Fig. 4: Anisotropic magnon transport.

a Magnon-generated nonlocal ISHE voltage hysteresis was measured as a function of the external in-plane electric field in devices with four different orientations labeled by the pseudocubic direction with respect to the orientation of the Pt wires. The power in the source electrode was fixed to 2 mW (I_ac= 1.7 mA). A depiction of the response of the Bi_0.85La_0.15FeO₃ unit cell to different poling fields is given in (b–d) for device orientations [010], \([\bar{1}10]\), and [110] respectively. k, E, P_net represent the propagation vector of the spin cycloid, in-plane electric field, and net in-plane ferroelectric polarization. The relation between P and k is drawn based on the experiment performed in Fig. 3, where P and k are mapped out, and are consistent with prior studies of the cycloid in BiFeO₃^{18, 26, 38}. In (b), the direction of P is set to [112] in a single domain after poling where \([1\bar{1}0]\) is the allowed spin cycloid direction (see Supplementary Note 6, Fig. 19). In the opposite poling configuration, the direction P is switched to \([1\bar{1}2]\) (see also Fig. 3(a–d)) where the corresponding spin cycloid propagation vector will be \([\bar{1}10]\). Similarly, a single domain device \(1\bar{1}0\) or [110] switched 180° in the in-plane (c.f. Fig. 3(m–o)) in opposite poling since the orientation of Pt electrodes decides the direction of P and underneath k. The schematics are only considered here for single-domain devices and for the multi-domain device [100] where the ISHE voltage change as a function of the electric field is found to be negligible (green data in (a)), more explanation is given in the Supplementary Note 6, Fig. 20.