Diffusion of Alkaline Metals in Two-Dimensional β₁-ScSi₂N₄ and β₂-ScSi₂N₄ Materials: A First-Principles Investigation

Abstract

The MA₂Z₄ family represents a class of two-dimensional materials renowned for their outstanding mechanical properties and excellent environmental stability. By means of elemental substitution, we designed two novel phases of ScSi₂N₄, namely β₁ and β₂. Their dynamical, thermal, and mechanical stabilities were thoroughly verified through phonon dispersion analysis, ab initio molecular dynamics (AIMD) simulations, and calculations of mechanical parameters such as Young’s modulus and Poisson’s ratio. Electronic structure analysis using both PBE and HSE06 methods further revealed that both the β₁ and β₂ phases exhibit metallic behavior, highlighting their potential for battery-related applications. Based on these outstanding properties, the climbing image nudged elastic band (CI-NEB) method was employed to investigate the diffusion behavior of Li, Na, and K ions on the material surfaces. Both structures demonstrate extremely low diffusion energy barriers (Li: 0.38 eV, Na: 0.22 eV, K: 0.12 eV), indicating rapid ion migration—especially for K—and excellent rate performance. The lowest barrier for K ions (0.12 eV) suggests the fastest diffusion kinetics, making it particularly suitable for high-power potassium-ion batteries. The significantly lower barrier for Na ions (0.22 eV) compared with Li (0.38 eV) implies that both β₁ and β₂ phases may be more favorable for fast-charging/discharging sodium-ion battery applications. First-principles calculations were applied to determine the open-circuit voltage (OCV) of the battery materials. The β₂ phase exhibits a higher OCV in Li/Na systems, while the β₁ phase shows more prominent voltage for K. The results demonstrate that both phases possess high theoretical capacities and suitable OCVs.

1. Introduction

Since the isolation of graphene in 2004, two-dimensional (2D) materials have attracted extensive research interest due to their unique structural, electronic, and mechanical properties [1]. Recent years have witnessed a surge in the exploration of novel 2D systems beyond conventional materials such as MoS₂ [2], WS₂ [3], and [4]. These materials are now widely studied for applications in electronics, optoelectronics, sensing, and energy storage. The emergence of atomically thin materials with tailored compositions has greatly enriched the 2D material family. Among them, the MA₂Z₄ (M = transition metal, A = group IV element, Z = N or other pnictogens/chalcogens) system represents a new frontier [5,6]. Notably, MoSi₂N₄ was synthesized experimentally via chemical vapor deposition (CVD) in 2020, marking the first realization of this class with high environmental stability and potential for integration in flexible electronic devices [7]. Although the majority of research to date remains theoretical, recent advances suggest that MA₂Z₄ materials could serve in real-world applications such as field-effect transistors, piezoelectric devices, and possibly as electrode materials in batteries [8], though experimental demonstrations remain limited.

The MA₂Z₄ family [9], including the archetypal MoSi₂N₄, has garnered increasing attention for its structural tunability via isoelectronic substitution. By systematically varying M (e.g., Mo, W, Cr, V, Sc), A (Si, Ge), and Z (N, P, As), researchers have predicted a wide range of stable configurations, each with distinct electronic, magnetic, or optoelectronic properties [10,11,12]. In particular, MSi₂N₄ [13,14] systems have shown promising potential, with members from early transition metals (e.g., groups III–VI) offering diverse physical behaviors due to d-orbital contributions. However, many of these studies are based solely on first-principles calculations, and caution must be exercised when extrapolating potential applications without experimental validation. Properties such as carrier mobility, band gap modulation, and magnetic ordering have been predicted, yet only a limited number of MA₂Z₄ structures have been synthesized and characterized to date. Among these, MoSi₂N₄ remains the most well-studied [15,16], underscoring the need to explore other group and period combinations experimentally to fully assess the potential of this materials platform.

