Large‐Scale Cooperative Sulfur Vacancy Dynamics in Two‐Dimensional MoS₂ From Machine Learning Interatomic Potentials

ABSTRACT

The formation of extended sulfur vacancies in MoS₂ monolayers is closely associated with catalytic activity and may also be the basis for its memristive behavior. Nanosecond‐scale molecular dynamics simulations using machine learning interatomic potentials (MLIPs) reveal key mechanisms of cooperative vacancy transport, including incorporation of vacancies into clusters of arbitrary size. The simulations provide a coherent atomistic explanation for irradiation‐induced vacancy patterns observed experimentally, especially the formation of line defects spanning tens of nanometers. Results and performance are compared of two MLIP frameworks: (i) on‐the‐fly learning with Gaussian approximation potential, and (ii) fine‐tuning of an equivariant foundation model.

Sulfur vacancy dynamics in monolayer MoS₂ are investigated using two machine learning interatomic potential (MLIP) approaches. Both accurately model vacancy migration, enabling nanosecond‐scale molecular dynamics simulations that uncover and explain mechanisms of cluster formation and line defect growth – in good agreement with recent experiments.

1. Introduction

Sulfur vacancy migration plays a central role in a wide range of phenomena in 2D MoS₂, with relevance to applications including catalysis, chemical sensing, photoluminescence, and electronic transport [1, 2, 3, 4, 5, 6]. In emerging device concepts for next‐generation computing, such as memtransistors and memristive elements based on 2D MoS₂ [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], the redistribution of sulfur vacancies within the conduction channel has been widely implicated as a key mechanism underlying their functional behavior [7, 10, 14, 21, 22, 23, 24, 25].

Understanding and ultimately controlling such device functionality requires quantitative insight into the dynamic mechanisms governing vacancy migration and aggregation. A detailed understanding of these mechanisms must account for the strong dependence of vacancy mobility on the local atomic environment, including variations in defect structure, charge state, and migration pathways [7, 26, 27, 28]. For example, sufficient mobility at room temperature has been shown to require interactions among adjacent vacancies, i.e. vacancy‐assisted transport, which can reduce migration barriers from 2–3 to 0.6–0.8 eV [7, 26, 27, 28]. However, even with these reduced barrier heights, the associated dynamics still occur on time scales that remain inaccessible to conventional molecular dynamics (MD) methods. Much longer simulation times are particularly desirable because the unusual vacancy dynamics in MoS₂ has experimentally been observed to lead to the agglomeration of vacancies into clusters, which preferentially form line defects [26]. At high vacancy concentrations typical for device applications, or elevated temperatures, vacancy clusters can span up to tens of nanometers [27]. These extended defect structures provide locally conductive pathways for charge transport and further reduce the barrier of vacancy migration [26, 27, 28, 29, 30, 31, 32], making them directly relevant to electronic device functionality.

In the past, several atomistic simulation methods have been applied to understand sulfur vacancy dynamics in MoS₂. Density functional theory (DFT) has been widely used to calculate electronic properties, formation energies, and migration barriers of selected defect configurations [26, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. While offering high accuracy, these methods are computationally expensive, limiting them to static analyses of migration barriers or ab initio MD simulations on picosecond timescales [43, 44]. Consequently, they are unable to fully capture the relatively slow dynamic evolution of sulfur vacancy arrangements. Purely static analyses of defect dynamics often fail to capture individual low‐energy migration pathways, resulting in discrepancies with experimental observations [36]. kinetic Monte Carlo (kMC) models based on these DFT energy barriers extend accessible time and length scales by orders of magnitude [45, 46, 47], but rely on predefined migration events [48]. In MoS₂, the strong dependence of vacancy migration barriers on the local atomic environment [46] and the cooperative character of vacancy transport make it extremely challenging to construct accurate models for anything beyond the simplest defect configurations. Other studies have addressed vacancy dynamics in MoS₂, using classical MD [28, 49, 50, 51, 52, 53] and tight‐binding MD [30, 54]. Classical and tight‐binding MD can simulate nanosecond‐scale dynamics in large systems. They have been valuable for exploring cluster formation and local rearrangements but rely on parameterized force fields that may not generalize to unforeseen defect configurations [48]. Although these classical force fields have been shown to achieve good accuracy for migration barriers around very small sulfur vacancy clusters, they can fail to predict energy barriers for atomic jumps within larger vacancy clusters at DFT accuracy [28]. For instance, a previous study addressing similar issues as the present work employed a classical reactive force field and successfully demonstrated the formation of triple‐line vacancies through the combination of a double vacancy and an additional vacancy [28]. However, it failed to account for the incorporation of additional vacancies required to form extended line defects. This limitation stems from the fact that the activation energies for low‐barrier vacancy jumps were consistent with DFT calculations only in the case of double vacancies. Despite their respective strengths and the insights kMC and classical MD approaches have provided, capturing the structural and dynamic evolution of extended defect structures remains challenging.

We address this challenge by performing MD simulations of MoS₂ using specifically trained machine learning interatomic potentials (MLIPs), an emerging approach for modeling large‐scale vacancy migration and defect evolution. MLIPs have recently demonstrated near‐DFT accuracy across nanosecond timescales in similar systems, offering a viable compromise between accuracy and computational efficiency [55, 56, 57, 58]. More specifically, recent studies have demonstrated that MLIPs can accurately capture effects of defects and atomic‐scale dynamics in a variety of materials [59, 60, 61, 62, 63]. Among the available approaches, the Gaussian approximation potential (GAP) [64] and the MACE graph neural network (GNN) [65] exhibit complementary strengths: GAP enables uncertainty‐based on‐the‐fly training [66, 67], while MACE supports transfer learning from general‐purpose foundation models [68, 69, 70]. In this work, we compare these methods by training GAP models during MD simulations of MoS₂ and fine‐tuning MACE models to MoS₂ DFT data starting from the MACE MP‐0 foundation model [68]. For both approaches, we evaluate different training data selection strategies and apply the resulting MLIPs to simulate sulfur vacancy migration over multiple nanoseconds, capturing structural evolution processes that remain inaccessible to conventional atomistic techniques. Our simulations reveal the underlying mechanism of sulfur vacancy dynamics in MoS₂ monolayers and provide an atomistic explanation for the experimentally observed aggregation of small defect clusters into extended line defects spanning tens of nanometers.

2. Results and Discussion

2.1. Training of MLIP Models

First of all, we outline the process of generating MLIPs that can later be applied to perform MD simulations of defective supercells on the time scale of nanoseconds with close to DFT accuracy. As described in detail in the Method Section 4.2, we employed two different machine learning architectures to construct MLIPs. The first is the equivariant GNN MACE [65] using the atomic cluster expansion to describe structural environments [71]. Second, we applied GAPs [64] using an on‐the‐fly training approach and atomic descriptors as implemented in VASP [66, 67, 72]. For each MLIP architecture, two different approaches for generating training data were tested. All models are fine‐tuned to DFT forces and energies, but differ in the number of structures in the training set and how these structures were obtained. The test errors of selected MLIPs models are reported in detail in Section 4.2.3. With one exception, the resulting force and energy test errors of the various MLIPs models are within the expected range for these MLIP architectures: a few meV/atom for the energies and a few 10 meV Å for the forces [65, 66]. Only a variant of the GAP model that has been fine‐tuned on supercells with four quite different defects shows comparably high test errors for forces. We interpret this as trade‐off between transferability and accuracy. We emphasize that these test errors are obtained using a deliberately stringent evaluation protocol based on equilibrium and non‐equilibrium configurations sampled from a MD simulation using the tested MLIP (Section 4.2.3), which systematically leads to higher errors than those typically reported in MLIP benchmarks focusing on near‐equilibrium structures. In recent complementary work, we have compared additional universal foundation and fine‐tuned MLIP architectures and show that all tested MLIPs require fine‐tuning to accurately predict MD, even though the corresponding foundation models achieve high accuracy in benchmarks dominated by equilibrium structures [63].

