Method for modeling the wafer shape and method for manufacturing a wafer

Abstract

A method for modeling a wafer profile by a function is provided, in which the function is used to calculate a displacement z in a thickness direction of a wafer and is a sum of multiple functions including a first function g(r), a second function Ar×h(Nθ), and a third function Br×i(M(θ-φ)). The first function g(r) indicates a first-order or higher-order polynomial with a distance r from the center of the wafer as a variable. The second function Ar×h(Nθ) indicates multiplying a sine or cosine function h(Nθ) by a first angle θ with respect to a predetermined position in a circumferential direction of the wafer as a variable and an integer N as a constant with a coefficient A and the distance r.The third function Br×i(M(θ-φ)) indicates multiplying a sine or cosine function i(M(θ-φ)) with the first angle θ as a variable, a second angle φ with respect to the predetermined position as a constant, and an integer M as a constant with a coefficient B and the distance r.

Description

TECHNICAL FIELD

[0001] The present invention relates to a method for modeling a wafer profile and a method for manufacturing a wafer. State of the art

[0002] A semiconductor wafer typically requires high flatness. Wafer flatness is typically determined by measuring a wafer profile using a capacitive or optical interferometric profiling device and evaluating the measurement with parameters such as warp and nanotopography.

[0003] As a technique for evaluating wafer flatness, Patent Literature 1 discloses a method for accurately evaluating nanotopography by excluding processing stress at a periphery as a disturbance in measuring a wafer profile using a capacitive profile measuring device.

[0004] In addition, Patent Literature 2 discloses a method for generating data of a surface profile of the entire wafer by expressing each profile along a radial direction of the wafer by a polynomial with a position r in the radial direction of the wafer as a variable and repeating the expression for every predetermined angle θ. CITATION LISTPATENT LITERATURE(S) Patent Literature 1: Japanese Patent No. 5862492 Patent Literature 2: Japanese Patent No. 6899080 SUMMARY OF THE INVENTION TASK(S) TO BE SOLVED BY THE INVENTION

[0005] However, the method described in Patent Literature 1 does not allow the evaluation of waviness at the periphery of the wafer because the processing stress at the periphery of the wafer is excluded to perform the evaluation by nanotopography.

[0006] However, the method described in Patent Literature 2, which can determine a profile in the radial direction of the wafer, cannot determine circumferential waviness.

[0007] An object of the invention is to provide a method for modeling a profile of a wafer, the method capable of replicating the circumferential waviness over the entire wafer and accurately evaluating the waviness of the wafer, and a method for manufacturing the wafer. MEANS FOR SOLVING THE TASK(S)

[0008] One aspect of the invention provides a method for modeling a wafer profile using a function, the function being used to calculate a displacement z in a thickness direction of a wafer, the function indicating a sum of a plurality of functions, the plurality of functions including: a first function g(r) being a first-order or higher-order polynomial with a distance r from a center of the wafer as a variable; a second function Ar × h(Nθ) indicating multiplying a sine or cosine function h(Nθ) by a first angle θ with respect to a predetermined position in a circumferential direction of the wafer as a variable and an integer N as a constant with a coefficient A and the distance r;and a third function Br × i(M(θ - φ)) which indicates the multiplication of a sine or cosine function i(M(θ - φ)) with the first angle θ as a variable, a second angle φ with respect to the predetermined position as a constant and an integer M as a constant with a coefficient B and the distance r.;

[0009] In the method for modeling a wafer profile according to the above aspect, the second function may be represented by A1r × h4(N1θ) + A2r × h2(N2θ) +... + A n r × h n (Nnθ) (n is an integer of 1 or more), and the third function can be represented by B1r × i1(M1(θ - φ)) + B2r × i2(M2(θ - φ)) +... + B m r × i m (M m (θ - φ)) (m is an integer of 1 or more).

[0010] In the wafer profile modeling method according to the above aspect, the predetermined position may be a reference position for indicating a crystal orientation, and the second angle φ may be formed between a cutting feed direction for cutting the wafer from an ingot and a straight line connecting the reference position to a center of the wafer.

[0011] In the method for modeling a wafer profile according to the above aspect, the first function may be a third-order polynomial, the second function may be Ar × sin4θ, and the third function may be B1r × cos2(E) - φ) + B2r × cos3(θ - φ).

