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SEMI MF1811-0704 © SEMI 2003, 2004 9 individual est imates of the power spectrum , it condenses the data into a few in trinsic surface parameters, an d provides a mechanism for e xtrapolating the measured d ata outside t…

SEMI MF1811-0704 © SEMI 2003, 2004 8

or deterministic values. Also known as random

roughness.

4.2.31.1 Discussion — For example, the profile

)2cos()( φ+

= xfAxZ

(7)

is deterministic if φ = const., but random if φ has a

finite-width probability distribution function P(φ). Fin-

ish parameters and functions such as Z(x)

, are then the

values of those quantities averaged over P(φ).

4.2.32 restoration — signal-processing procedure in

which measurements are compensated for a non-unit

measurement transfer function by passing them through

a digital filter that restores the effective measurement

function to unity over its bandpass.

4.2.32.1 Discussion — The measured profile can be

restored and the statistics of the restored profile can

then be estimated. The most common spatial- and

frequency-domain filters used for this purpose are

“inverse” and “Wiener” filters. This guide does not

discuss the details of such restoration processes, which

may be found in standard signal-processing texts such

as those listed in Section 7.

4.2.33 RMS profile roughness, R

, [nm] — square root

of the mean-square profile roughness.

4.2.34 RMS profile slope, ∆

— square root of the

mean-square profile slope.

4.2.34.1 Discussion — The slope is dimensionless,

although the fundamental unit is the radian. In practice

it may be convenient to express the rms slope of highly

polished surfaces in microradians.

4.2.35 sample interval, D, [µm] — distance between

adjacent measurements of the surface height along the x

axis. Also known as sampling interval.

4.2.35.1 Discussion — The sample interval is usually

chosen or recommended by the manufacturer of the

profile instrument being used. The sample interval

defines the Nyquist frequency and hence, the extreme

HFL of the measurement. This guide does not address

measurements with unequal sample intervals or those

made along nonlinear traces over the surface.

4.2.36 sampled profile, Z(x

), [nm] — surface height,

Z(x

), measured at N equally-spaced points along the x

axis.

4.2.36.1 Discussion — This guide uses the following

indexing convention for the position of the height

samples,

NnDnx

,,2,1,)1( L=−= (8)

Therefore the distance between the first and last points

in the profile trace is (N – 1)D.

4.2.37 sampled slope, m(x

) — surface slope, m(x

measured at N equally-spaced points along the x axis

using the same indexing convention as for the sampled

profile.

4.2.37.1 Discussion — Some instruments measure the

surface slope directly, while others, in effect, measure

the surface height at N + 1 points and generate N slope

values using the equation:

[]

NnxZxZ

nnn

,,2,1,)()(

)(

L=−=

(9)

4.2.38 slope power spectrum, S

′

[µm] — statistical

function that shows how the mean-square profile slope

is distributed over surface spatial frequencies as

follows:

()

)()2(d2exp)(

Lim)(

1 xx

xLx

fSfxxfixm

fS π=













π=

′

∫

−

∞→

(10)

4.2.38.1 Discussion — This simple connection

between the slope and roughness power spectra permits

one to be determined immediately in terms of the other.

The prime on S

, on the left denotes that this is the PSD

of the slope, while the unprimed S

on the far right is

the PSD of the height.

4.2.39 spatial frequency, f

,[1/µm] — frequency

parameter in the Fourier transform of the surface profile

Z(x).

4.2.39.1 Discussion — The parameter f

is related to

the spatial wavelength, d

through f

= 1/d

. Similar

quantities are defined for the y component, and the

magnitude of the two-dimensional spatial-frequency

vector,

fff += , that appears in the two-dimen-

sional power spectral density of an isotropically-rough

surface, S

(f).

4.2.40 spatial wavelength, d

, [µm] — reciprocal of the

spatial frequency, f

4.2.40.1 Discussion — The mechanical-engineering

community frequently uses the symbol λ for the spatial

wavelength, while the optical community reserves that

symbol for the radiation wavelength.

