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Effect of Molecular Dissociation Energy on the Sputtering of Molecular Targets† Christian Anders and Herbert M. Urbassek* Fachbereich Physik und Forschungszentrum OPTIMAS, UniVersita¨t Kaiserslautern, Erwin-Schro¨dinger-Strasse, D-67663 Kaiserslautern, Germany ReceiVed: June 30, 2009; ReVised Manuscript ReceiVed: October 14, 2009
Available theories of sputtering predict the sputter yield to be mainly dependent on the projectile energy and the target cohesive energy. We address the question in how far the dissociation energy of molecular targets influences the sputter yield. To this end we perform molecular-dynamics simulations of sputtering of a diatomic molecular target. We focus on cluster projectiles in which the spike contribution to sputtering is dominant. While our target is modeled as an O2 solid, we modify its dissociation energy, D, artificially from 0.1 (of the order of the intermolecular cohesive energy) up to 20 eV. We find that sputter yields are only little (by (20%) affected by D. However, a reduced value of D makes the target softer; hence, projectile incorporation and penetration depth of the cluster are increased, as are the crater sizes and the time during which sputtering is active. While the number of dissociated molecules increases strongly with reduced D, the total amount of energy dispensed in dissociation remains roughly constant. Finally, we find that the energy partitioning between the internal and translational degrees of freedom is little affected by D. Only for the highest values of D g 5 eV do we find vibrational excitation to have diminished. In general we find that sputter yields are high when the rovibrational excitation of sputtered molecules is small. I. Motivation Sputtering of molecular targets is reduced compared to atomic targets due to draining of energy into internal molecular degrees of freedom.1 The projectile impact energy is redistributed into rotational and vibrational excitation of the target molecules and into their center-of-mass (translational) energy. However, for stiff intramolecular bondssusually equivalent to a high dissociation energysvibrational excitation becomes difficult, since the corresponding high vibrational frequency requires a high collision velocity for excitation.2 Hence, molecules with small dissociation energy may become more strongly vibrationally excited and the energy available for sputtering is reduced. Additionally the breaking of the intramolecular bond consumes energy which would otherwise be available for translational motion. The translational energy enables the molecules to escape from the target and hence to contribute to the sputter yield. These effects will tend to reduce the sputter yield for small values of the dissociation energy. On the other side, the fraction of the projectile energy, which has been redistributed in internal degrees of freedom or which has been used for bond breaking, is still stored in the molecules and may at later times become again converted to translational energy and contribute to sputtering. The interplay of the two effects mentioned is not clear, and the overall effect on sputtering is an interesting question. The present paper performs a systematic study to assess the effects of the molecular dissociation energy on sputtering. II. Simulation Method By means of molecular dynamics simulation, we study the impact of a cluster into a molecular target. We take clusters as projectiles to create an extended hot region (spike).3 Both target * To whom correspondence should be addressed. E-mail: urbassek@ rhrk.uni-kl.de. URL: http://www.physik.uni-kl.de/urbassek/. † Part of the “Barbara J. Garrison Festschrift”.
