Ph 202

ACTAPOLYTECHNICASCANDINAVICAAPPLIED PHYSICS SERIES No. 202

Molecular Dynamics Simulations of Atomic Collisionsfor Ion Irradiation Experiments

KAI NORDLUND

University of HelsinkiDepartment of PhysicsAccelerator LaboratoryP.O.Box 43 (Pietari Kalminkatu 2)FIN-00014 University of HelsinkiFinland

Academic DissertationTo be presented, with the permission of the Faculty of Science of the University of Helsinki, for publiccriticism in Auditorium XII of the main building of the University, on November 11th, 1995, at 10 o’clock am.

HELSINKI 1995

2

K. Nordlund: Molecular dynamics simulations of atomic collisions for ion irradiation experi-ments. Acta Polytechnica Scandinavica, Applied Physics Series No. 202, Helsinki 1995, 72 pp. Published

by the Finnish Academy of Technology, ISBN 951-666-465-2. ISSN 0355-2721.

Classification (INSPEC): A3410

Keywords (INSPEC): Molecular dynamics method, ion implantation, radiation damage

ABSTRACT

An understanding of the way damage is formed during ion implantation of solids is very important in

many research and industrial applications. One of the best ways to study damage formation theoretic-

ally is provided by molecular dynamics simulations, in which the movement of atoms is followed by

solving numerically the equations of motion.

The purpose of work described in this thesis has been to develop and use molecular dynamics methods

to study damage production during ion irradiation.

Molecular dynamics simulations of full collision cascades have been used to studythe effect of the

interatomic potential on vacancy production in silicon. We found that the repulsive part of the potential

does not significantly affect vacancy production, whereas the form of the potential well has a strong

effect.

Ion range profiles obtained from simulations utilizing different silicon samplestructures have been

compared to experimental profiles to deduce the damaged sample structure of high-doseion-implanted

silicon. Comparison of experimental and simulated mean ranges have been used tostudy electronic

stopping powers in silicon and metals.

3

CONTENTS

ABSTRACT 2

1 INTRODUCTION 5

2 PURPOSE AND STRUCTURE OF THIS STUDY 6

3 DEFECT PRODUCTION IN COLLISION CASCADES 9

3.1 Overview of damage creation processes: : : : : : : : : : : : : : : : : : : : : : : 9

3.2 General principles of MD simulations: : : : : : : : : : : : : : : : : : : : : : : : 10

3.3 Our MD studies of vacancy production in collision cascades: : : : : : : : : : : : : 13

4 CALCULATIONS OF LARGE-SCALE DAMAGE PRODUCTION 16

4.1 Overview: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16

4.2 Our MD studies of large-scale damage production: : : : : : : : : : : : : : : : : : 18

5 STUDIES OF THE ELECTRONIC STOPPING POWER 23

5.1 Overview of electronic slowing down: : : : : : : : : : : : : : : : : : : : : : : : 23

5.2 Our MD studies of electronic stopping powers: : : : : : : : : : : : : : : : : : : : 25

6 CONCLUSIONS 26

ACKNOWLEDGEMENTS 27

REFERENCES 28

5

1 INTRODUCTION

In the early twentieth century X-ray diffraction techniques showed that atoms are ordered in a regular

crystalline structure in many forms of solid matter [1, 2]. Fairly soon after this both theoretical and

experimental work led to the conclusion that naturally occurring crystals contain deviations from a

perfect lattice structure [3, 4]. These deviations, generally called “defects”, are the source of many

interesting and useful properties in solids.

Nowadays several techniques for introducing defects in solids are known [5, 6]. Among them, ion im-

plantation offers several advantages compared to many other methods, for instancegood control of the

implantation dose and depth profile [7–9]. Despite the fact that ion irradiation techniques are regularly

used in research and industrial applications, the mechanisms that generate damage in materials during

implantation are still not very well known. Since implantation damage occurs predominantly in bulk

material, it is very difficult to examine the damage creation processes directly [10]. Therefore, it is

important to develop realistic theoretical methods by which damage creation can be studied.

Due to the large number of atoms involved in damage creation mechanisms, the usefulness of analyt-

ical methods in this field is limited [11]. Nowadays computer simulations areregularly used to study

physical phenomena which can not be satisfactorily treated analytically [12].Molecular dynamics

(MD) simulations are especially well suited for studying defect creationprocesses dynamically [13].

In MD simulation methods the time evolution of a number of atoms is obtained by solving numerically

the equations of motion that govern the system [12, 14]. The forces that act between the atoms are

generally obtained either from classical interatomic potentials [15, 16], or from quantum mechanical

ab initio calculations [17–20]. These two main approaches are called classical and quantum molecular

dynamics, respectively.

If a simulation includes atoms that move at energies higher than a few hundred eV one must also in-

clude a description of the energy loss of the energetic atoms to electrons in the medium, the so called

6

electronic stopping power [21].

The interatomic forces and the electronic stopping power are typically the only physical input that is

given to an MD simulation. Therefore, if both are well known, MD methods can in principle be used

to study ion implantation processes in a realistic way by simulating the time evolution of the atomic

collision processes induced during implantation.

