Cutting and controlled modification of graphene with ion beams
Using atomistic computer simulations, we study how ion irradiation can be used to alter the morphology of a graphene monolayer by introducing defects of specific type, and to cut graphene sheets. Based on the results of our analytical potential molecular dynamics simulations, a kinetic Monte Carlo code is developed for modelling morphological changes in a graphene monolayer under irradiation at macroscopic time scales. Impacts of He, Ne, Ar, Kr, Xe and Ga ions with kinetic energies ranging from tens of eV to 10 MeV and angles of incidence between 0 and 88 are studied. Our results provide microscopic insights into the response of graphene to ion irradiation and can directly be used for the optimization of graphene cutting and patterning with focused ion beams.
pacs:61.80.Az, 61.48.Gh, 61.80.Jh
Recent reports on large-scale production of graphene [1, 2, 3], including growth of centimeter-size sheets on copper surfaces , have brought closer the utilization of graphene’s excellent electronic properties  in future electronic devices. In particular, quantum dots , nanoribbons , and antidot lattices [8, 9], which provide electron confinement within the graphene plane, have received considerable attention. Production of such structures is based on graphene patterning by various techniques such as electron beam lithography combined with reactive ion etching [6, 10, 11], chemical methods including unfolding of carbon nanotubes [12, 13] and graphene cutting with a focused electron beam .
Since focused ion beams (FIB) are already routinely employed in today’s semiconductor industry, this method could become an alternative approach. Indeed, cutting and patterning graphene using a FIB with a high spatial resolution was recently demonstrated [15, 16, 17]. The method is based on using a 30 keV He ion beam in a helium ion microscope to directly sputter carbon atoms from predetermined areas of graphene sheets, either suspended or deposited on a substrate. Etched gaps down to 10 nm have been reported, with sharp edges and little evident damage or doping to the sample. Concurrently with the experiments on He beam–based patterning of graphene, the possibility of using Ar ion irradiation has also been studied .
Efficient use of ion beams and optimization of the graphene cutting process require detailed microscopic knowledge of damage production mechanisms and types of defects created by the energetic ions in the sample. In order to get insight into the cutting process, the interaction of He ions with the target was modeled in Ref.  by a semi-empirical method  based on the binary-collision approximation, combined with statistical algorithms to calculate how a moving ion transfers its energy to the target atoms. This approach implemented in the TRIM software package  gives reasonable results for bulk materials. However, as has been pointed out [21, 22], the theory of irradiation effects developed for bulk materials does not always work at the nanoscale. In particular, it was recently demonstrated  that this approach cannot be applied to graphene, as the sample is treated as an amorphous matrix with a homogeneous mass density while the explicit account for the atomic structure is required for atomically thin targets.
In this study, we use analytical potential (AP) molecular dynamics (MD) simulations, a much more accurate method than the one implemented in the TRIM code, to study defect production in graphene under ion irradiation. Our ultimate goal is to provide the means for determining optimum parameters, such as ion mass, energy and angle of incidence for graphene cutting, which would enable the production of smooth edges with the minimum number of defects at faster cutting rates. We simulated more than two million impacts of energetic ions onto suspended graphene and gathered statistics on types and abundances of defects for He, Ne, Ar, Kr, Xe and Ga ions with energies ranging from tens of eV to 10 MeV. The role of the angle of incidence of the ions was studied in detail. To establish a direct link to the experiments, we further developed a kinetic Monte Carlo (kMC) code , which utilizes the statistics from the MD simulations for predicting the evolution of graphene under ion irradiation at macroscopic timescales. This allows modelling the behavior of irradiated graphene under realistic experimental conditions in, e.g., FIB systems.
In our MD simulations (a general introduction to this method can be found in Ref. ) the carbon-carbon interaction in graphene was modelled using the second-generation reactive empirical bond-order Brenner potential . The bond conjugation term, which is not expected to be important in irradiation processes , was omitted. This approach has previously been successfully used in modelling the irradiation response of graphene and other carbon nanostructures [23, 28, 29, 30, 31]. The interaction between energetic noble gas ions (He, Ne, Ar, Kr and Xe) and target carbon atoms was simulated using the Ziegler-Biersack-Littmark universal repulsive potential . In addition to the noble gas ions, irradiation with Ga ions was simulated using the same universal repulsive potential, as gallium is a typical ion species used in FIB systems. Chemical effects are expected to be important only at low ion energies ( eV) and they should not play any role at typical operating energies (30 keV) of FIB systems, especially in such a thin target, so that the use of the purely repulsive potential is justified also for the Ga–C interaction.
