Charge-state effects on charge transfer of helium ions in gold nanosheet

We investigate the charge-state effects on charge transfer of helium ions in gold nanosheet using the time-dependent density functional theory non-adiabatically coupled to the molecular dynamics. In order to characterize and extract the charge-state information of incident particles inside the nanosheet, we develop two novel computational methods. It is found that the charge transfer behavior of He ion in gold nanosheet is sensitive to its charge state at the time of incident. Analysis of these results allows us to gain new insights in the interaction between He ions and gold nanosheet. This work validates the ability of current methodology in dealing with ion collisions in materials.


Introduction
When an incident particle moves through a material, it is angularly dispersed and loses its kinetic energy due to collisions with nuclei and the electrons that compose the material. A detailed theoretical understanding of the interaction of particle radiation with materials is highly crucial for addressing a number of important problems in many fields of fundamental science and practical technology [1,2], such as nuclear industry [3][4][5][6] and space technology [7] applications, high-energy density physics [8], radio-therapy in medicine [9][10][11], fundamental-research laboratories, as well as manufacturing of integrated circuits.
On average, the ability of a material to slow down projectile particles traveling in its interior is characterized by the stopping power [12][13][14][15][16]. Conceptually, the stopping power of a material for a given type of projectile can be divided into two categories depending on the type of particles that compose the material: one the nuclear stopping [12,16,17] at low projectile velocities, in which the projectile particle mainly transfers energy to nuclei without electronic excitations, and another the electronic stopping [13,[17][18][19][20] at high projectile velocities, in which the projectile particle mainly transfers energy to electrons.
Although for bulk materials under ion irradiation, stopping power has been fairly well-studied experimentally [15,16,[21][22][23][24], and tabulated for a diverse range of target materials and incident projectiles, there is a fair amount of information of detailed stopping process that can be extracted from a fully atomistic first-principles simulations which is not available in the form of experimentally tabulated data. Indeed, several modern first-principles simulations [13,17,[25][26][27][28][29][30] have provided a detailed description of the energy and charge dynamics in bulk, as it offers greater spatial and temporal resolution than currently achievable experimentally.
In general, the interaction of incident particles with the matter may trigger numerous processes, involving two degrees of freedom of nuclei and electrons. In light of the present findings of charge exchange/transfer effects, some interesting open points in electronic interaction of slow ions can be understood. For instance, the excessive stopping can be explained as a consequence of charge exchange, which represents an additional energy loss channel [12]. Overall, a good understanding of these intriguing charge exchange behaviors of the stopping process requires extensive microscopic knowledge of how an incident particle loses its kinetic energy and transfers energy to its surrounding atoms near the surface of the material. In addition, the physics of the charge transfer process itself deserves further attention, and it is important to take into account the projectile's charge state inside the material in order to identify the charge exchange effects that take part in the shopping process. For this reason, in this work, He ion in gold is taken as an example. Here, as a point of reference, the stopping power of He ions in bulk gold has been studied extensively both experimentally [23,24,[31][32][33][34] and theoretically [35] providing opportunities to validate our results for the bulk limit. Another point in this respect is that there is no model calculations made in similar gold nanosheet conditions covering the whole range from low to intermediate energies for helium ions. The purpose of this work is to make a comparative study of the charge transfer for helium ions in single crystal nanosheets of gold, considering the effects of the ion charge [36] on the stopping power.
The rest of this paper is organized as follows. In section 2, this paper mainly introduces the method and model used in this calculation. In section 3 introduces the results of this simulation calculation and some analysis of these results. In section 4 summarizes the conclusions of this study. Atomic units (h = m e = e = 1) are used throughout unless explicitly stated.

Computational methods
In a recent study, we have used a model to compute the slowing down of ions in single crystal films, where we combine time-dependent density functional theory (TDDFT) calculations for electrons with molecular dynamics (MD) simulations for ions in order to obtain an unbiased microscopic insight into the ion-film interaction on the same footing in real-time [37,38]. This approach allows for a self-consistent coupling of electronic and nuclear motion. Here we briefly recall the theoretical framework and the mathematical relations. The motion of electrons is described by the time-dependent Kohn-Sham (TDKS) equation, where ψ i is the Kohn-Sham orbital (KSO), and the HamiltonianĤ KS is defined bŷ where m e is the mass of an electron, and the TDKS potential V KS is divided into the following terms: where V en is the electron-nuclei Coulomb potential using pseudopotential or all-electron approach in calculations with atomistic details, such as V en (r, R(t)) = − The Hartree potentials, V H , is given by and accounts for the electrostatic Coulomb interaction between electrons, where ρ(r, t) represents the electronic density given by: where N is the number of electrons in the system. And V XC is the exchange and correlation (XC) potential, which can be expressed as [39] where E hom XC is the XC energy density of a homogeneous electron gas of densitiesρ, which has an exact expression in [40].
