JPH03116385A - Particle simulation system - Google Patents

Abstract

PURPOSE:To attain analysis with physically high accuracy while sufficiently utilizing the feature of particle simulation by arranging super-particles in an analysis area where particle simulation is executed, and afterwards, handling the transportation of particles, which are remained in the area without being distributed to the super- particles, according to the transient analysis method of fluid simulation. CONSTITUTION:When the transportation of the plural particles is analyzed, the super- particles are arranged in the analysis area, to which the particle simulation is executed, and afterwards, the transportation of the particles, which are remained in the area without being distributed to the super-particles, is handled according to the transient analysis method of the fluid simulation. When the analysis is executed by using a window method, the transportation for the small number of the particles, which are remained in a window without being distributed to the super-particles, and the particles out of the window is handled by the transient analysis method of the fluid simulation and when the supplying source of the particles such as an ohmic electrode, etc., is included in the analysis area, the transient analysis is executed under a contour condition that the number of the particles is fixed in a contour with the supplying source. Thus, while effectively utilizing the feature of the particle simulation, the simulation can be executed with physically high accuracy.

Description

[Detailed Description of the Invention] [Object of the Invention] (Industrial Application Field) The present invention relates to a method for analyzing the transport characteristics of a plurality of particles, and in particular uses a computer to numerically determine each physical quantity within an analysis target. This paper relates to a method for performing characteristic analysis.

(Conventional technology) When analyzing transport phenomena by computer simulation, 2
There are two methods.

One is called particle simulation, in which the particles existing in the analysis region are represented by a practical number of virtual particles (hereinafter referred to as superparticles), and the transport of each superparticle is calculated using random numbers. (For example, Tetsuo Uemura et al.,
``Computer experiments on plasma using a superparticle model'' Journal of the Physical Society of Japan, Vol. 28, No. 12, p. 1019, 1973).

This method deals with the elementary processes of each superparticle such as drift and scattering in a field (electric field, magnetic field, etc.) without approximating it.
Physical accuracy is high. However, since a limited number of superparticles are moved using random numbers, in order to obtain a statistically stable distribution of physical quantities, one must use a sufficiently large number of superparticles, or move the superparticles for a sufficiently long period of time, and use the ergodic method to calculate the time average. It is necessary to take the collective average according to the hypothesis. Both require a huge amount of calculation time.

Another method is called fluid simulation, which treats transport as fluid motion, expresses the average value of physical quantities at each location as a differential and integral equation, and solves this numerically (for example, by S. , “Anal
ysis and sia+lation or 5
eIlliconductor devices”, S
Pinger-Verlag, 1984). This method deals with the transport of average values, so there is no statistical instability, and a solution can be obtained in a shorter time than particle simulation. However, since the only physical quantity information is the average value, it does not provide physically accurate results. I can't.

Conventionally, particle simulation has been used when physical accuracy is required rather than economic efficiency (for example, analysis of transport characteristics of electrons and holes in microscopic semiconductor devices).

One of the problems with conventional particle simulations is that if the analysis region contains an extremely small number of particles, no superparticles can be placed in that region, resulting in an unnatural physical quantity of zero. A distribution occurs. Further, even in a region where a sufficient number of superparticles are arranged, a limited specific number of particles cannot be distributed to representative superparticles, and information on the number of remaining particles and their physical quantities is lost.

For example, consider the case where particle simulation is applied to characteristic analysis of an electron conductive insulated gate transistor (hereinafter referred to as nMO8) shown in FIG. 4(a). B-B'
The electron concentration distribution in the cross section is as shown in FIG. 4(b). The dashed line is the result of fluid simulation. Normally, in particle simulation, superparticles are arranged using the results of fluid simulation as initial values. Due to calculation time constraints, only about 10,000 to 100,000 superparticles can be handled, so it is necessary to represent many electrons with one superparticle, as shown in Figure 4 (b).
) A small number of electrons that cannot be assigned to superparticles remain, as shown by the shaded area in ). In particular, B'-B' in Figure 4(b)
Since no superparticles are placed in the area, the electron concentration is zero, but this is physically impossible.

