Irradiation dynamics and effects on electronic structure
Three outcomes of the bombardment are observed: reflection, transmission, and adsorption of the incident hydrogen atom; no sputtering of any type was observed in our quantum-classical approach. Figure shows the probabilities of these processes as a function of incident H-atom energy. At 20 eV and above, transmission is the dominant process, as expected from the potentials in Figure . At the midrange energies of 1 to 10 eV, reflection is primarily observed, with a peak at 2 eV. At 1 eV, H still transfers enough kinetic energy to the target carbon atoms to allow its bonding in the wells of depth approximately 0.5 eV near the lattice points and bond centers. As the incident energy becomes comparable with the depth of this and smaller wells, adsorption becomes the dominant process as expected.
Probabilities of reflection, transmission, and adsorption as a function of incident kinetic energy.
Reflection of the incident hydrogen can occur at all points in the graphene lattice. As can be seen in Figure , the threshold for transmission is approximately 2.5 eV at the hexagon center. These atoms are still of insufficient energy to penetrate the barrier at the C-C bond position, so those that do not impact near the center of the hexagon are reflected (see Figure ).
Figure 4 Positions of reflection, transmission, and adsorption events for the quantum-classical calculations. In a representative graphene hexagon, using SCC-DFTB. Adsorption (left) shows clustering of hydrogen atoms around the lattice carbons. Reflection (center) (more ...)
By examining the position within the hexagon where incident atoms are reflected, transmitted, or adsorbed, one can infer the form of the many-body potential at nonsymmetrical parts of the lattice. Figure shows the hexagon-localized reflection, transmission, and adsorption for several energies. Lattice positions represented in Figure are the turning points for reflection, closest approach positions for transmission, and final x-y positions for adsorption. Adsorbed atoms are clustered around the carbon atoms, often showing some lateral vibration.
Reflection is distributed evenly around the perimeter of the hexagon, indicating that incident atoms are deflected away from the hexagon center due to the relatively low force experienced here. Also due to the weak interaction at the hexagon center, it is the most probable location for transmission to occur. Thus, atoms incident upon or deflected toward this position are both able to penetrate. These results agree with those from a previous study [30
], which found that reflection occurs at all points in the hexagon, and transmission is most probable near the hexagon center.
The scattering of incident noble gas atoms has been investigated at high energies (keV), where transmitted particles were found to have very little angular deflection while leaving the graphene relatively unaffected [8
]. Here, similar results are found for hydrogen at lower energies. Figure shows the angular cross sections of reflection and transmission for our SCC-DFTB results. Radii are normalized to unity for the purpose of comparison. Cross sections are calculated according to
Angular distributions of reflected (θ < 90°) and transmitted (θ > 90°) hydrogen atoms. Distributions found to fit the data are shown in black.
Here, 1/Nmax normalizes the distribution, and the differential solid angle dΩ becomes 2π sin θ dθ due to the azimultal symmetry of the problem. N(θ ± Δθ/2) is the number of atoms scattered into a bin of width Δθ centered at polar angle θ.
Small changes in the x- or y-components of an atom's linear momentum are much more visible for low incident energies, where these changes can be comparable to the initial momentum. In the SCC-DFTB simulations, atoms with such low incident energy tend to reflect when not adsorbed, and the reflected angular distribution shows much more scattering. Transmitting hydrogen atoms in these simulations tend to have higher incident energies, so the small x- or y-forces don't produce a significant angular displacement of their momenta. While atoms incident at 5 and 10 eV have a wider distribution than at the higher energies, they tend to penetrate only near the center, where the H-C interactions are weakest.
The dominance of adsorption in SCC-DFTB simulations at impact energies below 1 eV provides enough statistical weight for an investigation of the effects of H-adsorption on the El-h quantity of the affected graphene. However, roughly a third of the incident atoms are found to bond to the surface after initially being reflected at a large angle relative to their initial momenta. These 'wandering' hydrogen atoms, primarily seen at 0.5 eV incidence, generally drift above the graphene surface at a distance of about 3 Å for 2 to 5 fs before falling toward a lattice carbon and adsorbing. Roughly 10% of these 'wanderers' do not bond to a carbon within the simulation time. Therefore, while they are counted as adsorbed in Figure , they are ignored in the henceforth analysis to reduce uncertainties.
