highfor n 1 dx

dEn

dx

dE

Molecular Dynamics of damage and sputtering induced by swift heavy ions M. Beuve$&; N. Stolterfoht&; M. Toulemonde§; C. Trautmann$ ; H.M. Urbassek#

§ CIRIL (Centre Interdisciplinaire de Recherche Ion Laser), Caen, France; $ GSI (Gesellschaft für Schwerionenforschung), Darmstadt, Germany;

& HMI (Hahn-Meitner-Institut), Berlin, Germany; # University of Kaiserslautern, Germany

e.g.: n=1 for condensed Ar

n=2 for condensed O2

n=4 for LiF

n function of the target

Experiments

• Great number of analytical models

• Large number of approximations

• Based on thermodynamics considerations

e.g.: - Gas Flow

- Shock wave or pressure pulse

- Thermal spike

n depends on the models (1, 3/2, 2, 3)

Analytical theories

Analytical Thermal Spike (ATS)e.g.: P. Sigmund & C.Claussen (81); R.E. Johnson & R Evatt (80)

Principle

• Local atomic equilibrium + Thermodynamics principle

• Cylindrical symmetry T = T (r,t)

• Diffusive transport of energy

C(T) specific heat ; K(T) thermal conductivity

• Initial temperature profile = Gaussian

– Fixed width

– Total energy = Stopping power

• Sputtering = sublimation process

Result

2n dx

dE

n

Molecular Dynamics (MD)

Principle

Solving the Newton equation for N atoms

)Vdt

d

Mdt

d

Nii

i

ii

irrr

p

pr

r

,...,....,( 10

• Advantages (/Monte Carlo Simulation)

– Collective effects (Shock wave)

– Adapted to strong perturbations

– Anisotropic effects (focusson)

• Disadvantages

– Computing time is huge!!!

Small samples (N104 to 105 )

Simple system model (Lennard-Jones)

612

10 4)( )(2

1,...,....,(

rrrVwithrV)V JL

jiijJLNi

rrr

Applied within the same conditions - Condensed-gas solid : Argon

- Same initial temperature profile: (Cylinder or Gaussian profile)

MD and ATS comparison

Explanations• Velocity distribution Non Maxwellian

• Transport of energy Not Only diffusive

• Effect of pressure wake

• Energy transport to the surface… Tsurf>Tbulk

dx

dE

dx

dEhigh for

1

Analytical Thermal Spike = Incorrect

Molecular dynamics Analytical Thermal Spike

E0

R0

dx

dE

dx

dEhigh for

2

[H.M. Urbassek et al. 94]

[E.M. Bringa et R.E. Johnson 98-99]

Principle :

The energy is transferred to atoms (or eventually extracted from atoms) by a random isotropic kick processes:

A kick of energy E0 instantaneously changes the atom momentum according to the equations:

(1)

- vi and vf are respectively the velocity before and after the kick,- M is the atom mass,- E0 is the energy transferred to the atom and may be negative,- q is an isotropic momentum. It is randomly tossed up in respect for the set of equations (1).

Isotropic random kick

22

0 /2

/

if

if

vvME

M vvq

Time-dependent energy transferPrinciple

Energy is progressively and periodically introduced during a time with a period /N (N from 100 to 1000).

Each transfer of energy is equal to(Lz is the thickness of the target)

t0N

zL

dx

dE

N

1

For each energy transfer i, only the atoms that set in the solid and within the cylinder of axe, the ion trajectory, and of fixed radius R0 (R0 =2) received an energy E0(i). Then

(Nat(i) is the number of atoms in the solid setting in the cylinder)

)i(NatL dx

dE

N

1)i(0E z

R0

The energy transfer to the atoms is performed according to the previously described random kick process

Observations / Interpretations• For time scale lower than 10-12 s, the effect of a time-dependent energy transfer is irrelevant

• For a time scale of 10-11s :

• the power exponent is clearly modified.

