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[Ivan Kurniawan](https://orcid.org/0000-0001-5419-0047), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452), [Kazuhiro Hono](https://orcid.org/0000-0001-7367-0193)

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[Machine learning study of highly spin-polarized Heusler alloys at finite temperature](https://mdr.nims.go.jp/datasets/f6578f96-f9e1-4f8f-bed4-0597a386061e)

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Machine learning study of highly spin-polarized Heusler alloys at finite temperaturePHYSICAL REVIEW MATERIALS 6, L091402 (2022)LetterMachine learning study of highly spin-polarized Heusler alloys at finite temperatureIvan Kurniawan ,1,2 Yoshio Miura ,1,3,* and Kazuhiro Hono 1,21Research Center for Magnetic and Spintronics Materials, National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba 305-0047, Japan2Graduate School of Science and Technology, University of Tsukuba, Tsukuba 305-8577, Japan3Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University,Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan(Received 10 June 2022; accepted 31 August 2022; published 14 September 2022)A huge magnetoresistance (MR) ratio exceeding 2000% at cryogenic temperature that was reported forhalf-metallic Heusler alloy based magnetic tunnel junctions showed large degradation at room temperature,which impedes practical application of Heusler alloy based MR devices. This motivates us to explore alternativeHeusler alloys that show high spin polarization at finite temperatures. Here, we propose half-metallic Heusleralloys based on finite-temperature first-principles calculation via the disordered local moment method togetherwith machine learning. We found several prospective materials at room temperature such as Co2MnGa0.2As0.8and Co2FeAl0.4Sn0.6. We also investigated two combinatorial series, Co2MnGayAs1-y and Co2FeAlySn1-y, tounderstand the effect of alloy mixing on temperature dependence and found that Fermi level tuning significantlyimproved the spin polarization and its temperature dependence, especially in Co2FeAlySn1-y.DOI: 10.1103/PhysRevMaterials.6.L091402I. INTRODUCTIONMagnetoresistance (MR) in magnetic tunnel junctions(MTJs) is one of the most important phenomena for therealization of spintronics applications, such as magnetic ran-dom access memories [1] and magnetoresistive sensors [2].Recently, high-throughput calculations have been widely per-formed to obtain novel magnetic materials with high spinpolarization around Fermi level (so-called half-metals) [3–7],because the high spin polarization is crucial to obtainingsufficient magnetoresistance for the applications. In particu-lar, Heusler alloys, a family of A2BC compounds, have beenactively explored by machine learning due to the abundantvariety of atomic combinations and the relatively simple fab-rication process. However, the machine learning investigationof highly spin-polarized Heusler alloys with first-principlescalculations has been performed at zero temperature [7]. Thismade a significant discrepancy between physical propertiesdesigned by first-principles calculations and experimental re-sults of the predicted material.One of the most serious problems of half-metallic Heusleralloys in spintronic device application is the reduction ofmagnetoresistance at room temperature (the large tempera-ture dependence) in tunnel magnetoresistance (TMR) and thecurrent perpendicular to plane giant magnetoresistive (CPP-GMR) devices [8,9]. This means that simple ground-statecalculations of spin polarization at 0 K are not enough to pre-dict half-metallicity at ambient temperature. In our previouswork, we performed finite-temperature calculations of half-metallic electronic structures of Co-based full Heusler alloys*Corresponding author: miura.yoshio@nims.go.jpand found that the conduction bands of the Co d orbital sig-nificantly degrade the spin polarization at room temperature[10]. Experimentally, tuning the Fermi level