Competing sublattice short-range orders and gap state engineering in multicomponent transition-metal dichalcogenide

Abstract

Discovering new materials with desirable band gap and gap state is a central task in the semiconductor community, primarily relying on composition modulation. In this work, by employing atomic simulations, using transition-metal dichalcogenide Re_0.5Nb_0.5(S_0.5X_0.5)₂ (X=Se, Te) monolayer as an example, we present an alternative avenue for gap state engineering via leveraging diverse chemical short-range orders (SROs). It is found the electronic state contributed by the SRO motif tends to be occupied and may merge with the valence band, yielding a clean band gap in these multicomponent systems. On the contrary, the energy unfavorable local configurations, can produce localized states. The chemical environment in the chalcogen sublattice which has negligible influence on the band gap size can further fine-tune the gap states. The strong coupling of multiple short-range orders and gap states revealed in our work unlock the potential application of a vast family of multicomponent semiconductors.

Introduction

For decades, scientists and engineers in the academic and industry communities have devoted tremendous effects on the band structure modification of the semiconductors to meet the application demands in the electronic and optoelectronic fields. In addition to the tailoring of the band-gap size, by introducing additional energy states, in particular shallow level states, to the gap region of the band structure, the electronic behavior of the semiconductors in terms of carrier type and concentration can be tuned. Doping with elements containing different valence-electron configuration or creating intrinsic defects are classical methods for tailoring gap states^1,2, while their practical application requires nontrivial efforts especially to avoid adverse consequences such as deep level states that act as scattering/charge-recombination center as well as detrimental effects on the matrix^3,4. Besides, the dopant or defect design space is rather limited, hindering the exploration of new systems. Therefore, new concept with applicable technical strategy in the band gap and gap state engineering of semiconductors is highly desirable, especially to meet the increasing high-performance demands in next-generation devices.

In recent years, 2D semiconductors have become one of the key enablers in device miniaturization, among which transition-metal dichalcogenides (TMDCs) have distinguished themselves as the preeminent systems. With the formula MX₂ (M: transition metal; X: chalcogen atom, mostly S/Se/Te), TMDC family has abundant structural phases with diverse physical properties⁵. By element doping, the library of TMDC can be further extended⁶, and one appealing feature of certain multicomponent system is the continuously tunable band gap^7,8. However, for the doping with certain element at a relatively large amount into the semiconductor, the resultant numerous electronic states in the band gap might be extended and delocalized, turning the semiconductor into a semimetal/metal^{9,10,11,12,13}, which obviously blocks their applications in scenarios taking advantages of the semiconducting features. Therefore, it is a very challenging task to target the semiconductor with desired band structure, in particular controllable gap states, from an ocean of multicomponent systems. In addition to the traditional composition and defect design, fortunately, chemical short-range order (SRO) which can be manipulated simply via temperature in materials processing^14,15,16, has recently become a new degree of freedom for property optimization of multicomponent TMDC systems^{17,18,19,20,21,22}. Generally, SRO refers to a local chemical fluctuation, which represents a composition deviation from a fully random solid solution. The presence of SRO modifies the local structure, thereby enabling the modulation of material properties through the well-established composition-structure-property paradigm in materials science. However, previous researches mainly focus on the SRO in single sublattice and tuning of the band-gap size^17,18,20. Whether chemical SRO can be utilized in the rational design of gap states and the underling mechanism remain an open question, which is the main concern of the present work. One notable clue for this gap state engineering approach is that, to suppress the deep level states caused by the intrinsic Se vacancy in MoSe₂/WSe₂, Huang et al. found that by placing the chalcogen vacancy in the deliberately created local environment (specific Mo-W local composition) in the doped Mo_1-xW_xSe₂, the vacancy induced deep level states can become much shallower³. Although it implies that gap states should be tuned via manipulating the local atomic configuration, how to effectively “design” the local configuration in more complex materials in the real world remains unknown.

In the present work, the quaternary Re_0.5Nb_0.5(S_0.5X_0.5)₂ (X=Se, Te) are chosen as the model systems to explore the gap states engineering via SRO, which is based on the following considerations: diverse SRO in the metal/chalcogen sublattice might be formed in these quaternary systems, leaving a larger room for gap state design and meanwhile enabling the investigation of sublattice SRO interaction (competition or cooperation) which is rarely studied in the literature; besides, the equal concentration of metal/chalcogen atoms in the corresponding sublattices in Re_0.5Nb_0.5(S_0.5X_0.5)₂ (X=Se, Te) makes this material a representation of the multi-principle element material (or named as medium-/high- entropy material) that have garnered widespread attention owing to the huge exploration space^16,20,23,24. In this study, the theoretical tools employed include ab initio calculation (density functional theory), MC (monte carlo) simulation and machine learning analysis.

