4.1.2 Poisson Equation

The Gauß's law for magnetism (4.4) is satisfied by introduction of the vector potential $ \ensuremath{\mathitbf{A}}$ as:

$\displaystyle \ensuremath{\mathitbf{B}} = \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\times} \ensuremath{\mathitbf{A}}$ (4.5)

Inserting (4.5) into (4.2) yields

$\displaystyle \ensuremath{\mathitbf{E}} = - \frac{\partial \ensuremath{\mathitb...
...}{\partial t} - \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}} \psi.$ (4.6)

Substituting this into the relation of the electric displacement and the electric field

$\displaystyle \ensuremath{\mathitbf{D}} = \ensuremath{\epsilon}\cdot\ensuremath{\mathitbf{E}}$ (4.7)

results into

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremat...
...math{\ensuremath{\ensuremath{\mathitbf{\nabla}}}} \psi) = - \ensuremath{\rho}.$ (4.8)

The permittivity is considered to be homogenous, therefore the first term in (4.8) is zero due to the definition of $ \ensuremath{\mathitbf{A}}$ (4.5), so the conventional form of the Poisson equation is obtained:

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremat...
...math{\ensuremath{\ensuremath{\mathitbf{\nabla}}}} \psi) = - \ensuremath{\rho}.$ (4.9)

The space charge density $ \ensuremath{\rho}$ can be expressed as the product of the elementary charge $ \mathrm{q}$ and the sum of the electron $ n$ and hole $ p$ concentrations and the net concentration of ionized dopants $ C_\ensuremath{\mathrm{net}}$:

$\displaystyle \ensuremath{\rho}= \mathrm{q}\cdot(n - p - C_\ensuremath{\mathrm{net}}).$ (4.10)

Substituting (4.10) into (4.9) gives:

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremat...
...hitbf{\nabla}}}} \psi) = \mathrm{q}\cdot(n - p - C_\ensuremath{\mathrm{net}}).$ (4.11)

S. Vitanov: Simulation of High Electron Mobility Transistors