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

Chapter 9 Conclusions

A comprehensive hot-carrier degradation (HCD) model should encompass all aspects of defect creation i.e. the effect of single- and multiple-carrier mechanisms and all their superpositions. Thus, an important element of the HCD model is the distribution of carriers over energy. In this work, the distribution function (DF) is first determined analytically using the moments of the Boltzmann transport equation (BTE) obtained from the drift-diffusion (DD) simulations. The expression used to approximate the carrier distribution function consists of terms for both hot and cold carriers. To compare the distribution functions, the full solution of the BTE is obtained using the spherical harmonics expansion method (SHE). The two different approaches, DD- and SHE-based, have been applied to describe carrier transport in n-channel LDMOS transistors and the calculated carrier energy distributions are used as input to a physics based hot-carrier degradation model. The degradation of the linear and saturation drain currents as well as the threshold voltage shift predicted by the two versions of the model, for different combinations of drain and gate voltages, are compared with the measurement data and it is shown that both approaches can capture HCD. This leads to the conclusion that in the case of nLDMOS devices the DD-based variant of the model provides good accuracy and at the same time is computationally less expensive. This makes the DD-based version attractive for predictive HCD simulations of nLDMOS transistors. The validity of the model is proven beyond nLDMOS transistors by following similar procedure for pLDMOS transistors. The carrier distribution functions, interface state density profiles, and changes of the drain currents vs. stress time are calculated in pLDMOS transistor. The comparison with measurements shows very good agreement.

Particular attention is paid to the study of the role of the cold fraction of the carrier ensemble. The effect of cold carriers is checked by neglecting the low energy carriers in HCD modeling in the case of nLDMOS devices stressed at high voltages. In this work, the cold carriers are represented by the corresponding term in the analytic formula for the carrier distribution function as well as by the multiple-carrier process of Si-H bond dissociation. It is shown that even in high-voltage devices stressed at high drain voltages the thermalized carriers still have a substantial contribution to HCD.

Different analytic models for the carrier distribution function, namely the heated Maxw-ellian, the Cassi model, the Hasnat model, the Reggiani model, and the model proposed in this work, are analyzed based on their applicability to describe hot-carrier degradation in nLDMOS devices. The DFs evaluated with these models are used to simulate the interface state generation rates, the interface state density profiles, and changes of the linear and saturation drain currents as well as the threshold voltage shift. It is shown that the heated Maxwellian model underestimates HCD at long stress times. This trend is also observed for the Cassi and Hasnat models but in these models HCD is underestimated in the entire stress time window. While the Reggiani model provides good results in the channel and drift regions, it cannot properly represent the high-energy tails of the DF near the drain, and thus leads to a weaker curvature of the degradation traces. Finally, the model used in this work is capable of capturing DFs with very good accuracy and, as a result, the change of the device characteristics with stress time.

The limits of applicability of the drift-diffusion based model for hot-carrier degradation have also been studied. Although the simplified version of the HCD model is quite successful for LDMOS devices, the analysis of linear and saturation drain current degradation predicted by the DD-based HCD model shows that it starts to fail for gate lengths shorter than 1.5 m and becomes completely inadequate for devices shorter than 1.0 m. This limitation of the DD-based model stems from the fact that electron-electron scattering (EES) becomes significant in scaled devices and this effect is not incorporated in the simplified model.

Thus, the role of electron-electron scattering on the DF shape in short channel devices is elucidated. EES is known to dominate the high-energy tails of the DF. In this work, the energy which corresponds to the onset of such high-energy tail is called “knee energy” and is determined by the balance equation for in- and out-scattering rates. The distribution functions for scaled MOSFETs are evaluated by first finding an initial solution using the same approach as used for LDMOS devices and then refining the DF due to knee energies. The DF obtained from this extended analytic model and those obtained using the deterministic BTE solver ViennaSHE are almost identical. Furthermore, good agreement between the degradation characteristics obtained with the analytic approach and measurement data suggests that the extended model is well suited for describing hot-carrier degradation in decananometer devices. It is shown that the model can properly represent HCD in nMOSFETs with gate lengths in a range of 65-300 nm, while for longer devices a similar model but omitting EES is applicable. The main advantage of this model is that is uses the simple and fast DD scheme instead of the computationally demanding solution of the BTE.