2.2. Evalution of MLIP Predicted Potential Energy Curves for Cooperative, Vacancy‐Assisted Hopping

The subject of this work, the large‐scale dynamics, is determined by rather rare events. In contrast, the test errors reported in Table 1 are based on equidistantly selected snapshots from MLIP MD simulations, which were reevaluated using DFT. As a result, rare events—such as vacancy jumps—are unlikely to be included in the test set. Consequently, the test errors primarily reflect the accuracy of the MLIPs on low‐energy structures that frequently occur during MD simulations in the form of vibrations of atoms around their equilibrium positions and deformations of the local environment. Therefore and in order to assess the accuracy of our MLIPs in a more targeted way, we calculate and compare potential energy curves of a particular sulfur defect jump within the nudged elastic band (NEB) framework, a standard approach—usually of DFT—to find the transition state and the lowest energy barrier between two prescribed configurations [73]. Specifically, we look at the jump of a sulfur atom to an interstitial site neighboring a pair of two adjacent vacancies in one of the sulfur layers. This jump has previously been studied using high‐resolution transmission electron microscopy (TEM) and DFT [26] and is illustrated by the vertical solid arrow in the inset of Figure 1b. After the jump, the sulfur atom is positioned at an interstitial site centrally between three sulfur vacancies, with two of these vacancies present prior to the atom's jump and the third vacancy being at the site from which the atom originated. In our MD simulations and actual experiments, this jump is typically followed by another jump of the same sulfur atom from the interstitial site back to the original vacancy or to one of the previously existing vacancies, as indicated by the non‐vertical arrows. These subsequent jumps are characterized by the same potential energy curve as the first jump to the interstitial site due to the symmetry of the structure. We note in passing that the result of the two equivalent jumps can be considered as vacancy‐assisted hopping [74, 75]: a sulfur atom and a vacancy switch positions in the presence of another “assisting” vacancy.

Table 1.

Training and test errors (root‐mean‐squared error: RMSE) of selected MLIPs. For the MACE architecture, results are shown for two models. One trained on 50 000 snapshots from an AIMD simulation and the other on 2000 snapshots from an MD simulation using the MACE MP‐0 foundation model—both featuring two adjacent sulfur vacancies in the initial structure. The GAP‐based MLIPs were trained on‐the‐fly, either using four different defect structures (see Section 4.2) or a single structure with two ($S_2$) or three ($S_3$) adjacent sulfur vacancies. The test sets consist of 150 structures that were computed from 10 ns long MD simulations using the tested MLIP. The reference energies and forces were computed via DFT on these test structures. All MLIP simulations used the same initial structures as those employed in the training or fine‐tuning of the respective models. For the GAP MLIP trained on multiple defect structures, application tests were conducted using a structure containing three adjacent sulfur vacancies.

	Training RMSE		Test RMSE
	Energy (meV/atom)	Force (meV/Å)	Energy (meV/atom)	Force (meV/Å)
MACE	1.1	25.8	10	112
(test set 2000 structures)
MACE	0.2	16.1	4.8	42.0
(AIMD 50 000 structures)
GAP	4.9	75.3	26	142
(multiple defect structures)
GAP	0.9	70.2	29	73.8
($S_2$ vacancy)
GAP	0.9	75.7	23	85.9
($S_3$ vacancy)

Figure 1.

Potential energy curves for the defect jump shown in the inset of (b) calculated using the NEB method. A comparison between results calculated with DFT and MACE or GAP MLIPs trained on different datasets (see Section 4.2) is shown in (a) and (b), respectively. MACE MLIPs were trained on data from ab initio molecular dynamics (AIMD) simulations (label: “AIMD”) or from DFT calculations of structures from an MLIP MD simulation using the MACE MP‐0 foundation model (label: “test set”). The GAP models differ in the defect structure present during the on‐the‐fly training MD simulations (see Section 4.2 for more details). In addition to the NEB data points, we show spline interpolations through them as a guide for the eye. In the inset, sulfur atoms are displayed in yellow and molybdenum atoms in purple. The moving sulfur atom is indicated with a black ring, and the surrounding vacancies are indicated with red circles. The potential energy curves shown here correspond to the jump indicated by the vertical arrow. The non‐vertical arrows represent possible additional components of jumps that are observed in MD simulations and that share the same potential energy curve as the first component. The configuration coordinate is defined in Equation (1) in Section 4.3.

In the following, we compare the NEB potential energy curves for various different MLIPs versions to reference DFT calculations for this jump (full blue line in Figure 1a,b). The DFT energy difference between the initial and final configurations is 0.81 eV. The energy barrier for the jump to the interstitial site is 0.83 eV, which makes the latter weakly meta‐stable with a very small energy barrier of about 0.02 eV for the backward jump from the interstitial site back to the original lattice site or one of the other two equivalent lattice sites.

The results for MACE and GAP MLIPs are shown in Figure 1a,b, respectively. Most noticeably, the MACE foundation model (gray symbols) yields even qualitatively wrong results: It predicts the energy of the interstitial sulfur defect to be considerably lower than that of the initial configuration with only two vacancies. All fine‐tuned MACE models perform significantly better than the foundation model. However, the agreement with the DFT NEB calculations depends quite strongly on the choice of training set. Black and purple symbols mark results for MACE MLIPs fine‐tuned to test sets of 200 and 2000 DFT calculations for configurations obtained as snapshots from a MD trajectory calculated with the foundation model. While these models accurately reproduce the potential energy curve near its minimum, they significantly deviate from the DFT reference values for energy barriers (0.57 and 0.59 eV, respectively, for the forward jump and 0.38 and 0.24 eV, respectively, for the backward jump from the interstitial site into one of the vacancies). The poor description of the transition region is somewhat surprising, because as the ground state of the foundation model, it should actually be overrepresented in the training sets. A possible explanation lies in our observation that the MACE foundation MD trajectory includes numerous other unphysical transition states. For example, it predicts rather frequent sulfur diffusion through the molybdenum layer from one side of the MoS₂ sheet to the other.

Qualitatively almost correct and quantitatively, much better results are obtained by fine‐tuning the MACE model on 200, 2000, and 50 000 snapshots obtained from AIMD simulations (red, orange, and green symbols, respectively). For a completely different material system, this approach has been shown to yield highly accurate MLIPs over the whole configurational space using a relatively small fine‐tuning dataset of around 1 ps of AIMD data [76]. In the case of MoS₂, we find that the accuracy at regions farther away from the potential energy minimum depends sensitively on the size of the fine‐tuning set. Specifically, the MLIP fine‐tuned on 200 AIMD snapshots predicts an energy difference of 0.52 eV between the initial and final configuration, while the one fine‐tuned on 2000 snapshots yields a value of 0.73 eV. With a substantially larger fine‐tuning set size of 50 000 snapshots using every second snapshot of a 100 ps long AIMD trajectory with time step of 1 fs, the MACE model closely approaches the DFT value (0.81 eV) for this energy difference with 0.80 eV and generally predicts the potential energy curves very accurately. None of these three AIMD‐fine‐tuned MACE models predict the final configuration of the jump to be metastable. Given that DFT predicts a backward energy barrier of only $0.03\mathrm{eV}=k_{B}348\mathrm{K}$, this will cause only very minor errors in all our MD simulations, which were performed for much larger temperatures.

To summarize the results for MACE MLIPs shown in Figure 1a: (i) Using a large training set sampled from AIMD simulations, we are able to fine‐tune a model that very accurately describes these sulfur jumps and in particular predicts an activation energy that matches the reference DFT value. (ii) The MACE foundation model and all models fine‐tuned on structures taken from MD simulations using the foundation model strongly underestimate the potential energy of the transition state. This underprediction of migration barriers has recently been reported for a variety of chemical systems and multiple universal MLIPs, including the MACE MP‐0 foundation model by Deng et al. [69]. Our observations for MoS₂ fully confirm their emphasis on the importance of sampling non‐equilibrium structures during the training. However, Deng et al. [69] did not see, or at least did not report the difference between sampling structures for the fine‐tuning dataset from MD simulations using the foundational MLIP or more accurate but slower and more expensive AIMD simulations that we observed.