For a direct analogy with the experiments, we use the term ”ion” throughout this article, although the charge of the incoming ion is not explicitly considered, as this is beyond the AP-MD method, and more importantly, the effects of low charge states are negligible . The AP-MD approach is computationally efficient enough for running the massive number of simulations (more than two million runs in total) required to get statistically meaningful results in the large parameter space explored.
It is well known that electronic stopping is the main mechanism of energy transfer from an ion to any solid target at high ion energies . At the same time, experiments [33, 34] indicate that energy deposited into electronic degrees of freedom of graphite gives rise to defects only if electronic stopping power exceeds a critical value of 7 keV/nm. Taking into account the excellent charge and heat conductance of graphene, similar behaviour can be expected for this material . As the typical electronic stopping value for all the ion/energy combinations used in our study is less than 0.7 keV/nm (we calculated electronic stopping power using the approach of Ziegler, Biersack, and Littmark ), and even the highest values for high-energy Xe ( keV/nm) are well below the critical one, electronic stopping effects were not taken into account.
The simulated graphene target consisted of 800 carbon atoms. Since thermal excitations due to temperature-related atomic motion do not play a significant role in the momentum transfer between the impinging ion and target atoms (typical phonon energies are within tens of meV per atom), the initial target temperature was chosen to be 0 K. After ion impacts the system was cooled down using the Berendsen thermostat  at the edges, with fs as the time constant. Such a setup was used to model the dissipation of heat generated by the ion impact as would happen if the target was part of a larger graphene sheet. Additional simulations with the thermostat turned off showed, however, that the results are not sensitive to its parametrization, as the damage creation processes typically take place before the heat wave generated by the ion impact even reaches the system edges, barring a few cases in the high ion energy and high angle of incidence regime. Adaptive time step MD code parcas  was used for the simulations. Within the code, the simulation time step is dynamically adjusted based on the fastest moving particles in the system. This resulted in time steps ranging from the attosecond scale to close to one femtosecond.
The system was allowed to relax for one picosecond after the ion impact. During this time the system had typically reached a local energy minimum. However, any reasonable simulation time is too short for the system to always find the most stable local configuration. To address this, the resulting structures were annealed for another picosecond at 1500 K and eventually cooled down to 0 K to facilitate further relaxation of the system and removal of at least the most unstable atomic configurations. Over macroscopic time scales at experimentally relevant temperatures (room temperature and above) the created defects are expected to reconstruct into patches of non-hexagonal carbon rings similar to those presented in Ref. . Although the AP-MD method cannot capture the reconstruction processes, the sizes of these patches are defined by the initial damage which is statistically predicted by our results.
Impact points for the ions were chosen randomly within the minimum irreducible area of the graphene lattice. The direction of the incoming ion was determined by randomly selecting an inclination angle and azimuthal angle independently so that for each an average result could be generated over . A schematic illustration of the simulation setup is presented in Fig. 1 along with an example of the effect of changing the angle.
A link between the theoretical data on damage production and experiments can be provided within the framework of the kMC method. We have previously implemented such a model for comparing theoretical results to experiments in the cases of electron irradiation of carbon nanotubes  and hexagonal boron nitride monolayers . Here, we developed a kMC model describing the response of graphene to ion irradiation by using the results from our AP-MD simulations as the source of damage event rates. Within this model, the impact event rate is derived from ion beam current, and the rates of producing any type of defects are derived by multiplying the ion impact rate by defect creation probability for the selected combination of ion species, irradiation energy and angle of incidence. We employed data on the defect size (area initially covered by perfect hexagonal carbon rings transformed into other polygons, termed ”amorphized area” within our kMC model) and the number of sputtered carbon atoms for each parameter combination. Because the MD simulations were always carried out on a pristine target, one should be careful when interpreting the kMC results when the defects start to overlap. However, as it is easier to displace under-coordinated atoms, our kMC results can be used to estimate the lower limit for the number of sputtered atoms even for overlapping defects.
