JP2955305B2 - Semiconductor device simulation method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION [Purpose of the Invention] (Field of Industrial Application) The present invention relates to a method of simulating a semiconductor device using a numerical calculation using a computer.

(Prior art) As a method of analyzing the potential distribution, electron distribution, hole distribution, etc. in a semiconductor for the purpose of characteristic analysis and evaluation of a semiconductor device, three basic equations of Poisson equation and electron and hole transport equations are used. Device simulation techniques for solving numerically using a computer are known (for example, S.Se
lberherr, “Analysys and Simulation of Semiconducto
r Devices ", Springer-Verlag", 1984 b). This technology includes: (a) inputting the shape and material of an insulating film and an electrode constituting a semiconductor element, a voltage applied to the electrode, and data on impurity concentration distribution inside the semiconductor; and (b) inputting these data or discretization. Grid points for numerical analysis are generated based on the grid point generation information for the semiconductor device, impurity concentrations and the like are set on each grid point, and (c) Poisson's equation describing the potential distribution inside the semiconductor, And (d) recording and outputting the obtained numerical solution.

At the stage of obtaining a numerical solution of the equation, for example, in the case of a two-dimensional space, grid points for discretization are set corresponding to the element cross-sectional structure as shown in FIG. Potential ψ, electron concentration n, hole concentration p on branch connecting grid points
Is solved by simultaneously combining the following three basic equations.

Poisson's equation; ∇ (ω∇ψ) = - q (p-n + N D -N A) an electron-hole transport equation; δn / δt + ∇ (μ n n∇ψ-D n ∇n) = GR δp / δt + ∇ (μ _p p∇ψ−D _p ∇p) = GR In these equations, ψ is a potential, n is an electron concentration, p is a hole concentration, q
Elementary charge, N _D is the donor concentration, N _A is the acceptor concentration, mu _n is the electron mobility, the mu _p hole mobility, D _n is the electron diffusion coefficient, D _p
Is the diffusion coefficient of holes, and GR is the rate of generation of electrons or holes per unit time and per unit volume (multiple functions of μ _n , μ _p , ∇ψ). In order to find simultaneous solutions of such complicated equations, device simulation techniques usually use n,
It is expressed in the form of a numerical calculation program that calculates p and ψ.

To input the data of the element cross-sectional structure in two-dimensional space,
Relatively simple. For example, the MOS shown in FIG.
For the transistor structure, the semiconductor substrate 51, the insulating film 52,
53, 54, the gate electrode 55, the source and drain electrodes 56, 57, etc., by approximating a region made of the same material by a polygon as shown in FIG. What is necessary is just to input coordinates, physical constants in a polygon, and a substance name.

However, in recent years, with miniaturization of elements of a semiconductor integrated circuit, it has become necessary to solve the above-described basic equation in a three-dimensional space. As a data input method of the three-dimensional element shape, for example, a method of dividing a semiconductor element structure as shown in FIG. 6A into a polyhedral region as shown in FIG. TDLinton.Jr, PABlakey, “A Fast Ge
neral Three-Dimetional Device Simulation and Its
Application in a Submicron EEPROM Design Study ", IE
EE Trans.CAD, vol.8 No.5, 508-515, May 1989). That three-dimensional shape composed of a semiconductor substrate 61 and the insulating film 62 is divided into two rectangular 61 _1, 61 _2, two triangular prism 61 _3, 61 ₄ and the one of the four pyramids 61 ₅ for the semiconductor substrate 61, an insulating film two rectangular 62 ₁ for 62, 62 _2, two triangular prism 62 _3, 62 _4, and is divided into two quadrangular pyramid 62 _5, 62 _6. Then, for each of the polyhedral regions, a rectangular parallelepiped, a triangular prism, or a quadrangular pyramid is designated, the coordinates of the vertices are designated according to the respective rules, and a physical constant or a material name is input.

However, this method has a problem that the number of polyhedrons to be divided is large and therefore the number of input data is large.
This is because polyhedron division is performed according to the rule that a single connection region made of a certain material must be divided into polyhedrons so as not to overlap with each other, and polyhedrons representing regions made of different materials must not overlap each other. Because. As a result, especially when the element structure is complicated, the number of input data becomes very large, and the user must work very carefully when dividing the polyhedron.

