JP2002047096A - Method for simulating distribution of point defect in single crystal - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a simulation method in which the calculation value of the point defect distribution in a single crystal extremely well matches up to an actual measurement. SOLUTION: This simulation method comprises modeling a hot zone of a pulling-up machine into a mesh structure (a first step), inputting the physical values of respective members into a computer (a second step), obtaining the surface temperature distribution of each member based on the physical value or the like (a third step), obtaining the interior temperature distribution of each member based on the above surface temperature distribution and further obtaining the inside temperature of a melted liquid 12 taking account of convection (a fourth step), obtaining the shape of the interface of solid and liquid conforming to an isothermal line including the triple point S of the single crystal (a fifth step), repeating the third to fifth steps until the above triple point becomes the melting point of the single crystal (a sixth step), repeating the first to sixth steps while changing the length of single crystal pulling-up to find the coordinates and temperature of single crystal mesh, and solving a diffusion equation based on the diffusion coefficient and boundary condition of vacancy in the crystal and interstitial atom to obtain the concentration distributions of the vacancy and interstitial atom after the cooling of the single crystal.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for computer simulation of a point defect distribution of a single crystal such as silicon pulled by the Czochralski (CZ) method.

[0002]

2. Description of the Related Art Conventionally, as a simulation method of this kind, as shown in FIG. 4, a pulling machine 1 for pulling a silicon single crystal 4 by a CZ method using an integrated heat transfer simulator.
Predicting the internal temperature distribution of the silicon melt 2 by manipulating the thermal conductivity of the silicon melt 2 based on the hot zone structure in the inside and the pulling speed of the silicon single crystal 4,
The coordinates and temperature of the mesh of the silicon single crystal 4 are obtained from the internal temperature distribution, and the diffusion equation is solved based on the diffusion coefficients and boundary conditions of interstitial silicon and vacancies in the silicon single crystal 4, thereby obtaining the above lattice. There is known a method of obtaining the concentration distribution of intersilicon and voids using a computer. In this simulation method, each member of the hot zone is modeled by mesh division. In particular, the mesh of the silicon melt 2 is set to be relatively coarse at about 10 mm in order to shorten the calculation time.

[0003]

However, the conventional method of simulating the concentration distribution of interstitial silicon and vacancies does not take into account the convection of the silicon melt generated in an actual pulling machine. Since the mesh of the melt was relatively coarse, the concentration distribution of interstitial silicon and vacancies (FIG. 5B) was significantly different from the actually measured values (FIG. 5E). An object of the present invention is to provide a method for simulating a point defect distribution of a single crystal, in which a calculated value of a point defect distribution in the single crystal is in good agreement with an actually measured value.

[0004]

The invention according to claim 1 is
As shown in FIGS. 1 to 3, the single crystal 1
A first step of modeling the hot zone of the puller 11 of the single crystal 14 in a state in which the single crystal 4 has been pulled up to a predetermined length by a mesh structure; a mesh for each member of the hot zone; A second step of providing the physical property values of the single crystal 14 and the pulling speed of the single crystal 14 corresponding to the pulling length, and the surface temperature distribution of each member based on the calorific value of the heater and the emissivity of each member. And obtaining the internal temperature distribution of each member by solving the heat conduction equation based on the surface temperature distribution and the thermal conductivity of each member, and assuming that the melt 12 is turbulent. A fourth step of further calculating the internal temperature distribution of the melt 12 in consideration of convection by connecting and solving the turbulence model equation and the Navier-Stokes equation thus obtained; Fifth determining the solid-liquid interface shape of a crystal 14 and melt 12 in accordance with the isotherm containing triple point S of the single crystal
Step and the third to fifth steps are the triple point S
The sixth step of repeatedly calculating the temperature distribution in the puller 11 until the melting point of the single crystal 14 is reached, obtaining the coordinates and temperature of the mesh of the single crystal 14 and providing these data, and the pulling length of the single crystal 14 A seventh step in which the temperature distribution in the puller 11 is calculated by repeatedly changing the first step to the sixth step stepwise to obtain the coordinates and temperature of the mesh of the single crystal 14 and to provide these data;
An eighth step of providing data of the coordinates and temperature of the mesh of the single crystal 14, the diffusion coefficient of vacancies and interstitial atoms in the single crystal 14, and boundary conditions, respectively; A ninth step of obtaining a concentration distribution of vacancies and interstitial atoms after cooling of the single crystal 14 by solving a diffusion equation based on the diffusion coefficient and boundary conditions of the interstitial atoms. This is a simulation method.

