US20150363516A1

US20150363516A1 - Topography simulation apparatus, topographysimulation method and recording medium

Info

Publication number: US20150363516A1
Application number: US14/482,502
Authority: US
Inventors: Hirotaka Tsuda; Masanori Takahashi
Original assignee: Toshiba Corp
Current assignee: Toshiba Corp
Priority date: 2014-06-13
Filing date: 2014-09-10
Publication date: 2015-12-17
Anticipated expiration: 2034-09-10
Also published as: KR20160019033A; US9201998B1; KR101627044B1; TW201546880A

Abstract

In one embodiment, a topography simulation apparatus includes a division module configured to divide topography of a substance of a semiconductor device into first to n-th layers, where n is an integer of two or more. The apparatus further includes a flux calculation module configured to calculate, for each of the first to n-th layers, a flux of particles which reach a surface of the substance in each layer. The apparatus further includes a topography calculation module configured to calculate, for each of the first to n-th layers, an amount of change of the topography of the substance in each layer based on the flux.

Description

CROSS REFERENCE TO RELATED APPLICATION

This application is based upon and claims the benefit of priority from the prior U.S. Provisional Patent Application No. 62/011,836 filed on Jun. 13, 2014, the entire contents of which are incorporated herein by reference.

FIELD

Embodiments described herein relate to a topography simulation apparatus, a topography simulation method and a recording medium.

BACKGROUND

In a topography simulation for a semiconductor device, it is required to calculate a flux of particles that reach a surface of a substance of the semiconductor device and contribute to the processing of topography of the substance in short time. The flux is called surface incident flux. For example, there is known a method of calculating the flux by using an aspect ratio of a concave portion formed in the substance. This method can calculate the flux in short time by approximating topography of the concave portion formed in the substance with one columnar shape, and analytically calculating the flux on a bottom surface of the concave portion.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart showing a procedure of a topography simulation method of a first embodiment;

FIGS. 2A and 2B are cross sectional views showing an example of an initial structure of a substance in the first embodiment;

FIGS. 3A and 3B are cross sectional views showing an example of a topography change of the substance in the first embodiment;

FIG. 4 is a perspective view for illustrating an aspect ratio x;

FIG. 5 is a graph showing an example of a relationship between the aspect ratio x and a flux ratio Γ_out/Γ_inin FIG. 4;

FIGS. 6A to 6C are cross sectional views for illustrating a method of calculating surface incident fluxes Γ_out _— ₁to Γ_out _— ₈of the first embodiment;

FIG. 7 is a cross sectional view showing another example of the topography change of the substance in the first embodiment;

FIG. 8 is a cross sectional view for illustrating a topography simulation method of a second embodiment;

FIG. 9 is a perspective view for illustrating a topography simulation method of a third embodiment;

FIG. 10 is a graph for illustrating a relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inin FIG. 9;

FIG. 11 is a graph showing an example of angle distribution of ions used for processing a semiconductor device;

FIG. 12 is a perspective view for illustrating a relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inin a case of using the ions;

FIG. 13 is a graph for illustrating the relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inin the case of using the ions;

FIGS. 14A and 14B are cross sectional views for illustrating a topography simulation method of a modification of the first to third embodiments;

FIG. 15 is an external view showing a configuration of a topography simulation apparatus of a fourth embodiment; and

FIG. 16 is a block diagram showing a configuration of a controller in FIG. 15.

DETAILED DESCRIPTION

Embodiments will now be explained with reference to the accompanying drawings.
In one embodiment, a topography simulation apparatus includes a division module configured to divide topography of a substance of a semiconductor device into first to n-th layers, where n is an integer of two or more. The apparatus further includes a flux calculation module configured to calculate, for each of the first to n-th layers, a flux of particles which reach a surface of the substance in each layer. The apparatus further includes a topography calculation module configured to calculate, for each of the first to n-th layers, an amount of change of the topography of the substance in each layer based on the flux.

