X-ray Porosimetry

Christopher L. Soles, Hae-Jeong Lee,Eric K. Lin, and Wen-li Wu

960-13Special Publication 960-13

i

DEPARTMENT OF COMMERCE

UNITED STATES OF AMERICA

NIST Recommended Practice Guide

Special Publication 960-13

Pore Characterization

in Low-k Dielectric

Films Using X-ray

Reflectivity:

X-ray Porosimetry

Christopher L. Soles, Hae-Jeong Lee,

Eric K. Lin, and Wen-li Wu

NIST Polymers Division

June 2004

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Undersecretary for Technology

National Institute of Standards and Technology

Arden L. Bement, Jr., Director
ii

◆ Pore Characterization: X-ray Porosimetry

Certain commercial entities, equipment, or materials may be identified in

this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or

endorsement by the National Institute of Standards and Technology, nor is it

intended to imply that the entities, materials, or equipment are necessarily the

best available for the purpose.

National Institute of Standards and Technology

Special Publication 960-13

Natl. Inst. Stand. Technol.

Spec. Publ. 960-13

68 pages (June 2004)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2004

For sale by the Superintendent of Documents

U.S. Government Printing Office

Internet: bookstore.gpo.gov Phone: (202) 512–1800 Fax: (202) 512–2250

Mail: Stop SSOP, Washington, DC 20402-0001
iii

FOREWORD

The persistent miniaturization or rescaling of the integrated chip (IC) has

led to interconnect dimensions that continue to decrease in physical size.

This, coupled with the drive for reduced IC operating voltages and decreased

signal-to-noise ratio in the device circuitry, requires new interlayer dielectric

(ILD) materials to construct smaller and more efficient devices. At the current

90 nm technology node, fully dense organosilicate materials provide sufficient

ILD shielding within the interconnect junctions. However, for the ensuing

65 nm and 45 nm technology nodes, porous ILD materials are needed to

further decrease the dielectric constant k of these critical insulating layers.

The challenge to generate sufficient porosity in sub-100 nm features and films

is a significant one. Increased levels of porosity are extremely effective at

decreasing k, but high levels of porosity deteriorate the mechanical properties

of the ILD structures. Mechanically robust ILD materials are needed to

withstand the stresses and strains inherent to the chemical–mechanical

polishing steps in IC fabrication. To optimize both k and the mechanical

integrity of sub-100 nm ILD structures requires exacting control over the pore

formation processes. The first step in achieving this goal is to develop highly

sensitive metrologies that can accurately quantify the structural attributes of

these nanoporous materials. This Recommended Practice Guide is dedicated

to developing X-ray Porosimetry (XRP) as such a metrology. It is envisaged

that XRP will facilitate the development of nanoporous ILD materials,

help optimize processing and fabrication parameters, and serve as a

valuable quality control metrology. Looking beyond CMOS technology,

many attributes of XRP will be useful for the general characterization of

nanoporous materials which are becoming increasingly important in many

emerging fields of nanotechnology.

Foreword ◆
iv

◆ Pore Characterization: X-ray Porosimetry
v

Acknowledgments ◆

ACKNOWLEDGMENTS

The authors would like to thank the many individuals who over the past

several years directly contributed to our low-k dielectrics characterization

project. These individuals include Barry Bauer, Ronald Hedden, Da-Wei Liu,

Bryan Vogt, William Wallace, Howard Wang, Michael Silverstein, Gary Lynn,

Todd Ryan, Jeff Wetzel, and a long list of collaborators identified in reference [4].

In addition, we are also grateful for the support from the NIST Office of

Microelectronics Programs and International SEMATECH. Without their

financial backing, this work would not have been possible. Finally, a special

debt of thanks goes to Barry Bauer and Ronald Hedden who were especially

instrumental in completing this Recommended Practice Guide.
vi

◆ Pore Characterization: X-ray Porosimetry
vii

Table of Contents ◆

TABLE OF CONTENTS

List of Figures .....................................................................................ix

List of Tables ........................................................................................x

I. INTRODUCTION .............................................................................1

1.A. Basics of Porosimetry ..........................................................2

1.B. The Concept of X-ray Porosimetry .....................................5

1.C. Fundamentals of Specular X-ray Reflectivity ....................5

1.C.1. Reflectivity from a Smooth Surface ........................7

1.C.2. Reflectivity from a Thin Film on a

Smooth Substrate .....................................................9

II. EXPERIMENTAL ...........................................................................13

2.A. X-ray Reflectometer Requirements ..................................13

2.A.1. Resolution Effects...................................................17

2.A.2. Recommended Procedure for Sample

Alignment.................................................................19

2.B. Methods of Partial Pressure Control ................................24

2.B.1. Isothermal Mixing with Carrier Gases ..................24

2.B.2. Sample Temperature Variations in a

Vapor Saturated Carrier Gas ..................................28

2.B.3. Pure Solvent Vapor .................................................30

2.B.4. Choice of Adsorbate ...............................................31

III. DATA REDUCTION AND ANALYSIS ...........................................33

3.A. Reducing the X-ray Reflectivity Data ................................33

3.B. Fitting the X-ray Reflectivity Data .....................................35

3.C. Interpretation of the XRP Data ...........................................35
viii

◆ Pore Characterization: X-ray Porosimetry

3.D. Special Concerns ................................................................47

3.D.1. Quantitative Interpretation of the

Physisorption Isotherms ........................................47

3.D.2. Isothermal Control ..................................................49

3.D.3. Time Dependence of Desorption ...........................50

3.D.4. P versus T Variations of P/Po ................................52

IV. SUMMARY ....................................................................................53

V. REFERENCES ..............................................................................55
ix

List of Figures

Figure 1. Schematic illustration of the six main types of gas physisorption

isotherms, according to the IUPAC classification scheme. ..................3

Figure 2. Calculated X-ray reflectivity results from silicon surface with

different surface roughness. The solid and dashed curves

correspond to RMS surface roughness values of 1 Å and

10 Å, respectively .................................................................................8

Figure 3. Calculated X-ray reflectivity data for Si surfaces with different

linear absorption coefficients. The solid curve corresponds to

a nominal absorption coefficient of μ = 1.47×10–6 Å–1 for pure

Si, while the dashed curve indicates an order of magnitude

increase, μ = 1.47×10–5 Å–1 ................................................................................ 9

Figure 4. Part (a) displays the low Q theoretical X-ray reflectivity for a

3000 Å thick PS film supported on a Si wafer; the inset extends

the reflectivity to high Q. The marked decrease in the reflectivity

at Q = 0.022 Å–1 and 0.032 Å–1 corresponds to the Qc values of

PS and Si, respectively. The region between these two Qc values

is the wave-guiding region of the reflectivity curve. Part (b)

illustrates how a ten-fold increase in the linear absorption

coefficient μ of the film significantly affects the reflected

intensities in this wave-guiding region ................................................10

Figure 5. The diagram in part (a) depicts the relevant degrees of freedom

for an X-ray reflectometer. In this representation, θ defines

goniometer angles for both the incident X-ray beam and the

detector position (x-ray optics) while Ω, Φ, and z are the tilt,

yawl, and vertical translation (x and y would be the lateral

translations) of the sample stage. The X-ray goniometer and

sample stage move/rotate on separate goniometers, Gx and Gs

respectively, as shown in part (b), and it is advisable to make

these circles concentric. It is critical that the X-ray beam passes

through the center O of the Gx ; detector and X-ray source

positions DN and SN where the beam does not pass through the

origin are unacceptable, as discussed in the text ................................14

Figure 6. Calculated X-ray reflectivity curves for a 1 μm PS film with

instrumental angular divergences of (3×10–5, 6×10–5 and

1.2×10–4) rad in both the wave-guiding (part a) and high Q

(part b) regimes. Increasing angular divergence significantly

damps the interference fringes, especially at low Q within

the wave-guiding regime .....................................................................18

Table of Contents ◆
x

◆ Pore Characterization: X-ray Porosimetry

Figure 7. Theoretical X-ray reflectivity curves for a 1 μm PS film with two

different wavelength dispersions. The solid line corresponds to

a wavelength dispersion (wd) of δλ/λ = 6×10–5 while the dotted

line denotes δλ/λ = 6×10–4. The main body illustrates these

effects in the low Q wave-guiding region while the inset

emphasizes the high Q effects ............................................................ 19

Figure 8. Panels (a) through (e) illustrate the process required to align

the X-ray reflectometer, as described in the text. In each panel,

the dotted vertical line indicates the current zero position for

each degree of freedom while the dashed line indicates the new

position that should be reinitialized to zero. Data are presented

for illustrating the alignment procedure only, so the standard

uncertainties are irrelevant and not provided....................................... 20

Figure 9. A θ scan of a well-aligned reflectometer. The pronounced peak

(off the vertical scale of the plot) from the main beam defines

the θ = 0° condition. The linear increase in intensity with the

angle beyond the main beam peak comes from increasing the

fraction of the beam that is focused onto the sample (footprint

effect). Standard uncertainty in the intensity is less than the

line width .............................................................................................23

Figure 10. Schematics depicting the different methods by which the partial

pressure in the XRP sample chamber can be controlled. Part (a)

is based on bubbling a carrier gas through a solvent reservoir

while part (b) illustrates controlled flow of pure vapor into a

chamber that is continually being evacuated by a vacuum pump ....... 25

Figure 11. UV absorption spectra of mixtures of pure and toluene-saturated

air with a total flow rate of 100 sccm. Starting with the flat

featureless spectra showing nearly 0 absorbance (pure airflow),

each successive curve indicates a 10 sccm increment in the

flow rate of the toluene-saturated air and a 10 sccm decrement

in the pure airflow rate. The ratio of the integrated areas

between the curve of interest and the toluene-saturated curve,

using the pure air curve as a baseline, defines P/P0. The inset

shows a linear correlation between the desired set-point and the

measured P/P0 , with a least-squared deviation of χ2 = 0.0002 ........ 26

Figure 12. There are two ways in which rc can be varied in the Kelvin

equation. rc values from approximately (1 to 250) Å can be

obtained by mixing ratios of dry and toluene-saturated air at

20 °C. Likewise, a comparable range of pore sizes can be

achieved by flowing air saturated in toluene at 20 °C across

the sample and heating the film between 20 °C and 125 °C ............... 29
xi

Figure 13. X-ray reflectivity data from a well-aligned sample and the

intensity from the background as measured at an angle offset

from the specular angle. The background level is indicated in

the inset schematic. Standard uncertainties are smaller than

the size of the data markers ................................................................ 34

Figure 14. SXR curves of porous HSQ, porous MSQ, and porous SiCOH

thin films. The curves are offset for clarity. The standard

uncertainty in log (I/I0) is less than the line width............................. 36

Figure 15. Critical angle changes for the porous HSQ sample as P/P0

increases systematically from 0 (dry air) to 1 (toluene-saturated

air). Condensation of the toluene inside the pores results in

an appreciable and measurable change in Qc

2 ............................................. 39

Figure 16. Physisorption isotherms for the porous HSQ, MSQ, and SiCOH

films. The lines are smooth fits using the cumulative sum of a

sigmoidal and a log-normal function for porous HSQ and MSQ

films, and the sum of a Gaussian and a sigmoidal function for

porous SiCOH film. We generally recommend fitting these

physisorption isotherms with such arbitrary fit functions.

If the curve faithfully parameterizes the experimental data, it

will be helpful later in extracting a smooth pore size distribution

(discussed later in Figure 18). Estimated standard uncertainties

are comparable to the size of the data markers...................................40

Figure 17. Schematic pore structures for the porous HSQ, MSQ, and

SiCOH films ........................................................................................ 41

Figure 18. Approximate pore size distributions from the fits (not data)

through the physisorption isotherms in Figure 16, using Eq. (1)

used to convert P/P0 into a pore size. The distributions from

the adsorption branch (solid lines) can be significantly broader

and shifted to larger pore sizes than the corresponding desorption

branch, especially in those materials (like the MSQ film) with

a large distribution of mesopore sizes .................................................42

Figure 19. SANS data for the porous HSQ (circles), MSQ (diamonds),

and SiCOH (triangles) films under vacuum. Error bars, usually

small than the data markers, indicate the standard uncertainty

in the absolute intensity .......................................................................44

Table of Contents ◆
xii

◆ Pore Characterization: X-ray Porosimetry

Figure 20. XRP data for a low-k film comprised of 3 distinct layers.

Part (a) shows the reflectivity data for the dry and

toluene-saturated films, revealing both a high-frequency

periodicity due to the total film thickness and low-frequency

oscillations due to the thinner individual layers. Part (b)

shows the real space scattering length density profiles as

a function of distance into the film, revealing the thickness

and density of the individual layers ..................................................... 46

Figure 21. Physisorption isotherms generated by the T (squares) and P

(all other data markers) variation methodologies of controlling

P/P0, as described in the text. The two techniques do not produce

isotherms indicating that adsorption is not temperature invariant.

Notice the discontinuous nature of the desorption pathways of

the P variation technique. These discontinuities can be attributed

to insufficient equilibration times, as described in the text and

in reference to Figure 22. The estimated standard uncertainty

in Qc

2 is comparable to the size of the data markers .......................... 50

Figure 22. Time dependence of the Qc

2 variations in the porous HSQ

sample after P/P0 jumps from 1.0 to 0.36 (squares) and

1.0 to 0.28 (circles). Notice that Qc

2 continues to evolve

for several hours after the jump, indicating the equilibrium

is difficult to achieve on the desorption branch. The same

time dependence is curiously absent upon adsorption.

