## 5 Discussion
The situation is even more dramatic, because the ratio \(R_{\rm c}/R_{\rm 2D}\) (and \(R_{\rm c}/R_{\rm ch}\)) also increases with the current. Indeed, from Figure 9 it can be concluded that above the threshold \(R_{\rm c}\) increases faster than \(R_{\rm 2D}\). We recognize that this is the most important result of the present work. This behavior has many practical consequences, because a fast increase in \(R_{\rm c}\) depending on \(I\) can deteriorate the performance of electronic devices. The increase in the ratio \(R_{\rm c}/R_{\rm 2D}\) can be discussed in terms of stronger \(R_{\rm c}\) self-heating compared to that of \(R_{\rm ch}\). Excess noise temperature \(\Delta T_{\rm nc}\) in \(R_{\rm c}\) (Figure 6, squares) is more than four times stronger than \(\Delta T_{\rm n}\) in the total resistance \(R\) of 45 \(\mu\)m-long channel (circles). The term \(\Delta T_{\rm nc}\times 2R_{\rm c}\) in Equation (5) is not negligible, even for the 45 \(\mu\)m-long channel (it would be negligible for the infinitely long channel only). Therefore, \(\Delta T_{\rm n}\) of a 45 \(\mu\)m-long channel reaches \(\sim\)130 K at a \(\sim\)0.3 A, and it is still slightly higher than its \(\Delta T_{\rm ch}\). Thus, the relation \(\Delta T_{\rm nc}>\Delta T_{\rm n}\gtrsim\Delta T_{\rm nch}\) holds. This suggests that the temperature \(T_{\rm c}\) (and thus, the self-heating) of \(R_{\rm c}\) is higher than the temperature \(T_{\rm ch}\) of \(R_{\rm ch}\). Let us first discuss the option that the self-heating of \(R_{\rm c}\) is caused by the self-heating of \(R_{\rm c2D}\) (see Equation (7)), while \(\rho_{\rm c}\) self-heating is negligible. The current density in \(R_{\rm c2D}\) is  
Figure 9: The dependence of the contact resistance \(R_{\rm c}\) on the channel sheet resistance \(R_{\rm 2D}\) for different samples; AlGaN/GaN (squares), AlGaN/AlN/GaN (triangles and circles). Lines illustrate the slopes of the dependencies. Starting at \(R_{\rm 2D}\sim 750\)\(\Omega\)/sq\({}_{\rm J}\), the increase in \(R_{\rm c}\) becomes faster (dashed lines versus solid lines). smaller than in \(R_{\rm 2D}\), therefore, the stronger self-heating of \(R_{\rm c2D}\) compared to \(R_{\rm 2D}\), might seem surprising. The current profile is illustrated in Figure 1b where the horizontal current vector magnitude gradually decreases as the current branches toward the electrodes. A detailed description of the current dependence on the distance can be found elsewhere [22]. We could suppose, that \(R_{\rm c2D}>R_{\rm 2D}\) due to lower electron mobility or lower sheet electron density under the electrodes. In this case dissipated power per unit area \(P_{\rm 2D}\) at a given current density \(j\) will be higher because \(P_{\rm 2D}\propto R_{\rm c2D}j^{2}\). Higher dissipated power in the contact resistance results in higher self-heating of \(R_{\rm c2D}\). The next option is the self-heating of \(\rho_{\rm c}\). If the metal-like shunt resistances from the metal to the 2 DEG channel under the electrodes persists, the electron mobility in these shunt resistances is expected to be poor due to the high density of electrons and associated intensive electron-electron scattering. However, the high electron density in metals is what could make the specific resistance of those metal protrusions low. The current density in metal protrusion is high if its cross section is small, leading to high resulting values of the dissipated power density per unit volume. The self-heating model in terms of the two temperatures \(T_{\rm c}\) and \(T_{\rm ch}\) is oversimplified. In GaN-based 2 DEG channels (applicable to \(R_{\rm c2D}\) and \(R_{\rm 2D}\)), excess (hot) electron temperature \(\Delta T_{\rm e}\) is higher than the excess lattice temperature \(\Delta T_{\rm L}\) when the constant electric field is applied [34].