[0012] According to another aspect of the invention, a method for manufacturing a wafer comprises: cutting the wafer by cutting an ingot; grinding and polishing both surfaces of the wafer; modeling a profile of the wafer using the modeling method according to the above aspect to provide a model; and evaluating the wafer based on the model.

[0013] In the method for manufacturing a wafer according to the further aspect of the invention, the evaluating is performed on a basis of a size of each of the coefficients of the function obtained by the modeling method. BRIEF DESCRIPTION OF THE DRAWINGS Fig. 1 schematically illustrates a positional relationship between a wafer and a wire saw according to an exemplary embodiment of the invention. Fig. Figure 2 is a map illustrating the wafer waviness to be modeled by a first function. Fig. Figure 3A is a map illustrating the wafer waviness to be modeled by a second function. Fig. Figure 3B is another map illustrating the wafer waviness to be modeled by the second function. Fig. 4 is a graph used in a calculation procedure of coefficients of the first function. Fig. Figure 5 is a diagram illustrating measurement points for a displacement z when calculating coefficients of the second and third functions. Fig. 6 is a graph used in a calculation method of the coefficients of the second and third functions. Fig. 7 is a flowchart illustrating a wafer manufacturing method according to an exemplary embodiment of the invention. Fig. Figure 8A is a map generated in Example 1. Fig. Figure 8B is another map generated in Example 1. Fig. 9A is a map generated in Example 2. Fig. Figure 9B is another map generated in Example 2. DESCRIPTION OF THE EMBODIMENT(S)

[0014] An exemplary embodiment of the invention is described below with reference to the accompanying drawings.

[0015] A method for modeling a wafer profile according to the exemplary embodiment of the invention is a method of replicating, as a model using a function (numerical formula), a profile of each wafer cut from a monocrystalline ingot manufactured by the Choklarsky method (CZ method) or other methods.

[0016] The model is generated by the function for calculating a displacement in a thickness direction of the wafer with a distance from the center of the wafer, an angle with respect to a predetermined position in a circumferential direction of the wafer, and the like as variables.

[0017] The method for manufacturing a wafer is characterized by comprising: modeling the wafer profile for a wafer that has been ground to provide a model; and evaluating the wafer using the model.

[0018] First, a method for modeling a wafer profile is described. A wafer with a diameter of 300 mm is described, but this is not limited to this.

[0019] The wafer profile modeling method according to the exemplary embodiment is a wafer profile modeling method using function (1) below. Function (1) is used to calculate a displacement z in a thickness direction of the wafer to be modeled. The displacement z in the thickness direction of the wafer can be expressed by the following function (1), where a distance r from the center of the wafer and a first angle θ with respect to a notch position are variables. z=f(r,θ)=ar3+br2+cr+d+ Ar×sin4θ+ B1r×cos2(θ−φ)+B2r×cos3(θ−φ)

[0020] That is, the function (1) gives a sum of several functions that have a first function (ar 3 + br 2 + cr + d), a second function (Ar × sin4θ) and a third function (B1r × cos2(θ - φ) + B2r × cos3(θ - φ)).

[0021] Here, the displacement in the thickness direction of the wafer represents the minimum distance between any point on a wafer's thickness center plane and a best-fitting surface of the thickness center plane. The wafer's thickness center plane means a plane formed by a set of points located at the center of thickness at any points on the wafer, where a width in a vertical direction of the wafer placed on a horizontal surface in a natural state is defined as the thickness. The natural state means a state where no external force is applied to adsorb the wafer to a horizontal surface. The best-fitting surface of the thickness center plane means the least-squares plane to the thickness center plane.

[0022] As in Fig. As illustrated in Figure 1, in the exemplary embodiment, a position where a notch Nt is formed to indicate a crystal orientation (notch position, reference position) is defined as the reference for the angle θ. Here, the angle θ is positive in the counterclockwise direction from the notch position. The reference for the angle θ is not limited to the notch position, but may be a predetermined position conventionally defined for wafers, indicating the reference in the circumferential direction, for example, may be a position where an orientation plane is formed.

