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SEMI MF1811-0704 © SEMI 2003, 2004 2 Details of these and related subjects can be found in th e references listed in Section 7. 2.8 Raw measured profi le data generally contain trending compon ents that are independent o…

SEMI MF1811-0704 © SEMI 2003, 2004 1

SEMI MF1811-0704

GUIDE FOR ESTIMATING THE POWER SPECTRAL DENSITY

FUNCTION AND RELATED FINISH PARAMETERS FROM SURFACE

PROFILE DATA

This guide was technically approved by the Global Silicon Wafer Committee and is the direct responsibility

of the North American Silicon Wafer Committee. Current edition approved for publication by the North

American Regional Standards Committee on March 14, 2004. Initially available at

www.semi.org May

2004; to be published July 2004. Original edition published by ASTM International as ASTM F 1811-97.

Last previous edition SEMI MF1811-97(2002).

1 Purpose

1.1 There is some confusion in the roughness-

measurement community concerning the use of

estimators and the calculation of power spectral

densities (PSDs) from discrete data sets. Use of this

guide can eliminate these differences and result in the

use of consistent units for the PSD and related

parameters. It also provides a uniform reporting

procedure for digital roughness data that can facilitate

communication between different workers and different

laboratories.

2 Scope

2.1 This guide defines the methodology for calculating

a set of commonly used statistical parameters and

functions of surface roughness from a set of measured

surface profile data. Its purposes are to provide

fundamental procedures and notation for processing and

presenting data, to alert the reader to related issues that

may arise in user-specific applications, and to provide

literature references where further details can be found.

2.2 This guide is limited to the analysis of one-

dimensional or profile data taken at uniform intervals

along straight lines across the surface under test,

although reference is made to the more general case of

two-dimensional measurements made over a

rectangular array of data points.

2.3 The data analysis procedures described in this

guide are generic and are not limited to specific

surfaces, surface-generation techniques, degrees of

roughness, or measuring techniques. Examples of

measuring techniques that can be used to generate

profile data for analysis are mechanical profiling

instruments using a rigid contacting probe, optical

profiling instruments that sample over a line or an array

over an area of the surface, optical interferometry, and

scanning-microscopy techniques such as atomic-force

microscopy. The distinctions between different

measuring techniques enter this guide through various

parameters and functions that are defined in Sections 4

and 5, such as their sampling intervals, bandwidths, and

measurement transfer functions.

2.4 The primary interest here is the characterization of

random or periodic aspects of surface finish rather than

isolated surface defects such as pits, protrusions,

scratches or ridges. Although the methods of data

analysis described here can be equally well applied to

profile data of isolated surface features, the parameters

and functions that are derived using the procedures

described in this guide may have a different physical

significance than those derived from random or

periodic surfaces.

2.5 The statistical parameters and functions that are

discussed in this guide are, in fact, mathematical

abstractions that are generally defined in terms of an

infinitely-long linear profile across the surface, or the

“ensemble” average of an infinite number of finite-

length profiles. In contrast, real profile data are

available in the form of one or more sets of digitized

height data measured at a finite number of discrete

positions on the surface under test. This guide gives

both the abstract definitions of the statistical quantities

of interest, and numerical procedures for determining

values of these abstract quantities from sets of

measured data. In the notation of this guide these

numerical procedures are called “estimators” and the

results that they produce are called “estimates”.

2.6 This guide gives “periodogram” estimators for

determining the root-mean-square (rms) roughness, rms

slope, and power spectral density (PSD) of the surface

directly from profile height or slope measurements.

The statistical literature uses a circumflex to distinguish

an estimator or estimate from its abstract or ensemble-

average value. For example, Â denotes an estimate of

the quality A. However, some word-processors cannot

place a circumflex over consonants in text. Any

symbolic or verbal device may be used instead.

2.7 The quality of estimators of surface statistics are, in

turn, characterized by higher-order statistical properties

that describe their “bias” and “fluctuation” properties

with respect to their abstract or ensemble-average

versions. This guide does not discuss the higher-order

statistical properties of the estimators given here since

their practical significance and use are application-

specific and beyond the scope of this document.

SEMI MF1811-0704 © SEMI 2003, 2004 2

Details of these and related subjects can be found in the

references listed in Section 7.