The graphene band gap is often computed using a band structure or density of states calculation. However, the graphene system studied here is subject to thermal motion as well as bombardment, and the impinging particle should not be included in Brillouin zone integration. As discussed earlier, we simply define a quantity El-h by subtracting the energy of the highest occupied orbital from that of the lowest unoccupied orbital. The 1,000-K electronic temperature used creates a 'smearing' of the orbital occupations near the Fermi level. We use occupations of 1.8 for h (analogous to the HOMO) and 0.2 for l (analogous to the LUMO). This allows us to accomplish significant statistics while accounting for the different sites of adsorption and variety of vibrational states in which atom is adsorbed. The system is a 336-atom supercell, equivalent to an 18 × 18 × 1 k-point grid.
Figure shows contour plots of the two equivalent potential wells for hydrogen, corresponding to two adjacent C atoms of graphene. The depth of the wells is about -0.61 eV. Thus, when the kinetic energy of H is comparable to the well depth, excited vibrational motion is possible after adsorption; to account for this, we average the change in El-h over a number of time steps at the end of the simulation. When doing this time-averaging, it is important to avoid time steps at which some of the hydrogen atoms have not yet bound to the graphene surface. Figure displays the standard deviation of the hydrogen z-position distribution averaged over 1, 4, 12, and 24 fs of simulation time. In all cases, the mean value is within a single standard deviation of 1.2 Å.
A contour plot of the potential energy of a H-atom. In vicinity of the two adjacent carbon bonding centers (C) in graphene, Z being the direction orthogonal to the graphene. The depths of the wells in which hydrogen bonds are equal.
Standard deviation of hydrogen z-position distribution as a function of averaging time.
One can see that the standard deviation, representative of the distribution's width, is higher for low averaging times. Additionally, atoms incident with higher kinetic energy are adsorbed with greater vibrational energy, so they display a wider distribution of z-positions. As shown in Figure , the wider distributions that come with this higher vibrational energy produce a smaller change in the El-h on average.
Previous studies [31
] have found that a tuning of the graphene band structure can be achieved by partial or full hydrogenation of nanoribbons, achieving band gaps of 0.43 to approximately 4.0 eV. The results obtained here support this, showing a sensitivity of the graphene band gap to even a single stuck hydrogen atom. At the largest averaging time considered here, the average change in El-h
is 171.5 meV for 0.1 eV incidence, 165.1 meV for 0.2 eV incidence, and 157.7 meV for 0.5 eV incidence (Figure ). As discussed above, El-h
is not equal to the band gap, though it is correlated with it. There is a nonlinear relationship between the change in El-h
-position of hydrogen, which is the source of the difference between these results.
Mean change in El-h as a function of averaging time for three incident kinetic energies.
Figure shows the change in the El-h quantity as a function of adsorbate distance for an ideal graphene plane. Since the hydrogen is directly above a lattice carbon, the minimum of the potential well is located at 1.2 Å. The average minima and maxima of low-energy (0.1 eV incidence) and high-energy (0.5 eV incidence) oscillations are 0.1 and 0.3 Å, respectively. The change in El-h decreases with increasing adsorbate distance, and the flattening observed below 1 Å causes the minimum position in oscillation to affect the gap less than the maximum position. Thus, larger oscillation amplitudes produce, on average, a smaller change in the El-h quantity. These averages are shown as thick dashed lines and have a difference of 30 meV, which is on the order of the 14 meV difference between average changes induced by 0.1 and 0.5 eV bombardments.
Figure 9 Mean change in El-h as a function of adsorbate hydrogen distance. Displayed are maximum, minimum, and average changes for typical large and small oscillation amplitudes resulting from 0.5 and 0.1 eV bombardments, respectively. Calculations are performed (more ...)
Comparison with classical molecular dynamics
In the classical molecular dynamics approach, the physical accuracy of the simulation is determined mainly by the quality of the interatomic potentials. Like its predecessor, the REBO potential, AIREBO is a member of the classical bond-order family of potentials [20
] of the Tersoff-Brenner type, which provides a good description of the covalent bonds for nonpolar systems. The REBO potential is short ranged (< 2 Å) and, therefore, considerably less costly to use in computation but might not be suitable for collisions where long-range interactions are important, or for describing the coupling of adjacent graphene planes. REBO is also known for its poor treatment of conjugated couplings [20
]. The AIREBO contains improved descriptions of the torsional and long-range van der Waals interactions (< 11 Å) as well as improved bonding interactions. The ability to use a classical (if reactive) molecular dynamics approach for the bombardment problem is highly desirable since these approaches are orders of magnitude computationally cheaper than even SCC-DFTB.