First Embodiment

FIG. 1 is a flow chart showing a procedure of a topography simulation method of a first embodiment. The topography simulation method of the present embodiment is performed by using an information processing apparatus such as a personal computer and a workstation.
The topography simulation method of the present embodiment will be described below with reference to FIG. 1. In this description, FIGS. 2A to 3B will be also referred to as appropriate.
In the topography simulation method of the present embodiment, the initial structure of a substance of a semiconductor device is inputted into the information processing apparatus (step S1). Although examples of a method of inputting the initial structure include various possible formats, the present embodiment uses a method in which topography of a surface of the substance is represented in a point sequence, and the information processing apparatus reads this point sequence.
FIG. 2A is a cross sectional view showing an example of the initial structure of the substance in the first embodiment. The initial structure in FIG. 2A includes a substrate 1, a mask layer 2 formed on the substrate 1, and a concave portion 3 formed in the substrate 1 and the mask layer 2. Examples of the substrate 1 include a semiconductor substrate, and a workpiece substrate in which a workpiece layer is formed on a semiconductor substrate. Examples of the mask layer 2 include a hard mask layer and a resist mask layer. Topography of the concave portion 3 in FIG. 2A (initial topography) is a columnar shape. Reference character G denotes the central axis of the concave portion 3. Reference character R denotes the radius of the concave portion 3 (initial radius).
FIG. 2A shows an X direction and a Y direction that are parallel to a main surface of the substrate 1 and are perpendicular to each other, and a Z direction that is perpendicular to the main surface of the substrate 1. In the present specification, the +Z direction is treated as an upward direction, and the −Z direction is treated as a downward direction. For example, the positional relationship between the substrate 1 and the mask layer 2 is expressed that the substrate 1 is positioned below the mask layer 2.
The topography of the initial structure of the substance is divided into first to n-th layers L₁to L_n, where n is an integer of two or more (step S2). A block in the information processing apparatus for performing a process of step S2 is an example of a division module of the present disclosure.
FIG. 2B is a cross sectional view showing, similarly to FIG. 2A, the example of the initial structure of the substance in the first embodiment. FIG. 2B shows the example in which the topography of the initial structure of the substance is divided into the first to eighth layers L₁to L₈having the same thickness T. Reference character d denotes the depth of the concave portion 3. Reference characters r₁to r₈denote radii r₁to r₈of the concave portion 3 in the first to eighth layers L₁to respectively. All the values of the radii r₁to r₈in FIG. 2B are R. In the present embodiment, the start position of dividing the topography of the substance (the upper surface of the first layer L₁) is set at the aperture face of the concave portion 3, and the end position of dividing the topography of the substance (the lower surface of the eighth layer L₈) is set at the bottom surface of the concave portion 3.
Steps S3 to S6 of the topography simulation method in the present embodiment will be described below with reference to the initial structure in FIGS. 2A and 2B as an example. Therefore, the division number n of the topography of the substance is set at eight.
Step 3 is performed to calculate, for each of the first to eighth layers L₁to L₈, a flux of particles that reach the surface of the substance in each layer. The flux is called surface incident flux. In the initial structure in FIGS. 2A and 2B, step 3 is performed to calculate the flux of the particles that reach the side surface of the concave portion 3 in each layer. A block in the information processing apparatus for processing a process of step S3 is an example of a flux calculation module of the present disclosure.
Reference character Γ_indenotes the flux of the particles that reach the upper surface of a region of the concave portion 3 in the first layer L₁. Reference characters Γ_out _— ₁to r_out _— ₈denote the fluxes of the particles that reach the lower surfaces and the side surfaces of regions of the concave portion 3 in the first to eighth layers L₁to L₈, respectively. Therefore, the fluxes Γ_out _— ₁to Γ_out _— ₈correspond to the surface incident fluxes in the first to eighth layers L₁to L₈, respectively. A method of calculating the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈will be described later.
The particles treated in step S3 in the present embodiment are particles that contribute to the processing of the topography of the substance. Examples of such particles include ions or radicals that contribute to deposition, oxidation, nitriding, epitaxial growth, etching, aching or the like.
Step 4 is performed to calculate, for each of the first to eighth layers L₁to L₈, an amount of change of the topography of the substance in each layer based on the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈. Specifically, step 4 is performed to calculate the amounts of change Δr₁to Δr₈of the radii r₁to r₈of the concave portion 3 in the respective layers after the lapse of a time Δt from an initial time t₀. A block in the information processing apparatus for processing a process of step S4 is an example of a topography calculation module of the present disclosure.
In step S4 of the present embodiment, it is assumed that topography of a region of the concave portion 3 in each layer changes isotropically in a radial direction of a columnar shape, and the columnar shape changes into another columnar shape. In other words, it is assumed that the radius of the concave portion 3 in each layer changes in step S4 of the present embodiment. The amount of change Δr_mof the radius r_mof the concave portion 3 in an arbitrary m-th layer among the first to eighth layers L₁to L₈is provided by the following expression 1):
$\begin{matrix} Δ r_{m} = \frac{Γ_{out_m}}{Γ_{i n}} R_{p} Δ t & (1) \end{matrix}$
where m is an integer from one to eight, and reference character R_pdenotes the processing rate of the substance. In the present embodiment, since the flux Γ_inis set at one, the expression (1) is replaced with the following expression (2).
Δr _m=Γ_out _— _m)R _p Δt (2)
In the present embodiment, the processes of the step S3 and step S4 are repeated until a preset process time elapses (step S5). Thereby, the topography of the substance changes as shown in FIGS. 3A and 3B. FIGS. 3A and 3B are cross sectional views showing an example of a topography change of the substance in the first embodiment.
FIG. 3B shows the topography of the region of the concave portion 3 in each layer after the time t elapses from the initial time t₀(=0). FIG. 3B shows a state in which the topography of the region of the concave portion 3 in each layer has changed with maintaining the columnar shape. It is noted that since the values of the radii r₁to r₈have differently changed, the topography of the entire concave portion 3 in FIG. 3B has changed into a shape different from the columnar shape.
In this way, the present embodiment makes it possible to simulate a process that the concave portion 3 having the columnar shape changes into a shape different from the columnar shape. For example, the present embodiment makes it possible to simulate a process that the initial topography in FIG. 2A changes into the topography in FIG. 3A.
FIG. 3B shows a state in which the concave portion 3 expands to a new nin-th layer L₉. In the present embodiment, it is also possible to simulate the change in depth d of the concave portion 3 by applying the expression (2) to the bottom surface of the concave portion 3. In this way, the present embodiment may vary the division number n of the topography of the substance with the change of the topography of the substance with the lapse of the time t. Such a process will be described later in detail in a second embodiment.
