# Non-equilibrium dynamics of electron emission from cold and hot graphene
under proton irradiation

Yifan Yao  Department of Materials Science and Engineering, University of
Illinois, Urbana-Champaign, Urbana, IL 61801, USA  Arne Metzlaff  University
of Duisburg-Essen, Faculty of Physics and CENIDE, 47057 Duisburg, Germany
Andreas Wucher  University of Duisburg-Essen, Faculty of Physics and CENIDE,
47057 Duisburg, Germany  Lukas Kalkhoff  University of Duisburg-Essen, Faculty
of Physics and CENIDE, 47057 Duisburg, Germany  Lars Breuer  University of
Duisburg-Essen, Faculty of Physics and CENIDE, 47057 Duisburg, Germany  Marika
Schleberger  University of Duisburg-Essen, Faculty of Physics and CENIDE,
47057 Duisburg, Germany  André Schleife<EMAIL_ADDRESS>Department of
Materials Science and Engineering, University of Illinois, Urbana-Champaign,
Urbana, IL 61801, USA  Materials Research Laboratory, University of Illinois
at Urbana-Champaign, Urbana, IL 61801, USA  National Center for Supercomputing
Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801,
USA

###### Abstract

Secondary electrons that are emitted from materials under electron or ion
irradiation can serve as a detailed probe for electron-ion processes within
the target material. This requires a thorough understanding of their emission
characteristics, including intensity, duration, and kinetic energy
distribution. However, complicated multi-length and time scale relaxation
processes connect the initial non-equilibrium excited electron distribution
with their eventual emission as secondary electrons. The situation is pushed
even further into the non-equilibrium regime e.g. in two-dimensional materials
or when using beams with high ion flux. We combine first-principles real-time
electron dynamics simulations with modern experiments to characterize the
processes that are involved in secondary electron emission for proton-
irradiated graphene. From our data we identify signatures of kinetic and
potential emission mechanisms, by comparing secondary electron characteristics
for the entrance and exit side. Simulated yields on the exit side are higher
by a factor of up to 2.75, which agrees with experiment within a factor of two
and the simulated yield on the exit side of 0.53 electrons per ion also agrees
with a value of 0.4 measured for argon irradiated two-layer graphene. Our
simulated kinetic energy spectra match electron and ion irradiation
experiments and prior models, and provide detailed insight into the emission
dynamics. The duration of the emission pulse is only about 1.5 femto-seconds,
pointing at high time resolution when used as a probe. Finally, we show that
secondary electron emission is significantly increased from graphene with high
lattice temperature, whereas the influence of electronic temperature is
negligible.

##  I Introduction

Secondary electrons can be emitted from the surface of a target material under
irradiation conditions upon impact of high-kinetic energy ions or electron
projectiles. The energy spectra and spatial distribution of these emitted
electrons play a crucial role, for example in modern light-ion microscopy and
provide high-resolution surface morphology images of target materials with
minimal collateral damage [1, 2, 3]. While the electronic stopping power and
charge exchange between a projectile ion and the target material can be
experimentally probed, e.g. using ion transmission microscopy, the emerging
electron and ion dynamics within the target after irradiation mostly remain
elusive in experiments. At the same time, the intensity and kinetic energy
distribution of emitted secondary electrons is influenced by the electron and
ion response in the target material. Hence, emitted secondary electrons can
provide important insight into the target material itself [4, 3, 5].

To achieve such insight, a deep understanding of secondary electron emission
as a complex multi-length and time scale process is required, but presently
still being developed. It requires modeling the interaction between the
incident ion and the target material, including the dynamics of the charge
state of the projectile, the emission probability of secondary electrons, and
the electron-electron and electron-ion relaxation dynamics following the
initial excitation An analytical model is developed by Chung [6], based on the
work function as a barrier against emission and the electron-electron
scattering mean free path as the probability for excited electron to diffuse
from bulk to the surface. This model has shown success in describing the
kinetic energy distribution of secondary electrons emitted from metals under
electron irradiation. Later, a modified version [7] was introduced to describe
the kinetic energy distribution of the secondary electrons in the helium ion
microscope. These models rely on the static work function of the target
material as a single empirical parameter. However, for a highly excited
surface [5] the surface potential and, thus, the work function is strongly
perturbed, requiring more sophisticated modeling. On the other hand, to
describe the relaxation dynamics in the target, theoretical efforts [8, 9, 10]
focus on thermalization of the radiation-induced excited electrons towards a
Fermi-Dirac distribution through electron-electron scattering, and the
subsequent emergence of thermal equilibrium between nuclei and electrons
through electron-phonon scattering. The connection of these processes to
secondary electron emission is so far unexplored but important because it can
be directly probed in pump-probe experiments with ultrashort ion pulses and
also indirectly influence electron emission in experiments under irradiation
conditions that expose a sample surface to multiple successive ion impacts. A
direct real-time electron dynamical simulation that describes the influence of
thermalization on secondary electron emission during successive ion impact is
still missing. However, this is necessary as the time interval between
successive ion impacts in modern ion beams reaches the sub-picosecond regime,
i.e., prior to electron thermalization.

The non-equilibrium electron dynamics of secondary electron emission becomes
even more complicated in two-dimensional (2D) targets, for which the impacting
ion does not reach an equilibrium charge state before leaving the target,
leading to a response that is very different from that of bulk. For bulk
targets, electronic stopping has been extensively investigated [11, 12, 13,
14, 15, 16] and specifically for self-irradiated silicon, first-principles
simulations have shown that a highly charged ion projectile equilibrates its
charge state after just a few nanometers [17]. For thin materials, however,
different energy deposition rates [18], charge excitation [19, 18], or charge
capture processes [20, 18] have been reported. In addition, while secondary
electrons excited in bulk materials can undergo scattering and reverse their
momentum, in thin targets it is doubtful whether such scattering occurs [21].
The measured electron mean free path transverse to the graphene layer is
comparable to the layer thickness [22]. Similarly, the estimated mean free
path of electrons inside the target is around 1 nm for a wide variety of
materials [23], which is comparable to the layer thickness of stacked 2D
materials. As a result, secondary electrons preferably travel in the same
direction as the projectile, due to momentum conservation. First-principles-
based simulations can provide insight into such complex electron-ion dynamics
and lead to a better understanding of ion irradiation of 2D or thin-film
materials.

At the same time, the challenge of grasping such electron dynamics in solids
and two-dimensional materials also drives the advancement of new experimental
methods. While ion-surface or ion-solid interactions have been a subject of
research for decades, direct experimental investigations into the dynamics
induced by ion impact within the target material remain unfeasible to this
day. Only the initial state before impact and the final state several
nanoseconds after impact are directly measurable. For laser excitation
experiments this gap can be bridged through pump-probe experiments [24, 25],
where a single laser beam is split to establish a fixed temporal relationship
between the two resulting pulses. One part of this beam is then used to excite
the target, while the split-off beam is time-delayed and used as probe. By
varying the delay, the induced dynamics in the sample can be investigated. Due
to the development of lasers with pulse durations as short as a few
attoseconds [26, 27, 28], even the swiftest processes on the sub-femtosecond
timescale can be examined. Such inquiries have not been achievable using ion
pulses, given their charge and mass, which constrain how much the duration of
monoenergetic pulses can be reduced. However, recent developments are now
opening the door to such experiments with ion pulses lasting only a few
picoseconds [29, 30, 31, 32, 33]. With such pulses, it will be possible for
the first time to experimentally track the excitation duration of electrons in
2D materials, such as graphene, in the time domain.