Recent theoretical investigations have expanded the MA₂Z₄ family to include the ScSi₂N₄ system, a previously unexplored compound that exhibits promising characteristics for energy storage applications. To date, the known studies on ScSi₂N₄ include [17,18]. In this work, two structurally stable polymorphs—β₁-ScSi₂N₄ and β₂-ScSi₂N₄—were designed and evaluated using density functional theory (DFT). This study focused on their viability as anode materials for Li-, Na-, and K-ion batteries. The diffusion energy barriers for alkali metal ions were found to be exceptionally low: 0.38 eV (Li), 0.22 eV (Na), and 0.12 eV (K) for both β₁ and β₂ phases, indicating excellent ion mobility. These values are competitive with or superior to those of conventional 2D anode materials such as phosphorene or Ti₃C₂ MXenes [19]. The relevance of these low barriers underscores the potential of ScSi₂N₄-based materials in next-generation rechargeable battery technologies, especially considering their favorable open-circuit voltages and theoretical capacities. As such, the exploration of Sc-based MA₂N₄ systems could significantly broaden the applicability of this emerging materials family.

2. Computational Details

First-principles calculations were carried out within the framework of density functional theory (DFT) [20] using the Vienna Ab initio Simulation Package (VASP 5.4) [21]. The interaction between ions and electrons was treated using the projector augmented-wave (PAW) [22] method, and the exchange–correlation energy was described by the generalized gradient approximation (GGA) [23] in the form proposed by Perdew, Burke, and Ernzerhof (PBE) [24]. The HSE06 hybrid functional (Heyd–Scuseria–Ernzerhof) [25] was employed to achieve accurate bandgap predictions by mixing exact exchange with PBE exchange–correlation energy. To ensure computational accuracy, the plane-wave energy cutoff was set to 520 eV, and the energy convergence criterion was set to 10⁻⁶ eV. During structural optimization, all atomic positions were relaxed until the residual force on each atom was less than 10⁻³ eV/Å. A supercell with a vacuum layer of approximately 30 Å was constructed to eliminate spurious interactions between periodic images along the out-of-plane direction. For the analysis of dynamical stability, phonon dispersion relations were calculated using the Phonopy package (2.22.1) [26] based on the supercell approach. In addition, ab initio molecular dynamics (AIMD) simulations were conducted under the canonical (NVT) ensemble to evaluate thermal stability at various temperatures. Each simulation lasted for 10 ps with a time step of 2.0 fs, and temperature control was achieved using the Nosé–Hoover thermostat [27]. The elastic constants C₁₁, C₂₂, and C₆₆ for the 2D β₁-ScSi₂N₄ and β₂-ScSi₂N₄ structures were obtained by applying a series of small strains to the fully relaxed structures using the Vienna Ab initio Simulation Package (VASP) and calculating the resulting stress tensors. The elastic constants were then extracted from the linear stress–strain relationships using the VASPKIT package (1.5.1) [28]. The diffusion behavior of Li, Na, and K ions on the material surface was investigated using the climbing image nudged elastic band (CI-NEB) method to determine the migration pathways and corresponding energy barriers, revealing their potential for energy storage applications. To calculate the open-circuit voltage (OCV) of battery materials, we adopted a thermodynamic approach, first computing the Gibbs free energy change (ΔG) during Li/Na/K intercalation in the electrode material, then deriving the voltage via the Nernst equation. For consistency, all compared data were obtained using the PBE-GGA functional unless otherwise noted. Previous studies [25] have shown that while hybrid functionals such as HSE06 can affect the predicted band gap of MA₂Z₄ compounds, the differences in adsorption energy and ion diffusion barriers between PBE and HSE06 remain minor. Thus, our comparisons are considered valid within the accepted computational accuracy for battery material screening. To evaluate the practical competitiveness of ScSi₂N₄ as an anode material, we compared its theoretical electrochemical performance with several extensively studied two-dimensional materials reported in the literature [29,30]. Although these materials differ in structure and composition, they serve as valid references for assessing electrochemical performance. To further evaluate the electrochemical performance of ScSi₂N₄-based electrodes, we compared our calculated results with those reported for other two-dimensional anode materials. These references were selected because they provide relevant performance benchmarks. We note that the reported values for these materials were obtained using different DFT methods (e.g., PBE, PBE + U, HSE06), which can lead to quantitative differences in properties such as adsorption energies, diffusion barriers, and voltages. Therefore, these comparisons are intended as qualitative indicators of relative performance rather than exact numerical matches. This approach highlights the competitive advantages of the ScSi₂N₄ phase, particularly in terms of ion mobility and capacity, while also acknowledging the differences between the different research methods. The formulas for Young’s modulus and Poisson’s ratio are as follows: $Y_x={[C}_{11}{C}_{22}-C_{12}^2]/C_{22}$ and $Y_y={\displaystyle \frac{\left[C_{11}{C}_{12}-C_{12}^2\right]}{C_{11}}}$; $V_x={\displaystyle \frac{C_{12}}{C_{22}}}$, $V_y={\displaystyle \frac{C_{12}}{C_{11}}}$.

