Structure search of two-dimensional systems using CALYPSO methodology

Abstract

The dimensionality of structures allows materials to be classified into zero-, one-, two-, and three-dimensional systems. Two-dimensional (2D) systems have attracted a great deal of attention and typically include surfaces, interfaces, and layered materials. Due to their varied properties, 2D systems hold promise for applications such as electronics, optoelectronics, magnetronics, and valleytronics. The design of 2D systems is an area of intensive research because of the rapid development of ab initio structure-searching methods. In this paper, we highlight recent research progress on accelerating the design of 2D systems using the CALYPSO methodology. Challenges and perspectives for future developments in 2D structure prediction methods are also presented.

Graphical abstract

Keywords

two-dimensional (2D) systems / CALYPSO / structure prediction

1 Introduction

Reduced dimensionality is a versatile approach for discovering exotic materials, whose physical properties are entirely different from those of their bulk form. Two-dimensional (2D) systems, including 2D layered materials, surfaces, and interfaces, have attracted much attention over recent decades due to their desirable properties. Graphene is the first “modern” 2D material, first isolated in 2004 [1]. It has high carrier mobility [2, 3] and a room-temperature quantum Hall effect [4, 5], making it suitable for use in nanoelectronics and as a saturable absorber [6]. Other examples include hydrogen-terminated diamond surfaces, which are promising materials for field-effect transistor devices due to their generated 2D-hole gases [7], and the heterojunction interface between SrTiO₃ and LaAlO₃, which allows the rare coexistence of superconductivity and ferromagnetism [8, 9].

The atomic-level structure of a material plays a key role in the physical and chemical properties of 2D systems, so determination of their structures is central to the design and exploration of new 2D systems. Structure prediction relies on locating the global free-energy minimum of the potential energy surface (PES), defined as the function of the energy of a structure and its atomic coordination. Therefore, structure prediction can be mathematically formulated as a global optimization problem of the PES. Due to the exponential growth of local minima (metastable phases) with increasing numbers of atoms in a system, the number of possible metastable phases of the PES is astronomically large [10], so exhaustive ergodic search strategies are impossible, even for systems with only several atoms per unit cell. There therefore needs to develop efficient and smart computational approaches, particularly for 2D structure prediction, to quickly locate the globally stable structure in the vast structural space of the PES.

Various global optimization algorithms have been reported for structure predictions based on chemical composition by reducing the search space or enhancing the sampling efficiency of PES. These algorithms have aided the development of structure prediction methods and include simulated annealing [11], minima hopping [12], basin hopping [13], metadynamics [14], random sampling [15], the genetic algorithm (GA) [10, 16-20], and the particle-swarm optimization (PSO) algorithm [21-23]. Some of these methods have been used to search structures of 2D systems. For example, Chen et al. explored 2D ice structures using random sampling methods [24], Tikhomirova et al. proposed a hexagonal NaCl thin film based on the GA algorithm [25], and Zhu et al. identified the 2D structures of tellurium using the PSO algorithm [26].

The CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) method [22, 23] is a highly efficient numerical method for crystal structure searches based only on chemical compositional information. The method comprises several modules dealing with structure, such as structure generation with constraints of symmetry, structure characterization using a bond characterization matrix, and structure evolution using the swarm-intelligence algorithm. The CALYPSO method has been widely used to design and discover 2D systems, including 2D materials [27-29], surfaces [30] and interfaces [31]. Attractive features have been implemented in CALYPSO to resolve various structure prediction problems, including 3D solids [22, 23], 0D nanoclusters or molecules [32], solid–solid transition states [33], X-ray diffraction data-assisted structure searches [34], and the inverse design of novel functional materials (e.g., superhard, electride, and optical materials) [35, 36]. The details of CALYPSO methodology and its applications can be found in Ref [37-39].

To date, CALYPSO has been extensively used to design materials in 2D systems and many 2D materials with interesting properties have been discovered [40]. In this work, we focus on applications of CALYPSO methodology to structure searches of 2D systems, discussing 2D layered materials in Section 2, surface construction in Section 3, and solid–solid interfaces in Section 4.

2 Two-dimensional layered materials

Nowadays, CALYPSO methodology is able to design of various 2D materials, including predicting the structures of single layer, multi-layer, thin-film 2D materials and atomic adsorption on these materials. The slab model was adopted in the CALYPSO method and involves vacuum and layered atomic structures.

(i) For single-layer materials, the random atomic positions of 2D layered structures are achieved under the constraints of 2D planar symmetry. Meanwhile, a distortion perpendicular to the in-plane layer is allowed to predict the structures of buckled layers [Fig. 1(a)].

