JP2604924B2 - Method for analyzing impurities in silicon crystal - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an FZ semiconductor single crystal manufacturing method in which a raw polycrystal is partially heated and melted using a high-frequency induction heating coil and the melting zone is moved to grow a single crystal. The present invention relates to a method of analyzing impurities in a concentrated portion of a grown silicon single crystal, and thereby quantifying impurities as a central portion or an average value of a raw material polycrystal, an intermediate silicon polycrystal rod, and a single crystal.

[0002]

2. Description of the Related Art When growing a silicon single crystal by the FZ method, a polysilicon rod is supplied from above a high-frequency coil, melted by the high-frequency coil, and a silicon seed crystal is brought into contact with the coil from below the coil. A high-purity silicon single crystal is grown downward.

[0003] During the growth of the silicon single crystal straight body, a new silicon melt is constantly supplied from above, and, unlike other single crystal growth methods, silicon melt is replaced with a relatively low-purity structural material such as quartz. Since there is no contact, the impurity concentration in the silicon melt does not increase, and the impurity concentration in the silicon single crystal is kept lower than that in the silicon melt due to the segregation phenomenon.

For this reason, the FZ silicon single crystal has been used as a high-purity single crystal for studying the behavior of crystal defects and impurities in the crystal, and also for examining the degree of contamination in the semiconductor device manufacturing process. There is a demand for high purity, and therefore, it is necessary to quantify the central or average impurity concentration of each of the raw material polycrystal as the raw material and the intermediate polycrystal as the intermediate as well as the FZ method silicon single crystal. It has become.

However, since the impurity concentration of the raw material or the intermediate polycrystal as well as the straight body of the FZ silicon single crystal is very low, it is impossible to directly detect and quantify the impurity concentration.

However, in the process of growing a silicon single crystal by the FZ method, a new silicon melt is not supplied at the end of the growth of the single crystal, and the melted silicon melt is solidified by stopping heating from the high frequency coil. In the final solidified portion (tail portion) of the silicon single crystal, the same impurity as the impurity taken into the straight body portion of the single crystal is condensed at a high concentration, and in some cases, exceeds the solid solubility limit and is precipitated at the same time. Therefore, if a high-concentration portion of the impurity including the portion where the impurity is precipitated is collected, the impurity can be detected and quantified, and the impurity concentration in the single-crystal straight body can be calculated from the theoretical calculation. Also, in the intermediate polycrystalline rod in which the heating zone passes only once, impurities can be concentrated at a high concentration in the final solidified portion, and the impurity concentration of the raw material polycrystalline rod as well as the intermediate polycrystalline central portion is controlled. It can be quantified by analyzing the final solidified portion.

[0007] In the conventional method, the collected sample is activated and analyzed by using a nuclear reactor to quantitatively analyze elements.

[0008] However, although this method can analyze many kinds of elements, it requires a long time for analysis and is not suitable for real-time process control. Further, it is not always necessary to quantify many types of elements for process control.
Under such circumstances, an analysis method capable of performing quantitative analysis of impurity elements in a short time is required.

Furthermore, if a sample is taken including an extra silicon crystal portion with a low impurity concentration, impurities are diluted during analysis and the detection sensitivity is lowered, so that the volume of the sample to be taken is as small as possible. Better.

However, in the conventional method, since a sample is collected by a cutting machine such as a diamond cutter or the like, a large amount of impurities are contaminated from a blade or cooling water at the time of cutting, so that accurate determination of impurities cannot be performed. Under such circumstances, there is also a demand for a method of collecting a single crystal tail in which impurities are concentrated without contaminating an impurity deposition portion.

[0011]

SUMMARY OF THE INVENTION The present invention has been made in view of the above circumstances, and a sample is taken from a high-concentration portion of an impurity including an impurity deposition portion in a final solidified portion of an FZ silicon crystal, It is an object of the present invention to provide a method for analyzing impurities in a silicon crystal in which a collected sample is chemically and quantitatively analyzed.

