Method for measuring resistivity of semiconductor wafer, and computer program

Abstract

In this method for measuring the resistivity of a wafer, a point cloud or a mesh reflecting the shape of an edge portion of the wafer is generated on the basis of the overall shape of the wafer, the shape of the edge portion, a parameter relating to a probe, etc., and the measured value (resistivity) of the wafer measured by the four-probe method is corrected using a correction coefficient value calculated by numerical simulation, to calculate the resistivity. A computer program according to the present invention is capable of executing the procedures of said resistivity measurement method.

Description

Method and computer program for measuring resistivity of semiconductor wafers

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

[0002] One of the standard methods for measuring the resistivity of semiconductor wafers is the four-probe method specified in the Japanese Industrial Standards (hereinafter referred to as "JIS standard") (Non-Patent Document 1). Another method for measuring the resistivity of conductive plastics is the four-probe method specified in the JIS standard (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. This measurement method involves contacting four probes (electrodes) A ​​through D with the surface of a semiconductor wafer (sample) 1 whose resistivity is to be measured, as shown in Figure 5. 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 using the two outer probes A and D. Resistivity ρ is calculated using Equation 2, referencing the solution (potential distribution) u of the mathematical model when the current I between probes B and C shown in Equation 1 is set to 1. The solution to this equation can be expressed mathematically, especially for wafers with regular shapes such as rectangular parallelepipeds or cylinders.

[0004] [Formula 1]

[0005] [Formula 2]

[0006] In the above formula 1, Ω represents the area occupied by the sample, and δ(x) represents the Dirac δ function. Ｃ is called the resistivity correction factor, which is a constant that depends on the shape of the sample and the position of the probe, and plays an important role in resistivity measurements.

[0007] In recent years, there has been an increasing need for highly accurate and reliable resistivity measurements in the field of semiconductor wafers. To improve the yield rate of semiconductor wafers, highly accurate resistivity measurements near the outer periphery of the sample (hereinafter referred to as the "edge") are required.

[0008] The resistivity measurement methods based on the above-mentioned formulas 1 and 2 specified in JIS use specific formulas for the solution of the

[0004] term. These methods are based on the assumption that the sample has a regular shape such as a rectangular parallelepiped or a cylinder, and are therefore only suitable for measuring the resistivity of samples with regular shapes. On the other hand, actual semiconductor wafers are disk-shaped, and have complex shapes (not shown) such as an orientation flat E (FIG. 6(a)) and a notch F (FIG. 6(b)) at the edge, as well as fillets and chamfers. The resistivity correction factor F calculated according to the previous JIS Ｃ However, since the orientation flat E, notch F, and complex shapes of the edge of an actual semiconductor wafer are not reflected in the method, its reliability near the edge of a semiconductor wafer is questionable.

[0009] Incidentally, the resistivity correction coefficient calculated according to JIS, which is based on a disk shape, does not correspond to the shape of the edge portion, so when applied to an actual wafer, the results are as shown in Figure 1. In Figure 1, the smaller the distance d from the reference position, the closer it is to the edge.

[0010] JIS H 0602:1995 Resistivity measurement method for silicon single crystals and silicon wafers using the four-probe method JIS K 7194:1994 Resistivity test method for conductive plastics using the four-probe method

[0011] The present invention was developed to solve the problem that current resistivity measurement methods are unable to accommodate the shape of edge portions, and aims to provide a method for measuring the resistivity of semiconductor wafers that is capable of measuring the resistivity of semiconductor wafers, particularly the resistivity of edge portions, with high accuracy and reliability by using a numerical simulation that precisely reproduces the shape of edge portions such as orientation flats, notches, fillets, and chamfers of semiconductor wafers, in contrast to the conventional mathematical models that govern resistivity measurement.

