Method for Measuring Resistivity of Semiconductor Wafer and Computer Program

Abstract

The resistivity measurement method of the wafer of the present invention is a method of calculating the resistivity by generating a point group or mesh reflecting the shape of the edge portion of the wafer based on the overall shape of the wafer, the shape of the edge portion, the parameters of the probe, etc., and correcting the measured value (resistivity) of the wafer measured by the four-probe method with the correction coefficient value calculated by numerical simulation. The computer program of the present invention is a computer program capable of executing the procedure of the resistivity measurement method.

Description

Technical Field

[0001] The present invention relates to a method for measuring the resistivity of a semiconductor wafer and a computer program for resistivity measurement.

Background Art

[0002] As one of the standard methods for measuring the resistivity of a semiconductor wafer, there is a four-probe method defined in Japanese Industrial Standards (hereinafter referred to as "JIS standard") (Non-Patent Document 1). There is a four-probe method defined in the JIS standard as a method for measuring the resistivity of a conductive plastic (Non-Patent Document 2).

[0003] The four-probe method is widely known as a method for measuring electrical properties such as electrical resistance and electrical conductivity. In this measurement method, as shown in FIG. 5, four probes (electrodes) A to D are brought into contact with the surface of a semiconductor wafer (sample) 1 for which resistivity is to be measured, a fixed current I is passed through the two inner probes B and C, and the potential difference V on the sample surface is measured with the two outer probes A and D. At this time, the resistivity ρ is calculated by Equation 2 with reference to the solution (potential distribution) u of the mathematical model when the current I between the probes B and C shown in Equation 1 is set to 1. In particular, the solution of this equation can be expressed by a mathematical formula only for wafers having regular shapes such as rectangular parallelepipeds and cylinders.

[0004] [Equation 1] TIFF0007717419000001.tif35166

[0005] [Equation 2] TIFF0007717419000002.tif23150

[0006] In the above Equation 1, Ω represents the region occupied by the sample, and δ(x) represents the Dirac delta function. In particular, F C appearing in Equation 2 is called a resistivity correction coefficient, which is a constant depending on the shape of the sample and the positions of the probes, and plays an important role in resistivity measurement.

[0007] In recent years, in the field of semiconductor wafers, the need for highly accurate and reliable resistivity measurement has been increasing. In order to improve the yield rate of semiconductor wafers, highly accurate resistivity measurement near the outer peripheral edge of a sample (hereinafter referred to as the "edge portion") is required.

[0008] The resistivity measurement methods based on the above-mentioned Formulas 1 and 2 defined in JIS use the specific formulas of the solutions in

[0004] . Since these methods assume samples with regular shapes such as rectangular parallelepipeds and cylinders, they are only suitable for measuring the resistivity of samples with regular shapes. On the other hand, a real semiconductor wafer is disk-shaped, and in addition to the orifice E (Fig. 6(a)) and notch F (Fig. 6(b)) in the edge portion, it has a complex shape such as a fillet or chamfer (not shown). The resistivity correction coefficient F calculated according to the conventional JIS C does not reflect the orifice E, notch F, and complex shape of the edge portion of a real semiconductor wafer, so the reliability near the edge portion of the semiconductor wafer is questioned.

[0009] Incidentally, since the resistivity correction coefficient calculated according to JIS assuming a disk shape does not correspond to the shape of the edge portion, the result of applying it to an actual wafer is as shown in Fig. 1. In Fig. 1, the smaller the distance d from the reference position, the closer it is to the edge.

Prior Art Documents

Non-Patent Documents

[0010]

Non-Patent Document 1

Non-Patent Document 2

Summary of the Invention

Problems to be Solved by the Invention

[0011] The present invention was developed to solve the problem that the current resistivity measurement method cannot cope with the shape of the edge portion, and for the conventional mathematical model that governs resistivity measurement, a numerical simulation that precisely reproduces the shape of the edge portion such as the orifice, notch, fillet, and chamfer of the semiconductor wafer is used. An object of the present invention is to provide a method for measuring the resistivity of a semiconductor wafer that can measure the resistivity of the semiconductor wafer, particularly the resistivity of the edge portion, with high accuracy and high reliability.

