5.1.1 Anisotropic Distribution Function

In the current equation (2.76) a temperature tensor occurs. The diagonal components of this tensor are defined as

$$T_{xx} = \frac{1}{\mathrm{k}_\mathrm{B}} \, \frac{1}{n} \, \ensuremath{\langle v_x \, p_x \rangle}\ ,$$ (5.1)

$$T_{yy} = \frac{1}{\mathrm{k}_\mathrm{B}} \, \frac{1}{n} \, \ensuremath{\langle v_y \, p_y \rangle}\ ,$$ (5.2)

$$T_{zz} = \frac{1}{\mathrm{k}_\mathrm{B}} \, \frac{1}{n} \, \ensuremath{\langle v_z \, p_z \rangle}\ ,$$ (5.3)

and the temperature $T_n$ used in the energy transport model is defined through the mean carrier energy (cf. eqn. (2.94))

$$T_n = \frac{2}{3} \, \frac{1}{\mathrm{k}_\mathrm{B}} \, \frac{1}{n} \, \ensuremath{\langle \mathcal{E} \rangle}\ .$$ (5.4)

For parabolic bands the relation

$$T_n = \frac{1}{3} \, \bigl( T_{xx} + T_{yy} + T_{zz} \bigr)$$ (5.5)

holds.

Monte Carlo simulations of a one-dimensional $n^+$-$n$-$n^+$ test structure show that the temperature components parallel ($T_{xx}$) and normal ($T_{yy}$) to the direction of the current flow are quite different (Fig. 5.4). In particular, the transverse temperature component is smaller than the longitudinal component [64, Fig.7]. This encourages the derivation of a transport model which takes an anisotropic distribution function into account.

Figure 5.4: Components of the temperature tensor compared to the temperature $T_n$ from the mean energy obtained by Monte Carlo simulations.

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