JP3298187B2 - Shape simulation method and semiconductor element manufacturing method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a shape simulation method, for example, when a semiconductor device or the like is manufactured by a lithography process, the state of dissolution of a resist by a developing solution, a change in the shape of a semiconductor substrate due to etching, or a metal or polysilicon. It is suitable to be applied to a case where a shape change or the like due to vapor deposition is predicted.

[0002]

2. Description of the Related Art For example, when a semiconductor device or the like is manufactured by a lithography process, an image of a mask pattern is projected on a wafer coated with a resist, and then the wafer is developed. The resist in the portion with a large amount of exposure is stripped. A desired pattern is formed on the wafer by depositing aluminum or the like on the peeled portion. In this case, the surface shape of the developed resist changes variously depending on the defocus state of the wafer, the developing time, and the like when projecting the image of the mask pattern. Therefore, in order to form a pattern with high accuracy, it is desirable to previously obtain, by computer simulation, conditions under which the surface shape of the developed resist is in the best state.

A technique for estimating a change in the surface shape of a resist is known as a resist simulation, and one example thereof is disclosed in, for example, Japanese Patent Application Laid-Open No. 3-237710. Furthermore, a shape simulation method is also being developed which generalizes this and predicts a change in the shape of a semiconductor substrate during etching and a change in the shape of a deposit when depositing metal or polysilicon.

Conventionally, various methods have been proposed as shape simulation methods. First, when dealing only with a two-dimensional shape change, it is effective to express the shape by a series of minute line segments, and such an expression is called a string model. When the string model is extended three-dimensionally, the shape is represented by a series of minute surface elements instead of minute line segments. The moving direction of the minute line segment or minute plane element is a direction perpendicular to the surface. On the other hand, a method of finding the moving direction by solving a differential equation similar to the differential equation of a light ray is called a ray tracing model.

There is also known a cell model which is a method of dividing an object into an aggregate of minute cells and expressing a shape change by disappearance or attachment of cells on the surface of the object.
Further, there is also known a modified diffusion model which is a method of expressing an object shape by an isosurface of a distribution function and obtaining a time change of the shape by solving a diffusion equation (for example, Semicon NEWS, No.
7, pp. 36-41 (1989)).

[0006]

In the prior art as described above, a string model or a ray tracing
When a model is used, a string (a minute line segment or a minute surface element) may form a loop, and it is necessary to delete the loop. However, in the model extended to three dimensions, it is necessary to check the presence or absence of a loop for an enormous number of small plane elements, which results in an extremely complicated algorithm and a large increase in calculation cost. there were.

[0007] The cell model can cope with the same algorithm in both the two-dimensional case and the three-dimensional case. It is necessary to correct the moving speed according to the number of surfaces. However, at this time, it is difficult to create a model whose physical meaning is clear, so that there is a disadvantage that reliability is lacking.

In the modified diffusion model, the two-dimensional case and the three-dimensional case can cope with the same algorithm, but the movement of the object surface is obtained by solving a diffusion method that appears due to a heat conduction problem or the like. Therefore, there is an inconvenience that accuracy is poor because it does not correspond to an actual physical phenomenon.
The diffusion distance is proportional to the square root of the diffusion time. Therefore, if it is necessary to find the solution of the diffusion equation after T seconds to check the shape change after t seconds, for example, to check the shapes after 2 t seconds and 3 t seconds, the shapes after 4 T seconds and We have to find the solution of the diffusion equation after 9T seconds,
There is a disadvantage that the calculation time and the calculation cost increase rapidly.

In view of the above, an object of the present invention is to provide a shape simulation method capable of accurately predicting two-dimensional and three-dimensional shape changes in a short calculation time. Ma
The present invention uses such a shape simulation method.
To provide a semiconductor device manufacturing method
You.

[0010]

According to a first shape simulation method according to the present invention, a shape of an object surface which is changed by physically or chemically processing the surface of the object is determined by three.
Distribution function c with dimensional coordinates x, y, z and time t as variables
In the method of predicting a change in the shape of the surface of an object by expressing it on the isosurface of (x, y, z, t), the gradient vector <∇c> of the distribution function c and the moving speed of the surface of the object are indicated. Using the velocity vector <v>, it is assumed that the time change of the shape of the object surface satisfies the following partial differential equation.