[0012]

In order to solve the above-mentioned problems, in the method of the present invention, an impurity in a final solidified portion of a single crystal or an intermediate polycrystal is analyzed by an impurity analysis method in a silicon crystal grown by the FZ method. A sample is taken from the high-concentration portion of the impurity including the precipitated portion of the impurity, and the impurity element is chemically quantitatively analyzed by atomic absorption spectrometry, inductively coupled plasma emission spectrometry, or inductively coupled plasma mass spectrometry.
From the analytical values of, raw material polycrystalline, intermediate silicon polycrystalline rod or
Determine the impurity at the center or average value of silicon single crystal
It is obtained so as to amount.

More specifically, the collected sample is dissolved in an aqueous solution of hydrofluoric nitric acid, the solution is evaporated to dryness, and the volume is adjusted with high-purity 10% nitric acid. Alternatively, quantitative analysis of impurity elements is performed by inductively coupled plasma mass spectrometry.

It is preferable to selectively remove only the high-concentration portion of the impurity including the impurity-precipitated portion in the final solidified portion without contaminating it. The final solidified part
Is a tail portion shown in FIGS. 1 and 2 and a portion with a high impurity concentration.
Refers to the symbol S (sample) shown in FIGS.
You. In this part, high-concentration impurities are
Growing dry, low light reflectance compared to other parts
As a result, the gloss is low and can be visually confirmed.

Preferably, the sample is collected by hitting the periphery of the sampled portion without directly contacting the sampled portion of the final solidified portion.

In the present invention, at the final solidified end,
A steady heating state is maintained until the end, and the application of the high-frequency power to the coil is stopped when the periphery of the solidified single crystal reaches the upper end shoulder of the silicon rod. By continuing the FZ process to the final stage, impurities to be analyzed are localized,
Further, as described above, the localized portion is easily separated and collected by the impact. With the final solidified end of the silicon single crystal from which the sample is to be taken positioned downward with respect to the horizontal table, the growth axis of the single crystal is set to 1
It is preferable to perform the above-mentioned impact while maintaining the angle in the range of 0 to 20 degrees.

Using the quantitative value of the impurity in the silicon intermediate polycrystal determined by the above method, the impurity concentration C ₀ of the straight body portion of the raw material polycrystalline silicon rod can be calculated and determined based on the following formula (XII). it can. ^{_{C 0 = A · kw k-}} 1 / {1- (1-k) e -k (Lw) / w} (w-w 1) k ··········· (XII) ( upper In the formula, C ₀ : impurity concentration of the straight body portion of the raw material silicon polycrystalline rod, A: quantitative value of the impurity determined by the method of claim 1, k: segregation coefficient of the impurity, x: zoning from the FZ starting point The length from the boundary to the melting zone on the crystal side takes a value from 0 to L−w, L: crystal length, w: melting zone width, w ₁ : in the melting zone at the moment of separation. The position where the portion corresponding to the sampled sample portion when the end of the melting zone on the crystal side is the origin has started solidification.)

Using the quantitative value of the impurity of the silicon single crystal determined by the above method, the impurity concentration of the straight body portion of the raw silicon polycrystalline rod can be calculated and determined based on the following formula (XIII). ^{_{C 0 = A · kw k-}} 1 / [1-k (1-k) (Lw) e -kL / w / w + {2k-2 + (1-k) e -k} e -k (Lw ^{) / w} ] (ww ₁ ) ^k ··· (XIII)

Using the quantitative value of the impurity of the intermediate silicon polycrystal determined by the above method, based on the following equation (II), the impurity concentration C ₁ (x) at the position x of the straight body portion of the intermediate silicon polycrystal rod: Can be calculated and quantified. C ₁ (x) = C ₀ {1- (1-k) e- ^{kx / w} } (II)

Using the quantitative value of the impurity of the silicon single crystal determined by the above method, based on the following formula (X),
The impurity concentration C ₂ (x) at the position x of the straight body portion of the silicon single crystal rod can be calculated and quantified. C ₂ (x) = C ₀ [1-k (1-k) x · e -k (w + x) / w / w + {2k-2 + (1-k) e -k} e -kx / w ] (X)

[0021]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The present invention will be specifically described below with reference to the accompanying drawings. First, an experiment performed on a method for collecting a sample will be described. As shown in FIG. 1, the part of the tail part 11 of the tail part 11 of the FZ method silicon crystal 10 at the end of the straight body 12 was placed on a solid base 13 and the part away from the sampled part was hit with a plastic hammer. The locations indicated by reference numerals 4A, 5A and 6A were tried as the locations to be hit. Further, as a method of holding the tail portion 11, the angle θ formed between the crystal growth axis in the tail portion 11 and the base 13 was set to 0 degree, 10 degrees, 20 degrees, and 30 degrees. In this experiment, the table 13 and its peripheral portion were covered with a vinyl sheet so that the protruding sample S did not come into direct contact with it. Ten runs were performed for each run condition. The success rate was defined as A = (the number of times the desired sample portion jumped out) / (the total number of times = 10) × 100 (%).

