Point defect simulator, point defect simulation program, point defect simulation method, method for producing silicon single crystals and computer-readable storage medium

Abstract

Point defect simulator that calculates concentration profiles of vacancies and interstitial silicon atoms during the growth of a silicon single crystal using the Czochralski process and a convection-diffusion equation, wherein the convection-diffusion equation a concentration C V eq of vacancies in thermal equilibrium, represented by equation (A) shown below, and a concentration C I eq of interstitial silicon atoms in thermal equilibrium, which is represented by equation (B) shown below, the point defect simulator includes: an analysis unit used to fit computational results to experimental results, where a coefficient of a first stress coefficient a V f in equation (A) and a second stress coefficient a If in equation (B) as a fitting parameter for the difference of a V f and a I f is used CV eq ( T , P ) = C 0, V exp ( − EV f + a V f P k BT ) C leq (T, P) = C 0, l exp (−Elf + alf P k BT) where in equations (A) and (B) T is a temperature, P is a voltage, C 0,V and C 0,I Constants are, k B the Boltzmann constant is E V f a formative energy of a vacancy and E I f is the formation energy of an interstitial silicon atom.

Description

TECHNICAL FIELD

[0001] The present disclosure relates to a point defect simulator, a point defect simulation program, a point defect simulation method, a method for producing silicon single crystal, and a computer-readable storage medium. BACKGROUND

[0002] Traditionally, silicon wafers are used as substrates for semiconductor devices. For example, a silicon wafer can be obtained by subjecting a single-crystal silicon ingot (hereinafter referred to simply as a "crystal") grown using the Czochralski (CZ) process to a wafer processing process.

[0003] Recently, with the continued miniaturization and integration of semiconductor devices, silicon wafers used as substrates must be free of grown-in defects, i.e., they must be defect-free. Grown-in defects include cavity defects formed by the aggregation of vacancies, interstitial dislocation clusters formed by the precipitation of interstitial silicon atoms, etc., and would remain in a fabricated silicon wafer, leading to deterioration of a gate oxide film or leakage currents in a semiconductor device.

[0004] The behavior of vacancies and interstitial silicon atoms in a crystal is explained by the Voronkov model. Specifically, when the value of the ratio v / G of the pulling rate v of the crystal to the temperature gradient G in the pulling direction of a single-crystal silicon ingot near the solid-liquid interface is greater than the critical value (hereinafter also referred to as "critical v / G"), vacancies dominate; and when the value is smaller than the critical v / G, interstitial silicon atoms dominate (see, for example, VV Voronkov, J. Crystal Growth, 59, 625 (1982) (NPL 1)).

[0005] If the value of v / G is the critical v / G, a defect-free crystal can be obtained. Generally, the range (latitude) of the pulling rate v corresponding to the critical v / G is very narrow, and it must be controlled, for example, to within the critical v / G ± 2%.

[0006] Recently, it has been proposed that the thermal stress in a crystal near the solid-liquid interface affects the distribution of vacancies (V) and interstitial silicon atoms (I) in the crystal and consequently the value of the critical v / G (see, for example, J. Vanhellemont, J. Appl. Phys., 110, 063519 (2001) (NPL 2)). Because the increase in crystal diameter is expected to further increase the thermal stress in the crystals in the future, it is very important to evaluate the above-described effect of thermal stress on the behavior of point defects, such as vacancies and interstitial silicon atoms.

[0007] Against this background, K. Sueoka, E. Kamiyama and H. Kariyazaki, J. Appl. Phys, 111, 093529 (2012) (NPL 3) report that a first-principles calculation based on density functional theory suggests that compressive stress in a crystal near the solid-liquid interface increases the concentration of vacancies and consequently decreases the value of the critical v / G. Furthermore, K. Nakamura, R. Suewaka, B. Ko, ECS Solid State Letters, 3, N5 (2014) (NPL 4) present experimental evidence for the reduction of the value of the critical v / G due to compressive stress of a crystal near the solid-liquid interface. CITATION LISTSNon-patent literature NPL 1: VV Voronkov, J. Crystal Growth, 59, 625 (1982) NPL 2: J. Vanhellemont, J. Appl. Phys., 110, 063519 (2011) NPL 3: K. Sueoka, E. Kamiyama and H. Kariyazaki, J. Appl. Phys, 111, 093529 (2012) NPL 4: K. Nakamura, R. Suewaka, B. Ko, ECS Solid State Letters, 3, N5 (2014) SUMMARY (Technical Task)

[0008] NPL 3 and NPL 4 above describe that the strain in a crystal affects the value of the critical v / G; however, in these documents, the distribution of point defects, including vacancies and interstitial silicon atoms, is not determined by taking strain in the crystal into account.

