JP2001358138A - Method for simulating semiconductor oxidizing process, and program recording medium - Google Patents

Abstract

PROBLEM TO BE SOLVED: To (1) stably calculate the concentration distribution of an oxidization seed and the displacement of nodes by using the Newton's method at simulation of a semiconductor oxidizing process at the same time, and (2) to obtain a smooth impurity distribution by calculating the redistribution of impurities during oxidization, including an oxide film which has been newly formed through the oxidization. SOLUTION: By enabling the calculation of oxidization seed diffusion and the calculation of displacement and stress, on the same structure of the same mesh by performing the calculation of deformation and stress by forming nodes in the interface between silicon and the oxide film in double nodes and giving the amount obtained by subtracting the thickness of consumed silicon from the thickness of an oxide film newly formed between the nodes as a displacement difference and changing the shape, in accordance with the obtained displacement, and the forming the final shape by moving the silicon-side interface between the silicon and oxide film by the consumed quantity of the silicon and inserting the newly formed oxide film into the space produced between the oxide film and silicon. The smooth impurity distribution is obtained, by calculating the redistribution of impurities in the oxidizing process through diffusion calculation, including a moving boundary flux which is calculated from the oxidizing speed by representing the impurity concentration in the newly formed oxidized film by the concentration of the double nodes in the interface.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for simulating an oxidation step in semiconductor manufacturing.
More specifically, the present invention relates to a method for simulating the formation of an oxide film in an oxidation step and a method for simulating an impurity distribution in an oxidation step.

[0002]

2. Description of the Related Art In a silicon semiconductor manufacturing process, the surface of a silicon wafer or a polysilicon film formed on the wafer is partially thermally oxidized to electrically isolate elements from each other to form a silicon oxide film as an insulator. To form The oxidation simulation is a program for calculating the shape of an oxide film formed in such a thermal oxidation process. silicon
When (Si) changes to a silicon oxide film (SiO ₂ ) by thermal oxidation, the volume increases compared to before oxidation, but this volume expansion may cause stress in the device and adversely affect device characteristics. is there. In the oxidation simulation, the stress generated by such volume expansion during oxidation is also calculated.

[0003] First, thermal oxidation of silicon will be described. FIG. 10 is a view for explaining a thermal oxidation process of silicon by an electric furnace. As shown in FIG. 10, in the thermal oxidation of silicon, an oxidizing gas (O ₂ gas in the case of dry oxidation, H _{2 in} the case of pyrogenic oxidation) is placed in a heated electric furnace.
And a mixed gas of O ₂ and a silicon wafer inserted therein. The oxidizing species (O ₂ gas in the case of dry oxidation, H ₂ O in the case of pyrogenic oxidation) contained in the gas in the electric furnace is a nitride film as shown in FIG.
(Si ₃ N ₄ ) flows into the oxide film from the opening of the oxidation resistant mask patterned on the wafer formed of a material such as (Si ₃ N ₄ ), diffuses in the oxide film, reaches the interface between Si and SiO ₂ , Reacts to form new SiO ₂ .

FIG. 12 is a diagram showing movement of an interface due to oxidation. The newly formed oxide film pushes up the originally existing oxide film as shown in FIG. At this time, if a part of the oxide film is fixed, deformation occurs, and stress is generated.
When a new oxide film is formed, silicon is consumed at the same time. The volume ratio between the consumed silicon and the newly formed oxide film is 1 with respect to silicon α = 0.44.

The rate at which an oxide film is formed (oxidation rate) is Si
And determined by oxidizing species concentration at the interface between SiO _2. This is shown in FIG. The higher the concentration of oxidizing species at the interface, the higher the oxidation rate, and the lower the concentration of oxidizing species, the lower the oxidation rate. FIG. 13 shows a situation in which the concentration of the oxidizing species decreases under the oxidation-resistant mask, the oxidation rate decreases, and the resulting oxide film becomes thinner.

Next, the calculation procedure of the oxidation simulation will be described. FIG. 14 shows a calculation flow of the oxidation simulation. First, in the first step, the distribution of the concentration of the oxidizing species in the oxide film is calculated to determine the oxidation rate. The distribution of the oxidizing species concentration is determined by the diffusion equation of the oxidizing species in the oxide film (formula 1 below).
Solving for Where C is the concentration of the oxidizing species, t is the time, D
Is the diffusion coefficient of the oxidizing species. In an actual simulation, the diffusion of the oxidizing species is fast, and the concentration distribution almost reaches a steady state. From the concentration distribution of oxidizing species obtained by solving this equation, according to the oxidizing species concentration at the interface between Si and SiO ₂ ,
Calculate different oxidation rates for each location.

[0007]

(Equation 1)

When the oxidation rate at the Si / SiO ₂ interface is determined, a new SiO ₂ film can be formed according to the oxidation rate.
Insert at the i / SiO ₂ interface (there are several ways to insert this new oxide, as described below).

[0009] In the second step of the oxidation simulation, the displacement (the amount of material movement, which causes a strain at different locations) and stress caused by inserting a newly formed oxide film are derived from the principle of virtual work. Calculate by solving the governing equation. Generally, a finite element method is used as a numerical solution for solving a governing equation, but when a finite element method is used for a numerical solution, a discretized governing equation is

[0010]

(Equation 2)

## EQU1 ## Here, B is a strain displacement matrix, and the superscript T indicates a transposed matrix. f is a nodal force vector, which includes internal nodal forces and nodal reaction forces due to nonlinear distortion. σ is the element stress and ∫dΩ represents the area integral.
Although not explicitly shown in Equation 3, the unknown in the equation is the nodal displacement.

In the third step, the shape of the element is updated in accordance with the displacement obtained by solving the governing equation. By repeating the above procedure, the final shape of the oxide film and the stress distribution in the element can be obtained.

In the oxidation simulation, it is necessary to insert a new oxide film formed by oxidation between the originally existing oxide film and silicon. There are several methods for this. FIG. 15 shows a conventional method (transition region method).
1 shows a first method of inserting a new oxide film according to the present invention. In this first method, a transition region where silicon is newly changed to an oxide film is set according to the concentration of oxidizing species at the Si / SiO ₂ interface. Then, the transition region is expanded, and the displacement amount and the generated stress are calculated. This method will be referred to as “transition region method”.