Table 1 shows the results of the experiment. When hitting the portions indicated by reference numerals 4A and 6A, the entire tail is often largely split at any angle θ,
The success rate was less than 20% . On the other hand, when the portion indicated by the reference numeral 5A was hit, when the angle θ was 0 degree, the entire tail portion was greatly broken similarly to the above, and the success rate was 20%. Also, the angle θ
However, at 30 degrees, only the vicinity of the hitting point 5A was often broken, and the success rate was 40%. When the angle θ is 10 degrees or 20 degrees, the sampling desired portion S pops out as shown in FIG.
The success rate above was obtained.

[0023]

[Table 1]

As shown in FIG. 2, a sample S collected from the tail 11 of the FZ silicon single crystal 10 was weighed and 5 g of the sample S was weighed.
Is a mixed acid consisting of hydrochloric acid and hydrofluoric acid in a 1: 1 composition by volume.
The substrate was washed by immersion in 180 ml at 180 ° C. for 30 minutes. Thereafter, the entire sample was dissolved using a mixed acid of high-purity hydrofluoric acid and nitric acid prepared so that the volume ratio was 1: 1. The amount of the hydrofluoric acid used at this time was the amount until the entire sample was dissolved. After the solution is evaporated to dryness, the concentration is 10%.
, And the solution was injected into an atomic absorption apparatus for quantitative analysis. The results are shown in Table 2.

[0025]

[Table 2]

As an example, a hole was drilled at the tip of the raw material polycrystalline rod, and a single metal of each of Cu, Fe, and Al was charged at a predetermined weight as shown in Table 2 (A), and impurities of intentionally contaminated crystals were introduced. The analysis results are summarized in Table 2. Columns (B) and (E) show the number of impurity atoms per unit volume in the sample portion collected from the tail after 1-pass and 2-pass, respectively. The results of calculating the impurity concentration in the intermediate polycrystal or single crystal from the values in columns (B) and (E) by the above-described calculation method are shown in columns (D) and (G), respectively. In addition, the results of actual measurement of the impurity concentration in the intermediate polycrystal or single crystal by activation analysis are shown in columns (I) and (J), respectively. The numerical values in each column are well matched when compared.

Hereinafter, a method of calculating the impurity concentration in the crystal will be described. FZ twice to obtain FZ silicon single crystal
Was performed, but it was determined that there was no impurity contamination in all the steps.
FIGS. 3A, 3B, and 3C show schematic diagrams of the first FZ process, and FIGS. 4A, 4B, and 4C show schematic diagrams of the second FZ process. Indicated. FIG. 3A and FIG.
(A) shows the middle of the process, (b) of FIG. 3 and (b) of FIG. 4 show the moment when the raw material polycrystalline rod and the intermediate polycrystalline rod are cut off, respectively, and (c) of FIG. FIG. 4 (c) shows a state where the silicon melt 26 or 32 remaining after being separated is solidifying. The shape of C in FIGS. 3 and 4 is simply described to facilitate understanding of the derivation of the impurity relational expression of the present invention. In fact, as shown in FIGS. 1 and 2, the final solidified end has a protruding center. It becomes a curved surface.

