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Finding stable multi-component materials by combining cluster expansion and crystal structure predictions - npj Computational Materials

Introduction .
Knowing the initial positions of atoms, or an approximate crystal structure prototype based on previous knowledge is preferred when performing a quantum mechanical calculation. This is commonly not much of a problem for systems with one or two elemental components, e.g., unary, and binary systems, since these have been investigated thoroughly. Villars and Iwata estimated that all unary systems and 72% of the potential binary systems have been experimentally studied. However, the potential elemental combinations increase rapidly for higher-order systems, which are also reflected in the number of materials systems reported as only 16% of ternary, 0.6% of quaternary, and 0.03% of quinary systems have been experimentally studied 1 . This demonstrates that experimental exploration of multi-component systems is still in its cradle and the avenues ahead are close to infinite. Theoretical studies may here serve as a complementary approach by predicting material combinations being most promising for future synthesis, as exemplified for ternary nitrides 2 and quaternary borides 3 .
A certain group of materials that recently have attracted a large interest is atomically layered materials with interlayer interactions significantly stronger than van der Waals forces, hindering mechanical exfoliation. One such material group is MAX phases 4 , which are comprised of transition-metal carbide and nitride (MX) layers interleaved by an A-group element. This structural arrangement along with the included elements often results in combined metallic and ceramic properties 5 . Furthermore, selective etching, also known as chemical exfoliation, of the A-element from the MAX phases has spurred an increased interest and has emerged as an alternative route for the preparation of a two-dimensional (2D) family of materials known as MXenes 6 , 7 . The search for additional multi-component materials with the possibility for exfoliation is thus of great interest, and the discovery of in-plane chemically ordered MAX phase quaternaries, coined i -MAX 8 , 9 , 10 , has enabled the synthesis of MXenes with in-plane chemical or vacancy ordering, such as Mo 4/3 Y 2/3 C, Mo 4/3 C, and W 4/3 C, showing promise for energy storage and catalysis 9 , 10 , 11 , 12 .
Similar to MAX phases are the layered metal boride-based materials, referred to as MAB phases, which consist of a transition metal (M), an A-group element (A), and boron (B). The mixing of metals in known ternary MAB phases have through theoretical guidance revealed both out-of-plane ( o -MAB) and in-plane ( i -MAB) 3 , 13 chemical ordering, where the latter phases have been converted into 2D derivatives coined boridene 14 . For the o -MAB phases, the chemical ordering of metal atoms was achieved by taking advantage of two distinctly different crystallographic sites and performing theoretical stability predictions, which revealed eleven stable o -MAB phases out of which Ti 4 MoSiB 2 , of a tetragonal I 4/ mcm symmetry, was later synthesized. The i -MAB phases Mo 4/3 Y 2/3 AlB 2 and Mo 4/3 Sc 2/3 AlB 2 (space group \(R\bar 3m\) ) were predicted thermodynamically stable by relaxing over 3,000 phases and were subsequently verified through powder synthesis 13 . The synthetized i -MAB phase Mo 4/3 Sc 2/3 AlB 2 with \(R\bar 3m\) symmetry will from this point forward be referred to as Mo 4/3 Sc 2/3 AlB 2 , with fractional notations, whereas predicted phases of similar compositions will be referred to as Mo 4 Sc 2 Al 3 B 6 , with integer notations.
Different theoretical frameworks are utilized in the search for low-energy crystal structures. Examples of such are crystal structure prediction (CSP) frameworks 15 , 16 , 17 and cluster expansion (CE) methods 18 . CSP methods are independent of any a priori information of the crystal structure and typically employ evolutionary algorithms to explore the chemical phase space for low-energy basins and possibly stable candidate compounds 15 , 16 , 17 . For example, Rybkovskiy et al., studied the Mo-B system and predicted MoB 5 to be stable 19 , and Xu et al., found a stable binary multi-layered MnB 2 phase with properties resembling a hard multifunctional material 20 , both utilizing the CSP framework of USPEX 15 , 16 , 17 . Addition examples include CSP searches performed within the W-Cr-B ternary system where both W 4 CrB 3 and W 2 CrB 2 were found to be stable, which is in line with experimental findings, along with the yet to be synthesized WCrB 2 , WCrB, and WCr 2 B 21 . Wang et al. predicted and synthesized the first hexagonal MAB phase Ti 2 InB 2 by employing USPEX algorithms 22 . The latter discovery increased the field of synthesizable layered MAB phases from orthorhombic and tetragonal to also include hexagonal symmetries 13 . However, the computational demands of CSP increase drastically with the increased complexity of the investigated system 23 , 24 , 25 . For example, Naumova et al. used CSP to search for structures in the quaternary C-H-N-O system, including pressure, which required approximately 1,800,000 density-functional theory (DFT) relaxations 26 .
