are shown on Table S4. The optimized solutions for are depicted in Figure S3. { ( ), ( )} Table S4: Initial guess and optimized parameters for the MM3 discussed in the main text using equations (1+S3+S4+S5) to describe the nonlinear ion-drift-diffusion dynamics. The resulting fitting is depicted on Figure  5 Table S4. This solution was obtained from fitting MM3 onto the I-V characteristics of Au-Ti/TiO 2 /Ti-Au devices in vacuum and at room temperature (cf. Figure 5 Tables S1, S2, S3, and S4 present the values of the fitting parameters for the four memristive models discussed here and in the main text for the case in which Au-Ti/TiO 2 /Ti-Au is in vacuum and at room temperature. We can use some of these parameters to extract useful information about the characteristics of the device interfaces [5,6], e.g. to estimate the maximum Schottky barrier height, [7]. Assuming that the charge transport mechanism of the system at OFF state is Φ ( = 0) predominantly due to thermionic emission [8,9] through the Schottky barrier, the current is then given by where is the effective Richardson constant, is the ideality factor, is the temperature, is the * Boltzmann constant, and is the electron charge. is the cross-sectional area of the device, in our case, where is the diameter of the nanowire. For the purpose of estimation, we will = ( 2) 2 consider , with being the universal Richardson constant. By * ≈ 0 = 1.20173 × 10 6 2 2 0 comparing equation (S6) with equation (1) in the main text, for the zero-bias case, we identify = * 2 ( -Φ ) ( 7) from which the maximum Schottky barrier height can be estimated. Values of for both memristive descriptions shown in Table S1 result in a Schottky barrier height range of 0.25-0.3 eV using that our nanowire diameters range nm. This agrees with existent measured values for other TiO 2 -= 50 -100 based structures as reported in [10,11,12,13,14].
The last improvement in the memristive models adopted in this work was done by including a static rectifying current contribution (for thermionic effects) to the total current response function as given in equation (8) in the main text. The initial guess and optimized values for this improved model are shown on Table S5. A sample code for this improved description that reconstructs the I-V fitting of Figure 7 in the main text is provided in the supplemental information [15]. Table S5: Initial guess and optimized parameters for the improved current response function (equation (8) in the main manuscript), with the dynamical variable described by MM1+ (equations (1+S1)) to describe the nonlinear ion-drift dynamics. The resulting fitting is depicted on Figure 8 in the main manuscript which shows the I-V characteristics of the Au-Ti/TiO 2 /Ti-Au device in vacuum and at room temperature. The initial condition for the internal state variable was set at .

Fitting Parameters
Initial guess Optimized We tested the robustness of the fitting presented in Figure 7 (main text) by inducing variations in some of the initial values of the parameters in Table S5 and its outcome is shown in Figure S4. The black dots are experimental data and the red surface shell outlines the range where all fitting results -using the last improved memristive model discussed in the main text -fell.  Table S5 were induced and all fitting curves fall within the marked region. The fittings follow the dual CW orientation of the I-V hysteresis.
To demonstrate the range of applicability of the model in equation (8) in the main text, we investigated how the shape of the I-V curve of our Au-Ti/TiO 2 /Ti-Au device in vacuum can be modified upon independent variation of the parameters presented on Table S5. Figure S5 shows how the I-V hysteresis shape can be modulated by varying a given parameter in the model (while keeping others fixed at their optimum values as on Table S5). The wide variety of I-V shapes displayed in the figure reflects the multiple transport features enclosing this particular Au-Ti/TiO 2 /Au-Ti nanowire family that can be captured by our model. Finally, Table S6 presents the fitting parameters obtained for the device exposed to air as depicted in Figure 9(b) in the main manuscript. The optimization was done using MM1+ description using equation (S1) for [1] nonlinear ion-drift dynamics. Table S6: Initial guess and optimized parameters for the MM1+ discussed in the main text using equations (1+S1) to describe the nonlinear ion-drift dynamics. The resulting fitting is depicted on Figure 9(b) in the main manuscript which shows the I-V characteristics of the Au-Ti/TiO 2 /Ti-Au device in air and at room temperature. The initial condition for the internal state variable was set at .

I-V characteristics at distinct temperatures
A detailed investigation on the shape of the I-V characteristics of the Au-Ti/TiO 2 /Ti-Au devices in vacuum was carried out in a wide temperature range of 260-370 K with 10 K step differences, which is shown in Figure S7. For all temperatures, a bipolar CW switching with increased hysteresis windows were observed as the temperature increased. The I-V curves evidence temperature dependency in both forward and reverse bias loops. This behaviour can be a result of the diffusivity of oxygen ions (vacancies) which typically depends on the temperature [16]. At low temperatures (< 300 K), the oxygen ions have less mobility, and this would explain the low levels of current found in those I-V curves. At high temperatures, however, the mobility of the oxygen vacancies increases and those can drift rapidly across the bulk of the TiO 2-x channel. Current levels increase with temperature with the underlying I-V shape being dictated by the nonlinear ion-drift-diffusion with retention terms.