Wafer shape modeling method and wafer manufacturing method

Abstract

To accurately evaluate undulation of a wafer by reproducing the undulation of the wafer in a circumferential direction over the entire wafer.SOLUTION: A modeling method for a wafer shape is provided, in the method for modeling the wafer shape with a function, the function is for calculating displacement (z) of a wafer in a thickness direction and is a sum of a plurality of functions including: a first function g(r) which is a primary or more polynomial defining a distance (r) from a center of the wafer as a variable; a second function Ar×h (Nθ) defining a first angle θ, which defines a predetermined position of the wafer in a circumferential direction as a reference, as a variable and resulting from multiplying, by a coefficient A and the distance (r), a sine function or a cosine function h (Nθ) defining an integer N as a constant; and a third function Br×i (M(θ-φ)) defining the first angle θ as a parameter and resulting from multiplying, by a coefficient B and the distance (r), a sine function or cosine function i (M(θ-φ)) defining a second angle φ, which defines the predetermined position as a reference, and an integer M as constants.SELECTED DRAWING: Figure 7

Description

[Technical Field]

[0001] The present invention relates to a method for modeling a wafer shape and a method for manufacturing a wafer. [Background technology]

[0002] Conventionally, semiconductor wafers have been required to have a high degree of flatness, which is generally evaluated by measuring the wafer shape using a capacitance or optical interference type shape measurement device and evaluating parameters such as warp and nanotopography.

[0003] As a technology for evaluating wafer flatness, Patent Document 1 discloses a method for accurately evaluating nanotopography by excluding processing distortions at the outer periphery as disturbances when measuring the wafer shape using a capacitance-type shape measuring device. Furthermore, Patent Document 2 discloses a method for converting the surface shape of the entire wafer into data by expressing the shape along the radial direction of the wafer using a polynomial with the position r in the radial direction of the wafer as a variable, and repeating this for every predetermined angle θ. [Prior art documents] [Patent documents]

[0004] [Patent Document 1] Patent No. 5862492 [Patent Document 2] Patent No. 6899080 Summary of the Invention [Problem to be solved by the invention]

[0005] However, in the method described in Patent Document 1, processing strain at the outer periphery of the wafer is excluded in order to perform evaluation using nanotopography, and therefore waviness at the outer periphery cannot be evaluated. Furthermore, the method described in Patent Document 2 can grasp the shape of the wafer in the radial direction, but cannot grasp the waviness in the circumferential direction.

[0006] An object of the present invention is to provide a wafer shape modeling method and a wafer manufacturing method that can reproduce circumferential waviness of a wafer over the entire wafer and accurately evaluate the wafer waviness. [Means for solving the problem]

[0007] The wafer shape modeling method of the present invention is a method for modeling the wafer shape using a function, wherein the function is a sum of a plurality of functions including a first function g(r), which is a polynomial of first order or higher and has a distance r from the center of the wafer as a variable, a second function Ar×h(Nθ), which is a sine or cosine function h(Nθ), which has a first angle θ based on a predetermined position in the circumferential direction of the wafer as a variable and an integer N as a constant, multiplied by a coefficient A and the distance r, and a third function Br×i(M(θ-φ)), which is a sine or cosine function i(M(θ-φ)), which has a second angle φ based on the predetermined position as a variable and an integer M as a constant, multiplied by a coefficient B and the distance r.

[0008] In the wafer shape modeling method, the second function is A1r×h1(N1θ)+A2r×h2(N2θ)+...+A n r×h n (N n θ) (n is an integer of 1 or more), and the third function is expressed as 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).

[0009] In the above-described wafer shape modeling method, the predetermined position may be a reference position for indicating a crystal orientation, and the second angle φ may be an angle between a cutting feed direction when cutting the wafer from an ingot and a straight line connecting the reference position and the center of the wafer.

[0010] In the wafer shape modeling method, the first function may be a cubic polynomial, the second function may be Ar×sin4θ, and the third function may be B1r×cos2(θ−φ) + B2r×cos3(θ−φ).

[0011] The wafer manufacturing method of the present invention is characterized by comprising a slicing step of obtaining the wafers by cutting an ingot; a grinding step of polishing both surfaces of the wafers; a modeling step of modeling the wafer shape using any of the wafer shape modeling methods described above; and an evaluation step of evaluating the wafers from the obtained model.