• for y 1 energy diffuses and can no longer produce an efficient sputtering

• for y >> 1 less energy is taken away by non linear mechanisms (pressure pulse...). The sputtering is then more efficient

Scaling to other materials Lennard Jones potential Time scaling

e.g.: Cu: tCu= 0.17 tAr

Cu atom dynamics is faster Time effect should appear sooner

Such time scales match, for instance, with F-centre creation in alkali halides.

[N. Itoh et K. Tanimura 90]

= 0 s t = 20 ps

= 10 ps t = 20 ps

Principle :

Based on two coupled equations:

• one for the atomic subsystem- Similar to the ATS model

• one for the electronic subsystem consisting of:

- A(r,t), takes into account :i) Electronic excitations by the incident ion (10-18 s)

ii) Transport of the excitations by electronic cascades (10-18 ~ 10-15 s)

- A heat equation dealing with thermal electrons

• g(Te-Tat) represents energy transfers due to electron-phonon coupling (g is constant)

Thermal spike model of Toulemonde et al.

F. Seitz et J.S.Koehler (23) ; M. Toulemonde et al. (92)

)T-g(T r

T)(TrK

r

1

t

T)(T )(

),()T-g(T r

T)(TrK

r

1

t

T)(T )(

ateat

atatat

atat

atee

eee

ee

rCII

trAr

CI

Limits:

- Assumes local atomic equilibrium

- Does not take account for any mechanism such as shock wave, focus on....

- Considers the target as an infinite medium

- Assumes sputtering as an evaporation process

Advantages ( / Analytical Thermal spike):

- i) Includes electron dynamics instead of a simple fixed profile of deposited energy.

- ii) Propose a process of energy transfer from electrons to atoms: electron-phonon coupling

- iii) Takes account for phase transformation (melting, superheating…)

Effect of energy deposition …

Parameter of the simulation:

- Atomic subsystem : Solid argon

Potential : Lennard-Jones

- Electron subsystem : Insulator

C(T) = 1 J cm-3 K-1

K(T) = 2 J cm-1 s-1 K-1

- Electron phonon-coupling : Very strong

- g = 0 for Tat > Te

- g = 2 10 14 W cm-3 K-1 for Tat < Te

Close to the value for SiO2 [Toulemonde et al. 2000]

The Idea:

To check the effects of a more realistic energy deposition beyond the fixed radial profile, we consider the thermal spike developed by Toulemonde et al.

Some details:- The sample is decomposed in n cylinders Ci of axe, the ion trajectory and of radius Ri = iR0 , i [0, n-1]

- n-1 domains Di are defined as Di = Ci \ Ci-1. A last one is defined as Dn = sample \ Cn-1

- At each time step t of the whole simulation (electron and atomic dynamics) and for each domain,

- the atomic temperature Tat is evaluated

- a transfer of energy is performed between both subsystems according to the formulae

(Vi volume of Di )

The energy is given to (or taken from) the atoms according to the described random kick process.

Note: outside the sample, g is set equal to 0 to solve the electron dynamics even far away from the ion trajectory

Dn

)T-g(T ate itV

Effect of potential?

Lennard-Jones n=1

Other potentials n1

Morse potential [E.M. Bringa et al. 00]

- 1 parameter more to change stiffness

- Obtained n~1.5 but:

- Potential ~ hard sphere

- Shockwave

- Artificial

Effect of energy deposition ?Experimental argument

=> Velocity effect

Theoretical argument

=> Work of Toulemonde et al.

....In this work , we investigated some effects of the energy deposition

?

Why does MD give n =1 ?

Observations / Interpretations Effect of a rather realistic energy deposition:

• a velocity effect can clearly be observed

• the power exponent is hugely modified

• the area concerned by sputtering is quite large. Its radius reaches a value up to 11 (larger than the Bohr adiabatic radius)

• although the atomic temperature stays lower than the sublimation temperature (Tsub= 640 K = kB

-1.Us=55 meV.kB-1),

the yield can reach very large value (e.g.: 150 at/ion)

• for high dE/dx, the sputtering occurs in a collective way. A block of matter suddenly flows out (in less than 20 ps).

