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Modelling and Measurement Uncertainty Estimation for Integrated AFM-CMM Instrument
Modelling and Measurement Uncertainty Estimation for Integrated AFM-CMM Instrument H.N. Hansen (2), P.Bariani, L. De Chiffre (1) Department of Manufacturing Engineering and Management Technical University of Denmark, Kgs. Lyngby, Denmark
Abstract This paper describes modelling of an integrated AFM - CMM instrument, its calibration, and estimation of measurement uncertainty. Positioning errors were seen to limit the instrument performance. Software for offline stitching of single AFM scans was developed and verified, which allows compensation of such errors. A geometrical model of the instrument was produced, describing the interaction between AFM and CMM systematic errors. The model parameters were quantified through calibration, and the model used for establishing an optimised measurement procedure for surface mapping. A maximum uncertainty of 0.8% was achieved for the case of surface mapping of 1.2x1.2 mm2 consisting of 49 single AFM scanned areas. Keywords: Atomic force microscopy (AFM), Coordinate measuring machine (CMM), Measurement uncertainty
1 INTRODUCTION Surface functionality and characterisation has been the subject of intensive research within ClRP for many years [I-21. The work has covered functionality [3], metrology [4] as well as instrumentation. Usually, these areas are interacting and overlapping in such a way that the problem of surface characterisation and analysis must be seen as an integrated problem. The increasing focus on metrology for micro- and nanotechnology has resulted in new developments dealing for example with traceability of SPM and AFM measurements [5] and with new instrument designs for CMMs [6-71. This paper describes modelling and measurement uncertainty estimation of an integrated AFM-CMM instrument developed at the authors' university [8-91. The integrated instrument is based on an AFM scanner mounted on the Z axis of a three axis CMM. The set-up has been previously described [8-101. The current AFM scanner has a range of 200 pm x 200 pm x 15 pm and the CMM has a working volume of 400 mm x 100 mm x 75 mm. The advantage of the instrument is to be able to probe large surface areas with nanometre resolution. A calibration procedure of the instrument was discussed in [9] but the practical use of the instrument was limited by manual stitching operations and a missing model for the total measurement uncertainty. This paper describes a solution of these issues. 2 STITCHING SOFTWARE The stitching principle is based on the adoption of an overlapping area between two consecutive scans to be merged in a single data file. This is achieved by displacing the sensor between single scans, by amounts of just a few micrometers smaller than the scan range. The aim is to have some surface portion represented in both images to be stitched. Best matching relative positioning of the two images is found, in the overlapping region, and then the images are stitched. Matching is based on the cross correlation function, calculated in the overlapping area, whereby any dependence on translational accuracy of the positioning system is avoided. The amount of overlap needed depends on the accuracy of the positioning device. However, this does not affect the resulting image as the
final stitching position is based on matching rather than CMM readouts [ I l l . The possibility of enlarging the measurement range is limited only by the current computer capacity in handling heavy data files and the finite working volume size of the CMM. Measurements are eventually performed in a commercially available software package on a data file resulting from a "mosaic" of single AFM scans. The automatic stitching procedure performs the following operations: a) Coarse vertical alignment of two images at a time, b) Determination of the best matching position, c) Fine vertical alignment, d) Stitching, The stitching routines were tested using software gauges. Accuracies better than 0.5 % were obtained for the lateral scaling factors when stitching up to 64 single scans [ I I 121. 3 INSTRUMENT MODELLING The integrated instrument was modelled assuming that the AFM is a measuring device that delivers three-dimensional data sets (X,Y,Z). These coordinates are associated to each probed point, with respect to the AFM reference coordinate system shown in Figure 1. The X-axis is aligned with the cantilever. The Y-axis is orthogonal to X and lying on the horizontal plane and the Z-axis is the vertical tip offset, computed from the centre of the dynamic range. The origin of the coordinate system is located in the centre of the scan range. In Figure 1, a hypothetical specimen is centred in the scanner dynamic range. The CMM reference system is illustrated in Figure 2. It corresponds to a standard CMM representation of a kinematic chain X-Y-Z. It is assumed that the two coordinate systems are aligned in such a way that the X-axes, Y-axes and Z-axes are coinciding. This definition of axes requires that the physical alignment of the axes between AFM scanner and CMM is of a high accuracy. With reference to Figure 2, the AFM scanner is mounted on the CMM Z-axis. Above the scanner, an optical microscope is mounted (not represented in the sketch). The microscope, together with a CMM rotary table and a calibration artefact featuring straight edges served for realizing the alignment of the two coordinate systems. This was achieved by recursive AFM scans over the specimen followed by manual adjustments of the AFM scanner angular orientation.
translational errors of the axes. A describes rotational errors and squareness errors. Finally AT describes rotational errors induced by probe offset XT. Integrated model The combined instrument is used in such a way that the CMM is used to position the AFM scanner in a desired position, called the origin of the mosaic. Then the first AFM scan is performed. After this, the probe is relocated through the CMM table and a new scan is performed. The position vector X can be written as:
x=x,+x, Figure 1: AFM scanner reference coordinate system. The x axis is aligned with the cantilever. The origin of the z axis is located in the middle of the scanner dynamic range.
Xp is the position of the zero point of the mosaic with respect to the CMM reference coordinate system. Xm is any further displacement vector which locates the centre of a new AFM scan with respect to the mosaic zero point. See Figure 3 for illustration. When equations (1) and (3) are introduced into equation (2) the following expression is derived:
e = P + A . X p +A.X,+AT.C.X*
Figure 2: CMM reference coordinate system The mathematical model of the integrated instrument is based on two approaches. The AFM is in this context considered a probe system mounted on a conventional CMM with three axes. AFM model The AFM scanner used in this integrated instrument is a tube scanner. It is not possible to identify three physical axes, realizing the instantaneous positioning of the probe. The raster scan is realized by actuating the tube in a proper way. However, for each coordinate set (X,Y) a Zvalue is obtained over a rectangular domain. The measurements can be modelled taking into account lateral scaling factors, XY-coupling and vertical scaling factors [12]. This results in the following linear model:
In Equation ( I ) , the calibration matrix C allows determination of the metric coordinates X through correction of measured coordinates X*. Matrix C is here extended to cover also the Z-direction in order to describe the scanner in three dimensions. In contrast to ordinary calibration procedures for AFMs it is therefore necessary to determine the coupling factors Cxz and Cyz in order to be able to complete C [13]. A higher order model than the linear equation (1) should be considered for improved AFM calibration [12]. CMM model A traditional model of the geometry of a CMM describes the real probe position in the working volume as a function of translational and rotational errors of the CMM axes [14]. This results in the following expression of the resulting Errors Vector e in the measuring volume of the CMM:
e = P + A. X
+ AT . X T
(2)
For each point in the measuring volume, X indicates the position of the CMM reference probe relative to the CMM coordinate system. P expresses positioning errors and
(3)
(4)
In Equation (4) the first two terms describe error components related to locating the zero point of the mosaic. The last two components are connected to the combined errors, within the mosaic. These are induced by AFM imaging and scanner relocations through the CMM. When using the integrated instrument in connection with the stitching software only the translational errors are assumed to be corrected for within an accuracy corresponding to the capability of the stitching algorithms. It should be noted that thermal drift of the system is not accounted for by the model.
Figure 3: Coordinate system of the integrated system Illustration of vectors Xp and Xm. 4 SYSTEM CALIBRATION In order to determine the necessary input for the model, calibration of both the AFM and the CMM was carried out. The CMM was calibrated using a combination of laser interferometers and straight edges. The results are reported in detail in [I21 and they agree with the calibration results reported in [9]. The calibration tasks connected to the AFM are: off-line non-linearity correction, lateral, vertical and vertical squareness calibrations according to equation (1). Routine calibration procedures yield Cx, Cy and Cxy using a standard calibration grid. Cz can be determined by measuring a step height standard by the so-called I S 0 5436 method [lo]. There is at present no automatic detection and correction of systematic angular deviations in the vertical planes Cxz and Cyz. However, a procedure based on a commercially available software has been developed [12].
In Figure 4, a profile of a non symmetrical triangular sample is shown. The image should not be corrected by planar fitting only, since this would induce a systematic error. With reference to Figure 4a, a, and p, are the measured slopes, calculated relative to the horizontal axis. The vertical axis is in theory orthogonal to the horizontal axis. The measured slopes, because of the so-called "installation slope" and the squrareness error, do not equal the sidewall angles of the cusp. Sidewall angles and the installation slope are defined in Figure 4b. The squareness deviation of the axes is noted by y. Upon tilting the specimen by an amount of 180 degrees around the vertical axis, and repeating the measurement, the profile of Figure 4c can be obtained. The installation slopes are not identical, since this angle is not reproduced upon repositioning of the sample; they are noted as el and e2. For the squareness calibrations 14 profiles were traced, each of them containing 5 triangles. An over determined linear system can be written with 140 equations and 3 unknowns: yxz , a , p and yyz , a , p respectively. Determination of y and the corrected slopes: a and p can be accomplished with the proposed reversal strategy. In this way coefficients for equation (1) can be determined (Cxz=tan(yxz);Cxy=tan(y,)).
Figure 4: Tilted profiles of a non symmetrical triangular microstructure. a) measured slopes. b) effective slopes differ from the nominal ones due to the installation slope el. y is the squareness error of the AFM scanner. c) Reversed sample positioning with different installation slope e2. On top of the correction terms discussed above, an issue is given by thermal drift related distortions. Drift velocities are known to be free of rotational components [10,13]. However drift effects should be conveniently described by considering both short term effects and long term ones. Surface mapping may require AFM operation for hours, involving stitching of several single AFM images. Thermal drift determines distortion in the single image. On a longer time scale, after a few probe relocations, thermal drift results in probe offset from the nominal position. In this work thermal distortion affecting single images are referred to as short term thermal drift. Long term thermal drift is considered to be the probe motion relative to indicated position after the time required to take at least one full scan. While short-term drift distortions can be corrected by image processing, long-term drift should be accounted for when setting the nominal overlap to be used in the stitching operation: because of drift the effective overlap may become too small. Lateral drift velocities in the order of hundredths of nanometres per second were observed through an experimental investigation described in [12].
5 UNCERTAINTY ESTIMATION The errors, determined through calibration, were introduced into the equations described above. Figure 5 shows the X-component of Xp, mapped over the entire CMM XY-table. Similar error components are found in Ydirection, but with smaller absolute values of the error [12]. It is clear that the CMM possesses large positioning errors. Figure 6 illustrates the error X-direction within a mosaic composed of 7x7 images, with 180 pm scan side length and origin within the CMM working volume (position x=234 mm. y=90 mm). Such systematic deviation was calculated under the assumption of stitching AFM images which are already calibrated (C,=l). The stepped distribution seen is mainly due to the deviation from XY-squareness of the CMM, revealed by calibration. The number of steps reflects the number of Y-axis displacements for building up the mosaic, which indeed was 7 for this simulation. It can be concluded that the errors on Figures 5 and 6 differ by several orders of magnitude. Since it is assumed that error contributions of the kind described in Figure 6 are compensated for by the stitching algorithms, it is expected that, under the hypothesis of effective AFM calibration, only errors in the nanometre-range are encountered when using the integrated instrument. The integrated system was tested by measuring a calibration standard containing several line patterns, covering the millimeter range. The artefact was the same as used in previous works [9]. The strategy adopted for building up the mosaic is described hereafter. The CMM table was displaced in order to locate the AFM according to the simulation above. A square array of 49 single AFM scans (7x7), was taken over the sample. The scan range was set to 200 pm x 200 pm. An overlap of 10% was kept between the neighbouring images. In Figure 7 the final image, resulting from stitching the 49 single scans, is shown: no discontinuities are discernible between single data sets. Drift correction and subtraction of the bow have been performed before stitching. Levels of brown in the image are linear with topography elevation (dark colours indicate low levels). The chromium lines are higher than the glass substrate as expected. It is worth to notice how the glass regions between two neighbouring lines are darker than the glass surface away from the chromium lines. This is obviously an imaging error, arising from the levelling procedure used to get rid of vertical drift. Single profiles are levelled with the mean vertical offset and as a consequence, those profiles which contain the chromium lines appear to have a bottom plane below the mean glass surface level. Line patterns are noted with Roman letters.
Figure 5: Mosaic zero point error in X-direction (epx) mapped over the whole CMM working volume.
The other deviations correspond in magnitude to the errors predicted by the model of the integrated instruments and the knowledge of the stitching software performance. 6 SUMMARY This paper describes modelling of an integrated AFM CMM instrument, its calibration, and estimation of measurement uncertainty. A geometrical model of the instrument was produced, describing the interaction between AFM and CMM systematic errors. The model parameters were quantified through calibration, and the model used for establishing an optimised measurement procedure for surface mapping. A maximum uncertainty of 0.8% was achieved for the case of surface mapping of 1.2x1.2 mm2 consisting of 49 single AFM images. Such uncertainty is mainly due to residual non-linearity of AFM images after calibration.
7 Figure 6: Errors within the AFM map in X-direction (emx). For the position x=234 mm. y=90 mm, 49 images stitched together to cover a range larger than 1.2 mm.
Figure 7: 3D plot of the resulting AFM image resulting from stitching 49 single scans together.
Figure 8: Deviations from reference values of dimensions measured on the image resulting from stitching. Distances within the stitched image, were measured, and compared to reference values obtained by calibration using length measurement machines and a metrology AFM. Deviations to reference values are shown in Figure 8. Perpendicular distances between 14 lines were measured on patterns Ill-Vll, while distances between 4, 7 and 8 lines were measured on patterns VIII, IX and X respectively. Sub micrometer deviations were seen for patterns Ill, IV, V, VI, IX and X over distances ranging from 222 pm - 635 pm. The most inaccurate dimension was obtained on pattern VII. There a deviation of 4.6 pm was observed on a reference distance of 555 pm, corresponding to a percent error of 0.8%. Such deviation was imputed to the nonlinearity of the scanner. This hypothesis was verified in a subsequent investigation [12].
REFERENCES [I1 Peters J., Bryan J. B., Estler W.T., Evans C., Kunzmann H., Lucca D.A., Sartori S., Sat0 , Thwaite E.G., Vanherck P., 2001, Contribution of CIRP to the development of metrology and surface quality evaluation during the last fifty years, Annals of the CIRP, 50/2:471-488. De Chiffre L. Kunzmann H., Peggs G.N., Lucca D.A., 2003, Surfaces in precision engineering, microengineering and nanotechnology, Annals of the CIRP, 52/2:561-577. [31 Evans C.J., Bryan J.B., 1999, "Structured", "textured" or "engineered" surfaces, Annals of the CIRP, 48/2:54 1-556. [41 Lonardo P.M., Trumpold H., De Chiffre L., 1996, Progress in 3D surface microtopography characterization, Annals of the CIRP, 45/2:589-598. [51 Leach R. et al, 2001, Advances in traceable nanometrology at the National Physical Laboratory, Nanotechnology 12:Rl-R6. [GI Peggs G.N., Lewis A.J., Oldfield S., 1999, Design for a compact high-accuracy CMM, Annals of the CIRP, 48/1:417-420. [71 Brand U., Kleine-Besten T., Schwenke H., 2000, Development of a special CMM for dimensional metrology of microsystem components, ASPE 15th annual meeting, Scottsdale, Arizona, p.542-546. De Chiffre, L., Hansen, H.N., Kofod, N., 1999, Surface topography characterization using an atomic force microscope mounted on a coordinate measuring machine, Annals of the CIRP, 48/1:463466. [91 Hansen H.N., Kofod N., De CHiffre L., Wanheim T., 2002, Calibration and industrial application of instrument for surface mapping based on AFM, Annals of the CIRP, 51/1:471-474. [ I 01 Kofod N., 2002, Validation and industrial application of AFM, Ph.D. Thesis, Dept. Manufacturing Engineering and Management and Danish Institute for Fundamental Metrology, Technical University of Denmark. [ I 11 De Chiffre L., Marinello F., Bariani P., Hansen H.N., 2004, Surface mapping with an AFM-CMM integrated system and stitching software, Proc. of 4th euspen Int. Conference, Glasgow, Scotland (UK), May-June 2004. [I21 Bariani P., 2005, Dimensional metrology for microtechnology, Ph.D. thesis, Dept. Manufacturing Engineering and Management, Technical University of Denmark. [I 31 Garnzs J., 2004, Personal communication. K., Kunzmann H., Waldele F., 1987, ~ 4 Busch 1 Calibration of coordinate measuring machines. Precision Engineering, 7 (3): 139-144.
Abstract This paper describes modelling of an integrated AFM - CMM instrument, its calibration, and estimation of measurement uncertainty. Positioning errors were seen to limit the instrument performance. Software for offline stitching of single AFM scans was developed and verified, which allows compensation of such errors. A geometrical model of the instrument was produced, describing the interaction between AFM and CMM systematic errors. The model parameters were quantified through calibration, and the model used for establishing an optimised measurement procedure for surface mapping. A maximum uncertainty of 0.8% was achieved for the case of surface mapping of 1.2x1.2 mm2 consisting of 49 single AFM scanned areas. Keywords: Atomic force microscopy (AFM), Coordinate measuring machine (CMM), Measurement uncertainty
1 INTRODUCTION Surface functionality and characterisation has been the subject of intensive research within ClRP for many years [I-21. The work has covered functionality [3], metrology [4] as well as instrumentation. Usually, these areas are interacting and overlapping in such a way that the problem of surface characterisation and analysis must be seen as an integrated problem. The increasing focus on metrology for micro- and nanotechnology has resulted in new developments dealing for example with traceability of SPM and AFM measurements [5] and with new instrument designs for CMMs [6-71. This paper describes modelling and measurement uncertainty estimation of an integrated AFM-CMM instrument developed at the authors' university [8-91. The integrated instrument is based on an AFM scanner mounted on the Z axis of a three axis CMM. The set-up has been previously described [8-101. The current AFM scanner has a range of 200 pm x 200 pm x 15 pm and the CMM has a working volume of 400 mm x 100 mm x 75 mm. The advantage of the instrument is to be able to probe large surface areas with nanometre resolution. A calibration procedure of the instrument was discussed in [9] but the practical use of the instrument was limited by manual stitching operations and a missing model for the total measurement uncertainty. This paper describes a solution of these issues. 2 STITCHING SOFTWARE The stitching principle is based on the adoption of an overlapping area between two consecutive scans to be merged in a single data file. This is achieved by displacing the sensor between single scans, by amounts of just a few micrometers smaller than the scan range. The aim is to have some surface portion represented in both images to be stitched. Best matching relative positioning of the two images is found, in the overlapping region, and then the images are stitched. Matching is based on the cross correlation function, calculated in the overlapping area, whereby any dependence on translational accuracy of the positioning system is avoided. The amount of overlap needed depends on the accuracy of the positioning device. However, this does not affect the resulting image as the
final stitching position is based on matching rather than CMM readouts [ I l l . The possibility of enlarging the measurement range is limited only by the current computer capacity in handling heavy data files and the finite working volume size of the CMM. Measurements are eventually performed in a commercially available software package on a data file resulting from a "mosaic" of single AFM scans. The automatic stitching procedure performs the following operations: a) Coarse vertical alignment of two images at a time, b) Determination of the best matching position, c) Fine vertical alignment, d) Stitching, The stitching routines were tested using software gauges. Accuracies better than 0.5 % were obtained for the lateral scaling factors when stitching up to 64 single scans [ I I 121. 3 INSTRUMENT MODELLING The integrated instrument was modelled assuming that the AFM is a measuring device that delivers three-dimensional data sets (X,Y,Z). These coordinates are associated to each probed point, with respect to the AFM reference coordinate system shown in Figure 1. The X-axis is aligned with the cantilever. The Y-axis is orthogonal to X and lying on the horizontal plane and the Z-axis is the vertical tip offset, computed from the centre of the dynamic range. The origin of the coordinate system is located in the centre of the scan range. In Figure 1, a hypothetical specimen is centred in the scanner dynamic range. The CMM reference system is illustrated in Figure 2. It corresponds to a standard CMM representation of a kinematic chain X-Y-Z. It is assumed that the two coordinate systems are aligned in such a way that the X-axes, Y-axes and Z-axes are coinciding. This definition of axes requires that the physical alignment of the axes between AFM scanner and CMM is of a high accuracy. With reference to Figure 2, the AFM scanner is mounted on the CMM Z-axis. Above the scanner, an optical microscope is mounted (not represented in the sketch). The microscope, together with a CMM rotary table and a calibration artefact featuring straight edges served for realizing the alignment of the two coordinate systems. This was achieved by recursive AFM scans over the specimen followed by manual adjustments of the AFM scanner angular orientation.
translational errors of the axes. A describes rotational errors and squareness errors. Finally AT describes rotational errors induced by probe offset XT. Integrated model The combined instrument is used in such a way that the CMM is used to position the AFM scanner in a desired position, called the origin of the mosaic. Then the first AFM scan is performed. After this, the probe is relocated through the CMM table and a new scan is performed. The position vector X can be written as:
x=x,+x, Figure 1: AFM scanner reference coordinate system. The x axis is aligned with the cantilever. The origin of the z axis is located in the middle of the scanner dynamic range.
Xp is the position of the zero point of the mosaic with respect to the CMM reference coordinate system. Xm is any further displacement vector which locates the centre of a new AFM scan with respect to the mosaic zero point. See Figure 3 for illustration. When equations (1) and (3) are introduced into equation (2) the following expression is derived:
e = P + A . X p +A.X,+AT.C.X*
Figure 2: CMM reference coordinate system The mathematical model of the integrated instrument is based on two approaches. The AFM is in this context considered a probe system mounted on a conventional CMM with three axes. AFM model The AFM scanner used in this integrated instrument is a tube scanner. It is not possible to identify three physical axes, realizing the instantaneous positioning of the probe. The raster scan is realized by actuating the tube in a proper way. However, for each coordinate set (X,Y) a Zvalue is obtained over a rectangular domain. The measurements can be modelled taking into account lateral scaling factors, XY-coupling and vertical scaling factors [12]. This results in the following linear model:
In Equation ( I ) , the calibration matrix C allows determination of the metric coordinates X through correction of measured coordinates X*. Matrix C is here extended to cover also the Z-direction in order to describe the scanner in three dimensions. In contrast to ordinary calibration procedures for AFMs it is therefore necessary to determine the coupling factors Cxz and Cyz in order to be able to complete C [13]. A higher order model than the linear equation (1) should be considered for improved AFM calibration [12]. CMM model A traditional model of the geometry of a CMM describes the real probe position in the working volume as a function of translational and rotational errors of the CMM axes [14]. This results in the following expression of the resulting Errors Vector e in the measuring volume of the CMM:
e = P + A. X
+ AT . X T
(2)
For each point in the measuring volume, X indicates the position of the CMM reference probe relative to the CMM coordinate system. P expresses positioning errors and
(3)
(4)
In Equation (4) the first two terms describe error components related to locating the zero point of the mosaic. The last two components are connected to the combined errors, within the mosaic. These are induced by AFM imaging and scanner relocations through the CMM. When using the integrated instrument in connection with the stitching software only the translational errors are assumed to be corrected for within an accuracy corresponding to the capability of the stitching algorithms. It should be noted that thermal drift of the system is not accounted for by the model.
Figure 3: Coordinate system of the integrated system Illustration of vectors Xp and Xm. 4 SYSTEM CALIBRATION In order to determine the necessary input for the model, calibration of both the AFM and the CMM was carried out. The CMM was calibrated using a combination of laser interferometers and straight edges. The results are reported in detail in [I21 and they agree with the calibration results reported in [9]. The calibration tasks connected to the AFM are: off-line non-linearity correction, lateral, vertical and vertical squareness calibrations according to equation (1). Routine calibration procedures yield Cx, Cy and Cxy using a standard calibration grid. Cz can be determined by measuring a step height standard by the so-called I S 0 5436 method [lo]. There is at present no automatic detection and correction of systematic angular deviations in the vertical planes Cxz and Cyz. However, a procedure based on a commercially available software has been developed [12].
In Figure 4, a profile of a non symmetrical triangular sample is shown. The image should not be corrected by planar fitting only, since this would induce a systematic error. With reference to Figure 4a, a, and p, are the measured slopes, calculated relative to the horizontal axis. The vertical axis is in theory orthogonal to the horizontal axis. The measured slopes, because of the so-called "installation slope" and the squrareness error, do not equal the sidewall angles of the cusp. Sidewall angles and the installation slope are defined in Figure 4b. The squareness deviation of the axes is noted by y. Upon tilting the specimen by an amount of 180 degrees around the vertical axis, and repeating the measurement, the profile of Figure 4c can be obtained. The installation slopes are not identical, since this angle is not reproduced upon repositioning of the sample; they are noted as el and e2. For the squareness calibrations 14 profiles were traced, each of them containing 5 triangles. An over determined linear system can be written with 140 equations and 3 unknowns: yxz , a , p and yyz , a , p respectively. Determination of y and the corrected slopes: a and p can be accomplished with the proposed reversal strategy. In this way coefficients for equation (1) can be determined (Cxz=tan(yxz);Cxy=tan(y,)).
Figure 4: Tilted profiles of a non symmetrical triangular microstructure. a) measured slopes. b) effective slopes differ from the nominal ones due to the installation slope el. y is the squareness error of the AFM scanner. c) Reversed sample positioning with different installation slope e2. On top of the correction terms discussed above, an issue is given by thermal drift related distortions. Drift velocities are known to be free of rotational components [10,13]. However drift effects should be conveniently described by considering both short term effects and long term ones. Surface mapping may require AFM operation for hours, involving stitching of several single AFM images. Thermal drift determines distortion in the single image. On a longer time scale, after a few probe relocations, thermal drift results in probe offset from the nominal position. In this work thermal distortion affecting single images are referred to as short term thermal drift. Long term thermal drift is considered to be the probe motion relative to indicated position after the time required to take at least one full scan. While short-term drift distortions can be corrected by image processing, long-term drift should be accounted for when setting the nominal overlap to be used in the stitching operation: because of drift the effective overlap may become too small. Lateral drift velocities in the order of hundredths of nanometres per second were observed through an experimental investigation described in [12].
5 UNCERTAINTY ESTIMATION The errors, determined through calibration, were introduced into the equations described above. Figure 5 shows the X-component of Xp, mapped over the entire CMM XY-table. Similar error components are found in Ydirection, but with smaller absolute values of the error [12]. It is clear that the CMM possesses large positioning errors. Figure 6 illustrates the error X-direction within a mosaic composed of 7x7 images, with 180 pm scan side length and origin within the CMM working volume (position x=234 mm. y=90 mm). Such systematic deviation was calculated under the assumption of stitching AFM images which are already calibrated (C,=l). The stepped distribution seen is mainly due to the deviation from XY-squareness of the CMM, revealed by calibration. The number of steps reflects the number of Y-axis displacements for building up the mosaic, which indeed was 7 for this simulation. It can be concluded that the errors on Figures 5 and 6 differ by several orders of magnitude. Since it is assumed that error contributions of the kind described in Figure 6 are compensated for by the stitching algorithms, it is expected that, under the hypothesis of effective AFM calibration, only errors in the nanometre-range are encountered when using the integrated instrument. The integrated system was tested by measuring a calibration standard containing several line patterns, covering the millimeter range. The artefact was the same as used in previous works [9]. The strategy adopted for building up the mosaic is described hereafter. The CMM table was displaced in order to locate the AFM according to the simulation above. A square array of 49 single AFM scans (7x7), was taken over the sample. The scan range was set to 200 pm x 200 pm. An overlap of 10% was kept between the neighbouring images. In Figure 7 the final image, resulting from stitching the 49 single scans, is shown: no discontinuities are discernible between single data sets. Drift correction and subtraction of the bow have been performed before stitching. Levels of brown in the image are linear with topography elevation (dark colours indicate low levels). The chromium lines are higher than the glass substrate as expected. It is worth to notice how the glass regions between two neighbouring lines are darker than the glass surface away from the chromium lines. This is obviously an imaging error, arising from the levelling procedure used to get rid of vertical drift. Single profiles are levelled with the mean vertical offset and as a consequence, those profiles which contain the chromium lines appear to have a bottom plane below the mean glass surface level. Line patterns are noted with Roman letters.
Figure 5: Mosaic zero point error in X-direction (epx) mapped over the whole CMM working volume.
The other deviations correspond in magnitude to the errors predicted by the model of the integrated instruments and the knowledge of the stitching software performance. 6 SUMMARY This paper describes modelling of an integrated AFM CMM instrument, its calibration, and estimation of measurement uncertainty. A geometrical model of the instrument was produced, describing the interaction between AFM and CMM systematic errors. The model parameters were quantified through calibration, and the model used for establishing an optimised measurement procedure for surface mapping. A maximum uncertainty of 0.8% was achieved for the case of surface mapping of 1.2x1.2 mm2 consisting of 49 single AFM images. Such uncertainty is mainly due to residual non-linearity of AFM images after calibration.
7 Figure 6: Errors within the AFM map in X-direction (emx). For the position x=234 mm. y=90 mm, 49 images stitched together to cover a range larger than 1.2 mm.
Figure 7: 3D plot of the resulting AFM image resulting from stitching 49 single scans together.
Figure 8: Deviations from reference values of dimensions measured on the image resulting from stitching. Distances within the stitched image, were measured, and compared to reference values obtained by calibration using length measurement machines and a metrology AFM. Deviations to reference values are shown in Figure 8. Perpendicular distances between 14 lines were measured on patterns Ill-Vll, while distances between 4, 7 and 8 lines were measured on patterns VIII, IX and X respectively. Sub micrometer deviations were seen for patterns Ill, IV, V, VI, IX and X over distances ranging from 222 pm - 635 pm. The most inaccurate dimension was obtained on pattern VII. There a deviation of 4.6 pm was observed on a reference distance of 555 pm, corresponding to a percent error of 0.8%. Such deviation was imputed to the nonlinearity of the scanner. This hypothesis was verified in a subsequent investigation [12].
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