This thesis deals with mechanical phenomena mainly caused by electro-thermal
stress conditions. Since the electrical burden produces heat and the heat
non-negligible volume expansion, the mechanical part has to be considered as
The basic equation used for TCAD purposes is HOOKE's2.27 law
which has been originally introduced by the words ``Ut tensio sic
vis''2.28. The corresponding
where the absolute value of the applied force
to a body is proportional to
. Here the constant
determines the stiffness
of the body.
More generally, HOOKE's law can be formulated for local quantities in a body
where the local stress tensor
is associated to the
GREEN2.29 tensor (local strain tensor)
for a given
where the proportionality factor is determined by the
and the strain is defined according to CAUCHY2.30via local displacements
is the displacement or deformation vector and
the local position.
Using the VOIGT2.31 notation [96,97,98],
the ranks of the tensors involved in (2.108) can be reduced due to
the symmetry of the material and due to the symmetry according to energy conservation
laws . Thus, the number of independent tensor entities
by material symmetry and further reduces to
independent tensor entities due to energy
Therefore, equation (2.108) can be expressed
are the vector-valued quantities for the
mechanical stress and strain in the VOIGT notation, respectively.
represents the stiffness matrix of
rank also in the VOIGT notation.
In TCAD applications of modern devices it is often sufficient to deal with
static stresses, only. In that cases, the speed of involved particles can be
The mechanical equations
have to fulfill general conservations laws [96,97] for energy,
momentum, angular momentum, and mass.
Thus, the mechanical subsystem can be described by the
local conservation laws of energy, momentum, and mass
the local energy density is denoted by
, the energy flux density by
represents the mechanical power
density. The latter equation represents the mechanical analogon of the POYNTING
Equation (2.113) is the momentum conservation equation, where
is the momentum density,
is the local force density,
the velocity of the moving particles, and
is the momentum flux density which is often called pressure tensor.
Equation (2.114) presents the local mass continuity equation, where the
specific mass density is denoted as
If mass fluxes have to be considered, for instance in electro-migration
analysis, the kinetic pressure tensor
where the specific mass density is denoted by
is the stress
is the speed of the moving particles.
can be also used as a scalar-valued quantity
when the simplified VOIGT notation is used:
If the flux of mass has not to be considered, the associated velocity of the
and the hydrostatic pressure can be determined
The definitions of hydrostatic pressure
and (2.117) can be used as a metric which provides a possibility to
visualize, to compare within measurements, or to define a figure of merit in an
optimization loop .
If moving particles are considered, the mechanical analogon to the electrical
continuity equation is the mass continuity equation (2.114) and can be
treated with the EULERian2.32 continuity
which is the mass conservation equation (2.114).
A mathematical coupling between the mass flux and the mechanical stress can be
obtain by using the first law for continuity mechanics from
represents the externally applied force density.