In practice, computer capacity limits what can be simulated. Therefore,for different physical pro-

cesses one must choose the most realistic simulation method that is able to calculate the desired res-

ults within a reasonable amount of time. Quantum molecular dynamics (QMD) methods have the best

description of atom-atom interactions [18], but they are far slower than MD methods that are based

on classical potentials. Even classical calculations of very high-energetic processes are very time-

consuming. Hence, in practical cases it is necessary to develop simulationmethods in which the num-

ber of interactions and atoms that are involved in the simulations is reduced.

2 PURPOSE AND STRUCTURE OF THIS STUDY

The purpose of the current work has been to develop and use molecular dynamics methods to study

damage created during ion implantation. Emphasis has been laid on creating simulation methods

which can be applied to analyze measurements performed at our laboratory. Throughthe use of MD

methods, we have attempted to obtain a better understanding of damage production and related phe-

nomena which occur during ion irradiation.

The thesis is organized as follows. In this section, the original publications arelisted, along with a

brief summary of their contents. In the text, the papers will be referred to bytheir Roman numbers.

The original papers are appended to the end of the text.

In sections 3 – 5, different aspects of this thesis are presented. In the firstsubsection(s) of each section,

a short overview of the field is given to help the reader place the work performed atour laboratory in

7

its proper context. In the last subsection our work is presented.

Section 3 contains a discussion of defect production during ion implantation. The emphasis is on how

MD simulation methods can be used to study defect production in detail. In section 4 methods with

which large-scale damage production can be studied are presented. In section 5the role and importance

of the electronic stopping power in keV ion implantation processes is discussed. Theconclusions are

given in section 6.

The publications included in this thesis are except for paper III the result of group work. The experi-

ments have been performed at the Accelerator Laboratory of the University of Helsinki, and some of

the theoretical work in cooperation with the Department of Technical Physics at Helsinki University

of Technology. All the simulations and some of the data analysis presented in the papers have been

performed by the author.

Summaries of the original papers

Paper I: Effect of the Interatomic Si-Si-potential on Vacancy Production during Ion Implanta-tion of Si, K. Nordlund, J. Keinonen, and A. Kuronen,Physica Scripta T54, 34 (1994).

MD simulations of full collision cascades produced by secondary knock-on atoms are used to

study the effect of different interatomic potentials on the average number of vacancies pro-

duced. The results indicate that the repulsive part of the potential does not affect vacancy pro-

duction significantly, whereas the form of the potential well has a large effect.

Paper II: First-principles simulation of collision cascades in Si to test pair-potentials for Si-Siinteraction at 10 eV – 5 keV, J. Keinonen, A. Kuronen, K. Nordlund, R. M. Nieminen, and A. P.

Seitsonen,Nucl. Instr. Meth. Phys. Res. B 88, 382 (1994).

The effect of different repulsive interatomicpotentials on damage production and the range pro-

file in ion-beam amorphized silicon is studied. We introduce a new interatomic potential cal-

culated from first-principles calculations for the Si-Si interaction, and show that it reproduces

experimental range profiles better than two commonly used repulsive potentials.

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Paper III: Molecular dynamics simulation of ion ranges in the 1 – 100 keV energy range,K. Nordlund,Comp. Mat. Sci. 3, 448 (1995).

An efficient molecular dynamics method for calculating ion ranges and deposited energies in

the keV recoil energy region is presented. The energy range in which interactions between lat-

tice atoms can be ignored in range calculations is examined. Our MD method is compared to

conventional binary collision approximation (BCA) range calculations.

Paper IV: Range profiles in self-ion-implanted crystalline Si, K. Nordlund, J. Keinonen, E. Rau-

hala, and T. Ahlgren,accepted for publication in Phys. Rev. B (1995).

The structure of initially single-crystal silicon damaged by 50 and 100 keV self-ion implanta-

tion is examined. MD range calculations are performed for various sample structures, and the

effect of the damaged sample structure on the range profile is demonstrated. Comparison of

experimental and simulated range profiles suggest a polycrystalline sample structure.

Paper V: A low-level detection system for hydrogen analysis with the reaction 1H(15N;αγ)12C,P. Torri, J. Keinonen, and K. Nordlund,Nucl. Instr. Meth. Phys. Res. B 84, 105 (1994).

We present an experimental setup by which the distribution of low concentrations of hydrogen

in silicon can be measured using nuclear techniques. Measured range profiles of silicon im-

planted with 40-keV hydrogen are compared with simulated ones. We find that the commonly

used electronic stopping powers given by J. F. Ziegler, J. P. Biersack, and U. Littmark [The

Stopping Powers and Ranges of Ions in Matter (Pergamon, New York, 1985), Vol. 1], has to be

multiplied by 1.21 to reproduce the experimental range profile.

Paper VI: Stopping of 5 – 100 keV helium in tantalum, niobium, tungsten, and AISI 316L steel,P. Haussalo, J. Keinonen, and K. Nordlund,submitted for publication in Nucl. Instr. Meth. Phys. Res.

B (1995).

