Monolayer and Bilayer MoS₂ Under Proton Irradiation: Electronic Stopping and Charge Capture Revealed by Real-Time TDDFT

Abstract

Monolayer and few-layer MoS₂ are promising two-dimensional electronic materials, but proton irradiation can trigger ultrafast electronic excitation and charge transfer before defect formation. Here, real-time time-dependent density functional theory (RT-TDDFT) is used to investigate proton-induced electronic stopping and localized charge capture in monolayer and bilayer MoS₂ under normal incidence. Four impact positions are examined in monolayer MoS₂, namely, the hollow channel, the Mo–S bond center, and two trajectories close to Mo and S atoms. Under hollow channel incidence, the stopping power shows a non-monotonic dependence on proton velocity. When comparing the different trajectories, the hollow channel path gives the lowest stopping power, whereas the Mo–S bond center path gives the highest values, indicating strong sensitivity to the in-plane valence charge distribution. By contrast, the time-averaged localized captured charge decreases with increasing velocity and is generally largest for the close to Mo trajectory. Under the same hollow channel condition, the monolayer stopping power exceeds the bilayer value in the main stopping region, whereas the bilayer generally shows slightly enhanced localized charge capture. These results show that electronic stopping and localized charge capture are distinct but coupled microscopic components of proton-induced electronic response in MoS₂ and provide first-principles insight relevant to ion-beam processing and radiation-tolerant two-dimensional devices.

1. Introduction

Atomically thin molybdenum disulfide, MoS₂, is a representative two-dimensional transition-metal dichalcogenide semiconductor for electronic and optoelectronic applications. Owing to its layer-dependent band structure, sizable bandgap, strong light–matter interaction, and mechanical flexibility, MoS₂ has been widely investigated for field-effect transistors, photodetectors, flexible electronics, valleytronic devices, and related nanoscale device platforms [1,2,3,4,5,6,7,8,9]. In such applications, the atomic thickness that enables strong electrostatic control also makes the material highly sensitive to external perturbations, including point defects, local strain, charged impurities, interfaces, and energetic-particle irradiation. Understanding how energetic ions interact with monolayer and few-layer MoS₂ is therefore important for both device reliability and controlled ion-beam modification of two-dimensional electronic materials.

Ion irradiation can modify two-dimensional materials through electronic excitation, nuclear collisions, defect production, local reconstruction, and charge redistribution [10,11,12]. In MoS₂ and related transition-metal dichalcogenides, ion beams have been used for nanopatterning, defect engineering, and electrical tuning, while proton and heavy-ion irradiation are also relevant to radiation environments such as space electronics and radiation-tolerant sensors [10,11,12,13,14,15,16]. In particular, proton irradiation experiments on MoS₂ field-effect transistors have shown that the irradiation response can depend on layer thickness, suggesting that monolayer and few-layer MoS₂ may exhibit different microscopic energy-loss and charge-transfer behavior [15]. However, the ultrafast electronic processes that occur during the initial stage of a single ion impact remain difficult to access directly in experiments.

Recent studies have further demonstrated that irradiation can be used not only as a degradation source but also as a tool for defect engineering and functional-property tuning in two-dimensional materials. For example, Yao et al. investigated nonequilibrium electron emission from cold and hot graphene under proton irradiation, showing that ultrafast electronic excitation and emission processes in atomically thin targets can be highly sensitive to irradiation conditions [17]. In monolayer MoS₂, high-energy electron-beam irradiation has been reported to generate optically active defect states and induce valley depolarization, indicating the strong sensitivity of excitonic and valley responses to irradiation-induced defects [18]. Recent theoretical work has also shown that ion irradiation can engineer sulfur-vacancy formation in two-dimensional MoS₂, thereby modifying its electronic properties and hydrogen-evolution activity [19]. In addition, swift-heavy-ion irradiation has been used to tailor the band gap, structural, and optical properties of CVD-grown few-layer MoS₂ [20]. These recent advances confirm that irradiation-induced electronic excitation, defect formation, and charge redistribution are central to controlling the properties of MoS₂-based materials.

