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[Heusler_Acta_SM.pdf](https://mdr.nims.go.jp/filesets/cf35ed92-8e47-4837-9950-78c74cc3f6b1/download)

## Creator

[Guangzong Xing](https://orcid.org/0000-0002-8299-8585), [Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Terumasa Tadano](https://orcid.org/0000-0002-8132-2161), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[Chemical-substitution-driven giant anomalous Hall and Nernst effects in magnetic cubic Heusler compounds](https://mdr.nims.go.jp/datasets/c36f1535-0641-432a-b800-8e96c8ca7aa7)

## Fulltext

”Chemical-substitution-driven giant anomalous Hall and Nernst effects inmagnetic cubic Heusler compounds”Guangzong Xing,1, ∗ Keisuke Masuda,1, 2, † Terumasa Tadano,1, 2 and Yoshio Miura1, 21Research Center for Magnetic and Spintronic Materials,National Institute for Materials Science, Tsukuba, Ibaraki, 305-0047, Japan2Digital Transformation Initiative Center for Magnetic Materials,National Institute for Materials Science, Tsukuba, Ibaraki, 305-0047, JapanA. Comparison of the band structureΓ XU|K Γ L W X−1.0−0.50.00.51.0ε−ε F(eV)Fe2ScAl0.9Si0.1DFTWannierFigure S1. Comparison of the band structure of Fe2ScAl0.9Si0.1 obtained based from DFT calculation (blackcurves) and Wannierization (red dots), respectively.B. Detailed parameters of WannierizationIn this study, we generate the Wannier functions using the selected columns of the densitymatrix (SCDM) method. Two pivotal parameters, namely µSCDM and σSCDM are automatically∗ XING.Guangzong@nims.go.jp† MASUDA.Keisuke@nims.go.jpmailto:XING.Guangzong@nims.go.jpmailto:MASUDA.Keisuke@nims.go.jp2determined through our developed Python scheme. Since the bands are entangled, we exclude 24semi-core states and proceed to extract the target Wannier functions from twice as many Blochstates. In the disentangle procedure, two energy windows (outer and inner) are applied. Wehave found that choosing the appropriate outer window is crucial to obtaining accurate Wannierfunctions. The initial lower (upper) bound ϵmax (ϵmin) of the outer window is chosen to be thelowest (highest) eigenvalue of the target states. In addition, the inner window is fixed to be[ϵmin : ϵF + 5] eV, where ϵF represents the Fermi energy. Subsequently, the energy differencedenoted as ∆ϵ (as detailed in the main text) is evaluated. If ∆ϵ > 2 meV, the Wannierization willbe repeated with an increase in the upper bound of the outer window with a step of 2 eV. Ultimately,it is worth noting that ∼98% of the investigated candidates meet the predefined criterion.C. Anomalous transport properties associated with the space group216 225Space group0100020003000|σxy|(Scm−1)(a)216 225Space group0.02.55.07.510.0|αxy|(Am−1K−1)(b)Figure S2. Calculated absolute values of anomalous Hall conductivity σxy (a) and anomalous Nernst con-ductivity αxy (b) for regular (space group Fm3̄m # 225) and inverse (space group F 4̄3m # 216) Heuslercompounds.D. Validity of the band-filling approachTo quantify the difference between the band-filling and the virtual crystal approximation (VCA)approaches, we introduced two metrics, ∆ITxy and ITxy . They are defined as follows:∆ITxy =∫ 0.1−0.1∣∣∣T stoichioxy (ϵ+ δ)− T substixy (ϵ)∣∣∣ dϵ (S1)3−0.2 0.0 0.2ε− εF (eV)500100015002000σxy(Scm−1)(a)(Co0.9Ni0.1)2MnGaBand fillingVCA−0.2 0.0 0.2ε− εF (eV)−3000−2000−10000(b)(Co0.9Ni0.1)2FeSn−0.2 0.0 0.2ε− εF (eV)−2000−1500−1000−50005001000(c)Rh2Co0.7Fe0.3InFigure S3. Comparison