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## Creator

[Naoki Chiba](https://orcid.org/0009-0003-0296-9026), [Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

## Rights

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Yoshio Miura et al., Appl. Phys. Lett. 124, 212408 (2024) and may be found at https://doi.org/10.1063/5.0206878[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Prediction of giant anomalous Nernst effect in Sm(Co,Ni)5](https://mdr.nims.go.jp/datasets/f2064a43-2cf3-4d02-8882-1fc214997554)

## Fulltext

Microsoft Word - Chiba_Sm(Co1-xNix)5_revised_manuscript_final.docx 1 Prediction of giant anomalous Nernst Effect in Sm(Co,Ni)5 1 Naoki Chiba1,2, Keisuke Masuda1, Ken-ichi Uchida1,2, and Yoshio Miura1,3,4,a)* 2  3 AFFILIATIONS 4 1 Research Center for Magnetic and Spintronic Materials, National Institute for Materials 5 Science, Tsukuba 305-0047, Japan 6 2 Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, 7 Japan 8 3 Faculty of Electrical Engineering and Electronics, Kyoto Institute of Technology, Kyoto 9 606-8585, Japan 10 4 Center for Spintronics Research Network, Osaka University, Osaka 560-8531, Japan 11 a) Author to whom correspondence should be addressed: MIURA.Yoshio@nims.go.jp 12  13  14 ABSTRACT 15 Sm-Co bulk alloys are well known permanent magnets having large remanent 16 magnetizations and coercive forces and are widely used in many industrial products. 17 Recently, a large transverse thermoelectric conversion was observed for SmCo5 over a wide 18 temperature range in the absence of magnetic fields. The large anomalous Nernst coefficient 19 (ANC) was also confirmed by the first-principles density functional theory (DFT) 20 calculations. In this study, we predicted further enhancement of the ANC by including Ni in 21 Co site of SmCo5. We showed that the ANC of Sm(Co1-xNix)5 increases with increasing the 22 Ni ratio and takes the maximum value αxy = 11.3 A K-1 m-1 around x = 0.08 at 300 K, which 23 is about 77% enhancement of αxy = 6.4 A K-1 m-1 in SmCo5. We clarified that the band-24 proximity points near the nodal line of Sm(Co0.92Ni0.08)5 are the main contributing factor to 25 the large Berry curvature, providing the steep slope of the energy dependence in the 26 anomalous Hall conductivity around the Fermi energy. 27  28  2  29  30 Thermoelectric generation is based on the conversion of heat into electricity driven by the 31 Seebeck effect,1 which can be applied to stand-alone power sources for IoT devices and the 32 reuse of waste heat.2,3 However, there are some issues that prevent the technology from 33 widely being used because of the complexity of the three-dimensional structure of the 34 integrated elements in the device and the inclusions of rare and toxic elements in the 35 materials that show excellent thermoelectric conversion performance.4,5 On the other hand, 36 transverse thermoelectric conversion by the anomalous Nernst effect (ANE) has attracted 37 much attention in recent years.6-8 The ANE is a thermoelectric effect unique to magnetic 38 materials and generates an electric filed in the direction perpendicular to the temperature 39 gradient and the magnetization.9-12 This means that the electric voltage generated by ANE 40 can be increased simply by increasing the length of the device plane perpendicular to the 41 temperature gradient.6,7 However, compared to Seebeck-type thermoelectric conversion, the 42 thermoelectric capacity is significantly insufficient, and the search for new materials is 43 strongly required. 44 To enhance the thermoelectric performance of ANE, we need materials with large 45 anomalous Nernst coefficient 𝑆 , which is expressed as7 46 𝑆 𝜌 𝛼 𝜌 𝛼 .                                                          