Next, we turn toward the performance of the GAP MLIPs, see Figure 1b. As described in Section 4.2, these models were trained using an on‐the‐fly learning approach during MD simulations of different defect configurations. The models trained starting with a $S_2$ vacancy pair (orange symbols) or a $S_3$ vacancy triplet (green) neatly match the DFT potential energy curve (blue) near the energy minimum and show only minor deviation near the maximum: Both predict the interstitial position to be meta‐stable. The energy barriers of 0.74 and 0.78 eV for the forward jump are close to the DFT value (0.83 eV), but slightly less accurate than the best MACE model (0.80 eV). The better performance of the vacancy‐triplet‐trained GAP MLIPs may be due to the fact that its on‐the‐fly learning involved more jumps. We checked that the jumps occurring in the on‐the‐fly training runs for these MLIPs involve the same process as studied in the NEB calculations. In contrast, for the model trained starting with only a single vacancy in the structure ($S_1$, purple) no jumps at all occurred during the training. Not surprisingly, it shows much larger deviations from the DFT potential energy curve than the previous two. Also the MLIP trained on multiple, structurally different vacancy clusters (red symbols) is accurate near the energy minimum, but shows a large deviation of about 0.4 eV in the high‐energy transition state, and is therefore not suitable for accurately describing the defect dynamics in MD simulations.

2.3. Discussion of Other Hopping Processes

Wang et al. [46] conducted a comprehensive analysis of DFT‐calculated energy barriers for sulfur jumps into a vacancy for the use in kMC calculations. They considered all 256 possible configurations of the eight neighboring sites surrounding both the initial and final positions of the jumping sulfur atom. Wang et al. identified the jump used in our evaluation in Figure 1 as having one of the lowest energy barriers, alongside with related configurations where additional neighboring sites are vacant. The relatively low barrier is attributed to the presence (“assistance”) of a vacancy adjacent to the destination site, which allows the sulfur atom to access the interstitial site. The latter is energetically favorably only if it is not directly above a molybdenum atom in the layer below, which is the case for every second of these interstitial sites as illustrated in Figure 2.

Figure 2.

Top (a) and side (b) view of a MoS₂ monolayer in the hexagonal 2H phase. Sulfur and molybdenum are displayed in yellow and purple, respectively. The annotations in (a) mark the final jump discussed in Section 2.3: A sulfur atom hopping into a line of three vacancies (red circle: vacancy, black ring: jumping atom on the intermediate, interstitial site).

Two defect jumps similar to the one shown in Figure 1 were previously studied by Ostadhossein et al. [49]. These jumps also originate from a double vacancy configuration, like the one considered in our work. However, in contrast to the jump analyzed here, the mobile sulfur atom in their study starts from a site farther from the center of the double vacancy, directly adjacent to only one of the two vacancies. The DFT‐predicted energy barrier for the jump investigated in our work is lower than those of these alternative jumps, which were reported to have barriers of 1.35 and 2.32 eV, depending on the sulfur atom involved [49]. In earlier studies, ReaxFF reactive force fields have been developed for MoS₂ [49, 50, 51, 52, 53]. Ostadhossein et al. [49] evaluated the accuracy of their force field by comparing the predicted potential‐energy profiles of these jumps to DFT results. Although the types of the jumps differ, we note that the energy barrier errors predicted by ReaxFF are larger than those from our best MLIPs, when benchmarked against DFT.

We also calculated the potential energy curve for the jump of a sulfur atom into a single sulfur vacancy. The minimum energy path for this jump into a single vacancy does not run through the center of the triangle spanned by three sulfur lattice sites, cf. inset of Figure 1b. Instead, it keeps more distance to the now occupied lattice site and thus runs closer to the neighboring molybdenum site. Compared to the case of two adjacent vacancies discussed earlier, the energy barrier for this jump is much larger: Our most accurate MACE model yields a value of 1.81 eV. Thus, according to the exponential Arrhenius law and considering the 1 eV barrier‐height difference, the jump of a sulfur atom into one of two adjacent vacancies is far more likely than into a single vacancy at any technologically relevant temperature.

Finally, we calculated using our best MACE MLIP model potential energy curves for the jump of one of the sulfur atoms into to a straight line of three vacancies, see Figure 2b. The sulfur atom can jump either into the center of the vacancy line or into one of the outer vacancies. We find an energy barrier of about 0.8 eV just as the jump into two neighboring vacancies shown in Figure 1 for both of these alternatives. However, the potential energies of the two possible final configurations differ significantly: The energy of the final configuration with the sulfur atom in the center of the row is 0.28 eV higher than that of the configuration with a sulfur atom occupying one of the outer vacancies. Consequently, the energy barrier for the reverse jump is 0.28 eV lower, i.e. approximately 0.52 eV. Energy barriers for jumps within a three‐vacancy cluster have previously been calculated using DFT and a ReaxFF force field [28]. The values predicted by our MLIP ($\sim 0.8$ eV) are in good agreement with those DFT results. In contrast, the ReaxFF force field drastically underestimates the barrier for the sulfur jump into the central vacancy (0.30 eV). Simultaneously, it overestimates the barrier for sulfur jumps into one of the outer vacancies, predicting 1.37 eV compared to 0.85 eV from DFT [28]. Notably, Gao et al. show that for only two adjacent vacancies, their ReaxFF model yields energy barriers in good agreement with DFT. As a result, their classical MD simulations provide valuable insights into the evolution of double vacancies. However, as the cluster size increases, the ReaxFF force field fails to accurately capture the correct dynamics due to its inaccurate barrier predictions, as will be discussed below in more detail.

2.4. Evaluation of MLIP‐Predicted Radial Distribution Functions

Radial distribution function (RDF) of sulfur $g_{S-S}(r)$ in a single layer of MoS₂ provide another way to assess the quality of the MLIPs for MD simulations. Results using both, DFT and MLIPs are shown in Figure 3. The first peak at about 3.2 Å corresponds to both, the distance between neighboring sulfur atoms within one of the two sulfur layers visible in Figure 2b and the distance between two sulfur atoms from the two different layers that are on top of each other in Figure 2a. The RDFs show good agreement between the two AIMD simulations performed using CP2K and VASP (see Section 4.1 for details), which serve as the reference, and the GAP and MACE MLIPs, which produced the most accurate potential energy curves. Only the MACE foundation model is unable to predict the RDF accurately. The AIMD trajectory generated using CP2K was employed as training data to fine‐tune the MACE model, while the VASP DFT implementation was used in the on‐the‐fly learning of the GAP model. Differences observed in the resulting RDFs from these two AIMD simulations arise from the distinct DFT implementations, as described in Section 4.1. However, the difference in peak area between the AIMD RDFs is significantly smaller than the difference with the MACE foundation model, which also fails to produce the correct potential energy surface.

Figure 3.

Time‐averaged S‐S radial distribution function $g(r)$ calculated from MD simulations of a MoS₂ monolayer with three sulfur vacancies in a row using both, AIMD and MLIPs. The GAP and MACE fine‐tuned MLIPs examined here are those with the best agreement with DFT in terms of potential energy curves in Figure 1a,b. The RDFs are calculated as described in Sections 4.4.

2.5. Sulfur Vacancy Dynamics

After generating MLIPs, capable of predicting potential energy curves and RDFs with satisfactory accuracy, we applied them to determine the mean squared displacement (MSD) and defect jump rates of various defect structures in monolayer MoS₂ on timescales not accessible with AIMD. The much extended accessible simulation times also allow for the identification of cooperative effects, which are discussed in the next section.

Figure 4a shows the sulfur MSD of a structure containing three sulfur vacancies computed from MD simulations for $T=1000$ K calculated with the MACE foundation model and the best fine‐tuned MACE (green line) and GAP (orange) models shown in Figure 1. As described in Section 4.5, these MSDs are corrected for the displacement due to atomic vibrations and thus show only the displacements of mobile sulfur atoms. For purely diffusive motion, all the conventional MSD and its variants (summed of atoms, corrected for vibrations) would grow linearly with time.