In this work, we develop a computational first-principles description of the
real-time electron emission dynamics of proton-irradiated graphene, as a
prototypical 2D material, and make connections with experiment. A combination
of real-time time-dependent density functional theory (RT-TDDFT) and Ehrenfest
dynamics has been successfully applied in numerous studies [5, 19, 18, 20, 34,
35, 36, 37] to simulate femto-second electron dynamics under external
irradiation, providing a temporal and spatial resolution that is still
experimentally difficult to achieve. Despite that, no first principles study
comprehensively examines the duration and kinetic energy spectra of secondary-
electron emission. Additionally, as the time interval between successive
projectile ions approaches a few picoseconds, the effect of non-vanishing
electron and lattice temperature, e.g. caused by excitations of preceding ion
impacts, must be considered. The understanding of such effects is lacking in
the literature, but crucial when using secondary electrons as a probe for
thermalization processes in the target material. Here we aim to develop a
description of how the local, pre-excited target material responds to external
ion irradiation and how this affects intensity and duration of secondary
electron pulses. Also, we compute the kinetic energy spectrum of the secondary
electrons and quantify the effect of kinetic energy and trajectory of incident
protons. This comprehensive understanding of secondary electron pulses from
ground-state and pre-excited monolayer graphene offers critical insights into
the local, and temperature-dependent material properties, enriching our
knowledge of the underlying electron dynamics in graphene.

##  II Computational Methods

In this work we perform real-time time-dependent density functional theory
simulations. Converged ground-state single-particle Kohn-Sham (KS) states from
DFT were used as the initial condition for the time-dependent KS equations

$i\frac{\partial}{\partial
t}\phi_{j}(\mathbf{r},t)=\left(-\frac{\nabla^{2}}{2}+V_{\mathrm{KS}}(\mathbf{r},t)\right)\phi_{j}(\mathbf{r},t).$
(1)

Here, $\phi_{j}$ are the single-particle KS orbitals evolving in a time-
dependent effective potential $V_{\mathrm{KS}}$ which is a functional of the
electron density

$n(\mathbf{\mathrm{r}},t)=\sum_{j}f_{j}\left|\phi_{j}(\mathbf{\mathrm{r}},t)\right|^{2}$
(2)

and $f_{j}$ are orbital occupations. The Kohn-Sham states are represented
using a plane-wave basis with a cutoff kinetic energy of 50 Hartree. Exchange
and correlation effects are treated using the adiabatic local density
approximation [38, 39]. The electron-ion interaction is described by norm-
conserving Hamann, Schlüter, and Chiang pseudo-potentials [40]. We utilize the
enforced time-reversal symmetry (ETRS) method with a time step of one
attosecond to propagate Eq. (1) in real-time.

Figure 1:  (a) Schematic of the simulation cell used in this work. The proton
is initially placed 25 $a_{0}$ away from the graphene and travels normally to
the graphene plane with fixed velocity. The green region is where the complex
absorbing potential (CAP, brown curves) is $<0$, see Eq. (3). (b) Both the
channeling (A) and centroid (O) impact points of the proton in monolayer
graphene are studied in this work.

We employed a simulation cell with 112 carbon atoms and 100 $\mathrm{a_{0}}$
of vacuum (see Fig. 1). This allows to use only the $\Gamma$ point for
Brillouin zone sampling. All these parameters remain consistent with those
used in previous works [19, 41]. We choose proton projectiles in our
simulations to avoid challenges arising when simulating the acceleration of
ions with partially filled valence states. Real-time propagation begins with
the proton placed 25 $\mathrm{a_{0}}$ away from the graphene. The proton
approaches and traverses the graphene with a constant velocity along a
trajectory normal to the graphene plane, and we compare the channeling (A) and
centroid (O) proton trajectories (see Fig. 1). We keep all carbon nuclei fixed
in this work since the femtosecond time scale of proton impact is too short
for them to move appreciably and the longest trajectory in this work is only
about 12 fs. This would neglect secondary electron emission occuring on longer
time scales, e.g. due to plasmon decay or carbon ion recoil.

As the proton impacts graphene, secondary electrons are emitted from both
sides and propagate outwards. We use absorbing boundary conditions as
implemented in the Qb@ll code [42, 43], to avoid the unphysical re-entering of
emitted electrons into a periodic image of the simulation cell. The complex
absorbing potential used in this work is represented as

$V_{\mathrm{CAP}}(z)=-i\cdot
W\cdot\sin^{2}\left(\frac{(z-\mathrm{z_{s}})\cdot\pi}{2\cdot dz}\right),$ (3)

for $\mathrm{z_{s}}<z<\mathrm{z_{s}}+2dz$, where $\mathrm{z_{s}}$ is the front
boundary of the CAP, and $dz$ is its half width [44, 41]. In our simulations,
we set $z_{s}$=$40~{}a_{0}$, $dz$=$10~{}a_{0}$, and $W$=$15$ Hartree. To avoid
artificial re-entering of the proton due to periodic boundary conditions, we
remove it once it reaches the cell boundary. An explicit comparison between
the secondary electron yield computed in this work (with absorbing boundary
conditions) and previous results (without absorbing boundary conditions) [19]
is shown in Fig. S1 of the supplemental information.

The number of emitted electrons in the entrance- and exit-side vacuum is
determined using an artificial detector surface, placed 10.5 $a_{0}$ away from
the graphene. The total number of emitted electrons $N_{\mathrm{e}}$ is
determined by integrating the current density passing through the detector
surface $\Omega$ over time,

$N_{\mathrm{e}}=\int_{\Omega}j(\mathbf{r},t)dtd\mathbf{r}.$ (4)

The number of electrons captured by the proton is determined by integrating
the charge density over a sphere with a radius of 5 $a_{0}$ around the proton.
Except for computing kinetic energy spectra in Figs. 4 and 5, we subtract
these captured electrons from the total emitted electrons in this work,
because they would not be measured in experiment as secondary electrons.
Subsequently, taking the time derivative of the number of secondary electrons
allows us to acquire the secondary electron current. These steps enable us to
gain insights into the temporal behavior and pulse duration of secondary
electron emission from the proton-irradiated graphene.

To obtain the kinetic energy spectrum of the electrons at each time step, we
devise the following method: First, we calculate their velocity from the
current density $j(\mathbf{r},t)$ and electron density $n(\mathbf{r},t)$,

$v(\mathbf{r},t)=\frac{j(\mathbf{r},t)}{n(\mathbf{r},t)}.$ (5)

The kinetic energy distribution at time $t$ can then be obtained (in atomic
units where $m_{e}$=1) as the number of electrons $n(\mathbf{r},t)d\mathbf{r}$
in the entire vacuum with a specific kinetic energy $v^{2}(\mathbf{r},t)/2$

$P(\mathrm{KE},t)=\int_{V_{\mathrm{vac}}}n(\mathbf{r},t)d\mathbf{r}\delta\left(\mathrm{KE}-\frac{v^{2}(\mathbf{r},t)}{2}\right),$
(6)

where $d\mathbf{r}$ is the volume attributed to each point of the real-space
grid that discretizes the electron density, and $V_{\mathrm{vac}}$ is the
entire vacuum. In practice, at each time step, we are histogramming the number
of electrons corresponding to the specific kinetic energy, obtaining an
instantaneous kinetic energy spectrum. Finally, we integrate these
instantaneous spectra over time to include all emitted electrons

$P_{\mathrm{acc}}(\mathrm{KE})=\int P(\mathrm{KE},t)dt.$ (7)

In Fig. S4 of the SI we compare the instantaneous and the accumulated spectra.
Since charge and current densities are observable in RT-TDDFT, we can directly
obtain the velocity of the electrons. Hence, we do not make assumptions for
the time that electrons are emitted nor the distance that electrons travel as
is done in the usually used time-of-flight (TOF) analysis [45].