In the method for simulating the point defect distribution of a single crystal according to the present invention, the internal temperature distribution of the single crystal is determined in consideration of the convection of the melt.
Since the point defect distribution in the single crystal 14 was determined based on the internal temperature distribution of No. 4 and in consideration of the diffusion of the point defect in the single crystal 14, the calculated value of the point defect distribution in the single crystal 14 was extremely different from the actually measured value. Good agreement.

[0006]

Embodiments of the present invention will now be described with reference to the drawings. As shown in FIG. 3, a quartz crucible 13 for storing a silicon melt 12 is provided in the chamber of the silicon single crystal pulling machine 11. This quartz crucible 1
3 is connected to the crucible driving means via a graphite susceptor and a support shaft (not shown), and the crucible driving means is a quartz crucible 13.
Is configured to rotate and move up and down. Further, the outer peripheral surface of the quartz crucible 13 is surrounded by a heater (not shown) at a predetermined interval from the quartz crucible 13, and this heater is surrounded by a heat retaining tube (not shown). The heater heats and melts the high-purity polycrystalline silicon charged into the quartz crucible 13 to form a silicon melt 12. Although not shown, a cylindrical casing (not shown) is connected to the upper end of the chamber, and the casing is provided with pulling means. The pulling means is configured to pull while rotating the silicon single crystal 14.

A method of simulating the point defect distribution of the silicon single crystal 14 in the silicon single crystal pulling machine 11 configured as described above will be described with reference to FIGS.
First, as a first step, each member of the hot zone of the silicon single crystal pulling machine 11 in a state where the silicon single crystal 14 is pulled up to a predetermined length L ₁ (for example, 100 mm),
That is, the chamber, the quartz crucible 13, the silicon melt 12, the silicon single crystal 14, the graphite susceptor, the heat retaining cylinder, and the like are modeled by mesh division. Specifically, the coordinate data of the mesh points of each member of the hot zone is input to a computer. At this time, a part or all of the mesh of the silicon melt 12 in the radial direction of the silicon single crystal 14 and directly below the silicon single crystal 14 of the silicon melt 12 (hereinafter referred to as a radial mesh). 0.01
To 5.00 mm, preferably 0.25 to 1.00 mm. The mesh of the silicon melt 12 in the longitudinal direction of the silicon single crystal 14 and a part or all of the mesh of the silicon melt 12 (hereinafter referred to as a longitudinal mesh) is 0.01 to 5.00 mm. , Preferably set to 0.1 to 0.5 mm.

[0008] The radial mesh is 0.01 to 5.00 mm
The reason is that the calculation time is extremely long when the length is less than 0.01 mm, the calculation becomes unstable when the length exceeds 5.00 mm, and the shape of the solid-liquid interface is not fixed even when the calculation is repeated. is there. The reason why the longitudinal mesh is limited to the range of 0.01 to 5.00 mm is that
If the length is less than 1 mm, the calculation time becomes extremely long, and 5.00 m
If it exceeds m, the calculated value of the solid-liquid interface shape does not match the actually measured value. Note that a part of the radial mesh is set to 0.
In the case of limiting to the range of 01 to 5.00, it is preferable to limit the silicon melt 12 near the outer peripheral edge of the silicon single crystal 14 in the silicon melt 12 immediately below the silicon single crystal 14 to the above range, 0.01 part of mesh
When the range is limited to the range of ~ 5.00, the silicon melt 1
It is preferable that the vicinity of the liquid surface and the vicinity of the bottom of No. 2 are limited to the above ranges.