The estimated standard uncertainty in Qc

2 is comparable

to the size of the data markers ............................................................ 51
xiii

List of Tables

Table 1: A summary of the atomic compositions, dielectric constants (k),

and structural characteristics of different porous low-k films.

The estimated standard uncertainties of the atomic compositions,

Qc

2, densities, porosities and pore radii are ± 2 %, 0.05 Å–2,

0.05 g/cm3, 1 %, and 1 Å, respectively ..............................................39

Table of Contents ◆
xiv

◆ Pore Characterization: X-ray Porosimetry
1

Introduction ◆

1. INTRODUCTION

Increased miniaturization of the integrated chip has largely been responsible

for the rapid advances in semiconductor device performance, driving the

industry’s growth over the past decade(s). Soon the minimum feature size

in a typical integrated circuit device will be well below 100 nm. At these

dimensions, interlayers with extremely low dielectric constants (k) are

imperative to reduce the cross-talk between adjacent lines and also enhance

device speed. State-of-the-art non-porous, silicon-based low-k dielectric

materials have k values on the order of 2.7. However, k needs to be

further reduced to keep pace with the demand for increased miniaturization.

There are a number of potential material systems for these next generation

low-k dielectrics, including organosilsesquioxane resins, sol-gel based silicate

materials, chemical-vapor deposited silica, and polymeric resins. It is not yet

evident as to which material(s) will ultimately prevail. Nevertheless, it has

become apparent that decreasing k beyond current values universally requires

generating significant levels of porosity within the film; sufficiently low enough

dielectric constants are not feasible with fully dense materials.

The demand of increased porosity in reduced dimensions faces many

challanges. To generate extensive porosity in sub-500 nm films and/or

features requires exacting control of the pore generation process. The first

step towards achieving this control, before addressing materials concerns,

is developing metrologies that quantitatively characterize the physical pore

structure with accuracy and reproduciblity in terms of porosity, average pore

size, pore size distribution, etc. Porosimetry, that is pore characterization from

a measurable response to liquid intrusion into the porous material, is a mature

field. There are several techniques, such as gas adsorption, mercury intrusion,

mass uptake, etc., capable of characterizing pores significantly smaller than

100 nm. However, these traditional methods lack the sensitivity to quantify

porosity in thin, low-k films. The sample mass in a 500 nm thick film will be

less than a few mg and the usual observables (i.e., pressure in a gas adsorption

experiment or mass in an gravimetric experiment) exhibit exceedingly small

changes as the pores are filled with the condensate. Thin film porosimetry

requires extraordinary sensitivity.

Currently, there are only a few techniques suitable for the on-wafer

characterization of the pore structures in low-k dielectric films. The most

widely known include positronium annihilation lifetime spectroscopy (PALS),[1, 2]

ellipsometric porosimetry (EP),[3] and scattering-based techniques utilizing

either neutrons or X-rays.[4] Each of these techniques comes with inherent

strengths and weaknesses. PALS is well-suited for average pore size

measurements when the pores are exceedingly small (i.e., 2 nm to 20 nm
◆ Pore Characterization: X-ray Porosimetry

2

in diameter) and capable of quantifying closed pores not connected to the

surface. However, PALS is also a high-vacuum technique (more difficult

to implement) and non-trivial in terms of quantifying the total porosity.

The neutron scattering measurements can be powerful but are severely

limited by access to neutron scattering facilities. EP is very similar to the

X-ray porosimetry technique described herein but critically relies upon the

index of refraction for the sample and condensate being either known or

approximated. This guide is dedicated to developing X-ray porosimetry

as a powerful tool for characterizing porosity in thin, low-k dielectric films,

both in terms of the requisite experimental set-up and the data interpretation.

However, it is also important to realize that the utility of X-ray porosimetry

extends beyond the semiconductor industry and the field of low-k dielectrics.

The technique is generally applicable to porous media supported on a smooth,

flat surface. Relevant or related applications include membranes, filters,

catalyst supports, separation media, and other nanoporous materials.

1.A. Basics of Porosimetry

The field of porosimetry is a mature science with vast documentation in

the literature. Traditional forms of porosimetry rely upon the surface

adsorption and/or condensation of vapor inside the porous media.

For example, it has been known for many years that charcoal (a naturally

porous material) will adsorb large volumes of gas. The first quantitative

measurements of this phenomenon appeared in the late 1700’s [detailed

in references 5, 6]. Over the next 200 or so years, the quantitative and

scientific advances in these gaseous uptake measurements evolved

into the modern-day field of porosimetry. To fully appreciate the current

status of the porosimetry field, we recommend the following texts [7–9].

Today, the most common form of porosimetry is the nitrogen physisorption

isotherm. In this method, the porous sample is sealed in a vessel of fixed

volume, evacuated, and cooled to the boiling point liquid N2. The sample

vessel is then filled with a known quantity of pure N2 gas. The fixed

volume of the vessel combined with the known volume of gas should

define the partial pressure. However, at low-dosing pressures some of

the N2 condenses on the surface of the sample, resulting in a reduction

of the “predicted” pressure that can be measured with an accurate

pressure transducer. From the pressure reduction and the volume,

the amount of adsorbed N2 onto the pore surfaces can be calculated.

At higher pressures, after a few monolayers of the gas adsorb onto the

pore walls, N2 begins to condense inside the smallest pores even though

the vapor pressure in the system may be less than the liquid equilibrium

vapor pressure. Zsigmondy[10] was the first to illustrate this effect and
3

described the process, using concepts originally proposed by Thompson

(Lord Kelvin), as capillary condensation.[11] The classic Kelvin equation

relates the critical radius for this capillary condensation, rc, to the partial

pressure P, equilibrium vapor pressure P0, liquid surface tension γ, and

molar volume Vm through:

(1)

where T is the absolute temperature. This relation demonstrates that

the critical (maximum) pore size for capillary condensation increases

with the partial pressure, until the equilibrium vapor pressure of liquid

N2 is achieved and rc approaches infinity. In the porosimeter apparatus,

capillary condensation leads to a noticeable pressure drop that

directly yields the amount of adsorbed vapor. The volume of the

adsorbed/condensed vapor is measured as the partial pressure first

increases to saturation and then decreases back to zero. A plot of the

volume of adsorbed condensate versus the partial pressure defines a

physisorption isotherm. The International Union of Pure and Applied

Chemistry (IUPAC) proposes that all physisorption isotherms are

classified into six general types,[12] schematically depicted in Figure 1.

Introduction ◆

( 0

m c RT P P

V

r ln

2 1

− = γ

)

Figure 1. Schematic illustration of the six main types of gas physisorption isotherms,

according to the IUPAC classification scheme.[12]
◆ Pore Characterization: X-ray Porosimetry

4

Before discussing these isotherms, we must establish the accepted pore

size nomenclature. IUPAC classifies pores by their internal width, w,

according to the following definitions[12, 13]:

A wealth of information about the pore structure can be obtained from

the general isotherm classifications in Figure 1. For example, a strong

increase at low partial pressures, like in Types I, II, IV, and VI, is usually

indicative of enhanced or favorable adsorbent-adsorbate interactions as

opposed to weak interactions in Type III or V. Initial uptakes that rapidly

plateau, like Type I and the mid-stages of Type IV, are usually indicative

of micropore filling, i.e., pores that are comparable in dimension to the

adsorbate molecule. The other possibility is that larger pores exist,

and after an initial monolayer plateau, continued uptake occurs. This

continued adsorption can be either gradual (Type II) or discrete in steps

of additional monolayers (Type VI). There can also be hysteresis loops

between the adsorption and desorption pathways at moderate partial

pressures as seen in Type IV and V. Hysteresis loops are a signature

of capillary condensation in mesopores. Finally, diverging uptakes near

saturation, seen in Types II and III, are an indication of macropore filling.

A more detailed description of these physiosorption isotherms can be

found in the general porosimetry literature.[7–9, 12]

The preceding text discussed porosimetry in the context of N2

adsorption isotherms, with the amount of adsorbed gas determined

from the pressure drop. One can also envisage monitoring other

parameters to determine the uptake of adsorbate, such as the mass or

heat of adsorption. However, these measurements become exceedingly

difficult in thin films where there is limited sample mass. For example,

a 500 nm thick, low-k dielectric film might have an average density of

1 g/cm3 and a porosity of approximately 50 % by volume. This means

that on a 7.62 cm (3O) diameter wafer, there will be approximately

1×10–3 cm3 of pore volume in comparison to the several grams of Si

from the supporting substrate. Traditional porosimetry techniques lack

the sensitivity to register the pressure, mass, or calorimetric changes that

occur when condensing such a small amount of vapor; metrologies with

higher sensitivity are critically needed to quantify the porosity in thin, low-k

micropores: w < 20 Å

mesopores: 20 Å ≤ w ≤ 500 Å 

macropores: w > 500 Å
5

dielectric films. It is also desirable that the characterization be in-situ

on the supporting Si wafer. This circumvents sample damage and the

inevitable errors (i.e., the creation of rough surfaces, damage to existing

pore structures, etc.) that result from scraping large quantities of low-k

material off multiple wafers. In-situ techniques further retain the

possibility of serving as on-line or process control checks in industrial

fabrication settings. To this end, we have developed X-ray porosimetry

as a viable and powerful tool for characterizing the pore structures of thin,

low-k dielectric films.

1.B. The Concept of X-ray Porosimetry

In the following, we describe a form of porosimetry with the requisite

sensitivity to quantify the small pore volumes in a thin, low-k dielectric

film. The technique, known as X-ray porosimetry (which we abbreviate

as XRP), combines specular X-ray reflectivity measurements with the

mechanism capillary condensation. As will be described below, specular

X-ray reflectivity is a means to characterize the density of thin films

supported on a Si substrate. If the environment surrounding the film

is gradually enriched with an organic vapor, like toluene, the vapor will

condense in the pores with radii less than the critical radius for capillary

condensation. This condensation increases the density of the sample

appreciably. Recall earlier estimates that the density of a low-k film is

on the order of 1 g/cm3 with approximately 50 % porosity (implying wall

density on the order of 2 g/cm3). Given that most condensed organic

vapors also have densities on the order of 1 g/cm3, complete condensation

leads to a significant (i.e., on the order of 100 %) change in the total film

density. This provides an effective means to monitor the vapor uptake as

a function of partial pressure, mapping out physisorption isotherms similar

to Figure 1. Before elaborating on the interpretation of these isotherms,

we first need to review the relevant fundamentals of X-ray reflectivity.

1.C. Fundamentals of Specular X-ray Reflectivity

Specular X-ray reflectivity (SXR) is an established technique for

measuring the thickness, density, and roughness of thin films.[14–18] For

X-ray wavelengths of a few tenths of a nanometer, the refractive index of

most materials is less than one. This implies that there is a critical angle,

θc, below which there is total external reflection of the incident X-ray

radiation. In the typical SXR measurement, the reflected intensity is

collected as the incident angle and detector position are scanned through

equal angles θ (the specular condition) from just below to well beyond θc.

The electron density profile perpendicular to the film surface can be

Introduction ◆
◆ Pore Characterization: X-ray Porosimetry

6

deduced by modeling the SXR data with a one-dimensional Schrödinger

equation.[14, 19] Free-surface roughness, interfacial roughness between

the layers, and density variations perpendicular through the film can be

extracted from the reflectivity data by using computers to systematically

vary model electron-density profiles to best fit the experimental data

(see reference [19] for one such fitting algorithm). It is important to

realize that the angles θ can be quite shallow with respect to the surface

parallel. This means that the footprint of the X-ray beam on the film is

rather large, typically several cm2; data is collected from a large area

of the sample. However, the coherence length of the X-ray beam

(depending on the optics) is typically on the order of 1 μm or less, implying

that the thickness, density, and roughness within the large footprint are

sampled with only μm resolution. This is equivalent to using a 1 μm ruler

to characterize a region of several cm2; locally the topology may appear

flat, even if larger scale roughness exists. An example would be a film

with visible, low-periodicity (period of say one mm) roughness but smooth

surfaces on the μm length scale; SXR would perceive a smooth film

with a thickness that was actually an average over all the low-frequency

thickness variations.

At X-ray energies significantly higher than the absorption edges for

any of the constituent elements in the sample, the electrons can be

considered “free” in that anomalous X-ray scattering can be ignored.

This assumption is valid for most laboratory based Cu-Kα sources

(used herein) because the typical elements encountered in the low-k

dielectric films are H, C, O, Si and possibly F. The refractive index, n,

of a material with respect to the X-ray radiation is:

(2)

where r0 is the classical electron radius (2.818×10–13 cm), ρe is the

electron density of the material or the total number of electrons per unit

volume, and λ is the X-ray wavelength. The imaginary component of

the refractive index stems from X-ray absorption, and μ is the linear

absorption coefficient of the material. For the 1.5416 Å Cu-Kα X-rays

used in this work, the surface reflection is practically independent of the

polarization of the incident beam, i.e., the p-wave and s-wave reflections

are almost identical. For samples comprised of multiple elements, ρe

can be expressed as:

Σ (3) = i

i

i i e a m A

n Z

ρ N ρ

δ

π

μλ

π

ρ λ

+ = 2 41 1

2

i

r

n i 0 e = − ) ( β − −
7

where Na is Avogadro’s number and ni is the number fraction of the

ith element with atomic weight Ai and atomic number Zi. Notice that

converting ρe to the mass density ρm requires knowledge of atomic

composition. If this composition is not known, ni of thin films can be

determined through a series of ion beam scattering experiments that

include low energy forward scattering, traditional backscattering (i.e.,

Rutherford backscattering), and forward recoil elastic scattering.[4a, 20]

The optical constant δ in Eq. (2) is typically on the order of 10–6 for

most materials. This means that the refractive index for X-rays is slightly

less than unity, the refractive index of air or a vacuum. From Snell’s Law,

it is easy to demonstrate that for grazing incident angles θ < θc ≈ (2δ)1/2

the radiation is totally internally reflected. The extent to which the

X-rays are reflected from the surface or interface at higher θ depends

upon the difference in electron densities ρe across the interface.