[0009] It might therefore be helpful to provide a point defect simulator, a point defect simulation program, a point defect simulation method, a method for producing a silicon single crystal, and a single crystal pulling apparatus that make it possible to determine the distribution of point defects in a silicon single crystal while taking into account the thermal stress in the silicon single crystal during growth. (Solution to the task)

[0010] We propose the following features to address the above challenge. [1] Point defect simulator that calculates concentration profiles of vacancies and interstitial silicon atoms during the pulling of a silicon single crystal with the Czochralski process using a convection-diffusion equation, where the convection-diffusion equation a concentration C V eq of vacancies in thermal equilibrium, which is represented by an equation (A) shown below, and a concentration C I eq of interstitial silicon atoms in thermal equilibrium, which is represented by equation (B) shown below, where The point defect simulator includes: an analysis unit used to fit results of the calculation to experimental results, wherein one of a first stress coefficient a V fin equation (A) and a second stress coefficient a I f in equation (B) as a fitting parameter for the difference of a V f and a I f is used, C'Veq(T,P)=C0,V exp(−EVf+aVfPkBT) Cleq(T,P)=C0,l exp (−EIf+alfPkBT) where in equations (A) and (B) T is a temperature, P is a voltage, C 0,V and C 0,I Constants are, k B is the Boltzmann constant, E V f a formation energy of a vacancy and E I f is a formation energy of an interstitial silicon atom. [2] Point defect simulator according to [1] above, wherein a solid-liquid interface between the silicon single crystal and a silicon melt has an upwardly convex shape. [3] Point defect simulator according to [1] or [2] above, wherein the silicon single crystal has a diameter of 300 mm or more. [4] A point defect simulation program for using a computer as the point defect simulator according to at least one of [1] to [3] above. [5] A point defect simulation method used to determine concentration profiles of vacancies and interstitial silicon atoms in a silicon single crystal using the point defect simulator according to at least one of [1] to [3] above. [6] Point defect simulation method according to [5] above, in which the solid-liquid interface between the silicon single crystal and the silicon melt has an upward convex shape. [7] Point defect simulation method according to [5] or [6] above, wherein the silicon single crystal has a diameter of 300 mm or more. [8] A method for producing a silicon single crystal, the method comprising: for at least one single crystal pulling apparatus, determining concentration profiles of point defects, including vacancies and interstitial silicon atoms, in a silicon single crystal pulled up using the single crystal pulling apparatus using one of the point defect simulator according to at least one of [1] to [3] above and the point defect simulation method according to at least one of [5] to [7] above; designing the single crystal pulling apparatus based on the thus determined concentration profiles of vacancies and interstitial silicon atoms; and producing a defect-free silicon single crystal using the designed single crystal pulling apparatus. (Beneficial effect)

[0011] The present disclosure can provide for the determination of the distribution of point defects in a silicon single crystal taking into account the thermal stress of the silicon single crystal during growth. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] In the attached drawings: is Fig. 1 is a diagram illustrating an example of a point defect simulator according to the present disclosure; Fig. 2 a graph illustrating the relationship between the stress coefficient and the critical v / G; Fig. 3 a graph illustrating the relationship between v / G and defect distribution; Fig. 4A is a graph illustrating the distribution of defects in a crystal obtained by experiments; Fig.4B is a diagram illustrating the distribution of defects in a crystal according to the conventional example; Fig. 4C is a graph illustrating the distribution of defects in a crystal according to Comparative Example; Fig. 4D is a diagram illustrating the distribution of defects in a crystal according to the example; Fig. 5A is a diagram showing the position of the boundary between a P I -range and an L / DL range in the Fig. 4A illustrates the defect distribution; Fig. 5B is a graph showing the relationship between the value of the stress coefficient and the profile of the boundary between the P I range and the L / DL range found without taking into account the concentration of oxygen atoms; and Fig.5C is a graph showing the relationship between the value of the stress coefficient and the profile of the boundary between the P I range and the L / DL range found taking into account the concentration of oxygen atoms. DETAILED DESCRIPTION (Point Defect Simulator)

[0013] Embodiments of the present disclosure will now be described with reference to the drawings. Fig. 1 is a schematic diagram illustrating an example of a point defect simulator according to the present disclosure. Fig. The point defect simulator 1 shown in Figure 1 includes an input unit 11, a display unit 12 and an analysis unit 13.

