## 5 Discussion
In the pulsed mode, \(\Delta T_{\rm L}\) is lower when compared to the DC mode; therefore, at short voltage pulses a nearly isothermal condition is maintained (\(\Delta T_{\rm L}\ll T_{\rm L}\)), and the self-heating effects related to acoustic phonons are mitigated. Furthermore, in the pulse mode, \(\Delta T_{\rm e}\) is higher due to the thermal quenching effect when the hot-electron temperature decreases along with the voltage pulse due to the electron mobility decrease [37]; the shorter the pulse, the weaker the thermal quenching. Mainly because of lower \(\Delta T_{\rm L}\) in the pulsed mode, higher electric fields and currents can usually be reached without the thermal breakdown of the channel. It is also important to mention, that even in the pulsed mode, at higher electric fields the LO-phonon-related self-heating is present. For 2 DEG densities, typical for GaN-based heterostructures, the hot-phonon effect is important, as non-equilibrium LO-phonons (hot-LO-phonons) strongly impact the hot-electron transport and temperature [25; 35]. At high fields under conditions of strong hot-electron-hot-phonon coupling, when the system is strongly non-equilibrium, the hot-electron temperature \(T_{\rm e}\) is close to the effective hot-LO-phonon temperature \(T_{\rm LO}\). In this case, different temperatures \(T_{\rm e}\), \(T_{\rm LO}\) and \(T_{\rm L}\) are introduced, for which the relation \(T_{\rm e}\gtrsim T_{\rm LO}\gg T_{\rm L}\) holds [51]. The hot-electron excess temperature \(\Delta T_{\rm e}\) is controlled by electron energy relaxation time \(\tau_{\rm en}\), \(\Delta T_{\rm e}\propto\tau_{\rm en}P_{\rm e}\propto\tau_{\rm en}P_{\rm V}/n \propto\tau_{\rm en}\rho j^{2}/n\), where \(P_{\rm e}\) is the dissipated power per electron and \(n\) is the electron density [60]. The dependence of excess noise temperature on dissipated power per electron in the 45 \(\upmu\)m long channel \(P_{\rm e}=U_{\rm a}I/N\) is shown in the inset of Figure 8 (open circles), where \(N\) is electron number in the channel available from the Hall measurements. The line in the inset is plotted for constant \(\tau_{\rm en}\) in the channel. At a higher power per electron, \(\tau_{\rm en}\) decreases due to enhanced electron-LO-phonon scattering, and experimental points (circles) deviate from that line. If the electron mobility in the 2 DEG channel under the electrodes is lower than in the 2 DEG channel between electrodes (corresponding to \(R_{\rm c2D}\) and \(R_{\rm 2D}\), respectively), then at a given current, \(\Delta T_{\rm e}\) in the former channel will be higher. Indeed, at a given current, the electric field (and thus, \(P_{\rm e}\)) will be higher for the 2 DEG channel with a lower electron mobility. A higher \(P_{\rm e}\) results in a higher \(\Delta T_{\rm e}\) (see inset of Figure 8). For the metal-like conductor, the hot-electron effect is not well expressed due to poor electron mobility; therefore, \(T_{\rm e}\approx T_{\rm L}\). In this case, \(\Delta T_{\rm e}\approx\Delta T_{\rm L}\propto(1/C_{\rm V})\rho j^{2}\), where \(C_{\rm V}\) is the heat capacity per unit volume. As such, qualitatively both cases (GaN-based channels and metal-like conductors) are similar in that \(\Delta T_{\rm e}\propto\rho j^{2}\). This is the reason why it is difficult to differentiate between the self-heating of resistances \(R_{\rm c2D}\) and \(\rho_{\rm c}\) from the temperature-current measurements. Assuming that \(R_{\rm c2D}\approx R_{\rm 2D}\) and \(\rho_{cf}\approx\rho_{c}\), the data in the inset of Figure 9 indicate the strong self-heating of \(\rho_{c}\). To conclude, we investigated the pulsed current-voltage (I-V) and hot-electron microwave noise (at 10 GHz frequency) characteristics of the TLM samples composed of AlGaN/(AlN)/GaN heterostructures with annealed Ti/Al/Ni/Au ohmic contacts to the 2 DEG channel. From the I-V measurements, the increase in the contact resistance \(R_{\rm c}\) dependent on the current was found for all samples, with a change up to six times as the current density approached \(\sim\)0.6 A/mm.