and projectile consist of diatomic molecules of the same species. The molecular dissociation energy, D, is varied around the dissociation energy of oxygen, D0 ) 5 eV, by a factor of between 0.02 and 4. In each specific simulation setup, all moleculessin the projectile and in the targetshave the same dissociation energy D. The target contains 67 412 molecules. This number has been chosen such that the energized zone lies well within the target. Only for the softest targets with D/D0 ) 1:20 and 1:50 does the energized zone comes close to the boundaries of the simulation box. For these two binding energies the simulations were repeated with a bigger target containing 114 601 molecules. Then sputter yields are increased by roughly 10% for the larger target. Since we consider this a small effect, the harder targets (D/D0 > 1:20) were simulated in the smaller simulation volume only. The projectile consists of 100 molecules. Both target and cluster were amorphized and relaxed before the simulation was started. The roughly spherical cluster hits the surface at normal direction at a fixed total energy of E ) 1600 eV; we know from our previous work that this energy is sufficient to produce considerable sputtering and a crater which does not exceed the target size.1 We damp the lateral and lower borders of the target4 in order to mimic the energy transport out of the simulation volume; the upper surface exposed to the cluster impact feels no additional boundary conditions. Our simulation employs the Verlet algorithm in velocity form for time integration5 and runs until the crater sizes and sputter yields reach a constant value after around 45-55 ps. The sputter yield is evaluated dynamically as a function of time by counting all those atoms that escaped farther than the potential cutoff radius rc ) 8.32 Å from the remaining target;6 in this way we take the dynamically evolving topography of the surface into account. Since target and projectile are made of the same species all (projectile and target) atoms meeting the distance criterion are counted to contribute to the sputter yield. Note that our sputter yield gives the number of sputtered
10.1021/jp9061384 2010 American Chemical Society Published on Web 11/18/2009
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atoms rather than molecules. A modified version of this cluster detector with a smaller detection radius (1.7 Å) serves to identify atoms belonging to the same molecule. This value is chosen larger than the equilibrium distance (r0 ) 1.2 Å) of the molecule so that vibrating molecules are still identified as intact molecules. We calculate the crater volume as follows.7 All atom positions of the initial target where no atom is found (within a radius equal to the average intermolecular distance, 3.64 Å, of the solid) are counted as crater atoms. We thus define the crater volume, C, in analogy to the astronomical definition8 as the number of atoms missing below the original surface of the bombarded solid. For each value of D, we only perform a single impact, since we know from our previous simulations1,3,9,10 that the fluctuations in sputtering are small for cluster impact. This has been checked for the projectile energy of 1600 eV in particular in ref 1, where five impact events were simulated. While previous simulations of the sputtering of an oxygen target11-13 were based on pair potentials, we employ here the reactive empirical bond order (REBO) potential developed by Brenner et al.14,15 It is a many-body potential which allows to describe the breaking and recombination of molecular bonds. Its analytical form is given by
Etot )
1 2
∑ {fc(rij)[Vrep(rij) - bijVatt(rij)] + VLJ(rij)}
(1)
i,j
Here, rij is the distance between atoms i and j, fc is a cutoff function which decreases smoothly to zero at the cutoff radius of 3 Å, and bij is the bond-order function, which introduces many-body binding effects into the interaction and has been taken unchanged from ref 14. To allow for intermolecular bonding the Lennard-Jones potential
[( σr ) - ( σr ) ]
VLJ(r) ) 4ε
12
6
(2)
is added; it gives the solid a cohesive energy of U ) 107 meV/ molecule. At small distances, r e 1.75 Å, the Lennard-Jones potential is splined smoothly to zero in order not to interfere with the bond-order potential. Also at large distances, r > 7.32 Å, this potential is smoothly splined to zero, such that it vanishes at rc ) 8.32 Å. The attractive and repulsive parts of the bond-order potential decay exponentially and give the molecular bonding a Morselike form:
Vrep(r) )
Vatt(r) )
D exp[-R√2S(r - r0)] S-1
(3)
[
(4)
SD exp -R S-1
]
2 (r - r0) S
For close collisions r < 0.5 Å, the bond-order potential is splined to the ZBL potential.16 The bond-order potential has originally been designed to describe oxygen with a dissociation energy of D0 ) 5 eV, which is close to that of ‘real’ oxygen (5.165 eV).17 From this potential, we derive a series of modified potentials, describing pseudooxygen species, which retain all characteristics of real oxygen but have a reduced D, which varies from 0.1 to 20 eV (D/D0 ) 0.02...0.4). The lowest value is of the same order of magnitude as the cohesive energy of the molecular solid, U ) 107 meV.
Figure 1. Bond-order part of the dimer potential, eq 5, for various dissociation energies as used in this investigation.
