Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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208 Part B Chemical and Microstructural Analysis

Table 5.1 (continued)

Methodology Subject under consideration Material type Atom

arrangement

Category

Method

Ref.

Structural order

Defects and impurities

Molecular architecture

Microstructure

Finite structures

Metal

Semiconductor

Insulator

Others

Crystalline

Amorphous

Requirements

Miscellaneous

Density

measurements

Gupta

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

Quartz crystal

microbalance

Gupta

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

Radioactivation

analysis

Alfassi

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Limitted

nuclear

species

SIMS Proﬁle 6 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ UHV

Q-Mass 4 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ UHV

LC-MS

5.4.3, 4.1 ◦ ◦ ◦ ◦ ◦ ◦ ◦

GC-MS

5.4.3, 4.1 ◦ ◦ ◦ ◦ ◦ ◦ ◦

GPC/GFC

5.4.3, 4.1 ◦ ◦ ◦ ◦ ◦ ◦ ◦

ICP

4.2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

AAS/AES

6 ◦ ◦ ◦ ◦ ◦ ◦ ◦ UHV

Electrophoresis 5.4.3 ◦ ◦ ◦ ◦ ◦ ◦

Electron

Holography

Tonomura

◦ ◦ ◦ ◦ ◦ ◦

FRET

5.4.3 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

Pump & Probe Shah

, 11 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Ultrafast

pulse laser

Chapter(s) and section(s) describing the detail or literature to which the interested readers should refer

Including macromolecules

Including phase distribution and texture

e.g., dendrites, cells, bones, ﬁbres, nanoparticles, device structures etc.

Isolated molecules and molecular solids

High vacuum

Ultrahigh vacuum

Including anomalous

Including HRTEM

Including HAADF-STEM

Including back scattering EM, SEM-EBIC, SEM-CL, scanning reﬂection electron microscopy

Including etch pits observation

Laser scanning confocal microscopy

Total internal reﬂection ﬂuorescence microscopy

[5.1]

Including photoacoustic spectroscopy, resonance-enhanced multiphoton ionization

Including pure rotation spectroscopy

Part B 5

210 Part B Chemical and Microstructural Analysis

Table 5.1 (continued)

Including DLTS, photocapacitance

Including optically-detected magnetic resonance

Including optical rotary dispersion and magnetic circular dichroism

Electron nuclear double resonance

Extended x-ray absorption ﬁne structure

Including XPS,UPS

Including internal friction, anelasticity

Including SEM-based and (S)TEM-based

Including EXELFS (extended energy loss ﬁne structure)

Electron probe micro analyzer

Flurometry

[5.2]

Including nonlinear ﬂuorescence (two-photon) microscopy and second-harmonic generation microscopy

[5.3]

[5.4]

Liquid chromatography-mass spectroscopy

Gas chromatography-mass spectroscopy

Gel permeation (ﬁltration) chromatography

Including ﬂame photometry (FAAS/FAES/FAFS (ﬂame atomic absorption (emission, ﬂuorescence) spectrometry))

Atomic absorption (emission) spectrometry

[5.5]

Fluorescence resonance energy transfer

[5.6]

Table 5.2 Sample requirements in some common methods in their conventional form

Method Minimum center density Minimum sample amount Maximum molecular size

Element analysis

GC-MS 10 ppb ≈1ml Boiling temperature < 400

◦

Radioactivation analysis 0.1 ppb 0.01 ng −

X-ray ﬂuorescence 10ppm 100 mg

−

Structure analysis

Single crystal XRD − 0.1mm

< 100 atoms

> 600 atoms

or 10

in molecular weight

Powder XRD − 0.2–0.5mm Several 10

atoms

NMR ≈ 1% 1mM,severalmg 3×10

in molecular weight

Defect analysis

ESR

–10

−3g

< cavity size −

or 10

–10

−3h

Positron annihilation 5×10

–5×10

−3i

No serious limitation ≈ 10 vacancies

Raman scattering 0.01–1% (several ppb

) Several μl (liquid) −

FT-IR 0.01% sub μg

−

PL 0.1 ppm > mg −

Capacitance spectroscopy 10

−3

1mm −

In crystal structure analysis, the size of atom assembly (molecule) occupying a lattice point

Reduced by orders of magnitude when a synchrotron radiation source is used

0.02 mm when a synchrotron radiation source is used

Part B 5

Nanoscopic Architecture and Microstructure 5.1 Fundamentals 211

Table 5.2 Sample requirements in some common methods in their conventional form, cont.

