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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Cell Molecular Dynamics for Cascades (CMDC): A new tool for cascade simulation Jean-Paul Crocombette ⇑, Thomas Jourdan CEA, DEN, Service de Recherches de Métallurgie Physique, F-91191 Gif-sur-Yvette, France

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Article history: Received 17 June 2014 Received in revised form 13 November 2014 Accepted 3 December 2014 Available online 20 December 2014 Keywords: Molecular dynamics Cascade modeling Acceleration of calculations

a b s t r a c t We present a new Molecular Dynamics (MD) scheme for the simulation of cascades: Cell Molecular Dynamics for Cascades (CMDC). It is based on the decomposition of the material in nanometric cells which are added and removed on the ﬂy from the MD simulation and the dynamics of which are treated with a local time step. An acceleration of several orders of magnitude is observed compared to standard calculation. The capacity of the method is demonstrated on the test cases of 60 keV He implantation and self-cascades in iron up to 1.8 MeV. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The evolution of materials under irradiation depends on the exact nature of the damage directly created by the atomic displacement cascades initiated by the fast moving neutrons or ions. Accurate cascade modeling is therefore of paramount importance in radiation effect simulations. It can divided in two types of calculations: Binary Collision Approximation (BCA) and Molecular Dynamics (MD) simulations [1]. BCA is a method designed especially for cascade simulations. It proves very fast but lacks the collective motion of atoms. It therefore basically describes only the ballistic phase. To obtain an accurate description of the primary state of damage, one therefore uses MD simulations. The procedure is then to consider a large (periodically repeated) box and to accelerate one atom inside the box. The method is thus rather straightforward. Except for a speciﬁc short range potential, a variable timestep and some temperature control at the border of the box, it is a standard MD calculation. It proves to describe at the best of the empirical potential the ballistic and thermal phases with all the collective atomic movements and the quenching of the thermal spike. However it faces some difﬁculties mainly related to the size of the simulation box. If it proves too small to contain the whole cascade, one observes an overlap of the cascade with itself. Moreover because of the temperature control on its border, any cascade which comes close to the borders is spoiled and must therefore be put to garbage after calculation. To avoid such situation one may

⇑ Corresponding author. Tel.: +33 169089285; fax: +33 169086867. E-mail address: [email protected] (J.-P. Crocombette). http://dx.doi.org/10.1016/j.nimb.2014.12.009 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.

increase the size of the box, but the computer power needed to calculate high energy events becomes prohibitive. All in all this strongly limits the energy of calculated cascades and the number of cascades that are performed in a given study, thus giving poor statistics on the nature and amount of damage. To circumvent this problem we propose a modiﬁed MD simulation of cascades: Cell Molecular Dynamics for Cascades (CMDC) which uses a MD algorithm speciﬁcally designed to model a displacement cascade. The goal of the method is to accelerate as much as possible the calculation of cascades by MD without loss of accuracy of the results, considering the speciﬁcities of the cascade unfolding. It is based on the observation that many parts of the usual MD boxes do not take part in the cascade and are just present in case the cascade would go there. The core principle of CMDC is to perform a regular MD simulation but just where and when necessary to properly describe the cascade unfolding. The principles of CMDC are detailed in the ﬁrst section. Demonstrating examples of the ability of the method on the test cases of iron PKA and 60 keV He implantation in iron are then presented. The last part is devoted to discussions of the method.

2. CMDC principles A crystal, having a periodically repeated structure can easily be built by the addition of cells containing atoms. In CMDC such cells are added or removed on the ﬂy from the MD simulation. Moreover each cell evolves according to its own time-step. The basic cell is chosen as the smallest supercell of the crystalline unit cell whose dimension are larger than the cut-off of the empirical potential

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J.-P. Crocombette, T. Jourdan / Nuclear Instruments and Methods in Physics Research B 352 (2015) 9–13

