Potassium cation storage and diffusion in SnS, SnS₂, and at SnS/SnS₂ interfaces

Abstract

Due to their promising performance, tin sulfide and tin disulfide have been investigated as anode materials in various types of batteries, such as Li-, Na-, and K-ion batteries. Understanding the thermodynamics and kinetics of processes involving metal ions at the atomistic level, and how these processes differ between tin sulfide and tin disulfide, is crucial for improving their electrochemical performance in respective applications. However, a direct comparison between these two materials during battery operation has been limited so far. Here, we report on potassium cation diffusion barriers in bulk tin sulfide and tin disulfide, as well as parallel and perpendicular to several SnS/SnS₂ interfaces by means of density functional theory calculations. We also investigate the thermodynamics of potassium storage in these materials. Our results demonstrate that while K⁺ diffusion in SnS occurs through elemental processes involving lower energy barriers, potassium storage in SnS₂ is thermodynamically more favorable. These observations suggest strategies to improve the overall electrochemical performance of SnS/SnS₂ heterostructures in battery applications.

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1. Introduction

Tin sulfide (SnS) and tin disulfide (SnS₂) have had a wide range of uses, ranging from earlier optoelectronic applications, such as solar cells^1–4 and photodetectors,^5–9 to ongoing investigations into their potential as battery materials. They were first proposed as conversion-type anode materials for lithium-ion batteries (LIBs), due to their high theoretical capacity and layered structure.^10–16 SnS₂ has also been studied as an anode material for sodium-ion batteries (SIBs), particularly due to its increased interlayer spacing, making it more suitable for accommodating larger Na⁺ ions.^12,17–19 Both SnS and SnS₂ have recently been studied as electrode materials for potassium-ion batteries (PIBs)^20–26 and lithium–sulfur batteries (LSBs)^27–30 as well. It has been shown that tin-based heterostructures,³¹ SnS₂ nanosheets anchored to doped MXene sheets,³² or composite materials out of SnS₂ and reduced graphene oxide^33,34 or doped carbon nanofibers²¹ could exhibit an improve electrochemical performance as anodes for potassium-ion batteries, partially due to metallic nature of the underlying carbonaceous material.^35,36 However, in their battery-related applications, challenges arise due to their volume expansion during battery operation, low electrical conductivity, and sluggish kinetics.³⁷ SnS/SnS₂ heterostructures have been proposed to provide better structural stability,^6,29,38 while anchoring these heterostructures to reduced graphene oxide is believed to enhance overall electronic conductivity and rate performance.^39–41

However, poor kinetics of metal ion diffusion in these structures still poses a major challenge, especially in the case of PIBs involving large potassium ions. In order to improve the kinetics of cation intercalation and diffusion, it is necessary to gather an atomistic picture on the underlying fundamental processes. In addition, a deeper comparative understanding of the thermodynamics of metal cation storage in SnS and SnS₂ will help further optimize the morphology of SnS/SnS₂ heterostructure for more efficient batteries. In this communication, we study the potassium cation diffusion in bulk SnS and SnS₂, as well as at SnS/SnS₂ interfaces by means of quantum-chemical calculations. Moreover, the thermodynamics of potassium storage in the bulk materials and the charge distribution at the interface are investigated and discussed.

2. Methodology

We constructed supercells of bulk SnS (2 × 4 × 4) and SnS₂ (4 × 4 × 4) from experimental crystallographic data^42,43 (Fig. 1). In this study, VMD⁴⁴ is used for visualization of all systems.

Fig. 1 Structures of bulk SnS (left) and SnS₂ (right) together with potassium ions considered in this work for ab initio molecular dynamics and nudged elastic band calculations. Tin atoms, sulfur atoms, and potassium ions are shown in gray, yellow, and purple. The same color coding is used throughout this paper.

We further constructed SnS/SnS₂ interfaces using pymatgen⁴⁵ from a (302) SnS surface slab and a (110) SnS₂ surface slab with two different terminations, as shown in Fig. 2.

Fig. 2 Two model SnS/SnS₂ interfaces formed by a (302) surface slab of SnS and a (110) surface slab of SnS₂ with two different terminations, labeled as interface 1a (left panel) and interface 1b (right panel).

Finally, a SnS/SnS₂ interface was constructed from a (020) surface slab of SnS and a (110) surface slab of SnS₂ (Fig. 3).

Fig. 3 A model SnS/SnS₂ interface formed by a (020) surface slab of SnS and a (110) surface slab of SnS₂, labeled as interface 2.

The lattice constants and atomic coordinates of all structures were fully relaxed; the resulting configurations are shown in the ESI.† Subsequently, four K⁺ ions were added to each system, followed by a second relaxation step. Atomic coordinates of the relaxed systems are given as ESI.† ⁴⁶ The resulting structures were used in density functional theory (DFT)-based ab initio molecular dynamics (AIMD) simulations to identify possible diffusion paths in each system.