We obtain the stopping powers for 5 – 100 keV helium implanted in metals by comparing exper-

imental range profiles measured with the elastic-recoil-detection-analysis method to simulated

ones. We present a method by which the total (nuclear + electronic) stopping power can be ob-

tained from MD simulations, and discuss the difference between BCA and MD range results.

9

3 DEFECT PRODUCTION IN COLLISION CASCADES

3.1 Overview of damage creation processes

The slowing down of ions in solid materials is conventionally interpreted to be due to two separate

processes, electronic and nuclear slowing down (stopping) [21, 22]. Electronic slowing down is dis-

cussed in detail in section 5. Here we only note that at keV energies the electronic slowing down does

not significantly contribute to the production of lattice defects [5]. Defect production is chiefly caused

by nuclear stopping, i.e. elastic collisions between a recoiling ion and atoms in the medium.

When an energetic ion collides with an atom in a crystal lattice and departs enough energy to it, the

lattice atom will collide with other lattice atoms, resulting in a large number of successive collisions.

All the atomic collisions initiated by a single ion are called a collision cascade.

A collision cascade can be divided into three phases [11]. The initial stage, during which atoms collide

strongly, is called thecollisional phase, and typically lasts about 0.1 – 1 ps. As a result of the collisions,

one can assume that all atoms near the initial ion path are in thermal motion at a high temperature. The

high temperature will spread and be reduced in the crystal by heat conduction. This phase is called the

thermal spike, and lasts roughly 1 ns.

When the thermal spike has cooled down, there will usually be left a large quantity ofdefects in the

crystal. The defects can be of several different shapes, ranging from vacancies and interstitial atoms

to complex interstitial-dislocation loops and volume defects [23–25]. If the lattice temperature is high

enough, many of these defects will relax by thermally activated migration [26]. This is the so called

relaxation phase of the collision cascade.

During ion implantation, electronic slowing down dominates the stopping of an implanted ion at high

ion energies (higher than roughly 1 keV/amu). However, when the ion has slowed down sufficiently,

10

nuclear slowing down will always sooner or later become significant. Thus, at least near the ion end-

of-range (EOR) collision cascades will be present and produce lattice damage.

The type of damage produced during ion implantation may be very complex (see eg. [27–30]), and

varies a great deal for different ion types, sample materials and implantation conditions. However,

some general characteristics can be identified for implantation into semiconductors and metals.

At very low doses ion implantation results in a number of isolated defects. The defect concentration

can be roughly estimated from the Kinchin-Pease equation

NV = FDn

2Ed(1)

whereFDn is the energy deposited into the sample by atomic collisions andEd is the displacement

threshold energy (in the literature, this equation is frequently also given asthe so called “modified

Kinchin-Pease equation” [10], but the “modification” can be easily embedded in the value of the con-

stantEd). Values forEd given in the literature range between 10 and 30 eV for simple solids [31–33].

Some controversy exists on whether the Kinchin-Pease equation is suitable for predicting the number

of Frenkel pairs produced at keV energies [10].

At a certain implantation dose (roughly 1014 ions=cm2 for silicon [34, IV]) the sample becomes

amorphous. At even higher doses (roughly 1016 ions=cm2 in silicon [35]) the sample may under suit-

able implantation conditions recrystallize or become polycrystalline [29, 36, 37].

3.2 General principles of MD simulations

In molecular dynamics simulations the time evolution of a system of atoms is calculated by solving

the equations of motion numerically.

Since the movement of each individual atom involved in a collisions cascade can be followed in MD

simulations, they offer the most realistic way of examining defect formation during ion implantation.

11

One of the very first uses of molecular dynamics methods was in fact simulation of collision sequences

in metals [38, 39]; since then MD simulations have been used to study a large variety of phenomena

in collision cascades (for recent examples see eg. refs. [25, 40–44]).

The molecular dynamics simulation process starts by calculating the forceFi acting on each atomi in

the system. The equations of motion for the system are solved using some suitable algorithm [45, 46].

The solution yields the change in the atom positions, velocities and accelerations over a finite time

step∆t. The smaller the time step, the more accurate is the solution of the equations of motion. After

the atoms have been moved the simulation continues by recalculating the forces in the new positions.

The atoms that are included in the calculation are usually placed in an orthogonalsimulation cell.

The forces governing the simulation can be obtained from classical or quantum mechanical calcula-

tions. Although promising advances have recently been made using tight-binding moleculardynam-

ics (TBMD) methods, quantum molecular dynamics methods are still far too time-consuming to allow

simulation of full collision cascades [18, 20, 47, 48]. Therefore classical MD simulations have to be

used in the foreseeable future for descriptions of energetic collision cascades.

In classical MD methods and the Newtonian formalism the forceFi acting on an atomi in the system

is calculated as

Fi(ri) = ∑j 6=i

Fi j(ri j) =�∑j 6=i

∇Vi j(ri j); (2)

whereFi j is the force acting between atomsi and j, ri j the distance between the atoms, andVi j(ri j) is

a potential energy function. The sum overj is taken over all atoms whose interaction with atomi is

stronger than a threshold valueVmin [12]. In practice, the threshold is usually implemented by defining

a cut-off radiusrc for the interactions.

