Journal of Nuclear Materials 419 (2011) 32–38

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Journal of Nuclear Materials

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Statistical Molecular Dynamics study of displacement energies in diamond

Diego Delgado ⇑, Rafael VilaUnidad de Materiales, Laboratorio Nacional de Fusion, CIEMAT, Madrid, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 September 2010Accepted 18 August 2011Available online 31 August 2011

0022-3115/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jnucmat.2011.08.035

⇑ Corresponding author.E-mail address: diego.delgado@ciemat.es (D. Delga

Molecular Dynamics simulations in bulk diamond using AIREBO potential have been used to calculateminimum displacement energy. A statistical approach has been applied calculating displacement proba-bility curves along the main crystallographic directions. With these curves a minimum displacementenergy of around 30 eV can be obtained, and a weighted average energy of around 70 eV. This valuehas been estimated as more accurate for its use in BCA displacement equations to obtain Frenkel pairsat moderate temperatures. This work also includes a study of defect states whose analysis reveals inter-esting results concerning the evolution of primary damage at higher PKA energies.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Diamond has a valid reputation of radiation hard material thatmakes it suitable for extreme environments, including among oth-ers, radiation detectors and dielectric/optic window for magneticconfinement fusion applications. In this field it can be used as coat-ing material and of course as dielectric windows in the diagnosticand heating systems [1,2]. For next generation fusion reactors, effectof neutron radiation must be carefully studied to obtain the lifetimeof all the components. One approach is obviously damage calcula-tion and this work provides important parameters to model the bin-ary collision processes of neutron and carbon ions. The physicaldescription of lattice damage in solids by neutron radiation beginswith the generation of the primary recoil atom or PKA (PrimaryKnock-on Atom). Propagation of PKA kinetic energy through the hostlattice normally results in atom displacements and therefore todamage production. In this way, the threshold displacement energy(hereafter Td) is a key parameter in the quantification of the damagedynamics; it characterizes the material potential damage and it isused as input parameter for the mathematical description of suchphenomenology by means of the Binary Collision Approximation(BCA) or the Collision Cascade Models (CCM) [3]. Its value is deter-mined both experimentally by means of particle (ion, electron orneutron) bombardment and computationally, for instance, viaMolecular Dynamics simulations (MD). Most of the references in thistopic manage a single Td value for each material, however, numericalstudies [4,5] support the existence of displacement probabilitycurves which depend on PKA energy and crystallographic direction,in case it applies.

In a numerical approach, several authors [5,6,4] discuss thisissue and calculate different threshold displacement values foreach crystallographic direction and the existence of an uncertainty

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do).

band between the lower and higher values for the displacementenergies1; or even a displacement energy surface [7].

In the case of synthetic diamond, comparable results for Td val-ues could be found in the literature with different methodologies.In a computational way, Wu and Fahy [8] obtained a MD valueTd = 52 eV in (100) direction; coherent with previous results [9]using different approaches. Kalish et al. [10] got the same in off-axis direction but obtained a different value in specific crystallo-graphic directions (60 eV in (100) and (110), 45 eV in (111)). Bothresults share the Tersoff interatomic potential in their calculations.Wu also studies the adiabatic energy barrier expected for carbonescape with result of 25 eV, this energy difference is explained asirreversible energy loss. Quite different values have been reportedin an experimental way, Clark et al. [11] deduced Td = 80 eV aftersample electron irradiation and, by means of a similar technique,Koike et al. [6] determines Td in type IIa natural diamond withresults of Td = 37.5 eV (100), Td = 45.0 eV (111) and Td = 47.6 eV(110), in this case crystallographic direction differences wereexplained interestingly on basis of elongated recombination vol-umes in close packed directions and the stability of (100) splitinterstitial over the corresponding (110) and (111) ones.

