JP4047833B2 - Change prediction method - Google Patents

Description

    The present invention relates to a simulation method for an object in which a problem that actually occurs is probabilistic, and further relates to a change prediction method to which this method is applied.

    As is well known, a semiconductor device manufacturing process includes a process of forming a thin film on a substrate. In many cases, a chemical vapor deposition (CVD) method is employed for producing these thin films.

  Recently, with the advancement of technology, it is necessary to make the circuit pattern on the substrate 1 μm or less, and the thin film generated on this pattern also needs to be miniaturized accordingly. When the density of the device is increased in this way, the embedding characteristic (step coverage) in the pattern step (trench) becomes a problem.

For example, using a low pressure CVD method, using silane (SiH ₄ ) as a source gas, argon (Ar) as a carrier gas, a wafer temperature of 700 ° C., a pressure of 25 Torr, an Ar flow rate of 4.5 liters / minute, an SiH ₄ flow rate of 1.0 liters / minute. When a polycrystalline silicon film is formed on the inner surface of a trench having an aspect ratio of about 10 on the condition of minutes, the film thickness is almost constant inside the trench, but an overhang is observed at the top of the trench. . This suggests the existence of a film-forming species having a low adhesion probability and a film-forming species having a high adhesion probability. That is, it is considered that at least two types of film formation species contribute to the film formation. It is considered that SiH ₄ gas reacts in the gas phase as expressed by equation (1) and affects film formation.

Since silylene (SiH ₂ ) produced by the reaction of formula (1) is considered to have a high reactivity and an adhesion probability of almost 1, it can be assumed that the film-forming species with a large adhesion probability is SiH ₂ . Here, the adhesion probability and density ratio of the film forming species in the vicinity of the trench are regarded as parameters that determine the film forming shape. For example, since the adhesion probability of SiH ₂ is almost 1, a molecule with such a large adhesion probability becomes a film at the entrance of the trench and cannot penetrate into the depth of the trench. Even a small amount of such reaction intermediates degrades coverage.

  By the way, the method of determining the film thickness and the film shape by performing the experiment while changing the conditions as described above lacks applicability, and it is difficult to find the optimum conditions that satisfy the requirements. For this reason, it is necessary to develop a prediction method for accurately and quickly predicting the thickness and shape of the film formation.

  Non-Patent Document 1, Non-Patent Document 2, and the like are known as documents disclosing this prediction method.

  These prediction methods use a combination of a rare gas flow analysis and a corresponding film growth simulation. Note that a direct simulation Monte Carlo (DSMC) method may be used for dilute gas flow analysis, and a string model or cell model may be used for simulation of film growth.

  A specific procedure for predicting film growth is to conduct a dilute gas flow analysis in the computational domain, and then follow a series of simulations to grow the film according to the volume of molecules attached. Repeated until the end.

However, in such a conventional simulation method, since the time integration of the rare gas flow analysis is performed, for example, when predicting the thin film embedding inside the trench described above, even if a relatively high speed computer is used In particular, when a film-forming species with a low adhesion probability is handled, it takes one month to one year or more to make a prediction once. By employing vector calculation or parallel processing, a certain amount of time can be shortened. However, when a film-forming species having a very low adhesion probability is used, it still takes a long time. For this reason, the conventional method has a problem that the time required for the prediction must be shortened by significantly sacrificing the accuracy or range of the prediction.
Tatsuta et al: Simulation of CVD Step Coverage for SiH4 using Parallel Processing of DSMC Method, Computational Mechanics '95 (Proceedings of International Conference on Computational Engineering Science) Vol.1, pp.592-597, Springer-Verlag (1995) Toshiki Iino: Simulation of semiconductor film formation shape, Proceedings of the 68th General Meeting of the Japan Society of Mechanical Engineers, Vol.D, 1994, pp. 447-449

  As described above, in the conventional simulation method for an object in which a problem that actually occurs is probabilistic, depending on the conditions, a large amount of time is required to perform prediction once, and prediction accuracy deteriorates if the time is reduced. There was a problem that the analysis range was limited.