of energy-dependent σxy for (Co0.9Ni0.1)2MnGa (a) and (Co0.9Ni0.1)2FeSn (b), andRh2Co0.7Fe0.3In (c), obtained based on the VCA and band-filling approach, respectively.0 200 400∆Iσxy (eV S cm−1)100101102103Count(a)ZYX0 1 2∆Iαxy (eV A m−1 K−1)100101102103Count(b)22 23 24 25 26 27 28NXYv0100200300400I σxy(eVScm−1)(c)〈Iσxy〉〈∆Iσxy〉22 23 24 25 26 27 28NXYV0.00.51.01.5I αxy(eVAm−1K−1)(d)〈Iαxy〉〈∆Iαxy〉Figure S4. Histogram of ∆Iσxy(a) ∆Iαxy(b) with chemical substitution at X, Y , and Z sites, respectively.Calculated Iσxy (c) and Iαxy (d) with respect to the number of valence electrons at Y and Y sites NXYv .The red and blue lines show the average values of ∆ITxy and ITxy (see text).ITxy =∫ 0.1−0.1∣∣∣T substixy (ϵ)∣∣∣ dϵ (S2)4where ∆ITxy quantifies the discrepancy in the energy-dependent curves of transport quantities (referto the gray area shown in Fig. S3 for σxy), with Txy representing σxy or αxy, between the VCAand band-filling approaches within and energy range of [−0.1:0.1] eV around the Fermi energy.ITxy represents the integrated magnitude of the transport quantities around the Fermi energy. Theenergy shift δ, due to chemical substitution, is determined within the stoichiometric compoundsusing the band-filling approach.As shown in Fig. S4(a) and (b), the ∆Iσxy and ∆Iαxy values tend to be small when the substi-tution occurs at the Z site, to which the band-filling approach potentially remains applicable. Bycontrast, a substitution at the X or Y site tends to give larger ∆Iσxy and ∆Iαxy values. This isexpected because the density of states for p states near the Fermi energy from the Z site is muchsmaller than that for d states from the X and Y sites.In the following, we focus on the cases where a substitution occurs either at the X or Y siteand show how ∆Iσxy and ∆Iαxy behave with the total number of valence electrons at these sites,NXYv . Figures S4(c) and (d) present the scatter plots of the Iσxy and Iαxy (green circles), andthe average values of ITxy (blue circles) and ∆ITxy (red circles) as a function of NXYv . Here, wehave excluded candidates with NXYv ≤ 22, as the values of σxy and αxy are small in that region.The non-zero values of ⟨∆ITxy⟩ for all candidates, which are observed across all NXYv , clearlyhighlight the discrepancies in the anomalous transport quantities predicted by the band-filling andthe VCA approaches. In particular, the highest values of ⟨∆ITxy⟩ were observed at NXYv ∼ 26.5,as shown in Figs. S4(c) and (d), indicating complete failure of the band-filling approach in thisregion. Moreover, unlike the singular peak value of ⟨∆Iσxy⟩, at NXYv ∼ 26.5, multiple peak valuesof ⟨∆Iαxy⟩ were observed (refer to Fig. S4(d) for NXYv ∈ [24:27]), making the band-filling approachhardly justifiable for predicting αxy.The non-zero values of ⟨∆ITxy⟩ alone already demonstrate the limitations of the band-fillingapproach. However, since our study primarily targets chemically substituted candidates exhibit-ing significant anomalous transport quantities, the ⟨ITxy⟩ values are also relevant. As shown inFigs. S4(c) and (d), notable ⟨ITxy⟩ were observed with NXYv values around 23, 25 and 26.5. Thisobservation suggests a strategic emphasis on the candidates with these NXYv values to achievesubstantial σxy and αxy values. Particularly noteworthy are the simultaneously large ⟨ITxy⟩ and⟨∆ITxy⟩ with NXYv around 26.5, indicating promising candidates not predictable by the band-fillingapproach. For