FIG. 16 shows a conventional method (forcible displacement method).
2 shows a second method of inserting a new oxide film according to the present invention. In this second method, a forced displacement is given to the Si / SiO ₂
Calculate the stress. _Assuming that the oxidation rate is v _ox , the amount by which the oxide film originally formed by the oxide film formed during the time Δt is pushed up is (1−α) v _ox Δt. This pushing amount is given as a forced displacement to the interface of the oxide film, and the deformation amount and the stress are calculated. At this time, the silicon region does not move as a rigid body. In this case, the stress in silicon is not obtained, but if you want to know the stress distribution in silicon,
The node reaction force on the Si / SiO ₂ interface obtained when calculating the deformation of the oxide film is applied to the Si region, and the displacement and stress are calculated only in the Si region. This method will be referred to as a “forced displacement method”.

FIG. 17 shows a conventional method (push-back method).
3 shows a third method of inserting a new oxide film according to the present invention. In the third method, a newly formed oxide film is virtually added to an originally existing oxide film. In addition, silicon is cut by the amount consumed by oxidation. With this alone, the regions overlap by the volume expansion, but the deformation amount and the stress are calculated by applying the boundary condition so that the overlap does not occur. This method will be referred to as a “push-back method”.

The above three methods have problems.
In the actual oxidation process, the diffusion of the oxidizing species and the oxidation reaction are affected by stress. The diffusion of oxidizing species and the oxidation reaction at the Si / SiO ₂ interface are suppressed by compressive stress. These effects are typically modeled as follows.

[0017]

(Equation 3)

Here, D is the diffusion coefficient of the oxidizing species (specifically,
D ₀ is the diffusion coefficient of the oxidizing species in the absence of stress, p is the hydrostatic pressure in the oxide film, k
Is the Boltzmann constant, T is the temperature, and V _d is the activation volume of the oxidizing species diffusion coefficient. k _s is the surface reaction rate constant between the oxide species and the silicon, k _s0 interfacial reaction rate constant when the stress does not work, s _n is the vertical stress to the interface, V _s is interfacial reaction rate constant Is the activation volume of

As described above, the diffusion coefficient and the interface reaction rate constant, which are parameters necessary for solving the diffusion of the oxidizing species, are parameters dependent on the stress value. (Therefore, nodal displacement amount) is required. Therefore, a discretized system of equations for solving oxidizing species diffusion is

[0020]

(Equation 4)

## EQU2 ## Here, Y is a set of N equations (N is the number of nodes), C is the concentration of oxidizing species at the nodes, a column vector of N rows, and a is a dimension number × N rows of node displacement vectors.
The reason why the diffusion equation of the oxidizing species includes not only the oxidizing species concentration C but also the nodal displacement a is that the diffusion coefficient and the interface reaction rate constant depend on the stress.

On the other hand, in the displacement / stress calculation, the amount of deformation / stress is calculated according to the thickness of the newly formed oxide film. The thickness of the newly formed oxide film depends on the concentration C of the oxidizing species. Therefore, in order to perform displacement / stress calculation, an oxidizing species concentration distribution is required. Therefore, the equation system discretized to calculate displacement and stress is

[0023]

(Equation 5)

## EQU2 ## Here, Φ is a set of dimensions × N equations. Here, Equation 7 is the same as Equation 3. The equation of displacement / stress includes not only the nodal displacement a but also the oxidizing species concentration C because the amount of deformation depends on the thickness of the newly formed oxide film.

FIG. 18 shows an image of a discretized equation system. There are two methods for obtaining the density distribution C and the nodal displacement amount a that simultaneously satisfy both Expressions 6 and 7. The first method is called the “method by numerical relaxation”. FIG. 19 shows a calculation procedure by numerical relaxation. First, the node displacement a is fixed, and only Equation 6 is solved to determine the correction amount of the oxidizing species concentration C. The superscript (n) represents the value in the n-th iteration. Next, the oxidizing species concentration C is updated using the corrected amount, and the corrected amount of the nodal displacement amount a is calculated by solving Equation 7 using the updated C. Further, the node displacement a is updated using the obtained correction amount, and the correction amount of the oxidation head ° C is obtained again using the updated node displacement. These operations are repeated, and when the solution no longer changes, it is determined that a convergent solution has been obtained, and the calculation is terminated. If the correction amount is added as it is at each iteration, the convergence of the solution is poor, so that a correction smaller than the obtained correction amount is usually performed (relaxation). This method is easy to implement in a program, but has the disadvantage of poor convergence.

The second method is a method called "Newton method". FIG. 20 shows a calculation procedure by the Newton method. Newt
In the on method, equations 6 and 7 are simultaneously solved to determine the amount of solution correction. This method has the advantage of good convergence.

Next, a method of simulating impurity diffusion in the oxidation step of the semiconductor manufacturing process will be described. The general-purpose process simulator is a simulator that handles not only an oxidation process but also a comprehensive process process including ion implantation and thermal diffusion. The oxidation step involves a heat treatment, during which the impurities are redistributed. Therefore, when the oxidation process is simulated by the general-purpose process simulator, a change in the element shape is calculated by the oxidation simulation, and a diffusion simulation for solving the diffusion equation of the dopant impurity is performed.

FIG. 21 shows a general procedure for calculating the redistribution of impurities in the oxidation step by diffusion simulation.
Shown in FIG. 21 shows an example in which a transition region method is used as a method for inserting a new oxide film. First, FIG.
For the initial shape of (a), a transition region is set as shown in FIG. 21 (b) using the oxidizing species concentration distribution obtained as a result of the oxidizing species diffusion calculation. At this time, the impurity concentration in the transition region is set by interpolation from the impurity concentration in silicon.

Subsequently, as shown in FIG. 21C, diffusion calculation is performed including the transition region. Finally, as shown in FIG. 21D, the shape is updated using the result of the displacement / stress calculation. At the time of shape update, the concentration of impurities in the transition region is reduced so that the total amount of impurities is preserved according to the volume expansion of the transition region.