  In view of the above, an object of the present invention is to provide a change prediction method capable of making the calculation time substantially constant without being influenced by the probability in the case of an object in which a problem that actually occurs is probabilistic.

According to this invention,
When predicting the change by performing a numerical simulation using a direct simulation Monte Carlo method for an object whose state or property changes due to the action of a factor depending on the probability, the number of times of the action of the factor depending on the probability or the above When dealing with the amount of change in the object, the number of actions or amount of change handled in the calculation is changed to the actual number of actions or change without changing the factor or factor history calculation method in the direct simulation Monte Carlo method. There is provided a change prediction method characterized by handling more than the amount.

Moreover, according to this invention,
When predicting the change by performing a numerical simulation using a direct simulation Monte Carlo method on an object or phenomenon that changes depending on the probability due to the action of a moving object colliding with another object, When calculating the number of actions of a factor depending on the probability or the amount of change of the object or phenomenon in the calculation, the calculation method of the factor or the history of the factor in the direct simulation Monte Carlo method is not changed. There is provided a change prediction method characterized in that the number of times of action of a factor or the amount of change of an object or phenomenon is handled more than the number of times of action of an actual factor or the amount of change of an actual object or phenomenon.

Furthermore, according to the present invention,
When predicting the change by performing a numerical simulation using a direct simulation Monte Carlo method on an object or phenomenon that changes depending on the probability due to the action of a moving object colliding with another object, In the calculation of the number of action of the factor depending on the probability or the amount of change of the object or phenomenon in the calculation, the calculation method of the factor or the history of the factor in the direct simulation Monte Carlo method is not changed. There is provided a change prediction method characterized in that at least a part of the case where the action does not act, or in the case where the object or phenomenon does not actually change, is treated as acting or changed in calculation.

  As described above, according to the change prediction method of the present invention, it is possible to predict, for example, the thickness and shape of a thin film that is finally generated in a short time. For this reason, for example, simulation of film growth in a semiconductor manufacturing apparatus can be performed efficiently, which can contribute to the optimization setting of the manufacturing method and apparatus.

  The basic idea inherent in the present invention can be applied to numerical simulations for various objects in which problems that actually occur are probabilistic. For example, applicable applications include thin film embedding, thin film generation, etching, lithography, mirror finishing, plasma application systems, metal non-metal evaporation, liquid solidification, chemical synthesis / separation, radiation application systems, images Processing, device simulation, river engineering, space engineering, audience rating survey, route search, cleaning, operations research, fusion analysis, ion implantation, etc. Applicable flow analysis fields include dilute flow, continuous flow, molecular orbital method, molecular dynamics, quantum mechanics, powder fluid analysis, radiation analysis, ray tracing, intermediate flow, network flow, graph theory, cluster analysis, Stability analysis, plasma analysis, multivariable control, etc. can be mentioned. Examples of applicable shape change models include a string model, a cell model, and a network model. Examples of applicable particles include electrons, atoms, molecules, clusters, fine particles, photons, nuclei, protons, ions, elementary particles, dusts, powders, and solids. Applicable phenomena include adhesion, collision, chemical reaction, energy conversion, generation, etching, and the like. Examples of the product at this time include amorphous, crystal, thin film, polycrystal, and solid. Examples of the carrier gas that can be used at this time include argon atoms, hydrogen gas molecules, and oxygen molecules. Further, examples of reactive gas that can be used at this time include silane, silylene, and disilane.

  Before describing specific embodiments, the features of the present invention will be described first. Now, by colliding or adhering particles such as atoms and molecules to object surfaces such as semiconductor substrates, glass, and gas or liquid particles, it is possible to generate amorphous, crystals, polycrystals, thin films, etc. on the object surface. When predicting the thickness, shape, chemical properties, physical properties, other physical properties, etc. of thin films that are finally generated or etched by numerical simulations for processes that perform etching, etc. Take as an example.

  For these predictions, a numerical analysis method such as continuous fluid flow or dilute gas flow analysis and a simulation of shape change such as film growth corresponding thereto are usually used together or individually.