Figure 4.

(a) Mean squared displacement (MSD) summed over all sulfur atoms calculated as described in Section 4.5 from MD simulations of a MoS₂ monolayer starting with three sulfur vacancies in a row, using three different MLIPs. The GAPs and MACE fine‐tuned MLIPs applied here are those with the best agreement with DFT in terms of potential energy curves in Figure 1. For reference, we also show the results of MD simulations using the MACE foundation model, which shows poor agreement with DFT on the potential energy curve in Figure 1a and is thus expected to dramatically overestimate the mobility of sulfur vacancies. (b) MSD of mobile sulfur atoms calculated from MD trajectories with two (dashed lines) or three (solid lines) adjacent sulfur vacancies in the initial structure using the same MACE fine‐tuned (green) and GAP (orange) MLIPs as in (a). The MACE calculation with three vacancies was performed over a significantly longer simulation time (about 35 ns) to demonstrate that it eventually reaches a similar plateau as the simulations with two vacancies (10 ns simulation time).

The two MoS₂‐specific MLIPs show reasonably good agreement with each other. Both predict the mobile sulfur atoms to move by about 3 Å on average over a time span of 1 ns. Based on the simulation data, only three atoms were observed to migrate between distinct lattice sites. In contrast, the MACE foundation model (blue line in Figure 4a) predicts much larger displacements in general, more mobile atoms, and a more or less linear increase in the MSD, which would suggest unrestricted diffusion of sulfur atoms on this time scale. Again, we attribute this failure to the systematic softening of the foundation model [69], which leads to an incorrect reduction not only of the energy barrier shown in Figure 1 but also of other jump barriers for sulfur atoms. In particular, in this MACE foundation MD simulation, we observe several jumps of sulfur atoms from the upper layer through the molybdenum layer to the bottom layer and vice versa that do not occur in MD simulations based on DFT or MoS₂‐specific MLIPs on this timescale. As shown earlier, the MACE foundation model also predicts a wrong ground state configuration for the sulfur atoms close to a double sulfur vacancy. However at the simulation temperature of 1000 K, sulfur atoms are not trapped in this configuration due to the relatively low energy barrier of the backward jump or equivalent jumps—$\sim 0.5$ eV according to the MACE foundation model as shown Figure 1. This leads to the huge MSD values shown in Figure 4a which continue to grow even after $\tau =3$ ns.

Figure 4b shows the averaged MSDs of mobile sulfur atoms obtained from MACE fine‐tuned and GAP MD simulations, starting from structures with two or three adjacent vacancies. When only a single vacancy is present, both MLIPs predict that sulfur atoms do not undergo lattice jumps, resulting in negligible mobility. In this case, the only displacement of sulfur atom results from their thermal vibration about the equilibrium position, yielding MSDs of roughly 0.1 Å. In contrast, when two or three vacancies are present, significantly larger displacements of sulfur atoms are observed, corresponding to one and three mobile atoms, respectively.

For both cases, the MSDs of mobile atoms reach a plateau at approximately 5 Å (see Figure 4b). This indicates that the motion of sulfur atoms is spatially constrained in this finite system at the ns timescale. Notably, the system with two vacancies exhibits a faster convergence of the MSD, reaching its plateau after around 0.5 ns. In contrast, the system with three vacancies continues to show an increasing MSD for several nanoseconds before leveling off. Thus, in the system with two vacancies, the single mobile sulfur atom appears to be confined to a smaller region, leading to a more rapid convergence of the MSD. We examine this behavior in greater detail later, in the discussion of Figure 6. We emphasize that these studies can neither be done within the computationally too expensive AIMD framework nor with the inaccurate MACE foundation model without fine‐tuning.

Figure 6.

Quantitative visualization of sulfur atom jumps observed in MD simulations with $T=1000$ K using our most accurate MACE MLIP presented in Figure 1. The initial positions of sulfur vacancies are indicated by red circles. Jumps between two sites are indicated by gray cylinders, with cylinder radii scaling logarithmically with the number of jumps between the two sites during the MD simulation. Panels (a) to (d) result from 10 ns long simulations. To highlight the importance of long simulation times, panels (e) and (f) show jump rates from only the first 100 ps – a typical timescale accessible to AIMD simulations – of the simulations fully shown in (c) and (d).

We observe only very limited dynamics of molybdenum atoms in our MD simulations, even in the presence of sulfur vacancies. The molybdenum MSD exhibits a rapid initial increase associated with thermal vibrations and subsequently reaches a plateau at approximately $0.085{\mathrm{\AA }}^2$ for longer lag times, indicating the absence of diffusive motion. Here, we employ the standard MSD definition given in Equation (4), rather than the alternative normalizations discussed in Figure 4 and Section 4.5, as no molybdenum vacancies are present in the system. Thus, the reported value corresponds to the MSD averaged over all molybdenum atoms and over all time windows for a given lag time.

The capability to run highly accurate nanosecond‐scale MD simulations now allows us to analyze the statistics of the relatively rare sulfur atom jumps. We performed 20 ns long simulations at temperatures of 800, 900, 1000, 1100, and 1200 K using our best fine‐tuned MACE model with initial structures containing two or three vacancies arranged in a row. Figure 5 shows the number of sulfur jumps per nanosecond, i.e. the jump rate, and error estimates as a function of temperature. The number of jumps was determined by projecting each atom to its nearest lattice site and counting the changes of the nearest lattice site between snapshots that were sampled at 50 fs intervals; the statistical error was estimated by dividing the 20 ns long MD trajectories into four parts, and calculating the jump rate for each part separately. Note that only about 50 jumps occurred during the 20 ns long double‐vacancy simulation at 800 K.

Figure 5.

Arrhenius plot for the jump rate in 20 ns long MD simulations of MoS₂ sheets with two or three adjacent sulfur vacancies as shown in Figure 6 at different temperatures. The simulations were performed with our most accurate MACE MLIP that was trained on AIMD data of defective MoS₂. The uncertainties of the data points are roughly estimated by dividing the MD trajectories into four parts each and calculating the mean of the jump rate in each part and its standard error.

In accordance with the Arrhenius law, the logarithm of the jump rate depends linearly on the inverse temperature, allowing us to estimate an “effective” energy barrier of sulfur jumps from the slopes in Figure 5. For two vacancies in the structure (blue symbols), the “effective” energy barrier of 0.81 eV agrees very well with the “static” value from NEB calculations (0.83 eV). In contrast, for three vacancies (green symbol) the energy barrier of 0.63 eV derived from MD simulations by the Arrhenius law fit differs by 0.22 eV from the value of 0.85 eV obtained from NEB calculations. This discrepancy highlights the significance of dynamical effects which are not captured in static NEB calculations, such as vibrational contributions from neighboring atoms, and cooperative mechanisms involving simultaneous jumps. As a result, these effects would not appear in typical kMC simulations based on NEB‐calculated energy barriers but are taken into account in MD simulations. At temperatures lower than those shown in Figure 5 ($T<800$ K), the rate of sulfur jumps decreases to a point where it becomes very challenging, if not impossible, to extract statistically meaningful information from MD simulations, even when using MLIPs. By simulating the system at elevated temperatures, we expect to sample all jump mechanisms that are also present at room temperature, while the characteristic time scales of the dynamical processes differ.

Having focused on temporal and averaged aspects of the sulfur‐vacancy dynamics, we analyze next the site‐dependent jump probabilities near small vacancy clusters. This analysis is based on 10 ns long MACE MD simulations with different initial vacancy configurations, see Figure 6. For all cases and consistent with earlier findings of Wang et al. [46], the sulfur‐atom mobility is predominantly governed by the jump mechanism discussed in detail in the context of Figure 1. As discussed above, this jump mechanism requires the presence of at least two adjacent sulfur vacancies. Furthermore, the jumping sulfur atom must be a first neighbor to both vacancies, and the interstitial site along the jump path must not be blocked by a molybdenum atom in the underlying atomic layer. Henceforth, we shall refer to these conditions as “hopping rules” or “vacancy‐assisted hopping.”

