Yao N. Focused Ion Beam Systems: Basics and Applications

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A slow ion (with velocity v lower than the velocity of the conduction

electrons of the target) represents a strong perturbation to the collision

system. A linear response theory for fast ions is not valid to treat the slow

ion–target interaction [32]. The recent nonlinear response theory of the

electronic stopping power expresses the electron excitation in a free electron

gas, in terms of elastic scattering of the electrons at the Fermi energy, E

with the intruding ion [33]. According to the nonlinear response theory, the

inverse mean free path (MFP) of projectile ion or recoiled target atoms to

excite the electron of interest is calculated as:

inel

3n”

2”

l¼0

m¼0

ð2l þ 1Þð2m þ 1Þ

· f1  cos 2–

ðE

Þcos 2–

ðE

Þþcos½2ð–

ðE

Þ–

ðE

ÞÞg

1

ð1  xÞ

1=2

ðxÞP

ðxÞdx; ð4:9Þ

where, –

l,m

are the phase shifts for the scattering of a conduction electron at

by the potential of the intruding ion or the recoiled target atom at rest.

The quantities n and P

l,m

are the density of conduction electrons and the

Legendre polynomials, respectively. Figure 4.5 shows the energy dependence

of the calculated MFPs of a Ga ion, and recoiled target atoms in aluminum

(Al) and gold (Au). The inverse MFP is directly proportional to v (or the

square root of the energy) and n, so that the number of excited electrons

increases with increasing v and depends upon the electron shell structure of

the target atoms, and the atomic number, Z

Ga in Al

Ga in Au

Al in Al

Au in Au

Ener

(eV)

10 10

–1

MFP, l

inel

(nm)

Figure 4.5 Energy dep endence of MFPs for electron excitation by a Ga ion,

and recoiled Al and Au atoms in Al and Au [34]. The MFPs are calculated

using the nonlinear response theory developed by Echenique et al.[32].

Focused ion beam systems98

Transport of secondary electrons to solid surfaces

The second stage in kinetic emission is the transport of excited SEs to the

surface. Energetic electrons may lose their original energy in a sequence of

inelastic collisions to other target electrons, including an electron multi-

plication by the cascade process.

For a metal with conduction electrons the electron excitations are very

efﬁcient in the slowing down processes. The stopping power for electrons in

metals has a maximum between 100 eV and 1 keV (see Figure 4.4), and

depends strongly on the density of conduction electrons. For electron exci-

tation by a primary electron, the differential inverse MFP, (E, !), of an

electron with an energy E is described by the complex dielectric response

function, Im[1/e(!, k)], i.e.,

ðE; !Þ¼

E

kþ

k

Im 

"ð!; kÞ



; ð4:10Þ

and the integration of (E, !) over the allowed values of k yields the inverse

MFP. Here ! and k are the energy and momentum transfers to conduction

electrons, and k± ¼2

1/2

(E !)

1/2

] in atomic units. Optical data of the

target material for the k ¼0 limit, "(!, 0), can be connected to "(!, k)

according to an ‘‘optical-data’’ model developed by Penn [35], and by Ashley

[36]. Since the optical data are based on the experimental results, it includes

complicated processes of the inter-band, intra-band, and some other transi-

tion mechanisms automatically, in addition to the single and bulk-plasmon

excitation of conduction electrons, which can be treated by the Lindhard

dielectric function. Figure 4.6 shows the energy dependence of the calculated

MFPs of electrons in Al and Au.

The electron energy loss is substantially inﬂuenced by the possible existence

of an energy gap E

, i.e., whether or not the material is an insulator. Elec-

trons, which possess less energy than E

, cannot lose energy by electronic

excitation. Therefore, the energy loss of SEs in the insulator takes place via

inefﬁcient collisions with core atoms though their density ﬂuctuation, i.e.,

phonons.

The elastic scattering is caused by the core atoms, but electron–electron

collisions may lead to directional changes as well. The frequent collisions

lead to a completely isotropic distribution of SEs in the solid. The MFP for

elastic scattering can be calculated using the partial wave expansion cross

section with appropriate solid-state potential [37] or using screened

Rutherford formula where the energy-dependent screening parameters are

used [38].

Imaging using electrons and ion beams 99

Escape of secondary electrons in vacuum

The third stage, the ejection over the surface barrier, accounts for the

selection of the SEs that are emitted. Since the majority of the SEs have an

energy in the ranges of eV and tens of eV, the SE emission is strongly affected

by the surface barrier, the magnitude of which ranges from a few eV up to

20 eV. The commonly accepted treatment implies that the SEs have to pass a

barrier of height, U ¼8 þE

, which is the distance from the bottom of the

conduction band to the vacuum level, in such a manner that the energy

component parallel to the surface is conserved. The component perpendi-

cular to the surface is reduced with U, so that the internal threshold energy

for escape is E ¼U.