During proton irradiation, two primary microscopic processes are especially important: electronic stopping and charge capture. Electronic stopping describes the rate at which the projectile transfers kinetic energy to the electronic subsystem of the target material through electronic excitations [13,14]. Charge capture, by contrast, describes the transient transfer of electrons from the target to the projectile, which modifies the effective charge state of the moving proton and can influence the subsequent ion–matter interaction [21]. These two processes are coupled but not equivalent. Electronic stopping is governed mainly by the excitation efficiency of the target electrons along the projectile path, whereas charge capture is strongly affected by interaction time, local electronic availability, and the spatial localization of the induced density around the projectile. A simultaneous description of both quantities is therefore necessary for understanding proton-induced electronic response in MoS₂.

The electronic response of atomically thin MoS₂ differs from that of conventional bulk materials in several ways. First, the projectile interacts with the target over an extremely short out-of-plane distance, so the energy deposition process is highly localized in both space and time. Second, the in-plane electronic density is strongly inhomogeneous because of the hexagonal lattice, Mo–S bonding network, and the hybridization between Mo d states and S p states. As a result, protons passing through hollow channel regions, Mo–S bond centers, or regions close to Mo and S atoms may experience substantially different local electronic environments. Third, the reduced dimensionality and dielectric screening of a monolayer can alter the efficiency of electronic excitation and transient charge transfer [22]. These features make trajectory-resolved analysis essential for connecting local atomic structure with electronic stopping and charge-capture behavior.

Layer number provides an additional degree of freedom in MoS₂ electronic materials. Many practical devices are based not only on ideal monolayers but also on bilayer or few-layer structures, where interlayer coupling, dielectric screening, and stacking configuration modify the electronic structure [15,16,23,24]. The presence of an additional layer can change the spatial distribution of electronic states and the screening environment sampled by a traversing proton. It is therefore not sufficient to infer the irradiation response of bilayer MoS₂ directly from monolayer results. A direct comparison between monolayer and bilayer systems is required to clarify how interlayer coupling affects electronic energy deposition and transient electron capture.

Building on this context, studies published between 2024 and 2026 have made significant strides in exploring the radiation tolerance and defect engineering of transition metal dichalcogenides under extreme environments [25,26]. Concurrently, advancements in computational methods have further elucidated the long-term structural evolution of 2D materials under various energy beams [27,28,29]. A critical comparison nevertheless reveals a remaining gap: while these contemporary works predominantly focus on macroscopic device degradation, steady-state defect structures, or long-term thermalization, the initiating event—the ultrafast, sub-femtosecond non-adiabatic electronic excitation and transient charge capture—remains largely unresolved.

First-principles real-time time-dependent density functional theory (RT-TDDFT) provides a suitable approach for resolving this nonequilibrium electronic stage of ion irradiation [30,31]. By propagating the electronic wave functions in real time in the presence of a moving charged projectile, RT-TDDFT can describe femtosecond-scale electronic excitation, energy transfer, and charge redistribution without relying on empirical stopping models. Previous studies have demonstrated the capability of RT-TDDFT to describe electronic stopping in metals, semiconductors, and insulators, including velocity and trajectory effects [32,33,34,35,36,37]. Nevertheless, systematic first-principles studies of proton-induced electronic stopping and charge capture in two-dimensional transition-metal dichalcogenides remain limited, particularly with respect to atomic-scale trajectory dependence and monolayer–bilayer differences.

In this work, RT-TDDFT simulations are used to investigate proton-induced electronic stopping and transient charge capture in monolayer and bilayer MoS₂. Normal-incidence proton trajectories are considered over a range of projectile velocities. For monolayer MoS₂, four representative impact positions are selected, including a hollow channel path, a Mo–S bond center path, and two centroid-like paths close to Mo and S atoms, respectively, to resolve the role of in-plane electronic inhomogeneity. A comparison between monolayer and bilayer MoS₂ is then performed under channeling conditions to evaluate layer-dependent effects. The results clarify how projectile velocity, atomic-scale trajectory, and interlayer coupling jointly control electronic energy loss and electron capture in proton-irradiated MoS₂, providing microscopic insight relevant to ion-beam processing and radiation response of two-dimensional electronic materials.

2. Computational Models and Methods

The integrated computational workflow for modeling the proton-induced electronic response in monolayer and bilayer MoS₂ is summarized in Figure 1, where the procedure is explicitly divided into three stages: atomistic system preparation, trajectory-resolved real-time propagation, and post-processing analysis of energy dissipation and charge exchange.