Means for Solving the Problems

[0012] In the method for measuring the resistivity of a semiconductor wafer according to the present invention, in the resistivity measurement using four probes, the potential difference U between probes B and C BC and the fixed current I between probes A and D AD The ratio U of BC / I AD The measured value obtained as (hereinafter referred to as "four-probe measurement value") is corrected by a resistivity correction coefficient obtained by a numerical calculation method (numerical simulation), and this resistivity correction coefficient is multiplied by the four-probe measurement value to obtain the resistivity. In particular, a large number of nodes (point groups) used for reproducing the shape of the wafer are provided on the semiconductor wafer (hereinafter simply referred to as "wafer"), and the nodes are arranged according to the shape and potential distribution characteristics of the wafer near the measurement point, or the wafer is divided into a large number of meshes (also referred to as division, discretization) by connecting these nodes to reproduce the wafer shape. Further, by reproducing the contour shape as a curve (including a straight line or a piecewise continuous straight line) or a curved surface (including a plane or a piecewise continuous plane) by arranging a part of the nodes at the edge portion of the wafer, the resistivity correction coefficient of the wafer is obtained.

[0013] The computer program of the present invention can execute point group or mesh generation so as to precisely represent the end shape of the wafer by the point group or mesh generation, the coarseness adjustment of the point group or mesh, and the nodes arranged for expressing the contour shape of the edge portion of the wafer, and can execute the calculation of the resistivity correction coefficient of the four-probe method by numerical simulation using the generated point group or mesh.

Advantages of the Invention

[0014] The resistivity correction coefficient used in the method for measuring the resistivity of a wafer according to the present invention is the resistivity correction coefficient of the four-probe method obtained by numerical simulation using a point cloud or mesh generated for the reproduction of the wafer shape. Since the shape of the edge portion such as the orifice, notch, fillet, chamfer, etc. of the wafer is precisely reflected, the resistivity of the wafer, particularly the resistivity of the edge portion of the wafer, can be measured with high precision, and highly reliable measurement is possible.

[0015] The computer program of the present invention can execute point cloud or mesh generation so as to precisely represent the edge portion shape by nodes arranged for generating a point cloud or mesh on the surface of the wafer, adjusting the coarseness of the point cloud or mesh, and representing the contour shape of the edge portion of the wafer. Since the calculation of the resistivity correction coefficient of the four-probe method can be executed by numerical simulation using the generated point cloud or mesh, the resistivity of the wafer, particularly the resistivity of the edge portion of the wafer with a complex shape, can be measured with high precision, and highly reliable measurement is possible.

Brief Description of the Drawings

[0016]

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Embodiments for Carrying Out the Invention

[0017] Embodiments of the method for measuring the resistivity of the wafer of the present invention and the computer program capable of executing the same will be described below. The following description is merely an example of the present invention. The present invention is not limited to this embodiment and can be designed and changed within the scope that can solve the problems.

[0018] In the method for measuring the resistivity of the wafer of the present invention, a resistivity correction coefficient is obtained when measuring the resistivity. The resistivity correction coefficient can be obtained by a computer according to the procedure in Fig. 2. Fig. 2 is an example of the case where the resistivity correction coefficient is obtained (calculated) by a computer. The procedure will be described below.

