ASTM F1811-97(2002)
(Practice)Standard Practice for Estimating the Power Spectral Density Function and Related Finish Parameters from Surface Profile Data (Withdrawn 2003)
Standard Practice for Estimating the Power Spectral Density Function and Related Finish Parameters from Surface Profile Data (Withdrawn 2003)
SCOPE
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 measu...
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Standards Content (Sample)
NOTICE: This standard has either been superceded and replaced by a new version or discontinued.
Contact ASTM International (www.astm.org) for the latest information.
Designation: F 1811 – 97 (Reapproved 2002)
Standard Practice for
Estimating the Power Spectral Density Function and Related
Finish Parameters from Surface Profile Data
This standard is issued under the fixed designation F 1811; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope that are generally defined in terms of an infinitely-long linear
profile across the surface, or the “ensemble” average of an
1.1 This practice defines the methodology for calculating a
infinite number of finite-length profiles. In contrast, real profile
set of commonly used statistical parameters and functions of
data are available in the form of one or more sets of digitized
surface roughness from a set of measured surface profile data.
height data measured at a finite number of discrete positions on
Its purposes are to provide fundamental procedures and nota-
the surface under test. This practice gives both the abstract
tion for processing and presenting data, to alert the reader to
definitions of the statistical quantities of interest, and numerical
related issues that may arise in user-specific applications, and
procedures for determining values of these abstract quantities
to provide literature references where further details can be
from sets of measured data. In the notation of this practice
found.
these numerical procedures are called “estimators” and the
1.2 The present practice is limited to the analysis of one-
results that they produce are called “estimates”.
dimensional or profile data taken at uniform intervals along
1.6 This practice gives “periodogram” estimators for deter-
straight lines across the surface under test, although reference
mining the root-mean-square (rms) roughness, rms slope, and
is made to the more general case of two-dimensional measure-
power spectral density (PSD) of the surface directly from
ments made over a rectangular array of data points.
profile height or slope measurements. The statistical literature
1.3 The data analysis procedures described in this practice
uses a circumflex to distinguish an estimator or estimate from
are generic and are not limited to specific surfaces, surface-
its abstract or ensemble-average value. For example, Â denotes
generation techniques, degrees of roughness, or measuring
an estimate of the quality A. However, some word-processors
techniques. Examples of measuring techniques that can be used
cannot place a circumflex over consonants in text. Any
to generate profile data for analysis are mechanical profiling
symbolic or verbal device may be used instead.
instruments using a rigid contacting probe, optical profiling
1.7 The quality of estimators of surface statistics are, in
instruments that sample over a line or an array over an area of
turn, characterized by higher-order statistical properties that
the surface, optical interferometry, and scanning-microscopy
describe their “bias” and “fluctuation” properties with respect
techniques such as atomic-force microscopy. The distinctions
to their abstract or ensemble-average versions. This practice
between different measuring techniques enter the present
does not discuss the higher-order statistical properties of the
practice through various parameters and functions that are
estimators given here since their practical significance and use
defined in Sections 3 and 5, such as their sampling intervals,
are application-specific and beyond the scope of this document.
bandwidths, and measurement transfer functions.
Details of these and related subjects can be found in References
1.4 The primary interest here is the characterization of
(1–10) at the end of this practice.
random or periodic aspects of surface finish rather than isolated
1.8 Raw measured profile data generally contain trending
surface defects such as pits, protrusions, scratches or ridges.
components that are independent of the microtopography of the
Although the methods of data analysis described here can be
surface being measured. These components must be subtracted
equally well applied to profile data of isolated surface features,
before the difference or residual errors are subjected to the
the parameters and functions that are derived using the
statistical-estimation routines given here. These trending com-
procedures described in this practice may have a different
ponents originate from both extrinsic and intrinsic sources.
physical significance than those derived from random or
Extrinsic trends arise from the rigid-body positioning of the
periodic surfaces.
part under test in the measuring apparatus. In optics these
1.5 The statistical parameters and functions that are dis-
displacement and rotation contributions are called “piston” and
cussed in this practice are, in fact, mathematical abstractions
“tilt” errors. In contrast, intrinsic trends arise from deliberate or
accidental shape errors inherent in the surface under test, such
This practice is under the jurisdiction of ASTM Committee F01 on Electron-
icsand is the direct responsibility of Subcommittee F01.06 on Silicon Materials and
Process Control. The boldface numbers in parentheses refer to the list of references at the end of
Current edition approved June 10, 1997. Published August 1997. this practice.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
NOTICE: This standard has either been superceded and replaced by a new version or discontinued.
