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Spin-wave (SW) spectra have theoretically been studied in a thin film of aferromagnet (FM) on a substrate from a paramagnet (PM) (an FM above thecritical temperature) or from superconductor (SC). A spin wave propagating inthe FM induces the dynamic magnetization and superconducting current in theunderlying PM and SC, respectively, which affect the SW propagation by theirmagnetic fields. As a result of this interaction, the SW spectrum becomesnonreciprocal to depend on the sign of the SW wave vector q. We show that thenonreciprocal contribution to the SW spectra in FM/PM and FM/SC systems isgiven by the frequency shift of{\Delta}{\omega}(q)={\omega}(q)-{\omega}(-q)=a(T)({\tau}{\cdot}q) with{\tau}=(n{\times}M) being the toroidal magnetic moment, M the FM magnetization,n the unit vector normal to the FM/PM(SC) interface, and a(T) thetemperature-dependent constant of a dipole nature, whose sign depends on thesubstrate type. As the {\Delta}{\omega}(T) dependence is strong at temperaturesT close to the critical temperature Tc for the FM-PM or normal metal-SCtransition, one gets a possibility to control the frequency SW nonreciprocitywith temperature variation near Tc. The dipolar mechanism we propose for SWfrequency nonreciprocity is promising for introducing this property of SWpropagation into functional devices.

Temperature-sensitive spin-wave nonreciprocity induced by interlayer dipolar coupling in ferromagnet/X (X=paramagnet, superconductor) hybrid systems