[0012] In the resistivity measurement method of the semiconductor wafer of the present invention, in the resistivity measurement using four probes, the potential difference U between probes B and C is ＢＣ and the fixed current I between probes A and D ＡＤ Relative to U ＢＣ / I ＡＤA resistivity correction factor for correcting a measurement value (hereinafter referred to as a "four-probe measurement value") obtained by a numerical calculation method (numerical simulation) is calculated, and the resistivity is calculated by multiplying the four-probe measurement value by this resistivity correction factor. In particular, a semiconductor wafer (hereinafter simply referred to as a "wafer") is provided with a large number of nodes (point clouds) used to reproduce the wafer shape, and the nodes are arranged according to the characteristics of the wafer shape and potential distribution near the measurement points, or the wafer is divided (also called division or discretization) by connecting the nodes into a large number of meshes to reproduce the wafer shape. In addition, the resistivity correction factor of the wafer is calculated 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 some of the nodes at the edge of the wafer.

[0013] The computer program of the present invention can generate a point cloud or mesh so as to accurately represent the edge shape of a wafer by generating the point cloud or mesh, adjusting the density of the point cloud or mesh, and placing nodes to represent the contour shape of the edge portion of the wafer, and can calculate the resistivity correction factor for the four-probe method through numerical simulation using the generated point cloud or mesh.

[0014] The resistivity correction factor used in the wafer resistivity measurement method of the present invention is a resistivity correction factor for the four-probe method obtained by numerical simulation using a point cloud or mesh generated to reproduce the wafer shape. Because the edge shape of the wafer, such as the orientation flat, notch, fillet, and chamfer, is precisely reflected, the resistivity of the wafer, particularly the resistivity of the edge of the wafer, can be measured with high precision, enabling highly reliable measurements.

[0015] The computer program of the present invention can generate a point cloud or mesh on the surface of a wafer, adjust the density of the point cloud or mesh, and generate a point cloud or mesh so as to accurately represent the edge shape using nodes placed to represent the contour shape of the wafer's edge.It can also calculate the resistivity correction coefficient for the four-probe method through numerical simulation using the generated point cloud or mesh, thereby enabling highly accurate and reliable measurement of the resistivity of a wafer, particularly the resistivity of the edge of a wafer with a complex shape.

[0016] 1 is a comparison diagram of the resistivity correction factor according to JIS and the resistivity correction factor of the four-probe method calculated by the numerical simulation of the present invention.

[0034] FIG. 1 is a schematic explanatory diagram of the wafer resistivity measurement method of the present invention.

[0035] FIG. 2 is a detailed explanatory diagram of the wafer resistivity measurement method of the present invention.

[0036] FIG. 3 is an explanatory diagram of a method for calculating the correction factor value in the wafer resistivity measurement method of the present invention.

[0037] FIG. 4 is an explanatory diagram of the four-probe measurement method. (a) is an explanatory diagram of the wafer orientation flat, and (b) is an explanatory diagram of the notch. (a) is an example of the potential distribution on the line connecting two of the four probes, A and D (FIG. 5), that are in contact with the sample surface, and (b) is a contour plot of the potential distribution in (a).

[0038] FIG. 5 is an explanatory diagram of the node placement on the edge of the wafer.

[0039] FIG. 6 is an explanatory diagram of the coordinates of the probe and the node on the wafer. (a) is an explanatory diagram of the division around the probe in contact with the wafer, and (b) is an explanatory diagram of the division of the entire wafer.

[0039] FIG. 7 is an explanatory diagram of the mesh of the entire wafer generated by combining the mesh inside (center) of the wafer with the point cloud or mesh of the edge.

[0039] FIG. 8 is an explanatory diagram of the division of the entire wafer with an orientation flat. An explanatory diagram of node adjustment near the notch of a wafer. An explanatory diagram of points for evaluating the Laplace operator of the electric potential distribution.

[0017] The following describes an embodiment of a wafer resistivity measurement method and a computer program capable of executing the method according to the present invention. The following description is merely an example of the present invention. The present invention is not limited to this embodiment, and design modifications are possible within the scope of solving the problems.

[0018] In the wafer resistivity measurement method of the present invention, a resistivity correction factor is calculated when the resistivity is measured. The resistivity correction factor can be calculated by a computer using the procedure shown in Figure 2. Figure 2 shows an example of how to calculate (calculate) the resistivity correction factor using a computer. The procedure is described below.

