Structure and property exploration of two-dimensional, bulk, and cluster lithium sulfide using the IM²ODE method

Abstract

Lithium sulfide (Li₂S) plays an important role in fields such as energy, environment and semiconductors. Exploration of the microstructure of Li₂S has significant implications for developing new materials and optimizing related material properties. In this work, the inverse design of materials by the multi-objective differential evolution (IM²ODE) method combined with density functional theory (DFT) calculations was used to predict the two-dimensional (2D), three-dimensional (3D), and cluster structures of Li₂S. Their structural stabilities and electronic properties were further investigated. Novel monolayer and double-layer hexagonal structures of 2D Li₂S are predicted. The double-layer structure has better thermal stability and a wider band gap of 3.5 eV than the single-layer structure. Various novel structures of 3D Li₂S are predicted. Some structures are similar to 1T-MoS₂ and the double-layer hexagonal structure of 2D Li₂S. With increasing number of atoms, the (Li₂S)_n clusters converge into a cage-like structure and their average binding energies decrease. The second-order energy differences of (Li₂S)_n clusters show an odd–even oscillation rule. The ionization potentials, electron affinities, electronegativities, and chemical hardnesses also decrease. These findings should improve theoretical understanding of the properties and behavior of new 2D, 3D, and cluster functional materials.

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1. Introduction

As a typical alkali metal sulfide with a wide band gap, lithium sulfide (Li₂S) has received much attention in recent years.^1–3 It has been widely used in ceramics, semiconductors and energy materials.^4–6 Li₂S can serve as a cathode material with a theoretical capacity of 1166 mA h g⁻¹ for lithium–sulfur (Li–S) batteries.⁷ However, certain deficiencies, such as low conductivity, a shuttle effect, and volume expansion, limit its performance and practical applications.^8–13 Nanoscale Li₂S has unique advantages and has facilitated many advances.^14,15 Since the discovery of graphene, remarkable progress has been made in the field of two-dimensional (2D) materials, which has led to a wide range of applications in several fields.^16–21 The structures of (Li₂S)_n clusters have also been reported.²² These can be used as electrolyte and cathode materials to achieve high energy density and long life for Li-ion or Li–S batteries.^23–25

Focusing on the prediction of lithium sulfide material structures, this study aims to explore and reveal the effectiveness and feasibility of these new structures in enhancing the overall performance of lithium–sulfur batteries. Through a combination of the inverse design of materials by the multi-objective differential evolution (IM²ODE) method and density functional theory (DFT) calculations, novel two-dimensional (2D), three-dimensional (3D), and cluster structures of Li₂S were predicted. Their stabilities and electronic properties were investigated. The discovery of new Li₂S structures may deepen our understanding of the essential properties and behavior of new functional materials.

2. Computational details

IM²ODE is a structural search method based on a multi-objective differential evolutionary algorithm, which is capable of searching for structures with specific properties.²⁶ To date, IM²ODE software has demonstrated powerful search and optimization capabilities, being applied to Au_n and TiO₂.^27,28 Here, IM²ODE was used to perform structural searches for 2D, 3D, and clusters of Li₂S composed of Li and S in a 2 : 1 stoichiometry. For the structure search of 2D Li₂S, the thickness of the vacuum layer was set to 15.0 Å and the projected area was set to 20.0 Ų. For the structural search of 3D Li₂S, the box volume and the number of atoms were correspondingly increased. For the cluster search, the ball mode was used. With increasing number of (Li₂S)_n cluster atoms, both the initial box volume and the cluster radius were increased appropriately.

Structure optimization and electronic property calculations for all predicted Li₂S structures were further performed using the Vienna ab initio simulation package (VASP) and DS-PAW, which are based on DFT.^29,30 The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional was adopted.³¹ The DFT-D3 method was used for dispersion correction.^32,33 The convergence criteria of the total energy and final force on all relaxed atoms were 10⁻⁶ eV and 0.01 eV Å⁻¹, respectively. For 2D and 3D Li₂S structures, default values were used for K points. For the Li₂S clusters, the gamma point was used for K points.

3. Results and discussion

3.1. Structures and properties of 2D Li₂S

3.1.1. Double- and single-layer hexagonal 2D Li₂S. The stability of 2D Li₂S structures can be assessed by the average binding energy (E_b), which is defined according to eqn (1):where E(Li₂S)_n is the energy of the Li₂S structure, and E(Li) and E(S) are the energies of the Li and S atoms, respectively. Fig. 1 shows the E_b ranking of all structures found in the IM²ODE search. There are three energy plateaus. The first plateau is located between −2.05 and −1.99 eV. Among these structures, there are some similar structures with different space groups. The second plateau is located between −1.93 and −1.82 eV and demonstrates relatively stable structures slightly higher in energy. The third plateau is between −1.42 and −1.40 eV, in which the structures of higher energy are unstable. Here, a novel double-layered hexagonal structure of 2D Li₂S with the lowest E_b of −2.15 eV was found, as shown in Fig. 2a and b. During extensive exploration, a single-layer hexagonal structure of 2D Li₂S was also investigated, as shown in Fig. 2c and d.