An alternative to the computationally demanding CSP is the use of methods such as CE which, in contrast to CSP, requires an a priori defined crystal structure. The expansion is carried out on one or multiple sublattices where parameterization is used to express the configurational dependence of physical properties such as energy 27 , band gap 28 , 29 , and magnetic interactions 30 . CE has also been used to explore the configurational space in layered hexagonal MAX phases upon mixing of metals in 41 quaternary systems 31 where the alloys were classified into three regimes—phase separation, weak ordering, and strong ordering. It was found that (V 2/3 Zr 1/3 ) 2 AlC should phase segregate into V 2 AlC and Zr 2 AlC 31 . This result, however, contradicts the experimental discovery of the in-plane ordered (V 2/3 Zr 1/3 ) 2 AlC i -MAX phase 8 . Note the structural differences between the MAX phase (which is the input for the CE model) and i -MAX phases, where the former has all atoms (V?+?Zr, Al, C) occupying hexagonal lattices while the latter combines honeycomb (for V), hexagonal (for Zr and C) and Kagomé-like (for Al) lattices. Furthermore, it has been demonstrated that the transformation from a hexagonal Al lattice to a Kagomé-like lattice gives rise to a significant decrease in energy for the material system 8 , 32 . This combined with an unsuccessfully truncated CE model due to a limited number of considered training structures may explain the discrepancy in why CE fails to predict the correct outcome in ref. 31 . Such inconsistency demonstrates the importance of using an appropriately truncated CE model when studying the mixing of metals in MAX phases 31 , 33 , 34 , 35 , 36 .
In this work, we apply both CE and CSP methods to search for low-energy basins within the quaternary material system (Mo x Sc 1– x ) 2 AlB 2 where 0?≤? x ?≤?1. Since we only vary the Mo to Sc ratio, while keeping Al and B fixed, the material system can be viewed as pseudo-binary. The reason for choosing Mo-Sc-Al-B as a model system is motivated by the recent discovery of i -MAB phase Mo 4/3 Sc 2/3 AlB 2 , comprised of in-plane chemical ordering of Mo and Sc. An additional motivation is the structural resemblance with the ordered i -MAX phases 8 and discrepancies reported when investigating MAX phases with CE 31 . The question is if CE and/or CSP can be used to verify the synthesized Mo 4/3 Sc 2/3 AlB 2 ( \(R\bar 3m\) ) compound discovered in ref. 13 while covering the phase space thoroughly with greater efficiency.
We herein demonstrate and discuss the possibilities and limitations of using CE and CSP independently when searching for low-energy structures and evaluating their thermodynamic stability. Despite CE being computationally efficient it is limited by its dependence on an a priori defined input structure, which limits the considered chemical phase space. CSP on the other hand does not require any structural input but demands significant computational resources. However, combining CE and CSP offers an efficient framework that yields reliable results when searching for low-energy basins as the drawbacks of each framework cancel out. The CSP is herein initially used to identify input structures for the CE models whereas the latter is used to explore the mixing and/or stability tendencies in (Mo x Sc 1– x ) 2 AlB 2 . Low-energy configurations found within the CE models are further used as seed structures within a final variational composition CSP search covering the complete phase space of (Mo x Sc 1– x ) 2 AlB 2 . Not only do we verify the stable and ordered i -MAB phase Mo 4/3 Sc 2/3 AlB 2 \(\left( {R\bar 3m} \right)\) , discovered in ref. 13 , but also predict Sc 4 Mo 2 Al 3 B 6 \((R3m)\) as stable and nine additional (Mo x Sc 1– x ) 2 AlB 2 structures as close to stable when the two frameworks are combined. We also identify two low-energy ternary MAB phase structures, Sc 2 AlB 2 ( Pmmm ) and Mo 2 AlB 2 ( P 4/ mbm ). Finally, the possibility of modifying the suggested framework for being applicable to any n dimensional material system with minor adjustments is discussed, i.e., by initiating a CSP search on the n –1 dimensionality subsystems followed by designing CE models on the obtained low-energy structures which results are then used to inspire a final CSP seach on the n dimensional material system.
Results .
