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SEMI MF1811-0704 © SEMI 2003, 2004 4 average over a local frequency range usi ng a particular weighting function. 4.2.1.1 Discussi on — The averaged quan tities must include the same range of surface spatial frequencies.…
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
SEMI MF1811-0704 © SEMI 2003, 2004 5
computer data analysis package (see Section 5.4.2 for
details).
4.2.13 finish parameters and functions — numbers or
functions that characterize surface height fluctuations.
4.2.13.1 Discussion — The values and forms of finish
parameters and functions may vary depending on the
bandwidth of surface frequencies that they contain, and
the shapes of the transfer functions of the measurement
instruments involved. These quantities are represented
by their ensemble-average values derived from
measurements using specific estimation routines. In
general, the finish parameters and functions of an area
are different from those of profiles taken across the
surface. In the case of surfaces that are statistically
isotropic, however, the area and profile statistics have a
one-to-one relationship. Except for incidental remarks,
this guide is concerned exclusively with the properties
of surface profiles.
4.2.14 fluctuation error — general term denoting the
deviation of a quantity from its mean, average or
detrended value.
4.2.14.1 Discussion — Fluctuation errors are usually
measured in terms of their mean-square or rms values.
For example, R
q
is the rms fluctuation error in the
surface height and ∆
q
is the rms fluctuation error in the
profile slope. In turn, the estimates of R
q
and ∆
q
have
their own fluctuation errors. The magnitudes of these
higher fluctuation errors not discussed in this guide.
4.2.15 high-frequency limit (HFL) [1/µm] — highest
spatial frequency contained in a profile data set or
specification.
4.2.15.1 Discussion — The HFL of a measurement is
determined by the details of the measurement process,
and its value in specifications is determined by the user.
If the sampling interval in the measurement process is
D, the extreme value of the HFL is given by the
Nyquist criterion, HFL = 1/(2D). However, other
electrical, mechanical, or optical filtering mechanisms
may further limit the HFL. Examples of such
mechanisms are the stylus tip radius, projected
measurement pixel size, optical resolution, and
electrical and digital filters, all of which contribute to
the high-frequency roll-off of the instrument transfer
function. If the Nyquist frequency is used to determine
the HFL, care should be taken to determine that the true
HFL is not reduced by these additional mechanisms.
4.2.16 intrinsic surface or finish parameters —
characteristics such as the rms roughness or rms slope
that contain all surface spatial frequencies from zero to
infinity.
4.2.16.1 Discussion — Intrinsic parameters are
statistical abstractions that cannot be measured or
estimated directly since real measurements are sensitive
to only limited ranges of surface spatial frequencies.
They can, however, be inferred from real measurements
by augmenting measurements with a-priori information
about very low and very high spatial frequencies
contained in physically-based models of the PSDs of
the surfaces involved. All measured finish parameters
are finite, but their corresponding intrinsic values need
not be. The important distinction between intrinsic and
measured (band-width limited) finish parameters is not
always made in the literature.
4.2.17 impulse response, of a profile measuring system
— measured shape of an impulse or infinitely-sharp
ridge lying perpendicularly to the profile direction.
4.2.17.1 Discussion — In the case of a linear
measuring system the impulse response is the Fourier
transform of the system transfer function. The impulse
response of a perfect measuring system would be an
infinitely sharp spike or delta function. In contrast, the
impulse response of real measuring systems has a finite
width.
4.2.18 isotropic surface — surface whose intrinsic
finish parameters and functions are independent of the
rotational position of the surface about its surface
normal. Also known as statistically-isotropic surface.
4.2.18.1 Discussion — The rms roughness of profiles
taken across an isotropically rough surface is
independent of the profile directions, and equals the
rms roughness of the surface area. The rms slope of an
isotropically rough surface is also independent of the
profile direction and equals 1/√2 of the rms area
gradient. The one-dimensional or profile power spec-
trum of an isotropic surface is also independent of the
direction of the profile on the surface, and is related to
the two-dimensional spectrum of the surface area by an
integral transform. Examples of this are given in
Section 4.2.41.
4.2.19 linear systems, linear measurement system — a
signal-processing concept more precisely described as a
linear, shift-invariant system.
4.2.19.1 Discussion — For the present purposes, a
linear measurement of the surface profile is the true
profile convolved with the impulse response of the
measuring system, or equivalently, the Fourier
amplitude spectrum of the measurement is the true
amplitude spectrum times the measurement transfer
function as indicated in Section 2.9. All practical
measurement systems are taken to be linear over their
operating ranges.