[0023] The reference for the angle θ is not limited to the notch position Nt or the orientation plane position, but can be changed as needed based on a mechanism for forming wafer waviness or the like.

[0024] φ (second angle) is an angle formed between a cutting feed direction FD for cutting wafers from a monocrystalline ingot using a wire saw and a straight line L connecting a notch position to the center of each wafer W. The cutting feed direction FD by the wire saw is a direction orthogonal to an extension direction of a wire Wi of the wire saw.

[0025] In function (1) “ar 3 + br 2 + cr + d" (referred to as the first function) is a polynomial that approximates a warpage of the wafer. a, b, c, and d are coefficients calculated by a calculation method described below and are parameters that determine a warpage of a model obtained mainly by the function (1).

[0026] The first function, which is a third-order polynomial in the exemplary embodiment, is not limited to this, but may be changed based on the accuracy required for a model, etc. For example, the first function may be a first-order polynomial when wafer warpage is assumed to be simple, or a fourth-order or higher polynomial when wafer warpage is assumed to be complex.

[0027] In the function (1), “Ar × sin4θ” (referred to as the second function) is a sine function that approximates the profile of the wafer W which is corrugated four times in the circumferential direction (sinusoidal profile oscillating four times in a circumference, or cross-profile waviness appearing on the wafer), as shown in Fig. 2. "A" in the second function is a coefficient calculated by the calculation method described below, and is a parameter that determines a size of circumferential waviness of a model obtained mainly by the function.

[0028] The inventors found that grinding wafers produced a ripple, as in Fig. 2, and incorporated the second function into function (1) to reflect this waviness in the model. Examples of grinding processes that cause such waviness include resin-applied grinding (see, for example, JP 2011-249652 A) and single-sided grinding.

[0029] The second function is "Ar × sin4θ" in the exemplary embodiment. However, the ripple can be approximated by using a cosine function (cos), without limitation to a sine function (sin), as the trigonometric functions, because the cosine function (cos) is the same as the sine function (sin) in terms of periodic changes. In the exemplary embodiment, the number of ripples is four. However, the number of ripples can be changed depending on the profile of the generated ripple.

[0030] In other words, it is sufficient that the second function is the function Ar × h(Nθ), which represents the multiplication of a sine or cosine function h(Nθ) with the first angle θ as a variable and the integer N as a constant with the coefficient A and the distance r.

[0031] The number of trigonometric functions can be increased to model the wafer profile based on further insights. In particular, the second function can be A1r×h1(N1θ)+A2r×h2(N2θ)+...+Anr×hn(Nnθ) represented (n is an integer of 1 or more).

[0032] In the function (1) “B1r × cos2(E) - φ) + B2r × cos3(θ - <p)“ (als die dritte Funktion bezeichnet) eine Kosinusfunktion, die das Profil des Wafers approximiert, der zweimal in der Umfangsrichtung gewellt wird (sinusförmiges Profil, das zweimal in einem Umfang oszilliert), und das Profil des Wafers, der dreimal gewellt wird (sinusförmiges Profil, das dreimal in einem Umfang oszilliert), wie in Fig. 3A and Fig. 3B illustrates.

[0033] “B1 and B2” in the third function are coefficients calculated by the calculation method described below, and, similar to “A” in the second function, are parameters that determine a size of waviness of a wafer model obtained mainly by the function.

[0034] The inventors found that the waviness, as described above, was generated depending on the cutting feed direction FD of the wire saw when a monocrystalline ingot was cut using the wire saw to obtain wafers, and incorporated the third function into the function (1) to reflect this waviness in the wafer model.

[0035] The third function is "B1r × cos2(θ - φ) + B2r × cos3(θ - φ)" in the exemplary embodiment. However, the ripple can be approximated using a sine function (sin), without being limited to a cosine function, as the trigonometric functions. In the exemplary embodiment, the number of ripples is two or three. However, the number of ripples can be changed depending on the profile of the generated ripple. The number of trigonometric functions in the third function can be one.

[0036] In other words, it is sufficient that the third function is the function Br × i(M(θ - φ)), which represents the multiplication of a sine or cosine function i(M(θ - φ)) with the first angle θ as a variable and the second angle φ and an integer M each as a constant with a coefficient B and the distance r.

