JPH1192286A - Production of silicon single crystal - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a method for producing a large-sized silicon single crystal in excellent productivity by stabilizing the flow of a raw material melt in a crucible even in the large volume of the melt in growing the silicon single crystal by a Czochralski method. SOLUTION: An inner diameter of a crucible used in the production of a silicon single crystal is D(m), the number of revolutions W (rpm) of the crucible per minute during crystal growth satisfies W>0.8×D<-7/2> or further the number of revolutions of the crucible is increased and the relation of the equation, W>4.5×d<9.4> ×ΔTv <3/4> ×D<-2> is satisfied when the depth of the melt is d(m) and the temperature difference between the melt surface of the melt of the central axis of the crucible and the bottom of the crucible is ΔTv ( deg.C) so as to grow a silicon 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 growing a silicon single crystal by the Czochralski method, and more particularly to the production of a large-diameter and large-weight crystal.

[0002]

2. Description of the Related Art In recent years, the demand for silicon wafers having a larger diameter has been increasing in response to the increase in the area of semiconductor devices, which requires a large crystal having a diameter exceeding 200 mm and a weight exceeding 150 kg. The production of such a crystal is necessarily limited to the Czochralski method, and an unprecedented large crucible is used.

[0003] However, the convection of the silicon melt in the crucible of the Czochralski method becomes stronger and more turbulent as the crucible diameter and the melt depth increase. Melt convection turbulence has a negative effect on crystal growth due to the following three reasons.

[0004] First, convection turbulence causes irregular temperature fluctuations at the crystal growth surface and at the same time breaks the axial symmetry of the melt temperature distribution. As a result, the heat balance at the crystal growth interface is disturbed, and crystal growth as a single crystal is hindered.

Second, the enhanced convection promotes the transport of impurities from the bottom of the degraded quartz crucible, which contacts the crystal and causes the crystal to polycrystallize.

Third, turbulence in convection causes uneven oxygen and dopant concentrations in the melt, and even if a dislocation-free crystal is successfully grown, the quality of oxygen, resistance, etc., is reduced on the wafer cut from the crystal. It causes unevenness.

Therefore, in the current single crystal manufacturing apparatus, the initial raw material charge is suppressed to less than 150 kg, and the weight of a single crystal that can be stably manufactured is limited to about 130 kg. It is considered difficult to simply scale up and mass-produce large crystals.

The extent to which the turbulence of the flow increases with the increase in the diameter and depth of the crucible is determined by the equation of motion of the fluid based on the buoyancy due to thermal expansion, the Coriolis force due to rotation of the crucible, and the viscous force of the melt. It can be expected by looking at the function of the inertial force of the flow itself. The equation of motion that expresses the relationship between the forces acting per unit volume of the melt is expressed in a coordinate system that rotates the same as the crucible
(1)

[0009]

(Equation 1)

The symbols here represent the following.

Ρ: density of the melt, Ω: rotational angular velocity of the crucible, α:
Volume expansion coefficient of the melt P: pressure in the melt, g: gravitational acceleration, ν:
The kinematic viscosity of the melt u: the flow velocity of the melt T: the temperature of the melt ∇: the derivative of the position coordinate with the differential operator In equation (1), the first term on the left side is the time change of the velocity, the second Term is inertial force, the first term on the right side is Coriolis force, the second term is pressure,
The third term indicates buoyancy, and the fourth term indicates viscous force.

With respect to the silicon melt of the Czochralski method, only the buoyancy acts as an accelerating force and only the viscous force acts as a decelerating force. Evaluated by force magnitude ratio.

In the Czochralski method, since crystals grow on the liquid surface, a temperature difference always occurs between the surface of the crucible and the bottom of the crucible, so that the melt is directly driven vertically upward by buoyancy. Such convection is called forced convection. The dimensionless number indicating the ratio of buoyancy to viscosity in this case is defined by the following equation (2) as the Rayleigh number Ra. Since the velocity U of the forced convection is substantially the same as the temperature transmission velocity, it is derived by the order calculation from the equation (1) as U = K / d.

[0014]

(Equation 2)

Here, d is the depth of the silicon melt, K is the temperature conductivity of the silicon melt, and ΔT _v is the temperature difference between the surface of the melt on the central axis of the quartz crucible and the bottom of the crucible.