via alloying ofdifferent elements, such as Co2Fe(Al0.5Si0.5) [11], is also im-portant to improve the large temperature dependence of spinpolarizations. These previous experimental and theoreticalstudies suggest that for realization of spintronics applications,machine learning with finite-temperature first-principles cal-culations is necessary to predict novel half-metallic Heusleralloys at room temperature.The purpose of the present work was to search foralternative half-metallic Heusler alloys having high spin po-larization at room temperature using machine learning andthe finite-temperature first-principles calculations. Due to themultielement character of Heusler alloys, machine learningis a suitable material informatics tool for multidimensionalanalysis. Recently, using a deep neural network, Hu et al.predicted >100 highly spin-polarized and six prospectivehalf-metal from >10 000 Heusler A2BC candidates at 0 K [7].In contrast to the previous work, we incorporate the finite-temperature effect via the disordered local moment (DLM)method to clarify the spin polarization of ternary, quater-nary, and quinary Heusler alloys at finite temperature [10,12].We successfully predict alternative quaternary Heusler al-loys on the basis of finite-temperature machine learning andfirst-principles calculations, such as Co2MnGa0.2As0.8 andCo2FeAl0.4Sn0.6, which show high spin polarization even atroom temperature.II. METHODWe prepare a list of candidates of Heusler compounds withthe general composition of A2(BxB′1−x )(CyC′1−y) (x and y are2475-9953/2022/6(9)/L091402(8) L091402-1 ©2022 American Physical Societyhttps://orcid.org/0000-0001-5419-0047https://orcid.org/0000-0002-5605-5452https://orcid.org/0000-0001-7367-0193http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevMaterials.6.L091402&domain=pdf&date_stamp=2022-09-14https://doi.org/10.1103/PhysRevMaterials.6.L091402KURNIAWAN, MIURA, AND HONO PHYSICAL REVIEW MATERIALS 6, L091402 (2022)coefficients from 0.0 to 1.0 with a 0.2 interval). Elementsfor A, B (B′) and C (C′) sites are taken from the distributionof components predicted by Hu et al. (A = Fe, Co, Ru, Rh;B, B′ = Sc, Ti, V, Cr, Mn, Fe, Y, Zr, Nb, Mo; C, C′ = Al,Si, P, Ga, Ge, As, In, Sn, Sb) [7]. These combinations leadto 73 440 different compositions. Each candidates assumesthe L21 structure, which has four atoms located on interpen-etrating fcc sublattices, consisting of two A atoms at (0.25,0.25, 0.25) and (0.75, 0.75, 0.75), the B (B′) atoms at (0.00,0.00, 0.00), and the C (C′) atoms at (0.50, 0.50, 0.50). Thelattice constants of ternary alloys were taken from the HeuslerDatabase of The University of Alabama [13] and the supple-mentary information from Hu et al. [7]. Using Vegard’s law[14], the lattice constants of quaternary and quinary Heuslercomposition were extrapolated.The computational procedures to calculate spin polar-ization at finite temperature are divided into two steps.First, spin-polarized electronic structure calculations at zerotemperature were performed by multiple-scattering Green’sfunction formalism in the Korringa-Kohn-Rostoker (KKR)method [15,16] implemented in the HUTSEPOT code [17]. Lo-cal spin density approximation (LSDA) by Perdew and Wang[18] was selected for the exchange-correlation functional anda 20×20×20 k mesh was used. The core and valence electronswere treated within the scalar-relativistic approximation andthe angular momentum expansion of scattering matrices ina basis of spherical harmonics is truncated at lmax = 3. Todeal with the atomic disorder, we used coherent potentialapproximation (CPA) [19]. The Kohn-Sham potential was de-termined using atomic-sphere approximation (ASA). Second,after obtaining the ground-state potential from the first step asthe “frozen” potential, we introduced the finite-temperatureeffect using the DLM method [20]. This effect assumes thatmagnetic compounds consist of local moments that fluctuateat finite temperature and have a longer stabilization timescalethan electron motion [20]. Therefore, the scalar-relativisticKohn-Sham-Dirac equation was solved non-self-consistentlywithin the ASA until it reached the convergence of the it-erative temperature value with the starting parameter of theWeiss field as the “average mean field.” Details of the tech-nical description of the DLM approach are given elsewhere[10,12,20].Total spd (total) spin polarization at zero and finite temper-ature is evaluated byPspd = D↑spd (EF) − D↓spd (EF)D↑spd (EF) + D↓spd (EF), (1)where D↑spd (EF) and D↓spd (EF) correspond to the total spddensity of states (DOS) on the Fermi level of majority and mi-nority spins, respectively. Despite the dependency of the MRratio on many factors such interface quality, defect, vacancy,lattice mismatch, etc., we only focus on the reduced bulk spspin polarization at finite temperature in this study. It has beendemonstrated that sp spin polarization has good agreementwith spin polarization extracted from the experimental TMRratio using Julliere’s model [21,22]. Furthermore, experimen-tal studies of CPP-GMR suggested that CPP-GMR ratios aremore properly explained by sp spin polarization due to thenegligible contribution of localized character d electrons tothe transport properties [23]. Therefore, we calculated sp spinpolarization, which is defined asPsp = D↑sp(EF) − D↓sp(EF)D↑sp(EF) + D↓sp(EF), (2)where D↑sp(EF) and D↓sp(EF) correspond to the sp DOS on theFermi level of majority and minority spin, respectively.Our main purpose was to find an alternative Heusler al-loy composition that retains the high spin polarization atfinite temperature. Due to the complex relation between theoutput (high spin polarization) and the input (composition),this problem can be assumed as a black-box function, andBayesian optimization is adopted to find its solution withhigh efficiency. In this work, the open-source PYTHON librarycalled COMBO [24] was employed to perform the optimizationprocess.Aside from the optimization method, there are other threerequired components for the machine learning process: thedescriptor, the evaluator, and the calculator [25]. We set thedescriptor to identify each composition A2(BxB′1−x )(CyC′1−y)as a concatenation of integer numbers following the ruleA2(BxB′1−x )(CxC′1−x ) → P|GA + P|GB + nx + P|GB′+ n1−x + P|GC + ny + P|GC′ (3)where P|G is a set of the number of period and group in theperiodic table to identify the element (A, B, B′, C, C′) andn indicates the stoichiometry coefficient of B, B′, C, and C′.For example, Co(Si) is in the fourth (third) period and ninth(14th) group; thus we express the P|GCo = 4|9 ⇒ 49 (P|GSi =3/14 ⇒ 314), respectively. The stoichiometry coefficient ofnx and ny were described as follows, x = 0.2 ⇒ nx = 02,and y = 1.0 ⇒ ny = 10, and so on. If there is no B′(C′)element, we set P|GB′ ⇒ 00 (P|GC′ ⇒ 00) and n1−x = 00(n1−y = 00), respectively. By concatenating all of these pa-rameters, each composition will be uniquely described witha 20-digit integer descriptor. For example, the Heusler alloyRu2(Mn0.6Fe0.4)(P0.2Sb0.8) can be expressed by “58 47 06 4804 315 02 515 08” due to P|GRu = 58, P|GMn = 47, nx = 06,P|GFe = 48, n1−x = 04, P|GP = 315, ny = 02, P|GSb = 515,and n1−y = 08.The framework of the ab initio calculations and MaterialsInformatics (MI) procedures is illustrated in Fig. 1. First, wedefine the set of candidates [see Fig. 1(a)], and the first 20compositions were chosen randomly to train the Bayesianregression as the machine learning model [see Fig. 1(b)]. Theperformance of each candidate was evaluated with the scoredefined byscore = TconvPsp(Tconv). (4)The Tconv and Psp(Tconv) were the converged temperatureand the sp spin polarization at Tconv, which were obtainedbased on the DLM calculations at finite temperature [seeFig. 