When direct method is used

When heavy atoms substitution method works

The sensitivity is quite dependent on what signal is detected. Here the signal is microwave absorption. Even a single spin can be detected if

a special technique is used [5.7]

In insulators

In semiconductors

For negatively charged vacancies at room temperature no serious limitation

Above which the vacancy clusters become indistinguishable from the surface

When resonant Raman scattering is employed

When combined with a chromatographic separation preprocess

5.1 Fundamentals

This section deals with the fundamental (minimum)

knowledge that the reader should have in advance. This

section is also intended to guide the reader to be able

to select properly the appropriate techniques for their

purpose by overviewing the range of energy, the length

scale, the spatial resolution and the time scale that are

covered by the entire range of these respective tech-

niques. Since it appears that diffraction methods are not

well covered in other parts of this handbook, the basic

physics of diffraction and the principles of microscopic

techniques (mainly TEM) based on beam diffraction

will be described in some depth. We leave most of the

basics of spectroscopic techniques to Chap. 11.

5.1.1 Diffraction and Scattering Methods

The scattering of particles by an interacting target

is a common phenomenon usually interpretable in

terms of classical mechanics with a minor correction.

Generally, the waves (electromagnetic, de Broglie, vi-

brational, etc.) are scattered with an efﬁciency that can

vary widely depending on the nature and the size of the

scatterers and the wavelength. The diffraction of waves

is a term used to describe the constructive interference

of waves coherently scattered by multiple scatterers.

Scattering and diffraction are, thus, inseparable con-

cepts that are sometimes treated together in textbooks.

Absorption is an apparently different aspect of waves

in which some wave energy is consumed by excitation

of a quasiparticle of different nature. In some cases,

however, absorption is an aspect of wave scattering in

which the excited state is derived from the interference

of the scattered waves. The wavelength dependence of

absorption is the central subject of various schemes of

spectroscopy, which are addressed mainly in Sect. 5.1.2.

The scatterers of electromagnetic waves from the

visible light to the x-ray regime are electrons, and those

in the infrared regime are phonons or molecular vi-

brations. Although the penetration of electromagnetic

waves through solids can vary signiﬁcantly from ma-

terial to material due to the wavelength dependence of

the absorption coefﬁcient, hard x-rays have a relatively

high penetrability through solids as long as the con-

stituent elements are light. In contrast, since electrons

have a negative electric charge, they interact so strongly

with matter that they practically cannot penetrate even

thin solids unless the energy is higher than 100 keV. On

the contrary, neutrons have no electric charge and so

can penetrate deeply into solids. Neutrons have a spin

magnetic moment and therefore interact with electrons,

which also have a magnetic moment. This fact pro-

vides unique experimental approaches for studies of

magnetism in solids. Furthermore, neutrons also inter-

act with nuclei with scattering cross sections that are

quite different from those of x-rays and electrons; for

example, the detection of protons that are difﬁcult to

detect with x-rays and electrons but that are detectable

by neutrons due to the large scattering cross section.