FIG. 22 shows a change in the impurity concentration distribution along the line AA ′ in FIG. 21 during the above procedure. FIG.
(A) is an impurity distribution at the time of completion of the oxidation species diffusion calculation. The interface between the silicon and the oxide film is a double node, causing a concentration difference due to segregation. FIG. 22B shows the impurity concentration after setting the transition region according to the oxidizing species concentration at the Si / SiO ₂ interface. The impurity concentration in the transition region is interpolated from the impurity concentration in silicon. At this time, the interface between the newly formed transition region and the silicon region is also changed to a double node.

Subsequently, diffusion calculation including the transition region is performed using the density distribution of FIG. 22B as an initial distribution. In this diffusion calculation, the impurity diffusion coefficient in the transition region has the same value as that of the oxide film. Also, between the transition region and the double node of silicon,
Consider the impurity flux F _seg due to segregation shown in the following equation.

[0032]

(Equation 6)

Here, C _A ^trans and C _A ^Si are the concentrations of the impurity A on the transition region side and the oxide film region side at the interface between the transition region and silicon. H and m are the mass transport coefficient and the segregation coefficient, respectively. FIG. 22C shows the concentration distribution at the end of the impurity diffusion calculation. A concentration gap occurs due to segregation at the interface between the transition region and silicon. Further, the concentration gap at the interface of the oxide film originally formed in the transition region is left as it is. Subsequently, FIG. 22D shows the concentration distribution after the shape is updated using the result of the displacement / stress calculation. At this time, according to the volume expansion of the transition region, the impurity concentration in the transition region is reduced so that the total amount of impurities is preserved.

[0034]

In the conventional oxidation simulation, the "method of inserting a new oxide film"
Any of the three methods has the following problems. First, in the case of the transition region method, comparing the region at the time of performing the oxidized species diffusion calculation with the region at the time of performing the displacement / stress calculation, a transition region is added to the latter, and the silicon region is also oxidized species. The shape of the region has changed compared to when the diffusion calculation was performed. When a region is added or the shape of the region changes, it is necessary to regenerate a mesh used for numerical calculation. Because regenerating a mesh takes a long time, it is not practical to regenerate a mesh at each iteration. Redrawing the mesh in the middle of the iteration means that the number of nodes changes in the middle of the iteration (that is, the number of unknowns also changes), which makes it difficult to perform the iterative calculation. Also, it is expected that the convergence will be adversely affected if the mesh changes at each iteration. Therefore, when the transition region method is used, the stability is high.
There is a problem that the Newton method cannot be applied.

Next, in the case of the forced displacement method, there are the following problems. Consider a floating polysilicon (polysilicon is floating in an oxide film) structure as shown in FIG.
An (gate) oxide film is formed on a silicon substrate, a gate electrode made of polysilicon is formed thereon, and an oxide film has already been formed on the upper portion and the side wall of the gate electrode.

Consider the change in the portion surrounded by the broken line in FIG. 23 when the structure is further subjected to additional oxidation. When the structure of FIG. 23 is oxidized,
The shape of the broken line should be exactly as shown in FIG. That is, the silicon in contact with the Si / SiO ₂ interface is consumed to the position indicated by the broken line by the oxidation reaction, and the volume expands about 1 / α times. Similarly, at the interface between the polysilicon forming the gate electrode and SiO ₂ , the polysilicon is consumed up to the position indicated by the broken line to form an oxide film, and the volume expands by about 1 / α. The gate electrode is pushed up by the newly formed oxide film between the gate and the substrate silicon. An oxide film is newly formed also on the upper surface of the gate.

However, when the forced displacement method is used, the calculated shape is as shown in FIG. Also in this case, silicon and polysilicon are consumed up to the positions indicated by broken lines. However, since silicon and polysilicon are not deformed (and do not move), a forced displacement is applied to the interface on the oxide film side, so that the gate electrode is not pushed up by the newly formed oxide film. Therefore, the distance between the gate electrode and the substrate silicon becomes narrower than it actually is. Further, since the originally existing oxide film is prevented from expanding in the vertical direction, a larger stress is generated. This is different from the actual situation. Therefore, the forced displacement method has a problem that the oxidation of the floating polysilicon layer cannot be properly handled.

The problems in the push-back method are the same as those in the transition area method. When the push-back method is used, it is necessary to extend the oxide film region once according to the oxidation rate calculated from the oxidizing species concentration. In this case, the positions of the nodes change, and in some cases, it is necessary to regenerate the mesh. Therefore, there is a problem that the stable Newton method cannot be used for the same reason as in the case of the transition region method.

Next, the "calculation method of impurity diffusion" during oxidation in the conventional simulation method of the oxidation step of the semiconductor manufacturing process has the following problems. First,
In order to reduce the impurity concentration in the transition region so that the total amount of impurities is preserved, a gap occurs in the concentration distribution during oxidation as shown in FIG. 22D, and the distribution in the oxide film is changed every time the oxidation calculation is repeated. The problem of sticking occurs. Further, in the impurity diffusion calculation, the Si / SiO ₂ shown in FIG.
Considering segregation only at nodes on the interface, FIG.
There is also a problem that the segregation is not taken into account for impurities taken into the transition region in the process of (b) interpolation. Therefore,
In order to more accurately calculate the impurity diffusion during the oxidation step and obtain a smooth impurity distribution, a better impurity diffusion calculation method was required. The present invention has been made to solve the above-mentioned conventional problems, and has as its object to obtain an improved method for simulating an oxidation step in a semiconductor manufacturing process.

[0040]

According to a first aspect of the present invention, there is provided a method of simulating a semiconductor oxidizing step, which simulates the formation of an oxide film in a semiconductor manufacturing process. The seed concentration distribution is obtained, (2) the oxidation rate is calculated from the concentration distribution of the oxidizing species at the interface between the silicon and the oxide film, and (3) the thickness of the newly formed oxide film is calculated based on the calculated oxidation rate. Calculating, (4) inserting the newly formed oxide film between the existing oxide film and silicon, and (5) inserting the newly formed oxide film into the material and the displacement of the material. In the oxidation simulation to calculate the stress generated by the simulation, (6) the finite element method is used for the numerical solution for the diffusion of oxidized species and the calculation of displacement and stress.
(7) Using the nodes on the interface between silicon and the oxide film as double nodes, the displacement difference is the amount obtained by subtracting the thickness of the consumed silicon from the thickness of the newly formed oxide film between these double nodes. Calculate displacement and stress by giving
(8) The object to be analyzed is deformed based on the obtained displacement amount,
(9) Calculation of moving the interface between the original silicon and the oxide film by the thickness of the consumed silicon, and (10) filling all the gaps between the generated oxide film and the silicon with the newly formed oxide film. The method is characterized in that a simulation of oxide film formation is performed by a method.