[0011]

(Equation 3) Then, by extracting an isosurface where the distribution function c obtained by solving the partial differential equation has a predetermined value, the processed shape of the object surface is obtained. When the partial differential equation (Equation 3) is specifically solved by numerical analysis, the first shape simulation method can be paraphrased as follows. That is, in the first shape simulation method, a first step (step 101) for giving an initial value of the distribution function c and a second step (step 107) for obtaining a velocity vector <v> indicating the moving velocity of the object surface. ) And the gradient vector <∇c> of the distribution function c.
And a third step (step 108) of obtaining a change amount Δc of the predetermined function c from the following calculation using the time step amount Δt.

Δc = − <∇c> · <v> Δt Further, the present invention has a fourth step (step 104) of obtaining new values of the distribution function c and the time t by the following calculation. c = c + Δc, t = t + Δt Then, using the distribution function c obtained by repeating the second to fourth steps a predetermined number of times, extracting an isosurface where the distribution function c has a predetermined value. Thus, the shape of the object surface after processing is obtained.

Further, the second shape simulation method of the present invention provides a method of physically or chemically processing the surface of an object to change the shape of the surface of the object into three-dimensional coordinates x,
distribution function c (x, y,
In the method of predicting the shape change of the surface of the object by expressing it on the isosurface of (z, t), it is assumed that the moving speed of the surface of the object is proportional to the gradient vector <∇c> of the distribution function c. Using the gradient vector <∇c> of the distribution function c and the signed magnitude V of the moving speed of the object surface, the temporal change in the shape of the object surface satisfies the following partial differential equation.

[0014]

(Equation 4) Then, by extracting an isosurface where the distribution function c obtained by solving the partial differential equation has a predetermined value, the processed shape of the object surface is obtained. Further, when the partial differential equation (Equation 4) is specifically solved by numerical analysis, the second shape simulation method can be paraphrased as follows. That is, in the second shape simulation method, the first step (step 101) of giving the initial value of the distribution function c, and the moving speed of the object surface is proportional to the gradient vector <∇c> of the distribution function c. The second step (step 102) of obtaining the signed magnitude V of the moving speed, the absolute value | <∇c> | of the gradient vector <∇c> of the distribution function c, and the time step amount A third step (step 103) of obtaining a change amount Δc of the distribution function c from the following calculation using Δt.

Δc = V | <∇c> | Δt Further, a fourth step (step 104) of obtaining new values of the distribution function c and the time t by the following operation is provided. c = c + Δc, t = t + Δt Then, using the distribution function c obtained by repeating the second to fourth steps a predetermined number of times, extracting an isosurface where the distribution function c has a predetermined value. Thus, the shape of the object surface after processing is obtained. In addition,
The semiconductor device manufacturing method according to Ming uses a lithography process.
And a method for manufacturing a semiconductor device, comprising:
After machining the object surface using a simulation method
It includes a step of obtaining a shape.

[0016]

The principle of the present invention will be described with reference to FIG.
First, in the present invention, it is assumed that an isosurface determined by c = c1 (where c1 is a constant) represents an object shape using a distribution function c (x, y, z, t). That is, the object shape to be obtained is a surface in which coordinates (x, y, z) satisfying the following expression are connected.

[0017]

C (x, y, z, t) = c1 Therefore, as shown in FIG. 3, an isosurface satisfying (equation 5) at time t is defined as a surface S. From that state, the time t + Δ
Let the isosurface at t be surface S '. Also, a point P on the surface S
(X, y, z) is a point P ′ (x + Δx,
(y + Δy, z + Δz), the following expression is satisfied at the point P ′.

[0018]

C (x + Δx, y + Δy, z + Δz, t + Δt) = c1 Here, the velocity vector <v> when moving from the point P to the point P ′ can be defined as follows.

[0019]

(Equation 7) Using each component of the velocity vector <v>, Δx, Δ
y and Δx are as follows.

[0020]

Δx = (dx / dt) Δt = v _x Δt Δy = (dy / dt) Δt = v _y Δt Δx = (dz / dt) Δt = v _z Δt Expressed in the form, the following equation is established by subtracting (Equation 5) from both sides.

[0021]

(Equation 9) By substituting (Equation 8) into (Equation 9), the following equation is obtained.

[0022]

(Equation 10) By dividing both sides of (Equation 10) by Δt, the following equation is obtained.