When the process time has elapsed, the final topography of the substance is outputted (step S6), and the calculation is finished.
(1) Method of Calculating Surface Incident Fluxes Γ_out _— ₁to Γ_out _— ₈
In the present embodiment, the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈are calculated by using an aspect ratio of the region of the concave portion 3 in each layer. The method of calculating the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈of the present embodiment will be described below with reference to FIGS. 4 to 6C.
FIG. 4 is a perspective view for illustrating an aspect ratio x.
FIG. 4 shows an example of the concave portion 3 having a columnar shape. Reference characters Γ_inand Γ_outdenote the fluxes of the particles that reach the upper surface and the lower surface of the concave portion 3, respectively. Reference character H denotes the height of the concave portion 3. Reference character W denotes the aperture width (diameter) of the concave portion 3. Reference character x denotes the aspect ratio of the concave portion 3. The aspect ratio x of the concave portion 3 is given by the expression of x=H/W.
FIG. 5 is a graph showing an example of a relationship between the aspect ratio x and a flux ratio Γ_out/Γ_inin FIG. 4. In the present embodiment, since the flux Γ_inis set at one, the flux ratio Γ_out/Γ_inis the same value as that of the flux Γ_out.
FIG. 5 shows the relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inin the case where the flow of the particles in the concave portion 3 is represented by Knudsen transport model. Knudsen transport model represents the flow of the particles in a case where a mean free path of the particles is sufficiently longer than the aperture width W (molecular flow). In this case, the flow of the particles is not much influenced by the collision between the particles, but is influenced by the collision between the particles and the inner surface of the concave portion 3.
The relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inin Knudsen transport model is given by the following expression (3).
$\begin{matrix} \frac{Γ_{out}}{Γ_{i n}} = K (x) = {(1 + \frac{3}{4} x)}^{- 1} & (3) \end{matrix}$
K(x) in the expression (3) is called Clausing factor representing a transmission probability that the particles pass from the upper surface to the lower surface of the concave portion 3.
In the present embodiment, the surface incident fluxes Γ_out _— ₁to F_out _— ₈are calculated assuming that the flow of the particles in each of the first to eighth layers L₁to L₈is represented by Knudsen transport model of the expression (3).
FIGS. 6A to 6C are cross sectional views for illustrating a method of calculating the surface incident fluxes Γ_out _— ₁to F_out _— ₈of the first embodiment.
FIG. 6A shows a method of calculating the surface incident flux L_out _— ₂of the second layer L₂. Reference characters x₁and x₂denote the aspect ratios of the regions of the concave portion 3 in the first and second layers L₁and L₂, respectively. The aspect ratios x₁and x₂are given by T/2r₁and T/2r₂, respectively. Reference character T represents the thickness of each layer shown in FIGS. 2A to 3B.
Reference character K₁denotes the transmission probability that the particles pass from the upper surface to the lower surface of the region of the concave portion 3 in the first layer L₁. Reference character K₂denotes the transmission probability that the particles pass from the upper surface to the lower surface of the region of the concave portion 3 in the second layer L₂. The transmission probabilities K₁and K₂are respectively given by the following expressions (4) and (5) by using the expression (3).
$\begin{matrix} K_{1} (x_{1}) = {(1 + \frac{3}{4} x_{1})}^{- 1} & (4) \\ K_{2} (x_{2}) = {(1 + \frac{3}{4} x_{2})}^{- 1} & (5) \end{matrix}$
As seen from the above, the transmission probabilities K₁and K₂are functions of the aspect ratios x₁and x₂, respectively.
Reference character K_out _— ₂denotes the transmission probability that the particles pass through the regions of the concave portion 3 in the first and second layers L₁and L₂. The transmission probability K_out _— ₂is given by the following expression (6) of the sum of an infinite series, taking the incidence and reflection of the particles in each layer into consideration as shown in FIG. 6A.
K ₁ K ₂
K ₁ K ₂ +K ₁ K ₂(1−K ₁)(1−K ₂)
K ₁ K ₂ +K ₁ K ₂(1−K ₁)²(1−K ₂)²
. . .
K ₁ K ₂ +K ₁ K ₂(1−K ₁)^p(1−K ₂)^p (6)
By calculating the sum of the infinite series of the expression (6), the transmission probability K_out _— ₂is given by the following expression (7).
$\begin{matrix} K_{out_2} = \frac{K_{1} K_{2}}{K_{1} + K_{2} - K_{1} K_{2}} & (7) \end{matrix}$
Therefore, the surface incident flux Γ_out _— ₂in the second layer L₂is given by the following expression (8) obtained by substituting the expression (7) into the expression (3).
$\begin{matrix} \frac{Γ_{out_2}}{Γ_{i n}} = K_{out_2} = \frac{K_{1} K_{2}}{K_{1} + K_{2} - K_{1} K_{2}} & (8) \end{matrix}$
It is noted that the transmission probability K_out _— ₁that the particles pass through the region of the concave portion 3 in the first layer L₁is the same value as that of the transmission probability K₁(K_out _— ₁=K₁). Therefore, the surface incident flux Γ_out _— ₂of the first layer L₁is given by the expression (9) obtained by substituting K_out _— ₁=K₁into the expression (3).
$\begin{matrix} \frac{Γ_{out_1}}{Γ_{i n}} = K_{out_1} = K_{1} & (9) \end{matrix}$
FIG. 6B shows a method of calculating the surface incident flux Γ_out _— ₃in the third layer L₃. Reference character x₃denotes the aspect ratio of the region of the concave portion 3 in the third layer L₃. Reference character K₃denotes the transmission probability that the particles pass from the upper surface to the lower surface of the region of the concave portion 3 in the third layer L₃. The transmission probability K₃is given by the following expression (10) by using the expression (3).
$\begin{matrix} K_{3} (x_{3}) = {(1 + \frac{3}{4} x_{3})}^{- 1} & (10) \end{matrix}$
As seen from the above, the transmission probability K₃is a function of the aspect ratio x₃.
Reference character K_out _— ₃denotes the transmission probability that the particles pass through the regions of the concave portion 3 in the first to third layers L₁to L₃. When the transmission probability K_out _— ₃is calculated, the first and second layers L₁and L₂are considered to be one layer L₂′ having the transmission probability K_out _— ₂. Therefore, the transmission probability K_out _— ₃is given by the following expression (11) as with the expression (7).
$\begin{matrix} K_{out_3} = \frac{K_{out_2} K_{3}}{K_{out_2} + K_{3} - K_{out_2} K_{3}} & (11) \end{matrix}$
Therefore, the surface incident flux Γ_out _— ₃of the third layer L₃is given by the expression (12) obtained by substituting the expression (11) into the expression (3).
$\begin{matrix} \frac{Γ_{out_3}}{Γ_{i n}} = K_{out_3} = \frac{K_{out_2} K_{3}}{K_{out_2} + K_{3} - K_{out_2} K_{3}} & (12) \end{matrix}$
FIG. 6C shows a method of calculating a surface incident flux Γ_out _— _kin a k-th layer L_kwhere k is an integer from two to n (=8). Reference character x_kdenotes the aspect ratio of the region of the concave portion 3 in the k-th layer L_k. Reference character K_kdenotes the transmission probability that the particles pass from the upper surface to the lower surface of the region of the concave portion 3 in the k-th layer L_k. The transmission probability K_kis given by the following expression (13) by using the expression (3).
$\begin{matrix} K_{k} (x_{k}) = {(1 + \frac{3}{4} x_{k})}^{- 1} & (13) \end{matrix}$
As seen from the above, the transmission probability K_kis a function of the aspect ratio x_k. The aspect ratio x_kis an example of a topography parameter representing the topography of the substance in the k-th layer.
Reference character K_out _— _kdenotes the transmission probability that the particles pass through the regions of the concave portion 3 in the first to k-th layers L₁to L_k. When the transmission probability K_out _— _kis calculated, the first to (k−1)-th layers L₁to L_k-1are considered to be one layer L_k-1′ having the transmission probability K_out _— _k-1. Therefore, the transmission probability K_out _— _kis given by the following expression (14) as with the expression (7).