1  47 ρxx (ρxy) and αxx (αxy) are the diagonal (off diagonal) components of the electric resistivity 48 and thermoelectric conductivity tensors, respectively. The first term of Eq. (1) originates 49 from the transverse thermoelectric conductivity 𝛼  and is regarded as an intrinsic part of 50 ANE. The second term is attributed to a contribution of the longitudinal thermopower i.e., 51 the Seebeck effect, which is usually smaller than the first term due to small anomalous Hall 52 angle (𝜌 /𝜌 ) and/or Seebeck coefficient. Thus, the material design with large 𝛼  will be 53 required to enhance 𝑆 . Recently, large 𝑆  were observed in ferromagnetic Heusler 54 compound Co2MnGa13,14 and Co2MnAl1-xSix,15 which can be attributed to the large 𝛼  due 55 to the topological nature of the band dispersion. 56 Recent studies have indicated that multilayered superlattice is effective in increasing the 57 ANE. Various magnetic/nonmagnetic combinations in multilayer systems show an increase 58 in ANE by varying parameters such as the number of layers and the number of stacks.16-18 59  3 Seki, et al., demonstrated the enhancement of ANE owing to the formation of Ni/Pt 60 superlattices. They found an optimized transverse thermoelectric conductivity, reaching 61 𝛼  4.8 A K−1 m−1 for thickness of Ni layer 𝑡  4.0 nm, which was the remarkable 62 enhancement compared to the bulk Ni.18 Furthermore, in our recent study, we demonstrated 63 an effectiveness of multilayer formation to obtain larger 𝛼  for Co/Ni multilayer system 64 based on Bayesian optimization and first-principles calculations.19 We clarified that the 65 multilayer formation with one monolayer alloying Co and Ni causes a fine modulation in 66 band structure and the proximity of bands with different states of parity near the Fermi 67 energy. Such band modulation can enhance 𝛼  up to ~10 A K-1 m-1, indicating an 68 effectiveness of the multilayer formation.  69 Another recently reported system which shows relatively large αxy is permanent magnets. 70 It was found that the well-known rare-earth SmCo5 permanent magnets show the anomalous 71 Nernst coefficient |𝑆 | ~5 μV K-1 at >400 K.20 The |αxy | of SmCo5 is more than one order 72 of magnitude greater than those for typical ferromagnetic metals and the figure of merit ZT 73 above room temperature is comparable to that in Co2MnGa.13,14 Furthermore, the 𝑆  of 74 the SmCo5 magnets monotonically increases with increasing the temperature and shows the 75 highest value of ZT reported so far at T > 500 K.21 We calculated the αxy of SmCo5 by the 76 linear response theory combined with the first-principles calculations including the orbital 77 polarization correction and obtained relatively large αxy corresponding to 6 A K-1 m-1 at the 78 Fermi level.20 The calculations suggest that αxy of SmCo5 has a maximum value not at the 79 Fermi energy but at the chemical potential μ = 0.04 eV.20 These results indicate that the 80 substitution effect is also important in enhancing 𝛼  of SmCo5-type permanent magnets. 81 Since permanent magnets are widely used in a society, the addition of thermoelectric 82 conversion functionality will contribute to the development of various energy-harvesting 83 technologies. Furthermore, SmCo5-type permanent magnets have large remanent 84 magnetization and coercive force, stable transverse thermoelectric conversion can be 85 realized over a wide temperature range in the absence of magnetic fields.22 In this work, we 86 theoretically investigate and discuss the enhancement of 𝛼  of SmCo5-type  permanent 87 magnets. Here, we focus on the intrinsic part of ANE, especially the transverse 88 thermoelectric conductivity 𝛼 , because the previous experiments indicated that the large 89 AEE and ANE of SmCo5 originated from the intrinsic properties of 𝛼  in a wide 90  4 temperature range.20,21 Since Ni-doped SmCo5 has never been synthesized in experiments, 91 it is very important for future studies to experimentally investigate not only the intrinsic 92 effects but also the extrinsic effects.23 We mainly discuss the dependence of 𝛼  on 93 substituting Ni for Co of SmCo5 because our previous study indicates the importance in 94 controlling the Fermi level of SmCo5 to enhance 𝛼 .20 We found that the substitution of Ni 95 for Co of SmCo5 can enhance 𝛼  more than 10 A K-1 m-1. 