When the physical properties of the silicon melt are applied, the equation (2) can be expressed as the following equation.

[0017]

(Equation 3)

From this, it can be seen that buoyancy acts in proportion to the cube of the depth of the melt per unit volume of the melt, and that an increase in the melt depth leads to a significant acceleration of the melt convection.

The temperature difference between the side of the crucible and the center of the crucible also gives a non-uniform driving force (buoyancy) distribution in the melt, thereby driving convection. This is called free convection. The dimensionless number indicating the ratio of buoyancy to viscosity in free convection is defined by the following equation (4) as the Grashof number Gr. Since the speed U of the free convection is almost equal to the transmission speed due to the viscosity, it can be obtained from the equation (1) as U = ν / d.

[0020]

(Equation 4)

[0021] Here, R represents the radius of the crucible, the [Delta] T _h is the temperature difference between the quartz crucible side wall temperature and the crucible center of the melt surface.

Thus, the crucible diameter per unit volume of the melt is
It can be seen that buoyancy acts in proportion to the third power, and that an increase in the diameter of the crucible also causes a significant flow acceleration, which makes it difficult to grow a crystal as a single crystal.

On the other hand, since the Coriolis force is a force acting perpendicular to the flow direction, it never accelerates the convection. However, the barotropic wave (Ba
It causes a flow form called roclinic wave, which greatly distorts the temperature distribution of the melt. In recent years, this phenomenon has also occurred in Czochralski crucibles, and it has been clarified that not only the increase in buoyancy but also the increase in Coriolis force significantly distorts the temperature distribution on the melt surface from axial symmetry (K
ishida et al., J. Crystal. Growth 130, 34, (1993),
JP-A-5-94075 and JP-A-8-259381).

The change in the flow form of thermal convection caused by the Coriolis force is almost predicted by the following two dimensionless numbers (Greenspan (1968) p125). One is Taylor number Ta
Where this is the square of the ratio of the Coriolis force to the buoyancy and is defined as:

[0025]

(Equation 5)

By applying the value of the kinematic viscosity coefficient ν of the silicon melt to the number of rotations of the crucible per minute as W (rpm) and the diameter of the crucible as D (m), the equation (5) can be expressed as Can be expressed.

[0027]

(Equation 6)

Another dimensionless number is the Rosby number Ro, which is the ratio between the Coriolis force and the inertial force, and is defined by the following equation.

[0029]

(Equation 7)

The representative flow rate U of the inertial force is estimated based on the following equation in consideration of the crucible rotation.

[0031]

(Equation 8)

Equation (8) indicates that the Coriolis force and buoyancy are large (thermal wind equilibrium: Green
Since nspan (1968) p125) is assumed, if the value of Ro defined by equation (7) is larger than 10, the physical meaning is weak.

When organized by two dimensionless numbers, Ro and Ta, the form of thermal convection in a rotating system can be drawn as shown in FIG. 2 and divided into three regions. This figure shows an experiment using mercury such as Fein (J. Fluid.
Mech., 75, 81, (1976)). Although the experiments by Fein et al. Are scientific, it is believed that the physical phenomena are the same in the Czochralski crucible. Figure
The convection form in each area of 2 is as follows.

In region A, since both the Coriolis force and the inertia force are weak, the convection is axially symmetric with respect to the rotation axis of the seed crystal (which coincides with the rotation axis of the crucible in the Czochralski method), and the temperature distribution on the melt surface is also axially symmetric. Becomes This is a very good condition for growing crystals. If the crucible has a diameter of several inches, it is possible to place the convection of the melt in this state by devising the heating conditions.However, since the Ta number increases with the fourth power of the crucible diameter, the crucible diameter is reduced to 10 inches. If it exceeds, the state of region C must be entered.

In the region B, the convection becomes turbulent because the Coriolis force is weak and the inertial force acts strongly. At this time, it is known that the crystal growth is impossible in this state because the temperature distribution is strongly disordered. At this time, the rotation speed of the crucible is extremely low (0.1 rpm) or less, which is a region where the meaning of rotating the crucible in the Czochralski method is lost.