The physical parameter of fundamental importance

in scattering and diffraction of waves or quantum beams

is the wavelength or wave number. Figure 5.1 shows

the wavelength of various beams plotted as a func-

tion of the quantum energy of the beam, the energy

quantum hν of a photon and the kinetic energy of an

electron and of a neutron. Since, as explained later,

diffraction occurs most signiﬁcantly when the wave-

length is close to the separation of the scatterers, particle

beams with a wavelength of the order of 0.1 nm are

most suitable for diffraction studies of atomic arrange-

ment. Among electromagnetic waves, x-rays of 10 keV

Part B 5.1

212 Part B Chemical and Microstructural Analysis

–2

–4

–2

Wavelength (nm)

Energy (eV)

Photon

Electron

Neutron

Fig. 5.1 Wavelength versus energy relations in various

quantum beams

are in this range, whereas electrons of 100 eV also have

such a wavelength. The fast neutrons emitted from nu-

clear reactors or accelerator targets have energies of

the order of MeV but can be slowed down to energies

below 1 keV after passing through a moderator consist-

ing of materials such as water and graphite. Among

the slow neutrons, those whose energy is near room

temperature (0.025 eV) and those below room temper-

ature are called thermal neutrons and cold neutrons,

respectively. These are the neutrons used for diffrac-

tion studies. Neutron diffraction can be an alternative

to x-ray diffraction when x-rays do not detect low-Z

elements well, to distinguish near-Z elements or to pen-

etrate thick samples.

Since a comprehensive description of the funda-

mentals of crystal diffraction is found in many good

textbooks [5.8–12], only the essence is summarized

here so that the reader may understand the princi-

ples of the experimental techniques described in later

sections.

In cases in which the interaction of waves with mat-

ter is weak (typically in x-ray and neutron diffraction),

it is a good approximation to use the kinematical theory

which assumes that the wave can be scattered only once

in the matter. However, when the matter is crystalline

and the crystal is thick and nearly perfect, diffraction

may have to be treated by the dynamical theory that rig-

orously takes into account multiple diffractions in the

crystal. The same situation also arises in the diffraction

of charged particles such as electrons that interact very

strongly with solids. Nevertheless, although the quan-

titative interpretation of some phenomena requires the

dynamical theory, many of the phenomena encountered

in experiments can be understood in the framework of

the kinematical theory. For the dynamical theory, the

minimum knowledge is given in Sects. 5.1.2 and 5.3.2

where it is required.

In real space, a strong diffraction of a wave of wave-

length λ is induced by a set of net planes (Note that

atoms on each net plane do not have to be regularly

arrayed.) regularly separated by a distance a (Fig. 5.2)

when the wave is incident on the net planes at an angle θ

satisfying the Bragg condition

2a sin θ =nλ (n : integer) . (5.1)

In the reciprocal space, the Ewald sphere is the most

commonly used graphical construction of wave diffrac-

tion, whether for x-ray or de Broglie waves. Precisely,

the sphere of radius 1/λ is three-dimensional but usu-

ally represented in two dimensions by a circle touching

the origin of the reciprocal lattice, as shown in Fig. 5.3.

The Bragg condition is satisﬁed when the sphere passes

through a reciprocal lattice point other than the origin.

The wave vector of the incident wave k

and that of

the diffracted wave k are indicated by the arrows drawn

Bragg condition:

2asinυ =nμ

2υ

K = g ⏐g⏐=

Fig. 5.2 Diffraction by a set of net planes periodically

separated by a distance a. Constructive interference takes

place under the Bragg condition. Note that atoms on each

net plane are not necessarily periodic for specular reﬂec-

tion to occur

Part B 5.1

Nanoscopic Architecture and Microstructure 5.1 Fundamentals 213

(0,0,0)

K = g

Fig. 5.3 Graphical reconstruction of wave diffraction by

Ewald sphere in the reciprocal lattice. K = g corresponds

to the Bragg condition

from the center of the sphere to the origin and to the

reciprocal lattice point, respectively. Thus, the Bragg

condition (5.1) is equivalent to

K = g , (5.2)

where K ≡k−k

is called the scattering wave vector,

and g(

=n/a) is the reciprocal lattice vector normal

to the net planes. (Throughout this chapter, we fol-

low the convention that the factor 2π is not included

in the deﬁnition of reciprocal vectors g and wave vec-

tors k.)