1(c)]. We adopted the spin polarization of sp states Pspat finite temperature as the evaluator (score) of the Bayesianoptimization, because the Psp is more suitable than the Pspd(total spin polarization) to describe the spin-dependent trans-port in TMR and CPP-GMR devices.The obtained score is used to estimate the black-box func-tion which is expensive to calculate. This estimation modelL091402-2MACHINE LEARNING STUDY OF HIGHLY … PHYSICAL REVIEW MATERIALS 6, L091402 (2022)FIG. 1. The schematic workflow for finding prospective candidates with highly spin-polarized Heusler alloys at finite temperature. (a)Set of candidates to be investigated in this study, (b) implemented Bayesian optimization procedure to find the prospective candidate, and(c) finite-temperature calculation to obtain the sp spin polarization at converged temperature T using starting parameter of Weiss field h.based on the Bayesian statistics will predict the next candi-date that needs to be evaluated. First, we randomly selected20 structures with respective descriptors for initial calcula-tions. The scores of these 20 structures are evaluated viafirst-principles calculation, and used to train the Bayesianregression model. Then, in every round the next ten candidatesare chosen by this estimation model and the correspondingscores of these ten candidates are added, repetitively, to im-prove the estimation model, until a fixed number of 2200candidates (220 rounds) are evaluated. In this study, the bestcandidates are chosen according to the Thompson samplingcriterion. Since Bayesian optimization is a widely used ma-chine learning framework, details are written elsewhere [24].III. RESULTS AND DISCUSSIONSIn Fig. 2, we show the score of the Bayesian optimiza-tion as a function of the number of calculated structuresfor the Psp of Heusler alloys, indicating the performance ofBayesian optimization. As shown in Fig. 2, after several iter-ation processes, Bayesian optimization converges to the bestcandidates with the largest score, which corresponds to theHeusler alloy compositions that retain high spin polarizationat finite temperature.In Table I, we show the top 30 compounds having highscores in the Bayesian optimization together with the latticeconstant a, the spin moment mtotal, total and sp spin polar-izations Pspd and Psp at the converged temperature Tconv, andalso the formation energy Eform, which are outputs of thefirst-principles calculations except for the lattice constants.Scores of (PspTconv) are used in predictions of black-box func-tion in Bayesian optimization. The formation energy Eform ofA2(BxB′1−x )(CyC′1−y) was calculated by the following equa-tion,Eform = E totalA2(BxB′1−x )(CyC′1−y ) − [2E totalA + xE totalB+ (1 − x)E totalB′ + yE totalC + (1 − y)E totalC′](5)FIG. 2. Bayesian optimization performance for the calculatedHeusler structures.L091402-3KURNIAWAN, MIURA, AND HONO PHYSICAL REVIEW MATERIALS 6, L091402 (2022)TABLE I. Summary of potential highly spin-polarized Heusler alloys at finite temperature suggested by Bayesian optimization.System a (Å) mtotal (μB) Tconv (K) Pspd (%) Psp(%) TconvPsp Eform(eV/f.u.)Co2Fe1.0Al1.0 5.700 4.86 352 39.0 86.7 30521 –1.71Co2Fe1.0Ga1.0 5.720 4.94 329 25.1 82.4 27134 –3.11Co2Fe1.0In1.0 5.980 5.10 305 11.4 79.8 24347 –3.38Co2Mn1.0As1.0 5.796 5.96 278 46.0 70.8 19684 –0.71Co2Mn1.0P1.0 5.638 5.67 223 4.6 77.1 17230 –1.79Co2Mn1.0Si1.0 5.630 5.00 299 79.0 88.0 26326 –2.44Co2Fe1.0Al0.4Sn0.6 5.876 5.29 334 –1.7 85.0 28419 –3.09Co2Fe1.0Ga0.2Ge0.8 5.736 5.32 280 –23.7 76.9 21485 –1.61Co2Fe1.0Ga0.2In0.8 5.928 5.06 310 15.3 80.4 24920 –3.18Co2Fe1.0Ga0.4Ge0.6 5.732 5.26 322 –7.4 82.4 26502 –2.00Co2Fe1.0Ga0.4In0.6 5.876 5.03 315 18.7 80.9 25476 –3.04Co2Fe1.0In0.4Sn0.6 5.988 5.40 324 –2.5 83.3 27009 –4.00Co2Mn1.0Al0.2As0.8 5.777 5.61 300 66.8 89.0 26681 –0.97Co2Mn1.0Al0.2Ge0.8 5.728 4.80 212 74.8 83.1 17589 –1.48Co2Mn1.0Al0.2Sb0.8 5.954 5.55 276 60.5 87.8 24250 –0.26Co2Mn1.0Al0.2Si0.8 5.644 4.80 237 80.5 88.2 20909 –2.29Co2Mn1.0Ga0.2As0.8 5.781 5.61 291 65.1 88.4 25711 –1.23Co2Mn1.0Ga0.2Ge0.8 5.731 4.81 208 73.3 82.2 17088 –1.75Co2Mn1.0Ga0.2Sb0.8 5.958 5.56 268 58.2 86.8 23254 –0.45Co2Mn1.0Ga0.4As0.6 5.765 5.20 264 71.2 86.4 22836 –1.73Co2Mn1.0Ga0.4Sb0.6 5.898 5.21 246 67.6 83.6 20525 –0.89Co2Ti0.2Mn0.8Ge1.0 