[0037] Similar to the second function, the number of trigonometric functions can be increased to model the wafer profile based on further insights. In particular, the third function can be B1r×i1(M1(θ−φ))+B2r×i2(M2(θ−φ))+...+Bmr×im(Mm(θ−φ)) (m is an integer of 1 or more). Calculation method of coefficients of the first function (third-order polynomial)

[0038] Next, a calculation method of coefficients (a, b, c, d) in a polynomial is described. (1) Displacements z r are measured at multiple points at each distance r from the center of a wafer and are averaged in a circumferential direction.

[0039] Considering that the wafer is a modeling target, the displacements z r measured at several points at each distance r from the center of the wafer. The displacements z rof the wafer, which are values of displacements measured by a measuring device, can be measured, for example, by a capacitive profile measuring device. The plurality of points on the circumference are preferably set at an equal distance and can, for example, be measured every one degree at 360 points. An average value of the displacements z r The displacement zavg measured at the multiple points is defined as a displacement zavg at the distance r from the center. Then, the displacement zavg is calculated for each distance r from the center. The distance r to be adjusted is preferably as short as possible. In the exemplary embodiment, the distance r is set from r = 0 (center of the wafer) to r = 146 (near a periphery of the wafer) in increments of 1 mm.

[0040] (2) An approximate expression for the displacements zavg with respect to the distances r from the center is obtained.

[0041] The distances r and the displacements zavg are plotted on a graph as in Fig. 4, plotted to obtain an approximate expression of a third-order polynomial. The approximate expression can be obtained using the least squares method. The coefficients (a, b, c, d) of the obtained approximate expression are the coefficients of the first function.

[0042] As described above, the approximate expression is not limited to the third-order polynomial. Calculation method of coefficients of the second and third functions

[0043] Next, a calculation method of coefficients (A, B1, B2) of the second and third functions is described.

[0044] (1) The distance r from the center of the wafer is determined at an arbitrary value.

[0045] The distance r from the center of the wafer is determined at an arbitrary value that provides a position for measuring the displacements z r in a thickness direction of the wafer. In the exemplary embodiment, the distance r is set to 146 mm. The distance r from the center of the wafer can be any value. However, since the wafer has greater waviness at the periphery, the distance r is preferably also a large value.

[0046] (2) The displacements z r in the thickness direction of the wafer are measured.

[0047] At the distance r, the displacements z r measured in the thickness direction of the wafer at any angle in the circumferential direction (actually measured). The arbitrary angle can be, for example, 22.5 degrees, as in Fig. 5. In this case, the number of measuring points is 16 at equal distances in the circumferential direction. The displacements z r are measured counterclockwise from the position θ = 0 degrees.

[0048] (3) An approximate expression of the displacements z r in terms of values obtained by multiplying the distance r by a trigonometric function is obtained.

[0049] The values obtained by multiplying the distance r by a trigonometric function (sin4θ, cos2(θ - φ) or cos3(θ - φ)) and the displacements z r are plotted on a graph as in Fig. 6, to obtain a first-order approximate expression.

[0050] As described above, φ is an angle determined depending on the cutting feed direction FD and can be, for example, -45 degrees.

[0051] For example, to determine the coefficient A of the second function (Ar × sin4θ) the horizontal axis indicates r × sin4θ and the vertical axis indicates the displacement z r For example, to determine the coefficient A of the second function (Ar × sin4 θ) and measure the displacement z r in the wafer thickness direction every 22.5 degrees, three different values of r × sin4 θ (horizontal axis values) are obtained (0 (in Fig. 5 marked with ∈ (triangle)), r (in Fig. 5 marked with ○ (circle)) and -r (in Fig. 5 marked with ∈ (square)). Depending on the trigonometric function to be determined and any angle described above, the number of different values of r × the trigonometric function obtained and the number of points of displacement plotted at each value on the horizontal axis will vary.

[0052] It should be noted that an approximate expression can be obtained using the least squares method. The coefficients of the obtained approximate expression are the coefficients (A, B1, B2) of the second and third functions.

[0053] Through the above steps, the coefficients a, b, c, d, A, B1, and B2 are calculated to obtain function (1). An operator can draw a model of the wafer by inputting function (1) into a computer or the like to determine a profile (warpage, waviness) of the wafer.