Then the motion of the Ith nuclei is determined by Newton's equations of motion: where M I and Z I are the mass and charge of the Ith nucleus, respectively. R(t) ≡ {R 1 (t), . . . , R L (t), . . . } denote ionic coordinates. We will utilize density functional theory to initialize the electronic orbitals of the projectile/target system, while TDDFT will be employed to simulate the electron dynamics. In our model of bare He 2+ ion collisions, He 2+ ion is simply given an initial velocity v. However, in the case of He + ion with one bound electron, the preparation of the initial state must be carefully considered in order to avoid an unphysical initial decoupling of the projectile's nucleus from the bound electron [41]. In order to achieve this, we included a phase factor e ik·r before KSO of the projectile's bound electron at time t = 0. This was done to ensure that the projectile's bound electron can move along with the projectile's nucleus [42], that is, ψ i (r, 0) −→ e ik·r ψ i (r, 0), with k = m e v/h. This phase factor, commonly referred to as the electron translation factor, causes the He + ion KSO to carry an extra linear momentum along with their kinematic energy [43].
When a projectile collides with a target, charge transfer takes place alongside the creation and breaking of bonding or antibonding states in the immediate vicinity of the target [44]. Charge transfer involving neutralization and reionization of ions is typically a localized process. The electron localization function (ELF) [44] is a tool used for the quantitative analysis of the complex bonding characteristics of electronic systems in their ground state, which is widely used in the search for atomic shells, covalent bonds, lone pairs, hydrogen bonds, ionic bonds, metal bonds and many others. Furthermore, the time-dependent ELF (TDELF) [45] serves as an extension of the ELF for the study of time-dependent processes, which enables the observation of the time-evolving bonding properties of an electronic system, taking into account the dynamics of excited electrons. The specific expression for TDELF is as follows: this equation compares the local value of D(r, t) with that of the Hartree-Fock homogeneous electron gas of the same local density D 0 (r, t) (a handy reference system against which to compare the actual value of D(r, t)). And D(r, t) is an effective local description of localization, the specific expression is as follows, with j(r, t) = |j(r, t)|, where j(r, t) is the electric current density, given by wherer denotes the operator of space coordinate r. In order to look at the electronic localization of systems as a whole, the average ELF, weighted according to the density, has been defined [46] as where Ω denotes the volume integral over a sphere of radius r Ω , centered at O Ω . Inspired by this, we defined average radial ELF as where Θ is the spherical shell of thickness ∆r Θ and radius r Θ , centered at O Θ . ELF is a dimensionless value. ELF is within the range of [0,1]. A large ELF value means that electrons are greatly localized, indicating that there is a covalent bond, a lone pair or inner shells of the atom involved. When the gold nanosheet is not disturbed by any disturbance, the ELF sketch is shown in figure 1(b). The calculations were done using the OCTOPUS code [47,48], with the numerical details and basic aspects summarized below. The wave function, density, and potential are discretized in a real space grid of a rectangular simulation cell under periodic boundary conditions in the x and y directions. The simulation cell size in the x and y directions was chosen so as to minimize the spurious effects of the repetition while maintaining manageable computational costs. Atoms of the system can be thought of as consisting of valence electrons and ionic cores, which consist of nuclei and core electrons. The interactions between valence electrons and ionic cores are described by means of the norm-conserving pseudopotentials of Troullier-Martins [49] type. In this research, two pseudopotential models are employed to substitute ionic cores of the system, one for the nucleus and the core electrons [Xe]4f 14 of the gold atom, and the other for the nucleus of He atom. In order to numerically propagate the equations by discretizing time, we use the velocity Verlet algorithm for time-stepping Newton's equations of motion, and an enforced time-reversal symmetry algorithm [50] for the TDKS equations, using the instantaneous local density approximation [51][52][53] to XC potential. Since the time step is controlled by the mesh spacing, we use an optimized time step of 0.01 a.u. for a selected mesh spacing of 0.45 a.u. under the condition that the time propagation is unitary. The mesh spacing corresponds to a cutoff energy of 47.91 Ry in the plane-wave expansion. We take the coordinate system shown in figure 1(a). A cubic unit cell of gold crystal adopted from [54], containing four atoms with a lattice parameter of 7.705 a.u., is expanded into a supercell, placed within a simulation box of size 15.41 a.u. × 15.41 a.u. × 61.65 a.u. The He ions begin at a distance of 15 a.u. from the gold nanosheet and travel through it along a channeling trajectory. A 2 × 2 × 1 grid of Monkhorst-Pack k points is used to sample the Brillouin zone.