Fig. 1 The average binding energies of all predicted 2D structures.

Fig. 2 Top (a and c) and side (b and d) views of double- and single-layer hexagonal 2D structures. The yellow and purple balls represent S and Li atoms, respectively.

The single-layer hexagonal structure of 2D Li₂S has space group 164 (P3̄m1). It is a sandwich structure, with the lithium atoms in the outer layers and the sulfur atoms in the inner layer, similar to that of 1T-MoS₂.^34,35 The distances between Li and three S atoms are 2.384 Å. The linear distances between Li and three neighboring Li atoms are 2.697 Å. In the novel double-layer hexagonal structure of 2D Li₂S with space group 164 (P3̄m1), the upper and lower layers are similar to the single-layer hexagonal structure. These two single-layer structures are alternately arranged, connected through Li–S chemical bonds. In the double-layer structure, the distances between Li and three S atoms are 2.385 Å. The distances between Li and three neighboring Li atoms are 2.759 Å, respectively. The lengths of the S–Li bonds between the double layers are 2.448 Å. Their geometrical parameters, E_b and band gap (E_g) of double- and single-layer hexagonal structures are shown in Table 1.

Table 1 The parameters of double- and single-layer hexagonal structures of 2D Li₂S

Structure	Space group	Lattice parameter a, b and c (Å)	Thickness (Å)	E b (eV)	E g (eV)
Double-layer	P3̄m1	3.976, 3.976, 24.264	4.633	−2.153	3.50
Single-layer	P3̄m1	3.932, 3.932, 11.953	1.457	−2.052	3.14

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As shown in Fig. 3a and b, the double-layer hexagonal structure of 2D Li₂S has a wider band gap than the monolayer hexagonal structure. The single-layer hexagonal structure has a direct band gap of 3.14 eV and the double-layer hexagonal structure has a direct band gap of 3.50 eV. The spin-up and spin-down densities of states (DOSs) are uniformly distributed. The spin-up DOSs of double-layer and monolayer 2D Li₂S are shown in Fig. 3c and d. In the DOS on the left of the Fermi energy, the Li 2s and S 3p orbitals of 2D Li₂S make the main contributions. On the right, the Li 2p and S 4s orbitals of 2D Li₂S make the main contributions. Due to strong chemical bonds between the two monolayers, the gap between the valence and conduction bands further broadens.

Fig. 3 Band structures (a and b) and densities of states (c and d) of double- and single-layer hexagonal structures of 2D Li₂S.

Phonons are quantized vibrations of a crystal lattice, which have a discrete frequency spectrum.³⁶ Phonons are important in the thermodynamic properties of solids, as in heat and sound propagation. The phonon spectrum of 2D Li₂S was predicted by density functional perturbation theory.³⁷ The phonon spectra of the single- and double-layer hexagonal structures of 2D Li₂S are shown in Fig. 4. For example, the single-layer hexagonal structure has 9 phonon modes. There are no imaginary frequencies throughout the Brillouin zone, which means that these two structures are stable.

Fig. 4 Phonon spectra of double- (a) and single-layer (b) 2D Li₂S.

3.1.2. Other monolayer 2D Li₂S. Besides the single- and double-layer hexagonal structures mentioned above, a number of other 2D structures were predicted by IM²ODE. Fig. 5 shows some predicted 2D structures of Li₂S. The first four are flat monolayer structures of 2D Li₂S. Structure 2D-a is a flat, single-layer hexagonal array with S atoms occupying the center of the hexagon. Structure 2D-b is a flat monolayer rectangular array formed by four S atoms and four Li atoms. Structure 2D-c is a flat, single-layer octagonal array. Structure 2D-d is a flat, single-layer chain structure. The latter four structures are twisted monolayer hexagonal structures of 2D Li₂S. Both 2D-e and 2D-f are twisted hexagonal structures. The difference between them is that the atoms are not twisted in the same position. Structure 2D-g has a twisted rectangular form, while structure 2D-h shows a regular wavy form.

Fig. 5 Other flat 2D-a–2D-d (a)–(d) and twisted 2D-e–2D-h (e)–(h) monolayer structures of 2D Li₂S.