According to a second aspect of the present invention, there is provided a method of simulating a semiconductor oxidizing step, which simulates an impurity distribution in an oxidizing step of a semiconductor manufacturing process. Simulation, which is obtained by solving a diffusion equation by using the finite element method or the finite element method;
2) The concentration distribution of the oxidizing species is obtained by solving the diffusion equation of the oxidizing species, (23) the thickness of the newly formed oxide film is calculated from the obtained oxidizing species concentration, and (24) the newly formed oxidation The film is inserted between the existing oxide film and the silicon, and (25) a combination of the amount of material displacement caused by inserting the newly formed oxide film and the oxidation simulation for calculating the stress generated by the material displacement. (26) In the simulation for calculating the redistribution of impurities during the oxidation step, (27) the node giving the impurity concentration on the interface between the silicon and the oxide film is a double node, and the impurity flow generated by segregation between these nodes. The diffusion equation is solved in consideration of the flux and the flux of impurities determined from the oxidation rate and flowing out of the system from these double nodes. (28) The newly formed oxide film Is set to a value equal to the oxide concentration on the oxide film side obtained as a result of the diffusion calculation, and (29) the impurity concentration at the silicon side node newly formed on the interface due to consumption of silicon is The method is characterized in that the impurity distribution is simulated by a calculation method in which interpolation is performed based on the impurity distribution in silicon before the interface moves.

According to a third aspect of the present invention, there is provided a method of simulating a semiconductor oxidizing step, which simulates an oxidizing step in a semiconductor manufacturing process.
(32) solving the diffusion equation of the oxidizing species by a numerical solution using the finite element method to obtain an oxidizing species concentration distribution;
(33) calculating the oxidation rate from the concentration distribution of the oxidizing species at the interface between the silicon and the oxide film; and (33) setting the node on the interface between the silicon and the oxide film as a double node and newly forming between the double nodes. Calculating a displacement amount and a stress by giving an amount obtained by subtracting the thickness of the consumed silicon from the thickness of the oxide film as the displacement difference;
4) deforming the object to be analyzed based on the obtained displacement amount, and moving the interface between the original silicon and the oxide film by the thickness of the consumed silicon; and (35) forming the oxide film and the silicon And filling the gaps with a newly formed oxide film. A simulation of oxide film formation is performed.

According to a fourth aspect of the present invention, in the method of simulating a semiconductor oxidation step, the step of calculating the displacement is performed under a condition that the nodal reaction forces acting on the double nodes are balanced. It is characterized by the following.

According to a fifth aspect of the present invention, there is provided a method of simulating a semiconductor oxidizing step, which simulates an impurity distribution in an oxidizing step of a semiconductor manufacturing process, wherein (51) an impurity concentration on an interface between silicon and an oxide film is given. Nodes are double nodes, considering the impurity flux generated by segregation between them and the impurity flux determined from the oxidation rate and flowing out of the system from these double nodes,
Calculating an impurity diffusion by solving a diffusion equation by a numerical method using a finite difference method or a finite element method;
(52) a step of obtaining a concentration distribution of the oxidizing species by solving a diffusion equation of the oxidizing species; (53) a step of calculating a thickness of a newly formed oxide film from the obtained oxidizing species concentration; (54) Inserting a newly formed oxide film between the existing oxide film and silicon; (55)
Setting the impurity concentration in the newly formed oxide film to a value equal to the interface concentration on the oxide film side obtained as a result of the diffusion calculation; and (56) on the interface newly formed by the consumption of silicon. Interpolating and setting the impurity concentration at the silicon side node from the impurity distribution in the silicon before the interface is moved, thereby simulating the impurity distribution.

According to a sixth aspect of the present invention, there is provided a program recording medium for simulating an oxidation step in a semiconductor manufacturing process, wherein (61) a diffusion equation of an oxidizing species is solved by a numerical solution method using a finite element method. (62) a process of calculating the oxidation rate from the concentration distribution of the oxidizing species at the interface between the silicon and the oxide film, and (63) a node on the interface between the silicon and the oxide film as a double node. (64) obtaining a displacement amount and a stress by giving as a displacement difference an amount obtained by subtracting the thickness of the consumed silicon from the thickness of the oxide film newly formed between the double nodes of the step (64). A process of deforming the analysis target based on the obtained displacement amount and moving the interface between the original silicon and the oxide film by the thickness of the consumed silicon;
5) A computer-readable program storing a program for causing a computer to execute a process of simulating the formation of an oxide film including a process of filling the gap between the generated oxide film and silicon with a newly formed oxide film. It is a program recording medium.

According to a sixth aspect of the present invention, there is provided a program recording medium for simulating an impurity distribution in an oxidation step of a semiconductor manufacturing process, wherein (71) two nodes giving an impurity concentration on an interface between silicon and an oxide film. Using the finite difference method or the finite element method in consideration of the impurity flux generated by segregation between these nodes and the impurity flux determined from the oxidation rate and flowing out of the system from these double nodes. (72) a process of calculating the impurity diffusion by solving the diffusion equation by a numerical solution method, (72) a process of obtaining the concentration distribution of the oxidizing species by solving the diffusion equation of the oxidizing species, and (73) a process of newly obtaining from the obtained oxidizing species concentration. A process for calculating the thickness of the oxide film to be formed; (74) a process for inserting the newly formed oxide film between the existing oxide film and silicon; A process of setting the impurity concentration in the formed oxide film to a value equal to the interface concentration on the oxide film side obtained as a result of the diffusion calculation; and (76) a process for forming a new interface by consuming silicon. Computer-readable recording of a program for causing a computer to execute a process of simulating an impurity distribution including a process of setting an impurity concentration of a silicon side node by interpolation from an impurity distribution in silicon before the interface moves. It is a recording medium.