Region C is a region where baroclinic waves occur due to strong Coriolis force. At this time, the rotational axis symmetry of the temperature distribution on the melt surface is broken and becomes unstable (FIG. 3 (a)). This phenomenon adversely affects the growth of a single crystal because the instability of the temperature distribution is directly linked to the instability of the crystal growth rate. Even in the state where this baroclinic wave is occurring,
A single crystal can be grown if the symmetry of the temperature distribution at the center of the crucible for growing the crystal can be kept to a somewhat small width throughout the growth of the crystal by the method shown in 259381 and the like. However, also in this region, the axial symmetry becomes worse as it enters the left side of the region where the Coriolis force is more effective, which is a severe condition for growing a crystal.

From the above, the action of the Coriolis force is defined as the Ta number.
When organized by Ro number, it is understood that baroclinic wave phenomenon by Czochralski method cannot be avoided. Currently, the production of single crystals with a diameter of 150 mm or 200 mm is in a situation where barely stable production conditions are obtained by devising the heating conditions of the melt. As can be seen from equation (6), increasing the diameter of the crucible means increasing the Coriolis force. Therefore, it is considered that increasing the diameter of the crucible more than the current situation will be a more severe condition for crystal growth.

Under these circumstances, for the production of large crystals, Japanese Patent Publication No. Sho 58-50951 and Japanese Patent Application Laid-Open No.
82, a method of applying a static magnetic field to the melt and suppressing the flow via Lorentz force is considered to be effective, as shown in JP-A-8-333191. However, in this method, it is necessary to provide a large-scale magnet that can accommodate a pulling furnace between the magnetic poles, which requires considerable capital investment and running costs. Further, as the furnace becomes larger, the distance between the magnetic poles becomes longer, and it becomes impossible to achieve a predetermined magnetic field intensity required for suppressing the flow of the melt, so that there is a limit as equipment.

[0039]

SUMMARY OF THE INVENTION In order to solve the above-mentioned problems, the present invention stabilizes the melt flow without using large-scale auxiliary equipment such as a magnetic field applying device in the Czochralski method, and provides a large-sized product with good productivity. It is an object of the present invention to provide a method for producing a silicon single crystal.

[0040]

In order to achieve the above object, the present invention relates to a method for growing a silicon single crystal by the Czochralski method, in which the inner diameter D (m) of a crucible to be used for growing a silicon single crystal is determined.
The relationship between the number of crucible rotations per minute W (rpm) and W> 0.
Only by designating that 8 × D ^−7/2 is satisfied, a crystal is grown in a state set so that geostrophic turbulence occurs in the melt.

At this time, the number of rotations of the crucible is further increased, the depth d (m) of the melt is suppressed, or the heat insulation and the heating method in the furnace are devised so that the surface of the melt on the central axis of the crucible and the bottom of the crucible are lowered. The temperature difference ΔT _v (° C.) of W> 4.5 × d
If the relationship of ^9/4 × ΔT _v ^3/4 × D− ² can be satisfied, a better effect can be obtained.

In the present invention, if the number of rotations of the crucible is excessively increased, the liquid surface shape becomes parabolic due to centrifugal force, which hinders the growth of stable crystals, and a mechanical problem is caused by rotating a heavy object at a high speed. Therefore, the crucible rotation speed is desirably 100 rpm or less.

On the other hand, if the diameter of the quartz crucible used is too large, the radiant heat flux from the melt surface becomes large, so that it is necessary to strengthen the heating and keep the crucible side walls at a high temperature. At this time, the temperature of the quartz crucible reaches its own weight softening temperature of 1630 ° C., and there is a risk of causing deformation. For this reason, it is desirable to use a crucible having a diameter of 1300 mm or less.

[0044]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The present invention is practiced by rotating a crucible at a required number of revolutions or more according to the diameter of the crucible during the period of growing a silicon single crystal by the Czochralski method. It is all about making Coriolis force in the room better than buoyancy.

At this time, the convection in the melt in the crucible is in a state as described below.

The force acting on the melt in the crucible, that is, the magnitude of the inertial force, Coriolis force, buoyancy, and viscous force, is determined by taking the representative flow velocity of convection as U and the radius of the crucible as R as the representative length. The comparison based on equation (1) is as follows.

