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SEMI MF1811-0704 © SEMI 2003, 2004 6 4.2.20 low-frequency limit ( LFL ) [1/ µ m] — lowest spatial frequency contained in a profile d ata set or specification. 4.2.20.1 Discussion — Th e min imu m LFL in a profile measure…
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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
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periodogram. In general, they differ from the
periodogram in that they incorporate different types and
degrees of a-priori physical or mathematical
information about the original data set. The
periodogram, in contrast, includes the maximum
number of degrees of freedom and is always used for
first-cut evaluation and analysis. Details of the
correlation and other spectral estimation methods are
discussed in the literature listed in Section 7. A
mathematical variant of the periodogram estimator is
the correlation method. This is a two-step process that
requires the estimation of an intermediate function, the
autocovariance function, which is then Fourier
transformed to obtain the periodogram estimate of the
power spectrum. This method is not discussed in this
guide since it is indirect, but when properly applied
gives identically the same results as the direct transform
method recommended in this guide.
4.2.29 power spectral density (PSD) — statistical
function that shows how the mean-square (rms)
2
of a
given quantity is distributed among the various surface
spatial frequencies inherent in the profile height. Also
known as power spectrum.
4.2.29.1 Discussion — The two conventional measures
of surface roughness, R
a
and R
q
do not carry any
information about the transverse scale of the surface
roughness. That is, they are independent of how much
the surface profile is squeezed or stretched parallel to
the surface plane. The PSD is the simplest statistic that
carries that important additional information.
4.2.29.2 profile or one-dimensional PSD of the surface
height, [µm
3
] — a function of the spatial frequency, f
x
,
in units of inverse micrometers, µm
–1
, defined as
follows:
0,d)(
2
Lim)(
2
2/
2/
2
1
>
=
∫
+
−
π
∞→ x
L
L
xfi
Lx
fxexZ
L
fS
x
(5)
4.2.29.2.1 Discussion — The subscript “x” on “f
x
”
corresponds to the direction of the profile on the surface
and can be omitted if no confusion is involved. In this
definition the spatial frequency, f
x
, is always positive
and greater than zero. The value at f
x
= 0 corresponds
to the average value of the profile height, which is zero
for detrended profiles. The factor of 2 accounts for the
equal contribution from negative frequencies and en-
sures that the area under the positive-frequency profile
spectrum equals the rms-squared (mean-square) profile
height.
4.2.29.3 area or two-dimensional PSD of the surface
height [µm
4
] — a function of the spatial frequencies in
both the x and y directions on the surface, f
x
and f
y
, in
units of inverse micrometers, µm
–1
, defined as follows:
(){}
,dd2exp),(
1
Lim),(
2
2
+π=
∫∫
∞→
yxyfxfiyxZ
A
ffS
A
yxAyx
+∞<<−∞
yx
ff , (6)
4.2.29.3.1 Discussion — The spatial frequency ranges
included in this definition cover the entire frequency
plane and are not limited to positive frequencies only as
in the case of the profile spectrum. In the case of an
isotropically rough surface the area spectrum is a
function only of the magnitude of the two-dimensional
frequency vector: f = √(f
x
2
+ f
y
2
). The profile spectrum
can be derived from the area spectrum, but the area
spectrum cannot, in general, be derived from the profile
spectrum. Uniaxial and isotropically rough surfaces are
exceptions.
4.2.29.4 Discussion of Units: The surface height
fluctuations of optical surfaces are usually measured in
units of nanometers (1 nm = 10
–3
µm), or the non-SI
units of Ångstroms (1 Å = 10
–4
µm). Values of the
PSDs estimated using height data in these units can be
converted to the recommended units by multiplying by
the following conversion factors:
• To convert S
1
in units of nm
2
µm to units of µm
3
multiply it by 10
–6
,
• To convert S
1
in units of Å
2
µm to units of µm
3
multiply it by 10
–8
,
• To convert S
2
in units of nm
2
µm
2
to units of µm
4
multiply it by 10
–6
, and
• To convert S
2
in units of Å
2
µm
2
to units of µm
4
multiply it by 10
–8
.
If the sample interval is given in millimeters instead of
micrometers, the conversion factors for S
1
should be
multiplied by an additional factor of 10
3
, and those for
S
2
should be multiplied by an additional factor of 10
6
.
4.2.30 radius of curvature,
x
R
)
— radius of a circle
fitted to the measured surface profile.
4.2.30.1 Discussion — When the radius us large
relative to the profile length its magnitude is most
easily determined from the quadratic term in the
detrending polynomial. If the average surface profile is
written as Z(x) = a + bx + cx
2
, the estimate of the radius
of curvature in the x direction is ).2/(1 cR
x
=
)
If Z and x
are expressed in micrometers,
x
R
)
is in micrometers.
Since the radii of curvature of nominally flat surfaces
can be quite large, other reporting units, such as meters
or kilometers, may be more appropriate.
4.2.31 random surface profile — surface height profile
that involves parameters that are distributed according
to statistical distribution laws rather than having fixed