Nuclear Instruments and Methods in Physics Research B 180 (2001) 99±104
www.elsevier.nl/locate/nimb
Angular dependence of the sputtering yield from a cylindrical track
E.M. Bringa *, R.E. Johnson
Engineering Physics, University of Virginia, Charlottesville, VA 22903, USA
Abstract
The dependence of the sputtering yield on the incident angle, H, is determined using molecular dynamics (MD)
simulations for a cylindrical track produced by a fast ion. For a `small' spike radius and for the mean energy in the
track, Eexc , smaller than the binding energy, U, a …cos H† 1:7 dependence is found, close to the linear collision cascade
(LCC) result and to some thermal spike models. On the other hand, when Eexc > U , the incident angle dependence is
…cos H† 1 . For a larger spike radius we obtain a …cos H† 1:6 dependence for both high and low energy densities.
Analytic spike models based on di€usive transport are shown not to give satisfactory results. In addition, at low energy
densities we see correlated atom ejection ignored in analytic models. Applying the MD results to the experimental data
for electronic sputtering of solid O2 at large excitation densities suggests that the e€ective spike radius is larger than the
initial Bohr adiabatic radius indicating that energy is rapidly transported from the initially narrow track. Ó 2001
Elsevier Science B.V. All rights reserved.
PACS: 61.80; 71.15.D; 79.20.R
Keywords: Sputtering; Energy transport; Thermal spikes; Oblique incidence
1. Introduction
Ejection of atoms and molecules from a solid
by ion bombardment (sputtering) is a process
which can test our ability to describe non-equilibrium, molecular scale processes in solids. Because of their simplicity, equilibrium models like
the thermal spike model, are often used to
describe the sputtering of metals and insulators
[1±7,20]. Such models are used to calculate the
average yield (the number of atoms or molecules
*
Corresponding author. Tel.: 1-804-924-4344; fax: 1-804924-3104.
E-mail address: [email protected] (E.M. Bringa).
ejected per ion incident), but they also are used to
predict the dependence of the yield on the angle
of incidence and the angular and energy distribution of the ejecta. In this paper, we present
results from molecular dynamics (MD) simulations of the ejection of atoms from an energized
track in an atomic solid. These are used to test
the predictions from spike models for angular
distribution of the ejecta and the incident angle
dependence of the sputtering yield from a track
produced by a fast ion.
Earlier we compared spike results to MD simulations of the dependence of the yield on the energy deposited per unit path length, …dE=dx†, and
the ejected atom energy distribution for normal
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 4 0 2 - 5
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E.M. Bringa, R.E. Johnson / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 99±104
incidence. We showed that spike models could be
used to understand aspects of the MD simulations
of the yield at low excitation densities but failed
dramatically at high excitation densities as melting
and the radial pressure pulse controlled the energy
transport. In spite of the complexity of the nonlinear energy transport processes [19,22] we
showed that the yield for ®xed spike radius had a
simple, nearly linear dependence on …dE=dx† at
high excitation densities. Here we show that the
incident angular dependence can di€er in these two
regimes and it di€ers from spike model predictions
in both regimes.
A problem of particular interest to us is the
sputtering of weakly bound condensed gas solids.
In astrophysics, this process can be a source of the
ambient neutral atoms and molecules around icy
objects [8]. Whereas low energy ions deposit most
of their energy in knock-on collisions, which directly produce sputtering, in condensed-gas solids
incident fast light ions produce a `track' of ionizations and excitations. The subsequent non-radiative relaxation processes release energy which
leads to particle ejection in these weakly bound
solids. The dependence of the yield on the angle of
incidence has been measured both for refractory
materials, in which knock-on sputtering dominates, and for MeV ion sputtering of solid O2 [9]
and solid CO [10].
In this paper MD simulations have been carried
out for an atomic, condensed gas solid. Due to
scaling, these results are applicable to a broad
range of materials, including molecular solids
[12±14]. Here we calculate the incident angle
dependence of the yield and the ejecta angular
distribution from a cylindrically energized track at
both high and low excitation densities. As we have
shown earlier, this energized `track' can be representative of either knock-on or electronic sputtering of a solid. We compare the results to
predictions from models for knock-on sputtering
and to spike models for sputtering. We will refer to
standard spike model results. By this we mean
models in which only a di€usion equation with a
well-behaved heat conductivity is used to obtain
the temperature pro®le, numerically or analytically, and then use this pro®le to calculate the loss
of surface material.