[0014] The input unit 11 is an input interface for influencing the point defect simulator 1 of the present disclosure, and may be composed of, for example, a keyboard, a pen tablet, a touchpad, a mouse, etc. The input unit 11 may be a touch panel integrated with a display unit 12 described below.

[0015] The display unit 12 is a device for displaying outputs such as simulation results and may be formed by a given display, for example a liquid crystal display or an organic light emitting diode (OLED) display.

[0016] The analysis unit 13 is a device used to calculate the concentration profiles of vacancies and interstitial silicon atoms during the pulling of a silicon single crystal using the CZ process. In the present disclosure, the analysis unit 13 is used to determine the concentration profiles of vacancies and interstitial silicon atoms based on the Voronkov model.

[0017] The convection-diffusion equations of vacancies and interstitial silicon atoms based on the Voronkov model can be represented by the following equations (1) and (2), respectively: ∂CV∂t=−∇JV−v(∂CV∂z)−KVI(CVCl−CVeqCleq) ∂Cl∂t=−∇Jl−v(∂Cl∂z)−KVI(CVCl−CVeqCleq)

[0018] On the right-hand side of equation (1) (equation (2)) from the top, the first term is a diffusion term describing the diffusion effect of the concentration gradient of vacancies (interstitial silicon atoms), the second term is an advective term due to the growth of a crystal, and the third term is a pair annihilation term involving the reaction between vacancies and interstitial silicon atoms. Furthermore, C V a concentration of vacancies, C I is a concentration of interstitial silicon atoms, t is a period of time, J V is a diffusion flow of vacancies, J I is a diffusion flow of interstitial silicon atoms, v is the growth rate of the silicon single crystal (pulling rate), K VI is the reaction coefficient of pair annihilation between vacancies and interstitial silicon atoms, C V eq is the concentration of vacancies in thermal equilibrium and C Ieq is the concentration of interstitial silicon atoms in thermal equilibrium.

[0019] Furthermore, the diffusion flow J V in equation (1) and the diffusion flow J I in equation (2) can be represented by equation (3) or (4): JV=−DV∇CV−DVCVQVkBT2∇T Jl=−Dl∇Cl−DlClQlkBT2∇T where D V is a diffusion coefficient of vacancies, D I is a diffusion coefficient of interstitial silicon atoms, Q V is the reduced heat transport of vacancies and Q I is the reduced heat transport of interstitial silicon atoms, which is defined as the heat flow per unit flux of a component atom in the absence of a temperature gradient. T is the absolute temperature and k B the Boltzmann constant. Equations (3) and (4) above describe the effect of the temperature gradient on diffusion.

[0020] Furthermore, the reaction coefficient K VI in equations (1) and (2) are represented by equation (5): KVl=4πac(DV+Dl)exp(−ΔGlVkBT) wherein a C the critical distance between vacancies and interstitial silicon atoms at which the pair annihilation reaction can occur (= 0.543 nm), and ΔG VI is the barrier energy for annihilation found by Sinno et al. (see, for example, J. Electrochem. Soc. 145 (1998) 302).

[0021] Here the concentration C V eq of vacancies in thermal equilibrium and the concentration C I eq of interstitial silicon atoms in thermal equilibrium is conventionally determined using equations (6') and (7'): CVeq(T,P)=C0,V exp(−HVfkBT)=C0,V exp (−EVfkBT) Cleq(T,P)=C0,I exp (−HIfkBT)=C0,I exp (−EIfkBT) where C 0,V and C0,I Constants are, H V f is an enthalpy of formation of a vacancy, E V f is a formation energy of a vacancy, H I f an enthalpy of formation of an interstitial silicon atom and E I f is a formation energy of an interstitial silicon atom.

[0022] As described above, it has been found that the thermal stress in a crystal near the solid-liquid interface has an effect on the distribution of vacancies and interstitial silicon atoms. However, the effect of the above-mentioned thermal stress is not considered for the above equations (6') and (7'). Accordingly, in the present disclosure, the concentration C V eq of vacancies in thermal equilibrium and the concentration C I eqof interstitial silicon atoms in thermal equilibrium using equations (6) and (7): CVeq(T,P)=C0,V exp (−EVf+aVfPkBT) Cleq(T,P)=C0,l exp (−Elf+alfPkBT) where P is the stress in a crystal, which represents a tensile stress when the value is positive and a compressive stress when the value is negative. Furthermore, a V f and a I f Stress coefficients.