3. Results and Discussion

3.1. Structural Information and Stability of 2D β₁-ScSi₂N₄ and β₂-ScSi₂N₄

Both β₁-ScSi₂N₄ and β₂-ScSi₂N₄ have a hexagonal lattice structure consisting of seven atomic layers, as shown in Figure 1. It can be regarded as the insertion of a layer of MoS₂-type ScN₂ into a layer of InSe-type Si₂N₂ monolayer; that is, sandwiching the ScN₂ layer between two SiN layers to form a hexagonal ScSi₂N₄ monolayer structure, which belongs to the P6m1 space group. The optimized lattice constant of β₁-ScSi₂N₄ is a = b = 3.00 Å, and the optimized lattice constant of β₂-ScSi₂N₄ is a = b = 2.98 Å. Both unit cells contain one Sc atom, two Si atoms, and four N atoms, and each Sc atom and Si atom are surrounded by six and four adjacent N atoms, respectively. β₁-ScSi₂N₄ belongs to the P-3m1 (164) space group, and β₂-ScSi₂N₄ belongs to the P3m1 (156) space group.

The stability calculation of two-dimensional materials is the bridge between theoretical predictions and experimental applications, and directly determines whether the material can be synthesized, whether it can be used in practice, and whether it has unique physical and chemical properties. In the phonon calculation, the phonon dispersion of β₁-ScSi₂N₄ and β₂-ScSi₂N₄ does not exhibit any imaginary frequency, confirming their dynamic stability, as shown in Figure 2a,b. The maximum frequency of 2D β₁-ScSi₂N₄ is 27.7 THz (925 cm⁻¹), while 2D β₂-ScSi₂N₄ is 28.6 THz (954 cm⁻¹), which is slightly lower than for VSi₂N₄ (992 cm⁻¹) [14] but sufficiently higher than MoS₂ (473 cm⁻¹) [31], showing the high stability of 2D β₁-ScSi₂N₄ and β₂-ScSi₂N₄. Ab initio molecular dynamics (AIMD) simulations were performed using a 3 × 3 × 1 supercell for the β₁-ScSi₂N₄ and β₂-ScSi₂N₄ monolayers at 300 K. Within 10 ps, there is no obvious displacement of the atomic positions in the structure, no obvious breakage of the chemical bonds, and the structure remains intact, indicating good thermal stability, as shown in Figure 2c,d. Cohesive energy is a physical quantity that describes the binding strength between atoms or molecules in a material. It reflects the material’s stability: the more negative the value, the stronger the atomic bonding and the more stable the material. The cohesive energy of β₁-ScSi₂N₄ is −7.78 eV/atom. The cohesive energy of β₂-ScSi₂N₄ is −7.79 eV/atom. The cohesive energies of β₁-ScSi₂N₄ and β₂-ScSi₂N₄ are more negative than those of YSi₂N₄ [32] (−4.294 eV/atom), Cr₂BN [33] (−4.06 eV/atom), and Fe₂Si [34] (−4.10 eV/atom). The negative cohesive energies indicate structural stability, demonstrating the experimental feasibility of synthesizing β₁-ScSi₂N₄ and β₂-ScSi₂N₄.