To extend these studies, we carefully calculate threshold dis-placement energy curves for bulk diamond in the four main crys-tallographic directions (100) (110) (111) ð�1 �1 �1Þ by means of MDcalculation and using the AIREBO potential [12]. The probabilitynature of such displacement events leads to two main results: (a)an interesting rich variability in the escape events and (b) thenecessity of coherent statistical data analysis for reliable results.The first part of this article is devoted to present the statisticalframework to support the study. In the second one we presentthe MD results for each crystallographic direction; accuracy of

1 This uncertainty band is defined between the minimum displacement energyneeded to get any atom displacement and the threshold energy to get a suredisplacement.

D. Delgado, R. Vila / Journal of Nuclear Materials 419 (2011) 32–38 33

previous works could be compared with our analysis. Latticedynamics of displacements result in an interesting variety of finallocal configurations and displacements ranges; a systematic classi-fication of configurations and ranges for the energy bands anddirections studied are presented and differences, correlations anddetailed analysis are finally discussed.

2. Method

A review of the available threshold displacement energy studiesin several insulator materials shows the statistic nature of such pro-cesses. This fact leads to different magnitude definitions dependingon the viewpoint or the experimental considerations to simulate[7,5,4,8]. All the authors, however, remark the statistical fluctuationof displacement events although the exact underlying mechanismsare not absolutely clear. In summary and in chronological order, Wuet al. choose to define a sharp threshold displacement energy in thethree main directions (100) (110) (111) without any reported sta-tistics in their simulations. In second place, Malerba and Perlado [5](in SiC) resorts to the definition of an uncertainty band per maincrystalline direction but the statistics reported in the simulationsis poor (three events in the same direction are enough) as we willshow later. Mota [4] calculates a displacement probability curve asfunction of the recoil energy and reports 24 events per energy; asthey work with amorphous silica, crystalline directions are not de-fined but are randomly selected. Finally Nordlund et al. [7] in thiscase with Fe, chooses to calculate an angle resolved displacementenergy surface. They discuss the displacement definitions and definethe lower displacement energy as the valid one for their simulations,it is worth noting the several million simulations performed for theangle resolved surface.

As a first conclusion the statistic nature of such magnitude, asmany others in material science, is clear and the differentapproaches produce results with quite different error bands.Therefore taking into account the uncertainty band introduced byMalerba (interpreted as an escape probability curve by Mota andothers) and the differences with the initial crystallographic PKAdirection mainly, we designed our computational framework withthe aim of getting a statistical coherent magnitude. Each displace-ment event is treated in a statistical way. From N independentevents we get the displacement probability from the quotientpN ¼ displacements

N with uncertainty given by the confidence interval(usually 95%) of our numerical experiment.

The confidence interval is calculated by means of the tree eventdeveloped in Fig. 1. The statistical recount of events defines the dis-crete probability envelope of possible results. Supposing an arbi-trary undiscovered theoretical probability (pth), the probabilitydistribution curve is defined through the values:

mðpthÞN�ið1� pthÞ

i ði ¼ 0;1;2; . . . ;NÞ ð1Þ

With N the number of events, m a multiplicity factor and i the dis-crete position of each possible result. These are the probabilities of anumerical estimation at values given by:

N � iN

ði ¼ 0;1;2; . . . ;NÞ ð2Þ

Such expressions define the probabilistic curve of possiblenumerical results and hence the mentioned confidence interval.The multiplicity factor m weights the paths that reach a deter-mined point inside the discrete tree event. It is well known thatthe m factor follows a Pascal’s triangle sequence with points given

by the binomial coefficient Ni

� �; hence:

m ¼N

i

� �ð3Þ

Again being N the corresponding number of events and i the po-sition of each case. The subsequent tree could reach any arbitraryevent number (see Fig. 1) whose envelope defines the confidentinterval (95%) of getting a numerical result (displacement proba-bility). Fortunately as the number of events simulated gets higher,the confidence interval gets bound and smaller; 25 PKA events getsan uncertainty of about 40% and above one hundred events confi-dence reaches reasonable values. A statistical coherent result ofsuch magnitude requires accumulating a considerable number ofevents with its correspondent computational effort.

