Giant anomalous Nernst signal in the antiferromagnet YbMnBi2

A large anomalous Nernst effect (ANE) is crucial for thermoelectric energy conversion applications because the associated unique transverse geometry facilitates module fabrication. Topological ferromagnets with large Berry curvatures show large ANEs; however, they face drawbacks such as strong magnetic disturbances and low mobility due to high magnetization. Herein, we demonstrate that YbMnBi2, a canted antiferromagnet, has a large ANE conductivity of ~10 A m−1 K−1 that surpasses large values observed in other ferromagnets (3–5 A m−1 K−1). The canted spin structure of Mn guarantees a non-zero Berry curvature, but generates only a weak magnetization three orders of magnitude lower than that of general ferromagnets. The heavy Bi with a large spin–orbit coupling enables a large ANE and low thermal conductivity, whereas its highly dispersive px/y orbitals ensure low resistivity. The high anomalous transverse thermoelectric performance and extremely small magnetization make YbMnBi2 an excellent candidate for transverse thermoelectrics.


Fig. S1
Temperature dependence of (a) resistivity and (b) thermal conductivity in the ab-plane and c-axis. The insets of (a) are magnified views showing the Néel temperature TN.

Magnetism measurement
The magnetic field dependence of the magnetization curves indicates a canted antiferromagnetic feature of YbMnBi2. Along c-axis, a saturation is observed at low temperatures at ~4-6 T ( Fig. S2(a)), demonstrating the non-collinear antiferromagnetic structure of YbMnBi2. Additionally, since it is extremely hard to push all the spin in the ab-plane, no saturation is observed until 7 T in the ab-plane, as shown in Fig. S2(b). Fig. S2(c)-(d) shows the hysteresis taking 160 K as an example.

ANE/AHE measurements
For the measurements of thermal transport properties, the sample was fixed on a piece of alumina heater sink, with a strain gauge heater attached to the other end to apply a temperature gradient. Two sets of chromel-constantan thermocouples were mounted at two points along the temperature gradient to measure the temperature difference, and chromel thermocouples were used to measure the voltage difference.
Anomalous Nernst/Hall effects shows up when the magnetic field is along a // (100), while no anomalous Nernst thermopower (Fig. S3) or Hall signal (Fig. S4) is observed when the magnetic field is along c // (001). These results are highly consistent with the theoretical predication, indicating that the origin of the ANE is the intrinsic Berry curvature.  On the other hand, at low temperatures, the Hall resistivity is non-linear change from negative to positive from low to high magnetic field (Fig. S4(c)), indicative of a two-types of charge carrier behavior, and electrons dominate at low fields. More importantly, the slope of the Hall resistivity goes through a negative to positive change, indicating a n-to p-type transition between 50-100 K.  The Gerlach sign convention of the measured Nernst thermopower and Hall resistivity are shown in Fig. S5. As shown in Fig. S5(a), in the condition with a heater on the left side (hot) and a heat sink on the right side (cold), and a magnetic field B out of paper, the bottom side of the sample is wired to the positive voltage (V+) and the upper side of the sample is wired to the negative voltage (V-) of the voltmeter. Similarly, as shown in Fig. S5(a), in the condition with a current input (Iin) on the left side and a current output (Iout) on the right side, and a magnetic field B out of paper, the bottom side of the sample is wired to the positive voltage (V+) and the upper side of the sample is wired to the negative voltage (V-) of the voltmeter. Assuming a canting angle of zero, the band structure shows that there is no contribution from the hole pocket at Г point, as shown in Fig. S6. By varying the canting angle from 8 º to 18 º, it is found that the valence band at Г point contribute more and more to the conduction, as shown in Fig. S7. Topological bands contributed by the p orbitals of Bi hardly changes with canting angle, but a larger canting angle results in an obvious change in the bands at Г point since they are contributed by the hybridization of the pz orbitals of Bi and the d orbitals of Mn. Most importantly, the joint of the bands at Г point results in a larger positive Berry curvature, which is advantageous for stronger ANE/AHE signals.

MR measurement
Magnetoresistance along different directions are shown in Fig. S8. Maximum MR is obtained when B is along c // (001) and current is along a // (100), as shown in Fig. S8(a). Because of this sharp band dispersion, the ab-plane resistivity of our high quality YbMnBi2 single crystal and undergoes clear Shubnikov-de Haas (S-dH) oscillations at B > 5 T from 2 K to 10 K (Fig. S8(a)). By subtracting a smooth background, Fig. S8(d) illustrates the oscillation amplitudes of resistivity against 1/B, which shrinks with increasing temperature. Fig. S8(e) show that the temperature dependence of the oscillation amplitudes can be well fitted by the Lifshitz-Kosevich (LK) formula, [1] which denotes an effective mass m * of ~0.23 m0. This value is close to a previous study [2] and it is quite small compared to ferromagnets.  Seebeck coefficient along different directions are shown in Fig. S9. The turn up of Seebeck coefficient along c-axis happened at 50-100 K, which is also because of the shift of Fermi level, similar to Hall but more complicated as it depends on not only the conductivity but also the Seebeck coefficient of the two types of charge carriers. It is known that for semimetals, SSeebeck,i = (Se,iσe,i +Sh,iσh,i)/(σe,i +σh,i), [3] in which Se and Sh is the Seebeck coefficient of the electrons and holes, and the subscribe i represents ab plane or c-axis. Hence the total Seebeck coefficient depends on the competition between Seσe,i and Shσh,i (note Sh is always positive while Se is always negative). At low temperatures where electrons dominate the conduction, |Seσe,i| would always larger than Shσh,i, leading to negative Seebeck coefficient. While at higher temperatures, Shσh,i can be larger than |Se|σe,i with more holes join in the conduction, which is particularly effective in the c-axis as σe,c is very low, again due to the extremely flat dispersion of the topological bands along c-axis. This results further demonstrate the significant shift of Fermi level with temperature. On the other hand, co-exist of positive and negative Seebeck coefficient in different crystallographic directions are observed, which may indicate that YbMnBi2 is a goniopolar material. [4] Anomalous Hall/Nernst conductivity Experimentally, σAHE (taking cb as an example) is calculated by: where ≫ . The anomalous Nernst conductivity αANE in cb (V // c) is calculated as: αcb= Scbσcc + Sbbσcb, (2) where Scb and Sbb are the SANE in cb and Seebeck coefficient along the b-axis, respectively. In our results, because Scbσcc ≫ Sbbσcb, Scbσcc makes the dominant contribution to αANE. Because of the extremely high σbb (~35 times higher than σcc at 80 K) and comparable Sbc, high αANE is achieved in bc rather than cb. Such large difference between σbb and σcc originates from the strong anisotropy of the Fermi surface, in which the electron pockets are highly dispersive in ab-plane but have nearly no dispersion along c-axis. Fig. S10 shows the first-principles calculated AHE conductivity as a function of Fermi energy and ANE conductivity as a function of temperature at different canting angles. The calculated ANE conductivity increases with temperature up to 250 K. The absolute values of the ANE conductivity show no obvious change by varying the canting angle or shift the Fermi energy.

The ratio of anomalous Nernst conductivity to anomalous Hall conductivity
As argued by Behnia et al. recently, [5] the ratio of anomalous Nernst conductivity (αij) to anomalous Hall conductivity (σij) would reach ~kB/e at 300 K if both AHE and ANE are dominated by the intrinsic Berry curvature contribution. Therefore, we analyzed the scaling ratio of the αij/σij of YbMnBi2, as shown below in Fig. S11. It is found that the ratio in cb (αcb/σcb) is within a few kB/e, however, there is a surprisingly large violation from kB/e in bc (αbc/σbc). The reasons for the large violation can be probably understood from four aspects as discussed below.

Fig. S11
The ratio of αij/σij in both cb and bc. The extremum value of αij/σij in other compounds are shown to compare, including Co2MnGa (300 K), [5] Co3Sn2S2 (70 K), [5] Ga0.96Mn0.04As (60 K), [6] UCo0.8Ru0.2Al (50 K), [7] Mn3Sn (250 K), [5] Mn3Ge (300 K), [5] Fe3Ga (300 K), [8] and MnBi (120 K). [9] First, there can be an extrinsic contribution to AHE and ANE in YbMnBi2, so that the AHE and ANE conductivities are not simply determined by the Berry curvature. Second, since the empirical observation of ~kB/e in the model is constructed based on a two-dimensional circular isotropic Fermi surface, [5,10] it is reasonable that YbMnBi2 is an exception since its Fermi surface are highly anisotropic (highly dispersive in ab-plane but presents almost no dispersion along c-axis). Moreover, the topological band in YbMnBi2 is highly linear, which is very different from a parabolic band as the model requires. [10] A precise description of the Fermi surface in the 3-dimensional Brillouin zone by considering the particular conductions, for example, the anisotropy of the band structure, the deviations of the band dispersion from parabolic spectrum, etc. is essential in the case of YbMnBi2. In YbMnBi2, |αij/σij| ≈ |Sij*σii|/|(ρij/ρiiρjj)| = Sij*ρjj/ρij. Since the ratio of the resistivity between ab-plane and c-axis is very large, while both Sij and ρij are almost at the same order in ab-plane and c-axis, the large difference in |αij/σij| originates from the significant difference of ρbb and ρcc. In fact, for most of the materials with large ANE observed so far, a large difference in ρii and ρjj has never been observed, even including Mn3Sn and Mn3Ge. Third, it maybe also because that there are more than one band so that the single-band approach on longer stands. At last, the empirical ration of ~kB/e is a high temperature limit near 300 K, which YbMnBi2 may never reach since its canting temperature is ~250 K. As can be seen in Fig. S11, most of the materials approach the empirical value of ~kB/e at ~300 K, while there can be a violation at low temperatures below 300 K, for example, Ga0.96Mn0.04As (60 K), MnBi (120 K), UCo0.8Ru0.2Al (50 K) show a ratio of few ~kB/e, and Co3Sn2S2 (70 K) shows a ratio two orders lower than ~kB/e (only ~0.01 ~kB/e).

Single crystal characterization
Laue X-ray diffraction pattern of the sample demonstrates the high quality of the single crystal, as shown by the distinct spots in Fig. S12. Additionally, the single crystals tend to grow in a layered structured with the surface being ab-plane. The backscattered electrons (BSE) image and Yb, Mn, Bi mapping confirm the uniform distribution of the elements (Fig. S13(a)). Further energy-dispersive X-ray spectroscopy (EDX) analysis ( Fig. S13(b)) indicates the composition ratio of Yb:Mn:Bi as 25.6:24.9:49.5, which agrees well with the stoichiometric 1:1:2 ratio. With a comparable ANE thermopower of few microvolts per Kelvin, in the bc configuration, YbMnBi2 shows a lower resistivity and much lower thermal conductivity than other compounds, as shown in Table S1.

Table S1
Resistivity and thermal conductivity of YbMnBi2, with a comparison to other ferromagnets with high ANE performance. Since different materials show the highest ANE performance at different temperatures, values at 300 K are adopted here for a rough comparison.