In this work, we perform simulations of secondary electron emission from
ground-state graphene and compare it to graphene with an elevated lattice as
well as electronic temperature. We represent an elevated electronic
temperature using a Fermi-Dirac distribution of KS occupation numbers within
Mermin DFT [46]. An elevated lattice temperature is represented using atomic
displacements that we create using the approach described in Ref. [47]. We
superimpose harmonic phonon modes with random phases and amplitudes according
to classical statistics. The mean-square amplitude of the $j$-th phonon mode
follows from

$\left<\left|Q_{j}\right|\right>=\frac{kT}{\omega^{2}_{j}}.$ (8)

All phonon eigenmodes are calculated via the finite-difference method as
implemented in the Vienna _Ab-initio_ Simulation Package (VASP). The
projector-augmented wave (PAW) method is used to describe the electron-ion
interaction. We use a relaxed 112-atom supercell with reduced vacuum size to
39.3 $a_{0}$, a $4\times 4\times 1$ $\mathbf{k}$-point grid, and 530 eV as the
kinetic energy cutoff of the plane-wave basis. We then use a single snapshot
resulting from this method to represent the atomic geometries in our Qb@ll RT-
TDDFT simulation of graphene with a large lattice temperature for multiple
proton trajectories.

##  III Experimental Methods

Figure 2:  Schematic of the experimental set-up. The ion beam is directed onto
a free standing graphene film in transmission geometry. The emitted electrons
are accelerated via increasing voltages towards the PIPS detector. The
potential of the detector is high enough to deflect the transmitted ions.

Experimentally we investigated the number of electrons emitted from a free-
standing graphene target under singly charged ion bombardment. The
experimental setup to investigate the electron emission is depicted in Fig. 2.
It is based on a design originally developed by the Aumayr group that is
described in great detail in Refs. [48, 49]. The experiment operates in
transmission geometry with free-standing graphene targets consisting of
multiple layers (2-4), see details of the sample preparation in Ref. [50]. A
singly charged argon ion produced by a rare gas ion source (Atomica
Duoplasmatron) hits the graphene target with kinetic energies between 2.5 keV
and 12.5 keV, leading to the emission of electrons. It is important to note
that due to the rather low velocity of the projectile, no stripping effects
are expected while passing the target and the singly charged argon ions can
thus be compared to the protons in our simulations. Also in experiments with
up to approximately 100 keV argon projectiles impacting thin carbon foils, the
average equilibrium charge state is less than one [51]. A proton with the same
velocity as a 100 keV argon ion has a kinetic energy of about 2.5 keV. Hence,
we expect that irradiation with $\sim 6.2$ keV protons, as simulated in this
work (see below), should produce comparable charge and velocity conditions for
secondary electron emission as in [Wittkower_charge_1973].

The emitted electrons are accelerated with a high voltage of 25 kV towards a
passivated implanted planar silicon (PIPS, Mirion Technologies) detector. It
is in principle an energy resolved detection device with a resolution of
approximately 3 eV defined by the direct band gap in the doped silicon chip.
Therefore, a single electron hitting the PIPS detector with a kinetic energy
of 25 keV will produce about 8000 electron-hole pairs, resulting in a well-
defined current pulse at the output of the detector. From this, we precisely
determine the number of electrons emitted from the graphene target per ion
impact, because $n$ electrons will exactly produce $n$ times the pulse of a
single electron.

The experiment is not time-resolved and therefore not capable of directly
following the dynamics in graphene triggered by the ion impact. Nevertheless,
it provides crucial experimental information about the total number of emitted
electrons per ion impact from graphene, supporting the results from
simulations and vice versa.

##  IV Results and Discussion

We first analyze the dependence of intensity and duration of the secondary
electron current on the proton impact point and velocity, as a comparison with
prior work [19]. To do so, we choose proton velocities ranging from 6.2 to 80
keV for which Ref. [19] reported large secondary electron yields on the
entrance and exit sides. In addition, we include another velocity that is
around the kinetic emission threshold, which is estimated to be 0.13 – 0.19 eV
[19], to investigate the effect of potential emission. Potential emission of
secondary electrons occurs due to the potential energy release as the
projectile neutralizes its charge state in the target. Although this process
is more commonly discussed for highly charged ions [52, 5], the potential
energy stored in the proton (13.6 eV) is almost three times the work function
of graphene (4.56 eV) [53, 54], which exceeds the minimum requirement for
Auger neutralization as a mechanism for potential emission [55].

Figure 3:  Secondary electron current on the (a) entrance side and (b) exit
side (see Sec. IV for details on how these are extracted and how the captured
electrons are removed). The proton is exactly in the graphene plane at
“impact”.

The peak intensity of the secondary electron current in Fig. 3 follows the
same trend with proton kinetic energy as the secondary electron yield reported
in Ref. [19]. It increases as the proton kinetic energy decreases for the
entrance side and the opposite is true for the exit side. The overall peak
intensity is higher at the exit side compared to the entrance side, which also
agrees with previous results [19]. Specifically, the peak intensity reaches a
maximum at around 10 keV of proton kinetic energy for the entrance side and 80
keV at the exit side. This difference in maximum yield on the entrance and
exit sides hints at different emission mechanisms. Potential emission is
expected to be symmetric with respect to entrance and exit side, but secondary
electrons due to kinetic emission preferably travel in the same direction as
the proton due to momentum conservation. In Ref. [56], ratios of the yield at
exit and entrance side range from 1 to 1.5 in a comparable velocity range for
hydrogen irradiated thin carbon foils. In our simulation, the ratio of ranges
from 1 to 2.75, which is slightly larger than the experiment. For the slowest
proton studied in this work, the kinetic energy of about 0.2 keV is less than
the kinetic emission threshold [19], and Fig. 3 shows a very small intensity
of the secondary electron pulse corresponding to a small secondary electron
current. Remarkably, even for this slowest proton, the yield ration is about
2.5 in our simulations.

Furthermore, our results in Fig. 3 show that for protons with kinetic energies
above the kinetic emission threshold, secondary electron emission from the
entrance and exit side ends after only about 1 – 1.5 fs, regardless of the
proton kinetic energy and impact point. A comparable duration of the electron
emission is also reported for graphene under highly charged ions irradiation
[5]. We note that a fluctuation of the secondary electron current occurs at
around 0.5 fs after the impact for the proton with 10 keV at the centroid
trajectory (O) and 6.2 keV at the channeling trajectory (A), see Fig. 1. Our
simulations show that this is due to the exchange between secondary electrons
and the captured electrons, and does not change the duration of the pulse.
Also, despite the similar duration of emission, at the entrance side, the
secondary electron current emerges earlier for trajectory O, than for the
channeling trajectory A. This indicates that secondary electrons due to
protons on trajectory O have larger kinetic energy than those emitted due to
channeling protons, as we will confirm below using peak positions of the
corresponding kinetic energy spectra.

Our data in Fig. 3(a) further indicates that irradiation with protons with a
kinetic energy of around 0.2 keV, i.e. around the kinetic emission threshold
[19], leads to pre-impact emission. In this regime, the potential emission is
expected to be the primary contribution. In both the entrance and exit side,
the secondary electron yield for this case is around 12 % of the maximum yield
in our simulations, see Fig. S1. Following the expression derived in Ref.
[57], which shows fair agreement with experiment [58], we obtained the
predicted potential emission yield around 0.09 (see Sec. VI.2 in the SI for
the procedure), which is comparable to our simulation (around 0.027 at
entrance side, and 0.065 at exit side). Additionally, the capture of electrons
is most notable for these slow proton projectiles [19], further indicating
that potential emission plays a role in this regime. However, due to the small
amount of potential energy stored in the proton, the corresponding secondary
electrons yield is small compared to slow but highly charged ions [52, 5].