of interactions. With such a choice for the basic cell, an atom inside a cell interacts through the potential only with atoms in its own cell and the 26 ﬁrst neighboring cells. For instance in iron, the conventional bcc unit cell parameter is 2.85 Å. The empirical potential we use [2] has a cutoff of 5.3 Å. The basic cell is therefore cubic with a cell parameter of 5.7 Å. It contains 16 atoms. The ﬁrst point of the CMDC code is to build the MD box during the unfolding of the cascade. A cascade is an event during which some atoms are accelerated high above their usual thermal agitation. The dynamics of atoms inside a cell will be disturbed by the cascade when they start to interact with fast moving atoms. In cell language that is translated as: when a fast moving atom enters a central cell, all the neighboring cells that were still inactive are turned on and become active. The actual turning on of the neighboring cell is driven by thresholds on the maximal kinetic energy of the atoms in the central cell and on its temperature (i.e. the average kinetic energy of the atoms of the central cell). When either of these thresholds is superseded, the neighboring cells are turned on. At that time the atoms in the turned-on cells are given some initial velocity and random displacement corresponding to the equilibrium temperature assigned to the material. The MD algorithm (in practice velocity Verlet) is then applied in the active cells. The simulation thus always starts with 27 active cells. As soon as any ‘‘hot’’ atom (most of the times the PKA) leaves the central cell and enters a neighboring one, the ‘‘sleeping’’ neighbors of this cell are waken up, etc. Symmetrically, cells are turned off, when the cascade is locally over. Temperature thresholds of the same nature as for the waking up of cells are used to determine which cells can be turned off. At this time the positions of the atoms inside the cell are frozen. The number of active cells in a simulation therefore starts at 27, rises and eventually decreases down to zero when all the cells have been turned off indicating the end of the simulation. The values of the turning on and turning off thresholds are the major parameters of the method. When high values are chosen for these thresholds, only highly perturbed cells are turned on and they are quickly turned off. In this case CMDC basically describes only the ballistic phase. At the opposite, when these thresholds are low, CMDC describes not only the ballistic phase but also the thermal phase which follows. In both cases, this method results in a decrease of the number of atoms in the simulation compared to full MD calculations, and therefore to a decrease of computation time. The second point accelerating the calculation is the use of a space variable time step for the simulation. We generalize the usual variable time step procedure by applying a different time step for each active cell. The time step of a central cell is calculated based on the maximum velocity of atoms in the central cell and its ﬁrst neighbors. Note that this local time step applies only to the force calculation, a global time step calculated as the minimum

of all local time steps is used for the movement of atoms. The atomic positions are evolved at each global time step but the forces are updated at each local time step. CMDC includes the slowing down of moving atoms by electronic losses as extracted from SRIM [3]. This slowing down has a double use. Indeed, beyond modeling in an approximate way the electronic losses, it constantly pumps out part of the kinetic energy of moving atoms. The total kinetic energy of the atoms in the simulation therefore constantly decreases which ensures that the simulation will indeed come to an end. Due to the abrupt freezing of atoms in turned off cells, the ﬁnal structure at the end of the simulation may be trapped in an unstable conﬁguration. To allow for some relaxations to take place, all the atomic positions in the cells that were active at some point of the cascade unfolding are eventually relaxed at the end of the simulation with a so-called fast quenching algorithm with asynchronous turning off of cells as in the cascade simulation. The CMDC method has been implemented in an eponym fortran 90 code. Proﬁling of typical runs has shown that, as in any MD simulation, the vast majority of CPU time is spent in the force calculations routines. A surface version of the code exists to describe external ion irradiations. 3. Example of CMDC studies 3.1. Self PKAs in iron Cascades in iron have been calculated for many years. Their interest is obvious regarding the use of steels in the nuclear industry. The energy of the PKAs has increased over the years with the CPU power available. They have recently reached a few hundreds keVs (e.g. 200 keV in Suzudo et al. [4], 500 keV in Zarkadoula et al. [5]). Using CMDC, we performed calculations of cascades at 17 energies following a geometrical series from 180 eV to 1.8 MeV using the potential recently developed by M-C Marinica [2]. At each energy, 100 cascades were calculated. Note that these PKA energies correspond to lower purely ballistic losses as indicated in Fig. 2. They span a ballistic energy range from 155 eV to 560 keV. The threshold average kinetic energies per cell for turning on and off the cells are 7.8 10 2 and 4.33 10 2 eV/atom, i.e. 900 K and 500 K respectively. The thresholds on maximum kinetic energies for each atom are set to ﬁve times the ones on average kinetic energies. One can note that these values are extremely low compared to the threshold displacement energies used in BCA models which are of the order of a few dozens of eV (about 40 eV for iron). Fig. 1 exhibits the atoms belonging to active cells at three different instants of a 1.8 MeV cascade as well as the ﬁnal distribution of defects. Fig. 2 shows the number of created Frenkel pairs expressed as a ratio compared to the NRT formula with the average displacement

Fig. 1. Active cells in a 1.8 MeV cascade in iron. From left to right Snapshots of atoms in active cells at 0.1, 0.4 and 0.7 ps and ﬁnal defects (interstitials in blue and vacancies in yellow). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

J.-P. Crocombette, T. Jourdan / Nuclear Instruments and Methods in Physics Research B 352 (2015) 9–13

Fig. 2. Surviving ratio of Frenkel pairs compared to NRT prediction in iron cascades. Black circles and blue diamonds are calculated in the present work with CMDC and standard MD calculations respectively. Red squares are data from Stoller et al. as summarized in [6] and Zarkadoula et al. [5]. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Table 1 Measured CPU time for CMDC calculations of one cascade, estimated acceleration compared to standard MD calculations and estimated CPU time for MD calculation of one cascade. Cascade type