Contact ASTM International (www.astm.org) for the latest information.
F 1811 – 97 (2002)
as a circular or parabolic curvature. In the absence of a-priori methodology of digital restoration is instrument specific and
information about the true surface shape, the intrinsic shape
this practice places no requirements on its use.
error is frequently limited to a quadratic (parabolic) curvature
1.12 This practice requires that any data on surface finish
of the surface. Detrending of intrinsic and extrinsic trends is
parameters or functions generated by the procedures described
generally accomplished simultaneously by subtracting a de-
herein be accompanied by an identifying description of mea-
trending polynomial from the raw measured data, where the
suring instrument used, estimates of its low- and high-
polynomial coefficients are determined by least-squares fitting
frequency limits, LFL and HFL, and a statement of whether or
to the measured data.
not restoration techniques were used.
1.9 Although surfaces and surface measuring instruments
1.13 In order to make a quantitative comparison between
exist in real or configuration space, they are most easily
profile data obtained from different measurement techniques,
understood in frequency space, also known as Fourier trans-
the statistical parameters and functions of interest must be
form, reciprocal or spatial-frequency space. This is because
compared over the same or comparable spatial-frequency
any practical measurement process can be considered to be a
regions. The most common quantities used to compare surfaces
“linear system”, meaning that the measured profile is the
are their root-mean-square (rms) roughness values, which are
convolution of the true surface profile and the impulse response
the square roots of the areas under the PSD between specified
of the measuring system; and equivalently, the Fourier-
surface-frequency limits. Surface statistics derived from mea-
amplitude spectrum of the measured profile is the product of
surements involving different spatial-frequency ranges cannot
that of the true profile and the frequency-dependent “transfer
be compared quantitatively except in an approximate way. In
function” of the measurement system. This is expressed
some cases measurements with partially or even nonoverlap-
symbolically by the following equation:
ping bandwidths can be compared by using analytic models of
A ~ f ! 5 A ~ f ! · T ~f !
meas x true x x
the PSDs to extrapolate the PSDs outside their measurement
bandwidth.
where:
1.14 Examples of specific band-width limits can be drawn
A = the Fourier amplitudes,
T( f ) = instrument response function or the measurement from the optical and semiconductor industries. In optics the
x
transfer function, and so-called total integrated scatter or TIS measurement technique
f = surface spatial frequency.
leads to rms roughness values involving an annulus in two-
x
–1
dimensional spatial frequencies space from 0.069 to 1.48 μm
This factorization permits the surface and the measuring ;
that is, a dynamic range of 1.48/0.069 = 21/1. In contrast, the
system to be discussed independently of each other in fre-
range of spatial frequencies involved in optical and mechanical
quency space, and is an essential feature of any discussion of
measurement systems. scanning techniques are generally much larger than this,
frequently having a dynamic ranges of 512/1 or more. In the
1.10 Figure 1 sketches different forms of the measurement
–1
latter case the subrange of 0.0125 to 1 μm has been used to
transfer function, T( f ):
x
discuss the rms surface roughness in the semiconductor indus-
1.10.1 Case (a) is a perfect measuring system, which has
try. These numbers are provided to illustrate the magnitudes
T ( f ) = 1 for all spatial frequencies, 0 # f # ‘ . This is
x x
and ranges of HFL and LFL encountered in practice but do not
unrealistic since no real measuring instrument is equally
constitute a recommendation of particular limits for the speci-
sensitive to all spatial frequencies. Case (b) is an ideal mea-
fication of surface finish parameters. Such selections are
suring system, which has T (f )=1for LFL # f # HFL and
x x
application dependent, and are to be made at the users’
T (f ) = 0 otherwise, where LFL and HFL denote the
x
discretion.
low-frequency and high-frequency limits of the measurement.
