Certain elements of integrated circuits like busses or memory cells make use of periodic structures
[116,117].
As example a part of an interconnect bus is shown in Fig. <6.22>. Often the capacitance
between the wires must be extracted which requires the calculation of the electric field, where the
wires are connected alternately to 0
V and to
V [118]. An appropriately fine
resolution of the simulation area is important for the accuracy and leads unfortunately for such domains
as in Fig. <6.22> to the generation of a huge number of nodes. Therefore the simulation
process will demand a lot of memory, and the simulation duration can be unacceptably long. Considering
the periodicity of the structure of Fig. <6.22> in direction of the x axis there is a
smart way to solve the problem by investigating only one geometrical period of the structure. A
possible geometrical period is the area shown in Fig. <6.23>. The electrodes for
the capacitance extraction consist of one interconnect line connected to 1 V and two parts of the
neighbor lines connected to 0 V, respectively, which can be seen in the top view in
Fig. <6.24>. The interconnect bus of Fig. <6.22> can be created
by periodic spatial continuation of the area of Fig. <6.23> along the x axis.
Therefore it is not necessary to simulate the whole area of Fig. <6.22>. It is sufficient
to simulate a single cell of periodicity as in Fig. <6.23>. The simulation has been
performed by our software Smart Analysis Programs [119].
It is based on the finite
element method using tetrahedral meshes.
The algorithm for the linear algebraic equations arising from the finite element discretization
is based on the iterative conjugate gradient method which uses an incomplete Cholesky pre-conditioning
technique to speed up the iteration process [120,121,122,123].
This application is a typical electro-static problem described by the Laplace equation (4.12)
for the electric potential
. The corresponding numerical analysis is based on scalar finite
elements and was discussed in Chapter 4.
The solution of the Laplace equation in the
dielectric is completely extracted from the data defined on the dielectric boundary. Usually the
electrodes are modeled by Dirichlet boundary conditions for the electrostatic potential and
the outer boundary of the simulation
domain by homogeneous Neumann conditions which can be implemented with the finite element method in a quite natural way.
Homogeneous Neumann conditions on a planer surface effect the field in the simulation area in such way,
as if the simulation area
would be mirrored at the respective boundary face. Such ``mirroring conditions'' can be exploited for
the simulation of symmetric structures and periodic structures which exhibit symmetry. However for
general periodic structures proper boundary conditions must be implemented which require special
treatment.
In this application two faces
and
are defined as periodic
boundary, if:
If due to the discretization
points are
created,
the electric potential
in dielectric
is approximated
as (4.13) with (4.22) by the sum
In (6.26) the unknown nodes are numbered from
to
. The Dirichlet
(known) nodes are numbered from
to
. Thus the finite element method leads to a linear equation
system for the first
coefficients
(the unknown ones with
).
In general the mesh generation software does not order the simulation nodes as in (6.26).
To implement the desired node ordering a supplemental auxiliary index array with the length
is allocated.
This additional index array is used by the assembling procedure. The first
entries of this index
array refer to the nodes in
without the nodes on the Dirichlet boundary.
The remaining entries refer to the nodes on the Dirichlet boundary (from
to
).
The additional index assignment of the simulation nodes gives advantages to the implementation of the
periodic boundary conditions. Each two corresponding points of the plains
and
get the same index in the additional index array. Thus, they are assembled
to the same row in the linear equation system. Due to the element-by-element processing of the simulation volume
each periodic point has not only its neighbor nodes but it is also connected to the neighbor nodes of the corresponding
periodic point.