[0019] [Calculation of Resistivity Correction Coefficient] 1. Before starting measurement (FIG. 2), various conditions necessary for generating a point cloud or mesh, such as the wafer dimensions, wafer edge dimensions, probe position to be contacted with the wafer, and the size, shape, and number of the point cloud or mesh M formed by the numerous nodes P (FIGS. 8 to 13) generated on the wafer surface, or the size, shape, and number of the mesh M connecting the numerous nodes P to partition the wafer, are input into a computer (FIGS. 2, 2-1). Mesh M can be a two-dimensional shell mesh, a three-dimensional solid mesh, or other meshes. The following example of the present invention illustrates a case where a numerical simulation is performed using a three-dimensional solid mesh, but the present invention can also perform a numerical simulation without using a mesh, using only the arrangement of the point cloud of nodes P. In the present invention, the number, density, shape, etc. of the point cloud or mesh M generated can be set arbitrarily, but a larger number and higher density lead to improved accuracy of the measurement results. The mesh shape can be square or any other three-dimensional shape (e.g., columnar). 2. The computer generates a point cloud or mesh on the wafer 1 according to the input conditions (FIGS. 2 and 2-2), as shown in FIG. 10(b). In this case, the point cloud or mesh is generated so that the wafer edge shape can be accurately represented using multiple nodes (FIG. 8) located near the outer periphery of the wafer edge. To represent the wafer edge shape, the curve or curved surface of the wafer cross section can be represented by referencing these nodes. In the present invention, to represent the curve or curved surface of the cross section of the wafer edge, as shown in FIG. 8, for example, multiple nodes P (10 points on the outer periphery in FIG. 8) are set near the outer periphery of the wafer edge, and the wafer edge shape is represented (reproduced) by curves or curved surfaces that pass through or pass near these nodes P (curves or curved surfaces that reference nodes P). In this case, the number of nodes P set is not limited to 10. The more nodes set, the more faithfully the actual cross-sectional curve or curved surface of the wafer can be represented. 3. A point cloud or mesh is generated based on the potential distribution (local potential distribution around the probe) according to each measurement point and probe placement (Fig. 2, 2-2).Specifically, while the potential distribution across the entire wafer is unknown, the potential distribution around the probe exhibits drastic changes in potential as shown in Equation 4 or Figures 7(a) and 7(b). Therefore, more nodes or meshes are arranged around the probe. In Figure 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. Examples of node or mesh arrangements corresponding to Figures 7(a) and 7(b) are shown in Figures 10(a) and 10(b). 4. The generated point cloud or mesh is used to perform a numerical simulation to calculate the resistivity correction factor for the four-probe method (Figures 2 and 2-3). For example, a numerical simulation is performed on the mathematical model shown in Equation 1, setting the resistivity ρ to 1, to obtain an approximate solution u of the solution (potential distribution) u of the mathematical model. ｈ It should be noted that the resistivity correction factor does not depend on the resistivity ρ. The approximate solution u of the mathematical model obtained ｈ Using this, the resistivity correction coefficient F is calculated using the following equation 3: Ｃ 5. Output the resistivity correction coefficient obtained by calculation (Figure 2, 2-4).

[0020] The above steps 2 to 5 are executed by the computer based on the conditions entered in step 1. In Figure 10(b), mesh dividing lines are displayed on the wafer, but in reality, the dividing lines are not displayed on the wafer, but are stored (memorized) as data in the computer.

[0021] [Formula 3]

[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 procedure for generating the mesh and point cloud will be described below with reference to FIG.

[0023] In generating the mesh and point cloud, a point cloud (FIGS. 10(a) and 10(b)) consisting of many nodes P is generated in a three-dimensional space consisting of the wafer's interior and surface in order to capture changes in the electric potential inside the wafer, and a mesh is created by connecting the generated nodes P as required for the numerical simulation. Since the accuracy of the numerical simulation depends on the density of the point cloud or mesh, it is important to set the density appropriately according to the mathematical model (Equation 1) and the wafer shape. In the present invention, a uniform point cloud is basically created for nodes P located far from the probe, but point clouds are generated at the boundary and around the probe using the methods described in the following 2, 3, and 4 (see FIGS. 10(a) and 10(b)).

