Standard Practice for Estimating the Power Spectral Density Function and Related Finish Parameters from Surface Profile Data
|Publication Date:||10 June 1997|
|ICS Code (Properties of surfaces):||17.040.20|
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 3 and 5, 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, Â 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
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-estimati
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:
A = the Fourier amplitudes,
T (fx) = instrument response function or the measurement transfer function, and
fx = surface spatial frequency.
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, T(fx):
1.10.1 Case (a) is a perfect measuring system, which has T (fx) = 1 for all spatial frequencies, 0 ≤ fx ≤ ∞. This is unrealistic since no real measuring instrument is equally sensitive to all spatial frequencies. Case (b) is an ideal measuring system, which has T (fx) = 1 for LFL ≤ fx ≤ HFL and T (fx) = 0 otherwise, where LFL and HFL denote the low-frequency and high-frequency limits of the measurement. The range LFL ≤ f x≤ HFL is called the bandpass or bandwidth of the measurement, and ratio HFL/LFL is called the dynamic range of the measurement. Case (c) represents a realistic measuring system, since it includes the fact that T (fx) need not be unity within the measurement bandpass or strictly zero outside the bandpass.
1.11 If the measurement transfer function is known to deviate significantly from unity within the measurement bandpass, the measured power spectral density (PSD) can be transformed into the form that would have been measured by an instrument with the ideal rectangular form through the process of digital "restoration." In its simplest form restoration involves dividing the measured PSD by the known form of ¦T(fx)¦2 over the measurement bandpass. Restoration is particularly relevant to measuring instruments that involve optical microscopes since the transfer functions of microscope systems are not unity over their bandpass but tend to fall linearly between unity at T(0) = 1 and T(HFL) = 0. The need for, and methodology of digital restoration is instrument specific and this practice places no requirements on its use.
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, LFL and HFL, and a statement of whether or not restoration techniques were used.
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 PSD between specified surface-frequency limits. Surface statistics derived from measurements involving different spatial-frequency ranges cannot be compared quantitatively except in an approximate way. In some cases measurements with partially or even nonoverlapping bandwidths can be compared by using analytic models of the PSDs to extrapolate the PSDs outside their measurement bandwidth.
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 µm−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 µm−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 LFL encountered in practice but do not constitute a recommendation of particular limits for the specification of surface finish parameters. Such selections are application dependent, and are to be made at the users' discretion.
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 PSD by a factor equal to zero for spatial frequencies outside the desired bandpass and unity within the desired bandpass, as shown in Case (b) in Fig. 1. This is called a top-hat or binary filter function. Before the ready availability of digital frequency-domain processing as employed in this practice, bandwidth limits were imposed by passing the profile data through analog or digital filters without explicitly transforming them into the frequency domain and multiplying by a top-hat function. The two processes are mathematically equivalent, providing the data filter has the desired frequency response. Real data filters, however, frequently have Gaussian or RC forms that only approximate the desired top-hat form that introduces some ambiguity in their interpretation. This practice recommends the determination of rms roughness and slope values using top-hat windowing of the measured PSD in the frequency domain.
1.16 The PSD and rms roughness are surface statistics of particular interest to the optics and semiconductor industries because of their direct relationship to the functional properties of such surfaces. In the case of rougher surfaces these are still valid and useful statistics, although the functional properties of such surfaces may depend on additional statistics as well. The ASME Standard on Surface Texture, B46.1, discusses additional surface statistics, terms, and measurement methods applicable to machined surfaces.
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.