[0023]

[Equation 11] In this case, the vector operator ∇
Is used, the gradient vector <∇c> of the distribution function c is defined as follows.

[0024]

(Equation 12) Therefore, by substituting (Equation 7) and (Equation 12) into (Equation 11), (Equation 11) is converted into a partial differential equation including the following inner product operation.

[0025]

(Equation 13) This is a partial differential equation representing the time change of the surface shape of the object, and is the same as that used in the first shape simulation method of the present invention. The method of solving this partial differential equation by numerical analysis to obtain the surface shape of the object is as described above. Next, in the second shape simulation method of the present invention, it is assumed that the moving direction of the object surface is perpendicular to the surface. Such an assumption is effective, for example, in resist simulation. Under this assumption, the velocity vector <v> is the gradient vector <∇c> of the distribution function c.
Is proportional to Therefore, the absolute value of the gradient vector <∇c> |
By using <∇c> | and a variable V representing the magnitude of the velocity vector, the following expression can be used.

[0026]

<V> = − V <∇c> / | ∇c | where the variable V is a function of position and time, and V = V
(X, y, z, t). (Equation 14) is replaced by (Equation 1)
By substituting into 3), the following equation is obtained.

[0027]

(Equation 15) This is the same as the partial differential equation used in the second shape simulation method of the present invention. The method of solving this partial differential equation by numerical analysis to obtain the surface shape of the object is also as described above. The method of the present invention for obtaining a shape change by solving the partial differential equation of (Equation 13) or (Equation 15) can cope with the same algorithm for both two-dimensional and three-dimensional. Further, as is clear from the above description, since the partial differential equation corresponding to the actual shape change is used, the shape change can be obtained with high accuracy. Further, since the moving speed vector <v> can be arbitrarily set, the present invention can be applied to, for example, simulation of anisotropic etching, selective etching, thin film growth, and the like. In addition, the real time that appears in the actual phenomenon can be used as the calculation time.

[0028]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of a shape simulation method according to the present invention will be described below with reference to the drawings. In this example,
The present invention is applied to a case where the shape of a photoresist after being dissolved by development is determined in a process of manufacturing a semiconductor element in a lithography process.

In this case, the surface shape of the resist is represented by an iso-surface of the distribution function c (x, y, z, t). However, it is assumed that there is no shape change in the y direction. Since the change in the resist shape can be considered to proceed in a direction perpendicular to the surface, the above (Equation 15) is used. That is, if the gradient vector <∇c> of the distribution function c and the magnitude of the moving speed of the surface shape of the resist, that is, the development speed, are expressed by V (x, y, z, t), 16) holds. However, the sign of the development speed V in this case is positive outside the surface shape.

[0030]

(Equation 16) Next, an example of a procedure for obtaining a resist shape after development in this example will be described with reference to the flowchart of FIG. 1 and FIG. First, in step 101 of FIG.
At time t = 0, that is, the value of the distribution function c corresponding to the initial surface shape of the resist before the start of development. As an initial surface shape, as shown in FIG.
A plane of z = d is set in a region of −P / 2 ≦ x <P / 2. In the region 2 where z ≦ d, the initial value of the distribution function c is 0. In the region 1 where 0 ≦ z <d, the initial value of the distribution function c is 1.
Set to 0. Next, in step 102, the magnitude of the moving speed of the shape surface at the time point t (currently t = 0), that is, the developing speed V is obtained.

Since the developing speed V is determined by the amount of exposure, for example, as shown in FIG. 2B, if the amount of exposure EL is large at the periphery in the x direction and small at the center in FIG. As shown in the figure, the developing speed V1 at the peripheral portion in the x direction is larger than the developing speed V2 at the central portion. As described above, the development speed V is obtained from the exposure amount EL, and the development speed V varies depending on the place and the time. In addition, since the distribution of the exposure amount EL changes depending on the positional deviation between the resist layer of the wafer and the best imaging plane of the projection optical system, that is, the defocus amount, the distribution of the development speed V also changes depending on the defocus amount.

In this case, since the direction of the gradient vector <∇c> on the resist surface is directed toward the inside of the resist, the sign of the developing speed V is negative. Therefore, V = −V
By converting a variable from V to V ′ by '(V ′> 0),
(Equation 16) is as follows.

