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SEMI MF1811-0704 © SEMI 2003, 2004 10 where: x N = the position of the N th or last point in the measurement (with x 1 assumed at ze ro), and D = 1 the sampling interval (see Section 4.2.15. 1). The periodogram estimate …

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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)

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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)

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where M

(and also M

and M

) are evaluated using the

general moment expression,

∑

nZn

)(

(23)

where:

P = 0, 1, or 2.

5.1.3 Piston and tilt detrending is as follows:

][)()( nbanZndZ

)

+−= (24)

where:

n = 1, 2, … , N,

)

]3)12[(

MMN

−+⋅

−

+ , and

)













−

5.1.4 Piston, tilt, and curvature detrending is as

follows:

][)()(

ncnbanZndZ

)

++−= (25)

where:

n = 1, 2, … , N,

)

],10)12(6)233[(

)2)(1(

210

MMNMNN

++−++

−−

)

and],)1(30)118)(12(2

)12)(2)(13[(

)4)(1(

MNMNN

MNNN

++++−

+++

−−

−

)

.6)1(6)2)(1[(

)4)(1(

210

MMNMNN

++−++

−−

NOTE 1: The estimated values of the coefficients depend on

the degree of the polynomial being detrended. For example,

the value of the coefficient a

)

for piston detrending and

piston-plus-tilt detrending derived from the same data set are

generally different.

NOTE 2: Despite these apparent differences, the mean values

of each of the detrended profiles given by

∑

ndZ

0)(

Mean value

)

(26)

vanishes in all cases. This means that the “dc” value of the

estimated power spectrum of the detrended profile is zero,

which offers a convenient numerical check on the numerical

processing routines used.

NOTE 3: Least-squares fitting routines are available in many

computer packages. Analytic results are given above for

reference and checking.

5.2 RMS Roughness

5.2.1 There are two different estimators for the rms

roughness, R

— one expressed in configuration space,

and the other in frequency space, as follows:

∑

ndZ

)(

)Config(

))

(27)

and

∑

)(

)Freq(

)

(28)

1 where )(

)

is the periodogram estimate of the PSD

based on )(ndZ

)

(see Section 5.4).

NOTE 4: These two estimates of R

are mathematically

identical if the periodogram is evaluated using a unit data

window, W(n) = 1. Although a unit window function is not

recommended for general use, the numerical identity of the

Equation 27 and Equation 28 in that case offers a convenient

check on the programming of the periodogram estimator.

NOTE 5: If a non-unit data window is used in the calculation

of the PSD the two estimates of the rms roughness given will

not, in general, be numerically identical for a particular pro-

file measurement. On the other hand, the two estimates are

identical for an ensemble average over a large number of pro-

file measurements. In other words, the two estimates of R

are statistically the same.

NOTE 6: The first estimator has the advantage of familiarity

and simplicity since it is expressed directly in terms of the

detrended values of the measured profile data. Its disad-

vantage is that it involves, perforce, the transfer function of

the measuring apparatus, and in a nonobvious way. In con-

trast, the frequency-space form may be more complicated to

evaluate but has the advantage that it permits the bandwidth to

be included in the rms value to be varied by selecting the

range of m values included in the frequency sum. In addition,

it permits the effects of a non-unit instrumental transfer

function within that bandpass to be examined directly, and to

be divided out by restoration processes, if its form is known

independently.

NOTE 7: The spectra of real surfaces frequently tend to

diverge at low spatial frequencies so that the values of the rms

roughness obtained using either estimator may depend signi-

ficantly on the value of the LFL of the measurement process,

or chosen as a reference value. In some cases, the presence of

a non-vanishing LFL can give a finite value of the profile

roughness when its intrinsic value is infinite or undefined.