Egerton R.F. Electron Energy-Loss Spectroscopy in the Electron Microscope

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118 3 Physics of Electron Scattering

Fig. 3.5 Angular distribution of elastic scattering from a single C atom (solid curve)andadia-

mond crystal (vertical bars) whose c rystallographic axes are parallel to the incident beam. For

comparison, the angular distributions of inelastic scattering by outer-shell and K-shell electrons

are shown as Lorentzian functions corresponding to energy losses of 35 and 300 eV, respectively.

In a crystalline solid, it is difﬁcult to apply t he concept of a mean free path for

elastic scattering; the intensity of each Bragg spot depends on the crystal orientation

relative to the incident beam and is not proportional to crystal thickness. Instead,

each reﬂection is characterized by an extinction distance ξ

, which is typically in

the range of 25–100 nm for 100-keV electrons and low-order (small θ

) reﬂections.

In the ideal two-beam case, where Eq. (3.12) is approximately satisﬁed for only one

set of reﬂecting planes and where effects of inelastic scattering are negligible, 50%

of the incident intensity is diffracted at a crystal thickness of t = ξ

/4 and 100% at

t = ξ

/2. For t >ξ

/2, the diffracted intensity decreases with increasing thickness

and would go to zero at t = ξ

(and multiples of ξ

) if inelastic scattering could be

neglected. This oscillation of intensity gives rise to the “thickness” or Pendellösung

fringes that can be seen in TEM images.

3.1.4 Electron Channeling

Solution of the Schrödinger equation for an electron moving in a periodic potential

results in wavefunctions known as Bloch waves: plane waves whose amplitude is

modulated by the periodic lattice potential. Inside the crystal, a transmitted elec-

tron is represented by the sum of a number of Bloch waves, each having the same

total energy. Because each Bloch wave propagates independently without change

of form, this representation can be preferable to describing electron propagation

in terms of direct and diffracted beams, whose relative amplitudes vary with the

depth of penetration. In the two-beam situation referred to in Section 3.1.3, there are

only two Bloch waves. The type-2 wave has its intensity maximum located halfway

between the planes of Bragg reﬂecting atoms and propagates parallel to these planes.

3.1 Elastic Scattering 119

The type 1 wave propagates in the same direction but has its current density peaked

exactly on the atomic planes. Because of the attractive force of the atomic nuclei,

the type 1 wave has a more negative potential energy and therefore a higher kinetic

energy and larger wavevector than the type-2 wave. Because of this difference in

wavevector between the Bloch waves, their combined intensity exhibits a “beating”

effect, which provides an alternative but equivalent explanation for the occurrence

of thickness fringes in the TEM image.

The relative amplitudes of the Bloch waves depend on the crystal orientation.

For the two-beam case, both amplitudes are equal at the Bragg condition, but if the

crystal is tilted toward a “zone axis” orientation (the angle between the incident

beam and the atomic planes being less than the Bragg angle) more intensity occurs

in Bloch wave 1. Conversely, if t he crystal is tilted in the opposite direction, Bloch

wave 2 becomes dominant. Away from the Bragg orientation the current density

distributions of the Bloch waves become more uniform, so that they more nearly

resemble plane waves.

Besides having a larger wavevector, the type 1 Bloch wave has a greater proba-

bility of being scattered by inelastic events that occur close to the center of an atom,

such as inner-shell and phonon excitation (thermal-diffuse scattering). Electron

microscopists refer to this inelastic scattering as absorption, meaning that the scat-

tered electron is absorbed by the angle-limiting objective aperture that is commonly

used in a TEM to enhance image contrast or limit lens aberrations. The effect is

incorporated into diffraction-contrast theory by adding to the lattice potential an

imaginary component V

= v/2λ

, where λ

is an appropriate inelastic mean free

path. The variation of this “absorption” with crystal orientation is called anomalous

absorption and is characterized by an imaginary potential V

. In certain directions,

the crystal appears more “transparent” in a bright-ﬁeld TEM image; in other direc-

tions it is more opaque because of increased inelastic scattering outside the objective

aperture. This behavior is analogous to the Borrmann effect in x-ray penetration

and similar in many respects to the channeling of nuclear particles through solids.

