In order to analyze the TiO-based memristive circuit (Fig. 3.2), the nonlinear model is used for each TiO memristive switch and thus, coupled with the equation Eq. 3.4, Eq. 3.13–Eq. 3.22 are numerically solved for both and . Fig. 3.6 shows the modulation of the tunnel barrier widths and during the implication operation for all possible input patterns (State 1–State 4) described in Table 3.2. It illustrates that for pulse durations between 1–10 ms, only in State 1, the target memristor (T) is switched and in all other cases both and are left unchanged. Accordingly, correct logic behavior is achieved for all input states and the logic result is stored as the final resistance state of . Here, the initial tunnel barriers are and which are equivalent to and at the readout voltage of 0.2 V (Fig. 3.4). The circuit parameters , , and are optimized to minimize the SDE as is explained below.

According to Fig. 3.6, the dominant SDs occur in State 1 (Fig. 3.6a) in () and in State 3 (Fig. 3.6c) in (). Therefore, maximizing the modulation of the voltage between State 3 and State 1 () minimizes the possible SDEs in . Fig. 3.7 shows as a function of for different values of where

| (3.23) |

As follows from Fig. 3.7, the optimum corresponds to the maximum which maximizes the modulation of the voltage drop on between State 3 and State 1 and thus minimizes the SDEs in shown in Fig. 3.6. Therefore, it is uniquely defined by the memristor’s properties, and . By using Fig. 3.7, an optimum is obtained for each value of and then one can optimize to minimize the gate error (Fig. 3.8).

In fact, the voltage modulation increases with increased and minimizes the SD in memristor . However, an increase in results in an increasing error on memristor S, because it tends to switch in State 1 () and State 2 (). Therefore, there is an optimum for which the total state drift () defined as normalized root mean square error as shown in Fig. 3.8. Optimum and are determined at any by

| (3.24) |

TiO memristive switches enable stateful implication logic by serving simultaneously as non-volatile memory and logic gates. Although the digital data is stored in the high- and low-resistance state of the memristive device, the internal state variable shows analog behavior (Fig. 3.6). Therefore, during the logic operations the voltage drops on and tend to push toward , also when their switching is undesired. This causes the state drift error, which accumulates in sequential logic steps and results in a one-bit error after a certain number of implication operations. Thus, refreshing circuitry is required to avoid this error [162]. Fig. 3.9 shows the cumulative SD in during 20 implication operations with 1 ms pulse duration when and are in high and low resistance states, respectively (State 3). It illustrates that after 14 steps the sate variable is equal to the median value of 1.65 nm which can be readout either as high- or low-resistance state. Whereas any resistance switching in State 3 is considered as an undesired switching, the initial logic state of has to be rewritten before reaches 1.65 nm. It is worth mentioning that the linear model predicts a SD of 48.9% [162] for a particular design example which means a refreshing is required after each implication operation. Compared to the nonlinear ionic drift model, the linear drift model exhibits higher state drift values since it assumes that the state drift is directly proportional to the current or voltage of the memristive devices. However, according to experimental data, the ionic drift velocity shows an exponential dependence on the applied current or voltage [148] which is taken into account in the nonlinear model by Eq. 3.20 and Eq. 3.21. Once again one has to note that, as high switching voltages are used for (high-speed) computing, the memristor nonlinear model has to be used to take the tunneling effect and dynamical memristor behavior into account.

Fig. 3.10 shows only a slight increase of the optimized pulse amplitude with the implication switching time decreased in contrast to the linear model. This results in large power consumption benefits at higher IMP speed (Fig. 3.11) and shows a good agreement between simulations based on the nonlinear model and the available experimental data (see Fig. S5(b) in Supplementary Information of [75]) which demonstrates a decrease in switching energy of a TiO memristive device with the pulse duration decreased. In fact, Fig. 3.11 demonstrates that when the linear model is used, event the trend of the average implication energy consumption () is wrongly predicted as it shows an increase with the IMP speed increased.

| (3.25) |

where denotes the implication energy consumption when the memristive devices and are initially in State .