For the predicted flat and twisted monolayer structures of 2D Li₂S, the geometrical parameters, E_b and E_g, are shown in Table 2. According to Table 2, the E_b of 2D Li₂S is inversely proportional to its band gap. The maximum band gap is 3.94 eV and the minimum band gap is 1.57 eV. Among the twisted monolayer structures, the hexagonal structure (Fig. 5e) with the band gap of 3.30 eV is the most stable. Another hexagonal structure (Fig. 5f) has a band gap of 3.69 eV.

Table 2 The parameters of other 2D Li₂S

Structure	Space group	Lattice parameter/Å (a, b, c)	Thickness (Å)	E b (eV)	E g (eV)
2D-a	P6/mmm	4.162, 4.162, 10.937	Planar	−1.930	3.94
2D-b	P4/mmm	4.508, 4.508, 8.367	Planar	−1.845	2.69
2D-c	P2/m	14.105, 4.405, 5.477	Planar	−1.602	1.83
2D-d	Pmmm	3.745, 6.297, 7.032	Planar	−1.665	1.57
2D-e	P21/m	13.592, 6.359, 3.951	1.98	−2.055	3.30
2D-f	P21/m	6.532, 4.097, 12.824	2.84	−1.994	3.69
2D-g	P21212	5.336, 6.300, 10.114	1.42	−1.852	3.16
2D-h	Pmma	8.132, 4.633, 8.935	2.75	−1.851	3.01

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3.2. Structure and properties of 3D Li₂S

3.2.1. Typical 3D Li₂S. As described for 2D Li₂S, the E_b of all predicted structures of 3D Li₂S were ranked. In Fig. 6, the red dots indicate the E_b of the structures in the Materials Project (MP) database. The blue dots indicate the E_b of the structures identified in the IM²ODE search that match those in the MP database. The black dots indicate the E_b of novel structures predicted by IM²ODE. There are three distinct plateaus, which are located at −2.25 to −2.19 eV, −2.17 to −2.07 eV, and −1.94 to −1.87 eV. In Fig. 6, three typical structures of 3D Li₂S predicted by IM²ODE match the MP database. Their structures are shown in Fig. 7. Their lattice parameters, space groups, and band gaps are listed in Table 3.

Fig. 6 The average binding energies of the predicted 3D structures.

Fig. 7 Three typical structures, T-a(a), T-b (b) and T-c of 3D Li₂S.

Table 3 Predicted lattice parameters, space groups, and band gaps of 3D Li₂S structures that match the MP database

Structure	Lattice parameter a, b and c (Å)	Cell volume (Å^3)	Space group
T-a	5.716, 5.716 5.716	186.771	Fm3̄m
T-b	6.297, 3.811, 7.235	173.615	Pnma
T-c	3.941, 3.941, 7.086	95.296	P63/mmc

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Fig. 8 presents the band structures and DOS for specific space group structures. Their band gaps are 3.40, 3.94, and 3.20 eV, which are consistent with the MP values of 3.40, 3.93, and 3.17 eV.³⁸ From the DOS, it can be seen that the Li 2s and S 3p orbitals of 3D Li₂S make the main contributions.

Fig. 8 Band structures and densities of states for three typical structures, T-a(a), T-b (b) and T-c (c).

3.2.2. Other 3D Li₂S. Here, the highest E_b of the three structures of 3D Li₂S in the MP database was used as a cut-off criterion. The structures below this energy were further investigated and divided into two parts, A and B, as shown in Fig. 6. On plateau A, the structures are essentially the same as those in the library with the space group Fm3̄m. Most of the novel structures are found on plateau B. Excluding similar structures, the structures on plateau B can be categorized into eight types, as shown in Fig. 9. The geometrical parameters of the structures are shown in Table 4. The crystal systems can be categorized in terms of space groups. Structure N-a has a hexagonal crystal form. Structures N-b and N-h have orthorhombic crystal forms. Structures N-c and N-g have monoclinic crystals. Structures N-d, N-e, and N-f have tetragonal crystal forms. An interesting phenomenon is that a similar sandwich structure to that of 1T-MoS₂ also exists in the 3D structures, as shown in Fig. 9a and b.

Fig. 9 Eight structures (a–h) of 3D Li₂S.