From an experimental perspective, reflectivity measurements are

straightforward. Radiation impinges upon a film surface at a grazing

angle θ, and the intensity of reflected X-rays is measured at an equal

angle θ. Measuring the ratio of the reflected (I) to the incident (I0) beam

intensities (the so called reflectivity, which is usually presented on either

logarithmic scale or as the logarithm of the ratio) down to I/I0 = 10–8 is

typically feasible with laboratory X-ray equipment, provided that low-noise

(low background level) detectors are employed. X-ray reflectivity yields

the Fourier transform of the electron density concentration gradient within

the specimen. By nature of the inverse problem, this leads to inherent

ambiguity in the interpretation in that more than one model concentration

profile could describe the observed reflectivity. Therefore, it is always

useful to use supporting information (and common sense) to corroborate

the validity of the fitted interpretation.

1.C.1. Reflectivity from a Smooth Surface

Low-k dielectric films are typically supported on Si wafers. Therefore,

single crystal Si is used herein for our example of reflectivity off a

smooth substrate. Figure 2 displays theoretical SXR data as a function

of the momentum transfer vector Q (Q = (4π/λ)sinθ) from two Si

surfaces, each with 1 Å (solid line) or 10 Å (dashed line) of surface

roughness. This roughness is equivalent to a root mean square (RMS)

value. However, within the context of SXR the surface roughness is

not treated precisely since it can only be defined along the surface

normal; the lateral nature of the roughness is ill-defined. Nevertheless,

Figure 2 clearly illustrates that increased roughness diminishes the

reflectivity, especially at high Q.

Introduction ◆
◆ Pore Characterization: X-ray Porosimetry

8

Figure 2. Calculated X-ray reflectivity results from silicon surface with different

surface roughness. The solid and dashed curves correspond to RMS surface

roughness values of 1 Å and 10 Å, respectively.

Notice that the position of the critical angle (where the reflectivity ratio

rapidly decreases from 1.0) and the reflectivity data near the critical

angle are largely unaffected by surface roughness. The corresponding

Q at the critical angle θc is defined as Qc. The magnitude of Qc

2 relates

to the total electron density ρe through the expression:

(4)

If this expression is recast in terms of angle θ, rather than the wave

vector Q, one obtains:

(5)

Most materials display moderate X-ray absorption, even at wavelengths

far from the absorption edges of their constituent elements. For

Cu-Kα X-rays, Si has a linear absorption coefficient μ = 1.47×10–6 Å–1.

Increasing μ by a factor of 10 significantly affects the reflectivity data

c 0 e Q 16π r ρ 2 =

θc = λ ρe r0 /π
9

as shown in Figure 3; the Si critical edge at Q = 0.032 Å–1 becomes

less well-defined while the magnitude of reflectivity decreases in the

low Q region. However, these effects become less noticeable at

higher Q. Figure 3 illustrates the importance of incorporating the

correct absorption coefficients in the data reduction and fitting

algorithms. The reflectivity is relatively insensitive to small changes

in μ, but a factor of ten error leads to noticeable effects at low Q.

1.C.2. Reflectivity from a Thin Film on a Smooth Substrate

A thin polystyrene (PS) film is used to simulate the X-ray reflectivity

data for the nanoporous thin films on Si substrates since the electron

density ρe and absorption coefficient μ are well documented. The ρe

of PS is also comparable to the average ρe of most candidate low-k

materials. The μ for PS will be slightly lower than most low-k

materials since the latter typically contain Si and O, elements with

higher atomic absorptions than the C and H in PS. Nevertheless,

PS is still an illustrative example, and Figure 4 shows the theoretical

reflectivity for a smooth, 3000 Å PS film supported on a thick Si

substrate. The main body of the figure emphasizes the low Q portion

Introduction ◆

Figure 3. Calculated X-ray reflectivity data for Si surfaces with different linear

absorption coefficients. The solid curve corresponds to a nominal absorption

coefficient of μ = 1.47×10–6 Å–1 for pure Si, while the dashed curve indicates an

order of magnitude increase, μ = 1.47×10–5 Å–1.
◆ Pore Characterization: X-ray Porosimetry

10

Figure 4. Part (a) displays the low Q theoretical X-ray reflectivity for a 3000 Å

thick PS film supported on a Si wafer; the inset extends the reflectivity to high Q.

The marked decrease in the reflectivity at Q = 0.022 Å–1 and 0.032 Å–1 corresponds

to the Qc values of PS and Si, respectively. The region between these two Qc values

is the wave-guiding region of the reflectivity curve. Part (b) illustrates how a

tenfold increase in the linear absorption coefficient μ of the film significantly

affects the reflected intensities in this wave-guiding region.
11

of the data while the inset of part (a) extends the Q range. Notice

that the reflectivity goes through two sharp drops in intensity, first

near Q = 0.022 Å–1 and then again near 0.032 Å–1. The latter drop is

identical to those depicted in Figures 2 and 3, corresponding to the Qc

of the Si substrate. The drop near Q = 0.022 Å–1 corresponds to

the Qc of the PS layer. The Qc of PS is smaller than the Qc of Si,

owing to the lower ρe of PS.

Multiple oscillations in the reflected intensity occur between the Qcs of

PS and Si, and a careful inspection reveals that the oscillations are not

periodic; their frequency decreases towards higher Q. The physical

origin of these oscillations is different from the high Q oscillations

that occur with regular periodicity (discussed below). In the regime

between these two Qcs, the momentum transfer vector perpendicular

to the film is sufficient for the X-rays to penetrate the air/film interface,

but insufficient to penetrate the film/Si interface. Each minimum in this

regime corresponds to an optical coupling condition, and the incident

X-ray is transmitted via the thin film in a manner similar to an optical

wave-guide. The phase and the amplitude of these oscillations relates

to the electron density profile of the film. It is imperative to accurately

fit data in this regime in order to obtain quantitatively correct information

regarding the electron density profile of the film.

The absorption coefficient of the film plays an important role in

shaping the reflectivity in this wave-guiding regime. As illustrated

in Figure 4b, a tenfold increase of the PS absorption from

μ = 4.49×10–8 Å–1 to 4.49×10–7 Å–1 dramatically decreases the

reflectivity in the wave-guiding regime. The difference diminishes

at high Q, similar to the effect seen in Figure 3. This reflects the fact

that in the wave-guiding region, the incident beam is localized inside the

film as an evanescent wave as opposed to simple transmission through

the film at higher Q. The effective path length through the film in the

wave-guiding region is significantly greater than the single reflection

path that decreases in length at high Q, magnifying absorption effects

in the former.

Beyond the Qc of the Si, X-rays reflect from both the air/PS interface

and the PS/Si interface. At periodic angles of incidence, the two

reflected beams give rise to constructive and destructive interference

patterns (oscillatory behavior) known as Kiessig fringes.[16–19]

The total film thickness D is inversely proportional to the fringe

periodicity ∆Q (D ≈ 2π/∆Q). Fourier transform methods can also

be used to convert the characteristic frequency of the Kiessig fringes

into a thickness.[4g] More detailed structural information can further

Introduction ◆
◆ Pore Characterization: X-ray Porosimetry

12

be extracted from the SXR data, including the electron density as

a function of distance from the substrate and the roughness of the

film interfaces. However, these parameters must be obtained by

systematically fitting various model electron density profiles to the

reflectivity data. X-ray reflectivity data reduction and analysis

are discussed later, but here it is important to appreciate that SXR

can quantify thickness in films ranging from a few angstroms to

approximately a micrometer thick, with angstrom-level resolution

and without knowledge regarding the optical constants of the material.
13

2. EXPERIMENTAL

2.A. X-ray Reflectometer Requirements

Currently, there are several companies producing X-ray instruments

capable of functioning as reflectometers. Some of these instruments are

designed to function specifically as reflectometers, but it is also possible

to use an X-ray diffractometer provided that the proper optics and

degrees of freedom in sample positioning are available. In the following,

we describe the necessary instrument requirements for high-resolution

reflectivity experiments. SXR data are always collected with the grazing

incident angle equal to the detector angle, as the term specular implies.

(This Recommended Practice Guide does not address off-specular

reflectivity, where the incident and detector angles are not equal. This

can be used to probe lateral structure within the plane of the film[21]).

Typical values for θ in a reflectivity scan range from 0.05° to 2.0°,

depending on the film thickness. The manner in which the incident and

detector angles are maintained equal through the reflectivity scan depends

upon the instrument. Conceptually, the simplest instruments maintain the

sample flat (no sample tilt; Ω = 0°) while the incident beam and detector

simultaneously move through equal angles θ. This scenario is illustrated

schematically in Figure 5a. Other instruments are designed (primarily

those which were intended to be diffractometers) with an incident beam

that does not move. In this case, the specular condition is maintained by

rotating the sample through an angle θ while the detector simultaneously

moves 2θ. In this θ-2θ configuration, the sample tilt angle Ω in Figure 5

effectively becomes θ, while the detector angle becomes 2θ. For simplicity,

the discussion herein uses the notation described in Figure 5a. However,

extension to a θ-2θ configuration is straightforward.

Alignment of the X-ray source (S) and detector (D) on the X-ray

goniometer (Gx) is crucial for reflectivity, more so than diffraction.

The condition where S is focused directly at D defines the θ = 0° state,

and the beam must pass precisely (within a few μm) through the origin

of the Gx under these conditions (see Figure 5b). This condition is not

automatically achieved or ensured with a factory-aligned diffractometer.

It is not trivial to ensure that D and S are focused perpendicular to

the tangent of the goniometer circle Gx (i.e., at the origin O). Most

instruments provide angular degrees of freedom such as α (see Figure 5b)

to change the incident (source) or acceptance (detector) angles relative

to a tangent of Gx. This leads to the potential situation SN-DN where the

detector and source are directly focused at each other, but the X-ray

beam does not pass through the origin. Defining SN-DN as θ = 0° would

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

14

Figure 5. The diagram in part (a) depicts the relevant degrees of freedom for an

X-ray reflectometer. In this representation, θ defines goniometer angles for both the

incident X-ray beam and the detector position (X-ray optics) while Ω, Φ, and z are

the tilt, yawl, and vertical translation (x and y would be the lateral translations) of

the sample stage. The X-ray goniometer and sample stage move/rotate on separate

goniometers, Gx and Gs respectively, as shown in part (b), and it is advisable to

make these circles concentric. It is critical that the X-ray beam passes through the

center O of the Gx; detector and X-ray source positions DN and SN where the beam

does not pass through the origin are unacceptable, as discussed in the text.
15

lead to incorrect angular values of the goniometer. In a diffractometer,

this type of misalignment is corrected by measuring the diffraction from a

known crystal. The various degrees of freedom (α and θ ) are adjusted until

the experimentally obtained diffraction peaks coincide with the theoretical

values. However, this level of alignment is usually insufficient for reflectivity

measurements. The positions of the high Q diffraction peaks are not

extremely sensitive to the X-ray beam passing through the origin; further

refinements in the alignment of the X-ray optics are usually required.

For the θ -θ instrument depicted in Figure 5, the process of focusing the

incident beam precisely through the center of Gx is non-trivial and typically

requires fabricating a non-standard alignment device. Such a device

might operate on the fact that a tightly collimated beam (either ribbon or

pin-hole) of X-rays focused at the origin of Gx will not move laterally

across a flat surface (if that surface also passes through the origin of Gx)

when the incident θ is scanned over large angles. The position of the

beam can be visualized using a phosphorous screen, photographic film,

or a 2-D detector array. Typically, the large flat surface is brought into

the beam at θ = 0° near the origin O and rotated about the sample stage

goniometer (Gs) until the plane of the film lies parallel to the X-ray beam

(the process by which this is done is described below in reference to

Figure 8, parts (a) through (c)). With the sample plane and beam now

coplanar, the incident θ is scanned over a large range (i.e., 10° through

90°). If the beam is focused off the center of the Gx, the point of focus

will move across the flat surface with θ. If this is the case, the angle α

on the incident X-ray source is adjusted in the appropriate direction,

and the process is repeated until the beam trace does not move with θ.

For θ-2θ instruments, the alignment is easier if the sample goniometer

can be rotated by 180° (this is usually not possible in θ -θ instruments).

In this scenario, the θ = 2θ = 0° condition that passes through the origin

of the goniometer can be tested by moving a knife edge on the sample

stage into the beam until it cuts the intensity in half (similar to Figure 8b,

discussed later). If the focus point is at the origin, rotating the sample by

180° results in the same final intensity as the sample moves from blocking

the bottom half of the beam to the top half; it should be symmetric.

For specular reflectivity, it is not critical that the centers of Gx and Gs

coincide. As long as (i) the beam passes directly through the origin of Gx,

(ii) the sample surface is coplanar with the beam at θ = 0°, and (iii) the

incident beam can be focused onto the sample, small off-sets of Gx and

Gs will not significantly affect the X-ray reflectivity. However, it is

advisable to make Gx and Gs as concentric as possible since off-sets

will make the angles in rocking curves (Ω rotations, sometimes used in

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

16

off-specular reflectivity) inaccurate. Likewise, this would also affect the

footprint correction that is discussed below. The procedure by which Gx and

Gs are made concentric is non-trivial and beyond the scope of this Practice

Guide, but it is similar to focusing the incident beam through the origin of Gx.