[0012] In the above-described wafer manufacturing method, the evaluation may be performed based on the magnitude of the coefficient of the function obtained by the wafer shape modeling method. [Brief explanation of the drawings]

[0013] [Figure 1] FIG. 2 is a schematic diagram illustrating the positional relationship between a wafer and a wire saw according to an embodiment of the present invention. [Figure 2] 1 is a map illustrating wafer waviness modeled by a first function. [Figure 3] 10 is a map illustrating wafer waviness modeled by a second function. [Figure 4] 10 is a graph used in a method for calculating coefficients of a first function. [Figure 5] FIG. 10 is a diagram illustrating measurement points of displacement z when calculating coefficients of the second and third functions. [Figure 6] 10 is a graph used in a method for calculating coefficients of the second and third functions. [Figure 7] 1 is a flowchart illustrating a method for manufacturing a wafer according to an embodiment of the present invention. [Figure 8] 1 is a map created in Example 1. [Figure 9] 10 is a map created in Example 2. DETAILED DESCRIPTION OF THE INVENTION

[0014] Hereinafter, an embodiment of the present invention will be described with reference to the drawings. The wafer shape modeling method according to an embodiment of the present invention is a method for reproducing the shape of a wafer sliced ​​from a single crystal ingot manufactured by the CZ method (Czochralski method) or the like as a model using a function (mathematical formula). The model is created using a function that calculates the displacement in the thickness direction of the wafer using variables such as the distance from the center of the wafer and the angle relative to a predetermined position in the circumferential direction of the wafer.

[0015] The wafer manufacturing method is characterized in that the shape of the wafer is modeled after the grinding process, and the wafer is evaluated using the model that is created.

[0016] First, a method for modeling the wafer shape will be explained. In the following, the wafer diameter will be assumed to be 300 mm, but this is not limitative. The wafer shape modeling method of this embodiment is a method of modeling the wafer shape using the following function (1). The function (1) is used to calculate the displacement z in the thickness direction of the modeled wafer. The displacement z in the thickness direction of the wafer can be expressed by the following function (1) in which the variables are the distance r from the center of the wafer and a first angle θ based on the notch position.

[0017] z=f(r,θ) =ar 3 +br 2 +cr+d + Ar×sin4θ + B1r×cos2(θ-φ) + B2r×cos3(θ-φ) ···(1)

[0018] That is, the function (1) is the 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)).

[0019] Here, the displacement in the thickness direction of the wafer represents the minimum distance from any point on the wafer's thickness midplane to the best-fit plane of the thickness midplane. The wafer's thickness midplane refers to a plane formed as a set of points located at the center of the thickness at any point on the wafer, where the thickness is the vertical width of the wafer when the wafer is placed on a horizontal surface in its natural state. The natural state refers to a state in which no external force is applied to adsorb the wafer to the horizontal surface. The best-fit plane of the thickness midplane refers to a least-squares plane relative to the thickness midplane. As shown in FIG. 1, in this embodiment, the reference for angle θ is the position (notch position, reference position) where a notch Nt is formed to indicate the crystal orientation. Here, the angle θ is positive in the counterclockwise direction from the notch position. The reference for angle θ is not limited to the notch position, but can be a predetermined position that is commonly determined for the wafer and indicates a reference in the circumferential direction, such as a position where an orientation flat is formed. Furthermore, the reference for the angle θ is not limited to the position of the notch Nt or the orientation flat, but may be changed as appropriate based on the mechanism of waviness formation in the wafer.

[0020] φ (second angle) is the angle between the cutting feed direction FD when slicing wafers from a single crystal ingot using a wire saw and the line L connecting the notch position and the center of the wafer W. Here, the angle φ is positive in the counterclockwise direction from the notch position. The cutting feed direction FD of the wire saw is perpendicular to the direction in which the wire Wi of the wire saw extends.

[0021] In function (1), "ar 3 +br 2 "(1) + cr + d" (called the first function) is a polynomial that approximates the warpage shape of the wafer. The first function is a cubic polynomial with one variable, r, which is the distance from the center of the wafer. a, b, c, and d are coefficients calculated by a calculation method that will be described later, and are parameters that mainly determine the warpage shape of the model obtained by function (1).

[0022] In this embodiment, the first function is a third-order polynomial, but is not limited to this and can be changed based on the accuracy required for the model, etc. For example, if the warpage of the wafer is considered to be simple, a first-order polynomial may be used, and if the warpage of the wafer is considered to occur in a complex manner, a fourth-order or higher polynomial may be used.

[0023] In function (1), "Ar×sin4θ" (referred to as the second function) is a sine function that approximates the shape of the wafer W undulating four times in the circumferential direction (the shape of a sine wave that oscillates four times per revolution, a cross-shaped undulation that appears on the wafer) as shown in Figure 2. "A" in the second function is a coefficient calculated by a calculation method described later, and is a parameter that mainly determines the magnitude of the circumferential undulation of the model obtained by the function.