(ii) For multi-layer materials, a van der Waals gap parameter is used to separate two adjacent layers to build the models of multilayered 2D systems [Fig. 1(b)].

(iii) For thin-film systems, the atomic structure is generated by 3D symmetry constraints within a given thickness [Fig. 1(c)].

(iv) For atoms adsorbed on 2D-layered materials, the structural model involves the adsorption region on either one or both sides of a given 2D substrate [Fig. 1(d)].

The PSO algorithm in the 2D structure search models of CALYPSO [27, 28] is constrained in 2D space to evolve the generated structures.

The CALYPSO method has been widely applied in predicting various 2D materials with exotic structures and properties. Here, we focus on partial applications of the CALYPSO method to different types of 2D materials such as borophene [42, 169, 170], boron-based superconductors, hypercoordinate compounds, ReWCl₆, Si thin-films, and hydrogenated/oxidized graphene.

Borophenes. Since the discovery of graphene, boron, adjacent to carbon on the periodic table, has been the subject of significant effort to discover its 2D sheet structures. Boron possesses three valence electrons distributed in four available orbitals, providing a notably different bonding character to that of carbon. Diversified 2D borophenes [42, 169, 170] with triangular and hexagonal units or pure hexagonal motifs are predicted by the CALYPSO structure prediction method, and several have been synthesized [171]. Boron-based compounds such as B-C [28], B-P [117, 172], Fe-B [50], and Ti-B [71] with desirable properties have also been investigated using the CALYPSO method and will aid the design of other 2D compounds.

2D boron-based superconductors. 2D boron-based compounds have been widely explored as promising superconductors using the CALYPSO structure search method. Qu et al. identified the t-MnB₃ and h-MnB{}_3 superconductors and showed that h-MnB{}_3has a maximum superconducting transition temperature (T_{\mathrm{c}}) of 34 K at 2% tensile strain [66]. Tao et al. identified several orthorhombic and hexagonal MB₆ compounds (M = Mg, Ca, Ti, Y) [58] and showed that the T_{\mathrm{c}} of hexagonal GaB{}_6is{}_{}22.6–28.4 K under a compressive strain of 3%. Dong et al. predicted δ-BS and B₂C sheets with T_{\mathrm{c}} values of about 21.56 K [59] and 21.20 K [59], respectively. Other 2D boron-based superconductors, such as B₂O [62, 63], XB{}_6(X = Ga, In) [64], W₂B₂ [61] and Mo₂B₂ [57] compounds, with T_{\mathrm{c}} values of <20 K, have also been designed using the CALYPSO method. Except for boron-based compositions, many superconductors with high T_{\mathrm{c}} values (shown in Table 1) based on other elements have been proposed using the CALYPSO method.

Hypercoordinate materials. Chemical bonding is fundamental to compound formation. The design of materials with rule-breaking chemical bonding is desirable because of their possible unique properties. CALYPSO has been widely used to design new materials with unique chemical bonds. Li et al. designed a quasi-planar BeC₂ monolayer with hypercoordinate C atoms, in which each C atom binds to six Be atoms [68]. Zhang et al. predicted a graphene-like FeB₆ monolayer with hexa- or octa-coordinate Fe atoms [70] and found the first planar hypercoordinate transition metal in extended systems. Yang et al. proposed a planar Cu₂Si monolayer comprising the hexacoordinate transition metal Cu and the group IV element Si [69]. More hypercoordinate 2D materials are shown in Table 1.

ReWCl₆ multiferroic materials. Magnetoelectric materials are highly desirable for technological applications such as magnetocapacitors, high-performance information storage, and processing devices [173, 174]. However, there are few strong magnetoelectric materials due to the inherent exclusion between ferroelectricity and ferromagnetism. The 2D{}_{} monolayer magnetoelectric material ReWCl{}_6was proposed using the CALYPSO methodology to provide a general mechanism for understanding the electrical control of magnetism [136]. Many magnetic materials (including several multiferroic materials with coupled ferroelasticity–ferroelectricity [135, 137], or ferromagnetic–ferroelectric materials [133]) have also been designed using the CALYPSO method.

Silicene thin-film photovoltaic materials. Silicene holds promise for advanced applications in the photovoltaic industry due to its outstanding properties. Monolayer and multilayer silicenes have been synthesized on the surfaces of various metal substrates [175-182] and slices of these materials are moderately stable in air [183, 184]. However, these silicenes are unlikely to be useful as photovoltaic materials because of their metallic or semi-metallic features on metal substrates [185]. A quasi-direct gap semiconducting tri-layer silicene structure with a photovoltaic efficiency of 29% at 1.0  μm was designed using the CALYPSO method [152] and to our knowledge, exhibits the highest conversion efficiency of thin-film solar cell absorbers to date, being comparable to the conversion efficiency of bulk GaAs.

Hydrogenated/Oxidized graphene semiconductors. Graphene has attracted interest because of its exotic properties (e.g., high electronic and thermal conductivities) [186]. It has a zero band gap and is therefore not suitable for use in electronic devices, but the adsorption of H atoms efficiently opens its band gap [187]. Gao et al. tuned the electronic structures of graphene by adsorbing H atoms using a structure search of atomic adsorption models in the CALYPSO package [29]. Oxidized graphene has also attracted attentions [188]. The CALYPSO method predicted a novel structure of graphene monoxide [189] that is 32 meV/O energetically more stable than the previously reported mix-mode [190]. CALYPSO has also been applied to several other atom-adsorbed 2D layered materials used as electrodes: Li on B/N-doped graphene [191] and O, F, or OH on four typical MXenes: Ti₂CT_x, Ti₃C₂T_x, Nb₂CT_x, and Nb₄C₃T_x (T = O, F, and OH) [192].

3 Surface reconstruction

The flowchart and slab model for CALYPSO surface structure search module were shown in Fig. 3. CALYPSO uses a slab model to explore surface atomic structures, in which surface structures are represented by finite slabs separated by a vacuum horizontal to the surface. Each slab has two surfaces. The bottom one is generally passivated by hydrogen atoms with integer or partial charges for covalent systems and kept unchanged for metal systems. The top surface of the slab consists of three regions, namely the bulk region with 4–8 layers, the unreconstructed surface with 2–4 layers, and the reconstructed surface. The atoms in the bulk region are fixed to keep the nature of the bulk material. And the atoms in the unreconstructed surface region are enabled to be relaxed in the process of geometric optimization. The atom positions in the reconstructed surface region are updated during the structure swarm evolution.

In the initial structural swarm-generation process, physical and chemical information, such as the plane group and electron counting rule, help to enhance surface configuration sampling efficiency. Furthermore, to improve structural diversity and ensure unbiased structure generation and coverage of the entire configuration space, a changeable number of candidate structures with lowest fitness (typically 20%) are eliminated and the same number of structures are randomly created.

Clean and chemically adsorbed diamond (100) surfaces.Diamond has both high electron and hole mobility and is one of the most promising candidates for next-generation wide-band-gap semiconductors [193]. The decreasing size of transistors results in the transistor surface playing an increasingly significant and even decisive role in determining the overall functionality of devices. Diamond films grown by techniques such as chemical vapor deposition (CVD) typically have a (100) surface, resulting in intensive theoretical and experimental studies to elucidate its properties.

Pristine diamond (100) surface. Under normal conditions, a pristine or clean diamond (100) surface has the well-known dimer surface in which the two adjacent C atoms form a symmetric dimer configuration. This structure was theoretically proposed in 1981 [194] and was obtained by resonant electron injection in 2001 [195]. The dimer surface [Fig. 4(a)] was easily located by the CALYPSO code as were more complicated reconstructions such as the formation of tube-like configurations. The TA1 surface [Fig. 4(b)] was energetically degenerate relative to the dimer surface under ambient conditions, but becomes energetically more favorable under compress straining or heating.

Oxygenated diamond (100) surface. Oxygenation is another commonly used technique for enhancing diamond properties. Raman and/or IR spectra show the presence of both carbonyl (C=O) and epoxide (C–O–C) moieties on oxidized diamond (100) surfaces [197, 198]. Surface structures were constructed of ketone and ether, which are fully covered by carbonyl and epoxide functional groups, respectively [199-201]. However, accurate density-functional-theory calculations indicate that the ketone surface is neither statically nor dynamically stable. The newly developed CALYPSO surface structure prediction method proposed three oxidized surfaces [Figs. 4(c)–(e)] more energetically favorable in the whole oxygen chemical-potential range [196]. The most stable surface configuration, the methoxyacetone surface, has almost no surface states in the bulk band gap, providing a perfect state for nano-sized diamond electronics.

4 Solid–solid interfaces

Solid–solid interfaces or grain boundaries (GB) are present in a broad range of materials such as heterojunctions and polycrystalline materials and often play a significant role in governing the physical properties of solids. The CALYPSO interface structure prediction method involves two main steps: (i) To reduce interfacial stress, an automatic search scheme was developed for a superlattice with minimal lattice-mismatch strain between the two bulk materials. This searched superlattice is used to create the simulated model for the subsequent interface structure search. (ii) Two simulated models have been adopted in this step. One is a slab model [Fig. 5(a)] containing a vacuum region, two bulk regions, and an interface region. The other model [Fig. 5(b)] consists of two bulk regions and two equivalent interfaces. This model effectively prevents exposure of the upper and bottom surfaces to vacuum and the influence of the dipole in a vacuum. Constraints on interatomic distance and atomic coordination number have been introduced to generate the initial interface structure to enhance search efficiency. Bulk crystal structures are fixed in the bulk region, and only the atomistic structure in the interface region is updated by a PSO algorithm designed for the interface system. Rigid-body displacement between the two bulk phases is used as a search dimension.

TiO₂ grain boundaries in photovoltaic materials. Although TiO₂ is a promising photocatalytic material, its photoreaction efficiency is limited by its large intrinsic band gap. Recently, it has been reported that introducing Ti³⁺ can improve photocatalytic efficiency [202]. The CALYPSO methodology has been used to determine the distributions of Ti³⁺ ions at the GB of the rutile TiO{}_2{}^{}\mathrm{\Sigma }5(210) twin boundary, as shown in Figs. 6(a)–(c) [31]. The calculated results reveal that the GB can be used to engineer the electronic properties and visible-light photoactivity of TiO₂.

Interface in all-solid-state batteries. Compared to traditional Li-ion batteries, all-solid-state batteries (ASSB) have greatly improved safety and cycle stability [203, 204] due to replacement of liquid electrolyte with solid electrolyte (e.g., polymers and inorganic materials). However, high interfacial resistance between the electrode and the solid electrolyte has been reported for many ASSB systems, limiting their practical application [205]. The microscopic origin of this interfacial resistance has been investigated using the CALYPSO interface structure prediction module to study a representative interface, the LiCoO₂(104) cathode/β-Li₃PS₄(010) solid electrolyte interface [206]. The predicted energetically favorable structures [Fig. 6(d)] indicate the formation of an interface reaction layer with both Co \leftrightarrow P and O \leftrightarrow S ion mixing. Calculation of the site-dependent Li chemical potentials indicate several interfacial Li sites with high Li chemical potential, resulting in the dynamical Li-depleted layer in the initial stage of charging, as determined from the calculated Li migration barriers across the interface. The calculated densities of states show the presence of interfacial electronic states, thus enhancing Li depletion at the interface. This study provides comprehensive insights into the microscopic mechanisms of interfacial Li⁺ transport and resistance in ASSBs.

Li₇La₃Zr₂O₁₂ (LLZO) is another important solid electrolyte due to its high Li conductivity and compatibility with a Li metal anode, resulting in significant research on the LLZO/Li interface. However, several studies have reported that the LLZO/Li interface is unstable [207-209], and an in-depth understanding of instability issues is lacking, limiting further optimization of the LLZO/Li battery. The CALYPSO interface structure prediction module has been used to investigate the stability of the LLZO/Li interface [210]. Extensive sampling of the terminations of low-index (100), (001), (110), and (101) surfaces of the LLZO showed wide distribution of coordinatively unsaturated Zr (coordination number <6) sites on the surface. The energetically stable surface terminations of LLZO and Li metal were used to search the interface configurations [Fig. 6(e)]. The results indicate that the interfacial adhesion energies are dependent on surface terminations, suggesting inhomogeneous contact between LLZO and Li, which may contribute to Li dendrite growth. Under-coordinated Zr at the interfaces results in partial occupation of the Zr-4d conduction band, remarkably reducing LLZO. This study thus provides a theoretical explanation of the instability issues of the LLZO/Li interface.

5 Conclusion

CALYPSO methodology has been applied to structure searches of 2D systems, including 2D layered materials and atomic adsorption on these layers, surface construction, and solid–solid interfaces, and has identified several novel 2D materials with unique electrical, magnetic, or optical properties.

Despite substantial progress on 2D structure prediction of systems comprising tens of atoms, larger systems remain challenging for the current algorithm. One of the main obstacles is the computational cost of energy calculations for large systems using the first-principles method. Machine learning is a promising solution to address this as its accuracy can reach the same order of magnitude as that of the first-principles method with much lower computational cost. An accelerated CALYPSO structure prediction method based on machine learning potential has recently been developed for structure searches of large cluster systems [211]. The method reduces computational cost by one to two orders of magnitude, and the structure search is fully consistent with first-principles calculations. Other challenges, such as structure prediction at finite temperature, will be addressed in future iterations of the CALYPSO structure prediction method.

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