The resolution of an X-ray reflectometer depends upon both the angular

resolution of the goniometers that move the incident beam/detector and

the quality of the beam conditioning optics. Since the periodicity of the

Kiessig fringes varies inversely with film thickness, a greater resolution

is required to measure thicker films. As a practical upper bound, it is not

possible to quantify films thicker than a few micrometers using a fine focus

copper X-ray tube. A standard X-ray diffractometer utilizing a Cu-Kα

radiation filter through a Ni foil filter, a detector with a 100 eV band pass

filter to crudely discriminate the Cu-Kα reflections, and simple Soller slit

pack collimation (standard on most base-model diffractometers) on the

incident and reflected beams should be able to generate reflectivity curves

on films 1000 Å to 2000 Å thick. To measure reflectivity from thicker

films requires non-standard upgrades or additions to increase the resolution.

We recommend conditioning the incident Cu-Kα beam with a four-bounce

germanium (Ge [220]) monochromator. It is also helpful to further

condition the reflected beam with a three-bounce germanium Ge [220]

channel cut crystal. This configuration ensures that only specularly

reflected X-rays are detected (moreso than 100 eV band pass filter), and

results in a Cu-Kα beam with a fractional wavelength spread of 1.3×10–4

and an angular divergence of 6×10–5 rad (12 arc seconds). The motion

of the goniometer arms that control the incident beam and detector

position should also be highly accurate. Using a closed-loop active servo

control system, an angular reproducibility of 0.0001° should be attainable.

Utilizing these high precision settings in both the X-ray optics and the

goniometer control, one should be able to detect the very narrowly

spaced interference fringes from films on the order of 1 μm thick.

For an SXR measurement, it is recommended to cover as wide a

Q range as possible to increase the confidence in model fits to the data.

A wide Q range also facilitates quantifying exceedingly thin films

whose Kiessig fringes become widely separated in Q space. This often

necessitates obtaining reflectivities as low as 10–8.

 Given that the typical

incident beam flux is approximately 107 counts/s for most laboratory

instruments with a highly monochromated and tightly collimated beam,

the photon flux at the detector is often 1 count/s or less. Therefore, the

X-ray detector must have extremely low noise or “dark current” levels,

preferably far below 1 count/s. Enhanced intensities can be gained

by incorporating a focusing mirror in the incident beam, effectively
17

directing a greater number of X-rays into the four-bounce Ge [220]

monochromator. Depending on the noise level of the detector, it should

be feasible to quantify the thickness in films as thin as (25 to 50) Å using

SXR. However, the density and roughness measurements are very

difficult in such thin films. There are so few Kiessig fringes (in these thin

films) that it is impossible to establish the high Q decay behavior that is

related to film roughness. Likewise, the significantly reduced path length,

even in the wave-guiding region, leads to negligible absorption or an

inability to resolve the critical edge of the film. The X-ray reflectometer

used in our laboratory contains all of the resolution-enhancing components

described above, including the four-bounce and three-bounce Ge [220]

monochromators, the focusing mirror, the low-noise detector, and the

closed-loop active servo control goniometer positioning.

2.A.1. Resolution Effects

The thickness of most low-k films range from 100 nm to 1 μm.

For a 1 μm thick film, the spacing between interference fringes will

be approximately 6×10–4 Å–1 beyond the wave-guiding regime.

However, the spacing between the first few oscillations in the

wave-guiding region is even smaller, as demonstrated in Figure 4.

This imposes very stringent requirements on both the wavelength

dispersion and angular divergence of the instrument in addition to the

step size of the goniometer. To resolve interference fringes with a

spacing of 6×10–4 Å–1 requires approximately 10 data points within the

Q range of a single fringe. This corresponds to a minimum repeatable

angular step size of approximately 0.0004°. Angular divergence is also

critical in defining the resolution of the reflectometer, as shown in

Figure 6; the theoretical SXR reflectivity curves for a smooth,

1 μm PS film on Si at three different angular divergences are given.

The solid line corresponds to δλ/λ = 6×10–5 with an angular divergence

of δθ = 3×10–5 rad, while the dashed and dotted lines indicate angular

divergences increased by ×2 and ×4, respectively. The angular

divergence of 6×10–5 rad is appropriate for instruments used in our

laboratory and the reflectivity data presented in this guide. Figure 6a

indicates that angular divergence has a large impact on the reflectivity

in the low Q wave-guiding regime; the first few oscillations right above

the Qc for PS become almost indistinguishable at δθ = 6×10–5 rad (×2).

At δθ = 1.2×10–4 rad (×4) the oscillations nearly dissapear. In Figure 6b,

one can see that the amplitude of the interference fringes in the high

Q regime are also dampened with increasing angular divergence, but

to a lesser extent in comparison to the low Q regime.

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

18

The wavelength dispersion of the instrument also plays a critical role in

the quality of the SXR data, similar in effect to the angular divergence.

Figure 7 shows the theoretical X-ray curves for the 1 μm thick PS

film with wavelength divergences of δλ/λ = 6×10–5 (solid lines) and

δλ/λ = 6×10–4 (×10, dotted lines), both with an angular divergence

of δθ = 6×10–5 rad. The wavelength dispersion of our reflectometer,

δλ/λ = 1.3×10–4, is intermediate between the two values depicted in

Figure 7. Greater wavelength dispersions tend to dampen the amplitude

of all the oscillations throughout the entire Q region. The simulation

Figure 6. Calculated X-ray reflectivity curves for a 1 μm PS film with instrumental

angular divergences of (3×10–5, 6×10–5 and 1.2×10–4) rad in both the wave-guiding

(part a) and high Q (part b) regimes. Increasing angular divergence significantly

damps the interference fringes, especially at low Q within the wave-guiding regime.
19

in Figure 7 is based on an instrument with a four-bounce Ge [220]

monochromator on the incident beam. Adding the three-bounce

Ge [220] to the reflected beam will increase the resolution but only

marginally. Replacing these Ge optics with a single crystal Si-based

monochromator can further enhance the wavelength resolution, but

this comes at the price of a decreased photon flux.

2.A.2. Recommended Procedure for Sample Alignment

It is always critical to precisely align the sample in any SXR or XRP

experiment. The first requirement is that the sample is flat with a

mirror-like surface. A RMS surface roughness greater than 100 Å

will degrade the Kiessig interference fringes to the point where XRP

is not possible. Likewise, large-scale bowing or curvature in the

sample is undesirable as this leads to different regions of the film being

“aligned” at different angles. Since the footprint of the illumination in

a reflectivity experiment is large, the sample should be flat over an

Figure 7. Theoretical X-ray reflectivity curves for a 1 μm PS film with two different

wavelength dispersions. The solid line corresponds to a wavelength dispersion (wd)

of δλ/λ = 6×10–5 while the dotted line denotes δλ/λ = 6×10–4. The main body

illustrates these effects in the low Q wave-guiding region while the inset emphasizes

the high Q effects.

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

20

area of several cm2. Once the flat sample is mounted into the X-ray

reflectometer, the optics need to be aligned. This is described below

with the help of Figure 5 which describes the requisite degrees of

freedom for a typical reflectometer.

The first step is to determine the θ = 0° condition for the reflectometer.

The procedure described below is predicated on the assumption that

Figure 8. Panels (a) through (f) illustrate the process required to align the

X-ray reflectometer, as described in the text. In each panel, the dotted vertical line

indicates the current zero position for each degree of freedom while the dashed line

indicates the new position that should be reinitialized to zero. Data are presented

for illustrating the alignment procedure only, so the standard uncertainties are

irrelevant and not provided.
21

X-ray optics are calibrated such that at θ = 0° the beam passes

precisely though the origin of the goniometer Gx. Lower the sample

height in the z direction sufficiently so that the sample is not in the

beam. Then scan the range of θ values on either the incident or

reflected beam (scanning only one while the other is maintained

stationary) until a maximum intensity is achieved (see Figure 8a).

This maximum corresponds to the condition where the X-ray source is

focused directly at the detector. According to Figure 8a, θ needs to be

offset by approximately 0.0014° so that the θ = 0° position corresponds

to the actual peak intensity (indicated by the shift from the dotted to

dashed lines). The mechanism by which this scan is achieved may

differ slightly between reflectometers. In some units the position of the

X-ray source is fixed and the detector moves. In this situation θ is half

the angle that the detector moves, defining a 2θ scan. In other units the

source and detector move in concert, each by an amount θ, in what is

known as a θ scan. As a word of precaution, these scans to determine

the zero angle should be done with an understanding of the intensity

limits of the detector. At full power (typically on the order of 40 mA

at 40 kV for most laboratory X-ray sources), the X-ray flux is often

sufficient to saturate the detector and induce irreversible damage.

It is prudent to locate the θ = 0° condition at a reduced power

setting on the X-ray generator or with an attenuator inserted in the

X-ray beam.

After the θ = 0° condition is determined, the stage should be scanned in

the z direction until the sample enters the beam. This will be obvious by

the rapid decrease in the count rate on the detector as shown in Figure 8b.

Drive the sample into the beam up to the point where the count rate is

cut in half as compared to the open beam, and tentatively assign this

position as z = 0. In Figure 8b, the current z = 0 position (dotted line)

needs to be shifted up by 0.075 mm to coincide with the physical beam

center (dashed line). Now scan Ω about its current position over a

sufficiently large range (typically an angular range greater than 0.5°).

If the plane of the sample surface is parallel to the vector between

the source and the detector at θ = 0°, this scan produces a maximum

intensity at the position approximately Ω = 0°. If the maximum does

not coincide with the Ω = 0°, i.e., dotted line in Figure 8c, drive Ω to

the position of the maximum (dashed line), and set this new position as

Ω = 0°. In Figure 8c, this corresponded to a Ω shift of –0.05°. It is

possible that the Ω scan in Figure 8c does not reveal a maximum, rather

the intensity increases or decreases monotonically. This indicates that

a larger angular range of Ω is needed for the scan. It is also possible,

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

22

and likely, that a larger Ω scan also tilts the sample out of the beam.

This is indicated by a plateau in the intensity rather than a maximum.

In this case, drive Ω to the highest intensity, somewhere in the plateau

region. Then increase the sample height until the open beam intensity

is cut in half, and repeat the Ω scan. It is not uncommon to go through

several iterations of these sample height adjustments and Ω scans

before the Ω = 0° condition is located, especially the first time a

new sample is mounted.

At this point the sample is crudely aligned. To fine-tune the alignment,

it is necessary to increase θ (both source and detector) to a sub-critical

value where total reflection of the incident X-rays occurs. For most

polymer or low-k dielectric films θ = (0.1° to 0.2°), or 2θ = (0.2° to 0.4°),

is sufficient to achieve these sub-critical conditions. At these specular

reflectivity conditions there should be a moderate intensity reaching

the detector. If the power was reduced or attenuators placed in the

beam at θ = 0°, it is now safe to undo these precautions. Fine-tuning

the alignment is an iterative process of systematically adjusting z, Ω,

and Φ. We typically start by scanning z over a range of 0.2 mm to

0.4 mm about the current position as shown Figure 8d. If the current

z = 0 position (dotted line) does not correspond to the pronounced

maximum (dashed line), drive z to the maximum intensity, and reset

this position to z = 0 (an offset of –0.05 mm in Figure 8d). If no

maximum is observed it is necessary to increase the range, but this

should generally not be the case if the crude alignment described

above is done correctly. Once the maximum in z is found, scan Ω

about its current position over a range of approximately 0.05°.

Notice that this range is significantly smaller than the previous range

of 0.5° recommended at θ = 0°; the Ω scan is considerably more

sensitive at these specular reflectivity conditions. This scan should

produce a strong, sharp peak as shown in Figure 8e. Once again, if the

Ω = 0° (dotted line) does not correspond to the maximum (dashed line),

reset Ω to the proper zero (shift of –0.010°) and proceed to maximize

Φ. We recommend scanning Φ over a large range of 3° as shown in

Figure 8f. Do not expect a pronounced maximum since this is the least

sensitive of the alignment adjustments. As shown in Figure 8f, the angle

Φ = 0 ° needs to be offset by 0.4° to properly align the reflectometer.

The fine-tuning procedure described in this paragraph should be

reiterated several times until the peak positions no longer change

between successive iterations; in the notation of Figure 8 this means

that the dashed and the dotted vertical lines coincide. When this

condition is achieved, initialize Ω = Φ = z = 0, and the reflectometer

is ready for a measurement.
23

Figure 9 displays a θ scan for a well-aligned reflectometer. The

intense peak of the main beam defines the zero angle condition where

Ω = θ = 0°. Beyond the main beam peak, the reflected intensity

increases linearly with θ because of changes in the area of the X-ray

beam footprint with angle. The beam height at grazing incidence

angles geometrically leads to large variations in the exposed film area

(footprint effect). At very shallow incident angles, the aerial footprint

of the X-rays can illuminate an area that is larger than the area of the

sample. As θ increases, the area of this footprint decreases, and more

X-rays are focused onto the aligned sample. This results in an increase

of the reflected intensity with θ from purely geometric considerations.

In the section below, we describe how the reflectivity curves must be

corrected for this footprint effect before proper interpretation of fits

to the data can be carried out. As θ continues to increase, the critical

angle of the sample is eventually reached, and the X-rays start to

penetrate the films, giving rise to the pronounced decrease in the

reflected intensity and the onset of the fringes.

