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

Chapter 6 Comparison of Drift-Diffusion Based Models

As already discussed in Section 4, several different approaches exist which can be employed to obtain the carrier DF in a simple manner. In this chapter, the different analytic models proposed for the carrier distribution function, namely the heated Maxwellian, the Cassi model, the Hasnat approach, the Reggiani model, and the model used in this work (Section 4.3), are compared. The applicability of these models for describing hot-carrier degradation in nLDMOS devices is verified. For reference, the carrier distribution functions are evaluated from the direct solution of the Boltzmann transport equation using the spherical harmonics expansion method. The simulation framework is the same as in Section 5.2 comprising of ViennaSHE for SHE simulations, ViennaMesh for generating the meshes, Sentaurus Process simulator for process simulations, and Minimos-NT for device simulations. The DFs obtained from the different models are used in the HCD model (Section 4.4) 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 shifts. These degradation curves are compared with experimental data and a conclusion on the validity of each model is drawn. Since all models use the same parameter set, the differences in the results can be directly traced back to inaccuracies in the approximation of the DF.

6.1 The Maxwellian Model

The heated Maxwellian distribution is a frequently used approach to mimic the DFs of non-equilibrium carriers [28]:

(6.1) \begin{equation} f(\varepsilon )=A\mathrm {exp}(-\varepsilon /k_{\mathrm {B}}T_{n}).\label {eq:Maxwellian} \end{equation}

An example of the DFs obtained using the Maxwellian function for different regions of the nLDMOS device (described in Section 5.1) is shown in Figure 6.1. One can see that the DFs evaluated with the heated Maxwellian approach are close to equilibrium in the channel and drain regions of the device, while in the bird’s beak region, the carriers are rather hot and the corresponding DFs are severely non-equilibrium.

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Figure 6.1: The electron DFs (coupled with the density of states to show the carrier occupancy) in nLDMOS obtained with the heated Maxwellian approach compared with those simulated with ViennaSHE for \( V_{\mathrm {gs}} \) = 2 \( \, \)V and for \( V_{\mathrm {ds}} \) = 18 \( \, \)V calculated at the drain, bird’s beak and channel regions. For comparison, the DFs evaluated with the DD-based model in Section 4.3 (light grey lines) are also plotted.