All other potential parameters have been left unchanged in our series of potentials from their original values:14,15 S ) 1.8, r0 ) 1.2 Å, R ) 2.7 /Å. Figure 1 shows the interatomic potential
V(r) ) fc(r)[Vrep(r) - Vatt(r)]
(5)
which governs the bonding in a dimer (bond-order function b ≡ 1), for various values of D used in this paper. III. Results A. Fate of the Projectile. The reflection of projectile particles from the target, Rp, in Figure 2 shows a significant dependence on D. At early times the vibrationally harder material (high D) reflects more atoms of the projectile. This trend is continued, but in a weaker form, until the final state. The penetration depth or range of a cluster is not as uniquely defined as the range of an ion. We measure it here by considering the time evolution of the center of mass of the projectile cluster; its maximum is the penetration depth, Rcm, of the cluster; for further discussion see refs 18 and 19. The high reflection and the low penetration ranges of the projectile shown in Figure 2b in the hard high-D material leads to energy deposition in the vicinity of surface; this will induce high sputter yields, as we already observed in previous work.10 Reducing D for molecular targets has a similar softening effect as the reduction of the cohesive energy in atomic systems.19,10 Softer materials are more deeply penetrated by the projectile; as a consequence, the projectile energy is deposited farther from the surface and the sputter yield is reducedsat least initially. This effect is compensated at later times, when the energy stored in the internal degrees of freedom of molecular targets can be used to refuel sputtering. Figure 2c displays the energy loss dE/dx (stopping force) of the cluster projectile in our targets. We employ two methods to measure it. As a rough estimate, we may divide the cluster energy per atom, Eatom, by the cluster range, Rcm. A more refined procedure measures the time dependence of the center-of-mass velocity of the cluster. Its slope gives the cluster deceleration and is directly connected to the cluster stopping force.20 As deceleration changes with time, we plot its maximum value, which occurs immediately after the cluster has fully penetrated the target surface. Figure 2c shows a clear trend that the harder materials with high D stop the cluster more efficiently. This feature is evident from both cluster stopping measurements presented here, but more pronounced in the simplified scheme, dE/dx ) Eatom/Rcm.
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Figure 2. Fate of the projectile as a function of D for different times after impact: (a) Number of re-emitted atoms Rp of the projectile (original size 200 atoms). (b) Penetration depth of the center of mass of the projectile cluster [fit line: (16.8 ( 0.6)(D/D0)-0.21(0.02 and (11.0 ( 0.3)(D/ D0)-0.88(0.05 Å in the steep regime] and time at which the center of the projectile cluster reaches its maximum depth [fit line: (0.78 ( 0.04)(D/ D0)-0.42(0.03 ps]. (c) Energy loss dE/dx (stopping) evaluated at the time of maximal stopping [fit line: (0.489 ( 0.021)(D/D0)0.10(0.03 eV/Å], and evaluated as the mean energy loss per atom during the full penetration range, Eatom/Rcm [fit line: (0.44 ( 0.02)(D/D0)0.19(0.02 eV/Å].
Figure 3. Energy partitioning of molecular kinetic energy into rotational energy, Erot, vibrational energy, Evib, and translational energy, Etrans, as function of D. Average over all molecules in the simulation volume. (a) 1 ps after impact. (b) Final.
For the highest dissociation energies, stopping dramatically increases as the projectile range becomes very short. B. Excitation of the Target. Figure 3 shows how the total kinetic energy, Ekin, in the sample is partitioned into rotation, vibration, and translation as a function of dissociation energy. In this plot, an average over the entire simulation volume is presented. For diatomic molecules the procedure of partitioning is straightforward: The velocities of the two atoms constituting the molecule can be decomposed to give the center-of-mass (or translational) and the relative velocity; the latter is projected onto the molecular axis and gives the vibrational energy, while the rest is rotation. Figure 3a shows the partitioning at an early
time, 1 ps after impact, where vibration has been excited by the impact especially for the low dissociation energies, while Figure 3b shows the energy partitioning at the end of the simulation. Immediately after impact molecules with low D obtain a high vibrational excitation, while stiffer molecules (high D) are less excited. A power law Evib/Ekin ) a(D/D0)b with a ) 0.040 ( 0.005 and b ) -0.49 ( 0.04 can be fitted to the decreasing trend. Etrans correlates with early time yields, cf. Figure 7 below. At 1 ps the target has been excited but sputtering has not yet set in, while at 5, 10, and 15 ps the correlation coefficients between the yield Y and the translational energy Etrans amount
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Figure 4. (a) Number of dissociations occurring after projectile impact. Both the maximal number of dissociations and the final number of dissociated molecules exhibit a moderate decay for D/D0 j 0.2 and a steeper decay at larger D. The lines are fits and read (41 ( 5)(D/D0)-0.91(0.04 [(28 ( 4)(D/D0)-0.72(0.05] for the maximum (final) number of dissociations in the moderate regime, and (8.3 ( 1.4)(D/D0)-1.9(0.2 [(6.4 ( 0.5)(D/D0)-1.8(0.1] in the steep regime. (b) Final number of exchanged bonds at the end of the simulation. Fit line: (117.8 ( 2.7)(D/D0)-1.037(0.007. For D/D0 > 1, the number of dissociations and exchanges is zero.