We finally choose a value of 150 PKA events (simulations) foreach energy and crystallographic direction that, in the worst possi-ble case of pth = 0.5, has an uncertainty of about 15% for a (95%)confidence. For each energy and direction the displacement proba-bility is found from PN ¼ displacements

N , with N = 150 the number ofevents simulated and ‘displacements’ the successful displacementscounted in each case.

Each displacement event was completely independent fromthe others in the sense that thermal fluctuations and initial timeare random. Numerical simulations were performed in 8 � 8 � 8unit cells (4096 atoms) and using periodic boundary condition.In order to simulate the bulk material a thermostatting scalingwas applied to the boundary atoms to absorb the energy excessand the PKA atoms were selected approximately in the middleof the simulation box. Our work uses the (Adaptive Intermolecu-lar Reactive Bond Order Potential) AIREBO [12] potential insteadthe Tersoff one of previous MD works, both of them included inthe Large-Scale Atomic/Molecular Massively Parallel Simulator(LAMMPS) molecular dynamics package [13]. As an extension ofthe last release of Brenner hydrocarbon potential, AIREBO is aspecialized version of covalent potentials for carbon and hydro-carbons with origin in the Tersoff one; in the case of AIREBO, itincludes a large range Lennard-Jones and Torsional terms fornon-bonded interactions, with little effect in Diamond and ne-glected in this work. The basic mathematical form that definesthe potential could be written as:

Eb ¼X

i

Xj>i

½VRðrijÞ � bijVAðrijÞ� ð4Þ

where VR(rij) and VA(rij) are the repulsive and attractive termsrespectively with rij the interatomic i to j distance and finally bij

the bond order term that account for different bonding configura-tion; it is given by the expression:

bij ¼12

br�pij þ br�p

ji

h iþPRC

ij þ bDHij ð5Þ

The first two terms br�pij ; br�p

ji with specific corrections, areroughly similar to their equivalent in the Tersoff potential, thethird PRC

ij , accounts for the conjugation and radical character inCarbon bonds and the fourth bDH

ij , represents local dihedrical effectsamong atoms. In order to describe short range interactions, typicalof energetic recoils similar to the displacements here simulated,we add the universal ZBL short range potential [14] for the repul-sive region of the interaction and smoothly fitted by means of acubic spline to the AIREBO one, set for intermediate (near to equi-librium distances) and large range interactions; it was performedfollowing a similar methodology from previous works [15].

All the PKA events follow the same numerical scheme. It con-sists in an initial equilibration stage of approximately 2.5 ps or10,000 time steps until a random PKA is selected. The first partof the production stage was run with a time step as small asdt = 0.05 fs for 5000 computational steps (0.5 ps) and last onewith a larger time step (dt = 0.25 fs) and 20,000 integrations(5 ps). This scheme allows both a affordable integration time evenat the higher energy displacements, and a detailed description ofthe initial sequence of the simulation (in order to analyze the ini-

Fig. 1. Tree map for statistical displacement events in a selected case (pth = 0.6). In the horizontal plane the number of independent events versus possible probabilisticmeasurement. Vertical axis weights the occurrences. A wide and discrete distribution is progressively replaced by a smooth and continuous one.

34 D. Delgado, R. Vila / Journal of Nuclear Materials 419 (2011) 32–38

tial dynamics). Due to the large amount of runs and output data,the final configurations were analyzed in an automatic way basedon the final coordinates of PKA atom and its final equilibrium en-ergy; when a displacement was found, the local configuration andPKA range was saved and analyzed. Statistics were thenperformed.

Fig. 2. Displacement probability curves as function of initial PKA energy for the fourcrystallographic directions of this study (100) (110) (111) ð�1 �1 �1Þ.

3. Results and discussion

In the previous section we explained how the AIREBO potentialwas used to obtain displacement energies. As seen in Fig. 1, a largeamount of numerical data is needed to estimate the minimum dis-placement energy (or threshold energy) with enough statisticalcoherence; such accumulation of simulations not only leads to dis-placement probability curves as shown but also includes many dis-placement data whose analysis reveal interesting results as we willshow in this section.

