Fig. 1: BFO/LSMO heterostructures and spin-wave transport.

a A schematic diagram of the interface of the BiFeO₃(BFO)/La_0.7Sr_0.3MnO₃(LSMO) heterostructure. The inset shows the atomically resolved interface of BFO/LSMO by annular dark field image. The yellow arrow indicates the pseudo-cubic [110] crystalline orientation of LSMO and BFO. b A schematic illustration of magnon–phonon coupling in the BFO/LSMO heterostructure. The system consists of two layers: The top layer BFO is multiferroic being both antiferromagnetic (AFM) and ferroelectric (FE) and hosts a longitudinal acoustic (LA) wave mode. The bottom layer consists of the ferromagnetic (FM) LSMO layer with an in-plane uniaxial anisotropy. The phonons in BFO and magnons in LSMO are hybridized via magnetoelastic coupling at the interfacial layer owing to the interfacial exchange coupling between LSMO and BFO and magnetoelastic interaction within the BFO. c A scanning electron microscope (SEM) image of a magnonic device for measuring spin-wave transmission. The field is applied in the plane and rotated with respect to the [110] crystalline orientation of BFO (LSMO). The inset is a zoom-in image showing two integrated gold antennas NSL1 and NSL2 with a center-to-center distance \(d=6\,{{\upmu }}{{{{{\rm{m}}}}}}\). d Spin-wave transmission spectra \({S}_{21}\) (spin waves excited at NSL1 and detected at NSL2), where mode X and Y are marked. A single spectrum is extracted at \({{{{{\rm{\theta }}}}}}=90^\circ\), from which a frequency span \(\triangle f\) is extracted for the derivation of the spin-wave group velocity used in Eq. (1). The data displayed in the map is the imaginary part of the transmission spectra, and the color bar denotes the amplitude of the imaginary part, which is in arbitrary units.