to 0.97, 0.91 and 0.84, respectively. The correlation between sputtering and translational energy decreases with time, since the energy, which is first confined to the sputter zone, later is distributed among all molecules in the simulation volume and so there is less correlation with the sputtered molecules. At the end of the simulation, the dissociation energy has no longer an effect on the energy partitioning, see Figure 3b. An exception is made by the highest dissociation energies D/D0 g 1. Here, vibration is only little activated, as the molecular bond is too stiff. Consequently, a high fraction of the molecular energy is in the translational degrees of freedom. The kinetic energy which has gone into the internal degrees of freedom is no longer available for sputtering. However, the energy which has been spent to break bonds also is no longer available for the sputter process. Figure 4 displays the number of dissociated molecules; not astonishingly, it is higher for reduced D. The impact destroys molecules whose atoms can recombine with new partners later on, so we display two measurements: The maximal number of dissociated molecules (this occurs shortly after impact), and the final number of dissociated molecules at the end of the simulation. Both distributions show a ‘knee’ occurring at D/D0 = 0.2, i.e., D = 1 eV; for small D, the number of dissociations roughly decreases like 1/D, while for higher D the slope is steeper, roughly like 1/D2. Note that for our impact energy of 1600 eV, each impacting molecule has an energy of 16 eV; for high D, dissociations become impossible; this argument qualitatively explains the position of the ‘knee’ in Figure 4a. As a consequence of the apparent 1/D decay of the number of dissociated molecules, the amount of energy spent into dissociation is rather independent of D and amounts to 150 eV for the maximum number, and 76 eV for the final number. If all the impact energy were used for dissociations and all radicals reacted, a maximum number of E/D of bond breaks (and exchanges) could be expected in the target. A more detailed calculation,21 including collision-cascade statistics and the fact that in a two-molecule collision only part of the impact energy can be used for bond breaking, decreases this number by a factor of 4. For the oxygen target (D ) 5 eV) and at the impact energy of E ) 1600 eV, this (upper) estimate gives 80 bond breaks. Figure 4a predicts a value of 40 from the extrapolation of the 1/D fit line of the maximum number of bond breaks valid at low values of the dissociation energy. Thus our estimate semiquantitatively explains the 1/D decrease of the bond breaks with dissociation energy.
Note that the number of dissociated molecules is considerably smaller than the number of the actually occurring bond-breaks; this observation is known from previous work.1 Figure 4b displays this total number of bond changes; again, it decreases roughly like 1/D with increasing dissociation energy. The consequence of the recombinations is that potential energy becomes again available as kinetic energy and may thus contribute to the late time evolution of the sputtering, as long as the particles are hot and mobile enough. After the target has cooled down some potential energy is still kept in the fragments (radicals) frozen in the surface, which may have consequences for prolonged irradiation. The energized fragments may react and form ozone in an oxygen target and, hence, lead to chemistry not captured in our model.22-25 We note that for more complex, e.g., organic, molecular targets, radicals formed during irradiation and left at or close to the crater surface may lead to a larger variety of chemical reactions. Also the energy partitioning between translational and internal degrees of freedom may be different in polyatomic molecules where the number of internal degrees of freedom is larger than in our diatomic model system. C. Sputtering. The impact of the cluster creates a hot region (spike) in the target material from which molecules and atomic fragments evaporate and leave a crater in the target material (Figure 5). Figure 6 displays the temporal evolution of the sputter yield, Y, for various values of D. Obviously the samples with small dissociation energy, i.e., softer molecules, show less sputtering at early times but this changes later. Figure 7a analyzes the dependence of Y for different times. At early times (5, 10, and 15 ps) there is only little sputtering for lower D, but the yield rises more or less monotonically with D. The lines connecting the points are to guide the eye; for t ) 10 ps, a power law Y ) a(D/D0)b with a prefactor a ) 1170 ( 33 and an exponent b ) 0.72 ( 0.03 can be fitted to the yield values for D/D0 e 1, and a ) 864 ( 54, b ) 0.45 ( 0.05 for all data points. Until 15 ps the yield keeps rising for all targets, but at this time it has almost saturated for D J 0.5D0. For the latest time, 46 ps, the yield becomes almost independent of the dissociation energy. Upon closer inspection, we observe a slightly convex curve with high yields for low and high dissociation energy and a slight reduction for the values in between. Interestingly, the sputter yield for an extremely soft target, where the dissociation energy has a similar size as the cohesive energy, is comparable to the sputter yield for a hard target, where
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Figure 5. Sputtering and temperature distribution (blue, < 9 K; red, > 90 K) in the target at 5 ps after cluster impact in targets with different molecular dissociation energy. (a) D/D0 ) 1. (b) D/D0 ) 1:20.
