Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

6.5 Results and Discussion

To check the validity of the different HCD models, the bond breakage rates, the interface state density profiles, and the change of the linear and saturation drain currents as well as the threshold voltage shifts are calculated. The data obtained from the different versions of the model are compared against the HCD results measured in nLDMOS transistors.

6.5.1 Distribution Functions and Interface State Densities

The heated Maxwellian DFs are plotted along with the DFs calculated with ViennaSHE in Figure 6.1. As expected, the Maxwellian model is only valid in the initial part of the channel region where the carriers are almost in equilibrium. It overestimates the DFs at high energies near the bird’s beak and fails completely in the drain region of the nLDMOS device. The heated Maxwellian distribution leads to reasonable $N_{\mathrm {it}}(x)$ profiles only in the

channel and source regions when compared with the $N_{\mathrm {it}}(x)$ profiles computed using the DF from ViennaSHE, see Figure 6.5. This approach overestimates the $N_{\mathrm {it}}$ values in the bird’s beak region and also fails in the drain region.

The Cassi model shows an improvement for the non-equilibrium case but cannot describe the DFs near the drain. This is because the DFs evaluated with this approach have a fixed curvature. Figure 6.5 summarizes the $N_{\mathrm {it}}(x)$ profiles obtained using the DFs from the Cassi model and from ViennaSHE. These $N_{\mathrm {it}}(x)$ profiles from the two approaches are only comparable in the bird’s beak region. Such a behavior is also consistent with the difference

in the DFs computed with this model and that from the BTE solution.

The Hasnat method cannot represent the DFs in neither the bird’s beak nor the drain region, as shown in Figure 6.3. The corresponding interface trap density is plotted in Figure 6.6, which does not match those calculated with ViennaSHE in the entire $X$ coordinate range. The $N_{\mathrm {it}}$ values are severely underestimated inside the channel and close to the drain region.

The Reggiani approach appears promising because it can describe the DFs in LDMOS

devices with good accuracy except for the drain region. The reason is that within the Reggiani model the DFs are linked to the local electric field, and thus hot carriers which form the high-energy tail of the DF are not properly described. From Figure 6.4 one can see that these tails are underpopulated if the Reggiani approach is used. As a result, this discrepancy translates also into an error in $N_{\mathrm {it}}$ which is underestimated in the drain, as is evident from Figure 6.6.

The DFs computed with the model developed in this work (Section 4.3) are also shown

in Figures 6.1, 6.2, 6.3, and 6.4 as light gray curves. The model was successful in representing the carrier distributions along the entire device.

6.5.2 Degradation Traces

The changes in $\Delta I_{\mathrm {d,lin}}(t)$ , $\Delta I_{\mathrm {d,sat}}(t)$ , and $\Delta V_{\mathrm {t}}(t)$ calculated with the different model versions are summarized and compared to the experimental data in Figures 6.7 $-$ 6.18.

The heated Maxwellian approach leads to a saturation of $\Delta I_{\mathrm {d,lin}}(t)$ , $\Delta I_{\mathrm {d,sat}}(t)$ , and $\Delta V_{\mathrm {t}}(t)$

at $\sim$ 500 $\,$ s, see Figures 6.7 $-$ 6.9. This is related to the drain $N_{\mathrm {it}}$ peak which is formed by hot carriers and which determines short term hot-carrier degradation [26, 147, 203]. One can see in Figure 6.5 that the interface state density $N_{\mathrm {it}}$ is saturated already at short stress times, and thus at long stress times, HCD is driven by colder carriers which contribute to the multiple-carrier process [26, 147, 203]. As a result, if the effect of cold carriers is underestimated, the change of the device characteristics saturates at longer stress.

Figures 6.2 and 6.5 show that the Cassi model highly underestimates the DFs and $N_{\mathrm {it}}$

values near the drain region. As a consequence, the degradation of all device characteristics is underestimated as well, especially at short stress times as in Figures 6.10 $-$ 6.12. This is because the most prominent discrepancy between the DFs obtained with ViennaSHE and the Cassi model is visible for the drain region of the device. The same behavior is typical also for the $N_{\mathrm {it}}$ drain maximum, which — as already discussed — determines short-term HCD.