example, the VCA predicts significant σxy values for (Co0.8Ni0.2)2FeZ (Z = Sn, Ge)5at NXYv = 26.4—values unattainable via the band-filling approach. This discrepancy further un-derscores the VCA’s importance in accurately calculating anomalous transport properties.E. Anomalous transport properties in Fe- and Co-based Heusler alloysAs emphasized in the main text the substantial αxy shown in Fig. S6(d) can be distinctlyattributed to two significant Berry curvatures (BC) depicted in Fig. S6(a), found at 0.08 eV abovethe Fermi energy along the X–L–W high-symmetry path. Here, we discuss the principal factorcontributing to the relatively modest αxy observed in the stoichiometric compound. The finiteBC in the stoichiometric compound, illustrated by the green curve in Fig. S6(b), is only foundalong the L–W high-symmetry path at 0.17 eV above ϵF. As a result, a relatively small ∂σxy(ϵ)/∂ϵemerges within the energy range of [0.1 : 0.17] eV above the Fermi energy. This gives rise to alocal minimum of −310.49 S cm−1 [refer to the asterisk in Fig. S6(c)]. and consequently leads toa smaller local maximum αxy of 4.24 Am−1K−1, located at 0.13 eV above ϵF. [indicated by theasterisk in Fig. S6(d)].Furthermore, an additional local minimum of −353.61 S cm−1 at 0.3 eV above ϵF for the stoi-chiometric compound predominantly originates from substantial BC along the X–L high-symmetrypath, as indicated by the red curve in Fig. S6(b). In essence, the reason for the observed smaller−0.2 −0.1 0.0ε− εF (eV)Figure S5. Nodal line network of Fe2Mn0.7Fe0.3P formed by two majority spin bands (see main text) thatgenerate large σxy.6−0.4 −0.2 0.0 0.2 0.4ε − εF (eV)−1000010002000σxy(Scm−1 )Fe2Mn1−xFexPx = 0.0x = 0.3−0.4 −0.2 0.0 0.2 0.4ε − εF (eV)−50510αxy(Am−1 K−1 )0.00.2ε−ε F(eV)X L W−3000−2000−10000−Ωz k(Å2 )Fe2Mn0.7Fe0.3P0.00.20.4ε−ε F(eV)X L W−3000−2000−10000−Ωz k(Å2 )Fe2MnP**(a) (b)(c) (d)Figure S6. (a) Berry curvature of Fe2Mn0.7Fe0.3P (a) and Fe2MnP (b) along the high-symmetry path. TheBerry curvature for Fe2Mn0.7Fe0.3P is evaluated at 0.08 eV above the Fermi energy, while for Fe2MnP it isevaluated at 0.17 (red) and 0.3 eV (blue) above the Fermi energy, respectively. Energy-dependent σxy (c)and αxy (d) of Fe2Mn1−xFexP with x = 0, and 0.3, respectively.−0.4 −0.2 0.0 0.2 0.4ε− εF (eV)−3000−2000−100001000σxy(Scm−1)(a)(Co1−xNix)2FeSnx = 0.0x = 0.1x = 0.2x = 0.3−0.4 −0.2 0.0 0.2 0.4ε− εF (eV)−50510αxy(Am−1K−1)(b)Figure S7. Energy-dependent σxy (a) and αxy (b) of (Co1−xNix)2FeSn with x = 0, 0.1, 0.2, and 0.3,respectively.local maximum αxy value observed at 0.13 eV above ϵF in Fe2MnP can be attributed to thedivergence of anti-crossing points into two distinct energy levels. This divergence hinders the si-multaneous generation of significant BC. In contrast, the simultaneous and substantial BC observed7in Fig. S6(a) lead to a noteworthy αmaxxy of 8.12 Am−1K−1 at 0.04 eV above ϵF. Additionally, theαmaxxy of 5.32 Am−1K−1 was observed at 0.03 eV above the Fermi energy, which is attributed tothe slope found around the Fermi energy in Fe2MnP.F. Optimum anomalous transport properties through chemical substitutionTable S1. Co-based doped candidates with promising σxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)](Co0.8Ni0.2)2FeSn 225 5.33 −2567.78 8.27 −3224.09 (0.02) 11.99 (−0.03)(Co0.8Ni0.2)2FeGe 225 5.31 −1867.45 8.20 −3152.20 (0.02) 10.90 (−0.03)Co2Mn0.9Fe0.1Ga 225 4.20 1701.59 2.43 1701.59 (0.00) 5.15 (0.04)(Co0.7Ni0.3)2FeSi 225 