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SEMI MF1811-0704 © SEMI 2003, 2004 3 of the PSD s to extrapolate the PSD s outside their measurement b andwidth. 2.14 Examples of specific band-width limi ts can be drawn from the optical and semiconducto r industrie s. …
SEMI MF1811-0704 © SEMI 2003, 2004 2
Details of these and related subjects can be found in the
references listed in Section 7.
2.8 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.
2.9 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:
)()()(
xxtruexmeas
fTfAfA = (1)
where:
A = the Fourier amplitudes,
T(f
x
) = instrument response function or the measure-
ment transfer function, and
f
x
= surface spatial frequency.
This factorization permits the surface and the mea-
suring system to be discussed independently of each
other in frequency space, and is an essential feature of
any discussion of measurement systems.
2.10 Figure 1 sketches different forms of the
measurement transfer function, T(f
x
):
2.10.1 Case (a) is a perfect measuring system, which
has T ( f
x
) = 1 for all spatial frequencies, 0 ≤ f
x
≤ ∞.
This is unrealistic since no real measuring instrument is
equally sensitive to all spatial frequencies.
2.10.2 Case (b) is an ideal measuring system, which
has T (f
x
) = 1 for LFL ≤ f
x
≤ HFL and T(f
x
) = 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.
2.10.3 Case (c) represents a realistic measuring system,
since it includes the fact that T (f
x
) need not be unity
within the measurement bandpass or strictly zero
outside the bandpass.
2.11 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
()
f
x
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 guide places no
requirements on its use.
2.12 This guide 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.
2.13 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
SEMI MF1811-0704 © SEMI 2003, 2004 3
of the PSDs to extrapolate the PSDs outside their
measurement bandwidth.
2.14 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.
2.15 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
Figure 1. This is called a top-hat or binary filter
function. Before the ready availability of digital
frequency-domain processing as employed in this
guide, 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 guide recommends the determination of rms
roughness and slope values using top-hat windowing of
the measured PSD in the frequency domain.
2.16 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.
2.17 The units used in this guide are a self-consistent
set of SI units that are appropriate for many
measurements in the semiconductor and optics industry.
This guide 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.
NOTICE: This standard does not purport to address
safety issues, 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 or other limitations prior
to use.
3 Referenced Standards
3.1 SEMI Standards
SEMI ME1392 — Practice for Angle Resolved Optical
Scatter Measurements on Specular or Diffuse Surfaces
SEMI MF1048 — Test Method for Measuring the
Effective Surface Roughness of Optical Components by
Total Integrated Scattering
3.2 ASTM Standard
E 284 — Terminology Relating to Appearance of
Materials
1
3.3 ANSI Standard
ANSI/ASME B46.1 — Surface Texture (Surface
Roughness, Waviness and Lay)
2
NOTICE: Unless otherwise indicated, all documents
cited shall be the latest published versions.
4 Terminology
4.1 Introduction — This section provides the
definitions of special terms used in this guide, and
includes the mathematical definitions of different
profile statistics in terms of continuous, infinitely-long
profiles. The corresponding estimators of those
statistics based on linear, sampled, finite-trace-length
data are given in Section 5. Definitions of terms not
included here can be found in SEMI ME1392, SEMI
MF1048, ASTM Terminology E 284, or ANSI/ASME
B46.1.
4.2 Definitions
4.2.1 averaging, aperture or local — smoothing of an
estimate of the power spectral density function (PSD)
by replacing its value at a given spatial frequency by its
1 Annual Book of ASTM Standards, Vol 6.01, ASTM International,
100 Barr Harbor Drive, West Conshohocken, PA 19428. Telephone:
610-832-9500, Fax: 610-832-9555, Website:
www.astm.org
2 Available from the American National Standards Institute, New
York Office: 11 West 42nd Street, New York, NY 10036, USA.
Telephone: 212.642.4900; Fax: 212.398.0023 Website:
http://www.ansi.org
.
SEMI MF1811-0704 © SEMI 2003, 2004 4
average over a local frequency range using a particular
weighting function.
4.2.1.1 Discussion — The averaged quantities must
include the same range of surface spatial frequencies.
