JP2980160B2 - Calculation method of deformation of viscoelastic body - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for calculating deformation of a viscoelastic material used for calculating deformation of an oxide film in an oxidation step of semiconductor manufacturing.

[0002]

2. Description of the Related Art A viscoelastic body has the property of relaxing applied strain and stress by viscous flow inside. Since the relaxation phenomenon does not occur instantaneously but progresses gradually, in order to reproduce the deformation of the viscoelastic body by numerical calculation, it is necessary to track changes in strain and stress over time. This calculation is performed according to a procedure as shown in FIG. 3 by discretizing the time. After initializing the time t in step S21, the time step Δt until the next time in S22
Set and update the time. In the next step S23, a dynamic equilibrium equation at a new time t is solved. In general, the equation is solved by dividing the analysis region into meshes as shown in FIG. 4 and describing the equations as simultaneous equations in which the displacement u between the mesh points Δt is unknown. In a viscoelastic body, when a relational expression between stress and strain is given, a method of obtaining a mechanical equilibrium equation with an unknown displacement is described in, for example, IEEE Journal of Solid State Circuits, Vol. 1985), Issue 1, p. 5
2-60 (IEEE Journal of Solid-State Circuits, Vol.SC
-20, No. 1, pp. 52-60, 1985). In the next step S24, it is determined whether or not the analysis has been completed up to a desired time _tmax.
Return to 22.

[0003] Such a calculation of deformation of a viscoelastic body is used for estimating the shape of an oxide film in an oxidation step of semiconductor manufacturing. When oxidized, silicon, a typical semiconductor material, becomes an insulator and expands in volume at the same time. By calculating the shape deformation due to the volume expansion, it is possible to predict the film thickness, the residual stress, and the like. It is known that a silicon oxide film behaves as a viscoelastic body at a general oxidation temperature, and it is necessary to calculate the deformation of the viscoelastic body for accurate prediction.

FIG. 5 shows a typical oxidation process, LOCOS.
FIG. 4 schematically shows a cross-sectional view near a substrate surface in a process. FIG. 6 shows a procedure for simulating such an oxidation step. The oxidation simulation is also calculated by discretizing the time. In step S31, initial settings relating to the time T are made, and in step S32, time intervals are set. In step S33, the concentration of the oxidizing species at the oxidizing interface 4 (the interface between the oxidized layer 2 and the oxidized layer 1, that is, where the oxidation reaction occurs) is determined. Oxidizing species are substances to be oxidized (LOCO
In the S step, it is a substance that reacts with and oxidizes with silicon or polysilicon. In the silicon oxidizing step, it is oxygen. At the beginning of the oxidation step, the oxide layer 1 on the surface is thin,
The oxidation rate is fast because sufficient oxygen is supplied. However, as the oxidation progresses and the oxidation interface 4 becomes deeper, the oxygen concentration at the oxidation interface decreases and the oxidation rate slows down. In the next step S34, the oxidation rate is determined assuming that the oxygen concentration determined in S33 does not change between the time T and ΔT, and a region 5 to be newly oxidized is determined within this time step. Therefore, the time interval ΔT in FIG. 6 should be controlled so that the oxidation rate can be obtained with sufficient accuracy.

In the LOCOS step, the surface of the substrate in a region where oxidation is to be suppressed is protected by a protective layer 3 that does not allow oxidized species to pass through. Therefore, the thickness of the region 5 to be oxidized varies from place to place. Since the oxidizing species also diffuses inside the oxide layer 1 in the horizontal direction, the layer to be oxidized located near the end of the protective layer 3 is oxidized. However, this region has the protective layer 3 above and cannot be freely extended upward due to its rigidity. Therefore, high stress is generated near this region, and viscous flow occurs.

In step S35, a deformation calculation including such a viscous strain is performed. There are various methods for giving the initial condition for the deformation calculation, but the simplest is ΔT
During the instant T occurs instantaneously at time T, and the oxide film in the newly oxidized region 5 starts only in a state where only the elastic strain is generated so that the volume becomes the same.

The deformation calculation is performed in accordance with the procedure shown in FIG. 3, where t _max corresponds to ΔT in FIG. Normal,
As the oxidation reaction proceeds and the oxidation rate and its time change decrease, ΔT is modified to a large value in order to shorten the calculation time. Depending on the simulator, ΔT and Δt in FIG. 3 may always be the same, but there is no basis that the deformation calculation can be performed at a large time interval following ΔT. Δt is originally to be determined so as to guarantee the accuracy of the deformation calculation, but its method has not been sufficiently examined, and it is a major problem that it relies on an empirical method of setting Δt. .