[0024] 1. The various conditions required for generating a point cloud or mesh, such as the wafer dimensions, wafer edge dimensions, the position of the probe in contact with the wafer, and the size, shape, and number of points or meshes to be generated on the wafer surface, are input into a computer (Figure 3, 3-1). 2. Generating a Point Cloud in the Proximal Area of ​​the Probe (Figure 3, 3-2). In the area surrounding the probe (also referred to as the proximal area or the vicinity of the probe), the solution of mathematical model Equation 1 exhibits a characteristic (singularity) that changes rapidly depending on the distance from the probes A and D (Figure 5). When performing a numerical simulation including singularities, in order to calculate a highly accurate resistivity correction factor, it is necessary to generate a point cloud by placing many nodes P in accordance with the characteristic of the solution of mathematical model Equation 1 in the vicinity of the probe, i.e., the singularity that changes rapidly depending on the distance as shown in Equation 4. As an example, nodes P are placed as shown in Figures 10(a) and 10(b).

[0025] [Formula 4]

[0026] 3. Generating a Point Cloud for the Wafer's Inner Region Based on the Orientation Flat and Notch Shape (Figures 3 and 3-3) The coordinates of each node P in the point cloud are expressed as polar coordinates, with the midpoint O on the line connecting tips A and D as the origin, as shown in Figure 9 . The argument Θ of node P(r, Θ) is the angle between the line connecting tips A and D and the line connecting origin O and node P, and the radius vector r is the distance from origin O to node P. In point cloud generation, nodes are placed concentrically around the origin. The greater the distance from the origin, the greater the distance between the placed node and its neighboring nodes. To represent the orientation flat and notch shapes, the positions of the concentrically placed nodes are adjusted based on their relative positions to the orientation flat or notch. As a specific example, if a node is located outside the orientation flat or notch, it is placed on the contour of the orientation flat or notch. If a node is located inside the orientation flat or notch, the node close to the orientation flat or notch is deleted (see Figures 12 and 13). 4. Generating a Point Cloud for the Edge (Figures 3, 3-4) To generate a point cloud for the wafer edge, the edge shape curve (Figure 8) or surface is created based on the conditions set in step 1, and the point cloud for the wafer edge surface is generated by referencing this shape curve or surface. Note that if there are nodes that the wafer edge shape curve or surface does not pass through, these nodes may not be referenced in subsequent steps. 5. Generating a Mesh (Figure 3, 3-5) Depending on the needs of the numerical simulation, the region may be divided by creating internal nodes or creating a polyhedron from adjacent nodes P. 6. The point clouds generated in steps 2, 3, and 4, or the mesh created in step 5, are combined to create a point cloud or mesh for the entire wafer (see Figure 11). 7. Outputting the Point Cloud or Mesh Data (Figures 3, 3-7).

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

[0028] [Numerical Simulation] In the resistivity measurement method of the present invention, the resistivity correction factor is calculated by utilizing an approximate solution to Equation 1 obtained by a numerical simulation technique based on point cloud or mesh data. Regarding the procedure for numerical simulation, this section explains the procedure for a general simulation technique with reference to FIG. 4, and section

[0029] explains the specific procedure for performing the numerical simulation of the present invention using the finite difference method. 1. Input the mesh data generated in FIG. 3 into a computer (FIG. 4, 4-1). 2. Create a stiffness matrix (FIG. 4, 4-2). Obtaining an approximate solution to Equation 1 using a numerical simulation technique boils down to solving the following linear equation 5: [Equation 5] Here, K in Equation 5 is called the stiffness matrix and is related to the differential 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. Furthermore, f in Equation 5 is called the load vector corresponding to the right side of the upper equation of Equation 1 and is known. The point cloud or mesh data input in Step 1 includes the coordinates of nodes P that reproduce the wafer shape and their connection information as required for the numerical simulation. By referencing the point cloud or mesh data, the stiffness matrix K in Equation 5 is created based on the relationship between the value of potential u at each node and coefficient vector c (a discretized equation of Equation 1 using numerical differentiation or variational equations) as determined by the numerical simulation method used. Furthermore, the load vector f on the right side of Equation 5 is created according to the positions of probes A and D.

