thmotiv.html

有限元学习研究用源代码(老外的),供科研人员参考

 
 <ul>
 <li> In high temperature steps we have a dopant diffusion through the
boundaries.  That's why we need the dopant concentrations also in
other regions. These concentrations are discontinuous on the
boundaries.
 
 <li> Sometimes we need also boundary concentrations of the dopants
(on the silicon surface in epitaxy)
 
 <li> We need other concentrations because the nonlinear diffusion
depends on these concentrations (vacancy and interstitial
concentration in silicon).
 
 <li> We have dopants in different states, with different diffusion
behaviour: cluster effect in silicon, grain and boundary concentration
in polysilicon.
 
 <li> For the simulation of mechanical deformation we need the
components of mechanical stress.
 
 <li> Subdivision of regions and materials may be necessary. In some
models there may be different equations in different materials which
may be considered as the same material in other models (different
subtypes of polysilicon and silicon oxide).
 
 </ul>


A special problem is that it is not a priori defined which functions
we need.  This depends on the special physical model we use. For
example, if we use a viscous model of mechanical transformation, we
don't need to save the stress from step to step. For the elastic or
viscoelastic model it is necessary to save the stress. If the reaction
between the clustered and the free dopant concentration is in
equilibrium we don't need to save different concentrations, else we
need. So, for the device description it must be easy to change the
number of functions we have to save.
  <p> There have to be trivial possibilities for the initialization of
the device.  Usually as the initial geometry a silicon region (the
substrate) and gas will be used.
  <p> There must be also possibilities to save the current state of the
device into a file after every technology step. This file can be used
to restart the technology simulation or as the input for device
simulation.
<h3><a NAME="stdeposit">Deposit</a></h3>


<b>Deposit</b> is an useful step for changing the geometry. To simulate
a deposit step we have to consider a transport of the deposit material
through the gas phase and a chemical reaction on the surface. The most
trivial model is the model of a constant growing rate. This model can
be used for a reaction-controlled deposit process. The other
possibility is the diffusion-controlled deposit process. Here the
growing rate has to be computed by a special diffusion equation in the
gas phase.
  <p> Already for the most trivial simulation --- the constant growing
rate --- we obtain topological problems. The topology can change and
really changes in the technological process. It is not a priori
defined when these changes occur --- this depends on the simulated
concentrations. So we have to compute these changes automatically.
  <p> Using the deposit step it is also possible to create very thin
layers. These thin layers are a problem especially for 3D grid
generators if the thickness of the layer is smaller than the usual
point density in this region.
<h3><a NAME="stetching">Etching</a></h3>


<b>Etching</b> is another process step used to create nontrivial
structures. There are two standard methods for these etch
processes. The older, easier method is the <b>chemical
etching</b>. Here we have usually an isotropic etch rate on the surface
which depends on the surface material.
  <p> But today usually other methods as <b>ion beam etching</b> will be
used. An ion beam strikes the surface of the device. To simulate these
methods we have to consider etch rates depending on the direction of
the ion beam. For a more detailed simulation we also have to consider
shadowing and reflection effects of the ion beam. So we obtain a more
complex space dependent etch rate. The resorbtion of the sputtered
material may lead to a local deposit process on another place. Since
the ideal etch process is a "selective etching", in all of these cases
the etch rate depends strongly on the material of the surface.
  <p> Changes of the topology are usual in this process step. Small
oscillations in the results of the simulation may lead to a big number
of very small regions.  Sometimes they may be of interest, but often
it seems better to ignore them.  So, it is sufficient (and may be
sometimes also necessary) to compute the topology only up to an
user-defined accuracy (topological smoothing).
<h3><a NAME="stimplantation">Implantation</a></h3>


