## 5 Discussion  
The increase in the contact resistance with the current can be treated in terms of self-heating. The nearly linear dependence of \(\Delta R_{\rm c}\) on dissipated power density \(P_{\rm c}\) is obtained in Figure 4b. In fact, the dissipated power in the contacts controls the change in \(R_{\rm c}\), which is evident from Figure 7, where the relative contact resistance change is plotted. The dependencies of different samples almost coincide when they are plotted versus \(P_{\rm c}\). Self-heating of the contact resistance also follows from the noise experiment (see Figure 6) under the assumption that noise temperature \(T_{\rm nc}\) is a measure of electron gas temperature \(T_{\rm ec}\) in the contact resistance, i.e. \(T_{\rm nc}\approx T_{\rm ec}\). The excess noise temperature of contact resistance (squares) approaches \(\sim\)700 K.  
The assumption \(T_{\rm nc}\approx T_{\rm ec}\) is realistic because it is based on the experiments with the GaN-based 2 DEG channels, where the microwave noise temperature of hot electrons \(T_{\rm n}\) is approximately equal to hot-electron temperature \(T_{\rm n}\approx\ T_{\rm e}\)[51]. The dependence of channel resistance change \(\Delta R_{\rm 2D}=R_{\rm 2D}-R_{\rm 2D0}\) on dissipated power density \(P_{\rm 2D}=R_{\rm 2D}(I/W)^{2}\) in the channel is presented in Figure 8 (closed symbols). It was found, that qualitatively, the change in contact and channel resistances is similar and follows a linear trend at higher power in both cases (see open symbols in Figures 4b and 8, respectively). The contact resistance \(R_{\rm c}\) can be described as [50]:  
\[R_{\rm c}=\sqrt{\rho_{\rm c}\times R_{\rm 2D}}/W \tag{7}\]  
where \(\rho_{\rm c}\) is the specific contact resistance. In actuality, \(\rho_{\rm c}\) is the resistance from the metal to 2 DEG under the electrodes. In Figure 1b, the current flow in \(\rho_{\rm c}\) and \(R_{\rm 2D}\) is shown as vertical and horizontal current vectors (in shaded areas only), respectively. Therefore, the observed self-heating of \(R_{\rm c}\) is related with the self-heating of both resistances \(\rho_{\rm c}\) and \(R_{\rm 2D}\). The contact-end (three-terminal) measurement method was developed for the estimation of the \(\rho_{\rm c}\) and \(R_{\rm 2D}\) values [22; 50]. The contact-front (two-terminal in this work) measurements, which are standard measurements of TLM, assume that \(R_{\rm 2D}=R_{\rm 2D}\). The specific resistance obtained by contact-front method are denoted here as \(\rho_{\rm cf}\). It was found that \(R_{\rm 2D}\
eq R_{\rm 2D}\) and the temperature dependencies of \(\rho_{\rm c}\) and \(\rho_{\rm cf}\) were qualitatively different [52]. Disregarding those concerns, \(\rho_{\rm cf}\) was estimated and the results are shown  
Figure 7: The dependence of the relative contact resistance change \((R_{\rm c}-R_{\rm c0})/R_{\rm c0}\) on the dissipated electric power density \(P_{\rm c}\) in contacts for different samples; AlGaN/GaN (squares), AlGaN/AlN/GaN (triangles and circles). in the inset of Figure 9. If we assume that the properties of the channel under the electrodes are similar to those between the electrodes (\(R_{\text{c2D}}\approx R_{\text{2D}}\)), then the contact-front and contact-end methods give the same results, i.e., \(\rho_{\text{cf}}\approx\rho_{\text{c}}\) and \(L_{\text{x}}\approx L_{\text{T}}\). Reiterating, in the present work, \(R_{\text{c}}\), \(R_{\text{2D}}\), \(\rho_{\text{cf}}\) and \(L_{\text{x}}\) increase with an increase in the current, while \(\rho_{\text{c}}\) and \(R_{\text{c2D}}\) are not known with confidence. We also can consider an _unintentionally_ formed field-plate-like structure near electrode, which, at higher voltages in the channel, could induce a local high-resistance region near the edge of electrode which also would appear as a increase in \(R\). In this case, the open symbols in Figure 4a would correspond to the resistance of the high-resistance region and not to the contact resistance change.