JPH07169936A - Compatible mesh generating method - Google Patents

Abstract

PURPOSE:To generate compatible meshes by a method wherein the coefficient, corresponding to the charging rate of local impurity in the position of each mesh branch, is set and the position of mesh point is changed in such a manner that the condition of static equilibrium is satisfied by considering each mesh branch as a spring in which set coefficients is retarded as spring constant. CONSTITUTION:The initial delta mesh is generated. The coefficients K71, K73 and K78 corresponding to the difference in impurity density between the set (7 and 1), (7 and 3), (7 and 8)... of mesh point 7, for example, of mesh point sets of (i, J) and (jnot equal to i) coupled by each mesh branch are provided. Each mesh position (x1, y1) is changed in such a manner that the condition of static, prescribed by the formulas of condition SIGMAKij(Xi-Xj)=O and equilibrium SIGMAij(Xi-Xj)= O and SIGMAil(yi-yj)=O by considering each mesh branch as a spring constant is satisfied. As a result, the compatible mesh for semiconductor device simulation can be generated with a fixed number of meshes maintained.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a conforming mesh generating method, and more particularly to a conforming mesh generating method used as a device simulator.

[0002]

2. Description of the Related Art Generally, a device simulator is used as a means for calculating a physical quantity inside a semiconductor transistor using a computer and calculating electrical characteristics such as a terminal current and a threshold voltage of the transistor. By using a device simulator when optimizing the design of a transistor so that the semiconductor device exhibits the best electrical characteristics, the cost involved in the development and design of the semiconductor device can be reduced compared to the case where the semiconductor device is actually prototyped. It is possible to significantly reduce each / period. In the device simulator, by calculating the physical quantity inside the semiconductor transistor, the behavior of the electrons and holes inside the semiconductor can be investigated in a precise manner. It can also be used to clarify the cause of the impact ionization phenomenon.

In this device simulator,
In order to obtain the physical quantity inside the semiconductor transistor, partial differential equations such as Poisson's equation and current continuity equation representing the relation between the potential and the carrier concentration are solved. The method of solving this partial differential equation is described in the article "Analys" by S. Selbe-rherr.
is and Simulation of Semiconductor Device ”(S-pr
As described in inger-Verlag) pp.149-201, a method of dividing a semiconductor device into small rectangular or triangular elements (mesh) and discretizing a partial differential equation is generally used.

Generally, the calculation accuracy of the semiconductor device simulator largely depends on the method of dividing the mesh, and the accuracy is improved as the number of meshes is increased.
On the other hand, since the calculation time increases in a super-linear manner with respect to the number of meshes, a mesh generation method that generates a “matching mesh” that achieves high calculation accuracy at as few mesh points as possible has been studied in the past. There is. In this conventional mesh generation method, a method has been used in which a coarse initial mesh is generated in advance and then mesh points are successively added to necessary portions to subdivide the mesh elements. For example, the article “Adadptive” by WMCoughram.Jr.
Grid Generation for VLSI Device Sim-uration ”(IEE
E Trans. On Computer-Aided Design, vol.CAD-10, pp.
1259 to 1275, 1991), a method of subdividing a triangular element by adding mesh points on a mesh branch that connects the local impurity concentration, the rate of change, and the mesh points with large errors in the partial differential equations is proposed. Proposed. Figure 6
(A), (b) and (c) schematically show the state of this subdivision.

[0005]

In the conventional mesh generation method in the above-described semiconductor device simulation, in the above-mentioned method for generating meshes on the mesh branch connecting between the local impurity concentration, the rate of change, and the mesh points having large errors in the partial differential equation. , A method of subdividing a triangular element by adding mesh points has been proposed. However, in such a method of adding mesh points, the number of mesh points increases too much depending on the semiconductor device structure, and the number of mesh points is predetermined. There is a drawback that the result exceeds the upper limit of the number of mesh points, and the calculation time is very long.

According to a first aspect of the present invention, there is provided a adaptive mesh generation method, wherein in a semiconductor device simulation, a first processing step of generating an initial triangular mesh and a set of mesh points connected by each mesh branch. Coefficient k according to the impurity concentration difference between [i, j] (j ≠ i)
_ij is set for each mesh branch, and each mesh branch is regarded as a spring having the coefficient k _ij as a spring constant, and a static equilibrium condition defined by the following conditional expression is set. So that each mesh point position (x _i ,
It is characterized by having at least a third processing step for changing y _j ), and Σk _ij (x _i −x _j ) = 0 Σk _ij (y _i −y _j ) = 0.