96 The transverse transport coefficients of Sm(Co1-xNix)5 were calculated by the linear 97 response theory24,25 combined with the first-principles calculations. We calculated the 98 electronic structures and the matrix elements of momentum operator of Sm(Co1-xNix)5 by 99 means of the full-potential linearized augmented plane wave method (FLAPW) including 100 the full-relativistic effect, which is implemented in the WIEN2k code.26 The generalized 101 gradient approximation proposed by Perdew, Burke, and Ernzerhof was adopted for the 102 exchange and correlation energies.27 We used the virtual crystal approximation28 (VCA) for 103 Co-Ni disordering in SmCo5. Here, we considered a homogeneous doping of Ni in Co site 104 of SmCo5 because we used the VCA for Co-Ni disordering. The VCA has been applied to 105 the calculation of 𝛼  in previous studies,29 where a dopant concentration dependence of 106 𝛼  by VCA shows a good agreement with experimental results, indicating the effectiveness 107 of VCA in transverse transport coefficients. 108 The atomic structure of SmCo5 with the hexagonal unit cell is shown in the left-upper inset 109 in Fig. 1. The lattice constant of the primitive unit cell of SmCo5 (SmNi5) was fixed to a = 110 4.982 (4.926) Å and c = 3.975 (3.980) Å. Then, we set the lattice constant of Sm(Co1-xNix)5 111 according to the Vegard’s rule, where the atomic positions are fixed at each lattice constant. 112 In the calculation, we considered the orbital polarization effect in the spin-orbit Hamiltonian 113 in order to satisfy the Hund rule in f-orbitals of Sm atom, where the additional potential 114 having the form 𝑉 𝑐 〈𝐿 〉𝑙  was added to the system.30-32 Here, 𝑐  is the orbital 115 polarization parameter, 〈𝐿 〉 is projection of the orbital momentum on the magnetization 116 direction M, and 𝑙  is single electron orbital momentum component z parallel to M. Then, 117 we considered the on-site Coulomb interaction U and the Hund coupling J for f-orbitals of 118 Sm and d-orbitals of Ni atoms because of the strong electron correlation33,34. Here, we chose 119 U =9.00 eV and J = 0.75 eV for Sm atom and U = 3.9 eV and J = 1.1 eV for Ni atom which 120 satisfy orbital magnetic moment according to the Hund rule.35,36 We obtained an anti-121  5 ferromagnetic state for the spin moments between Sm and Co atoms from the self-consistent-122 field calculation with 19 × 19 × 21 k-points, which is energetically more stable than the 123 ferromagnetic state. We checked the U dependence of the orbital magnetic moment of Sm 124 in SmCo5. We found that the orbital magnetic moments of Sm strongly depend on the U 125 value, which are around 2 μB at U = 0.00 eV and 4 μB at U = 5.00 eV. On the other hand, it 126 is around 5 μB at U = 9.00 eV, which satisfy the second Hund’s rule. Once the correct orbital 127 magnetic moment around 5 μB according to the second Hund rule for Sm is obtained, the 128 DFT+U calculation gives the reasonable electronic structures within DFT.35 Thus, we 129 adopted U = 9.00 eV for Sm f orbital, which yields a large 𝛼  around 6 K-1Am-1. Another 130 method for dealing localized f orbitals is the Quasiparticle self-consistent GW (QSGW) 131 method.37 The QSGW method approximately treats quasiparticle states, and thus can 132 correctly calculate the localized and excited states. Compared to the DFT+U calculation, the 133 QSGW calculation generally reduces the bandwidth of d and f states, leading to the 134 enhancement of 𝜎  and 𝛼  around the Fermi level. Individual discussions on the QSGW 135 calculations for Ni-doped SmCo5 will be future tasks. 