4.83 −1418.63 2.50 −2036.72 (0.19) 7.93 (−0.06)Co2MnAl0.9Si0.1 225 4.08 1271.53 0.24 1618.76 (−0.01) 3.69 (0.09)Co2Mn0.9Cr0.1In 225 4.53 1132.95 3.60 1546.55 (−0.28) 4.09 (−0.27)(Co0.7Ni0.3)2MnP 225 5.48 −1130.86 4.02 −1328.90 (0.01) 8.44 (−0.05)(Co0.9Fe0.1)2CrGa 225 2.82 1069.31 1.86 1246.06 (−0.01) −5.01 (−0.28)(Co0.8Fe0.2)2CrAl 225 2.57 1018.37 0.75 1777.19 (−0.08) −4.21 (−0.24)(Co0.7Ni0.3)2MnSb 225 5.50 −938.50 −0.05 −1260.77 (−0.02) 5.80 (−0.09)Co3Ga 225 4.27 747.47 2.04 892.29 (−0.02) 3.36 (0.05)Co3Al 225 4.21 714.23 1.52 1001.91 (0.27) −3.34 (0.23)Co3In 225 4.42 633.37 1.47 800.33 (0.26) 3.03 (0.29)(Co0.8Fe0.2)3Ge 225 4.63 473.19 1.38 658.98 (−0.08) −3.45 (−0.18)(Co0.8Fe0.2)3Sn 225 4.79 470.19 0.84 570.95 (0.30) 4.82 (0.30)(Co0.8Fe0.2)2NiAl 216 3.71 456.14 0.82 1151.49 (0.22) −4.30 (0.17)8Table S2. Fe-based doped candidates with promising σxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)](Fe0.8Mn0.2)2MnSn 225 6.50 1598.79 0.76 1623.15 (−0.01) 4.04 (0.06)(Fe0.9Mn0.1)2CoIn 225 7.08 1583.29 2.47 1646.07 (−0.01) 7.27 (0.07)Fe2ScIn0.8Sn0.2 225 3.98 1473.78 −4.17 1769.44 (0.13) 5.63 (0.16)(Fe0.8Mn0.2)3Sn 225 7.15 1472.18 3.30 1787.16 (−0.03) 5.01 (0.04)Fe2MnAs0.8Ge0.2 225 3.78 1431.72 −0.71 1431.72 (0.00) 5.45 (0.08)Fe2Mn0.7Cr0.3Sb 225 3.68 1387.34 −0.14 1398.77 (0.01) 4.88 (0.07)Fe2MnP0.9Si0.1 225 3.87 1379.60 0.99 1389.77 (−0.01) 5.27 (0.07)Fe2Co0.9Ni0.1Si 225 4.86 1042.26 −1.16 1125.41 (0.03) −3.87 (−0.07)(Fe0.7Mn0.3)2CoGa 225 6.91 1035.39 1.84 1143.59 (−0.01) 5.15 (0.08)(Fe0.8Mn0.2)3Al 225 6.68 997.30 −0.15 1156.80 (−0.27) 4.15 (−0.30)Fe2Cr0.8Mn0.2Sn 225 4.69 928.93 1.14 1654.31 (−0.19) −6.19 (−0.24)Fe2Co0.9Fe0.1In 216 6.28 772.06 1.57 1493.68 (−0.24) 4.77 (−0.20)(Fe0.7Co0.3)2MnSi 225 3.57 701.82 −5.05 1634.78 (−0.24) 7.12 (0.21)Fe2Co0.7Fe0.3Ga 216 5.83 618.72 1.55 740.52 (−0.22) 2.80 (−0.29)(Fe0.7Mn0.3)2NiIn 216 5.66 576.99 1.53 1171.76 (−0.03) 1.98 (0.03)(Fe0.7Co0.3)3As 225 5.97 560.68 2.79 704.92 (−0.01) 3.32 (0.02)Fe2VSi0.7P0.3 225 1.11 550.52 1.57 713.11 (−0.04) 2.63 (0.04)(Fe0.7Co0.3)2CrGa 225 1.60 493.52 −2.94 1536.44 (0.08) −2.94 (0.00)(Fe0.8Co0.2)2CrAl 225 1.39 413.74 −2.65 1730.62 (0.11) −4.02 (0.06)(Fe0.8Co0.2)2ScAl 225 1.53 −397.96 −1.97 −691.30 (−0.13) 4.67 (−0.24)(Fe0.8Co0.2)2CoP 216 5.74 391.97 −0.86 −623.42 (0.29) 4.45 (0.12)(Fe0.8Mn0.2)3Si 225 5.32 356.54 3.28 1524.39 (−0.11) 5.77 (−0.06)(Fe0.7Co0.3)2ScGa 225 1.33 297.87 −1.09 −571.70 (−0.25) 6.64 (−0.30)9Table S3. Mn-based doped candidates with promising σxy at the Fermi energy. Contents in the table arethe space group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermienergy, and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermilevel. ∆ϵ denotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)]Mn2Ti0.9V0.1Ga 225 2.90 1448.19 1.20 1448.19 (0.00) 3.99 (0.04)Mn2Fe0.8Mn0.2Ga 225 7.20 1396.36 −0.60 1586.36 (−0.02) −3.94 (−0.05)Mn2CoGe0.9As0.1 225 7.64 1365.31 −0.47 1369.37 (0.01) 3.99 (0.06)(Mn0.8Cr0.2)2ScGe 225 3.38 1161.17 0.10 1562.43 (−0.26) −3.81 (−0.06)Mn2Ti0.9V0.1Al 225 2.90 1044.91 2.37 1544.45 (−0.01) 4.40 (0.03)(Mn0.7Fe0.3)2NiAl 225 6.79 989.08 −2.21 1463.94 (0.04) −4.04 (−0.06)(Mn0.7Fe0.3)2ScAl 225 3.87 946.70 −4.16 1304.23 (0.07) 5.81 (0.18)(Mn0.7Cr0.3)2ScIn 225 6.46 813.71 1.41 1104.70 (0.06) −3.25 (−0.12)Mn2Ti0.8Sc0.2In 225 3.46 785.57 −1.22 966.26 (0.08) −1.88 (−0.05)(Mn0.7Cr0.3)2ScSi 225 3.58 692.26 −3.34 1242.33 (0.03) −3.87 (−0.02)(Mn0.8Cr0.2)3Si 225 0.38 −626.83 −1.38 −885.94 (−0.05) −3.07 (0.07)Mn2Ti0.8Sc0.2Ge 225 2.18 385.65 0.93 813.23 (−0.30) 3.49 (−0.25)Mn2Ni0.7Co0.3In 225 8.22 −323.33 1.59 −357.25 (0.02) 2.33 (−0.03)10Table S4. Rh-based doped candidates with promising σxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)](Rh0.8Ru0.2)2MnIn 225 3.95 1950.49 −2.14 1950.49 (0.00) −6.67 (−0.05)(Rh0.9Ru0.1)2MnGa 225 3.94 1894.95 −3.98 2148.66 (0.04) −7.09 (−0.07)Rh2MnAl 225 4.11 1742.50 2.47 2027.10 (−0.04) 5.97 (0.09)Rh2Cr0.9Mn0.1In 225 3.27 1344.07 0.95 1529.13 (−0.12) −5.51 (−0.17)(Rh0.9Ru0.1)2CrGa 225 2.83 1316.51 −0.89 1316.51 (0.00) −3.89 (−0.06)Rh2CrAl 225 3.01 1285.60 1.63 1794.87 (−0.03) −5.15 (−0.29)Rh2NiSn0.8Sb0.2 225 0.99 −1104.01 6.15 −1722.29 (0.11) 7.91 (−0.06)Rh2NiSi0.7P0.3 225 0.92 −770.92 8.48 −1595.03 (0.06) 8.79 (−0.02)Rh2FeAl0.8Si0.2 225 4.19 764.20 0.37 863.80 (−0.01) 4.83 (0.17)(Rh0.7Ru0.3)2CoIn 225 3.50 670.14 3.62 −1060.04 (0.11) 6.13 (0.05)(Rh0.7Ru0.3)2FeGe 225 4.20 623.32 3.01 −996.44 (0.02) 8.67 (0.29)Rh2Fe0.8Co0.2In 225 4.06 −595.57 5.10 −1252.68 (0.05) 5.12 (0.01)(Rh0.9Ru0.1)2FeGa 225 4.47 511.23 −1.64 511.23 (0.00) 4.60 (0.28)(Rh0.7Ru0.3)2CoGa 225 3.39 505.69 1.91 1074.60 (0.03) 5.27 (0.06)Rh2CoAl0.7Si0.3 225 2.90 435.46 −1.62 506.76 (−0.29) 3.62 (0.23)(Rh0.9Ru0.1)2NiGa 225 2.01 420.61 0.81 953.41 (−0.30) 4.59 (0.29)(Rh0.9Ru0.1)2NiAl 225 1.98 363.69 1.25 1085.13 (−0.29) 4.32 (−0.27)11Table S5. Ru-based doped candidates with promising σxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)](Ru0.8Rh0.2)2FeP 225 4.21 1436.35 −1.36 1436.35 (0.00) 4.41 (0.11)(Ru0.8Rh0.2)2CoAl 225 2.91 1426.00 1.48 2219.34 (−0.07) 7.33 (0.09)Ru2Co0.7Fe0.3Ge 225 3.15 1370.43 2.54 1370.43 (0.00) −6.49 (−0.18)(Ru0.7Rh0.3)2FeSb 225 4.36 1146.57 −1.02 1241.76 (0.08) 6.54 (0.15)Ru2Mn0.9Cr0.1Sb 225 3.93 1140.69 −1.12 1161.79 (0.01) −4.66 (−0.07)(Ru0.7Rh0.3)2CrSb 225 3.61 1133.04 −3.74 2032.25 (−0.07) 4.76 (0.26)Ru2Mn0.9Fe0.1P 225 4.04 1086.88 0.47 1086.88 (0.00) −4.59 (−0.11)(Ru0.9Rh0.1)2MnAs 225 4.13 1069.22 3.04 1100.74 (−0.04) −5.08 (−0.13)(Ru0.8Rh0.2)2FeAs 225 4.29 1044.04 −2.77 1381.84 (0.07) 3.99 (0.12)Ru2FeSi 225 3.93 1014.70 1.33 1014.70 (0.00) 3.43 (0.06)Ru2Co0.7Ni0.3P 225 1.84 989.33 1.16 1026.25 (−0.03) −4.83 (−0.13)Ru2Fe0.9Mn0.1Ge 225 3.94 985.44 0.84 985.44 (0.00) −3.87 (−0.12)Ru2Fe0.7Mn0.3Sn 225 3.80 889.24 −1.00 946.16 (0.01) −3.73 (−0.07)(Ru0.7Rh0.3)2FeGa 225 3.72 884.07 −2.44 1099.03 (0.05) 3.64 (0.11)(Ru0.7Rh0.3)2FeAl 225 3.67 859.74 −2.34 1184.26 (0.07) 4.13 (0.14)(Ru0.8Rh0.2)2CoSi 225 2.66 814.11 2.68 1070.31 (−0.02) 3.84 (0.04)(Ru0.7Rh0.3)2MnSn 225 3.70 758.11 −4.56 1286.89 (0.07) −4.61 (−0.01)Ru2CrSn0.7In0.3 225 1.87 −666.38 0.45 −666.38 (0.00) 2.93 (−0.05)(Ru0.8Rh0.2)2CrP 225 3.36 457.30 −1.55 1306.74 (0.11) 3.90 (0.29)(Ru0.7Rh0.3)2MnGe 225 3.64 430.01 −3.63 1322.52 (0.12) −3.80 (0.02)(Ru0.9Rh0.1)2CrAs 225 3.12 408.33 −0.40 1313.59 (0.18) −5.18 (0.13)(Ru0.7Rh0.3)2VSi 225 1.54 378.21 −0.92 1123.37 (0.21) −2.02 (0.06)12Table S6. Cr-based doped candidates with promising σxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)]Cr2Ti0.7Sc0.3As 225 3.26 951.99 0.15 951.99 (0.00) −2.18 (−0.05)(Cr0.7Mn0.3)2TiSb 225 2.40 853.77 −0.23 853.77 (0.00) 2.81 (−0.14)(Cr0.7V0.3)2ScSi 225 5.17 613.27 −1.22 613.27 (0.00) −2.17 (−0.06)(Cr0.8Mn0.2)2ScP 225 3.60 559.48 −1.88 1297.23 (0.03) 2.75 (0.08)Cr2MnGe0.8Ga0.2 225 4.80 544.09 −0.02 544.09 (0.00) 1.77 (0.26)Cr2Fe0.7Mn0.3P 216 0.70 −504.21 −1.75 −508.96 (0.01) −2.93 (0.04)Cr2Ti0.7Sc0.3Si 225 4.01 400.80 −0.66 −979.00 (−0.27) −2.20 (−0.05)(Cr0.8V0.2)2TiGe 225 3.51 257.39 0.46 −518.83 (−0.18) 3.05 (−0.24)Table S7. Ni-based doped candidates with promising σxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)](Ni0.8Co0.2)2MnIn 225 4.80 −456.14 −0.85 −774.35 (−0.04) 1.63 (−0.10)(Ni0.8Co0.2)2FeSb 225 3.47 −396.06 0.34 −508.19 (0.04) 1.95 (0.29)(Ni0.9Co0.1)2MnAl 225 4.43 −318.29 0.53 −398.26 (−0.02) 2.40 (−0.07)Table S8. V-based doped candidates with promising σxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (S cm−1) (Am−1K−1) (S cm−1) [(eV)] (Am−1K−1) [(eV)]V2Cr0.7V0.3Al 225 0.54 352.33 −1.08 −1090.72 (−0.17) −5.68 (−0.12)(V0.7Cr0.3)2TiSb 225 3.12 234.24 −0.40 1255.83 (0.18) 2.22 (0.22)13Table S9. Co-based doped candidates with promising αxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)](Co0.7Ni0.3)3Sn 225 3.15 −30.66 8.56 −2614.20 (0.06) 11.48 (0.03)(Co0.9Ni0.1)2FeSn 225 5.59 −158.04 8.31 −3351.26 (0.10) 12.65 (0.04)(Co0.8Ni0.2)2FeGe 225 5.31 −1867.45 8.20 −3152.20 (0.02) 10.90 (−0.03)(Co0.8Ni0.2)2MnP 225 5.84 86.84 7.36 −1548.89 (0.12) 8.41 (0.03)(Co0.7Ni0.3)3Ge 225 3.03 156.11 6.23 −1477.30 (0.18) 7.43 (0.03)(Co0.8Ni0.2)2FeSi 225 5.12 −605.89 5.80 −1944.35 (0.28) 6.55 (−0.02)(Co0.8Ni0.2)2MnSb 225 5.82 −546.77 5.41 −1217.02 (0.04) 5.43 (−0.01)Co2MnGa0.8Ge0.2 225 4.24 831.12 4.66 1368.59 (−0.03) 4.68 (0.01)Co2MnAl0.8Si0.2 225 4.17 950.80 4.20 1454.80 (−0.07) 4.41 (0.02)Co2MnIn 225 4.76 447.94 3.87 1924.80 (−0.24) 5.03 (−0.06)(Co0.9Ni0.1)3Ga 225 3.98 435.88 3.59 961.37 (−0.09) 3.67 (0.02)(Co0.9Ni0.1)3Al 225 3.93 368.36 3.57 1562.10 (0.19) −4.18 (0.15)(Co0.7Fe0.3)2CrGa 225 2.41 701.27 3.36 2043.27 (−0.04) −8.17 (−0.14)(Co0.9Fe0.1)2CrAl 225 2.77 629.76 3.02 1168.69 (−0.01) 3.55 (0.02)(Co0.9Ni0.1)3In 225 4.09 308.08 2.73 724.61 (−0.08) 6.27 (0.29)(Co0.9Fe0.1)2NiAl 216 3.46 215.79 1.99 1234.34 (0.16) −3.75 (0.12)14Table S10. Fe-based doped candidates with promising αxy at the Fermi energy. Contents in the table arethe space group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermienergy, and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermilevel. ∆ϵ denotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)]Fe2CoIn0.9Sn0.1 225 6.82 802.75 6.81 1815.55 (0.27) 6.81 (0.00)Fe2Mn0.7Fe0.3P 225 4.30 745.39 6.08 1400.34 (−0.12) 8.19 (0.04)(Fe0.8Mn0.2)2MnAs 225 3.57 1207.02 −5.99 1408.86 (0.06) 7.62 (0.23)(Fe0.7Co0.3)2MnSi 225 3.57 701.82 −5.05 1634.78 (−0.24) 7.12 (0.21)(Fe0.7Co0.3)2CrAl 225 1.59 413.14 −4.90 1695.66 (0.08) −5.40 (0.03)Fe2Mn0.9Cr0.1Sn 225 5.94 929.58 4.70 2006.10 (−0.09) 5.16 (−0.02)(Fe0.9Mn0.1)3Sn 225 6.95 715.93 4.64 1849.10 (−0.09) 5.05 (−0.02)Fe2Co0.8Ni0.2Ga 225 6.18 212.84 −4.59 1700.37 (0.23) 6.00 (−0.11)(Fe0.8Mn0.2)2CoSi 225 5.10 330.44 −4.59 1162.46 (0.10) −4.68 (0.01)Fe2ScIn0.9Sn0.1 225 4.01 654.43 −4.58 1437.36 (0.12) 4.85 (0.17)(Fe0.7Mn0.3)3Al 225 6.83 480.17 −3.70 2170.11 (−0.29) 6.16 (−0.25)(Fe0.7Co0.3)2CoP 216 5.60 273.62 3.65 −848.66 (0.28) 4.14 (0.02)(Fe0.7Co0.3)2ScAl 225 1.30 −78.92 −3.61 −893.14 (−0.19) 5.61 (−0.29)(Fe0.8Co0.2)2CrGa 225 0.73 316.34 −3.53 868.19 (0.04) 4.18 (0.18)(Fe0.8Mn0.2)3Si 225 5.32 356.54 3.28 1524.39 (−0.11) 5.77 (−0.06)Fe2Cr0.7Mn0.3Sn 225 4.85 910.52 3.16 2103.23 (−0.22) −8.33 (−0.25)(Fe0.7Mn0.3)2MnSb 225 5.42 473.06 2.94 2309.91 (−0.22) 6.54 (−0.14)Fe2CoIn0.9Sn0.1 216 6.14 301.25 2.89 1122.48 (−0.29) 4.20 (−0.27)(Fe0.7Co0.3)3As 225 5.97 560.68 2.79 704.92 (−0.01) 3.32 (0.02)(Fe0.7Mn0.3)2ScGa 225 2.65 287.33 2.54 1473.84 (−0.15) 4.70 (−0.09)Fe2Ni0.7Co0.3In 216 5.53 186.71 2.44 1006.76 (−0.10) 3.02 (−0.03)Fe2Co0.8Fe0.2Ga 216 5.75 425.83 2.20 753.58 (−0.29) 2.20 (0.00)(Fe0.7Co0.3)2VSi 225 1.47 −5.26 1.99 621.25 (−0.08) 2.50 (−0.03)15Table S11. Mn-based doped candidates with promising αxy at the Fermi energy. Contents in the table arethe space group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermienergy, and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermilevel. ∆ϵ denotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)](Mn0.8Cr0.2)2FeGa 225 7.03 463.05 −4.94 2130.25 (0.08) 4.98 (0.13)(Mn0.9Fe0.1)2TiGa 225 2.82 400.87 4.71 1431.43 (−0.06) 4.81 (−0.01)(Mn0.7Fe0.3)2CoGe 225 7.37 508.27 4.55 1315.63 (−0.07) 4.55 (0.00)(Mn0.9Fe0.1)2TiAl 225 2.81 121.81 4.39 1711.06 (−0.07) 4.83 (−0.02)(Mn0.7Fe0.3)2ScAl 225 3.87 946.70 −4.16 1304.23 (0.07) 5.81 (0.18)(Mn0.7Cr0.3)2ScSi 225 3.58 692.26 −3.34 1242.33 (0.03) −3.87 (−0.02)Mn2NiAl0.7Si0.3 225 7.05 −52.13 −3.29 900.54 (0.09) −3.64 (0.02)(Mn0.7Cr0.3)2ScGe 225 3.58 494.56 −3.26 1142.28 (0.06) −3.35 (−0.01)(Mn0.9Cr0.1)3Si 225 0.68 −248.65 −2.88 −966.70 (−0.15) −3.44 (−0.03)(Mn0.8Cr0.2)2ScIn 225 6.36 440.14 2.86 1006.45 (−0.10) −2.95 (−0.15)Mn2Ti0.8V0.2In 225 3.78 375.80 2.27 828.78 (−0.14) 2.41 (−0.02)Mn2Ni0.7Co0.3In 225 8.22 −323.33 1.59 −357.25 (0.02) 2.33 (−0.03)Mn2TiGe 225 1.98 41.97 1.52 −378.26 (0.08) 3.79 (−0.29)16Table S12. Rh-based doped candidates with promising αxy at the Fermi energy. Contents in the table arethe space group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermienergy, and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermilevel. ∆ϵ denotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)]Rh2Co0.7Fe0.3In 225 3.43 394.57 8.57 −1649.50 (0.10) 9.74 (0.03)Rh2NiSi0.7P0.3 225 0.92 −770.92 8.48 −1595.03 (0.06) 8.79 (−0.02)Rh2NiSn 225 1.15 −545.48 7.56 −1728.71 (0.14) 7.56 (0.00)Rh2Co0.7Fe0.3Ga 225 3.44 284.36 6.96 −1844.04 (0.10) 9.35 (0.04)Rh2MnGa0.8Ge0.2 225 4.31 1012.05 6.47 2374.07 (−0.13) −8.67 (−0.22)Rh2Co0.7Fe0.3Al 225 3.47 344.62 6.29 −1778.14 (0.11) 9.67 (0.05)Rh2MnAl0.8Si0.2 225 4.28 1178.80 5.95 2187.48 (−0.12) 5.95 (0.00)Rh2Mn0.9Cr0.1In 225 4.29 1413.77 5.87 1985.98 (−0.05) −6.72 (−0.13)(Rh0.7Ru0.3)2CrIn 225 2.51 123.31 −5.74 1295.38 (0.04) −5.87 (0.01)Rh2Fe0.8Co0.2In 225 4.06 −595.57 5.10 −1252.68 (0.05) 5.12 (0.01)(Rh0.8Ru0.2)2CrAl 225 2.61 775.06 −4.11 1384.58 (0.04) −4.12 (−0.01)(Rh0.7Ru0.3)2CrGa 225 2.43 300.99 −3.80 1446.90 (0.23) −4.12 (0.02)Rh2FeGa0.7Ge0.3 225 4.08 50.33 3.30 −907.58 (0.07) 3.93 (0.29)(Rh0.7Ru0.3)2FeGe 225 4.20 623.32 3.01 −996.44 (0.02) 8.67 (0.29)Rh2Fe0.9Mn0.1Al 225 4.43 262.70 −2.20 824.08 (0.09) 5.29 (0.26)Rh2Ni0.9Co0.1Al 225 2.02 254.27 2.11 −791.36 (0.29) 4.72 (0.28)Rh2Ni0.9Co0.1Ga 225 2.02 313.48 1.62 738.58 (−0.27) 6.18 (0.29)17Table S13. Ru-based doped candidates with promising αxy at the Fermi energy. Contents in the table arethe space group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermienergy, and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermilevel. ∆ϵ denotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)]Ru2CoAl0.8Si0.2 225 2.40 406.58 6.22 1480.71 (0.23) −9.82 (0.19)Ru2Mn0.9Cr0.1P 225 3.85 498.27 −5.41 1190.01 (0.09) 5.56 (0.17)Ru2Mn0.9Cr0.1As 225 3.83 579.02 −5.05 1177.03 (0.07) −5.11 (−0.01)Ru2Mn0.7Cr0.3Sb 225 3.73 592.00 −4.77 1298.23 (0.07) −4.77 (0.00)Ru2FeP 225 4.26 339.75 −4.68 1591.51 (0.07) −5.27 (0.02)(Ru0.8Rh0.2)2CrSb 225 3.41 525.47 −4.66 