Anomalous absorption is also responsible for the Kikuchi bands that appear in the

background to an electron-diffraction pattern (Kainuma, 1955; Hirsch et al., 1977).

The orientation dependence of the Bloch-wave amplitudes also affects the inten-

sity of inner-shell edges visible in the energy-loss spectrum. As the crystal is tilted

through a Bragg orientation, an ionization edge can become either more or less

prominent, depending upon the location (within the unit cell) of the atoms being

ionized, relative to those that lie on the Bragg-reﬂecting planes (Taftø and Krivanek,

1982a). Inner-shell ionization is followed by de-excitation of the atom, involving the

emission of Auger electrons or characteristic x-ray photons. So as a further result of

the orientation dependence of absorption, the amount of x-ray emission varies with

crystal orientation, provided the incident beam is sufﬁciently parallel (Hall, 1966;

Cherns et al., 1973). This variation in x-ray signal is utilized in the ALCHEMI

method of determining the crystallographic site of an emitting atom (Spence and

Taftø, 1983).

In a more typical situation in which a number of Bragg beams are excited simul-

taneously, there are an equally large number of Bloch waves, whose current density

120 3 Physics of Electron Scattering

Fig. 3.6 Amplitude of the fast-electron wavefunction (square root of current density) for (a) a 100-

keV STEM probe randomly placed at the {111} surface of a silicon crystal, (b) these electrons after

penetration to a depth of 9.4 nm, and (c) after penetration by 47 nm. Channeling has concentrated

the electron ﬂux along 111 rows of Si atoms. From Loane et al. (1988), copyright International

Union of Crystallography, available at http://journals.iucr.org/

distribution can be relatively complicated. The current density associated with each

Bloch wave has a two-dimensional distribution when viewed in the direction of

propagation; see Fig. 3.6.

Decomposition of the total intensity of the beam-electron wavefunction into a

sum of the intensities of separate Bloch waves is a useful approximation for chan-

neling effects in thicker crystals. In thin specimens, however, this independent Bloch

wave model fails because of interference between the Bloch waves, as observed

by Cherns et al. (1973). For even thinner crystals (t  ξ

) the current density is

nearly uniform over the unit cell and the incident electron can be approximated

by a plane wave, allowing atomic theory to be used to describe inelastic processes

(Section 3.6).

3.1.5 Phonon Scattering

Because of thermal (and zero-point) energy, atoms in a crystal vibrate about their

lattice sites and this vibration acts as a source of electron scattering. An equiv-

alent statement is that the transmitted electrons generate (and absorb) phonons

while passing through the crystal. Since phonon energies are of the order of kT

(k = Boltzmann’s constant, T = absolute temperature) and do not exceed kT

= Debye temperature), the corresponding energy losses (and gains) are below

0.1 eV and are not resolved by the usual electron microscope spectrometer sys-

tem. There is, however, a wealth of structure in the vibrational-loss spectrum, which

has been observed using reﬂected low-energy electrons (Willis, 1980; Ibach and

Mills, 1982) and by high-resolution transmission spectroscopy (Geiger et al., 1970;

Fig. 1.9).

3.1 Elastic Scattering 121

Except near the melting point, the amplitude of atomic vibration is small com-

pared to the interatomic spacing and (following the uncertainty principle) the

resulting scattering has a wide angular distribution. Particularly, in the case of

heavier elements, phonon scattering provides a major contribution to the dif-

fuse background of an electron-diffraction pattern. This extra scattering occurs

at the expense of the purely elastic scattering; the intensity of each elastic

(Bragg-scattered) beam is reduced by the Debye–Waller factor: exp(−2M), where

M = 2(2π sin θ

/λ

)u

, λ is the electron wavelength, and u

is the component of

mean-square atomic displacement in a direction perpendicular to the corresponding

Bragg-reﬂecting planes.

The total phonon-scattered intensity (integrated over the entire diffraction plane)

is speciﬁed by an absorption coefﬁcient μ

= 2V

/(v), where V

is the phonon

contribution to the imaginary potential and v is the electron velocity. An inverse

measure is the parameter ξ

/2π = 1/μ

, which is roughly equivalent to a mean free

path for phonon scattering. Typical values of 1/μ

are shown in Table 3.1. Unlike

other scattering processes, phonon scattering is appreciably temperature dependent,

increasing by a factor of 2–4 between 10 K and room temperature. Like elastic

scattering, it increases with the atomic number Z of the scattering atom, roughly as

3/2

; see Table 3.1.

Rez et al. (1977) calculated the phonon-loss intensity distribution in the diffrac-

tion pattern of 100-keV electrons transmitted through 100-nm specimens of Al, Cu,

and Au. Their results show that the phonon-loss intensity is sharply peaked around

each Bragg beam (FWHM < 0.1 mrad) but with broad tails that overlap and con-

tribute a background to the diffraction pattern, a situation quite similar to that arising

from the inelastic scattering due to valence-shell excitation (Fig. 3.5). Because the

energy transfer is a small fraction of an electron volt, phonon scattering is con-

tained within the zero-loss peak and is not removed from TEM images or diffraction

patterns by energy ﬁltering.

Inelastic scattering involving plasmon and single-electron excitation is also con-

centrated at small angles around each Bragg beam. However, typically 20% or more

occurs at scattering angles above 7 mrad (Egerton and Wong, 1995), so inelastic

scattering from the atomic electrons also makes a substantial contribution to the dif-

fuse background of an electron-diffraction pattern. Because it involves an average

energy loss of some tens of electron volts, this electronic component is removed

by energy ﬁltering. Unlike phonon scattering, it varies only weakly with atomic

number, and for light elements (Z < 13) it makes a larger contribution than phonon

scattering to the diffraction pattern background (Eaglesham and Berger, 1994).

Table 3.1 Values of phonon mean free path (1/μ

, in nm) calculated by Hall and Hirsch (1965)

for 100-keV electrons

T (K) Al Cu Ag Au Pb

300 340 79 42 20 19

10 670 175 115 57 62

122 3 Physics of Electron Scattering

3.1.6 Energy Transfer in Elastic Scattering

We now revert to an atomic model of elastic scattering and consider an electron (rest

mass m

) of kinetic energy E

deﬂected through an angle θ by the electrostatic ﬁeld

of a stationary nucleus (mass M). Conservation of energy and momentum requires

that the electron must transfer to the nucleus an amount of energy E given by

E = E

max

sin

(θ/2) = E

max

(1 −cos θ)/2 (3.12a)

max

= 2E

+2m

)/(Mc

) (3.12b)

Here E

max

is the maximum possible energy transfer, corresponding to θ = π rad.

For a typical TEM incident energy (≈ 100 keV) and the scattering angles involved

in electron diffraction and TEM imaging (< 0.1 rad), Eq. (3.12a)givesE < 0.1 eV;

therefore, the energy loss is not measurable in a TEM-EELS system. If the atom is

part of a solid, this small amount of energy transfer can be used to generate phonons.

In the case of a high-angle collision, E approaches E

max

, whose value depends on

and on the atomic weight (or atomic number) of the target atom but ranges from

a few electron volts for heavy atoms to several tens of electron volts for light atoms.

The energy loss associated with elastic backscattering (θ>π/2 rad) of 40-keV

electrons can be measured and has been proposed as an analytical method (Went

and Vos, 2006).