4.2.2 averaging, data — numerical averaging of
statistical estimates of the PSD, the mean-square
surface roughness or the mean-square profile slope
derived from different measurements, in order to obtain
a single, composite result.
4.2.2.1 Discussion — For example, a rectangular or
square array of measurements can be separated into a
set of parallel profile measurements which can be
analyzed separately and the results averaged. Again,
the averaged quantities must include the same range of
surface spatial frequencies.
4.2.3 bandwidth, bandwidth limits — range of surface
spatial frequencies included in a measurement or
specification.
4.2.3.1 Discussion — Bandwidth is specified by a
high-frequency limit (HFL) and a low-frequency limit
(LFL). The bandwidth and the measurement transfer
function over the bandwidth must be taken into account
when measurements or statistical properties are
compared. Different measuring instruments are
generally sensitive to different ranges of surface spatial
frequencies; that is, they have different bandwidth
limits. Real bandwidth limits are necessarily finite
since no measuring instrument is sensitive to infinitely-
low or to infinitely-high surface spatial frequencies.
4.2.4 bias error — average deviation between an
estimate of a statistical quantity and its true value.
4.2.4.1 Discussion — The periodogram estimator of
the power spectral density (PSD) given in this guide is a
zero-bias or unbiased estimator of the PSD. On the
other hand, local averaging of the periodogram can
introduce bias errors in regions where the spectrum
varies rapidly with frequency.
4.2.5 deterministic profile — surface profile that is a
known function of surface position, with no random
dependencies on position.
4.2.5.1 Discussion — In contrast, a random profile is
known only in terms of a probability distribution
function.
4.2.6 detrended profile, Z
d
(x) — the raw or measured
profile after removing instrumental and surface trends.
The detrended profile is the input for the statistical
estimation routines described in Section 5.
4.2.6.1 Discussion — If the parametric form of the
trend is known, its least-squares-fitted form can be
subtracted from the measured profile data. Otherwise a
generic power-series form can be used. This guide
describes the procedures for removing a zero-, first- or
second-order polynomial in the trace distance. A zero-
order polynomial removes piston; a first-order
polynomial removes piston and tilt; and a second-order
polynomial removes piston, tilt and quadratic curvature.
In each case the detrended data set has zero mean. The
coefficients of constant and linear terms correspond to
the rigid-body orientation of the part being measured
and need not be recorded. However, the coefficient of
the quadratic term represents the intrinsic curvature of
the surface being measured and should be recorded.
4.2.7 dynamic range — ratio of the high- to low-
frequency limits of the bandwidth of a given
measurement technique.
4.2.7.1 Discussion — The ratio HFL/LFL is a useful
single-number characteristic of a measuring apparatus.
It completely describes the measurement effects on
surfaces with power-law power spectra.
4.2.8 ensemble — infinitely large collection (infinite
ensemble) of quantities, the properties of which are
governed by some statistical distribution law.
4.2.8.1 Discussion — For example, ensembles include
surface profiles, and rms roughness values.
4.2.9 ensemble average value — value of a particular
surface parameter or function averaged over the
appropriate distribution functions.
4.2.9.1 Discussion — The ensemble average value of
the quantity A is denoted by <A>. Estimates of
ensemble-average quantities based on a finite collection
of measurements (finite ensemble) can deviate from
their infinite-ensemble values by fluctuation and bias
errors.
4.2.10 estimate — ensemble-average value of a
roughness statistic from a finite set of measured profile
data.
4.2.10.1 Discussion — In this guide, a circumflex is
used to distinguish estimates from the corresponding
ensemble-average quantities (see also Section 2.6).
4.2.11 estimator — algorithm or mathematical
procedure for calculating an “estimate.”
4.2.12 fast fourier transform or FFT — algorithm for
calculating the Fourier transform (discrete Fourier
transform or DFT) of a set of numerical data.
4.2.12.1 Discussion — The discovery of the FFT is
generally attributed to Cooley and Tukey, although it
was used and reported in the earlier literature by a
number of others, including Gauss, two centuries
before. It is now ubiquitous and can be found in any