Figure 4:  The kinetic energy spectrum computed for emitted electrons (see
details in text), including both the secondary electrons and electrons
captured by the incident proton, for (a) entrance side and (b) exit side.
Vertical dashed lines in (b) denote the electron kinetic energy corresponding
to the proton velocities.

Next, in Fig. 4 we show the kinetic energy spectrum of the emitted electrons.
We only include the kinetic energy of electrons within 1.5 fs after the proton
impact, because Fig. 3 shows that the secondary electron pulse ends within 1.5
fs after the proton impact. The kinetic energy spectrum of the emitted
electrons serves as a probe of the local electronic structure of the target
and of the local escape probability of the electrons near the proton beam
impact point. For instance, within the scattering model from Chung [6], the
first peak in the kinetic energy spectrum lies at $w_{0}/3$ with the work
function $w_{0}$.

The kinetic-energy spectrum we compute for the exit side (see Fig. 4b) shows a
clear peak that corresponds to electrons with a kinetic energy that matches
the proton velocity. Those electrons are mostly located around the proton and
we compare the kinetic energy spectrum of electrons within 5 $a_{0}$ of the
proton to the entire spectrum in Fig. S3 of the SI. From this, we conclude
that these electrons are captured by the proton and would not be detected in
the experimentally measured secondary electron spectrum. While it is possible
that captured electrons could be detached if they are weakly bound, this
phenomenon is not observed in our few-fs RT-TDDFT simulations. Also, those
captured electrons occupy the 1$s$ orbital of the proton [18], indicating that
most of them are strongly bound. In contrast, highly charged ions can lose
captured electrons due to autoionization [59, 60].

Second, the kinetic energy spectra in Fig. 4 also show a characteristic peak
around 3 – 4 eV. The overall energy position of this peak and the general
shape of the spectra in this kinetic energy region are very similar among all
trajectories and proton velocities on both the entrance and exit sides. The
variation of incident proton energy and trajectory only influences the amount
of emitted electrons, but not their kinetic energy distribution. This implies
that the emission process is predominantly governed by intrinsic material
properties of the graphene target, such as the mean free path and the work
function.

Moreover, Fig. 4 provides a more detailed understanding of the secondary
electron pulse observed in the real-time evolution shown in Fig. 3. From the
real-time data of emitted electrons at the detector surface (see Fig. 3) it is
hard to disentangle which of the emitted electrons travel faster than others
and when they were emitted. The kinetic energy spectrum provides this
information. For a given kinetic energy, a proton on the centroid trajectory
(O) results in a peak position that is about 0.75 – 1.22 eV higher in energy
than for the channeling trajectory (A), and a larger magnitude of the tail.
This matches the trend in Fig. 3, where the secondary electron pulse for the
centroid trajectory (O) occurs around 0.1 fs earlier than that for the
channeling trajectory (A). Thus, our kinetic energy spectrum supports that the
earlier arrival of the secondary electrons for the centroid trajectory (O) is
due to their higher kinetic energy, but not due to the earlier occurrence of
emission.

The short time scale of secondary electron emission shown in Fig. 3 indicates
that delayed electron emission, resulting from the relaxation of the energy
stored during ion impact, is not observed in our simulations. In particular,
while Fig. S2 in the SI shows a distinct peak at around 12 eV, indicating
excitation of plasmonic oscillation in our proton-irradiated graphene, plasmon
decay is not seen in our real-time profile of the secondary electron emission.
Plasmon decay [5] with an intrinsic plasmon lifetime [61] of about 12 ps, is
much slower than the timescale of our simulations. If the time-scale in our
RT-TDDFT simulation was long enough to capture such delayed processes we might
be able to observe a shoulder-like feature as was seen in the kinetic energy
spectrum of the secondary electrons from ion-irradiated metallic surfaces [55,
62]. Plasmon-assisted emission was also reported as a local minimum in the
derivative of the kinetic energy spectrum near $\omega_{p}-\omega_{0}$, where
$\omega_{p}$ is the plasmon frequency, and $\omega_{0}$ is the work function
of host materials [55, 62, 63, 64]. In the future, we envision to compare with
experiments in more detail to quantify the electron yield due to the delayed
process, such as plasmon decay.

Figure 5:  Comparison of our kinetic energy spectrum from Fig. 4 (centroid
trajectory, kinetic energy of 80 keV) with a measured spectrum of highly
charged $\mathrm{Xe^{40+}}$ ions impacting graphene [52], a measured spectrum
under electron irradiation in scanning electron microscopy (SEM) experiments
[65], and an analytical expression derived by Chung [6] (see details in the
text). The $y$ axis is normalized to the same peak intensity.

In Fig. 5 we show that the general shape of our kinetic energy spectrum agrees
well with experimental data and with the analytical expression developed by
Chung [6]. For this comparison, we choose the centroid trajectory (O) and a
proton kinetic energy of 80 keV as an example. The Chung model [6] relates the
position of this secondary electron peak to the work function of the material.
For graphene, with a work function of approximately 4.56 eV [53, 54], it
yields $w_{0}/3=4.56/3=1.52$ eV which is about 2 eV less than what we observe
in our data. The scanning electron microscopy (SEM) data [65] is obtained in
reflection geometry and should thus be compared to our entrance side
simulations. For this data, the Chung model captures peak position and shape
very accurately (see Fig. 5). The general shapes of our kinetic energy spectra
also agree with previous classical Monte Carlo and RT-TDDFT simulations [20],
however, our spectra reproduce the peak in the low-energy regime and agree
closer with experimental data shown in Fig. 5. This could be partially
explained by the inaccurate description of the electrostatic interaction in
the classical approach and the effects due to periodic boundary conditions in
the RT-TDDFT simulations.

For ion irradiation experiments as well as our proton simulation, the kinetic
energy peak appears shifted to higher kinetic energies by about 2 eV relative
to Chung’s model. In Fig. 5 we also include data from highly charged Xe40+
irradiation [52, 5], which is measured in forwarding direction and should be
compared to our exit-side simulations. Although there is a modified version of
the Chung model [7, 65] that extends it to helium irradiation, these
modifications rely on fitting the energy distribution of the excited electrons
to the experimental data. Those parameters might then be material dependent,
and hence, we do not compare with the modified Chung model in this work. In
addition, the constant value of the work function used in Chung’s model will
be modified by the radiation-induced electron dynamics [5]. Other processes,
such as attraction from the positively charged incident ion before impacting
the target, might lead to further deviations for ion-induced emission since
Chung’s model is derived for electron radiation.

We note that the two emission mechanisms discussed above, i.e. potential and
kinetic emission, contribute differently for different projectiles. For
example, Xe40+ is a highly charged ion and is expected to be dominated by
potential emission [52], while potential emission in our case is less
important, as discussed above. The good agreement of the kinetic energy
spectra for these very different projectiles indicates that electron emission
is an intrinsic property of the target that is not influenced by the different
types of projectiles. We also note that our simulated kinetic energy spectrum
results in a longer tail than the experimental spectrum. One possible error in
our approach is due to the finite size of the vacuum in our simulation, any
single instantaneous kinetic energy spectrum is not capable of capturing all
emitted electrons with all ranges of kinetic energy. To solve this problem, we
accumulate the instantaneous kinetic energy spectrum over time until the pulse
ends at 1.5 fs, since this is when all secondary electrons are emitted into
vacuum. This procedure may not fully account for long-term secondary electron
dynamics near the surface due to finite-size effects of our finite vacuum
size. In experiment, all emitted electrons are collected using a high voltage
(see Fig. 2).