[0019] [Calculation of Resistivity Correction Coefficient] 1. At the start of measurement (Figure 2), input to the computer various conditions necessary for generating a point group or mesh, such as the dimensions of the wafer, the dimensions of the edge portion of the wafer, the probe positions in contact with the wafer, the size, shape, and number of the point group formed by a large number of nodes P (Figures 8 to 13) on the surface of the wafer or the mesh M that divides the wafer by connecting a large number of nodes P (Figure 2, 2-1). The mesh M includes two-dimensional shell meshes, three-dimensional solid meshes, and other meshes. The following example of the present invention is a case where numerical simulation is performed using a three-dimensional solid mesh. However, in the present invention, numerical simulation can also be performed by arranging only the point group of nodes P without using a mesh. The number, density distribution (density), shape, etc. of the point group or mesh M generated in the present invention can be arbitrarily set. However, a larger number and higher density lead to an improvement in the accuracy of the measurement results. The mesh shape can be a square or any other arbitrary three-dimensional shape (e.g., columnar). 2. The computer generates a point group or mesh on the wafer 1 as shown in Figure 10(b) according to the input conditions (Figure 2, 2-2). In this case, generate the point group or mesh so that the shape of the edge portion of the wafer can be precisely represented by a plurality of nodes (Figure 8) arranged near the outer peripheral edge of the edge portion of the wafer. To represent the shape of the edge portion of the wafer, the curve or surface of the wafer cross-section can be represented with reference to these nodes. In the present invention, to represent the curve or surface of the cross-section of the edge portion of the wafer, as an example, a plurality of nodes P (10 points on the outer peripheral edge in Figure 8) are set near the outer peripheral edge of the edge portion of the wafer as shown in Figure 8, and the shape of the edge portion of the wafer is represented (reproduced) by a curve or surface that passes through or near these nodes P (a curve or surface with reference to the nodes P). At this time, the number of set positions of the nodes P is not limited to 10 points. The more the set positions, the more faithfully the actual cross-sectional curve or surface of the wafer can be represented. 3. Generate a point cloud or mesh based on the potential distribution corresponding to each measurement location and the probe arrangement (local potential distribution around the probe) (Figs. 2, 2-2). Specifically, although the potential distribution across the entire wafer is unknown, there are significant changes in potential around the probe as shown by Equation 4 or Figs. 7(a) and (b). Therefore, more nodes or meshes are arranged around the probe. In Fig. 7(a), which shows the potential distribution around the probe, the horizontal axis represents the horizontal axis of the wafer, and the vertical axis represents the potential distribution around the probe. Also, examples of node or mesh arrangements corresponding to Figs. 7(a) and (b) are shown in Figs. 10(a) and (b). 4. Use the generated point cloud or mesh to perform numerical simulation to calculate the resistivity correction factor of the four-probe method (Figs. 2, 2-3). For example, for the mathematical model shown in Equation 1 above, perform numerical simulation with the resistivity ρ set to 1 to obtain an approximate solution û of the solution (potential distribution) u of the mathematical model. Note that the resistivity correction factor does not depend on the resistivity ρ. Use the obtained approximate solution û of the mathematical model to calculate the resistivity correction factor F using the following Equation 3. h of the mathematical model. Note that the resistivity correction factor does not depend on the resistivity ρ. Use the obtained approximate solution û h of the mathematical model to calculate the resistivity correction factor F C using the following equation. 5. Output the resistivity correction factor obtained by the calculation (Figs. 2, 2-4).

[0020] The operations in steps 2 to 5 above are executed by a computer based on the conditions input in step 1. In Fig. 10(b), mesh division lines are shown on the wafer, but in reality, the division lines are not displayed on the wafer and are stored (in memory) as data in the computer.

[0021] [Equation 3] TIFF0007717419000003.tif35130

[0022] [Generation of Point Cloud or Mesh] In the resistivity measurement method of the present invention, as described above, a point cloud or mesh that reproduces the wafer shape is generated. The procedures for mesh and point cloud generation are described below based on Fig. 3.

[0023] In the generation of the mesh and the point cloud, in order to capture the change in potential inside the wafer, a point cloud (Figs. 10(a) and (b)) consisting of many nodes P is generated in the three-dimensional space composed of the inside and the surface of the wafer, and according to the necessity in numerical simulation, a mesh is created by connecting the generated nodes P. Since the accuracy of numerical simulation depends on the coarseness or fineness of the point cloud or the mesh, it is important to appropriately set the coarseness or fineness according to the mathematical model formula 1 and the wafer shape. In the present invention, the nodes P located at positions far from the probe basically create a uniform point cloud, but in the boundary part and the periphery of the probe, the point cloud is generated by the methods described in 2, 3, and 4 below (see Figs. 10(a) and (b)).

[0024] 1. Input into the computer various conditions necessary for the generation of the point cloud or the mesh, such as the dimensions of the wafer, the edge dimensions of the wafer, the probe position in contact with the wafer, the size, shape, and number of the point cloud or the mesh to be generated on the surface of the wafer, etc., which are the various conditions necessary for the generation of the point cloud or the mesh (Figs. 3 and 3-1). 2. Generation of the point cloud in the vicinity of the probe (Figs. 3 and 3-2) In the periphery of the probe (also referred to as the vicinity of the probe or near the probe), the solution of the mathematical model formula 1 has a characteristic (singularity) that changes rapidly depending on the distance from the probes A and D (Fig. 5). When simulating including the singularity in numerical simulation, in order to calculate a high-precision resistivity correction coefficient, it is necessary to arrange many nodes P according to the aforementioned characteristic of the solution of the mathematical model formula 1 near the probe, that is, the singularity that changes rapidly depending on the distance as shown in formula 4, to generate a point cloud. As an example, the nodes P are arranged as shown in Figs. 10(a) and (b).