2.1. Material Models and Ground-State Properties

To investigate the layer-dependent electronic response under proton irradiation, we constructed atomistic models of monolayer and bilayer MoS₂ strictly based on the thermodynamically stable 2H phase [7,8,9]. While realistic experimental irradiation or high-strain conditions may eventually induce localized structural transformations to metastable phases (such as the 1T phase), restricting our dynamic simulations to the pure 2H phase is physically justified by the strict timescale separation of the interaction. The 2H-to-1T phase transition requires the collective intra-layer gliding of sulfur atoms, a nuclear process occurring on the scale of hundreds of femtoseconds. Conversely, the primary proton transit and the accompanying electronic excitation occur in less than 5 fs. Consequently, dynamic structural phase transitions cannot occur during the initial energy deposition window. Furthermore, because the 2H phase is the globally predominant matrix in functional MoS₂ electronic devices, evaluating its isolated response provides the fundamental baseline necessary for understanding femtosecond-scale electronic stopping and initial charge capture before any long-term structural degradation occurs.

The monolayer system was modeled using a $3\times 3\times 1$ periodic supercell containing 9 Mo atoms and 18 S atoms in a trigonal prismatic coordination. This supercell dimension yields a lateral distance of approximately 9.5 Å between periodic images. To eliminate spurious interactions between periodic replicas along the out-of-plane direction, a total vacuum region of 16 Å was introduced. The finite-size convergence of the lateral supercell, including $2\times 2\times 1$ through $5\times 5\times 1$ benchmark cells, is discussed below together with the numerical benchmark tests.

For the bilayer configuration, two MoS₂ layers were arranged in the 2H-AB stacking sequence [16], representing the most energetically favorable structure observed experimentally. This specific stacking geometry allows us to explicitly capture layer-dependent electronic coupling and dielectric screening effects, which play critical roles in modulating charge capture during ion traversal.

The ground-state electronic structures of these material models were obtained via density functional theory (DFT) using the real-space grid-based OCTOPUS code [38,39,40,41]. The interaction between valence electrons and ionic cores was described by norm-conserving pseudopotentials from the PseudoDojo library (nc-sr-04_pbesol standard set) [42], ensuring high transferability for structural and electronic property calculations. The exchange-correlation interactions were treated within the local density approximation (LDA), employing the Perdew–Wang [43] and Perdew–Zunger [44] parameterizations. This functional form provides a reliable description of the valence electronic structure relevant to energy-loss processes in low-dimensional systems. Furthermore, the electronic wave functions were represented on a uniform real-space grid with a spacing of 0.18 Å. This fine grid resolution is physically necessary to accurately capture both the highly localized d-orbital states of the transition metal (Mo) and the high-momentum components of the electronic density fluctuations that will be induced by the fast-moving proton in subsequent dynamic simulations. The ground-state self-consistent field (SCF) calculations were performed with a strict convergence criterion of ${10}^{-6}$ for the relative density difference. To properly integrate the Brillouin zone of the $3\times 3\times 1$ supercell, a $\Gamma$-centered Monkhorst–Pack k-point mesh was employed. For the subsequent RT-TDDFT dynamic propagation, the k-point sampling was restricted to the $\Gamma$ point to maintain computational feasibility while capturing the highly localized electronic response.

2.2. Real-Time Electron Dynamics Under Proton Irradiation

The interaction between energetic protons and MoS₂ layers involves highly non-equilibrium processes where electronic excitations and charge redistribution occur on a femtosecond timescale. To resolve proton-induced nonequilibrium electronic response from first principles, we employed RT-TDDFT [45,46] simulations with prescribed proton trajectories and a frozen-lattice approximation. Formally, the RT-TDDFT/Ehrenfest framework couples the evolving electronic density to ionic coordinates through

$$M_I{\displaystyle \frac{d^2{\mathbf{R}}_I}{d{t}^2}}=-{\nabla }_{I}E[\rho (\mathbf{r},t),\left\{{\mathbf{R}}_I\right\}]$$

(1)

where $M_I$ and ${\mathbf{R}}_I$ denote the mass and position of ion I, respectively, and E is the instantaneous total energy functional of the system. In the present constant-velocity implementation, this coupled framework is reduced to a prescribed normal-incidence proton trajectory with fixed MoS₂ lattice ions; the electron density is then propagated quantum mechanically according to the time-dependent Kohn–Sham (TDKS) equations:

$$i\hslash {\displaystyle \frac{\partial {\psi }_i(\mathbf{r},t)}{\partial t}}=\left(-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2+V_{\mathrm{KS}}\left[n\right](\mathbf{r},t)\right){\psi }_i(\mathbf{r},t)$$