[0024] The inventors discovered that grinding a wafer generates waviness as shown in Fig. 2, and incorporated a second function to reflect this waviness in the model. Grinding methods that generate such waviness include, for example, resin-attached grinding (see JP 2011-249652 A, etc.) and single-side polishing grinding.

[0025] The second function in this embodiment is "Ar×sin4θ", but since the trigonometric function is not limited to a sine function (sin), and is the same in that it changes periodically, a cosine function (cos) may also be used to approximate the shape of the undulations. Also, in this embodiment, the number of undulations is set to four, but the number of undulations may be changed based on the shape of the undulations that are generated. That is, the second function may be a function Ar×h(Nθ) obtained by multiplying a sine or cosine function h(Nθ) with the first angle θ as a variable and an integer N as a constant by a coefficient A and the distance r.

[0026] Additionally, the number of trigonometric functions may be increased to model the wafer shape based on further knowledge, i.e., the second function may be: A1r×h1(N1θ)+A2r×h2(N2θ)+…+A n r×h n (N n θ) (n is an integer greater than or equal to 1) It can be a function expressed as:

[0027] In function (1), "B1r × cos2(θ-φ) + B2r × cos3(θ-φ)" (called the third function) is a cosine function that approximates a two-times undulating shape in the circumferential direction of the wafer (a sine wave shape that oscillates twice per revolution) and a three-times undulating shape (a sine wave shape that oscillates three times per revolution), as shown in Figures 3(A) and 3(B).

[0028] "B1, B2" of the third function are coefficients calculated by a calculation method described later, and, like "A" of the second function, are parameters that mainly determine the magnitude of the waviness of the wafer model obtained by the function. The inventors discovered that when a single crystal ingot is sliced ​​using a wire saw to obtain wafers, the above-mentioned waviness occurs corresponding to the cutting feed direction FD of the wire saw, and incorporated a third function to reflect this waviness in the wafer model.

[0029] The third function in this embodiment is "B1r × cos2(θ-φ) + B2r × cos3(θ-φ)", but the shape of the undulations may be approximated using not only a cosine function (cos) but also a sine function (sin) as the trigonometric function. Also, in this embodiment, the number of undulations is set to 2 or 3, but the number of undulations may be changed based on the shape of the undulations to be generated. Also, the number of trigonometric functions in the third function may be one. That is, the third function may be a function Br×i(M(θ-φ)) obtained by multiplying a sine or cosine function i(M(θ-φ)) with the first angle θ as a variable and the second angle φ and integer M as constants by a coefficient B and a distance r.

[0030] Similarly to the second function, the number of trigonometric functions may be increased to model the wafer shape based on further knowledge. That is, the third function may be expressed as: B1r×i1(M1(θ-φ))+B2r×i2(M2(θ-φ))+…+B m r×i m (M m (θ-φ) (m is an integer greater than or equal to 1) It can be a function expressed as:

[0031] [Calculation method of the coefficients of the first function (cubic polynomial)] Next, a method for calculating the coefficients (a, b, c, d) of the polynomial will be described. (1) Displacement z at multiple points at each distance r from the center of the wafer r is measured and averaged in the circumferential direction. For the wafer to be modeled, displacement z is measured at multiple points on the circumference at a distance r from the center of the wafer. r Measure the wafer displacement z r is a displacement value measured by a measuring device, and can be measured, for example, by a capacitance-type shape measuring device. It is preferable to set multiple points on the circumference at equal intervals, and for example, measurements may be made at 360 points every 1°. The displacement z measured at multiple points r The average value of the displacement z at a distance r from the center avg Then, the displacement z at the distance r from the center avg The calculation is performed for multiple distances r. The finer the distance r is set, the more preferable it is, and in this embodiment, it is set in 1 mm increments from r=0 (center of the wafer) to r=146 (near the outer periphery of the wafer).

[0032] (2) Displacement z relative to distance r from the center avg Find an approximate formula for As shown in Figure 4, the distance r and displacement z are plotted on the graph. avg Plot the equations and find the approximate equation of a third-order polynomial. The approximate equation can also be found using the least squares method. The coefficients (a, b, c, d) of the obtained approximate equation become the coefficients of the first function. As mentioned above, the approximation formula is not limited to a third-order polynomial.