4.2.41 spectral model — analytic expression for the

power spectral density which contains a number of

adjustable parameters called finish parameters.

4.2.41.1 Discussion — The values of the finish

parameters are obtained by fitting estimates of the PSD

of the surface height or slope fluctuations to the model.

The fitting process performs a number of important

functions: it averages out the fluctuations appearing in

SEMI MF1811-0704 © SEMI 2003, 2004 9

individual estimates of the power spectrum, it

condenses the data into a few intrinsic surface

parameters, and provides a mechanism for extrapolating

the measured data outside the measurement bandwidth.

4.2.41.2 One example of a spectral model is the ABC

model, which has the following form for the profile

PSD:

2/2

])(1[

)(

= (11)

and for the two-dimensional spectrum of an isotrop-

ically-rough surface:

2/)1(2

])(1[

)(

′

fS (12)

where:

)2/(

2/)1(

A ⋅

+Γ

⋅

′

The finish parameters in this model are A, B and C,

which have the dimensions of µm

, µm

, and µm

. This

model is sometimes called the K-correlation model, and

the quantity B/(2π), the correlation length.

4.2.41.3 Another example of a spectral model is the

fractal model, which has the following form for the

profile PSD:

)( = (13)

and for the two-dimensional spectrum of an

isotropically-rough surface:

)(

fS (14)

where:

)2/(

2/)1(

K ⋅

+Γ

⋅

The finish parameters in this model are K

and C, which

have the dimensions of µm

(3–C)

and µm

. The

dimensionless number C usually lies between 1 and 3

but need not be an integer. The quantity K

sometimes referred to as the spectral strength, and the

parameter, C, the spectral index.

4.2.41.4 The fractal model is the limiting case of the

ABC model when the finish parameter B becomes very

large. The value of the intrinsic mean-square profile

and area roughness of the ABC model, obtained by

integrating the ABC spectrum over all frequencies, is as

follows:

d)(2d)(

ffSfffSR

xxq

′

−

=π==

∫∫

∞∞

(15)

which is finite for C > 1. The intrinsic value of the

mean-square roughness of the fractal model is always

infinite because of its divergence at low spatial

frequencies. In contrast, the measured roughness

values, obtained by integrating only over the measure-

ment bandpass, are finite for the ABC model for any

value of C, and for the fractal model.

4.2.41.5 A third example of a spectral model is the

periodic model, which is for a surface consisting of a

periodic structure and has the following forms for the

profile PSD:

∑

∞=













−δ⋅=

xkx

fAfS

)( (16)

and for the two-dimensional spectrum:

∑

+∞=

−∞=

δ⋅













−δ⋅=

yxkyx

fAffS )(

),(

||2

(17)

where the k = 0 is excluded. The finish parameters of

this model are the A

' s, the Fourier amplitudes of the

periodic profile, and d

, the fundamental spatial

wavelength of the periodicity, both expressed in µm.

/2 is the mean-square roughness of the k

harmonic

of the profile (k = 1 is the fundamental), and δ(F) is a

unit-area function that is sharply peaked about the point

F = 0. The value of the intrinsic mean-square profile

and area roughness of the periodic model is as follows:

∑

∫∫∫

∞

+∞

∞−

+∞

∞−

∞

===

d),(d)(

yyxxxxq

AfffSdfffSR (18)

In contrast, the measured value is the right-hand side

summed over those spectral lines that fall within the

measurement bandpass.

4.2.41.6 Finally, it is possible to develop a composite

model that is made up of a sum of terms involving

different models or models with different parameters, or

both.

4.2.42 trace length, L [µm] — total length of the

surface sampled by a linear profile measurement. Also

known as profile length.