rotational degrees of freedom obtain less energy than the vibrational mode. These features remind us that sputtered molecules stem from a strongly nonequilibrium ejection process, which is far from thermodynamic equilibrium. D. Translational Energy Distribution. In Figure 9 we present the translational energy distributions of sputtered monomers for the targets investigated. The distributions have been normalized to unit area. For comparison, a Thompson distribution as applicable for collision-cascade sputtering26
f(x) ) Figure 6. Time evolution of Y vs time for molecular targets with varied D.
vibrations are only poorly excited. We explain this finding by noting that in these cases, the target mainly responds as an ensemble of atoms: the difference between intra- and intermolecular interactions vanishes for small D, while molecules act as hard units for large D. For intermediate values of D, however, molecular influences are largest. A similar trend appears for C, defined as the number of atoms missing below the original surface, cf. Figure 7b. Initially, soft materials with lower D exhibit a smaller crater, since the crater is still filled with the hot molecules, cf. Figure 5b. The hot molecules remaining in the crater zone store energy which is temporally not available for sputtering and crater formation. In the final state there are generally somewhat smaller craters for materials with high values of D. Excluding the point at D/D0 ) 1/50, a power law with a prefactor a ) 5496 ( 39 and exponent b ) -0.15 ( 0.01 can be fitted to the final crater values. Figure 8 plots the energy partitioning of the sputtered free molecules which have left the target. A fraction of 10-40% of the kinetic energy of sputtered molecules is internal energy, while 60-90% is in translational motion. Note that it is for the vibrationally hard molecules (high D/D0 g 1) that 90% of the molecular energy is in translational motion. Etrans shows a similar dependence on D as Y in Figure 7. A statistical analysis reveals that the correlation coefficient between Y and Etrans of the free sputtered molecules amounts to 0.90. This demonstrates that it is the translational energy that determines the sputter yield. As an aside we note that the distribution of the energy into the various degrees of freedom shown in Figure 8 is far from thermal equilibrium. In statistical equilibrium rotation would receive twice as much kinetic energy as vibration. The sputtered molecules have an excess of translational energy; the two
2x (1 + x)3
(6)
has been included, where x ) Etr/U. Note that eq 6 applies for collision-cascade sputtering,26 and not for sputtering by cluster impact where a strong spike develops. Nevertheless, we use the Thompson distribution as a reference against which we discuss our simulation results. We see that the energy distributions differ both at high and at low energies from the Thompson distribution. At the highenergy end, the small amount of detected molecules can be explained by the possibility of molecule dissociation. Molecules which are emitted with high translational energies are on average also highly excited in their rotational and vibrational degrees of freedom and hence prone to dissociation; accordingly, they are missing in the spectrum of intact monomers plotted in Figure 9. For the highest value of D, i.e., the stiffest molecules, there is an astonishing agreement with the Thompson distribution. Due to the normalization of the data to unit area, the ‘missing’ contribution at high energies for the soft molecules is reflected by an excess contribution at low energies. For the smaller values of D, a broad maximum at Etr/U = 0.5 becomes visible. E. Discussion. a. Quantum Effects. This study was performed with classical dynamics while in reality the excitation of, in particular, vibrations is governed by quantum mechanical effects. The vibrational excitation energy is quantized with pω. For the potential used in this simulation, eq 5, the vibrational angular frequency of the dimer is ω ) R2D0/µ. With R ) 2.7 /Å taken from eq 3 and the reduced mass of the oxygen molecule µ ) 8 amu, we obtain ω ) 2.965 × 1014 Hz and a vibrational excitation energy of 0.196 eV. Energy transfers larger than this value can approximately be treated classically, while for smaller energy transfers quantum mechanical effects become more important. Therefore, for many collisions, in particular in the spike phase after the collision cascades, vibrational excitation
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Figure 7. (a) Y and (b) C vs D for different times after impact.