[0025] [Formula 4] TIFF0007717419000004.tif55161

[0026] 3. Generation of the point cloud in the inner part of the wafer based on the orifice and notch shapes (Figs. 3 and 3-3) The coordinates of each node P of the point group are, for example, as shown in FIG. 9, polar coordinates with the midpoint O on the line connecting the probes A and D as the origin. The declination Θ of the node P(r, Θ) is the angle between the line connecting the probes A and D, the origin O, and the line connecting the origin O and the node P, and the radius vector r is the distance from the origin O to the node P. In point group generation, the nodes are arranged concentrically from the origin. At this time, the greater the distance from the origin, the greater the distance between the arranged node and the node adjacent to it. When expressing the orifice shape and the notch shape, the position of the point is adjusted according to the positional relationship between the nodes arranged concentrically and the orifice or the notch. As a specific example, when the node is outside the orifice or the notch, the node is arranged on the contour line of the orifice or the notch. When the node is inside the orifice or the notch, the node close to the orifice or the notch is deleted (see FIGS. 12 and 13). 4. Generation of Point Group for Edge Portion (FIGS. 3, 3-4) The generation of the point group for the edge portion of the wafer creates the shape curve (FIG. 8) or the curved surface of the edge portion based on the various conditions set in Step 1, and generates the point group on the surface of the edge portion of the wafer with reference to the shape curve or the curved surface. Note that if there are nodes that the edge shape curve or the curved surface of the wafer does not pass through, these nodes may not be referred to in the subsequent steps. 5. Mesh Generation (FIGS. 3, 3-5) Depending on the necessity of numerical simulation, the region may be divided by creating internal nodes or creating a polyhedron from adjacent nodes P. 6. When the point group generated in Steps 2, 3, and 4 or the mesh is created in Step 5, the meshes are combined to form the point group or the mesh of the entire wafer (see FIG. 11). 7. Data Output of Point Group or Mesh (FIGS. 3, 3-7).

[0027] The operations 1 to 7 are also executed by a computer based on the conditions input in 1.

[0028] [Numerical Simulation] In the resistivity measurement method of the present invention, the resistivity correction coefficient is calculated by using an approximate solution of Equation 1 by a numerical simulation method based on point group or mesh data. Regarding the procedure of numerical simulation, the procedure in a general simulation method will be described based on FIG. 4 in this section,

[0029] and the specific procedure when the numerical simulation in the present invention is performed by the finite difference method will be described in this section. 1. Input the mesh data generated in FIG. 3 into the computer (FIG. 4, 4-1). 2. Creation of the stiffness matrix (FIG. 4, 4-2) Obtaining an approximate solution of Equation 1 by a numerical simulation method is reduced to solving the following linear equation 5. [Equation 5] TIFF0007717419000005.tif2387 Here, K in Equation 5 is called the stiffness matrix and is related to the derivative appearing on the left side of the upper equation of Equation 1 and is known. c in Equation 5 represents a coefficient vector related to the approximate solution of u in Equation 1 and is unknown. Also, f in Equation 5 is called the load vector corresponding to the right side of Equation 1 and is known. The point group or mesh data input in step 1 includes the coordinates of the nodes P that reproduce the wafer shape and their connection information according to the necessity in numerical simulation. Referring to the point group or mesh data, the stiffness matrix K of Equation 5 is created according to the relationship between the value of the electric potential u at each node determined by the numerical simulation method used and the coefficient vector c (the discretized equation of Equation 1 by numerical differentiation or variational formula), and further, the load vector f on the right side of Equation 5 is created according to the positions of the probes A and D.

[0029] As an example, in the numerical simulation by the finite difference method when the distance between nodes is a certain amount h, a list of the values of the electric potential u at each node P of the mesh corresponds to the coefficient vector c, and the electric potentials at the nodes p1, p2, ···, p6 around the node p0 are set as c1, c2, ···, c6 respectively (see FIG. 14), then the discretized equation of Equation 1 at the node p0 becomes the following Equation 6. [Equation 6] The horizontal vector (-6, 1, 1, 1, 1, 1, 1) on the right side of Equation 6 in TIFF0007717419000006.tif44149 corresponds to a part of K, and the vertical vector corresponds to a part of the coefficient vector c. Also, the load vector f sets the values of the nodes P corresponding to the probes A and D to 1 and -1 respectively, and the values are 0 at other nodes P. However, depending on the numerical simulation method used, the horizontal vector (-6, 1, 1, 1, 1, 1, 1) / h of Equation 6 2 or the load vector may be in other dimensions or have other values. 3. Solving the Discretized Equation (Figure 4, 4-3) The system of linear equations in Equation 5 can be solved by methods such as Gaussian elimination. Each component of the coefficient vector c in Equation 5 is an approximation of the potential value at each point. Let the approximate solution of the potential distribution represented by the coefficient vector c be u h be. 4. Calculating the Resistivity Correction Coefficient (Figure 4, 4-4) Referring to the potential distribution u h calculated by the above Procedure 3, the potential values at the probes B and C are obtained, and the resistivity correction coefficient F C is obtained by Equation 3. 5. Output of the Resistivity Correction Coefficient (Figure 4, 4-5).