[0054] The wafer profile can also be determined from the obtained coefficients without drawing a model. For example, the operator can judge that the waviness at the periphery of the wafer is larger when the coefficients A, B1, and B2 are larger. Method for producing a wafer

[0055] Next, a method for manufacturing a wafer using the above wafer profile modeling method will be described.

[0056] As in Fig. 7, the method for manufacturing a wafer includes a cutting step S1, a grinding step S2, a modeling step S3, an evaluation step S4, an etching step S5, and a mirror polishing step S6.

[0057] In the cutting step S1, a monocrystalline ingot (or an ingot obtained by cutting a monocrystalline ingot) is cut using a cutting device such as a wire saw to cut out a plurality of wafers. A plurality of wafers, each having a thickness of, for example, about 1 mm, are thus obtained. A cutting feed direction FD by a wire saw can be determined, for example, by referring to a notch position.

[0058] It should be noted that a lapping step may be performed between the cutting step S1 and the grinding step S2. A chamfering step of an outer edge of the wafer, which is not specifically described, may be performed including the primary chamfering after the cutting step S1 and the secondary chamfering after the grinding step S2, wherein the secondary chamfering is larger in chamfering amount than the primary chamfering.

[0059] In grinding step S2, both sides of the wafer are ground using a grinding device. Examples of grinding methods include resin-applied grinding, single-side grinding, and simultaneous double-side grinding.

[0060] In the modeling step S3, the wafer that has undergone the grinding step S2 is modeled using a wafer profile modeling method. Specifically, the coefficients a, b, c, d, A, B1, and B2 of the function (1) are determined by measuring the displacement z. r in a thickness direction of the wafer at each distance r using a capacitive profiler or the like. The obtained function (1) is used to draw a model of the wafer using a computer or the like.

[0061] In evaluation step S4, the operator evaluates the wafer by referring to the wafer model obtained in modeling step S3. The operator can understand the wafer's warpage and waviness by referring to the model. Based on the wafer's warpage and waviness, the operator can re-grind the wafer as needed and assess whether the wafer is satisfactory or unsatisfactory.

[0062] The etching step S5 is a step of performing chemical etching to remove processing damage caused on a surface of the wafer in the previous steps.

[0063] The mirror polishing step S6 is a step of polishing both sides of the wafer using a polishing jig.

[0064] In the etching step S5 and the mirror polishing step S6, the etching process and the polishing process can be adjusted based on the results of the evaluation step S4.

[0065] According to the above wafer profile modeling method, since the trigonometric functions are included in the function (1) and the circumferential waviness of the entire wafer can be replicated due to these trigonometric functions, the waviness of the wafer can be accurately evaluated.

[0066] In particular, the circumferential waviness assumed to be caused by the wire saw and the waviness assumed to be caused by the grinding step can be replicated independently.

[0067] The modeling accuracy can also be increased by increasing the order of the first function and the number of trigonometric functions.

[0068] According to the above wafer manufacturing method, it is possible to evaluate the wafer based on the warpage and waviness of the wafer before the etching step S5 and the mirror polishing step S6 are performed. Example 1

[0069] Next, the invention will be described in more detail with reference to examples. In Example 1, a map of a wafer was generated using only measured values and a map was generated using a model obtained by the wafer profile modeling method of the invention, and the two maps were compared.

[0070] First, a profile of a wafer was measured using a capacitive profile measuring device, and a Fig. The map illustrated in Figure 8A was generated based on the measured values.

[0071] In addition, a function was obtained by the wafer profile modeling method of the invention, and one as shown in Fig. The map illustrated in Figure 8B was generated using a model obtained by this function. Table 1 shows the coefficients of the function. Table 1 a µm / mm 2 ) b (µm / mm 2 ) c (µm / mm) d (µm) A (µm / mm) B1 (µm / mm) B2 (µm / mm) ϕ(deg) -1,02 × 10 -7 -5,27 × 10 -4 -1,35 × 10 -2 7,01 1,55 × 10 -2 5,91 × 10 -3 5,99 × 10 -3 -44,3

[0072] As can be seen from a comparison between the Fig. 8A and Fig. As can be seen from Figure 8B, the map generated using only the measured values and the map generated using the model agree, particularly with regard to the waviness at their peripheries. Example 2

[0073] Next, Example 2 of the invention will be described. In Example 2, two wafers subjected to the grinding step were modeled using the wafer profile modeling method.

