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Hiroshi Mizuseki, [Ryoji Sahara](https://orcid.org/0000-0003-0788-2985), [Kenta Hongo](https://orcid.org/0000-0002-2580-0907)

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[Valence electron concentration-dependent stability of L1<math display="inline">  <msub>    <mrow></mrow>    <mrow>      <mn>2</mn>    </mrow>  </msub></math>, D023, and D022 ordered phases in high-entropy alloys](https://mdr.nims.go.jp/datasets/1574a935-7be9-4ef8-9338-919956207c7c)

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Valence electron concentration-dependent stability of L1[formula omitted], D023, and D022 ordered phases in high-entropy alloysComputational Materials Science 259 (2025) 114114 0n Contents lists available at ScienceDirectComputational Materials Sciencejournal homepage: www.elsevier.com/locate/commatsci  Full length articleValence electron concentration-dependent stability of L12, D023, and D022ordered phases in high-entropy alloysHiroshi Mizuseki a , Ryoji Sahara b , Kenta Hongo c ,∗a Korea Institute of Science and Technology (KIST), Seoul 02792, Republic of Koreab National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japanc Research Center for Advanced Computing Infrastructure, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, JapanA R T I C L E  I N F OKeywords:Semi-ordered atomic arrangementsMulticomponent alloysMulti-principal element alloysOrder–disorder competitionFirst-principles calculations A B S T R A C TWe investigate the valence electron concentration (VEC) dependence of semi-ordered phases (SOPs) in high-entropy alloys (HEAs) via first-principles calculations. Fifteen equiatomic quaternary alloys composed of Al, Fe, Co, Ni, Cu, and Zn, along with non-equiatomic CrFeCoNi alloys, are analyzed. Formation energies of L12, D022, D023, and random solid solution (RSS) phases are evaluated. The results reveal that SOPs consistently exhibit lower formation energies than RSS. Although D023 phases have not yet been experimentally observed in HEAs, they are predicted to stabilize in specific intermediate VEC regions depending on composition, bridging the stability regimes of L12 and D022. These findings clarify VEC-dependent stability trends and provide insights into conditions favoring D023 formation in HEAs.1. IntroductionOrdered phases such as L12 and D022 in intermetallic compounds are known to correlate closely with valence electron concentration (VEC). A clear VEC-dependent competition between L12 and D022phase stability has been demonstrated in binary alloys like Pd3𝑋 and Pt3𝑋 (𝑋 = 3d transition metals) [1] and pseudobinary alloys like (Pt,Rh)3V, (Pd,Rh)3V, and Pt3(V,Ti) [2]. However, the stability of or-dered phases in high-entropy alloys (HEAs) remains largely unexplored.HEAs are traditionally considered random solid solutions (RSS) [3–7]. Nonetheless, experimental studies recently suggest local ordering, particularly involving L12 or D022 precipitates enhancing mechanical properties [8–12], though these studies mainly focus on multi-phase HEAs rather than single-phase ordered structures. As for the single-phase case, Niu et al. [13] first identified Cr-based L12 ordering in CrFeCoNi both experimentally and theoretically, while our previous work theoretically demonstrated stabilization of L12 and D022 phases in 3d transition-metal HEAs [14]. In binary alloys, it has been well known that the long-period D023 structure consisting of alternating stacking of L12 and D022 layers emerges [15]. However, the existence of long-period D023 phases in HEAs remains unexplored. Since direct experimental observation of atomic-scale ordering in such phases is challenging, this study aims to clarify their stability and the conditions under which they form through first-principles simulations.