As a second step, meshes are grouped for each member in the hot zone, and the physical property values of the members are input to the computer for the grouped meshes. For example, if the chamber is made of stainless steel, the thermal conductivity, emissivity, viscosity, volume expansion coefficient, density, and specific heat of the stainless steel are input to the computer. Further, a pulling length of the silicon single crystal 14, a pulling speed of the silicon single crystal 14 corresponding to the pulling length, and a random number parameter C of a random number model formula (1) described later are input to the computer.

As a third step, the surface temperature distribution of each member in the hot zone is obtained by using a computer based on the calorific value of the heater and the emissivity of each member. That is, the calorific value of the heater is arbitrarily set and input to the computer, and the surface temperature distribution of each member is obtained from the emissivity of each member using the computer. Next, as a fourth step, the internal temperature distribution of each member is obtained by solving the heat conduction equation (2) using a computer based on the surface temperature distribution and the thermal conductivity of each member in the hot zone. Here, an xyz rectangular coordinate system is used for simplicity of description, but a cylindrical coordinate system is used in actual calculation.

[0011]

(Equation 1) Here, ρ is the density of each member, c is the specific heat of each member, T is the absolute temperature at each mesh point of each member,
t is time, λ _x , λ _y, and λ _z are the x, y, and z components of the thermal conductivity of each member, and q is the heating value of the heater. On the other hand, with respect to the silicon melt 12, after the internal temperature distribution of the silicon melt 12 is obtained by the heat conduction equation (2), the silicon melt 12 is turbulent based on the internal temperature distribution of the silicon melt 12. The turbulence model equation (1) and the Navier-Stokes equations (3) to (5), which are obtained assuming that there is, are connected to obtain the internal flow velocity distribution of the silicon melt 12 using a computer.

[0012]

(Equation 2) Here, kappa _t is turbulent thermal conductivity of the silicon melt 12, c is the specific heat of the silicon melt 12, Pr _t is the Prandtl number, [rho is the density of the silicon melt 12,
C is a turbulence parameter, d is a distance from the wall of the quartz crucible 13 storing the silicon melt 12, and k is a sum of squares of a fluctuation component with respect to an average flow velocity of the silicon melt 12.

[0013]

(Equation 3)

Here, u, v and w are the silicon melt 12
Are the x, y and z-direction components of the flow velocity at each mesh point, ν _l is the molecular kinematic viscosity coefficient (physical value) of the silicon melt 12, and ν _t is due to the turbulence effect of the silicon melt 12 F _x , F _y, and F _z are the kinematic viscosity coefficients, and are the x, y, and z direction components of the body force acting on the silicon melt 12.
The turbulence model equation (1) is called a kl (Kel) -model equation, and a turbulence parameter C of the model equation is 0.4 to 0.4.
Preferably, any value within the range of 0.6 is used. The reason why the turbulence parameter C is limited to the range of 0.4 to 0.6 is that if it is less than 0.4 or exceeds 0.6, there is a problem that the calculated interface shape does not match the actually measured value. . The Navier-Stokes equations (3) to (5) are equations of motion when the silicon melt 12 is a fluid that is incompressible and has a constant viscosity.
By solving the thermal energy equation (6) based on the obtained internal flow velocity distribution of the silicon melt 12, the internal temperature distribution of the silicon melt 12 in consideration of the convection of the silicon melt 12 is further obtained using a computer. .

[0015]

(Equation 4)Here, u, v and w are each mesh of the silicon melt 12.
X, y, and z components of the flow velocity at a point, where T is
Is the absolute temperature at each mesh point of the melt 12 and ρ is
C is the density of the silicon melt 12, and c is the density of the silicon melt 12.
Specific heat, κ _lIs the molecular thermal conductivity (physical property value), κ_t
Is the turbulent thermal conductivity calculated using equation (1).