$\begin{matrix} K_{out_k} = \frac{K_{out_k - 1} K_{k}}{K_{out_k - 1} + K_{k} - K_{out_k - 1} K_{k}} & (14) \end{matrix}$
Therefore, the surface incident flux Γ_out _— _kin the k-th layer L_kis given by the following expression (15) obtained by substituting the expression (14) into the expression (3).
$\begin{matrix} \frac{Γ_{out_k}}{Γ_{i n}} = K_{out_k} = \frac{K_{out_k - 1} K_{k}}{K_{out_k - 1} + K_{k} - K_{out_k - 1} K_{k}} & (15) \end{matrix}$
In this way, the surface incident flux Γ_out _— _kin the k-th layer L_kof the present embodiment is calculated based on the surface incident flux Γ_out _— _k-1(=K_out _— _k-1) in the (k−1)-th layer L_k-1adjacent to the k-th layer L_k, and the transmission probability K_kin the k-th layer L_k.
The surface incident fluxes Γ_out _— ₁to Γ_out _— ₈calculated in this way are respectively used for calculating the amounts of change Δr₁to Δr₈in the radii r₁to r₈in step S4.
It is noted that the values of the transmission probabilities K₁to K₈and the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈change with the lapse of the time t. The reason for this is that since the radii r₁to r₈change with the lapse of the time t, the aspect ratios x₁to x₈also change with the lapse of the time t.
(2) Details of Method of Calculating Surface Incident Fluxes Γ_out _— ₁to Γ_out _— ₈
The method of calculating the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈of the present embodiment will be described in detail with reference to the same FIGS. 6A to 6C.
As described above, the present embodiment calculates the surface incident fluxes Γ_out-1to Γ_out _— ₈using Knudsen transport model. Knudsen transport model well fits the behavior of the particles moving isotropically like radicals. Furthermore, Knudsen transport model well fits the behavior of the particles moving in a low-pressure atmosphere close to vacuum. Therefore, the method of calculating the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈of the present embodiment is suitable for, for example, simulating a plasma process in manufacturing a semiconductor device. An example where the present embodiment is applied to the particles moving anisotropically like ions will be described later.
The present embodiment can analytically calculate the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈by using Knudsen transport model as shown by the expression (15) for example. However, the present embodiment may use a model other than Knudsen transport model. In this case, the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈may be given by a form other than analytical formula, for example, by a form of an empirical formula obtained by experimentation.
In the present embodiment, the initial topography of the concave portion 3 is set to have a columnar shape. However, the initial topography of the concave portion 3 of the present embodiment may be set to have a shape other than the columnar shape. Examples of such an initial topography of the concave portion 3 include a prism shape having a quadrilateral or another polygonal cross section, and a cylindrical shape having an annular cross section.
In the present embodiment, the change of the topography of the region of the concave portion 3 in each layer is calculated such that it changes from a columnar shape into another columnar shape. However, the present embodiment is applicable to other topography changes by introducing amounts of change of the topography other than the amounts of change Δr₁to Δr₈of the radii r₁to r₈, and topography parameters other than the aspect ratios x₁to x₈. Examples of such amounts of change of the topography include an amount of change of the length of an edge of the concave portion 3 having a prism shape. In addition, examples of such topography parameters include a ratio of the length of the edge of the concave portion 3 to the height (thickness) of the concave portion 3.
In the present embodiment, the thicknesses of the first to eighth layers L₁to L₈are set at the same thickness T. However, the thicknesses of the first to eighth layers L₁to L₈of the present embodiment may be set to be different from one another. An example will be described later in which the thicknesses of these layers are set to be different from one another.
In the present embodiment, the radii r₁to r₈of the initial topography of the concave portion 3 in the first to eighth layers L₁to L₈are set at the same value. However, the radii r₁to r₈of the initial topography of the concave portion 3 of the present embodiment may be set to be different from one another. In other words, in the initial topography of the concave portion 3 in the present embodiment, the cross sectional shapes of the concave portion 3 in the first to eighth layers L₁to L₈may be set to be different from one another.
(3) Modification of First Embodiment
FIG. 7 is a cross sectional view showing another example of the topography change of the substance in the first embodiment.
Reference character U denotes the surface of the concave portion 3 at the time t. Arrows A denote directions in which the surface U of the concave portion 3 changes with the lapse of the time t. Reference character B denotes an end portion of the surface U of the concave portion 3.
The present embodiment can simulate a process that the side surface of the concave portion 3 is etched in a horizontal direction by calculating the amounts of change Δr₁to Δr₈of the radii r₁to r₈. For example, in a dry etching by using fluorine-containing etching gas, it is known that the substrate 1 is etched isotropically both in the depth direction and the horizontal direction. As shown in FIG. 7, the present embodiment makes it possible to simulate such dry etching. FIG. 7 shows the result of simulating an undercut process that the substrate 1 below the mask layer 2 is etched by using the mask layer 2 which has a processing rate lower than that of the substrate 1.
As described above, the topography simulation method of the present embodiment divides the topography of the substance into the first to n-th layers L₁to L_n, calculates the surface incident fluxes Γ_out _— ₁to Γ_out _— _nfor the respective layers, and calculates the amounts of change Δr₁to Δr_nof the radii r₁to r_nof the concave portion 3 for the respective layers.
Therefore, the present embodiment makes it possible to handle the topography change of the side surface of the concave portion 3, for example, in the case where the topography of the concave portion 3 changes from the columnar shape into a shape other than the columnar shape. In other words, the present embodiment makes it possible to handle the topography change of the substance that is to be processed into various shapes.
Furthermore, the present embodiment makes it possible to calculate the surface incident fluxes Γ_out _— ₁to Γ_out _— _nin short time, because the surface incident flux in each layer is calculated by using the aspect ratio of each layer. In other words, the present embodiment makes it possible to handle various topography changes of the substance, while shortening the calculating time of the surface incident fluxes Γ_out _— ₁to Γ_out _— _n. This is also applied to the case where a topography parameter other than the aspect ratio is used.
In the present embodiment, it is assumed that the flux on the side surface of the region of the concave portion 3 in each layer is the same as the flux (Γ_out _— ₁to Γ_out _— _n) on the bottom surface of the region of the concave portion 3 in each layer. Therefore, each of the fluxes Γ_out _— ₁to Γ_out _— _nin the present embodiment is used as the surface incident flux on the side surface of the region of the concave portion 3 in each layer. The precision of such an approximation is considered to become finer by setting the division number n of the topography of the substance to be sufficiently large and making the thickness T of the layers sufficiently small. However, the division number n and the thickness T in the present embodiment can be arbitrarily set in accordance with the precision required for individual topography simulations.

