3.3.4 Normalization

When implementing the analytical expressions for the distribution function and the supply function into a device simulator it is necessary to assure consistency: the carrier concentration defined by the analytical distribution function must match the carrier concentration from the transport model used. Therefore, the normalization prefactor $A$ has to be evaluated from

$$n = \langle 1 \rangle = \frac{1}{4\pi^3} \int f(\mathbf{k}) \, \ensuremath {\mathrm{d}}^3 k \ .$$ (3.37)

This equation can be transformed to spherical coordinates using $k = (k_x^2 + k_y^2 + k_z^2)^{1/2}$

$$n = \frac{1}{4\pi^3} \int_{-\pi}^{\pi}\,\ensuremath {\mathrm{d}}\... ...suremath {\mathrm{d}}\theta \int_0^\infty f(k)k^2\,\ensuremath {\mathrm{d}}k\ .$$ (3.38)

For a parabolic dispersion relation we have $\ensuremath {\mathrm{d}}k = \ensuremath{m_\mathrm{eff}}/k\hbar^2 \ensuremath {\mathrm{d}} {\mathcal{E}}$ which finally leads to

$$n = \int_0^\infty f({\mathcal{E}}) \frac{4\pi\sqrt{2\ensuremath{m... ...{eff}}^3}}{h^3} \sqrt{{\mathcal{E}}} \,\ensuremath {\mathrm{d}}{\mathcal{E}}\ ,$$ (3.39)

where the integration is performed from the conduction band edge $\ensuremath {{\mathcal{E}}_\mathrm{c}}=0$. For a MAXWELLian or heated MAXWELLian distribution (expressions (3.20) or (3.24)), the normalization constant evaluates to

$$A= \frac{nh^3}{\displaystyle 4\pi ({\mathrm{k_B}}T_\nu)^{3/2} \, \Gamma\left(\frac{3}{2}\right) \sqrt{2\ensuremath{m_\mathrm{eff}}^3} } \,$$ (3.40)

where $T_\nu$ is either the lattice temperature (for the assumption of a MAXWELLian distribution) or the carrier temperature (for the assumption of a heated MAXWELLian distribution). Using the non-MAXWELLian distribution (3.27) the normalization constant evaluates to

$$A = \displaystyle \frac{n h^3 b}{4\pi {\mathcal{E}_\mathrm{ref}}^... ...( \displaystyle \frac{3}{2b} \right) \sqrt{2\ensuremath{m_\mathrm{eff}}^3}} \ ,$$ (3.41)

while for expression (3.28) it is

$$A=\frac{nh^3}{\displaystyle 4\pi\left( \frac{{\mathcal{E}_\mathrm... ...Gamma\left(\frac{3}{2}\right)\right) \sqrt{2\ensuremath{m_\mathrm{eff}}^3}} \ .$$ (3.42)

A. Gehring: Simulation of Tunneling in Semiconductor Devices