Figure shows a comparison of the refitted [22
] AIREBO and REBO H-graphene potentials with that of SCC-DFTB shown in Figure . The CMD potentials were calculated using a 480-atom graphene cluster
, i.e., no periodic boundary conditions. All three potentials display a well near 1.2 Å for incidence upon a lattice carbon, and the subsequent repulsive barriers agree well as they are all fit to ZBL [22
]. However, REBO and AIREBO predict 0.5 and 1.0 eV barriers, respectively, before the potential wells. As these barriers are not present in the SCC-DFTB potential, it is expected that REBO and AIREBO result in different dynamics at low-energy bombardment. The dissimilarities are even more distinct for the other positions in the lattice. AIREBO predicts a potential barrier of over 20 eV, peaking at about 1.35 Å, for incidence on the C-C bond center. Neither DFTB nor REBO agree with this barrier, which is produced by the long-range Lennard-Jones terms in AIREBO, since the AIREBO and REBO results are indistinguishable at distances less than 1 Å. Another peak of 20 eV height is found at z =
0 for AIREBO and REBO, only 2 eV higher than the corresponding SCC-DFTB curve. REBO is consistently 2 to 5 eV more repulsive than DFTB, but qualitatively very similar. The most distinctive difference between these three potentials is their treatment of the graphene π-orbitals. Clearly, the AIREBO Lennard-Jones interactions coming from the six adjacent carbons produce a potential barrier at the graphene hexagon center that is more than 60 eV (525%) higher than the potential in its predecessor, which is in turn roughly 10 eV (380%) higher than DFTB. The REBO potentials clearly agree much more with the DFTB calculations than those of AIREBO, which indicates that the Lennard-Jones interactions which produce the observed potential barriers likely overestimate the hydrogen-graphene interaction.
Potential energy of the H-graphene interaction at canonical points in the lattice. As calculated in DFTB (solid), AIREBO (single dash), and REBO (double dash).
Ito et al. [6
], Nakamura et al. [5
], and Saito et al. [33
] have done a comprehensive study of the response of a single graphene sheet (reflection, transmission, and absorption) to the impact of hydrogen atoms and its isotopes in an energy range below 200 eV. They used classical molecular dynamics simulations with the short-range (< 2 Å) modified Brenner (REBO) potential. Unfortunately, the distribution of barriers and wells is not clear for their modification of the potential; however, their calculation of the reflection, transmission, and adsorption probabilities upon normal impact shows good qualitative agreement with our AIREBO calculations, as illustrated in Figure . Classical MD AIREBO and REBO results are significantly different than the quantum-classical SCC-DFTB results mainly due to the presence of the potential barriers observed in Figure . The AIREBO and REBO graphene potentials are more repulsive than those of SCC-DFTB, which results in a 15 eV higher threshold for transmission to occur. However, all methods converge to 100% transmission at high energies, as has been observed in previous studies [8
]. While REBO shows a much higher peak in adsorption probability, the presence of the aforementioned barriers in both potentials result in a dominance of reflection at low energies, inconsistent with the results of SCC-DFTB presented above.
Figure 11 Probabilities of reflection, transmission, and adsorption as calculated by AIREBO and REBO [5,30]. The presence of potential barriers before potential wells (see Figure 10) results primarily in reflection at low incident energies.
Another result of the increased repulsiveness of AIREBO is the occurrence of physical carbon sputtering upon impact of hydrogen. Figure shows the sputtering yield as a function of incident kinetic energy for AIREBO calculations. If Ed
is a carbon atom displacement energy from the rapheme, then the kinetic energy of the impact atom in the head-on binary collision is
The known energy for displacing one atom from a pristine rapheme is 22.2 eV, which yields
(H) = 78.2 eV. Consistently, the sputtering yields in Figure for all sputtered species are zero at 20 eV and start rising from 50 eV impact energy (where the sputtering yield is approximately 0.002). After a peak of 0.0325 at about 200 eV, they then decrease with higher incident energy. Chemical sputtering, i.e., production of CH, which is a second-order process (breaking of a carbon bond followed by capture of H by the carbon atom) here, is quite improbable, and its yield stays well below 0.005. We note again that no sputtering, physical or chemical, is measured in the SCC-DFTB simulations in the considered range of impact H energies (< 200 eV).
Sputtering yields of C and CH as determined by AIREBO simulations.
Lastly, the CMD calculations result in larger angular scattering effects than SCC-DFTB, as can be seen in Figure . The reason for the markedly different distribution is again in the potential barriers that arise from the Lennard-Jones interactions. The stronger interaction produces a more significant change in the incident hydrogen x- or y-momentum, resulting in a cos(θ-θo) distribution with maximum at roughly 37°. Transmitting atoms also interact with the potential barriers on the opposite side of the surface, which are responsible for deflecting these atoms and producing the much wider distribution than observed in the SCC-DFTB calculations.
Angular distributions of reflected (θ < 90°) and transmitted (θ > 90°) hydrogen atoms. As calculated in AIREBO CMD simulations.