[0033] [Calculation method of coefficients of the second and third functions] Next, a method for calculating the coefficients (A, B1, B2) of the second and third functions will be described. (1) Determine an arbitrary value for the distance r from the center of the wafer. Displacement z in the thickness direction of the wafer r An arbitrary value is determined for the distance r from the center of the wafer, which is the position where the waviness is measured. In this embodiment, the distance r is set to 146 mm. The distance r from the center of the wafer can be any value, but since the waviness of the wafer is larger on the outer periphery, it is preferable to set the distance r to a large value. (2) Displacement z in the thickness direction of the wafer r Measure. At a distance r, the displacement z in the thickness direction of the wafer for each angle in the circumferential direction r As shown in FIG. 5, the arbitrary angle can be set to, for example, 22.5°. In this case, the number of measurement points is 16, evenly spaced in the circumferential direction. Displacement z r is measured counterclockwise starting from the position θ=0°.

[0034] (3) Distance r multiplied by a trigonometric function and displacement z r Find an approximate formula for As shown in Figure 6*, the distance r multiplied by a trigonometric function (sin4θ, cos2(θ-φ), or cos3(θ-φ)) and the displacement z are plotted on a graph. r Plot and find a linear approximation. As mentioned above, φ is the angle determined by the cutting feed direction FD, which may be, for example, −45°.

[0035] For example, when calculating the coefficient A of the second function (Ar×sin4θ), the horizontal axis is r×sin4θ and the vertical axis is the displacement z r For example, when calculating the coefficient A of the second function (Ar×sin4θ), the displacement z in the thickness direction of the wafer is calculated every 22.5°. r When measuring, there are three values ​​of r×sin4θ (values ​​on the horizontal axis): 0 (△ mark in Figure 5), r (○ mark in Figure 5), and -r (□ mark in Figure 5). However, the number of values ​​of r×trigonometric function and the number of displacements plotted at each horizontal axis value will change depending on the trigonometric function being sought and the arbitrary angle mentioned above. The approximate equation may be obtained using the least squares method. The coefficients of the obtained approximate equation become the coefficients (A, B1, B2) of the second function and the third function.

[0036] Through the above process, the coefficients a, b, c, d, A, B1, and B2 are calculated, and function (1) is obtained. The operator can input function (1) into a computer to draw a model of the wafer and grasp the shape of the wafer (warpage, waviness). Furthermore, the shape of the wafer can be grasped from the calculated coefficients without drawing a model. For example, it can be determined that the larger the coefficients A, B1, and B2, the larger the waviness of the wafer's outer periphery.

[0037] [Wafer Manufacturing Method] Next, a method for manufacturing a wafer to which the above-described wafer shape modeling method is applied will be described. As shown in FIG. 7, the wafer manufacturing method includes a slicing step S1, a grinding step S2, a modeling step S3, an evaluation step S4, an etching step S5, and a mirror polishing step S6.

[0038] The slicing step S1 is a step in which a single crystal ingot (or a block obtained by slicing the single crystal ingot) is cut into multiple wafers using a cutting means such as a wire saw. This results in multiple wafers with a thickness of, for example, about 1 mm. The cutting feed direction FD of the wire saw can be determined, for example, based on the notch position.

[0039] A rough polishing (lapping) step may be performed between the slicing step S1 and the grinding step S2. Although a step of chamfering the outer edge of the wafer is not specifically shown, the chamfering step may involve performing primary chamfering after the slicing step S1 and performing secondary chamfering with a larger chamfering amount after the grinding step S2.

[0040] The grinding step S2 is a step of grinding both sides of the wafer using a grinding device. Grinding methods include resin-attached grinding, single-side polishing grinding, and double-side simultaneous grinding.

[0041] In the modeling step S3, a model of the wafer that has been subjected to the grinding step S2 is created using a wafer shape modeling method. Specifically, a capacitance-type shape measuring device or the like is used to measure the displacement z in the thickness direction of the wafer. r The coefficients a, b, c, d, A, B1, and B2 of function (1) are determined by measuring at each distance r, and the obtained function (1) is used to draw a model of the wafer using a computer or the like.

[0042] In the evaluation step S4, the operator evaluates the wafer by referring to the wafer model obtained in the modeling step S3. By referring to the model, the operator can understand the warpage and waviness of the wafer. Based on the warpage and waviness of the wafer, the operator can perform re-grinding as necessary or determine whether the wafer is good or bad.

[0043] The etching step S5 is a step of performing chemical etching to remove damage caused by machining on the wafer surface in the previous step. The mirror polishing step S6 is a step of polishing both surfaces of the wafer using a polishing machine. In the etching step S5 and the mirror polishing step S6, the etching method and the polishing method can be tuned based on the results of the evaluation step S4.

[0044] According to the above-described wafer shape modeling method, function (1) includes a trigonometric function, which can reproduce circumferential waviness over the entire wafer, thereby enabling wafer waviness to be evaluated with high accuracy. In particular, it is possible to independently reproduce circumferential waviness that is expected to occur in a wire saw and waviness that is expected to occur in a grinding process. Moreover, by increasing the degree of the first function or the number of trigonometric functions, the accuracy of the model can be improved.