Figure 9. A θ scan of a well-aligned reflectometer. The pronounced peak (off

the vertical scale of the plot) from the main beam defines the θ = 0° condition.

The linear increase in intensity with the angle beyond the main beam peak comes

from increasing the fraction of the beam that is focused onto the sample (footprint

effect). Standard uncertainty in the intensity is less than the line width.

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

24

Realignment of the reflectometer between successive measurements

is sometimes required, depending on several factors. The first is

the stability of the goniometers. If the motors that move the sample

and/or the goniometer tend to drift, frequent alignment will be required.

If the temperature of the stage is varied, thermal expansion is often

sufficient to require readjustments of the sample height. Typically, the

infusion of adsorbents inside the pores does not require realignment

provided that the positioning motors are stable and the sample exhibits

minimal swelling (< 1 % in thickness). We generally recommend noting

the count rate at a sub-critical angle like θ = 0.1° and Ω = Φ = z = 0.

If this count rate decreases by more than 10 % at any point, we

recommend sample realignment.

2.B. Methods of Partial Pressure Control

XRP is predicated on the ability to control the partial pressure of an

organic solvent in the sample environment. There are several ways in

which this environmental control can be achieved. We discuss three

such methods below and comment on their attributes and liabilities.

These techniques include: (i) the controlled isothermal mixing of

solvent-free and vapor-saturated carrier gases; (ii) sample temperature

variations in a vapor saturated carrier gas; and (iii) pure solvent vapor

exposure. We demonstrate these ideas using toluene as a model organic

vapor but emphasize that any number of different organic solvents can

also be employed.

2.B.1. Isothermal Mixing with Carrier Gases

The P/P0 ratio can be varied to probe a range of rc values through

a controlled, isothermal mixing of streams of pure (P/P0 = 0) and

toluene-saturated (P/P0 = 1) carrier gas. Figure 10a schematically

indicates the construction of such a device. We typically use dry air

for carrier gases 1 and 2. Using 6.35 mm (1/4 inch) stainless steel

tubing (other gages can be utilized) and standard Swagelok[22] fittings,

we connect the air cylinders to mass flow control (MFC) valves.

We utilized MKS Instruments 1179A MFC valves configured for a

maximum flow rate of 500 standard cm3/min (sccm). The size of the

sample chamber, the desired speed of the system, and the size of the

tubing primarily determine the maximum flow rate. If the flow rate is

too high relative to the conductance of the system, the carrier gas will

pressurize the system and affect the equilibrium partial pressure. For a

similar environmental control set-up for our smaller neutron scattering
25

sample cells, 1.59 mm (1/16 inch) stainless steel tubing is used, and the

total flow rate is reduced to 10 sccm in order to prevent pressurization.

It is recommended that the MFCs be fitted with stainless steel valve

seats. Care should be taken to ensure that the wetted surfaces of the

control apparatus are compatible with the solvent vapors being used.

The flow rates of our MFC valves are computer controlled through a

MKS Instruments 146C unit configured with two MFC cards. After

passing through the MFC valves, the carrier gas streams are fed into

bubblers that contain the solvent of interest. In our system, 500 cm3

stainless steel bubblers from Epichem (Model 212, ordered “empty”)

with liquid level sensors were utilized. Other sizes or standard glass

bubblers can be used; we opted for stainless steel after breaking

several glass bubblers in prototype controllers. After the air streams

pass through the two the bubblers, they are mixed together and enter

the X-ray reflectometer sample chamber. This chamber is in a

vacuum-tight aluminum housing, sealed with a Viton o-ring, and has X-ray

transparent beryllium windows to allow the X-rays to enter and exit.

Figure 10. Schematics depicting the different methods by which the partial pressure

in the XRP sample chamber can be controlled. Part (a) is based on bubbling a

carrier gas through a solvent reservoir while part (b) illustrates controlled flow of

pure vapor into a chamber that is continually being evacuated by a vacuum pump.

Experimental ◆

Environmental Control Methods
◆ Pore Characterization: X-ray Porosimetry

26

There are also two ISO-25 QF ports that allow the carrier gas to enter

one side of the chamber, flow across the sample, and then exit on the

other side of the sample. Far downstream on the exit line, the toluene

vapor is removed from the carrier gas with a Kinetics Thermal System

VT890A cold trap. Finally, notice in Figure 10 the manual shut off

valves inserted between all components. These are useful in preventing

unwanted back-flow of organic vapors or cross-contamination between

the two carrier gas lines when the system is static.

The operation of the system illustrated in Figure 10a is straightforward.

Bubbler 1 is typically left empty (dry air) while bubbler 2 is filled with

approximately 250 cm3 of toluene. Using the 146C control unit, the

two MFC valves are set in the “totaling mode” for a 100 sccm total

flow rate. In the totaling mode, a flow rate is manually entered for

the independent MFC channel (arbitrarily chosen to be the dry air),

and the flow rate for the dependent MFC (in this case the toluene) is

automatically adjusted to total 100 sccm. For example, if a flow rate

Figure 11. UV absorption spectra of mixtures of pure and toluene-saturated air

with a total flow rate of 100 sccm. Starting with the flat featureless spectra showing

nearly 0 absorbance (pure airflow), each successive curve indicates a 10 sccm

increment in the flow rate of the toluene-saturated air and a 10 sccm decrement in

the pure airflow rate. The ratio of the integrated areas between the curve of interest

and the toluene-saturated curve, using the pure air curve as a baseline, defines P/P0 .

The inset shows a linear correlation between the desired set-point and the measured

P/P0, with a least-squared deviation of χ2 = 0.0002.
27

of 35 sccm is entered for the dry air, the toluene line automatically

adjusts to 65 sccm to give a total flow rate of 100 sccm and a toluene

partial pressure of P/P0 = 0.65. By varying the dry air flow rate

from 0 sccm to 100 sccm, P/P0 values from 1 to 0 can be achieved.

We generally find that this controlled mixing of dry and toluene-saturated

air provides adequate partial pressure control. This is based on two

separate control experiments. First, the XRP data we obtain using a

toluene flow rate of 100 sccm (P/P0 = 1.0) is identical to the data from

a static system where the sample chamber is sealed with an open

beaker of toluene inside the chamber. Second, we have performed

separate experiments to corroborate that the partial pressures are

accurate at set-points where P/P0 < 1.0. Toluene has a strong

ultra-violet (UV) absorption between 230 nm and 280 nm, as

shown in Figure 11. The integrated absorption in this regime,

in reference to the dry air spectrum, defines the partial pressure.

Figure 11 shows a series of UV absorption spectra for toluene flow

rate of 0 sccm to 100 sccm, in 10 sccm increments, using the totaling

mode of operation where the total flow rate is 100 sccm. In this

experiment, the mixed gas line was connected not to the X-ray

reflectometer, but used to purge the sample cell of a Perkin Elmer

UV spectrometer. The partial pressures were calculated from the

integrated absorption between 230 nm and 280 nm, and the inset of

Figure 11 shows an excellent correlation between measured P/P0

and the set-point. The data fit to a linear function with a slope of

1.008 ± 0.004, intercept of –0.005 ± 0.002, and χ2 = 0.0002.

It is critical to control the temperature of the sample stage since

variations, under certain circumstances, can affect dramatic changes

in the adsorption behavior; typical room temperature fluctuations of a

few degrees can lead to irreproducible results. We typically operate

at a set point of 25.0 °C, and thermal fluctuations in the sample stage

of less than ± 0.1 °C are sufficient to achieve smooth isotherms with

minimal fluctuations. We also recommend regulating the temperature

of the bubblers since evaporation cools the toluene. The equilibrium

vapor pressure is temperature dependent, and cooling the toluene will

lead to decreased levels of saturation. Our system circulates water

maintained at 25 °C around the exterior of the bubblers to prevent

cooling. The sample stage and bubbler are the only components of

our system that are temperature controlled. Efforts were not made

to regulate the temperature of the gas handling lines, air supplies, etc.

simply because adequate data was obtained (see Figure 11) without

such precaution. This, however, may not be true under all conditions.

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

28

If the sample is heated above ambient, it would also be necessary to

heat all the components in the vapor handling system. Otherwise,

toluene vapor would condense on the ambient (cooler) surfaces,

decreasing the partial pressure of the environment at the sample.

Likewise, large temperature fluctuations in the environment or possibly

the use of other solvents may require more extensive thermal

regulation.

2.B.2. Sample Temperature Variations in a Vapor-Saturated

Carrier Gas

Two thermodynamic parameters in Eq. (1), either T or P, can be

used to vary the critical radius for capillary condensation. In the

preceding section, we described how different values of rc can be

probed isothermally by mixing streams of dry and toluene-saturated

air (from here on we refer to this as the “P variation method”).

Analytically, this isothermal method is attractive since both γ and Vm

are constant. However, the experimental set-up for isothermal mixing

can be somewhat elaborate since accurate mass flow control valves

are required. It is easier to simply flow a stream of air saturated with

toluene at ambient temperature over the sample and then vary film

temperature (referred to as the “T variation method”). When the film

is also at ambient temperature, P/P0 = 1.0 (where P is current vapor

pressure of the toluene in the atmosphere and P0 is the equilibrium

vapor pressure), and toluene condenses in all of the accessible pores.

If the sample is heated, the equilibrium vapor pressure P0 of the

condensed toluene increases, but the toluene content in the airstream

(i.e., P) remains the same because it was equilibrated at room

temperature. This effectively reduces P/P0 and thereby rc.

This non-isothermal approach is complicated by the fact that the

temperature dependence of γ and Vm need to be accounted for.

From a thermodynamic perspective, though, it is immaterial whether

T or P is used to probe the range of critical pore sizes; the two should

yield identical pore size distributions. However, we demonstrate later

that this is not always the case.

Figure 12 graphically illustrates how the T and P variation methods

probe a range of rc values from approximately 1 Å to 250 Å when

toluene is used as the condensate vapor. The vertical axis indicates

the critical radius rc while the upper and lower horizontal axes

correspond to the P and T variation methods, respectively. The solid

line (reading from the upper horizontal axis) shows how rc varies

isothermally (20 °C) with P/P0, achieved by mixing dry and
29

toluene-saturated vapor also at 20 °C. Likewise, the dotted line

(reading from the lower horizontal axis) depicts the rc variations by

heating the sample in the presence of a stream of air that has been

saturated with toluene at 20 °C. In the simple model of the Kelvin

equation, it should be possible to compare the non-isothermal

physisorption data with the isothermal data. This requires first

accounting for the temperature dependence in the non-isothermal data.

This is simply done by equating rc and accounting for the T dependence

of γ and Vm:

(6)

By correcting the T variation physisorption data this way, it should be

possible to compare directly with isothermal phyisorption data obtained

with the P variation method. We elaborate on this later, ultimately

Figure 12. There are two ways in which rc can be varied in the Kelvin equation.

rc values from approximately (1 to 250) Å can be obtained by mixing ratios of dry

and toluene-saturated air at 20 °C. Likewise, a comparable range of pore sizes can

be achieved by flowing air saturated in toluene at 20 °C across the sample and

heating the film between 20 °C and 125 °C.

Experimental ◆

°

° °

= 293

20

20 , 20

T T

C C

V

Vm T

0 C 0 T P

P

P

P γ

γ
◆ Pore Characterization: X-ray Porosimetry

30

demonstrating that the two techniques are not equivalent; the simple

Kelvin equation neglects thermal changes in the energy or heat of

adsorption.

2.B.3. Pure Solvent Vapor

Thus far, the methods of controlling the partial pressure require using

a carrier gas. However, it is possible to construct an environmental

control system that utilizes pure solvent vapors. This may be

advantageous in instances where the presence of the carrier gas

affects the adsorption. Figure 10b schematically depicts how to

construct a carrier gas-free environmental control system. In this case,

the solvent reservoir is a 500 cm3 stainless steel cylinder with two

threaded ports (available from Swagelock). One of the ports is capped

while the other is fitted with standard Swagelock fittings for 6.35 mm

(1/4 inch) stainless steel tubing. Approximately 250 cm3 of toluene is

poured into the reservoir before connecting it to the pressure control

valve (PCV). Our system used a MKS Instruments (Model 248A)

pressure control valve with stainless steel valve seats. From the

pressure control valve, flexible stainless steel vacuum tubing is used

to connect to the ISO QF-25 flange on the sample chamber. A tee

is added at this junction, and a MKS Instruments 626A 13TDE

capacitance manometer is attached to read the absolute pressure of

the sample chamber. Both the PCV and the capacitance manometer

are connected to the MKS Instruments 146C control unit. The exit

ISO QF-25 flange on the sample chamber is connected to an oil-free

vacuum pump through the Kinetics Thermal System VT890A cold trap.

An adjustable bleed valve is inserted into the vacuum line between

the sample chamber and the vapor trap. There are also mechanical

shut-off valves conveniently placed between each of the components,

as indicated in Figure 10b.

The operating principle for this system is simple. After loading the

toluene and sealing the Swagelock fittings, a dewar of liquid nitrogen is

used to freeze the toluene in the solvent reservoir. The entire system

is then evacuated to remove the air. Valve V1 is then closed while

the frozen toluene is allowed to thaw. We recommend repeating this

freeze–thaw cycle a few more times to degas the toluene. Once the

toluene is sufficiently degassed, valve V1 is closed, and the entire system

is evacuated one final time. The vacuum pump is left on and the bleed

valve is partially closed to keep a modest pumping rate on the sample

chamber. Then a pressure set point is entered into the 146C. When

manual shut-off valve V1 is opened, toluene vapor enters the sample
31

chamber, regulated by the PCV. We recommend gently warming the

toluene reservoir (to approximately 50 °C) to generate a slight pressure

difference; this temperature is chosen based on the temperature

dependence of the toluene equilibrium vapor pressure and the minimum

pressure difference across the valve specified by the manufacturer for

good control. Since the vapor in the sample chamber is pure toluene,

the partial pressure is read directly off of the capacitance manometer.