Displacement probability curves as a function of PKA energy,calculated for main crystallographic directions are presented inFig. 2. Lower energy values compare quite well among differentdirections but higher ones presents clear differences. As discussed,the statistic coherence of results was checked. Fig. 3 presents a typ-ical evolution of displacement probability with the number ofevents; as expected, when the number of accumulated cases in-creases, the result stabilizes towards the right value and its oscilla-tions are damped. This graph represents a practical example of theconfidence process described in the previous section and demon-strates that at least 100 events are needed to obtain a reliablevalue.

Although BCA does not strictly applies for the energies computedin this work, distance of PKA to first colliding atom provides an initialclue to the displacement energy needed in each case. In suchapproach, the values obtained for [111] direction are a consequenceof the energetic collision with its first neighbor (at 0.156 nm) thatleads to a much more difficult displacement than for ½�1�1�1� direction(collision with its fourth neighbor at 0.617 nm). Finally, [100] and[110] directions give rise to intermediate probabilities in the whole

PKA energy band and correspond to paths interrupted by its third(0.356 nm) and second (0.256 nm) neighbor respectively.

Presented displacement probability curves describe a wideuncertainty probability band for displacements; this probabilisticinterpretation is based on the statistic nature of such events as dis-cussed by several authors [4]. In spite of this, large scale mathemat-ical models for radiation damage need a single value of displacementenergy used as a parameter. Some examples are models based onBCA approximation, as SRIM [16] or MARLOWE [17], and phenome-nological models used in some kind of estimations as CCM. Thiscrude approximation leads to single displacement energies as foundin most of previous works with values mainly from 37.5 eV to 52 eVand in some cases higher (70–80 eV), the first ones in the lowerenergy segment of the bands here reported. As a matter of fact, inthese cases, minimum or threshold displacement energy is definedwithout any statistical significance, that is just by its ability to

Fig. 3. Statistical convergence process in a selected case. As the number ofsimulated events gets higher the results gets defined and the fluctuations aredamped.

D. Delgado, R. Vila / Journal of Nuclear Materials 419 (2011) 32–38 35

produce displacements but with unclear displacement probability.From our results, we do not found straightforward to get a single va-lue from the wide probability bracket described for displacement;reported values are usually placed in the lower energy segment, cir-cumstance already noted in a previous numerical work [5] in a dif-ferent material (SiC).

Having the probability curves we still need a rule to obtain a sin-gle value. One approach can be to take the lower displacementenergy (where displacements start to be significant) as defined byMalerba [5]. From our results this value is found just above 30 eVand fairly coincident between crystallographic directions; thereforethe minimum energy to really observe displacements could be esti-mated between 35 and 40 eV, which agrees quite well with experi-mental results of Koike. It is even well noting that in the lower limit,the [111] direction is not the most difficult one, but on the contraryit requires somewhat less energy for displacement production.

Nevertheless this lower energy value neglects the wholedisplacement bracket obtained. At this minimum Td range, displace-ments are statistically possible but with a very low probability; theydo not grow significantly until PKA energy values are higher andleads to strong direction differences. Hence, the lower displacementenergy clearly undervalues the magnitude under study and we needa statistical value that really summarizes the several directions andprobability curves obtained in diamond lattice. Therefore we willapply a rule to obtain a single value. For this purpose, in Table 1we present arithmetical and weighted averages of displacementprobability curves (obtained from Fig. 2). The number of equivalentcrystallographic directions weights the energies (min, max and mid-dle) in each direction to obtain a single weighted energy. Arithmet-ical averages are also presented. The weighted average could berounded to 70 eV. This effective displacement value takes intoaccount all directions and weights, with probability values around

Table 1Averages for numerical displacement energies estimated (eV).