5.756 4.39 216 66.7 88.0 18998 –1.62Co2Fe0.8Mo0.2As0.2Sb0.8 5.979 4.60 267 37.4 81.1 21680 0.60Co2Fe0.8Mo0.2In0.2Sb0.8 6.037 4.80 264 44.9 82.4 21737 –0.13Co2Fe0.8Mo0.2Sn0.4Sb0.6 6.039 4.78 265 47.0 83.2 22002 –1.27Co2Fe0.8Nb0.2In0.6Sb0.4 6.032 4.81 244 31.2 80.6 19692 –1.70Co2Mn0.8Fe0.2Ge0.4Sb0.6 5.902 5.51 230 –7.0 75.4 17302 –0.20Co2Mn0.8Fe0.2In0.4Sb0.6 6.001 5.40 270 54.4 83.5 22585 –1.40Co2Mn0.8Fe0.2Sn0.6Sb0.4 6.000 5.54 275 45.9 86.3 23748 –2.79Co2Mn0.8Nb0.2Ge0.2As0.8 5.773 4.90 226 74.9 89.0 20158 –0.63Co2Mn0.8Nb0.2Ge0.4As0.6 5.763 4.78 207 76.8 88.0 18251 –0.77Co2Mn0.8Zr0.2P0.2Ge0.8 5.760 4.50 203 37.8 83.1 16836 –1.22where E total is the total energy for each system for Heusleralloys A2(BxB′1−x )(CyC′1−y), and single elements A, B, B′, C,and C′ are calculated using first-principles calculation at zerotemperature.As shown in Table I, despite considering many elementsin the A and B sites, the majority of prospective candidateshave Co occupy the A site, to be specific, Co2Fe- and Co2Mn-based Heusler alloys which consist of three-, four-, andfive-element compounds. The magnetic moments of thesecandidates are pretty high, more than 4.5 μB, which leadsto converged temperature around 200–350 K using the samestarting parameter Weiss field. Interestingly, these candidateshave various value of Pspd spanned over the range –20%–80%,despite the relatively high Psp being more than 70%. Thisresult implies a conventional approach to find the highly spin-polarized material based on merely Pspd value excluding manyprospective candidates. Note that we also found that almostall potential candidates except Co2Fe0.8Mo0.2As0.2Sb0.8have negative formation energy, which confirms thethermodynamical stability of most of the proposedcompounds.In order to understand why the most prospective highlyspin-polarized Heusler alloys suggested by Bayesian op-timization are Co-based materials, we also evaluated theground-state properties of all candidates by high-throughputcalculations. Note that despite calculations of all candidates(73440 candidates in the system) were conducted, only 41612calculations (∼56%) converged and obtained the solution.In Figs. 3(a)–3(c), we show the distributions of the con-verged calculation results for ground-state Pspd and Psp valuesmapped on a two-dimensional plane over the A elements, thenumber of elements, and spin moments at 0 K, respectively.The conventional approach of finding highly spin-polarizedmaterial was done by “vertically scanning” over the candi-dates in the high-Pspd area (specified by sky blue rectangles).However, as mentioned previously, Psp is more suitable toexplain the magnetoresistance effect, so here we propose “hor-izontally scanning” over the high-Psp area (specified by redrectangles). Based on Fig. 3(a), it is clear that the high-Psparea is filled by the Co-based Heusler alloy. These candidatesmainly consist of quaternary and quinary compounds as aconsequence of the increase of the number of combinationswith increasing the number of elements [see Fig. 3(b)]. Itturned out that significant portions of these candidates alsoexhibit a large spin moment (4–6 μB) at 0 K [see Fig. 3(c)]which may imply high Curie temperature and therefore theL091402-4MACHINE LEARNING STUDY OF HIGHLY … PHYSICAL REVIEW MATERIALS 6, L091402 (2022)FIG. 3. The results of high-throughput calculation for ground-state properties of all candidates as (a) distribution over A element,(b) distribution over number of elements, and (c) distribution over 0 K moment.robustness of spin polarization of Co-based Heusler alloys atfinite temperature [26].In order to understand the importance of including thefinite-temperature effect, we picked some ternary compoundswith relatively large magnetic moments (3–6 μB) and demon-strated Psp > 70% at 0 K and plotted the Psp at finitetemperature in Fig. 4. We show Psp of Co2CrAl (3.00 μB),Fe2MnP (4.00 μB), and Ru2MnSb (4.00 μB) as representa-tive cases of the large temperature dependence of Psp.These materials show Psp larger