(2)

where ${\psi }_i(\mathbf{r},t)$ represents the time-dependent Kohn–Sham orbitals. In the equation, $\mathbf{r}$ and t represent the spatial and time coordinates, respectively. The instantaneous electron density at time t is given by

$$n(\mathbf{r},t)=\sum _i{\left|{\psi }_i(\mathbf{r},t)\right|}^2$$

(3)

The instantaneous electron density is given by the sum of the single-electron probabilities. The Kohn–Sham effective potential $V_{\mathrm{KS}}$ can be expressed as follows:

$$V_{\mathrm{KS}}\left[n\right](\mathbf{r},t)=V_{\mathrm{ext}}[\mathbf{R}\left(t\right),\mathbf{r}]+V_{\mathrm{H}}\left[n\right](\mathbf{r},t)+V_{\mathrm{xc}}\left[n\right](\mathbf{r},t),$$

(4)

$V_{\mathrm{ext}}[\mathbf{R}\left(t\right),\mathbf{r}]$ represents the interaction between the ions and the electrons. $V_{\mathrm{H}}\left[n\right](\mathbf{r},t)$ represents the classical electrostatic interaction between the electrons. $V_{\mathrm{xc}}\left[n\right](\mathbf{r},t)$ is a crucial term in density functional theory (DFT) that accounts for the quantum mechanical effects arising from the exchange and correlation interactions between electrons.

To propagate the electronic wave functions in time, we utilized the Enforced Time-Reversal Symmetry (ETRS) propagator [38]. Given the rapid electronic response associated with ion-induced excitations, a stringent time step of 0.001 fs was adopted to ensure the numerical stability and unitary nature of the integration.

The projectile was modeled as a bare proton (${\mathrm{H}}^{+}$) by setting the total charge of the simulation cell to $+1$. The proton was initially positioned in the vacuum region and assigned a normal-incidence velocity directed along the z-axis. To systematically probe the in-plane inhomogeneity of the electronic environment, four representative trajectories were assigned: the channeling path (hollow region), the bond center path (Mo–S bond), and centroid-like paths passing near Mo and S atoms, respectively.

A key assumption in our model is the frozen-lattice approximation, where the lattice atoms of the target remain fixed throughout the simulation. This is physically justified by two quantitative factors. First, SRIM estimates confirm that for protons in the 0.5–2.5 a.u. velocity range (∼6–150 keV), the nuclear stopping power ($S_n$) is typically less than 3.2% of the electronic stopping power, making the nuclear contribution to the direct energy loss negligible. Second, there is a stark time-scale separation: at typical velocities (e.g., 1.0 a.u. or ∼2.18 Å/fs), the proton traverses the effective atomic thickness of the monolayer in less than 3 fs. Since the fastest optical phonon periods in MoS₂ are on the order of 80–100 fs, the atomic cores do not have sufficient time to undergo significant displacement during the primary interaction window. Therefore, while subsequent electron–phonon coupling governs long-term thermalization and structural damage, the instantaneous trajectory-dependent electronic response and charge capture dynamics are robustly captured within the frozen-lattice framework.

It is important to note that while the deep core electrons are frozen within the norm-conserving pseudopotentials, all valence electrons of the MoS₂ lattice are explicitly treated as active. These valence electrons are fully available to interact with the projectile and are propagated in real time via the TDKS equations. This approach allows us to decouple the electronic stopping power from nuclear recoil effects, providing a direct measure of non-adiabatic energy dissipation into the active electronic degrees of freedom [31,47].

2.3. Evaluation of Stopping Power and Charge Capture

The electronic stopping power, $S_e$, characterizes the rate at which the incident proton dissipates its kinetic energy into the electronic system of the MoS₂ target. This non-adiabatic energy transfer is a primary driver for localized heating and potential defect generation in irradiated microelectronics. In the present RT-TDDFT simulations, the stopping power was extracted from the increase in the electronic energy $E_{\mathrm{el}}\left(t\right)$ during proton traversal.

$$S_e=-{\displaystyle \frac{d{E}_{\mathrm{el}}}{dx}}=-{\displaystyle \frac{1}{v}}{\displaystyle \frac{d{E}_{\mathrm{el}}\left(t\right)}{dt}}$$

(5)

where v is the constant velocity of the projectile. This constant-velocity approximation is well justified because the total electronic energy loss during the traversal of the atomically thin target (typically tens of eV) is orders of magnitude smaller than the initial kinetic energy of the proton (tens of keV for the velocities considered here), resulting in a negligible change in velocity (<0.2%). To mitigate numerical noise arising from instantaneous high-frequency electronic oscillations, the energy-loss curves were processed using moving average smoothing and linear regression, yielding statistically robust stopping power values for each incident velocity [30,31].