[0047]

DESCRIPTION OF THE PREFERRED EMBODIMENTS Embodiment 1 A method for simulating the formation of an oxide film in a semiconductor manufacturing process according to the first embodiment of the present invention will be described. FIG.
5 is a flowchart of a simulation of oxide film formation according to the present embodiment. FIG. 2 is a view for explaining a method of inserting a new oxide film. 1 and 2
As shown in FIG. 1, first, the diffusion calculation of the oxidizing species is performed to obtain the concentration distribution of the oxidizing species (FIG. 1, step 31), and the oxidation rate at the interface is calculated (FIG. 1, step 32) (see FIG. 2 (a)). ). Assuming that the size of one time step for updating the shape is Δt and the local oxidation rate is v _ox , the thickness of the newly formed oxide film during one shape update is v _ox Dt .
At this time, the thickness of the consumed silicon is av _ox Dt,
The amount by which the originally existing oxide film is pushed up is (1-a) v _ox
Dt.

In the method of the present invention, nodes on the oxidized Si / SiO ₂ interface (or the interface between polysilicon and SiO ₂ ) are double nodes as shown in FIG. A constraint condition (Equation 9 below) is imposed that the difference is (1-a) v _ox Dt, and a balance condition between nodal forces R _i and R _j acting on nodes i and j (Equation 10 below). Also impose. The displacement and stress are calculated under these constraint conditions (FIG. 1, step 33) (see FIG. 2 (b)).

[0049]

(Equation 7)

The oxide film is deformed according to the obtained displacement as shown in FIG. Furthermore, in the silicon side region, according to the oxidation rate v _ox obtained by the calculation of the oxidation species, FIG.
The interface is moved by αv _ox Δt as shown in FIG.
Step 34) (see FIG. 2 (c)). Finally, the gap formed between the silicon region and the oxide film region is filled as an oxide film region as shown in FIG. 2D, thereby completing the final oxide film shape (FIG. 1, step 35) (FIG. 1). 2 (d)).

(From the original problem solving means) As described above, the simulation method of the present invention is particularly characterized by the following steps. That is, in the simulation method according to the present invention, (a) nodes on the interface between the silicon and the oxide film where oxidation has occurred are set as double nodes, and (b) displacements of these nodes are caused by the newly formed oxide film. Thick (1
-α) times (α is the volume ratio of silicon to the oxide film) and a condition that the nodal reaction forces acting on these nodes balance each other to obtain the amount of displacement (step 33), c) The shape is updated according to the obtained displacement amount (Step 34), and (d) a newly formed oxide film is inserted into the gap generated by the shape update (Step 3).
5). This method is referred to as a “displacement difference method”.

When an oxidation simulation is performed by the "displacement difference method" of the present invention, both the calculation of diffusion of oxidized species and the calculation of displacement / stress are performed on the structure shown in FIG. Therefore, there is no need to redraw the mesh in the process of Newton iteration, and the stable distribution of oxidized species is achieved using the Newton method.
There is an effect that C and the nodal displacement a can be obtained at the same time.

When a newly formed oxide film is formed by the “displacement difference method” of the present invention, the region for calculating the diffusion of oxidized species and the region for calculating the displacement and stress can be set to be the same. There is an effect that the oxidized species concentration distribution and the nodal displacement can be simultaneously obtained by using the stable Newton method. Further, according to the method of the present invention, since the displacement / stress calculation is performed without fixing the interface on the silicon (or polysilicon) side, even when the oxidation calculation of the structure including the floating polysilicon region is performed, As the polysilicon region is pushed up by the newly formed oxide,
There is an effect that it can be handled correctly.

A method similar to the present invention is introduced in the following reference: References: S. Isomae et. Al, IEEE Tran
s. Computer-Aided-Design, CAD-6, (1987), p. 410
(Hereinafter simply referred to as references). In this reference as well, the nodes at the Si / SiO ₂ interface are regarded as double nodes, and a difference in the amount of displacement is given between these nodes. However, the differences between the reference and the method of the present invention are as follows. The first difference is
In the reference, the stress is calculated after changing the consumed oxide film portion to an oxide film, whereas in the method of the present invention, the deformation / stress calculation is performed with the consumed silicon region remaining as silicon, and then Is that the silicon region consumed in the step is changed to an oxide film. If the silicon portion consumed before performing the deformation / stress calculation is changed to an oxide film, the shape of the region changes, so that it is necessary to renew the mesh. If the mesh is re-stretched, the number of nodes changes, so that it is impossible to simultaneously solve the oxidizing species concentration distribution and the stress distribution.

The second difference is that this reference uses the boundary element method for diffusion of oxidizing species and calculation of deformation / stress, whereas the method of the present invention uses the finite element method. The boundary element method has an advantage that it is not necessary to divide the inside of the region into elements because diffusion and deformation / calculation of oxidized species are performed only at nodes on the region boundary. On the other hand, since the boundary element method is a solution in which a homogeneous material is symmetric, it cannot be applied when the diffusion coefficient differs depending on the place. Further, in the oxidation simulation, the oxide film is treated as a viscoelastic body, but the boundary element method cannot deal with the viscoelastic problem in the case where stress is initially present. On the other hand, according to the method of the present invention, the finite element method is used for the calculation of the diffusion of oxidizing species and the deformation / stress, so that the case where the diffusion coefficient differs depending on the place can be handled, and the stress is previously stored in the region. There is an advantage that viscoelastic analysis can be performed even when it exists.

In this embodiment, a two-dimensional oxidation simulation has been described as an example, but it goes without saying that the same technique can be applied to one-dimensional and three-dimensional oxidation simulations. Further, in this embodiment, the description has been made using the pair of silicon and the oxide film, but the pair of polysilicon and the oxide film, or a similar interface between the first material and another second material, A reaction that forms a second material by reacting the first material with a reactive species that has diffused through the material and reached the interface with the first material (for example, Ti and Si
Of reaction to form titanium silicide by reaction and reaction of Co and Si to form cobalt silicide).