Crystal structure prediction for ternary systems .
Cluster expansion is an efficient approach for studying alloying in (Mo x Sc 1– x ) 2 AlB 2 but requires an input structure to build a model upon. The hexagonal Ti 2 InB 2 with P \(\bar 6\) m 2 symmetry and the orthorhombic Cr 2 AlB 2 , Mn 2 AlB 2 and Fe 2 AlB 2 with Cmmm symmetry 13 , 36 are examples of experimentally known MAB phases with M 2 AB 2 composition. Note that these two prototype structures may not necessarily be of the lowest energy for Sc 2 AlB 2 and Mo 2 AlB 2 compositions and exploration for possible additional structures is thus motivated. We initiate the search for low-energy structures considering the Sc 2 AlB 2 and Mo 2 AlB 2 compositions by performing CSP searches. The corresponding evolution of the CSP search is shown in Fig. 1 a, b where the formation enthalpy, using Eq. 2 , is shown as a function of the number of generations. Identified low-energy structures are shown in Fig. 1c–f including the known P \(\bar 6\) m 2 and Cmmm symmetries.
Fig. 1: The evolutionary trajectory of the formation enthalpy, Δ H cp . The Δ H cp of the ternary phases Mo 2 AlB 2 ( a ) and Sc 2 AlB 2 ( b ) along the evolutionary trajectory where lowest-energy structure in each generation is represented by a black circle and those at higher energies by gray crosses. Schematic illustration of predicted crystal structures of lowest energy corresponding to c Mo 2 AlB 2 in space group P 4/ mbm and d Sc 2 AlB 2 in space group Pmmm . Known M 2 AB 2 prototype structures are illustrated with space group e Cmmm and f \(P\bar 6m2\) . Each structure is shown from two crystallographic different directions. Mo or Sc, Al, and B are represented by blue, gray, and green atoms.
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The stability for Mo 2 AlB 2 , displayed in Fig. 1a , shows that the predicted lowest crystal structure after 72 generations is P 4/ mbm (illustrated in Fig. 1c ) and it is close to stable with Δ H cp ?=?5?meV/atom. This structure is obtained after 42 generations and has previously been reported for W 2 CrB 2 in ref. 21 with Mo/W and Al/Cr atoms in Wyckoff positions 4h and 2a, respectively, and originates from the V 3 B 2 prototype structure. The structure found with the next-lowest energy is Cmmm (Fig. 1e ), i.e., the Cr 2 AlB 2 prototype structure, at Δ H cp ?=?16?meV/atom. Interesting to note is that the Cmmm structure have been reported as an intergrowth in the stable MoAlB structure of Cmcm symmetry 37 . Additionally, Pmmm (identified as the low-energy structure for Sc 2 AlB 2 in Fig. 1b ) and \(P\bar 6m2\) structures are found at Δ H cp ?=?148 and 89?meV/atom, respectively.
The evolution of the formation enthalpy for Sc 2 AlB 2 is shown in Fig. 1b and illustrates two low-energy structures being close to stable with space group Pmmm (Fig. 1d ) and Cmmm (Fig. 1e ) at Δ H cp ?=?22 and 42?meV/atom, respectively. The Pmmm structure in Fig. 1d is comprised of an extra arrangement of Sc-Al and Sc-B layers constituting a layered B-Sc-B-Sc-Al-Sc-Al-Sc-Al-Sc-B-Sc-B stacking order. This is in contrast to the typical M 2 AB 2 Cmmm structure, shown in Fig. 1e , with an Al-Sc-B-Sc-Al-Sc-B-Sc-Al stacking order. A noteworthy difference between the orthorhombic Pmmm and Cmmm and the hexagonal \(P\bar 6m2\) structures are the appearance of their B-layers, which have a zig-zag structure for the former while being planar for the latter. Additionally, the P 4/ mbm (identified as the low-energy structure for Mo 2 AlB 2 in Fig. 1a and illustrated in Fig. 1c ) and \(P\bar 6m2\) (Fig. 1f ) structures are found at Δ H cp ?=?139 and 98?meV/atom, respectively. Detailed structural information related to structures shown in Fig. 1c–f is found in Table 1 .
Table 1 Space group, lattice parameters, Wyckoff position, and fractional atomic coordinates of predicted low-energy structures for Sc 2 AlB 2 and Mo 2 AlB 2 . Full size table
Cluster expansion for identification of mixing tendencies .