In the case of small vacancy clusters, the hopping rules constrain sulfur‐atom mobility to a localized region surrounding the vacancy cluster. These turn out to be equilateral triangles, as illustrated in Figure 6a,b for two or three vacancy clusters in the structure. According to the hopping rules and confirmed by the MD simulations, all sulfur‐atom jumps occur in a triangular pattern involving 3 or 6 sulfur lattice sites in total. This explains the difference in MSD between these two cases that has been mentioned earlier in the context of Figure 4. Notably, on the longer timescales accessible with the MLIP, all jumps within the vacancy triangle occur with comparable frequency, as indicated by the gray connecting lines in Figure 6. Sulfur atoms outside of the outer triangles are not mobile because they either neighbor only one potentially vacant lattice site and/or a molybdenum atom blocks the path to adjacent vacancies.

The hopping rules observed here closely resemble so‐called kinetically constrained models, which have long been used to investigate the dynamics of glassy systems and continue to be highly relevant in recent years [74, 77]. In particular, the model of one‐vacancy‐assisted hopping on a triangular lattice of Jäckle and Krönig [78, 79] differs from our hopping rules for MoS₂ only by the additional constraints imposed by the underlying molybdenum layer. Jäckle and Krönig show that for their model of two‐vacancy‐assisted hopping, certain vacancy configurations can and will grow with probability 100% across the whole system. They argue that for an infinitely large system this can lead to regular diffusion after a possibly very long cross‐over time of subdiffusive propagation [78, 79]. Similarly, we will show that sulfur vacancy clusters in Mo₂ can capture vacancies just outside their triangular mobile region, creating a larger triangular mobile region and thereby facilitating sulfur vacancy mobility potentially across the whole system. As illustration of this mechanism, Figure 6c shows the jump probabilities resulting from a 10 ns long MD simulation starting with three adjacent vacancies in a row (as in panel (b)) and an additional vacancy, to be incorporated later. During the simulation, the three vacancies move within the 6‐site mobile region of the cluster (c.f. panel (b)) and as soon as at least one of them is adjacent to the vacant site outside that region, a sulfur atom from outside the mobile region can jump. This leads to cluster agglomeration, resulting in an expanded triangular mobile region of the cluster that now contains a total of $N_{\mathrm{vac}}·(N_{\mathrm{vac}}+1)/2=10$ sites. For such an agglomeration to occur, the vacancy to be captured must be adjacent to the mobile region of an existing cluster. The location of this additional vacancy determines the direction in which the mobile region of the cluster is expanded.

To demonstrate how this mechanism can lead to the formation of larger vacancy clusters at comparatively low vacancy densities, we performed a MD simulation for the vacancy arrangement shown in Figure 6d. Initially, only vacancies 1 and 2 are nearest neighbors, allowing for the sulfur jump to occur with a low energy barrier only within the dashed, red triangle. Then, the next nearest neighboring vacancy 3 is captured similar to the process shown in Figure 6(c), resulting in a cluster of edge length 3, which is now adjacent to vacancy 4. In this way, all vacancies are captured one by one, spanning the mobile region of the cluster through the periodic boundary of the supercell back to the initial vacancy 1, as indicated by the site‐dependent jump rates shown in Figure 6.

This vacancy‐capture mechanism has not been observed in a previous MD study of sulfur‐vacancy dynamics by Gao et al. [28], most likely due to the inaccurate energy barriers of their classical reactive force field, as mentioned earlier in Subsection 2.3. Specifically, the ReaxFF force field overestimates the barrier for a sulfur jump into the edge of a three‐vacancy line and underestimates the barrier for jumps into the center of the line—both by several tenths of an eV. In contrast, the MACE MLIP model developed here predicts these energy barriers with good accuracy compared to DFT, enabling it to reliably reproduce the collective vacancy dynamics over long timescales and thus reveal the mechanisms behind the experimental observation of extended sulfur vacancy clusters [26, 27], as will be discussed in detail below.

The panels (e) and (f) of Figure 6 are added to highlight the need for accurate MD simulations capable of reaching timescales far beyond those accessible by AIMD. These panels present the site‐resolved jump rates extracted from only the first 100 ps of the same MLIP‐based MD simulations shown in panels (c) and (d), respectively. The 100 ps timescale is representative of what can typically be achieved with AIMD simulations. For our typical MoS₂ supercells and using our CP2K setup (see Section 4.1), a single iteration in the DFT self‐consistent field (SCF) loop takes approximately 2.5 s on 24 CPU cores, with each MD step requiring around seven SCF iterations to converge the forces. Consequently, simulating a 100 ps trajectory with a 0.5 fs timestep would require roughly 40 days computation time. As illustrated in Figure 6e,f, only a very small number of atomic jumps are observed within this timescale, predominantly near regions with a high initial vacancy concentration. In contrast, about 1 ns of MD can be computed within a day with both MLIP architectures employed in this study.

To ensure the robustness of the findings shown in Figure 6, we performed MD simulations with different, arbitrary initial vacancy positions inside of the mobile region of the clusters. For sufficiently long simulations, this leads to the same distribution of jump probabilities within mobile regions as shown in Figure 6.

2.6. Discussion of Sulfur Vacancy Aggregation

The concept of vacancy‐assisted hopping expressed by the simple hopping rules explain the formation of regions with mobile sulfur atoms and sulfur vacancies. However, it is not sufficient to explain experimental facts such as the frequent observation of extended straight vacancy lines using TEM [26, 27], because it includes neither energy differences between different vacancy arrangements nor arrangement‐dependent differences of the transition‐barriers heights. Both these aspects are, of course, accounted for by the MLIP MD simulations and are discussed in detail in the following.

The observed sequence of atomic jumps leading to the incorporation of additional vacancies into an existing cluster is depicted in Figure 7, based on the same MD simulation as Figure 6c. In this case, the fourth vacancy, initially located outside the mobile region of the original three‐vacancy cluster, is captured relatively fast, i.e. within approximately 15 ps. This results in the formation of a vacancy line with a 60 kink (Figure 7b). The resulting configuration persists in a dynamic equilibrium with other similar structures, most notably the one shown in Figure 7c. Both jumps indicated in Figure 7c lead to configurations that are symmetry‐equivalent to the one shown in panel (b). Notably, once the buckled four‐vacancy line forms, no dissociation into smaller clusters is observed. Eventually, after around 3 ns, the cluster forms a straight line of four vacancies located along the edge of the triangular mobile region visible in Figure 6c. This configuration remains comparably stable thereafter, with only occasional, brief jumps of vacancies back out of the line, followed by their prompt return.

Figure 7.

Illustration of a vacancy cluster of size 3 capturing a fourth vacancy, as observed in the MD simulation underlying Figure 6c. Following the capture after about 15 ps, the resulting four‐vacancy cluster transitioned repeatedly between the configurations shown in panels (b) and (c) or similar ones, persisting in this dynamic state for a relatively long duration ($\sim$ 3 ns) before eventually forming a straight line defect (d). The jumping sulfur atoms (yellow) are indicated by arrows.

The tendency of vacancy clusters to align in straight lines along the edge of their mobile region is consistently observed across our MD simulations. To explore the driving force behind this behavior, we analyze the total DFT energy difference $E_{\mathrm{tot}}(N_{\mathrm{vac}})-E_{\mathrm{tot}}(N_{\mathrm{vac}}-1)\mathrm{between}$ vacancy lines of lengths $N_{\mathrm{vac}}$ and $N_{\mathrm{vac}}-1$, which differs only by the sulfur chemical potential ${\mu }_S$ from the differences in the formation energy; see Section 4.6. The energy difference can be interpreted as the binding energy of an additional vacancy to a preexisting line of $N_{\mathrm{vac}}-1$ vacancies. Figure 8 shows that the vacancy binding energy decreases monotonically with increasing vacancy line length, almost certainly approaching a finite value in the large‐$N_{\mathrm{vac}}$ limit. Therefore, the formation energy is a concave curve and consequently longer vacancy lines are energetically more favorable than multiple shorter lines with the same total number of vacancies. This general result is in full agreement with the conclusions of Le et al. [29] and Komsa et al. [26], who compared the formation energies of vacancy lines of varying lengths at a specific value of the sulfur chemical potential.