Results

Atomic arrangements in Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayer

To determine the stable atomic configurations of H-phase Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayer, six independent MC simulations are conducted at 923 K, a temperature comparable to that used in the experimental growth of typical TMDC²⁵. These 6 MC simulations start with SQS structures having slightly different degree of atom-occupation randomness, in order to minimize the impact of input structure on the MC results. Each simulation consists of approximately 1000 steps, meaning 9 swap trials per atom. The variation of the system energy per atom ($\Delta {\rm{E}}$) during MC simulation, as computed using ab initio calculation, is shown in Fig. 1a. Here $\Delta {\rm{E}}$ is defined as the energy difference of initial SQS structure referenced to the configuration with lowest energy identified during MC simulations. Although starting from different input structures, all six MC simulations converge to systems having comparable energy, with the discrepancy less than 5 meV per atom, confirming the reliability of present DFT + MC simulation. Figure 1b shows the energy evolution during one MC simulation that manages to locate the lowest energy state (#mc_4), in which only the energy of the structure generated by accepted atom-exchange is displayed. A significant energy decrease can be seen within the first 100 MC steps, then two plateaus are observed between steps 200 and 500, corresponding to the local minima on the potential energy surface. The system eventually reaches an equilibrium state, characterized by energy fluctuations within a narrow range. The energy evolution in other MC simulations is provided in Supplementary Fig. S2.

Fig. 1: Energy evolution and representative configurations of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ obtained from Monte Carlo simulations.

a Energy and the corresponding variance for Re_0.5Nb_0.5(S_0.5Se_0.5)₂ system in six independent MC simulations. b Profile of the energy variation in one typical MC simulation (#mc_4 in (a)), along with top views of the initial and most stable atomic structures of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ found in the simulation.

Using simulation #mc_4 as the example, atomic arrangements in the sampled structures that have diverse energy are analyzed. Firstly, the dynamic stability of the most stable structure found in simulation #mc_4 is confirmed using AIMD method, by a quantitative comparison of the structure features before/after AIMD simulations at 300 K, as shown in Fig. S3 which demonstrates that the atomic geometry remain unchanged and stable in the AIMD simulation. Figure 2a, b depict the first-nearest neighbor shell (1-NNS) of one central metal atom and chalcogen atom, respectively: in Re_0.5Nb_0.5(S_0.5X_0.5)₂, each metal atom has six equivalent neighbors in 1-NNS in both of the metal sublattice and chalcogen sublattice, while each chalcogen atom has three equivalent neighboring atoms in the 1-NNS of the metal sublattice as well as seven inequivalent neighboring atoms in the chalcogen sublattice. More NNS details can be found in Table S2. Then the short-range order parameter as defined in Eq. (2) is calculated. Since Re_0.5Nb_0.5(S_0.5X_0.5)₂ contains two types of sublattices and the interactions inter-/intra-sublattice are obviously different, here three types of short-range orders are considered: short-range order within metal sublattice (${{\rm{SRO}}}_{{{\rm{M}}}_{1},{{\rm{M}}}_{2}}^{{\rm{k}}}$), short-range order within chalcogen sublattice (${{\rm{SRO}}}_{{{\rm{S}}}_{1},{{\rm{S}}}_{2}}^{{\rm{k}}}$), and short-range order between metal and chalcogen sublattice (${{\rm{SRO}}}_{{\rm{M}},{\rm{S}}}^{{\rm{k}}}$/${{\rm{SRO}}}_{{\rm{S}},{\rm{M}}}^{{\rm{k}}}$). The SRO parameters for 1-NNS (${{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{1}$) in the input SQS structure and most stable structure from MC simulation #mc_4, are shown in Figs. 2c, d, respectively. For SQS structure, it can be seen that the value of ${{\rm{SRO}}}_{{{\rm{M}}}_{1},{{\rm{M}}}_{2}}^{{\rm{k}}}$, ${{\rm{SRO}}}_{{{\rm{S}}}_{1},{{\rm{S}}}_{2}}^{{\rm{k}}}$ and ${{\rm{SRO}}}_{{\rm{M}},{\rm{S}}}^{{\rm{k}}}$ are close to 0.5, indicating a random atomic distribution. Meanwhile the value of ${{\rm{SRO}}}_{{\rm{S}},{\rm{M}}}^{{\rm{k}}}$ deviates from 0.5, and part of the reasons is that in the generation of SQS, only the randomness within the sublattice is enforced, while the inter-sublattice randomness can not be guaranteed. It is important to note that the change of SRO parameter among various structures rather than its absolute value is more meaningful here. From Fig. 2e, it can be seen that as the energy decreases from the SQS structure to the most stable configurations, the corresponding changes of ${{\rm{SRO}}}_{{{\rm{M}}}_{1},{{\rm{M}}}_{2}}^{1}$ are notably pronounced, followed by ${{\rm{SRO}}}_{{\rm{S}},{\rm{M}}}^{1}$, while the changes of ${{\rm{SRO}}}_{{{\rm{S}}}_{1},{{\rm{S}}}_{2}}^{1}$ are minimal. This indicates that the thermodynamic stability of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ is strongly correlated with the interactions (SRO) between metal atoms, i.e., attraction (repulsion) between dissimilar (identical) metals, while other types of interactions play a minor role. Similar phenomena are found in the SRO variance for other MC simulations shown in Fig. S4. Notably, the maximum value of ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ observed in the MC simulation is 0.667, which corresponds to either a linear or zigzag arrangements of one metal element (Re or Nb) as displayed in Fig. S5. While the structure with ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ = 0.667 obtained here exhibits both linear and zigzag local features as can be seen in the inset of Fig. 1b.