[0029] For example, in a numerical simulation using the finite difference method when the distance between nodes is a constant amount h, the list of values ​​of the potential u at each node P in the mesh corresponds to the coefficient vector c, and the node p ０ Node p in the vicinity of １ , p ２ , ..., p ６ The potential at １ , c ２ , ..., c ６ (See FIG. 14) and the node p ０ The discretized equation of Equation 1 in the above equation is Equation 6 below. [Equation 6] The horizontal vector (-6,1,1,1,1,1,1) on the right side of Equation 6 corresponds to part of K, and the vertical vector corresponds to part of the coefficient vector c. In addition, the values ​​of the load vector f are set to 1 and -1 at the nodes P corresponding to the probes A and D, respectively, and the value is set to 0 at the other nodes P. However, the horizontal vector (-6,1,1,1,1,1,1) / h in Equation 6 may vary depending on the numerical simulation method used. ２ The load vectors may have other dimensions or other values. 3. Solving the discretized equations (Figure 4, 4-3) The linear equation system of Equation 5 can be solved by techniques such as Gaussian elimination. Each component of the coefficient vector c in Equation 5 is an approximation of the potential value at each point. The approximate solution of the potential distribution expressed by the coefficient vector c is called u. ｈ 4. Calculation of resistivity correction factor (Fig. 4, 4-4) The potential distribution u calculated by the above procedure 3 is ｈ The potential values ​​at probes B and C are calculated by referring to the formula, and the resistivity correction coefficient F Ｃ 5. Output of resistivity correction coefficient (Figure 4, 4-5).

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

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

[0032] The computer program of the present invention can be used in wafer resistivity measurement devices. In particular, the method of the present invention provides highly accurate resistivity correction factors for wafers with edge features such as orientation flats, notches, fillets, and chamfers, which were not fully covered by the previous Japanese Industrial Standards.

[0033] When measuring resistivity using the four-probe method, it is possible to correct the measured resistivity values ​​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 problem can be solved. Furthermore, as a numerical simulation, a discretization method such as the finite element method or the boundary element method can be used regardless of the use of a mesh.

[0035] 1 Wafer A Probe B Probe C Probe D Probe E Orientation flat F Notch M Mesh O Origin P Node

Claims

1. A method for measuring the resistivity of a semiconductor wafer using a four-probe method, comprising: calculating the resistivity of the semiconductor wafer by multiplying the measured value obtained by the four-probe method by a resistivity correction coefficient obtained by a numerical simulation that is adapted to the characteristics of the three-dimensional area shape of the wafer.

2. The method for measuring the resistivity of a semiconductor wafer according to claim 1, wherein a plurality of nodes are set on the outer periphery of the wafer, the shape of the edge of the wafer is reproduced using a curve or a curved surface that references these nodes, the shape of the edge is partitioned, and a point cloud or mesh of the edge is generated, and the resistivity correction coefficient is calculated by numerical simulation using the generated point cloud or mesh according to Equation 3. [Equation 3] A method for measuring the resistivity of a semiconductor wafer.

3. A method for measuring the resistivity of a semiconductor wafer according to claim 2, characterized in that the generated point cloud or mesh is adjusted in coarseness and density, and the point cloud or mesh after the coarseness and density adjustment is used in the numerical simulation.

4. A method for measuring the resistivity of a semiconductor wafer as claimed in claim 1, characterized in that the conditions required for generating a point cloud or mesh, such as the overall dimensions of the semiconductor wafer, edge dimensions, position of the probe to be brought into contact with the wafer, mesh size or density, shape and number of point clouds, are input into a computer, the point cloud or mesh is generated by the computer on the semiconductor wafer in accordance with said input, and the wafer shape is reproduced by the computer using the generated point cloud or mesh.

5. A method for measuring the resistivity of a semiconductor wafer according to claim 1, characterized in that a planar wafer is simulated in a two-dimensional area, or a wafer having a certain volume is simulated in a three-dimensional area.

6. A computer program for generating a point cloud or mesh in the method for measuring the resistivity of a semiconductor wafer according to any one of claims 1 to 5, calculating a resistivity correction factor for the four-probe method through numerical simulation using the generated point cloud or mesh, and measuring the resistivity of a semiconductor wafer using the resistivity correction factor.

7. A computer program as claimed in claim 6, which is capable of adjusting the density of a point cloud or mesh around the probe and generating a point cloud or mesh that reproduces the shape of the edge of the wafer.

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