The four structures identified with low energy upon the CSP search in Fig. 1 ( Pmmm , P 4/ mbm , Cmmm , and \(P\bar 6m2\) ) are used as model structures in the search for low-energy structures within the pseudo-binary (Mo x Sc 1– x ) 2 AlB 2 system. The Al and B sublattices are considered spectator atoms as the alloying is focused on mixing Mo and Sc on the metal lattice. The lattice constants used to construct the CE models for each considered symmetry were set to an average value of the relaxed Mo 2 AlB 2 and Sc 2 AlB 2 compositions. The CLEASE code offers three different methods for generating new configurations: (i) randomly, (ii) ground-state search, and (iii) probe structures. Within the ground-state search, the CE model aims at generating a new structure of low energy using a simulated annealing technique based on a pool of clusters from prior generations that are on or near the convex hull 38 , 39 . With this approach, the focus is on the exploration of low-energy regions within assigned compositional and structural boundaries. On the contrary, with the probe technique, structures are generated in order to increase the variance within the pool of configurations to consist of as independent structures as possible 40 , 41 . This approach may be utilized when exploring the complete chemical space of the system. Figure 1 illustrates the isostructural formation enthalpy Δ H iso as a function of x in (Mo x Sc 1– x ) 2 AlB 2 when using the three different generating strategies available in CLEASE and here applied to the \(P\bar 6m2\) system to show the similarities and differences between the techniques.
The outcome using randomly generated structures in Fig. 2a show less variation in Δ H iso as compared to ground-state and probe generated structures in Fig. 2b, c . Δ H iso for probe generated structures shows an increased variance but lacks in number of low-energy structures. The ground-state search gives the greatest number of low-energy structures, and this at a fairly low computational cost (~150 DFT calculations), but does not yield as many structures at higher energy as the probe does. Note that both probe and random mode will eventually find the same low-energy structures as the ground-state search but requires additional training configurations to be considered to improve the CE models.
Fig. 2: The alternative techniques for generating configurations in CLEASE applied to the \(P\bar 6m2\) system. The training set is denoted by circles and predicted energies as crosses with at least 100 a randomly, b ground-state, and c probe generated structures.
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Since we are interested in predicting low-energy structures, the results shown in Fig. 2 demonstrate the ground-state search technique to be most suitable, which motivates its use to pave the pathway towards the discovery of low-energy symmetries at a reduced computational effort rather than surveying the complete phase space with DFT calculations. All CE models considered herein are thus assembled using a iterative procedure with the ground-state search formalism. A database of different configurations of (Mo x Sc 1– x ) 2 AlB 2 alloys are thus initially established for each considered CE model consisting of 25 randomly structures followed by at least 100 ground-state structures generated iteratively with each generation composed of 25–50 new generated configurations. Figure 3a–d shows the isostructural formation enthalpy Δ H iso as a function of x in (Mo x Sc 1– x ) 2 AlB 2 for the generated and relaxed ground-state configurations. We use Δ H iso to illustrate possible mixing tendencies among the four model systems. In addition, the phase stability of each ground-state configuration within the CE models is shown in Fig. 2e–h by calculating the formation enthalpy Δ H cp .
Fig. 3: Cluster expansion models applied to identify symmetries from the initial crystal structure prediction. a – d Isostructural formation enthalpy Δ H iso and e , f formation enthalpy Δ H cp of (Mo x Sc 1– x ) 2 AlB 2 based on M 2 AlB 2 in the ( a , e ) P 4/ mbm , ( b , f ) Pmmm , ( c , g ) Cmmm , and ( d , h ) \(P\bar 6m2\) crystal structure where M?=?Sc, Mo. CE generated structures are represented by gray circles.
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Neither P 4/ mbm (Fig. 3a ) nor Cmmm (Fig. 3c ) is found to have any strong preference for mixing Mo and Sc on the M-site, as indicated by the majority of structures with Δ H iso ?>?0. This is in contrast to Pmmm (Fig. 3b ) and \(P\bar 6m2\) (Fig. 3d ) which both show mixing tendencies as the majority of these structures have Δ H iso ? Corresponding stability for the four model systems, using Eq. 2 , is displayed in Fig. 3e–h where the formation enthalpy Δ H cp is shown as a function of x in (Mo x Sc 1– x ) 2 AlB 2 . Systems based on P 4/ mbm (Fig. 3e ) and Cmmm (Fig. 3g ) now show even fewer tendencies for mixing Mo and Sc. For Pmmm , the Mo-rich region, which in Fig. 3b demonstrated mixing tendencies, is now found to be far from stable. For the \(P\bar 6m2\) symmetry in Fig. 3h , 6 structures are found stable in the region 0.5?≤? x ?≤?0.67.