In addition, in conventional particle simulation, there is a method of limiting the area to which particle simulation is applied in order to save calculation time (for example, P. Venturt et al.
A general purpose of' dev
ice slmulatlon couphng po
lsson and Monte Carlo tra
nsports vlth application
o deep submlcron N.

5FET's, IEEE Transaction
on Con+puter-Aided Design
, Vol. 8. No, 8. p, 3[io, 19
89) Again, let us take as an example the characteristic analysis of the nMOS as shown in FIG.

What governs the electrical characteristics of this device is the electron transport characteristics directly under the gate oxide film [2] between the n+ diffusion layers 804 and 605. Therefore, if a physically highly accurate analysis is required, it is economical to apply particle simulation only to the area within the window 609 and apply fluid simulation to the other areas. This method is generally a window method, a hybrid method, or a domain Monte method.
This is called the Carlo method.

Hereinafter, the calculation procedure of the window method will be explained using the flowchart of FIG.

First, in step P1, information on the analysis structure is read.

Next, in step P2, the coordinates of the window 609 to which particle simulation is applied are read.

Next, step P3f'C--perform fluid simulation in the entire region.

Next, in step P4, boundary conditions such as the number of superparticles at the boundary of the window 609 and the inflow velocity are set using the results of the fluid simulation.

Next, in step P5, using the results of the fluid simulation, the electrons existing within the window 609 are represented by superparticles, and the initial state of the superparticles is determined.

Next, in step P6, a particle simulation is executed for Δ seconds.

Next, in step P7, the superparticles that have gone outside the window 809 are deleted, and new superparticles are placed at the boundary of the window 609 so as to satisfy the boundary conditions determined in step P4.

If the steady state has not been reached in step P8, the Poisson equation is recalculated in step P9 using the superparticle distribution obtained in step P7, and the process returns to step P6.

If the steady state has been reached in step P8, step PI
At O, transport coefficients such as mobility are extracted from the particle simulation results.

Next, perform a fluid simulation in the entire region using those transport coefficients at step pH to obtain the terminal current values.

The processing contents of the fluid simulation in step P3 and step pH are as shown in FIG. 7, in which combinations of basic equations are made to be simultaneous and solved by a matrix solution method.

As shown in FIG. 8, the processing content of the particle simulation in step P6 is to track the drift of each superparticle due to the electric field and scattering with lattices and impurities using random numbers. ΔT is the time interval during which the particle simulation is performed, and within this ΔT seconds, each superparticle repeats free flight (drift) and scattering for 61 seconds determined by a random number multiple times.

T is the elapsed time of the superparticle movement within 61 seconds.

However, when analysis is performed using the conventional window method as described above, there is a problem in that a physically unnatural distribution occurs at the window boundary, which adversely affects the overall transport characteristics.

For example, the distribution of superparticles and their velocity distribution in the cross section AA' in FIG. 9(a) are as shown in FIGS. 9(b) and (C) at the time the process of step P5 is completed. Become. Next, after the particle simulation is executed for Δ seconds in step P6, the results are as shown in FIGS. 9(d) and 9(e). The number of superparticles decreases near the window boundary, resulting in a large velocity distribution toward the outside of the window, which becomes unnatural. This is because there is no inflow of particles from outside the window.

In the next step P7, new superparticles are placed at the window boundary as shown in Figures 9(r) and (g) so as to satisfy the boundary conditions determined in step P4. Only the boundary part has particle number and velocity distribution. It is clear that when the Poisson equation is recalculated using the distribution shown in FIG. 9(f') in order to update the electric field distribution, the electric field value near the window boundary becomes unnatural. This unnatural distribution near the window boundary adversely affects the overall transport properties finally obtained.

To avoid the window border issue mentioned above,
Shorten the calculation time Δt of particle simulation (Δt~
IQ - 15 seconds), a method can be considered to place a new superparticle before an extremely unnatural distribution occurs at the window boundary. However, the number of iterations required to reach steady state increases,
Computation time increases. Moreover, it is not a fundamental solution.

A similar problem also occurs when a particle supply source such as an ohmic electrode is included in the particle simulation analysis area.

For example, when analyzing the characteristics of a semiconductor device, it is physically appropriate to keep the number of electrons and holes constant at the boundary where the semiconductor and ohmic electrode are in contact. In fluid simulation, it is sufficient to use a fixed boundary condition in which the number of electrons and holes are constant at the electrode, but in order to achieve this electrode condition in conventional particle simulation, after running the particle simulation for △ seconds, , the required number of superparticles must be placed at the electrode section before recalculating the Poisson equation. For Δ seconds, the superparticles only flow out unilaterally toward the electrode, so the distribution and velocity distribution of the superparticles become unnatural, similar to the conventional window boundary.

(Problem to be solved by the invention) As described above, in the conventional particle simulation technology,
There were problems in that superparticles could not be placed in areas where the number of particles was small, resulting in physical quantities becoming zero, and the distribution of physical quantities becoming unnatural near window boundaries and particle sources.

In order to solve the above problems, the present invention incorporates into the calculation a small number of particles that cannot be distributed into superparticles in particle simulation, and provides a method for rationally realizing the inflow of particles from outside the window and from the particle source. The purpose is to provide

[Configuration of the Invention (Means for Solving the Problems) The particle simulation method of the present invention is a computer-based particle simulation method that tracks particles using random numbers in order to analyze transport phenomena. a step of reading information; a step of executing a fluid simulation within the analysis structure; and a step of executing a particle simulation for a part that mainly contributes to physical phenomena within the analysis structure; The present invention is characterized by comprising the step of calculating, by fluid simulation, the transport of particles that are not distributed among virtual particles representing a plurality of particles existing in the calculation region in a portion that does not contribute to the calculation.

In analyzing the transport of multiple particles, the present invention uses a fluid simulation transient analysis method to analyze the transport of particles that are not distributed to the superparticles and remain within the region after placing superparticles in the analysis region where particle simulation is performed. 〇Also, when performing a similar analysis using the window method, the transport of a small number of particles that are not distributed into superparticles and remain within the window, and particles outside the window, is handled by the transient analysis of fluid simulation. It is characterized by being handled by methods.

Furthermore, when a particle supply source such as an ohmic electrode is included in the particle simulation analysis area, a transient analysis of the fluid simulation is performed under a boundary condition in which the number of particles is constant at the boundary with the supply source.

Note that the maximum number of superparticles that can be distributed may be arranged at each location, or a part of the maximum number may be arranged.

(Function) According to the present invention, when analyzing the transport of a plurality of particles, after placing superparticles in the analysis region where particle simulation is performed, the transport of particles that are not distributed to the superparticles and remain in the region is reduced. It is handled using the transient analysis method of fluid simulation. Therefore, a distribution in which the physical quantity is zero due to no superparticles being placed within the analysis region of the particle simulation does not occur.

When performing a similar analysis using the window method, the small number of particles that are not distributed into superparticles and remain within the window, and the transport of particles outside the window are handled by the transient analysis method of fluid simulation, and the transport of particles outside the window is handled by the transient analysis method of fluid simulation. When a particle supply source such as an ohmic electrode is included in the analysis domain, a transient analysis of the fluid simulation is performed under a boundary condition in which the number of particles is constant at the boundary with the supply source. No unnatural distribution occurs.

Therefore, it is possible to perform a physically highly accurate simulation that takes advantage of the characteristics of particle simulation.

(Example) An example of the present invention will be described below with reference to FIGS. 1 and 2.

This example shows an example of analyzing an electron transport phenomenon inside an nMOS using a window method.

First, the structure of the nMOS shown in FIG. 2(a) will be explained. A source 204 and a drain 205 as an n+ diffusion layer are provided in the P-type semiconductor substrate 201.
A source electrode 206 is in ohmic contact with the source electrode 04, and a drain electrode 207 is in ohmic contact with the drain. source 204
A gate insulating film 202 is provided on the channel region between and the drain 205, and a gate electrode 203 is provided on the gate insulating film 202. The electrodes 206 and 207 and the gate electrode 203 are insulated by an oxide film 208.

FIG. 1 is a flowchart showing a specific calculation procedure of the particle simulation method of the present invention.