As described above, in the method of simulating the formation of an oxide film according to the present invention, the nodes on the interface between silicon and the oxide film are double nodes, and the consumption of the oxide film newly formed between these nodes is reduced. By giving the amount obtained by subtracting the thickness of the silicon as the displacement difference,
Perform stress calculation. Then, the shape is changed in accordance with the obtained displacement amount, the interface between the silicon and the oxide film on the silicon side is moved by the consumption amount of silicon, and a newly formed oxide film is inserted into a gap created between the oxide film and the silicon. To form the final shape. This makes it possible to perform the diffusion calculation of the oxidation species and the displacement / stress calculation on the same structure of the same mesh, and as a result, it becomes possible to apply the stable Newton method.

Next, a description will be given of a computer process for simulating the formation of an oxide film according to the present invention. The simulation of the present invention described above is executed by a computer. For this purpose, each step or each process is described as a computer-readable program. The present invention also provides a recording medium on which such a computer-readable program is recorded. Since each step or each process of the program for executing the present invention is as described above, repeated description will be omitted.

Embodiment 2 Hereinafter, a method for simulating the impurity distribution in the oxidation step of the semiconductor manufacturing process according to the second embodiment of the present invention will be described. FIG. 4 is a flowchart of a simulation of impurity distribution according to the present embodiment. FIG. 5 is a diagram for explaining a method of calculating impurity diffusion. In the present embodiment, first, impurity diffusion is calculated for the initial shape of FIG. 5A (FIG. 4, step 51) (see FIG. 5A). At this time, at the interface between the silicon and the oxide film, a node that solves the impurity diffusion is a double node. In addition to the segregation flux (formula 11 below), the two nodes from the outside of the analysis region are added between the double nodes. Consider a moving boundary flux F _mb (Equation 12 below) that flows in (exactly flows out because it is a negative value).
FIG. 6 is a diagram for explaining a diffusion calculation method including the moving boundary flux. In Equations 12 and 13 below, C _A ^{SiO 2} and C _A ^Si are the concentrations of the impurity A on the oxide film side and the silicon side on the Si / SiO ₂ interface, respectively.

[0060]

(Equation 8)

Next, the concentration distribution of the oxidizing species is obtained by solving the diffusion equation of the oxidizing species (FIG. 4, step 52).
The thickness of the newly formed oxide film is calculated from the oxidation rate (FIG. 4, step 53). Subsequently, as shown in FIG. 5B, the oxide film shape is updated based on the nodal displacement,
The side interface is moved (see FIG. 5B). Finally, a new oxide film is formed in the gap between the originally formed oxide film and silicon (FIG. 4, step 54). The impurity concentration in this oxide film is set to a value equal to the impurity concentration on the oxide film side on the Si / SiO ₂ interface at the end of the impurity diffusion calculation (FIG. 4, step 5).
5) (see FIG. 5 (c)). At this time, the impurity concentration at a new node formed on the moved Si interface is determined by interpolation from the impurity concentration in silicon (FIG. 4, step 56).
FIG. 7 shows a change in impurity distribution along BB ′ in FIG. 7A and 7B are diagrams showing a method of setting the impurity concentration in the oxide film newly formed by the moving boundary flux method. FIG. 7A shows the state at the end of the impurity diffusion calculation, and FIG.
Indicates the concentration distribution after the oxide film is inserted. As shown in FIG. 7B, the impurity concentration in the newly formed oxide film is set to the oxide film-side interface concentration C _A ^{SiO 2} at the end of the diffusion calculation. The impurity distribution in the silicon region is interpolated from the impurity distribution after the diffusion calculation.

As described above, the simulation method of the present invention is particularly characterized by the following steps. That is,
In the simulation method according to the present invention, (a) a node at the interface between the silicon and the oxide film is a double node, (b) the concentration in the newly formed oxide film at the node on the oxide film side is represented, (c) The impurity diffusion is calculated in consideration of the moving boundary flux generated by the formation of the oxide film (FIG. 4, step 5).
1), (d) The finally obtained interface concentration on the oxide film side is set as the concentration in the newly formed oxide film (FIG. 4,
Step 55). This method is referred to as “moving boundary flux method”.

In the method of calculating impurity diffusion according to the “method of moving boundary flux” of the present invention, the concentration of impurities contained in a newly formed oxide film is included in the analysis. At the same time, it is possible to obtain an impurity distribution in which the total amount is preserved and changes smoothly at the Si / SiO ₂ interface,
There is an effect that a more accurate impurity distribution can be calculated by reflecting all the impurity concentrations in the newly formed oxide film in the segregation flux.

Next, setting of impurity concentration in the case of two-dimensional calculation
The method will be described. FIG. 8 shows how to set the impurity concentration in this case.
FIG. 8A is a diagram for explaining the method._TwoInterface transfer
FIG. 8B shows a state before the movement and after the movement. In FIG. 8 (b)
As shown, the moved Si / SiO _TwoNodes on the interface must be the first
Si / SiO_TwoSince there is a corresponding node on the interface,
The setting of the pure substance concentration can be performed in the same manner as the one-dimensional calculation. Here vs
A corresponding node is a set of double nodes with a displacement difference. one
, New Si / SiO_TwoNodes on interfaces and newly formed acids
For the nodes inside the oxide film, the initial Si / SiO_TwoOn the interface
Does not have a corresponding node. In such a case,
The first one closest to the node for which you want to set the density
Si / SiO_TwoSet the impurity concentration of the node on the interface to the node being set.
Shall be set as the density.

The method of the present invention (moving boundary flux method) is based on the following considerations. FIG. 9 is a diagram for explaining the principle of the impurity diffusion calculation. First, as shown in FIG. 9, considering the control volumes Ω _SiO2 and Ω _si on the oxide film side and the silicon side node, the impurity concentration in the newly formed oxide film is represented by the impurity concentration C _A ^{SiO 2} on the oxide film on the interface side. . At this time conservation law of the amount of impurities including the oxide film newly formed oxide velocity as v _ox,

[0066]

(Equation 9)

Is obtained. Here, J _SiO2 and J _Si are an impurity flux flowing out of the control volume Ω _SiO2 to the oxide film side and an impurity flux flowing out of the control volume Ω _Si to the silicon side, respectively. The third term on the left side of Expression 13 is the total amount of impurities contained in the newly formed oxide film corresponding to the hatched portion (shaded portion) in FIG. By dividing Equation 13 into a continuous equation in each region of the oxide film and silicon, and adding a flux F _seg due to segregation, the following Equation 14 is obtained.
And Equation 15: The sum of the first term on the left side in Expressions 14 and 15 is the impurity flux corresponding to the moving boundary flux.