4.2.42.1 Discussion: In the indexing used in this guide

DNxL

)1( −

(19)

where:

= the position of the N

or last point in the

measurement (with x

assumed at zero), and

D = 1 the sampling interval (see Section

4.2.15.1).

The periodogram estimate is based on a Fourier

representation of the surface profile. The basic

periodicity of that expansion is ND rather than the

literal profile length L = (N–1)D. Depending on the

type of FFT used in the practical evaluation of the PSD,

N may be required to be a power of 2, such as 1024,

although in general, there is no restriction on N in this

guide.

4.2.43 transfer function — function of spatial

frequency having a magnitude between zero and one

which describes the sensitivity of a linear measuring

system to the amplitudes of different spatial-frequency

components in the profile being measured. Also known

as measurement transfer function.

4.2.43.1 Discussion — The ideal transfer function is

unity within the measurement bandpass and zero for

frequencies outside the bandpass. Real-world

measurement transfer functions can deviate

significantly from this. The transfer function is the

Fourier transform of the impulse response function of

the measuring apparatus.

4.2.44 uniaxial surface — surface whose roughness is

confined to a particular direction or lay, so that it can be

completely characterized by profile measurements

perpendicular to the lay direction. Surfaces that display

harmonic lines are frequently uniaxial. Also known as

grating-like surface.

4.2.44.1 Discussion — In contrast, an isotropic surface

can also be completely characterized by profile

measurements made in one direction, but there is no

preferred direction as there is for uniaxial surfaces.

Surfaces that are neither uniaxial nor isotropic can be

characterized using the procedures described in this

guide, although profile measurements taken on many

directions across the surface may be needed to generate

a complete statistical description of the surface under

test.

4.2.45 window function, W(x

) — bell-shaped or

smooth-edged function that multiplies the detrended

profile data set before it is inserted into the period-

ogram estimation routine. Also known as data window.

4.2.45.1 Discussion — The window function

“smoothes out” possible discontinuities at the ends of

the measured, finite-length data set in order to eliminate

the spurious oscillations that those discontinuities

would otherwise generate in the spectral estimate. As

long as the window function performs its function of

reducing the contributions from the ends of the data

record and has the proper normalization, its shape is of

secondary importance.

4.2.46 zero padding — procedure of adding zero

values to a data set to bring the total number of data

points, N, to a power of two to facilitate the evaluation

of the FFT appearing in the periodogram spectral

estimate.

4.2.46.1 Discussion — The window functions should

be applied to the data set before zero padding. Zero

padding is less important with the ready availability of

arbitrary- N FFT computing packages.

5 Calculations

5.1 Detrending

5.1.1 Introduction — The estimators defined in this

section are based on the analysis of a data set Z(n)

consisting of N discrete values of the surface profile

Z(x

= (n–1)D) measured at equally-spaced locations

along a straight line of length L, where n = 1 to N. If

Z(n) is the measured profile, the detrended profile is

given by:

][)()(

ncnbanZnZd ⋅+⋅+−= (20)

where the quantity in the square bracket is the quadratic

detrending polynomial. The estimated values of the

polynomial coefficients a, b, and c, denoted by a

)

, b

)

and c

)

, respectively, are determined by least-squares

fitting of the polynomial to the measured profile data as

now described. The degree of the detrending

polynomial is chosen by the following considerations:

Removing piston only (zeroth-order polynomial, a) is

useful for instructional purposes but is inadequate in

practice. It affects only the zero-frequency or “dc” term

in the power spectral density. Removing piston and tilt

(first-order polynomial, a + b·n) is sufficient for

removing uncertainties in the rigid-body positioning of

a nominally flat sample in the measurement apparatus.

Removing piston, tilt and curvature (second-order

polynomial, a + b·n + c·n

) removes an additional

quadratic term in the profile that may result from

instrumental (extrinsic) effects or true (intrinsic)

curvature in the surface being measured.

5.1.2 Piston detrending is as follows:

anZndZ

)

−= )()( (21)

and

)

(22)