Results and discussion
The stopping power S is used to measure a material's ability to stop incoming projectiles, defined as the energy loss per unit path length of charged ions [55]. It is calculated as S = −dE/dx [23,29,[56][57][58][59][60]. In the simulation of a gold nanosheet, the S is approximated as ∆E/∆x, where ∆E is the difference in kinetic energy before and after the projectile penetrates the nanosheet, and ∆x is the displacement of the projectile [23,56,61]. The size of the stopping power capability is represented by the Scattering Stop Cross section (SCS), calculated as ε = S/n, where n is the atomic number density of the target [23,56,61]. In this study, we calculated the ε value of He + and He 2+ particles in gold nanosheet during channeling and off-channeling. Figure 2 presents the simulation results along with reference values. The reference data comprises the SRIM-2013 curve (represented by a green solid line) [62], TDDFT calculations conducted by Zeb et al [35], and experimental values obtained from Martínez-Tamayo et al [24], Semrad et al [63], and Markin et al [23]. These references were selected due to their coverage of both high-velocity and low-velocity regions. It can be seen that the ε value trend of the two ions in channeled and off-channeled channels is consistent with the SRIM-2013 curve [62]. In the velocity range of 0.1-0.3 a.u., it is consistent with the experimental values of  [35], the black ▼ represents the experimental values of Martínez-Tamayo et al [24]. The experimental values of Semrad et al [63] are represented by the gray ◀, while the experimental value of Markin et al [23] is depicted as a gray ▶. It should be noted that in both experiments, the ε value refers to electronic stopping cross section. The inset displays typical incident points of channeling and off-channeling situations.
Markin et al [23], and the theoretically calculated values of Zeb et al [35]. All these indicate that the model parameters used in this simulation are reasonable. However, when the velocity is greater than 1.2 a.u., the calculated value of this simulation is lower than the reference value, which is caused by the sampling position of the incident point. The inset in figure 2 shows the incident points used in simulation, representing both channeling and off-channeling typical situations. Here, the off-channeling ε value is obtained by averaging three off-channeling incident points. It can be seen that for He + , the values of channeling and off-channeling are in line with the variation trend of the SRIM-2013 curve [62], and the value of off-channeling is greater than that of channeling. However, for He 2+ , the energy loss becomes gentle and decreases due to resonance charge transfer within the velocity interval of 0.6-1.0 a.u., regardless of channeling or off-channeling conditions. When the velocity is 0.6 a.u., the value of the off-channeling is lower than that of the channeling. To summarize, there is a notable disparity in the energy loss values between He + and He 2+ in gold nanosheet, which is partly due to the different charge transfer mechanisms exhibited by He ions with different charge states in the material.
