3.3.2 MAXWELL-BOLTZMANN Distribution

For non-degenerate semiconductors the FERMI energy is located below the conduction band edge. Therefore, $\ensuremath{{\mathcal{E}}_\mathrm{min}}- \ensuremath{{\mathcal{E}}_\mathrm{f}}\gg {\mathrm{k_B}}T$ holds in expression (3.13) and the FERMI-DIRAC distribution (3.15) can be approximated by a MAXWELL-BOLTZMANN (or MAXWELLian) distribution

$$f({\mathcal{E}}) = \exp \left( \displaystyle \frac{\ensuremath{{\mathcal{E}}_\mathrm{f}}- {\mathcal{E}}}{{\mathrm{k_B}}T} \right) \ .$$ (3.20)

This expression is compared to the FERMI-DIRAC distribution in Fig. 3.3. It can be seen that only for energies well above the FERMI energy the expressions deliver equal results.

Figure 3.3: Comparison of the FERMI-DIRAC and the MAXWELL-BOLTZMANN distribution on a linear scale (left) and on a logarithmic scale (right). At energies above the FERMI energy the expressions yield similar results.

Using this expression, $\xi$ in (3.14) becomes

$$\xi_i=\displaystyle\int_0^\infty f_i({\mathcal{E}}) \,\ensuremath... ...{k_B}}T}\right)\, \ensuremath {\mathrm{d}}{\mathcal{E}}_\rho \qquad\qquad i=1,2$$ (3.21)

which evaluates to

$$\xi_i = \displaystyle {\mathrm{k_B}}T \exp \left( -\frac{{\mathca... ...math{{\mathcal{E}}_{\mathrm{f},i}}}{{\mathrm{k_B}}T} \right) \qquad\qquad i=1,2$$ (3.22)

and yields a supply function of

$$N({\mathcal{E}}_x) = {\mathrm{k_B}}T \left( \exp \left( -\frac{{\... ...- \ensuremath{{\mathcal{E}}_\mathrm{f,2}}}{{\mathrm{k_B}}T} \right) \right) \ .$$ (3.23)

A. Gehring: Simulation of Tunneling in Semiconductor Devices