[0008]

SUMMARY OF THE INVENTION In the conventional analysis method,
Only at a finite number of discrete times, the equations describing the force balance are solved. That is, there is no guarantee that the balance of the force is satisfied at a time between the time t and the time t + Δt. Therefore, when the time step Δt is increased, an error is generated due to ignoring the balance of the force in the intermediate state, and it becomes impossible to correctly obtain the distortion and the stress. If the time step is shortened, the error is reduced. However, if the time step is shortened more than necessary, the calculation time only increases, so some criterion is required. In addition, if a formulation can be performed in consideration of the balance of the force in the intermediate state, there is a possibility that the calculation can be performed in a time step larger than that of the conventional method without losing accuracy.

SUMMARY OF THE INVENTION An object of the present invention is to provide a method for calculating the deformation of a viscoelastic body at high speed while maintaining accuracy. To provide a calculation method that enables the use of type steps and reduce the analysis time, and to provide a calculation method that can maintain an appropriate time step by examining the satisfaction of force balance in an intermediate state. It is in.

[0010]

According to a first aspect of the present invention , there is provided a method for calculating a temporal change in the shape of a viscoelastic body, comprising solving a dynamical equilibrium equation at each discrete time. Alternatively, a viscoelasticity is obtained by calculating a displacement amount that minimizes an amount obtained by integrating the residual of the equilibrium equation at each time obtained over a time step with the assumption that the strain rate in the time step is constant. The body deformation calculation method is obtained.

According to a second aspect of the present invention , there is provided a method for calculating a temporal change of a shape of a viscoelastic body, comprising: obtaining a displacement amount after a certain time step has elapsed; Is obtained by assuming that the residual of the equilibrium equation at each time obtained assuming that is constant is integrated over time within the time step.If this amount exceeds a predetermined allowable amount, the calculation of the displacement amount is invalidated. If the time step is made smaller and the process returns to the displacement calculation, and if the amount is equal to or less than the allowable amount, the displacement after the time step has elapsed is calculated, and the calculation of the displacement at the next time is made larger by increasing the time step. A method for calculating the deformation of the viscoelastic body is obtained.

[0012]

The deformation of a viscoelastic body according to the first aspect of the present invention is calculated.
In the method , step S23 of FIG.
Is replaced by a calculation method derived by the following procedure. In step S23, the following equation 1 representing the balance of the force at a certain time t, which is discretized, is solved.

[0013]

(Equation 1) Here, u represents the displacement amount of each mesh point between the time steps when the analysis region is divided into meshes as shown in FIG. A and b are a coefficient matrix and a constant vector describing the mechanical equilibrium equation at the analysis time t, respectively, and each element changes over time. Normally, this equation is solved only at time t, and then at time t + Δt. However, originally, the equation should be solved so that the force balance is maintained even at time t + αΔt (0 <α <1). Intermediate time t + α
Also at Δt, the coefficient matrix A (α) of the dynamic equilibrium equation and the constant vector b
(Α) is required. Assuming that the strain rate from time t to t + Δt, that is, the rate of change of the displacement over time is constant, the displacement at time t + αΔt is represented by αu, so the equation to be satisfied at time t + αΔt is Equation 2

[0014]

(Equation 2) The degree to which this equation is satisfied from time t to t + Δt can be evaluated by the following equation (3).

[0015]

(Equation 3) Here, w (α) is a normalized weight, and represents the importance of each intermediate time. "Normalized" means that the following equation (4) is satisfied. The simplest weight is an even weight, ie, w (α) = 1 (0 <α
<1).

[0016]

(Equation 4) There is a displacement amount u that minimizes ε given by Expression (3), and it is obtained as a solution of the following Expression 6 obtained from Expression 5 below.

[0017]

(Equation 5)

[0018]

(Equation 6) Here, the integral with respect to a matrix and a vector represents a matrix and a vector whose components are the results of performing integration on each component. In the viscoelastic body deformation calculation method described in claim 1, the equation obtained as described above is solved (FIG. 1, step S3). U obtained by solving this equation is a solution optimized in consideration of not only the times t and t + Δt, but also the intermediate state. Therefore, if the same time interval as that of the conventional method is used, the calculation method of the present invention can calculate with higher accuracy. From another point of view, the time step for obtaining the same level of accuracy is large, and there is a possibility that the calculation can be performed faster than the conventional method.