136 Then, we calculated the anomalous Hall conductivity 𝜎  by following equations,38,39 137 𝜎 𝐸𝑒ℏ𝑑 𝑘2𝜋Ω 𝐤,𝐸 . (2) 138 ΩZ(k,E) is the Berry curvature given by 139 Ω 𝐤,𝐸 2 Ω 𝐤,𝐸 , (3) 140 Ω 𝐤,𝐸ℏ𝑚𝑓𝐤 𝐸 𝑓𝐤 𝐸Im⟨𝐤𝑚|𝑝 |𝐤𝑛⟩ 𝐤𝑛 𝑝 𝐤𝑚  𝜀𝐤 𝜀𝐤, (4) 141 where m and n are the occupied and unoccupied band indices, 𝑝  (𝑝 ) is the x (y) component 142 of the momentum operator, |𝐤𝑛⟩ is the eigenstate with the eigenenergy 𝜀𝐤 , and 𝑓𝐤 𝐸  is 143 the occupation function for the band n and wave-vector k at the energy E relative to the 144 Fermi energy. In the σxy calculation, the M direction was set to be along the c axis of the unit 145 cell, and the uniform k-point mesh of 68 × 68 × 73 was used for the Brillouin-zone (BZ) 146 integration providing good convergence for 𝜎 . The 𝛼  value at temperature T was 147 obtained from the Boltzmann transport theory by integrating 𝜎  with energy E,20 148  6 𝛼1𝑒𝑇𝑑𝐸𝜕𝑔𝜕𝐸𝐸 𝜇 𝜎 𝐸 , (5) 149 where 𝑔 1 exp 𝐸 𝜇 𝑘 𝑇⁄ 1⁄  is the Fermi distribution function with the chemical 150 potential μ. The absolute value of 𝛼  for T = 300 K at the Fermi energy, μ = 0 eV, was 151 adopted as the calculation results of 𝛼 . 152  We show in Fig. 1 total density of states (TDOS) of Sm(Co1-xNix)5 for x = 0.0, 0.1, 0.4, 0.5, 153 and 1.0. In the TDOS of Sm(Co1-xNix)5, we can find several sharp peaks around E - EF = 3.0, 154 4.5, and 6.5 eV for the majority-spin state, and E - EF = -7.5, -6.3, -5.0, 1.5, and 2.0 eV for 155 the minority-spin state. These sharp peaks are due to local density of states (LDOS) of f-156 orbital of Sm, which are well localized far from the Fermi level. Since LDOS around the 157 Fermi level is mainly composed of Co and Ni orbitals in Sm(Co1-xNix)5, the transport 158 properties can also be attributed to electronic states of Co1-xNix sites. The magnifications of 159 the TDOS around the Fermi level are shown in the right-lower inset in Fig. 1. If we focus on 160 the minority-DOS around the Fermi level, we can confirm the rigid band shift of TDOS 161 peaks toward the lower energy regions with increasing x up to x = 0.4 because of one more 162 valence electron of Ni than Co. However, the peaks and their shifts around the Fermi level 163 due to the substitution of Ni for Co disappear and split into two peaks for x = 0.5, indicating 164 that the electronic structures of Sm(Co1-xNix)5 cannot be described by the rigid band model 165 for x  0.5. 166  We show in Figs. 2(a) and (b) the 𝜎  and 𝛼  values of Sm(Co1-xNix)5 as a function of 167 the Ni ratio x up to x = 0.10, i.e., within the rigid band model. For comparison, the x-168 dependence of 𝜎  and 𝛼  values for Sm(Co1-xFex)5 are also shown up to x = 0.10 in Figs. 169 2(a) and (b). The 𝜎  values of Sm(Co1-xNix)5 and Sm(Co1-xFex)5 decrease with increasing x 170 due to the Fermi level shift, and the sign changes from plus to minus around x = 0.05 and 171 0.10, respectively. The 𝛼  value of Sm(Co1-xNix)5 increases with increasing the Ni ratio and 172 reaches the maximum value 𝛼  = 11.3 A K-1 m-1 around x = 0.08. This value is an 173 improvement of about 77% compared to 𝛼  = 6.4 A K-1 m-1 in SmCo5. On the other hand, 174 the 𝛼  value of Sm(Co1-xFex)5 decreases with increasing the Fe ratio, clearly indicating the 175 chemical trend of  𝛼  in this system and the importance of Ni substitution for Co. In Fig. 176 3(a) and (b), the composition dependences of 𝜎  and 𝛼  are shown for the promising Ni 177 substitution case up to x = 0.4. Here, we calculated 𝜎  and 𝛼  up to x = 0.4 for Sm(Co1-178  7 xNix)5, because we do not expect a further increase of the Nernst coefficient in Ni 179 concentration. We confirmed that the Ni substitution for Co enhances the 𝛼  up to x = 0.14. 180 The monotonic reduction of 𝛼  by further Ni doping can be attributed to the decrease of Co 181 d LDOS by electron doping, because the Co d states give the major contribution to the 𝜎  182 and 𝛼 . To clarify the origin of the enhancement of 𝛼 , we show in Figs. 3(c) and (d) the 183 energy E dependence of the 𝜎  and μ dependence of the 𝛼  for x = 0.00 and x = 0.08. 