3 Results and Discussion
3.1 Dynamical simulations of defect production
When analyzing the results of the MD simulations, we categorized the defect structures produced in graphene into single vacancies (one missing atom from otherwise intact structure), double vacancies (two missing atoms and no further damage), complex vacancies (defect structures with missing atoms other than single and double vacancies) and amorphous regions (defect structures with no missing atoms, the simplest example of such a structure being the Stone-Wales defect ). Experimental examples of these structures are presented in a recent review article , except for complex vacancies, examples of which are presented in Ref. . We calculated the probabilities of producing each of these defect types for all ion species and the whole range of energies (35 eV – 10 MeV) and angles of incidence () considered, as presented in Fig. 2. Also the probabilities of creating any defect was calculated.
As is evident from Fig. 2, varying any of the parameters has a drastic effect on the produced defect types and their abundances. At energies in the keV range, single vacancies are the typical defects when the angle of incidence is perpendicular to the graphene sheet. Upon tilting the ion beam, first double vacancies become the dominant defect type after which complex vacancies become the most common type of defect. This can be attributed to the increased projected density of graphene, when viewed from a grazing angle. The maxima in single vacancy production are shifted towards higher ion energies as the angle of incidence is increased, which can be useful when ion irradiation with a fixed energy is used for cutting graphene. The location of the maxima in complex vacancy production first moves toward lower energies with increasing angle of incidence after which it again moves to higher energies. Therefore, if graphene is to be used as a membrane in external beam experiments as suggested in Ref. , the resilience of the membrane can be improved by slightly tilting it. For example, in the case of 2 MeV Ne ions, tilting the graphene sheet by will result in a twofold decrease in the sputtering rate, as will be discussed in more detail later in this article.
A range of parameters where no defects are created corresponds to the large angle, low energy region of the graphs in Fig. 2. Low defect production has the same origin as under ion channeling conditions in crystalline solids. When coming in at a grazing angle, the ion interacts with a long row of atoms, and if the energy is low enough, depending on the angle of incidence, none of the target atoms receives enough momentum to be displaced. Instead, the ion is reflected away from the graphene sheet. However, if ion energy is increased, the ion will penetrate the sheet, and it typically creates significant damage along the way, as can be seen from the steeply rising complex vacancy probabilities in the low energy–large angle regimes in Fig. 2. Amorphization events are observed mostly at low ion energies ( 1 keV), where the ions have barely enough kinetic energy to displace a target atom, but the displaced atom remains bonded to the sheet as an adatom or a part of a local amorphization.
3.2 Statistical model for continuous defect production
General trends and dominant defect types for specific irradiation conditions can be directly obtained from Fig. 2. However, making quantitative predictions on the evolution of a graphene target under ion irradiation is not straightforward, as both variations in the defect sizes and the number of sputtered atoms must be taken into account. Also, the probability distribution for creating each defect is not Gaussian, which means that mean size and standard deviation are not sufficient for describing the damage caused to graphene. For this reason, we developed a kMC code, which directly uses the statistics on the sizes of the defects and number of sputtered atoms, extracted from the MD simulations. This code can be used for predicting the evolution of a graphene sheet under ion irradiation at macroscopic time scales. The code has a web-based interface  and is available for public use. Both a visual representation of the sample and quantitative estimates of the total amorphized area and the number of sputtered atoms can be obtained using the program. As the original data is collected for individual ion impacts on pristine graphene, we stress that the code underestimates the produced damage at high ion doses due to a drop in the displacement threshold energy for under-coordinated atoms .