In x-ray diffraction, the entities that scatter x-rays

in matter are electrons. Electrons forced to oscillate by

the alternating electric ﬁeld of the electromagnetic wave

emit an electromagnetic wave of the same photon en-

ergy (elastic scattering or Thomson scattering) or of

a slightly reduced energy (inelastic scattering or Comp-

ton scattering), the latter constituting the background

for the coherent interference of the former, elastically

scattered, waves. Generally, waves incident with inten-

sity I

and wave vector k

are scattered by a collection

of electrons or an electron cloud of density ρ(r)toin-

terfere to yield a far-ﬁeld electromagnetic wave with

a wave vector k and intensity

I(K) = I





ρ(r)exp

(

2πiK ·r

)



. (5.3)

If the scatterers are electrons only in a single atom,

the scattering intensity is given by

(K) = I

f (K)

, (5.4)

with the atomic scattering factor deﬁned by

f (K) ≡



(r)exp

(

2πiK ·r

)

dv, (5.5)

the magnitude of which is determined by the atomic

electron density ρ

(r) and hence speciﬁc for the atomic

element. The atomic scattering factor is a complex

quantity and may also vary depending on the beam en-

ergy, bringing about the anomalous dispersion effect

(Sects. 5.2.1 and 5.2.3).

In the Born approximation, or the kinematical the-

ory, in which a crystal is thin enough for x-ray scattering

to take place at most once in the solid, the intensity of

the x-ray scattered by the assembly of atoms is simply

given by

I(K) = I





n, j

(K)exp



2πiK ·







≡ I

·G(K)·

F(K)

. (5.6)

Here, as shown in Fig. 5.4, R

= n

a +n

b +n

c de-

notes the position of the nth crystalline unit cell (the

vectors a, b and c are the unit-cell translations while

, n

and n

are integers), r

is the relative position of

the jth atom within the unit cell, and f

is the atomic

scattering factor of the jth atom. The factor

F(K) ≡

unit cell



(K)exp



2πiK ·r



(5.7)

Fig. 5.4 X-ray diffraction by a crystal lattice the unit cell

of which is indicated with a paralellogram

Part B 5.1

214 Part B Chemical and Microstructural Analysis

is called the structure factor and characterizes the

atomic arrangement in the unit cell. Another factor

G(K) ≡





exp

(

2πiK ·R

)





exp (2πik



exp (2πik



exp (2πik



(5.8)

is called the interference function, which is written

in the second formula in terms of the components of

the vector K = (k

, k

) projected onto the unit-cell

translation vectors. In diffraction of a monochromatic

wave from a sufﬁciently large single crystal (n

, n

and n

are all large), G(K) is nonzero (n

·n

) only

when

a =h, k

b =k and k

c =, (5.9)

for integers h, k and . This is again equivalent to the

Bragg condition (5.2), the condition for constructive

interference. In such cases, the diffraction intensity is

sin υ

400

300

200

100

–1.0 –0.5 0.0 0.5 1.0

N =20

N =10

N =5

Fig. 5.5a,b As the sharpness of diffraction decreases with the decreasing size of a grating in a monochromator (a),

diffraction peaks in small crystals show broadening, which is represented by the diffuseness of reciprocal lattice points

(b)

simply given by |F(g)|

times the number of unit cells

in the sample. In small crystals, however, diffraction

also occurs in directions other than those satisfying (5.2)

with intensities that depend on the interference function,

G(K), which reﬂects the size and external shape of the

crystal.

Thus, we have to consider two cases: the case of

large crystals for which the effects of G(K) can be ne-

glected, and the case of small crystals for which the

morphological information contained in G(K) is our

main concern.

Combining (5.5)and(5.7)weﬁndthatF(K) is re-

lated to the electronic density ρ(r) through a Fourier

transformation. This means that for large crystals we

could deduce the electronic density ρ(r) or the atomic

arrangement in the crystalline unit cell by inverse

Fourier transformation of the structure factors that can

be found from the experimental scattering intensity.

This forms the basis of structural determination by

x-ray diffraction from crystalline samples (Sects. 5.2.1

and 5.4.1).