184 Figure 3(c) shows that the large slope of 𝜎  in SmCo5 is shifted to just above EF due to Ni 185 substitution for Co. According to the Mott formula obtained from the Sommerfeld expansion 186 of Eq. (5), the absolute value of 𝛼  is proportional to 𝜕𝜎 𝜕𝐸⁄ , i.e., 𝛼  can be attributed 187 to the energy derivative of 𝜎  at each μ. Therefore, large 𝛼  is obtained for 188 Sm(Co0.92Ni0.08)5 where the steep slope of 𝜎 𝐸  is shifted directly above EF (μ=0) [Figs. 189 3(c) and (d)]. The comparison of the band dispersion of SmCo5 and Sm(Co0.92Ni0.08)5 also 190 clarified the shift of the Fermi energy by the Co substitution of SmCo5. 191  The large 𝛼  (large energy dependence of 𝜎 ) at the Fermi level in Sm(Co0.92Ni0.08)5 192 is attributed to the fact that 𝜎  shows a large positive value (~+1000 S/cm) just below the 193 Fermi level and a large negative value (~ -1000 S/cm) just above the Fermi level. To clarify 194 the origin of the 𝜎  behavior, the k-resolved Berry curvatures of Sm(Co0.92Ni0.08)5 just 195 below (E - EF = -0.05 eV) and just above (E - EF = +0.05 eV) are shown in Fig. 4. In Figs. 196 4(a) and (b), the Berry curvatures at E - EF = -0.05 eV and E – EF = +0.05 eV are mapped 197 onto their iso-energy surfaces of 𝜀𝐤  in the BZ. From Fig. 4(a), the large Berry curvature can 198 be confirmed near the lines connecting the L and K high-symmetry points and the H and M 199 high-symmetry points. Figure 4(b) shows a relatively large Berry curvature near the line 200 connecting the L and H points. Figures 4(c) and (d) show the band dispersion just on the 201 high-symmetry lines and the Berry curvatures along the dispersion 𝜀𝐤   for E - EF = ±0.05 202 eV. The large Berry curvatures are observed on the K-L and M-H high-symmetry lines just 203 below the Fermi level (E – EF = -0.05 eV) and on the L-H high-symmetry line just above the 204 Fermi level (E – EF = +0.05 eV), respectively. These behaviors of the Berry curvature in 205 Sm(Co0.92Ni0.08)5 contribute to the abrupt change in the 𝜎  near the Fermi level shown in 206 Fig. 4(e). 207  8 To obtain more detailed understanding on the origin of 𝜎 , we focus on the formulation 208 of the Berry curvature given by Eq. (4). The Berry curvature consists of matrix elements of 209 the momentum operator in the numerator and difference between arbitrary eigen-energies in 210 the denominator. Due to the selection rule of dipole transitions, the matrix elements of the 211 momentum operator have non-zero values for transitions between orbital states (s, p, d) with 212 different parity, such as s-p and p-d transitions, which depends on the orbital hybridization 213 in occupied and unoccupied eigenstates. On the other hand, the difference of eigen-energies 214 in the denominator reflects the band proximity points, i.e., the so-called nodal-line in the 215 band dispersion, which contributes significantly to the Berry curvature. To estimate the 216 contribution of the nodal line, we calculated the following quantity for Sm(Co0.92Ni0.08)5, 217                                         𝑁 𝐤,𝐸 2𝑓𝐤 𝐸 𝑓𝐤 𝐸  𝜀𝐤 𝜀𝐤,                                            6  218 which is the sum of Eq. (4) without the matrix elements and contributes to the Berry 219 curvatures around the band proximity points. Figure 4 shows the mapping of the Berry 220 curvature Ω 𝐤,𝐸  and the nodal line 𝑁 𝐤,𝐸  onto the Fermi surface for three bands 221 crossing the Fermi energy. The plotting Eqs. (4) and (6) on the Fermi surface can show the 222 relationship between the Berry phase and nodal points in the Brillouin zone. For a further 223 understanding, it is necessary to analyze in detail the DOS projected onto the band that 224 creates the nodal line.40 Comparison of (a) with (d), (b) with (e), and (c) with (f) in Fig. 5 225 shows that the Berry curvature and nodal line have large peaks in the same region on the 226 Fermi surface in BZ. This suggests that there are many band proximity points in the vicinity 227 of the Fermi energy, which determines the energy dependence of the Berry curvature in 228 Sm(Co0.92Ni0.08)5. 