2.8 Raw measured profile data generally contain

trending components that are independent of the

microtopography of the surface being measured. These

components must be subtracted before the difference or

residual errors are subjected to the statistical-estimation

routines given here. These trending components

originate from both extrinsic and intrinsic sources.

Extrinsic trends arise from the rigid-body positioning of

the part under test in the measuring apparatus. In optics

these displacement and rotation contributions are called

“piston” and “tilt” errors. In contrast, intrinsic trends

arise from deliberate or accidental shape errors inherent

in the surface under test, such as a circular or parabolic

curvature. In the absence of a-priori information about

the true surface shape, the intrinsic shape error is

frequently limited to a quadratic (parabolic) curvature

of the surface. Detrending of intrinsic and extrinsic

trends is generally accomplished simultaneously by

subtracting a detrending polynomial from the raw

measured data, where the polynomial coefficients are

determined by least-squares fitting to the measured

data.

2.9 Although surfaces and surface measuring

instruments exist in real or configuration space, they are

most easily understood in frequency space, also known

as Fourier transform, reciprocal or spatial-frequency

space. This is because any practical measurement

process can be considered to be a “linear system”,

meaning that the measured profile is the convolution of

the true surface profile and the impulse response of the

measuring system; and equivalently, the Fourier-

amplitude spectrum of the measured profile is the

product of that of the true profile and the frequency-

dependent “transfer function” of the measurement

system. This is expressed symbolically by the

following equation:

)()()(

xxtruexmeas

fTfAfA = (1)

where:

A = the Fourier amplitudes,

T(f

) = instrument response function or the measure-

ment transfer function, and

= surface spatial frequency.

This factorization permits the surface and the mea-

suring system to be discussed independently of each

other in frequency space, and is an essential feature of

any discussion of measurement systems.

2.10 Figure 1 sketches different forms of the

measurement transfer function, T(f

2.10.1 Case (a) is a perfect measuring system, which

has T ( f

) = 1 for all spatial frequencies, 0 ≤ f

≤ ∞.

This is unrealistic since no real measuring instrument is

equally sensitive to all spatial frequencies.

2.10.2 Case (b) is an ideal measuring system, which

has T (f

) = 1 for LFL ≤ f

≤ HFL and T(f

) = 0

otherwise, where LFL and HFL denote the low-

frequency and high-frequency limits of the

measurement. The range LFL ≤ f

≤ HFL is called the

bandpass or bandwidth of the measurement, and ratio

HFL/LFL is called the dynamic range of the

measurement.

2.10.3 Case (c) represents a realistic measuring system,

since it includes the fact that T (f

) need not be unity

within the measurement bandpass or strictly zero

outside the bandpass.

2.11 If the measurement transfer function is known to

deviate significantly from unity within the measurement

bandpass, the measured power spectral density (PSD)

can be transformed into the form that would have been

measured by an instrument with the ideal rectangular

form through the process of digital “restoration.” In its

simplest form restoration involves dividing the

measured PSD by the known form of

| |

()

over the

measurement bandpass. Restoration is particularly

relevant to measuring instruments that involve optical

microscopes since the transfer functions of microscope

systems are not unity over their bandpass but tend to

fall linearly between unity at T (0) = 1 and T(HFL) = 0.

The need for, and methodology of digital restoration is

instrument specific and this guide places no

requirements on its use.

2.12 This guide requires that any data on surface finish

parameters or functions generated by the procedures

described herein be accompanied by an identifying

description of measuring instrument used, estimates of

its low- and high-frequency limits, LFL and HFL, and a

statement of whether or not restoration techniques were

used.

2.13 In order to make a quantitative comparison

between profile data obtained from different

measurement techniques, the statistical parameters and

functions of interest must be compared over the same or

comparable spatial-frequency regions. The most

common quantities used to compare surfaces are their

root-mean-square (rms) roughness values, which are the

square roots of the areas under the PSD between

specified surface-frequency limits. Surface statistics

derived from measurements involving different spatial-

frequency ranges cannot be compared quantitatively

except in an approximate way. In some cases

measurements with partially or even nonoverlapping

bandwidths can be compared by using analytic models

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

3.3 ANSI Standard

ANSI/ASME B46.1 — Surface Texture (Surface

Roughness, Waviness and Lay)

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:

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2 Available from the American National Standards Institute, New

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