1.15 The limits of integration involved in the determination
The range LFL # f # HFL is called the bandpass or
x
bandwidth of the measurement, and ratio HFL/LFL is called of rms roughness and slope values from measured profile data
the dynamic range of the measurement. Case (c) represents a are introduced by multiplying the measured PSD by a factor
realistic measuring system, since it includes the fact that T (f ) equal to zero for spatial frequencies outside the desired
x
need not be unity within the measurement bandpass or strictly bandpass and unity within the desired bandpass, as shown in
zero outside the bandpass.
Case (b) in Fig. 1. This is called a top-hat or binary filter
function. Before the ready availability of digital frequency-
1.11 If the measurement transfer function is known to
domain processing as employed in this practice, bandwidth
deviate significantly from unity within the measurement band-
limits were imposed by passing the profile data through analog
pass, the measured power spectral density (PSD) can be
or digital filters without explicitly transforming them into the
transformed into the form that would have been measured by
frequency domain and multiplying by a top-hat function. The
an instrument with the ideal rectangular form through the
process of digital “restoration.” In its simplest form restoration two processes are mathematically equivalent, providing the
data filter has the desired frequency response. Real data filters,
involves dividing the measured PSD by the known form of
however, frequently have Gaussian or RC forms that only
?T ~ f ! ? over the measurement bandpass. Restoration is par-
x
ticularly relevant to measuring instruments that involve optical approximate the desired top-hat form that introduces some
ambiguity in their interpretation. This practice recommends the
microscopes since the transfer functions of microscope systems
are not unity over their bandpass but tend to fall linearly determination of rms roughness and slope values using top-hat
between unity at T (0) = 1 and T(HFL) = 0. The need for, and windowing of the measured PSD in the frequency domain.
NOTICE: This standard has either been superceded and replaced by a new version or discontinued.
Contact ASTM International (www.astm.org) for the latest information.
F 1811 – 97 (2002)
1.16 The PSD and rms roughness are surface statistics of not included here will be found in Terminology E 284, Practice
particular interest to the optics and semiconductor industries E 1392, Test Method F 1048 or ANSI/ASME B46.1.
because of their direct relationship to the functional properties
3.2 aperture averaging, local averaging, data
of such surfaces. In the case of rougher surfaces these are still
averaging—As used here, aperture and local averaging mean
valid and useful statistics, although the functional properties of
that an estimate of the power spectral density function (PSD)is
such surfaces may depend on additional statistics as well. The
“smoothed” by replacing its value at a given spatial frequency
ASME Standard on Surface Texture, B46.1, discusses addi-
by its average over a local frequency range using a particular
tional surface statistics, terms, and measurement methods
weighting function. Data averaging means the numerical aver-
applicable to machined surfaces.
aging of statistical estimates of the PSD, the mean-square
1.17 The units used in this practice are a self-consistent set
surface roughness or the mean-square profile slope derived
of SI units that are appropriate for many measurements in the
from different measurements, in order to obtain a single,
semiconductor and optics industry. This practice does not
composite result. For example, a rectangular or square array of
mandate the use of these units, but does require that results
measurements can be separated into a set of parallel profile
expressed in other units be referenced to SI units for ease of
measurements which can be analyzed separately and the results
comparison.
averaged.
1.18 This standard does not purport to address all of the
3.2.1 Discussion—The averaged quantities must include the
safety concerns, if any, associated with its use. It is the
same range of surface spatial frequencies.
responsibility of the user of this standard to establish appro-
3.3 bandwidth, bandwidth limits—The range of surface
priate safety and health practices and determine the applica-
spatial frequencies included in a measurement or specification.
bility of regulatory limitations prior to use.
It is specified by a high-frequency limit (HFL) and a low-
frequency limit (LFL).
2. Referenced Documents
3.3.1 Discussion—The bandwidth and the measurement
2.1 ASTM Standar
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