Figure 8.

Difference in DFT total potential energy between linear and zigzag arrangements of $N_{\mathrm{vac}}$ and $N_{\mathrm{vac}}-1$ sulfur vacancies. As detailed in Section 4.6, the values presented here correspond to the difference in defect formation energies between lines of length $N_{\mathrm{vac}}$ and $N_{\mathrm{vac}}-1$, offset by the sulfur chemical potential.

Figure 8 also includes these energy differences for zigzag configurations as a function of the number of vacancies $N_{\mathrm{vac}}$ (green symbols). In the zigzag configuration, the angle between any three adjacent vacancies is 120, as shown for $N_{\mathrm{vac}}=4$ in Figure 7c. The zigzag energy differences are consistently larger than their straight‐line counterparts (blue symbols). Taken together, these two observations explain the tendency of vacancy clusters to organize into long lines along the edge of their mobile region as observed in our MD simulations. This formation of extended vacancy lines has also been observed experimentally using TEM [26, 27] and is believed to play a crucial role in the functional behavior of memristive elements [10].

We need to address three more aspects of vacancy dynamics, before we compare to what is experimentally known. The first aspect concerns cluster growth. We saw that clusters can grow by capturing additional vacancies located adjacent to their mobile region (cf. Figure 7). For the following argument, which is adapted from Ref. [78], we ignore any energetic aspects and consider only the hopping rules and argue that for any finite vacancy concentration $c$ and infinite systems, there will be with probability 100% a mobile region that can grow across the whole system. The chances that for triangular mobile region with edge length $N_{\mathrm{vac}}$ at least one of its $3(N_{\mathrm{vac}}+1)$ neigboring sites is vacant, is $1-{(1-c)}^{3(N_{\mathrm{vac}}+1)}$. This factor approaches 1 exponentially, as soon as $N_{\mathrm{vac}}\gg 1/(3c)$. Completely analogously to the growth of rain drops in oversaturated air, it is difficult for a condensation nucleus to reach a critical size; but once the critical size has been reached, further growth is almost certain. For example, the mobile region of a vacancy line with length $N_{\mathrm{vac}}=4$ has 15 adjacent sulfur lattice sites. Experimentally, vacancy densities of ${\rho }_{\mathrm{vac}}=N_{\mathrm{vac}}/N_{\mathrm{S}}\mathrm{sites}=0.1$ can be achieved by irradiation while retaining structural properties of MoS₂ [80, 81]. Under these conditions, the probability that at least one of the 15 neighboring sites is vacant—thus allowing the cluster to eventually capture an additional vacancy—is $1-0.9^{15}\approx 80\%$.

The second aspect concerns the mobility of line defects: We saw that most likely a straight line is the energetically most advantageous configuration of a cluster involving a given number of vacancies (cf. Figure 8). However, at finite temperature in MD simulations or sample growth, entropy will disfavor this configuration. Indeed, in MD simulations, we typically observe a dynamic equilibrium in which the vacancy cluster is located near the edge of the mobile region. There, buckled configurations—similar to those shown in Figure 7b,c—continue to form before transitioning back to a straight line. Over longer timescales, on the order of several nanoseconds for small $N_{\mathrm{vac}}$, the vacancy line can even relocate to a different edge of the mobile region via a sequence of many individual jumps. The most straightforward such sequence obeying the hopping rules is illustrated in Figure 9. Noticeably, during some MD runs, we observe successive jumps—following for example the mechanism illustrated in Figure 9b,c—occurring within a short time span ($\sim 1$ ps). In these cases, a sulfur atom initially blocking the jump of a second sulfur atom relocates just prior to or concurrently with the second jump, enabling both jumps to proceed in a correlated manner. This observation highlights the value of MLIP MD simulations over extended timescales, as other approaches—such as kMC models—require prior knowledge of such correlated vacancy dynamics in order to accurately capture cooperative vacancy jumps.

Figure 9.

Illustration of the most direct series of low‐energy jumps leading to the flip of a row of four vacancies (red) within its mobile region (cf. Figure 6c).

The third aspect concerns the question whether the energetically advantageous line defects break into pieces at finite temperature in order to increase entropy. This is a purely academic question for very long chains, because their formation along the lines described above takes too long, both in the computer and in real life. In our MLIP MD simulations with $N_{\mathrm{vac}}\lesssim 6$, we do not observe lines splitting apart after their initial formation. Occasionally, individual sulfur atoms jump into vacancies near the center of a line, transiently dividing it into two segments, but they return to their original sites before further jumps can stabilize the new configuration and lead to a permanent line reconfiguration. To investigate the energetic stability of vacancy lines, we performed further DFT calculations comparing atomic configurations to assess the driving forces opposing line fragmentation beyond the data already shown in Figure 8. For straight line defects with $N_{\mathrm{vac}}=5$ ($N_{\mathrm{vac}}=10$), the energetic costs for a sulfur atom to jump into the center of the line are about 0.40 eV (0.59 eV) where jumps into one of the terminal vacancies cost only about 0.15 eV (0.20 eV). For $N_{\mathrm{vac}}=10$, a partial 60 flip of half the vacancy line (Figure 9)—dividing it into two segments of five vacancies—yields a configuration 0.68 eV higher in energy than the single straight line. In summary, in the regime $N_{\mathrm{vac}}\lesssim 10$, static DFT calculations confirm the presence of rather strong, energetic driving forces that counteract the entropically driven disintegration of vacancy lines, explaining and confirming our observation of stable vacancy lines in nanosecond‐scale MD simulations.

The non‐negligible variation in formation energies and energy barriers across different vacancy configurations—as discussed here and illustrated in Figure 8—underscores a key advantage of the MLIPs developed in this work over previously used kMC models for studying sulfur dynamics in MoS₂.

The formation of vacancy clusters has been analyzed experimentally at the atomic scale using TEM imaging. Komsa et al. [26] observed the agglomeration of irradiation‐induced vacancies into a large number of line defects at room temperature. According to our MD simulations, this can be attributed to small vacancy clusters incorporating a nearby vacancy through a short sequence of low‐energy jumps, as illustrated in Figure 7. But with increasing cluster size, this takes more and more time, as it requires a specific sequence of individual jumps to occur in the right order for the vacancies to hop to another edge of their mobile region to capture further vacancies. As a result, many moderately sized vacancy clusters form close to room temperature, generally aligning along straight lines [26, 35] that are found to increase the conductivity in these regions [35, 82]. This can also be observed in MoS₂ bi‐layers, where compact, electron–irradiation induced vacancy clusters are observed to slowly evolve into more linear arrangements in sequences of TEM images [33]. At $T=800$ K in situ heating and, thus, larger hopping rates, TEM measurements reveal the emergence of substantially longer vacancy lines, extending over tens of nanometers [27]. This corresponds to the low end of the temperature range discussed here in the frame of Figure 5. Insights from our MLIP MD simulations strongly suggest that these elevated temperatures enhance the mobility of moderately long lines by allowing them to overcome the energetic barriers associated with intermediate configurations during the line‐flip process (see Figure 9). Consequently, as clusters grow, longer lines can still incorporate additional vacancies adjacent to their triangular mobile regions, as illustrated for a small vacancy cluster in Figure 7. These temperature‐dependent constrained defect dynamics may lead to glass‐like behavior with history‐dependent properties, as seen in the kinetically constrained models briefly discussed above [74, 77, 78, 79].