  In shape change simulation, shape modeling is a string model that treats the surface as a continuous line segment, a cell model that treats the shape as a polygon or polyhedron, or a combination of them, or a network that connects multiple points. It is expressed by a model or the like, and is performed by changing the shape according to the volume or area at the time of attachment.

  Since this simulation is equivalent to taking a further time integration of the flow analysis, it usually takes a lot of time for the calculation. For example, in the case of predicting the embedding of a thin film in a trench in a semiconductor device manufacturing process, as described above, the calculation time may require one month to one year or more.

  Therefore, here, the analysis method of the present invention will be described with reference to an example of predicting embedding of a thin film inside a trench in a semiconductor device manufacturing process. In order to clarify the difference from the conventional analysis method, the analysis method of the present invention is referred to as the Stick-at-All-the-Reflection-Poits method, or SARP method for short. The SARP method is also effective when parallel processing is performed by a parallel computer or the like, and analysis can proceed in parallel for each particle or each calculation region.

  In the SARP method, for example, the following handling is performed as modeling for calculation. That is, taking a thin film embedding by CVD as an example, a rare gas analysis method using the DSMC method is used for flow analysis, and a modified Nanbu method is used for intermolecular collision. And as shown in FIG. 1, the left half of a trench is made into a calculation area | region, and the right-and-left boundary is handled as specular reflection. Molecules, for example, carrier gases such as argon gas atoms, hydrogen gas molecules, and oxygen molecules, and reactive gases such as silane, silylene, and disilane flow only from the upper boundary, and reaction of the film-forming species due to intermolecular collisions within the calculation domain. Is nothing.

  Molecules that collide with the trench adhere (probability η) or reflect (probability 1-η) according to the reaction probability such as the adhesion probability (η). Change. In other words, such an object can be said to be an event that occurs depending on the probability, or a state or property that changes due to the action of a factor that depends on the probability.

Here, the SARP method is based on the following new concept. That is, in the simulations conducted so far, when dealing with the adhesion of molecules to the film formation surface by the CVD method, the adhesion probability η and the random number are used to adhere when the random number is smaller than η, and to reflect when the random number is larger. Is generally. This is shown in FIG. Probabilistically speaking, molecules are attached after η reflections. For example, if η is 10 ⁻⁵ , the molecules will finally attach once after 10 ⁵ reflections.

On the other hand, in the SARP method, all reflection of molecules on the adhesion surface is handled as adhesion. That is, as shown in FIG. 3, for the movement of molecules, a simulation using the same method as the conventional one is performed, and only the way of attaching molecules is changed. In other words, in the calculation, it is treated as having more molecules attached than in the conventional method. When this method is used, the expected value of the number of attached molecules becomes η times the conventional value, and the calculation time is simply shortened in proportion to this. Hereinafter, it will be described that the conventional method and the SARP method give the same calculation result to the distribution of the number of attached molecules on the trench. As an example, consider the problem shown in Table 1.

[1] Regarding the number of attachments First, consider the difference in the number of attachments of molecules. Considering that the molecules enter the trench as shown in FIG. 2 and FIG. 3 and that there is no outflow of molecules outside the calculation region,
Adhesion probability: η
Number of incident molecules: N
In the i-th reflection on the adhesion surface,
Number of molecules attached: n _i
Probability of sticking: p _i
Then, in the conventional method,

You can leave. In other words, the probability of adhesion by the i-th reflection is expressed by equation (3). If the molecular trajectories are the same and equation (3) is the same even in the SARP method, the same film formation shape should be realized as a result. Hereinafter, in the SARP method, * is attached. In the conventional method, a molecule that finally attaches in the (i + 1) th and subsequent reflections is treated as an attachment in the SARP method.

It becomes.
Comparing the conventional method with the SARP method, as can be seen from the equations (2) and (4), the SARP method is only 1 / η times larger, and the others show exactly the same results as the conventional method. It can be seen that it leads to the adhesion of molecules similar to conventional methods. That is, the same result can be obtained with respect to the distribution of the number of adhesions when the number of film formation types is one.