Fig. 2: Local atomic environments and SRO parameters of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayers.

a The first nearest-neighbor shell surrounding a central metal atom. b the first nearest-neighbor shell surrounding a central chalcogen atom. c SRO parameter for the SQS structure. d SRO parameter for the most stable structure identified through MC simulations (#mc_4). e The difference in SRO parameters between the structures in (c) and (d).

To quantitatively evaluate the correlation between diverse SRO parameters and energetics of the systems, the negative Pearson correlation coefficients (NPCC) between SRO parameters and energy per atom of the structures generated in 6 MC simulation (including both the accepted and rejected structures) are calculated, as shown in Fig. 3a. As the larger NPCC value represents a stronger correlation, the statistical analysis in Fig. 3a reveal that ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ has the most substantial impact on the thermodynamic stability, furthermore, ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ parameter is almost linearly correlated with the average energy of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ structures having the same ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$, with a coefficient of 0.996 as shown in Fig. 3b, indicating a dominate role of the interactions and atomic arrangements within the metal sublattice in determining the stability of the systems. One the contrary, atomic arrangements inside the chalcogen sublattice as measured by ${{\rm{SRO}}}_{{\rm{S}},{\rm{Se}}}^{2}$ in Fig. 3a has negligible effects on the system energy. Meanwhile, ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$, both of which represent inter-sublattice interactions, have NPCC exceeding 0.65, indicating that interactions between metal and chalcogen atoms also have a quite significant impact on the energetic stability of the systems. As the two sublattices interpenetrate within the monolayer, one type of SRO may interplay with others. Here we aim to figure out the interactions among SROs that matter most for the energy of the system, including ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$, ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$. Figure 3c illustrates a linear regression between ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ parameter and corresponding average values of ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$/${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$, and the R² scores for ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$-${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$-${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$ regression are 0.830 and 0.901, respectively. This demonstrates that in the present material systems the evolution of the SRO in the metal sublattice (${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$) is strongly correlated with that between metal and chalcogen sublattice (${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$), which both contribute to the stability of the systems. Additionally, in Fig. 3c, the slope of ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ variation with ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ is smaller than that of ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$, indicating that as ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ changes, compared to Se-Nb, fewer S-Nb configurations will be broken and re-configured, consistent with the fact that S has stronger interactions with metal atoms due to its larger electronegativity than Se. Further Bader charge analysis reveals that S undergoes a larger electron transfer compared to Se, supporting the notion that S forms stronger bonds with the metal atoms, and details of the electronic analysis will be presented below. For all the structures with ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ = 0.667 which corresponds to a nearly fixed arrangement of metal atoms, their energy changes with ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ exhibits a local minimum as can be seen in Fig. 3d. This demonstrates the interplay between atomic arrangement in the metal sublattice and that of chalcogen sublattice: prior to reaching the local energy minima, the values of ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$ tend to increase to lower the system energy, while their further increase beyond the local minima is restricted as it would lead to an increase in energy. This restriction should stem from the fixed atomic arrangements in the metal sublattice and the interactions with the chalcogen atoms.

Fig. 3: Relationships between various SRO parameters and the energy of Re_0.5Nb_0.5(S_0.5Se_0.5)₂.

a The negative Pearson correlation coefficient among diverse SROs and energy of Re_0.5Nb_0.5(S_0.5Se_0.5)_2. b Relationship between ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ and the system energy. c ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ versus the mean value of ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$. d Value of ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{Se}},{\rm{Nb}}}^{1}$ against the energy per atom in the structures with ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ = 0.667.

SRO modulation via temperature

In the current metropolis MC simulation, the temperature determines the acceptance probability of the configuration after an atom swap as indicated in Eq. (1), and with increasing temperature, configurations with a larger energy increasing can be adopt in the simulation. This means that local atomic interactions such as SRO can be easily disrupted in the simulation at high temperature, consistent with the concept of thermal annealing in real word. In this section, we first investigate the temperature effect on SRO in Re_0.5Nb_0.5(S_0.5Se_0.5)₂. In addition to the simulation at 923 K discussed above, here the much lower and higher temperature (200 K and 1600 K) are selected for comparison, and Fig. 4a–f shows the energy evolution profile and the corresponding SRO for the converged structure at the end of the simulation. It can be seen that at the low temperature of 200 K in Fig. 4a, the simulation rapidly converges into a low-energy state, as high energy configuration is unlikely accepted due to the low possibility of acceptance at this temperature. In contrast, at the high temperature of 1600 K in Fig. 4c, within the predefined 1000 steps the energy of the accepted structures fluctuates within a certain range, and the overall energy of the sampled structures are higher compared to those at 200 K and 973 K. For the SRO in the stable structures after convergence, as shown in Fig. 4d–f, it can be seen that all SRO parameters vary towards 0.5, suggesting that heating will break the short-range order in Re_0.5Nb_0.5(S_0.5Se_0.5)₂. By utilizing the kinetics of atomic diffusion in the materials, the SRO tuning via temperature (including slow-cooling and ice-quenched) can be more effective in the experiment, as recently demonstrated in the medium-entropy alloys²⁶.

Fig. 4: Energy and SRO variation of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ as obtained by MC simulations at different temperature.

a–c Energy variation of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayer in MC simulation at 200 K, 923 K (#mc_4), and 1600 K, respectively. Pink dots represent the energies of structures generated during MC searches, among which the points connected by blue line are MC accepted structures. d–f SRO parameters of structures after convergence in simulation at 200 K, 923 K (#mc_4), and 1600 K, respectively.