Second Embodiment

FIG. 8 is a cross sectional view for illustrating a topography simulation method of a second embodiment.
Reference character U denotes the surface of the concave portion 3 in the initial structure of the substance. Arrows C denote directions in which the surface U of the concave portion 3 changes with the lapse of a time t. Reference character D denotes a point of closure of the concave portion 3. FIG. 8 shows a state that the concave portion 3 is closed by embedding an insulating material in the concave portion 3 while a void 3 a is left. FIG. 8 further shows a process that the insulating material is deposited on the substrate 1 outside the concave portion 3. Examples of the concave portion 3 include an isolation trench.
FIG. 8 illustrates five layers L_Ato L_E. The layer L_Ais positioned at a height where the layer L_Aincludes the point of closure D. The layers L_Band L_Care positioned below the layer L_A. The layers L_Dand L_Eare positioned above the layer L_A. The layer L_Dis positioned at a height where the layer L_Dincludes the main surface of the substrate 1, and the layer L_Eis positioned above the main surface of the substrate 1.
In the present embodiment, when the concave portion 3 is closed at the point of closure D, it is useless to perform the processes of steps S3 and S4 in FIG. 1 to the layers L_Band L_C.
Therefore, if the concave portion 3 is closed at the point of closure D at the time t, a process of removing the layers L_Band L_Cbelow the layer L_Ais performed between steps S3 and S4 at the time t and steps S3 and S4 at the time t+Δt.
In the present embodiment, if the insulating material is deposited on the substrate 1 outside the concave portion 3, it is desirable that the layers L_Dand L_Eincluding this insulating material are added. Therefore, when the insulating material is deposited on the substrate 1 outside the concave portion 3 at the time t, a process of adding the layer L_D(as well as the layer L_Eif needed) between steps S3 and S4 at the time t and steps S3 and S4 at the time t+Δt.
The timing to add the layers L_Dand L_Emay be arbitrarily set. For example, the layer L_Emay be added at the time when the surface of the insulating material reaches the inside of the layer L_E, or may be added at the time when the surface of the insulating material reaches the middle point between the upper surface and the lower surface of the layer L_E. This is also applied to the timing to add the above-described nin-th layer L₉(FIG. 3B).
As described above, the topography simulation method of the present embodiment varies the division number n of the topography of the substance with the change of the topography of the substance with the lapse of the time t. Therefore, the present embodiment makes it possible to realize a faster and more precise topography simulation by deleting a useless layer and adding a required layer.

