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

5.3 Spherical Harmonics Expansion Solution

ViennaSHE is a deterministic BTE solver which considers energy exchange mechanisms such as the electric field, impact ionization, electron-electron and electron-phonon scattering, scattering at ionized impurities, surface scattering, and full-band effects [18, 20, 200]. A cell-centered discretization scheme used in ViennaSHE simplifies the treatment of material interfaces and can be used on arbitrary grids, not requiring the Delaunay property [46]. As a result, ViennaSHE provides smooth DF curves spreading over many orders of magnitude and over several electronvolts.

Figure 5.3 shows two sets of electron DFs obtained for \( V_{\mathrm {gs}} \) = 2 \( \, \)V and \( V_{\mathrm {ds}} \) = 20 \( \, \)V and for different positions in the nLDMOS device, i.e. one set corresponds to the bird’s beak region, while the second one summarizes DFs evaluated for the drain region.

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Figure 5.3: The electron DFs obtained with ViennaSHE and with the DD-based model for \( V_{\mathrm {ds}} \) = 20 \( \, \)V, \( V_{\mathrm {gs}} \) = 2 \( \, \)V, calculated for different positions near the drain and close to the bird’s beak region of the nLDMOS transistor.

One can see that the carrier DFs are severely non-equilibrium. While at the drain DFs have a cold Maxwellian tail at low energies, the DFs calculated for the bird’s beak do not have this contribution and their shape is completely different. In all cases the DFs have extended high-energy tails. The DFs for the pLDMOS transistor obtained for \( V_{\mathrm {ds}} \) = \( - \)50 \( \, \)V, \( V_{\mathrm {gs}} \) = \( - \)1.5 \( \, \)V are shown in Figure 5.4. The DFs calculated with ViennaSHE for the STI corner and the drain region have a considerable amount of high energy carriers. These DFs are used as a benchmark for the DD-based approach.