[0047]

(Equation 9)

Usually, the crucible rotation in the Czochralski method is
Rotational angular velocity Ω is on the order of ¹ × 10 ^-1 rad / sec several times per minute, and kinematic viscosity ν of silicon is 3 × 10 ^-7 m
² / s, and the volume expansion coefficient α of molten silicon is
It is of the order of 1x10 ^-5 and the gravitational acceleration g is about 1x10 ¹
m / s ² , the radius R of the crucible used is of the order of ¹ × 10 ⁻¹ m, and the maximum temperature difference T inside the melt is ¹ × 10 ¹
° C, and the flow rate U of the convection of the melt does not exceed the rotation speed ΩR of the crucible even if it is largely estimated, and 1x10 ^-2 m / sec.
Taking into account the degree, the size of each term in equation (9) is estimated as follows.

[0049]

(Equation 10)

Therefore, in the melt in the crucible of the Czochralski method, the viscous force has little effect on the flow form, and the former three forces are almost in opposition to each other. Should be. In such a situation, it can be seen that intentionally increasing and excelling the Coriolis force acting on the melt against the increasing buoyancy and inertia force is practical and extremely effective in controlling the flow.

The convection in a container that is not originally rotating rises in the vertical direction near the crucible wall and also falls in the vertical direction at the center, so that the rotation axis of the convection is in the horizontal direction. However, when the Coriolis force acts by rotating the crucible, the convection rises in the direction slightly deviated from the vertical direction near the crucible wall, so the convection rotation axis is inclined from the horizontal direction to the vertical direction A tendency appears. Therefore, in the melt in the crucible in the method of the present invention in which the Koori force acts extremely strongly, the convection in the crucible does not go in the vertical direction and moves almost horizontally, that is, the rotation axis in the crucible is vertical. It will be filled with vortices that face in the direction (FIG. 1). This state corresponds to what is called geostrophic turbulence in hydrodynamics.

Conventionally, it has not been clarified at all whether or not such geostrophic turbulence occurs in a crucible melt of Czochralski, and under what conditions it occurs. According to the results of experiments conducted by the inventors in a furnace used for producing crystals, the temperature distribution on the surface of a small-diameter crucible having a diameter of 16 inches or less and having a low rotation speed is well known, as shown in FIG. 3 (a). However, geostrophic turbulent flow could not be confirmed because the baroclinic unstable wave pattern constantly appeared, but it was only when the crucible rotation speed was increased with a large crucible with a diameter of 22 inches or more. The temperature pattern as shown in (b) appeared and the geostrophic turbulence was achieved. The method of confirming the geostrophic turbulence state will be described later.

Since the purpose of the present invention is to overwhelmingly increase the Coriolis force, the conditions for the occurrence of turbulent geostrophic flow are most suitably expressed by the numbers of Ro and Ta as in FIG. Therefore,
Observation of crucible melt characteristics at various rotation speeds in crucibles of various diameters shows that the conditions under which geostrophic turbulent flow was actually achieved are shown on the Ro-Ta diagram. The turbulence region was located on the left side of the baroclinic wave region, and it was found that the boundary was the following straight line. (Fig. 5)

[0054]

[Equation 11]

[0055] In this equation, (5) (7) applies Property values of the molten silicon as a constant in the equation, regarded as constant temperature gradient [Delta] T _h / R of the radial direction in the equation (7), the crucible diameter D
(m) and the number of rotations of the crucible is W (rpm).
0.8 x D- ⁷ / 2. Therefore, the conditions necessary for causing the geostrophic turbulence to be aimed by the present invention are as follows. This is the condition of the crucible rotation number required by the present invention.

[0056]

(Equation 12)

In crucibles of various diameters, this inequality (12)
Calculating the crucible rotation speed required to satisfy
It looks like a row.

[0058]

[Table 1]

In practice, even if the condition of inequality (12) is satisfied, if the vertical convection is not sufficiently suppressed, the crystal may be polycrystallized.

This can be dealt with by lowering the Rayleigh number Ra of the melt or increasing the number of rotations of the crucible to make the Coriolis force more effective. To get this effect, Chandras
theoretical analysis of Benard convection of ekhar (S. Chandrasekh
ar: "Hydrodynamics and Hydromagnetic instability",
By analogy with p120, Oxford University. Press, (1961)), it is conceivable that the Taylor number should be increased by 2/3 to the increase in Rayleigh number. According to experiments performed by the inventors, it has been found that in the crystal growth by the Czochralski method, setting the value of the proportional coefficient to 0.5 has a sufficient effect. This can be expressed by equation (13).