Next, as a fifth step, the shape of the solid-liquid interface between the silicon single crystal 14 and the silicon melt 12 is indicated by a point S in FIG. 2 and is a triple point S of silicon (tri-junction of solid, liquid and gas). ) Is determined using a computer in accordance with the isotherm including As a sixth step, the heating value of the heater input to the computer is changed (increased gradually), and the third to fifth steps are repeated until the triple point reaches the melting point of the silicon single crystal 14.
The temperature distribution in the inside is calculated to determine the coordinates and temperature of the silicon single crystal mesh, and these data are stored in a computer.

Next, as a seventh step, after adding δ (for example, 50 mm) to the pulling length L ₁ of the silicon single crystal 14 and repeating the first to sixth steps, the temperature distribution in the pulling machine 11 is reduced. The coordinates and temperature of the mesh of the silicon single crystal 14 are obtained by calculation, and these data are stored in a computer. This seventh step is performed until the pulling length L ₁ of the silicon single crystal 14 reaches L ₂ . When the pulling length L ₁ of the silicon single crystal 14 reaches L ₂ , the process proceeds to an eighth step, in which the diffusion coefficient and boundary conditions of interstitial silicon and vacancies in the silicon single crystal 14 are input to a computer. Further, the concentration distribution of the interstitial silicon and the vacancies after cooling the silicon single crystal 14 is obtained by solving the diffusion equation based on the diffusion coefficients of the interstitial silicon and the vacancies and the boundary conditions.

More specifically, the following equation (7) is used to calculate the concentration C _i of interstitial silicon, and the following equation (8) is used to calculate the concentration C _v of vacancies. In equations (7) and (8), the thermal equilibrium between interstitial silicon and vacancies is maintained at the side, top, and solid-liquid interfaces of the crystal to calculate the evolution of concentration C _i and concentration C _v over time. Assume that

[0019]

(Equation 5) Here, K ₁ and K ₂ are constants, E _i and E _v are the formation energies of interstitial silicon and vacancies, the superscript e is the equilibrium amount, k is the Boltzmann constant, and T is the absolute temperature. The equilibrium equation is differentiated with respect to time, and the following equations (9) and (10) are obtained for interstitial silicon and holes.

[0020]

(Equation 6) D _i and D _v of each of the first term on the right side of formula (9) and (10) represents the Fickian diffusion with diffusion coefficient as shown in the following equation (11) and (12).

[0021]

(Equation 7) Here △ E _i and △ E _v is the activation energy of each interstitial silicon and vacancy, d _i and d _v are each constant. Also, the second term on the right side of each of the equations (9) and (10)

[0022]

(Equation 8) Is the activation energy of interstitial silicon and vacancies due to thermal diffusion, and k _{iv in} the third term on the right side of equations (9) and (10) is the recombination constant of the interstitial silicon and vacancy pairs. .

The point defect distribution of the silicon single crystal 14 obtained by the above calculation substantially matches the actually measured value. As a result, at the design stage of the pulling machine 11, the distribution of point defects in the silicon single crystal 14 pulled by the pulling machine 11 can be predicted. In addition, the structure of the pulling machine 11 can be considered at the design stage. In this embodiment, a silicon single crystal has been described, but a GaAs single crystal, In
P single crystal, ZnS single crystal or ZnSe single crystal may be used.

[0024]

Next, examples of the present invention will be described in detail together with comparative examples. <Example 1> As shown in FIG. 3, a point defect distribution in a silicon single crystal 14 when pulling a silicon single crystal 14 having a diameter of 6 inches from a silicon melt 12 stored in a quartz crucible 13 is shown. 1 and a simulation method based on the flowchart of FIG. That is, the hot zone of the silicon single crystal pulling machine 11 was modeled with a mesh structure. Here, the silicon single crystal 1 of the silicon melt 12
The mesh in the radial direction of the silicon single crystal 14 just below 4 is set to 0.
The mesh in the radial direction of the silicon single crystal 14 other than immediately below the silicon single crystal 14 of the silicon melt 12 was set to 75 mm. Further, the mesh in the longitudinal direction of the silicon single crystal 14 of the silicon melt 12 is 0.25 to 5 m.
m, and 0.45 was used as the turbulence parameter C of the turbulence model equation. Under such conditions, the internal temperature distribution of the silicon single crystal 14 is determined in consideration of the convection of the silicon melt 12, and the diffusion of point defects based on the internal temperature distribution of the silicon single crystal 14 and within the silicon single crystal 14 is performed. In consideration of the above, the point defect distribution in the silicon single crystal 14 was obtained.

