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

$\displaystyle T_{xx}$ $\displaystyle = \frac{1}{\mathrm{k}_\mathrm{B}} \, \frac{1}{n} \, \ensuremath{\langle v_x \, p_x \rangle}\ ,$ (5.1)
$\displaystyle T_{yy}$ $\displaystyle = \frac{1}{\mathrm{k}_\mathrm{B}} \, \frac{1}{n} \, \ensuremath{\langle v_y \, p_y \rangle}\ ,$ (5.2)
$\displaystyle T_{zz}$ $\displaystyle = \frac{1}{\mathrm{k}_\mathrm{B}} \, \frac{1}{n} \, \ensuremath{\langle v_z \, p_z \rangle}\ ,$ (5.3)

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

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

For parabolic bands the relation

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


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