Beyond energy dissipation, quantifying the transient charge exchange is critical for evaluating the radiation tolerance of 2D electronic devices. Unlike bulk materials, the reduced dielectric screening in atomically thin MoS₂ makes it highly sensitive to local charge fluctuations, which can trigger transient leakage currents or single-event upsets (SEUs). To capture this dynamic process, we evaluated the transiently captured charge using a real-space density difference approach rather than a standard orbital projection:

$$\Delta n(\mathbf{r},t)=n(\mathbf{r},t)-n_0\left(\mathbf{r}\right)$$

(6)

where $n(\mathbf{r},t)$ is the instantaneous electron density during the collision and $n_0\left(\mathbf{r}\right)$ is the ground-state reference density of the pristine MoS₂. The time-averaged localized captured charge is then obtained by integrating this density difference within a spherical region centered at the instantaneous position of the moving proton, ${\mathbf{R}}_p\left(t\right)$:

$${\overline{Q}}_{\mathrm{cap}}={\int }_{|\mathbf{r}-{\mathbf{R}}_p\left(t\right)|<R_c}\Delta n(\mathbf{r},t)d^{3}r$$

(7)

The cutoff radius was set to $R_c=2.0$ Bohr to isolate the localized charge tightly bound to the projectile while minimizing spurious contributions from the surrounding host electron density [48]. Physically, this radius (∼1.06 Å) encompasses the spatial extent of a 1s-like hydrogenic bound state while remaining sufficiently smaller than the Mo–S bond length (∼2.4 Å) to exclude background charge-density fluctuations from the host lattice. The finite-size convergence tests discussed below further evaluate possible periodic-image overlap in the charge-integration analysis. This time-resolved measure provides direct physical insight into the efficiency of charge transfer under varying trajectory and layer-coupling conditions.

To guarantee the physical fidelity and numerical stability of the simulated electron-ion dynamics, key computational parameters were rigorously benchmarked. Quantitative convergence tests were performed using the hollow channel trajectory at a representative velocity of $v=1.0$ a.u. Decreasing the real-space grid spacing from 0.18 Å to 0.15 Å altered the calculated electronic stopping power ($S_e$) by less than 3.1%. Similarly, reducing the propagation time step from 0.001 fs to 0.0005 fs resulted in an $S_e$ deviation of under 0.7%, while ensuring strict total energy conservation. For localized charge extraction, increasing $R_c$ from 2.0 Bohr to 2.5 Bohr changed ${\overline{Q}}_{\mathrm{cap}}$ by only 4.3%, and additional sensitivity analyses over 1.5–2.5 Bohr confirmed that the extracted charge-capture trends remain robust within this physically reasonable range. Thus, aside from the separate lateral finite-size effect evaluated below, the chosen grid, time-step, and charge-integration parameters provide a stable and reliable description of both the energy-loss curves and the localized charge capture phenomena.

To rigorously evaluate the finite-size effect, benchmark calculations were expanded to include $2\times 2\times 1$, $3\times 3\times 1$, $4\times 4\times 1$, and $5\times 5\times 1$ supercells at $v=1.0$ a.u. under hollow channel incidence. The calculated electronic stopping powers ($S_e$) were 9.228, 9.211, 9.991, and 10.036 eV/Å, respectively. This non-linear convergence behavior originates from the competition between long-range dynamic screening and short-range localized binding. In the smaller cells ($2\times 2\times 1$ and $3\times 3\times 1$), the artificial overlap of transient polarization clouds between periodic images suppresses the long-range collective excitations. Upon expanding to $4\times 4\times 1$ and $5\times 5\times 1$, the lateral distance becomes sufficient to decouple these mirror wake effects, allowing $S_e$ to reach a physical plateau (with a negligible 0.45% difference between the $4\times 4$ and $5\times 5$ cells). Conversely, the localized captured charge (${\overline{Q}}_{\mathrm{cap}}$) relies on deep, short-range Coulomb binding and is relatively immune to boundary perturbations, showing a maximum deviation of less than 3.6% across all cell sizes. Due to the extreme computational cost of real-time Ehrenfest dynamics, the $3\times 3\times 1$ supercell was retained for all production runs, providing a balanced and tractable description of the localized ultrafast dynamics.