SRO-featured energy model

In previous sections, statistical analysis established the decisive role of SRO, particularly within the metal sublattice, in governing the energetic stability of the systems. However, whether a quantitative relationship exists between various SRO parameters and the system energy remains an open question. To address this, in this section, we construct a machine learning model solely based on SRO, to predict the system energy. There are two merits for this model: the SRO parameter (Eq. (2)) used as the feature can be readily obtained for a given structure; the SRO parameter derived from the local atomic environment has clear physical meaning which will enhance the interpretability of the machine learning model, in analogy to the utilizing of composition-based descriptors in interpretable models in previous studies^{21,27,28,29,30}. To build such a model, firstly, a dataset consisting of 1350 Re_0.5Nb_0.5(S_0.5Se_0.5)₂ structures, derived from MC simulation at 923 K supplemented with some randomly generated structures, is constructed, which is split into training and testing sets (with a ratio of 8:2). The energy distribution for the dataset structures can be found in Fig. S6. For the features (SRO), initially the SRO parameter up to the third-nearest neighboring shell is considered. With feature engineering, SRO parameters within the second-nearest neighboring shell (2-NNS) are sufficient and yield better prediction performance. Figure 5a shows the correlation coefficients between the SROs that used for the features, with all absolute values below 0.75, indicating that they are rather independent. As for the model training algorithm, KNeighbors, Gradient Boosting and Random Forest algorithms are employed. Using grid search strategy to optimize the combination of features and algorithms, it is found that Random Forest combined with the SROs up to 2-NNS have superior performance with highest R² score and lowest RMSE, as shown in Fig. 5b, c. More details about the grid search results are provided in Fig. S7. The trained SROs-energy model works well within the present dataset, and its generalization capability is further tested. The unseen structures used to test the model is derived from the MC simulations conducted at 200 K and 1600 K, and moreover, MC simulation at a superhigh temperature of 2800 K is performed to generate additional out-of-distribution data. The comparison of the energy obtained by model prediction and DFT calculation can be seen in Fig. S8, and the results for the case of 2800 K is shown in Fig. 5d. Overall, it can be seen that the model effectively captures the trend of the energy variance curve, while the deviation still exists which is partly due to the fact that the SRO parameter is only a two-body descriptor lacking many-body information. Nevertheless, the SRO-based model demonstrates potential as an energy estimation tool for structure screening.

Fig. 5: Machine learning model based on SRO parameters for energy prediction.

a Pearson correlation coefficients between SRO parameters used as descriptors. b Performance of Random Forest model with SRO features up to first, second and third nearest neighbor shells. c Random Forest algorithm predictions in comparison with DFT-calculated energies for the test set in the model training. d Energy prediction using machine learning model based on Random Forest (RF) for the structures sampled in the DFT-MC simulation at 2800 K.

Effects of SRO on the electronic structure of Re_0.5Nb_0.5(S_0.5Se_0.5)₂

The effects of short-range order ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ on the electronic property is concentrated in this section for two reasons: one is that the preceding analysis has demonstrated the importance of ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ in determining the thermodynamic stability of Re_0.5Nb_0.5(S_0.5Se_0.5)₂; besides, Re (5d⁵6s²) and Nb (4d⁴5s¹) have distinct valance electron configurations, and their local arrangement may have nontrivial effects on the electronic behavior of the matrix. Thus electronic properties of the structures with varying ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ parameters (0.500, 0.537, 0.574, 0.611, 0.648, 0.667) are calculated, the energy of which decreases gradually (Table S3). Here, the mechanical stability of these systems has been confirmed according to the previously defined mechanical stability criterion based on the elastic constants Cij: C11 > 0, C11*C22 > C12*C12, det(Cij)>0³¹. And for the present systems the calculated elastic constants are summarized in Table S4. Figure 6a shows the electronic density of states (DOS) for various Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayers obtained using standard DFT calculation. To consider the effects of electron localization, Hubbard U correction is applied to the d orbital of Nb (U = 3 eV), and the resulting DOS profiles are shown in Fig. S9. As can be seen, the inclusion of Hubbard U does not significantly alter the DOS compared to the standard DFT result. Thus in the following electronic structure investigation, only standard DFT calculation is employed. A direct conclusion can be drawn from Fig. 6 is that the electronic states of Re_0.5Nb_0.5(S_0.5Se_0.5)₂, particularly those near the Fermi level, can be significantly modified with the change of ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$. As a result, the monolayer can exhibit as metal, semimetal and semiconductor, depending on the SRO parameter. Figure 6a shows that the electronic states near the Fermi level are predominantly contributed by Re and/or Nb. The electronic states near the Fermi level can be broadly categorized into three types: states primarily contributed by Nb, which are typically occupied (denoted as type I); states predominantly contributed by Re, which are generally unoccupied due to their higher energy (type II); states arising from the combination of Re and Nb contributions, including some hybridized states, which are (partially) occupied (type III). One trend is that as the value of ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ increases, representing the neighboring of more Nb atoms with Re, the third type of Re-Nb hybridized states become occupied and shift to lower energy level, which may eventually merge with valence band and lead to a clean band gap. More importantly, as demonstrated in the bottom two panels in Fig. 6a, the number and the position of the gap states can also be tuned when ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ changes. These gap states are composed of the first two types of states/peaks classified above, i.e., by the independent contribution of Nb (type I, near the valence band maximum, VBM) or Re (type II, near the conduction band minimum, CBM). Therefore, in the present multicomponent Re_0.5Nb_0.5(S_0.5Se_0.5)₂ system, the energy favorable SRO motif leads to the formation of a rather clean band gap, with the SRO-induced electronic states confined in the valence band, while the energy-unfavorable local configuration such as Nb or Re cluster, can generate gap states with tunable energy levels. In real materials, variations in the gap states, including the number of gap states and their energy level, can significantly affect both electron conductivity and the underlying conduction mechanism. These changes can shift the conduction mechanism from band conduction to hopping conduction or introducing a mixed mechanism depending on the nature and distribution of these gap states.