∗ Corresponding author.E-mail address: kenta_hongo@mac.com (K. Hongo).This study systematically examines the stability of FCC semi-ordered phases (SOPs: L12, D022, and D023) in HEAs via first-principles cal-culations. We evaluate (i) 15 equiatomic quaternary alloys across a VEC range of 7.50 to 10.50 and (ii) Cr(Fe,Co,Ni)3 non-equiatomic alloys (Cr backbone) across a more controlled VEC range of 7.78 to 8.72 for more detailed analysis, clarifying VEC-dependent trends and identifying conditions favorable for D023 stabilization. In high-entropy alloy (HEA) systems, VEC has been reported to correlate with the stable crystal structure [16,17]: a single FCC phase tends to form when VEC exceeds 8.00, a single BCC phase appears when VEC is below 6.87, and a mixture of FCC and BCC phases is observed in the intermediate range (6.87 < VEC < 8.00). Accordingly, in this study, all investigated compositions are assumed to adopt a single FCC structure, and the possible formation of BCC phases is neglected.2. Computational methodsWe performed spin-polarized density functional theory (DFT) cal-culations using VASP [18,19] with the PBEsol exchange–correlation functional [20] and projector-augmented wave pseudopotentials [21,22]. Crystal structures were modeled using 4 × 4×4 FCC supercells containing 256 atoms. Ten random configurations were generated per composition to account for statistical variations [14,23]. Brillouin zone sampling used a 1 × 1×1 Monkhorst–Pack k-grid and cut-off energies https://doi.org/10.1016/j.commatsci.2025.114114Received 9 June 2025; Received in revised form 5 July 2025; Accepted 9 July 202927-0256/© 2025 The Authors. Published by Elsevier B.V. This is an open access artc-nd/4.0/ ). 5icle under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by- https://www.elsevier.com/locate/commatscihttps://www.elsevier.com/locate/commatscihttps://orcid.org/0000-0003-0973-7693https://orcid.org/0000-0003-0788-2985https://orcid.org/0000-0002-2580-0907mailto:kenta_hongo@mac.comhttps://doi.org/10.1016/j.commatsci.2025.114114https://doi.org/10.1016/j.commatsci.2025.114114http://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/H. Mizuseki et al. Computational Materials Science 259 (2025) 114114 Fig. 1. Phase stability comparison of SOPs and RSS in AlCoCuZn. Formation energies of 12 semi-ordered structures (SOPs) and a random solid solution (RSS) for the equiatomic AlCoCuZn alloy. SOPs are constructed by assigning each of the four constituent elements to the ordered backbone sites in L12, D023, and D022 phases. Error bars represent standard deviations from 10 random configurations. In this case, SOP1, SOP2, and SOP3 correspond to Co-D023, Co-L12, and Co-D022, respectively.were set to the default values, validated by convergence tests. The convergence criteria for energy and force criteria were set to 10−4 eVand 10−3 eV/Å, respectively. Initial magnetic moments were assigned as follows: Al and Zn were treated as non-magnetic, Fe, Co, and Ni as ferromagnetic, and Cr as antiferromagnetic.We modeled 15 equiatomic quaternary alloys composed of Al, Fe, Co, Ni, Cu, Zn, considering 12 possible SOPs (L12, D022, D023) and one RSS per alloy. The remaining 75% of the sites were occupied by the other three elements based on the simplified Warren-Cowley short-range order (WC-SRO) method [24] adopted in our previous study [14], thereby completing the structural models.  For simplicity, SOPs are denoted using the label ‘‘X-SOP’’ (e.g., Co-D023, Al-L12 in Fig.  1), where the element X occupies the ordered sublattice positions, while the remaining elements are randomly distributed over the disordered sites. For instance, ‘‘Co-D023’’ represents a D023 structure in which Co atoms form the backbone of the ordered sublattice.Additionally, we constructed CrFeCoNi non-equiatomic alloys by varying Fe, Co, Ni ratios while fixing the Cr content at 25 at.% to control the VEC. The ordered sites in SOPs were occupied by Cr atoms, and the remaining sites were randomly occupied by Fe, Co, and Ni atoms. Although the WC-SRO method was not applied here, we verified that the simple pseudo-random assignments yield formation energies (defined below) consistent with those obtained via WC-SRO within the statistical error, as confirmed for the equiatomic CrFeCoNi alloy.In this study, the VEC is defined as the average number of valence electrons per atom for a given alloy, calculated as follows: VEC =∑𝑖𝑥𝑖𝑣𝑖, (1)where 𝑥𝑖 is the atomic fraction and 𝑣𝑖 the number of valence electrons of element 𝑖. The conventional valence counts used were: Al (3), Cr (6), Fe (8), Co (9), Ni (10), Cu (11), and Zn (12). Using the DFT-based total energy for each structure, the formation energy per atom were obtained via: 𝐸𝑓 = 𝐸(HEA) −∑𝑖𝑥𝑖𝐸(𝑋𝑖), (2)where 𝐸(HEA) is the total energy per atom of the alloy, and 𝐸(𝑋𝑖) the energy per atom of element 𝑖 in its ground-state crystal structure (FCC Al, BCC Cr, BCC Fe, HCP Co, FCC Ni, FCC Cu, HCP Zn) [25,26].3. Results and discussionFig.  1 shows the formation energies of AlCoCuZn as a representative case. Across all 15 equiatomic alloys, SOPs exhibit lower formation 2 energies than RSS (details for others are Figures S2 in Supporting Information). Specifically, Co-D023 emerges as the most stable (SOP1), followed by Co-L12 (SOP2) and Co-D022 (SOP3).Hereafter, for convenience, the three SOPs corresponding to each HEA composition are labeled SOP1, SOP2, and SOP3, in ascending order of formation energy. For instance, in Fig.  1, SOP1, SOP2, and SOP3 in AlCoCuZn correspond to Co-D023, Co-L12, and Co-D022, respectively.To further investigate the VEC-dependence, we introduce relative formation energy differences defined as: 𝛥𝐸𝑓 = 𝐸𝑓 (SOP𝑖) − 𝐸𝑓 (SOP2); (𝑖 = 1 or 3), (3)where SOP2 is the reference, making 𝛥𝐸𝑓  negative for SOP1, zero for SOP2, and positive for SOP3, enabling intuitive comparison.Fig.  2 displays the relative formation energies 𝛥𝐸𝑓  for SOP1 and SOP3. L12 stabilizes at 7.75 ≤ VEC ≤ 8.50 and VEC ≥ 9.50, while D023 emerges at intermediate VECs (8.75, 9.00). D023 is predicted to stabilize between the L12 and D022 phases in a narrow VEC window, providing a new insight into semi-ordering in HEAs. This finding sug-gests VEC-guided pathways for engineering short-range order in FCC HEAs without changing the elemental types. Thus, as VEC increases, SOP1 transits as D022 → L12 → D023 → L12. This observation prompted a more detailed analysis of the conditions under which the D023 phase emerges.To this end, CrFeCoNi non-equiatomic alloys were examined by varying Fe, Co, Ni compositions, keeping Cr at 25 at.%. This model can be regarded as a ternary alloy on the remaining 75 at.% sites. Note that previous studies confirmed equiatomic CrFeCoNi (VEC = 8.25) stabilizes Cr-L12 [13,14].Fig.  3(a) shows that Cr-L12 and Cr-D022 coexist for 8.28 ≤ VEC ≤ 8.31, with Cr-L12 dominant at lower VEC (< 8.25) and Cr-D022 at higher VEC (> 8.47); Cr-D023 emerges around VEC = 8.28–8.41. The fine VEC resolution reveals transitions from Cr-L12 to Cr-D023 and then Cr-D022; large error bars near VEC = 8.35 reflect transition regions. Fig. 3(b) maps compositional regions favoring each SOP1 phase: Cr-L12 and Cr-D022 dominate Fe-rich (low VEC) and Ni-rich (high VEC) regions, respectively, whereas Cr-D023 appears in intervening Co-rich (middle VEC) regions.To verify generality, our results are compared with binary inter-metallic compounds and pseudo-binary alloys (Table S2 in SI) [1,2,15,27]. Experimental data show L12 stabilizes at VEC = 8.25 and 9.25 <VEC < 11.00, and D022 at VEC = 8.50–8.75 [15]. Theoretical studies, including pioneering band-filling work by A. Bieber et al. [27], also demonstrate similar VEC dependencies. Our non-equiatomic results align with these trends.H. Mizuseki et al.Fig. 2. VEC-dependent stability trend of SOPs across 15 equiatomic HEAs. Relative formation energies of SOP1 (circles) and SOP3 (crosses) with respect to SOP2 for 15 equiatomic quaternary HEAs. Marker colors denote SOP types: red for L12, purple for D023, and blue for D022. SOP1 transitions from D022 to L12 to D023 and then to L12 again with increasing VEC.Fig. 3. Compositional and VEC effects on SOP stability in CrFeCoNi non-equiatomic HEAs. (a) Relative formation energies of SOP1 and SOP3 with respect to SOP2 for CrFeCoNi alloys with varying Fe-Co-Ni ratios (Cr fixed at 25 at.%). Error bars denote standard deviations. (b) Compositional map showing SOP1 stability regions projected onto the Fe-Co-Ni ternary plane. Marker colors represent phase type (red: Cr-L12, purple: Cr-D023, blue: Cr-D022), and marker size corresponds to SOP2. Solid circles indicate statistically significant energy differences beyond the standard deviation, while open circles indicate overlaps.Computational Materials Science 259 (2025) 114114 3 H. Mizuseki et al. Computational Materials Science 259 (2025) 114114 In contrast, D023 is experimentally reported only at low VEC (VEC = 3.25) [15], with no reports in the studied range. This may raise doubts about its predicted stability; however, similar discrepancies between theory and experiment are well-known in binary systems. [28,29]. First-principles calculations consistently show small energy dif-ferences (< 10 meV/atom) among the three phases. For instance, first-principles calculations for Al3Ti predict D023 as the most stable, whereas experiments often observe D022 [15]. This is attributed to the fact that experimental results are influenced by kinetic factors and processing histories, often leading to the appearance of metastable D022 phases. Our predictions call for future experimental efforts, such as low-temperature annealing and advanced diffraction or microscopy techniques, to detect D023-like ordering in HEAs. Although our cal-culations predict D023 stability, experimental confirmation requires carefully controlled processing to selectively stabilize this phase.4. ConclusionIn summary, our first-principles calculations for equiatomic qua-ternary alloys composed of Al, Fe, Co, Ni, and Cu, as well as for non-equiatomic CrFeCoNi alloys, revealed that the semi-ordered phases (L12, D023, D022) consistently exhibit lower formation energies than the random solid solution (RSS) in high-entropy alloys (HEAs). VEC-dependent stability trends indicate that D023 phases bridge the stability ranges of L12 and D022. The stabilization of D023 between L12 and D022phases is supposedly attributed to its crystal structure incorporating features of both L12 and D022 (See Figure S1 and Table S1 for your eye guide). These results align with experimental trends observed in binary systems, reinforcing the role of VEC in governing ordered phase stability. Our findings extend these trends to HEAs and predict D023 stabilization in previously unexplored VEC regions. This finding suggests VEC-guided pathways for engineering short-range order in FCC HEAs without changing the elemental types. This study highlights VEC as a key factor governing the stability of ordered phases in HEAs and provides guidance for future experimental exploration of long-period ordered structures. Given that experimentally resolving atomic-scale arrangements in alloys requires significant effort, the insights obtained from this study are of considerable importance.CRediT authorship contribution statementHiroshi Mizuseki: Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Ryoji Sahara: Writing – review & editing, Validation, Resources, Methodol-ogy, Investigation, Funding acquisition, Formal analysis. Kenta Hongo: Writing – review & editing, Writing – original draft, Supervision, Soft-ware, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.Declaration of Generative AI and AI-assisted technologies in the writing processDuring the preparation of this manuscript, the authors used Chat-GPT o4 in order to improve its language and readability. After using this tool, all the authors reviewed and edited the content as needed and takes full responsibility for the content of the publication.Declaration of competing interestThe authors declare that they have no known competing finan-cial interests or personal relationships that could have appeared to influence the work reported in this paper.4 AcknowledgmentsThis study was supported by the computational resources of the HPCI system [Project ID: hp230037, hp230468, hp240027], the JHPCN system [Project IDs: jh230038 and jh240026], Institute for Materials Research, Tohoku University [Proposal No. 202312-SCKXX-0502] and Numerical Materials Simulator at National Institute for Materials Sci-ence. The computations in this work have been partially performed us-ing the facilities of the Research Center for Advanced Computing Infras-tructure (RCACI) at JAIST. 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Supplementary data Data availability References