Transcript MD simulation of sputtering - UMR5822 Institut de Physique

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
Abstract
Due 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
Why does MD give n =1 ?
Effect of potential?
Lennard-Jones
n=1
Other potentials
n1
?
Morse potential [E.M. Bringa et al. 00]
- 1 parameter more to change stiffness
e.g.: P. Sigmund & C.Claussen (81); R.E. Johnson & R Evatt (80)
• Diffusive transport of energy
T 1  
T 

rK(T)
t r r 
r 
t  0 : T  T0 (r)
C (T)
e.g.: n=1 for condensed Ar
n=2 for condensed O2
n=4 for LiF
T 1  
Te 
( I ) Ce (Te ) e 
rK
(T
)
 g(Te - Tat )  A(r , t )
e e


t r r 
r 
T 1  
Tat 
( II ) Cat (Tat ) at 
rK
(T
)
 g(Te - Tat )
at at
t r r 
r 
 Simple system model (Lennard-Jones)
n
n2
12
6



 
1






V0 (r1 ,...., ri ,...rN )   VL  J (rij ) with VL  J (r )  4       
 r   r  
i j 2


Isotropic random kick
q / M  v f  v i

2
2
2
E
/
M

v

v
 0
f
i
Principle
 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)

0
(1)
Kick
t
N
E0(i) 
vi
Before
1 dE
Lz
N dx
 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).
1 dEThen
q
After
N dx
L z Nat(i)
(Nat(i) is the number of atoms in the solid setting in the cylinder)
- 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).
 = 0 s t = 20 ps
R0
The energy transfer to the atoms is performed according to
the previously described random kick process
Scaling to other materials
• For time scale lower than
the effect of a
time-dependent energy transfer is irrelevant
t 
:
energy deposition beyond the fixed radial profile,
we consider the thermal spike developed by
Toulemonde et al.
• for y >> 1 less energy is taken away by non
linear mechanisms (pressure pulse...). The
sputtering is then more efficient
M at

Such time scales match, for instance, with Fcentre creation in alkali halides.
[N. Itoh et K. Tanimura 90]
Potential : Lennard-Jones
Observations / Interpretations
C(T) = 1 J cm-3 K-1
Effect of a rather realistic energy deposition:
K(T) = 2 J cm-1 s-1 K-1
• a velocity effect can clearly be observed
- Electron phonon-coupling : Very strong
- g = 0 for Tat > Te
• the power exponent is hugely modified
- g = 2 10 14 W cm-3 K-1 for Tat < Te
• the area concerned by sputtering is quite large. Its radius
reaches a value up to 11 (larger than the Bohr adiabatic
radius)
Close to the value for SiO2 [Toulemonde et al. 2000]
Some details:
Advantages ( / Analytical Thermal spike):
- i) Includes electron dynamics instead of a
simple fixed profile of deposited energy.
- n-1 domains Di are defined as Di = Ci \ Ci-1. A last one is
defined as Dn = sample \ Cn-1
• 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).
- ii) Propose a process of energy transfer from
electrons to atoms: electron-phonon coupling
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
A very large value of electron-phonon coupling
 Energy transfer occurs before:
Dn
- At each time step t of the whole simulation (electron
and atomic dynamics) and for each domain,
• any thermal diffusion of the electronic energy
y
• any local thermodynamic equilibrium for the atomic
subsystem. The notion of specific heat could not be used
here to describe the atomic evolution
- the atomic temperature Tat is evaluated
- a transfer of energy is performed between both
subsystems according to the formulae
g(Te - Tat )tVi
Ion
Energy [MeV/n]
dE/dx [eV/Å]
(Vi volume of Di )
H
0.5
3.78
158.5
y
The energy is given to (or taken from) the atoms
according to the described random kick process.
H
0.3
5.23
219
H
0.2
6.5
272. 5
H (*)
0.11
7.47
313
He
6.5
7.42
311
He
3.25
11.2
470
He
1.8
15.5
650
He (*)
0.6
21.7
910
Bragg Peak (*)
Note: outside the sample, g is set equal to 0 to solve the electron
dynamics even far away from the ion trajectory
General conclusion
(a)
- 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.
diff. sputter yield (F, Li)
?
- In particular, for the first time, non-linear
relations between yield and stopping power were
showed with Molecular Dynamic for high
stopping power.
Au (210)
?
3
10
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
I (150)
2
10
Ni (70)
q=0°
45°
1
10
Both these phenomena would produce a more isotropic
distribution.
a60-75°
0
H (0.11 MeV) t = 30 ps
Other observations / Suggestions
• for He(0.6MeV/n), more atoms seems to be ejected at
large angles,
10
10
A shock wake takes away energy from the track
• for H(0.11MeV), a significant part of the sputtered atoms
would be ejected perpendicularly to the surface
4
- We showed that a more realistic time-dependent
and space-dependent energy deposition may
drastically modify the sputtering yield.
[E.M. Bringa et R.E. Johnson 98]
The analysis of some movies seems to indicate that:
5
10
- 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..
 = 10 ps t = 20 ps
Parameter of the simulation:
- The sample is decomposed in n cylinders Ci of axe, the
ion trajectory and of radius Ri = iR0 , i [0, n-1]
- iii) Takes account for phase transformation
(melting, superheating…)