than 70% and almost half-metallic electronic structures at 0 K. However, Psp of thesecompounds around 300 K is smaller than 50%. On the otherhand, Co2MnAs (5.98 μB) and Co2FeIn (5.09 μB) showslow decay of Psp with increasing temperature, preservinghigh spin polarization over 70% around 300 K. These re-sults indicate that the inclusion of the finite-temperatureeffect in the first-principles calculations is necessary to findprospective Heusler alloys with machine learning. Note thatin Fig. 4, one may find spin polarization for Co2CrAl andRu2MnSb practically reduced to zero at the correspond-ing calculated Curie temperature. Due to the mean-fieldapproximation, it is clear that the calculated Curie temper-ature is overestimated. However, we confirm that for theFIG. 4. The temperature dependence of Psp of some ternaryHeusler alloys that demonstrated large magnetic moment (3−6 μB)and spin polarization >70% at 0 K.Co-based Heusler alloys, calculated Curie temperature isproportional with both experimental Curie temperature andmagnetic moment (not shown here); therefore this DLMmethod is still adequate to discuss the finite-temperatureproperties.Then we systematically investigated the combinatorialsystem for four prospective candidates (Co2MnGa0.2As0.8,Co2MnAs, Co2FeAl0.4Sn0.6, and Co2FeAl) to understand therelation between electronic structure and temperature depen-dence of spin polarization. In Figs. 5(a)–5(f), we show thetemperature dependence of Pspd and Psp of Co2MnGayAs1-y,and the density of states (DOS) for total and sp states, andPspd and Psp around the Fermi level.Recently, Co2MnGa was reported to show giant anoma-lous Nernst effect in the L21 structure [27]. Meanwhile,theoretical calculations of Co2MnAs suggest a very largemagnetic moment of 6 μB [28]. However, both of these com-pounds and their combinatorial alloy are not widely exploredin terms of spin polarization and its application in magne-toresistance. Thus, Co2MnGa0.2As0.8 (and Co2FeAl0.4Sn0.6)obtained in this study by the Bayesian optimization withfinite-temperature calculations are materials having high Psparound room temperature. As shown in Fig. 5(a), despite thepretty high value of Pspd of Co2MnAs at 0 K, Ga dopingimproves it further. This can be explained by consideringthe Fermi level tuning from Co2MnAs (near the conduc-tion band edge) to the Co2MnGa (near the valence bandedge) [see Figs. 5(b) and 5(c)]. Similarly, the effect of Gadoping is also observed in Psp [see Figs. 5(d)–5(f)], whichsuggests Co2MnGa0.2As0.8 and Co2MnGa0.4As0.6 have su-perior spin polarization among the series. Note that despitethe significant shift of the Fermi level for y from 0.0 to0.8 for Co2MnGayAs1-y, the temperature dependence of Pspdoes not change depending on y. This is because significantincrease of minority spin states happens at the conductionband rather than the valence band edge. For y = 0.0, the Fermilevel is already quite distant from the conduction band edge,therefore shifting further from the conduction band edge asincreasing y does not affect the temperature dependence thatmuch. This phenomenon is also observed experimentally inCo2MnAlySi1-y with y < 0.4 by Sakuraba et al. [29].Figure 6(a) shows the temperature dependence of Pspdfor Co2FeAlySn1-y, (b) spd electronic structure at 0 K, (c)energy dependence of Pspd calculated at 0 K, and (d–(f) itsL091402-5KURNIAWAN, MIURA, AND HONO PHYSICAL REVIEW MATERIALS 6, L091402 (2022)FIG. 5. (a) The temperature dependence of Pspd on value of y from Co2MnGayAs1-y, (b) spd electronic structure at 0 K, (c) energydependence of Pspd calculated at 0 K, and (d)–(f) its counterpart for Psp and sp electronic structure. The reference of the energy E is theFermi energy.FIG. 6. (a) The temperature dependence of Pspd for Co2FeAlySn1-y, (b) spd electronic structure at 0 K, (c) energy dependence of Pspdcalculated at 0 K, and (d)–(f) its counterpart for Psp and sp electronic structure. The reference of the energy E is the Fermi energy.L091402-6MACHINE LEARNING STUDY OF HIGHLY … PHYSICAL REVIEW MATERIALS 6, L091402 (2022)counterpart for Psp and sp electronic structure. As shown inFigs. 