System and method of curve fitting

[0001] The present application is a non-provisional U.S. patent application of provisional application Ser. No. 60/481,642, filed Nov. 14, 2003.

[0002] The field of the invention is data processing, and more particularly relates to fitting a curve to a plurality of data points.

[0003] Fitting a curve to a plurality of data points is commonly performed in engineering applications where characterization of the plurality of data points is required in order to obtain information regarding a response of a system. For example, curve fitting of data points is performed in the field of semiconductor processing such as in controlling focus for a photolithographic tool. Image size measurement data points generated from images produced by the photolithographic tool for various focus settings can be analyzed to fit a curve to the data points in order to determine a “best focus” condition for the photolithographic tool.

[0004] According to one conventional method, a 2^nd-order polynomial (i.e. parabolic model) is fit to image size measurement data points. Unfortunately, parabolic modeling is resistant to classical point-tossing methodologies, usually based upon model residuals, making it difficult to detect and discard inaccurate measurement data. The reasons for this difficulty include:

• • [0005] a) parabolic models are weighted heavily towards their extremes because of the squaring of the independent variable. In other words, the outermost data points have the most influence on the modeled parameters; and
  • [0006] b) the outermost data points that have the most influence on the modeled parameters are also the most likely to be inaccurate. This is due to image quality being poor at the extremes of focus of the photolithographic tool, thus making the images difficult to measure.

[0007] In order to improve confidence in the parabolic model, and to minimize the effects of inaccurate image size measurement data points, over-sampling of the photolithographic system occurs. In a manufacturing environment, this decreases productivity which increases manufacturing costs.

[0008] Another conventional method is to fit the image size measurement data to a 2nd-order polynomial and remove data points based upon the individual residual errors. Data at the extremes of focus of the photolithographic tool typically do not fit parabolic behavior due to poor image quality, thus making the images difficult to measure. Yet the 2^nd-order polynomial method gives excessive weighting to the data at the extremes of focus resulting in a reduction in accuracy in the determination of best focus. For example, FIG. 1 illustrates a 2^nd-order polynomial 30 fit to a plurality of measurement data points (denoted by X). Data points 35 at the extremes of focus are given excessive weighting such that data points 40 are erroneously removed and are not included in the determination of the 2^nd-order polynomial 30.

[0009] Yet another conventional method is to fit the image size measurement data to a 4th-order polynomial or fit the image size measurement data to a 4^th-order polynomial and remove data points based upon the individual residual errors. The 4^th-order polynomial method accommodates the data points at the extremes of focus at the expense of the center data points where best focus is typically located. As described above with reference to the 2^nd-order polynomial, data at the extremes of focus of the photolithographic tool typically do not fit parabolic behavior due to poor image quality. Thus, the 4^th-order polynomial method also gives excessive weighting to the data at the extremes of focus resulting in a reduction in accuracy in the determination of best focus.

[0010] What is required is a method to fit a curve to a plurality of data points where the effect of a one or more inaccurate data points is reduced.

[0015] An embodiment of the invention relates to an iterative process to fit a curve to a plurality of data points where the effect of one or more inaccurate data points is reduced. Inaccurate data is detected and sequentially eliminated from the inventive calculation to determine a best fit curve for the plurality of data points. The invention determines a “seed curve” from a first subset of the plurality of data points, and then tests remaining data points (i.e. data points which are not included in the first subset) for fitness of inclusion in the seed curve. Thus, the seed curve is augmented with more accurate data points in order to determine a best fit curve. Although an embodiment of the invention will be described in terms of generating a “parabolic” best fit curve for a plurality of data points, it should be understood to those skilled in the art that this invention can be applied to generate a best fit curve having other responses such as, for example, a line, a circle, a semi-circle and other such curves described by a polynomial of any order.