[0033]

∂c / ∂t = −V ′ | ∇c | Next, in step 103, the change amount Δc of the distribution function c after the time change of Δt is obtained. From (Equation 17),
The Δc is calculated by the following equation using the gradient vector <∇c> at the time t.

[0034]

Δc = −V ′ | ∇c | Δt Next, in step 104, Δt is added to the current time t, and Δc is also added to the distribution function c. Then, steps 102 to 104 are repeated until the time t reaches the development end time T. As an example, Δt is set to 0.
The development time T was 2 seconds and the development time T was 60 seconds. Then step 1
At 06, an iso-surface having a value of 0.8 in the distribution function c at time t = T was obtained. The value 0.8
Is the value chosen to match the result with the string model. In addition, if the equivalent surface of c = 0.8 is extracted assuming that the value continuously changes from 0 to 1 in the initial value of the distribution function c, the surface becomes z = d and there is no problem.

A curve 3A in FIG. 4A shows the shape of the resist surface calculated according to the flowchart of FIG. 1 assuming that the defocus amount ΔF of the wafer is -0.7 μm. Curves 3B and 3C of b) and (c) are
The defocus amount ΔF is set to 0.0 μm and +0.7, respectively.
This shows the final shape of the resist surface obtained assuming μm. 4A to 4C, the horizontal axis is the x-axis, and the vertical axis is the z-axis. In addition, for comparison, FIGS.
Under the same conditions as above, resist shapes obtained by a conventional string model are shown by curves 4A to 4C in FIGS. 4D to 4F, respectively. By comparing these, it can be seen that the method of the present embodiment provides virtually the same result as using the string model. Moreover,
In the case of the present embodiment, the time required for the calculation is greatly reduced.

For reference, the results of determining the resist shape using a conventional diffusion model are shown in FIGS. 5 (a) to 5 (e).
Shown in 5 (a), (b), (c), (d) and (e) show that the defocus amount ΔF is -1.0,-
0.8, -0.6, -0.4 and -0.2 are shown, wherein the curves 5A to 5E are the resist shapes when the string model is used, and the curves 6A to 6E are the cases when the diffusion model is used. 3 shows the resist shape. From this result, it is understood that the error increases in the diffusion model when the defocus amount ΔF is more negative than −0.4 μm. On the other hand, in this example, the resist shape can always be obtained quickly and accurately regardless of the value of the defocus amount ΔF.

In the above embodiment, if the direction of the developing speed is not necessarily assumed to be perpendicular to the resist surface, the gradient vector <∇c> of the distribution function c and the moving speed of the resist surface shape, that is, the developing speed Using the velocity vector <v (x, y, z, t)>
Is numerically analyzed to determine the final shape.

[0038]

[Equation 19] Specifically, in this case, the velocity vector <v (x, y, z, t)> of the surface of the shape is obtained as shown in step 107 instead of step 102 in FIG. Then, in order to obtain the variation Δc of the distribution function c, the following calculation is performed as shown in step 108 instead of step 103 in FIG.

[0039]

Δc = − <∇c> · <v> Δt The other steps are the same as in the above embodiment. Although the above-described embodiment has been described with respect to a two-dimensional shape for comparison with a string model, the present invention can be similarly applied to a case where a three-dimensional shape is obtained. Further, for example, the shape of a contact hole pattern or the like can be calculated. It goes without saying that the present invention can be applied not only to a semiconductor manufacturing process but also to a shape simulation in an arbitrary field.

As described above, the present invention is not limited to the above-described embodiment, and can take various configurations without departing from the gist of the present invention.

[0041]

According to the first shape simulation method of the present invention, two-dimensional and three-dimensional shapes can be calculated very accurately in a short calculation time by, for example, numerical analysis of a first-order partial differential equation. is there. Further, even when the shape change has anisotropy or selectivity, it can be dealt with by appropriately giving the velocity vector <v>. Further, it is possible to cope with not only etching and dissolution but also calculation of growth of a thin film and a crystal, and the application range is extremely wide.

According to the second shape simulation method of the present invention, especially, the moving speed of the surface of the object is controlled by the distribution function c.
In the field where the gradient vector is proportional to the gradient vector, for example, numerical analysis of the first-order partial differential equation has an advantage that two-dimensional and three-dimensional shapes can be calculated very accurately in a short calculation time.

[Brief description of the drawings]

FIG. 1 is a flowchart showing one embodiment of a shape simulation method according to the present invention.

