Computer-aided design and analysis of circuits and semiconductor – Nanotechnology related integrated circuit design
Reexamination Certificate
2001-06-25
2003-12-16
Niebling, John F. (Department: 2812)
Computer-aided design and analysis of circuits and semiconductor
Nanotechnology related integrated circuit design
C703S001000, C703S007000
Reexamination Certificate
active
06665849
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates to a method and apparatus for simulating fields especially electromagnetic fields, particularly useful in the context of analysis of interconnect structures, is presented.
BACKGROUND OF THE INVENTION
Many problems in engineering, physics and chemistry require solving systems of partial differential equations of the type:
∇
→
·
J
→
(
k
)
+
∂
ρ
(
k
)
∂
t
=
S
(
k
)
;
k being a positive whole number
In this equation (equation 1), J represents a flux of a substance under consideration whose density is given by &rgr; and S represents some external source/sink of the substance. To mention a few examples:
Electrical engineering:
&rgr; is the charge density,
J is the current density,
S is the external charge source (recombination, generation, . . . )
Structural engineering
Computational Fluid Dynamics:
&rgr;
(i)
=u
(i)
the components of the fluid velocity field,
J
(
i
)
=
P
-
μ
3
∇
·
u
(
i
)
the
pressure
tensor
,
S
(
i
)
=
μ
ρ
∇
2
u
(
i
)
+
F
(
i
)
m
the
external
force
,
∂
ρ
∂
t
->
∂
ρ
∂
t
+
∇
->
·
u
->
the
convective
derivative
.
The list is not exhaustive and there exist many examples where problems are reformulated in such a form that their appearance is as in equation (1). An important example is the Laplace equation and Poisson equation, in which
{right arrow over (J)}={right arrow over (∇)}&psgr;
is the derivative of a scalar field.
There have been presented a number of methods for solving a set of partial differential equations as given above. All numerical methods start from representing the continuous problem by a discrete problem on a finite set of representative nodes in the domain where one is interested in the solution. In other words a mesh is generated in a predetermined domain. The domain can be almost anything ranging from at least a part of or a cross-section of a car to at least a part of or a cross-section of a semiconductor device. For clarification purposes, the discussion is limited here to two-dimensional domains and two-dimensional meshes. This mesh comprises nodes and lines connecting these nodes. As a result, the domain is divided into two-dimensional elements. The shape of the elements depends amongst others on the coordinate system that is chosen. If for example a Cartesian coordinate system is chosen, the two-dimensional elements are e.g. rectangles or triangles. Using such a mesh, the domain can be introduced in a computer aided design environment for optimization purposes. Concerning the mesh, one of the issues is to perform the optimization using the appropriate amount of nodes at the appropriate location. There is a minimum amount of nodes required in order to ensure that the optimization process leads to the right solution at least within predetermined error margins. On the other hand, if the total amount of nodes increases, the complexity increases and the optimization process slows down or even can fail. Because at the start of the optimization process, the (initial) mesh usually thus not comprise the appropriate amount of nodes, additional nodes have to be created or nodes have to be removed. Adding nodes is called mesh refinement whereas removing nodes is called mesh coarsening. Four methods are discussed. As stated above, for clarification and simplification purposes the ‘language’ of two dimensions is used, but all statements have a translation to three or more dimensions.
The finite-difference method is the most straightforward method for putting a set of partial differential equations on a mesh. One divides the coordinate axes into a set of intervals and a mesh is constructed by all coordinate points and replaces the partial derivatives by finite differences. The method has the advantage that it is easy to program, due to the regularity of the mesh. The disadvantage is that during mesh refinement many spurious additional nodes are generated in regions where no mesh refinement is needed.
The finite-box method, as e.g. in A. F. Franz, G. A. Franz, S. Selberherr, C. Ringhofer and P. Markowich “Finite Boxes-A Generalization of the Finite-Difference Method Suitable for Semiconductor Device Simulation” IEEE Trans. on Elec. Dev. ED-30, 1070 (1983), is an improvement of the finite-difference method, in the sense that not all mesh lines need to terminate at the domain boundary. The mesh lines may end at a side of a mesh line such that the mesh consists of a collection of boxes, i.e. the elements. However, numerical stability requires that at most one mesh line may terminate at the side of a box. Therefore mesh refinement still generates a number of spurious points. The issue of the numerical stability can be traced to the five-point finite difference rule that is furthermore exploited during the refinement.
The finite-element method is a very popular method because of its high flexibility to cover domains of arbitrary shapes with triangles. The choice in favor of triangles is motivated by the fact that each triangle has three nodes and with three points one can parameterize an arbitrary linear function of two variables, i.e. over the element the solution is written as
&psgr;(
x,y
)=
a+b.x+c.y
In three dimensions one needs four points, i.e. the triangle becomes a tetrahedron. The assembling strategy is also element by element. Sometimes for CPU time saving reasons, one performs a geometrical preprocessing such that the assembling is done link-wise, but this does not effect the element-by-element discretization and assembling. The disadvantage is that programming requires a lot of work in order to allow for submission of arbitrary complicated domains. Furthermore, adaptive meshing is possible but obtuse triangles are easily generated and one must include algorithms to repair these deficiencies, since numerical stability and numerical correctness suffers from obtuse triangles. As a consequence, mesh refinement and in particular adaptive meshing, generates in general spurious nodes.
The finite-element method is not restricted to triangles in a plane. Rectangles (and cubes in three dimensions) have become popular. However, the trial functions are always selected in such a way that a unique value is obtained on the interface. This restriction makes sense for representing scalar functions &psgr;(x,y) on a plane.
In the box-integration method, each node is associated with an area (volume) being determined by the nodes located at the closest distance from this node or in other words, the closest neighbouring node in each direction. Next, the flux divergence equation is converted into an integral equation and using Gauss theorem, the flux integral of the surface of each volume is set equal to the volume integral at the right hand side of the equation, i.e. equation 1 becomes
∫
∂
Ω
n
J
->
(
i
)
·
ⅆ
->
s
=
∫
Ω
n
(
S
(
i
)
-
∂
ρ
(
i
)
∂
t
)
ⅆ
n
x
The assembling is done node-wise, i.e. for each node the surface integral is decomposed into contributions to neighboring nodes and the volume integral at the right-hand side is approximated by the volume times the nodal value. The spatial discretization of the equation then becomes
∑
k
J
1
k
∂
Ω
1
k
h
1
k
=
(
S
1
(
i
)
-
∂
ρ
l
(
i
)
∂
t
)
Δ
Ω
1
The advantages/disadvantages of the method are similar as for the Finite element method because the control volumes and the finite elements are conjugate or dual meshes. Voronoi tessellation with the Delaunay algorithm is often exploited to generate the control volumes.
However, forming and refining the mesh
Magnus Wim
Meuris Peter
Schoenmaker Wim
Interuniversitair Microelektronica Centrum vzw
Knobbe Martens Olson & Bear LLP
Niebling John F.
Whitmore Stacy A
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