6.2.2 Bulk Case

By neglecting in the six moments transport model (eqns. (2.178) to (2.183)) all terms containing derivatives, equations for the homogeneous case are obtained

$$\ensuremath{\boldsymbol{\mathrm{J}}}_n = \mathrm{q}\, \mu_n \, \ensuremath{\boldsymbol{\mathrm{E}}}\, n \ ,$$ (6.4)

$$\ensuremath{\boldsymbol{\mathrm{S}}}_n = - \frac{5}{2} \, \mathrm{k}_\mathrm{B}\, \frac{\tau_S} {\tau_m} \, \mu_n \, \ensuremath{\boldsymbol{\mathrm{E}}}\, n \, T_n \ ,$$ (6.5)

$$= \ensuremath{\boldsymbol{\mathrm{E}}}\ensuremath{\cdot}\ensurema... ...2} \, \mathrm{k}_\mathrm{B}\, n \frac{T_n - T_\mathrm{L}}{\tau_\mathcal{E}} \ ,$$ 0 (6.6)

$$= - 2 \, \mathrm{q}\, \ensuremath{\boldsymbol{\mathrm{E}}}\ensure... ...}_\mathrm{B}^2 \, n \, \frac{T_n^2 \, \beta_n - T_\mathrm{L}^2}{\tau_\beta} \ .$$ 0 (6.7)

Inserting eqn. (6.4) into eqn. (6.6) and eqn. (6.5) into eqn. (6.7) yields the equations

$$\mathrm{q}\, \mu_n \, E^2 = \frac{3}{2} \, \mathrm{k}_\mathrm{B}\, \frac{T_n - T_\mathrm{L}}{\tau_\mathcal{E}} \ ,$$ (6.8)

$$\mathrm{q}\, \frac{\tau_S}{\tau_m} \, \mu_n \, E^2 \, T_n = \frac{3}{4} \, \mathrm{k}_\mathrm{B}\, \frac{T_n^2 \, \beta_n - T_\mathrm{L}^2}{\tau_\beta} \ ,$$ (6.9)

which can further be reduced to a bulk relation for the non-MAXWELLian parameter

$$\beta_{n, \mathrm{bulk}} = \frac{T_\mathrm{L}^2}{T_n^2} + 2 \, \f... ...tau_\beta}{\tau_\mathcal{E}} \, \Bigl( 1 - \frac{T_\mathrm{L}} {T_n} \Bigr) \ .$$ (6.10)

Unfortunately, the models for the relaxation times are not accurate enough to obtain a realistic estimate for $\beta_{n, \mathrm{bulk}}$. Therefore the following fit to Monte Carlo data has been developed

$$2 \, \frac{\tau_S}{\tau_m} \, \frac{\tau_\beta}{\tau_\mathcal{E}}... ... x_1 \Bigl( 1 - \exp \Bigl( - x_2 \, \frac{T_\mathrm{L}}{T_n} \Bigr) \Bigr) \ ,$$ (6.11)

with $x_0 = 0.69$, $x_1 = 1.34$, and $x_2 = 1.89$. This expression is accurate for doping concentrations around $10^{18} \, \mathrm{cm}^{-3}$ but the doping dependence of $\beta_{n, \mathrm{bulk}}$ is only relevant at lower temperatures (Fig. 6.3).

Figure 6.3: Kurtosis $\beta _n$ as a function of the temperature $T_n$ for bulk silicon with the doping concentration as a parameter together with the analytical expression for $\beta_{n, \mathrm{bulk}}$.

However, the presented formulation for $\beta_{n, \mathrm{bulk}}$ did not solve the problem of anomalous SOI simulation results--presumably because the decrease of the kurtosis with increasing temperature occurs too slowly.

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF