Fig. 5: Long-distance propagation of magnon–phonon hybridized mode at the BFO/LSMO interface.

Transmission amplitude \({S}_{21}\) plotted in a logarithmic scale as a function of the distance \(d\) between excitation and detection antennas. The left inset shows the SEM image of Sample Z13 with \(d=150\) µm as an example. The red open circles are data points extrapolated for mode Y from different samples as shown in Fig. 2a–e. The red solid line is a fit considering an exponential magnon decay, which yields a magnon decay length \(\lambda\) to be ~5 μm for mode Y. The black squares (triangles) are data points taken from samples with NSL (CPW) antennas for mode X (Supplementary Figs. 17 and 18). The black solid line is a fit based on Eq. (2) considering the contributions from a pure magnon mode with a short decay length λ_m ≈ 5 μm in superposition to a magnon–phonon hybridized mode with a long decay length of \({\lambda }_{{{{{{\rm{mp}}}}}}}=1\,{{{{{\rm{mm}}}}}}\). The right inset shows angle-dependent transmission spectra \({S}_{21}\) measured on Sample Z13 with an applied field of 100 mT and a propagation distance \(d=150\,{{\upmu }}{{{{{\rm{m}}}}}}\), where the black arrows indicate two branches of mode X attributed to the magnon–phonon hybridized mode.