Order–disorder competition in equiatomic 3d–transition–metal quaternary alloys: phase stability and electronic structure

ABSTRACT

We use high-throughput first-principles sampling to investigate competitive factors that determine the crystal structure of high-entropy alloys (HEAs) and the energetics dependence of the stable phase on the atomic configuration of ‘semi-ordered’ L1₂, D0₂₂, and random solid solution (RSS) phases of equiatomic quaternary alloys comprising four of the six constituent elements (Cr, Mn, Fe, Co, Ni, and Cu). Note that, generally, an FCC lattice consists of four L1₂/D0₂₂ sublattices. In this study, we call ‘semi-ordered’ phase a FCC lattice where one of the L1₂/D0₂₂ sublattices is fully occupied by a certain element, whereas the others are randomly occupied by the other elements like RSS. Considering the configurational entropy, we demonstrate that valence electron concentration (VEC) and temperature are crucial to determine the phase stability of HEAs at finite temperatures, wherein the ‘rather enthalpy-driven’ ordered phases are energetically more favorable than ‘rather entropy-driven’ RSS phases. Some D0₂₂ phases with high VEC are energetically more stable than L1₂ phases, though both phases are metastable. Furthermore, we explore magnetic configurations to identify the origin of the enthalpy term. The calculations reveal that ordered phases comprising antiferromagnetic atoms surrounded by ferromagnetic atoms are energetically stable. Relationships between magnetic ordering and atomic arrangements are also discussed.

GRAPHICAL ABSTRACT

KEYWORDS:

• High-entropy alloys
• high-throughput simulations
• order–disorder phases
• enthalpy-driven stabilization
• entropy-driven stabilization

1. Introduction

High-entropy alloys (HEAs) [1,2] have been gaining increasing attention because of their novel material properties that cater to a wide range of applications. Although HEAs were originally applied to structural materials that required both strength and ductility [3–7], their recent applications have been extended to other functional materials, such as catalysis [8], superconductors [9], radiation resistance materials [10], and magnetic materials [11,12]. These extensions of their applications represent a new direction of materials design based on the fundamental and key concept of ‘high-entropy’. Along with this new direction, the comprehensive understanding of atomic arrangements on HEA crystal structures is crucial for exploiting their potential applications [13]. This implies that the atomistic characterization of HEA structures is challenging for both ‘experiments’ and ‘theories/computations’, in terms of ‘spatial resolutions’ [14] and ‘computational costs for vast amounts of possible atomic configurations’ [15,16], respectively.

It is considered that HEAs have random solid solutions (RSSs) instead of ordered phases. Although disordered face-/body-centered cubic (FCC/BCC) phases (A1/A2) have been reported as single HEA phases [17], ordered FCC/BCC (L1₂/B2) and hexagonal close-packed (HCP, A3) have also been found [18–20]. For example, the L1₂ phase of HEA means a FCC lattice having of four L1₂ sublattices where one of the L1₂ sublattices is fully occupied by a certain element, whereas the others are randomly occupied by the other elements like RSS. In this study, we call this kind of ordered HEA phases ‘semi-ordered’. Although there are a few examples of semi-ordered phases, perfect randomness in some types of HEAs has been excluded in several studies. For example, short range order (SRO) in HEAs has been reported [21–24]; however, the existence of long-range order (LRO) is rarely discussed and remains a controversial topic [25,26].

In the present study, we systematically elucidate atomistic randomness in HEAs using a simple model of LRO. To the best of our knowledge, there exist several examples. For instance, one of the typical ordered HEA phases is CrFeCoNi, with the L1₂ structure [26], wherein the antiferromagnetic Cr atoms occupy the ordered lattice sites surrounded by randomly distributed ferromagnetic Ni, Fe, and Co atoms. This can be interpreted as a spin-driven stabilization of the atomic configuration, which indicates that enthalpy enhancement by magnetization cooperates partial randomness and then overcomes the entropy because of perfect randomness. This was theoretically predicted [26–28] and verified experimentally only for the ‘surface’ using scanning transmission electron microscopy (STEM) [26]; experimental verifications for the ‘bulk’ have not been feasible because the constituent atoms are adjoining elements in the periodic table and have similar chemical properties and comparable radii, which makes them indistinguishable when using spectroscopic methods [14]. Thus, the question arises if the spin-driven stabilization is unique for CrFeCoNi or if it generally holds for other quaternary alloys. To address this issue from a microscopic perspective, instead of experiments, first-principles approaches are the most relevant for identifying atomistic structures of HEAs and investigating their various properties [29–32], though the previous studies dealt with only single composites.

We systematically focus on 3d-transition-metal-based quaternary alloys with Cr, Mn, Fe, Co, Ni, and Cu as the constituent elements by ‘high-throughput’ first-principles sampling: a total of $15\left(=\left(\begin{array}{c}6\\ 4\end{array}\right)=:\frac{6!}{4!2!}\right)$ different types of composites with nine phases are considered (shown later in Table 1), for each of which 100 structures are generated. These elements are the most commonly used magnetic HEA constituents [19], and all quaternary alloy combinations that results from them can be speculated to have the FCC phase [33]. Therefore, they are expected to be disordered/RSSs as their atomic sizes and the nature of chemical bonds formed by them are similar. Our first-principles study revealed that differences in their magnetic properties with respect to CrFeCoNi lead to different magnetic orderings: our target phases were not restricted to only RSSs and L1₂ ordered phases; they extended to the D0₂₂ ordered phases as well (shown later in Figure 1), motivated by well-known studies on order–disorder phase competition in ordered alloys [34–39]. The L1₂ and D0₂₂ phases are ordered crystal structures for 25:75 at.% FCC alloys. As prototype systems for the ordered phases, their phase competition has been well investigated for conventional alloys in terms of valence electron concentration (VEC) [34,35]. For example, the L1₂ phase of intermetallic compounds is observed for VEC $\ge$ 7.5. However, to the best of our knowledge, there is no previously reported study on their counterpart for HEAs; in HEAs, VEC values are known to correlate well with structure types (FCC/BCC structures prefer higher/lower VEC values) [40–43]. Thus, we analyze the RSS–L1₂–D0₂₂ thermodynamical competition in the 3d-transition-metal quaternary system in terms of VEC. In order to systematically understand the RSS–L1₂–D0₂₂ competition for the 15 different composites, we performed the high-throughput first-principles sampling from one RSS phase and four types of L1₂ and D0₂₂, i.e. 8 semi-ordered phases. For example, in the case of MnCoNiCu case, Mn, Co, Ni, and Cu can be occupied on the ordered sites in semi-ordered phases for L1₂ and D0_22, respectively. (See the horizontal axis of Figure 2). We found that composites with lower VEC values tend to have semi-ordered phases over wide temperature ranges; however, for higher VEC values, RSSs emerge as the temperature increases. In addition, we discuss the distribution of the magnetic moment at the first nearest neighbor site and the constituent elements in our HEAs.