FIG. 2A shows a distribution function c prior to resist shape calculation.
FIG. 7B is a diagram showing an example of setting of initial conditions, and FIG. 7B is a diagram showing an example of distribution of an exposure amount on a resist surface.

FIG. 3 is a diagram illustrating the principle of the present invention.

FIGS. 4A to 4C are cross-sectional views showing resist shapes calculated for different defocus amounts by the method of the embodiment of the present invention, and FIGS. 4D to 4F are respectively (a). ~
FIG. 14 is a cross-sectional view showing a resist shape calculated using a string model under the same conditions as in (c).

FIG. 5 shows a resist shape and a string / string calculated using a conventional diffusion model under various defocus amounts.
FIG. 9 is a cross-sectional view illustrating a resist shape calculated using a model.

[Explanation of symbols]

 1 A region where the initial value of the distribution function c is 1 2 A region where the initial value of the distribution function c is 0 S, S 'isosurface

Claims (6)

(57) [Claims] 1. A distribution function c (x, x) having three-dimensional coordinates x, y, z and time t as variables, the shape of the object surface changing by physically or chemically processing the object surface.
y, z, t) in the method of estimating the shape change of the object surface by expressing the gradient vector <∇c> of the distribution function c and the velocity vector <v>, the time change of the shape of the surface of the object satisfies the following partial differential equation: A shape simulation method characterized by obtaining a processed shape of the object surface by extracting an isosurface where the distribution function c obtained by solving the partial differential equation has a predetermined value. 2. A distribution function c (x, x) having three-dimensional coordinates x, y, z and time t as variables, which is changed by physically or chemically processing the object surface.
a method for predicting a change in shape of the surface of the object by expressing the shape change of the surface of the object by expressing the surface of the object as an iso-surface of (y, z, t); A second step of obtaining a vector <v>, a third step of obtaining a change amount Δc of the distribution function c from the following calculation using a gradient vector <∇c> of the distribution function c and a time step amount Δt. And a fourth step of obtaining a new value of the distribution function c and the time t by the following operation: c = c + Δc, t = t + Δt The second step Using the distribution function c obtained by repeating the process from the process to the fourth process a predetermined number of times, extracting an isosurface where the distribution function c has a predetermined value to obtain a processed shape of the object surface A shape simulation method characterized by the following. 3. A distribution function c (x, x, y, z) with time t as a variable, wherein the shape of the object surface which is changed by physically or chemically processing the object surface is changed.
y, z, t) in a method for predicting a change in shape of the object surface by expressing the object surface shape change assuming that the moving speed of the object surface is proportional to the gradient vector <∇c> of the distribution function c. Using the gradient vector <∇c> of the distribution function c and the signed magnitude V of the moving speed of the object surface, the temporal change in the shape of the object surface satisfies the following partial differential equation: Equation 2 A shape simulation method characterized by obtaining a processed shape of the object surface by extracting an isosurface where the distribution function c obtained by solving the partial differential equation has a predetermined value. 4. A distribution function c (x, x, y, z) and a time t, which are variables obtained by physically or chemically processing the surface of the object.
a method for predicting a change in shape of the surface of the object by expressing the shape change on the surface of the object by the isosurface of (y, z, t), wherein a first step of providing an initial value of the distribution function c; A second step of obtaining the signed magnitude V of the moving velocity as being proportional to the gradient vector <∇c> of the function c; and the absolute value | <∇ of the gradient vector <∇c> of the distribution function c.
c> | and a step amount Δt of time to obtain a variation Δc of the distribution function c from the following calculation: Δc = V | <∇c> | Δt And c = c + Δc, t = t + Δt using the distribution function c obtained by repeating the second to fourth steps a predetermined number of times. A shape simulation method characterized by obtaining a processed shape of the surface of the object by extracting an isosurface where the distribution function c has a predetermined value. 5. The method according to claim 1, wherein the processing of the surface of the object comprises dissolving a resist with a developer, etching a semiconductor substrate, vapor deposition, growing a thin film, or growing a crystal.
The shape simulation method according to any one of claims 1 to 4. 6. A method for manufacturing a semiconductor device using a lithography step, comprising a step of obtaining a processed shape of the surface of the object using the shape simulation method according to claim 1. A method for manufacturing a semiconductor device, comprising: