Yao N. Focused Ion Beam Systems: Basics and Applications

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the FIB is in use, providing for higher levels of accuracy in the creation of

cross sections [20]. The SEM’s damage-free imaging is especially useful in the

creation of samples for the TEM, since using an FIB alone causes sample

damage while imaging the process. It can also be very useful in the locali-

zation of integrated circuit failures, so that more damage is not caused to the

sample than is necessary, and the system can combine ion milling, deposition,

and SEM imaging to characterize the failures [21].

The two-beam system also improves the deposition of metal or insulating

layers. In the case of insulating layers, ion beams often leave the layer with

poor insulating properties due to the incorporation of gallium ions in the

deposition; however, in the two-beam system, the SEM beam can be used to

induce deposition, ensuring a high insulating quality. One such case is that of

SiO

, where the resistance of the layer is two orders of magnitude higher than

when an ion beam is used for the deposition [22].

One more advantage of using the two-beam system is the creation of

smaller diameter holes. Using the FIB system, a hole can only be accurately

milled to a diameter of 10 nm without having material along the sides par-

tially ﬁll in the hole. When the two-beam system is used, however, the

nanoscale holes can ﬁrst be milled using the ion beam and then gently ﬁlled

using electron beam deposition so that the diameter of these holes can be

further reduced by 50%. The ability to have smaller diameter holes has

potential applications for single molecule studies, DNA sequencing, and

ultra-high-resolution single atom doping.

Yet another important application of the two-beam FIB system is TEM

sample preparation. The FIB completes electron transparent samples by

thinning out a region of a bulk material, making samples as thin as possible

in signiﬁcantly less time than older methods of sample preparation. When an

SEM is used in addition to an FIB the basic techniques of lift-out and micro-

pillar sampling are vastly improved.

In addition to machining and TEM sample preparation, we also look into

the many imaging capabilities of the FIB in the two-beam system. Whereas

electrons in an SEM beam generate far less damage to the material and

provide much better resolution, the ion beam of the FIB offers much better

sensitivity to details such as crystal orientation and grain structure as well as

better contrast. The two-beam system is then an exceptionally practical

machine, combining the crisp nondestructive imaging of the scanning electron

microscope with the milling capabilities of the focused ion beam. Three-

dimensional material information is also available through the two-beam

system by shaving off thin layers and imaging them with the SEM, obtaining

a series of useful two-dimensional illustrations. Graphs and images provided

Focused ion beam systems28

only by the combined SEM and FIB systems can then be easily interpolated

with the two-dimensional data. If secondary ion mass spectrometry (SIMS)

is then performed to yield the elemental composition of the material, the set

of data can be complete. In summary, the two-beam FIB has immense

relevance as one of the most important tools in the study of nanotechnology

today.

Acknowledgements

The author is grateful to his students Austin Akey, Alex Epstein, and Franz

Sauer, who have partially contributed in preparation of the manuscript, and

to the National Science Foundation-MRSEC program and New Jersey

Commission of Science and Technology for their support.

References

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[3] J. Orloff, M. Utlaut and L. Swanson. High Resolution Focused Ion Beams: FIB

and its Applications , (New York: Kluwer Academic/Plenum Publishers, 2003),

pp. 21–77.

[4] J. Zhou. Handbook of Microscopy for Nanotechnology, ed. N. Yao and Z. L.

Wang, (New York: Springer/Kluwer Academic Publishers, 2005), pp. 287–321.

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[6] R. Gerlach and M. Utlaut. Proc. SPIE Int. Soc. Opt. Eng., 4510 (2001), 96–105.

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[8] K. Van Doorselaer, M. Van den Reeck, L. Van den Bempt et al. Proc. 19th Int.

Symp. Testing and Failure Analysis (1993), pp. 405–14.

[9] J. Orloff, M. Utlaut and L. Swanson. High Resolution Focused Ion Beams: FIB

and its Applications , (New York: Kluwer Academic/Plenum Publishers, 2003),

pp. 147–52.

[10] J. M. Lafferty. J. Appl. Phys., 22 (1951), 299–309.

[11] M. Futamoto, M. Nakazawa and U. Kawabe. Sur. Sci. 100 (1980), 470–80.

[12] D. B. Williams and C. B. Carter. Transmission Electron Microscopy: A Textbook

for Materials Science, (New York: Plenum Press, 1996), 76–7.

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(1986), L507–9.