First, the balance of the amount of impurities in the melting zone 22 of the raw material polycrystal 21 in the first FZ step is shown in FIG.
In (a), ds = (C ₀ −kC _L ) dx (I) Where s is the amount of impurities in the molten zone 22 and C
₀ is the impurity concentration in the raw material polycrystalline rod 21, C _L is the impurity concentration in the melting zone 22, x is the position from the FZ starting point 24 to the melting zone 22, and k is the impurity segregation coefficient. Since C _L = s / w, the equation (I) becomes ds / (C ₀ −ks / w) = dx. When both sides are integrated, log (C ₀ −ks / w) = − k (x + a) / W,
(A: integration constant) ∴a · e− ^{Kx / w} = C ₀ −ks / w. As a boundary condition, when x = 0, s = C ₀ • w, so that a = C ₀ (1-k ) Also, since the impurity concentration at the position x after the first FZ is C ₁ (x) = kC _L = ks / w, the following expression is obtained: ∴C ₁ (x) = C ₀ ｛1- (1-k) e ^{-kx / w} ｝ ・ ・ ・ ・ ・ ・ (II)

Next, as shown in FIG. 3B, the silicon melt 2 cut off at the tail 25 of the raw material polycrystalline rod 21.
6 is solidified from the side of the intermediate polycrystal 28 as shown in FIG. 3C, so that the impurity concentration C (x) at the position x of the solidified portion 28 is C (x) = − ds / dx... (III) Here, s indicates the amount of impurities in the melt 28 left undissolved.

Further, apart from the equation (III), since there is a relation of C (x) = kC _L = ks / (w−x), dx / (w−x) = − ds / s... integrating the ·········· (IV) (IV) wherein as a boundary _{condition, s = C 0 'w (} x
= 0), C (x) = kC ₀ ′ (w−x) ^k−1 / w ^k−1 (V) Here, C ₀ ′ indicates the impurity concentration in the melt 26 at the moment when the tail 25 of the raw material polycrystalline rod is cut off.

Next, the molten zone 3 in the second FZ step
4 is (w / k) dC ₂ (x) = ｛C ₁ (x + w) −C ₂ (x)｝ dx, ∴dC ₂ (x) / dx + ( (k / w) C ₂ (x) = (k / w) C ₁ (x + w) (VI).

Here, C ₂ (x) indicates the impurity concentration in the single crystal rod 30, and C ₁ (x + w) indicates the impurity concentration in the intermediate polycrystalline rod 29 grown in the first FZ step.

Using the constant change method in equation (VI), the solution at dC ₂ (x) / dx + (k / w) C ₂ (x) = 0 is given by C ₂ (x) = ae− ^{kx / w} ( a: integral constant) Here, a = Z (x), and C ₂ (x) = Z (x) e− ^{kx / w} (VII) Differentiating equation (VII) with respect to x, C ₂ '(x) = Z' (x) e- ^{kx / w-} kZ (x) e- ^{kx / w} / w (VIII) (VII) Comparing equation (VIII) and equation (VI) using the equation, Z ′ (x) e ^{−kx / w} = (k / w) C ₁ (x + w), and using equation (II), Z ′ ( _{x) = C 0 (k /} w) {e kx / w - (1-k) e -k} Furthermore, when integrating the equation for x, Z (x) = C 0 (k / w) {(w ^{· e kx / w / k)} -x (1-k) e -k + a} Therefore, (from VII) formula, C ₂ (x) = As _{0 {1- (k / w)} (1-k) x · e -k (w + x) / w + a · e -kx / w} ····· (IX) boundary conditions,

(Equation 1) Is applied to the equation (IX). A = 2k−2 + (1−k) e− ^k Therefore, C ₂ (x) = C ₀ [1−k (1−k) x · e− ^{k (w + x ) / w / w + {2k} -2 + (1-k) e -k} e -kx / w ] · · · · · · (X)

In the second FZ step, solidification of the remaining melt 32 after cutting off the intermediate polycrystalline tail 31 can be considered in the same manner as in the first time. Therefore, the impurity concentration in the solidified portion 34 is given by the formula (V). Can be

Next, a method of calculating the impurity concentration in the intermediate polycrystalline cylinder from the quantitative value of the impurity element determined from the crystal tail will be described. When the quantitative value of the amount of impurities is A,
From equation (V),

(Equation 2) Is obtained.

First, the case where impurities are quantified from the tail portion of the intermediate polycrystal will be described. Here, w ₁ indicates the position where the portion corresponding to the sampled sample portion when the crystal side end of the melting zone is set as the origin in the melting zone at the moment of separation, at which solidification has started. Further, since C ₁ (x) and C (x) are continuous at a distance x from the FZ start point 24 of the intermediate polycrystalline rod, C (0) = C ₁ (L−w) {kC ₀ ′ = C ₀ }. ^{1- (1-k) e -k} (Lw) / w} ^{_{Therefore, C 0 = A · kw k}} -1 / {1- (1-k) e -k (Lw) / w} (w-w 1 ) ^K (XII) Therefore, C ₀ is obtained from the quantitative value using the formula (XII),
From the formula (II), the impurity concentration in the intermediate polycrystal is obtained.