Recoil direction Multiplicity Lower Middle Upper

(100) 6 30 70 90(110) 4 30 65 90(111) 12 30 95 150ð�1 �1 �1Þ 4 30.0 55 85Arithmetical average 30.0 78.46 116.92Weighted average 30.0 71.25 103.75

50%. As MD simulations have been performed at room temperature,it also includes temperature effects (higher temperatures enhancesion recombination and decrease the Frenkel pairs number; this ef-fect is sometimes simulated in BCA model by just increasing the dis-placement energy or even by means of a recombination volume). Allthese reasons suggest that the weighted average value is the mostaccurate one. This value is higher than previously proposed values(70 eV instead of around 40–50 eV from most previous authors),but at the same time it should correspond to a more realistic energymainly because it is supported by a coherent analysis with the statis-tic nature of the phenomenon, aspect neglected in previous works.The average value obtained could be introduced as a parameter inother models, for instance the CCM; this kind of simple expressionsare deduced on the basis of only few representative parameters ofthe system; therefore these parameters must be carefully chosento incorporate the main physical processes; for instance, Td has anessential influence on the number of Frenkel Pairs that are produced,in the simple case this magnitude is given by [18]:

mNRT ¼ 0:8EK=2Td ð6Þ

where EK is the damage energy (equal to PKA energy if electronic en-ergy loss is not included). Previous works, not only the numerical butalso the experimental ones, resorts to different displacement defini-tions from the present work with result in lower values; it would beinteresting to test all these insights by comparison with collision cas-cades studies that give some clue about what criteria remains valid,however virtually no references exits in this field in Diamond.

The information from Fig. 2 does not resolve the physical mech-anism involved in the statistics. Displacement events described inthis work are not accurately described by simple binary collisionmodels (extensively employed at high energy regimes, for instancecollision cascades), as many body effects are quite relevant in thisdynamics. Therefore, the displacement path mentioned abovemust be supported not only by pure binary collisions but also bygeometric and many body description of displacements; the ener-gies involved in the displacement usually are not high enough todisplace more than one atom, the energy interchange in collisionsare quickly damped by intermediate neighbors and, depending ondirection, two and even more collisions are virtually simultaneous.Wu and Fahy [8], for instance, described this process as reversibleand irreversible energy interchange between lattice and energeticatoms. Additional atomistic studies in this type are under develop-ment. In this work this complexity is addressed through the studyof displacement ranges and final damage states using MD numer-ical simulations with emphasis on their possible correlation. Toillustrate this, energies versus displacement ranges are presentedin Fig. 4. Displacement ranges as a function of initial PKA energyare plotted for the four directions studied. It can be shown thatthe lower initial PKA displacements correspond to shorter rangesand growing initial PKA energies results in larger displacements,as could be expected, but interesting differences between crystal-lographic directions could also be observed. Five bins classify theranges in the graph. First and second bins are set for short rangedisplacements (<0.08 nm and <0.15 nm); Third and fourth classifymiddle range displacements (<0.25 nm and <0.35 nm). Finally, thefifth is for large range displacements (>0.35 nm). Note thatdistances agree with neighbor levels in diamond. The discussionis obvious, as initial energy increase the energetic atom not onlyis able to pass the adiabatic barrier to displace [8] but sometimesreach larger ranges and this occurs more often as higher energiesare involved. These larger ranges generate more complex damageand higher formation energy defects. What is important to men-tion is that transition energies from short ranges to middle andlarge ones show important differences between directions. Mostof the displacements at lower energies in (100) direction areapproximately shared between the two short ranges at around

Fig. 4. Displacement ranges (nm) as function of initial PKA energy in the directions studied (100) (110) (111) ð�1 �1 �1Þ. The magnitude range represents the distance fromlattice position where energetic recoil displaces.

Fig. 5. Displacement energies (eV) versus final local states in the directions studied (100) (110) (111) ð�1 �1 �1Þ. Damage states, presented as final states in the calculations aredescribed in table (2).