[0023] Conventionally, the diffusion coefficient D V of vacancies and the diffusion coefficient C I of interstitial silicon atoms, for example, using equations (8') and (9') (see, for example, K. Nakamura, “Research on point defect diffusion and secondary defect formation in silicon single crystal growth process”, PhD thesis, Tohoku University, 2001 (in Japanese)): DV(T,P)=D0,V exp (−EVmkBT) Dl(T,P)=D0,I exp (−EImkBT) where D 0,V and D 0,I Constants are, E V m is a diffusion activation energy of a vacancy and E I m is a diffusion activation energy of an interstitial silicon atom.

[0024] For equations (8') and (9') above, the effect of thermal stress is not taken into account, just as in equations (6') and (7') above. The effect of thermal stress is taken into account by transforming equations (8') and (9') above into equations (8) and (9): DV(T,P)=D0,V exp(−EVm+aVmPkBT) Dl(T,P)=D0,l exp(−Elm+almPkBT) wherein a V m and a I m stress coefficients are.

[0025] The above equations (6) to (9) include new variable stress coefficients a V f , a I f , a Vm and a I m to describe the effect of thermal stress. Accordingly, the values ​​for the four stress coefficients a V f , a I f , a V m and a I m can be determined from above by solving the convection-diffusion equations (1) and (2) to determine the concentration profiles of vacancies and interstitial silicon atoms in the silicon single crystal. However, these values ​​cannot be unambiguously determined by comparison, for example, with experimental results.

[0026] The inventor of the present disclosure has carefully studied ways to obtain the four voltage coefficients a V f , a I f , a V m and a I mfrom above. As a result, he found that the concentration profiles of vacancies and interstitial silicon atoms in a silicon single crystal can be determined by a V f - a I f is determined so that the defect distribution obtained from experiments can be well reproduced with the defect distribution obtained by simulations using a V m and a I m as constants (for example zero) and a V f - a I f were obtained as fitting parameters. The background to this finding is described below.

[0027] First of all, the critical v / G (P) = ξ cri (P) mentioned above is represented by an equation (10): ξcri(P)=Dl(Tmp,P)Cleq(Tmp,P)(H+Ql)−Dv(Tmp,P)Cveq(Tmp,P)Cveq(Tmp,P)(H+Qv)kBTmp2(Cveq(Tmp)−Cleq(Tmp)) where T mpis the melting point of silicon (1685K) and H is the mean enthalpy of formation H V f a vacancy and the enthalpy of formation H I f of an interstitial silicon atom, as described by equation (11): H=Hvf+Hlf2 where the enthalpy of formation H V f a vacancy and the enthalpy of formation H I f of an interstitial silicon atom can be determined in accordance with equations (12) and (13): Hlf=Ev+avfP Hlf=El+alfP

[0028] The change of ξ cri (P) above represents a change in the behavior of point defects introduced due to the effect of thermal stress, and the effect of thermal stress on the behavior of point defects can be determined by comparing ξ cri (P) and ξ cri (0) can be predicted. The value of ξ cri (0) is here 0.163 mm2 / min / K.

[0029] Now ξ cri determined using equation (10) above, taking into account the values ​​of the physical properties specified in NPL 4, which are given in Table 1 below, and the values ​​of the stress coefficients (a V f = 0.154 meV / MPa, a I f = -0.07 meV / MPa, a V m = 0.03 meV / MPa, a I m = -0.038 meV / MPa). The value of the stress P is set here to -10 MPa, which is a typical value for a crystal used as a 300 mm diameter wafer. [Table 1] Physical property blank space Interstitial silicon atom C0 (cm -3 ) 3,9175 × 10 26 6,2303 × 10 26 TO f (house) 3,940 4,050 C0 (cm 2 s -1 ) 3,5105 × 10 -4 2,4547 × 10 -1 TO m (house) 0,300 0,900 Q (eV) 0,000 1,010

[0030] The results of the calculations above show that the value of ξ cri (-10) 0.163 mm 2 / min / K is when a V m = 0.03 meV / MPa, a I m = -0.038 meV / MPa (where a V f = a I f= 0 meV / MPa); and the value of the stress (P) is set to -10 MPa and that the value of ξ cri (-10) equal to the value of ξ cri (0). This result shows that the value of ξ cri (P) to a suitable value, for example zero, in a way independent of the values ​​of a V m and a I m can be set.