Next, a case where impurities are quantified from the tail portion of the single crystal will be described. Similarly, for a single crystal rod, C
_{Since 2} (x) and C (x) are continuous, C (0) = C ₂ (L−w) Therefore, C ₀ = A · kw ^k−1 / [1-k (1-k) (L−w ) E- ^{kL / w} / w + {2k-2 + (1-k) e- ^k } e- ^{k (Lw) / w} ] ( ^ww - ₁ ) ^k ... (XIII) Therefore, C ₀ is obtained from the quantitative value using the formula (XIII), the impurity concentration in the intermediate polycrystal is obtained from the formula (II), and the impurity concentration in the single crystal is obtained from the formula (X).

FIG. 5 is a profile of the impurity concentration in the growth direction showing the result of calculating the impurity concentration in the single crystal rod from the numerical values in Table 2 using the theoretical calculation formula of the present invention. In the specific example described above, the weight of the sample S is converted into a volume, and further divided by the cross-sectional area of the single crystal to obtain a dimension of the length, whereby (w−w ₁ ) is obtained. In addition, taking into account the segregation coefficient k, the length L of the single crystal, and the length w of the heat-melting zone, the values of the impurity concentration obtained by putting the numerical values of the impurity concentrations in the column of Table 2 (F) into the formula (X) are given. FIG. 5 shows the concentration distribution in the crystal growth direction.

[0039]

As described above, according to the present invention, by calculating the profile of the impurity concentration, it is possible to selectively collect the impurity-contaminated sample having a concentration that cannot be determined by the ordinary method from the single crystal. Thus, it is possible to investigate the electrical influence of the type and concentration of the impurity on the crystal and the generation behavior of crystal defects at a higher purity level. Further, when the crystal is contaminated at a high concentration, the segregation coefficient of the contaminated impurity can be estimated by combining the crystal with the impurity profile in the growth direction by another quantitative method. Further, since the impurity concentration of the single crystal can be predicted from the result of the impurity concentration of the intermediate polycrystal, there is a merit that the quality control accuracy is improved. Furthermore,
The degree of contamination in the FZ process can be grasped from the difference between the predicted value and the actually measured impurity value of the final single crystal, and the process can be quantitatively controlled.

[Brief description of the drawings]

FIG. 1 is a schematic view showing an experimental method for collecting a sample for impurity analysis from the tail of an FZ method silicon crystal.

FIG. 2 is a schematic view showing a method of collecting a sample for analyzing impurities from the tail of an FZ method silicon crystal.

3A and 3B are schematic diagrams of a first FZ step, wherein FIG. 3A is a schematic view of a first FZ step, and FIG. 3B is a schematic view of a first FZ step.
FIG. 3C is a schematic diagram when the separated silicon melt is solidified in the Z step, and FIG. 3C is a schematic diagram when the separated silicon melt is solidified.

4A and 4B are schematic diagrams of a second FZ step, in which FIG. 4A is a schematic view of a second FZ step, and FIG. 4B is a schematic view of a second FZ step.
FIG. 3C is a schematic diagram when the separated silicon melt is solidified in the Z step, and FIG. 3C is a schematic diagram when the separated silicon melt is solidified.

FIG. 5 is a graph showing a profile in a growth direction of an impurity concentration of intentionally contaminated crystals calculated from a quantitative value by a chemical analysis method and a theoretical calculation formula.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 10 Silicon single crystal 11 Tail part 12 Straight body end part 21 Raw material polycrystal rod 22 Melt zone 23 Intermediate polycrystal 24 FZ starting point 25 Raw material polycrystal tail part cut off 26 Silicon melt at the moment of separation 27 Remaining silicon melt 28 Normal Part solidified by freezing 29 Intermediate polycrystalline rod 30 Single crystal rod 31 Separated intermediate polycrystalline tail 32 Silicon melt at the moment of separation 33 Remaining silicon melt 34 Part solidified by normal freezing S Sampling sample