A high-angle collision is comparatively rare but if E exceeds the displacement

energy of an atom (20 eV for elemental copper and 80 eV for diamond), it gives

rise to knock-on displacement damage in a crystalline specimen, atoms being dis-

placed from their lattice site into an interstitial positions, to form vacancy–interstitial

(Frenkel) pairs; for example, Oen (1973) and Hobbs (1984). Alternatively, a

high-angle collision with a surface atom may transfer energy in excess of the

surface-displacement energy (generally below 10 eV), giving rise to electron-

induced sputtering (Bradley, 1988; Egerton et al., 2010). These processes occur only

if the incident energy exceeds a threshold value given by

= m

{[1 +(M/2m

)(E

)]

1/2

−1}

= (511 keV){[1 +AE

/(561 eV)]

1/2

−1}

(3.12c)

where A is the atomic weight (mass number) of the target atom and E

is the bulk

displacement or surface-binding energy.

At high scattering angles, the screening effect of the atomic electrons is small and

the differential cross section for elastic scattering is close to the Rutherford value,

Eq. (3.3). The Rutherford formula can be integrated between energy transfers E

max

and E

min

to yield a cross section:

= (0.250 barn)FZ

[(E

max

min

) −1] (3.12d)

3.1 Elastic Scattering 123

where 1 barn = 10

−28

and F = (1−v

)/(v

) is a relativistic factor. Using

min

= E

and E

max

given by Eq. (3.12b), a cross section for the displacement

process can be obtained from Eq. (3.12d). In the case of sputtering, the specimen-

thinning rate can then be estimated as J

monolayers per second for a current

density as J

electrons per area per second.

In principle, greater accuracy is achieved by using Mott cross sections that

include the effects of electron spin. Such cross sections have been tabulated by Oen

(1973) and Bradley (1988). For lighter elements (Z < 28) an approximation due to

McKinley and Feshbach (1948) can be used: the Rutherford cross section is mul-

tiplied by a correction factor and can be integrated analytically to give (Banhart,

1999)

= (0.250 barn)[(1−β

)/β

]{X +2παβ X

1/2

−[1+2 παβ+(β

+παβ)ln(X)]}

(3.12e)

where α = Z/137, β = v/c, and X = E

max

min

= sin

(θ

max

/2)/sin

(θ

min

/2).

For light elements, Eq. (3.12e) gives cross sections close to the true Mott cross

sections but for heavier elements (Z > 28) the values are too low and the Rutherford

formula gives a better approximation. A computer program S

IGDIS that evaluates

Eqs. (3.12d) and (3.12e) is described in Appendix B.

In the case of electron-induced sputtering, it is possible that only the compo-

nent of transferred energy perpendicular to the surface is used to remove a surface

atom, which corresponds to a planar escape potential rather than a spherical one.

The Rutherford and McKinley–Feshbach–Mott cross sections are then obtained by

using E

min

= (E

max

)

1/2

rather than E

min

= E

in Eq. (3.12d)or(3.12e). A typ-

ical situation probably lies somewhere between these two extremes, depending on

the directionality of the atomic bonding, for example, Egerton et al. (2010). The

MATLAB program S

IGDIS described in Appendix B calculates Rutherford and

Mott cross sections for both of these escape potentials.

Because Eq. (3.12a) relates each energy loss to a particular scattering angle θ,

the integrated Rutherford cross section can alternatively be expressed in terms of

the minimum and maximum scattering angles involved:

= (0.250 barn)FZ

[(sin

max

/2)/sin

(θ

min

/2) −1] (3.12f)

The equivalent McKinley–Feshbach–Mott cross section is given by Eq. (3.12e) with

X = sin

(θ

max

/2)/sin

(θ

min

/2). Such cross sections ignore screening of the atomic

nucleus and are valid only if θ

min

considerably exceeds some characteristic angle,

given by the Lenz model as θ

= Z

1/3

. For 60 keV electrons, θ

= 27 mrad

for Z = 6 and 66 mrad for Z = 92. These angles are also large enough to ensure

that diffraction effects do not greatly affect the angle-integrated signal. A computer

program S

IGADF that evaluates σ

and σ

as a function of angle is described in

Appendix B.