Fig. 6: Density of states (DOS) for Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayers with ${{\rm{SRO}}}_{{\rm{Re}},{\rm{Nb}}}^{1}$ values of 0.5, 0.537, 0.571, 0.611, 0.648, and 0.667.

a Partial DOS for each element, in which examples of three types of electronic states are illustrated. b Total DOS and inverse participation ratio (IPR). c Partial DOS for the dxy, dz² and d(x²-y²) orbitals of Nb. d Partial DOS for the dxy, dz² and d(x²-y²) orbitals of Re. The vertical line indicates the position of Fermi level.

Furthermore, for the structures with diverse SRO parameters, the inverse participation ratio is calculated according to Eq. (1)³².

$$\mathrm{IPR}({{\rm{\psi }}}_{{\rm{n}}})=\frac{{\sum }_{{\rm{i}}=1}^{{\rm{N}}}{\left|{{\rm{\psi }}}_{{\rm{n}}}({\vec{{\rm{r}}}}_{{\rm{i}}})\right|}^{4}}{{\left[{\sum }_{{\rm{i}}=1}^{{\rm{N}}}{\left|{{\rm{\psi }}}_{{\rm{n}}}({\vec{{\rm{r}}}}_{{\rm{i}}})\right|}^{2}\right]}^{2}}$$

(1)

In Eq. (1), N denotes the number of atoms in the system, and ${{\rm{\psi }}}_{{\rm{n}}}({\vec{{\rm{r}}}}_{{\rm{i}}})$ represents the projected wave function of the electronic state ${\rm{n}}$ at the site ${\vec{{\rm{r}}}}_{{\rm{i}}}$. A higher IPR indicates a greater degree of electron localization within the system. As shown in Fig. 6b, generally, the electronic states near the Fermi level are more localized than those staying deeper in the valence band. For the electronic states jointly contributed by Nb and Re, the Re component exhibits a higher IPR compared to Nb, which is consistent with the expectation that 5 d electron (Re) should be more localized than the 4 d electron (Nb). Then the electronic states are projected onto the five d orbitals. As shown in Fig. 6c, d, the electronic states of Re and Nb near the Fermi level are mainly contributed by three of their five d-orbitals: dxy, d(x²-y²), and dz². In Nb, the dz² orbital predominantly contributes to the states (type I) near the Fermi level. For Re, the unoccupied states (type II) above the Fermi level are primarily attributed to the dz² orbital, while the occupied states show a significant increase in contribution from the dxy and d(x²-y²) orbitals. These occupied dxy and d(x²-y²) states are components of the hybridized states formed by Re and Nb (type III), and as the SRO becomes more pronounced, their contribution at the VBM increases. This suggests that the occupation of the dxy and d(x²-y²) orbitals of Re plays a crucial role in mediating the attractive interaction between Re and Nb, ultimately leading to the formation of Re-Nb SRO. To visualize the orbital interactions in real space, the partial charge distributions associated with three typical electronic states near the Fermi level as labeled in Fig. 6a is examined. As can be seen in Fig. 7a and Fig. 7b, it is evident that the electronic states contributed by one single element, referred to as type I and type II states as mentioned above, mainly originate from the atomic clusters composed of that element. Here type I and type II states are primarily contributed by Nb and Re clusters, respectively. On the other hand, the type III states arise from the adjacent Re and Nb cluster, and the nearest-neighboring Re and Nb atoms should contribute to the hybridized states. It can be inferred that more nearby Re and Nb atoms (energetically favorable SRO motif) lead to more hybridized states, as seen in the VBM in Fig. 6a.

Fig. 7: Mapping between local structural motifs and partial charge density associated with the electronic states near the Fermi level.

a–c Partial charge density associated with type I-III electronic states near the Fermi level, respectively, as labeled in Fig. 6a. Isosurface values are set to be 0.002 e/ų.