36 D. Delgado, R. Vila / Journal of Nuclear Materials 419 (2011) 32–38

Fig. 6. Scheme of representative point defects generated in the simulations following the order described in Table 2. Displaced atoms are presented in orange, empty latticesites in white and in some cases affected Carbons and empty sites are labeled. Finally, arrows illustrate the main displacements.

Table 2Point defects and damage states observed in displacements simulations.

Num Site Description

1 Site Broken bond or local distorsion in thevicinity of PKA site

2 Replacement PKA atom is found in replacement position3 Interstitial Undefined interstitial configuration for

PKA atom4 BC Interstitial Bond Centered interstitial configuration.5 Dumbbell (100) (100) split interstitial configuration6 Dumbbell (110) (110) split interstitial configuration7 Interstitial C Complex intertitial configuration.

Multiple sp2 bonds found

D. Delgado, R. Vila / Journal of Nuclear Materials 419 (2011) 32–38 37

60 eV energy, middle ranges become important and finally fewlarge displacements appear. The (110) direction presents some dif-ferences, most of shorter displacements are concentrated in thefirst bin (<0.08 nm) with a transition approximately at 50 eV(10 eV before); large displacements are, in this case, significanttoo. Displacement probability curves are more regular in bothcases. Shorter ranges in (111) are dominant in the energy bracketstudied; transitions to middle and large ones are now observednear 100 eV and are still incomplete even at 150 eV. Finally,according to probability results ð�1�1�1Þ translate from short to mid-dle ranges at about 45 eV and medium ones dominate the end ofthe energy bracket presented (85 eV). General features from dis-placement probability curves are reproduced and better under-stood by this presentation of ranges and differences betweendirections are still present under this point of view. This bar graphshows that different displacement dynamics takes place as PKAenergy and direction changes.

In order to get a better understanding of our calculations, finaldamage states versus initial PKA energy are presented in Fig. 5,for each crystallographic direction. Seven different configurationsobtained in MD simulations are described and a scheme is shownin Fig. 6; among them we obtain the main self-interstitial sites pre-viously studied in Diamond [19] and some unidentified configura-tions classified on the basis of their common characteristics:

1. Site: Damage configuration around PKA lattice point basedonly in a local distortion consequence of the energetic event.

2. Replacement: accounts for all the replacements eventsgenerated.

3. Interstitial simple interstitial configurations not included inthe following cases.

4. Interstitial BC: describes the bond centered interstitial con-figuration got by the recoil.

5. Dumbbell (100): is used for split interstitial structures in the(100) and equivalent directions.

6. Dumbbell (110): is noted for dumbbell (110) structure andequivalent ones.

7. Finally Interstitial C configuration is used for more complexunidentified interstitial structures with the presence oftwo and three sp2 bonds around an interstitial atom in ano-dumbbell compatible configuration and probably asearly stages of extended defects.

Damage states here described are summarized in Table 2 andnoted in the graphs in the same order for clarity. In our opinionthese states does not necessarily represents final stable configura-tions but initial damage states or defect precursors as result of theenergetic event registered. Although a systematic study of energiesor local configurations is out of the scope of this letter some inter-esting correlations deserve to be discussed, Fig. 5 shows clearlyhow defect complexity increases with the energy; an example isthat local distortion (type 1 defect) is dominant at lower energiesand type 7 becomes only important at higher ones. In particular,the variety of damage sites in (100) direction, especially at lowerenergies, is absent in the other three directions studied; specifi-cally in (110), with similar displacement probability curve; inthese three directions final sites appear more as less specific siteslike replacements and unidentified simple interstitials instead ofthe known interstitial BC and Dumbbells; finally complex intersti-tials (site 7) always appear at higher energies and become statisti-cally important for all directions. These results give us some cluesabout how the initial recoil energy is damped into the diamond lat-tice. These precursor sites also correlate with the displacementranges discussed before. Fig. 7 presents this information in a con-densed way. Shorter ranges correlate with simple states (simpleinterstitials, local distortion and replacements) and larger oneswith complex configurations. In the graph, shorter ranges corre-spond mainly with local sites as mentioned, middle ranges are ingeneral populated with variety of interstitial configurations witha soft transition to more complex sites found at larger ranges.Crystallographic differences seems to be averaged in this graph,indicating a quite general behavior.