[0031] Next, the inventor studied intensively how the values ​​of a V f and a I f can be determined, where the values ​​of a V m and a I m were set to zero. These values ​​of a V f and a I f However, they cannot be clearly determined by comparison with, for example, experimental results.

[0032] Against this background, the inventor determined the value of ξ criwith equation (10) above, while the values ​​of a V f and a I f were set differently. This showed that the same ξ cri , ie, the same value of the critical v / G was obtained even if the absolute values ​​of a V f and a I f were different when a V f - a I f is held.

[0033] Fig. 2 illustrates the relationship between the value of a V f - a I f and the value of the critical v / G (= ξ cri ) and illustrates cases where the value of a V f was set to -0.2 meV / MPa, 0 meV / MPa and 0.2 meV / MPa. As can be seen from Fig. 2, for a given a V f - a I f the same value of the critical v / G (= ξ cri ) even if the absolute value of a Vf (and a I f ) is different. This result shows that the concentration profiles of vacancies and interstitial silicon atoms can be determined by a V f - a I f is determined so that the defect distribution obtained from experiments can be well reproduced by the defect distribution obtained by simulations using a V f - a I f as fitting parameters. In detail, this can be done by using one of a V f and a I f is set as a fixed value and the other is used as a fitting parameter. For example, the value of a I be set to -0.07 meV / MPa, which was determined using a first-principles calculation and V f can be used as a fitting parameter.

[0034] It has been reported that when fitting using the value of a V f - a I f as fitting parameter ΔC V , which is represented by equation (14) below, obtained by solving the convection-diffusion equations (1) and (2), ie the difference ΔC V between the excess concentration of (C V -C V eq ) of vacancies and the excess concentration of (C I -C I eq ) of interstitial silicon atoms at 1,273 K, which corresponds to the defect distribution in the crystal (see, for example, TY Tan et al. and U. Gosele, Appl. Phys. A37, 1 (1985)). ΔCv=(Cv−Cveq)−(Cl−Cleq) at 1273K

[0035] Fig.Figure 3 illustrates the relationship between the value of v / G and the defect distribution in a silicon single crystal, and the horizontal axis represents the position in the silicon single crystal in the diameter direction. As shown in Fig. 3, a region 41 of COP formation, that is, a crystal region where COPs are formed, dominates in the single-crystal silicon when the value of v / G is large; and when the value of v / G is lower, a region 42 of potential OSF nuclei, which appears as a ring-shaped OSF region when subjected to a certain oxidation heat treatment, dominates a region 43 of promoting oxygen precipitation (hereinafter, it may also be referred to as "P V (1) region”), which is a crystal region where oxygen precipitation is likely to occur and COPs are not detected, and an oxygen precipitation promotion region 44 (hereinafter it may also be referred to as “P V(2) region”), i.e. a crystal region in which oxygen precipitates are present and COPs are not detected.

[0036] If v / G is even lower, an area 45 of oxygen precipitation inhibition (hereinafter referred to as “P I region”), that is, a crystal region in which oxygen precipitation hardly occurs and COPs are not detected, and a crystal region dislocation cluster region (hereinafter, it may also be referred to as “L / DL region”) 46, that is, a crystal region in which interstitial dislocation clusters are detected.

[0037] Among the silicon wafers obtained from a silicon single crystal having the defect distribution described above depending on the value of v / G, the silicon wafers obtained from the crystal region of a silicon single crystal having the P V (1)-Area 43, the P V (2)-area 44 and the P I-area 45 or a combination of these areas, defect-free silicon wafers without crystal defects.

[0038] As described above, it was reported that the ΔC represented by equation (14) above V corresponds to the defect distribution in a crystal; however, the position can be determined with a ΔC V -value of -0.2129 × 10 13 / cm 3 the border between the P I region 45 and the L / DL region 46 (see, for example, K. Nakamura, “Research on point defect diffusion and secondary defect formation in silicon single crystal growth process”, PhD thesis, Tohoku University, 2001 (in Japanese)).

[0039] Accordingly, a V f - a I f determined so that in the defect distribution in a crystal obtained by simulations, the position with ΔC V = -0.2129 × 10 13 / cm 3Regarding the defect distribution in the crystal obtained by the simulations, the boundary between the P I -region 45 and the L / DL region 46 in an experimentally obtained defect distribution. Because at least one of a V f and a I f can be determined by a suitable calculation approach, such as a first-principles calculation, the value of a V f and a I f be determined.