In β₁-ScSi₂N₄, the elastic constants are C₁₁ = 325.96 N/m, C₂₂ = 121.65 N/m, and C₆₆ = (C₁₁ − C₁₂)/2 = 102.16 N/m. For β₂-ScSi₂N₄, these values are C₁₁ = 381.83 N/m, C₂₂ = 137.80 N/m, and C₆₆ = (C₁₁ − C₁₂)/2 = 122.01 N/m. These elastic constants satisfy the Born criteria for hexagonal systems [35,36] (C₁₁ > 0 and C₁₁ > |C₁₂|), confirming their mechanical stability. Figure 3 demonstrates that the angle-dependent Young’s modulus and Poisson’s ratio exhibit remarkable isotropy, indicating consistent mechanical properties in all directions, which facilitates practical applications of these materials. The magnitude of Young’s modulus reflects material stiffness—higher values correspond to greater resistance to deformation. The β₁-ScSi₂N₄ monolayer shows a Young’s modulus of 281 N m⁻¹, while β₂-ScSi₂N₄ exhibits a higher value of 332 N m⁻¹. These values surpass those of Ti₂C (130 N m⁻¹) [37] and MoS₂ (119 N m⁻¹) [38]. The Poisson’s ratio, describing the transverse strain response to longitudinal stress, is 0.37 for β₁-ScSi₂N₄ and 0.36 for β₂-ScSi₂N₄, comparable to CrAsS₄ (0.34) [39] and MoS₂ (0.31) monolayers. These results confirm that both β₁-ScSi₂N₄ and β₂-ScSi₂N₄ possess excellent tensile/compressive resistance within their elastic limits.

3.2. Electronic Properties of 2D β₁-ScSi₂N₄ and β₂-ScSi₂N₄

Next, the spin-polarized band structures were calculated using the PBE and HSE06 methods to explore the electronic properties of the β₁-ScSi₂N₄ and β₂-ScSi₂N₄ monolayers (Figure 4). In the PBE calculations, we found that the spin-up and spin-down channels of β₁-ScSi₂N₄ and β₂-ScSi₂N₄ cross the Fermi level, indicating their intrinsic metallic nature. However, the PBE functional typically underestimates band gaps. Therefore, we further calculated the band structures of the β₁-ScSi₂N₄ and β₂-ScSi₂N₄ monolayers using the HSE06 method.

With the inclusion of HSE06 (Figure 4c,d) we observed that the spin-down band of β₁-ScSi₂N₄ crosses the Fermi level, exhibiting metallic behavior, while the spin-up band lies below the Fermi level. The valence band at the Γ point is primarily contributed by N atoms, whereas the conduction band at the M point is mainly dominated by Sc atoms. A band gap of approximately 4.0 eV exists between the conduction and valence bands, resulting in an overall half-metallic character. The presence of half-metallicity implies vacant states in the band structure, leading to spin asymmetry and partially filled valence electron arrangements. Due to the significantly higher electronegativity of N compared with Sc and Si, its ability to attract electrons is much stronger. Based on the Bader charge [40] analysis shown in Table S1, we calculated the charge distribution of Sc, Si, and N atoms. The results show that N atoms are negatively charged, indicating their strong electron-withdrawing ability, while Sc and Si atoms are positively charged, indicating that electrons are mainly transferred to N atoms, supporting their strong electronegativity. Additionally, through orbital-projected band structure analysis, we demonstrated that the half-metallic nature primarily originates from the N atoms, as illustrated in Figure 5.