While this principle is simple, accurate pressure control with this

system is difficult. We recently designed and built this system

because of the attractive option of no carrier gases. However,

unlike the previous methods, this system has an active control loop.

As with any dynamic system, establishing the proper proportional,

integral, and derivative gains to achieve stable control can be a

challenge. This added level of complexity should be considered when

choosing a method of environmental control. An alternative would be

to use a quasi-static mode of operation that does not require closed

loop control. With the vacuum pump on and the bleed valve properly

set, the PCV could be manually opened by incremental amounts.

For each setting of the PCV, a new equilibrium will be established by

the balance of the toluene input and output flow rates. In each case,

the equilibrated partial pressure would simply be read off of the

manometer.

2.B.4. Choice of Adsorbate

In the preceding discussions, toluene is used as the organic vapor.

However, the toluene can be replaced by any number of organic

solvents or adsorbates. Gurvich’s rule[23] states that the pore volume

equals the volume of the adsorbed/condensed liquid at saturation,

assuming that the molar volume of the adsorbate is the same as pure

liquid at the temperature of interest. In this respect, the total porosity

should not depend upon the solvent choice. This principle has been

generally demonstrated in a wide range of mesoporous solids and

adsorbates,[7] including a few low-k dielectric films studied by the

EP technique.[3a] However, deviations from Gurvich’s rule should be

an important concern, especially in microporous solids where the pore

size approaches the size of the adsorbate molecules. Geometric size

restriction, steric limitations imposed by directional interactions with

the walls and/or other molecules, or changes in the fundamental

properties of the fluid/condensate can become appreciable. There

is a large body of literature dedicated to understanding how fluids

confined in nanopores change their fundamental properties (e.g., see

Experimental ◆
◆ Pore Characterization: X-ray Porosimetry

32

references 24–43). For example, neutron diffraction measurements

of the radial distribution function of benzene condensed in pores with

diameters from 24 Å to 65 Å show that excluded volume effects

increase as the pore size decreases.[27] This leads to a decrease of

the fluid density in the nanopore as compared to bulk, suppression

of the melting temperatures, and possibly even inhibited crystallization

of the adsorbate.[28] One must also be aware that the different

adsorbates display different affinities for the porous matrix material.

We have seen fluorinated low-k dielectric films where toluene would

not wet the wall material and, therefore, experienced problems infusing

the film with adsorbate. However, strong interactions with the wall

material might induce swelling, or restructuring of the pore network.

There are many factors that should be considered when choosing

an adsorbate vapor. Toluene was selected here because: (1) the

temperature dependence of the vapor pressure is appropriate to probe

a reasonable range of pore sizes; (2) the low polarity liquid wets most

(except the fluorinated example cited above) low-k materials; and

(3) most low-k materials do not swell appreciably in the presence

of toluene.
33

3. DATA REDUCTION AND ANALYSIS

3.A. Reducing the X-ray Reflectivity Data

The raw reflectivity data taken directly from the instrument is reduced

to a normalized reflectance defined as the ratio of the measured intensity

at a given angle to the intensity of the direct beam (I/I0). In addition,

the specular angle, θ, should be reduced to Q through Q = (4π /λ)sinθ.

These normalizations allow the data to be interpreted independent of the

X-ray wavelength and/or varying X-ray beam intensities. Typically, the

data are plotted as the logarithm of the reflectivity as a function of Q

because the relative peak intensities of the Kiessig fringes decay very

rapidly, as shown in earlier sections. There are two main considerations

needed to properly normalize a reflectivity curve. First, one needs to

correct for variations in I0 due to varying illumination areas that arise

from the height of the X-ray beam at these shallow grazing incidence

angles (the footprint effect). Second, the background intensity needs

to be subtracted in order to ensure correct reflectivity values. Figure 9

shows the raw intensity data from an aligned sample in an X-ray

reflectometer. The data in the low Q region before the critical angle

show increasing reflected intensity with increasing angle. The finite

height of the beam leads to an illuminated area at low specular angles

that is often much larger than the sample size. Given the beam height,

simple geometric formulas are sufficient to describe the relative area

of illumination (assuming the sample is perfectly flat). For example, for

a beam height of 0.5 mm (using the full width at half maximum from

the derivative of a direct beam scan shown in Figure 8b), a beam width

of 1 mm, and an incident angle θ = 0.1°, the area of illumination at the

sample surface is approximately 286.5 mm2. Changes in this illumination

area with increasing angle are well approximated with a linear function

because the geometric formula is a sine function at very small angles.

Therefore, the process for calculating I0 as a function of θ is

straightforward as follows:

1. Perform a linear fit to the footprint region of the X-ray reflectivity

data before the first critical angle.

2. Calculate the θ where the beam footprint area reaches the size

of the sample.

3. I0 increases linearly with θ up to this angle, then remains constant

for larger θ.

4. Normalize the measured X-ray intensity by the complementary

I0 value.

Data Reduction and Analysis ◆
◆ Pore Characterization: X-ray Porosimetry

34

For many X-ray reflectivity systems, the maximum of I0 can be taken

to be the intensity value at the first critical angle. Nevertheless, it is

recommended that all instrumental parameters be identified so that any

sample size considerations can be coded into the data reduction software.

Figure 13. X-ray reflectivity data from a well-aligned sample and the intensity

from the background as measured at an angle offset from the specular angle.

The background level is indicated in the inset schematic. Standard uncertainties

are smaller than the size of the data markers.

The second correction needed to properly reduce the X-ray reflectivity

data is the subtraction of any background intensity that may arise from

incoherent scattering or other instrumental effects (detector noise).

If a sample is assumed to be flat (no lateral correlations), then the

baseline from an off-specular rocking curve described earlier (i.e.,

Figure 8e over a larger range — see inset of Figure 13) provides a good

measure of the background intensity. Figure 13 shows both the specular

X-ray reflectivity data (Ω = 0) and the background level collected by

repeating the reflectivity measurement with Ω sufficiently off-set so

that the intensities are in the flat background region indicated in the

inset. For X-rays, the background intensities from absorption and

incoherent scattering are mostly negligible relative to the specular peak.

Nevertheless, the background intensity should be evaluated to ensure
35

that this assumption is reasonable for the particular system and sample

being used. It is also important to perform the background subtraction

before normalizing the intensities with I0 .

3.B. Fitting the X-ray Reflectivity Data

A complete analysis of X-ray reflectivity data requires modeling

the reduced data using well-established models and procedures.

The equations needed to model specular reflectivity data are

straightforward and known for many years.[14–19] The basic procedure

involves constructing a model electron density depth profile consistent

with well-known scattering length density parameters for the material

or materials of interest, discretizing the model profile into material slices

defined by their thickness and electron density, solving a recursion formula

for the expected X-ray reflectivity data, then iterating the model profile

until a satisfactory fit is obtained with the experimental reflectivity curve.

The biggest limitation in the analysis of X-ray reflectivity data is the need

for a model profile to fit to the data. Direct inversion of the reflectivity

data to obtain the electron density profile is not possible because one

measures the intensity of the reflected beam but not the phase of this

reflection (the inverse problem). There are many different software

packages available to fit X-ray reflectivity data, and it is beyond the scope

of this practice guide to evaluate the options. References [17–19] provide

a good review of the existing literature on this active field of research.

For XRP, it is often sufficient to measure just the critical angle of the

sample because changes in the amount of adsorption are proportional to

the changes in the sample electron density. For more detailed analysis,

it may become necessary to perform full data fits of the XRP reflectivity

curves for situations: for example, where the film swells, if the pore size

distribution is not uniform through the film, or if the initial film itself is not

uniform. The following sections show how the X-ray reflectivity data

can be used to determine important information about the structure of

nanoporous, thin films.

3.C. Interpretation of XRP Data

In this section, we illustrate how to interpret and analyze XRP by way of

example, focusing on films from three different classes of porous low-k

dielectric material. The films will be characterized in terms of critical

structural information, such as the average film density, average wall

material density, porosity, and the pore size distribution. By way of

introduction, the first class of material is a hydrogensilsesquioxane (HSQ)

film prepared by the evaporation of a high boiling point solvent after

Data Reduction and Analysis ◆
◆ Pore Characterization: X-ray Porosimetry

36

gelling the HSQ resin.[4a] The second is a methylsilsesquioxane (MSQ)

film generated through a templated vitrification of low-molecular mass,

siloxane oligomers.[44] The third material is deposited by plasma

enhanced chemical vapor deposition (PECVD) from a mixture of an

organic precursor (the porogen) and tetramethylcyclotetrasiloxane.[45]

This last film will be referred to as porous SiCOH. The reported

dielectric constants of the porous HSQ, MSQ, and SiCOH films are 2.20,

1.85 and 2.05, respectively. In light of the nominally similar dielectric

constants, we contrast the detailed pore structures in these films since the

latter also affect critical aspects like the mechanical properties of the film.

The ensuing discussion effectively demonstrates how XRP is able to

perceive subtle differences in the pore structures of these otherwise

similar low-k films. Later, the characteristics of the porous films obtained

from XRP are compared with similar data obtained through scattering,

specifically the SANS/SXR techniques described elsewhere.[4]

Figure 14 displays experimental SXR data for the three porous low-k

films both under vacuum and in the presence of toluene-saturated air.

The data are presented as the logarithm of the reflectivity (I/I0) as a

function of Q. As the incident angle (i.e., Q) increases, the X-rays first

penetrate the film at the critical wave vector Qc, at which point the

Figure 14. SXR curves of porous HSQ, porous MSQ, and porous SiCOH thin films.

The curves are offset for clarity. The standard uncertainty in log (I/I0) is less than

the line width.
37

reflectivity drops sharply. The square of this critical angle is proportional

to average electron density of the film, as discussed with Eqs. (4) and (5),

which is then used to calculate the average mass density from the atomic

composition. The atomic composition of these films is determined

experimentally via ion beam scattering, using the combination of the

Rutherford backscattering, forward scattering, and forward recoil

scattering techniques discussed elsewhere.[4a, 20]

Comparing the reflectivity curves under vacuum in Figure 14 shows that

Qc occurs at lower Q for the films with lower k, qualitatively indicating

that the density of porous thin film decreases with the dielectric constant.

It is also clear that Qc shifts to higher Q in the presence of the saturated

toluene vapor. Capillary condensation of the toluene inside the accessible

pores results in an appreciable increase of the density. A larger difference

between Qc of the vacuum and the toluene-saturated curves indicates

a greater uptake of toluene, and thus a greater porosity. From these

two curves alone, one can calculate the total amount of toluene adsorbed,

thus the total porosity P, as well as the density of the wall material ρwall

separating the pores.[46] A simple rule of mixtures leads to two equations

and two unknowns:

(7)

(8)

In addition to P and ρwall, ρdry is the total average density of the dry

(evacuated) film, ρsat is the average density of the film in the presence

of the saturated vapor, and ρliquid is the density of the condensed liquid.

ρdry and ρsat are defined by the Qc

2 values of the reflectivity experiments

in the dry and vapor-saturated environments, respectively, while ρliquid

is assumed to be bulk-like. To convert the Qc

2 values into ρdry and ρsat,

use the elemental composition of the low-k film, the elemental composition

of the toluene condensate, and Eqs. (3) and (4) to convert the electron

densities into mass densities. Knowing ρdry and ρsat leaves two equations

that can be simultaneously solved for the remaining two unknowns,

P and ρwall.

Eqs. (7) and (8) do not account for pores that may be inaccessible

to the condensing vapor; such porosity is folded into the average wall

density of the material. Pores might not be accessible if they are isolated,

not interconnected to the surface, too small to accommodate the vapor

molecules, or excluded because of an unfavorable surface energy.

Data Reduction and Analysis ◆

( ) dry wall ρ = 1−P ρ

( ) sat liquid wall ρ = Pρ + 1−P ρ
◆ Pore Characterization: X-ray Porosimetry

38

Our experience with a broad range of low-k dielectric materials indicates

that toluene is able to access most pores; toluene has low polarity and

wets most low-k materials, which by their nature must be engineered with

low polarity (i.e., polar groups lead to higher dielectric constants, and

these are designed to be low-k films). The only exception, however, was

the highly fluorinated materials that the toluene did not wet or penetrate

the wall material.

Eqs. (7) and (8) also do not account for swelling of the low-k film

by the condensate. XRP is, however, well suited to address swelling

because X-ray reflectivity provides a highly accurate measure of film

thickness. Using toluene vapor, we rarely see evidence of film swelling

greater than 1 % in thickness for most non-organic low-k materials.

Sometimes polymeric films will swell appreciably in the presence of

toluene, but for most low-k materials this is not the case. In some

respects, a small amount of film swelling may in fact be desirable.

If the condensate can gently swell the film, without altering the pore

structure, it is reasonable to assume that all the pores are accessible.

Swelling means that the condensate diffuses through the wall material,

ensuring that all pores are accessible. In the presence of swelling,

Eq. (8) should be modified such that the porosity is given by:

(9)

where tdry and tsat are the film thickness values of the dry and

swollen films. This equation assumes that the volume of liquid toluene

is conserved (rule of mixtures) upon absorption into the film and the

wall material. Once P is determined, Eq. (7) can be used to extract

ρwall for the dry material.