The interatomic potentialVi j from which the forces are obtained is conventionally divided into a re-

pulsive part governing high-energetic (kinetic energy' 1 eV) collisions, and the potential well gov-

erning equilibrium phenomena.

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At very small distances between the nuclei, the repulsive interaction can beregarded as essentially

Coulombic. At greater distances, the electron clouds screen the nuclei from each other. Thus the

repulsive potential can be described by multiplying the Coulombic repulsion between nuclei with a

screening functionφ(r),V(r) = 1

4πε0

Z1Z2

rφ(r); (3)

whereφ(r) �! 1 whenr �! 0. HereZ1 andZ2 are the charges of the interacting nuclei, andr the

distance between them.

A large number of different repulsive potentials and screening functions have been proposed over the

years, some determined semi-empirically, others from theoretical calculations. A much used repulsive

potential is the one given by Ziegler, Biersack and Littmark [21], the so called ZBL repulsive potential.

It has been constructed by fitting a universal screening function to theoretically obtained potentials

calculated for a large variety of atom pairs [21].

The standard deviation of the fit of the universal ZBL repulsive potential to the theoretically calculated

potentials is 18 % above 2 eV [21]. In cases where experimental range profiles can bedetermined with

a good depth resolution, it is crucial to have a repulsive potential with a better accuracy than that given

by the ZBL potential.

A more accurate repulsive potential can be obtained from self-consistent totalenergy calculations us-

ing density-functional theory and the local-density approximation (LDA) for electronic exchange and

correlation [18, 49]. In this approach the total energy, including both the electronicpart and the inter-

nuclear Coulomb repulsion, is obtained as a function of the distance between atoms ina dimer bond. In

practice, we have used the DMol package [50] well tested in calculations of energetics and structures

of small molecules to obtain the interatomic potentials [II, IV, VI].

When one wants to describe a solid with covalent bonds it is necessary to describethe interactions

governing the system in a way which takes into account the angular dependence of the bonds. This

13

can be achieved by using potentials which depend on the positions of three or more atoms [51–55]. In

this case, a sum over multiple atom combinations must be added in equation 2.

3.3 Our MD studies of vacancy production in collision cascades

The application of MD algorithms to calculations of collision cascades includesa few pitfalls. One is

that the recoiling ions that cause the cascade introduce a very high temperature toparts of the simu-

lation cell. Therefore, some mechanism for letting the thermal energy diffuse out of the cell without

affecting the heart of the thermal spike is required. We have solved the problem by placing the cascade

at the heart of the simulation cell, and scaling down the atom temperatures near the cell borders [I].

The time step∆t is usually fixed in simulations of systems in thermal equilibrium. In collision cascade

calculations the initial time step must be very short. Therefore, using a fixedtime throughout the sim-

ulation is very ineffective. Instead, we choose the length of the time step dynamically from two main

criteria [III]. Firstly, the time step is made inversely proportional to the velocity of the fastest mov-

ing atom in the cell. Thus, the time step becomes longer as the recoil atom slowsdown, significantly

reducing the calculation time. Secondly, the time step is made drastically shorter during very strong

collisions. Although this somewhat slows down the simulations, it is a necessaryprecaution to ensure

stable solution of the equations of motion.

Since the interatomic potential determines the behaviour of molecular dynamics simulations, it is im-

portant that the potentials and their effect on the simulation outcome are well known.

In papers I and II we have simulated collision cascades in silicon produced by single 300-eV and 1-

keV silicon recoils. We used the Stillinger-Weber interatomic potential [51] and scaled down the cell

temperature of the outermost 3 atom layers to 300 K to dissipate energy out of the simulation cell.

We studied the number of vacancies produced in several collision cascades. We foundthat the number

of vacancies varied strongly between individual recoil events. This shows that if onewishes to make

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0 50 100 150

Time (fs)

0

2

4

6

8

NS

i(E

>15

eV)

0

50

100

150

NV

MoliereZBLDMol

one 1 keV Si c-Si

Figure 1: Average numbers of recoiling Si atoms with energies greater than 15 eV (NSi(E > 15eV))and number of vacancies (NV ) produced in the slowing down process of one 1-keV Si atom in crystal-line silicon [I,II].

quantitativeconclusions on defect production from MD simulations of collision cascades, it is essential

to simulate a large number of events in order to obtain a statistically significant average of the number

of defects produced.

Several repulsive potentials have been proposed over the years for the Si-Si interaction. We calculated

the total number of vacancies produced in the initial phase of the collision cascade for the ZBL poten-

tial [21], a modified Moliere potential [56] and a potential calculated withthe DMol program [II]. In

Fig. 1 the number of recoiling silicon atoms with energies greater than the threshold displacement en-

ergy and the total number of vacancies are shown. The numbers are averages calculated from about

50 simulations of recoil events.

From Fig. 1 it is evident that the strength of the repulsive potential is reflectedin the number of high-

energy recoils produced during the collisional phase of the cascade. Despite this, the number of va-

cancies produced in the cascades is affected only very weakly by the choice of therepulsive potential.

This indicates that the number of vacancies is primarily determined by the totalnuclear deposited en-

ergy, which is almost the same for all the repulsive potentials (the electronic stopping power accounts

for less than 10 % of the total energy loss in this case).