[0074] The Fig. 9A and Fig. Figure 9B shows maps generated based on actual measurements of the two prepared wafers W1 and W2. From the maps, it can be seen that the profiles of the respective wafers differ significantly from each other. Both of the wafers used are after they have undergone the grinding step (resin-applied grinding).

[0075] Table 2 shows coefficients for each of the two wafers, where the coefficients are calculated using the wafer profile modeling method. Table 2 a (µm / mm 3 ) b (µm / mm 2 ) c (µm / mm) d(µm) A (µm / mm) B1 (µm / mm) B2 (µm / mm) ϕ (deg) Wafer 1 -9,71 × 10 -7 -1,60 × 10 -5 -3,15 × 10 -2 4,42 1,67 × 10 -2 2,76 × 10 -3 4,84 × 0 -3 -45 Wafer 2 7,03 × 10 -3 2,04 × 10 -5 -1,91 × 10 -2 1,54 1,68 × 10 -2 6,84 × 10 -3 4,35 × 0 -3 -45

[0076] The coefficient A of wafer W1 is approximately the same as the coefficient A of wafer W2, indicating that the degrees of waviness presumably caused by the grinding step are similar. Furthermore, since the coefficients a, b, c, and d of the polynomial differ significantly between wafer W1 and wafer W2, it is understood that a significant difference between the two maps is influenced by the wafer warpage. Explanation of the codes

[0077] According to the present invention, there is provided a method for manufacturing a wafer, the method comprising the following steps: L...straight line connecting a notch position and a center of a wafer, Nt...notch, r...distance from a center of a wafer, W...wafer, z...displacement in a thickness direction of a wafer, θ...first angle with respect to a notch position, φ...second angle defined by a cutting feed direction, S1...cutting step, S2...grinding step, S3...modeling step, S4...evaluation step, S5...etching step, S6...mirror polishing step QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited patent literature

[0000] JP 6899080

[0004]

Claims

[1] A method for modeling a wafer profile using a function, the function being used to calculate a displacement z in a thickness direction of a wafer, the function specifying a sum of a plurality of functions, the plurality of functions including: a first function g(r) which is a first or higher order polynomial with a distance r from a center of the wafer as a variable; a second function Ar × h(Nθ) which represents the multiplication of a sine or cosine function h(Nθ) with a first angle θ with respect to a predetermined position in a circumferential direction of the wafer as a variable and an integer N as a constant with a coefficient A and the distance r; and a third function Br × i(M(θ - φ)) which indicates the multiplication of a sine or cosine function i(M(θ - φ)) with the first angle θ as a variable, a second angle φ with respect to the predetermined position as a constant and an integer M as a constant with a coefficient B and the distance r. [2] A method for modeling a wafer profile according to claim 1, wherein the second function by A1r × h1(N1θ) + A2r × h2(N2θ) +... + A n r × h n (Nnθ) (n is an integer of 1 or more), and the third function by B1r × i1(M1(θ - φ)) + B2r × i2(M2(θ - φ)) +... + B m r × i m (M m (θ - φ)) (m is an integer of 1 or more). [3] A method for modeling a wafer profile according to claim 1 or 2, wherein the predetermined position is a reference position for indicating a crystal orientation, and the second angle φ is formed between a cutting feed direction for cutting the wafer from an ingot and a straight line connecting the reference position to a center of the wafer. [4] A method for modeling a wafer profile according to claim 3, wherein the first function ar 3 + br 2 + cr + d is, the second function is Ar × sin4θ, and the third function is B1r × cos2(θ - φ) + B2r × cos3(θ - φ). [5] A method of manufacturing a wafer, comprising: Cutting the wafer by cutting an ingot; Grinding and polishing both surfaces of the wafer; Modeling a profile of the wafer using the modeling method of claim 1 to provide a model; and Evaluate the wafer based on the model. [6] A method of manufacturing a wafer according to claim 5, wherein the evaluating is performed on a basis of a magnitude of each of the coefficients of the function obtained by the modeling method.

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