A very large value of electron-phonon coupling

Energy transfer occurs before:

• any thermal diffusion of the electronic energy

• any local thermodynamic equilibrium for the atomic subsystem. The notion of specific heat could not be used here to describe the atomic evolution

[E.M. Bringa et R.E. Johnson 98]

A shock wake takes away energy from the track

Other observations / SuggestionsThe analysis of some movies seems to indicate that:

• for H(0.11MeV), a significant part of the sputtered atoms would be ejected perpendicularly to the surface

• for He(0.6MeV/n), more atoms seems to be ejected at large angles,

Indeed, in the latter case, a block of matter, which contains a great number of atoms, undergoes an evolution in the vacuum soon after its ejection:

- a kind of explosion occurs because the atoms, surrounding it in the solid, are now missing

- the highly perturbed surface attracts the last ejected atoms and then deviates them to higher angles

Both these phenomena would produce a more isotropic distribution.

These suggestions, which are qualitatively in agreement with measurements performed for LiF target (except for the sharp peak at 0°), have to be confirmed by angular distribution calculations.He (0.6 MeV/n) t = 30 psH (0.11 MeV) t = 30 ps

Ion H H H H (*) He He He He (*)

Energy [MeV/n] 0.5 0.3 0.2 0.11 6.5 3.25 1.8 0.6dE/dx [eV/Å] 3.78 5.23 6.5 7.47 7.42 11.2 15.5 21.7

y 158.5 219 272. 5 313 311 470 650 910

Bragg Peak (*)

General conclusion- From this work, we learned that it may not be realistic to model swift-ion interaction with solids depositing suddenly a fixed part of the stopping power in a cylinder of fixed radius.

- We showed that a more realistic time-dependent and space-dependent energy deposition may drastically modify the sputtering yield.

- In particular, for the first time, non-linear relations between yield and stopping power were showed with Molecular Dynamic for high stopping power.

- This work opens directions to new extensive studies. Indeed, the whole process of swift-ion interaction with solid (including electronic excitations, transport, eventual trapping…), has, a priori, to be considered..

AbstractDue to its complexity, modelling of damage induced in

solids by swift heavy ions was first undertaken using analytical theories based on model concepts such as thermal spike or pressure pulse model. About 10 years ago, Molecular Dynamics (MD) simulations were started in this field [1], reducing the number of approximations in the theoretical description. However, the electronic processes, which govern the primary ion-solid interaction and also the subsequent energy transfer from the target electrons to the lattice, are introduced in these simulations in an ad hoc manner. For instance, sputtering predictions are generally obtained considering that a fixed fraction of the projectile energy loss is deposited inside a cylinder of fixed radius. Thus, effects related to the radial extension of the electron cascade (e.g. velocity effect) can not be taken into account. Furthermore, for high-energy depositions, the sputtering yield is predicted to be proportional to the stopping power which is in disagreement with experimental data.

The aim of our MD calculations is to explore the influence of the time and space dependent energy transfer from the electrons to the lattice atoms on observable quantities (e.g., sputtering and track size). For this purpose, we use the electron-phonon coupling mechanism developed by Toulemonde et al. [2] in the thermal spike model. The evolution of the electronic temperature Te(r,t) is described by a continuum equation that is coupled to the Newton equations of the target atoms (MD). We can assess, for example, the influence of the projectile velocity and of the electron phonon coupling on the atom dynamics.

1 H. M. Urbassek, H. Kafemann, R. E. Johnson, Phys. Rev. B 49, 786 (1994)

2 M.Toulemonde, J. M. Costantini, Ch. Dufour, A. Meftah, E. Paumier and F.Studer Nucl. Instr. Meth. B116 (1996)37

q=0°

a60-75°

45°

-80 -40 0 40 80100

101

102

103

104

105

Au (210)

I (150)

Ni (70)

angle q

diff

. spu

tter

yie

ld (

F, L

i)

(a)

?

?

ion

at

N

N

Before Kick Aftervi v

f

q

(r)TT : 0 t

r

TrK(T)

r

1

t

T(T)

0

rC

y

St,

(T)dSdtY

U

MMt atat