For non-degenerate semiconductors the FERMI energy is located below the
conduction band edge. Therefore,
holds in expression
(3.13) and the FERMI-DIRAC distribution (3.15) can be
approximated by a MAXWELL-BOLTZMANN (or MAXWELLian) distribution
![$\displaystyle f({\mathcal{E}}) = \exp \left( \displaystyle \frac{\ensuremath{{\mathcal{E}}_\mathrm{f}}- {\mathcal{E}}}{{\mathrm{k_B}}T} \right) \ .$](img312.png) |
(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,
in (3.14) becomes
![$\displaystyle \xi_i=\displaystyle\int_0^\infty f_i({\mathcal{E}}) \,\ensuremath...
...{k_B}}T}\right)\, \ensuremath {\mathrm{d}}{\mathcal{E}}_\rho \qquad\qquad i=1,2$](img316.png) |
(3.21) |
which evaluates to
![$\displaystyle \xi_i = \displaystyle {\mathrm{k_B}}T \exp \left( -\frac{{\mathca...
...math{{\mathcal{E}}_{\mathrm{f},i}}}{{\mathrm{k_B}}T} \right) \qquad\qquad i=1,2$](img317.png) |
(3.22) |
and yields a supply function of
![$\displaystyle N({\mathcal{E}}_x) = {\mathrm{k_B}}T \left( \exp \left( -\frac{{\...
...- \ensuremath{{\mathcal{E}}_\mathrm{f,2}}}{{\mathrm{k_B}}T} \right) \right) \ .$](img318.png) |
(3.23) |
A. Gehring: Simulation of Tunneling in Semiconductor Devices