CMDC CPU time (hours)

Acceleration

Standard MD CPU time (hours)

10 keV Fe in Fe 100 keV Fe in Fe 1 MeV Fe in Fe 60 keV He in Fe

0.1 0.67 98 46

215 983 1.7 105 6.9 105

140 8.8 103 1.7 106 3.2 107

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defects than Stoller’s cascades. For these energies we performed a few (25 at each energy) cascade calculations using standard MD. These values fall closer to the ones of CMDC than form the ones obtained by Stoller. It appears that the surviving ratio in this energy range depends highly on details of the calculation. For high energies our results are consistent with the results of Zarkadoula et al. considering that the latter are averaged on two cascades only. The obtained agreement is satisfactory in view of the acceleration of the calculation comparing CMDC and regular MD. To assess this acceleration we compared the number of force calculation in CMDC and in MD for the same type of cascades. The average number of force calculation in CMDC is an output of the code. For MD it is obtained by the product of the number of atoms in the equivalent MD box multiplied by the average number of iterations to complete cascade. The volume of the equivalent MD box is estimated to be equal to the maximum over all CMDC cascades of the tetragonal volume containing all the ﬁnal state defects. This volume is the minimum one should use to deﬁne an MD box containing all the defects at the end of the cascade. Considering that the spatially variable time step allows for an additional factor of 3 acceleration, one ends up with the accelerations compared to standard MD indicated in Table 1, along with the actual CPU times, for selected PKA energies. 3.2. 500 keV Fe implantation in iron The damage created by Fe surface implantation in iron hac been calculated to allow comparisons with SRIM (full cascade) calculations. The results are shown Fig. 3. One can see that the implanted ion proﬁles are comparable. However CMDC predicts the existence of a long tail down to 1 lm below the surface which corresponds to channeling events. Such events are not taken into account in SRIM as this code considers a random (non crystalline) structure. At the opposite the implantation peak is slightly shifted toward the surface for CMDC compared to SRIM. The vacancy distributions follow the same trends as the implantation proﬁles. Note however that the number of created vacancies is much lower with CMDC than SRIM. This is consistent with the low surviving ratio discussed in the previous section. As far as SRIM is concerned the number of predicted vacancies is about two times larger than the NRT value which is expected for such ‘‘full cascade’’ simulations. It is indeed quite tricky to obtain a number of vacancies consistent with the NRT law using SRIM as recently discussed by Stoller et al. [7]. 3.3. 60 keV helium implantation in iron

Fig. 3. Iron and defect proﬁles (integral normalized to one) obtained with CMDC and SRIM for 500 keV Fe implantation in iron.

energy of 37.2 eV known for the empirical potential. One can see that the surviving ratio decreases at low energies from about 1 to stabilize around 0.30 at high energies which is the expected behavior. The values are in good agreement with the ones obtained by Stoller [6] at both ends of the energy spectrum. A slight discrepancy appears around 10–20 keV where CMDC produces more

Helium implantation is the ﬁrst step in helium desorption experiments whose aim is to probe the stability of small vacancy-helium clusters created during implantation or the subsequent isothermal or isochronal annealing [8]. One easily understands that the evolution of helium and defects depends highly on the exact nature and spatial repartition of damage created by implantation [9]. We used CMDC to obtain a representative picture of the primary state of damage after 60 keV helium implantation in iron. The helium-iron interaction was represented by the Juslin– Nordlund potential [10]. One thousand separate CMDC calculations were performed to obtain satisfactory statistics. The crystalline orientation of iron was randomly chosen and the helium velocity was set perpendicular to the surface (normal implantation). The temperature and energy thresholds were the same as in the previous example. The distribution of defects as a function of types and depths are given in Fig. 4. To detect clusters, the following approach is used. Self-defects are considered as part of the same cluster if they are closer than the sum of their effective radii [11]. In addition, vacancies and interstitials are removed if they

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Fig. 4. Left: helium and defect proﬁles obtained with CMDC averaged on 1000 implantation events. Right: comparison with SRIM. (Integrals are normalized to one).