  [2] As described above with respect to the generation number, the SARP method increases the adhesion number by 1 / η compared to the conventional method. This becomes a problem when there are two or more types of film formation.

In the example of Table 1, since the molecule B adheres 10 ⁶ times more and the molecule C does not change, an incorrect result is produced when the number of attachments is not corrected. As a solution, it is conceivable to increase the number of depositions by η for each film formation type, but it cannot be adopted because the advantage of speeding up by the SARP method is lost. Therefore, the SARP method corrects the number of molecules generated.

Conventionally, the generation of molecules is performed in proportion to the concentration ratio (partial pressure ratio) of the gas to be handled. For example, 10 ⁶ molecules A and B are generated for each generation of one molecule C, and sorting is performed by generating random numbers.

However, in the case of the SARP method, it is inconvenient to adopt the same method, so that it is generated not at a simple concentration ratio but at a ratio of P _k × η _k . However, P _k is the concentration ratio of the film-forming species k, and η _k is the adhesion probability of the film-forming species k. Therefore, the ratio of the number of film formation species is as follows.

However, when a carrier gas (inert gas) is present, it cannot be determined because the adhesion probability of the carrier gas is zero. Although this can take any value, it is desirable that it is not so far away from the number of other gas molecules generated from the viewpoint of calculation accuracy. Therefore, equation (6) becomes

For example, in the example of Table 1, A: B: C is 1/2: 1/4: 1/4. As a result, n _{j in} equation (4) becomes n _j ^* η _k , so the situation where the number of attached molecules becomes 1 / η times that of the conventional method is canceled and the number of attached molecules (equation (4)) Probability (equation (5)) can be obtained in the same way as the conventional method.

[3] Intermolecular collisions Next, consider molecular motion. In the molecular motion, the molecular orbital itself excluding intermolecular collision is the same in both the conventional method and the SARP method. This can be easily imagined because the SARP method is not directly related to molecular motion. However, since the generation of molecules is performed according to equation (7), the calculated ratio of the number of molecules is different from the actual ratio of the number of molecules. For this reason, intermolecular collisions must be corrected by manipulation. If one of the collision molecules is i, the other is j, and the collision probability of i to j per unit time is G _ij , the probability that the i molecule collides with any molecule per unit time is

You can. However, the collision partner molecule j is generated by an equation (7) different from the actual method, and the number of molecules j is different from the true number.

And shall be. Here, the film-forming species to which the molecule j belongs is k (j), R _{k (j)} is the true concentration ratio, r _{k (j)} is the calculated concentration ratio based on the equations (6) and (7). (Occurrence ratio). However,

It is. Therefore, from the equations (8) and (9), the probability of intermolecular collision in the SARP method is

Can be obtained as It should be noted that the molecular velocity after collision in the case of collision according to equation (12) is handled as usual.

[4] Finally, consider the acceleration rate achieved by the SARP method. For this purpose, the ratio of the number of adhesions per hour by the conventional method and the number of adhesions by the SARP method (that is, how many times the adhesion number is increased) is used. Conversely, since the acceleration rate is directly proportional to the number of STEPs (or the ratio of the calculation progress time) until a certain number of deposits are deposited, the film formation shape at the same number of steps can be changed by changing the calculation conditions. Care must be taken when making a comparison or when the time of film formation is a problem. In addition, since the number of calculated STEPs and the calculation time are approximately proportional to each other, the acceleration rate here can be said to be an acceleration rate with respect to the conventional method of the time required for the calculation in practice. Assuming that there is no outflow of molecules out of the calculation area and carrier gas is present, the number of molecules generated in the conventional method is

The adhesion is the same ratio as this,

It occurs in. Where k is the film formation type.

On the other hand, in the SARP method, as shown by the equations (6) and (7), it is generated by the product (r _k ) of the concentration ratio and the adhesion probability. Since deposition occurs at a rate that is further 1 / eta _k times the r _k · _{η k,} the use of a carrier gas was 1/2 (7), the ratio of occurrence of adhesion is

It becomes. In other words, the conclusion is that the ratio between Eqs. (14) and (16) indicates the acceleration rate,

Get. For example, under the conditions in Table 1, it is about 330,000.

  What is characteristic is that this has the same value for any film forming species in the mixed gas, so that the acceleration rate changes depending on the film forming species regardless of the composition gas, that is, the SARP method is used. Does not lead to false results.

  The above is the concept of the SARP method, which is the basic idea of the present invention. In summary, the handling of molecules is different from the conventional method in three points: (1) adhesion on the film formation surface, (2) number of generations, and (3) intermolecular collision.

  Next, an analysis method of the present invention will be described with reference to FIG. 4, taking as an example the prediction of embedding of a thin film by CVD in the trench in the manufacturing process of the semiconductor device.

  First, assuming a model as shown in FIG. 1, the flow field (calculation region) is divided into a plurality of cells (step S1). Next, a molecule is generated and placed in each cell (step S2). Next, the molecular flow analysis (step S3) including the inflow of molecules according to Eq. (7) when the molecules are moved by Δt time using DSMC method (Step S3) and intermolecular collision (modified Nanbu method) ) Is calculated (step S4). Then, a film growth simulation corresponding to the adhesion of molecules to the trench as shown in the equation (4) is operated to grow the film (step S5). Steps S3 to S5 are repeated until the film grows to a desired level. Then, the calculation ends when the growth has reached the desired level.

  In order to efficiently perform parallel calculation on a parallel computer during the above calculation, the amount of communication between processors is reduced, and the calculation load of each processor is dynamically averaged (dynamic load distribution) according to the load. It is necessary, but if parallelization is performed for each molecule and molecular information is allocated to each processor separately, the information of molecules in the same cell is divided into several processors, which is complicated when extracting collision pairs. It is necessary to exchange information through communication between processors.

  Therefore, a method of dividing the calculation area into the number of processors and assigning each divided area to each processor in a one-to-one manner (area division method) is generally used, but the wall moves as in the object described here, When the analysis region changes as the calculation progresses, the dynamic load distribution method tends to be complicated. Therefore, in the case of such an object, it is preferable to employ a cell division method that can cope with a complicated change in the analysis region.

The cell division method utilizes the fact that the calculation of intermolecular collision is performed independently in each cell, and the entire region is divided into cells and parallelized by assigning them to each processor one after another. Data exchange is performed only between one parent processor and other child processors, and data management is performed by the parent processor and calculation is performed by the child processor. Each child processor runs the same program and simply repeats reception, calculation, and transmission. The parent processor repeats the process of giving data one cell at a time starting from the first cell to the child processor that has not performed the calculation or has completed the calculation, and receiving the return of the calculation result. When the calculation results of all the cells are received, one step is completed, and the process returns to the same operation from the first cell again. Since the load on the child processor is automatically averaged without the operator's awareness, parallelization is performed in the same way regardless of the number of processors and analysis areas, regardless of whether it is programmed or executed. Easy. In addition, no problem arises when there is anisotropy in the number of movements of molecules or when there are variations in performance among processors. Further, if the child processor is set so that only one cell is calculated at a time, the storage capacity can be suppressed to one cell, which can cope with a large-scale calculation as compared with the area division method. Next, the results of prediction of thin film embedding actually performed using the SARP method will be described. First, Table 2 shows the CVD conditions used in the simulation.

The conditions different from those in Table 1 are that simulation is practically impossible with the conventional method because the calculation time is enormous, and the adhesion probability is high under the conditions in Table 2, so This is because it can be compared with the SARP method. The film deposition contribution rate is set to be the same for both SiH ₄ and SiH ₂ . The deposition contribution rate is defined by S _k = η _k · P _k (17) for each deposition type.

  This film formation contribution rate is an index indicating the contribution of film formation types to film formation, and the degree of contribution of each film formation type to film formation can be relatively compared. FIG. 5A shows the result (film formation shape) analyzed by the conventional method based on the conditions of Table 2, and FIG. 5B shows the analysis by the SARP method based on the conditions of Table 2. The result (film formation shape) is shown. As can be seen from both figures, the conventional method and the SARP method show almost the same results, and it can be seen that there is no change in the results even with the SARP method. The acceleration rate under this condition is about 78 from equation (16).

Table 3 shows a comparison of calculation times obtained by actually measuring until a total of 1000 representative molecules adhere to the trench, and Table 4 shows a comparison of the number of repeated calculations.

In this case, in order to simplify the problem, only Ar and SiH _{4 in} Table 2 are treated, and the adhesion probability of SiH ₄ is used as a variable. In the table, (calculation time of the conventional method / acceleration rate of the SARP method) is a result obtained by dividing the calculation time or the number of times by the conventional method by the acceleration rate by the SARP method represented by the equation (16). This is a prediction of an ideal acceleration based on a conventional method. The SARP method and (computation time of the conventional method / acceleration rate of the SARP method) almost coincide with each other, and the validity of the above-described theory, particularly, the equation (17) is proved. The reason why the conventional method is not proportional to 1 / η is that there is a calculation for a certain period of time that is not related to the adhesion probability (the SARP method is slightly higher than (calculation time of the conventional method / acceleration rate of the SARP method)). This is probably because the number of molecules in the system is smaller when the probability of adhesion is large than when it is small (the number of molecules disappearing due to adhesion), and when the probability of adhesion is 1, the conventional method matches the SARP method. This is probably because the generation of molecules is performed by the product of the concentration ratio and the adhesion probability in the SARP method (that is, the generation ratio of attached molecules is higher in the SARP method). FIG. 7 shows a simulation result close to reality.

  This is the result of applying the SARP method to the CVD conditions in Table 5. The acceleration rate is 781250 from the equation (16), and the conventional method is predicted to require 1 to 10 years or more. As a whole, it can be seen that an almost accurate simulation is performed.

  The above is an example in which the present invention is applied to the prediction of thin film formation by CVD, but the present invention can be applied to the various fields described above by performing appropriate correction for each field.

The figure for demonstrating an example of the model used with the prediction method of this invention Illustration for explaining the basic concept used in the conventional prediction method Diagram for explaining the basic concept used in the prediction method of the present invention The flowchart which shows the process sequence by the prediction method of this invention. (a) is a diagram showing a result (film formation shape) analyzed by a conventional method, and (b) is a diagram showing a result (film formation shape) analyzed by a method according to the present invention. The figure which shows the result (film-forming shape) analyzed by the method concerning this invention

Claims (3)

When predicting the change by performing a numerical simulation using a direct simulation Monte Carlo method for an object whose state or property changes due to the action of a factor depending on the probability, the number of times of the action of the factor depending on the probability or the above When dealing with the amount of change in the object, the number of actions or amount of change handled in the calculation is changed to the actual number of actions or change without changing the factor or factor history calculation method in the direct simulation Monte Carlo method. A change prediction method characterized by handling more than quantity. When predicting the change by performing a numerical simulation using a direct simulation Monte Carlo method on an object or phenomenon that changes depending on the probability due to the action of a moving object colliding with another object, When calculating the number of actions of a factor depending on the probability or the amount of change of the object or phenomenon in the calculation, the calculation method of the factor or the history of the factor in the direct simulation Monte Carlo method is not changed. A change prediction method characterized in that the number of times of action of a factor or the amount of change of an object or phenomenon is handled more than the number of times of action of an actual factor or the amount of change of an actual object or phenomenon. When predicting the change by performing a numerical simulation using a direct simulation Monte Carlo method on an object or phenomenon that changes depending on the probability due to the action of a moving object colliding with another object, In the calculation of the number of action of the factor depending on the probability or the amount of change of the object or phenomenon in the calculation, the calculation method of the factor or the history of the factor in the direct simulation Monte Carlo method is not changed. A change predicting method characterized in that at least a part of a case where no action occurs or an object or phenomenon does not actually change is treated as acting or changing in calculation.

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