2. MD simulation
MD simulations were performed to follow the
response of an atomic solid to the excitation of a
cylindrical region. These simulations were carried
out using a Lennard±Jones (L±J) (6±12) potential
6
12
V …r† ˆ 4e‰…r=r†
…r=r† Š with a cut-o€ radius
rcut ˆ 2:5r to describe an fcc atomic solid. This
cut-o€ radius includes up to ®fth nearest neighbors
for a density n ˆ 1:046=r3 (a total of 78 neighbors). More details on our previous L±J calculations can be found in [11±13].
Since the equations of motion can be fully
scaled using the L±J parameters the results also
scale with these parameters, r for length and e for
energy. The mass will only change the timepscale

through the dimensionless time, t ˆ r m=e.
Since the binding energy of a fcc solid, U 8e, and
the mean particle spacing, l ˆ n 1=3 , are proportional to the L±J parameters, they can replace r
and e in the scaling of our results. Earlier we
showed that the yield versus energy density deposited in the track also scales with U and l for
more realistic potentials [13] and discussed the
di€erences for amorphous materials.
In order to calculate the energy transport and
sputtering from a cylindrically excited region of
radius rcyl , each non-radiative de-excitation event
was simulated by giving an atom a kinetic energy
Eexc in a random direction. We call …dE=dx†eff the
total kinetic energy release per unit path length in
the cylinder. This is usually presumed to be proportional to the true stopping power of the material but, of course, does not have to be [14].
Therefore, in our fcc solid …dE=dx†eff is equal to
Nexc Eexc =d, where d is the interlayer spacing, Nexc is
the average number of excited particles per layer in
the track of radius rcyl . For a large radius or an
amorphous material this becomes …dE=dx†eff 2
nprcyl
Eexc with Eexc the average non-radiative energy release per atom in the track.
The dependence of the yield on the angle of the
ion beam respect to the surface is studied here. We
use H as the incidence angle measured respect to
the surface normal. The beam is assumed here to
be in the xz plane coming from x > 0 towards
x < 0 and the intersection of the track with the
surface is centered at x ˆ 0 with the surface at
E.M. Bringa, R.E. Johnson / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 99±104
z ˆ 0. A yield Yi is calculated for each simulation.
These are then averaged to obtain the average
yield Y and the `error' bars in Y represent the
standard deviation.
We studied the yield from a [0 0 1] surface.
Yields from other crystallographic orientations
have been shown to have roughly the same dependence on …dE=dx†eff [12]. Of course, the focused collision component of the ejecta, discussed
earlier [15] will di€er for di€erent crystallographic surfaces. The beam was directed along
the h1 0 0i direction in the xz plane. Some simulations were run rotating the sample, so the beam
would not be directed along any low index
crystallographic direction, and the yield did not
change within the standard error. Simulations
were carried out for rcyl ˆ 2r and rcyl ˆ 5r. The
yield at normal incidence and large …dE=dx†eff
was shown to scale roughly linearly with rcyl
[12,14], and we expect this to remain valid for
angular incidence.
3. Incident angle dependence of the sputtering yield
In order to explain the experimental results for
the sputtering of solid oxygen at high deposited
energy density, a spike model with a constant radius rcyl 2r was used in [7]. For this spike radius,
the average MD yield, Y, as a function of scaled
stopping power, x ˆ …dE=dx†eff (n 1=3 =U ) is shown
in Fig. 1 for normal incidence (H ˆ 0), from [12],
and for H ˆ 60. The yields are seen to exhibit the
same overall dependence on …dE=dx†eff . There is a
threshold regime at the lowest values of x and
there is a roughly linear regime at the larger values
x, both discussed extensively before [12,13]. A ®t at
large x of the form Y / xn gives n ˆ 1:1 0:2 for
H ˆ 0 (from [12]) and n ˆ 1:3 0:3 for H ˆ 60,
indicating the dependence of the yield on
…dE=dx†eff does not change signi®cantly with incident angle. In the following, two cases are considered in detail, one in the `threshold' regime,
Eexc ˆ 0:8U …x 10†, and another in the large x
regime, Eexc ˆ 4U …x 50†. The Eexc < U case is
equivalent to the spike criterium for low excitation
2
density, ‰…dE=dx†eff =…prcyl
nU † < 1Š, with the opposite being true for Eexc > U .