are distant from each other by less than some recombination distance. One can see that while most of the created defects are isolated vacancies or interstitials, some clusters of defects are formed, including very uncommon large clusters with up to 8 vacancies and 9 interstitials. As far as calculation details are concerned, due to the turning on-turning off procedure of CMDC, the maximum number of atoms in active cells, i.e. the maximum number of moving atoms at any given time, is on average 29,700. The total number of atoms that were moving at some point during the simulation averages at 420,000. Using the same procedure as the previous example, we estimate the acceleration be higher than three orders of magnitude compared to standard MD. A comparison with SRIM is presented on the right panel of Fig. 4. One can see that the implantation proﬁle obtained with CMDC is slightly more spread than the one obtained with SRIM. More vacancies are predicted by CMDC than SRIM especially at high depth. In this comparison n-vacancy clusters are treated as n monovacancies. 4. Discussion CMDC is one of a few attempts to accelerate MD simulations of cascades. It proves however different in principles from the various BCA-DM mixed approaches where a fast (usually BCA) approach is linked to subsequent MD simulations. Unlike these approaches (such as ComoD) [12] CMDC does not rely on a spatial decomposition of the cascade into ‘‘subcascades’’ and treats all the atoms taking part in the cascade on the same foot. CMDC is based on the on the ﬂy addition and removal of atoms in the simulation. In that respect, it is conceptually close to BCA where only moving atoms are added in the simulation after each collision and removed when their energy become small. CMDC is however quite different as it is not based on a sequence of collisions. Moreover entire cells are added and removed and not only atoms and the atomic positions are evolved with an MD algorithm. Because of this algorithm CMDC remains incomparably slower than pure BCA approaches which routinely calculate cascades of any energy in a couple of seconds. CMDC algorithm can in fact be thought as a bold extension of the ‘‘MD range’’ [13] and equivalent [14] concepts that exists for the calculation of the implantation proﬁle of ions in materials. The obvious difference being that in these approaches, the active cells are centered on the implanted ion and just follow its movement. The numbers of active cells and atoms in the simulation are therefore constant. Branching and cascade development are

therefore not accounted for. As a consequence such codes give much less information than CMDC as they do not produce information on the number or repartition of produced defects. At the opposite, CMDC does not treat the PKA or implanted ion in a speciﬁc way so that branching, cascade and defect formation are taken into account as well as local heating. As they are dealing only with the PKA, codes like MD range are much faster than CMDC, with calculation times for cascade comparable to BCA codes. The main objectives we had when we started the development of CMDC have been achieved. On one hand, while not reaching the lightning speed of BCA (or MD range) simulations, CMDC calculations prove orders of magnitude faster than standard MD simulations. The examples presented above, prove that it can be used to consider irradiation conditions almost inaccessible to standard MD simulations. Within reasonable CPU times, ion energies of a few MeV can actually be dealt with. The speed of calculations also allows calculating many events thus allowing for rare damage types to occur and giving access to reasonable statistics, which is known to be important for the simulation of microstructure evolution. The level of description of the cascade events is on the other hand much higher than in BCA simulations. Collective atomic movements are described as well as the thermal phase and the corresponding slow quenching of the cascade. Some weaknesses remain however. The detailed description structure of atomic defects is still imperfect. For instance it has not been possible yet to obtain a correct distribution of SIA clusters in iron cascades. CMDC predicts SIA to be spread and more isolated than observed in standard MD simulations. This may come from the fact that these SIA clusters are formed by the interaction of pressure waves and centers of subcascades [15]. Such pressure waves created during the cascade travel faster than thermal waves in the material and thus move away from the cascade centers quicker than the corresponding thermal waves. These waves are dealt with in standard MD simulations, even if some spurious reﬂections and transmissions take place at the borders of the periodically repeated box. In CMDC in its present form, the criteria to turn cells on and off are entirely based on temperature. Pressure waves are therefore not taken into account in the simulation. A natural path of improvement for CMDC is then to build a pressure or stress criterion to turn the cells on and off while limiting the additional calculation time. 5. Conclusion We have presented a new method to model displacement cascade or implantation in crystalline materials at the MD level. It is

J.-P. Crocombette, T. Jourdan / Nuclear Instruments and Methods in Physics Research B 352 (2015) 9–13

based on the decomposition of the material in nanometric cells which are added and removed on the ﬂy from the MD simulation and the dynamics of which are treated with a local time step. Compared to standard MD the speed up of CMDC simulations is huge. It allows simulating many events at energies beyond the reach of usual MD simulations. Beyond the present methodological development, we believe that considering cells in cascade simulations is an efﬁcient way to design MD simulations speciﬁc to cascades, a path which remains largely unexplored in our opinion. For instance cell decomposition appears to us as a natural way to deal with pressure waves or advanced electronic loses models [16]. References [1] J.P. Crocombette, Cascade modeling, in: S. Yip (Ed.), Handbook of Materials Modeling, Springer, Berlin, 2005, p. 987. [2] L. Malerba et al., Comparison of empirical interatomic potentials for iron applied to radiation damage studies, J. Nucl. Mater. 406 (2010) 19–38. [3] J.F. Ziegler, J.P. Biersack, M.D. Ziegler, SRIM – The Stopping and Range of Ions in Matter, Ion Implantation Press, 2008. [4] T. Suzudo et al., Annealing simulation of cascade damage in a-Fe â€’’ damage energy and temperature dependence analyses, J. Nucl. Mater. 423 (2012) 40–46.

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