In the adaptive mesh generating method of the second invention, in a semiconductor device simulation, a first processing step for generating an initial triangular mesh and a set of mesh points [i, j] connected by each mesh branch. ]
A second processing step of setting a coefficient k _ij according to the impurity concentration difference between (j ≠ i) for each mesh branch, and each mesh branch is regarded as a spring having the coefficient k _ij as a spring constant. Then, a third processing step of changing each mesh point position (x _i , y _j ) so as to satisfy the static equilibrium condition defined by the following conditional expression, and Σk _ij (x _i −x _j ) = 0 Σk _ij (y _i −y _j ) = 0 In the third processing step, the fourth processing step of connecting the mesh branches of the mesh points whose positions have been changed to generate a new triangular mesh is performed. Characterized by having at least.

[0007]

DESCRIPTION OF THE PREFERRED EMBODIMENTS Next, the present invention will be described with reference to the drawings.

FIG. 1 is a diagram showing a flow chart in the first embodiment of the present invention.

In FIG. 1, first, in process step 101, an initial triangular mesh is generated. Next, in processing step 102, a coefficient k _ij is set for each mesh branch according to the impurity concentration difference between the mesh point pairs [i, j] (j ≠ i) connected by the mesh branches. . Finally, in processing step 103, each mesh branch is regarded as a spring having k _ij as a spring constant, and each mesh point position (x _i , y _j ) is set so as to satisfy the static balance condition according to the following equation. change.

Σk _ij (x _i −x _j ) = 0 Σk _ij (y _i −y _j ) = 0 Regarding the relationship between the spring constant and the impurity concentration difference, for example, the following definition may be used. it can. k _ij = a | sinh ^-1 [(N _Di- N _Ai ) / c] -sinh ^-1 [(N _Dj -N _Aj ) / c] | + b In the above equation, N _Di and N _Ai are mesh points, respectively. i is the donor and acceptor concentration at i, a,
b and c are appropriate constants.

FIG. 2 is a diagram showing an example of a data structure for realizing the present embodiment on a computer. The data required to realize a spring model for a mesh branch on a computer requires the coordinates and impurity concentration of each mesh point, the number of mesh points that are connected to it around the mesh branch via mesh branches, and those numbers. Is the spring constant between. The coordinates of the mesh points and the impurity concentration are stored in an array using the mesh point number as a key, and the head address thereof is stored in the mesh point information array.

A specific example of these data will be described with reference to FIG. The mesh point 7 has coordinates (x ₇ ,
y ₇ ), and the donor concentration and the acceptor concentration are N _d7 and N _a7 , respectively. The mesh point 7 is connected to the surrounding mesh points 1, 3, 8, 10, and 15, and their spring constants are k ₇₁ , k ₇₃ , k ₇₄ , k ₇₈ , k ₇₁₀ , k.
_{It is 715} .

Next, a second embodiment of the present invention will be described. FIG. 3 is a diagram showing a flowchart in the embodiment. First, in processing step 301, an initial triangular mesh is generated. Then process step 302
In, the coefficient k _ij is set for each mesh branch according to the impurity concentration difference between the mesh points connected by the mesh branch. Then, in process step 303,
Each mesh branch is regarded as a spring whose spring constant is k _ij set in the processing step 302, and each mesh point position (x _i , y _j ) is satisfied so as to satisfy the above equation of the static equilibrium condition. To change. And processing step 30
In 4, the mesh branches of the mesh points whose positions have been changed in the third processing step are reconnected to generate a new triangular mesh. In this embodiment,
In the processing step 303, when the mesh position is changed to form a very flat triangle,
It is effective in eliminating them.