Comparative Example 1 As shown in FIG. 4, a point defect distribution in a silicon single crystal 4 when a silicon single crystal 4 having a diameter of 6 inches is pulled from a silicon melt 2 stored in a quartz crucible 3. Was determined by a conventional simulation method. That is, the hot zone of the silicon single crystal pulling machine 1 was modeled with a mesh structure. Here, silicon melt 2
The mesh in the radial direction of the silicon single crystal 4 was set to 10 mm, and the mesh in the longitudinal direction of the silicon single crystal 4 of the silicon melt 2 was set to 10 mm. In addition, the convection of the silicon melt 2 was not taken into account (the turbulence model equation and the equation connecting the Navier-Stokes equations were not used). A simulation was performed using a computer in the same manner as in Example 1 except for the above.

<Comparative Example 2> A simulation was performed using a computer in the same manner as in Example 1, except that the convection of the silicon melt was taken into account, but the diffusion of point defects in the silicon single crystal was not taken into account. went. Comparative Example 3 A computer was simulated in the same manner as in Example 1 except that neither the convection of the silicon melt nor the diffusion of point defects in the silicon single crystal was taken into account.

<Comparative Test and Evaluation> The point defect distribution of the silicon single crystal was determined by the simulation method of Example 1 and Comparative Examples 1 to 3. The results are shown in FIGS. 5A to 5D together with measured values of the point defect distribution of the silicon single crystal (FIG. 5D). As is clear from FIG. 5, the point defect distributions (FIGS. 5B to 5D) of the silicon single crystals obtained by the simulation methods of Comparative Examples 1 to 3 are actually measured values (FIG. 5C).
On the other hand, the point defect distribution (FIG. 5A) of the silicon single crystal obtained by the simulation method of Example 1 was found to substantially match the measured value.

[0028]

As described above, according to the present invention, the internal temperature distribution of a single crystal is determined in consideration of the convection of the melt, and the point defect in the single crystal is determined based on the internal temperature distribution of the single crystal. The point defect distribution in the single crystal was determined in consideration of the diffusion of, and the calculated value of the point defect distribution in the single crystal agrees very well with the actually measured value. As a result, at the design stage of the single crystal pulling machine, it is possible to predict the distribution of point defects in the single crystal pulled by the pulling machine, and conversely, to obtain the desired distribution of point defects in the single crystal pulled by the pulling machine. The structure can be considered at the design stage.

[Brief description of the drawings]

FIG. 1 is a flowchart showing the first half of a method for simulating a point defect distribution of a silicon single crystal according to an embodiment of the present invention.

FIG. 2 is a flowchart showing the latter half of the method for simulating the point defect distribution of the silicon single crystal.

FIG. 3 is a sectional view of an essential part of a silicon single crystal pulling machine having a mesh structure of a silicon melt according to the present invention.

FIG. 4 is a sectional view of a main part of a conventional silicon single crystal pulling machine having a mesh structure of a silicon melt.

FIG. 5 is a longitudinal sectional view showing distributions of interstitial silicon and vacancies in a silicon single crystal actually measured in Example 1, Comparative Example 1 (conventional example), Comparative Examples 2 and 3, and actually measured.