(Problems to be Solved by the Invention) As described above, in the conventional device simulation for a three-dimensional shape, there is a problem that the number of polyhedrons to be divided is very large, and the work is also difficult.

SUMMARY OF THE INVENTION The present invention has been made in view of the above circumstances, and has as its object to provide a semiconductor device simulation method capable of describing a complicated three-dimensional shape with a small amount of input data.

[Configuration of the Invention] (Means for Solving the Problems) The present invention relates to a simulation of a semiconductor device.
When dividing the three-dimensional shape into a plurality of polyhedron regions representing a single connection region, a positional relationship is allowed such that a certain polyhedron has a common part with another polyhedron. On the other hand, real values indicating priorities are set in the polyhedron, and if there is a part commonly included in a plurality of polyhedrons, the material is determined based on the real values indicating the priority for the part. In other words, priorities are determined in advance by materials, and when a polyhedron with a higher priority and a polyhedron with a lower priority are divided with a common part, the designation of the polyhedron with a higher priority takes precedence over the common part. To do.

(Operation) According to the present invention, the number of polyhedrons for dividing a three-dimensional shape is reduced as compared with the conventional method, and the burden imposed on the user when inputting data is greatly reduced. In addition, when the data for designating the priority order is a real number type, when the element structure whose shape is already defined is slightly changed, a new polyhedron region is added or the numerical value of the already defined priority is simply changed. , A new element structure can be easily specified.

(Example) Hereinafter, an example of the present invention will be described.

FIGS. 1 (a) and 1 (b) show how a three-dimensional element structure is divided into polyhedral regions, corresponding to FIGS. 6 (a) and 6 (b). That is, a three-dimensional device structure and the semiconductor substrate 11 shown in FIG. 1 (a) made of an insulating film 12, as shown in FIG. (B), expressed in a rectangular solid 11 ₁ for the semiconductor substrate 11, an insulating film 12 is divided into two rectangular parallelepipeds 12 ₁ and 12 ₂ and two triangular prisms 12 ₃ and 12 ₄ .
The rectangular parallelepiped 11 ₁ representing the semiconductor substrate 11 is cut out including the portion of the insulating film 12, and thus the rectangular parallelepiped 12
₁ , 12 ₂ and triangular prisms 12 ₃ , 12 ₄ have common parts. In this case, by setting the priority of the insulating film region higher than that of the semiconductor region, the designation of the rectangular parallelepipeds 12 ₁ and 12 ₂ and the triangular prisms 12 ₃ and 12 ₄ is given priority over the rectangular parallelepiped 11 _{1 of the} semiconductor region for this common portion. Therefore, desired element structure data can be input. As is apparent from a comparison between FIG. 1B and FIG. 6B, in this embodiment, the number of divided polyhedrons is less than 1/2 of the conventional method.

FIG. 2 is a flowchart showing a data input method and a processing method after the three-dimensional element structure is divided into polyhedrons as described above. First, set the number of the polyhedron to be entered (S1, S
2) Read all relevant data of the k-th polyhedron (S3). Specifically, the shape of the k-th polyhedron (triangular pyramid,
Data that specifies quadrangular pyramids, triangular prisms, cuboids, etc.)
(K) is read, and NUM (k) pieces of coordinate data X (k, I), Y (k, I), Z (k, I) (I = 1,2,. And reads data MATER (k) representing the material of the polyhedron. If the k-th polyhedron is an electrode area, the electrode number I ELECTR (k) is read. After that, the real number data PRIORT (k) indicating the priority of the k-th polyhedron is read (S4). The order of data input so far does not matter. If the input method using a keyword is used, these data readings may be mixed. Then, it is determined whether or not the above data reading has been completed for all the polyhedrons (S5). If NO, similar data input is performed for the next polyhedron. By repeating the above scanning and inputting the above data for all polyhedrons, based on real number data representing the next priority, for example, the polyhedron numbers are arranged in order from the polyhedron region having the lowest priority, and the array data Store in I ORDER (k) (S
6). Even if the arrangement order is reversed, it suffices if the processing is appropriate later.

Next, other data necessary for the simulation is read (S7). That is, data representing the impurity concentration distribution inside the semiconductor, data designating the voltage to be applied to the electrodes, physical constant data, optional parameters related to input / output, and the like are read. Then, based on the coordinates of the vertices of the read polyhedron, NXYZ grid points X (1), X for discretization in a three-dimensional space
(2), ..., X (NX), Y (1), Y (2), ..., Y (NY),
Z (1), Z (2),..., Z (NZ) (NXYZ = NX * NY * NZ) are generated (S8).