The Hasnat method behaves similarly to the Cassi model where the DF values are un-

derestimated for most of the device regions, see Figures 6.13 $-$ 6.15. As a result, the degradation of both linear and saturation drain currents as well as the threshold voltage shift are massively underestimated in the entire experimental stress time window.

As for the Reggiani model, the model underestimates the interface trap density near the drain. This peculiarity results in weaker curvatures of $\Delta I_{\mathrm {d,lin}}(t)$ , $\Delta I_{\mathrm {d,sat}}(t)$ , and $\Delta V_{\mathrm {t}}$ traces as shown in Figures 6.16 $-$ 6.18. Although the degradation curves are close to the experimental ones at long stress times, they are unable to predict the correct degradation

for the entire stress time slot.

Finally, Figures 6.7 $-$ 6.9 show very good agreement between measured $\Delta I_{\mathrm {d,lin}}(t)$ , $\Delta I_{\mathrm {d,sat}}(t)$ , and $\Delta V_{\mathrm {t}}$ curves and those simulated with the DD-based method developed in this work for the whole stress time window. Both the SHE and DD variants of the HCD model use the same set of parameters for different combinations of stress voltages and for the degradation traces of different device characteristics.

To conclude, in this section a comparison between various HCD models, which employ different approaches to approximate the solution of the Boltzmann transport equation, has been performed using hot-carrier degradation data measured on nLDMOS devices. In several realizations of the HCD model, carrier energy distribution functions obtained with the heated Maxwellian, Cassi, Hasnat, Reggiani and our model are used. These different versions are compared in terms of carrier DFs, interface state density profiles, and degradation of the linear and saturation drain currents, as well as the threshold voltage. The heated Maxwellian neglects the cold carrier fraction of the carrier ensemble and thus the $N_{\mathrm {it}}(x)$ profiles are adequate to some extend only in the channel and source regions. As a result, the degradation traces ( $\Delta I_{\mathrm {d,lin}}(t)$ , $\Delta I_{\mathrm {d,sat}}(t)$ , and $\Delta V_{\mathrm {t}}(t)$ ) show a spurious saturation at longer stress times. The Cassi and Hasnat models underestimate the values of the carrier DFs in most of the device regions. As for the $N_{\mathrm {it}}(x)$ profiles, the former approach leads to somewhat reasonable $N_{\mathrm {it}}$ values only near the bird’s beak, while the interface state densities simulated with the latter approach have substantially lower values (as compared to those evaluated using the full BTE solution) in the entire range of the lateral coordinate. Therefore, the $\Delta I_{\mathrm {d,lin}}(t)$ , $\Delta I_{\mathrm {d,sat}}(t)$ , and $\Delta V_{\mathrm {t}}(t)$ degradation characteristics computed using the Cassi and Hasnat models are underestimated within the entire stress time slot. The Reggiani model can represent the DFs for most device regions except the drain where the agreement deteriorates. As a result, the curvature of the degradation traces is weaker within the Reggiani model. Finally, the proposed DD-based method can mimic the DFs in the entire device with a slight discrepancy in the drain region for the high energy tails where the magnitude of DF has already dropped by $\sim$ 20 orders. This discrepancy does not translate to an error in the results evaluated using this approach. This means that the agreement between the $N_{\mathrm {it}}(x)$ profiles and the $\Delta I_{\mathrm {d,lin}}(t)$ , $\Delta I_{\mathrm {d,sat}}(t)$ , and $\Delta V_{\mathrm {t}}$ degradation curves simulated with the SHE- and DD-based realization of the model is very good. It can, therefore, be concluded that in long-channel LDMOS transistors HCD can be modeled with very good accuracy even with a DD-based formalism provided a good approximative model for the distribution function is used.