[0068]

(Equation 10)

In this method, since the diffusion is solved by representing the impurity concentration in the newly formed oxide film by the concentration at the node of the interface, the operation of reducing the impurity concentration by volume expansion is not required, and the newly formed oxide film becomes unnecessary. Even after setting the impurity concentration in the formed oxide film, no gap is formed in the impurity distribution as shown in FIG. In addition, since the segregation flux is calculated by representing the impurity concentration in the newly formed oxide film by the concentration at the interface node, this region can be obtained without actually providing a newly formed oxide film region. There is an effect that the diffusion calculation can be performed in consideration of the effect of segregation for all the impurities taken into the semiconductor device.

In this embodiment, the diffusion of dopant impurities has been described as an example, but it goes without saying that the method of the present invention can be applied to the diffusion of interstitial silicon, vacancies, impurity clusters, and the like.

As described above, in the simulation of the impurity distribution according to the present invention, the impurity concentration in the newly formed oxide film is represented by the concentration of the double node on the interface, and the movement calculated from the oxidation rate. By performing the diffusion calculation including the boundary flux, the redistribution of the impurities during the oxidation step can be calculated to obtain a smooth impurity distribution.

Next, a description will be given of a computer process for simulating the impurity distribution according to the present invention. The simulation of the present invention described above is executed by a computer. For this purpose, each step or each process is described as a computer-readable program. The present invention also provides a recording medium on which such a computer-readable program is recorded. Since each step or each process of the program for executing the present invention is as described above, repeated description will be omitted.

[0073]

As described above, according to the method of inserting a newly formed oxide film in the method of the present invention, a new oxide film is formed in a gap where a double node on the interface between silicon and an oxide film can be given a displacement difference. By adopting the method of inserting a film, it becomes possible to calculate the diffusion of oxidized species and the calculation of displacement stress for the same structure of the same mesh. There is an effect that the node displacement amount can be calculated stably. Further, in the method of moving boundary flux in the method of the present invention, by including the moving boundary flux calculated from the oxidation rate in the impurity diffusion calculation, the impurity diffusion calculation considering the impurities contained in the newly formed oxide film is performed. And it is possible to obtain a smooth impurity distribution.

[Brief description of the drawings]

FIG. 1 is a flowchart of a simulation (method of displacement difference) of oxide film formation according to the first embodiment of the present invention.

FIG. 2 is a view for explaining a method of inserting a new oxide film by the method of the present invention (method of displacement difference).

FIG. 3 is a diagram showing a displacement difference given to a double node.

FIG. 4 is a flowchart of a simulation (method of moving boundary flux) of impurity distribution according to the second embodiment of the present invention.

FIG. 5 is a view for explaining a method of calculating impurity diffusion in the method of the present invention (method of moving boundary flux).

FIG. 6 is a view for explaining a method of diffusion calculation including a moving boundary flux in the method of the present invention.

FIG. 7 is a view showing a method of setting an impurity concentration in a newly formed oxide film by the method of the present invention (moving boundary flux method).

FIG. 8 is a diagram illustrating a method for setting an impurity concentration according to the present invention.

FIG. 9 is a view for explaining the principle of impurity diffusion calculation in the present invention.

FIG. 10 is a diagram illustrating a thermal oxidation process of silicon by an electric furnace.

FIG. 11 illustrates an oxidation process of silicon.

FIG. 12 illustrates movement of an interface due to oxidation.

FIG. 13 is a graph showing the relationship between the concentration distribution of oxidizing species and the amount of oxidation.

FIG. 14 is a diagram showing a calculation flow of an oxidation simulation.

FIG. 15 is a view showing a method of inserting a new oxide film by a conventional method (method of a transition region).

FIG. 16 is a view showing a method of inserting a new oxide film by a conventional method (method of forcible displacement).

FIG. 17 is a view showing a method of inserting a new oxide film by a conventional method (push-back method).

FIG. 18 is a diagram showing an image of a discretized equation system.

FIG. 19 is a diagram showing a calculation procedure by numerical relaxation.

FIG. 20 is a diagram showing a calculation procedure by the Newton method.

FIG. 21 is a diagram illustrating a method of calculating impurity diffusion during an oxidation step in a conventional method.

FIG. 22 shows a method for setting an impurity concentration in a newly formed oxide film by a conventional method.

FIG. 23 illustrates a floating polysilicon structure.

FIG. 24 is a diagram showing a correct shape change when a floating polysilicon structure is oxidized.

FIG. 25 is a diagram showing a change in shape of a floating polysilicon structure when a forced displacement method is used.

[Brief description of reference numerals]

 S31 Step of obtaining an oxidizing species concentration distribution; S32 Step of obtaining an oxidation rate; S33 Step of calculating a displacement amount and stress; S34 Step of moving an interface; S35 Step of inserting an oxide film; S51 Step of obtaining an impurity diffusion; Obtaining a concentration distribution of the oxidizing species; S53 obtaining an oxide film thickness; S54 inserting an oxide film; S55 setting an oxide film concentration; and S56 setting a silicon concentration.

