## 3 Results
The I-V characteristics in Figure 2a do not reflect the sample steady state electrical behavior because Joule heat keeps increasing the internal temperature of the sample. To minimize Joule heating, steady-state I-V characteristics must be recorded upon the application of a train of short (milliseconds long) voltage pulses with increasing amplitude. Furthermore, to prevent cumulative effects due to residual heat, the time between pulses should be at least 1 s long. This method is referred to herein as a pulsed I-V method. Figure 2b compares two I-V curves, one recorded with the pulsed method and the other recorded with a relatively slow voltage ramp speed (0.1 V/s). The remarkable difference between the I-V curves clearly shows the impact of self-heating on the electrical properties of the devices. The magnitude of the negative differential resistance (NDR) region in the I-V curve can be characterized by the ratio of the peak current to the valley current. This study found that the magnitude of the NDR region can be controlled by adjusting the voltage ramp speed. Additionally, the application of Joule heating pulses can be used to program the magnitude of the NDR. Figure 3 illustrates the programming of the NDR region using voltage ramps. When a series of seven I-V loops were consecutively recorded, the magnitude of the NDR increased by approximately 15 times. It is important to note that between each curve, the sample was left to rest (without applied bias) to dissipate the heat generated by the previous I-V loop. Figure 3: The behavior of consecutive I-V loops on the NDR magnitude. The voltage scan rate was 1 V/s. In between each loop, the sample rested for 10 s to dissipate the heat generated by the Joule effect. The magnitude of NDR causes long-lasting changes in the sample's behavior. As shown in Figure 4, it took around 20 h for the I-V curve to return to its original state after the sample was left to rest. It is difficult to quantify the time constant of this decay using non-steady-state I-V curves (curves recorded with a voltage ramp). However, by measuring the pulsed I-V curve in between each heat cycle, we can gain insight into the underlying physics and quantify the phenomenon. The inset in Figure 4 shows the decay of the current measured at 1 kV/cm plotted on a log-log scale. The decay follows a power law, \(i(t)=(1/t)^{\alpha}\), with a small exponent of \(\alpha=0.03\), indicating a relatively slow decay. To demonstrate this long-lived memory behavior, the sample was subjected to a series of constant current steps of increasing duration (from 1 s to 10 s). The resulting changes in conductance, caused by Joule heat, were monitored by recording the steady-state I-V curves. To ensure that all the I-V curves were recorded at a constant internal sample temperature, the sample was allowed to rest for 30 s before each pulsed I-V curve measurement to allow the internal heat generated by the pulse to dissipate. Figure 5 shows the increase in the sample's conductance when well-defined Joule heating pulses were applied (I = 20 \(\mu\)A). The inset shows the incremental conductance as a function of pulse duration, with only pulses longer than 4 s causing an increase in conductance. Like the decay described above, these changes also follow a power law with exponent \(\alpha=0.03\) and were relatively long-lived. The sample's ability to be temporarily programmed at conductance levels close to one another, combined with the long-lived memory of these changes, has a significant effect on negative differential resistance (NDR) behavior. Even small changes in conductance can lead to a significant increase in current and Joule heating, which drives the NDR properties. The non-linear electrical behavior, with its long-term plasticity under successive excitation, is like that of neural synapses. This suggests that these materials could potentially be explored for use in neuromorphic applications [20]. The current-voltage characteristics of a pyrogallol-formaldehyde sample without ZrO\({}_{2}\) nanoparticles were also studied.