PBE notoriously underestimates bandgaps, while HSE06 shows excellent agreement with experimental values, justifying its use for electronic structure analysis. Consequently, β₁-ScSi₂N₄ holds promising potential for applications in magnetism, optics, and related fields. In the HSE06 calculations, the spin-up and spin-down bands of β₂-ScSi₂N₄ both cross the Fermi level, maintaining their metallic nature. However, while the spin-up and spin-down states were initially degenerate, the inclusion of HSE06 leads to band splitting, with the spin-down valence band shifting to higher energies. Based on the HSE06 results, β₁-ScSi₂N₄ exhibits a half-metallic character, whereas β₂-ScSi₂N₄ remains metallic, thereby expanding the potential applications of the MA₂Z₄ family. COHP (crystal orbital Hamilton population) analysis helps to distinguish bonding, anti-bonding, and non-bonding interactions by evaluating the energy-resolved overlap of electronic states between atom pairs. The ionic bonding character is confirmed as the near-zero ICOHP values (≈0) observed in the COHP curves near the Fermi level (−2 to 2 eV) indicate negligible electronic state overlap in the Sc-N/Si-N bonding (Figure 6c,d). Furthermore, differential charge density and crystal orbital Hamiltonian population (COHP) analyses confirm that the Sc-N and Si-N bonds in both β₁-ScSi₂N₄ and β₂-ScSi₂N₄ exhibit ionic characteristics, as illustrated in Figure 6.

3.3. β₁-ScSi₂N₄ and β₂-ScSi₂N₄ Performance as Anode Materials

The evaluation of binding energies for Li/Na/K atoms on substrate materials is crucial for assessing the performance of ion batteries. In this study, to investigate the surface adsorption behavior on β₁-ScSi₂N₄, we selected four typical adsorption sites on the outermost AZ layer: the top site of the N atom (S1), the top site of the Si atom (S2), the bridge midpoint between N and Si atoms (S3), and the center site of the honeycomb ring (S4). Similarly, for the surface adsorption study on β₂-ScSi₂N₄, four characteristic sites on the outermost AZ layer the adsorption points are: the top site of the Sc atom (S1), the top site of the Si atom (S2), the bridge midpoint between Sc and Si atoms (S3), and the center site of the honeycomb ring (S4) (as shown in Figure S1).

The binding energy can be calculated using the following formula:

$$∆E_b=E_{tot}\left(Sc{Si}_2{N}_4\right)+E_{tot}\left(M\right)-E_{tot}\left(MSc{Si}_2{N}_4\right),\left(M=Li,Na,K\right)$$

(1)

In the binding energy calculations, $E_{tot}\left(Sc{Si}_2{N}_4\right)$ represents the total energy of the pristine structure without adsorbed atoms, $E_{tot}\left(MSc{Si}_2{N}_4\right)$ denotes the total energy of the system after Li/Na/K atom adsorption, and $E_{tot}\left(M\right)$ corresponds to the total energy of an isolated Li/Na/K atom (show in Table S2). The research results demonstrate that for the β₁-ScSi₂N₄ material, Li/Na/K atoms initially positioned at the S3 site spontaneously migrate to the S2 site. In contrast, for the β₂-ScSi₂N₄ material, Li/Na/K atoms at the S3 site relax to the S1 site. This indicates that the S1, S2, and S4 sites can serve as effective adsorption sites for Li/Na/K atoms in both materials. The binding energy data presented in Table S1 show that a larger $∆E_b$ value corresponds to a stronger interaction between the ScSi₂N₄ material and Li/Na/K atoms (take Li as an example). Consequently, in the β₁-ScSi₂N₄ material, the S4 site exhibits the most stable adsorption for Li/Na/K atoms, while in the β₂-ScSi₂N₄ material, the S1 site demonstrates optimal adsorption stability.