[Explanation of symbols]

 11 silicon single crystal pulling machine 12 silicon melt 14 silicon single crystal S triple point of silicon

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[Procedure amendment]

[Submission Date] April 13, 2001 (2001.4.1
3)

[Procedure amendment 1]

[Document name to be amended] Statement

[Correction target item name] Claim 1

[Correction method] Change

[Correction contents]

[Procedure amendment 2]

[Document name to be amended] Statement

[Correction target item name] 0004

[Correction method] Change

[Correction contents]

[0004]

The invention according to claim 1 is
As shown in FIGS. 1 to 3, the single crystal 1
A first step of modeling the hot zone of the puller 11 of the single crystal 14 in a state in which the single crystal 4 has been pulled up to a predetermined length by a mesh structure; a mesh for each member of the hot zone; computer pull-up length of the physical properties of the single crystal 14 and the pulling rate of the single crystal 14 corresponding to the pull-up length, respectively
And a third step of obtaining the surface temperature distribution of each member based on the calorific value of the heater and the emissivity of each member.
Turbulent flow obtained by assuming that the melt 12 is turbulent after obtaining the internal temperature distribution of each member by solving the heat conduction equation based on the step and the surface temperature distribution and thermal conductivity of each member A fourth step of further calculating the internal temperature distribution of the melt 12 in consideration of convection by connecting and solving the model equation and the Navier-Stokes equation; and forming the solid-liquid interface shape of the single crystal 14 and the melt 12 The fifth step, which is determined according to the isotherm including the triple point S, and the third to fifth steps are repeated until the triple point S reaches the melting point of the single crystal 14, and the temperature distribution in the puller 11 is calculated to calculate the single crystal. these data respectively co sought coordinates and temperature of 14 mesh
The sixth step to be input to the computer and the first step to the sixth step are repeated by gradually changing the pulling length of the single crystal 14 to calculate the temperature distribution in the pulling machine 11 to calculate the coordinates of the mesh of the single crystal 14 and A seventh step of obtaining temperatures and inputting these data to a computer, and data of the coordinates and temperature of the mesh of the single crystal 14 and the single crystal 1
An eighth step of inputting the diffusion coefficients and boundary conditions of the vacancies and interstitial atoms in the computer 4 to the computer, respectively. the computer includes a ninth step of obtaining a concentration distribution of vacancies and interstitials after cooling of the single crystal 14 by solving the diffusion equation based
This is a method of simulating a point defect distribution of a single crystal using the method.

[Procedure amendment 3]

[Document name to be amended] Statement

[Correction target item name] 0009

[Correction method] Change

[Correction contents]

As a second step, meshes are grouped for each member in the hot zone, and the physical property values of the members are input to the computer for the grouped meshes. For example, if the chamber is made of stainless steel, the thermal conductivity, emissivity, viscosity, volume expansion coefficient, density, and specific heat of the stainless steel are input to the computer. The inputs and pulling speed of the silicon single crystal 14 corresponding to the pull-up length and the pull-up length of the silicon single crystal 14, and a turbulence parameter C turbulence model expression (1) described below to the computer.

[Procedure amendment 4]

[Document name to be amended] Statement

[Correction target item name] 0010

[Correction method] Change

[Correction contents]

As a third step, the surface temperature distribution of each member in the hot zone is obtained by using a computer based on the calorific value of the heater and the emissivity of each member. That is, the calorific value of the heater is arbitrarily set and input to the computer, and the surface temperature distribution of each member is obtained from the emissivity of each member using the computer. Next, as a fourth step, the internal temperature distribution of each member is obtained by solving the heat conduction equation ( 1 ) using a computer based on the surface temperature distribution and the thermal conductivity of each member in the hot zone. Here, an xyz rectangular coordinate system is used for simplicity of description, but a cylindrical coordinate system is used in actual calculation.

[Procedure amendment 5]

[Document name to be amended] Statement

[Correction target item name] 0011

[Correction method] Change

[Correction contents]

[0011]

(Equation 1) Here, ρ is the density of each member, c is the specific heat of each member, T is the absolute temperature at each mesh point of each member,
t is time, λ _x , λ _y, and λ _z are the x, y, and z components of the thermal conductivity of each member, and q is the heating value of the heater. On the other hand, with respect to the silicon melt 12, after the internal temperature distribution of the silicon melt 12 is obtained by the heat conduction equation ( 1 ), the silicon melt 12 is turbulent based on the internal temperature distribution of the silicon melt 12. The turbulence model equation ( 2 ) and the Navier-Stokes equations (3) to (5), which are obtained assuming that there is, are connected to obtain the internal flow velocity distribution of the silicon melt 12 using a computer.