Finally, we investigate the extent to which pre-existing electronic
excitations in the graphene target influence subsequent electron emission.Non-
equilibrium excited electrons, e.g. due to proton irradiation, will thermalize
within a ps or even less through electron-electron scattering and then heat
the lattice through electron-phonon scattering. This timescale is comparable
to the time interval between successive ion impacts in modern ion beams,
suggesting that finite lattice and electronic temperatures should be
considered. In addition, our RT-TDDFT simulations revealed that the duration
of the secondary electron pulse is only around 1.5 fs (see Fig. 3). This is a
fast time scale compared to thermalization, suggesting that the intensity of
secondary electron emission can serve as a probe of the thermalization process
that occurs in the target material following the excitation by the preceding
incident ions [5]. To separate the effects from the elevated electron and
lattice temperature, we investigated proton irradiation on a cold lattice with
elevated electronic temperature, and on an elevated lattice temperature with
cold electrons. We use electron and lattice temperatures from previous works
based on two temperature models [66, 67, 68].

Figure 6:  Electron yield $\gamma$ from graphene samples with different layer
numbers as a function of the projectile energy. The yield was calculated from
the electron spectrum measured with the PIPS detector by fitting a Furry
distribution to the data.

From the experiments performed with the PIPS detector we calculate the
electron yield $\gamma$. Fig. 6 shows the dependence of the electron yield for
different numbers of graphene layers, namely two, three and four. The
resulting spectra are taken for Ar+ ions with different kinetic energies and
are fitted with a Furry distribution [69, 70], which is a Polya distribution
where $b=1$, to evaluate the respective electron yield as shown in Fig. 6. We
find that there is no strong dependence on the kinetic energy of the
projectile. The main variation of the different yields is observed for
different layer numbers. For the two layer graphene a yield of approximately
$\gamma=0.4$ electrons per ion, $\gamma=1.8$ electrons per ion for the three
layer graphene and $\gamma=0.7$ electrons per ion for the four layer graphene
sample can be observed.

Figure 7:  Number of secondary electrons, computed as described in Sec. IV, at
elevated electron temperature of $10,000$ K or elevated lattice temperature of
$1,000$ K, compared to electron emission in the ground state of $T$=0 K. The
proton is exactly in the graphene plane at “impact”.

In Fig. 7, we compare the number of emitted secondary electrons from ground-
state graphene to that from finite electronic or lattice temperature. Compared
to our experiments on two layer graphene under $\mathrm{Ar}^{+}$ irradiation,
which results in a secondary electron per ion of $\gamma\approx 0.4$ (see Fig.
6), our proton irradiation results in a slightly larger secondary electron
yield, where the maximum yield at the exit side is around $\gamma=0.53$, see
Fig. S1.

Next, we compute the secondary electron yield for an electronic temperature of
$10,000$ K, which results in $2.76\cdot 10^{20}\,\mathrm{cm^{-3}}$ excited
electrons. This electronic temperature is exaggeratedly high for ion
irradiation after 1 ps, but is achievable within the first 100 fs after the
impact, as predicted in Ref. [66, 67]. Nevertheless, our simulations show that
the secondary electron yield is very similar to that from the ground state
(see Fig. 7), and the effect of even such a high electron temperature is
minor.

Contrary, our data shows that secondary electron yield with a lattice
temperature of $1,000$ K is significantly increased (see Fig. 7). The increase
is about 10 – 13 % at the entrance side and about 3 – 5 % at the exit side,
relative to the ground state. Moreover, this enhanced emission is also
observed in the channeling trajectory (see Fig. S8). Such a lattice
temperature was reported after about 400 fs from molecular dynamics
simulations of ion-irradiated Fe [67]. Besides secondary electron yield, we
also simulate the kinetic energy spectrum at this elevated lattice
temperature, see Fig. S5. The peak position and the shape of the spectrum are
similar to that of the ground state. Hence, we only identify the enhanced
secondary electron yield as an indicator of the lattice temperature in the
materials.

Figure 8:  Comparison of the planar integrated charge density difference
between the (a) elevated electron ($T$=$10,000$ K) or (b) lattice ($T$=$1,000$
K) temperature and ground state graphene, at the initial state before the
real-time propagation starts. The black circles denote the atomic positions of
graphene at an elevated electron or lattice temperature. The black diamond
denotes the proton impact point of the channeling trajectory (A) and the black
X denotes the centroid trajectory (O).

To explain the difference in secondary electron yield for hot electrons and a
hot lattice we analyze the electron density distribution in both cases in Fig.
8. Even though an electronic temperature of $10,000$ K is very high, it merely
results in a small charge density difference compared to the ground-state
electron density of graphene. Conversely, the electron density at an elevated
lattice temperature is about two orders of magnitude higher. We note that as
the long-range Coulomb interaction between proton and electrons decays as
$1/r$, the enhancement of electron emission is not related to the charge
density difference exactly at the impact point. The charge density at the
elevated lattice temperature at the exact impact point is actually lower than
in the ground state. On the contrary, the charge density at the impact point
of the centroid trajectory (O) at the ground state is higher than the
channeling trajectory (A), resulting in an enhanced yield (see Fig. S1). We
also note that in addition to the difference in the electronic density, the
atomic displacements at high lattice temperatures also change the exact
position of the proton impact point relative to the carbon atoms. The
structure and impact points at finite lattice temperature are shown in Fig. S7
of the supplemental information. This might introduce a slight variation of
the secondary electron emission.

##  V Conclusion

We employed a combination of experiments and real-time time-dependent density
functional theory to quantitatively investigate the non-equilibrium dynamics
of secondary electron emission in proton-irradiated graphene, including high
electronic and lattice temperatures. While the secondary electron yield
sensitively depends on the trajectory and the kinetic energy of the
projectile, the shape of the secondary electron pulse and the kinetic energy
spectra agree well among different impact parameters. Our simulated kinetic
energy spectrum agrees qualitatively with literature reports for electron and
ion projectiles, with peak positions varying by about 1 – 2 eV. The predicted
secondary electron pulse concludes within 1.5 fs for all trajectories and
velocities studied here. This is much shorter than the thermalization process
of non-equilibrium electrons, which is on the order of ps. Additionally, pre-
excited graphene during successive ion irradiation shows an enhanced electron
emission due to electron density perturbations that are present for high
lattice temperatures. Further experimental work is needed to verify the
prediction of the electron yield at a non-vanishing lattice or electron
temperature. Nevertheless, we conclude that the characteristics of the
secondary electron yield can serve as a probe for thermalization in the target
material, which is the basis of the high promise of recently developed ion-
pump-probe experiments.

###### Acknowledgements.

We thank Zhihao Jiang and Alina Kononov for fruitful discussions. This
material is based upon work supported by the National Science Foundation under
Grant Nos. OAC-2209857 and OAC-1740219. The authors acknowledge the German
Research Foundation (DFG) for funding through project 278162697 within the
SFB1242 “Non-Equilibrium Dynamics of Condensed Matter in the Time Domain”. An
award of computer time was provided by the Innovative and Novel Computational
Impact on Theory and Experiment (INCITE) program. This research used resources
of the Argonne Leadership Computing Facility, which is a DOE Office of Science
User Facility supported under Contract DE-AC02-06CH11357. This work made use
of the Illinois Campus Cluster, a computing resource that is operated by the
Illinois Campus Cluster Program (ICCP) in conjunction with the National Center
for Supercomputing Applications (NCSA) and which is supported by funds from
the University of Illinois at Urbana-Champaign.

