## 3 Results  
The DC measurements of the Cu-TiN\({}_{x}\)O\({}_{y}\) film sheet resistance produced a value of 220 \(\pm\) 8 \(\Omega\)/\(\square\), the resistor thickness extracted from TEM data was 22 \(\pm\) 1 nm, and its effective resistivity was 480 \(\pm\) 20 \(\mu\Omega^{*}\)cm. In order to explore the stability of resistors, we measured their T-dependence in two sets of measurements separated by 18 months. Shown in Figure 2 is the temperature dependence of the extrinsic (total), Cu-TiN\({}_{x}\)O\({}_{y}\) sheet (\(R_{SH}\)) and TiN\({}_{x}\)O\({}_{y}\)/Ti/Cu contact (\(R_{C}\)) resistances extracted from the extrinsic values using the following expressions:  
\[\left\{ \begin{array}{l}R_{SH}=\left(R_{3 qq}-R_{1 sq}\right)/2\\ R_{C}=\left(3\cdot R_{1 sq}-R_{3 sq}\right)/4\end{array} \right. \tag{1}\]  
Figure 1: RF resistor fabricated from 22 nm thick Cu-doped TiN\({}_{x}\)O\({}_{y}\) film grown by ALD. The resistor body is connected to ground-source-ground (GSG) copper routing designed for DC/RF measurements. Titanium was used as the copper adhesion layer. The structure is schematically broken into S1–S6 sections for high-frequency modeling. The fabrication process flow diagram is shown below the resistor image. where \(R_{1 sq}\) and \(R_{3 sq}\) are the measured extrinsic resistances of 1-square and 3-square long resistors. The above data suggest there was an ageing effect present in both resistors. Indeed, the measurements in the years 2019 and 2021 revealed that the differential resistance at zero external bias increased by 10-15%. There are several aspects of ageing implied by the data from Figure 2b: the data become noisier, the main contribution to the resistance increase comes from the contacts degradation, R(T) is strongly non-linear and the sheet resistance of the resistor body becomes essentially flat (within experimental error) below 200 K. However, the contact resistance almost tripled at room temperature and increased up to five times at low temperature due to the ageing. All \(I\)-\(V\) curves of resistors were linear at zero bias, but as the applied voltage increased, the differential resistance decreased. For low-frequency large signal applications, we quantified this non-linearity by introducing the U\({}_{10}\) parameter, which was the voltage that caused a 10% decrease in the differential resistance compared to its value at zero bias, as shown in Figure 3a. The non-linearity of \(I\)-\(V\) curves was also affected by the ageing phenomena. Figure 3b compares the U\({}_{10}\) T-dependence in short and long resistors measured at different times. The non-linearity of resistors has notably increased at low temperatures as the associated parameter U\({}_{10}\) has decreased. More detailed characterizations of resistors' non-linearity including an analytical expression for \(I\)-\(V\)-dependencies at different temperatures are needed for high-frequency circuits design. Therefore, we fitted the experimental \(I\)-\(V\) curves by the following polynomial function where we only kept significant terms:  
\[I(V)=B_{1}V+B_{3}V^{3}+B_{5}V^{5}+B_{7}V^{7} \tag{2}\]  
and extracted its coefficients at different temperatures. In the function (2) \(I\) and \(V\) are the current and voltage across the resistor and the coefficient \(B_{1}\) is the conductivity of the resistor, while the other coefficients do not have simple physical meaning and are rather mathematical entities. The non-linear current polynomial coefficients were calculated for resistors based on \(I\)-\(V\) curves measurements, their absolute values are summarized in Figure 4. All coefficients were positive except \(B_{5}\), which was strictly negative. The fitting fidelity was high, as evidenced by the adjusted R-square value of 0.99997. The fitting parameters extracted from the non-linear \(I\)-\(V\) curve show some differences that seem to increase as temperature decreases. We believe that the origin of these differences is related to the age dependent contact resistance.