First, in step M1, information on the analysis structure is read.

Next, in step M2, the coordinates of the window 209 to which particle simulation is applied are read.

Next, in step M3, a fluid simulation is performed in the entire region.

Next, in step M4, using the results of the fluid simulation, the electrons existing within the window 209 are represented by superparticles, and the initial state of the superparticles is determined.

Next, in step M5, particle simulation is executed for Δ seconds.

On the other hand, in step M6, the window 20
The distribution of electrons left inside and outside 9 after a second is determined by transient analysis of fluid simulation.

In step M7, if necessary, step M5,
Add up the results of M6 and output.

If the output value has not reached a steady state in step M8, the superparticles that have come out of the window 209 are fluidized (state M9), and the number of electrons in the electronic fluid that has flowed into the window 209 is equal to or more than one superparticle. The fluid is made into superparticles at the location where the fluid is present (step M10).

Next, the results of step M9 and MIO are added together, the Poisson equation is recalculated using the distribution, the electric field distribution is updated (step M11), and the process returns to steps M5 and M6.

If a steady state has been reached in step M8, transport coefficients such as mobility are extracted from the particle simulation results in step M12, and fluid simulation is executed in the entire region in step M13 to obtain terminal current values.

The above processing will be explained using the movement of superparticles and fluid when viewed in a cross section taken along line AA' in FIG. 2(a).

As a result of the process in step M3, an electron distribution as shown in FIG. 2(b) is obtained.

As a result of the processing in step M4, FIGS. 2(e) and (d)
It can be divided into superparticle and fluid distributions.

As a result of steps M5 and M6, the distribution of superparticles and fluid after Δ seconds is as shown in FIGS. 2(e) and 2(f).

The superparticles that went outside the window 209 in step M7 are
The fluid is made into a fluid as shown in FIG. 2(e), and the fluid flowing into the window 209 in step M8 is made into superparticles as shown in FIG. 2(r).

For the superparticles and fluid redistributed as shown in FIGS. 2(g) and (h), Δ is again calculated for seconds in steps M5 and M6.

Note that if an ohmic electrode is included in the window 209, step M6 is executed using a boundary condition that fixes the electron concentration at the portion where the electrode and the semiconductor are in contact.

Alternatively, the coordinates of the entire analysis area may be read in step M2. In that case, in step M6, the electron distribution and the ohmic electrode portion that are left behind without being completely distributed into superparticles within the analysis region are treated as a fluid. Then, as a result of the fluid simulation in step M6, if a large number of electron distributions are generated that can be distributed to superparticles, step M9
to make it into super particles.

Moreover, the process of step M2 may be performed after step M3. That is, there is also a method of detecting a region where an important transport phenomenon occurs in the analysis region from the results of the fluid simulation in step M3 and setting a window. The detection process may be performed by a human or automatically by a computer.

The processing content of the fluid simulation in steps M3, M6, and Ml3 is to solve the combination of basic equations as shown in FIG. 7 by matrix solving, including time differential terms. That is, in step F1, coefficient matrices of the discrete equations of the basic equations, such as the Poisson equation, the continuous electron current equation, and the continuous hole current equation, are created. In step F2, the above-mentioned simultaneous-order equations are solved by a matrix solution method. Next, in step F3, the potential, electron concentration, and hole concentration are updated. In step F4, if the result in step F3 does not converge, return to step F1 and repeat the above steps until convergence occurs in step F4.

Further, the processing contents of the particle simulation in step M5 are as shown in FIG. In step R1, T
is set to 0, and in step R2, the free flight time δ of the particle
Find T using a random number. Then, in step R3, the particles are allowed to fly freely for a period of δT.

Next, in step R4, the particle scattering mechanism is determined using random numbers, and in step R5, the state after scattering is determined using random numbers. In step R6, T + ΔT is set to T, and in step R7, if the time interval ΔT during which the particle simulation is executed is not greater than T (the elapsed time of superparticle motion within 61 seconds), the process returns to step R2, and T becomes ΔT. Repeat until.

When T becomes ΔT in step R7, step R8
If the particle simulation has not been completed for all particles, the process returns to step R1, and steps R1 to R7 are executed for all particles.