[0016] According to an embodiment of the invention, a plurality of data points is analyzed to determine a best fit curve for the data points. The data can be representative of image size measurements of structures formed in a photoresist layer on a semiconductor substrate at various settings of a photolithographic tool such as, for example, focus and/or exposure. The image size measurement data typically includes a subset of “well-behaved” data representing image size measurements from structures which are formed correctly and with no significant measurement error introduced by a measurement system used to provide the measurement data. The image size measurement data also typically includes a subset of more variable data (i.e. “less well-behaved”) representing image size measurements from structures which are not formed correctly (i.e. out of focus), or from structures which are formed correctly but there is a significant measurement error introduced by the measurement system, or from structures which are not formed correctly and there is also a significant measurement error introduced by the measurement system.

[0017] Referring to FIG. 2, a method according to an embodiment of the invention includes, for example, a first group of steps 200 (i.e. steps 210-245) to determine a “seed curve” from a first subset of the data points, and a second group of steps 250 (i.e. steps 255-275) to determine whether a second subset of the data points can be included with the first subset of data points to form a best fit curve representing all of the data points. In step 210, any obvious inaccurate data points (i.e. “flyers”) are eliminated by methods known in the art. For example, a check of the data points corresponding to extreme x-coordinate values (i.e. data points corresponding to the lowest and highest x-coordinate values) to ensure that each extreme data point is not a minimum value can be performed. If a minimum value is determined, the individual data point is eliminated from the plurality of data points. Then, a 4th-order polynomial is fit to the remaining plurality of data points and a residual error for each data point is calculated according to Equation 1. Any individual data point that has a residual error that does not satisfy a pre-determined limit is removed from the plurality of data points. The pre-determined limit can be defined by a user.
Residual Error=|y_i−y_ie|  Equation 1

[0018] where i=1 to n data points;

[0019] y_i represents the measured value of a data point; and,

[0020] y_ie represents the estimated value of the data point from the fit curve.

[0021] A next step 215 determines a “seed curve” such as, for example, a “seed parabola”. A selection is made of n data points from the well behaved data points to form an initial seed parabola. In step 220, a fit curve such as, for example, a 2^nd-order polynomial is fit to the n data points. An RMS (Root-Mean-Square) value of the difference between the measured value and the estimated value from the fit curve for each of the n data points is calculated according to Equation 2. According to an embodiment of the invention, a quality of fit (QoF) is defined as equaling the RMS value.
QoF=SQR ROOT[Σ(y_i−y_ie)²/n],  Equation 2

[0022] where i=1 to n;

[0023] y_i represents the measured value of a data point; and,

[0024] y_ie represents the estimated value of the data point from the fit curve.

[0025] This is followed by step 225 that determines if the QoF is acceptable. When the QoF satisfies a pre-determined limit set by the user, the n data points are accepted as the seed parabola in step

[0026] Optionally, an r-squared analysis known in the art can be performed according to Equation 3 as a check to ensure that the seed curve has the correct shape (i.e. parabolic).
r²=1−[Σ(y_i−y_ie)²/Σ(y_i−y_ave)²]  Equation 3

[0027] where i=1 to n;

[0028] y_i represents the measured value of a data point;

[0029] y_ie represents the estimated value of the data point from the fit curve; and,

• • [0030] y_ave represents the average of the measured value of the n data points.

[0031] It is noted that Equations 1-3 are equations known in the art and additional details regarding the equations can be found in references such as, for example, “Introduction to Probability and Statistics”, Fourth Edition, W.H. Freeman and Company, San Francisco, Calif., 1968, herein incorporated by reference.

[0032] If the QoF does not satisfy a pre-determined limit, a search of the well-behaved data points is performed in steps 235 and 240 to determine a subset “n−1”, “n−2”, etc. of the well-behaved data points. For example, a subset of n−1 data points is tested in step 240 as described above with respect to steps 220 and 225 for acceptability as a seed parabola. In an embodiment of the invention, all combinations of the n−1 data points are analyzed and a QoF is calculated for each of the n−1 data point combinations. When one subset of the n−1 data point combinations produces a QoF which satisfies the pre-determined limit then the one subset of n−1 data points is accepted as the seed parabola. If more than one of the subsets of the n−1 data point combinations each have a QoF which satisfies the pre-determined limit then the combination of n−1 data points that has a QoF that satisfies a second condition is accepted as the seed parabola. For example, the second condition can be that the n−1 data point combination which produces the lowest QoF is accepted as the seed parabola.

