## 4 Discussion
Indeed, given the values of the maximum current, the voltage at the maximum current (Figure 5) and the resistor body dimensions, one may calculate that such a resistor can sustain power density in the low resistivity state up to \(\sim\)5 kW\({}^{\text{*}}\)cm\({}^{-2}\), with currents in the range of tens of mA corresponding to the effective current densities in excess of 2 MA\({}^{\text{*}}\)cm\({}^{-2}\). The enhanced robustness of Cu-TiN\({}_{\text{x}}\)O\({}_{\text{y}}\) resistors was demonstrated by their ability to survive several 10 s open-air incandescence tests (Figure 6a,b), sustaining up to 100 mW or 40 W\({}^{\text{*}}\)mm\({}^{-2}\) power for about 10 s. At higher power levels, the resistors switch to high-resistivity state due to the nitrogen/copper desorption and local oxidation (Figure 6c) before eventually breaking open explosively at voltages over 150 V due to the copper routing sublimation. Therefore, these devices may have a strong potential as fusible resistors for tens mA currents. A peculiar pattern of the N-to-O phase transition in TiN\({}_{\text{x}}\)O\({}_{\text{y}}\) resistor body starting from the middle and propagating towards the high-bias terminal (shown in the inset of Figure 6a) may be understood by the copper routing heat removal mechanism. Since copper has much higher thermal conductivity than TiN\({}_{\text{x}}\)O\({}_{\text{y}}\) and Sitall, effectively--together with the measurement probes--it serves as a heat sink. Since the middle part of the resistor body is the farthest point from the copper heat sink, it has the highest temperature and N-to-O transition outsets from there. Once a tiny N-depleted gap of O-rich semi-insulating TiN\({}_{\text{x}}\)O\({}_{\text{y}}\) is formed in the middle of the resistor body (Figure 6c), it is instantly exposed to a high field, which may cause avalanche electric breakdown phenomena in the newly formed O-rich semi-insulating phase micro-regions of TiN\({}_{\text{x}}\)O\({}_{\text{y}}\) film and, thus, generate the observed current inrush spikes (Figure 6d). The light emission may originate from localized heating or hot electron-hole radiative recombination as well. Indeed, an interesting reproducible oscillatory current spiking behavior has been found at substantially high overvoltage ramps and high voltages. Judging by the associated abrupt changes of color from grey to transparent and the localized microscopic light-flashes synchronized with the observed current spikes, we believe that these current spikes should be related to the high-field avalanche currents and localized phase transition from the high-conductance semi-metallic state of Cu-doped N-rich composition to the low-conductance semi-insulating state of O-rich composition of TiN\({}_{\text{x}}\)O\({}_{\text{y}}\) film. The high frequency S-parameters' data showed relatively low mismatch of resistors across the wafer, which was 0.1% at 10 MHz and gradually increased to 10% at 26 GHz. The simulated S-parameters reproduce the experimental data for long resistors and were slightly off for the short ones at frequencies over 1 GHz (Figure 7). This effect may be due to the increased fringe fields between the resistor body contacts, which are much closer to one another in the short resistor. Full 3D EM simulations are planned in the future in order to account for this mismatch, as the fringe fields' effects may play a significant role in smaller structures, and EM resonances may appear in larger structures. Both effects are not captured by the cascaded chain of the quasi-2D coplanar lines approximation. Thus, a somewhat higher sheet resistance value of 250 \(\Omega\)/\(\square\) extracted from the simulations of the short resistor is likely an overestimate due to oversimplified consideration of the fringe fields in simulations. The sheet resistance of 215 \(\Omega\)/\(\square\) extracted from simulations of the long resistor matched well with the experimental DC value due to reduced AC-coupling of the farther separated terminals in the case of the long resistor.