3. Conclusion

Large sulfur vacancy clusters in MoS₂ form and undergo dynamic changes even at elevated temperatures on timescales that are inaccessible to AIMD calculations. The underlying processes depend on details of the cluster configuration in a way that is too complex for kMC simulations. This leaves MD based on MLIPs as, arguably, only alternative. However, such calculations are still challenging because the transition states and energy barrier heights between (meta‐) stable positions must be described quantitatively. We find that MLIPs can accurately describe the potential energy surface near stable, minimum‐energy configurations of MoS₂ with only minimal fine‐tuning, see Figure 1. In contrast, a careful construction and selection of appropriate training datasets is needed for a qualitatively and quantitatively correct description of the transition state.

To investigate this issue, we compared two MLIP architectures: GAP and the equivariant GNN MACE [64, 65, 66, 67, 72]. We assessed different dataset generation strategies by evaluating potential energy profiles for the lowest‐energy sulfur atom jump, as predicted by MLIPs trained on different datasets, and compared them to reference DFT calculations.

The GAP framework inherently supports on‐the‐fly training strategies, allowing it to dynamically filter atomic structures for inclusion in the training set. This leads to efficient training requiring relatively few DFT calculations and no “manual” selection of representative structures for the training set to achieve accurate results. In contrast, the accuracy of MACE was found to be more sensitive to the size and representativeness of the training dataset. We successfully trained a MACE MLIP on a large dataset sampled from AIMD simulations. This approach yielded highly accurate predictions for both the transition‐state energies for the important “vacancy‐assisted hopping” and the RDF. Finally, both approaches reliably reproduce the underlying potential energy surfaces that govern vacancy migration. This suggests that usability—understood as simplicity of application and efficiency in data requirements—may outweigh minor differences in accuracy when it comes to determining which MLIP approach will be most widely adopted in the future.

Importantly, training accurate MLIP models allowed us to perform MD simulations far beyond the typical time scales that have previously been accessible at ab initio accuracy and thus enabled us to gain new insights into the local dynamics of sulfur vacancies in MoS₂. These dynamics are driven by a specific low‐energy sulfur jump mechanism (“vacancy‐assisted hopping”) occurring adjacent to at least two vacancies [26, 29, 46]. By analyzing MSDs, jump rates, and site‐dependent jump probabilities from nanosecond‐long MLIP–MD simulations, we observed new mechanisms behind the collective mobility of sulfur vacancies at the atomic level. These atomistic processes are inaccessible to static DFT calculations or kMC models, highlighting the added value of long‐timescale, high‐accuracy simulations.

At low vacancy densities, only small clusters form. The mobility of such clusters is confined to a triangular region, with an edge length approximately equal to or less than the number of vacancies within the cluster due to the specific sulfur jump governing the vacancy mobility [26, 29, 46]. By computing “effective” energy barriers via the Arrhenius relation, we find that the mobility within these clusters is strongly size‐dependent. This analysis of MD simulations at different temperatures also highlights the importance of dynamical effects—such as vibrational contributions from neighboring atoms, and cooperative mechanisms involving simultaneous jumps—which could not be captured by the combination of static NEB calculations and kMC simulations previously used to study sulfur dynamics in MoS₂ [45, 46, 47].

Furthermore, we demonstrate that small clusters can capture vacancies located adjacent to their restricted area of motion, thereby expanding their range. This mechanism could in principle lead to large and even infinite, mobile vacancy clusters at experimentally relevant vacancy densities, but would need very long to do so.

MD simulations and static DFT calculation indicate that within these mobile regions, vacancies preferentially form straight lines in order to minimize their energy. Taken together, these findings explain the experimental observation of a large number of moderately long vacancy lines at room temperature, and fewer—but significantly longer—vacancy lines, spanning several tens of nanometers, at elevated temperatures [26, 27]. The mobility of small vacancy clusters observed here suggests that vacancies could preferentially agglomerate at grain boundaries, where a higher defect density would permit longer‐range diffusion. This highlights the need for further theoretical modeling of vacancy dynamics in polycrystalline MoS₂ systems.

We expect future research to explore the interwoven aspects of concentration dependence, history‐dependent glassy behavior, and memristivity on large scales in more detail. These all originate from the non‐linearity intrinsic to any vacancy‐assisted (vacancy) propagation. Likely, effective continuum models will be developed for that purpose based on concentration‐ and history‐dependent mobilities derived from MLIP MD.

4. Methods

4.1. Ab Initio Energies and Molecular Dynamics

The AIMD data used to fine‐tune MACE models was calculated with the CP2K software package [83] using the Quickstep algorithm [84] and efficient orbital transformation [85] with Gaussian basis sets [86] and pseudopotentials [87] at a kinetic energy cutoff of 500 eV. To generate training data for GAP MLIPs and calculate potential energy curves, we also performed DFT calculations using the Vienna Ab initio Simulation Package (VASP 6.4) [88, 89] within the projector augmented‐wave method [90] with a kinetic energy cutoff of 430 eV. The PBE exchange correlation functional [91] was used throughout this study. A single $\mathbf{k}$ vector at the $\mathrm{\Gamma }$ point is expected to suffice for all integrations in the reciprocal $\mathbf{k}$ space because the studied MoS₂ supercells are large in real space.

The supercells typically contain 108 atoms in a monolayer of hexagonal 2H‐MoS₂ parallel to the x‐y plane. The z dimension of the supercell was chosen to be 40 Å in order to minimize the interaction between replica of the monolayer originating from periodic boundary conditions. A larger supercell with 432 atoms is used in order to reduce the in‐plane interaction between periodic images of the 4 vacancy clusters shown in Figure 6c,e.

All calculations presented in this work were performed without charge compensation, i.e., as “undoped” systems, because we found the neutral sulfur vacancy to be energetically favorable for electron chemical potentials near the center of the band gap when comparing supercells with sulfur vacancies for different numbers of electrons and compensating background charges. This is in agreement with previous DFT results [39].

The MD simulations were performed in the NVT ensemble using a Nosé–Hoover chain thermostat [92, 93, 94] to maintain the temperature at 1000 K, unless stated otherwise. The thermostat was applied using CP2K's default settings, corresponding to a chain length of 3 and a thermostat time constant of 1 ps. All simulations were done within the Born–Oppenheimer approximation. We chose a MD time step of 0.5 fs for production runs and 1 fs for simulations used to generate training data for MLIPs.

4.2. Training of MLIP Models

4.2.1. MACE

As a starting point for training MACE models [65], we use the MACE MP‐0 foundation model [68] which has been pre‐trained on DFT data of about 150 000 crystals extracted from the materials project database [95]. To generate an accurate MACE model for MoS₂, we employed two different approaches for generating training data used to fine‐tune the foundation model. (i) We performed a 10 ns long MD simulation using the MACE foundation model with two adjacent sulfur vacancies in the initial structure. Then, we calculated DFT energies and forces on a series of equidistantly selected snapshots. This test dataset for the foundation model was then used to fine‐tune the MACE model. (ii) Alternatively, we extracted DFT energies and forces directly from a 90 ps long AIMD trajectory with the same initial structure.

For both approaches, we compared MLIPs fine‐tuned to different numbers of training structures, which have been sampled equidistantly from the respective MD simulations. As discussed in the Results Section, we find the latter approach to yield a more accurate MLIP and thus used it in our production runs. Additionally, we find that increasing the fine‐tuning set size to a large number of 50 000 structures yields noticeably better results. In the future, this number could be reduced by sampling the training data for rare events, possibly without affecting the accuracy of the MLIP. During each training process, 10 % of the training data was reserved for validation. Since these MLIP models are designed for use in MD simulations and therefore require accurate atomic forces, our loss function contained a weight of the force error that is 100 times that of the energy error. The MACE MLIP fine‐tuning was done with a initial learning rate of 0.01 for 200 epochs during which we observe a convergence of the force and energy errors.