Next, the polyhedrons are selected in ascending order of priority (S9), and necessary data processing is performed on all grid points included in the L-th polyhedron (S10). Here, first, three-dimensional coordinates (X (I),
It is determined whether or not the grid point P represented by Y (J), Z (K)) is included in the L-th polyhedral region, and if so, an array representing the material of the grid point and a physical constant are determined. In the array that represents
The material data MATER (L) and physical constants of the polyhedron are stored. The method of determining whether or not the grid point of interest is included in the L-th polyhedral region is as follows. First, NUM (L) vertex coordinates X (N), Y (N), Z of the polyhedron region
(N) (N = 1,2, ..., NUM (L)), minimum value Xmi
Find n, Ymin, Zmin and maximum values Xmax, Ymax, Zmax. And (a) the maximum value of the subscript I that satisfies Xmin ≦ X (I) ≦ Xmax
Find Imax and minimum value Imin.

(B) Maximum value of subscript J satisfying Ymin ≦ Y (J) ≦ Ymax
Find Jmax and minimum value Jmin.

(C) Maximum value of subscript K satisfying Zmin ≦ Z (K) ≦ Zmax
Find Kmax and minimum value Kmin.

Based on the values thus obtained, if I is in the range from Imin to Imax, J is in the range from Jmin to Jmax, and K is in the range from Kmin to Kmax, the point P (X (I), Y (J) , Z (K)) are completely contained in the L-th polyhedron. Then, the material and the physical constant of the L-th polyhedron are stored in the array representing the material and the physical constant of the lattice point P.

It is determined whether such data processing has been performed for all the polyhedrons (S11), and if NO, the same processing is repeated for the next polyhedron. Then, when the same data processing is performed on another polyhedron selected later and having a higher priority, if a lattice point included in the previously processed polyhedron region is included, the subsequent data having a higher priority is used. , The data at the previous grid point is rewritten.

In more detail in the example of Fig. 1 (b), a rectangular parallelepiped 12 ₁ which is an insulating film region with respect to a rectangular parallelepiped 11 ₁ which is a semiconductor region, 1
2 ₂ , the triangular prisms 12 ₃ , 12 ₄ are given higher priority. Therefore, the data processing is first performed on the region body 11 ₁ of the semiconductor region, and the material of the lattice points included in the region is assumed to be a semiconductor. Thereafter, the data of the rectangular parallelepipeds 12 ₁ , 12 _{2 and the} like which are the insulating film regions are obtained. in the process, the material of the grid point overlapping the rectangular solid 11 ₁ of the semiconductor region is rewritten as an insulating film.
This allows accurate data entry even though the polyhedral divisions include common parts.

After that, it is checked whether or not data is set in the array representing the material and physical constant of each grid point, and if there is a grid point for which data is not yet written, appropriate error processing is performed. (S13). Subsequently, impurity concentrations are set for lattice points inside and on the surface of the semiconductor according to a normal device simulation program (S1).
4) Solve the basic equation (S15) and output the calculation result (S16).

As described above, according to this embodiment, by allowing the division of the polyhedral region including the common part under the condition of setting the priority, the three-dimensional shape of the semiconductor element can be reduced with very little input data compared to the conventional method. By describing, numerical simulation of the element can be performed.

In the above embodiment, the case where the orthogonal grid is used as the grid for discretization has been described. However, the present invention can also be applied to the case where discretization using elements made up of tetrahedrons is used.
However, in this case, the method of determining whether a certain point P is included in the L-th polyhedron needs to be modified. This uses a method of determining whether or not the point P and an appropriate point of the polyhedron are on the same side with respect to the surface constituting the polyhedron.

FIG. 3 specifically shows a determination algorithm when the polyhedron is a triangular pyramid (tetrahedron). A is the vertex of the triangular pyramid
(XA, YA, ZA), B (XB, YB, ZB), C (XC, YC, ZC), D (XD, YD,
First, to determine whether the point P (XP, YP, ZP) is on the same side as the point D with respect to the plane ABC, the determinants DETP and DETD represented by the following equations are calculated (P1).

