Simulation method

Details Simulation method Simulation method

1999-05-05

2003-08-12

Lim, Krisna (Department: 2123)

Data processing: structural design, modeling, simulation, and em

Electrical analog simulator

Of physical phenomenon

C703S002000, C702S057000, C702S075000

Reexamination Certificate

active

06606586

ABSTRACT:

BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a simulation method for obtaining a physical quantity such as a potential inside an electrical element such as a semiconductor device, or parasitic element constants such as capacitance, resistance and inductance by numerical analysis.
2. Description of the Background Art
As an electrical element such as a semiconductor device becomes smaller, an effect that a parasitic element produces on the electrical element becomes relatively greater, so a quantitative grasp on the effect is required. It is generally difficult to obtain the parasitic element constant which quantitatively indicates the effect of the parasitic element by measurement of a practical device, and therefore the parasitic element constant is obtained by a simulation of numerical analysis with a computer.
Through this simulation, a numerical analysis is performed on a physical quantity (e.g., potential, temperature and the like) in a given position inside the electrical element, accompanying computation of the parasitic element constant. As a method of numerical analysis, the FEM (finite element method), the FDM (finite difference method) or the BEM (boundary element method) is often used.
Discussion will be made below on an exemplary case where a value of parasitic capacitance of a conductor in a two-dimensional analysis region is numerically analyzed by the finite difference method using the control volume method.
An direct object of the finite difference method is to obtain a physical quantity in a point of intersection (referred to as “node”) of mesh (hereinafter, for simple discussion, a potential is adopted as a specific example of the physical quantity, and discussion is based thereon unless otherwise specified).
FIG. 9
illustrates a structure of a region to be analyzed. This figure shows a partial region VP of an electrical element and the region VP consists of a conductor region CR
1
and a dielectric region DR
1
having a dielectric constant of ∈. The region VP is divided into meshes on expression (the conductor region CR
1
is not divided into meshes since uniform potential exists in an electrostatic field inside the conductor region CR
1
). The conductor region CR
1
has material boundaries C
1
and C
2
and is separated thereby from others. The dielectric region DR
1
has material boundaries C
2
and C
3
and analysis boundaries D
1
an D
2
and is separated thereby from others. In the material boundary C
3
, the dielectric region DR
1
adjoins a not-shown conductor region. Herein, the analysis boundary refers to a limit of an analyzable region and the material boundary refers to a boundary where different materials adjoin.
It is assumed in
FIG. 9
that for example, the potentials at the nodes on the material boundaries C
2
and C
3
and the normal components of electric fields (derivative of potential to position) relative to interfaces (hereinafter, referred to as “normal differential of potential”) at the nodes on the analysis boundaries D
1
and D
2
are given as boundary conditions. A node having an unknown quantity to be obtained is represented by a blank circle (◯), a node given a value of potential as a boundary condition is represented by a solid circle (&Circlesolid;) and a node given a value of normal differential of potential as a boundary condition is represented by a double circle ((⊚). In a case of
FIG. 9
, the unknown quantity to be obtained is a potential at the node represented by ◯ and there are 36 nodes of ◯ in the whole region VP. Values of potentials at the 36 points should be obtained under the above boundary conditions.
From the potential distribution thus obtained, the amount of electric charges that the conductor region CR
1
and other not-shown conductor regions have under the respective boundary conditions is calculated, to obtain the parasitic capacitance of the conductor through numerical analysis.
As another exemplary case for obtaining the parasitic capacitance of the conductor in the two-dimensional analysis region, numeric analysis by the boundary element method will be discussed below.
FIG. 10
illustrates a structure of a region to be analyzed. This figure shows the same region VP as
FIG. 9
by different expression. In the boundary element method, insince the boundary is expressed like a polygonal line and an object is to obtain the potential or its normal differential at each segment line (hereinafter, referred to as “boundary element”), the inside of the dielectric region DR
1
is not expressed in a form of mesh. In
FIG. 10
, the boundary elements are represented by the nodes (&Circlesolid;, ⊚) which are points of intersection of the boundary elements, like in FIG.
9
.
In the boundary element method, at each boundary element, the value of either one of the potential and its normal differential is given as a boundary condition and the value of the other is the unknown quantity to be obtained. Therefore, in the case of
FIG. 10
, the potential is unknown at the node ⊚ and the normal differential of potential is unknown at the node &Circlesolid;, and there are 35 unknown quantities in the whole region VP.
Next, a method for obtaining an unknown quantity in this region VP and its theoretical ground will be discussed. Hereinafter, a region to be analyzed (referred to as “analysis space”) is represented by V and a close boundary existing inside the analysis space V is represented by S. When the analysis space V is assumed to be an electrical element, the inside of the boundary S is conductive and the outside is dielectric.
From the Gauss' law,
div {right arrow over (D)}=&rgr;
  (1)
is true. Further, from the definition of electric field and dielectric flux density,
{right arrow over (D)}=∈{right arrow over (E)}=−∈ grad &psgr;
  (2)
is also true where {right arrow over (D)}, &rgr;, ∈, {right arrow over (E)} and &psgr; represent a dielectric flux density vector, a charge density, a dielectric constant, an electric field vector and a potential, respectively. An arrow above the reference sign indicates a vector sign.
Substituting Eq. 2 into Eq. 1, the Poisson's equation as expressed by
div
⁡
(
grad
⁢
 
⁢
ψ
)
=
Δψ
=
-
ρ
ϵ
(
3
)
is obtained. If &rgr;=0. Eq.3 is the Laplace's equation.
The Green's function G({right arrow over (x)}, {right arrow over (x′)}) is defined as a function satisfying
&Dgr;
G
(
{right arrow over (x)},{right arrow over (x
′)})=−&dgr;(
{right arrow over (x)}−{right arrow over (x
′)})  (4)
G
(
{right arrow over (x)},{right arrow over (x
′)})=0 (where |
x
′|→∝)  (5)
where {right arrow over (x)}, {right arrow over (x′)}, &Dgr; and &dgr; represent a position vector for a given position in the analysis space V, a position vector for a given point on the boundary S in the analysis space V, Laplacian with respect to {right arrow over (x)} and Dirac's delta function, respectively. The Green's function G({right arrow over (x)}, {right arrow over (x′)}) is a solution to the boundary problem with respect to {right arrow over (x)} when the unit charge is put on the position of {right arrow over (x′)}, from physical interpretation.
The Dirac's delta function &dgr; satisfies, from its definition,
∫
v
dV
′&psgr;(
{right arrow over (x′)})&dgr;(
{right arrow over (x)}−{right arrow over (x
′)})=&psgr;(
{right arrow over (x)}
)  (6)
where dV′ represents a variable of integration in the analysis space V.
Spatial dimension for {right arrow over (x)} and {right arrow over (x′)} is arbitrary. It is known that for example, in a case of two dimensions, the Green's function G({right arrow over (x)}, {right arrow over (x′)}) is expressed as
G
⁡
(
x
→
,
x
′
→
)
=
-
1
2
⁢
π
⁢
ln
⁢
&L

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