4.1.3 Continuity Equations

In order to obtain the continuity equations from the first Maxwell equation (4.1) the current density $ \ensuremath{\mathitbf{J}}$ has to be split in an electron and hole component $ \ensuremath{\mathitbf{J_\ensuremath{\mathrm{n}}}}$ and $ \ensuremath{\mathitbf{J_\ensuremath{\mathrm{p}}}}$. Assuming the net concentration of ionized dopants $ C_\ensuremath{\mathrm{net}}$ is time-invariant the following equation is obtained:

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremat...
...}_\ensuremath{\mathrm{p}}) + \mathrm{q}\cdot\frac{\partial}{\partial t}(p-n)=0.$ (4.12)

(4.12) can be expressed in two different ways with the help of the quantity $ R$:
$\displaystyle \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremat...
...itbf{J}}_\ensuremath{\mathrm{n}} - \mathrm{q}\cdot\frac{\partial n}{\partial t}$ $\displaystyle =$ $\displaystyle \mathrm{q}\cdot R,$ (4.13)
$\displaystyle \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremat...
...itbf{J}}_\ensuremath{\mathrm{p}} + \mathrm{q}\cdot\frac{\partial p}{\partial t}$ $\displaystyle =$ $\displaystyle -\mathrm{q}\cdot R.$ (4.14)

This formulation gives $ R$ the meaning of the net generation or recombination of electrons and holes. As such, it can be modeled for the respective recombination/generation mechanisms, thereby (4.13) and (4.14) can be considered as two equations.

S. Vitanov: Simulation of High Electron Mobility Transistors