For small crystals, we have broadening of diffrac-

tion spots as happens in the diffraction of light by an

optical grating of ﬁnite size (Fig. 5.5a). This diffraction

broadening could be regarded as due to the aug-

Part B 5.1

Nanoscopic Architecture and Microstructure 5.1 Fundamentals 215

Real space Reciprocal space

a) b)

c) d)

Fig. 5.6a–d Broadening and sharpening of reciprocal lat-

tice points (b),(d) due to the external shape of the crystals

(a),(c)

mentation of the reciprocal lattice points (Fig. 5.5b).

Mathematically, G(K) is again the Fourier transform

of the external shape of the crystal, so thin crystals

elongated in one direction, as shown in Fig. 5.6a, yield

sharpening of the reciprocal lattice points along that

direction but broadening perpendicular to it, as shown

in Fig. 5.6b. Similarly, plate-like ﬂat crystals (Fig. 5.6c)

broaden the diffraction peaks along the surface normal,

as shown in Fig. 5.6d. This fact makes it possi-

ble to analyze the medium-range order (Sect. 5.2.2),

to determine experimentally the small particle size

(Sect. 5.5.3), and to conduct high-resolution TEM ob-

servations (Sect. 5.1.2).

5.1.2 Microscopy and Topography

Microscopy allows one to study material structures with

spatial resolution. The resolving power of microscopes

depends on the beam that is used to construct images.

Most microscopic methods may be grouped into one

of two types: ordinary microscopy, which uses certain

waves for sample illumination and a set of lenses for

construction of an image; and scanning microscopy,

which uses a tiny probe that is scanned over the sam-

ple to construct a microscopic image synchronously

displayed on a two-dimensional screen. In the former

type of microscopes, the dominant factor governing the

spatial resolution is the wavelength of the wave or the

aberration of the objective lens (the convex lens near-

est to the object being observed). Even if the lens is

free of aberration, its focus is not a perfect point but is

blurred by the effect of wave diffraction due to the ﬁnite

lens diameter (Fig. 5.7). Owing to the parallel imaging

capability in the former type of microscopes, one may

observe in situ the dynamic change or motion of objects,

which could be recorded by a video camera, although

the same may also be possible in the latter type if the

scanning rate is high enough to follow the events. Fig-

ure 5.8 shows the range of scales that is covered by var-

ious microscopes. The lower limit and the upper limit

of the lateral scale indicate the lateral resolution and the

maximum size of the specimen that can be surveyed,

respectively. The lower and upper limits of the vertical

scale indicate the sample thinness required for micro-

scopic observations and the sample thickness allowed

for experiments, respectively. Other features of micro-

scopic techniques are also summarized in Table 5.3.

Optical Microscopy (OM)

We mention here only brieﬂy three types of optical

microscopy (OM): interference microscopy, which is

mature but plays important roles in the assessment

of material microstructures; laser scanning confocal

Point resolution:

= 0.67

n sinα:

numerical aperture

n sinα:

Fig. 5.7a–c Point resolution in microscopes limited by

wave diffraction. In lenses of ﬁnite size, even if aberration-

free, the focal point is blurred due to the diffraction effect.