## References

  * Fox  _et al._ [2013] D. Fox, Y. B. Zhou, A. O’Neill, S. Kumar, J. J. Wang, J. N. Coleman, G. S. Duesberg, J. F. Donegan, and H. Z. Zhang, Helium ion microscopy of graphene: Beam damage, image quality and edge contrast, Nanotechnology 24, 335702 (2013).
  * Iberi  _et al._ [2015] V. Iberi, I. Vlassiouk, X.-G. Zhang, B. Matola, A. Linn, D. C. Joy, and A. J. Rondinone, Maskless Lithography and in situ Visualization of Conductivity of Graphene using Helium Ion Microscopy, Scientific Reports 5, 11952 (2015).
  * Hlawacek  _et al._ [2014] G. Hlawacek, V. Veligura, R. van Gastel, and B. Poelsema, Helium ion microscopy, Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 32, 020801 (2014).
  * Werner  _et al._ [2020] W. S. M. Werner, V. Astašauskas, P. Ziegler, A. Bellissimo, G. Stefani, L. Linhart, and F. Libisch, Secondary Electron Emission by Plasmon-Induced Symmetry Breaking in Highly Oriented Pyrolytic Graphite, Physical Review Letters 125, 196603 (2020).
  * Niggas  _et al._ [2022] A. Niggas, J. Schwestka, K. Balzer, D. Weichselbaum, N. Schlünzen, R. Heller, S. Creutzburg, H. Inani, M. Tripathi, C. Speckmann, N. McEvoy, T. Susi, J. Kotakoski, Z. Gan, A. George, A. Turchanin, M. Bonitz, F. Aumayr, and R. A. Wilhelm, Ion-Induced Surface Charge Dynamics in Freestanding Monolayers of Graphene and MoS2 Probed by the Emission of Electrons, Physical Review Letters 129, 086802 (2022).
  * Chung and Everhart [1974] M. S. Chung and T. E. Everhart, Simple calculation of energy distribution of low-energy secondary electrons emitted from metals under electron bombardment, Journal of Applied Physics 45, 707 (1974).
  * Petrov and Vyvenko [2011] Yu. Petrov and O. Vyvenko, Secondary electron emission spectra and energy selective imaging in helium ion microscope, in  _SPIE Defense, Security, and Sensing_, edited by D. E. Newbury, S. F. Platek, D. C. Joy, and T. K. Maugel (Orlando, Florida, United States, 2011) p. 80360O.
  * Bernardi  _et al._ [2014] M. Bernardi, D. Vigil-Fowler, J. Lischner, J. B. Neaton, and S. G. Louie,  _Ab_ _Initio_ Study of Hot Carriers in the First Picosecond after Sunlight Absorption in Silicon, Physical Review Letters 112, 257402 (2014).
  * Brown  _et al._ [2016] A. M. Brown, R. Sundararaman, P. Narang, W. A. Goddard, and H. A. Atwater, Nonradiative Plasmon Decay and Hot Carrier Dynamics: Effects of Phonons, Surfaces, and Geometry, ACS Nano 10, 957 (2016).
  * Brown  _et al._ [2017] A. M. Brown, R. Sundararaman, P. Narang, A. M. Schwartzberg, W. A. Goddard, and H. A. Atwater, Experimental and _Ab Initio_ Ultrafast Carrier Dynamics in Plasmonic Nanoparticles, Physical Review Letters 118, 087401 (2017).
  * Ullah  _et al._ [2018] R. Ullah, E. Artacho, and A. A. Correa, Core Electrons in the Electronic Stopping of Heavy Ions, Physical Review Letters 121, 116401 (2018).
  * Ullah  _et al._ [2015] R. Ullah, F. Corsetti, D. Sánchez-Portal, and E. Artacho, Electronic stopping power in a narrow band gap semiconductor from first principles, Physical Review B 91, 125203 (2015).
  * Shukri  _et al._ [2016] A. A. Shukri, F. Bruneval, and L. Reining,  _Ab Initio_ electronic stopping power of protons in bulk materials, Physical Review B 93, 035128 (2016).
  * Lee and Schleife [2019] C.-W. Lee and A. Schleife, Hot-Electron-Mediated Ion Diffusion in Semiconductors for Ion-Beam Nanostructuring, Nano Letters 19, 3939 (2019).
  * Correa [2018] A. A. Correa, Calculating electronic stopping power in materials from first principles, Computational Materials Science 150, 291 (2018).
  * Schleife  _et al._ [2015] A. Schleife, Y. Kanai, and A. A. Correa, Accurate atomistic first-principles calculations of electronic stopping, Physical Review B 91, 014306 (2015).
  * Lee  _et al._ [2020] C.-W. Lee, J. A. Stewart, R. Dingreville, S. M. Foiles, and A. Schleife, Multiscale simulations of electron and ion dynamics in self-irradiated silicon, Physical Review B 102, 024107 (2020).
  * Kononov and Schleife [2020] A. Kononov and A. Schleife, Pre-equilibrium stopping and charge capture in proton-irradiated aluminum sheets, Physical Review B 102, 165401 (2020).
  * Kononov and Schleife [2021] A. Kononov and A. Schleife, Anomalous Stopping and Charge Transfer in Proton-Irradiated Graphene, Nano Letters 21, 4816 (2021).
  * Vázquez  _et al._ [2021] H. Vázquez, A. Kononov, A. Kyritsakis, N. Medvedev, A. Schleife, and F. Djurabekova, Electron cascades and secondary electron emission in graphene under energetic ion irradiation, Physical Review B 103, 224306 (2021).
  * Kononov  _et al._ [2022a] A. Kononov, A. Olmstead, A. D. Baczewski, and A. Schleife, First-principles simulation of light-ion microscopy of graphene, 2D Materials 9, 045023 (2022a).
  * Geelen  _et al._ [2019] D. Geelen, J. Jobst, E. E. Krasovskii, S. J. van der Molen, and R. M. Tromp, Nonuniversal Transverse Electron Mean Free Path through Few-layer Graphene, Physical Review Letters 123, 086802 (2019).
  * Ramachandra  _et al._ [2009] R. Ramachandra, B. Griffin, and D. Joy, A model of secondary electron imaging in the helium ion scanning microscope, Ultramicroscopy 109, 748 (2009).
  * Elzinga  _et al._ [1987] P. A. Elzinga, F. E. Lytle, Y. Jian, G. B. King, and N. M. Laurendeau, Pump/probe spectroscopy by asynchronous optical sampling, Applied Spectroscopy 41, 2 (1987), https://doi.org/10.1366/0003702874868025 .
  * Sokolowski-Tinten and von der Linde [2004] K. Sokolowski-Tinten and D. von der Linde, Ultrafast phase transitions and lattice dynamics probed using laser-produced x-ray pulses, Journal of Physics: Condensed Matter 16, R1517 (2004).
  * López-Martens  _et al._ [2005] R. López-Martens, K. Varjú, P. Johnsson, J. Mauritsson, Y. Mairesse, P. Salières, M. B. Gaarde, K. J. Schafer, A. Persson, S. Svanberg, C.-G. Wahlström, and A. L’Huillier, Amplitude and Phase Control of Attosecond Light Pulses, Phys. Rev. Lett. 94, 033001 (2005).
  * Hentschel  _et al._ [2001] M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Attosecond metrology, Nature 414, 509 (2001).
  * Agostini and DiMauro [2004] P. Agostini and L. F. DiMauro, The physics of attosecond light pulses, Reports on Progress in Physics 67, 813 (2004).
  * Breuers  _et al._ [2019] A. Breuers, M. Herder, P. Kucharczyk, M. Schleberger, K. Sokolowski-Tinten, and A. Wucher, A concept to generate ultrashort ion pulses for pump-probe experiments in the keV energy range, New Journal of Physics 21, 053017 (2019).
  * Kucharczyk  _et al._ [2020] P. Kucharczyk, A. Golombek, and A. Wucher, Generation of ultrashort ion pulses in the keV range: Numerical simulations, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 483, 41 (2020).
  * Golombek  _et al._ [2021] A. Golombek, L. Breuer, L. Danzig, P. Kucharczyk, M. Schleberger, K. Sokolowski-Tinten, and A. Wucher, Generation of ultrashort keV Ar+ ion pulses via femtosecond laser photoionization, New Journal of Physics 23, 033023 (2021).
  * Kalkhoff  _et al._ [2023a] L. Kalkhoff, A. Golombek, M. Schleberger, K. Sokolowski-Tinten, A. Wucher, and L. Breuer, Path to ion-based pump-probe experiments: Generation of 18 picosecond keV ${\mathrm{Ne}}^{+}$ ion pulses from a cooled supersonic gas beam, PRRESEARCH 5, 033106 (2023a).
  * Mihaila  _et al._ [2023] M. C. C. Mihaila, G. L. Szabo, A. Redl, M. Goldberger, A. Niggas, and R. A. Wilhelm, Generation of picosecond ion pulses by ultrafast electron-stimulated desorption, (2023), arXiv:2310.02631 [physics.app-ph] .
  * Gruber  _et al._ [2016] E. Gruber, R. A. Wilhelm, R. Pétuya, V. Smejkal, R. Kozubek, A. Hierzenberger, B. C. Bayer, I. Aldazabal, A. K. Kazansky, F. Libisch, A. V. Krasheninnikov, M. Schleberger, S. Facsko, A. G. Borisov, A. Arnau, and F. Aumayr, Ultrafast electronic response of graphene to a strong and localized electric field, Nature Communications 7, 13948 (2016).
  * Ueda  _et al._ [2016] Y. Ueda, Y. Suzuki, and K. Watanabe, Quantum dynamics of secondary electron emission from nanographene, Physical Review B 94, 035403 (2016).