Gap states fine-tuning via chalcogen-sublattice SRO in Re_0.5Nb_0.5(S_0.5Te_0.5)₂

For the Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayer studied above, the chalcogen sublattice does not exhibit apparent SRO, which can be caused by the similarity of the chemical character of S/Se³³ or the lack of sufficient freedom to occupy lattice site due to the influence of the metal sublattice as previously discussed. To introduce more chemical diversity into the chalcogen sublattice, by replacing Se with Te that has larger difference compared to S, Re_0.5Nb_0.5(S_0.5Te_0.5)₂ system is studied in this section. Details of three independent MC simulation for Re_0.5Nb_0.5(S_0.5Te_0.5)₂ systems are provided in Fig. S10, with the SRO analysis displayed in Fig. S11. Figure 8a, b presents the SRO parameters for 1-NNS in one representative simulation, comparing the input SQS structure and most stable structure obtained from MC simulation, respectively. For SQS structure, it can be seen that the value of ${{\rm{SRO}}}_{{{\rm{M}}}_{1},{{\rm{M}}}_{2}}^{{\rm{k}}}$, ${{\rm{SRO}}}_{{{\rm{S}}}_{1},{{\rm{S}}}_{2}}^{{\rm{k}}}$ and ${{\rm{SRO}}}_{{\rm{M}},{\rm{S}}}^{{\rm{k}}}$ are all close to 0.5, indicating a random atomic distribution. Figure 8c shows the change of SRO parameters between the SQS and stable configuration. Compared to Re_0.5Nb_0.5(S_0.5Se_0.5)₂ (Fig. 2e), here the variation of ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ is less pronounced, while a strengthening of ${{\rm{SRO}}}_{{\rm{S}},{\rm{Nb}}}^{1}$ can be seen in Fig. 8c. Additionally, Se-Nb exhibits a stronger tendency to be a neighbor compared to Te-Nb, which can be attributed to the higher electronegativity of Se. Similar as the case of Re_0.5Nb_0.5(S_0.5Se_0.5)₂, the SRO in the metal sublattice, ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ in particular, also governs the atomic arrangements in Re_0.5Nb_0.5(S_0.5Te_0.5)₂ (Fig. S10e). However, as can be seen from Fig. 8d, quantitatively, the determinative role of ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ in the system energy is diminished, since compared to Re_0.5Nb_0.5(S_0.5Se_0.5)₂, the linear fitting of ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$-energy in Re_0.5Nb_0.5(S_0.5Te_0.5)₂ exhibits a lower R² score and a more significant error bar (deviation). Surprisingly, when ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ reaches 0.667, the total energy increases, indicating a disruption of short-range order of Re-Nb in the metal sublattice, which underscores the intricate interplay between various SROs.

Fig. 8: Effects of SRO on the energetic and electronic properties of Re_0.5Nb_0.5(S_0.5Te_0.5)₂ and Re_0.5Nb_0.5(S_0.5Se_0.5)₂ monolayers.

a–c SRO parameters for Re_0.5Nb_0.5(S_0.5Te_0.5)₂ system in SQS configuration, the most stable configuration found in MC simulation, and the corresponding variation, respectively. d ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ as a function of energy per atom, along with a linear fitting model demonstrating the relationship between SRO parameter and energy of Re_0.5Nb_0.5(S_0.5Te_0.5)₂. e Partial DOS for all elements in six structures with ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}\,$= 0.5, 0.537, 0.571, 0.611, 0.648, and 0.667. f, g Charge transfer for structures with diverse SRO in Re_0.5Nb_0.5(S_0.5Te_0.5)₂, in (g) the Bade charges of Nb and Re are shown. h, i Charge transfer in Re_0.5Nb_0.5(S_0.5Se_0.5)₂ structures with diverse SRO, in (i) the Bade charges of Nb and Re are shown.

To investigate the electronic consequence of the reconfigured SRO after the replacement of Se using Te in Re_0.5Nb_0.5(S_0.5Te_0.5)₂, similar as the previous procedure, DOS of six Re_0.5Nb_0.5(S_0.5Te_0.5)₂ structures with distinct ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ values (0.500, 0.537, 0.574, 0.611, 0.648, 0.667) are computed and plotted in Fig. 8e. The energy of these six structures decreases with the increasing SRO value except the case of 0.667 which shows an increase in energy compared to structure with ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ of 0.648, as detailed in Table S5. Again, it can be observed that SRO enables a transformation in the electronic feature of Re_0.5Nb_0.5(S_0.5Te_0.5)₂, from a semimetal/metal, to a semiconductor with gap states and finally to a semiconductor with a clean band gap. In contrast to the case of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ in Fig. 5a, as ${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$ increases to 0.574, Re_0.5Nb_0.5(S_0.5Te_0.5)₂ has already transformed to a typical defect-containing semiconductor with isolated electronic states close to VBM, from the metallic states with extended gap states in the case of random configurations. Notably, the previously identified three types of electronic states near the Fermi level, contributed by the metals, are weakened, and fewer isolated gap states near the Fermi level are formed, indicative of weaker metal-metal interactions in Re_0.5Nb_0.5(S_0.5Te_0.5)₂. The connection between gap states and local atomic motifs, as illustrated in Fig. S12, is consistent with the behavior observed in Re_0.5Nb_0.5(S_0.5Se_0.5)₂: Nb clusters give rise to localized gap states below the Fermi level, while Re clusters produce states above it. When Nb and Re clusters neighbor each other, they generate delocalized states near the Fermi level due to the hybridization between adjacent Nb and Re atoms. Meanwhile, in Re_0.5Nb_0.5(S_0.5Te_0.5)₂, chalcogen sublattice has more significant impact on the metals, fostering enhanced chalcogen-metal and chalcogen-chalcogen interactions, which result in a substantial presence of electronic states of the chalcogen atoms within the valence band, as evidenced by the projected DOS profile.