3. Results and Discussion

The atomic models and representative proton impact positions used in the present RT-TDDFT simulations are summarized in Figure 2. For monolayer MoS₂, four normal-incidence trajectories are considered, namely the hollow channel, the Mo–S bond center, and two impact positions close to Mo and S, respectively. For the monolayer–bilayer comparison, the same hollow channel geometry is adopted so that the influence of layer number can be examined under equivalent impact conditions.

3.1. Reference Electronic Stopping Under Hollow Channel Incidence

We first consider monolayer MoS₂ under hollow channel incidence and compare the RT-TDDFT stopping powers with Stopping and Range of Ions in Matter (SRIM) reference values, as shown in Figure 3. Over the extended velocity range, the RT-TDDFT stopping power increases from the low-velocity regime, reaches a maximum in the intermediate-velocity range, and then decreases again at higher velocities. This non-monotonic dependence reflects the competition between increasing electronic-excitation probability and decreasing interaction time as the proton velocity increases.

The SRIM curve exhibits the same qualitative velocity dependence but gives systematically larger absolute values than the RT-TDDFT results. Such a difference is expected because SRIM provides an empirical bulk-material reference, whereas the present RT-TDDFT simulations explicitly resolve the finite-thickness monolayer geometry, the selected hollow channel trajectory, and the nonequilibrium electronic response during proton traversal. The comparison therefore serves as a physically meaningful reference rather than a strict validation against a bulk-like model.

It should be noted that, based on the four-supercell finite-size benchmarks described in Section 2.3, the absolute electronic stopping power values obtained using the production $3\times 3\times 1$ supercell carry a systematic finite-size uncertainty (error margin) of approximately 8% relative to the converged large-supercell plateau. However, because this finite-size constraint applies systematically to the underlying host matrix, it acts primarily as a global quantitative offset. Consequently, the relative physical trends discussed herein—specifically the trajectory-dependent ordering and the non-additive layer effects—are fully robust and independent of this systematic error margin.

3.2. Trajectory-Dependent Electronic Response in Monolayer MoS₂

Figure 4 combines the trajectory dependence of the electronic stopping power and the localized captured charge in monolayer MoS₂. As shown in Figure 4a, the stopping power increases with projectile velocity over the range of 0.5–0.9 a.u. for all four trajectories, but its magnitude depends strongly on the impact position. The hollow channel trajectory consistently yields the lowest stopping power, whereas the Mo–S bond center trajectory gives the highest values throughout the investigated velocity interval. The close to Mo and close to S trajectories show intermediate behavior, with only a weak change in their relative ordering as the velocity increases.

This pronounced trajectory dependence indicates that proton-induced energy deposition in monolayer MoS₂ is governed by the in-plane inhomogeneity of the local electronic environment. A proton moving through the hollow region interacts with relatively low projected valence charge density and therefore induces weaker electronic excitation. In contrast, the Mo–S bond center trajectory samples a charge-richer bonding region, leading to stronger coupling with the electronic subsystem and a larger stopping power. The two near-atomic trajectories probe intermediate local environments and therefore produce intermediate stopping strengths.

A different trend is observed for the localized captured charge in Figure 4b. For all four trajectories, ${\overline{Q}}_{\mathrm{cap}}$ decreases monotonically with increasing projectile velocity, even within the same velocity interval where the stopping power rises. In addition, the trajectory ordering of the captured charge is not identical to that of the stopping power. The close to Mo trajectory generally gives the largest values, whereas the hollow channel trajectory shows the strongest reduction at higher velocity and becomes the lowest at 0.8–0.9 a.u.

These results show that electronic stopping and localized charge capture should not be regarded as interchangeable descriptors of proton-induced electronic response. The stopping power mainly reflects the efficiency of electronic excitation and energy transfer along the trajectory, whereas the localized captured charge is more sensitive to interaction time and to the local availability of electrons that can transiently accumulate around the moving proton.