1542.99 (0.10) −4.77 (0.01)Ru2Co0.7Fe0.3P 225 2.80 285.38 4.59 1246.46 (−0.13) 4.81 (−0.02)(Ru0.7Rh0.3)2MnSn 225 3.70 758.11 −4.56 1286.89 (0.07) −4.61 (−0.01)Ru2CoGe0.9As0.1 225 2.21 499.65 −4.16 979.90 (0.17) −4.98 (−0.03)Ru2Cr0.8V0.2As 225 2.71 92.75 −3.97 1165.73 (0.27) −4.13 (0.22)Ru2Co0.9Fe0.1Si 225 2.40 136.82 −3.90 942.81 (−0.29) −4.17 (0.02)(Ru0.7Rh0.3)2MnGe 225 3.64 430.01 −3.63 1322.52 (0.12) −3.80 (0.02)Ru2Fe0.9Co0.1As 225 4.20 793.02 −3.59 1232.81 (0.04) 4.11 (0.15)Ru2Fe0.7Mn0.3Ge 225 3.74 619.29 −3.54 1021.12 (0.08) −3.88 (−0.04)Ru2Fe0.7Mn0.3Si 225 3.71 513.30 −3.49 1122.02 (0.12) 3.74 (0.18)(Ru0.8Rh0.2)2FeGa 225 3.53 384.30 −3.15 1090.70 (0.12) 3.48 (0.18)Ru2Fe0.9Mn0.1Sn 225 4.03 431.43 2.70 893.07 (−0.07) −3.95 (−0.16)(Ru0.7Rh0.3)2FeAl 225 3.67 859.74 −2.34 1184.26 (0.07) 4.13 (0.14)Ru2Cr0.8V0.2P 225 2.76 89.01 −2.26 1162.72 (0.27) −3.38 (0.22)Ru2Cr0.7V0.3Sn 225 1.78 24.53 1.92 393.88 (−0.05) 1.92 (0.00)(Ru0.9Rh0.1)2FeSb 225 4.39 241.05 −1.82 1056.85 (0.05) 2.35 (0.17)(Ru0.8Rh0.2)2VSi 225 1.35 247.16 −1.40 1006.25 (0.26) 3.17 (0.29)18Table S14. Cr-based doped candidates with promising αxy at the Fermi energy. Contents in the table arethe space group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermienergy, and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermilevel. ∆ϵ denotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)]Cr2Fe0.8Mn0.2P 216 0.79 −31.04 −3.21 654.09 (0.29) −3.21 (0.00)(Cr0.7Mn0.3)2ScP 225 3.39 514.02 2.54 1280.13 (−0.03) 2.81 (0.02)(Cr0.9Mn0.1)2TiAs 225 2.81 619.69 2.42 758.96 (−0.08) −2.77 (−0.12)(Cr0.9V0.1)2TiSb 225 3.21 79.48 2.31 1367.10 (−0.05) 2.33 (−0.01)Cr2Mn0.7Fe0.3Ge 225 4.78 285.88 1.97 552.04 (−0.06) −2.42 (0.28)(Cr0.7V0.3)2TiGe 225 3.34 109.03 −1.75 580.18 (−0.17) 2.20 (−0.23)Cr2Ti0.7V0.3Si 225 3.16 175.84 −1.34 1059.06 (0.12) 2.26 (0.17)(Cr0.7Mn0.3)2ScSi 225 4.39 257.95 1.23 709.41 (−0.01) −2.65 (0.27)Table S15. Ni-based doped candidates with promising αxy at the Fermi energy. Contents in the table arethe space group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermienergy, and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermilevel. ∆ϵ denotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)]Ni2FeSb0.8Sn0.2 225 3.40 −210.64 −2.73 −456.81 (−0.06) 4.00 (0.29)(Ni0.7Co0.3)2MnIn 225 5.01 −382.82 1.69 −795.55 (0.05) 2.12 (−0.27)Ni2Mn0.7Fe0.3Al 225 4.13 −186.71 −1.15 −530.25 (−0.22) 3.07 (−0.24)Table S16. V-based doped candidates with promising αxy at the Fermi energy. Contents in the table are thespace group (SG), the calculated magtic moment m per formula unit, the σxy and αxy at the Fermi energy,and the maximum value σmaxxy and αmaxxy obtained in an energy window 0.3 eV around the Fermi level. ∆ϵdenotes the energy difference of the maximum values with respect to the Fermi energy.Candidates SG m σxy αxy σmaxxy [∆ϵ] αmaxxy [∆ϵ](µB/f.u.) (Scm−1) (Am−1K−1) (Scm−1) [(eV)] (Am−1K−1) [(eV)](V0.8Ti0.2)2TiSb 225 0.21 −83.71 3.13 942.64 (−0.06) −5.09 (−0.10)V2CrAl0.8Si0.2 225 1.17 239.69 −2.35 437.09 (0.06) 2.93 (0.16) "Chemical-substitution-driven giant anomalous Hall and Nernst effects in magnetic cubic Heusler compounds" Comparison of the band structure Detailed parameters of Wannierization Anomalous transport properties associated with the space group Validity of the band-filling approach Anomalous transport properties in Fe- and Co-based Heusler alloys Optimum anomalous transport properties through chemical substitution