Figure 1. Crystal structures of (a) L1₂ (Cu₃Au-type, space group $Pm\bar{3}m$, no. 221), (b) D0₂₂ (Al₃Ti-type, space group $I4/mmm$, no. 139), and (c) RSS for quaternary alloys. Labels indicate the first, second, third, and fourth nearest neighbor sites from the red site located at the cubic corner.

Figure 2. Free energy of formation at 1,000 K for MnCoNiCu. Error bars indicate the standard deviations (SD) for each phase. The free energies of formation for the other 14 alloys are shown in Figure S-3 (Supplementary Information).

Table 1. Temperature dependence of the most stable structures for 15 cases of equiatomic quaternary systems (column name: ‘Composite’) investigated in the present study. VEC_ave corresponds to the average valence electron concentration of the composite, where Cr, Mn, Fe, Co, Ni, and Cu have VEC values of 6, 7, 8, 9, 10, and 11, respectively. ${\bar{T}}_m$ represents the average melting temperature of the four elements that comprise the composite. OP₀ represents the semi-ordered phase possessing the lowest energy at 0 K. For example, Cr-L1₂ indicates the semi-ordered phase wherein the semi-ordered lattice positions are occupied by Cr. (NB: This does not necessarily mean the semi-ordered phase has a lower energy than the RSS.) $T_s(X)$ (X = L1₂, D0₂₂, RSS) indicates the temperature at which phase $X$ is stabilized, i.e. its free energy of formation becomes zero. $T_c$ represents a ‘crossover’ temperature between the semi-ordered phase OP₀ and RSS. $F_c$ (eV/atom) denotes the free energy of formation at $T_c$. $\mathrm{\Delta }{T}_s$ represents the temperature range up to $T_m$, wherein either one of the semi-ordered phases or the RSS phase is realized as the most stable solid phase (MSSP). All temperatures are listed in Kelvin (K).

				$T_s$
Composite(#)	VECave	${\bar{T}}_m$	OP0	L12	D022	RSS	$T_c$	$F_{\mathrm{c}}$	MSSP ($\mathrm{\Delta }{T}_s$)
CrMnFeCo(1)	7.50	1,820	Cr-L12	380	683	638	1,016	−0.045	L12 (380 $\sim$ 1,016)
									RSS (1016 $\sim$ ${\bar{T}}_m$)
CrMnFeNi(2)	7.75	1,810	Cr-L12	448	737	657	963	−0.037	L12 (448 $\sim$ 963)
									RSS (963 $\sim$ ${\bar{T}}_m$)
CrMnCoNi(3)	8.00	1,799	Cr-L12	282	553	574	1,003	−0.051	L12 (282 $\sim$ 1,003)
									RSS (1,003 $\sim$ ${\bar{T}}_m$)
CrMnFeCu(4)	8.00	1,717	Cr-L12	2,594	2,755	1,820	687	0.135	Unstable
CrFeCoNi(5)	8.25	1,872	Cr-L12	0	127	494	1,333	−0.100	L12 (0 $\sim$ 1,333)
									RSS (1,333 $\sim$ ${\bar{T}}_m$)
CrMnCoCu(6)	8.25	1,706	Cr-L12	2,251	2,425	1,649	768	0.105	RSS (1,649 $\sim$ ${\bar{T}}_m$)
CrFeCoCu(7)	8.50	1,779	Cr-L12	1,913	1,917	1,469	819	0.078	RSS (1,469 $\sim$ ${\bar{T}}_m$)
CrMnNiCu(8)	8.50	1,696	Cr-L12	1,881	1,950	1,398	691	0.085	RSS (1,398 $\sim$ ${\bar{T}}_m$)
MnFeCoNi(9)	8.50	1,706	Mn-L12	0	128	333	970	−0.076	L12 (0 $\sim$ 970)
									RSS (970 $\sim$ ${\bar{T}}_m$)
CrFeNiCu(10)	8.75	1,769	Cr-D022	1,786	1,546	1,223	750	0.074	RSS (1,223 $\sim$ ${\bar{T}}_m$)
MnFeCoCu(11)	8.75	1,614	Mn- L12	1,597	1,665	1,218	663	0.066	RSS (1,218 $\sim$ ${\bar{T}}_m$)
CrCoNiCu(12)	9.00	1,758	Cr-D022	1,711	1,594	1,117	417	0.092	RSS (1,117 $\sim$ ${\bar{T}}_m$)
MnFeNiCu(13)	9.00	1,604	Mn-D022	1,395	1,205	956	592	0.057	RSS (956 $\sim$ ${\bar{T}}_m$)
MnCoNiCu(14)	9.25	1,593	Mn-D022	1,497	1,204	981	653	0.060	RSS (981 $\sim$ ${\bar{T}}_m$)
FeCoNiCu(15)	9.50	1,666	Fe-L12	421	474	469	539	−0.008	L12 (421 $\sim$ 539)
									RSS (539 $\sim$ ${\bar{T}}_m$)

The paper is organized as follows: Computational details in the present study are given in ‘Methods’. In ‘Results’, we start with our main finding ‘Order–disorder competition’, followed by ‘Energetics’, ‘Magnetic ordering’, and ‘Relationship between structures and properties’. ‘Summary and future prospects’ summarizes our findings and their significance towards further investigation in HEAs.

2. Methods

2.1. HEA structural models

HEAs have been regarded as solid solutions, wherein every site in the crystal structure is randomly occupied by the constituent elements. However, it is difficult to reproduce such a random configuration with a small simulation cell size. Therefore, finding an appropriate method to mimic the configuration of HEAs under a limited periodic condition is crucial for theoretical investigations. This problem can be solved by evaluating the correlation functions of the atomic configurations in HEAs. For instance, Zunger et al. proposed the concept of special quasirandom structures (SQSs) [44], which are supercell approximations for a disordered system optimized to mimic the random local atomic environment. Therefore, the SQS approach is widely applied to HEA research because it provides a good approximation for the RSS phase. However, it is difficult to discuss the energetics and phase stability of HEAs in detail using only one configuration obtained from the SQS model. It has previously been reported that applying several SQS configurations of HEAs leads to non-negligible energy fluctuations (0.01–0.02 eV/atom) [1,45]. For example, several sampling calculations are required to discuss the phase stability of HCP and FCC phases in CrCoNi and CrMnFeCoNi (cf. Figure 3 (a,b) in Ref [45]). We performed high-throughput first-principles sampling to provide insights regarding the phase stability of HEAs with high accuracy and reliability for overcoming the disadvantage of using a single configuration in the SQS approach. For detail, see the next Section 2.2.

Figure 3. Free energy of formation of CrMnCoNi quaternary alloy as a function of temperature up to its melting temperature (1,799 K). Blue, red, and green lines indicate the RSS, Cr-D0₂₂, and Cr-L1₂ semi-ordered phases, respectively.

In addition to the RSS phases, our main concern is to investigate semi-ordered phases, L1₂ and D0₂₂. Crystal structures of the ordered and disordered phases are shown in Figure 1, which were drawn using the VESTA software [46]. The L1₂ and D0₂₂ phases have typical ordered crystal structures in the 25:75 (at.%) FCC binary alloys. For example, the L1₂ structure can be observed in many alloys with a 3:1 composition, such as Cu₃Au, Ni₃Al, and TiPt₃; similarly, the D0₂₂ structure can be observed in trialuminides Al₃ (Sc, Ti, V) and Ni₃V. Although the appearance of the L1₂ and D0₂₂ phases in FCC binary alloys has been discussed previously, only a few studies have focused on the configuration of the neighboring sites in multiprincipal-element alloys [26]. The number of first, second, third, and fourth nearest neighbor sites in the L1₂ and D0₂₂ phases are 12, 6, 24, and 12, respectively (shown later in Table 2). The 25 at.% element in the L1₂ and D0₂₂ phases is not located at the first nearest neighbor sites, and thus, they can be isolated from each other and surrounded by the 75 at.% elements. For instance, Au atoms in the L1₂ phase of Cu₃Au are not located at their first nearest neighbor sites and are surrounded by 12 Cu atoms. In our equiatomic quaternary alloys, the 25 at.% element occupies the ordered sublattice site, whereas the 75 at.% elements consist of three different 25 at.% elements occupying atomic sites besides the ordered sites, similar to the RSS phases. Thus, instead of using the SQS approach, their structural models were also determined by our procedure shown below.

Table 2. Comparison of the site occupancies of L1₂, D0₂₂, and RSS phases containing 25 at.% of ordered elements. For each neighbor site therein, the number of the ordered elements over the total number of the sites is given with its percentage in brackets. For the RSS phase, the site occupancy for any site is 25 at.% on average (shown in brackets).

Neighbor site	L12	D022	RSS
First nearest	0/12 $⟨0⟩$	0/12 $⟨0⟩$	$⟨25⟩$
Second nearest	6/6 $⟨100⟩$	4/6 $⟨67⟩$	$⟨25⟩$
Third nearest	0/24 $⟨0⟩$	8/24 $⟨33⟩$	$⟨25⟩$
Fourth nearest	12/12 $⟨100⟩$	4/12 $⟨33⟩$	$⟨25⟩$

Two-, three-, and four-body clusters were selected for a correlation function, with a short-range order (SRO) parameter [47,48]. Since the SRO parameter is defined for a pair of elements, an alloy with $N$ component is characterized by the set of $N(N-1)/2$ SRO parameters. Therefore, our quaternary alloy system involves six kinds of SRO parameters in general. In the present study, we dealt with a pair of same or different elements as four kinds of constituent elements to reduce the number of SRO parameters: The present study adopted a simplified Warren-Cowley SRO (WC-SRO) parameters (${\alpha }_{ij}^m$) [49] as our SRO parameters, which is defined as

(1) ${\alpha }_{ij}^m=1-\frac{c_{ij}^m}{c_i{c}_j},$(1)

where $c_{ij}^m$ is the probability of finding an $(i,j)$ pair of $i$-th and $j$-th elements in the $m$-body cluster, and $c_i$ is the concentration of $i$-th elements. To describe the WC-SRO parameters for semi-ordered phases, L1₂ and D0₂₂, of equiatomic quaternary alloys, we set $c_i$ and $c_j$ to be the concentrations of same and different elements with $0.25$ and $0.75$, respectively. The range of the WC-SRO parameters was restricted to the fourth nearest neighbor site for two-body clusters and to the third nearest neighbor site for three- and four-body clusters to reduce the computational time. The tetrahedron-octahedron approximation in the cluster expansion method for phase diagram calculation has been well established for FCC [50], where only first and second nearest neighbor sites are considered. The present correlation function takes into account sites with more longer distances than those in the approximation. For the RSS phase, the configurations of four elements were explored to realize a better correlation function using the Metropolis algorithm [51]. For both the semi-ordered and RSS phases, four elements were considered for evaluating the correlation function. In the L1₂ semi-ordered phase, one element was fixed at the cubic corner (CC) sites in the FCC lattice, while the other three elements in the quaternary system occupied the face-centered sites to realize an ideal L1₂ semi-ordered phase with a good correlation function. The D0₂₂ phase was also constructed in a similar manner.

A wide range of site occupancies was used to elucidate the influence of the number of first nearest neighbor sites on the formation energy. Accordingly, a large number of atomic configurations were randomly generated for RSS structures (shown later in Figure 4). To compute the formation energies and distributions of magnetic moments, our structural models of the quaternary alloys were set to be $2\times 2\times 2$ supercells with 32 sites (each element occupies 8 sites). To investigate dependence of the formation energies of semi-ordered and RSS phases on site occupancy, we also used $4\times 4\times 4$ supercells with 256 sites (each element occupies 64 sites). We confirmed that their formation energy difference between the $2\times 2\times 2$ and $4\times 4\times 4$ supercells is much less than 0.001 eV/atom in the formation energy (e.g. 0.0002 eV/atom for CrFeCoNi). Note that the $2\times 2\times 2$ supercell with less computational costs was used to achieve statistically reliable results, while the $4\times 4\times 4$ supercell with higher costs was used to evaluate physical properties involving longer spatial correlations.

Figure 4. Dependence of the formation energies of semi-ordered and RSS phases in the CrFeCoNi alloy on the site occupancy of the Cr–Cr pairs at the (a) first and (b) second nearest neighbor sites. The red dots represent the RSS phase obtained using 1,500 samples, whereas the blue lines serve as guidelines. The blue and green dots indicate the Cr-L1₂ and Cr-D0₂₂ semi-ordered phases, respectively.

2.2. High-throughput first-principles sampling

The free energy of formation (shown later in Figure 2) for each phase was evaluated by the average and its standard deviation over 100 samples (configurations with a $2\times 2\times 2$ supercell). This statistical evaluation is more reliable than one sampling based on SQS. The same sampling dataset was used to obtain distributions of magnetic moments of the most stable semi-ordered phases for the 15 quaternary alloys (shown later in Figure 5 for (a) CrFeCoNi, (b) CrMnCoNi, (c) MnFeCoNi, and those for all alloys are given in Supplementary Information (SI)). To the best of our knowledge, we have, for the first time, discussed the phase stabilities in HEA from the viewpoint of the first-principles sampling. Figure S-2 shows the formation-free energies obtained using 100 configurations. The dependence of the formation energy on the site occupancy (shown later in Figure 4) and the bond-length distributions (shown later in Figure 6) for the CrFeCoNi quaternary alloy were obtained based on 1,500 and 100 configurations, respectively, using a $4\times 4\times 4$ supercell.

Figure 5. Distribution of magnetic moments as a function of the average magnetic moment at the first nearest neighbor site in the most energetically stable semi-ordered phase for (a) CrFeCoNi, (b) CrMnCoNi, and (c) MnFeCoNi listed in Table 1 as OP₀. Red, orange, green, light blue, and dark blue indicate Cr, Mn, Fe, Co, and Ni, respectively. $m$ and $m_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{N}\mathrm{N}}$ represent the magnetic moment for each atom and the average magnetic moment of the 12 first nearest neighbor sites, respectively. The corresponding distributions for the other 12 alloys are shown in Figure S-4 (Supplementary Information).

Figure 6. Relative frequency distribution of bond-lengths fitted with Gaussian function for each element pair in the (a) RSS phase and (b) Cr-L1₂ semi-ordered phase of the CrFeCoNi quaternary alloy after structure optimization. The bond lengths in the legends indicate the average values obtained upon fitting with the Gaussian function.

We carried out the high-throughput spin-polarized first-principles simulations based on DFT [52] using the Vienna ab initio simulation package (VASP) by inputing all generated HEA models above [53,54]. The PBEsol functional [55] was selected for the electronic structure calculations and geometry relaxation. Projector-augmented wave (PAW) [56,57] potentials were used to consider the interactions between the ion cores and valence electrons. The Brillouin zone was integrated using the Monkhorst–Pack method [58] with a $3\times 3\times 3$ grid mesh for both the $2\times 2\times 2$ and $4\times 4\times 4$ supercells; this grid size was chosen by testing the convergence for $3\times 3\times 3$, $5\times 5\times 5$, $7\times 7\times 7$, and $9\times 9\times 9$; their differences are less than 0.001 eV/atom in the formation energy. In addition, the convergence of the relative energies were confirmed for cut-off energy, and the default values are large enough to achieve the 0.001 eV/atom accuracy with regard to the formation energy. For structural optimization, the convergence criteria for energy and force were set to be ${10}^{-3}$ eV and ${10}^{-2}$ eV/Å, respectively. The lattice parameter was set at 3.495 Å as an initial condition for structural optimization. As an initial condition, Fe, Co, Ni, and Cu were set as ferromagnetic, whereas Cr was set as antiferromagnetic. For systems without Cr, Mn was set as antiferromagnetic; otherwise, Mn was set as ferromagnetic as an initial condition because its magnetism is affected by Cr. The present computational method based on DFT is only eligible for obtaining the magnetic moments of constituent elements at 0 K. Therefore, we note that magnetization at finite temperature is beyond the scope.

2.3. Formation energy and free energy of formation

To discuss phase stability, the formation energy $E_f(\mathrm{H}\mathrm{E}\mathrm{A})$ is defined as

(2) $E_f(\mathrm{H}\mathrm{E}\mathrm{A})=E(\mathrm{H}\mathrm{E}\mathrm{A})-\sum _i{x}_{i}E(X_i).$(2)

Here, $E(\mathrm{H}\mathrm{E}\mathrm{A})$ denotes the total energy per atom of the HEA with the alloying elements; $x_i$ represents the fractions of the alloying elements; and $E(X_i)$ denote the energies per atom of the alloying elements $x_i$ in their ground state structures, i.e. BCC Cr, $I\bar{4}3m$ Mn, BCC Fe, HCP Co, FCC Ni, and FCC Cu, which are estimated by referring to the Open Quantum Materials Database [59,60]. A negative value of the formation energy implies that the HEA is stable, whereas a positive value implies that it is less stable than the ground states of the pure alloying elements.

Temperature is one of the most important factors that governs the phase stability of materials because most materials are used in a finite temperature range instead of being used at absolute zero [61]. Here, the Helmholtz free energy is defined as

(3) $F(V,T)=E_{\mathrm{e}\mathrm{l}}(V)+E_{\mathrm{v}\mathrm{i}\mathrm{b}}(V,T)-T{S}_{\mathrm{e}\mathrm{l}\mathrm{e}}(V,T)-T{S}_{\mathrm{v}\mathrm{i}\mathrm{b}}(V,T)-T{S}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}(V,T),$(3)

where $E_{\mathrm{e}\mathrm{l}}(V)$ denotes the internal energy, and the phonon vibration contribution includes the lattice vibration energy, i.e. $E_{\mathrm{v}\mathrm{i}\mathrm{b}}(V,T)$, and lattice vibration entropy, i.e. $S_{\mathrm{v}\mathrm{i}\mathrm{b}}(V,T)$. Further, $S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}(V,T)$ represents the atomic configurational entropy, and the last term $S_{\mathrm{e}\mathrm{l}\mathrm{e}}(V,T)$ denotes the electronic-scale entropy contribution, which includes thermal excitation and spin polarization. In the current work, the total energy of each cell was computed within the DFT framework at 0 K, and no vibrational entropy was considered, because the vibrational contributions for both the semi-ordered and RSS phases were estimated and found to be comparable [24,62]. In previous studies [24,63,64], contributions from the electronic (magnetic) entropy term were found to be negligibly small compared with the configurational entropy. Thus, the present study does not consider these entropy terms. $S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}$ in the configurational entropy term is given by $S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}=R{\sum }_i{c}_{i}ln{c}_i$, where $c_i$ denotes the ratio between the number of atoms of a disordered component and the total number of disordered atoms, and $R$ represents the gas constant. In this study, we introduced an equiatomic quaternary system and calculated $S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}=Rln(3)\times 0.75=0.824R$ for the semi-ordered phases and $S_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}=Rln(4)=1.386R$ for the RSS phase. In other words, the RSS phase has a greater configurational entropy contribution. Therefore, the free energy of formation is calculated by

(4) $F_f(\mathrm{H}\mathrm{E}\mathrm{A})=E_f(\mathrm{H}\mathrm{E}\mathrm{A})-T{S}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}.$(4)

By definition, $F_f(\mathrm{H}\mathrm{E}\mathrm{A})>0$ means that the corresponding phase is thermodynamically unstable (see Table 1), thereby giving rise to the phase separation or the appearance of the other phases rather than those investigated in the present study: Although one of the investigated semi-ordered phases, L1₂, has been observed experimentally [1,26], there could exist the occurrence of phase separation and then the appearance of other phases with ordered structures in equiatomic high entropy alloys. In this sense, this study addresses the relative stability restricted to the RSS and the two semi-ordered phases.

3. Results

3.1. Order–disorder competition

One of the main objectives of this study is to investigate whether a semi-ordered phase (OP₀) is more stabilized than the corresponding RSS for each studied quaternary alloy. That is, we intended to determine the ‘crossover’ temperature ($T_c$) at which the phase transition between the OP₀ and RSS occurs for each composite of the quaternary alloy. Therefore, it is important to consider the configurational entropy term to discuss phase stability in HEAs in the finite temperature region. It is imperative to quantitatively analyze the influences of VEC and temperature on the stable HEA crystal structures for investigating the order–disorder competition and phase stability. This is because the enthalpy term defined at the absolute zero temperature correlates with VEC, whereas the entropy effect is temperature dependent; the most stable phase is determined by free energy, i.e. the sum of the enthalpy and entropy terms. Guo et al [40,41]. reported several HEA systems wherein the single FCC phase was stable in the region of VEC $>8.00$, the single BCC phase was stable in the region of VEC $<6.87$, and the mixed FCC and BCC phases appeared in the region of $6.87<\mathrm{V}\mathrm{E}\mathrm{C}<8.00$. The average VEC values of magnetic HEAs are greater than 7.50, and therefore, they possess the FCC structure. The present study considers only single FCC phases in order to systematically elucidate a VEC dependence of their relative stability within the FCC crystal system, thereby clarifying our discussion. Hence, this study does not focus on the lattice type of magnetic HEAs, but on their magnetic orderings.

As a starting point, we shall look at the VEC-dependent stabilities within the semi-ordered phases, L1₂ and D0₂₂. Figure 7 indicates the difference of formation energy at 0 K between L1₂ and D0₂₂ for each composite as a function of VEC: $\mathrm{\Delta }{E}_f=E_f(\mathrm{L}{1}_2)-E_f(\mathrm{D}{0}_{22})$, where $E_f$(L1₂) and $E_f$(D0₂₂) are, respectively, formation energies for L1₂ and D0₂₂, as given in Equation (2)(2) $E_f(\mathrm{H}\mathrm{E}\mathrm{A})=E(\mathrm{H}\mathrm{E}\mathrm{A})-\sum _i{x}_{i}E(X_i).$(2) . Note that $E_f$(L1₂) and $E_f$(D0₂₂) involve error bars because each of them was averaged over 100 samples of the atomic configurations. From this figure, we can point out a trend that the D0₂₂ phases have lower energies than the L1₂ ones as the VEC value increases. At finite temperatures, the entropy terms contribute to this trend as temperature effects.

Figure 7. Difference of formation energy at 0 K between the L1₂ and D0₂₂ semi-ordered phases for each composite as a function of VEC.

Table 1 lists the quaternary alloy composites investigated in this study (Composite), VEC and melting temperature averaged over the four constituent elements in the alloy (VEC_ave and ${\bar{T}}_m$), semi-ordered phase with the lowest energy at 0 K (OP₀), temperature at which stability is attained for each phase (T_S(X) (X = L1₂, D0₂₂, RSS)), ‘crossover’ temperature between the semi-ordered phase OP₀ and RSS ($T_c$), and the corresponding free energy of formation at $T_c$ ($F_{\mathrm{c}}$). The most stable solid phase (MSSP) of each composite below ${\bar{T}}_m$ and the corresponding temperature range ($\mathrm{\Delta }{T}_s$) are also listed in the table. To estimate the free energy of formation in Equation (4)(4) $F_f(\mathrm{H}\mathrm{E}\mathrm{A})=E_f(\mathrm{H}\mathrm{E}\mathrm{A})-T{S}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}.$(4) , we can consider the semi-ordered phases as a 75 at.% equiatomic ternary alloy for the configurational entropy term because one of the elements is completely located at a specific site (Figure 1(a,b)). Consequently, one element of the semi-ordered phases can be ignored for the number of the elements considering the configurational entropy term. Alternatively, the RSS phases need to be treated as a quaternary alloy. Accordingly, the slope of the free energy of formation for the RSS phase is steeper than that for the semi-ordered phases. The semi-ordered phases consequently appear gradually as the temperature increases (rather enthalpy-driven stabilization), and this is followed by the appearance of the RSS phases at higher temperatures (rather entropy-driven stabilization). The above discussion presupposes thermal equilibrium. Note that semi-ordered phases that thermally equilibrate at higher temperatures can appear even at ambient temperatures by non-equilibrium processing, i.e. rapid solidification and cooling [65].

Herein we investigate the temperature dependence of the free energy of formation, $F(T)$, for the semi-ordered and disordered phases. As an example, $F(T)$ of CrMnCoNi is shown in Figure 3; the results for the other alloys as well as CrMnCoNi are shown in Figure S-1 (Supplementary Information). For the both semi-ordered phases, Cr-L1₂ and Cr-D0₂₂ are the most stable phase of each semi-ordered phase (Figure S-3 (3)). Therefore, we compare the free energies of formation of the Cr-L1₂ and Cr-D0₂₂ semi-ordered phases and the RSS phase up to ${\bar{T}}_m$ (1,799 K). The temperature dependence can be seen as follows:

1. Within the low-temperature region (up to 282 K), all semi-ordered and RSS phases exhibit a positive free energy of formation, and they are less stable than the corresponding ground state of each pure element.

2. At 282–1,003 K, Cr-L1₂ shows a negative free energy of formation, which is lower than those of the Cr-D0₂₂ and RSS phases. Therefore, Cr-L1₂ is the most stable phase in this temperature region.

3. Beyond 1,003 K, the RSS phase shows a lower free energy of formation than Cr-L1₂ and becomes a stable phase in the high-temperature region.

Meanwhile, the Cr-D0₂₂ phase does not appear as the most stable phase in the entire temperature range. Consequently, in the case of the CrMnCoNi quaternary alloy ($\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.00$), the L1₂ crystal structure appears as a semi-ordered phase in a specific temperature region.

Further, as shown in Figure S-1 and Table 1, the CrMnFeCo (1; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=7.50$), CrMnFeNi (2; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=7.75$), CrFeCoNi (5; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.25$), MnFeCoNi (9; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.50$), and FeCoNiCu (15; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=9.50$) quaternary alloys exhibit the same phase transition between the L1₂ semi-ordered phase and RSS phase at a specific temperature. However, the CrFeCoNi (5; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.25$) and MnFeCoNi (9; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.50$) quaternary alloys exhibited the L1₂ structure at 0 K. Furthermore, the CrMnCoCu (6; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.25$), CrFeCoCu (7; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.50$), CrMnNiCu (8; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.50$), CrFeNiCu (10; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.75$), MnFeCoCu (11; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.75$), CrCoNiCu (12; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=9.00$), MnFeNiCu (13; VEC${ }_{\mathrm{a}\mathrm{v}\mathrm{e}}=9.00$), and MnCoNiCu (14; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=9.25$), quaternary alloys show the RSS phase only below the melting temperature because of the small difference in the free energy of formation at 0 K (Figures S-1 (6)–(8) and (10)–(14)). The CrMnFeCu (4; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.00$) quaternary alloy is unstable in the entire temperature range.

The results of this study are summarized as follows: The alloys without Cu as a constituent prefer the L1₂ semi-ordered phases for VEC values $\le 8.25$. The Mn-L1₂ semi-ordered MnFeCoNi (9) and the Fe- L1₂ semi-ordered FeCoNiCu (15) are also stable under a specific temperature. The free energy of formation of the D0₂₂ semi-ordered phases are lower than L1₂ semi-ordered phases for VEC values $\ge 8.50$, except for MnFeCoCu (11) and FeCoNiCu (15). However, D0₂₂ semi-ordered phases do not appear as the most stable structure, because there is no condition that the free energy of formation is negative and smaller than the RSS in the entire temperature range. In contrast, alloys with large VEC values and Cu as a constituent exhibit the RSS phases.

3.2. Energetics

A clear understanding of the energetics of each phase is important to determine the free energy of formation at a finite temperature for discussing the phase stability of HEAs. Figure 2 shows the average free energies of formation over 100 different configurations, with the standard deviation at 1,000 K for the MnCoNiCu alloy, each of which has eight types of semi-ordered phases and the RSS phase. Semi-ordered phases with the lowest energy at 0 K (OP₀) for each quaternary alloy are listed in Table 1. For all quaternary alloys (Figure S-1), the free energy of formation of at least one semi-ordered phase is lower than that of the RSS phase at 0 K. Interestingly, the D0₂₂ semi-ordered phases exhibit a lower free energy of formation than the L1₂ semi-ordered phases in the high VEC region in some alloys. However, because the free energy of formation of D0₂₂ semi-ordered phases is not negative below the RSS value, the D0₂₂ phases are not stable phase in the entire temperature range.

Four alloys with relatively high VEC values, CrFeNiCu (10; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.75$), CrCoNiCu (12; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=9.00$), MnFeNiCu (13; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=9.00$), and MnCoNiCu (14; $\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=9.25$) exhibit D0₂₂ semi-ordered phases with relatively lower energies than that of the L1₂ semi-ordered phases (see Figure S-3). These alloys demonstrate that the free energy of L1₂ increases remarkably, as a result, these free energies of formation show a positive value at 1,000 K. This result means that these semi-ordered structures are thermodynamically unstable, however, we compare the free energies of L1₂ and D0₂₂ to get insight of their competitions. Similar competitive behavior between the phase stability of L1₂ and D0₂₂ for a wide range of VECs was previously reported for Ni₃V, Pd₃V, and Pt₃V, which are fundamental binary alloys, in contrast with HEAs [35]. The VEC at which the crossover of the energy difference between the L1₂ and D0₂₂ phases occurs in this study ($\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.50$) is comparable to the value reported for the previously studied binary system [35]. One key result obtained from Figure S-3 is that the systems comprising Cu have high free energies of formation. Moreover, quaternary alloys in which Cu forms the semi-ordered phase entail higher free energies of formation of the semi-ordered phase than those of the RSS phase. We attribute the high energies of formation associated with Cu to the positive formation energies exhibited by the least energetic structures of the binary compounds consisting of Cu and 3d transition metals (Cr, Mn, Fe, Co, and Ni) [66–68]. The present results are consistent with previous studies because as the number of bonding pairs between Cu and other 3d transition metals increased in the Cu-ordered phases (cf. Table 2).

3.3. Magnetic ordering

Figure 5 illustrates the distribution of magnetic moments as a function of the average magnetic moment at the first nearest neighbor sites in the most stable semi-ordered phase for the CrFeCoNi alloy (Cr-L1₂). The results of other quaternary alloys are shown in Figure S-4 (Supplementary Information). It is worth pointing out the present calculation scheme can produce the distribution of magnetic moments of CrFeCoNi that is the similar result by Niu et al [26]. In Figures 5 and S- 4, we observe certain common trends: ‘Cr’ and ‘Mn without Cr (see Figure 5(c))’ show antiferromagnetic properties, ‘Mn with Cr (see Figure 5(b))’, ‘Fe’, and ‘Co’ show ferromagnetic properties, and ‘Mn with a ferromagnetic moment’ shows values that can be compared to those of Fe. In contrast, Mn with antiferromagnetic properties behave similar to Cr. Therefore, the magnetic nature of Mn depends on the presence or absence of Cr. Ni and Cu exhibit nonmagnetic (paramagnetic) behavior in all systems. Ni in the MnFeCoNi (Figure 5(c)), MnFeNiCu (Figure S-4 (13)), and MnCoNiCu (Figure S-4 (14)) composites show a slightly negative magnetic moment as an exception. Although the magnetic moments are mostly similar, the magnetic moment distributions of some alloys show that the average magnetic moments at the first nearest neighbor sites are discrete; for example, at Cr and Fe in CrFeNiCu ($\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.75$), as shown in Figure S-4 (10). Such a discrete distribution can be attributed to the number of ferromagnetic or antiferromagnetic atoms located at the first nearest neighbor sites of the FCC lattice (Figure 7(b) in [26]). For example, the antiferromagnetic atoms in the semi-ordered phases are surrounded by four ferromagnetic atoms on average (see Table 2).

Niu et al [26] previously evaluated the magnetic moment of each element and the average magnetic moment of the 12 first nearest neighbor sites for the RSS and Cr-L1₂ semi-ordered phases of the CrFeCoNi quaternary alloy ($\mathrm{V}\mathrm{E}{\mathrm{C}}_{\mathrm{a}\mathrm{v}\mathrm{e}}=8.25$), and they are the same as that of the composite shown in Figure 5. The results obtained in this study demonstrate that Fe and Co exhibit ferromagnetic properties, whereas Cr exhibits anti-ferromagnetic properties. Since the results obtained are consistent with those reported previously [26], the validity of the calculation methods described herein can be confirmed. Moreover, Niu et al. reported a discrete distribution of the average magnetic moments at the first nearest neighbor sites in the Cr-L1₂ semi-ordered phase. They revealed that the magnetism of Cr in the CrFeCoNi alloy is strongly influenced by the magnetism of the first nearest neighbor sites [26]. For RSS phases of FCC-MnFeCoNi and FCC-CrMnFeCoNi alloys [69], their magnetic moments of the individual atoms were investigated: Fe has a higher magnetic moment than Co; Ni and Cr exhibit a near-zero magnetic moment and antiferromagnetism, respectively; Mn atoms exhibit both ferromagnetism and antiferromagnetism with a wider distribution.

Ghazisaeidi et al. reported a similar magnetism distribution in the hexagonal close packed (HCP) and FCC phases of CrCoNi, FeCoNi, MnFeNi, MnCoNi, and CrMnFeCoNi [45]. The magnetism of Cr and Mn atoms ranges from ferromagnetism to antiferromagnetism in the RSS phase. Fedorov et al [70] and Schönfeld et al [71] pointed out that the magnetic properties of the constituent elements are key factors governing phase stability and the formation of semi-ordered phases in 3d-transition-metal-based alloys.

3.4. Relationship between structures and properties

Figure 4 shows the relationship between the formation energy and site occupancy of Cr–Cr pairs at the first and second nearest neighbor sites in the RSS phase of the CrFeCoNi alloy. In the RSS phase, site occupancy by the same element can remain at 25 at.% on average at any nearest neighbor site. Meanwhile, the site occupancies for the semi-ordered phases depend on the nearest neighbor site, as can be seen in Table 2. For the first nearest neighbor sites (Figure 4(a)), the calculated results reveal a lowering of the energy with a decreasing site occupancy, such that the energy approaches the values corresponding to the L1₂ and D0₂₂ semi-ordered phases on extrapolation of the site occupancy to zero. However, for the second nearest neighbor sites (Figure 4(b)), an opposite trend is observed, i.e. the formation energies decrease with increasing site occupancy. The values for both semi-ordered phases can be found on the extrapolation line. These results imply that the configurations at the first and second nearest neighbor sites affect the formation energy of the CrFeCoNi alloy. Furthermore, the lower formation energies of the semi-ordered phases can be attributed to their unique site occupancies.

Figure 6(a,b) show the bond-length distributions in the RSS and Cr-L1₂ semi-ordered phases, respectively. In Figure 6, the solid lines indicate pairs of same elements, whereas the dotted and dashed lines indicate those of different elements. Many 3d transition metals, especially Cr, Fe, Co, and Ni, exhibit comparable atomic radii. However, the CrFeCoNi alloy shows a different bond-length distribution. These results indicate that these element pairs have different interatomic distances. As shown in Figure 6(a), the Cr–Cr pairs in the RSS phase exhibit a broad bond-length distribution and longer bond lengths than those corresponding to the first nearest neighbor sites, as estimated by the lattice constant. The bond lengths of the homogeneous element pairs are in a decreasing order of Cr, Fe, Ni, and Co. The same trend is observed for the Cr-L1₂ semi-ordered phase in Figure 6(b). Notably, there are no Cr–Cr pairs in the Cr-L1₂ semi-ordered phase because Cr atoms occupy the sites constructing the framework of the L1₂ semi-ordered phase, and they are not located at the first nearest neighbor sites (Table 2). Although the structure optimization process is applied to all atoms in the Cr-L1₂ semi-ordered and RSS phases, the bond-length distributions of the Cr-L1₂ semi-ordered phase are narrower than those of the RSS phase.

Let us consider the half widths at half maximum (HWHM) of the fitted Gaussian functions. The HWHMs of Fe–Fe, Co–Co, and Ni–Ni in the L1₂ semi-ordered phases are 64.5%, 71.2%, and 68.6%, respectively, of the corresponding RSS phases. It is presumed that the Cr occupancy at the second nearest neighbor sites results in a small bond-length distribution with regularity. Gao and Alman used ab initio molecular dynamics (AIMD) simulations [72] and reported that the Al_1.3CrFeCoNiCu senary alloy exhibits a different distribution for each element pair, which is similar to the results obtained in this study for 3d-transition-metal-based equiatomic quaternary alloys. However, the HfNbTaTiZr quinary alloys show almost the same distribution for all its constituent elements without a 3d transition metal [72]. In this study, we conclude that the bond-length distributions of the described systems are significantly different because each 3d transition metal has an individual magnetic behavior.

Summary and future prospects

Since 3d-transition-metal quaternary systems are base alloys for their extension to more multicomponent systems, a comprehensive understanding of their structures would serve as a stepping stone toward efficient HEA design. In this study, we performed first-principles calculations considering the configurational entropy term to investigate the effect of VEC and temperature on the stabilities of the two semi-ordered phases: L1₂ and D0₂₂. Further, we drew a comparison with the disordered atomic configurations in equiatomic quaternary alloys comprising Cr, Mn, Fe, Co, Ni, and Cu. We found that the VEC is a significant factor controlling phase stability in both semi-ordered phases. In the high-VEC region, the D0₂₂ semi-ordered phase is energetically more stable than the L1₂ semi-ordered phase, though both phases are metastable. The magnetic moments indicate that the anti-ferromagnetic Cr and Mn atoms are surrounded by the ferromagnetic Fe, Co, and Ni atoms located at the first nearest neighbor sites. Antiferromagnetic correlations tend to occupy positions more distant than the first nearest neighbor, which may be promoting ordering of L1₂/D0₂₂. Since the constituent elements corresponding to 25 at.% in the L1₂ and D0₂₂ semi-ordered phases exhibit no bonding between the first nearest neighbor sites, the extent of the combination of anti-ferromagnetic and ferromagnetic elements can be increased in the L1₂ and D0₂₂ phases. The bond-length distribution results reveal that the Cr–Cr distances are longer and the distributions are broader than those of other element pairs, which suggests that the high formation energy of the RSS phase can be attributed to the abnormal bond length between the anti-ferromagnetic atoms.

In this study, the enthalpy terms that describe structural properties were evaluated by first-principles calculations to obtain insight into an atomistic origin of the energetics of the 3d-transition-element HEAs. Since the entropy term is as important as the enthalpy term, the addition of a configurational entropy term in the density functional theory (DFT) calculations is effective for determining the crystal structure of HEAs. This enables us to estimate the transition temperatures between the semi-ordered and RSS phases, which is useful for obtaining the relationship between structure and processing. It is already widely recognized in the materials science field that the relationship among structures, properties, and processing methods is a crucial factor in realizing outstanding structural materials [73]. For example, Kenel et al. designed a 3d-transition-metal HEA using a 3D printing technique [74] with oxide nanopowder, which exhibits prominent mechanical properties at ambient and cryogenic temperatures [75]. Furthermore, it is necessary to predict an accurate phase diagram of HEA by combining the calculation of the phase diagram (CALPHAD) [76] and the first-principles based thermodynamic assessment [77]. We believe that the systematic investigation described herein will be helpful for realizing such an assessment and for predicting the microstructures of HEA by performing model simulations such as ‘first-principles phase field model’ without any thermodynamic empirical parameters [78].

Finally, we note that our theoretical approach is applicable to ‘ceramics’, ‘oxides’, and ‘carbide’, as well because the ‘high-entropy’ concept has been adopted even in them [79–86]. Furthermore, the approach can be extended to the constituent phases of even multiprincipal element alloys (MPEAs) [87,88] and multiphase compositionally complex alloys (CCAs) [89]. Thus, our finding and approach can encourage both experimentalists and theoreticians to investigate the microscopic identification of their structures.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27660400.2022.2153632.

Additional information

Funding

This study was supported by the computational resources of the HPCI system [Project ID: hp190014, hp190059, hp200040, hp210019, hp220017], the JHPCN system [Project IDs: jh210045 and jh220037], Institute for Materials Research, Tohoku University [Proposal No. 19S0501, 20S0505, 202012-SCKXX-0506], JAIST, and NIMS. H.M. thanks the Korea Institute of Science and Technology [Grant No. 2Z05840, No. 2E31731], Bilateral Collaboration Program through the National Research Foundation of Korea (NRF) [Grant No. 2021K2A9A2A0700009311] and No. 2021-0-02076 (IITP) funded by the Korea government (MSIT) for providing the financial support for this research. R.S. acknowledges the support from the Council for Science, Technology and Innovation under the Cross-ministerial Strategic Innovation Promotion Program. K.H. acknowledges the financial support from MEXT-KAKENHI [Grant Nos. JP19K05029, JP19H05169], JSPS Bilateral Joint Research Seminars with Korea NRF [JPJSBP220218802], and the Air Force Office of Scientific Research [Award Number: FA2386-20-1-4036].