In order to further describe the charge transfer process of the projectile inside the material, the probability P i 1s that the ith occupied KSO ψ i is projected onto the 1s orbital of He + ion is defined here as: where ϕ 1s represents the static wave function of He + at the position of R He . It is emphasized that the ϕ 1s is computed at each time t, for the reason that the positions of the He + ion are different from its initial ones. k can be expressed as where v(t) represents the velocity of projectile ions at time t. And the probability P i 1s that the ith occupied KSO does not project onto the 1s orbital of He + ion is expressed as, So the probability that there is only one electron in the 1s orbital of He + ion during the projectile's travel is, It is worth noting that the higher the excited state, the larger the volume occupied. In gold nanosheet, the space to accommodate electrons captured by projectiles is limited, so only the 1s ground state electrons are most likely to survive in collision processes. Generally, upon entering the material, the charge state of the projectile initially will undergo a pre-equilibrium process [55,64,65] before its equilibration occurs. In pre-equilibrium regime, the response of materials to projectile irradiation has been shown to depend on the charge state of the projectile. According to figure 3, the charge transfer process can be categorized into five distinct stages as outlined below: (a) pre-entering stage, where the He ion is located far from the front surface of the gold nanosheet; (b) entering stage, in which He ions enter the front surface of the gold nanosheets; (c) internal stage, where the He ions reside within the gold nanosheets; (d) leaving stage, in which the He ions leave the rear surface of the gold nanosheets; (e) post-leaving stage, where the He ion is located far from the rear surface of the gold nanosheet. Figure 3(a) shows the results of He + traveling through the gold nanosheet. It can be seen that P 1s can always remain 1 in stage (a), indicating that no charge transfer occurs around He + during this process. In stage (b), the probability of He + being in the 1s state will sharply decrease with increasing velocity. In stage (c), the probability of He + in the 1s state tends to change slowly. In stagr (d), there is a significant perturbation in the probability of He + in the 1s state. In stagr (e), the probability of He + in the 1s state remains unchanged. Figure 3(b) shows the results of He 2+ traveling through the gold nanosheet. It is evident that, in stage (a), the probability of He 2+ capturing an electron to the 1s state has consistently remained at 0. In stage (b), the probability of He 2+ capturing an electron to the 1s state increases sharply with increasing velocity. In stage (c), the probability of He 2+ capturing an electron to the 1s state undergoes gradual changes over time. In stage (d), there is a significant perturbation in the probability of He 2+ capturing an electron to the 1s state. In stage (e), the probability of He 2+ capturing an electron to the 1s state remains unchanged. Comparing the collision processes of He + and He 2+ projectiles, several interesting findings are as follows: (1) during the collision process at a lower velocity of 0.3 a.u., the electrons of He + projectile ions in the 1s state are almost not lost, while He 2+ projectile ions hardly capture electrons to the 1s state. (2) At a higher velocity of 0.9 a.u., the probability of He ions formed after collision capturing one electron to the 1s state tends to approach equality. (3) It is found that the charge transfer behavior of He ion in gold nanosheet is sensitive to its charge state at the time of incident.
It is a well-established fact that metal crystals contain free electrons that are not confined to individual metal atoms, but rather are shared across the entire crystal structure. Therefore, the outer valence electron of  metallic crystals are in a low localization state (see figure 1(b)). In figure 4, ELF diagrams of two He ions at different velocities at z = 0 a.u. are given. During the entire collision process, He ions are surrounded by a cluster of electrons that can move with the projectile, and these electrons are captured by He ions. The electron capture region is surrounded by a low localization and density electron hole region, resulting in the presence of electron holes along the trajectory. It clearly shows that the electron excitation regime is located around the He ion. The distribution of holes is concentrated along the trajectory of the projectile. The ELF value indicates the degree of localization of an electron, with higher values indicating a more bound state. In the case of He ions, the 1s state electrons exhibit the highest localization. Therefore, a higher ELF value for electrons captured by He ions implies a greater probability of the electron being in the 1s state. By comparing the cases of He + and He 2+ projectiles under identical conditions (velocity and position), it becomes evident that the probability of a bound electron of He + remaining in the 1s state is higher than the probability of He 2+ capturing an electron to the 1s state.
