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SEMI MF1811-0704 © SEMI 2003, 2004 5 computer data analysis p ackage (see Secti on 5.4.2 for details). 4.2.13 fini sh parameters an d functions — numbers or functions that characterize surface height fluctuations. 4.2.13…
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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
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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.
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4.2.20 low-frequency limit (LFL) [1/µm] — lowest
spatial frequency contained in a profile data set or
specification.
4.2.20.1 Discussion — The minimum LFL in a profile
measurement is the reciprocal of the length of the
surface profile. The estimated value of the PSD at this
value of the LFL is generally attenuated by the
detrending process. To avoid this effect the lowest
practical LFL is sometimes taken to be 3 to 5 times the
reciprocal of the scan length. The LFL in surface
specifications is determined by the user.
4.2.21 mean-square profile roughness, R
q
2
[nm
2
] —
ensemble-average value of the square of the height of
the detrended profile.
4.2.21.1 Discussion — This value is given by the
following relationship:
()
∫∫
+∞+
−
∞→
==
0
1
2/
2/
2
2
d)(d)(
1
Lim
xx
L
L
dLq
ffSxxZ
L
R (2)
where:
Z
d
(x) = detrended surface profile, and
S
1
(f
x
) = its power spectral density.
The intrinsic value of the mean-square roughness of an
isotropically-rough surface area equals the mean-square
roughness of any profile across it. The rms roughness,
R
q
, is distinct from the arithmetic-average roughness,
R
a
. The two are only related through a specific height-
distribution function. For example, for a Gaussian
height distribution,
qqa
RRR 798.0
2
=
π
= (3)
Note that the optics literature uses the symbol σ for R
q
.
4.2.22 mean-square profile slope, ∆
q
2
[units of choice]
— average value of the square of the slope of the
detrended profile.
4.2.22.1 Discussion — This expression, given by
∫∫
+∞+
−
∞→
π=
=∆
0
1
2
2/
2/
2
2
d)()2(d
d
d
1
Lim
xxx
L
L
d
Lq
ffSfx
x
z
L
(4)
assumes that the average slope has been removed in the
detrending process. The integrand in the frequency
integral on the right can be viewed as the slope power
spectral density. The mean-square surface slope of an
isotropically-rough two-dimensional surface is half the
mean-square gradient of the surface itself.
4.2.23 measured profile parameters and functions —
quantities derived from detrended profile data that
include the bandwidth and transfer function effects of
the particular measurement system used.
4.2.23.1 Discussion — Measured profile parameters
and functions can be used for comparing surfaces
quality providing the same measurement system is used
in all cases. In order to compare quantitative
measurements made by different measurement systems,
or to estimate intrinsic surface properties, the system
bandwidths and transfer functions must be taken into
account. In the early literature, measurement systems
were taken to be “perfect” in the sense of Section
2.10.1, and the effects of their bandwidths and transfer
functions were ignored.
4.2.24 Nyquist frequency [1/µm] — spatial frequency
equal to the reciprocal of twice the sampling interval.
See Section 4.2.15.1.
4.2.24.1 Discussion — The Nyquist frequency
represents the highest undistorted frequency involved in
a series of uniformly-spaced profile measurements.
Higher-frequency components in the surface appear at
lower-frequencies through the process of aliasing.
Unless the effects of aliasing are removed by anti-
aliasing mechanisms in the measurement process, they
corrupt the measured spectrum immediately below the
Nyquist frequency. In that case the HFL should be
taken to be a factor of 3 to 5 below the Nyquist
frequency.
4.2.25 periodic random roughness — modified version
of purely periodic roughness that has a definite
fundamental spatial frequency, but random variations in
its phase or amplitude (see Section 4.2.26).
4.2.26 periodic roughness — roughness with a definite
fundamental spatial frequency.
4.2.26.1 Discussion — The power spectra of periodic
and periodic random roughness appear as isolated peaks
in the power spectral density function. This pattern is
distinct from the broad variations appearing for purely
random surfaces. Random surfaces can be viewed as
periodic surfaces with a broad distribution of funda-
mental periods.
4.2.27 periodogram estimate — estimates of particular
finish parameters that are derived from the periodogram
estimator for the power spectrum.
4.2.28 periodogram estimator — a particular estimator
for the power spectral density that is proportional to the
square magnitude of the discrete Fourier transform of
the detrended data set.
4.2.28.1.1 Discussion — The periodogram is the
estimator for the PSD that is used in this guide. The
signal-processing literature contains many different
estimators of the power spectrum in addition to the