DFT^47,48 calculations were performed using the CP2K^49,50/QUICKSTEP^50–52 software together with a DZVP-MOLOPT-SR-GTH basis set⁵³ for the valence electrons, and Goedecker–Teter–Hutter (GTH) pseudopotentials.^54–56 To account for exchange and correlation (XC) effects, we applied the Perdew–Burke–Ernzerhof (PBE) functional,^57,58 together with the semi-empirical DFT-D3 method⁵⁹ with Becke–Johnson (BJ) damping⁶⁰ and revised damping parameters⁶¹ to correct for the long-range dispersion interactions. A plane-wave energy cutoff of 350 Ry and a relative cutoff of 40 Ry were chosen. Only the Γ point was considered due to the large supercells and the convergence criterion for self-consistent field cycles was set to 10⁻⁶. Periodic boundary conditions were used for all calculations.

AIMD simulations were carried out in a canonical ensemble (NVT) at 500 K for 40 ps to 100 ps using a Nosé–Hoover chain thermostat.^62,63 A time step of 2 fs was chosen for atomic coordinate propagation.

To calculate the migration barriers for each of the paths obtained through AIMD simulations, climbing image⁶⁴ nudged elastic band^65–67 (CI-NEB) calculations were performed. In the CI-NEB calculations, the number of replicas and the spring constant were set to 8 and 2.0 × 10⁻² a.u., respectively.

Layer-resolved sums of charges perpendicular to the interface (along z axis) were calculated using different approaches, namely density-derived atomic point charges (DDAPC),⁶⁸ Hirshfeld,⁶⁹ Mulliken,⁷⁰ and restrained electrostatic potential (RESP).⁷¹

3. Results and discussion

Our AIMD simulations at 500 K reveal that, in bulk SnS, K⁺ ions migrate via low-energy path (1) shown in Fig. 4(a). Moreover, we observe a reversible K ↔ Sn substitution, hindering the diffusion of potassium in the lattice. In bulk SnS₂, however, no K⁺ migration is observed. In the AIMD simulations of interfaces 1a and 1b (Fig. 2), no diffusion is detected either. However, we observe K⁺ migration at interface 2 via paths (7) to (10) [Fig. 4(g)–(j)]. The corresponding K⁺ diffusion coefficients computed from the mean square displacements (MSDs) using TRAVIS^72,73 are 81 pm² ps⁻¹ and 230 pm² ps⁻¹ for bulk SnS and interface 2, respectively.

Fig. 4 K⁺ migration paths and corresponding energy profiles in different structures: (a) path (1) in bulk SnS; (b) path (2) in bulk SnS₂; (c)–(f) paths (3)–(6) at SnS/SnS₂ interface 1b; (g)–(j) paths (7)–(10) at SnS/SnS₂ interface 2.

Altogether, we studied ten migration pathways: path (1) from the AIMD simulation for SnS [Fig. 4(a)], one theoretically constructed path (2) for SnS₂ [Fig. 4(b)], four theoretically constructed paths (3) to (6) for interface 1b [Fig. 4(c)–(f)], and four paths (7) to (10) from the MD simulation for interface 2 [Fig. 4(g)–(j)]. Atomic coordinates of the fully relaxed diffusion paths together with the respective energy profiles are provided as ESI.† ⁴⁶ Energy barriers are given in Table 1. Note that the asymmetry of migration barriers in SnS and SnS₂ is due to the presence of adjacent K⁺ ions. In principle, the initial and final K⁺ interstitial sites are crystallographically equivalent within paths (1) and (2).

Table 1 K⁺ migration barriers ΔE^M in SnS, SnS₂ and at SnS/SnS₂ interfaces for the forward (→) and backward (←) processes, along with jump distances d

System	Path no.	d/Å	ΔE^M→/eV	ΔE^M←/eV
Bulk SnS	(1)	1.76	0.21	0.23
Bulk SnS2	(2)	3.31	0.71	0.37
Interface 1b	(3)	1.88	0.05	0.06
	(4)	3.89	0.72	0.23
	(5)	6.17	2.42	1.17
	(6)	5.21	1.26	0.97
Interface 2	(7)	4.28	0.87	0.99
	(8)	3.86	1.44	0.85
	(9)	2.30	0.38	0.75
	(10)	2.51	0.24	0.09

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Path (2) is higher in energy than path (1), showing better K⁺ diffusion kinetics in SnS. At high charge/discharge rates, this could suggest better electrochemical performance of SnS-containing electrodes than of those involving SnS₂, since at a higher current rate the kinetics becomes more important. For Na-ion batteries, this improved rate performance for SnS has been measured.⁴⁰ We constructed two paths (3) and (4) parallel to interface 1b [Fig. 4(c) and (d)], as well as two paths (5) and (6) perpendicular to it [Fig. 4(e) and (f)]. K⁺ migration parallel to the interface is significantly more facile than perpendicular to it, with energy barriers comparable to bulk SnS and SnS₂. Note that all interfacial paths (3) to (10) alone cannot mediate long-range diffusion. This is particularly true for low-energy path (3), which must be followed by path (4) for effective migration in the y-direction. Higher diffusion barriers perpendicular to the interface indicate possibly hindered K⁺ motion between the two materials in a heterostructure. For interface 2, paths (7), (9) and (10) describe migration parallel to the interface and path (8) perpendicular to it. Paths (7) and (8) have rather high barriers, while paths (9) and (10) are again comparable to the bulk materials.