124 3 Physics of Electron Scattering

The high-angle elastic cross sections relate directly to the signal (I

electrons/s)

received by a high-angle annular dark-ﬁeld (HAADF) detector when an electron

beam (I electrons/s) passes through a STEM specimen (N atoms per unit area):

= NIσ

(3.12g)

In the case of an atomic-sized electron probe, Eq. (3.12g) allows discrimination

between atoms of different atomic numbers, based on the HAADF signal (Krivanek

et al., 2010). For a large inner angle θ

min

, the scattering is Rutherford like and I

∝

, as seen in Fig. 3.4.Forθ

min

= 0 and large θ

max

, I

∝ Z

4/3

according to

the Lenz model, Eq. (3.8). For a typical HAADF detector (θ

min

= 60 mrad and

max

= 200 mrad), I

∝ Z

1.64

has been observed for Z < 12 (Krivanek et al., 2010).

3.2 Inelastic Scattering

As discussed in Chapter 1, fast electrons are inelastically scattered by electrostatic

interaction with both outer- or inner-shell atomic electrons, processes that predom-

inate in different regions of the energy-loss spectrum. Before considering inelastic

scattering mechanisms in detail, we deal brieﬂy with theories that predict the total

cross section for inelastic scattering by the atomic electrons. In light elements, outer-

shell scattering makes the largest contribution to this cross section. For aluminum

and 100-keV incident electrons, for example, inner shells represent less than 15% of

the t otal-inelastic cross section, although they contribute over 50% to the stopping

power and secondary electron production (Howie, 1995).

3.2.1 Atomic Models

For comparison with elastic scattering, we consider the angular dependence of

inelastic scattering (integrated over all energy loss) as expressed by the differen-

tial cross section dσ

/d. By modifying Morse’s theory of elastic scattering, Lenz

(1954) obtained a differential cross section that can be written in the form (Reimer

and Kohl, 2008)

dσ

d

4γ



1 −

[1 +(qr

)

]



(3.13)

where γ

= (1 −v

)

−1

and a

= 0.529 × 10

−10

m, the Bohr radius; r

is a

screening radius, deﬁned by Eq. (3.4) for a Wentzel potential and equal to a

−1/3

according to the Thomas–Fermi model. The magnitude q of the scattering vector is

given approximately by the expression

= k

(θ

) (3.14)

3.2 Inelastic Scattering 125

in which k

= 2π/λ = γ m

v/ is the magnitude of the incident electron

wavevector, θ is the scattering angle, and

= E/(γ m

) is a characteristic angle

corresponding to an average energy loss

E. Comparison with Eq. (3.3) shows that

the ﬁrst term (4γ

Z/a

)inEq.(3.13) is the Rutherford cross section for scattering

by Z atomic electrons, taking the latter to be stationary free particles. The remaining

term in Eq. (3.13)isaninelastic form factor (Schnatterly, 1979).

Equations (3.13) and (3.14) can be combined to give a more explicit expression

for the angular dependence (Colliex and Mory, 1984):

dσ

d

4γ

(θ

)



1 −



+θ



(3.15)

where θ

= (k

)

−1

as in the corresponding formula for elastic scattering, Eq. (3.5).

Taking r

= a

−1/3

leads to the estimates

≈ 0.2 mrad and θ

≈ 20 mrad for a

carbon specimen, taking E

= 100 keV and E ≈ 37 eV (Isaacson, 1977).

In the angular range

<θ<θ

, which contains most of the scattering, dσ/d

is roughly proportional to 1/θ

, whereas above θ

it falls off as 1/θ

; see Fig. 3.7.