To gain deeper insights into the adsorption mechanism of Li, Na, and K atoms on the surface of MA₂Z₄ monolayer materials, a quantitative evaluation of the charge transfer between the metal atoms and the two-dimensional ScSi₂N₄ was carried out based on Bader charge analysis. The results indicate that Li acts as a typical electron donor during adsorption, transferring approximately 0.85 e and 0.9 e of charge to the β₁-ScSi₂N₄ and β₂-ScSi₂N₄ monolayers, respectively. For Na and K atoms, the amount of charge transferred to the β₁ configuration is approximately 0.88 e and 0.84 e, respectively, and the same values are observed for the β₂ configuration. These substantial charge transfers suggest that the adsorption of Li/Na/K atoms on the MA₂Z₄ surface is primarily governed by electron transfer from the metal atoms to the substrate, effectively filling the lowest unoccupied molecular orbitals (LUMO) of the ScSi₂N₄ monolayer. This process enhances both the adsorption stability and interfacial coupling strength.

To further elucidate the spatial distribution of the electronic structure during adsorption, the differential charge density (Δ$\rho$) was calculated for Li, Na, and K atoms adsorbed at the S4 site of β₁-ScSi₂N₄ and the S1 site of β₂-ScSi₂N₄. The results reveal evident regions of charge accumulation and depletion in the vicinity of the adsorption sites, indicating strong electronic coupling and electrostatic interactions between the adsorbed metal atoms and the ScSi₂N₄ substrate. These differential charge density maps not only corroborate the dominant role of electron transfer in the adsorption mechanism but also provide critical theoretical insight into the interfacial charge behavior of MA₂Z₄-based two-dimensional materials for applications in energy storage and related electrochemical systems.

$$\Delta \rho =\rho \left(MSc{Si}_2{N}_4\right)-\rho \left(Sc{Si}_2{N}_4\right)-\rho \left(M\right),M=Li,Na,K$$

(2)

Here, $\rho \left(MSc{Si}_2{N}_4\right)$ and $\rho (Sc{Si}_2{N}_4$) represent the electronic charge densities of the two-dimensional ScSi₂N₄ system with and without the adsorbed Li, Na, or K atoms, respectively. At the same time, $\rho \left(M\right)$ denotes the charge density of an isolated Li, Na, or K atom. In the differential charge density plots shown in Figure 7, the yellow and blue regions correspond to electron accumulation and depletion, respectively. A clear charge transfer from the Li, Na, and K atoms to the ScSi₂N₄ monolayer is observed, indicating significant electron redistribution upon adsorption. This charge transfer leads to the formation of stable chemical bonds (M–Sc) between the adsorbed metal atoms and the substrate. Such bonding interactions are beneficial for suppressing the aggregation of metal atoms into clusters, thereby enhancing the stability of single-atom dispersion and offering theoretical support for their application in energy storage and related fields.

To determine the maximum monolayer adsorption capacity of lithium and sodium atoms on two-dimensional ScSi₂N₄, structural optimizations were performed using a 2 × 2 × 1 supercell (as shown in Figure S1). Li and Na ions were incrementally added in pairs to the energetically favorable adsorption sites on the ScSi₂N₄ surface, followed by full geometric relaxation of each configuration. The final stable structures correspond to the fully adsorbed states of Li and Na. The results indicate that up to four Li or Na atoms can be stably adsorbed on each side (top and bottom) of both β₁-ScSi₂N₄ and β₂-ScSi₂N₄, resulting in a total of eight Li or Na atoms per supercell. This corresponds to a maximum stoichiometry of ScSi₂N₄-Li₂ and ScSi₂N₄-Na₂.