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Potential Width (Å) Depth (eV) Vacancies Monovac. Others

1. (unmodified) 0.903 2.16 27 15 8

2. (narrower) 0.847 2.16 16 12 4

3. (narrower, deeper) 0.847 2.28 19 13 6

4. (wider) 0.924 2.16 37 18 15

5. (wider, shallower) 0.924 2.10 36 19 12

6. (deeper) 0.903 2.28 32 15 12

7. (shallower) 0.903 2.10 23 13 7

Table 1: Widths and depths for the unmodified and six modified Stillinger-Weber potentials. The av-erage number of vacant sites, monovacancies and more complex vacancies produced in a collisioncascade by one 300-eV silicon ion recoiling in crystalline silicon are shown inthe three last columns[I].

To test the effect of the form of the potential well on vacancy production, we calculated the number of

different vacancy types produced by 300-eV silicon recoils in crystalline silicon [I]. We used the un-

modified Stillinger-Weber potential along with six modified versions, in whicheither the width, depth

or product of width and depth was kept the same as in the unmodified potential. Parameters for the

modified potentials are given in Table 1.

In Fig. 2 the number of vacancies produced with the different potentials are shown forthe first 1000 fs

in the collision cascade. After the first 500 fs the number of vacancies reachesa roughly constant level.

The average number of different types of vacancies calculated between 500 and 1000 fs are given in

Table 1. Comparison of the potential parameters and vacancy results show that large changes in the

number of vacancies may result from small changes in the form of the potential well.

15

The parameterEd in the Kinchin-Pease equation (1) has been determined empirically in silicon by

measuring the number of Frenkel pairs produced during ion implantation. The values obtainedrange

between 15 and 25 eV [32, 33], whence one 300-eV recoil should produce 6 – 10 vacancies.

Frenkel pairs can be understood to correspond to the number of monovacancies calculatedin our sim-

ulations. Thus, the number of vacancies produced in our simulations are too high compared with the

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0 200 400 600 800Time (fs)

0

10

20

30

40

50

Nu

mb

ero

fva

can

cies

4.3.2.1.

7.6.5.one 300 eV Si c-Si

Figure 2: Average number of vacancies produced during the first 1000 fs by one 300-eV Si atom forthe 7 modified Stillinger-Weber potentials given in Table 1. The numbers are averages calculated fromat least 30 simulations of recoil events [I].

experimental values, indicating that the Stillinger-Weber potential is not suitable for quantitative ana-

lysis of collision cascades.

4 CALCULATIONS OF LARGE-SCALE DAMAGE PRODUC-TION

4.1 Overview

In the previous section we have discussed how one can simulate full collision cascades with MD sim-

ulations to study defects in a small part of a sample. However, this is not always possible or practical

primarily due to limitations in computer capacity.

In many practical cases the measured quantities are of a large-scale character, i.e. describe macro-

scopic properties like the sample structure or concentration of impurities in it. A full understanding

of all details in a collision cascade is not always necessary for analysis ofsuch measurement results.

Instead, it is important to develop methods by which the measurable quantities canbe simulated.

For instance, using Rutherford backscattering spectrometry (RBS)/channeling techniques one can

17

measure the overall amount of damage in a sample [24, 57]. The damage creation processes can be

examined with MD methods by injecting into a small part of a sample an amount of recoil atoms which

corresponds to the implantation dose [28].

In high-energy ion implantation, a recoil ion will first loose energy primarily dueto electronic slowing

down, and hence not cause significant amounts of lattice damage (cf. section 5 ). Onlynear the ion end-

of-range will nuclear stopping become dominant. Hence one can estimate the locationand amount of

lattice damage from the ion range distribution (i.e. the concentration of implanted ions as a function

of depth). Furthermore, the form of the ion range distribution depends on the sample structure [II,

III, IV]. For these two reasons, simulations of ion range distributions can be useful in studying ion

irradiation induced damage.

Ion range distributions can be measured with several experimental techniques, like for instance sec-

ondary ion mass spectrometry (SIMS) [57], elastic recoil detection analysis (ERDA) [58, VI], and

nuclear resonance broadening (NRB) [59, II, IV]. These methods yield the concentration of impurity

atoms as a function of the depth, but no direct information about the sample structure. However, if one

can simulate range distributions for different types of sample structures, comparison of experimental

and theoretical range distributions can yield information on the damaged sample structure.

The conventional approach to simulating ion range distributions, which has been used since the

1960’s [60, 61], is based on the binary collision approximation (BCA). In BCA methods themove-

ment of ions in the implanted sample is treated as a succession of individual collisions between the

recoil ion and atoms in the sample. For each individual collision the classical scattering integral [22]

Θ = π�2Z ∞

rmin

pdr

r2q

1� V(r)Ec� p2

r2

(4)

is solved by numerical integration. HereΘ is the scattering angle in center-of-mass coordinates,Ec

the center of mass energy,p the impact parameter of the collision andV(r) is the interatomic repulsive

potential.

18

The impact parameterp in equation (4) is determined either from a stochastic distribution [21] or in a

way that takes into account the crystal structure of the sample [62].