From this calculation result, it is determined whether | DETP | is within a predetermined error ε (P2), and if NO, DETP *
It is determined whether or not DETD is negative (P3). If this is also NO, then it is expressed by the following equation to determine whether or not point P is on the same side as point C with respect to plane ABD. Determinants DETP, DETC
Is calculated (P4).

From this calculation result, it is determined whether or not | DETP | is within a predetermined error ε range (P5). If NO, DETP *
It is determined whether or not DETC is negative (P6). If this is also NO, then the following equation is used to determine whether or not point P is on the same side as point B with respect to plane ACD. Determinants DETP, DETB
Is calculated (P7).

From this calculation result, it is determined whether | DETP | is within a predetermined error ε range (P8), and if NO, DETP *
It is determined whether or not DETB is negative (P9). If this is also NO, then the following equation is used to determine whether or not the point P is on the same side as the point A with respect to the plane BCD. Determinants DETP, DETA
Is calculated (P10).

From this calculation result, it is determined whether or not | DETP | is within a predetermined error ε (P11).
* It is determined whether or not DETA is negative (P12). If negative, the point P is determined to be inside the triangular pyramid ABCD,
Then, data setting (INNER = 1) is made (P13). Steps
If YES in any of P3, P6, P9, and P12, it is determined that the point P is outside the triangular pyramid ABCD, and the data is set to that effect (INNER = -1) (P14). Steps P2, P5, P6, P
If YES in any of P8 and P11, point P is triangular pyramid AB
Judge that it is on the surface of the CD and set the data accordingly (INNER
= 0) (P15).

As described above, when the polyhedron is a cone, it can be easily determined whether or not the point of interest is inside the polyhedron. Even when the polyhedron has a shape like a triangular prism or a quadrangular prism, the determination can be easily made similarly by applying the algorithm of FIG. Experience shows that most shapes are triangular pyramids,
Since it can be approximated by a combination of a quadrangular pyramid, a triangular prism, and a quadrangular prism, if a determination program as shown in FIG. 3 that can be applied to these four types of polyhedrons is provided, there is no practical problem even with a complicated element structure. Data can be input by dividing the element.

[Effects of the Invention] As described above, according to the present invention, when a three-dimensional structure is divided into polyhedrons and described in a numerical simulation of a semiconductor device, priority is determined and polyhedrons have a common part. By permitting it, it is possible to describe the shape with very little data as compared with the related art. With a complicated device structure, the number of data can be reduced to about 1/3 to 1/5. It is also possible to change the already defined shape easily by changing the already specified priority or adding a few polyhedral regions.

[Brief description of the drawings]

1 (a) and 1 (b) are views for explaining a state of polyhedral division of an element structure according to an embodiment of the present invention, FIG. 2 is a flowchart showing a simulation procedure according to the embodiment, and FIG. FIG. 4 is a flowchart showing a procedure of a method of determining whether or not a point is inside a triangular pyramid; FIG. 4 is a diagram showing an example of a discretization grid in a numerical simulation of a semiconductor element in a two-dimensional space; FIGS. 6A and 6B are diagrams showing a two-dimensional structure of a semiconductor device and a polygonal region for designating the two-dimensional structure, and FIGS. 6A and 6B show a state of polyhedral division of a conventional three-dimensional device structure. It is a figure for explaining. 11 ... semiconductor substrate, 12 ... insulating film, 11 ₁ ... rectangular parallelepiped, 1
2 ₁ , 12 ₂ …… a cuboid, 12 ₃ , 12 ₄ …… a triangular prism.

Claims (1)

(57) [Claims] An electric characteristic of a semiconductor device is obtained by inputting a shape of a semiconductor device, a shape and a position of an electrode, a potential applied to an electrode, and a distribution of an impurity concentration in a semiconductor, and solving a basic equation inside the semiconductor by numerical calculation. In a simulation method of a semiconductor device to be analyzed or evaluated, a three-dimensional space region occupied by the semiconductor device is divided into a plurality of polyhedrons while allowing a positional relationship having a common part with respect to each other. Data is input with real values indicating priorities together with physical constants, and at a stage of data processing for solving a basic equation inside the semiconductor or data processing preceding it, a predetermined area forming the common part is formed by the real numbers indicating the priorities. A method of simulating a semiconductor device, wherein a material or a physical constant is determined.