The models which consider the thermal volume expansion have similar problems
with the material structure as the models for the electrical and electrical
conductivities. However, the thermal volume expansion coefficient is estimated
with an equation given for bulk material as
A rigorous mechanical investigation would
require the appropriate volume expansion coefficients for the
contributing materials. Those parameters can be either obtained from rigorous
Monte Carlo calculations or from measurements.
The coefficients given in Tab. 3.4 were obtained from
measurements of bulk materials and give sufficient approximations.
The microelectronic devices considered in this thesis show a slightly different
behavior because the volume expansion described in (3.22) is mechanically
constricted by additional material layers.
The mechanical equation system is applied to materials which are embedded in
rather stiff bulk material which normally do not allow expansion. This results
in mechanically highly stressed material regions, especially if high temperature
gradients are present.
The thinner the material layers are the more sensitive they are and tend to
relax by cracking. A crack in a layer often results in a fatal failure of the
complete system, which is assumed to be the worst case.
To avoid this type of failure, an appropriate investigation is required which
also includes the mechanical subsystem.
If a block consisting of different materials tends to
expand it is limited in its movement by the surrounding materials
in which it is embedded. Hence, mechanical stress develops according to the
values given in Tab. 3.4 for the thermal volume expansion coefficients.
The relation (2.110) considers expansion as well as contraction and torsion,
thus, all phenomena where atoms are moved.
However, if movements of atoms or atom clusters have to be considered, the mesh
for those regions is very critical and can cause tremendous problems. For
instance, in a region where a layer is contracted due to surrounded materials
with a higher stiffness, the mesh is contracted as well. Due to limited
numerical precision or approximations made in the material models, a single mesh
point might move through a mesh edge between two other mesh points.
This results in negative volumes and negative coupling coefficients in the finite
element method and causes numerical problems, because the resulting system
matrices are no longer positive definite and the solver have commonly severe
convergence problems or even fails.