(a) is equivalent to (b); (c) deﬁnes the point resolution

Part B 5.1

216 Part B Chemical and Microstructural Analysis

Table 5.3 Comparison of various microscopic techniques

Method Operation

mode

Object Resolution Sample

requirement

Test

environment

Merit Demerit

Optical

microscopy

Interference

microscopy

Steps < 1μm Transparent or

reﬂective

Ambient, liquid Easy operation,

wide survey

range,

dynamic

observation

Limited

magniﬁcation

LSCM Light scatterers,

ﬂuorochromes

TIRFM ﬂuorochromes

X-ray

topography

Internal

architecture

< afewμm

High quality

single crystal

Ambient,

gaseous, vacuum

Applicable to

bulky samples

Low

magniﬁcation

TEM Two-wave

diffraction

Dislocations,

planar defects

Several tens

Thickness

< 1 μm

Vacuum Crystallographic

analysis,

dynamic

observation

Narrow survey

range

Weak beam Dislocations 0.5nm Thickness

< 1 μm

Vacuum

HRTEM Extended

defects, layer

stacking

0.2nm Thickness

< 100 nm

Vacuum Atomic-level

resolution,

dynamic

observation

Critical sample

preparation

STEM HAADF Extended

defects,

impurities

< 0.2nm Thickness

< several tens

Vacuum Insensitivity to

focusing and

sample thickness

Slow image

acquisition

SEM SEM-SE Surface

morphology

< 1nm High SE

efﬁciency,

electrically

conductive

Vacuum, gas Easy operation,

wide survey

range, dynamic

observation

Little

information on

material

substance

SEM-EBIC Recombination

centers in

semiconductors

≈1 μm

Semiconductors

with electrodes

High Vacuum

(HV)

High sensitivity Needs electrical

contacts

SEM-CL Cathodo-

luminescent/

nonradiative

centers

≈1 μm

Luminescent

semiconductors

HV Nondestructive

spectroscopic

Limited to

luminescent

materials

EPMA Element

analysis

≈1 μm X-ray

ﬂuorescence

HV, Ultra-High

Vacuum (UHV)

Established Vulnerable to

contamination

FIM Surface atoms < 0.1nm Electrically

conductive thin

needles

UHV Atomic

resolution

Difﬁcult sample

preparation

STM Constant

current

Surface LDOS,

subsurface

defects

< 0.1nm Electrically

conductive

clean surface

Ambient, UHV,

liquid

Afﬁnity with

various micro-

spectroscopic

techniques

Limited imaging

speed, difﬁculty

in reproducible

tip preparation

AFM Noncontact Surface atoms,

surface

morphology

≈0.2nm

(depends)

Roughness

< 1 μm

Ambient, UHV,

liquid

Applicable to

nonconductive

samples

Limited imaging

speed

SNOM Va ri ab le Var ia bl e Several tens

Optically

active

Ambient,

UHV, liquid

Many potential

applications

Limited imaging

speed

Variable depending on the material and experimental conditions

Part B 5.1

Nanoscopic Architecture and Microstructure 5.1 Fundamentals 217

microscopy (LSCM) of the scanning type; and total

internal reﬂection ﬂuorescence microscopy (TIRFM),

a relatively new technique with expanding applications

particularly to biological substances. For the principles

of conventional OM, the interested reader should refer

to [5.13].

Interference microscopy is based on the phase shifts

of the illuminating light that take place when the light

passes through a transparent sample with different re-

fraction indices or when it is reﬂected at a surface

with steps of different heights. The interference of this

light with a reference light forms interference fringes

or, in differential interference Nomarski microscopes

(Fig. 5.9), the interference of two slightly displaced

beams is converted to different colors with high spa-

tial resolution, which allows the measurement of step

heights with a precision of 20 nm.

LSCM is classiﬁed as a scanning microscope, us-

ing a laser beam as a probe. The principle is shown in

Analyzer

Interference color

Half mirror

White light

Nomarski prism

Extraordinary ray

Backfocal point of the objective lens

Ordinary ray

Objective lens

Sample

Slightly displaced

Incident wave front

Reflected wave front

Sample surface

Phase shift

Polarizer

Fig. 5.9 (a) The principle of differential interference opti-

cal (Nomarski) microscopy. (b) Phase shift that occurs in

reﬂecting waves at a step edge

Fig. 5.10, where a parallel laser beam focused at an en-

trance aperture is further focused by an objective lens

to a small spot in the sample and the scattered light is

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

–11

–10

–9

–8

–7

–6

–5

–4

–3

–2

–1

Vertical scale (m)

Lateral scale (m)

Human

eye

SEM

PCM

STEM

TEM

HRTEM

FIM

STM AFM

SNOM

Fig. 5.8 Scale range covered by various types of microscopic tech-

niques. For the meaning of the horizontal and vertical axes, see the

text

Photo-

detector

Exit aperture

Half mirror

Objective lens

Sample

ScanRegion to be omitted

Laser beam

Entrance

aperture

Fig. 5.10 The principle of laser scanning confocal micro-

scopy

Part B 5.1