229 In summary, we studied the transverse thermoelectric coefficients of Sm(Co1-xNix)5 230 based on the linear response theory and the first-principles calculations. We found that 𝛼  231 of Sm(Co1-xNix)5 increases with increasing the Ni ratio x and takes the maximum value 𝛼  232 = 11.3 A K-1 m-1 around x = 0.08, which is about 77% improvement of 𝛼  = 6.4 A K-1 m-1 233 in SmCo5. We clarified that the rigid band shift of electronic structures of Sm(Co1-xNix)5 234 provides the steep slope of 𝜎 𝐸  above EF (μ=0), leading to the large 𝛼  value for 235 Sm(Co0.92Ni0.08)5. Furthermore, we revealed that the band proximity points around the nodal 236 line of Sm(Co0.92Ni0.08)5 are the main contributing factor to the Berry curvature and the 237  9 energy dependence of 𝜎 . Our results suggest the importance of the tunning of the Fermi 238 level by substituting Ni for Co of SmCo5 for obtaining large transverse thermoelectric 239 coefficients, and experimental verification will be expected. 240  241  See the supplementary material for the Ni composition x dependence of spin and orbital 242 magnetic moments of Sm(Co1-xNix)5. 243   244 ACKNOWLEDGMENTS 245 This work was partly supported by Grant-in-Aids for Scientific Research (Grant No. 246 JP22H04966, JP22H04965) from JSPS, CREST “Creation of Innovative Core Technologies 247 for Nano-enabled Thermal Management” (JPMJCR17I1) and ERATO “Magnetic Thermal 248 Management Materials” (JPMJER2201), from JST, Japan. 249  250 REFERENCES 251 1 H. J. Goldsmid, Introduction to Thermoelectricity (Springer, Berlin, 2009). 252 2 L. E. Bell, Science 321, 1457 (2008). 253 3 M. Haras and T. Skotnicki, Nano Energy 54, 461 (2018). 254 4 H. S. Kim, W. Liu, G. Chen, C-W. Chu, and Z. Ren, Proc. Natl. Acad. 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Commun. 239, 197 (2019). 313  314  315  316  317  318  319  320  321  322  323  324  12  325  326 Fig. 1 Total density of sates (TDOS) of Sm(Co1-xNix)5 (x=0, 0.1, 0.4, 0.5, and 1.0) as a 327 function of energy relative to the Fermi energy. The positive (negative) TDOS indicates 328 the majority (minority) spin states. (Left upper part) Schematic view graph of atomic 329 structure of SmCo5. (Right lower part) Magnification of TDOS around the Fermi energy. 330  331  332  333  334  335  13 Fig.2  (a),(b) The anomalous Hall conductivity 𝜎  and Nernst coefficient 𝛼  as a 336 function of x in Sm(Co1-xNix)5 and Sm(Co1-xFex)5. 337  338  339 Fig. 3 (a), (b) The Ni ratio dependence of anomalous Hall conductivity and Nernst 340 coefficient as a function of x in Sm(Co1-xNix)5. (c) The energy E dependence of 𝜎  for 341 SmCo5 and Sm(Co0.92Ni0.08)5. (d) The chemical potential μ dependence of 𝛼  for SmCo5 342 and Sm(Co0.92Ni0.08)5. 343  344  14  345 Fig. 4 (a), (b) Color map of the Berry curvatures Ω 𝐤,𝐸  at (a) E = -0.05 eV and (b) E = 346 +0.05 eV plotted on the iso-energy surface in the three-dimensional first Brillouin zone of 347 Sm(Co0.92Ni0.08)5, which were visualized with FERMISURFER.41 (c) The band dispersion 348 of Sm(Co0.92Ni0.08)5 along the high-symmetry line showing large Ω 𝐤,𝐸  at E = ±0.05. 349 (d) The Berry curvatures Ω 𝐤,𝐸  at E = ±0.05 along the corresponding high-symmetry 350 line. (e) The energy dependence of  𝜎  near the Fermi level. 351  352  353  354  15  355 Fig. 5 (a)-(c) The mapping of the Berry curvature Ω 𝐤,𝐸  onto the Fermi surface for 356 three bands crossing the Fermi energy, respectively. (d)-(f) The mapping of the nodal line 357 𝑁 𝐤,𝐸  onto the Fermi surface for three bands crossing the Fermi energy, respectively. 358 Visualizations were performed with FERMISURFER.41 359