[0061]

(Equation 13)

Here, the values of Ra and Ta are obtained from the equations (5) and (7), and the equation (13) is used to calculate the melt depth d (m), the crucible diameter D (m), the melt surface and the bottom of the crucible. When the crucible rotation speed W (rpm) is expressed as the temperature difference ΔT _v (° C.), the following expression (14) is obtained.

[0063]

[Equation 14]

In order to achieve the condition of the expression (14), it is effective to increase the number of rotations of the crucible. It is also effective to reduce the ΔT _v by adjusting the position of the heater described above, and to reduce the depth d of the melt by suppressing the amount of the silicon material to be dissolved.

In the fourth column of Table 1, the minimum number of crucible rotations defined by the equation (14) is shown for a typical diameter of a quartz crucible used for growing crystals. At this time, the depth d of the melt is 0.25 m, ΔT
_v was 50 ° C.

By increasing the number of rotations of the crucible so as to satisfy the equations (12) and (14), when a geostrophic turbulent state is achieved,
Although there is almost no axial symmetry of the temperature distribution in the melt, vortices as shown in Fig. 1 are formed inside the melt.
At this time, looking at the temperature fluctuation on the melt surface, the effect of making the temperature distribution uniform by the vortex acts, and although the distribution is present, the amplitude of the fluctuation becomes extremely small.

For this reason, the temperature fluctuation on the melt surface on which the crystal grows is greatly reduced as compared with before the geostrophic turbulence is achieved, and it is extremely easy to pull up the crystal without deformation. Along with this, the crystal pulling speed is also improved, and the productivity can be improved.

When the geostrophic turbulent flow can be achieved in the melt, the vertical motion in the melt is suppressed, the instability of the solidification interface temperature due to boiling is suppressed, and the temperature from the bottom of the crucible increases. Polycrystalline crystallization can be prevented by reducing the transport of undissolved silicon and impurities from the deteriorated quartz crucible.

It should be noted that whether or not a state of geostrophic turbulence was realized was determined by the temperature of the melt surface during the immersion of the seed crystal before the start of crystal growth, as shown in JP-A-8-259381. The distribution and its time variation can be performed by using a video camera or the like to record and record the image data as radiation light.

When the geostrophic turbulence state is achieved, the temperature distribution on the melt surface photographed by the camera becomes a state in which a plurality of high-temperature regions are distributed in an island shape as shown in FIG. 3B). However, a similar temperature distribution is observed even when the vertical convection is strong and a vertical vortex group as shown in FIG. 1 is not formed, so that it is not sufficient to judge only with the image. Since geostrophic turbulence is a two-dimensional turbulence, it is necessary to confirm that it has a steep continuous energy spectrum based on the -4 power law of frequency f which is peculiar to two-dimensional turbulence. Specifically, any one of the video images
The time series data of the temperature fluctuation at the pixel is taken out and the power spectrum is obtained from the Fourier transform. In the case of geostrophic turbulent flow, as shown in Fig. 4, it can be confirmed that a gradient part along the -4th power law appears over the range of 0.1 Hz to 1 Hz.

[0071]

EXAMPLE 150 kg of raw silicon was dissolved in a crucible having a diameter of 24 inches (0.6 m), and crystal growth was attempted under various crucible rotation conditions. The number of rotations of the crucible, the root-mean-square value of the melt surface temperature fluctuation measured 10 minutes immediately before immersion of the seed crystal, and a diameter of 320 m
Table 2 shows the percentage of single crystal grown when a crystal weighing 130 kg was grown in m.

[0072]

[Table 2]

In Comparative Examples 1 and 2, the crucible rotation was low.
At 1 rpm and 3 rpm, the condition of the formula (12) of the present invention is not satisfied. In each case, the temperature fluctuation of the melt was large, exceeding 6 ° C. in root-mean-square, and polycrystallization often occurred during growth. However, under the conditions of the crucible rotation speed of 6 rpm or more in Examples 1 to 4, since the condition of the expression (12) is satisfied, the single crystal can be easily grown, and the single crystal can be grown at a high rate of 90% or more. Was completed. From Example 1
In 4, the melt was extremely stable, and no crystal deformation occurred, and the pulling speed in Comparative Example 2 could be increased to 0.75 mm / sec, compared to 0.40 mm / sec. In particular, in Examples 3 and 4, since the condition of the condition of the equation (14) was also satisfied at the same time, the melt surface was extremely stable, and the root-mean-square value of the melt temperature fluctuation was 3 ° C. or less, and each of 10 times. No polycrystallization occurred once during the growth.