On the other hand, in the method for calculating the deformation of the viscoelastic body according to the first aspect of the present invention, it is determined whether or not the time step Δt is appropriate using ε given by the equation (3) as an index. FIG. 2 shows the procedure of the calculation. The deformation calculation method may be a conventional method or a method described in claim 1 (step S1).
3). In any case, ε represents a deviation from the dynamic equilibrium equation over the entire time within the time step.

If ε is below the allowable value, the time step Δt is reset to a larger value (FIG. 2, step S1).
5, S16). When the value exceeds the allowable value, the time step is reduced and the calculation is performed again (FIG. 2, step S18).
According to this procedure, the deviation from the mechanical equilibrium state can be suppressed, so that highly accurate deformation calculation can be performed.

[0021]

FIG. 7 shows an example of an analysis of a structure in which two one-dimensional viscoelastic body elements 6 are connected in series and both ends are fixed. The viscoelastic element comprises a spring representing elasticity and a dashpot representing viscosity. Two viscoelastic elements a,
It is assumed that Young's modulus of b is equal. When a step-like strain is given at time 0, the viscoelastic element generates an initial stress σ ₀ at time 0, and then decays exponentially to σ
Asymptotic to _∞ . The time when the stress reaches (σ ₀ −σ _∞ ) / l is called a stress relaxation time constant, and the stress relaxation time constant of a is τ _a =
Τ _b = 300 sec for the stress relaxation time constant of 30 sec, b
And It is assumed that σ 両_者 is the same for both. When the same compression distortion is given to both at step 0 as a step function,
Since they consist only of elastic strain and have the same Young's modulus, the initial stress σ ₀ is the same for both, and the balance of the force at time 0 is satisfied. As the time elapses, the stress of the element a is relaxed earlier, and the stress is smaller than that of the element b.
In order to balance the forces, element a is compressed and b expands.

FIG. 8 shows the change over time of the distortion of the element a, with the initial distortion being 1, and the time steps are 6 seconds, 60 seconds, 120 seconds, 300 seconds, 600 seconds, and 120 seconds.
The calculation results for the case of 0 second are displayed together. The method of calculating the deformation is a conventional one, and solves an equation that satisfies the force balance only at each discrete time.
When the time step is 1200 seconds, the stress of both viscoelastic elements has been almost completely reduced by this time. If the intermediate state is not considered at all, the distortion hardly changes because only the balance of the force in the relaxed state is considered. In order to accurately track the change in strain, a time step of about 6 seconds, which is sufficiently smaller than the stress relaxation time constant of both elements, must be used.

FIG. 9 shows the result of calculating the temporal change of the distortion using the deformation calculation method according to the first embodiment of the present invention. Specifically, at each discretized time, equation (5) is solved. Here, the weight w (α) was set to 1 irrespective of the value of α, and the integration in Equation (5) was performed numerically. Calculated for the same time steps as in FIG. 8, they are shown in FIG. 9 using the same symbols. In this calculation method, since the force equilibrium state at an intermediate time is taken into account, a certain degree of change in distortion is observed even in a 1200-second time step in which elements are almost completely relaxed within one time step. Also,
It can be seen that the calculation accuracy is improved in all time steps as compared with the conventional calculation method. When the time step is about 60 seconds, the error with respect to the amount of change in distortion falls within 3%, and practical calculation is possible.

In equation (5), since the coefficient matrix A (α) is multiplied by the transposed matrix, the number of non-zero elements in the coefficient matrix increases. Therefore, it takes more time to solve the simultaneous linear equations than in the conventional method. However, for example, even if it takes three times as long to solve the equation, in this embodiment, if the time step is about 100 seconds or less, it is more advantageous to solve the equation (5). That is, the accuracy is higher than when the conventional method is applied by setting the time step to one third. The fact that the accuracy is improved as compared with the conventional method means that the calculation time for achieving the same accuracy is reduced from a different viewpoint.

Now, taking a closer look at the time change of the strain in FIG. 9, the difference between the strain values calculated at Δt = 6, 60, 120 seconds after 120 seconds and the difference between them after 240 seconds has elapsed. It can be seen that there is almost no difference from. This means that most of the calculation errors of Δt = 60, 120 seconds occur within the first 120 seconds, and that sufficient accuracy is maintained thereafter. Therefore, it can be said that the calculation time for performing the calculation while keeping the time step constant is wasteful.