101
Fig. 1. Sputtering yield as a function of scaled …dE=dx†eff ,
x ˆ …dE=dx†eff …n 1=3 =U †, for normal incidence (H ˆ 0o , up triangles) from [12] and for H ˆ 60o (squares). A line with slope
x1:3 is shown as a guide to the eye.
The yield, for rcyl ˆ 2r, as a function of the incident beam angle H is shown in Fig. 2, normalized
to the yield at H ˆ 0. In the `threshold' regime the
yield varies steeply with H, approximately as
…cos H† 1:7 . On the other hand, at large …dE=dx†eff ,
where the yield is nearly linear with x, it is seen to
vary approximately as …cos H† 1 . Earlier we showed
that the threshold regime could be roughly described using spike models but the large …dE=dx†eff
1
regime could not. On the other hand, the …cos H†
dependence found here for large …dE=dx†eff and the
nearly linear dependence shown on track radius [14]
indicates that the yield is determined by the amount
of material in the surface region attaining a critical
energy density prior to the radial cooling of the
spike. In this sense it resembles an ablation process
1:7
[16,17]. The reason for the steep …cos H†
dependence at Eexc ˆ 0:8U is the large sensitivity of the
surface ¯ux to the surface temperature when the
thermal energy is low respect to the sublimation
energy, as discussed in [12,18]. The dependence for
the Eexc ˆ 4U case is then less steep because at high
energy the above sensitivity is lost, leading to a
…cos H† 1 dependence that is related to the larger
surface of the track.
At normal incidence we showed earlier [14] that
the calculated yield for a molecular solid has
similar scaling laws to that for an atomic solid. We
also expect this to be true when the incident angle
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E.M. Bringa, R.E. Johnson / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 99±104
agreement [9]. However, they both exhibit a steeper dependence on incident angle than the results
from our MD simulations, although the analytic
model (Eq. (1)) does approach an inverse cosine
dependence at very large incident angles. This
disagreement with the MD simulations is consistent with our earlier work. That is, spike models
do not apply at large …dE=dx†eff and the results
from the MD simulations di€er from the measured
results for condensed gas solids for the typical
assumptions made about …dE=dx†eff and rcyl which
we will discuss below.
We note that the standard linear collision cascade model (LCC) for describing knock-on sputtering [24] also predicts a dependence on incident
Fig. 2. Y …H†=Y …0† for rcyl ˆ 2r, Eexc ˆ 0:8U (open circles) and
Eexc ˆ 4U (solid squares). The lines are a guide to the eye:
…cos H† 1:1 (solid), …cos H† 5=3 (dashed). Experimental points
for 2 MeV Heq‡ (charge equilibrated He‡ ) bombardment of
solid O2 from [11] are also included (solid triangles), as well as
the analytical ®t from the spike superposition model [23] (dotted line).
is varied. Therefore, the data for electronic sputtering of solid O2 bombarded with charge equilibrated 2 MeV Heq‡ [9] is also displayed.
Assuming rcyl is equal to the Bohr radius this data
is relevant to the large …dE=dx†eff regime. In comparing this data to models care should be taken
above H ˆ 60 as neither spike models nor our MD
model allow for backscattering or surface roughness. In addition, the analytic expression
Y ‰HŠ=Y ‰0Š ˆ ‰…4=p†…1= cos H† tan 1 …1= cos H†Š
…1†
for the incident angular dependence predicted for
high …dE=dx†eff by a spike model [23] is also
shown. The results for electronic sputtering of
solid O2 and the analytic expression from the spike
model were shown earlier to exhibit reasonable
Fig. 3. Y …H†=Y …0† for rcyl ˆ 5r, Eexc ˆ 0:8U (open circles),
Eexc ˆ 2U (open diamonds) and Eexc ˆ 4U (solid squares). Experimental points for 2 MeV Heq‡ (charge equilibrated He‡ )
bombardment of solid O2 from [11] are also included (solid
triangles). Functions of the form 1/cosn H with n ˆ 1:2, 1.6, 2
are included as a guide to the eye. Error bars are only shown for
Eexc ˆ 0:8U for clarity; they are 10% for the rest of the MD
data.