4.2.2. GAP

Each of the GAPs  [64, 66, 67, 72] was trained in an on‐the‐fly learning MD run with a simulation length of 1 ns and the MD parameters described in Section 4.1. The GAPs were refitted to new DFT data on the current structure of the MD simulation whenever the Bayesian error estimate for the force on at least one atom exceeded a given threshold. This threshold value was initialized at 2 meV ${\mathrm{A}}^{-1}$ and then dynamically updated during the on‐the‐fly training. For these update steps, the GAP to DFT errors at the previous 10 force field refits were taken into account. In this way, rare events are automatically overrepresented in the training dataset, thus requiring fewer structures in the training dataset compared to our most accurate MACE model. Typical GAP runs triggered roughly 600 on‐the‐fly DFT calculation with subsequent adjustment of the MLIP.

Again, we compared two different approaches for selecting training structures: (i) Four separate MLIPs were trained for four distinct defective MoS₂ structures: those containing one, two, and three sulfur vacancies in a row, as well as a MoS₃ vacancy, i.e., the defect resulting from the removal of one molybdenum atom along with its three adjacent sulfur atoms in one of the sulfur layers. (ii) Alternatively, a single MLIP was obtained by subsequent GAP calculations for these four defect structures, where the training run for any one of these defect structures was started with the MLIP resulting from training on the previous defect structure. Thus, the resulting MLIP had multiple, quite distinct defect structures in its training database.

4.2.3. Training and Test Errors

For both MLIP architectures, the test set was generated after the training process by sampling structures from an MD simulation performed using the trained model. DFT forces and energies were then computed for 150 equidistantly sampled snapshots from this trajectory. The resulting force and energy test errors for selected MLIPs models are presented in Table 1 and discussed in Section 2.1.

4.3. Potential Energy Curves and Activation Energies

Figure 1 shows activation barriers and potential energy curves as function of the so‐called configuration coordinate for a single sulfur atom moving from its initial position ${\mathbf{r}}_{\mathbf{0}}$ into an interstitial site at ${\mathbf{r}}_{\mathbf{1}}$ surrounded by three sulfur vacancies, as illustrated in the inset of Figure 1b. Due to the symmetric situation, the path of the sulfur atom is found to be very close to a straight line and, thus, well described by

$$\lambda =\frac{\left(\mathbf{r}-{\mathbf{r}}_{\mathbf{0}}\right)·\left({\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{0}}\right)}{\left({\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{0}}\right)·\left({\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{0}}\right)}$$ (1)

as configuration coordinate corresponding to the moving sulfur's position $\mathbf{r}$. To find the minimal energy path between the start and end point of the transition, the NEB method was applied [73]. To ensure comparability between the curves, ${\mathbf{r}}_{\mathbf{0}}$ and ${\mathbf{r}}_{\mathbf{1}}$ were always taken from the DFT geometry optimizations using VASP, although the initial and final configurations of the NEB were independently optimized for each method (CP2K, MACE, or GAP).

4.4. Radial Distribution Functions

The S‐S RDFs shown in Figure 3 were calculated according to

$$\begin{array}{ccc}\hfill g_{S-S}(r)& =& \frac{1}{4\pi r^2\rho }\frac{1}{N_S}\sum _{i,j=1}^{N_S}<\delta (|{\mathbf{r}}_i-{\mathbf{r}}_j|-r){>}_t\hfill \end{array}$$ (2)

$$\begin{array}{ccc}& =& \frac{V}{N_S^2}\frac{<n_{S-S}(r){>}_t}{4\pi r^2\mathrm{\Delta }r}\hfill \end{array}$$ (3)

where $\rho =N_S/V$, $N_S$ and $V$ denote the sulfur density, the number of sulfur atoms in the supercell, and its volume, respectively. The number of sulfur pairs with a given distance $n_{S-S}(r)$ was determined by constructing a histogram with a bin width of $\mathrm{\Delta }r=0.016$ Å. The time average was calculated over about 10 000 time steps of each MD simulation shown in Figure 3. The first picosecond of the simulation was left out.

4.5. Mean Squared Displacements

The starting point of the MSD calculation is the typical definition

$$\mathrm{MSD}(\tau )={〈{〈|{\mathbf{r}}_i^{(S)}(t_0+\tau )-{\mathbf{r}}_i^{(S)}(t_0){|}^2〉}_i〉}_{t_0}$$ (4)

Here, ${\mathbf{r}}_i^{(S)}(t)$ denotes the position of the i‐th sulfur atom at time $t$. The two averages are over all sulfur atoms in the system and over all time intervals with length $\tau$ starting from $t_0\in [0,t_{\mathrm{end}}-\tau ]$.

As described in the Results Section, the motion of sulfur atoms not close to vacancies is restricted to the relatively small surrounding. Thus, in our case, averaging over all sulfur atoms creates an artificial dependence of the MSD on the number of sulfur atoms in the supercell. To eliminate this dependence, we adjust the definition from Equation (4) in two different ways:

The MSDs in Figure 4a are not averaged but summed over sulfur atoms. We still refer to it as a MSD because the mean displacements over all time windows with a given lag time has been calculated. Additionally, the offset due to the atoms' vibrations about their equilibrium positions is subtracted. The contribution to the squared displacement due to these thermal vibrations was calculated as the MSD of a defect‐free supercell for long lag times. This way, only mobile atoms contribute to the MSD and the MACE foundation model is comparable to the other two MLIPs even though it predicts too many atoms to be mobile.

The MSDs in Figure 4b are not averaged over all sulfur atoms, but only over the mobile sulfur atoms. Atoms are categorized as mobile, if their maximum MSD during the span of the calculation exceeds half of the distance between two adjacent sulfur sites. Both the MLIPs shown here predict one atom to be mobile in a cluster with two vacancies and three atoms to be mobile in a cluster with three vacancies.

4.6. Defect Formation Energy

The stability of vacancy lines of different lengths $N_{\mathrm{vac}}$ can be assessed using their formation energies $E_F$. We define the energy for the formation of extended vacancy line defects by using the DFT energy of a single sulfur vacancy as reference:

$$E_F(N_{\mathrm{vac}})=E_{\mathrm{tot}}(N_{\mathrm{vac}})-E_{\mathrm{tot}}(1)+(N_{\mathrm{vac}}-1){\mu }_S$$ (5)

$E_{\mathrm{tot}}(N_{\mathrm{vac}})$ and $E_{\mathrm{tot}}(1)$ denote the total potential energies of a structure containing a vacancy line with length $N_{\mathrm{vac}}$ and of the reference structure containing just a single vacancy, respectively. The sulfur chemical potential ${\mu }_S$ depends strongly on the chemical environment of the MoS₂ sheet. In this case, in order to compare the stability of vacancy lines with varying length it is sufficient to analyze differences in the formation energy.

$$\begin{array}{cc}\hfill \mathrm{\Delta }{E}_F(N_{\mathrm{vac}})& =E_F(N_{\mathrm{vac}})-E_F(N_{\mathrm{vac}}-1)\hfill \end{array}$$ (6)

$$\begin{array}{cc}& =E_{\mathrm{tot}}(N_{\mathrm{vac}})-E_{\mathrm{tot}}(N_{\mathrm{vac}}-1)+{\mu }_S\hfill \end{array}$$ (7)

$$\begin{array}{cc}& =\mathrm{\Delta }{E}_{\mathrm{tot}}(N_{\mathrm{vac}})+{\mu }_S\hfill \end{array}$$ (8)

It is evident that both $\mathrm{\Delta }{E}_F(N_{\mathrm{vac}})$ and $\mathrm{\Delta }{E}_{\mathrm{tot}}(N_{\mathrm{vac}})$ have the same slope. This allows for a qualitative assessment of the curvature of $E_F(N_{\mathrm{vac}})$ without making any assumptions about the sulfur chemical potential, as discussed in the Results Section in the context of Figure 8. These calculations were carried out using the CP2K software package as described in Section 4.1.

Funding

Support through the projects “MemWerk” and “Ilmenau School of Green Electronics” (P2022‐00‐135) that were made possible by funding from the Carl–Zeiss–Stiftung is gratefully acknowledged.

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

The best‐performing GAP and MACE MLIP models, together with the corresponding AIMD training data as well as the MLIP training and AIMD input files, are publicly available in Ref.