──────────────────────────────────────────────────続 き Continuation of the front page (51) Int.Cl. ⁶ Identification code Agency reference number FI Technical indication location G01N 31/00 G01N 31/00 Z (72) Inventor Hirotoshi Yamagishi 2-13 Isobe, Annaka-shi, Gunma Prefecture No. 1 Shin-Etsu Semiconductor Co., Ltd. Isobe Laboratory

Claims (9)

(57) [Claims] In an impurity analysis method in a silicon crystal grown by the FZ method, a sample is collected from a high-concentration portion of the impurity including a portion where the impurity is precipitated in a final solidified portion of a single crystal or an intermediate polycrystal, and elemental chemical quantitative analysis by the atomic absorption method or induction coupled plasma emission spectrometry or inductively coupled plasma mass spectrometry, or the analysis value
Raw material polycrystalline, intermediate silicon polycrystalline rod or silicon single
A method for analyzing impurities in a silicon crystal, comprising quantifying an impurity as a central portion or an average value of the crystal. 2. A without directly contacting the sampling part of the final solidified portion, by striking the peripheral portion of the sample collection portion according to claim 1, characterized in that so as to collect the sample Method for analyzing impurities in silicon crystal. 3. A silicon single crystal to be sampled is
The said striking is performed by holding the solidified portion downward so as to abut on the horizontal table so that the growth axis of the single crystal is in the range of 10 to 20 degrees. 2
The method for analyzing impurities of a silicon crystal according to the above. 4. An impurity concentration C _{0 of} a straight body portion of a raw silicon polycrystalline rod based on the following equation (XII) using a quantitative value of impurities in the silicon intermediate polycrystal determined by the method according to claim 1. An impurity analysis method, characterized by calculating and quantifying the amount of impurities. C ₀ = A · kw ^k−1 / {1- (1-k) e− ^{k (lw) / W} } (ww− ₁ ) ^K (XII) (top) In the formula, C _O : impurity concentration of the straight body portion of the raw material polycrystalline silicon rod, A: quantitative value of the impurity determined by the method of claim 1, k: segregation coefficient of the impurity, x: zoning from the FZ starting point The length from the boundary to the melting zone on the crystal side takes a value from 0 to L−w, L: crystal length, w: melting zone width, w ₁ : in the melting zone at the moment of separation. The position where the portion corresponding to the sampled sample portion when the end of the melting zone on the crystal side is the origin has started solidification.) 5. Using the quantitative value of impurities in a silicon single crystal determined by the method according to claim 1, the following formula (XI)
An impurity analysis method comprising calculating and quantifying an impurity concentration in a straight body portion of a raw material silicon polycrystal bar based on II). C _O = A · kw ^k−1 / [1-k (1-k) (L−w) e− ^{kl / w} / w + ｛2k−2 + (1−k) e− ^k ｝ e− ^{k (Lw ) / w} (ww ₁ ) ^K ] (XIII) 6. The impurity at the position x of the straight body portion of the intermediate silicon polycrystal rod based on the following equation (II), using the quantitative value of the impurity of the intermediate silicon polycrystal determined by the method according to claim 1. An impurity analysis method comprising calculating and quantifying the concentration C ₁ (x). C ₁ (x) = C _o {1- (1-k) e ^{-kx / w} } (II) 7. The following formula (X) is obtained by using the quantitative value of impurities of the silicon single crystal determined by the method according to claim 1.
An impurity analysis method characterized by calculating and quantifying an impurity concentration C ₂ (x) at a position x of a straight body portion of a silicon single crystal rod based on the following formula. C ₂ (x) = C ₀ [1-k (1-k) x · e -k (w + x) / w / w + {2k-2 + (1-k) e -k} e -kx / w ] (X) 8. The method according to claim 1, wherein a quantitative value of impurities in the silicon single crystal determined by the method according to claim 1 is used.
An impurity analysis method characterized by calculating and quantifying an impurity concentration C ₁ (x) at a position x of a straight body portion of an intermediate silicon polycrystalline rod based on I). 9. The impurity concentration at the position x of the straight body portion of the silicon single crystal rod based on the equation (X), using the quantitative value of the impurity of the intermediate silicon polycrystal determined by the method according to claim 1. An impurity analysis method, wherein C ₂ (x) is calculated and predicted.