The <b>dopant implantation</b> is usually not connected with a change
of the topology. It is the basic step of creating a nontrivial dopant
concentration, usually in the substrate. In this step we have to
compute a new dopant concentration. This concentration may be
nontrivial on places with trivial dopant concentration before. Since
the dopant concentration is the most interesting output inside the
regions, local refinement usually has to be done in dependence of
these dopant concentrations. So an implementation usually is connected
with a new refinement step. Here we have to be careful with the
computation of the new concentration on new points. So it is not
possible to compute the new concentrations on the old grid and then to
interpolate these results onto the new grid. We have to compute the
"new" part of the concentration on the new grid and to add the result
to the "old" part which is interpolated from the old grid. The
resulting concentration is necessary input for the refinement
criteria, so it is necessary to organize the related data transfer in
the grid generation step.
  <p> An interesting variant which creates topological changes is the
oxygen implantation into silicon. In the parts of the silicon with
high oxygen concentrations the material properties change --- we
obtain silicon oxide. So we have to subdivide the silicon region into
a silicon part and a silicon oxide part dependent on a critical oxygen
concentration.
<h3><a NAME="stoxidation">Oxidation</a></h3>


This is a very complex process step. We have to compute here many
different subprocesses. At first we have to compute an oxygen
diffusion through the silicon oxide. Then we have an oxidation
reaction on silicon-oxide surfaces.  The growing rate on these
surfaces depends on the oxygen which is available there. So the
growing rate is space-dependent. The volume of the created oxide is
greater as the volume of the silicon used by the reaction. So we have
also a mechanical deformation. Topological changes are also possible
and have to be simulated.
  <p> Since oxidation is a "high temperature step" we also have a
redistribution of the dopant concentrations --- a nonlinear diffusion
with difficult boundary conditions on the changing boundary and a
convection defined by the mechanical deformation. In principle all
these subprocesses are coupled. So the mechanical stress and the
dopant concentration can influence the reaction rate. The reaction
rate often will be considered as isotropic, but depends in reality on
the crystal orientation of the silicon. So we have to consider also
anisotropic reaction rates. The mechanical properties of the silicon
oxide may depend on dopant concentrations.
<h3><a NAME="stepitaxy">Epitaxy</a></h3>


This step can be considered as a special deposit step. Here the
deposit material is silicon. But from point of view of simulation it
is more complex, because it is a high temperature step and we have to
compute the dopant redistribution. As in the oxidation we have to
consider a diffusion problem for the concentrations with difficult
boundary conditions on a changing boundary. We have also a special
effect --- auto-doping --- connected with a boundary concentration
which has an interaction with the concentration inside the silicon and
a high diffusion. This leads to the necessity to consider boundary
concentrations and equations for these concentrations.

<h2><a NAME="stconclusion">Conclusion</a></h2>


So, let's list the requirements for grid generation and geometry description we
get from our example application:

 
 <ul>
 <li> We have a lot of different regions (materials) and no a-priori
information about geometry and topology. Topological restrictions are
not allowed. The regions may not be convex, connected, simply
connected.
 
 <li> Topological changes are possible and have to be detected and
computed automatically.
 
 <li> Accurate computation of the boundary position is necessary to
obtain a correct velocity for the time-dependent boundaries.
 
 <li> Boundary data (boundary conditions, concentrations) have to be
handled (e.g. interpolated to the new boundary position).
 
 <li> It must be easy to create and modify the geometry description
even for a complicate geometries.
 
 <li> Grid generation must be fully automatic, and we obtain high
requirements to the stability of the grid generator.
 
 <li> Because a grid has to be created in every time step for a
changing geometry, the grid generation must be fast enough.
 
 <li> Grids have to be created not only for regions, but also for
boundaries. The grid of a region has to be connected along it's
boundaries with the related boundary grids and the region grids on the
other side.
 
 <li> Caused by the general necessity of point economy in 3D, but also
by the considerations of typical geometries, not only local, but also
anisotropic refinement is necessary.
 
 <li> Refinement is necessary even to describe the topology. That
means, to describe the geometry on a regular coarse grid before the
refinement step seems not good.
 
 <li> Interpolation and manipulation of functions on the grid must be
possible in grid generation time.
 
 <li> The problem of interpolation of functions from the old grid has
to be considered. An algorithm leading to nearly identical grids
inside the region even if the boundary changes has to be preferred.
 
 <li> To avoid negative concentrations numerical stability is very
important.  This leads to high requirements for numerical grid
quality.
 
 <li> Integrated 1D, 2D and 3D grid generation and geometry
description is useful.
 
 <li> The interfaces have to be designed very careful. They have to be
easy to use but powerful enough to allow the necessary influence of
the application.
 
 </ul>


The aim of this thesis is to find algorithms which allow to meet these
requirements.