The kinetic emission occurs if the projectile ion gives a conduction electron

sufﬁcient kinetic energy to overcome the surface barrier U. This leads to a

threshold energy for kinetic emission, E

¼(1/2)M

, where v

¼(1/2)

[(1 þ8E

)

1/2

1] and M

is the ion mass. The threshold energy for Ga ions

is 10.7, 18.8, and 26.6 keV for Al, Cu, and Au, respectively, which is incon-

sistent with experiments on the kinetic electron emission observed at only a

few keV. Such inconsistencies have already been observed for other heavy

ions [39,40]. This has been considered to be due to the electron promotion

mechanism modeled recently for electron excitation within kinetic emission

processes [41], including the inner shell electrons, as well as conduction

electrons. The mechanism promotes the electrons to higher and vacuum

levels during close collisions with material atoms. For simplicity, the kinetic

emission below an energy E

may be also treated as a reduction in the surface

–1

Ener

(eV)

MFP. l

inel

(nm)

e in Al

e in Au

Figure 4.6 Energy dependence of inelastic MFP of electrons in Al and Au

[34]. The MFPs are calculated using the optical-data model by Ashley [36].

Focused ion beam systems100

barrier energy U; e.g., 0.5E

þ8, which has been arbitrarily chosen. With

decreasing surface barrier energy, the SE yield increases and the threshold

energy is signiﬁcantly reduced. For electron-induced SE emission, however,

the reduction of the surface barrier energy causes a much smaller increase in

the SE yield [42].

For insulators the surface barrier U is determined by the electron afﬁnity

. Usually, the value of E

from 0 to 1 eV. The average electronic stopping

power of insulators for ions is similar for metals of comparable atomic

numbers, but the stopping power for the migration of low-energy SEs and the

magnitude of surface barrier are much smaller for the insulating material. As

a result, the SE yield from insulators is much larger than the yield for metals

of comparable atomic numbers.

4.2.3 Monte Carlo simulation of ion-induced kinetic secondary

electron emission

As mentioned above, the kinetic emission is modeled in SE excitation in a

solid, transport of the SEs to the surface, and their escape through a surface

barrier. The SE is excited by three collision processes; one due to collisions

between projectile ions and target electrons, one due to collisions between

recoiled target atoms and target electrons, and one due to collisions between

excited SEs and other target electrons.

The basic concept for a Monte Carlo model is to simulate trajectories of a

projectile ion penetrating into the target bulk and of recoiled target atoms

and excited SEs traveling towards the surface in the basis of the binary

collision approximation [43] with given mean free paths (MFPs) for elastic

and inelastic collisions. The motion of projectile ions and recoiled target

atoms are treated in the same way: the straight free ﬂight path is determined

from the total MFP, l

tot

, deﬁned as 1/l

þ1/l

inel

, using a random number.

The elastic MFP, l

, is ﬁxed at N

1/3

where N is the atomic density of the

target. Depending on each of the inverse MFPs, l

and l

inel

, either elastic

collision or inelastic collision (i.e., SE excitation) is chosen using another

random number.

If elastic collision is chosen, the scattering angle is determined using a so-

called scattering integral with appropriate interatomic potential, or any

asymptotic procedure [44]. The elastic energy loss is calculated within a clas-

sical collision scheme, and in each elastic collision, a new recoiled target atom

is generated. If inelastic collision is chosen, the particle loses its energy and

excites an SE. The energy of the excited SE is equal to the energy loss of the

particle, e.g., (4.8), which is calculated from a head-on collision of the particle

Imaging using electrons and ion beams 101

with a conduction electron. The initial directional angle of the excited elec-

tron is calculated using the energy and momentum conservation law.

The excited SEs interact with the target through elastic collisions with the

target core atoms, and through excitations of other target electrons. The

trajectory of each SE is chosen as a series of random numbers to determine

the path length between collision events, the type of collision that has taken

place and the energy loss or the scattering angle. In each inelastic process,

SEs are excited, so that an electron cascade is generated. The electron cascade

model can also be applied for the simulation of SE emission by electron

bombardment [45–51].

4.3 Comparison between secondary electron images in SIM and SEM

Contrast mechanisms in SIM images have been discussed [9,52,53] and they

fall largely into the categories of material contrast, topographic contrast, and

channeling contrast. Although the contrast mechanism present in the SIM

images is similar to that in SEM, there are some differences. In this section

SIM and SEM images are discussed from a viewpoint of interaction between

ion or electron beams and samples. While SIM advantages include high

contrast sensitivity to surface topography and crystal grain (channeling),

drawbacks include sample surface damage and ion implantation. Therefore,

beam charging for insulator samples and image deterioration due to surface

sputtering and contamination will be presented in this section.