[0030] The operations 1 to 5 above are also executed by a computer based on the conditions input in 1.

[0031] The computer program of the present invention can execute various operations required for the resistivity measurement method of a wafer, such as generating a point group or mesh in the resistivity measurement method of the wafer described above, particularly generating a point group or mesh that reproduces the shape of the edge portion of the wafer, adjusting the coarseness and fineness of the point group or mesh around the probe, calculating the resistivity correction coefficient of the four-probe method by numerical simulation using the generated point group or mesh, and measuring the resistivity of the wafer using the resistivity correction coefficient.

Industrial Applicability

[0032] The computer program of the present invention can be used in a resistivity measuring device for wafers. In particular, in the method according to the present invention, a high-precision resistivity correction coefficient is provided for wafers having an orifice, a notch, a fillet, chamfering, etc. at the edge portion, which could not be fully handled by the previous Japanese Industrial Standards.

[0033] During resistivity measurement by the four-probe method, it is possible to correct the resistivity measurement value in real time by performing a simulation simultaneously with the measurement.

[0034] The above embodiment is an example of the present invention. The present invention can be modified as long as the problems can be solved. Also, as numerical simulation, it is possible to use discretization methods such as the finite element method and the boundary element method regardless of the use of a mesh.

Explanation of Reference Numerals

[0035] 1 Wafer A Probe B Probe C Probe D Probe E Orifice F Notch M Mesh О Origin P Node

Claims

1. In resistivity measurement of a semiconductor wafer by the four-probe method, the resistivity of the semiconductor wafer is calculated by multiplying the measured value obtained by the four-probe method measurement by a resistivity correction coefficient obtained by numerical simulation adapted to the characteristics of the three-dimensional region shape of the wafer. A method for measuring the resistivity of a semiconductor wafer, characterized by the above.

2. In the method for measuring the resistivity of a semiconductor wafer according to Claim 1, a plurality of nodes are set on the outer peripheral edge of the wafer, the shape of the edge portion of the wafer is reproduced by a curve or a curved surface referring to these nodes, the shape of the edge portion is partitioned, and a point group or a mesh of the edge portion is generated. The resistivity correction coefficient is calculated by numerical simulation using the generated point group or mesh according to Equation 3. [Equation 3] A method for measuring the resistivity of a semiconductor wafer, characterized by the above.

3. In the method for measuring the resistivity of a semiconductor wafer according to Claim 2, the density of the generated point group or mesh is adjusted, and the numerically simulated point group or mesh after density adjustment is used. A method for measuring the resistivity of a semiconductor wafer, characterized by the above.

4. In the method for measuring the resistivity of a semiconductor wafer according to Claim 1, the overall dimensions, edge dimensions, probe positions in contact with the wafer, mesh size, or density, shape, and number of the point group of the semiconductor wafer, which are the conditions necessary for generating the point group or mesh, are input into a computer. A point group or mesh of the semiconductor wafer is generated by a computer according to the input. The shape of the wafer is reproduced by a computer using the generated point group or mesh. A method for measuring the resistivity of a semiconductor wafer, characterized by the above.

5. In the method for measuring the resistivity of a semiconductor wafer according to Claim 1, a planar wafer is simulated in a two-dimensional region or a wafer having a certain volume is simulated in a three-dimensional region. A method for measuring the resistivity of a semiconductor wafer, characterized by the above.

6. Generation of a point group or mesh in the method for measuring the resistivity of a semiconductor wafer according to any one of Claims 1 to 5, calculation of the resistivity correction coefficient of the four-probe method by numerical simulation using the generated point group or mesh, and execution of resistivity measurement of the semiconductor wafer using the resistivity correction coefficient. A computer program.

7. In the computer program according to Claim 6, A computer program that can perform rough and fine adjustment of point clouds or meshes around a probe, and generate point clouds or meshes that reproduce the shape of the edge portion of a wafer. A computer program.

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