2.4.2.2 The Energy-Transport Model

Taking the first four moments of (2.2) into account yields the hydrodynamic model which, however, incorporates convective terms difficult to handle in a numerical simulator. If they are neglected, the following equation system can be derived (the expressions for holes are analogous and have been omitted)

$${\mathbf{J}}_n = \mu_n {\mathrm{k_B}}\left(\ensuremath{{\mathbf{\nabla}}}\left(n... ...ght) + \frac{\ensuremath {\mathrm{q}}}{{\mathrm{k_B}}} {\mathbf{E}} n\right)\ ,$$ (2.8)

$${\mathbf{S}}_n = - \frac{\tau_S}{\tau_m} \left( \frac{5{\mathrm{k_B}}^2}{2\ensur... ...rac{5{\mathrm{k_B}}^2}{2\ensuremath {\mathrm{q}}} T_n {\mathbf{J}}_n \right)\ ,$$ (2.9)

$$\ensuremath{{\mathbf{\nabla}}}\cdot {\mathbf{J}}_n = \ensuremath {\mathrm{q}}\left( R + \partial_t n\right)\ ,$$ (2.10)

$$\ensuremath{{\mathbf{\nabla}}}\cdot {\mathbf{S}}_n = -\frac{3}{2} {\mathrm{k_B}}\partial_t(nT_n) + {\mathbf{E}}\cdot... ...{k_B}}n \frac{{\mathrm{T}}_n - T_L}{\tau_{\mathcal{E}}} + G_{{\mathcal{E}}n}\ .$$ (2.11)

This equation system is commonly known as energy-transport model. Here, ${\mathbf{S}}$ denotes the energy flux density, $T_{n}$ the electron temperature, $\tau_{\mathcal{E}}$, $\tau_S$, and $\tau_m$ the energy, energy flux, and momentum relaxation time, and $G_{{\mathcal{E}}n}$ the net energy generation rate.

A. Gehring: Simulation of Tunneling in Semiconductor Devices