4.3.1 Material contrast

Material contrast arises from differences in the yield of SEs as a function of

atomic number Z

of samples. In experiments, the SE emission in SIM shows

a decrease with increasing Z

[7,8,54]. Figure 4.7 shows the relative SE

intensities as a function of Z

for SIM and SEM images. The SE intensities

were measured for Al, Si, Cr, Fe, Ni, Cu, Ag, and Pt bombarded both by

30 keV Ga ions in the FIB (Hitachi FB-2000A) system and by 5 keV electrons

in the SEM (Hitachi S-4200). The SE intensities for the SIM broadly descend

with increasing Z

, being superimposed with a ﬁne structure. The SE inten-

sities for the SEM, on the other hand, show Z

-dependency increasing with

, commonly known as material contrast.

Recently, Monte Carlo (MC) simulations on the SE emission have revealed

the origin of the difference in the material contrast between SIM and SEM

images [4,34,42,54,55]. SEs are produced along the entirety of the trajectory

of a projectile particle penetrating into the sample material. Due to their low

Focused ion beam systems102

excitation energy and energy loss in inelastic collisions before reaching the

surface, only a small fraction of the SEs can overcome the surface barrier to

escape into the vacuum. The SE yield increases due to the existing back-

scattered electrons (BSEs), which scatter backwards inside the material and

approach the surface to excite additional SEs that can escape from the sur-

face barrier [11,34]. Since the electron backscattering coefﬁcient increases

with increasing Z

, the SE yields of high- Z

metals under electron bom-

bardment are dominated by the additional electron excitation by the BSEs on

their way out, as shown in Figure 4.8(a). Furthermore, it shows that the

general increase in the SE yield with increasing Z

, which has been observed

in experiments [7 ,8, 54], is mainly caused by the increase in the number of SEs

excited by the BSEs.

For bombardment with light ions, such as hydrogen (H) ions, the projectile

ion backscattering is effective in the kinetic emission as well [56], whereas for

relatively heavy, Ga ions, it is much less important, due to the small back-

scattering coefﬁcient. A more important factor in the kinetic emission due to

the bombardment of Ga ions is the electron excitation by recoiled material

atoms, which receive a large fraction of the total kinetic energy of the pro-

jectile ion in frequent elastic collision events. Therefore, the electron excita-

tion can be split into three components, as shown in Figure 4.8(b): one due to

excitation by a projectile ion, one due to excitation by the recoiled material

atoms and one due to excitation by the excited SEs, i.e., the electron cascade.

With increasing Z

, the component from the projectile ion, which dom-

inates the total SE yield, generally decreases while showing a ﬁne structure.

This general decrease is because a large energy transfer from the Ga ion to

100

Target atomic number, Z

100

d (arb. units)

g (arb. units)

g (Ga; 30 keV)

d (e; 5 keV)

Figure 4.7 Relative SE intensities as a function of Z

for the Ga-SIM and

SEM images [54].

Imaging using electrons and ion beams 103

the heavier material atoms causes the ion velocity to decrease steeply just

after incidence. The lowering of the ion velocity leads to a lowering in the

electron excitation probability or the inverse MFP for electron excitation (see

(4.9)). Due to the decreasing excitation energy with increasing mass of the

recoiled atoms, the component from the electron excitation by the recoiled

material atoms decreases with increasing Z

. As a result, a strong decrease in

the total SE yield with increasing Z

is obtained for Ga ion bombardment.

The component from the electron cascade is less important for such heavy

–1

–2

–3

–4

–5

–2

–3

40 50 60

Target atomic number, Z

Partial electron yields, d

and d

Partial electron yields, g

, g

and g

(trapped electrons)

(reflected electrons)

(Ga ions)

(recoiled atoms)

(electron cascade)

(b) 30 keV Ga

(a) 10 keV e

Figure 4.8 Parti al SE yields due to electron excitations (a) by primary

electrons backscattered from the surface and ﬁnally trapped inside the

material for 10 keV electron bombardment, and (b) by projectile ions,

recoiled material atoms , and electron cascades for 30 keV Ga ion

bombardment, as a function of the atomic number, Z

, of material atoms

[34]. The lines are only intended as a guide for the eye.

Focused ion beam systems104

ions. For light ions, however, most of the SEs are excited by the electron

cascades in the same manner as electron bombardment [57].

The Z

dependence is characterized by a ﬁne structure relating to the

periods of the periodic system. Furthermore, a different structure from the

case for electron bombardment was observed in Z

dependence for heavy ion

bombardment [58]. Such ﬁne structures are calculated for both Ga ion and

electron bombardments. For Ga ion bombardment, the components from

the electron excitation by the recoiled material atoms and the electron cas-

cade change with Z

in a different manner from the SE yield for electron

bombardment. The SE yields for electron bombardment correlate with the

density of conduction electrons, so that they are proportional to the stopping

power. This indicates the dominant contribution of bulk properties of

the target materials. However, due to large elastic energy loss of projectile

ions and surface effects (e.g., the work function), the SE yields for slow,

heavy ion bombardment are not directly proportional to the electronic

stopping power [59].