The differential cross section therefore approximates to a Lorentzian function with

an angular width

and a gradual cutoff at θ = θ

On the basis of Bethe theory (Section 3.2.2), cutoff would occur at a mean Bethe

ridge angle

≈

√

). In fact, these two angles are often quite close to one

another; for carbon and 100-keV incident electrons, θ

≈

≈ 20 mrad. Using

this value as a cutoff angle in Eqs. (3.53) and (3.56), the mean and median angles of

inelastic scattering are estimated as

θ  20

 4mradand

θ  10

 2mradfor

carbon and 100-keV incident electrons, respectively. Inelastic scattering is therefore

concentrated into considerably smaller angles than elastic scattering, as seen from

Figs. 3.5 and 3.7.

Fig. 3.7 Angular

dependence of the differential

cross sections for elastic and

inelastic scattering of

100-keV electrons from a

carbon atom, calculated using

theLenzmodel(Eqs.(3.50,

(3.7), and (3.15)). Shown

along the horizontal axis are

(from left to right)the

characteristic, median, mean,

root-mean-square and

effective cutoff angles for

total inelastic scattering,

evaluated using Eqs. (3.53),

(3.54), (3.55), and (3.56)

126 3 Physics of Electron Scattering

Integrating Eq. (3.15) up to a scattering angle β gives the integral cross section:

(β) ≈

8πγ

1/3



(β

)(θ

)

(β

+θ

)



(3.16)

Extending the integration to all scattering angles, the total inelastic cross section is

≈ (16πγ

1/3

)ln(θ

) ≈ (8πγ

1/3

)ln(2/

) (3.17)

where the Bethe ridge angle (2

)

1/2

has been used as the effective cutoff angle θ

(Colliex and Mory, 1984). Comparison of Eqs. (3.8) and (3.17) indicates that



≈ 2ln(2/

)/Z = C/Z (3.18)

where the coefﬁcient C is only weakly dependent on atomic number Z and inci-

dent energy E

. Atomic calculations (Inokuti et al., 1981) suggest that (for Z < 40)

E varies by no more than a factor of 3 with atomic number; a typical value is

E = 40 eV, giving C ≈ 17 for 50-keV electrons and C ≈ 18 for 100-keV

electrons. Experimental measurements on solids agree surprisingly well with these

predictions; see Fig. 3.8. This simple Z-dependence of the inelastic/elastic scattering

Fig. 3.8 Measured values of

inelastic/elastic scattering

ratio for 80-keV electrons, as

a function of atomic number

of the specimen. The solid

line represents Eq. (3.18)

with C = 20

3.2 Inelastic Scattering 127

ratio has been used to interpret STEM ratio images of very thin specimens; see

Section 2.6.6.

Measurements and more sophisticated atomic calculations of σ

differ from

Eq. (3.17) because of the variable number of outer-shell (valence) electrons, which

contribute a large part of the scattering, especially in light elements. Instead of a

simple power-law Z-dependence, the inelastic cross section reaches minimum val-

ues for the compact, closed-shell (rare gas) atoms and maxima for the alkali metals

and alkaline earths, where the weakly bound outer electrons contribute a substantial

plasmon component; see Fig. 3.9.

Closely related to the total inelastic cross section is the electron stopping power

S, deﬁned by (Inokuti, 1971)

S =

= n

Eσ

(3.19)

where E represents energy loss, z represents distance traveled through the specimen,

E is a mean energy loss per inelastic collision, and n

is the number of atoms per unit

volume of the specimen. Atomic calculations (Inokuti et al., 1981) show that

E has

a periodic Z-dependence that largely compensates that of σ

,givingS a relatively

weak dependence on atomic number.

In low-Z elements, inner atomic shells contribute relatively little to σ

(Ritchie

and Howie, 1977), but they do have an appreciable inﬂuence on the stopping power

Fig. 3.9 Total-inelastic cross section σ

and plasmon (outer-shell) cross section σ

for 200-keV

electrons, based on EELS data of Iakoubovskii et al. (2008b). The measurements include all scat-

tering up to 20 mrad but none beyond 40 mrad, so σ

will be too low for the heavier elements