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Transverse acoustic waves in an inhomogeneous medium are analogues toelectromagnetic waves and will exhibit topological behavior due to the Berrygauge potential in the momentum space due to inhomogeneity. The inhomogeneous(or gradient) medium can be created using an applied strain gradient in asemiconductor thin film (silicon) since the phonon frequency and dispersionwill be a function of the local strain along the strain gradient direction. Asa consequence, topological phonon mediated spin and heat transport can beengineered in the semiconductor thin films. Here, we present evidence of along-distance (100 um) spin transport in the freestanding Si thin film sampleunder an applied strain gradient using transverse spin-Nernst effectmeasurement. The long-distance spin transport was attributed to the topologicalspin-Hall effect of phonons in an inhomogeneous medium. The inhomogeneousmedium was validated using unidirectional magnetoresistance of phonons wherethe magnitude of the coefficient of the non-reciprocal response at roomtemperature was as large as reported in the BiTeBr at low temperatures. Thetopological phonons also manifested the topological Nernst effect. This worknot only enhances the current understanding of inhomogeneous systems but alsolays the foundation of the topological and spin phononics.

Topological phonons in an inhomogeneously strained silicon-1: Evidence  of long-distance spin transport and unidirectional magnetoresistance of  phonons