The porosities obtained by this method (no swelling, Eqs. (7) and (8))

for the porous HSQ, MSQ, and SiCOH films are (46 ± 1) %, (34 ± 1) %,

and (31 ± 1) %, respectively. The resulting wall densities of the porous

HSQ, MSQ, and SiCOH films are (1.83 ± .05) g/cm3, (1.35 ± .05) g/cm3

and (1.31 ± .05) g/cm3, respectively. Notice that the porous HSQ film

has the highest porosity, yet its dielectric constant is the largest. This

apparent contradiction is understood through the observation that the

porous HSQ film also has the highest wall density. It is also interesting

that the porous MSQ and SiCOH films have similar average densities,

wall densities, and atomic compositions, despite the fact they are prepared

from very different techniques. The elemental composition, detailed

dry

sat dry

liquid dry

sat sat dry dry

t

t t

t

t t

P − − − = ρ

ρ ρ
39

Table 1. A summary of the atomic compositions, dielectric constants (k), and

structural characteristics of different porous low-k films. The estimated standard

uncertainties of the atomic compositions, Qc

2, densities, porosities and pore radii

are ± 2 %, 0.05 Å–2, 0.05 g/cm3, 1 %, and 1 Å, respectively.

structural characteristics, and standard uncertainties in the pore

characteristics of the samples are summarized in Table 1.

The sensitivity and resolution of SXR are sufficient to reveal subtle

differences between the three films as the toluene gradually infiltrates the

pores as the partial pressure varies from P/P0 = 0 (pure air) to P/P0 = 1

(toluene saturated). In terms of the reflectivity data, Figure 15 illustrates

Data Reduction and Analysis ◆

Atomic

Porous Composition Qc

2 ρdry ρwall P

Film (Si:O:C:H) k (×10–4 Å–1) (g/cm3) (g/cm3) (%) XRP SANS

HSQ 25:48:7:20 2.20 4.25 0.98 1.83 46 13 9

MSQ 15:25:19:41 1.85 3.95 0.89 1.35 34 15 10

SiCOH 10:20:28:42 2.05 4.04 0.91 1.31 31 5 3

Figure 15. Critical angle changes for the porous HSQ sample as P/Po increases

systematically from 0 (dry air) to 1 (toluene-saturated air). Condensation of the

toluene inside the pores results in an appreciable and measurable change in Qc

2.

Pore Radius

(Å)
◆ Pore Characterization: X-ray Porosimetry

40

these changes in the porous HSQ film in the region of Qc. By assuming

that the mass density of adsorbed toluene is the same as that of the bulk

fluid, the increase in Qc

2 above the vacuum level can be used to calculate

the amount of toluene adsorbed into the film as a function of the relative

partial pressure. These data are presented as physisorption isotherms in

Figure 16, completely analogous to the examples discussed in Figure 1.

Figure 16. Physisorption isotherms for the porous HSQ, MSQ, and SiCOH films.

The lines are smooth fits using the cumulative sum of a sigmoidal and a log-normal

function for porous HSQ and MSQ films, and the sum of a Gaussian and a sigmoidal

function for porous SiCOH film. We generally recommend fitting these physisorption

isotherms with such arbitrary fit functions. If the curve faithfully parameterizes

the experimental data, it will be helpful later in extracting a smooth pore size

distribution (discussed later in Figure 18). Estimated standard uncertainties

are comparable to the size of the data markers.

In Figure 16 for the porous SiCOH film, there is a prominent uptake at

low partial pressures that quickly levels off. There is also no hysteresis

between the adsorption and desorption branches. This is characteristic

of filling micropores (pores less than 2 nm in diameter),[7–9, 12] indicating

that the PECVD process generates a large population of small pores.

In comparison, the porous MSQ film continues to adsorb over a broad

range of larger partial pressures. There is also a pronounced hysteresis

between the adsorption and desorption pathways for the porous MSQ
41

film, consistent with the filling of larger mesopores.[7–9, 12] This indicates

that the porogen in the PECVD film forms smaller domains than the

spin-cast MSQ film, leading to smaller pores in the porous SiCOH film.

This is striking because the porous SiCOH and MSQ films are otherwise

similar in terms of their atomic composition, total porosity, and wall

densities; the primary difference is in their pore size distributions (PSDs).

At low partial pressures, the adsorption process in both spin-on-glass

(SOG) films (porous HSQ and MSQ films) is gradual at first, but it

increases steeply at intermediate partial pressures due to capillary

condensation in the mesopores. After these mesopores are filled, the

adsorption isotherms level off. Note that capillary condensation and

evaporation do not take place at the same partial pressures, i.e., there is

an appreciable hysteresis. The hysteresis loop in the MSQ film is highly

asymmetric (broad on adsorption, narrow on desorption), suggesting that

pore connectivity (network) effects are important.[47] If larger pores can

only be accessed through neighboring or interconnected narrow pores, the

larger pores are not free to desorb at the relative pressure corresponding

to their critical capillary radius. The smaller pores block the desorption of

the larger pores in a manner that has been described as the “ink-bottle”

effect in the porosimetry literature.[7, 48] The result is a simultaneous

draining of both the small- and large-mesopores at a partial pressure

corresponding to the critical radius of the smaller pores. Hence the

sharper drop in the desorption curve as compared to the adsorption

branch in Figure 16.

Data Reduction and Analysis ◆

Figure 17. Schematic pore structures for the porous HSQ, MSQ, and SiCOH films.

By comparing the general shape of the adsorption/desorption isotherms

for the three different films, one can qualitatively arrive at the schematic

pore structures depicted in Figure 17. The PECVD process creates

small pores in very large quantities. The lack of a hysteresis loop

confirms that the dimensions of pores are on the order of a few toluene

molecules. These pores, as well as all the films studied here, are

apparently interconnected since toluene freely diffuses into and out of
◆ Pore Characterization: X-ray Porosimetry

42

the structures. The total porosity is, however, greater in the HSQ

material as indicated by the largest total toluene uptake in Figure 16.

The pores in the porous HSQ are also clearly larger than the PECVD

material’s pores. Like the pores in porous SiCOH, those in the porous

HSQ are fairly uniform in their size distribution as indicated by the

symmetric and relatively narrow hysteresis loop. This is in contrast to

the MSQ film with its very broad pore size distribution and the largest

average pore size (despite the lower porosity in comparison to the HSQ

film). As discussed above, the sharp desorption branch of the MSQ film

in contrast to the broad adsorption pathway indicates that pore blocking

effects are significant.

Figure 18. Approximate pore size distributions from the fits (not data) through the

physisorption isotherms in Figure 16, using Eq. (1) used to convert P/P0 into a pore

size. The distributions from the adsorption branch (solid lines) can be significantly

broader and shifted to larger pore sizes than the corresponding desorption branch,

especially in those materials (like the MSQ film) with a large distribution of

mesopore sizes.

This qualitative description can be quantified using Eq. (1) to convert

the partial pressures for capillary condensation in Figure 16 into pore

radii. Plotting the relative toluene uptake as a function of the pore size

establishes the pore size distribution (PSD). To illustrate this procedure,

we use the simplest example of the Kelvin equation (Eq. (1)) to convert
43

P/P0 into a critical radius for capillary condensation. However, we

acknowledge that alternative expressions to Eq. (1) may be more

appropriate. The Kelvin equation lacks terms that account for

interactions of the condensate with the pore surfaces and is not

particularly well suited for situations where the pore size approaches

the dimensions of the condensate molecule; these items are discussed

in greater detail in the Special Concerns section below. Nevertheless,

Figure 18 shows the resulting pore size distribution, with the total areas

beneath the curve scaled to a value consistent with the total P for

each film as reported in Table 1. It should be noted that the symbols

in Figure 18 do not represent actual data; they are generated from the

smooth fits described in the caption of Figure 16. Generally, we find

that the increments of P/P0 in experimental data of Figure 16 are not

sufficiently fine for a point-by-point differentiation when using a Kelvin

equation to generate Figure 18; significantly smoother and reasonable

distributions are obtained by transforming the arbitrary fit function.

As long as the empirical fit accurately parameterizes the physisorption

data, this is a reasonable procedure.

In the porosimetry literature, it is traditional to use the desorption branch

of the physisorption isotherms to report the PSDs. The average pore radii

in the porous HSQ, MSQ, and SiCOH films taken from the desorption

curves in Figure 18 are approximately 13 Å, 15 Å, and 5 Å, respectively.

However, it is important to realize that this average is biased towards the

minimum or constricting pore in regards to the ink-bottle effect; a more

complete interpretation of the pore structure always results from

analyzing both the adsorption and desorption branches. For example,

in Figure 18 notice that there is a relatively large difference between

adsorption and desorption pore size distributions for the MSQ materials.

While the average pore radius on desorption was approximately 15 Å, the

desorption branch leads to an approximate average pore radius of 27 Å.

By contrast, the discrepancy is much less for the HSQ film (ca. 13 Å on

desorption and 16 Å on absorption) and nearly the same (no discrepancy)

for the SiCOH film. This is consistent with the previous discussion of the

isotherms in Figure 16 and the schematic pore structures in Figure 17.

Small angle neutron scattering (SANS) is also capable of quantifying the

average pore size in these low-k dielectric films, providing a basis for

comparing for the XRP data. The SANS measurements discussed herein

are done on the NG1 beam line at the National Institute of Standards and

Technology Center for Neutron Research. The SANS data are collected

with the sample under vacuum, like the XRP, and shown in Figure 19.

Data Reduction and Analysis ◆
◆ Pore Characterization: X-ray Porosimetry

44

These scattering curves can be fit with the Debye model[49] to extract

an avrage pore size. A detailed discussion of the SANS data is beyond

the scope of this guide. Briefly, an increase in the scattering intensity

corresponds to increase in the pore size and/or porosity. In Figure 19

the porous SiCOH film shows noticeably weaker scattering than either

of the porous SOG films, which is consistent with the smaller pore size

and lower porosity. The greatest scattering intensities occur in the porous

MSQ film, which is also consistent with the greatest population of larger

pores. For the porous HSQ and MSQ films, the Debye model gives

reliable fits with average pore radii of 9 Å and 10 Å, respectively.

The radii are comparable to the XRP pore sizes reported in Table 1.

Unfortunately, the SANS intensities in the porous SiCOH film are too low

to reliably apply the Debye model; the scattering is not strong from this

low porosity film. If the Debye model were applied, the resulting pore

radius would be approximately 3 Å. This result is unphysical in light of

the implicit assumption in the Debye model that there is a distinct

interface between the pore and the wall material; 3 Å is comparable to a

single atomic radius, meaning that the model is inappropriate. Nevertheless,

there is reasonably good agreement between the SANS and XRP data for

Figure 19. SANS data for the porous HSQ (circles), MSQ (diamonds), and SiCOH

(triangles) films under vacuum. Error bars, usually smaller than the data markers,

indicate the standard uncertainty in the absolute intensity.
45

the higher porosity HSQ and MSQ films. We must also mention that there

are examples, not reported here, of where the SANS and XRP data do

not yield consistent data. The agreement, or lack of agreement, between

the XRP and SANS data here is not of particular significance. Recall the

discussion above of the oversimplifications of the Kelvin equation; this

was chosen as the simplest case example for demonstration. Likewise,

SANS also suffers from the “inverse problem” in that pore sizes can not

be obtained directly from inversion of the scattering data; a model (here

the Debye model was used) must be invoked. To a large extent, the

agreement between SANS and XRP will depend on the consistency of

the models chosen to analyze the pore size distributions.

More relevant than the quantitative agreement between the SANS and

XRP, it is important to realize that XRP is a highly sensitive porosimetry,

well suited for materials with very small pores. This is illustrated with the

porous SiCOH film. SANS is unable to perceive the porosity in this film

because pores are too small, below the resolution limit of the technique.

However, in Figure 14 there is still a very significant change in the raw

XRP data as toluene condenses inside the pores. Because of this

extraordinary sensitivity, XRP is superior to SANS or other scattering

techniques for thin films with very small pores.

One distinct advantage of XRP over other forms of porosimetry,

including the sister ellipsometric porosimetry technique,[3] is the potential

to vertically depth profile the density (and therefore porosity) through

the thickness of the film. Recall that the reflectivity data is fit using a

multi-layer model. In the preceding examples, the low-k films were

modeled with a single layer of uniform density, both under vacuum and

in the presence of toluene. However, Figure 20 shows an example of

a low-k dielectric film that has a multi-layer structure with different

porosities in the different layers. Figure 20a shows the reflectivity

data for this film both under vacuum and in the toluene-saturated

environment. Notice that the general shape of the reflectivity curves

is noticeably different from the preceding examples. In addition to the

high-frequency interference fringes reflecting the total film thickness,

there are also lower frequency oscillations in the data as roughly

indicated by the lines and arrows; a fourier transform of this data would

reveal multiple characterstic frequencies. The low frequency oscillations

stem from the mutli-layer nature of the structure. Fitting this reflectivity

data requires 3 distinct layers for the low-k film, as revealed in Figure

20b. Here the electron density of the layers as a function of distance

into the film (starting from the free surface) is plotted in terms of Qc

2.

Near the free surface (layer 1) the density is noticeably larger than the

Data Reduction and Analysis ◆
◆ Pore Characterization: X-ray Porosimetry

46

Figure 20. XRP data for a low-k film comprised of 3 distinct layers. Part (a)

shows the reflectivity data for the dry and toluene-saturated films, revealing both

a high-frequency periodicity due to the total film thickness and low-frequency

oscillations due to the thinner individual layers. Part (b) shows the real space

scattering length density profiles as a function of distance into the film, revealing

the thickness and density of the individual layers.
47

rest of the low-k film, the consequence of a plasma treatment to the

surface. Also note that the density appears lowest (layer 3) near the

Si substrate.