To demonstrate the potential of the kMC code, in Figs. 3 and 4 we present example cases simulated with the code. If graphene is uniformly irradiated with 650 eV Ne ions at normal direction to the sheet, nearly exclusive production of single vacancies is observed (Fig. 3a). Changing the ion to 400 eV Kr will result in uniform distribution of single and double vacancies with roughly a 1/1 ratio (Fig. 3(b)). For high energy ions with incident direction normal to the sheet the probability of creating any defect is greatly reduced, as is evident from Fig. 3(c) for the case of 2 MeV Ne: Although the fluence is increased by one order of magnitude as compared to the previous examples, the total number of defects is lower. The individual defects are much larger, however, which can be attributed to the fact that during the interaction of a target atom and the fast ion the target atom is practically immobile, resulting in symmetrical momentum transfer in the direction of the ion trajectory. This leads to recoiled atoms moving very close to a direction perpendicular to the ion trajectory. In the case of ion trajectory perpendicular to the graphene sheet, this leads to in-plane collision cascades and large defect structures. This effect is also demonstrated in Fig. 1. If the direction of the ions is tilted away from the normal of the graphene plane, the typical direction of the recoiled atoms is correspondingly tilted out of the graphene plane, which results in smaller defect structures due to the absence of collision cascades, as can be seen in Fig. 3(d). Thus the resilience of graphene under high energy ion irradiation can be improved by tilting it, as was suggested above.
3.3 Optimizing ion beam cutting of graphene
As experimentally demonstrated by Lemme et al.  and Pickard et al. , a focused He ion beam can be used to cut graphene with a very high precision. The used acceleration voltage was 30 kV and the cutting was performed with the ion beam pointed in the normal direction of the sample surface. The effects of such irradiation on graphene are shown in Fig. 3(e) assuming a focused spot with a Gaussian distribution with a full width at half maximum (FWHM) of 6 nm. Approximately half of the damage events lead to sputtering of target atoms and half of the events lead to amorphizations. If clean cutting is to be achieved, the amorphization events are not desirable. The situation can be improved by tilting the beam by approximately 60 (Fig. 3(f)). As the projected atomic density of the target is now increased, the defect creation probability is correspondingly higher. The percentage of sputtered atoms inside the spot area (illustrated with a white line along the FWHM edge) is increased with the same total dose even though the spot area is larger. Also, as demonstrated previously, tilting the beam at high ion energies leads to decreased probability of creating in-plane collision cascades and local amorphizations, in favor of single vacancies. This leads ultimately to cleaner edges in cuts made with a FIB.
To further illustrate what can be achieved with parameter optimization when cutting graphene with a FIB, three examples of the use of the kMC code in linear scan mode are presented in Fig. 4. In these examples, the beam spot is moved in the vertical direction. The spot width is 10 nm (FWHM) and the ion beam direction is tilted towards the direction of the spot movement. The ion energy and species are similar to what was used in Ref. . Line dose of 8 nC/cm was reported to not be adequate for making a cut in graphene. This observation is supported by our simulations (Fig. 4(a)): Only 6 % of the carbon atoms within the cut area are sputtered, although much of the area is amorphized. However, when the ion beam is tilted by 62.5 there is almost a threefold increase in the number of sputtered atoms (Fig. 4(b)). To compare the effectiveness of graphene cutting at different angles of incidence, a third simulation was conducted, where the line dose was decreased so that the number of sputtered atoms is approximately the same as in the first case. Comparing the results in Figs. 4(a) and 4(c) shows that tilting the He beam by 60 gives a threefold increase in the cutting efficiency of the beam, while the amorphized area outside the cut is decreased considerably.
To conclude, we studied the effects of ion irradiation on graphene using atomistic computer simulations. The role of the angle of ion incidence in a wide range of energies was investigated, and the types and concentrations of defects were identified for various ions. The dramatic effect of the angle of incidence on defect production demonstrates the fundamental difference of strictly two-dimensional graphene from traditional bulk targets. The peculiarities of graphene’s response to ion irradiation can be used to gain detailed control over produced defect types and their abundances. The presented publicly available computer code  enables one to make quantitative predictions of defect production in graphene under energetic ion bombardment. This information is needed in order to controllably create specific types of defects, or when graphene is to be nanomachined by a FIB. To illustrate the latter, we showed an example where tilting of the sample gave a threefold increase in sputtering and reduced amorphized areas in the sample. Further on, with respect to the possible use of graphene membranes in ion-beam analysis , we showed that the resilience of the membrane can be improved for high energy ions by choosing an optimum angle with regard to the direction of the ion beam. Although we looked specifically at noble gas irradiation, our results can also provide insights into the response of graphene to irradiation by other species, e.g., nitrogen, as in very recent experiments on irradiation mediated nitrogen doping of graphene .
We thank the Finnish IT Center for Science for generous grants of computer time. This work was supported by the Academy of Finland through several projects and the Centre of Excellence programme.
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