# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 3 Conclusions  
One-dimensional discrete phononic superlattices composed of alternating segments of two different harmonic crystals exhibit Bloch modes with specified handedness, with a finite amplitude for one direction and zero amplitude for the opposite direction. By using the transfer matrix method, i.e., solving the scattering of acoustic waves by the superlattice, we showed that the one-way transport nature of these eigenwaves arises from the periodicity of the system, even when the eigenvalues of the transfer matrix do not span the complete unit circle in the complex plane. The one-way propagation behavior of the discrete superlattice approaches that of a continuous superlattice in the long wavelength limits. A necessary condition for one-way propagation in our superlattice model is that the second harmonic crystal segment possesses a spatial extent that exceeds the lattice parameter. This condition enables Fabry-Perot-type resonances. In the limits of a superlattice composed of single lattice parameter segments, the model reduces to the SSH system, which does not exhibit one-way propagation. In this case, the eigenvalues of the transfer matrix span the complete unit circle in the complex plane and allow Bloch modes with non-zero amplitude to exist over the entire Brillouin zone. By comparing an infinite superlattice and a finite superlattice in the long wavelength limits, we also showed that reciprocity is always achieved. One-way propagation in the infinite superlattice results from the combined effects of translational periodicity, the non-trivial handedness of the associated eigenvector associated with a pseudo-spin, and the Fabry-Perot resonances of the second medium in the unit cell of the binary superlattice. Phononic superlattices supporting topologically protected acoustic or elastic waves that avoid back-reflections may provide attractive solutions for designing low-loss devices such as bulk or surface acoustic wave devices for telecommunication systems [21]. Another application of current interest is the use of one-way propagation for the management of thermal phonons. Specifically, recent advances in this context have been reported for phonon focusing [22], thermal switching [23], and thermal barriers [24]. Furthermore, binary superlattices that can be manufactured using thin-film deposition methods [25] or colloid-assembled nanocrystal superlattices [26] can serve as platforms for phonon engineering. These offer viable platforms for the experimental investigation of the effects discussed herein.  
**Author Contributions:** Conceptualization, P.A.D. and K.R.; methodology, P.A.D., K.R., A.K. and A.A.,: formal analysis, P.A.D., K.R., A.K. and A.A.,: writing--original draft preparation, P.A.D. and K.R.; writing--review and editing, A.K. and A.A.A.; funding acquisition, P.A.D., K.R., A.K. and A.A. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the US National Science Foundation (NSF) grant #2242925 through the Science and Technology Center New Frontiers of Sound (NewFoS).  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.