Overall, we conclude that K⁺ diffusion kinetics in bulk SnS is faster than in bulk SnS₂, suggesting that a heterostructure could benefit from more facile diffusion in SnS. In addition, migration parallel to SnS/SnS₂ interfaces shows energy barriers that range between SnS and SnS₂, neither affecting material performance positively nor negatively. Perpendicular to these interfaces, K⁺ diffusion is slowest, indicating that an optimal particle size and morphology must be experimentally determined for good anode kinetics. The SnS₂ particles should be small enough to prevent long-range bulk diffusion, while the bulk-to-interface ratio should be large enough to minimize excessive migration across the interfaces.

To study the thermodynamics of potassium intercalation, we calculated the enthalpy of the intercalation reactions as

where x = 1, 2 and c denotes the reaction involving a potassium atom (c = 0) or a cation (c = +1). In the calculation of enthalpies, volume expansion was assumed to have a negligible contribution. The difference between the K⁺ intercalation enthalpies in SnS and SnS₂, defined as , was obtained to be around −1.335 eV, while ΔΔH⁰ ≃ −1.762 eV in the case of a neutral potassium atom. This clearly shows a more thermodynamically favorable potassium storage in bulk SnS₂, resulting in a lower potential difference to the cathode material. Moreover, the theoretical capacities of SnS and SnS₂ for potassium storage, as reported in the literature, are 1136 mA h g⁻¹ and 733 mA h g⁻¹, respectively.²⁵ However, under experimental conditions, the specific capacities and cycling performances in the case of sodium storage are significantly closer.^38,40

Furthermore, we calculated the sum of charges per layer perpendicular to interface 1b using different theoretical approaches (Fig. 5). The layer thickness was set to 2.5 Å. Apart from discrepancies in values obtained via different methods, the formation of an interfacial electric field is predicted by all theoretical approaches. This built-in electric field could explain the comparatively higher diffusion barrier from the interface region into the SnS slab [path (5)] compared to the SnS₂ slab [paths (6) and (8)] through electrostatic repulsion and attraction, respectively.

Fig. 5 Layer-wise sum of charges perpendicular to the SnS/SnS₂ interface 1b calculated using different methods. The shaded region highlights the interface.

4. Conclusions

In this study, we report the thermodynamics of potassium storage and the kinetics of K⁺ migration in bulk SnS and SnS₂, as well as parallel and perpendicular to different SnS/SnS₂ interfaces, obtained through density functional theory calculations. Our findings reveal that bulk SnS₂ offers thermodynamically more stable potassium storage, while K⁺ diffusion is faster in bulk SnS due to atomistic processes with comparatively lower barriers. Therefore, at higher current rates, SnS is expected to outperform SnS₂. These results also suggest potential routes for designing SnS/SnS₂ heterostructures to improve overall battery performance. Diffusion barriers parallel to the SnS/SnS₂ interfaces in the heterostructure are comparable to those in the individual bulk materials. However, our findings show that, despite the formation of an interfacial electric field, K⁺ diffusion perpendicular to the interface is the slowest process in SnS/SnS₂ heterostructures overall. Therefore, we propose an experimental optimization of the SnS/SnS₂ heterostructures with respect to bulk-to-interface ratio. In terms of composition, increasing the SnS content leads to better rate performance, while SnS₂ should result in intercalation of potassium starting at lower voltages during the charging process. Our theoretical results hint at synthesis strategies to form SnS/SnS₂ heterostructures that maximize the bulk-to-interface ratio to minimize excessive migration across the interfaces between the two materials.

Author contributions

C. K. contributed to formal analysis, investigation, methodology, visualization, and writing (original draft, review & editing); D. S. contributed to supervision and validation; P. P.-A. contributed to conceptualization, funding acquisition, methodology, supervision, validation, and writing (review & editing).

Data availability

Data supporting this article have been included as part of the ESI.† Data for this article, including relaxed atomic coordinates of the studied systems and diffusion paths (.xyz files), as well as energy profiles for the latter (.txt files) are available at Zenodo at https://doi.org/10.5281/zenodo.15011135.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

P. P.-A. gratefully acknowledges DFG funding via projects 420536636 and 446879138, as well as the computing time made available on the high-performance computer at the NHR Center of TU Dresden via the project ‘p_oligothiophenes.’ The authors also thank Wenxi Wang and Yan Lu for fruitful discussions.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ma00230c

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