To depict a more detailed charge transfer behaviors, we conducted calculations to determine the average radial ELF distribution at various positions for He 2+ projectiles with different velocities (all with He 2+ as the center of integration, r Θ = 5 a.u., ∆r Θ = 0.3 a.u.), which can be used to describe the distribution of electrons captured by projectile ions. The corresponding results can be observed in figure 5. Several interesting findings are as follows: (1) the charge transfer process occurs in a sequential manner. Initially, electrons are captured by a distant, higher excited state, away from He 2+ . As the collision progresses, the quantity of trapped electrons in the higher excited state steadily rises, accompanied by the transfer of charge to the lower Figure 6. The ELF snapshot of He ions generated by He + and He 2+ projectiles at incident velocities of 0.3, 0.6, and 0.9 a.u., respectively. This phenomenon occurs when the projectile and target are widely separated after collision. excited state, which is in closer proximity to He 2+ . (2) The electronic state distribution of He ions formed after collision with He 2+ at three different velocities of 0.3, 0.6, and 0.9 a.u. can be observed in figure 5(d), revealing significant differences. Figure 6 illustrates the distribution characteristics of electronic states in He ions generated by He + and He 2+ projectiles at incident velocities of 0.3, 0.6, and 0.9 a.u., respectively, when projectile and target are widely separated after collision. It can be seen that the ELF distribution of He ions can be divided into two categories: high localization and low localization. High localization means that the electron is mainly in a lower excited state, while low localization means that the electron is mainly in a higher excited state. Interestingly, for a He 2+ projectile with a velocity of 0.3 a.u., a hollow He ion can be formed. Figure 7 shows the average number of electrons (n e ) of He ions generated by He + and He 2+ projectiles in the case of channeling and off-channeling incidence when projectile and target are widely separated after collision. Several interesting findings can be observed from figure 7(a) as follows: (1) The n e for He + projectile is almost not sensitive to velocity v within the range of 0.1 ⩽ v ⩽ 0.7 a.u. However, as the velocity increases within the range of 0.7 < v ⩽ 2.0 a.u., the n e decreases. (2) For He 2+ projectile, n e exhibits an obvious resonance peak in the range of 0.1 ⩽ v ⩽ 2.0. (3) When 0.1 ⩽ v < 1.0 a.u., there is a significant difference in the variation trend of n e between He + and He 2+ projectiles, indicating that their charge transfer mechanisms have significant differences. Figure 7(b) demonstrates that the change in n e with velocity remains consistent across various integration radii. This finding suggests that the integration radius employed in this simulation is indeed appropriate.
The charge transfer mechanism may be understood based on a phenomenological classical over-barrier model [66], in which, as an electron moves from a target to a projectile, it tends to occupy a projectile orbital where its single-particle energy E p = ε p i + ∆ closest to that E T = ε T j of the target, with ε p i /ε T j denoting the energy of the ith/jth single-particle orbital of the projectile/target at rest, and ∆ denoting the translational kinetic energy of the transferred electron traveling with the projectile. In this model picture, for the low/high incident energy case, the velocity of the projectile is rather low/high and the electron around the Fermi level of the target prefers to transfer into the weakly/strongly bound orbital of the projectile, leading to the formation of highly/lowly excited projectile after the collision. Clearly, electrons in metals can be divided into two categories, one is non-localized valence electrons that can cruise between atoms, and the other is localized core electrons bound near atoms. When the velocity of the projectile ion is low, the core electron cannot be excited, and only the valence electron can be excited. Therefore, the energy loss is mainly contributed by the valence electron excitation. In this case, the energy loss is not sensitive to the track of the projectile ion passing through the metal (i.e. not sensitive to the channeling and off-channeling situations), and of course not sensitive to whether the freedom of the core electron is opened in the model. When the velocity of projectile ions is high, both core electrons and valence electrons can be excited and contribute to the energy loss. The energy loss in this case is certainly sensitive to the trajectory of projectile ions passing through the metal (i.e. sensitive to channeling and off-channeling conditions) and is also sensitive to the freedom of opening the core electrons.

Summary
In this paper, the charge-state effects on charge transfer of helium ions in gold nanosheet has been investigated by using the TDDFT non-adiabatically coupled to the MD. In order to extract the probability of having one electron in the 1s orbital of He + ions during the projectile's travel, we have developed a state-selective projection probability method. In order to visually characterize charge transfer behavior during the projectile's travel, we developed an average ELF weighted according to the density. These two methods can be mutually validated and supplemented. The charge transfer process of He ion projectiles within gold nanosheet is investigated in detail, obtaining a combination of interesting findings: (1) The energy loss for He 2+ projectile does not change linearly with velocity, as there is a resonant charge transfer occurring within the velocity range of 0.6-1.0 a.u. (2) The charge transfer behavior of He ion in gold nanosheet is sensitive to its charge state at the time of incident. (3) The electron capture region is surrounded by a low localization and density electron hole region, resulting in the presence of electron holes along the trajectory. (4) The charge transfer process occurs in a sequential manner. (5) For He 2+ projectile, n e exhibits an obvious resonance peak in the range of 0.1 ⩽ v ⩽ 2.0. When 0.1 ⩽ v < 1.0 a.u., there is a significant difference in the variation trend of n e between He + and He 2+ projectiles, indicating that their charge transfer mechanisms have significant differences. Analysis of these results allows us to gain new insights that charge transfer is an important reason for the difference in stopping power between He ions with different charge states and the gold nanosheet. This work validates the ability of present methodology in dealing with ion collisions in nanosheet, which may have implications for the study of stopping power in more complex systems in the future. It is still an active field and many new results are expected in the future.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.