Electron cascades and secondary electron emission in graphene under energetic ion irradiation

Highly energetic ions traversing a two-dimensional material such as graphene produce strong electronic excitations. Electrons excited to energy states above the work function can give rise to secondary electron emission, reducing the amount of energy that remains in graphene after the ion impact. Electrons can be either emitted (kinetic energy transfer) or captured by the passing ion (potential energy transfer). To elucidate this behavior that is absent in three-dimensional materials, we simulate the electron dynamics in graphene during the first femtoseconds after ion impact. We employ two conceptually different computational methods: a Monte Carlo (MC)-based one, where electrons are treated as classical particles, and time-dependent density functional theory (TDDFT), where electrons are described quantum mechanically. We observe that the linear dependence of electron emission on deposited energy, emerging from MC simulations, becomes sublinear and closer to the TDDFT data when the electrostatic interactions of emitted electrons with graphene are taken into account via complementary particle-in-cell simulations. Our TDDFT simulations show that the probability for electron capture decreases rapidly with increasing ion velocity, whereas secondary electron emission dominates in the high-velocity regime. We estimate that these processes reduce the amount of energy deposited in the graphene layer by 15%–65%, depending on the ion and its velocity. This finding clearly shows that electron emission must be taken into consideration when modeling damage production in two-dimensional materials under ion irradiation.

Figure 1

Schematics of the electron dynamics before (top), during (middle), and after (bottom) the impact of a SHI in graphene. The approximate timescale of the dynamics shown here is on the order of femtoseconds.

Figure 2

Different snapshots from the femtosecond real-time evolution, computed by 2D-PIC simulations, for emitted electrons from MC-CDF. Top left panel includes the simulation setup and boundary conditions. In each panel, the graphene sheet lies along the bottom edge, and the ion travels along the left edge. Black dots show superparticles, which are used to track positions of electrons in vacuum. The color coding shows the electrostatic potential generated by the emitted electrons.

Figure 3

Number of electrons emitted (black curve), number of electrons captured (blue curve), and energy deposited (red curve) by a 25 keV ${\mathrm{H}}^{+}$ ion for different impact trajectories as calculated by TDDFT. The red dashed line shows the energy deposited by the same ion in a single layer (6.33 ${\mathrm{a}}_0$ thick [58]) of bulk graphite as calculated with SRIM [74]. The inset shows the impact points (cyan) for the different ion trajectories normal to the graphene (orange); point O lies at the centroid of the gray dashed triangle.

Figure 4

Energy deposition in graphene for different projectile ion species and velocities. Results from TDDFT (diamonds), MC-CDF (solid curves), and SRIM (circles) are compared. The text annotations correspond to the initial charge states of the ions in TDDFT; the MC-CDF calculations use an equilibrium effective charge given by the Barkas formula [30].

Figure 5

Normalized kinetic-energy spectra of the secondary electrons emitted from graphene after impact by (a) ${\mathrm{H}}^{+}$ 80 keV and (b) ${\mathrm{Si}}^{+4}$ 15 MeV. TDDFT with (green) and without (red) captured electrons is compared to MC-CDF before (blue) and after (orange) including PIC simulations. MC-CDF results were computed with the Barkas effective charge [30].

Figure 6

(a) Number of emitted secondary electrons in TDDFT (diamonds), MC-CDF (solid curves), and $\mathrm{MC}\text{−}\mathrm{CDF}+\mathrm{PIC}$ simulations (dots). The TDDFT calculations used the initial charge states indicated beside the data points, while the $\mathrm{MC}\text{−}\mathrm{CDF}(+\mathrm{PIC})$ calculations employed the Barkas effective charge [30]. (b) Electron emission vs energy deposited in TDDFT (green), MC-CDF (orange), and $\mathrm{MC}\text{−}\mathrm{CDF}+\mathrm{PIC}$ simulations (blue) for Si projectile ions with $v=2.92$ a.u. The TDDFT calculations used the initial charge states indicated beside the data points, while the $\mathrm{MC}\text{−}\mathrm{CDF}(+\mathrm{PIC})$) calculations employed effective charge states in ascending order from $+1$ to $+9$.

Figure 7

(a) Number of electrons emitted and captured in TDDFT for ${\mathrm{Si}}^{+4}$ ions of varying velocities. (b) Total energy deposited and dissipated via electron emission and electron capture for ${\mathrm{Si}}^{+4}$ ions of varying velocities in TDDFT simulations.

Figure 8

Percentage of the initially deposited energy that is lost via electron emission in TDDFT (diamonds), MC-CDF (solid curves), and $\mathrm{MC}\text{−}\mathrm{CDF}+\mathrm{PIC}$ (points). The text annotations indicate the initial charge states of the ions in TDDFT; the MC-CDF calculations use the equilibrium effective charge state given by the Barkas formula [30].