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2.1 MAXWELL's Equations

MAXWELL's equations determine a relation among the electric and magnetic field, the space charge density and the current density. They read [4, p.31]

$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\ensuremath{\times}\ensuremath{\boldsymbol{\mathrm{E}}}}$ $\displaystyle = - \ensuremath{\partial_{t} \, \ensuremath{\boldsymbol{\mathrm{B}}}}\ ,$ (2.1)
$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\ensuremath{\cdot}\ensuremath{\boldsymbol{\mathrm{B}}}}$ $\displaystyle = 0 \ ,$ (2.2)
$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\ensuremath{\times}\ensuremath{\boldsymbol{\mathrm{H}}}}$ $\displaystyle = \ensuremath{\boldsymbol{\mathrm{J}}}+ \ensuremath{\partial_{t} \, \ensuremath{\boldsymbol{\mathrm{D}}}}\ ,$ (2.3)
$\displaystyle \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}}\ensuremath{\cdot}\ensuremath{\boldsymbol{\mathrm{D}}}}$ $\displaystyle = \varrho \ ,$ (2.4)

where $ \ensuremath{\boldsymbol{\mathrm{E}}}$ and $ \ensuremath{\boldsymbol{\mathrm{D}}}$ are the electric field and displacement vectors, respectively, $ \ensuremath{\boldsymbol{\mathrm{H}}}$ and $ \ensuremath{\boldsymbol{\mathrm{B}}}$ are the magnetic field and induction vectors, respectively, $ \varrho$ is the total electric charge density, and $ \ensuremath{\boldsymbol{\mathrm{J}}}$ the conduction current density.

Eqns. (2.1) and (2.2) and eqns. (2.3) and (2.4) each build a subsystem of their own. They are linked together by the relations

$\displaystyle \ensuremath{\boldsymbol{\mathrm{D}}}$ $\displaystyle = \varepsilon \, \ensuremath{\boldsymbol{\mathrm{E}}}\ ,$ (2.5)
$\displaystyle \ensuremath{\boldsymbol{\mathrm{B}}}$ $\displaystyle = \mu \, \ensuremath{\boldsymbol{\mathrm{H}}}\ ,$ (2.6)

where $ \varepsilon$ and $ \mu$ are the permittivity and permeability, respectively. Eqn. (2.5) and eqn. (2.6) are valid only in materials where no piezoelectric or ferroelectric phenomena occur. In semiconductors the frequency dependence of $ \varepsilon$ and $ \mu$ can be neglected.


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