[0040] Although the concentration of oxygen atoms in the silicon single crystal for C 0,V is not taken into account in equation (1) above, C 0,V taking into account the concentration of oxygen atoms, as described below.

[0041] First, the concentration of vacancies in thermal equilibrium represented by equation (1) above is represented by equation (15) when the concentration of oxygen atoms is taken into account: Cveq(T,P,Oi)=C0,V,Oi exp (−HvfkBT)=C0,V,Oi exp(−Evf+avfPkBT) where Oi is the concentration of oxygen atoms defined in ASTM F121-1979 and C 0,V,Oi is a constant.

[0042] Furthermore, the effect of the concentration of oxygen atoms on the concentration of vacancies in thermal equilibrium at the melting point is given by equation (16): CVeq(Tmp,P,Oi)=CVeq(Tmp,P,0)+aOi where C V eq (T mp ,P,0) is the concentration of vacancies in thermal equilibrium when the concentration of oxygen atoms is 0, and a is a constant (for example, 4 × 10 -12 ).

[0043] From equations (15) and (16) we arrive at the following equation (17). C0,V,Oi​exp(−Evf+avfPkBTmp)=C0,V exp(−Evf+avfPkBTmp)+aOi

[0044] Using equation (17) above, C 0,V represented by equation (18): C0,V=C0,V,Oi−aOi⋅exp(Evf+avfPkBTmp)

[0045] Using equation (18) obtained as described above, the value of C 0,V , which reflects the consideration of the concentration Oi of oxygen atoms. This allows the fitting of a V f using various experimental results and can provide generality to the analysis. Furthermore, a point defect simulator according to the present disclosure can simulate the experimentally obtained shape of the boundary between the P Iregion 45 and the L / DL region 46 without using equation (18) above; however, as described in the examples below, the position of the crystal pulling direction may differ from that obtained experimentally. The use of equation (18) above also allows the position in the crystal pulling direction to coincide.

[0046] Furthermore, the point defect simulator according to the present disclosure can preferably determine the concentration profiles of vacancies and interstitial silicon atoms in the crystal with higher precision when the solid-liquid interface between the silicon single crystal and the silicon melt has an upwardly convex shape.

[0047] Furthermore, the point defect simulator according to the present disclosure makes it possible to determine the concentration profiles of vacancies and interstitial silicon atoms in the crystal with higher precision when the diameter of the silicon single crystal is 300 mm or more. When the diameter of a silicon single crystal is larger, the temperature difference between the interior and exterior of the crystal is larger, and the thermal stress in the crystal is higher. Accordingly, the concentration profiles of vacancies and interstitial silicon atoms in the crystal can be determined with higher precision when the diameter of the silicon single crystal is 300 mm or more.

[0048] Thus, when determining the concentration profiles of vacancies and interstitial silicon atoms in a crystal based on the above convection-diffusion equations (1) and (2), the concentration profiles of vacancies and interstitial silicon atoms in the crystal can be determined using the strain coefficients a V f or a I f in equations (6) and (7) above as fitting parameters for an experimentally obtained defect distribution. (Point defect simulation program)

[0049] The point defect simulation program according to the present disclosure is a program for making a computer operate as the above-described point defect simulator according to the present disclosure. This program can be stored in a memory unit of the computer, and a CPU in the computer can read the program containing processing instructions for executing the processing instructions, thus performing each step.

[0050] Furthermore, the program containing processing instructions may be distributed, for example, by selling, transferring, or renting portable media such as Blu-ray (registered trademark in Japan, other countries, or both) discs, DVDs, and CD-ROMs; or, for example, by storing such a program in a memory of a server on a network and transferring the program from the server to other computers via the network.

[0051] Furthermore, a computer executing such a program may, for example, temporarily store a program stored in a portable medium or a program transferred from a server in its memory. In another embodiment of this program, a computer may read the program directly from a portable medium and perform processing in accordance with the program; alternatively, each time the program is transferred from a server to the computer, the computer may sequentially perform processing in accordance with the received program. (Point defect simulation method)

[0052] The point defect simulation method according to the present disclosure is used to determine the concentration profiles of vacancies and interstitial silicon atoms in a silicon single crystal using the above-described point defect simulator according to the present disclosure. This makes it possible to determine the distribution of point defects in the silicon single crystal while taking thermal stress into account.

