Topical Review: the rise of Klein tunneling in low-dimensional materials and superlattices

We review recent advances in Klein and anti-Klein tunneling in one- and two-dimensional materials. Using a general tight-binding framework applied to multiple periodic systems, we establish the criteria for the emergence of Klein tunneling based on the conservation of an effective reduced pseudo-spin. The inclusion of higher-order terms in the wave vector leads to nontrivial matching conditions for wave scattering at interfaces. We further examine the emergence of multiple types of Klein tunneling in two-dimensional materials beyond graphene, including phosphorene and borophene, as well as in one-dimensional systems such as Su-Schrieffer-Heeger lattices. Finally, we discuss how these tunneling phenomena can be tested in both synthesized and artificial lattices, including elastic metamaterials, optical, photonic, phononic, and superconducting platforms, demonstrating the universality of Klein tunneling across different wave natures and length scales.