[Procedure amendment 6]

[Document name to be amended] Statement

[Correction target item name] 0017

[Correction method] Change

[Correction contents]

Next, as a seventh step, after adding δ (for example, 50 mm) to the pulling length L ₁ of the silicon single crystal 14 and repeating the first to sixth steps, the temperature distribution in the pulling machine 11 is reduced. The coordinates and temperature of the mesh of the silicon single crystal 14 are obtained by calculation, and these data are stored in a computer. The seventh step is carried out until the pull-up length L ₁ of the silicon single crystal 14 reaches a length L _2. Pull-up length L ₁ is the length L ₂ of the silicon single crystal 14
When reached, the process proceeds to the eighth step, the silicone single binding
The data of the coordinates and temperature of the mesh of the crystal 14 are input to a computer together with the diffusion coefficients and boundary conditions of interstitial silicon and vacancies in the silicon single crystal 14. Further, the concentration distribution of the interstitial silicon and the vacancies after cooling the silicon single crystal 14 is obtained by solving the diffusion equation based on the diffusion coefficients of the interstitial silicon and the vacancies and the boundary conditions.

[Procedure amendment 7]

[Document name to be amended] Statement

[Correction target item name] 0027

[Correction method] Change

[Correction contents]

<Comparative Test and Evaluation> The point defect distribution of the silicon single crystal was determined by the simulation method of Example 1 and Comparative Examples 1 to 3. The results are shown in conjunction with FIG. 5 (a) ~ <br/> measured values of point defect distribution of a silicon single crystal (d) (Fig. 5 (e)). 5 As is clear from, the point defect distribution of the silicon single crystal obtained by the simulation method of the Comparative Examples 1 to 3 (FIG. 5 (b) ~ (d) ) the actual value (Fig. 5 (e))
On the other hand, the point defect distribution (FIG. 5A) of the silicon single crystal obtained by the simulation method of Example 1 was found to substantially match the measured value.

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Claims (1)

[Claims] An apparatus for pulling a single crystal (14) in a state where the single crystal (14) is pulled up to a predetermined length by a puller (11).
The first to model the hot zone of (11) with a mesh structure
Step, a mesh for each member of the hot zone, and a pulling length of the single crystal (14) together with a physical property value of each member for the collected mesh and the single crystal (14) corresponding to the pulling length. A second step of providing a pulling speed of each member; a third step of obtaining a surface temperature distribution of each member based on a calorific value of a heater and an emissivity of each member; and a surface temperature distribution and thermal conductivity of each member. After solving for the internal temperature distribution of each member by solving the heat conduction equation based on, the turbulence model equation and the Navier-Stokes equation obtained by assuming that the melt (12) is turbulent are connected. A fourth step of further calculating the internal temperature distribution of the melt (12) in consideration of convection by solving the melt, and the solid-liquid interface shape of the single crystal (14) and the melt (12) Including triple point (S) A fifth step in accordance with the isotherm;
(S) repeatedly calculates the temperature distribution in the pulling machine (11) until the melting point of the single crystal (14) is obtained, obtains the coordinates and temperature of the mesh of the single crystal (14), and gives these data, respectively. A sixth step, wherein the pulling length of the single crystal (14) is changed stepwise, and the first to sixth steps are repeated to repeat the pulling.
A seventh step of calculating the temperature distribution in (11) to determine the coordinates and temperature of the mesh of the single crystal (14) and providing these data, respectively; and vacancies and interstitial atoms in the single crystal (14). An eighth step of respectively giving the diffusion coefficient and the boundary condition of: and solving the diffusion equation based on the coordinates and temperature of the mesh of the single crystal (14), the diffusion coefficient of the vacancies and the interstitial atoms, and the boundary condition. A ninth step of obtaining a concentration distribution of the vacancies and the interstitial atoms after cooling of the single crystal (14).