[0053] As described above, the solid-liquid interface between the silicon single crystal and the silicon melt preferably has an upwardly concave shape, and the diameter of the silicon single crystal is preferably 300 mm or more. (Method for producing a silicon single crystal)

[0054] A method for producing a silicon single crystal of the present disclosure includes: for at least one single crystal pulling apparatus, determining the concentration profile of point defects, including vacancies and interstitial silicon atoms, in a silicon single crystal pulled up using the single crystal pulling apparatus using the above-described point defect simulator according to the present disclosure or the above-described point defect simulation method according to the present disclosure; designing the single crystal pulling apparatus based on the thus-determined concentration profiles of vacancies and interstitial silicon atoms; and producing a defect-free silicon single crystal using the designed single crystal pulling apparatus.

[0055] The concentration profiles of point defects in a silicon single crystal pulled up using a predetermined single crystal pulling apparatus can be determined using the above point defect simulator according to the present disclosure or the above point defect simulation method according to the present disclosure. Thus, the pulling conditions under which a defect-free silicon single crystal can be obtained can be determined.

[0056] It should be noted that if the pulling conditions under which a defect-free silicon single crystal can be obtained cannot be determined from the obtained concentration profiles of point defects, the change in the design of the single crystal pulling apparatus and calculations of the concentration profiles of point defects can be repeated until the conditions under which a defect-free silicon single crystal can be obtained are found.

[0057] The "design of the single-crystal pulling apparatus" above refers to the design of all or some of the structures in the single-crystal pulling apparatus that primarily affect the thermal environment of the silicon single crystal during pulling. The structures in the single-crystal pulling apparatus include a heat shield and a water cooler surrounding the silicon single crystal being pulled, a heater, a crucible, a thermal insulation component arranged around and below the heater, and the silicon melt in the crucible. Accordingly, the "design of the single-crystal pulling apparatus" is intended to determine the size, shape, material, and relative positions of all or some of these structures.Furthermore, the point defect simulator according to the present disclosure can also be used for designing the shape of the chamber of the single crystal pulling apparatus, which affects the thermal environment of the silicon single crystal, and for setting the radiation factor in addition to the design of the structures in the single crystal pulling apparatus.

[0058] In this way, a defect-free silicon single crystal can be produced by pulling a silicon single crystal under pulling conditions under which a defect-free silicon single crystal can be obtained, using a single crystal pulling apparatus for which such conditions have been found. EXAMPLES

[0059] Examples will now be described in detail; however, the present disclosure is not limited to the examples. (Production of silicon single crystal)

[0060] A silicon single crystal (oxidation concentration: 10 to 15 × 10 17 atoms / cm 3 ) for a 300 mm diameter wafer was fabricated using the CZ process. For specific pulling conditions, the crystal growth rate was gradually reduced while pulling up the silicon single crystal while a transverse magnetic field was applied. The obtained silicon single crystal was cut in the pulling direction, and after Cu decoration at 750°C for 5 min, Wright etching, which is a selective etching, was performed. The defect distribution in the crystal was determined by locating each defect region under a microscope. The obtained defect distribution is shown in Fig. 4A illustrates. (Example)

[0061] Using a point defect simulator according to the present disclosure, the concentration profiles of vacancies and interstitial silicon atoms in a silicon single crystal were determined for a 300 mm diameter wafer, wherein the silicon single crystal was pulled up under the same conditions as in the preparation of the silicon single crystal above. In this context, while the values ​​of a V m in equation (8) and a I m in equation (9) were set to zero, the value of a I f in equation (7) is set to -0.07 meV / MPa and the a V f as a fitting parameter for the experimentally obtained defect distribution, which is Fig. 4A. The positions at which the value of ΔC given in equation (14) V then -0.2129 × 10 13 cm -3 would depend on the value of a V fchange, as in Fig. 5B. While the C given in Table 1 0,V was determined without considering the stress, the stress was considered for equation (6), and so the difference would have been caused because the stress was treated differently.