[0045] According to the above-described wafer manufacturing method, the warpage and waviness of the wafer can be grasped and the wafer can be evaluated at a stage before the etching step S5 and the mirror polishing step S6 are performed.

[0046] Example 1 Next, the present invention will be described in more detail with reference to examples. In Example 1, a wafer map was created using only measured values, while a map was created using a model obtained using the wafer shape modeling method of the present invention, and the two maps were compared.

[0047] First, the shape of the wafer was measured using a capacitance type shape measuring device, and a map as shown in FIG. 8(A) was created based on the measured values. Furthermore, a function was determined using the wafer shape modeling method of the present invention, and a map such as that shown in Fig. 8(B) was created using a model created using this function. Table 1 shows the coefficients of the function.

[0048] [Table 1]

[0049] As can be seen by comparing Figures 8(A) and 8(B), the map created using only the measured values ​​and the map created using the model match well, especially in the waviness on the outer periphery.

[0050] Example 2 Next, a description will be given of Example 2 of the present invention. In Example 2, two wafers that had undergone a grinding process were modeled using a wafer shape modeling method. Figures 9(A) and (B) are maps based on the actual measurements of the two wafers W1 and W2 prepared, and show that the shapes are significantly different. Both are wafers after the grinding process (resin-coated grinding). Table 2 shows the coefficients calculated for two wafers using the wafer shape modeling method.

[0051] [Table 2]

[0052] Since the coefficient A of wafers W1 and W2 is approximately the same, it can be seen that the strength of the waviness presumably caused by the grinding process is about the same. In addition, since the coefficients a, b, c, and d of the polynomials are significantly different, it can be seen that the large differences in the maps are due to the influence of the warpage shape of the wafers. [Explanation of symbols]

[0053] L...line connecting the notch position and the center of the wafer, N...notch, r...distance from the center of the wafer, W...wafer, z...displacement in the thickness direction of the wafer, θ...first angle based on the notch position, φ...second angle in the cutting feed direction, S1...slicing process, S2...grinding process, S3...modeling process, S4...evaluation process, S5...etching process, S6...mirror polishing process.

Claims

1. 1. A method for modeling a wafer shape by a function, comprising: The function calculates a displacement z in the thickness direction of the wafer, a first function g(r) that is a polynomial of first or higher order with a distance r from the center of the wafer as a variable; a second function Ar×h(Nθ) obtained by multiplying a sine function or cosine function h(Nθ) having a first angle θ relative to a predetermined position in the circumferential direction of the wafer as a variable and an integer N as a constant by a coefficient A and the distance r; A method for modeling a wafer shape that is the sum of a plurality of functions including a third function Br×i(M(θ-φ)) obtained by multiplying a sine function or cosine function i(M(θ-φ)) that has the first angle θ as a variable, a second angle φ relative to the predetermined position as a reference, and an integer M as constants, by a coefficient B and the distance r.

2. 2. The method for modeling a wafer shape according to claim 1, The second function is A １ r x h １ (N １ θ) + A ２ r x h ２ (N ２ θ) + ... + A ｎ r x h ｎ (N ｎ θ) (n is an integer of 1 or more), The third function is B １ r x i １ (M １ (θ-φ)) + B ２ r x i ２ (M ２ (θ-φ)) + ... + B ｍ r x i ｍ (M ｍ A method for modeling a wafer shape as a function expressed by (θ-φ) (m is an integer of 1 or more).

3. 3. The method for modeling a wafer shape according to claim 1 or 2, the predetermined position is a reference position for indicating a crystal orientation, A method for modeling a wafer shape, wherein the second angle φ is an angle between a cutting feed direction when cutting the wafer from an ingot and a line connecting the reference position and the center of the wafer.

4. 4. The method for modeling a wafer shape according to claim 3, The first function is ar ３ +br ２ +cr+d, the second function is Ar×sin4θ, The third function is B １ r×cos2(θ−φ) + B ２ A method for modeling the wafer shape, which is r×cos3(θ−φ).

5. a slicing step of cutting the ingot to obtain the wafers; a grinding step of polishing both surfaces of the wafer; a modeling step of modeling a wafer shape using the wafer shape modeling method according to claim 1; and an evaluation step of evaluating the wafer from the obtained model.

6. 6. The method for manufacturing a wafer according to claim 5, A method for manufacturing a wafer, wherein the evaluation is performed based on the magnitude of the coefficient of the function obtained by the wafer shape modeling method.

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