6(a)–6(c), the Pspd value and spd electronic structure ofCo2FeAl over a significant temperature range fail to explainthe giant tunneling magnetoresistance (TMR) demonstratedespecially in the Co2FeAl/MgO-based MTJ [30]. It waswidely understood that giant TMR effect in Fe/MgO-basedMTJ is due to the spin-filtering effect of single-crystallineMgO in which �1 symmetry Bloch states at the in-planek-vector �k|| = (0, 0) mainly propagate for one spin channelonly because of the half-metallic character of the �1 bandin bcc Fe [31]. Due to the fact that s, pz, dz2 atomic orbitalsare compatible with the �1 symmetry, here we can roughlyapproximate the strength of the spin-filtering effect as spinpolarization considering the sp electron. That explains whythe TMR effect could be properly described by Psp behaviorinstead of the very low value of Pspd [see Figs. 6(a) and 6(d)]which is consistent with the previous studies on Heusleralloy/MgO-based MTJ [21,22]. Our calculation shows thatPsp exhibits very low temperature dependence but still retainsa high value of spin polarization for Co2FeAl which alsois consistent with estimated spin polarization from Julliere’smodel of TMR effect of Cr/Co2FeAl/MgO/CoFe MTJ [32].The sp electronic structure and energy dependence of spinpolarization at 0 K shown in Figs. 6(e) and 6(f) indicated thatshifting the Fermi level further enough from the conductionband edge will lead to lower temperature dependence. Mixingthe Al with Sn will shift the Fermi level position toward theconduction band edge, resulting in lower spin polarization andstronger temperature dependence for high Sn-content compo-sition. However, for 0.4 � y � 1.0, the high spin polarizationand small temperature dependence is still retained.In summary, we successfully performed Bayesian opti-mization combined with finite-temperature calculation to findthe highly spin-polarized Heusler alloys around room temper-ature. We found Co2MnGa0.2As0.8 and Co2FeAlySn1-y (0.4 �y � 1.0) can show high sp spin polarization at around 300 K.Our study emphasized the importance of Psp instead of thePspd value to explain the magnetoresistance effect, and thealloy mixing to find the more prospective candidate with afour- or five-element based compound. However, most sareCo-based Heusler alloys, which is supported by the resultsfrom high-throughput screening. We also investigated twocombinatorial series, Co2MnGayAs1-y and Co2FeAlySn1-y,to understand the effect of alloy mixing on the tempera-ture dependence and superiority of Co2MnGa0.2As0.8 andCo2FeAlySn1-y (0.4 � y � 1.0) compared to other Co-basedHeusler alloys. Co2FeAlySn1-y has good lattice matching withfcc Ag and is promising as a ferromagnetic electrode forCPP-GMR with Ag spacer. Furthermore, Co2MnGayAs1-y isexpected to be an effective spin injection source into GaAssemiconductors because it is insensitive to diffusion of Gaand As. These results confirmed the importance of distancingthe Fermi level position from the conduction band edge viaalloy mixing to improve the temperature dependence of spinpolarization.ACKNOWLEDGMENTSWe are grateful to S. Mitani, Y. Sakuraba, T. Tadano, andK. Masuda of NIMS for valuable discussions on this work.We thank J. B. Staunton at the University of Warwick andC. E. Patrick at the University of Oxford for support on theDLM method. I.K. acknowledges NIMS for the provision ofa NIMS junior research assistantship. This work was partlysupported by Grants-in-Aid for Scientific Research (GrantsNo. 17H06152, No. 20H02190, and No. 22H04966) fromthe Japan Society for the Promotion of Science, Center forSpintronics Research Network (CSRN) of Osaka University,the Cooperative Research Project Program of the ResearchInstitute of Electrical Communication in Tohoku Univer-sity, and the Japan Science and Technology Agency (JST)CREST (Grant No. JPMJCR21O1). 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