Figure 8. Final energy partitioning of free sputtered molecules into rotation, vibration, and translation as function of D.
due to several collisions, while the quantization procedure should apply to each collision separately. b. Impact Energy Dependence. In this work we focused on the variation of the molecular dissociation energy while the impact energy was kept fixed. The specific energy of 1600 eV was chosen low enough to contain the spike well within the simulation volume and sufficiently high so that vibrational excitation became possible. The effect of changing the impact energy at fixed dissociation energy, D0, was studied in previous work.1 An important result of that work is that with low impact energy it is impossible to excite vibrations. The excitation of molecular vibrations is possible if the molecular collision energy is large compared to the dissociation energy due to the stiffness of the bond and the adiabaticity effect.2 In the present case of 16 eV/molecule impact energy, we found that vibrational excitation of target molecules becomes possible if D e D0, while it is quenched for D > D0. The fact that D0 is the dissociation energy of real oxygen is a coincidence of our choice of 16 eV/molecule impact energy. At higher impact energies, we expect that also stiffer molecules can be vibrationally excited; the threshold impact energy, E, for vibrational excitation can be assumed to scale linearly with D. IV. Conclusions
Figure 9. Normalized translational energy distribution of sputtered molecules. The theoretical line gives a Thompson distribution, eq 6.
will be described by quantum mechanics, and our classical analysis is only an approximation to the real dynamics. Note, however, that also in classical mechanics collisions with a high vibrational frequency (stiff systems) are difficult to excite due to the adiabaticity of slow collisions;2 we hence assume that our analysis is at least qualitatively correct. A rough estimate of the influence of quantization on the energy partitioning may be obtained by performing a simpleminded a posteriori quantization in the spirit of refs 27 and 28; only if the kinetic energy imparted to vibration is larger than the vibrational quantum pω is this energy attributed to vibration, otherwise it is assumed to contribute to translation. We verified that this assignment leads to an energy partitioning which is qualitatively similar to Figure 3. Note that this approach is only approximative, in particular since the data shown in Figure 3 apply to the summed-up excitation of molecules which may be
We varied the dissociation energy of a diatomic molecular model solid in order to study its effect on the sputtering induced by cluster impact. We find thatsin a first approximationsthe dissociation energy has surprisingly little influence on the sputter yield. This is in line with available theories of cluster-induced sputtering (of elemental materials), which predict the sputter yield to depend on the total impact energy and the cohesive energy of the solid. In detail, however, we see the following effects: 1. By the excitation of the internal molecular degrees of freedom, part of the impact energy is no longer available for sputtering. For high dissociation energies, vibrational excitation and molecule dissociation are reduced, resulting in an increased sputter yield. 2. The molecular target becomes very soft for small dissociation energy. As a result, projectiles penetrate deeper and the time scale for crater formation and sputtering is prolonged. 3. Not surprisingly, the number of dissociated molecules increases strongly with decreasing dissociation energy. However, the vast majority of dissociated molecules can form new molecules with new partners; the number of bond changes is up to an order of magnitude larger than the number of finally dissociated molecules. The net