Although the BCA methods have been successfully used in describing many physical processes (see

eg. [63–66]), they have some drawbacks that limit their usefulness in describing the slowing down

process of energetic ions.

Due to the basic assumption that collisions are binary, problems arise when trying to take multiple

interactions into account [61]. Also, in simulating crystalline materialsthe selection process of the

next colliding lattice atom and the impact parameterp always involves unphysical parameters, i. e.

program parameters that do not correspond to any physical quantity. Tests of BCA simulation pro-

grams show that these may affect the results by 10-20 % even for apparently quite reasonable choices

of the parameter values [61, 67]. This reduces the usefulness of BCA simulations in interpreting ex-

perimental results which have been measured with a better accuracy than this.

4.2 Our MD studies of large-scale damage production

To overcome the difficulties of BCA simulation methods, we have developed range calculation algo-

rithms which are based on molecular dynamics principles. The method and our program MDRANGE

[68] are presented in detail in paper III.

In MD simulation algorithms all interactions are taken into account simultaneously. Hence no prob-

lems arise in taking multiple collisions into account. Furthermore, although MDcalculations also in-

volve unphysical simulation parameters, these can usually be selected in an unambiguous way. For

instance, by selecting a short enough time step∆t and a long enough cut-off radiusrc one can assure

that their values do not affect the simulation results.

Molecular dynamics simulations of full collision cascades are far too inefficient for calculations of

range distributions at keV and higher energies. Although the time evolution of collision cascades in-

19

a) b)RS3

Figure 3: The principle by which simulation cell atoms are translated during thecalculation of ionranges. In the Figure a two-dimensional simulation cell is shown before (a) and after (b) the atomshave been moved. In Figure a) the recoil atom has come closer thanRS3 to the cell border. When thishappens, all atoms within the shaded area are moved away from the cell border, andnew atoms areplaced in front of the recoil atom [III].

duced by keV-energy ions can nowadays be simulated [43], obtaining a reliable distribution of ion

ranges requires simulation of at least about 1000 implantation events.

To make the simulations more effective, several special features have beenincluded in our range cal-

culation method [III]. A variable time step is used in the same manner as described in the previous sec-

tion. Furthermore, the simulation cell is made much smaller by only including atoms that are within a

distance that is not much larger than the cut-off radiusrc from the recoiling ion. To ensure that the re-

coiling ion is always surrounded by lattice atoms, the simulation cell is madeto “follow” the recoiling

ion (Fig. 3).

19

To further reduce the calculation time, we introduced the recoil interactionapproximation (RIA) in

paper III. In this approximation, no interactions between sample ions are calculated during the sim-

ulation. Instead, the positions of atoms that do not interact with the recoiling ion are kept fixed. By

extensive testing we were able to show that the RIA does not affect the range results at keV energies.

Since ion range calculations only follow the movement of the recoiling ion, they can not beused to

predict the type of damage produced during implantation. However, the MDRANGE program allows

simulation of implantation into samples with different types of crystal structures. By calculating ion

20

range profiles in different sample structures, and comparing these results with experimental ones, we

have been able to draw conclusions on the structure of the sample [IV].

In papers II and IV we have studied processes induced by silicon self-ion implantation at 10-100 keV

energies. Using the same element for both the implanted ion and sample material brings the advantage

that chemical reactions do not complicate the damage creation processes.

In paper II we studied implantation of 1�1016 10-keV30Si+ ions=cm2 into originally crystalline sil-

icon. The range profile of the implanted30Si was measured with a 620-keV proton beam using the

NRB technique [69]. Our focus was on examining the repulsive interatomic potential of the silicon-

silicon interaction.

It is well known that silicon “amorphizes” (gets strongly damaged) during keV self-ion-implantation

when the implantation dose is roughly 1014 ions=cm2 [30, 70]. Since the implantation dose used here

was much higher than the amorphisation threshold, all of the implanted ions could be assumed to slow

down in amorphized silicon. Therefore, the simulations were carried out in anamorphous silicon struc-

ture obtained fromab initio calculations [19, 71, 72].

We carried out the simulations using three different interatomic potentials.We found that the potential

obtained from DMol calculations reproduced the experimental range profile best.

The energy straggling of the ion beam used in NRB measurements reduces the resolution of the con-

centration measurement. In the 10-keV implantation the range of the implanted ionswas so small that

the form of the range profile was determined to a large extent by the proton beam straggling.

In paper IV we studied 50 and 100-keV silicon self-ion-implantation. Due to the longerranges of the

implanted ions, the effect of the proton straggling on the form of the range profile was reduced. This

allowed for a more detailed analysis of the measurement results. At very high-dose ion implantation

it is well known that the sample may recrystallize or obtain a more complex damage structure [35, 37,

21

73, 74]. Our objective was to study the type of damage produced in silicon self-ion-implantation at

very high doses (> 1016 ions=cm2 ), which have not been studied much in the literature.

The 50-keV28Si implantations were carried out at doses 5�1013 to 1�1018 ions=cm2 . To prevent

channeling the< 100> normal of the silicon sample was aligned 6� off the beam axis.