Taken together with the hollow-channel velocity scan in Figure 3, these monolayer results can be placed in the broader context of other 2D materials studied using similar first-principles methods. Our calculated stopping powers for MoS₂ (peaking between roughly 9 and 15 eV/Å) lie in the same order of magnitude as those reported for proton irradiation of freestanding graphene (typically peaking around 4 to 8 eV/Å) [45,49]. The higher absolute energy loss in MoS₂ is physically consistent with its larger effective thickness and higher valence electron density compared to a single carbon layer. Furthermore, the observed physical trends—specifically the strong trajectory dependence in Figure 4 and the non-monotonic velocity dependence in Figure 3—closely mirror the phenomena established in graphene and related 2D systems [50]. Our localized charge capture dynamics and energy-loss magnitudes are also conceptually aligned with recent experimental and TDDFT investigations of highly charged ion and proton interactions with MoS₂ [46]. This cross-material consistency supports the view that coupling between local in-plane electronic inhomogeneity and transient energy dissipation is a general feature of ion-irradiated 2D materials.

3.3. Layer-Dependent Electronic Response Under Hollow Channel Incidence

The effect of layer number is summarized in Figure 5, which compares monolayer and bilayer MoS₂ under hollow channel incidence. As shown in Figure 5a, both systems exhibit a similar overall velocity dependence of the stopping power: the stopping power first increases, then reaches a maximum, and finally decreases at higher velocity. However, the layer effect is clearly non-additive. At the lowest velocities considered, the bilayer gives comparable or slightly larger stopping values than the monolayer. Above approximately 0.7 a.u., the monolayer becomes systematically larger and shows a higher peak stopping power in the main stopping region.

This result indicates that the presence of a second layer does not simply increase the energy loss by providing more electrons. Instead, interlayer coupling and screening modify the spatial distribution of the induced electronic response and reduce the effective excitation efficiency sampled along the hollow channel path. Therefore, the lower stopping power of the bilayer in the main stopping region should be interpreted as a consequence of layer-dependent screening and finite-thickness response, rather than as a reduction in the total number of electrons.

As shown in Figure 5b, the localized captured charge decreases with increasing projectile velocity in both systems, while the bilayer generally shows slightly enhanced charge capture compared with the monolayer over most of the plotted range. This layer dependence differs from that of the stopping power, suggesting that the additional MoS₂ layer modifies the transient electronic redistribution around the moving proton rather than simply amplifying the monolayer response.

Taken together, Figure 5 demonstrates that stopping power and charge capture respond differently to the presence of an additional MoS₂ layer. The bilayer does not behave as a simple sum of two monolayers; instead, it exhibits a distinct balance between excitation, screening, and transient charge redistribution.

3.4. Mechanistic Interpretation from the Local Charge-Density Environment

The physical origin of the above trends is summarized schematically in Figure 6. In the top-view projected valence charge density environment, the hollow channel trajectory passes through a relatively electron-depleted region at the center of the hexagonal network, whereas the Mo–S bond center trajectory crosses a region with stronger bonding charge. The close to Mo and close to S trajectories probe intermediate local environments. This spatial picture establishes a direct physical correlation between the static charge density and the dynamic energy loss. The magnitude of the electronic stopping power (Figure 4a) scales with the integrated valence charge density (Figure 6a) sampled along the proton’s path. The Mo–S bond center and near-atomic trajectories feature higher local electron densities, increasing the probability of inelastic scattering events and yielding higher $S_e$. Conversely, the hollow channel lies in an electron-depleted region, resulting in the lowest excitation efficiency.

Furthermore, the local potential landscape and atomic orbital characteristics explain the distinct curve crossings observed in the transient charge capture (Figure 4b). Notably, the localized captured charge for the hollow channel drops precipitously with increasing velocity, crossing below the other trajectories. At lower velocities, the proton’s longer transit time allows it to induce long-range polarization and capture charge even in the hollow region. However, at higher velocities, kinematic matching becomes stringent; stabilizing transient charge around a fast-moving projectile requires strong, localized atomic potential wells. The hollow channel lacks nearby atomic cores to provide this stabilization. In contrast, trajectories passing close to Mo atoms benefit from the highly localized Mo d-orbitals, which serve as robust charge reservoirs capable of sustaining transient electron accumulation even under fast impact conditions.

The side-view schematic in Figure 6b further illustrates the mechanistic difference between the monolayer and bilayer environments. It is important to note that this difference does not arise from strongly overlapping static ‘interlayer states’ in the ground state, but rather from dynamic polarization and screening across the van der Waals gap. When the proton transits the bilayer, the second layer expands the spatial volume available for dielectric response. As visualized by the polarization field in Figure 6b, the interlayer coupling facilitates a cooperative transient screening effect. This expanded dynamic screening volume modifies the local electric field, which explains the reduced effective excitation efficiency (lower $S_e$) in the main stopping region and provides a broader environment for transient localized charge accumulation (larger ${\overline{Q}}_{\mathrm{cap}}$).