E.M. Bringa, R.E. Johnson / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 99±104
103
angle like that in the analytic spike model above [23].
This is due to the variation of the distance from the
incident particle track to the surface and the transport of energy to the surface in both cases. For small
incident angles (and Mtarget =Mprojectile 6 3 for LCC)
5=3
this dependence is approximated as …cos H†
in
both models which is also shown in Fig. 3. This
dependence is seen to agree, fortuitously, with our
MD results for the threshold regime even though the
sputtering process is non-linear and the dependence
at small angles is extrapolated to large angles. In
fact, in both models, as seen for the analytic spike
result in Fig. 3, the dependence on incident angle is
less steep at large angles. Therefore, care should be
taken in using the small angle result, cos 5=3 H.
Claussen [18] obtained an approximate result for a
spike having an initial width. He found that, as the
energy density decreases, the dependence of the
yield on incident angle rapidly becomes steeper than
5=3
…cos H†
. For the parameters used in the MD
simulations of the threshold regime Claussen pren
dicts …cos H† with n 3:75 for rcyl ˆ 2r, a dependence not found here.
Recently, we suggested [14] that the high energy
density deposited within the Bohr radius rapidly
expands until the mean energy per particle is close
to the binding energy of solid O2 . If correct, a
radius of about rcyl ˆ 5r should be used to compare with the experimental data in Fig. 2. This is
shown in Fig. 3, for several excitation energies.
The excitation energy in the atomic solid giving the
same …dE=dx†eff would be Eexc ˆ 2U [14] for
rcyl ˆ 5r. We see that the angular dependence in
both the linear and threshold regime can be
roughly approximated by a cos 1:6 H dependence,
in reasonable agreement with experiments on O2
and CO [9,10]. The steeper dependence occurs
because at large track radii the spike is sustained
longer so emission from deeper layers contributes
signi®cantly to the yield, as discussed in [21]. In
addition, considerable ejection of low energy particles occurs at all …dE=dx†eff for wide tracks.
ated by a fast ion in a solid. We calculated the
dependence of the sputtering yield on the angle of
incidence of the ions. The sputtering yield as a
function of the e€ective energy deposition,
…dE=dx†eff , was shown to have roughly the same
behavior with …dE=dx†eff for both normal incidence (H ˆ 0) and for H ˆ 60.
At ®xed energy deposition, the variation of the
yield with incident angle H is 1= cosn H with
n 1:6 for large spike radius (rcyl ˆ 5r). When
the spike radius is smaller (rcyl ˆ 2r), n 1 for
large …dE=dx†eff and n 1:7 for small …dE=dx†eff .
When describing electronic sputtering the assumption of a spike radius equal to the Bohr
radius does not appear to be consistent with the
experimental data. However, if the energy expands rapidly as we suggested [14], the spike radius increases giving an angular dependence that
is consistent with the experimental results for
electronic sputtering of a number of condensed
gas solids.
Even though one would like to use simple
analytic models to describe the sputtering yield
from cylindrical tracks those available appear to
fail. At low …dE=dx†eff , small variations in the
surface temperature distribution caused by the
change in beam angle and correlated emission
a€ect the yield. The discrepancies between our
results and spike models are principally due to
the di€erences in the surface temperature pro®les
given by the radial di€usion equation from those
obtained in the MD simulation [11]. Therefore,
although we showed that spike models can describe aspects of the sputtering in at low
…dE=dx†eff , they, apparently, cannot describe the
dependence of the yield on incident angle in either the high …dE=dx†eff or low …dE=dx†eff regimes. As a consequence, the rough agreement of
the analytic spike model with the data for electronic sputtering of solid O2 [9] is probably fortuitous.
4. Summary and conclusions
Acknowledgements
In this paper we studied the ejection of atoms
from a cylindrical energized region, like that cre-
The work was supported by the NSF Divisions
of Astronomy and Chemistry.
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E.M. Bringa, R.E. Johnson / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 99±104
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