Claims (7)

[Claims] 1. A method for simulating the formation of an oxide film in a semiconductor manufacturing process, comprising: (1) solving a diffusion equation of an oxidizing species to obtain an oxidizing species concentration distribution; and (2) oxidizing species at an interface between silicon and an oxide film. (3) calculate the thickness of the newly formed oxide film based on the calculated oxidation rate, and (4) replace the newly formed oxide film with the existing oxide film. (5) In an oxidation simulation for calculating the amount of material displacement caused by inserting a newly formed oxide film and the stress generated by the material displacement,
(6) The finite element method is used for the numerical solution for the diffusion of the oxidizing species and the calculation of the displacement / stress. (7) The nodes on the interface between the silicon and the oxide film are regarded as double nodes, and these double nodes are The displacement / stress calculation is performed by giving the displacement difference the amount obtained by subtracting the thickness of the consumed silicon from the thickness of the oxide film newly formed in between. (8) The analysis target is obtained based on the obtained displacement amount. And (9) the interface between the original silicon and the oxide film is moved by the thickness of the consumed silicon, and (10) the gap between the generated oxide film and the silicon is entirely replaced by the newly formed oxide film. A simulation method of a semiconductor oxidation process, wherein a simulation of an oxide film formation is performed by a calculation method of filling with a step. 2. A method for simulating an impurity distribution in an oxidation step of a semiconductor manufacturing process, comprising:
1) A diffusion simulation in which the time change of the impurity concentration distribution is determined by solving a diffusion equation using a finite difference method or a finite element method as a numerical solution, and (22) an oxidation species by solving a diffusion equation of the oxidation species. Calculating the concentration distribution, (23) calculating the thickness of the newly formed oxide film from the obtained oxidized species concentration, (24) inserting the newly formed oxide film between the existing oxide film and silicon, (25)
Combining the amount of material displacement caused by inserting a newly formed oxide film with the oxidation simulation for calculating the stress caused by the material displacement, (26)
In the simulation for calculating the redistribution of the impurities during the oxidation step, (27) the node that gives the impurity concentration on the interface between the silicon and the oxide film is a double node, and the impurity flux generated by segregation between these nodes and the oxidation The diffusion equation is determined in consideration of the velocity determined and the impurity flux flowing out of the system from these double nodes. (28) The impurity concentration in the newly formed oxide film is obtained as a result of the diffusion calculation. (29)
The simulation of the impurity distribution is performed by a calculation method in which the impurity concentration of the silicon side node on the interface newly formed by consumption of silicon is set by interpolation from the impurity distribution in silicon before the interface is moved. A method for simulating a semiconductor oxidation process, which is characterized by the following. 3. A method for simulating an oxidation step in a semiconductor manufacturing process, comprising: (31) obtaining a concentration distribution of oxidizing species by solving a diffusion equation of oxidizing species by a numerical solution using a finite element method; ) Calculating the oxidation rate from the concentration distribution of the oxidizing species at the interface between the silicon and the oxide film; and (33) forming a new node between the double nodes with the node on the interface between the silicon and the oxide film as a double node. Calculating the displacement and stress by giving the displacement difference the amount obtained by subtracting the thickness of the consumed silicon from the thickness of the oxide film to be processed; and (34) an analysis target based on the obtained displacement. (35) moving the interface between the original silicon and the oxide film by the thickness of the consumed silicon, and (35) renewing the gap between the generated oxide film and the silicon. A method for simulating the formation of an oxide film, the method including a step of filling the oxide film with a properly formed oxide film. 4. The simulation method according to claim 3, wherein the step of calculating the displacement amount includes a condition that a nodal reaction force acting on the double node is balanced. 5. A method for simulating an impurity distribution in an oxidation step of a semiconductor manufacturing process, comprising:
1) A node that gives an impurity concentration on an interface between silicon and an oxide film is a double node, and is determined from an impurity flux generated by segregation therebetween and an oxidation rate, and flows out of the system from these double nodes. Calculating the diffusion of impurities by solving the diffusion equation by a numerical solution method using the finite difference method or the finite element method in consideration of the impurity flux; and (52) solving the diffusion equation of the oxidation species by solving the diffusion equation of the oxidation species. Obtaining a concentration distribution; (53) calculating a thickness of a newly formed oxide film from the obtained oxidized species concentration; and (54) forming a newly formed oxide film between the existing oxide film and silicon. (55) setting the impurity concentration in the newly formed oxide film to a value equal to the interface concentration on the oxide film side obtained as a result of the diffusion calculation; Setting the impurity concentration at the silicon side node on the interface newly created by the consumption of the capacitor by interpolation from the impurity distribution in the silicon before the interface moves. Simulation method for a semiconductor oxidation process. 6. A process for simulating an oxidation process in a semiconductor manufacturing process, wherein (61) a process for solving a diffusion equation of an oxidizing species by a numerical solution using a finite element method to obtain an oxidizing species concentration distribution; ) A process of calculating the oxidation rate from the concentration distribution of the oxidizing species at the interface between the silicon and the oxide film, and (63) forming a new node between the double nodes by setting the node on the interface between the silicon and the oxide film as a double node. A process of calculating the amount of displacement and stress by giving, as a displacement difference, an amount obtained by subtracting the thickness of silicon consumed from the thickness of the oxide film to be processed, and (64) an analysis target based on the obtained amount of displacement. (65) to move the interface between the original silicon and the oxide film by the thickness of the consumed silicon, and (65) to fill the gap between the generated oxide film and the silicon with the newly formed acid. A computer readable program recording medium in which a program for causing a computer to execute a process of simulating the formation of an oxide film, including a process of filling with an oxide film, is recorded. 7. A process for simulating an impurity distribution in an oxidation step of a semiconductor manufacturing process, comprising:
1) A node that gives an impurity concentration on an interface between silicon and an oxide film is a double node, and is determined from an impurity flux generated by segregation therebetween and an oxidation rate, and flows out of the system from these double nodes. In consideration of the impurity flux, a process of solving the diffusion equation by the numerical solution method using the finite difference method or the finite element method to calculate the impurity diffusion, and (72) solving the oxidation equation of the oxidized species by solving the diffusion equation of the oxidized species. (73) calculating the thickness of a newly formed oxide film from the obtained oxidizing species concentration; and (74) forming the newly formed oxide film between the existing oxide film and silicon. (75) a process for setting the impurity concentration in the newly formed oxide film to a value equal to the interface concentration on the oxide film side obtained as a result of the diffusion calculation; and (76) silicon consumption. Ruko A program for causing a computer to execute a process of simulating the impurity distribution including a process of interpolating and setting an impurity concentration of a silicon side node on the interface newly formed by the interpolation from the impurity distribution in the silicon before the interface moves. A computer-readable program recording medium recording a program.