Similarly, the maximum adsorption capacity for potassium atoms was found to be two atoms per side, yielding a total of four K atoms per 2 × 2 × 1 supercell. This corresponds to a stoichiometry of ScSi₂N₄-K₁.₅. These findings provide important theoretical insights into the alkali metal storage capability of ScSi₂N₄ monolayers, which is crucial for evaluating their potential as electrode materials in alkali-ion batteries.

The calculation formula for the sequential binding energy is as follows:

$$E_{sb}\left(x\right)={\displaystyle \frac{\left[E\left(Sc{Si}_2{N}_4\right)+xE\left(M\right)-E\left(M_{x}Sc{Si}_2{N}_4\right)\right]}{x}},\left(M=Li,Na,K\right)$$

(3)

Here, $E\left(M_{x}Sc{Si}_2{N}_4\right)$, $E\left(Sc{Si}_2{N}_4\right)$, and $E\left(M\right)$ represent the total energies of the ScSi₂N₄ system with x adsorbed Li, Na, or K atoms, the pristine ScSi₂N₄ monolayer, and a single isolated Li, Na, or K atom, respectively. A decreasing trend in the sequential binding energy indicates that the adsorption becomes progressively less stable with increasing metal ion concentration. This can be attributed to the enhanced electrostatic repulsion between neighboring Li, Na, or K ions at higher coverages. The calculated sequential binding energies ($E_{sb}$) for Li and Na adsorption on both β₁- and β₂-ScSi₂N₄ phases remain positive across the full range of x $x$ values (1–8), confirming that these phases can stably host up to eight Li or Na atoms without reaching adsorption saturation. In contrast, potassium adsorption shows a marked drop in binding energy at $x$ = 8 (0.18 eV for β₁ and 0.38 eV for β₂), indicating that K approaches its saturation limit at high coverage and is less favorably accommodated than Li or Na. (show in Table S3) Overall, Li and Na exhibit strong and stable interactions with both ScSi₂N₄ polymorphs, whereas K shows limited adsorption capacity at high loading. At the same time, these results suggest that, throughout the entire adsorption process, Li, Na, and K atoms energetically prefer to bind to the 2D ScSi₂N₄ surface rather than aggregating into bulk metallic clusters, highlighting their favorable dispersion and potential for energy storage applications.

According to the maximum number of Li, Na, and K atoms stably adsorbed on 2D β₁-ScSi₂N₄ and β₂-ScSi₂N₄, the capacity is defined as follows:

$$C={\displaystyle \frac{cF}{M\left(Sc{Si}_2{N}_4\right)}}$$

(4)

where $c$ represents the maximum number of intercalated ions, $F$ is Faraday’s constant (26,801 mAh/mol), and $M$ denotes the molecular weight. By first principles calculation we calculated the theoretical storage capacities of the β₁ and β₂ phases of ScSi₂N₄. For lithium and sodium intercalation with Li (X = 8), the capacity reaches 343.45 mA h/g, while for potassium with K, it is 171.12 mA h/g. The calculated theoretical capacities surpass those of several typical anode materials, such as TiS₂ (240 mA h g⁻¹ for Li) [41], Ta₂CO₂ (132.1 mA h g⁻¹ for Li) [42], and YS₂ (350 mA h g⁻¹ for Na) [43]. These values are comparable to those of conventional anode materials, with the lithium and sodium capacities exhibiting excellent application potential.

The open-circuit voltage was also calculated to estimate the voltage output. The open-circuit voltage formula is:

$$V=-{\displaystyle \frac{\left[E\left(x_2\right)-E\left(x_1\right)-\left(x_2-x_1\right)E\left(M\right)\right]}{e\left(x_2-x_1\right)}},\left(M=Li,Na,K\right)$$

(5)