Furthermore, the Bader charge in these two systems are compared in Fig. 8f–i, where positive (negative) value means electron gain (loss). On the whole, the amount of charge transfer in Re_0.5Nb_0.5(S_0.5Se_0.5)₂ is larger than that in Re_0.5Nb_0.5(S_0.5Te_0.5)₂, meaning that Te-containing sublattice exerts a weaker oxidizing field on the metals and the reduction in ionic bonding within the material. Interestingly, Te acts like a metal by donating electrons to S. As seen in Fig. 8f, h, the variance in charge transfer of Re, Nb and Se/Te is rather large, while S always maintains a small variance. In Fig. 8g, i, for both systems, with rising ${{\rm{SRO}}}_{\mathrm{Re},{\rm{S}}}^{1}$ and ${{\rm{SRO}}}_{{\rm{Nb}},{\rm{S}}}^{1}$ values, the number of electrons lost by Re and Nb increase, which is understandable as more surrounding S atoms can attract more electrons from the metal. Surprisingly, in Re_0.5Nb_0.5(S_0.5Te_0.5)₂, the charge transfer variation for Re and Nb reaches up to one electron with a maximum Bader charge close to 0 e, whereas in Re_0.5Nb_0.5(S_0.5Se_0.5)₂, the charge transfer variation is limited to 0.5 electron with a maximum Bader charge of −0.5 e. Therefore, replacement of Se using Te not only significantly regulate the magnitude of the charge transfer in these two multicomponent materials, but also introduces substantial diversity in the number of electrons lost by individual metal atoms. This atom-to-atom diversity could be advantageous for catalytic applications of the material by creating a variety of active sites. Finally, the greater charge transfer in Re_0.5Nb_0.5(S_0.5Se_0.5)₂ can result in a larger electric fields in the systems, which can promote charge redistribution and orbital-orientation variation. These effects make the formation of the predefined three types of gap states more likely in Re_0.5Nb_0.5(S_0.5Se_0.5)₂, thereby inducing the difference in the band structure between Re_0.5Nb_0.5(S_0.5Se_0.5)₂ and Re_0.5Nb_0.5(S_0.5Te_0.5)₂.

Discussion

The present work investigates the SRO phenomena in monolayer Re_0.5Nb_0.5(S_0.5X_0.5)₂ (X=Se, Te) using density functional theory (DFT) calculation combined with Monte Carlo (MC) simulation, focusing on the interplay between various SROs and their impact on the gap states. Firstly, it is demonstrated that the SRO of the first-nearest neighboring shell in the metal sublattice (${{\rm{SRO}}}_{\mathrm{Re},{\rm{Nb}}}^{1}$) has the most significant impact on the thermodynamic stability of Re_0.5Nb_0.5(S_0.5X_0.5)₂, followed by the SRO between metal and chalcogen atom, such as ${{\rm{SRO}}}_{{\rm{X}},{\rm{Nb}}}^{1}$. The dominate role of the metal SRO in the atomic arrangement of the material and the influence on other types of SROs is elucidated. The strong correlation between SRO and the energy of the material facilities the development of a machine learning model that utilizes SRO parameter as the primary feature for energy prediction. More importantly, we present a physical picture that maps the local atomic motif to the position as well as the occupation of the electronic states near the Fermi level. Thus with this mapping, on-site design of the gap states and electronic properties of the materials can be realized, by regulating multiple SROs via technical methods such as thermal annealing. By diving into the electronic origin of the gap state tuning, it is found that interactions between dz² orbitals of the metals, as well as the involving of dxy/d(x²-y²) orbital of Re accounts for the diversity of the gap states. Furthermore, the introduction of chalcogen atoms with varying electronegativity modifies the charge transfer and ionic bonding within the system, allowing for further fine-tuning of the gap states. To conclusion, using quaternary Re_0.5Nb_0.5(S_0.5X_0.5)₂ as the example, our findings establish a fundamental relationship among local atomic arrangement (SRO), thermodynamic stability and gap states. Our findings not only offer a novel strategy for gap state engineering but also vastly expand the horizon of material systems suitable for gap state design in the semiconductor industry. This approach moves beyond conventional host materials to include a wide spectrum of multicomponent semiconductors, greatly enriching the toolbox for future semiconductor innovation.

Methods

In this work, we focus on the hexagonal (H-phase) Re_0.5Nb_0.5(S_0.5X_0.5)₂ (X=Se, Te), which has been identified as the most stable crystalline geometry^18,20. A supercell containing 108 atoms, with a vacuum region of 20 ${\rm{\mathring{\rm A} }}$, is constructed to model the Re_0.5Nb_0.5(S_0.5X_0.5)₂ monolayer.