There is much difference in the potential of electron multiplication

between the SEs excited by Ga ions and by primary electrons [42,54,57]. The

mean energies of SEs excited by 10 keV Ga ions are as low as approximately

7–15 eV, decreasing with an increasing Z

. The mean velocity of recoiled

target atoms also decreases with an increasing mass under identical energy.

The excited SEs with low energies have poor potential in the SE multi-

plication and also have less chance than the SEs to escape from the surface.

Therefore, the SE yield generally decreases with an increasing Z

. For elec-

tron impact in SEM, on the other hand, the mean energy of electrons excited

by 10 keV electrons is as high as approximately 40–60 eV, increasing with an

increasing Z

(contrary to the case of Ga ion impact). Then, the SE yield

increases with an increasing Z

, as it is contrary to the SE yield under Ga ion

bombardment.

4.3.2 Topographic contrast

Topographic contrast is familiar to anyone who has seen an SEM image of

an irregular object. In general, topographic contrast in SIM has been

explained in a similar manner to topographic contrast in SEM, in terms of

differences in the SE yield as a function of the angle of incidence of primary

beam relative to sample surface normal as the local inclination of the sample

surface varies.

At high energy bombardment, a simple geometric model [29], which

assumes constant electronic stopping along the ion trajectory in the SE

Imaging using electrons and ion beams 105

escape depth and disregards scattering of the ion, leads us to a simple law in

the form

ðÞ

ð0Þ

¼ðcos Þ

1

; ð4:11Þ

which is called the ‘‘inverse cosine law’’;  () and  (0) are the SE yields at an

incident angle,  measured from the surface normal and at normal incidence,

respectively. Figure 4.9 shows pseudo SE images calculated for a rod with a

circular cross section, assuming different incident-angle-dependences,

including the inverse cosine law [60]. Since the ‘‘inverse cosine’’ approaches to

inﬁnity at grazing angles, the boundary between the rod and the background

is enhanced in the pseudo image. However, at the energy ranges of tens of

keV or less, due to the breakdown of the assumptions in the model, the

incident angle dependence of the SE yield largely deviates from the inverse

cosine law and varies complicatedly, depending on the ion energy and solid

materials.

A systematic investigation on the incident angle dependence of the SE yield

for various ion species and materials has been performed by means of Monte

Carlo calculations at wide energy range of 100 eV–1 MeV [61,62,63]. The

deviation from the inverse cosine law was represented by a ﬁtting parameter f

to a modiﬁed form of (4.11) where the exponent of (–1) is replaced to (– f ). At

low energies, the ion energy loss within the SE escape depth contributes to

255.00

14.00

15.00

510

Pixels

(a)

(b)

Figure 4.9 Reconstruction of pseudo SE images from the incident angle

dependence of SE yield, (), assuming (a) f ¼1 and (b) f ¼2 where

()/(0) ¼(cos )

f

[60]. Random noise at S/N ¼10 is artiﬁcially added

to each pixel intensity. The images intensities saturate artiﬁcially at the

intensity of 255.

Focused ion beam systems106

the incident angle dependence negatively ( f < 1). The negative ( f < 1) and

positive ( f > 1) effects of the backscattered ions on the incident angle

dependence can be apparently seen at low and intermediate energies,

respectively. The positive effect is due to the additional excitation of SEs near

the surface when the ion was backscattered, while the negative effect is due to

the appearance of the ions backscattered from the surface just after incidence

without excitation of SEs. When the incidence is inclined, the positive

(negative) effect is enhanced at intermediate (low) impact energy. At high

energy, however, the ( )/ (0) approaches the inverse cosine ( f ¼ 1), because

the backscattering coefﬁcient becomes small as the impact energy increases.

For light ion impacts, as well as electron impact in the SEM, the back-

scattering effects are a major reason for the deviation from the inverse cosine

law (Figures 4.10(a) and b).

For Ga ion impact, although the contribution of backscattered ions to

kinetic emission is negligible, a large number of fast recoiled atoms are

generated. The fast recoiled atoms impose the same effects on the incident

0.1

100

1000

Energy (keV)

Proton

0.1 1 10 100 1000

Ener

(keV)

Fitting parameter, f

(b)

(a)

Electron

Figure 4.10 Calculated ﬁtting parameter f in ()/(0) ¼(cos )

f

and

–()/–(0) ¼(cos )

f

as a function of energy (a) for proton impact and

(b) electron impac t, respectively, on different materials [61].

Imaging using electrons and ion beams 107