To limit the number of structures chosen for further investigation, we introduce a cutoff at Δ H cp ? Fig. 4: Identified low-energy structures obtained from the CE models. The symmetries a Cmc 2 1 , b \(P\bar 1\) , and c \(R\bar 3m\) illustrated from the top and two side views. The atomic species are denoted as the following: blue is Mo, red is Sc, gray is Al, and green is B.
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Note that all structures in Fig. 4 bear the same layer sequence as the original \(P\bar 6m2\) symmetry in Fig. 1f but with a rearranged Al-layer (hexagonal lattice for the ternary M 2 AlB 2 \(P\bar 6m2\) ). The structure at x ?=?0.5 symmetry (Fig. 4a ), Δ H cp ?=??12?meV/atom, demonstrates an alternative stacking of in-plane ordered Mo-Mo-Sc-Sc interleaved by the Al-layer where Mo atoms are closer to the B layer while Sc atoms are closer to the Al-layer. The latter thus leads to a rearrangement of the Al-layer away from a hexagonal lattice. Similarly, the structure at x ?=?0.56 (Fig. 4b ), Δ H cp ?=??11?meV/atom, does not display a particular order of Mo and Sc but noteworthy is that Sc is closer to the Al layer than Mo. The structure at x ?=?0.67 is predicted stable with Δ H cp ?=??30?meV/atom (Fig. 4c ) and show clear in-plane order of Mo and Sc and where Al atoms form a Kagomé lattice (hexagonal lattice in \(P\bar 6m2\) ). The reason for the stable structure at x ?=?0.67 is further decomposed in Supplementary Fig. 1 where it is demonstrated that the major component for the decreased energy can be traced to the rearrangement of the Al atoms caused by the displacement of Sc towards the Al layer. Minor energy contributions are found from the Mo, B, and structural relaxations. Such secondary effects in spectator lattices are in agreement with similar observations demonstrated for the i -MAX phases reported in refs. 32 , 42 where it was demonstrated that metals (M’ and M”) with a significant size difference ( r M” ?>? r M’ ) in a ratio M’:M”?=?2:1 are mixed and ordered within the layered hexagonal MAX phase. This apparent in-plane order forces the larger M” to be displaced towards the Al-layer. This procedure is illustrated in Supplementary Fig. 1 where the Al lattice is observed to rearrange from a hexagonal lattice, which is the spectator lattice used within the CE model as defined by the input symmetry, to a Kagomé lattice when Sc is displaced towards the Al-layer.
The limitations of the cluster expansion formalism when used to pave the path towards stable and possibly synthesizable materials is the restriction defined by the input structure. Relying on an input structure may hinder the exploration of the phase space. Using CE alone is thus prone to miss valuable information hidden within the complete chemical space. An alternative approach that may circumvent this problem, and where the dependence of any initial structure is of lesser importance, or even neglected, is thus preferred.
Search for low-energy structures in a quaternary system .
Next, we focus on exploring the chemical phase space within the (Mo x Sc 1– x ) 2 AlB 2 system where 0?≤? x ?≤?1 through the use of the crystal structure prediction code USPEX. The aim is to identify stable low-energy structures and compare when no a priori information is used with seed structures retrieved from cluster expansion. Figure 5a illustrates the evolution of the CSP search for x ?=?0.67 where the formation enthalpy, using Eq. 2 , is shown as a function of the number of generations without any seed structures (black) and with seed structures applied after 5 (red), 25 (blue), and 50 (yellow) generations. Included seed structures are gathered from the ground-state CE models depicted in Fig. 3 .
Fig. 5: Compositional crystal structure prediction including seeds from the cluster expansion models. a Formation enthalpy Δ H cp of (Mo x Sc 1– x ) 2 AlB 2 , x ?=?0.67, along the evolutionary trajectory without (black) and with seed structures applied after 5 (red), 25 (blue), and 50 (yellow) generations. b Δ H cp as a function of x in (Mo x Sc 1– x ) 2 AlB 2 for predicted low-energy structures when performing the variational composition search. c Schematic illustration of the predicted stable structures located at x ?=?0.33 and x ?=?0.67 from two crystallographic directions.
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Figure 5a illustrates the formation enthalpy, Δ H cp , for x ?=?0.67 along the evolutionary trajectory. When no seeds are used, the predicted lowest-energy structure after 135 generations is an orthorhombic Amm 2 structure with Δ H cp ?=?29?meV/atom. This structure is illustrated in Supplementary Fig. 2 , with the corresponding structural information found in Supplementary Table 2 , and bears resemblance with the orthorhombic Cmmm and Pmmm structures, illustrated in Fig. 1d, e , but with double-layers of Al and Mo. This is in contrast to the evolutionary searches using seed structures retrieved from the CE ground-state search in Fig. 3 , which results in a predicted stable structure, Δ H cp ?=??30?meV/atom, with space group \(R\bar 3m\) . This structure is found immediately after applying seeds following 5, 25, and 50 generations. Note that the number of individual structures generated with lower energy is observed to increase when seeds are applied at a later generation. This observation may conclude that using seeds not only predicts the global energy minima structure with greater efficiency but also significantly reduces the number of generations required to find additional low-energy basins.
The set of ground-state search generated structures with Δ H cp ?variable composition search within the pseudo-binary (Mo x Sc 1– x ) 2 AlB 2 system for 0?≤? x ?≤ 1. Figure 5b shows the corresponding formation enthalpy Δ H cp as a function of x limited to structures with Δ H cp ? Common for the low-energy structures at x ?=?0.33 and 0.67 displayed in Fig. 5c are the flat Al-layers interleaved with M-B layers. The Mo 4 Sc 2 Al 3 B 6 ( x ?=?0.67) structure demonstrates more symmetric characteristics and higher order of Mo and Sc where Sc is located closer to the Al-layer while Mo is closer to the B-layer. Mo 2 Sc 4 Al 3 B 6 ( x ?=?0.33), on the other hand, yields less symmetric metal-layers, which is likely the reason for not being as stable Mo 4 Sc 2 Al 3 B 6 . Additional structural information for the stable Mo 4 Sc 2 Al 3 B 6 and Mo 2 Sc 4 Al 3 B 6 is found in Table 2 .
Table 2 Space group, lattice parameters, Wyckoff position, and fractional atomic coordinates of predicted stable within the (Mo x Sc 1– x ) 2 AlB 2 , 0?≤? x ?≤?1, system. Full size table
Combining cluster expansion and crystal structure prediction .
Using either cluster expansion (CE) or crystal structure prediction (CSP) alone when searching for low-energy structures in a multi-component system, like performed herein for the quaternary Mo-Sc-Al-B system, may yield limited and inadequate results. Even though CE is less computationally demanding out of the two approaches, it relies on a predefined input structure, which limits the considered chemical space when searching for low-energy basins. CSP searches, on the other hand, are independent of any initial structure but become more challenging when the dimensionality of the system is increased. The CSP searches applied to the ternary compositions consisted herein of an initial sampling of up to 200 DFT calculations followed by at least 50 generations, where each generation is comprised of 40 individual DFT calculations, before reaching desired convergence criteria. The CSP may occasionally yield a structure that is not the global energy minimum structure but rather a local minimum, as demonstrated in Fig. 5a . Users are thus promoted to be running several CSP searches on the same system in parallel to validate the stochastic results 43 which demands significant computational resources. Performing a variable composition search is arguably even more strenuous than considering a specific composition as the complexity of the system is increased. The variable composition search performed herein consists of 800 initially sampled structures followed by at least 50 generations each containing 100 new structures constructed by the variational operators.
The herein suggested solution for performing an efficient search in the quest for low-energy structures within a multi-component system is thus to combine CE and CSP. A proposed framework is illustrated in Fig. 6 for studying mixing and/or stability within higher order systems, such as the (Mo x Sc 1– x ) 2 AlB 2 system considered in this work.
Fig. 6: Framework for combining cluster expansion with crystal structure prediction. Gray labels represent computational procedures, yellow labels denote sections that require manual input and optional modules are denoted by dashed lines.
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Before constructing the CE model systems for an n -dimensional system, a CSP search should be performed on the lower n –1 order systems constituting the system in focus, in our case the ternary systems Mo 2 AlB 2 and Sc 2 AlB 2 , to identify possible low-energy structures to be used for constructing the CE models. These structures do not necessarily have to be stable, as exemplified herein for P 4/ mbm Mo 2 AlB 2 (5?meV/atom) and Pmmm Sc 2 AlB 2 (22?meV/atom). Additional low-energy symmetries, \(P\bar 6m2\) and Cmmm , found within the ternary CSP search are further considered. Note that the only CE model that resulted in strong mixing tendencies was the \(P\bar 6m2\) symmetry, which reflects the importance of considering alternative low-energy basin symmetries. Alternatively, additional low-energy symmetries may be obtained from literature, e.g., ref. \(P\bar 6m2\) 22 and Cmmm (Springer Materials: The Landolt-B?rnstein database), which have previously been reported as M 2 AB 2 structures. Next, CE models are constructed based on the pool of identified ternary low-energy symmetries from the CSP search. New structures are generated iteratively using the ground-state search technique for the exploration of low-energy basins within the studied chemical and structural-phase spaces. This analysis can be performed both in terms of isostructural enthalpy, Δ H iso , and with respect to competing phases by calculating the formation enthalpy Δ H cp . Configurations that are found close to stable, herein governed by the limit Δ H cp ?variable compositional CSP search on the higher order material system.
Discussion .
Approaches like crystal structure prediction and cluster expansion are in general useful methods for predicting low-energy crystal structures. CE offers a robust and computationally effortless procedure where energies of generated crystal structures may be predicted by mapping the cluster interactions of a training set onto a lattice of mixed atomic species. The model relies on a predefined lattice where mixing is to take place, which herein is performed on hexagonal, tetragonal, and orthorhombic symmetries of M 2 AB 2 structures, and thus only explores a fraction of the complete energy landscape. Still, a CE model is herein shown to serve as a good seed generator for a CSP search. We find multiple crystal symmetries at various compositions in (Mo x Sc 1– x ) 2 AlB 2 , from orthorhombic and tetragonal symmetries at the ternary end-points ( x ?=?0 and 1) to trigonal, orthorhombic, and monoclinic symmetries in between by inspiring the CSP variational operators with the close to stable configurations found within the CE models.
Covering the complete chemical phase space of a quaternary system is a daunting task even though the crystal structure prediction algorithms utilized in USPEX provide a strong framework. Ref. 26 impressively performed a CSP search on the C-H-N-O material system while also considering pressure, which required the consideration of roughly 1,800,000 structural relaxations. Although we herein limit ourselves to consider the (Mo x Sc 1– x ) 2 AlB 2 system, the structural variety is still extremely large. However, by combining CE with CSP, we show that the search for low-energy structures could be accelerated by utilizing the strengths of both frameworks.
We suggest that an iterative combination of the two methodologies, CE?+?CSP, could reduce the computational efforts while still thoroughly searching the energy landscape within higher order material systems for low-energy basins. The CE framework offers a computational effortless framework covering the initial grounds of the system, which may later be used as seed structures to further inspire the evolution of the CSP search. Multiple structures, including the i -MAB phase Mo 4/3 Sc 2/3 AlB 2 ( \(R\bar 3m\) ) comprised of in-plane chemical ordering of Mo and Sc, are identified as stable or close to stable. This framework is applicable to any material system after only minor adjustments. The suggested approach is to initiate the search on the n –1 dimensionality systems, which constitutes the higher order, n -dimensional, system. Herein being the Mo 2 AlB 2 and Sc 2 AlB 2 systems. Once the foundation is established, the mixing and phase stability of the n dimensionality system ((Mo x Sc 1–x ) 2 AlB 2 ) is motivated to be explored through variable compositional CSP searches with low-energy seed structures acquired from the CE models.
Methods .
Density-functional theory calculations .
All structural relaxations required for the cluster expansion and crystal structure predictions were performed within the DFT framework, using the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) to model the exchange correlation effects, and the all-electron projector augmented wave (PAW) method, as implemented in the Vienna Ab initio Simulation Package (VASP) version 5.4.1 44 , 45 , 46 , 47 . A plane wave energy cutoff set to 400?eV and a Monkhorst-Pack scheme used to sample the Brillouin zone with a k-point density of 2 π × 0.04?? ?1 was selected. The chosen settings are motivated by the convergence tests performed in ref. 48 , exploring quaternary metal borides, which yield energies accurate enough for efficient probing of the energy landscape. All structures were relaxed at ambient pressure and 0?K.
Evaluation of phase stability .
The thermodynamical phase stability is herein determined by the formation enthalpy calculated at 0?K. The formation enthalpy of a general MAB phase M x A y B z is calculated by comparing the energy of M x A y B z with respect to all possible linear combinations of competing phases within the M-A-B material system. The pseudo-binary (Mo x Sc 1– x ) 2 AlB 2 system considered herein is thus compared to all phases within the Mo-Sc-Al-B material system. The set of most competing phases constituting the minimum energy, which represents the decompositions within a quaternary Mo-Sc-Al-B system. The equilibrium simplex is obtained by solving
$${{{\mathrm{min}}}}\;E_{cp}\left( {b^{{{{\mathrm{Mo}}}}},b^{{{{\mathrm{Sc}}}}},\;b^{{{{\mathrm{Al}}}}},\;b^{{{\mathrm{B}}}}} \right) = \mathop {\sum}\limits_i^n {x_iE_i} ,$$
(1)
in which the left-handed side of Eq. ( 1 ) represents the equilibrium simplex energy for the given composition constraints b Mo , b Sc , b Al , and b B of elements Mo, Sc, Al, and B, respectively. The right-hand side denotes the set of linear combinations constructed of phase i with the amount x i and energy E i . The formation enthalpy Δ H cp is thus calculated as
$${\Delta}H_{cp}\left( {\left( {{{{\mathrm{Mo}}}}_x{{{\mathrm{Sc}}}}_{1 - x}} \right)_2{{{\mathrm{AlB}}}}_2} \right) = E\left( {\left( {{{{\mathrm{Mo}}}}_x{{{\mathrm{Sc}}}}_{1 - x}} \right)_2{{{\mathrm{AlB}}}}_2} \right) - {{{\mathrm{min}}}}E_{cp}\left( {b^{{{{\mathrm{Mo}}}}},\;b^{{{{\mathrm{Sc}}}}},\;b^{Al},\;b^{{{\mathrm{B}}}}} \right),$$
(2)
where \(E\left( {\left( {{{{\mathrm{Mo}}}}_x{{{\mathrm{Sc}}}}_{1 - x}} \right)_2{{{\mathrm{AlB}}}}_2} \right)\) is the calculated energy of an arbitrary phase within (Mo x Sc 1– x ) 2 AlB 2 and E cp is the energy of the equilibrium simplex. Furthermore, a phase is attributed as stable when Δ H cp ??0 is considered not stable or at best metastable. The set of competing phases considered herein are obtained from databases such as OQMD 49 , 50 , Materials project 51 , and Springer Materials (Springer Materials: The Landolt-B?rnstein database). The complete set of competing phases is found in Supplementary Table 3 and the identified set of most competing phases for each considered composition is shown in Supplementary Table 4 .
Cluster expansion .
The energy of a crystalline solid within the cluster expansion 18 (CE) formalism is composed as a function of the underlying grid of atomic sites or atomic arrangement of a given lattice. The herein used cluster expansion model is formulated using the cluster expansion in atomic simulation environment (CLEASE) code 52 motivated by its options to generate and predict new configurations. There are three techniques to generate configurations: randomly, through a ground-state search 38 , 39 , and probe 40 , 41 . A brief introduction to the CE formalism is found in Supplementary Methods 1 . The initial training set is composed of 25 randomly generated configurations. To improve the accuracy of the CE models and their predictability for exploration of low-energy regions a minimum of 100 configurations is generated iteratively through simulated annealing. The iterative procedure consists of generating a pool of 25–50 configurations within each generation. The evolution of the CE models upon the iterative procedure is shown in Supplementary Fig. 4 . The randomly generated configurations for each CE model were constructed to consist of up to a maximum of 50 atoms and up to 100 atoms for configurations generated using the ground-state search.
Crystal structure prediction .
All crystal structure prediction searches have employed the Universal Structure predictor: Evolutionary Xtallography (USPEX) code 17 , 53 where evolutionary algorithms are used to find a set of most optimal crystal structures within a given system. USPEX has been used for CSP searches within the ternary Mo 2 AlB 2 and Sc 2 AlB 2 systems and the quaternary (pseudo-binary) (Mo x Sc 1– x ) 2 AlB 2 system. The ternary systems were initiated with 200 randomly generated structures whereas succeeding generations each contained 40 structures. The CSP search for (Mo x Sc 1– x ) 2 AlB 2 was conducted by a variable composition search based on 800 randomly generated initial structures followed by succeeding generations containing 100 structures each. 60% of phases of the lowest energy in each generation were used as inspiration for the succeeding generation. 40% of each succeeding generation was generated by the heredity operator, 15% with lattice mutations, 15% with transmutation operations, 15% with soft-mode mutations, and 15% randomly. These settings apply to both CSP search scenarios, i.e., single, or variable composition. .

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