Klein tunneling in phononic crystals: A step case

In a recent paper [Jiang, et al., Science 370, 1447 (2020)], it was reported that zero reflection or Klein tunneling can be observed for normally incident quasiparticles upon a potential barrier constructed by two phononic crystals (PCs) with Dirac cone band structures. Here, we develop a first-principles approach for accurate computation of the reflection of quasiparticles by a potential step with two PCs at normal incidence. Strikingly, it is found that minimal reflection of quasiparticles ( Rm = 0.8%) occurs at an energy below the potential step, while moderate reflection ( Rd = 17%) remains at the center energy of the potential step, even with the PCs adopted by Jiang et al. A physical model is presented to understand such phenomena, where the zigzag interface in the step serves as a graded antireflection layer and reflection increases dramatically near the band gap of PCs. Two solutions are also shown for realizing lower Rd or even ideal Klein tunneling in PCs, with reducing the difference between the two PCs or enhancing the antireflection effect of the interface. Our work reveals the effects of the zigzag interface and band gap on Klein tunneling in PCs, which will be inspiring for exploring more fascinating phenomena of quasiparticles in classical wave systems.