Anisotropy of electronic stopping power in graphite

The rate of energy transfer from ion projectiles onto the electrons of a solid target is hard to determine experimentally in the velocity regime between the adiabatic limit and the Bragg peak. First-principles simulations have lately offered relevant new insights and quantitative information for prototypical homogeneous materials. Here, we study the influence of structural anisotropy on electronic stopping power with time-dependent density functional theory simulations of a hydrogen projectile in graphite. The projectile traveled at a range of angles and impact parameters for velocities between 0.1 and 1.4 a.u., and the electronic stopping power was calculated for each simulation. After validation with average experimental data, the anisotropic crystal structure was found to have a strong influence on the stopping power, with a difference between simulations parallel and perpendicular to the graphite plane of up to 25%, more anisotropic than expected based on previous work. The velocity dependence at low velocity displays clear linear behavior in general, except for projectiles traveling perpendicular to graphitic layers, for which a thresholdlike behavior is obtained. For projectiles traveling along graphitic planes, metallic behavior is observed with a change of slope when the projectile velocity reaches the Fermi velocity of the electrons.

Figure 1

(a) Graphite unit cell showing the projectile initial positions 1–4 and (b) trajectories for simulations of projectiles moving out of the plane of graphitic layers. The shaded triangle in (a) is the region of crystallographically unique positions. $\alpha$ is the angle of the trajectory from the graphite $c$ axis, with $\alpha =0^{\circ }$ perpendicular to the graphitic planes.

Figure 2

(a) Graphite unit cell showing the projectile trajectories for simulations of projectiles moving parallel to the graphitic layers ($\alpha ={90}^{\circ }$). $\beta$ is the angle of the trajectory from the $a$ axis of the unit cell. $\beta =0^{\circ }$ and ${60}^{\circ }$ are crystallographically equivalent. (b) Distance of projectile trajectories from the graphitic planes.

Figure 3

Electronic stopping power for a proton shooting through graphite at different angles relative to the graphite $c$ axis. The filled circle depicts the experimental data from the work by Käferböck et al. [6]. The uncertainty in each data point is $\pm 0.6$ eV/Å. RESP is the random electronic stopping power for $\alpha ={90}^{\circ }$, averaged over all impact parameters simulated for comparison with Ref. [10].

Figure 4

(a) Electronic stopping power of a hydrogen atom in graphite moving at angle $\alpha$ from the graphite $c$ axis at a velocity of 0.3, 0.5, and 1.0 a.u.. The data for ${90}^{\circ }$ were calculated from simulations with the projectile moving midway between two planes of atoms along $\beta =0^{\circ }$ (see Figs. 1 and 2). The error bars are due to the uncertainty in fitting to the slope of the energy plot. (b) Average electron density along the trajectory of the projectile for different $\alpha$ values. (c) Correlation between local electron density and $S_e$.

Figure 5

(a) Shows the electronic stopping power for a projectile moving parallel to the graphitic layers at angles $\beta$ from the $a$ axis of graphite, midway between the graphitic layers. (b) Compares the electronic stopping power for a proton moving at different distances from the graphitic layers as a fraction of the interplanar distance (Fig. 2). The electronic stopping power increases the closer the path of the projectile is to a graphitic layer, likely due to the higher electron density closer to the planes.

Figure 6

Comparison of electronic stopping power in graphite with a hydrogen projectile traveling at 0.1 to 1 a.u. with supercell sizes of $2\times 2\times 2$ and $4\times 4\times 2$ primitive unit cells. In these simulations, the projectile moved perpendicular to the graphitic planes along the shortest supercell dimension, through the center of a channel.