Fig. 7. Displacement ranges (nm) versus final local states in the directions studied (100) (110) (111) ð�1 �1 �1Þ. See correspondent Table (2) for details about the damage statespresented.

38 D. Delgado, R. Vila / Journal of Nuclear Materials 419 (2011) 32–38

4. Conclusions

A large collection of MD simulations have been performed andanalyzed to obtain displacement probability curves along maindirections in diamond lattice. From these curves a minimum dis-placement energy between 30 eV and 35 eV has been obtained,although a value around 70 eV has been estimated as more accu-rate for its use in displacement equations. This estimation takesinto account the probability that the displaced ion recombine rap-idly and therefore should lead to more correct values of Frenkelpairs considering moderate temperatures. The analysis of a largeamount of defect types has revealed interesting results concern-ing the evolution of primary damage at higher PKA energies,where displacement ranges increases and defect structures evolveto higher complexity. This study has been done with a differentinteratomic potential (ZBL-AIREBO), it would be interesting tocompute a similar statistical study by means of other carbon po-tential, Tersoff for instance, to test the possible effect of potentialon Td.

Acknowledgements

This work has been performed in the framework of the CIEMATproject for Nuclear Fusion Research supported by EU and SpanishMinistry of Science and Innovation under project ENE2008–06403-C06–01/FTN.

References

[1] A. Ibarra, M. Gonzlez, R. Vila, J. Moll, Diamond and Related Materials 6 (1997)856–859.

[2] M. Thumm, Diamond and Related Materials 10 (2001) 1692–1699.[3] M. Robinson, Journal of Nuclear Materials 216 (1994) 1–28.[4] F. Mota, Journal of Nuclear Materials 329 (2004) 1190–1193.[5] L. Malerba, J.M. Perlado, Physical Review B 65 (2002) 045202.[6] J. Koike, D.M. Parkin, T.E. Mitchell, Applied Physics Letters 60 (1992)

1450–1452.[7] K. Nordlund, J. Wallenius, L. Malerba, Nuclear Instruments and Methods in

Physics Research B 246 (2006) 322–332.[8] W. Wu, S. Fahy, Physical Review B 49 (1994) 3030–3035.[9] J.F. Prins, T.E. Derry, J.P.F. Sellschop, Physical Review B 34 (1986) 8870–8874.

[10] R. Kalish, A. Reznik, S. Prawer, D. Saada, J. Adler, Physica Status Solidi (a) 174(1999) 83–99.

[11] C.D. Clark, P.J. Kemmey, E.W.J. Mitchell, Discussions of the Faraday Society 31(1961) 96–106.

[12] S.J. Stuart, A.B. Tutein, J.A. Harrison, The Journal of Chemical Physics 112(2000) 6472–6486.

[13] S. Plimpton, Journal of Computational Physics 117 (1995) 1–19.[14] J. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids,

Pergamon, New York, 1985.[15] D. Schwen, E. Bringa, Nuclear instruments and methods in physics research

section b: beam interactions with materials and atoms, in: Atomic Collisions inSolids – Proceedings of the 22nd International Conference on Atomic Collisionsin Solids, 22nd International Conference on Atomic Collisions in Solids, vol.256, 2007, pp. 187–192.

[16] J. Ziegler, J. Biersack, M. Ziegler, SRIM: The Stopping and Range of Ions inMatter, Mc Graw-Hill, 2008.

[17] M.T. Robinson, Physical Review B 40 (1989) 10717–10726.[18] M. Norgett, M. Robinson, I. Torrens, Nuclear Engineering and Design 33 (1975)

50–54.[19] S.J. Breuer, P.R. Briddon, Physical Review B 51 (1995) 6984–6994.