# Thermal Modeling of the GaN-based Gunn Diode at Terahertz Frequencies
## 3 Result and Discussion
The temperaure profile within the device is a reflection of the distribution of the joule heating, which in turn reflects the electric field distribution. The peak temprature and heating occurs at where the field is a maximum. As shown in Figure 8, the electric field inside the notch layer enhances at larger N\({}_{\rm ac}\)/N\({}_{\rm notch}\), thus the lattice temperature peak gradually transfers from the anode side to the notch layer. The lattice temperature distributes most uniformly at the doping level of 2 \(\times\) 10\({}^{17}\) cm\({}^{-3}\). At the doping level of 2.5 \(\times\) 10\({}^{17}\) cm\({}^{-3}\), the lattice temperature peak completely transfers to the notch layer. This is why we get the highest \(\eta\) and P\({}_{\rm RF}\) at the doping level of 2 \(\times\) 10\({}^{17}\) cm\({}^{-3}\) and no oscillation can be obtained at the doping level higher than 2.5 \(\times\) 10\({}^{17}\) cm\({}^{-3}\). Figure 5: Electric field profiles derived inside one oscillation circle at the same time steps, when the diode is applied at different voltage: (**a**) V\({}_{\rm dc}\) = 50 V and (**b**) V\({}_{\rm dc}\) = 75 V under EB model; and (**c**) V\({}_{\rm dc}\) = 50 V and (**d**) V\({}_{\rm dc}\) = 55 V under NEB model. We also investigate the influence of transit region length L\({}_{\rm{ac}}\) on output performance of GaN Gunn diode, where L\({}_{\rm{ac}}\) changes from 0.8 to 1.4 \(\upmu\)m at a step of 0.1 \(\upmu\)m in this simulation. Figure 9 shows the influence of L\({}_{\rm{ac}}\) on dc-to-ac conversion efficiency and oscillation frequency under both EB and NEB models. The dc-to-ac efficiency \(\eta\) and the oscillation frequency f\({}_{\rm{osc}}\) decrease as L\({}_{\rm{ac}}\) increases under both  
Figure 8: Lattice temperature profiles in the active region at different N\({}_{\rm{ac}}\) under NEB model. Figure 6: I–V characteristics of GaN Gunn diode under NEB model and EB model, and the global lattice temperature at different N\({}_{\rm{ac}}\) under NEB model. Figure 7: (a) Simulation results of the oscillation frequency f\({}_{\rm{osc}}\) and dc-to-ac conversion efficiency \(\eta\) versus the doping level N\({}_{\rm{ac}}\) and (b) RF (Radio Frequency) output power P\({}_{\rm{RF}}\) versus N\({}_{\rm{ac}}\) under EB and NEB models. EB and NEB models, as illustrated in Figure 9. It is noteworthy that the oscillation frequency obtained under NEB model becomes larger than that obtained under EB model when L\({}_{\text{ac}}\) is larger than 0.9 \(\upmu\)m. We can explain the phenomenon by using extracted electric field profiles of 1.2-\(\upmu\)m-long device under EB and NEB model, as shown in Figure 2:0 Figure 10a shows that the Gunn diode operates at dipole domain mode under EB model. Figure 10b indicates that the device operates at accumulation mode as well as the dead zone is enlarged obviously when we take the self-heating effect into consideration under NEB model. The lattice temperature, particularly the temperature near the anode side, increases with L\({}_{\text{ac}}\). The dipole domain will become not able to sustain at higher lattice temperature, as shown in Figure 10b. Based on [9; 30], in order to allow the dipole domains forming in the Gunn diode, design criteria must be met: (N\({}_{\text{ac}}\)\(\times\) L\({}_{\text{dc}}\)) \(>\) (N \(\times\) L)\({}_{\text{o}}\)\(\equiv\) 3 \(\times\)\(\varepsilon\)\(\times\)\(\
u_{\text{peak}}\)/4\(\upmu_{\text{NDR}}\) (\(\varepsilon\): dielectric constant; \(\upmu_{\text{NDR}}\): the peak negative differential mobility). The numerical value is for GaN. Therefore, most references believe that the dipole domains prefer to form at the longer transit region with higher doping level, which may only hold in ideal situation without considering the joule heating. In our simulation, we draw an absolutely opposite conclusion: the dipole domain modes easily degenerate into accumulation modes due to the self-heating effect in the longer channel.