Sputtering of Molecular Targets energy taken out of the system by dissociations remains roughly constant. 4. The energy partitioning between the internal and translational degrees of freedom averaged over all molecules in the irradiated target is roughly independent of the molecular dissociation energy. Only when D is high (assuming its real value of 5 eV) do we observe a reduction in internal energy due to the difficulty of exciting molecular vibration. As a consequence, the molecular translational energies are high and so is the sputter yield. Sputtered molecules show their highest internal excitation for intermediate values of D. 5. The temporal structure of the collision spike in the material, and hence of sputtering and crater formation, are most strongly affected by D. Small values of D make the target material considerably softer; this is seen in the delayed formation of a crater and of sputtering. 6. The softer target material for small D also leads to an increased incorporation of the cluster into the target (less reflection), an increased penetration depth, a longer stopping time, and a decreased target stopping. References and Notes (1) Anders, C.; Urbassek, H. M. Phys. ReV. Lett. 2007, 99, 027602. (2) Levine, R. D. Bernstein, R. B., Molecular reaction dynamics and chemical reactiVity; Oxford University Press: Oxford, 1987. (3) Colla, T. J.; Aderjan, R.; Kissel, R.; Urbassek, H. M. Phys. ReV. B 2000, 62, 8487. (4) Beeler, J. R., Jr., Radiation effects computer experiments; NorthHolland: Amsterdam, 1983. (5) Heermann, D. Computer simulation methods in theoretical physics, 2nd ed.; Springer: Berlin, 1990. (6) Stoddard, S. D. J. Comput. Phys. 1978, 27, 291.
J. Phys. Chem. C, Vol. 114, No. 12, 2010 5505 (7) Aderjan, R.; Urbassek, H. M. Nucl. Instrum. Methods B 2000, 164165, 697. (8) Melosh, H. J. Impact Cratering; Oxford University Press: New York, 1989. (9) Anders, C.; Urbassek, H. M.; Johnson, R. E. Phys. ReV. B 2004, 70, 155404. (10) Anders, C.; Urbassek, H. M. Nucl. Instrum. Methods B 2005, 228, 84. (11) Banerjee, S.; Liu, M.; Johnson, R. E. Surf. Sci. Lett. 1991, 255, L504. (12) Dutkiewicz, L.; Johnson, R. E.; Vertes, A.; Pedrys, R. J. Phys. Chem. A 1999, 103, 2925. (13) Bringa, E. M.; Johnson, R. E. Surf. Sci. 2000, 451, 108. (14) Brenner, D. W.; Robertson, D. H.; Elert, M. L.; White, C. T. Phys. ReV. Lett. 1993, 70, 2174. (15) Brenner, D. W. MRS Bull. 1996, 21, 36. (16) Ziegler, J. F. Biersack, J. P., Littmark, U. The Stopping and Range of Ions in Solids; Pergamon: New York, 1985. (17) Ni, B.; Lee, K.-H.; Sinnott, S. B. J. Phys.: Condens. Matter 2004, 16, 7261. (18) Anders, C.; Urbassek, H. M. Nucl. Instrum. Methods B 2005, 228, 57. (19) Anders, C.; Urbassek, H. M. Nucl. Instrum. Methods B 2008, 266, 44. (20) Anders, C.; Urbassek, H. M. Nucl. Instrum. Methods B 2007, 258, 497. (21) Balaji, V.; David, D. E.; Tian, R.; Michl, J.; Urbassek, H. M. J. Phys. Chem. 1995, 99, 15565. (22) Stulik, D.; Orth, R. G.; Jonkman, H. T.; Michl, J. Int. J. Mass Spectrom. Ion Phys. 1983, 53, 341. (23) David, D. E.; Michl, J. Prog. Solid St. Chem. 1989, 19, 283. (24) Magnera, T. F.; Michl, J. Z. Phys. D 1993, 26, 93. (25) Fama, M.; Bahr, D. A.; Teolis, B. D.; Baragiola, R. A. Nucl. Instrum. Meth. B 2002, 193, 775. (26) Sigmund, P. In Sputtering by particle bombardment I; Behrisch R., Ed.; Springer: Berlin, 1981; p 9. (27) Snowdon, K.; Hentschke, R.; Heiland, W.; Hertel, P. Z. Phys. A 1984, 318, 261. (28) Urbassek, H. M. J. Phys. B 1987, 20, 3105.
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