Here, $E\left(x_2\right)$, $E\left(x_1\right)$, and $E\left(M\right)$ denote the total energies of $M{x}_{2}Sc{Si}_2{N}_4$, $M{x}_{1}Sc{Si}_2{N}_4$ and a single Li, Na, or K atom, respectively. The calculated open-circuit voltages (OCVs) for both β₁- and β₂-ScSi₂N₄ exhibit a monotonic decrease with increasing alkali-metal concentration for all three ions (Li, Na, K), as illustrated in Figure 8. Throughout the entire insertion range, all OCV values remain positive, indicating thermodynamically favorable adsorption without spontaneous metal release. Notably, the final OCVs at the maximum insertion concentration are still within the practical voltage window for anode applications. These results collectively suggest that both β₁- and β₂-ScSi₂N₄ possess the key electrochemical characteristics required for high-performance Li-, Na-, and K-ion battery anodes.

Another critical feature of high-performance anode materials for rechargeable batteries is their rate capability, which originates from the mobility of the intercalated ions. To further evaluate the charge/discharge performance of ion batteries, the diffusion behavior of ScSi₂N₄ was investigated using the climbing image nudged elastic band (CI-NEB) method. S4 and S1 sites were chosen for simulation based on the adsorption energy analysis. One of the most favorable diffusion pathways at this site is illustrated in Figure 9a,b. The migration path of β₁-ScSi₂N₄ follows S4 → S2 → S4, while that of β₂-ScSi₂N₄ follows S1 → S2 → S1. The calculated diffusion energy barriers for Li, Na, and K ions in both β₁- and β₂-ScSi₂N₄ are notably low (Figure 9c,d), indicating favorable ion mobility. For comparison, these diffusion barriers are lower than or comparable to several reported anode materials, such as GeP₃ (0.50 eV) [44] and T-graphene (0.44 eV) [29]. The diffusion barriers for Na and K are comparable to those of MoS₂ (0.28 eV) [30] and Ti₃C (0.18 eV) [45], respectively.

These results suggest that the 2D ScSi₂N₄ electrode offers excellent rate performance for rechargeable Li-, Na-, and K-ion batteries. In particular, Na ions exhibit faster diffusion than Li ions due to their larger atomic radius, which leads to a greater distance from the substrate and reduced repulsive interactions. Similarly, K ions diffuse even more rapidly than Na ions for the same reason.

To further clarify the influence of structural anisotropy on ion migration, three possible diffusion pathways were evaluated, and the lowest-barrier Path 1 was chosen as the reference. The migration energy barriers of Li⁺, Na⁺, and K⁺ along Path 2 and Path 3 in both configurations (see Figure S2 and Table S4) reveal that in β₁, the barriers differ significantly between directions, demonstrating distinct anisotropic behavior—Li⁺ migrates more easily along Path 2, while Na⁺ and K⁺ prefer Path 3. This directional preference originates from variations in channel width and local potential fields. In contrast, β₂ exhibits overall higher migration barriers and reduced directional differences, approaching isotropic diffusion. These findings demonstrate that the inherently low migration barriers of ScSi₂N₄ not only ensure excellent ion transport kinetics, but also highlight that the asymmetric channels in β₁ can further enhance diffusion along specific directions, while the high-symmetry β₂ suppresses such directional selectivity.

4. Conclusions

This study systematically investigates the structural stability, electronic properties, and potential applications of two-dimensional β₁-ScSi₂N₄ and β₂-ScSi₂N₄ as anode materials for rechargeable Li-, Na-, and K-ion batteries. Our first-principles calculations demonstrate that both phases possess low diffusion energy barriers for the considered alkali ions, indicating excellent ion mobility and suggesting potential for fast charge/discharge cycles. The materials also exhibit high theoretical capacities, reflecting their ability to accommodate a significant number of ions without structural degradation. Moreover, the predicted average open-circuit voltages lie within suitable ranges that align well with typical metal-ion battery operating voltages, indicating good compatibility with existing battery technologies. Taken together, these favorable structural and electrochemical properties highlight two-dimensional ScSi₂N₄ as a promising candidate for next-generation anode materials in diverse alkali-ion battery systems.