FIG. 3 shows the results obtained by the present invention using the process shown in FIG. 1 and the results obtained by the conventional method shown in FIG. This is a comparison of the results.

Figure 3(a) is a comparison of electron concentration distribution, and Figure 3(b)
is a comparison of electron velocity distributions. The solid line is the result of the present invention, and the broken line is the result of the conventional method. In the present invention, the window 209
It can be seen that no unnatural distribution occurs near the boundary. Therefore, unlike the conventional method, the unnatural distribution near the boundary of the window 209 does not adversely affect the overall transport characteristics.

Although the present invention has been explained by taking as an example the analysis of transport characteristics of electrons and holes within a semiconductor device, the particle simulation method of the present invention can also be applied to a method of manufacturing a semiconductor device. For example, in the growth of a semiconductor layer, particle simulation can be applied to a portion where a reactive gas contacts the semiconductor surface, and fluid simulation can be applied to other portions. Furthermore, when etching a semiconductor layer, particle simulation can be performed while etching is being performed, and fluid simulation can be performed during other steps.

[Effects of the Invention] As explained above, according to the present invention, in a particle simulation that analyzes the transport characteristics of a plurality of particles, a distribution in which no physical quantity exists within the analysis region or a distribution near the window boundary or a particle supply source occurs. This eliminates the occurrence of unnatural distributions and enables highly accurate physically analysis that takes full advantage of the characteristics of particle simulation.

[Brief explanation of drawings]

Fig. 1 is a flowchart showing the calculation procedure according to an embodiment of the present invention, Fig. 2 is a diagram showing the movement of superparticles and fluid in the embodiment, and Fig. 3 (a) and (b) are calculated by the embodiment. Figures 4(a) and 4(b) are diagrams comparing the electron concentration distribution and electron velocity distribution with the conventional technology.
Figure 5 is a structural cross-sectional view of 9MO8 to which the window method of conventional particle simulation is applied and a diagram showing the window coordinates, and Figure 6 is a diagram showing the arrangement of superparticles in the central cross section. A flowchart showing the simulation window calculation procedure, Figure 7 is a diagram showing an overview of the fluid simulation method, Figure 8 is a diagram showing an overview of the particle simulation method, and Figure 9 is a diagram showing the application of the conventional particle simulation window method. FIG. 3 is a diagram showing the movement of superparticles and changes in velocity distribution in the case of FIG. 201.501.801...P-type semiconductor substrate, 202
,502.602...gate oxide film, 203,503
, 603... gate electrode, 204.504. BO4-
--n+ diffusion layer (source), 205,505,805
...n+ diffusion layer (drain), 206, 506.6
08... Source electrode, 207.507,607...
-Drain electrode, 208.508,608...Oxide film, 209.809...Window.

Claims (4)

[Claims] (1) In a computer-based particle simulation method that tracks particles using random numbers in order to analyze transport phenomena, the steps include: reading information on an analysis structure; executing a fluid simulation within the analysis structure; and A particle simulation comprising the steps of: performing a particle simulation within a structure; and calculating transport of particles that are not distributed among virtual particles representing a plurality of particles existing within a calculation region using a fluid simulation. method. (2) In a computer-based particle simulation method that tracks particles using random numbers in order to analyze transport phenomena, the steps include: reading information on an analysis structure; executing a fluid simulation within the analysis structure; and Particle simulation is performed for parts of the structure that mainly contribute to physical phenomena, and virtual simulations that represent multiple particles existing in the calculation domain are performed for parts of the analysis structure that do not mainly contribute to physical phenomena. 1. A particle simulation method comprising the step of calculating transport of particles that are not distributed among particles by fluid simulation. (3) Applying particle simulation to a limited area,
2. Transport of particles that are not distributed to the virtual particles within the region and particles that exist outside the region is calculated by the fluid simulation.
Particle simulation method described in . (4) When a particle supply source is included in the analysis region to which particle simulation is applied, the fluid simulation may be executed with a boundary condition that fixes the number of particles at the boundary between the supply source and the analysis region. The particle simulation method according to claim 1, characterized in that: