Onsager Reciprocal Relation between Anomalous Transverse Coefficients of an Anisotropic Antiferromagnet

Whenever two irreversible processes occur simultaneously, time-reversal symmetry of microscopic dynamics gives rise, on a macroscopic level, to Onsager’s reciprocal relations, which impose constraints on the number of independent components of any transport coefficient tensor. Here, we show that in the antiferromagnetic ${\mathrm{YbMnBi}}_2$, which displays a strong temperature-dependent anisotropy, Onsager’s reciprocal relations are strictly satisfied for anomalous electric (${\sigma }_{ij}^A$) and anomalous thermoelectric (${\alpha }_{ij}^A$) conductivity tensors. In contradiction with what was recently reported by Pan et al. [Nat. Mater. 21, 203 (2022)], we find that ${\sigma }_{ij}^A(H)={\sigma }_{ji}^A(-H)$ and ${\alpha }_{ij}^A(H)={\alpha }_{ji}^A(-H)$. This equality holds in the whole temperature window irrespective of the relative weights of the intrinsic or extrinsic mechanisms. The ${\alpha }_{ij}^A/{\sigma }_{ij}^A$ ratio is close to $k_B/e$ at room temperature but peaks to an unprecedented magnitude of $2.9k_B/e$ at $\sim 150\mathrm{K}$, which may involve nondegenerate carriers of small Fermi surface pockets.

Figure 1

The basic properties of ${\mathrm{YbMnBi}}_2$. (a) The crystal structure of ${\mathrm{YbMnBi}}_2$ with two sets of Bi: One kind of Bi is bonded to four Mn atoms, while another forms a 2D intercalation network. The spin of Mn cants in the $xy$ plane. (b) Temperature dependence of magnetization $M$ in the $xy$ plane ($H\parallel xy$). (c) Temperature dependence of longitudinal resistivity ${\rho }_{yy}$ (red circles) and ${\rho }_{zz}$ (black squares). (d) Temperature dependence of Seebeck coefficients $S_{yy}$ (red squares) and $S_{zz}$ (black circles).

Figure 2

Transverse electric transport. (a) The setup for measuring Hall signals in the $zy$ and $yz$ configurations. The Hall resistivity is identical: ${\rho }_{zy}=-{\rho }_{yz}$. (b) The AHC ${\sigma }_{zy}({\sigma }_{yz})$ at 280 K. The Hall conductivity is also identical: ${\sigma }_{zy}=-{\sigma }_{yz}$. (c) The temperature dependence of anomalous Hall conductivity shows that Onsager’s reciprocal is true in the whole temperature range. It exhibits turning an anomaly around 50 K, concomitant with an anomaly in magnetization. (d) The temperature derivative of ${\sigma }_{ij}^A/T$, which shows a kink around 50 K.

Figure 3

Transverse thermoelectric transport. (a) The setup for measuring Nernst signals in the $zy$ and $yz$ configurations. The transverse thermopower shows an extreme anisotropy in both reciprocal configurations. (b) The anomalous Nernst conductivity is calculated. Its magnitude for the two configurations is identical within experimental error shown the established Onsager’s reciprocal relation. (c) The anomalous Nernst conductivity (ANC) ${\alpha }_{zy}({\alpha }_{yz})$ at 200 K. (d) The temperature dependence of ANC shows that Onsager’s reciprocal is true in the whole temperature range. It also exhibits a kink around 50 K. (e) The temperature derivative of ${\alpha }_{ij}^A/T$, which shows a kink around 50 K.

Figure 4

Scaling relation of the anomalous Hall conductivity. The longitudinal conductivity (a) ${\sigma }_{yy}$ for ${\mathrm{YbMnBi}}_2$ lines within the good-metal regime, and there is a crossover behavior to the skew scattering region in the lower temperature, suggesting the skew scattering will dominate the transverse transport. The flat AHC [${\sigma }_H^A=(\sigma {)}^0=\mathrm{const}$] is concomitant with the maximum of magnetization at 50 K in region I. A second observed intrinsic AHC can be attributed to the magnetization turning point at 15 K in region II. An extrinsic mechanism may play a role by below 15 K. (b) The ${\alpha }_{zy}^A/{\sigma }_{zy}^A$ ratio at different temperatures. Similar to other topological magnets, it approaches $86\mu \mathrm{V}/\mathrm{K}$ at room temperature, but there is a peak as large as $250\mu \mathrm{V}/\mathrm{K}$ at 150 K.