[14] R. P. Feynman, R. B. Leighton and M. Sands. The Feynman Lectures on Physics,

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[15] W. A. Lamberti. Handbook of Microscopy for Nanotechnology, ed. N. Yao and

Z. L. Wang (New York: Springer/Kluwer Academic Publishers, 2005),

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[16] M. W. Phaneuf. Introduction to Focused Ion Beams: Instrumentation, Theory,

Techniques and Practice, ed. L. A. Giannuzzi and F. A. Stevie (New York:

Springer, 2005), pp. 145–72.

Introduction to the focu sed ion beam system 29

[17] L. F. Dong, J. Jiao, D. W. Tugg le and S. Foxley. Microsc. Microanal., 7S2

(2001), 398–9.

[18] R. Hull, D. Dunn a nd A. Kubis. Microsc. Microanal., 7S2 (2001), 34–5.

[19] P. Gnauck, U. Zeile, W. Rau, G. Benner and A. Orchowski. Microsc.

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[21] J. M. Soden and R. E. Anderson. Proc. IEEE, 81, 5 (1993), 703–15.

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Focused ion beam systems30

Interaction of ions with matter

nobutsugu imanishi

Kyoto University

2.1 Introduction

When a beam of energetic particles enters a solid, several processes are

initiated in the area of interaction. A fraction of the particles a re back-

scattered from the surface layers, whilst the others ar e slowed down in the

solid. The collision in duce s secondary process es such as r ecoil and sput-

tering of constituent atoms, defect formation, el ectron excitation and

emission, and photon emission. Thermal and radiation-induced diffusion

contributes to various phenomena of mixing of constituent elements, phase

transformation, amorphization, c rystallization, track formation, permanent

damage, and so on. Ion implantation and sputtering changes the surface

morphology; craters, facets, g rooves, ridges, and pyramids and/or blister-

ing, exfol iation, and a spongy surface may develop. All those processes are

interrelated in a complicated wa y and several processes have to be included

for the understanding of individual phenomena. Therefore, it is necessary to

quantitativel y un dersta nd t he expe rimenta l obse rvati ons and to have

stringent design abilities for sop histic ated applicatio ns of these v ersati le

processes in the ﬁeld of nanotechnolo gy aiming at material modiﬁcation,

deposition, imp lanta tion, er osion , nano-fab ric ation , surface ana lysis, an d

so on.

This chapter is composed of basic processes and outline of theoretical

models, ion implantation and defect formation, sputtering, and surface

morphology. It is focused on the recent experimental ﬁndings in the ﬁeld of

interaction of ions with matter and theoretical models including various

simulation codes explaining the complicated experimental phenomena.

Focused Ion Beam Systems: Basics and Applications, ed. N. Yao.

Published by Cambridge University Press. ª Cambridge University Press 2007.

2.2 Basic processes and outline of theoretical models

2.2.1 General remarks

An energetic ion incident on a solid sequentially collides with constituent

atoms. These collisions basically contain a complicated many-body problem

because of the atomic composition of a nucleus and many electrons. For-

tunately in the interaction of an ion with an atom, the collision between the

ion and the nucleus can be treated separately from that of the ion and the

electrons because of the large mass difference between the nucleus and

the electron. The former collision is called an elastic or nuclear collision and

the latter is an inelastic or electronic collision. Kinetic energy and momentum

should be conserved during the nuclear collision by which the incident ion

recoils the target atom and is scattered at the same time. The electronic

collision results in excitation and ionization of the constituent electrons in the

atom. When the kinetic energy of the incident ion is not high enough to go

deep inside the atom, the nuclear charge is screened by the inner-shell elec-

trons. In this case, the electrons should be included in the nuclear collision for

taking into account the screening of the nuclear charge. This screened

interaction potential has been one of the fundamental debates for the

understanding of the nuclear collision process.

This subsection ﬁrst treats interatomic potentials, binary scattering process,

recoil energy, and scattering cross section concerning the nuclear collision

process dominating in a low energy region. Then, nuclear and electronic energy

losses are introduced based on several models. Finally, theoretical models and

simulation codes which will be presented in this chapter are listed.