  * Ueda  _et al._ [2018] Y. Ueda, Y. Suzuki, and K. Watanabe, Secondary-electron emission from multi-layer graphene: Time-dependent first-principles study, Applied Physics Express 11, 105101 (2018).
  * Tsubonoya  _et al._ [2014] K. Tsubonoya, C. Hu, and K. Watanabe, Time-dependent density-functional theory simulation of electron wave-packet scattering with nanoflakes, Physical Review B 90, 035416 (2014).
  * Zangwill and Soven [1980] A. Zangwill and P. Soven, Resonant Photoemission in Barium and Cerium, Physical Review Letters 45, 204 (1980).
  * Zangwill and Soven [1981] A. Zangwill and P. Soven, Resonant two-electron excitation in copper, Physical Review B 24, 4121 (1981).
  * Vanderbilt [1985] D. Vanderbilt, Optimally smooth norm-conserving pseudopotentials, Physical Review B 32, 8412 (1985).
  * Kononov  _et al._ [2022b] A. Kononov, C.-W. Lee, T. P. Dos Santos, B. Robinson, Y. Yao, Y. Yao, X. Andrade, A. D. Baczewski, E. Constantinescu, A. A. Correa, Y. Kanai, N. Modine, and A. Schleife, Electron dynamics in extended systems within real-time time-dependent density-functional theory, MRS Communications 12, 1002 (2022b).
  * Draeger and Gygi [2017] E. Draeger and F. Gygi, Qbox code, Qb@ll version, https://github.com/LLNL/qball (2017).
  * Schleife  _et al._ [2014] A. Schleife, E. W. Draeger, V. M. Anisimov, A. A. Correa, and Y. Kanai, Quantum Dynamics Simulation of Electrons in Materials on High-Performance Computers, Computing in Science & Engineering 16, 54 (2014).
  * De Giovannini  _et al._ [2015] U. De Giovannini, A. H. Larsen, and A. Rubio, Modeling electron dynamics coupled to continuum states in finite volumes with absorbing boundaries, The European Physical Journal B 88, 56 (2015).
  * Ullrich [2012] C. Ullrich,  _Time-Dependent Density-Functional Theory: Concepts and Applications_ , Oxford Graduate Texts (Oxford University Press, Oxford ; New York, 2012).
  * Mermin [1965] N. D. Mermin, Thermal Properties of the Inhomogeneous Electron Gas, Physical Review 137, A1441 (1965).
  * Åberg  _et al._ [2013] D. Åberg, P. Erhart, and V. Lordi, Contributions of point defects, chemical disorder, and thermal vibrations to electronic properties of Cd 1 - x Zn x Te alloys, Physical Review B 88, 045201 (2013).
  * Aumayr  _et al._ [1991] F. Aumayr, G. Lakits, and H. Winter, On the measurement of statistics for particle-induced electron emission from a clean metal surface, Applied Surface Science 47, 139 (1991).
  * Schrempf  _et al._ [2013] D. Schrempf, W. Meissl, and F. Aumayr, An ultra-compact setup for measuring ion-induced electron emission statistics, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 317, 44 (2013).
  * Kalkhoff  _et al._ [2023b] L. Kalkhoff, S. Matschy, A. S. Meyer, L. Lasnig, N. Junker, M. Mittendorff, L. Breuer, and M. Schleberger, Ultra-large polymer-free suspended graphene films (2023b), arXiv:2311.08137 [physics.app-ph] .
  * Wittkower and Betz [1973] A. B. Wittkower and H. D. Betz, Equilibrium-charge-state distributions of energetic ions (z $>$ 2) in gaseous and solid media, Atomic Data and Nuclear Data Tables 5, 113 (1973).
  * Schwestka  _et al._ [2019] J. Schwestka, A. Niggas, S. Creutzburg, R. Kozubek, R. Heller, M. Schleberger, R. A. Wilhelm, and F. Aumayr, Charge-Exchange-Driven Low-Energy Electron Splash Induced by Heavy Ion Impact on Condensed Matter, The Journal of Physical Chemistry Letters 10, 4805 (2019).
  * Yu  _et al._ [2009] Y.-J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, Tuning the Graphene Work Function by Electric Field Effect, Nano Letters 9, 3430 (2009).
  * Yan  _et al._ [2012] R. Yan, Q. Zhang, W. Li, I. Calizo, T. Shen, C. A. Richter, A. R. Hight-Walker, X. Liang, A. Seabaugh, D. Jena, H. Grace Xing, D. J. Gundlach, and N. V. Nguyen, Determination of graphene work function and graphene-insulator-semiconductor band alignment by internal photoemission spectroscopy, Applied Physics Letters 101, 022105 (2012).
  * Baragiola  _et al._ [2001] R. A. Baragiola, C. A. Dukes, and P. Riccardi, Plasmon excitation in ion–solid interactions, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 182, 73 (2001).
  * Ritzau and Baragiola [1998] S. M. Ritzau and R. A. Baragiola, Electron emission from carbon foils induced by keV ions, Physical Review B 58, 2529 (1998).
  * Kishinevsky [1973] L. M. Kishinevsky, Estimation of electron potential emission yield dependence on metal and ion parameters, Radiation Effects 19, 23 (1973).
  * Lakits  _et al._ [1990] G. Lakits, F. Aumayr, M. Heim, and H. Winter, Threshold of ion-induced kinetic electron emission from a clean metal surface, Physical Review A 42, 5780 (1990).
  * Wilhelm  _et al._ [2017] R. A. Wilhelm, E. Gruber, J. Schwestka, R. Kozubek, T. I. Madeira, J. P. Marques, J. Kobus, A. V. Krasheninnikov, M. Schleberger, and F. Aumayr, Interatomic Coulombic Decay: The Mechanism for Rapid Deexcitation of Hollow Atoms, Physical Review Letters 119, 103401 (2017).
  * Creutzburg  _et al._ [2020] S. Creutzburg, J. Schwestka, A. Niggas, H. Inani, M. Tripathi, A. George, R. Heller, R. Kozubek, L. Madauß, N. McEvoy, S. Facsko, J. Kotakoski, M. Schleberger, A. Turchanin, P. L. Grande, F. Aumayr, and R. A. Wilhelm, Vanishing influence of the band gap on the charge exchange of slow highly charged ions in freestanding single-layer MoS 2, Physical Review B 102, 045408 (2020).
  * Ni  _et al._ [2018] G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B.-Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, Fundamental limits to graphene plasmonics, Nature 557, 530 (2018).
  * Baragiola and Dukes [1996] R. A. Baragiola and C. A. Dukes, Plasmon-Assisted Electron Emission from Al and Mg Surfaces by Slow Ions, Physical Review Letters 76, 2547 (1996).
  * Riccardi  _et al._ [2003] P. Riccardi, A. Sindona, P. Barone, A. Bonanno, A. Oliva, and R. Baragiola, Bulk and surface plasmon excitation in the interaction of He+ with Mg surfaces, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 212, 339 (2003).
  * Riccardi and Dukes [2022] P. Riccardi and C. Dukes, Effects of the solid target on electronic excitations during binary atomic collisions in the interaction of Ne ions with Al surfaces, Vacuum 204, 111393 (2022).
  * Zhou  _et al._ [2016] Y. Zhou, D. S. Fox, P. Maguire, R. O’Connell, R. Masters, C. Rodenburg, H. Wu, M. Dapor, Y. Chen, and H. Zhang, Quantitative secondary electron imaging for work function extraction at atomic level and layer identification of graphene, Scientific Reports 6, 21045 (2016).
  * Duvenbeck  _et al._ [2004] A. Duvenbeck, F. Sroubek, Z. Sroubek, and A. Wucher, Computer simulation of low-energy electronic excitations in atomic collision cascades, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 225, 464 (2004).
  * Zarkadoula  _et al._ [2014] E. Zarkadoula, S. L. Daraszewicz, D. M. Duffy, M. A. Seaton, I. T. Todorov, K. Nordlund, M. T. Dove, and K. Trachenko, Electronic effects in high-energy radiation damage in iron, Journal of Physics: Condensed Matter 26, 085401 (2014).
  * Tsai and Jiang [2005] H.-L. Tsai and L. Jiang, Fundamentals of energy cascade during ultrashort laser-material interactions (Invited Paper), in  _Lasers and Applications in Science and Engineering_, edited by J. Fieret, P. R. Herman, T. Okada, C. B. Arnold, F. G. Bachmann, W. Hoving, K. Washio, Y. Lu, D. B. Geohegan, F. Trager, and J. J. Dubowski (San Jose, CA, 2005) p. 343.
  * Furry [1937] W. H. Furry, On Fluctuation Phenomena in the Passage of High Energy Electrons through Lead, Phys. Rev. 52, 569 (1937).
  * Dietz and Sheffield [2008] L. A. Dietz and J. C. Sheffield, Secondary electron emission induced by 5–30‐keV monatomic ions striking thin oxide films, Journal of Applied Physics 46, 4361 (2008).
  * Kim  _et al._ [2012] S. Kim, I. Jo, D. C. Dillen, D. A. Ferrer, B. Fallahazad, Z. Yao, S. K. Banerjee, and E. Tutuc, Direct Measurement of the Fermi Energy in Graphene Using a Double-Layer Heterostructure, Physical Review Letters 108, 116404 (2012).

