Fig. 4: Transverse thermoelectric effects in Mn₃X (X = Sn, Ge) and their theoretical analyses.

a–c Temperature dependence of the Nernst coefficient, \(- S_{ij}\), measured under B = 14 T. \(S_{zx}\) (red circle), \(S_{yz}\) (blue diamond), and \(S_{xy}\) (black triangle). The label definitions are modified from the original ones in ref. ¹⁵ so that they become consistent with the results obtained by other groups. a Zero-field anomalous transverse thermoelectric conductivity, \(- \alpha _{xz}\), with the best fit to the minimal model of Weyl fermions (solid line) (b), and the two contributions to \(- \alpha _{xz}\) (c) in Mn₃Ge. d Temperature dependence of \(- \alpha _{{\boldsymbol{xz}}}\) in Mn_3.06Sn_0.94 and Mn_3.09Sn_0.91 from ref. ⁹. e k-space distribution of the Weyl points (W₁–W₉) under \(B\parallel \left[ {0\bar 110} \right]\) predicted by the DFT calculation in ref. ³⁰. The red and blue frames mark the W₁ and W₃ pairs that are consistent with the WP1 (f) and WP3 (WP′3) (g) pairs revealed by our calculation. Contour plots of the theoretical Berry-curvature spectrum |Ω_z| in the k_x–k_y-plane under \(B\parallel \left[ {01\bar 10} \right]\) at k_z = 0 (Å⁻¹) (f) and at k_z = 0.137 (Å⁻¹) (g). The arrows indicate the Berry curvature arising from the WP1 (f) and WP3 (WP′3) (g) pairs. Each pair consists of two Weyl points with opposite chirality. Experimental \(- \alpha _{zx}/T\) (solid symbols) vs. theoretical curves obtained from the DFT calculations for Mn₃Sn (h) and Mn₃Ge (i). Calculations are made with the energy shifts in the chemical potential corresponding to the extra Mn in the off-stoichiometric samples, relative to the Fermi level E_F of the stoichiometric case (i.e., E = E_F). A convergence factor of γ = 0.02 and a coefficient AZ² = 5.0 eV⁻¹ are used in the calculation. The renormalized temperature T* = T/5 for Mn₃Sn is set accordingly to the ARPES results. A smaller renormalization factor T* = T/2 is chosen for Mn₃Ge given its larger bandwidth relative to that of Mn₃Sn. The inset of (i) shows the calculated Hall conductivity \(- \sigma _{zx}\) for Mn_3.06Ge_0.94 and Mn_3.03Ge_0.97 using the same parameters. Adapted from ref. ¹⁵, APS (a–d).