[0062] Against this background, C 0,V using equation (18) so that in accordance with the value of a V f the position of the boundary between the P I -range and the L / DL range with the position in the Fig. 4A depicted, experimentally obtained defect distribution, as in Fig. 5C. While the positions of the boundaries in the crystal center were aligned, a V f so that the peak position of the boundary between the P I -range and the L / DL range with the peak position in the Fig. 4A shown, experimentally obtained defect distribution (the position of 980 mm in Fig. 5A). As a result, the value of a V f to 0.12. Thus, the values ​​of the four stress coefficients a V f , a I f , a V m and a I m and the concentration profiles of vacancies and interstitial silicon atoms in the crystal were determined. The results are presented in Fig. 4D specified. (Conventional example)

[0063] As in the example, the concentration profiles of vacancies and interstitial silicon atoms in a silicon single crystal were determined, and the defect distribution in the crystal was determined. Note that instead of equations (6) to (9), equations (6') to (9') were used, for which the effect of strain was not considered. All other conditions were the same as those in the example. The obtained defect distribution in the crystal is shown in Fig. 4B illustrates. (Comparison example)

[0064] As in the example, the concentration profiles of vacancies and interstitial silicon atoms in a silicon single crystal were determined, and the defect distribution in the crystal was sought. It should be noted that of the four strain coefficients in equations (6) to (9), for the values ​​of a V f and a I fValues ​​were used that were found with the first-principles calculation described in NPL 3. All other conditions were the same as in the example. The obtained defect distribution in the crystal is shown in Fig. 4C illustrates.

[0065] A comparison of Fig. 4A, which illustrates the experimental results, with Fig. 4B to Fig. Figure 4D, illustrating the simulation results, shows that the position of the valley of the OSF region is too low and the peak of the P I -area in Fig. 4B, which corresponds to a conventional example, is not reproduced. Furthermore, Fig. 4C, which corresponds to the comparative example, the position of the value of the OSF area is too high, although the peak of the P I -area is reproduced. In contrast, in Fig. 4D, which corresponds to the example, the shapes of the OSF area and the P I -area well reproduced. INDUSTRIAL APPLICABILITY

[0066] The present disclosure provides the determination of the distribution of point defects taking into account the thermal stress in a silicon single crystal and is thus useful in the semiconductor industry. LIST OF REFERENCE SYMBOLS 1 point defect simulator 11 Input unit 12 Display unit 13 Analysis unit 41 Area of ​​COP education 42 Range of potential OSF cores 43 Area of ​​oxygen precipitation promotion 44 Area of ​​oxygen precipitation promotion 45 Area of ​​inhibition of oxygen precipitation 46 Dislocation cluster area

Claims

[1] Point defect simulator that calculates concentration profiles of vacancies and interstitial silicon atoms during the pulling of a silicon single crystal with the Czochralski process using a convection-diffusion equation, where the convection-diffusion equation a concentration C V eq of vacancies in thermal equilibrium, which is represented by equation (A) shown below, and a concentration C I eq of interstitial silicon atoms in thermal equilibrium, which is represented by equation (B) shown below, where the point defect simulator includes: an analysis unit used to fit calculation results to experimental results, wherein a coefficient of a first stress coefficient a V f in equation (A) and a second stress coefficient a If in equation (B) as a fitting parameter for the difference of a V f and a I f is used, CVeq(T,P)=C0,V exp(−EVf+aVfPkBT) Cleq(T,P)=C0,l exp(−Elf+alfPkBT) where in equations (A) and (B) T is a temperature, P is a voltage, C 0,V and C 0,I Constants are, k B is the Boltzmann constant, E V f a formation energy of a vacancy and E I f is a formation energy of an interstitial silicon atom. [2] The point defect simulator according to claim 1, wherein a solid-liquid interface between the silicon single crystal and a silicon melt has an upwardly convex shape. [3] The point defect simulator according to claim 1 or 2, wherein the silicon single crystal has a diameter of 300 mm or more. [4] A point defect simulation program for allowing a computer to serve as the point defect simulator according to at least one of claims 1 to 3. [5] A point defect simulation method used to determine concentration profiles of vacancies and interstitial silicon atoms in a silicon single crystal, wherein the point defect simulator according to any one of claims 1 to 3 is used. [6] The point defect simulation method according to claim 5, wherein the solid-liquid interface between the silicon single crystal and the silicon melt has an upwardly convex shape. [7] The point defect simulation method according to claim 5 or 6, wherein the silicon single crystal has a diameter of 300 mm or more. [8] A method for producing a silicon single crystal, the method comprising: for at least one single crystal pulling apparatus, determining concentration profiles of point defects including vacancies and interstitial silicon atoms in a silicon single crystal pulled up using the single crystal pulling apparatus using one of the point defect simulator according to any one of claims 1 to 3 and the point defect simulation method according to any one of claims 5 to 7; designing the single crystal pulling apparatus based on the thus determined concentration profiles of vacancies and interstitial silicon atoms; and producing a defect-free silicon single crystal using the designed single crystal pulling apparatus. [9] A computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to carry out the method according to claim 8.

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