3.5. Implications for Proton-Irradiated MoS₂ Electronic Materials

The present results have several implications for proton-irradiated MoS₂ electronic materials. First, the large variation among the four monolayer trajectories indicates that laterally averaged stopping values may obscure substantial atomic-scale differences in local energy deposition. Second, the comparison between monolayer and bilayer systems shows that layer number modifies the balance between energy deposition, screening, and charge accumulation in a nontrivial manner, so the bilayer response cannot be inferred by simply scaling the monolayer result. Third, the opposite velocity trends of stopping power and localized captured charge highlight that ultrafast energy deposition and transient charge transfer must be considered separately when discussing irradiation response, ion-beam processing, and charge-related degradation in MoS₂-based devices.

While the present RT-TDDFT simulations describe ultrafast femtosecond-scale electron dynamics, these microscopic insights qualitatively correlate with recent macroscopic experimental observations. For instance, proton irradiation experiments on MoS₂ field-effect transistors have shown that the irradiation response depends significantly on layer thickness [15]. Our finding that bilayer MoS₂ exhibits reduced effective stopping in the main stopping region together with slightly enhanced charge capture compared to the monolayer provides a fundamental electronic origin for these layer-dependent macroscopic behaviors. Furthermore, our computed localized charge transfer mechanisms are qualitatively consistent with experimental observations regarding transient charge exchange in 2D materials, which alters the projectile’s effective charge state and influences subsequent ion-matter interactions [51]. Thus, while direct quantitative comparisons of initial femtosecond energy loss remain experimentally challenging, our theoretical framework bridges the gap between atomic-scale impact geometry and the practical radiation response observed in MoS₂ devices.

Although the present simulations focus on the electronic component of proton irradiation and do not include nuclear recoil or long-time structural relaxation, they identify the primary femtosecond-scale electronic processes that precede defect formation. The results therefore provide a microscopic basis for understanding how local impact geometry and layer number influence proton-induced electronic response in two-dimensional MoS₂.

4. Conclusions

In this work, real-time time-dependent density functional theory (RT-TDDFT) simulations were used to investigate proton-induced electronic stopping and localized charge capture in monolayer and bilayer MoS₂. The results show that the proton-induced electronic response is jointly controlled by projectile velocity, atomic-scale impact position, and layer number. For hollow channel incidence over an extended velocity range, the electronic stopping power exhibits a non-monotonic dependence on proton velocity. Within the narrower velocity window used for trajectory comparison in monolayer MoS₂, the stopping power increases with velocity while retaining a pronounced dependence on the impact position.

In monolayer MoS₂, the hollow channel trajectory consistently gives the lowest stopping power, whereas the Mo–S bond center trajectory gives the highest values, with the close to Mo and close to S trajectories lying in between. This ordering indicates that proton-induced energy deposition is strongly governed by the local in-plane electronic environment. By contrast, the time-averaged localized captured charge decreases monotonically with increasing velocity for all trajectories and is generally largest for the close to Mo path. These results show that electronic stopping and localized charge capture should not be treated as interchangeable descriptors of proton-induced electronic response. The former is governed mainly by the efficiency of electronic excitation along the projectile path, whereas the latter is more sensitive to interaction time and local electron availability.

The comparison between monolayer and bilayer MoS₂ under hollow channel incidence reveals a non-additive layer effect. The bilayer gives comparable or slightly larger stopping values at the lowest velocities, whereas the monolayer becomes larger in the main stopping region. At the same time, the bilayer generally exhibits slightly enhanced localized charge capture. These opposite-layer trends indicate that energy deposition and transient charge accumulation respond differently to interlayer coupling and screening.

Overall, the present results show that trajectories sampling charge-rich bonding or near-atomic regions promote stronger electronic excitation and therefore larger stopping power, whereas lower projectile velocity favors larger localized charge capture because of the longer interaction time. These findings provide microscopic insight into proton-induced electronic response in MoS₂ electronic materials and may help interpret irradiation experiments and guide the design of ion-beam processing conditions and radiation-tolerant two-dimensional devices. Extending this framework to include nuclear recoil and longer-time structural evolution will be important for connecting the initial electronic response to permanent defect formation.