In the modification calculation method according to the second embodiment of the present invention, the time step is controlled with reference to ε given by equation (3). Since it has been found that a calculation method that minimizes ε improves accuracy, the validity of using ε as an index of calculation error is clear. The degree of modification of the time step is usually changed in accordance with the ratio of the magnitude of ε to the allowable value εmax, but it can be simpler. For example, Δt is doubled in step S16 in FIG.
Is halved. If the calculation is continued at the same time step, ε given by the equation (3) attenuates exponentially, so that there is a high possibility that the time step is doubled at each time. If this is possible, let N be the number of calculations in a given time step.
Only about log2 N calculations are required, and the calculation can be performed at an overwhelmingly high speed. Time control such as doubling the time step is an existing method, but the major difference from the conventional method is that errors are suppressed using the deviation of the force in the time step from the equilibrium state as an index. Therefore, high speed can be realized while maintaining accuracy.

In FIG. 2, when ε exceeds the allowable value, the previous calculation is discarded and the calculation is performed again. However, in order to avoid wasting the calculation time, it is advisable to perform the calculation at the next time by making the previous calculation valid within a range not significantly exceeding the allowable value. Also, in this figure, the flow for setting ε _max for each time step is shown.
_{If max} is a constant, it may go out of the loop.

[0028]

According to the deformation calculation method of the present invention , it is possible to increase the time step without deteriorating the accuracy as compared with the conventional method, and the calculation time required for one time step can be reduced. The total calculation time taken into account can also be reduced.

If the deformation calculation method described in claim 2 is used, time step control can be performed using an amount representing a deviation from the mechanical balance equation in the deformation calculation as an index of an error. Given the accuracy of the deformation calculation, a long time step can be automatically selected while maintaining the accuracy, and the calculation can be performed at high speed.

[Brief description of the drawings]

FIG. 1 is a flowchart showing a procedure of a deformation calculation method according to a first embodiment of the present invention.

FIG. 2 is a flowchart illustrating a procedure of a deformation calculation method according to a second embodiment of the present invention.

FIG. 3 is a flowchart showing a procedure of a conventional deformation calculation method .

FIG. 4 is a diagram showing a state in which an analysis region is divided into triangular meshes in order to calculate deformation of a viscoelastic body.

FIG. 5 is a diagram schematically showing a cross-sectional view near a semiconductor substrate surface in a LOCOS process which is a typical oxidation process in a semiconductor manufacturing process.

FIG. 6 is a flowchart showing a procedure in a case where a deformation calculation of a viscoelastic body is applied to a simulation of positive oxidation in a semiconductor manufacturing process.

FIG. 7 is a schematic view of two viscoelastic elements analyzed to show the effect of the present invention.

8 is a diagram showing a result of calculating a time change of strain of the viscoelastic element shown in FIG. 7 by a conventional method.

FIG. 9 is a diagram showing a result of calculating a time change of the strain of the viscoelastic element shown in FIG. 7 using the deformation calculation method according to the first embodiment of the present invention.

[Explanation of symbols]

 Reference Signs List 1 oxidized layer (already oxidized layer) 2 oxidized layer (unoxidized layer) 3 protective layer 4 oxidized interface 5 (oxidized in time step) region 6 one-dimensional viscoelastic element

──────────────────────────────────────────────────続 き Continued on the front page (58) Field surveyed (Int.Cl. ⁶ , DB name) H01L 21/66 H01L 21/316 H01L 21/76

Claims (2)

(57) [Claims] 1. A method for calculating a temporal change of a shape of a viscoelastic body.
A law, instead of solving the dynamic equilibrium equations at each time which is discretized, the residual of the balance equation at each time when the strain rate in the time step is obtained assuming constant over the time step Deformation calculation of viscoelastic body characterized by finding displacement amount that minimizes time integrated amount
How . 2. A method for calculating a temporal change of a shape of a viscoelastic body.
A method, after obtaining the displacement amount after the lapse of a certain time step, the amount of strain rate residuals balance equation at each time obtained by assuming a constant and the time integral over the time step in the time step If this amount exceeds the predetermined allowable amount, the calculation of the displacement amount is invalidated, the time step is reduced, and the process returns to the calculation of the displacement amount.If the amount is equal to or less than the allowable amount, the displacement after the time step has elapsed is calculated. A method for calculating the deformation of a viscoelastic body, wherein the calculation of the displacement at the next time is performed by increasing the time step assuming that the amount has been obtained.