For all the implanted samples, RBS/channeling measurements [75] were performed. It was found

that the irradiated samples were strongly damaged at implantation doses above 2�1014 ions=cm2 .

However, the damage level did not reach the amorphous level, indicating that some remains of a crys-

talline structure are left in the samples.

The widths of the RBS range profiles, determined as the FWHM of the displaced atom distributions,

was seen to grow logarithmically with the ion dose,

width(nm) = a+b log�dose (ions/cm2)

�(5)

In paper IV we showed that the logarithmic growth of the damage width can not be explained by as-

suming that the onset of a strong damage level occurs when the exponential tail of the rangeor de-

posited energy distributions exceed some threshold value. This is in good agreement with the recent

experimental results which indicate that the growth of the damaged layer is affected by migration of

vacancies to the interface between the damaged region and crystalline silicon [24, 74].

Resistance measurements on some of the samples also indicated the presenceof vacancies in the in-

terface layers. The resistivities of the damaged regions of the samples were found to be almost six

orders of magnitude higher than the resistivity of unimplanted parts. The high resistivities varied only

weakly with the implantation dose, which suggests that much of the resistance is due to carrier traps

in the interface between the damaged region and crystalline silicon.

For implantation doses higher than about 1016 ions/cm2 the concentration of implanted ions was high

enough to enable measurement of the range profiles of the implanted ions with the NRB technique

22

..............

....................0 50 100 150

Depth (nm)

Co

nce

ntr

atio

n(a

rb.u

nit

s)

200 nm50 nm30 nm10 nmExperiment50 keV Si -> Si (100)

Figure 4: Experimental and simulated range distributions for 50-keV Si self-ion implantation. Thesimulated range results are for polycrystalline silicon with domain sizesof 10, 30, 50 and 200 nm[IV].

[57, 69]. Therefore, in these samples 2�1016 50-keV 30Si+ ions=cm2 was implanted after the28Si

implantations.

The30Si range profiles turned out to be almost identical, indicating that the structure ofsilicon at im-

plantation doses between 2�1016 and 1�1018 ions=cm2 remains unchanged. Furthermore, the profiles

have a clear “tail” at large range values. Range profiles for implantation into amorphous materials are

well known to have a Gaussian form; hence the tail indicates the presence of some sort of crystallin-

ity in the samples. On the other hand, the RBS profiles clearly showed that the samples can not be

crystalline.

We simulated the experimental range profiles with our MD method using the DMol interatomic po-

tential for different sample structures. As was expected, the experimentalrange profile could not be

reproduced using an amorphous or crystalline sample structure.

There are literature results which indicate that high dose ion implantation in some cases can lead to

a polycrystalline sample structure in silicon [36, 76]. Also, the RBS/channelingspectrum of a poly-

crystalline structure resembles that of an amorphous structure [77, 78], and polycrystalline layers are

known to have a high resistivity [6]. Since the experimental results were consistent with a a polycrys-

23

talline sample structure, we simulated range profiles in polycrystalline silicon for different average

grain sizes. Results for some grain sizes are shown in Fig. 4 along with theexperimental range distri-

bution. The experimental range profile is well reproduced by a theoretical range distribution with an

average grain size of roughly 50 nm.

5 STUDIES OF THE ELECTRONIC STOPPING POWER

5.1 Overview of electronic slowing down

The electronic slowing down process has until now only been given a few fleeting references in this

thesis. Yet it is an intimate part of the slowing-down processes of energetic ions. Omitting it in simula-

tions even at low energies where the nuclear stopping power clearly dominates may lead to significant

changes in measurable quantities [79]. Therefore it is justifiable to discuss it in greater detail in this

context.

By electronic slowing down one means slowing down due to the inelastic collisions between electrons

in the medium and the ion moving through it [22]. The term inelastic is used to signifythat the colli-

sions may result in excitations in the electron cloud of the ion; therefore the collision can not be treated

as a classical scattering process between two charged particles. Incase the kinetic energy of a collid-

ing electron is greater than the ionization energy of electrons bound to the ion, the inelastic collision

may strip electrons from the recoiling ion. After several collisions the ionbecomes positively charged,

typically by several electron charges.

The electronic slowing down is usually treated by calculating a weighted average over different charge

states and scattering mechanisms to yield a simple function of energySe(E) [21, 22]. This func-

tion, called the electronic stopping power, gives the energy loss of the recoiling ionfor a unit dis-

tance it travels in a medium. The Bethe-Bloch formula, which has been derivedtheoretically, gives

24

log E

log

S =

log

dE/d

x

Nuclearstopping

Electronicstopping

Figure 5: Typical ratio between nuclear and electronic stopping power. The maximumof the nuclearstopping curve typically occurs at an energy around 1 keV/amu and the maximum of the electronicstopping power at about 100 keV/amu energies. For very light ions slowing down in heavy materials,the nuclear stopping is weaker than the electronic at all energies.

the electronic stopping power to an accuracy of a few % in the energy range above several hundred

keV/amu [80, 81].