Monte Carlo simulation

Canonical ensemble Monte Carlo simulation (MC)³⁴ is employed to sample and locate the stable atomic arrangements in Re_0.5Nb_0.5(S_0.5X_0.5)₂ monolayer, i.e., the atomic configuration that occurs with high probability at specific temperature. In the MC simulation, atoms will be swapped to sample the configuration space, and given that the system comprises a metallic sublattice and a chalcogen sublattice, two types of swaps exist: the exchange between Re and Nb atoms, and the exchange between S and X (X=Se, Te). A random number is generated to determine which atom will be chosen to exchange with another atom in the same sublattice. The acceptance probability P of atomic exchange is determined by the metropolis algorithm³⁵ which is calculated using Eq. (2),

$${\rm{P}}=\min \left\{1,\exp \left[\frac{{{\rm{E}}}_{1}-{{\rm{E}}}_{2}}{{{\rm{k}}}_{{\rm{B}}}{\rm{T}}}\right]\right\}$$

(2)

where ${{\rm{E}}}_{1}$ denotes the energy of original structure before the atom swap, ${{\rm{E}}}_{2}$ denotes energy of the new structure after the swap, ${{\rm{k}}}_{{\rm{B}}}$ is boltzman constant, and T means temperature (in Kelvin). The initial input structure for each MC simulation is generated using SQS algorithm³⁶ as in ATAT code³⁷, assuming random atomic occupation. The energy of the structure is evaluated using ab initio calculations.

Ab initio calculation

The ab initio calculations are performed in the framework of density functional theory (DFT) implemented in Vienna ab initio simulation package (VASP)³⁸. The projector augmented wave (PAW) pseudopotential³⁹ is adopted, and the Perdew-Burke-Ernzerh functional is used to describe the electron exchange-correlation effects. For the electronic structure calculations, DFT + U⁴⁰ method is further employed by setting U value of 3 eV for the d electrons of Nb^41,42,43, and for Re, the U correction is not applied, as DFT already well predicts the property of Re-based binary TMDC according to the literature⁴⁴. It turns out that U term correction has negligible effects in the density of states profile for our systems. The energy convergence criterion is set to 10⁻⁵ eV, and the relaxation is terminated when all the forces on the atoms are smaller than 0.01 ${\rm{eV\cdot }}{{\rm{\mathring{\rm A} }}}^{-1}$. Low-precision settings are used in the structural relaxations in MC simulation and ab initio molecular dynamic (AIMD) simulation, with a 1 × 1 × 1 K-point mesh and a cutoff energy of 320 eV, which is found sufficient to describe the relative energy between different configurations. Further structural relaxations and electronic structure calculations are conducted using high-precision settings, with a 5 × 5 × 1 K-point mesh, a cutoff energy of 400 eV, and an energy convergence criterion of 10⁻⁶ eV. This customized precision setting which has been validated in our previous work^16,22, leads to reasonable results in the present system, as shown in the test in Fig. S1.

Short-range order characterization

To quantify the short-range order in diverse structures, the parameter ${{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{{\rm{k}}}$, derived from Warren–Cowley parameter⁴⁵, is defined. Here k denotes the k-th nearest-neighbor shell, ${{\rm{e}}}_{1}$ represents the species of the central atom, and ${{\rm{e}}}_{2}$ denotes the species of neighboring atoms in the k-th nearest-neighbor shell. The value of ${{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{{\rm{k}}}$ is determined using Eq. (3).

$${{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{{\rm{k}}}=\frac{1}{{\rm{N}}}\mathop{\sum }\limits_{{\rm{i}}=1}^{{\rm{N}}}\frac{{{\rm{M}}}_{{{\rm{e}}}_{2}}^{{\rm{k}}}}{{{\rm{M}}}_{{\rm{total}}}^{{\rm{k}}}}$$

(3)

where i denotes the index of atom, N marks the total number of ${{\rm{e}}}_{1}$ atoms in the structure, ${{\rm{M}}}_{{\rm{total}}}^{{\rm{k}}}$ signifies the total number of atoms in k-th nearest neighbor shell of ${{\rm{e}}}_{1}$ atom, ${{\rm{M}}}_{{{\rm{e}}}_{2}}^{{\rm{k}}}$ indicates the number of ${{\rm{e}}}_{2}$ atoms in the k-th nearest neighbor shell of ${{\rm{e}}}_{1}$ atom. For the sublattice-equiatomic Re_0.5Nb_0.5(S_0.5X_0.5)₂ studied here, $0\le {{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{{\rm{k}}} < 0.5$ indicates that ${{\rm{e}}}_{1}$ and ${{\rm{e}}}_{2}$ atom tend to avoid neighboring with each other, ${{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{{\rm{k}}}=0.5$ represents a random arrangement, and $0.5\le {{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{{\rm{k}}} < 1$ suggests that two types of atoms prefer to be neighbors.

Machine learning

To quantitatively capture the relationship between SRO and the total energy of Re_0.5Nb_0.5(S_0.5Se_0.5)₂ system, machine learning models are built with short-range order parameters ${{\rm{SRO}}}_{{{\rm{e}}}_{1},{{\rm{e}}}_{2}}^{{\rm{k}}}$ as the key features. These features have clear physical meanings, corresponding to the coordination environment of the atoms, thus should have strong influence on the energy of each atom as well as the energy of the whole system. The crystal structure is represented using structure class in Pymatgen⁴⁶. KNeighbors⁴⁷, Gradient Boosting³³ and Random Forest⁴⁸ algorithms implemented in scikit-learn⁴⁹ are used to train and optimize the model. For comparison, identical hyperparameters (listed in Table S1) are applied to machine learning models featuring the short-range order within the first-, second- and third-nearest neighboring shells.