3.3.1 FERMI-DIRAC Distribution

In equilibrium the energy distribution function of electrons or holes is given by the FERMI^3.2-DIRAC^3.3 statistics

$$f({\mathcal{E}}) = \frac {1} {1 + \exp \left( \displaystyle \frac... ...thcal{E}}-\ensuremath{{\mathcal{E}}_\mathrm{f}}}{{\mathrm{k_B}}T} \right) } \ ,$$ (3.15)

which can be derived from statistical thermodynamics [100]. Separating the longitudinal and transversal energy components ${\mathcal{E}}= {\mathcal{E}}_x + {\mathcal{E}}_\rho$ and splitting the integral in (3.14) $N({\mathcal{E}}_x) = \xi_1({\mathcal{E}}_x) - \xi_2({\mathcal{E}}_x)$ the values of $\xi_1$ and $\xi_2$ become

$$\xi_i = \displaystyle \int_0^\infty f_i({\mathcal{E}}) \,\ensurem... ...}T} \right)}\,\ensuremath {\mathrm{d}}{\mathcal{E}}_\rho \qquad\qquad i=1,2 \ .$$ (3.16)

This expression can be integrated analytically using

$$\int \frac{\ensuremath {\mathrm{d}}x}{1+\exp(x)} = \ln \left( \frac{1}{1 + \exp(-x)} \right) + C \ ,$$ (3.17)

so expression (3.16) evaluates to

$$\xi_i = \displaystyle {\mathrm{k_B}}T\ln\left( 1 + \exp \left( -\... ...athcal{E}}_{\mathrm{f},i}}}{{\mathrm{k_B}}T} \right) \right) \qquad\qquad i=1,2$$ (3.18)

and the total supply function (3.14) becomes

$$N({\mathcal{E}}_x)={\mathrm{k_B}}T\ln\left( \frac{\displaystyle 1... ...- \ensuremath{{\mathcal{E}}_\mathrm{f,2}}}{{\mathrm{k_B}}T}\right)} \right) \ .$$ (3.19)

A. Gehring: Simulation of Tunneling in Semiconductor Devices