2.1.1 POISSON's Equation

By introducing a vector potential and a scalar potential MAXWELL's equations can often be rewritten in a more practical form. The vector potential $ \ensuremath{\boldsymbol{\mathrm{A}}}$ is defined by

$\displaystyle \ensuremath{\boldsymbol{\mathrm{B}}}= \ensuremath{\ensuremath{\en...
...l{\mathrm{\nabla}}}}\ensuremath{\times}\ensuremath{\boldsymbol{\mathrm{A}}}}\ ,$ (2.7)

which fulfills eqn. (2.2) since $ \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\ensuremath{...
...athrm{\nabla}}}}\ensuremath{\times}\ensuremath{\boldsymbol{\mathrm{A}}}}\bigr)}$ evaluates to zero for every vector field $ \ensuremath{\boldsymbol{\mathrm{A}}}$. Inserting eqn. (2.7) into eqn. (2.1) gives

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}...
...rm{\nabla}}}}\ensuremath{\times}\ensuremath{\boldsymbol{\mathrm{A}}}}\bigr)}\ .$ (2.8)

Interchanging the order of the time derivative and the curl operator,

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}...
...igl(\ensuremath{\partial_{t} \, \ensuremath{\boldsymbol{\mathrm{A}}}}\bigr)}\ ,$ (2.9)

and using the associative property of the curl operator,

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}...
...\ensuremath{\partial_{t} \, \ensuremath{\boldsymbol{\mathrm{A}}}}\bigr)}= 0 \ ,$ (2.10)

the argument of the curl operator can be substituted by the gradient of a scalar potential

$\displaystyle \ensuremath{\boldsymbol{\mathrm{E}}}+ \ensuremath{\partial_{t} \,...
... \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\, \psi}}\ ,$ (2.11)

since $ \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\ensuremath{...
...nsuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\, \psi}\bigr)}$ yields zero for every scalar field $ \psi$. The minus sign on the right hand side of eqn. (2.11) is introduced by convention based on historical reasons.

In the quasi-stationary case, which holds true for semiconductor devices2.1, the time derivative of the vector potential can be neglected

$\displaystyle \ensuremath{\boldsymbol{\mathrm{E}}}= - \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\, \psi}\ .$ (2.12)

POISSON's equation is finally obtained by inserting eqn. (2.12) into eqn. (2.5)

$\displaystyle \ensuremath{\boldsymbol{\mathrm{D}}}= - \varepsilon \, \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\, \psi}\ ,$ (2.13)

which is in turn inserted into eqn. (2.4)

$\displaystyle \boxed{\ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\n...
...{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\, \psi}}= - \varrho}\ .$ (2.14)

In the case of vanishing space charge density $ \rho$ POISSON's equation simplifies to the LAPLACE equation

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}...
...suremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\, \psi}}= 0 \ .$ (2.15)

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF