ASTM-F1811 › Standard Practice for Estimating the Power Spectral Density Function and Related Finish Parameters from Surface Profile Data (Withdrawn 2003)
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This standard was transferred to SEMI (www.semi.org) May 2003
1.1 This practice defines the methodology for calculating a set of commonly used statistical parameters and functions of surface roughness from a set of measured surface profile data. Its purposes are to provide fundamental procedures and notation for processing and presenting data, to alert the reader to related issues that may arise in user-specific applications, and to provide literature references where further details can be found.
1.2 The present practice is limited to the analysis of one-dimensional or profile data taken at uniform intervals along straight lines across the surface under test, although reference is made to the more general case of two-dimensional measurements made over a rectangular array of data points.
1.3 The data analysis procedures described in this practice are generic and are not limited to specific surfaces, surface-generation techniques, degrees of roughness, or measuring techniques. Examples of measuring techniques that can be used to generate profile data for analysis are mechanical profiling instruments using a rigid contacting probe, optical profiling instruments that sample over a line or an array over an area of the surface, optical interferometry, and scanning-microscopy techniques such as atomic-force microscopy. The distinctions between different measuring techniques enter the present practice through various parameters and functions that are defined in Sections and , such as their sampling intervals, bandwidths, and measurement transfer functions.
1.4 The primary interest here is the characterization of random or periodic aspects of surface finish rather than isolated surface defects such as pits, protrusions, scratches or ridges. Although the methods of data analysis described here can be equally well applied to profile data of isolated surface features, the parameters and functions that are derived using the procedures described in this practice may have a different physical significance than those derived from random or periodic surfaces.
1.5 The statistical parameters and functions that are discussed in this practice are, in fact, mathematical abstractions that are generally defined in terms of an infinitely-long linear profile across the surface, or the "ensemble" average of an infinite number of finite-length profiles. In contrast, real profile data are available in the form of one or more sets of digitized height data measured at a finite number of discrete positions on the surface under test. This practice gives both the abstract definitions of the statistical quantities of interest, and numerical procedures for determining values of these abstract quantities from sets of measured data. In the notation of this practice these numerical procedures are called "estimators" and the results that they produce are called "estimates".
1.6 This practice gives "periodogram" estimators for determining the root-mean-square (rms) roughness, rms slope, and power spectral density (PSD) of the surface directly from profile height or slope measurements. The statistical literature uses a circumflex to distinguish an estimator or estimate from its abstract or ensemble-average value. For example, denotes an estimate of the quality A. However, some word-processors cannot place a circumflex over consonants in text. Any symbolic or verbal device may be used instead.
1.7 The quality of estimators of surface statistics are, in turn, characterized by higher-order statistical properties that describe their "bias" and "fluctuation" properties with respect to their abstract or ensemble-average versions. This practice does not discuss the higher-order statistical properties of the estimators given here since their practical significance and use are application-specific and beyond the scope of this document. Details of these and related subjects can be found in References (1-10) at the end of this practice.
1.8 Raw measured profile data generally contain trending components that are independent of the microtopography of the surface being measured. These components must be subtracted before the difference or residual errors are subjected to the statistical-estimation routines given here. These trending components originate from both extrinsic and intrinsic sources. Extrinsic trends arise from the rigid-body positioning of the part under test in the measuring apparatus. In optics these displacement and rotation contributions are called "piston" and "tilt" errors. In contrast, intrinsic trends arise from deliberate or accidental shape errors inherent in the surface under test, such as a circular or parabolic curvature. In the absence of a-priori information about the true surface shape, the intrinsic shape error is frequently limited to a quadratic (parabolic) curvature of the surface. Detrending of intrinsic and extrinsic trends is generally accomplished simultaneously by subtracting a detrending polynomial from the raw measured data, where the polynomial coefficients are determined by least-squares fitting to the measured data.
1.9 Although surfaces and surface measuring instruments exist in real or configuration space, they are most easily understood in frequency space, also known as Fourier transform, reciprocal or spatial-frequency space. This is because any practical measurement process can be considered to be a "linear system", meaning that the measured profile is the convolution of the true surface profile and the impulse response of the measuring system; and equivalently, the Fourier-amplitude spectrum of the measured profile is the product of that of the true profile and the frequency-dependent "transfer function" of the measurement system. This is expressed symbolically by the following equation:
This factorization permits the surface and the measuring system to be discussed independently of each other in frequency space, and is an essential feature of any discussion of measurement systems.
1.10 Figure 1 sketches different forms of the measurement transfer function,
1.10.1 Case (
1.11 If the measurement transfer function is known to deviate significantly from unity within the measurement bandpass, the measured power spectral density (
1.12 This practice requires that any data on surface finish parameters or functions generated by the procedures described herein be accompanied by an identifying description of measuring instrument used, estimates of its low- and high-frequency limits,
1.13 In order to make a quantitative comparison between profile data obtained from different measurement techniques, the statistical parameters and functions of interest must be compared over the same or comparable spatial-frequency regions. The most common quantities used to compare surfaces are their root-mean-square (rms) roughness values, which are the square roots of the areas under the
1.14 Examples of specific band-width limits can be drawn from the optical and semiconductor industries. In optics the so-called total integrated scatter or TIS measurement technique leads to rms roughness values involving an annulus in two-dimensional spatial frequencies space from 0.069 to 1.48 m-1; that is, a dynamic range of 1.48/0.069 = 21/1. In contrast, the range of spatial frequencies involved in optical and mechanical scanning techniques are generally much larger than this, frequently having a dynamic ranges of 512/1 or more. In the latter case the subrange of 0.0125 to 1 m-1 has been used to discuss the rms surface roughness in the semiconductor industry. These numbers are provided to illustrate the magnitudes and ranges of HFL and
1.15 The limits of integration involved in the determination of rms roughness and slope values from measured profile data are introduced by multiplying the measured
1.16 The
1.17 The units used in this practice are a self-consistent set of SI units that are appropriate for many measurements in the semiconductor and optics industry. This practice does not mandate the use of these units, but does require that results expressed in other units be referenced to SI units for ease of comparison.
1.18 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.
Keywords
estimates; estimators; power spectral density; rms values; root mean square; roughness; slope; surface roughness; surface slope; surface statistics; ICS Number Code 17.040.20 (Properties of surfaces)
To find similar documents by ASTM Volume:
10.04 (Electronics; Declarable Substances in Materials; 3D Imaging Systems)
To find similar documents by classification:
17.040.20 (Properties of surfaces)
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Document Number
ASTM-F1811-97(2002)
Revision Level
1997 R02 EDITION
Status
Superseded
Modification Type
Withdrawn
Publication Date
Dec. 10, 2002
Document Type
Practice
Page Count
13 pages
Committee Number
F01.06