At energies lower than and around about 100 keV/amu it is very difficult to calculate a general expres-

sion for the electronic stopping power. A fairly good solution has been given by Brandt andKitagawa,

who calculated a general stopping power formula by forming an expression for the effective charge

of an ion [82]. A very much used electronic stopping power is the one given by Ziegler, Biersack and

Littmark (the “ZBL” stopping), which is based on the Brandt-Kitagawa model [21].

The collisions between the ion and electrons can lead to bond breaking in covalently bonded materials,

and chain breaking and cross-linking in polymers [83]. Since these effects seldomcause large scale

damage, and since they are absent in metals, it is common practice to say that the electronic slowing

down does not contribute to damage creation during ion implantation [5, 7]. However, it has become

known fairly recently that collisions with electrons may cause damage even in metals at MeV ener-

gies [84, 85].

Good knowledge of the strength of the electronic stopping power is vital in many applications. For

instance, the accuracy of the results of several nuclear measuring techniques (eg. ERDA, RBS and

25

NRB) depends directly on how well the electronic stopping power of the measuring ions isknown. In

addition, although electronic slowing down itself does not generate damage, the strength of the elec-

tronic stopping power affects the location of the range distribution, and thus the distribution of damage

during implantation.

5.2 Our MD studies of electronic stopping powers

Our MD method for calculating ion ranges described in the previous section can be applied to studying

electronic stopping powers in a straightforward way. In cases where simulated and experimental range

profiles do not have the same mean range and the target structure is known, one can assumethat either

the interatomic potential or electronic stopping power should be modified. Interatomicpotentials de-

termined fromab initio calculations can be considered to be fairly accurate compared to electronic

stopping powers, and the electronic stopping power dominates the slowing down process at energies

higher than roughly 1 keV/amu. Hence, in many cases it is justifiable to assume that the electronic

stopping power is responsible for the discrepancy between theory and experiment.

In paper V we studied 40-keV hydrogen ion implantation into silicon. The proton implantations

were carried out at doses of 2�1013 – 1�1017 ions=cm2 . The range profile for a low concentration

of implanted hydrogen was measured using the NRB technique with a specially designed low-level

background-radiation detection system.

The range profile was simulated using an early version of our MD method. Since the implantation

doses used in the measurements were low, we assumed that the hydrogen slowed down in crystalline

silicon. By comparing the experimental and theoretical range profile, we found thatthe electronic

stopping given by the ZBL model had to be multiplied by 1.21 in order to reproduce the experimental

range profile.

26

In paper VI we examined 5 – 100 keV He implantation of AISI 316L stainless steel,niobium, tan-

talum and tungsten. The experimental range profiles were measured using the ERDA method. The

simulations were carried out with the MDRANGE program for polycrystalline sample structures. The

repulsive potentials employed were calculated with the DMol program.

We found that the mean ranges for implantation into niobium and steel were well reproduced by MD

simulations using the ZBL electronic stopping power. However, in tantalum and tungsten the ZBL

electronic stopping power had to be multiplied by 1.4 and 1.1, respectively, in orderto reproduce the

experimental results.

In paper VI we also presented principles by which the total stopping power can be calculated from MD

simulations by derivating the average velocity of a large number of recoiling ions.

6 CONCLUSIONS

In this study we have discussed how molecular dynamics simulation methods can be applied to study

damage creation during ion irradiation.

We have approached the subject from two main viewpoints, namely the microscopic view of studying

the formation of single defects by simulations of full collision cascades, and the macroscopic view of

simulating the slowing-down of ions in different damaged sample structures. Bothapproaches where

shown to be fruitful in studying damage creation processes.

We have also discussed subjects related to damage creation, in particularthe role of electronic stopping

power in the ion slowing-down process.

In section 2 we stated that our goal has been to develop MD methods which would enableus to obtain

a better understanding of damage production processes during ion irradiation. We feel that this thesis

demonstrates that we have indeed been able to meet this goal.

27

ACKNOWLEDGEMENTS

I wish to thank Professor Mauri Luukkala, former head of the Department of Physics, for providing

me with the opportunity to work at the Department. I am deeply indebted to my supervisor, Professor

Juhani Keinonen, current head of the Department, for his inspiring and knowledgeable guidance, and

for placing the facilities at the Department at my disposal. I owe much of my insight and interest in

physics to the inspiring lectures and guidance given by Professor Dan-Olof Riska.

My sincere thanks are due to my colleague Dr. Antti Kuronen for his never ceasing patience in explain-

ing the mysteries of molecular dynamics simulations to me. I also wish to thank him and Kai Arstila,

M. Sc. for their advice on using the computer equipment and software which has been indispensable

during this work. I am grateful to Dr. Eero Rauhala, Pekka Haussalo, M. Sc. and T. Ahlgren, M. Sc.

for useful discussions on how to interpret and analyze the experimental data used in parts of this work.

For creating a pleasant working atmosphere I want to thank all the personnel at the Accelerator Labor-

atory and the Department of Physics. I am also thankful to the many friends who have shared with me

the moments of joy and encouraged me during the occasional bouts of frustration I have experienced

during this work.

Financial support from the Magnus Ehnrooth Foundation, Waldemar von Frenckell Foundation and

the Scandinavian Research Education Academy (NorFA) is gratefully acknowledged.

Helsinki, September 1995

Kai Nordlund

28

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