Third Embodiment

FIG. 9 is a perspective view for illustrating a topography simulation method of a third embodiment.
FIG. 9 shows an example of the concave portion 3 having a columnar shape, similarly to FIG. 4. Reference characters 17, and Γ_outdenote the fluxes of the particles that reach the upper surface and the lower surface of the concave portion 3, respectively. Reference character K is the Clausing factor, which denotes the transmission probability that the particles pass from the upper surface to the lower surface of the concave portion 3. Reference character S denotes a reaction probability that the particle react with the substance inside the concave portion 3 or outside the concave portion 3. The reaction probability S is also called sticking probability.
FIG. 9 shows the flow of the particles in a case where the flow of the particles in the concave portion 3 is represented by Coburn reactive transport model. Coburn reactive transport model is a compensation model in which the reaction probability S is introduced into Knudsen transport model. Knudsen transport model is equivalent to Coburn reactive transport model in which the reaction probability S is one.
FIG. 9 shows a flux SΓ_outthat reacts with the substance after passing through the concave portion 3, a flux (1−K)Γ_inthat is reflected without passing through the concave portion 3, and a flux K(1−S)Γ_outthat is reflected without reacting with the substance after passing through the concave portion 3. The following expression (16) holds among these fluxes.
Γ_in−(1−K)−_in −K(1−S)Γ_out =SΓ _out (16)
Therefore, a flux ratio Γ_out/Γ_inin Coburn reactive transport model is given by the following expression (17).
$\begin{matrix} \frac{Γ_{out}}{Γ_{i n}} = C (x) = \frac{K (x)}{K (x) + S - SK (x)} & (17) \end{matrix}$
The expression (17) indicates that the Clausing factor K(x) is replaced with a factor C(x) by replacing Knudsen transport model with Coburn reactive transport model. The relationship between the aspect ratio x and the Causing factor K(x) is expressed by the above-described expression (3).
FIG. 10 is a graph for illustrating a relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inin FIG. 9.
A curve K shows the flux ratio Γ_out/Γ_inwhen S=1. Curves C₁to C₄show flux ratios Γ_out/Γ_nwhen S=0.5, S=0.1, S=0.01 and S=0, respectively. The curve K is the same as the curve in FIG. 5.
(1) Topography Simulation Method of Third Embodiment
The topography simulation method of the present embodiment is performed in the same manner as the topography simulation method of the first embodiment. However, the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈in the present embodiment are calculated assuming that the flow of the particles in each of the first to eighth layers L₁to L₈can be represented by Coburn reactive transport model of the expression (17).
Therefore, the surface incident flux Γ_out _— _kin a k-th layer L_kof the present embodiment is given by the following expression (18) obtained by replacing K(x) of the expression (15) with C(x).
$\begin{matrix} \frac{Γ_{out_k}}{Γ_{i n}} = C_{out_k} = \frac{C_{out_k - 1} C_{k}}{C_{out_k - 1} + C_{k} - C_{out_k - 1} C_{k}} & (18) \\ C_{k} (x) = \frac{K_{k} (x)}{K_{k} (x) + S - {SK}_{k} (x)} & (19) \end{matrix}$
In this way, the surface incident flux Γ_out _— _kin the k-th layer L_kof the present embodiment is calculated based on the surface incident flux Γ_out _— _k-1(=C_out _— _k-1) in a (k−1)-th layer L_k-1adjacent to the k-th layer L_k, the transmission probability K_kof the k-th layer L_k, and the reaction probability S of the particles.
The surface incident fluxes Γ_out _— ₁to Γ_out _— ₈calculated in this way are used for calculating the amounts of change Δr₁to Δr₈of the radii r₁to r₈in step S4.
As described above, the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈in the present embodiment are calculated using Coburn reactive transport model. Therefore, the present embodiment makes it possible to simulate the topography change of the substance to be processed into various topography with high precision by taking the reaction probability S of the particles into consideration.