2.2.2 Interatomic potentials

In the atomic collision between two particles with charges Z

e and Z

e,an

accurate and versatile potential used frequently is the following Coulomb

potential multiplied by the screening function 8

deduced on the basis of

the Thomas–Fermi atomic model [1]:

VðrÞ¼

xðÞ; e

¼ 1:44 eVnm



ð2:1Þ

where r is a distance between the two particles and x is given by x ¼r/a. The

screening length a is obtained by

a ¼

3



2=3

h

1=3





0:8853a

1=3

; ð2:2Þ

Focused ion beam systems32

where a

is the Bohr radius and Z is obtained from the Firsov approximation

in the region r < 0.1nm [2],

Z ¼ Z

1=2

þ Z

1=2



: ð2:3Þ

The Thomas–Fermi screening function 8

is calculated from the following

equation:

xðÞ

3=2

xðÞ

1=2

: ð2:4Þ

Examples of approximations of the Thomas–Fermi screened potential are the

Moliere potential (applied in the range of 0 < x < 6) [3]:

VðrÞ¼

0:35 exp 0:3xðÞþ0:55 exp 1:2xðÞþ0:10 exp 6:0xðÞ½; ð2:5Þ

and from Lindhard et al.[4]:

VrðÞ¼



n1

; 1  n 1; ð2:6Þ

where 

¼ 2=2:7183 · 0:8853; x ¼ r=a

and a

¼ 0:8853a

ðZ

2=3

þ Z

2=3

1=2

Ziegler et al. proposed a universal screening function which accurately pre-

dicts the interatomic potential between atoms, as given by

VðrÞ¼

½0:1818 expð3:2xÞþ0:5099 expð0:9423xÞ

þ 0:2802 expð0:4029xÞþ0:02817 expð0:2016xÞ;

ð2:7Þ

where x ¼ r=a

; and a

¼ 0:8854a

=ðZ

0:23

þ Z

0:23

Þ [5].

2.2.3 Binary scattering and recoil

Collision of the atoms with masses M

and M

and respective charges of Z

and

can be treated in the center-of-mass system as the classical motion of the

particle with the mass of „ ¼M

/(M

þM

) in the potential of V(r). The

following scattering integral can be easily obtained from the equations of

motion in the framework of the energy and angular momentum conservations:

‡ ¼

Ldr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2„ E  VðrÞðÞðL

þ const; ð2:8Þ

where r is the distance and ‡ is the angle displayed in the polar coordinate

system, respectively. E and L are the energy of relative motion and the angular

Interaction of ions with matter 33

momentum respectively and are given by E ¼ „v

=2 ¼ M

=ðM

þ M

Þ and

L ¼ „r

d&=dt ¼ p

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2„E

: E

is the kinetic energy of the incident particle. The

scattering angle  is obtained as a function of impact parameter p from the

following integral equation:

 ¼   2

min

pdr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 

VðrÞ



 p

: ð2:9Þ

In the cases of a hard sphere with a radius of r

, the Coulomb potentials are

given by V(r) ¼Z

/r and V(r) ¼Z

and the scattering angles

are  ¼ 2arcsin(p/r

),  ¼ 2 arctan(2pE/Z

), and  ¼ ½1 ð1þ

ðZ

EÞÞ

1=2

, respectively. The energies E

and E

of the scattered and

recoiled particles are

 M

ðÞ

þ 4M

cos

ð=2Þ

þ M

ðÞ

; ð2:10Þ

and

sin

ð=2Þ

þ M

ðÞ

; ð2:11Þ

respectively.

2.2.4 Cross section

The cross section of the particle scattered to the polar angle of  is obtained

from the following relation:

ðÞ¼

p ðÞ

sin 

d



: ð2:12Þ

In the cases of the V ¼Z

/r and V ¼Z

potentials, the results are

ðÞ¼



sin

4





; ð2:13Þ

and

ðÞ¼

ð

=EÞ   ðÞ



2  ðÞ

sin 

; ð2:14Þ

respectively.

Focused ion beam systems34

Lindhard et al. approximately obtained the following differential scattering

cross section using the Thomas–Fermi potential applicable in a low energy

region [4]:

d ¼ a

3=2



1=2



; ð2:15Þ

where t ¼ "

sin

ð=2Þ: The reduced energy " is the ratio between the screening

radius a and the collision diameter b and is given by

" ¼

„v

þ M

ðÞZ

: ð2:16Þ

The function f ðt

1=2

Þ is approximated as



1=2



‚t

1=6

1 þ 2‚t

2=3

ðÞ

2=3

3=2

; ð2:17Þ

where ‚ ¼1.309.