Table 4 Geometrical parameters of the predicted 3D Li₂S structures

Structure	Lattice parameter a, b and c (Å)	Cell volume (Å^3)	Space group
N-a	3.982, 3.982, 6.787	93.216	P63mc
N-b	4.291, 6.464, 6.493	180.081	Cmc21
N-c	6.089, 5.297, 5.662	182.619	Abm2
N-d	5.518, 5.545, 7.985	186.289	C2
N-e	3.829, 3.829, 6.120	89.712	P4/nnm
N-f	5.304,5.304, 3.915	110.151	P42/mnm
N-g	4.429, 4.146, 6.481	107.801	P21/m
N-h	6.067, 5.978, 5.915	214.482	Cmma

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The band structures and DOS are shown in Fig. 10. Among these 3D structures, the direct band gaps of structures N-a, N-f, and N-h are 3.68, 2.80, and 2.89 eV, respectively. The indirect band gaps of structures N-b, N-c, N-d, N-e, and N-g are 2.87, 3.48, 3.38, 3.55, and 3.04 eV, respectively. As regards the DOS, both the valence and conduction bands are mainly contributed by the Li 2s, S 3s, and S 3p orbitals. In Fig. 10, the widths of the DOS at the Fermi level are in accordance with the gaps in the band structures.

Fig. 10 Band structures and densities of states of the other eight structures (a–h) of 3D Li₂S.

The phonon spectra of all these structures of 3D Li₂S were obtained. According to the calculation results, only the phonon spectrum of structure N-a and three typical structures (T-a, T-b and T-c) do not show imaginary frequencies, as shown in Fig. 11. These indicate that the typical structures T-a, T-b and T-c and the new N-a structures are stable.

Fig. 11 Phonon spectrum of the structures, T-a (a), T-b (b), T-c (c) and N-a (d).

3.3. Structures and properties of (Li₂S)_n clusters

The structures of (Li₂S)_n (n = 1–12) clusters were systematically searched by means of IM²ODE. This process aimed to identify the most stable structure with the lowest energy at different sizes, as shown in Fig. 12. It may be noted that the (Li₂S)_n (n = 1–12) clusters gradually converge to a cage-like structure as the number of atoms is increased. The stabilities and electronic properties of these cluster structures were analyzed in detail.

Fig. 12 Lowest energy structures of (Li₂S)_n clusters: (a)–(l) correspond to n = 1–12. The yellow and purple balls represent S and Li atoms, respectively.

3.3.1. Relative stability of (Li₂S)_n clusters. In general, the stability of a cluster can be judged by the average binding energy (E_b) and the second-order energy difference (Δ²E_n).^39,40 The variation in the E_b with cluster size is shown in Fig. 13a. As the cluster size increases, E_b monotonically decreases, which is consistent with previous results.²² The second-order energy difference can be defined as:

Δ²E_n = E_n+1 + E_n−1 − 2E_n (2)

Fig. 13 The average binding energies (a) and second-order energy differences (b) of the clusters.

As shown in Fig. 13b, the second-order energy difference clearly shows the relative stability of (Li₂S)_n (n = 1–12) clusters. In the interval of n = 2–11, Δ²E_n shows an alternating parity pattern. Clusters containing an even number of S atoms have a relatively higher Δ²E_n than those containing an odd number of S atoms. This phenomenon is known as the parity oscillation effect.⁴¹ It can be observed that the highest peak of second-order energy difference corresponds to the (Li₂S)₄ cluster.

3.3.2. Electronic properties of (Li₂S)_n clusters. The electronic properties of (Li₂S)_n (n = 1–12) cluster structures were calculated. These included the DOS, the energy gap (E_gap) between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), ionization potential (IP), electron affinity (EA), electronegativity (χ_M), and chemical hardness (η).

The DOS of the clusters reflects the distribution of the number of electrons in a certain energy level range.⁴² If the number of energy levels in the given interval is higher, the DOS is higher. The DOS of (Li₂S)_n (n = 1–12) clusters is shown in Fig. 14. As the cluster size increases, the peaks on either side of the Fermi energy level become larger. The bandwidth gradually broadens from 1 eV for Li₂S to 3 eV for (Li₂S)₁₂. The peaks between −4 and −1 eV are mainly due to the 3p state of the S atoms. The peak due to the 3p state of the S atom moves away from the Fermi energy level when n < 11.

Fig. 14 Densities of states of (Li₂S)_n (n = 1–12) clusters.

The energies (E_HOMO and E_LUMO) of the HOMO and LUMO levels reflect the ability of a cluster to gain or lose electrons.⁴³ In Table 5, it can be seen that the E_HOMO of the clusters was the smallest (absolute maximum) when n = 4. The E_gap between E_HOMO and E_LUMO is an important parameter in characterizing the electronic structure and stability of a cluster. E_gap corresponds to the ability of an electron to jump from an occupied orbital to a vacant orbital. To some extent, it also represents the ability of the molecule to participate in chemical reactions.⁴⁴ Higher energy gaps mean that higher energies are required to change the electronic structure of the clusters. The corresponding clusters are more chemically stable and less chemically reactive. In Table 5, it can be seen that the E_gap of the clusters gradually decreased from n = 1 to 3, and then stabilized for n ≥ 4.

Table 5 Orbital energy levels of (Li₂S)_n (n = 1–12) clusters

Cluster	E LUMO (eV)	E HOMO (eV)	E gap (eV)
Li2S	−1.385	−3.549	2.164
(Li2S)2	−1.614	−3.496	1.882
(Li2S)3	−1.831	−3.593	1.762
(Li2S)4	−1.213	−4.391	3.178
(Li2S)5	−1.179	−4.143	2.964
(Li2S)6	−1.051	−4.037	2.986
(Li2S)7	−1.020	−3.895	2.875
(Li2S)8	−0.787	−3.768	2.982
(Li2S)9	−1.022	−3.757	2.735
(Li2S)10	−0.844	−3.744	2.900
(Li2S)11	−0.758	−3.751	2.993
(Li2S)12	−1.326	−3.627	2.300

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Ionization potential (IP) is an important parameter that reflects the interdependence between the cluster size and electronic structure.⁴⁵ The IP of a system can be defined as:

IP = E(N − 1) − E(N) (3)

where N is the number of electrons, E(N) is the energy of the (Li₂S)_n cluster, and E(N − 1) is the energy of the cluster after losing an electron. Here, it is assumed that the spatial structure of the ionization system remains unchanged. Fig. 15a illustrates the IPs for n = 1–12 (Li₂S)_n clusters. They show an overall decreasing trend with increasing size.

Fig. 15 Ionization potentials (a), electron affinities (b), electronegativities (c), and chemical hardnesses (d) of (Li₂S)_n (n = 1–12) clusters.

Electron affinity (EA) refers to the energy that needs to be surmounted for an atom or molecule to gain an electron.⁴⁶ EA can be defined as:

EA = E(N) − E(N + 1) (4)

The lower the EA, the easier it is to gain an electron. Fig. 15b illustrates the variation in the EAs of (Li₂S)_n clusters with size. They show a general decreasing trend. Among the homologues, (Li₂S)₃ has the highest EA value.

Electronegativity (χ_M) combines the IP and the EA.⁴⁷ It indicates the relative strength of the ability to attract electrons when forming a chemical bond between two different atoms.⁴⁸ The electronegativity of atoms and molecules can be estimated according to eqn (6):

Fig. 15c shows the variation of the electronegativity of (Li₂S)_n with cluster size. In general, it decreases with increasing cluster size.

Chemical hardness (η) can be used to characterize the stability of clusters.⁴⁹ Theoretically, η can be calculated from the IP and the EA:

The calculated chemical hardnesses of (Li₂S)_n clusters are shown in Fig. 15d. The η of (Li₂S)_n decreases with increasing cluster size.

By investigating IP, EA, χ_M, and η, it became apparent that these properties of (Li₂S)_n (n = 1–12) generally decrease with increasing cluster size. This trend may be influenced by various factors, such as size effects, surface effects, and internal interactions. These findings further our understanding of the structures and properties of (Li₂S)_n clusters, and provide theoretical guidance for application in various fields.

4. Conclusions

In summary, the structures and electronic properties of 2D, bulk, and cluster Li₂S have been predicted by IM²ODE and DFT methods. Novel monolayer and double-layer hexagonal structures of 2D Li₂S have been predicted. The double-layer structure is more stable than the single-layer structure and has a wider band gap. These structures are predicted to be stable by phonon spectral calculations. The predicted structures of 3D Li₂S encompass three structures in the MP database and various novel structures. They also include a sandwich structure similar to that of 1T-MoS₂ and double-layer hexagonal structures of 2D Li₂S. These 3D Li₂S structures have a wide range of band gaps. A stable new 3D structure with the space group P6₃mc was found. With increasing number of atoms, the (Li₂S)_n clusters converge into a cage-like structure, and the average binding energy decreases. The second-order energy difference shows an odd–even oscillatory pattern. The ionization potentials, electron affinities, electronegativities, and chemical hardnesses decrease with increasing number of atoms. These findings should provide greater theoretical understanding of the properties and behavior of new 2D, 3D, and cluster functional materials.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (21703158, 21873073 and 12404265) and the Basic Research Project of Wenzhou City (L2023005, L20240001 and L20240002). We gratefully acknowledge HZWTECH for providing computation facilities.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp03587a

‡ Both authors contributed equally.

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