First Modification of First to Third Embodiments

In the first embodiment, the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈are calculated using Knudsen transport model. Knudsen transport model well fits the behavior of the particles moving isotropically like radicals. Therefore, when the present embodiment is applied to the particles moving anisotropically like ions, it is desirable that the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈are calculated using a model different from Knudsen transport model.
FIG. 11 is a graph showing an example of angle distribution of the ions used for processing a semiconductor device. The vertical axis in FIG. 11 denotes an ion intensity I. The horizontal axis in FIG. 11 denotes an angle θ. FIG. 11 shows an example that the angle distribution of the ions follows a normal distribution. Reference character σ denotes the standard deviation of the normal distribution. The unit of the angle θ and the standard deviation σ is radian, for example. A width 2σ shown in FIG. 11 corresponds to a half width of the angle distribution of the ions.
FIG. 12 is a perspective view for illustrating a relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inin the case of using the ions.
FIG. 12 shows an example of the concave portion 3 having a columnar shape, similarly to FIGS. 4 and 9. Reference characters Γ_inand Γ_outdenote the fluxes of the particles that reach the upper surface and the lower surface of the concave portion 3, respectively. Reference character x denotes the aspect ratio of the concave portion 3. An angle 2α shown in FIG. 12 corresponds to an angular aperture in a case where the aperture is looked up at from the bottom portion of the concave portion 3 having the aspect ratio x.
When the ions are used for processing the semiconductor device, there is a problem with the ratio of the angular aperture 2α, to the half width 2σ. In a case where the ratio α/σ is small, since the angular aperture 2α of the concave portion 3 is large relative to the half width 2σ, many ions can reach the bottom surface of the concave portion 3. In contrast, in a case where the ratio α/σ is large, since the angular aperture 2α of the concave portion 3 is small relative to the half width 2σ, many ions cannot reach the bottom surface of the concave portion 3. Therefore, when the ions are used for processing the semiconductor device, the surface incident fluxes Γ_out _— ₁to Γ_out _— ₈depend on the half width 2σ.
The flux ratio Γ_out/Γ_inin FIG. 12 is given by the following expression (20) when the angle distribution of the ions in FIG. 11 is applied:
$\begin{matrix} \frac{Γ_{out}}{Γ_{i n}} = \erf (\frac{\arctan (1 / (2 x))}{σ \sqrt{2}}) & (20) \end{matrix}$
where “erf” denotes an error function, “arc tan” denotes arc tangent, “σ” denotes the standard deviation of the normal distribution, and “x” denotes the aspect ratio.
FIG. 13 is a graph for illustrating the relationship between the aspect ratio x and the flux ratio Γ_out/Γ_inthe case of using the ions.
Curves C₁to C₄the flux ratios Γ_out/Γ_inwhen σ=0.1, σ=0.05, σ=0.02 and σ=0, respectively. When comparing FIG. 5 and FIG. 13, the flux ratios Γ_out/Γ_inwith the same aspect ratio x are larger in the case of FIG. 5 than in the case of FIG. 13. It is therefore understood that the ions are more likely to reach the bottom surface of the concave portion 3 than the radicals.
The first embodiment can be applied to the particles moving anisotropically like the ions by replacing the function form of the transmission probability K(x) of the expression (3) with a function form shown in FIG. 13. In this case, the transmission probability K(x) is a function of the aspect ratio and the half width 2α. The half width 2σ is an example of a distribution parameter representing the distribution of the particles.
In addition, the third embodiment can be applied to the particles moving anisotropically like the ions by using the function form shown in FIG. 13.

Second Modification of First to Third Embodiments

FIGS. 14A and 14B are cross sectional views for illustrating a topography simulation method of a modification of the first to third embodiments.
In FIGS. 2A and 2B, the main surface of the substrate 1 in the initial structure of the substance is set to be a flat plane. In contrast, in FIGS. 14A and 14B, the main surface of the substrate 1 in the initial structure of the substance includes a difference in level. Therefore, the mask layer 2 in FIGS. 14A and 14B includes a first upper surface which is a higher upper surface, and a second upper surface which is a lower upper surface. The first to third embodiments can be also applied to such an initial structure.
When the substance as shown in FIG. 14A is actually processed, the aperture is likely to be closed in the vicinity of the side surface of the mask layer 2. Therefore, when the processing of the topography of such a substance is simulated, it is desirable that a calculation with high precision is performed to the vicinity of the side surface of the mask layer 2. Therefore, in FIG. 14B, a thickness T₁of layers L₁to L₄close to the side surface of the mask layer 2 is set to be small, and a thickness T₂of layers L₅to L₉far from the side surface of the mask layer 2 is set to be large. In the present modification, the start position of dividing the topography of the substance (the upper surface of the first layer L₁) is set at the second upper surface (lower upper surface) of the mask layer 2, the end position of dividing the topography of the substance (the lower surface of the nin-th layer L₉) is set at the bottom surface of the concave portion 3.
When the substance as shown in FIG. 14A is actually processed, the topography change of the concave portion 3 is often affected by forming a deposition layer on the side surface and the mask layer 2 or etching the side surface of the mask layer 2. Therefore, the present modification may handle, as a target of the topography simulation, the topography change of the mask layer 2 as well as the topography change of the substrate 1. In this case, it is desirable to separately set, as a processing rate R_pin the expression (1), the processing rate of the substrate 1 and the processing rate of the mask layer 2.