##  VI Supporting Information

###  VI.1 Comparison of total emitted electrons with Literature

Figure S1:  Secondary electrons from (a) the entrance side and (b) and exit
side. The solid lines with open circles are the results of this work, with
absorbing boundary conditions. The cross markers are the results from Ref.
[19], with periodic boundary conditions.

###  VI.2 Potential emission yield in graphene

Following the expression derived in Ref. [57],

$\gamma=\frac{0.2}{E_{\mathrm{F}}}\cdot\left(0.8E_{\mathrm{i}}-2\omega_{0}\right)~{},$
(S1)

where $\mathrm{E_{i}}$ is the ionization potential (13.6 eV for proton), and
$\omega_{0}$ is the work function (4.56 eV for graphene) [53, 54].
$\mathrm{E_{F}}$ is the Fermi energy of graphene that can be calculated by
$E_{\mathrm{F}}=v_{\mathrm{F}}^{2}/2$ in the atomic unit, which is measured as
$v_{\mathrm{F}}=1.17\cdot 10^{6}\,\mathrm{m/s}$ [71]. Collecting all
parameters, we obtained the predicted potential emission yield as 0.09. We
also note that the Eq. S1 is most accurate in the range
$3\omega_{0}<E_{\mathrm{i}}<2(E_{\mathrm{F}}+\omega_{0})$. The ionization
energy $E_{\mathrm{i}}=13.6$ eV is slightly below the lower bound
$3\omega_{0}=13.68$ eV, which might result in a deviation.

###  VI.3 Dipole moment for potential plasmon excitation

We calculate the dipole moment by integrating the charge density

$d(t)=\int_{V}zn(r,t)dr^{3}$ (S2)

where $z$ is the position normal to the graphene surface, and $V$ is the
volume within $10.5\,\mathrm{a_{0}}$ of graphene. To remove the influence of
the electrons captured by the incident proton, we only include the time step
after proton leaves the graphene for 28 $\mathrm{a_{0}}$. To extract the
information on frequencies, we perform a Fourier transform of dipole moment.
Note that the coarse resolution is due to the limited time-scale and saved
charge density files.

Figure S2:  Fourier transform of the dipole moment, see Eq. (S2), of (a) the
channeling trajectory (A) and (b) centroid trajectory (O).

###  VI.4 Kinetic energy spectrum for secondary, captured and total emitted
electrons in the exit side

Figure S3:  The kinetic energy spectrum of the total emitted electrons at the
exit side of the centroid trajectory (O) and 25 keV proton kinetic energy is
shown in Fig. 4 in the main text. The secondary electrons are the total
emitted electrons subtracted by the captured electrons. The captured electrons
form a peak at the proton velocity, indicating that they are captured by the
proton.

###  VI.5 Kinetic energy spectrum at different time

Figure S4:  The time-dependent kinetic energy spectrum of the exit side of the
centroid trajectory (O) and 80 keV proton kinetic energy is shown in solid
lines. The kinetic energy spectrum accumulating over time is shown in Fig. 4
in the main text.

###  VI.6 Kinetic energy spectrum at elevated lattice temperature

Figure S5:  The kinetic energy spectrum of the emitted electrons of the
centroid trajectory (O) at lattice temperature 1000 K, compared with the
kinetic energy spectrum at the ground state 0 K lattice temperature. The
vertical dashed lines indicate the kinetic energy of the electrons moving at
the same speed as the proton.

###  VI.7 Charge density before proton irradiation, at 0 Kelvin, elevated
electron and elevated lattice temperature

To investigate the origin of the difference in the electron emission at
different electron and lattice temperatures, we compare the ground state
charge density among the elevated lattice temperature at 1000 Kelvin, elevated
electron temperature at 10,000 Kelvin, and those at 0 Kelvin.

Figure S6:  Ground state electron density of graphene. The black diamond
denotes the channeling trajectory (A), and the black X denotes the centroid
trajectory (O). The black circles denote the positions of carbon atoms in
graphene.

###  VI.8 Atomic positions near proton impact at different lattice temperature

The distortion of the atomic positions compared to its ground state at 0
Kelvin results in a shift of the exact positions of the proton impact point,
which adds another uncertainty to our comparison of the secondary electron
emission.

Figure S7:  Carbon atoms in graphene near proton impact points at $T$=$1,000$
K in (a) a top view and (b) a side view. Due to the distortion of the atomic
positions at 1,000 K lattice temperature, the impact point of the channeling
and centroid trajectories are not exactly identical compared to those at 0 K.

###  VI.9 Real-time electron emission at different temperatures

Figure S8:  Electron emission at the elevated lattice temperature of $1,000$
K, compared to emission at ground state 0 K, in both channeling trajectory (A)
and centroid trajectory (O). The proton is exactly in the graphene plane at
“impact”.