[0033] In the event that no combination of n−1 data points produces a QoF which satisfies the pre-determined value, the process proceeds iteratively to analyze all combinations of n−2, n−3, etc. data points until an acceptable seed parabola is determined or until too many data points have been eliminated (step 245).

[0034] For the case where no acceptable seed parabola has been determined, the plurality of n data points is not used to determine the seed parabola and another plurality of data points is analyzed according to the method of the invention in an attempt to determine the seed parabola. In most instances, such as when determining a “best focus” condition for a photolithographic tool, multiple sets of data points are typically available for analysis and, as such, when one set of data points can not be used to determine a seed parabola there are other sets of data points which can be analyzed and most likely at least one set of data points will produce an acceptable seed parabola.

[0035] When an acceptable seed parabola is determined according to the first group of steps 200, additional data points are then selected for inclusion with the data points that define the seed parabola according to the second group of steps 250. In step 255, a data point from the remaining data points that were not included in the determination of the seed parabola is selected as a candidate for inclusion with the m data points which form the seed parabola, where m represents the number of data points (i.e. n, n−1, n−2, etc.) from the first group of steps 200. In a next step 260, a fit curve such as, for example, a 2^nd-order polynomial curve is fit to the resulting m+1 data points, and the m+1 data points are tested for acceptability as described above with respect to step 220. Whether the QoF of the m+1 data points satisfies a pre-determined limit is determined in step 265. If the QoF satisfies the pre-determined limit then the 2^nd-order polynomial fit curve based on the m+1 data points is acceptable and the data point is included with the m+1 data points. If the QoF does not satisfy the pre-determined limit then the 2^nd-order polynomial fit curve based on the m+1 data points is not acceptable and the data point is not included with the m+1 data points (step 270). Steps 255, 260, 265, 270 are repeated until all remaining acceptable data points are included with the m data points to obtain a best fit parabolic curve for the plurality of data points.

[0036] Optionally, an r-squared analysis (see Equation 3) can be performed after all of the acceptable data points have been included with the m data points as a check to ensure that the best fit curve has a parabolic shape. Alternatively, the r-squared analysis can be performed after each acceptable data point is included with the m data points. The r-squared analysis can also be made to be an additional condition that must be satisfied in order for a data point to be included with the m data points. For example, when the r-squared analysis is performed after the QoF analysis for each data point, if a data point satisfies the pre-determined limit for QoF but does not satisfy a pre-determined limit for r-squared, the data point can be rejected for inclusion with the m data points.

[0037] An embodiment of the invention will be described with respect to determining a “best focus” condition for a photolithographic system. It should be understood to those skilled in the art that the invention can be applied to determining a best fit curve for any plurality of data such as, for example, data from a critical dimension (CD) system, film thickness measurement system and other such systems.

[0038] Integrated circuits continue to decrease in size resulting in improvements in circuit performance (i.e. speed) and a reduction in manufacturing costs. Decreasing the size of integrated circuits requires the structures such as gate conductors, interconnects (wires), trenches, etc. which form the integrated circuits to be reduced in size. An important step in forming smaller structures in a semiconductor process is photolithography. A photolithographic tool (i.e. NSR-S203B or 204B manufactured by Nikon Precision Instruments, Inc. or Micrascan 3 manufactured by Silicon Valley Group, Lithography Division (SVGL)) typically includes projection optics having a lens and a stage upon which a substrate is placed during the exposure step to “print” an image on a surface of the substrate. The ability of the photolithographic tool to form accurate images in exposed photoresist is critical for the continued reduction in integrated circuit size. Thus, control of the photolithographic tool is important to ensure that accurate images are formed, especially when the photolithographic tool is attempting to print the smallest possible structures.

[0039] One area of photolithographic tool control is focus control. As is known in the art, focus is the distance from the lens to an upper surface of a substrate placed on the stage of the photolithographic tool and images are exposed in a photoresist layer on the upper surface. Having the correct focus setting for the photolithographic tool is important to ensure that accurate images are formed on the upper substrate surface. Conventional focus-exposure matrices (FEMs) are performed to determine a reference focus setting (also referred to as “best focus”) for the photolithographic tool. FEMs generate a plurality of data points representing image size measurements corresponding to various focus and exposure (i.e. dose) settings of the photolithographic tool. Typically, an exposure setting is determined prior to performing the FEM resulting in data generated from the FEM that shows the focus response of the photolithographic tool for a fixed exposure setting as shown in FIG. 3. Data points 1-15 represent measured image size values (CD) corresponding to various focus settings (Focus) for the photolithographic tool graphed on an x-y plot. The units for both the x (Focus) axis and the y (CD) axis are micrometers (um).

[0040] The focus response data from the FEMs is analyzed to determine the “best focus” setting (described in more detail herein below). As the need to print smaller and accurate images increases so does the requirement to decrease the error in the determination of the best focus setting. In one operating condition, the photolithographic tool is required to have a focus setting for the photolithographic tool be set within a tolerance of +/−0.09 um (or +/−90 nm) of the best focus condition in order for accurate images to be printed by the photolithographic tool. The inclusion of inaccurate image size measurement data results in a reduction in accuracy in the determination of best focus setting. In a conventional analysis of the focus response, inclusion of inaccurate image size measurement data can introduce an error of as much as 20% of the focus tolerance (i.e. +/−20 nm) in the calculation of best focus which can result in an incorrect best focus setting for the photolithographic tool. With an incorrect best focus setting for the photolithographic tool, printed image quality is negatively affected resulting in reduced product yield and increased manufacturing costs.

[0041] An example to illustrate an embodiment of the invention will be discussed with respect to determining a “best focus” condition for a photolithographic tool. The example is meant to be illustrative of an embodiment of the method of the invention described with reference to the steps shown in FIG. 2 and, as such, details of calculations using Equations 1-3 described herein above are not shown for the sake of clarity.

[0042] Referring to FIG. 3, data points 1-15 include well behaved data points such as, for example, data points 6, 7, 8, and less well-behaved data points such as, for example, data points 14 and 15. For image size measurement data related to the photolithographic tool, well behaved data points usually have the lowest image size measurements (or the highest image size measurements for different process or tool settings). As described with respect to FIG. 2, in step 210 a check of data points 1-15 is performed to eliminate obvious inaccurate data. A check of data points 1 and 15 corresponding to extreme x-coordinate values reveals that neither data point 1 or 15 is a minimum value so data points 1 and 15 remain in the set of data points 1-15. Then, a 4^th-order polynomial is fit to data points 1-15 and a residual error (see Equation 1) is calculated for each data point. The residual error of data point 1 is determined to be greater than a pre-determined maximum limit of 100 nanometers (nm) so data point 1 is eliminated. Thus, after step 210, the remaining data points comprise data points 2-15.

[0043] In step 215, a selection of n=6 data points from well behaved data points is made to form an initial seed parabola. For example, data points 5-10 are selected based on having the lowest measurement values from the well behaved data points. In step 220, a 2^nd-order polynomial curve (not shown) is fit to data points 5-10. In step 225, a QoF value according to Equation 2 is calculated based on the measured and estimated CD values of data points 5-10 and a QoF is calculated to be less than a pre-determined maximum limit of 25 nm. Thus, data points 5-10 are accepted as the seed parabola in step 230. An r-squared analysis (see Equation 3) is performed on data points 5-10 and the r-squared value is calculated to be greater than or equal to a pre-determined minimum limit of 0.90 which indicates that the seed parabola is a parabolic shape.

[0044] In step 255, data points 2-4 and 11-15 which were not included in the determination of the seed parabola are selected for inclusion with data points 5-10 beginning with the data point with the lowest CD value. Thus, data point 4 is selected. In step 260, a 2^nd-order polynomial curve is fit to data points 4-10, and data points 4-10 are tested for acceptability as described above with respect to step 220. In step 265, a QoF for data points 4-10 is less than the pre-determined maximum limit of 25 nm so the corresponding 2^nd-order polynomial fit curve is acceptable and data point 4 is included with data points 5-10. Data point 11 is the next lowest CD value and is selected for inclusion with data points 4-10. Repeating steps 255-265 for data point 11, it is determined that a QoF for data points 4-11 is less than the pre-determined maximum limit of 25 nm so the corresponding 2^nd-order polynomial fit curve is acceptable and data point 11 is included with data points 4-10. Data point 3 is the next lowest CD value and is selected for inclusion with data points 4-11. Repeating steps 255-265 for data point 3, it is determined that a QoF for data points 3-11 is greater than the pre-determined maximum limit of 25 nm so the corresponding 2^nd-order polynomial fit curve is not acceptable and data point 3 is removed from data points 3-11 (see step 270). Data point 2 is the next lowest CD value and is selected for inclusion with data points 4-11. Repeating steps 255-265 for data point 2, it is determined that a QoF for data points 2 and 4-11 is greater than the pre-determined maximum limit of 25 nm so the corresponding 2^nd-order polynomial fit curve is not acceptable and data point 2 is removed from data points 2 and 4-11. Likewise, data points 12, 14, 15 and 13 are selected and it is determined that none of these data points are acceptable for inclusion with data points 4-11. Thus, in step 275, all acceptable data points 4-11 have been added to the seed parabola and a best fit curve 20 is fit to data points 4-11. An r-squared analysis is performed on data points 4-11 and the r-squared value is calculated to be greater than or equal to a pre-determined minimum limit of 0.90 which indicates that best fit curve 20 is a parabolic shape as shown in FIG. 3.

[0045] The best focus setting for the photolithographic tool can be obtained from best fit curve 20 by determining the focus value corresponding to the minimum 25 of CD value as shown in FIG. 3. Accordingly, for the example described above, the best focus setting is determined to be at a focus value (i.e. x-coordinate value) of about −0.103 um corresponding to minimum 25 of CD value (i.e. y-coordinate value). Thus, the distance (i.e. z-axis) of the stage of the photolithographic tool relative to the lens is adjusted to have a best focus, or reference focus, value of −0.103 um. It is noted that in a typical FEM, more than one best fit curve 20 will be generated with each best fit curve 20 representing data from multiple field locations in an exposed image field. This is the case for a Micrascan3 photolithographic tool where multiple best fit curves 20 are analyzed and the various best focus values are used to adjust not only the distance between the stage and the lens (i.e. z-axis) but also the tilt of the stage.

[0046] With respect to a photolithographic system, an advantage of the invention over the conventional over-sampling technique is a reduction in measurement time to obtain the data points resulting in increased photolithographic tool availability for manufacturing production. Implementation of an embodiment of the invention as the focus control technique for 248/193 nm photolithographic tools in a semiconductor manufacturing facility reduced measurement sampling by approximately 33% compared to conventional methods and without degrading measurement variation. Increasing the availability of the photolithographic tool for manufacturing production reduces manufacturing costs due to an increase in product throughput for the photolithographic tool. Another advantage of the invention over the conventional 2^nd-order and 4-th order polynomial methods is the invention preserves valid center data points where the condition of best focus is most likely to be found as was described with reference to FIG. 3.

[0047] As a practical matter, the invention is embodied in computer program code stored on a computer readable storage medium such as a tape, ROM, floppy disk, compact disc, and the like. A personal computer (PC) or RS6000 tool has a display, a central processor unit (CPU), memory (RAM), and input/output devices such that the medium can be read and the tool controlled as by the program code of the invention to carry out the operations described above. Alternatively, the invention could be run on a separate computer that has control inputs to the processing tool in question. The invention has been embodied in a program written in the “C” computer programming language, however those skilled in the art will realize that any computer programming language that can perform statistical calculations can be used to carry out the invention. The behavior of the program can be controlled with parameters that may be customized based on manufacturing requirements, and tool capabilities.

[0048] While the invention has been described above with reference to the preferred embodiments thereof, it is to be understood that the spirit and scope of the invention is not limited thereby. Rather, various modifications may be made to the invention as described above without departing from the overall scope of the invention as described above and as set forth in the several claims appended hereto.