Fourth Embodiment

FIG. 15 is an external view showing a configuration of a topography simulation apparatus of a fourth embodiment.
The topography simulation apparatus in FIG. 15 includes a controller 11, a display module 12, and an input module 13.
The controller 11 controls the operation of the topography simulation apparatus. For example, the controller 11 performs one of the topography simulation methods of the first to third embodiments. The controller 11 will be described in detail hereafter.
The display module 12 includes display devices such as a liquid crystal monitor. For example, the display module 12 displays a screen for inputting setting information for the topography simulation, and calculation results of the topography simulation.
The input module 13 includes input devices such as a keyboard 13 a and a mouse 13 b. For example, the input module 13 is used for inputting the setting information for the topography simulation. Examples of the setting information include information on calculation expressions, information on experimental values or predicted values, information on the structure of the substance, information on the flux of the particles, and instruction information on conditions or procedures of the topography simulation.
FIG. 16 is a block diagram showing a configuration of the controller 11 in FIG. 15.
The controller 11 includes a central processing unit (CPU) 21, a read only memory (ROM) 22, a random access memory (RAM) 23, a hard disk drive (HDD) 24, a memory drive 25 such as a compact disc (CD) drive or digital versatile disk (DVD) drive, and a memory interface (I/F) 26 such as a memory port or a memory slot.
In the present embodiment, a topography simulation program for one of the topography simulation methods of the first to third embodiments is stored in the ROM 22 or the HDD 24. When predetermined instruction information is inputted from the input module 13, the CPU 21 reads the program from the ROM 22 or the HDD 24, expands the read program on the RAM 23, and performs the topography simulation under this program. Various types of data generated with this process are held in the RAM 23.
In the present embodiment, a non-transitory computer readable recording medium in which the topography simulation program is stored may be prepared to install the topography simulation program to the ROM 22 or the HDD 24 from the recording medium. Examples of such a recording medium include CD-ROMs and DVD-ROMs.
In the present embodiment, the topography simulation program may be installed into the ROM 22 or the HDD 24 by downloading it via a network such as the Internet.
As described above, the present embodiment makes it possible to provide the topography simulation apparatus and the topography simulation program for performing the topography simulation methods of the first to third embodiments.
While certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the inventions. Indeed, the novel apparatuses, methods and media described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of the apparatuses, methods and media described herein may be made without departing from the spirit of the inventions. The accompanying claims and their equivalents are intended to cover such forms or modifications as would fall within the scope and spirit of the inventions.

Claims

1. A topography simulation apparatus comprising:

a division module configured to obtain data representing topography of a substance of a semiconductor device and to divide the topography of the substance in the data into first to n-th layers, where n is an integer of two or more;

a flux calculation module configured to calculate, for each of the first to n-th layers, a flux of particles which reach a surface of the substance in each layer and contribute to processing of the topography of the substance; and

a topography calculation module configured to calculate, for each of the first to n-th layers, an amount of change of the topography of the substance in each layer based on the flux,

wherein the flux calculation module calculates the flux in a k-th layer based on the flux in a (k−1)-th layer adjacent to the k-th layer and a transmission probability of the particles in the k-th layer, where k is an integer from 2 to n.

2. The apparatus of claim 1, wherein the division module divides the topography of the substance which includes a concave portion into the first to n-th layers.

3. The apparatus of claim 2, wherein topography of a region of the concave portion in each layer is a columnar shape, and

the topography calculation module calculates an amount of change of a radius of the concave portion in each layer.

4. (canceled)

5. The apparatus of claim 1, wherein the transmission probability is a function of a topography parameter representing the topography of the substance in the k-th layer.

6. The apparatus of claim 5, wherein the topography parameter is an aspect ratio of a region of a concave portion of the substance in the k-th layer.

7. The apparatus of claim 1, wherein the transmission probability is a function of a topography parameter representing the topography of the substance in the k-th layer and a distribution parameter representing distribution of the particles.

8. The apparatus of claim 7, wherein the distribution parameter is a half width of the distribution of the particles.

9. The apparatus of claim 1, wherein the division module varies a value of the division number n of the first to n-th layers with a change of the topography of the substance.

10. The apparatus of claim 1, wherein the flux calculation module calculates the flux in the k-th layer based on the flux in the (k−1)-th layer, the transmission probability of the particles in the k-th layer, and a reaction probability of the particles.

11. A topography simulation method performed by an apparatus comprising:

obtaining data representing topography of a substance of a semiconductor device by the apparatus, and dividing the topography of the substance in the data into first to n-th layers by the apparatus, where n is an integer of two or more;

calculating, for each of the first to n-th layers, a flux of particles which reach a surface of the substance in each layer and contribute to processing of the topography of the substance, by the apparatus; and

calculating, for each of the first to n-th layers, an amount of change of the topography of the substance in each layer based on the flux, by the apparatus,

wherein the flux in a k-th layer is calculated based on the flux in a (k−1)-th layer adjacent to the k-th layer and a transmission probability of the particles in the k-th layer, where k is an integer from 2 to n.

12. (canceled)

13. The method of claim 11, wherein the transmission probability is a function of a topography parameter representing the topography of the substance in the k-th layer.

14. The method of claim 13, wherein the topography parameter is an aspect ratio of a region of a concave portion of the substance in the k-th layer.

15. The method of claim 11, wherein the transmission probability is a function of a topography parameter representing the topography of the substance in the k-th layer and a distribution parameter representing distribution of the particles.

16. The method of claim 15, wherein the distribution parameter is a half width of the distribution of the particles.

17. The method of claim 11, further comprising varying a value of the division number n of the first to n-th layers with a change of the topography of the substance.

18. The method of claim 11, wherein the flux in the k-th layer is calculated based on the flux in the (k−1)-th layer, the transmission probability of the particles in the k-th layer, and a reaction probability of the particles.

19. A non-transitory computer-readable recording medium containing a topography simulation program which causes a computer to perform a topography simulation method, the method comprising:

obtaining data representing topography of a substance of a semiconductor device by the computer, and dividing the topography of the substance in the data into first to n-th layers by the computer, where n is an integer of two or more;

calculating, for each of the first to n-th layers, a flux of particles which reach a surface of the substance in each layer and contribute to processing of the topography of the substance, by the computer; and

calculating, for each of the first to n-th layers, an amount of change of the topography of the substance in each layer based on the flux, by the computer,

20. The medium of claim 19, wherein the flux in the k-th layer is calculated based on the flux in the (k−1)-th layer, the transmission probability of the particles in the k-th layer, and a reaction probability of the particles.