When this multi-layer film is exposed to the toluene environment, two

effects become immediately obvious. First, the density increases in all

three layers as toluene condenses in the individual pores. Second, the

total film thickness swells by approximately 4.4 %, meaning that Eq. (9),

not (8), must be used to extract the porosities. What may be less

obvious is that each of the 3 layers picks up different amounts of toluene.

Qualitatively, Figure 20b shows that layer 2 takes in the most toluene while

layer 1 picks up the least. Quantitatively, we can use Eq. (9) to predict

that the porosities in layers 1, 2, and 3 of this low-k film are approximately

13 %, 21 %, and 16 %, respectively. Despite the lower overall electron

density in layer 3, the greatest porosity is found in layer 2. This is

because the wall densities are also greater in layer 2, offsetting the

porosity effect in terms of the average film density. This example

demonstrates the power of XRP in extracting depth-dependent, detailed

pore information.

3.D. Special Concerns

3.D.1. Quantitative Interpretation of the Physisorption Isotherms

The shapes of the adsorption/desorption isotherms in Figure 16 readily

convey a wealth of qualitative information regarding the pore structure.

However, it can be a challenge to quantify these characteristics in

terms of both the average and distribution of pore sizes. This is

equally true for the thin porous films presented here as well as all

physisorption isotherms in general; this is not a specific limitation of

X-ray porosimetry. The adsorption/desorption process is complex and

affected by many factors. Consequently there are several ways to

interpret the isotherms, each focusing on different aspects or based

on different assumptions. Generally, these analysis schemes can be

grouped into three general classes. The first group emphasizes the

initial stages of adsorption, while the first monolayer(s) of adsorbate

adhere to the surface. Well-known examples include the Henry’s Law

type analyses, the Hill–de Boer equation,[50] the Langmuir theory,[51]

and the Brunauer–Emmett–Teller (BET) analysis.[52] These theories

have been developed primarily for adsorption onto non-porous solids

and are surface-area techniques.

Data Reduction and Analysis ◆
◆ Pore Characterization: X-ray Porosimetry

48

The low partial pressure regions of an isotherm are also where

micropore filling occurs. Micropores, with widths less than 2 nm,

are easily filled by a few monolayers of most adsorbents, and the

second group of equations or theories attempt to extract micropore

characteristics from the initial stages of the isotherm. These are

similar to the surface area techniques and include Henry’s Law

based interpretations, the Langmuir–Brunauer equation,[53] and the

Dubinin–Stoeckli based theories.[54, 55] In systems that have very small

pores, like the porous SiCOH film, these types of analysis are probably

more appropriate than the Kelvin equation.

The final group of equations focuses on the latter stages of adsorption,

where the mesopores (ca. 2 nm to 50 nm in width according to the

IUPAC definition[7, 13]) are filled. This is the region where capillary

condensation occurs, and the Kelvin equation is the simplest of these

interpretations. There are numerous variations on the Kelvin equation

that account for effects like multilayer adsorption prior to capillary

condensation (i.e., BJH method[56]), disjoining pressure effects in the

condensed liquid (i.e., DBdB method[57]), etc. However, it is not our

intention to ascertain which adaptation of the Kelvin equation is most

appropriate. This will vary significantly with the nature of the adsorbate

and the porous material. Rather, we applied the simplest form of the

Kelvin equation (see Eq (1)) to demonstrate the technique. These

variations of the Kelvin equation are easy to implement, but the

underlying assumptions should be carefully considered for each

different experimental consideration. For assistance in evaluating

which analysis method is most appropriate, we recommend the recent

textbook by Rouquerol, et al.[7]

Eq. (1) provides the simplest possible Kelvin equation conversion

between P/P0 and the pore sizes. Figure 18 displays the corresponding

pore size distributions extracted from data in Figure 16. Notice that

pore size distributions have been extracted from both the adsorption

(solid lines) and desorption (dotted lines) branches of the isotherm,

and that the distributions are always broader and shifted to a higher

average pore size for the adsorption branch. This discrepancy is

especially evident for the highly porous samples and consistent with

the pore-blocking effects discussed earlier. It is sometimes a “general

practice” in the porosimetry field to report pore size distributions for the

desorption branch of the isotherm. However, in the presence of pore

blocking this leads to artificially narrow and smaller distributions. It is

crucial to look at both the adsorption and desorption isotherms to obtain

a more comprehensive understanding of the pore structure. Once again,
49

though, this criticism is applicable to all forms of porosimetry and is not

unique for X-ray porosimetry.

The pore size distributions in Figure 18 are approximate and only

intended to demonstrate the technique. The average sizes are below

20 nm, which means that interpretations in terms of simple capillary

condensation may be in error by well over 100 %.[58–60] Generally,

the physics of capillary condensation in pores smaller than 20 nm is

poorly understood. Studies show that more accurate interpretations

require computer modeling and/or simulations of the isotherms.

Gage Cell Monte-Carlo (GCMC) simulations and non-linear density

functional theory (NLDFT) methods[58–60] have recently been

developed to interpret adsorption/desorption isotherms from nanoporous materials and reveal that all of the existing interpretations of

capillary condensation fail to extract reliable pore sizes. Currently,

these GCMC and NLDFT techniques are the only reliable methods

for understanding capillary condensation in pores significantly smaller

than 20 nm and should, therefore, be considered when studying low-k

dielectric films.

It is important to realize that this issue of the actual pore size is a

matter of interpretation, but the raw isotherms in Figure 16 are robust.

Likewise, the relative differences between the different materials are

obvious and reliable. Porosimetry in general is very model dependent.

To interpret the data one must assume a model and then verify how

well it fits the data. This does not ensure that the solutions obtained are

unique. To reach a proper interpretation requires supporting information,

like the nature of the interactions between the adsorbent and adsorbate,

feasible schemes for the pore architecture (i.e., isolated spheres,

interconnected channels, fractals), etc. This type of supporting data

will help facilitate a correct interpretation of the isotherms.

3.D.2. Isothermal Control

In the experimental section, we discussed the different methods by

which the P/P0 ratio can be varied, focusing specifically on the

isothermal mixing of dry and toluene-saturated airstreams (the P

variation method) as well as the controlled variation of the sample

temperature in the presence of air saturated at 25 °C with toluene

(the T variation method). With both of these techniques, accurate

temperature control of the sample is imperative because P0 is very

temperature sensitive; inaccurate physisorption isotherms will be

obtained if the temperature control is poor. A clear illustration of this

Data Reduction and Analysis ◆
◆ Pore Characterization: X-ray Porosimetry

50

is seen in Figure 21 below where several physisorption isotherms are

shown for the same porous HSQ material discussed earlier. Examining

the desorption pathways using the P variation technique, for which the

sample temperature was maintained at 25 °C, the triangles correspond

to a desorption curve where the control unit failed and the sample

was maintained at (25 ± 1.5) °C, while asterisks represent good

thermal control of (25 ± 0.1) °C. The two desorption pathways are

different and would lead to large differences if the average pore size

and distributions were calculated. The required accuracy of the

temperature control will depend upon how temperature sensitive

the vapor pressure is. For toluene we recommend the temperature

be controlled to within a tenth of a degree for reproducible

physisorption data.

3.D.3. Time Dependence of Desorption

In all of the physisorption isotherms presented thus far, approximately

30 min is allowed for equilibration after each P/P0 change, for both the

Figure 21. Physisorption isotherms generated by the T (squares) and P (all other

data markers) variation methodologies of controlling P/P0, as described in the text.

The two techniques do not produce identical isotherms indicating that adsorption

is not temperature invariant. Notice the discontinuous nature of the desorption

pathways of the P variation technique. These discontinuities can be attributed to

insufficient equilibration times, as described in the text and in reference to Figure 22.

The estimated standard uncertainty in Qc

2 is comparable to the size of the data markers.
51

T and P variations techniques. This is based on initial studies on the

time dependence of the Qc

2 variations after a step increase of P/P0

from 0 to 1.0. We generally found that Qc

2 would continue to increase

for approximately 15 min after such a step increase, finally reaching

a time invariant plateau; 30 min equilibration times were deemed

reasonable. This was further reinforced by the fact that the

measurements paused at a given P/P0 in the adsorption branch and

held over night would resume with the same Qc

2 value the next

morning. However, eventually we noticed that measurements

could not be paused reversibly in the region of steep descent on

the desorption branch of the hysteresis loop; Qc

2 would continue to

drop significantly in this region while P/P0 was held constant. This is

shown in Figure 21. On the desorption branch with the open triangles,

notice that the curve looks discontinuous in the region between the

two dashed vertical lines at P/P0 = 0.28 and P/P0 = 0.36. The location

of this discontinuity coincides with a 12 h pause in the data collection.

Likewise, a similar discontinuity is seen in the asterisks 3 points below

the P/P0 = 0.28 vertical line. The desorption branch with the asterisk

data markers starts turning concave up just below P/P0 = 0.28, but then

Data Reduction and Analysis ◆

Figure 22. Time dependence of the Qc

2 variations in the porous HSQ sample after

P/P0 jumps from 1.0 to 0.36 (squares) and 1.0 to 0.28 (circles). Notice that Qc

2

continues to evolve for several hours after the jump, indicating the equilibrium is

difficult to achieve on the desorption branch. The same time dependence is curiously

absent upon adsorption. The estimated standard uncertainty in Qc

2 is comparable

to the size of the data markers.
◆ Pore Characterization: X-ray Porosimetry

52

a discontinuous jump occurs at P/P0 = 0.20. This discontinuity also

coincides with a pause in the data collection. When the desorption

branch in this region is repeated, ensuring sufficient equilibration

after each P/P0 change, the large open circles and heavy dotted

line indicate the smooth equilibrium desorption path.

To explore the time dependence of the Qc

2 variations on

adsorption/desorption further, we performed the following

experiments. Using the P variation technique, P/P0 jumps from

1.0 to 0.28 and 1.0 to 0.36 were imposed onto the same porous

HSQ displayed in Figure 21. The evolution of Qc

2 was then tracked

over extended periods of time, as shown in Figure 22. Qc

2 decreases

very slowly, reaching a plateau only after several days (note that

in Figure 22 the time axis is logarithmic). Surprisingly, this time

dependence is not observed if the sense of the P/P0 jump is reversed,

i.e., from 0 to 0.28 or 0.36; under these conditions equilibrium is

rapidly achieved, consistent with the original 30 min equilibration times.

The reason for these differences between the adsorption and desorption

pathways is not understood. We suspect that it reflects diffusion of the

toluene through the matrix walls at low partial pressure as opposed to

adsorption. Regardless, this clearly means that equilibrium desorption

isotherms may be difficult to achieve. Simple parameters like the rate

of P/P0 change can affect the resulting pore size distributions. Care

must be exercised, especially on the desorption pathways, to ensure

that equilibration is achieved.

3.D.4. P versus T Variations of P/P0

Figure 21 also displays physisorption isotherms corresponding to the

P (circles — the equilibrium data) and T (squares) variation techniques.

The closed symbols denote the adsorption branch of the isotherm while

the open symbols indicate the desorption branch. In this presentation,

the non-isothermal T variations have been transformed into their

isothermal equivalent at 25 °C through Eq. (6). This allows us to

directly compare the true isothermal P/P0 variations to their T variation

counterpart. Notice that the physisorption isotherms are not equivalent.

This means that different results would be obtained if Eq. (1) is used

to quantify the pore size distributions. The reason for the discrepancy

lies in the simplistic and inappropriate assumptions within the Kelvin

equation. Factors like heats of the toluene adsorption and/or deviations

from the bulk physical properties for the condensed toluene are

probably significant. The exact reason for this failure has yet to be

identified. Obviously, a more sophisticated data analysis procedure is

required to fully comprehend these differences.
53

IV. SUMMARY

We present XRP as a powerful tool for characterizing the pore characteristics

in low-k dielectric or other smooth nanoporous film. The technique is based

on gradually increasing or decreasing the partial pressure of toluene or some

other adsorbate in the presence of the film and simultaneously monitoring

the changes in the critical angle for total X-ray reflectance. The change in

the critical angle can be converted into the volume of adsorbed condensate,

and thus porosity, if the density of the condensed fluid is known or assumed

to be bulk-like. Monitoring the amount of adsorbed condensate as a

function of the partial pressure defines a physisorption isotherm, the basic

starting point for any number of analytical interpretations. XRP directly

reveals the average porosity, average film density, and average wall density

of the material separating the pores. By further invoking a model of

condensation (for example, the Kelvin equation for capillary condensation),

it is straightforward to calculate the average pore size and pore size

distribution from the physisorption isotherm. XRP has the added advantage

that it directly monitors the thickness of the low-k film, with Å level resolution

and without knowledge of the optical constants of the film. Simultaneous

thickness measurements are invaluable for determining if the adsorbed

condensate swells the film. Finally, XRP has the distinct advantage of being

able to resolve the density profile through the thickness dimension of the film.

This is useful for perceiving porosity measurements at different depths into

the films in the case of layered structures.

Summary ◆
◆ Pore Characterization: X-ray Porosimetry

54
55

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◆ Pore Characterization: X-ray Porosimetry

56

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X-ray Porosimetry

Christopher L. Soles, Hae-Jeong Lee,Eric K. Lin, and Wen-li Wu

960-13Special Publication 960-13