The cross sections in the laboratory system can be obtained from those in

the center-of-mass system as



ðÞ¼

1 þ



þ2



cos 



3=2

1 þ



cos 

ðÞ; ð2:18Þ

and



ðÞ¼4 sin





ðÞ; ð2:19Þ

where 

and 

are the scattering and recoiled angles in the laboratory

system, respectively. The energies E

and E

of the scattered and recoiled

particles are respectively given by

cos 



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

 M

sin



þ M

; ð2:20Þ

and

cos



þ M

ðÞ

: ð2:21Þ

Interaction of ions with matter 35

2.2.5 Energy loss

Ions passing through matter collide with constituent atoms and lose their

kinetic energies by electronic excitation and ionization and by kinetic colli-

sions. The energy loss by the electron-related collision is called electronic or

inelastic energy loss and the kinetic-collision induced energy loss is called

nuclear or elastic energy loss. The former and the latter respectively work

mainly at a high and a low velocity range. The term of stopping power is

also used instead of the energy loss. At the high velocity range the ions lose

their energies by the process of photon generation and by inducing nuclear

reactions. In the following the nuclear and electronic energy losses will be

presented.

The most familiar energy loss formula is the following Bethe–Bloch

equation, which is derived on the basis of the plane wave Born approxima-

tion applicable at the high velocity range where the screening of the nuclear

charge and the nuclear energy loss need not be considered:

Ndx

4Z

2mv

 ln 1 







–



eV-cm

;

ð2:22Þ

where dE/dx is the energy loss per unit length, N the number of atoms in

unit volume, m the electron mass, v the ion velocity, I ameanenergyof

excitation and ionization, c the light velocity, C/Z

a shell correction term,

and –/2 a density effect [6,7]. The shell correction term is important in a

relatively low velocity range where the ion cannot ionize nor excite inner

shell electrons . The terms (v/c)

and –/2 contribute i n the relativisti c velocity

range.

In a keV energy range, the velocities of ions, especially of heavy ions are

very low and the Bet he –Bloch fo rmula is not applicable any mor e. Th e

ions capture elect rons fr om the const ituent a toms and as a resul t the

screening o f the nuclear charge becomes important. The nuclear collision

competes with and further predominates over the electronic colli sion

with decreasing ion v eloci ty. Lindhard , Scharff, and Schiott (LSS) derived

the uni ve rsal electronic and nu clear energy loss formula e based on the

Thomas–Fermi atomic model at a low velocity range [8]. The el ectronic

energy loss is given by

Ndx

¼ ·

8Z

NZv

; ðv < v

2=3

Þ; ð2:23Þ

Focused ion beam systems36

where ·  Z

1=6

; and a

and v

are the Bohr radius an d velocity, r es pec-

tively. The f ormula displayed by a reduced energy and length i s

d



¼ k

ﬃﬃﬃ

; ð2:24Þ

where " is the reduced energy given in (2.16); the length  is gi ve n by

 ¼ xNM

4a

=ðM

þ M

; a

is the Lindhard screening radius,

¼ 0:8853a

ðZ

2=3

þ Z

2=3

1=2

;andthecoefﬁcientk depends on Z

, Z

,andM

and is given by

k ¼

0:0793Z

2=3

1=2

þ M

ðÞ

3=2

2=3

þ Z

2=3



3=4

3=2

1=2

The nuclear collision is very important for the sputtering and defect

creation processes. Lindhard et al. derived the differential scattering cross

section formula of (2.15) on the basis of the Thomas–Fermi atomic model:

d ¼ a

ﬃﬃ



; ð2:25Þ

where t is a reduced variable relating to the reduced energy " and the scat-

tering angle  : t ¼ "

sin

ð=2Þ: Then, the nuclear energy loss is given by

d



ﬃﬃ



ﬃﬃ

: ð2:26Þ

As described above, the Bethe–Bloch and LSS equations are underlying

energy loss formulae in high- and low-energy ranges, respectively. However,

obtained experimental values of energy loss depend on incident ion and

energy, and the target atom. Ziegler et al. have checked very carefully vast

amounts of parameters necessary for obtaining the energy losses covering

any incident ion–target atom combinations over an energy range between

1 keV/u and 2 GeV/u [5]. The versatile program named SRIM developed by

Ziegler et al. is opened for public uses and covers the calculation of energy

losses, ranges, range stragglings, damage distributions, sputtering yields, and

so on [5].

2.2.6 Outline of theoretical models

Nanotechnology-aimed applications of phenomena induced by energetic ions

in matter need more detailed and quantitative theoretical understanding of

Interaction of ions with matter 37