Further, 180 kg of raw silicon was melted in a crucible having a diameter of 28 inches in a furnace in which the heat insulating ability was increased by using a high heat insulating material, and crystal growth was attempted under various crucible rotation conditions. The number of rotations of the crucible, the root-mean-square value of the melt surface temperature fluctuation measured 10 minutes immediately before immersion of the seed crystal, and the weight of 16
Table 3 shows the percentage of single crystal grown when 0 kg of crystal was grown.

[0075]

[Table 3]

In Comparative Example 3, the number of revolutions was low and did not satisfy the condition of equation (12). For this reason, the temperature fluctuation of the melt was large and the root mean square was close to 6 ° C., the melt was unstable, and polycrystallization often occurred during the growth, and only 40% of the single crystal could be grown. On the other hand, in Examples 5 to 9, in each case, the temperature difference ΔT _v in the melt was small due to the strengthened heat insulation, and the conditions of Equations (12) and (14) were simultaneously satisfied. Was sufficiently suppressed. Therefore, in each case, the growth of a single crystal could be completed without polycrystallization with a high probability of 90% or more.

Before starting the crystal growth under each test condition, radiation light from the surface of the melt having a diameter of 400 mm covering the crystal growth area was photographed with a video camera, and the surface of the melt shown in FIG. The temperature distribution diagram is an image formed by processing the image in the second embodiment. The image shows an island-like high temperature region peculiar to geostrophic turbulence. The power spectrum of the temperature fluctuation shown in FIG. 5 is also that of the second embodiment, and a gradient portion proportional to the −4 power of the frequency peculiar to the geostrophic turbulence appears.
This confirms that geostrophic turbulence has been achieved. These characteristics were confirmed in any of Examples 1 to 9, but were not confirmed in Comparative Examples 1 to 3.

[0078]

The present invention has a diameter of 200 mm or more and a weight of 150 kg.
In the production of the above large ingot, according to the initial silicon charge amount and the crucible diameter, the inner diameter of the crucible and the number of rotations of the crucible at the time of pulling are controlled only by setting within a predetermined range. There is no need to provide ancillary equipment such as electromagnets, and the manufacturing and maintenance costs of the device can be greatly reduced.

The following effects can be exerted during operation by utilizing the characteristics of geostrophic turbulence.

First, since temperature fluctuations on the melt surface are alleviated by the vortex group, it is extremely easy to pull up dislocation-free crystals without deformation.

Second, by stabilizing the melt, the pulling speed of the crystal is also improved, and the productivity can be improved.

Third, since the flow in the vertical direction in the melt is suppressed, the transport of undissolved silicon from the crucible bottom and impurities from the deteriorated quartz crucible to the solidification interface due to boiling can be reduced, and the crystal The possibility of polycrystallization is greatly reduced.

[Brief description of the drawings]

Fig. 1 Schematic diagram of geostrophic turbulent flow inside a crucible

FIG. 2 is a diagram showing the form of thermal convection in a rotating container arranged by Ro and Ta.

Fig. 3 (a) Isotherm distribution map of melt surface in baroclinic wave state, (b) Isotherm distribution map of melt surface in geostrophic turbulent flow state

Fig. 4 Power spectrum diagram of temperature fluctuation under geostrophic turbulence

FIG. 5 is a diagram showing the relationship between the Ro number and the Ta number at which geostrophic turbulence appears in a silicon melt in a crucible of the Czochralski method.

Claims (2)

[Claims] When growing a silicon single crystal by the Czochralski method, the relation between the inner diameter D (m) of the crucible used and the number of rotations W (rpm) of the crucible per minute is W> 0.
A method for producing a silicon single crystal, characterized in that the setting is made so as to satisfy 8 × D− ^{7 / 2} . 2. The number of rotations of the crucible per minute W (rpm)
And the difference ΔT (° C.) between the surface of the melt temperature on the central axis of the crucible and the bottom of the crucible and the inner diameter D (m) of the crucible, the maximum depth d (m) of the melt during crystal growth. , W> 4.5 × d ^9/4 × ΔT
_{2. The} method for producing a silicon single crystal according to claim 1, wherein _v ^3/4 × D ⁻² is satisfied.