FIG. 4 shows an M used as an application example of the present invention.
It is the figure which showed the structure and impurity distribution of OSFET typically. As shown in FIG. 4, the MOSFET is
The source electrode 41, the oxide film 42, the gate electrode 43, the drain LDD diffusion layer 44, the drain electrode 45, the drain diffusion layer 46, the base 47, the source LDD diffusion layer 48, and the source diffusion layer 59. This MOSFET
In contrast, the results obtained by applying the present invention are shown in FIG. FIG. 5 shows a MOSFE according to the present invention.
FIG. 6 is a diagram showing a triangular mesh corresponding to T, and as is clear from the comparison between FIGS. 4 and 5, the spring constant increases near the PN junction surface of the source electrode 41, the drain diffusion layer 46, and the substrate 47. You can see that the mesh is dense.

FIG. 7 is a diagram showing the results of comparing the drain current of the MOSFET in the weak inversion region in the case of using the present invention with the case of using the uniform mesh of the same number of points. As is clear from FIG. 7, when a uniform mesh is used (characteristics shown as 72 in FIG. 7), the current value converges slowly with an increase in the number of meshes, and the fluctuation characteristics of the current are oscillating. In contrast, in the case of the present invention (the characteristic shown as 71 in FIG. 7), the current value quickly converges with an increase in the number of meshes, and is clearly compatible with the case of a uniform mesh. It turns out that it has.

As is clear from the above description, according to the present invention, a coefficient corresponding to the local change rate of the impurity concentration at that location is set for each mesh branch, as compared with the conventional mesh generation method. However, it is characterized in that each mesh branch is regarded as a spring having these coefficients as spring constants and the mesh point positions are changed so as to satisfy the static equilibrium condition. , Set the coefficient corresponding to the local change rate of the impurity concentration at that place, and regard each mesh branch as a spring having these coefficients as spring constants, and set the mesh point position so as to satisfy the static equilibrium condition. Along with the change, the mesh branches of the mesh points whose positions have been changed are reconnected to generate a new triangular mesh.

[0017]

As described above, the present invention is applied to the mesh generation method corresponding to a semiconductor device, and at least corresponds to the change rate of the local impurity concentration at each location of each mesh branch. By setting the specified coefficients and changing each mesh branch position so that each mesh branch is regarded as a spring having these coefficients as spring constants and the static equilibrium condition is satisfied, the number of meshes is kept constant. In this state, there is an effect that a suitable mesh for semiconductor device simulation can be generated.

[Brief description of drawings]

FIG. 1 is a diagram showing a flowchart in a first embodiment of the present invention.

FIG. 2 is a diagram showing a data structure in the first embodiment.

FIG. 3 is a diagram showing a flowchart in a second embodiment of the present invention.

FIG. 4 is a schematic diagram of a structure and impurities of a MOSFET according to an application example of the present invention.

FIG. 5 is a diagram showing an example of a conforming mesh of a MOSFET obtained by applying the present invention.

FIG. 6 is a schematic diagram showing subdivision of a triangular mesh element by a conventional method.

FIG. 7 is a diagram showing the mesh number dependence of the drain current value in the weak inversion region of the MOSFET.

[Explanation of symbols]

 41 Source Electrode 42 Oxide Film 43 Gate Electrode 44 Drain LDD Diffusion Layer 45 Drain Electrode 46 Drain Diffusion Layer 47 Substrate 48 Source LDD Diffusion Layer 49 Source Diffusion Layer 101-103, 301-304 Processing Steps

Claims (2)

[Claims] 1. In a semiconductor device simulation, a first processing step for generating an initial triangular mesh, and a set of mesh points [i,
j] (j ≠ i), the coefficient k _ij according to the impurity concentration difference is
A second processing step set for each mesh branch, and each mesh branch is regarded as a spring having the coefficient k _ij as a spring constant, and a static equilibrium condition defined by the following conditional expression is satisfied. , Each mesh point position (x _i , y _j )
And a third processing step of changing Σk _ij (x _i −x _j ) = 0, and Σk _ij (y _i −y _j ) = 0. 2. In a semiconductor device simulation, a first processing step for generating an initial triangular mesh, and a set of mesh points [i,
j] (j ≠ i), the coefficient k _ij according to the impurity concentration difference is
A second processing step set for each mesh branch, and each mesh branch is regarded as a spring having the coefficient k _ij as a spring constant, and a static equilibrium condition defined by the following conditional expression is satisfied. , Each mesh point position (x _i , y _j )
And Σk _ij (x _i −x _j ) = 0 Σ k _ij (y _i −y _j ) = 0 in the third processing step. And a fourth processing step of reconnecting to generate a new triangular mesh, and: