Abstract
Highentropy alloys, with N elements and compositions {c_{ν = 1,N}} in competing crystal structures, have large design spaces for unique chemical and mechanical properties. Here, to enable computational design, we use a metaheuristic hybrid Cuckoo search (CS) to construct alloy configurational models on the fly that have targeted atomic site and pair probabilities on arbitrary crystal lattices, given by supercell random approximates (SCRAPs) with S sites. Our Hybrid CS permits efficient global solutions for large, discrete combinatorial optimization that scale linearly in a number of parallel processors, and linearly in sites S for SCRAPs. For example, a fourelement, 128site SCRAP is found in seconds—a more than 13,000fold reduction over current strategies. Our method thus enables computational alloy design that is currently impractical. We qualify the models and showcase application to real alloys with targeted atomic shortrange order. Being problemagnostic, our Hybrid CS offers potential applications in diverse fields.
Main
Complex solidsolution alloys (CSAs), which have a subset of nearequiatomic highentropy alloys^{1,2,3,4,5,6,7}, show remarkable properties for number of elements N ≥ 4 and set of elemental compositions {c_{ν = 1,N}}^{8}, and even for mediumentropy (N = 3) alloys^{9}. Such findings have encouraged research into CSAs for use in extremeenvironment technologies, such as aerospace and energy generation, for example by adding refractory elements for higher operational temperatures. In refractory CSAs, vacancy defects—which are ubiquitous when processing—can have a profound influence on stability and phase selection^{10}, thereby adding another design ‘element’. CSAs thus have a vast design space to create materials with novel or improved properties (for example, resistance to fatigue, oxidation, corrosion and wear), but many (especially bulk) properties, including resistivity, thermoelectricity, elasticity and yield strength^{6,7,8,9,10,11,12,13,14,15}, can alter rapidly with small compositional changes. As such, accurate, rapidly generated CSA models are needed to enable computational design and to identify trends in {c_{ν}}derived properties and thermal stability. Yet, models of CSAs have a design space that grows exponentially with number of elements N, number of pairs \(\frac{1}{2}N(N1)\) and number of sites S—a type of NPhard (NP, nondeterministic polynomial time) combinatorial problem.
To make computational alloy design practical, we employ a metaheuristic cuckoo search (CS)^{16,17} that follows the brood parasitism of female cuckoo birds in which they mimic the color and pattern of eggs for a few host species—an evolutionary algorithm (EA)^{18}. The advantages of CS are as follows: (1) it has a global convergence success that is higher than other EAs, (2) its local and global searches are controlled by a switching parameter and (3) Lévy flights scan the solution space more efficiently, with no random walks, so it is better than a Gaussian process^{16,17,19,20}. A CS yields approximate solutions (‘nests’) for intractable or gradientfree problems^{21} with little problemspecific knowledge—often only a ‘fitness’ function^{22}. For complex cases, fitness can be discontinuous (nondifferentiable to noisy). Related methods^{19} include simulatedannealing^{23}, geneticalgorithm^{24}, particleswarm^{25}, antcolony^{26} and bat^{27} methods.
Inspired by CS successes, including in materials design^{28}, we introduce a Hybrid CS that is more efficient and establish this for standard functions, where CS already bests most common EAs^{16,17}. Our Hybrid CS employs Lévy flights for global searches and Monte Carlo (MC) for local explorations of large multimodal space, and scales linearly with the number of processors (that is, doubling the number of processors in parallel halves the compute time). Selecting a best nest at each iteration (or cycle) ensures that solutions ultimately converge to optimality, while diversification via randomization avoids stagnation (that is, being trapped in local minima).
As CSA properties can vary rapidly with composition {c_{ν}}, Hybrid CS enables onthefly optimal model generation with a substantial reduction in solution times, scaling linearly with system size (in addition to the number of processors inherent to the CS). Hybrid CS constructs pseudooptimal (discrete) supercell random approximates (SCRAPs) for S sites occupied by N elements to mimic CSAs (Fig. 1) with target onesite {c_{ν}} and twosite (pair) probabilities for a crystal symmetry (for example, body (bcc) or face (fcc) centered cubic). The pair probabilities are atomic shortrange order (SRO) parameters that qualify a model’s fitness, and these can be measured^{29,30}. Each SRO pair takes values over R neighbor shells (say, 1–5) with Z_{R} atoms per shell, leading to a total number of parameters per site of \(\frac{1}{2}N(N1)\ {\sum }_{R}{Z}_{R}\) over which to optimize.
Solution spaces grow rapidly with N and S (see ‘Solution size and fitness’ section). For a fourelement, 128atom model (a 10^{73} solution space), Hybrid CS SCRAP is optimal in 0.8 min—a reduction of more than 13,000fold over current strategies. A fiveelement alloy with a 250 (500) atom model (with a space of 10^{169} (10^{415})) is optimal in only 1.6 (4.9) min. Thus, Hybrid CS optimizes large problems with a substantial reduction in time over current methods (Table 1), enabling computational design that is currently impractical.
After establishing the bona fides of Hybrid CS, we define the fitness and associated physical (and discrete model) bounds to eliminate stagnation of MC searches. Hybrid CSgenerated SCRAPs are presented for CSAs with targeted SRO in different crystal structures to show that solution times scale linearly with size and with number of processors, enabling rapid creation of optimal Hybrid CS SCRAPs (‘nests’). We then showcase alloy design and model assessment using electronic densityfunctional theory (DFT) to predict properties. For any random alloy, we discuss a symmetry requirement that permits a reduction in the number of DFT calculations for an alloy design. However, the Hybrid CS is problemagnostic, so potentially offers optimization applications in diverse areas.
Results
Hybrid CS versus CS
Hybrid CS reaps the benefits of MC for local optimization alongside those of CS for the global optimum using multiplenest explorations via Lévy flight. A ‘nest’ represents, for example, a function value or an alloy configuration (SCRAP). A global CS discards a fraction of nests, p_{a}, with the worst fitness (that is, the probability of finding an alien nest^{16}). In the Methods, we replace the local search in CS Algorithm 1 by standard MC and create a Hybrid CS Algorithm 2.
To show efficacy, we applied Hybrid CS and CS to onedimensional (1D) benchmark functions that are used in applied math (defined in the Methods). In Fig. 2, simulations (mean over 100 runs) are shown for both algorithms versus iterations to reach the optimum. The algorithms converge to optimal values but at different rates. Hybrid CS outperforms CS in all cases, reducing evaluations by factors of 1.75 (Michalewicz) to 8 (Rastrigrin). MC is thus a more efficient search of local minima than Lévy flights alone^{31}.
The CS has two parameters: (1) number of nests n and (2) discard probability p_{a}. Hybrid CS has two more: (3) fraction of top nests (#TopNests) chosen for an MC step and (4) number of MC steps (#MCiters). The number of iterations to convergence versus parameter values is tested in Fig. 3. Iterations are roughly constant after n = 20–30 (Fig. 3a) and increase linearly with p_{a} (Fig. 3b), while #MCiters increases roughly linearly after n = 10–20 (Fig. 3c) and the #TopNests passed in MC, with the rest untouched, has little effect on iterations (Fig. 3d). So, to achieve the least iterations to optimum, these results suggest n ≥ 15, #MCiters ≤ n and 0.1 < p_{a} < 0.4. The parameters were fixed for tests in Fig. 2 (n = 15, p_{a} = 0.25, #MCiters = 15 and #TopNests = 0.3).
For any function with appropriate fitness, Hybrid CS outperforms CS, which already bests most common EAs^{16,17}. We thus employ optimal Hybrid CS SCRAPs for materials design using the parameters found above.
Solution size and fitness
To assess Hybrid CS (pseudo)optimal SCRAPs, as in Fig. 1, we need a fitness and size estimate of the solution spaces in terms of S sites and N elements. We illustrate this with C cells built from cubes so that S = A ⋅ C^{3}, with A = 2(4) atoms for bcc (fcc). If C = N (the number of elements), SRO parameters can be exactly zero (homogeneously random) in a smallestcell solution. For a bcc equiatomic ternary (ABC), quaternary (ABCD) and quinary (ABCDE), S is a 54, 128 and 250atom cell, respectively. In terms of combinatorial coefficient, \({\,}^{S}{{\mathcal{C}}}_{S/N}\), the estimated configurations for site occupations are

\({\rm{ABC}}{:}\,\,^{54}{{\mathcal{C}}}_{18}{\times }^{36}{{\mathcal{C}}}_{18}\approx 1{0}^{23}\)

\({\rm{ABCD}}{:}\,\,^{128}{{\mathcal{C}}}_{32}{\times }^{96}{{\mathcal{C}}}_{32}{\times }^{64}{{\mathcal{C}}}_{32}\approx 1{0}^{73}\)

\({\rm{ABCDE}}{:}\,\,^{250}{{\mathcal{C}}}_{50}{\times }^{200}{{\mathcal{C}}}_{50}{\times }^{150}{{\mathcal{C}}}_{50}{\times }^{100}{C}_{50}\approx 1{0}^{169}\)

\({\rm{ABCDE}}{:}\,\,^{500}{{\mathcal{C}}}_{100}{\times }^{400}{{\mathcal{C}}}_{100}{\times }^{300}{{\mathcal{C}}}_{100}{\times }^{200}{{\mathcal{C}}}_{100}\approx 1{0}^{415}\)
A cell with bigger S at fixed N alters compositions in discrete but more refined ways, as evinced by the two N = 5 cell sizes shown above, but the times to render an optimal cell and use it are more challenging. The solution space increases as the number of pairs \(\frac{1}{2}N(N1)\) grows, requiring a solution for each atom and its pairs over the range R (1–5 shells).
We must define a fitness function for optimization. An Ncomponent CSA is characterized uniquely by N − 1 onesite (occupation) probabilities \({p}_{\nu }^{i}\) for species ν and by \(\frac{1}{2}N(N1)\) twosite (pair) probabilities per neighbor shell, with the following definitions (and sum rules):
Here, average compositions (\({\hat{p}}_{\nu }={c}_{\nu }\)) are given by the sum over all sites S (with all species conserved). The SRO parameters, \({\alpha }_{\nu \beta }^{ij}\), dictate pair probabilities \({p}_{nu\beta }^{ij}\) with ν (β) atoms at site i (j), and their values are bounded^{32}:
where α < 0 indicates orderingtype SRO (increased pair probabilities), whereas α > 0 indicates clustering of like pairs (decreased pair probabilities). The final SRO for all sites and pairs qualifies a model and so serves as the fitness. So, a SCRAP must be optimized with constraints for target SRO values, and, to avoid stagnation of solutions and senseless iterations (wasted computing), we place ‘stop’ conditions on MC searches when the SRO falls below discrete bounds set by N and S (Methods). Discrete limits on floor/ceiling SRO values are exemplified in the Supplementary Information for a noncubic SCRAP for a bcc equiatomic quinary.
Hybrid CS versus MConly models
With MC stagnation addressed, Hybrid CS enables onthefly generation of optimal SCRAPs to model CSAs with arbitrary concentrations, structures and targeted atomic distributions. For ease of plotting, we first use a ternary (N = 3) with S = 54 sites (no SRO) to compare Hybrid CS with MConly (Fig. 4). Cells with up to 2,000 sites and 10 elements are timed as shown in Table 1. Background on the MConly generated cells is provided in the Discussion.
There is a substantial difference in timing for Hybrid CS (0.3 min) versus MConly (1,440 min, or one day) from the ATAT code^{33} (Table 1), with it increasing dramatically with larger S and N. Hybrid CS in Fig. 4 was successful in every attempt to find the global (pseudo)optimum—zero SRO for all pairs over three shells for every site—irrespective of the initialization, albeit the iteration count varied. MConly failed to reach an optimum from stagnation in all cases but one (a random event). For Hybrid CS in larger cells with N = 3, 4 and 5, the SCRAPs have the targeted SRO and the distributions are Gaussian (Supplementary Information)—a general result for all system sizes.
Hybrid CS SCRAPs timings and scaling
The timings for Hybrid CScreated SCRAPs (Table 1) show markedly reduced times compared to MConly, which suffer stagnation as the values of S and N increase. For a binary 40atom cell, MConly needed 105 min while Hybrid CS in serial mode required 0.11 min (<0.01 min in parallel).
The Hybrid CS timings demonstrate that the algorithm scales linearly with the number of processors (n_{proc} ≤ n_{nests}) used for parallel solution (often called strong scaling in computational sciences); that is, we reduce the solution time by a factor of two for every doubling of processors used. In Table 1, we report timings for doubling from 12 to 24 processors (fifth column). By taking their ratio, the reduction factor is found. As an example, for a bcc fiveelement, 250atom cell with 12 (24) processors, the solution time is 3.57 (1.60) min and the ratio is 3.57/1.60—a 2.22 (222%) decrease by doubling the number of processors. For 24 nests using 24 processors, we find a reduction of 30 (sixth column), rather than the 24 given by linear scaling. (In the following we show that linear scaling is expected by limiting R, so this factor depends slightly on the number of competing internal nests.) We plot t versus n_{proc} in Supplementary Fig. 1 up to 24 (the total nests used here), showing these reduction factors graphically. If we check the ratio for any size SCRAP in Table 1, this reduction is confirmed. The parameters for the Hybrid CS were set to 10 optimization steps (typically converged in 3–5), with a solution for each step having up to 1,000 Lévy and 750 MC searches (iterations).
As the SRO qualifies the alloy and our solutions, we note that all the multinary SCRAPs in Table 1 have specified a value of zero for three shells about all sites (worst error of <10^{−3} for two pairs in the third shell in the 10element, 2,000site case). The Hybrid CS optimizes models in minutes (0.25 to 5 min on the processors used here) for the cell sizes (S = 54–500) typically considered.
Let us assess the scaling of execution times with a limit on the range R of the SRO. Typically, the range is limited in a solid solution, except near a phase transition where the SRO diverges (or if electronic Fermisurface nesting operates^{15}). Optimization at each site over a few shells with Z_{tot} atoms is then usually valid and should scale as the number of atoms S. For ease of analysis, we again use bcc (fcc) SCRAPs with S = A ⋅ N^{3} so that the SRO parameters for each pair (section ‘Solution size and fitness’) can be exactly zero (target value set and achieved in Table 1). The relative time for different symmetries (for example, bcc and fcc) with fixed N and R can then be estimated as \({t}_{\rm{rel}}^{N}={t}_{\rm{fcc}}^{N}/{t}_{\rm{bcc}}^{N}\approx \frac{{A}_{\rm{fcc}}\times {Z}_{\rm{tot}}^{\rm{fcc}}}{{A}_{\rm{bcc}}\times {Z}_{\rm{tot}}^{\rm{bcc}}}=\frac{4}{2}\times \frac{42}{26}=3.2\) for SRO over three shells. Checking the ratio of timings in Table 1, the ratio is indeed about 3. Similarly, a relative timing for two sizes of fixed symmetry (say bcc) cell should scale as \({t}_{\rm{rel}}^{\rm{sym}}={t}_{\rm{bcc}}^{{N}_{2}}/{t}_{\rm{bcc}}^{{N}_{1}}\approx \frac{{S}_{2}}{{S}_{1}}\times {[\frac{{N}_{2}}{{N}_{1}}]}^{1/4}\), as can be verified from the timings in Table 1. So, the Hybrid CS SCRAP optimization scales linearly with S, in addition to linearly with n_{proc}, as is inherent to the Hybrid CS.
Hybrid CS SCRAPs are obtained rapidly to address concentrationdependent CSA properties. Six bcc 250atom, 5element (quinary) SCRAPs, such as A_{x}(BCDE)_{1 − x} versus the x composition along a line in composition space, for example, are found in minutes. Furthermore, any SRO values may be targeted, as SRO in alloys can lower the enthalpy or drive elemental surface enrichment. Smaller Satom cells with larger N can be obtained, but zero SRO will not be possible at all compositions.
Note that DFT methods typically scale (Methods) as S^{3} ⋅ K_{pt}, where K_{pt} is the number of symmetrydistinct kpoints used to solve the electronic DFT eigenvalue problem. DFT solutions (on the processors used here) for a 54atom cell take ~6 min for a 2 × 2 × 2 kmesh (K_{pt} = 4) and 1.9 h for a 5 × 5 × 5 kmesh (K_{pt} = 63), with different iterations (10–40) to converge the charge densities and total energies. From Table 1, the MConly solution for a ternary 54atom cell takes ~1,440 min (24 h) to get a model for one composition. Similarly, for a quaternary 128atom cell, the solution time is ~10,000 min (one week), whereas each DFT iteration (with K_{pt} = 4) takes ~77 min. Thus, MConly model generation is more timeconsuming and becomes worse with larger S or N. By contrast, Hybrid CS requires 0.3 (0.79) min to yield an optimal 54atom (128atom) cell, so DFT is the design bottleneck.
Real alloy applications
We constructed SCRAPs to assess the formation energy (E_{form}) versus SRO parameters (observed or trial \({\alpha }_{\mu \nu }^{\rm{shell}}\)). We assessed the relative energy (E) versus lattice constants (a) and equilibrium values (\(\bar{a}\)), along with atomic displacement {u_{i}} distributions for binaries to quinaries. We employed an allelectron Korringa–Kohn–Rostoker (KKR) Green’s function method^{34,35} and the pseudopotential Vienna abinitio simulation package (VASP)^{36,37} to obtain E_{form} versus SRO and \(\bar{a}\), compared to experiments. For DFT, we used a Perdew–Burke–Ernzerhof (PBE) exchangecorrelation functional^{38} and Monkorst–Pack meshes for Brillouin zone integrations^{39}. See Methods for details and Supplementary Information for supporting results.
CSAs with SRO
Hybrid CS works for any \(\frac{1}{2}N(N1)\) SRO pairs, so we use fcc Cu_{3}Au (N = 2) as an example for ease of presentation (one CuAu SRO value per shell, α^{s}) and because there are experimental data available. SCRAPs with specified SRO (each optimized in 0.6 min, Table 1) are used to mimic (1) a homogeneously random state at 495 °C (α^{s} = 0), as well as alloys with observed α^{s} values^{30} at (2) 450 °C and (3) 405 °C. Figure 5 presents a plot of E_{form} versus SRO. For any N and S, atomic displacements from ideal sites have zero mean (Supplementary Figs. 2 and 3).
KKR and the experiment with no SRO agree well (3–5 meV atom^{−1} difference). Both methods show similar trends, but KKR includes known alloying corelevel shifts, explaining the higher values observed with VASP. A gain of 30 meV atom^{−1} gain is found with SRO (lower entropy). The KKR \(\bar{a}\) without SRO is 3.765 Å, which agrees with the 3.749 Å observed^{40,41,42}, with only a small 0.43% mismatch. With a disordered alloy with SRO this value is 3.755 Å, closer to the ordered alloy value of 3.743 Å. The VASP \(\bar{a}\) with no SRO is 3.823 Å (with SRO it is 3.816 Å), that is, a 2% mismatch.
Distributions and averages
To simplify presentation, we assessed VASP E versus a and displacements {u_{i}} for NbMoTa 54atom SCRAP (Fig. 6a–c). At \(\bar{a}\), the energy is −64.4 meV atom^{−1} when volumerelaxed (ideal sites) and −80.5 meV atom^{−1} when atomrelaxed (a reduction of 16 meV). Vector displacements {u_{x,y,z}} sum to zero individually and are Gaussian distributed, as required by CSA symmetry, giving \(\bar{a}\) as the diffraction value. Meansquared displacements determine the Debye–Waller factor (Supplementary Information), which describes the attenuation of Xray, neutron or electron scattering caused by thermal motion, providing background diffuse intensity from inelastic scattering. Diffraction on ‘large’ samples (for example, 1 cm^{3}) gives ‘selfaveraged’ properties, as the Avogadro’s number for each local configuration is sampled simultaneously. We find similar results for any N and S (Fig. 6d–g). For quaternary TaNbMoW, volumerelaxed (−63.3 meV atom^{−1}) and atomrelaxed (−74.5 meV atom^{−1}) energies show an 11 meV atom^{−1} reduction from displacements. For quinary TaNbMoWV, volumerelaxed (−105.5 meV atom^{−1}) and atomrelaxed (−126.3 meV atom^{−1}) energies show a larger gain in stability from displacements with vanadium addition (−21 meV). Displacements increase with complexity, but more with vanadium alloying (Fig. 6d,g), enhancing the stability, lattice distortions and mechanical behavior, as discussed in the following.
Configurations
SCRAPs provide good averages if a cell is large enough (‘infinite’ is exact, but impractical); otherwise, configurational averaging may be warranted. In principle, thermodynamically, all configurations should be sampled (‘good’, ‘bad’ and ‘ugly’, leading to an average ideal lattice)—not just relaxed, lowenergy (good) ones, as are often chosen in the literature, but higherenergy, unfavorable (bad) and metastable (ugly) ones too. SCRAPs (before relaxations) are used for arbitrary choices of atomic site occupations, so relate to just one representative configuration out of many, so a model must be qualified. To complete a model, atom types Nb, Mo or Ta must be assigned to A, B or C sites. For example, the formation energy after relaxations may indicate that the structure is stable (favorable negative values), but phonons may exhibit lattice instabilities, as indicated by phonon frequencies ω, making it dynamically unstable. (Phonons with ω^{2} ≥ 0 are stable and those with ω^{2} < 0 are unstable—with \(\omega \propto \sqrt{1}\) (that is, ‘imaginary’)—such as occurs when a pencil is stood on its point (unstable) as opposed to being held like a pendulum by its point (stable).)
A statistical average governs nature’s reality, and an instability is controlled by environments around each atom. So, to eliminate an instability, a larger SCRAP is necessary to improve the statistical ‘selfaverage’; alternately, a simple swap of atom types in a small SCRAP may eliminate a local instability. For example, if we assign Nb, Mo and Ta to A, B and C sites, respectively, we find a minimum energy and stable lattice (that is, positive phonon frequencies) (Fig. 6c), found from the PHONOPY code^{43} with DFT inputs (Methods). From these results we can assess the alloy properties; for example, the average \(\bar{a}\) is 3.248 Å. Yet, with A ↔ C (that is, Nb ↔ Ta), we find a higher (+0.05 eV) energy and unstable phonons (Fig. 6c), suggesting that this model is in general too small, and care must be taken.
Lattice distortions
Each atom in a CSA has a different chemical environment that can cause lattice distortion (for example, from atomic size differences)^{44}. However, the effect of lattice distortion on the CSA mechanical response has been explored less because of a lack of computationally efficient models. In SCRAPs, lattice distortion in refractory CSAs can be tuned by changing the local environment to enhance the mechanical response (as intimated in Fig. 6a,d,f), an effect that is observed in ultrastrong ternaries^{45}. Rather than a size difference, embodied to zerothorder in a solid solution’s electronic bandwidths (the electronic origin of HumeRothery’s sizeeffect rule^{46}), strength enhancement correlates with the electronegativity difference between elements (on the Allen scale for solids, with vanadium largest), where largest bond distortions occur around vanadium sites (Supplementary Fig. 4).
Discussion
Reducing DFT computational times
Having saved orders of magnitude in model generation, DFT computational time is a major issue as DFT methods typically scale as S^{3} ⋅ K_{pt}, whereas Hybrid CS SCRAPs generation scales as S. However, a savings in DFT time is possible. As displacements {u_{i}} must have zero mean in any disordered alloy, the equilibrium (average) volume must be mathematically identical to that of the ‘ideal’ (diffraction) lattice. An example of this was shown in Fig. 6a, where ideal and atomically relaxed SCRAPs have identical equilibrium volumes. So, relaxations for any sized multinary SCRAP with any SRO need only be performed at the equilibrium volume (found from SCRAPs with ideal atomic positions) to assess properties and trends.
Limitations
Our Hybrid CS can be built with an arbitrary cell created by using M_{1} × M_{2} × M_{3} smaller base units, but we must carefully limit the range R of the SRO parameters so as not to correlate them directly with distant sites due to periodic boundary conditions (true for any cellular technique). In addition, to exemplify the methods and analysis, we limited the alloy model generation in the implemented code to homogeneously disordered crystal structures (simple cubic, bcc, fcc and hexagonal closepacked (hcp)). However, there is no restriction in general, so, in the near future, we will generalize the code for more complex superstructures (like partially disordered compounds).
Related cellular techniques
A supercell to mimic random alloys is not a new idea. Structural models are often constructed by specifically occupying sites of a finitesized periodic cell. For Metropolis MC methods^{47}, including simulated annealing, potential energies serve as a fitness criterion for acceptance of a trial move, yet solutions for global optima often stagnate, even in problems that are not large^{31}. We have already discussed the fitness for SCRAPs configurational optimization, along with floor/ceiling bounds given by each S and N and SRO value in the MC optimization (a worked example is provided in the Supplementary Information).
The original special quasirandom structure (SQS) used Isinglike MC to find supercells that mimic zero atomic correlations in the alloy by arranging atoms in particular ordered layers depending on the number of sites and atom types^{48}. In some cases, there was more than one configuration for a fixed number of sites, thus requiring an average. Such SQS did not have proper lattice symmetry (like bcc), so atomic displacements could not sum to zero as required by symmetry, in contrast to SCRAPs. Recently, the SQS algorithm was implemented using a stochastic MC approach^{33} to determine a sample configuration allowing a supercell with an arbitrary number of base units, such as M_{1} × M_{2} × M_{3} bcc twoatom cells, as done for SCRAPs. However, as N or S increase, MConly times become impractical and solutions stagnate (Table 1). Other implementations of the MConly approach with an arbitrary number of base units have been done recently, although some of the results were correlated through the use of improper boundary conditions^{49}.
In principle, Hybrid CS and MConly schemas should get the same supercells for specific cases, but our Hybrid CS avoids stagnated solutions, and timings are markedly reduced in serial mode and substantially reduced in parallel mode (Table 1). Moreover, Hybrid CS can rapidly optimize any sized SCRAP for any number of elements and for any targeted disorder, that is, any values of SRO parameters.
Enabling design via machine learning
Our Hybrid CS optimal SCRAPs permit the design of arbitrary complex solidsolution alloys, predicting properties and trends, including for surfaces, catalysis and oxidation, that are currently impractical. To showcase this, we assessed the stability and properties of binary to quinary solid solutions and discussed the qualification of the models. However, notable DFT calculation resources are necessary to generate databases for a broad range of compositions and properties. To further accelerate design, SCRAPs is integrated with highthroughput DFT calculations to produce accurate but limited databases, possibly validated or supplemented with experimental data, then we utilize the data for machinelearning (physicsbased) models, an ongoing activity. Finally, Hybrid CS offers potential optimization improvements in other fields, such as manufacturing, commerce, finance, science and engineering, as long as an appropriate fitness can be defined that can be evaluated expeditiously.
Methods
Cuckoo search
The CS is based on the brood parasitism of a female cuckoo bird, which specialize in mimicking the color and pattern of a few host species. For this there are three idealized rules: (1) a cuckoo lays an egg in a randomly selected nest; (2) the nest with highestquality egg (fitness) survives and is forwarded to the next generation; (3) the host bird can discover the cuckoo egg with a probability p_{a} ∈ (0,1) and, once discovered, it dumps either the nest or the cuckoo egg. The key advantages of this process are listed in the main text.
Hybrid CS
Our Hybrid CS schema reaps the benefits of traditional MC for local optimization alongside the CS schema for global optimization utilizing multiple nest explorations via Lévy flight. A global CS removes a fraction of nests, p_{a}, with worst fitness (a nest represents a lattice configuration), and it signifies the probability of finding an alien nest^{16}. We replace the local search in CS Algorithm 1 with MC and create the Hybrid CS given in Algorithm 2 (shown in pseudocode below), where the global search uses multiplenest explorations. For Hybrid CS, a basic MC worked well, as embodied in Algorithm 2 between ‘begin MC’ and ‘end MC’, in performing the MC steps:
(1) Obtain a nest from the sample of nests.
(2) Randomly swap a pair of site occupations.
(3) If Fitness_{new} < Fitness_{old}, Accept Swap; or
(4) Else Reject; Switch; and Go to (1).
Algorithm 1
Cuckoo Search Algorithm.
Input: Fix input & identify optimization variables 
Output: Optimized solution 
Initialize nests 
while iteration < Global maximum number do 
Create new nests using Lévy Flight (Global Search) 
Calculate fitness F of the nests 
Choose a nest randomly 
if F_{old} < F_{new} then 
replace nest with the new cuckoo 
Discard fraction p_{a} of worst nests & build new ones 
Keep best nests with the best results 
Rank the solutions & find the current best 
Return the best solutions 
Algorithm 2
Hybrid CS Algorithm.
Input: Fix input and optimization function 
Output: Optimized solution 
Initialize nests 
while iteration < Global maximum number do 
Create new nests using Levy Flight (GLOBAL Search) 
Calculate fitness F of the nests 
Choose fraction of nests with best fitness (top nests) 
LOCAL Search via Monte Carlo 7D2 ≪ begin MC ≫ 
foreach nests ∈ top nests do 
acceptance = 0 
rejections = 0 
while iteration < Local iterations do 
Calculate delta step (δx = σ*randn) 
Perturb nests with x_{i} + δx 
Calculate fitness, F(x_{i} + δx) 
Calculate δF = F(x) − F(x_{i} + δx) 
if δF > 0 then 
Perform the switch 
acceptance += 1 
else 
rejections += 1 
if acceptance > mc_{1}* Local iterations then 
σ = σ*a 
if rejections > mc_{2}* Local iterations then 
σ = σ/b 
Discard fraction p_{a} of worst nests ≪ end MC ≫ 
Rank the solutions & find the current best 
Return the best solutions 
‘Local’ MC iteration chooses a fraction of nests to optimize based on a nest’s value of fitness and a fraction equal to top nests ∈ {0, 1}. Aside from ‘top nests’, the local MC depends on mc_{1} ∈ {0, 1} and mc_{2} ∈ {0, 1}, which are used to optimize the value of step size, δx = σ*randn, by altering the value of σ (randn is a value from a standard normal distribution). For local optimization, number of acceptances/rejections are counted and, depending on their value, the value of δx alters. The other parameters are a(b) > 1, the increase/decrease increment in σ. By collecting the number of acceptances/rejections, we increase/decrease ∣δx∣ to get a local optimized value faster.
Standard test functions
The Hybrid CS schema (Algorithm 2) and the CS schema using only Lévy flights (Algorithm 1) were competed using a standard set of 1D benchmark functions, as shown in Fig. 2. The function name, its global optimum f(x*), which occurs at x*, and the function definition are given below, where d is the dimension of the input parameters:
1. Michalewicz (d = 5): f(x*) = −4.6876
\(f(x)=\mathop{\sum }\nolimits_{i = 1}^{d}\sin ({x}_{i}){\sin }^{2m}\left(\frac{i{x}_{i}^{2}}{\uppi }\right)\)
2. Rosenbrock (d = 16): 0.0 at x* = (1, 1, ..., 1)
\(f(x)=\mathop{\sum }\nolimits_{i = 1}^{d1}100{({x}_{i+1}{x}_{i}^{2})}^{2}+{({x}_{i}1)}^{2}\)
3. De Jong (d = 16): 0.0 at x* = (0, 0, ..., 0)
\(f(x)=\mathop{\sum }\nolimits_{i = 1}^{d}{x}_{i}^{2}\)
4. Ackley (d = 16): 0.0 at x* = (0, 0, ..., 0)
\(f(x)=20\,\exp (0.2\sqrt{\frac{1}{d}\mathop{\sum }\nolimits_{i = 1}^{d}{x}_{i}^{2}})\)
5. Rastrigin: 0.0 at x* = (0, 0, ..., 0)
\(f(x)=10d+{\sum }_{i = 1}{\rm{d}}{x}_{i}^{2}10\cos (2\uppi {x}_{i})\)
6. Easom: −1 at x* = (π, π)
\(f(x)=\cos ({x}_{1})\cos ({x}_{2})\exp ({({x}_{1}\uppi )}^{2}{({x}_{2}\uppi )}^{2})\)
Bounded discrete searches—no stagnation
SCRAPs must be optimized with constraints for target SRO values:
\({\hat{\alpha }}_{\alpha \beta }^{s}\) refers to the average SRO for the sth shell for an (α, β) pair. For \(\frac{1}{2}N(N1)\) pairs, \({d}_{\alpha \beta }^{s}\) is the target SRO value. The final SRO for all sites and pairs qualifies the model.
To avoid stagnation of solutions we place ‘stop’ conditions on MC (local) searches when the SRO falls below the discrete bounds set by the cell N and S. Such criteria avoid senseless iterations (wasted computing), working well when combined with a CS that guarantees global (pseudooptimal) convergence in a range R. The discrete limits for SRO parameters from equations (1)–(4) are given by
with the radial distribution function (\({g}_{\alpha \beta }^{s}\)), number of atoms in shell s (n_{s}) and type α (n_{α}). ⌊⌋ and ⌈⌉ represents the decimal at the lower (floor) and higher (ceiling) integer values, respectively. We use distance of SRO \({\alpha }_{\alpha \beta }^{s}\) from one of these values for a ‘stop’ criteria, that is
where ϵ_{1} and ϵ_{2} are predefined values. Choosing S and N to set discrete \({p}_{\alpha }^{i}\) and specifying \(\frac{1}{2}N(N1)\) target values for \({p}_{\alpha \beta }^{ij}\), the final values of SRO for all sites and atom pairs qualify the model fitness. We exemplify discrete limits on floor/ceiling SRO values in a 3 × 3 × 5 bcc supercell for an equiatomic quinary in the Supplementary Information, with the values shown in Supplementary Table 1.
DFT
For Cu_{3}Au, the VASP results used a 108atom SCRAP with SRO. The structures were relaxed using a 350eV planewave energy cutoff, PBE exchangecorrelation functional^{38} and 3 × 3 × 3 Monkhorst–Pack kmesh^{39} for Brillouin zone integrations. Total energy calculations were done at a denser (7 × 7 × 7) kmesh. By definition, E_{form} = E_{tot} − ∑_{i}n_{i}E_{i}, where E_{tot} (E_{i}) is the total energy of the alloy (pure elements ‘i’) and n_{i} is the number of sites per element in a supercell. For the same kmeshes, KKR^{34} was also used for E_{form}. Selfconsistent charge densities were found using the Green’s function by complexenergy (Gauss–Legendre semicircular) contour integration with 24 energies in a sphericalharmonic basis, including s, p, d and f orbital symmetries^{35}. Core eigenvalues were from Dirac solutions, and the valence used a scalarrelativistic solution (no spin–orbit coupling).
Typically, DFT methods scale as S^{3}, as for any matrix eigenvalue solution. However, electronic DFT must be solved for all points on the kmesh used for convergence—that is, \({N}_{k}^{\rm{total}}={N}_{{k}_{1}}\times {N}_{{k}_{2}}\times {N}_{{k}_{3}}\)—which may be reduced to fewer symmetrydistinct points, \({K}_{\rm{pt}}\le {N}_{k}^{\rm{total}}\), for a given highsymmetry crystal, like bcc or fcc. Hence, total times depends on S^{3} ⋅ K_{pt}.
For ternary to quinary alloys, a 54atom TaNbMo SCRAP was relaxed in VASP using a 350eV planewave energy cutoff, 8 × 8 × 8 kmesh (K_{pt} = 256) and the PBE exchangecorrelation functional^{38}. Compared to ternary, the only difference for 128atom TaNbMoW and 250atom TaNbMoWV was the kmesh, that is, 5 × 5 × 5 and 2 × 2 × 2, respectively. After symmetry operations are applied, the 5^{3} and 2^{3} meshes have, respectively, 63 and 4 symmetrydistinct K_{pt} points.
For phonons, DFT energy and force convergence criteria were set to be very high (10^{−7} eV and 10^{−6} eV Å^{−1}, respectively). A finitedisplacement method (set to 0.03 Å) was employed using PHONOPY^{43}. Phonon dispersion was plotted along the highsymmetry Brillouin zone directions (ΓHNΓ). Unstable (imaginary) frequencies are plotted as negative frequencies for simplicity of presentation.
Data availability
Supporting data for all data plotted in the Figs. 1–6 (as well as Supplementary Figs. 1–4) are available as source data in spreadsheets, in the Supplementary Information (see additional information) and at Code Ocean^{50} and https://github.com/DuaneDJohnson/HybridCuckooSearch/. Source data are provided with this paper.
Code availability
Interactive opensource codes are available via Code Ocean for HybridCS SCRAPs^{50} and for Hybrid CS for 1D functions^{51}. For opensource codes (and data) for Hybrid CS SCRAPs or 1D functions, see https://github.com/DuaneDJohnson/HybridCuckooSearch/.
References
 1.
Yeh, J. W. et al. Nanostructured highentropy alloys with multiple principal elements: novel alloy design concepts and outcomes. Adv. Eng. Mater. 6, 299–303 (2004).
 2.
Cantor, B., Chang, I. T. H., Knight, P. & Vincent, A. J. B. Microstructural development in equiatomic multicomponent alloys. Mater. Sci. Eng. A 375–377, 213–218 (2004).
 3.
Senkov, O. N., Miller, J., Miracle, D. & Woodward, C. Accelerated exploration of multiprincipal element alloys with solid solution phases. Nat. Commun. 6, 6529 (2015).
 4.
George, E. P., Raabe, D. & Ritchie, R. O. Highentropy alloys. Nat. Rev. Mater. 4, 515–534 (2019).
 5.
Gao, M.C., Yeh, JW., Liaw, P. K., Zhang, Y. (Ed.), HighEntropy Alloys: Fundamentals and Applications, 1st ed., Springer Inter. Publishing, Switzerland, 2016, pp. 333–366.
 6.
Singh, P., Smirnov, A. V. & Johnson, D. D. Atomic shortrange order and incipient longrange order in highentropy alloys. Phys. Rev. B 91, 224204 (2015).
 7.
Singh, P. et al. Design of highstrength refractory complex solidsolution alloys. npj Comput. Mater 4, 16 (2018).
 8.
Miracle, D. B. & Senkov, O. N. A critical review of high entropy alloys and related concepts. Acta Mater. 122, 448–511 (2017).
 9.
Zhang, Y. et al. Influence of chemical disorder on energy dissipation and defect evolution in concentrated solid solution alloys. Nat. Commun. 6, 8736 (2015).
 10.
Singh, P. et al. Vacancymediated complex phase selection in high entropy alloys. Acta Mater. 194, 540–546 (2020).
 11.
Karati, A. et al. Ti_{2}NiCoSnSb—a new halfHeusler type highentropy alloy showing simultaneous increase in Seebeck coefficient and electrical conductivity for thermoelectric applications. Sci. Rep. 9, 5331 (2019).
 12.
Ding, Q. et al. Tuning element distribution, structure and properties by composition in highentropy alloys. Nature 574, 223–227 (2019).
 13.
Li, Z., Pradeep, K. G., Deng, Y., Raabe, D. & Tasan, C. C. Metastable highentropy dualphase alloys overcome the strengthductility tradeoff. Nature 534, 227–230 (2016).
 14.
Zhang, R. et al. Shortrange order and its impact on the CrCoNi mediumentropy alloy. Nature 581, 283–287 (2020).
 15.
Singh, P., Smirnov, A. V. & Johnson, D. D. TaNbMoW refractory highentropy alloys: anomalous ordering behavior and its intriguing electronic origin. Phys. Rev. Mater. 2, 055004 (2018).
 16.
Yang, X. S. & Deb, S. Cuckoo search via Lévy flights. In Proc. World Congress on Nature and Biologically Inspired Computing 210–214 (IEEE, 2009).
 17.
Yang, X. S. & Deb, S. Engineering optimisation by Cuckoo Search. Int. J. Math. Model. Numer. Optim. 1, 330–343 (2010).
 18.
Back, T., Fogel, D. & Michalewicz, Z. Handbook of Evolutionary Computation (Oxford Univ. Press, 1996).
 19.
Yang, X. S. Engineering Optimization: An Introduction with Metaheuristic Applications (Wiley, 2010).
 20.
Yang, X. S., Koziel, S. & Leifsson, L. Computational optimization, modelling and simulation: recent trends and challenges. Procedia Comput. Sci. 18, 855–860 (2013).
 21.
Blum, C. & Roli, A. Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput. Surv. 35, 268–308 (2003).
 22.
Ashlock, D. Evolutionary Computation for Modeling and Optimization 1st edn (Springer, 2016).
 23.
Kirkpatrick, S., Gelatt, C. D. Jr & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983).
 24.
Holland, J. H. Adaptation in Natural and Artificial Systems 1st edn (MIT Press, 1992).
 25.
Kennedy, J. & Eberhart, R. Particle swarm optimization. In Proc. ICNN’95 International Conference on Neural Networks Vol. 4, 1942–1948 (IEEE, 1995).
 26.
Dorigo, M., Maniezzo, V. & Colorni, A. Ant system: optimization by a colony of cooperating agents. IEEE Trans. Syst. Man Cybern. B Cybern. 26, 29–41 (1996).
 27.
Yang, X. S. Bat algorithm for multiobjective optimisation. Int. J. BioInspired Comput. 3, 267–274 (2011).
 28.
Sharma, A., Singh, R., Liaw, P. K. & Balasubramanian, G. Cuckoo searching optimal composition of multicomponent alloys by molecular simulations. Scrip. Mater. 130, 292–296 (2017).
 29.
Cowley, J. M. An approximate theory of order in alloys. Phys. Rev. 77, 669–675 (1950).
 30.
Moss, S. C. Xray measurement of shortrange order in Cu_{3}Au. J. Appl. Phys. 35, 3547–3553 (1964).
 31.
Gutowski, M. Lévy flights as an underlying mechanism for global optimization algorithms. Preprint at https://arxiv.org/pdf/mathph/0106003.pdf (2001).
 32.
Johnson, D. D. in Computation of Diffuse Intensities in Alloys, Characterization of Materials (ed. Kaufmann, E.) 346–375 (Wiley, 2012).
 33.
Van de Walle, A. et al. Efficient stochastic generation of special quasirandom structures. Calphad 42, 13–18 (2013).
 34.
Johnson, D. D., Nicholson, D. M., Pinski, F. J., Gyorffy, B. L. & Stocks, G. M. Densityfunctional theory for random alloys: total energy within the coherentpotential approximation. Phys. Rev. Lett. 56, 2088–2091 (1986).
 35.
Alam, A. & Johnson, D. D. Structural properties and stability of (meta)stable ordered, partially ordered, and disordered Al–Li alloy phases. Phys. Rev. B 85, 1441202 (2012).
 36.
Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561 (1993).
 37.
Kresse, G. & Hafner, J. Ab initio moleculardynamics simulation of the liquid–metal/amorphous–semiconductor transition in germanium. Phys. Rev. B 49, 14251–14269 (1994).
 38.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1994).
 39.
Monkhorst, H. J. & Pack, J. D. Special points for Brillouinzone integrations. Phys. Rev. B 13, 5188–5192 (1976).
 40.
Orr, R. L. Heats of formation of solid Au–Cu alloys. Acta Metall. 8, 489–493 (1960).
 41.
Flinn, P. A., McManus, G. M. & Rayne, J. A. Elastic constants of ordered and disordered Cu_{3}Au from 4.2 to 300 °K. J. Phys. Chem. Solids 15, 189–195 (1960).
 42.
Smallmann, R. E. & Nagan, A. H. W. Modern Physical Metallurgy 3rd edn (Butterworths, 1970).
 43.
Togo, A. & Tanaka, I. Firstprinciples phonon calculations in materials science. Scrip. Mater. 108, 1–5 (2015).
 44.
Yeh, J.W. Alloy design strategies and future trends in highentropy alloys. JOM 65, 1759–1771 (2013).
 45.
Sohn, S. S. et al. Ultrastrong mediumentropy singlephase alloys designed via severe lattice distortion. Adv. Mater. 31, 1807142 (2019).
 46.
Pinski, F. J. et al. Origins of compositional order in NiPt alloys. Phys. Rev. Lett. 66, 766–769 (1991).
 47.
Ceguerra, A. V. et al. Shortrange order in multicomponent materials. Acta Crystallogr. A 68, 547–560 (2012).
 48.
Zunger, A., Wei, S.H., Ferreira, L. G. & Bernard, J. E. Special quasirandom structures. Phys. Rev. Lett. 65, 353–356 (1990).
 49.
Song, H. et al. Local lattice distortion in highentropy alloys. Phys. Rev. Mater. 1, 023404 (2017).
 50.
Singh, R., Sharma, A., Singh, P., Balasubramanian, G. & Johnson, D. D. SCRAPs: a multicomponent alloy structure design tool; https://doi.org/10.24433/CO.0000024.v1
 51.
Singh, R., Sharma, A., Singh, P., Balasubramanian, G. & Johnson, D. D. HybridCS code for 1D test functions; https://doi.org/10.24433/CO.6419254.v1
Acknowledgements
R.S. was supported in part by D.D.J.’s F. Wendell Miller Professorship at ISU. Work at Ames Laboratory (by R.S., A.S., P.S. and D.D.J.) was funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science & Engineering Division. Ames Laboratory is operated for the US DOE by Iowa State University under contract no. DEAC0207CH11358. G.B. was funded by the National Science Foundation through award no. 1944040.
Author information
Affiliations
Contributions
D.D.J. proposed and supervised the project. R.S. wrote the SCRAPs generation code using the hybrid CS algorithm. R.S. and A.S. did initial testing. D.D.J. developed the linearscaling parallel algorithm and scaling analysis. P.S. and R.S. implemented SCRAPs optimization with parallelization and catalogged timings. P.S. completed DFT and phonon calculations, and performed analysis with D.D.J. P.S. got the code running on Code Ocean. R.S., A.S., P.S. and G.B. drafted the initial manuscript, then D.D.J. prepared the final manuscript with approval from all the authors.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Fernando Chirigati was the primary editor on this Article and managed its editorial process and peer review in collaboration with the rest of the editorial team. Nature Computational Science thanks the anonymous reviewers for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
About this article
Cite this article
Singh, R., Sharma, A., Singh, P. et al. Accelerating computational modeling and design of highentropy alloys. Nat Comput Sci 1, 54–61 (2021). https://doi.org/10.1038/s43588020000067
Received:
Accepted:
Published:
Issue Date:
Further reading

Pseudoelastic Deformation in Refractory (MoW) <sub>85</sub> Zr <sub>7.5</sub>(TaTi) <sub>7.5</sub> HighEntropy Alloy
SSRN Electronic Journal (2021)