M at
U
e.g.: Cu: tCu= 0.17 tAr
 Cu atom dynamics is faster
 Time effect should appear sooner
• for y  1 energy diffuses and can no longer
produce an efficient sputtering
 To check the effects of a more realistic
- Electron subsystem : Insulator
• g(Te-Tat) represents energy transfers due to
electron-phonon coupling (g is constant)
10-11s
Lennard Jones potential  Time scaling
• the power exponent is clearly modified.
The Idea:
• one for the electronic subsystem consisting of:
- A heat equation dealing with thermal
electrons
• For a time scale of
Effect of energy deposition …
- Atomic subsystem : Solid argon
i) Electronic excitations by the incident ion (10-18 s)
ii) Transport of the excitations by electronic cascades
(10-18 ~ 10-15 s)
[E.M. Bringa et R.E. Johnson 98-99]
Time-dependent energy transfer
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:
• one for the atomic subsystem
- Similar to the ATS model
- A(r,t), takes into account :
Explanations
• Velocity distribution Non Maxwellian
• Transport of energy Not Only diffusive
• Effect of pressure wake
• Energy transport to the surface… Tsurf>Tbulk
 Small samples (N104 to 105 )
10-12 s,
Principle :
 Based on two coupled equations:
Analytical Thermal Spike = Incorrect
– Computing time is huge!!!
t,S
 dE 
 
 dx 
2
dE
 dE 
    for high
dx
 dx 
[H.M. Urbassek et al. 94]
• Disadvantages
Observations / Interpretations
F. Seitz et J.S.Koehler (23) ; M. Toulemonde et al. (92)
1
dE
 dE 
    for high
dx
 dx 
 n depends on the models (1, 3/2, 2, 3)
Experimental argument
=> Velocity effect
Thermal spike model of
Toulemonde et al.
Molecular dynamics  Analytical Thermal Spike
– Anisotropic effects (focusson)
Y   dt dS  (T)
Result
- Condensed-gas solid : Argon
- Same initial temperature profile:
(Cylinder or Gaussian profile)
– Adapted to strong perturbations
• Sputtering = sublimation process
• Great number of analytical models
• Large number of approximations
• Based on thermodynamics considerations
e.g.:
- Gas Flow
- Shock wave or pressure pulse
- Thermal spike
Applied within the same conditions
– Collective effects (Shock wave)
• Initial temperature profile = Gaussian
– Fixed width
– Total energy = Stopping power
Analytical theories
R0
• Advantages (/Monte Carlo Simulation)
C(T) specific heat ; K(T) thermal conductivity
 n function of the target
Effect of energy deposition ?
....In this work , we investigated some
effects of the energy deposition
• Cylindrical symmetry T = T (r,t)
n
dE
 dE 
    1  n for high
dx
 dx 
E0
dri p i

dt M i



 
dp i
  ri V0 (r1 ,...., ri ,...rN )
dt
• Local atomic equilibrium + Thermodynamics principle
Experiments
MD and ATS comparison
Principle
 Solving the Newton equation for N
atoms 

Principle
- Obtained n~1.5 but:
- Potential ~ hard sphere
- Shockwave
- Artificial
Theoretical argument
=> Work of Toulemonde et al.
Molecular Dynamics (MD)
Analytical Thermal Spike (ATS)
N at
 
N ion
-80
-40
0
angle q
40
80
He (0.6 MeV/n) t = 30 ps
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.