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

TL;DR

This review synthesizes Klein and anti-Klein tunneling across low-dimensional and artificial lattices using a general tight-binding Bloch framework. It derives interface matching conditions via an effective reduced pseudospin and develops Fresnel-like coefficients and a transfer-matrix formalism to treat stratified media, unifying a wide range of KT phenomena. The work catalogs multiple KT variants beyond graphene—anomalous KT in anisotropic 2D materials, anti-KKT in phosphorene, super-KKT in pseudo-spin-1 systems, 1D SSH chains, and valley-cooperative KT in Kekulé graphene—demonstrating universal principles across electronic, photonic, phononic, and acoustic waves. It further discusses experimental platforms from synthesized lattices to metamaterials and topolectrical/photonic systems, highlighting potential applications in ultrafast electronics, electron optics, and wave-based devices, all governed by conservation of an effective pseudospin rather than the Dirac equation alone.

Abstract

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 pseudospin. 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.

Figures (15)

• Figure 1: (a) Anisotropic hexagonal lattice that represents to strained graphene and simplified phosphorene structure, which is formed by two triangular sublattices. The arrows correspond to lattice vectors and nearest neighbors of atomos within the unit cell. The hopping parameters are $t_1$ and $t_2$ for quinoid deformation. (b) Electronic band structure with the values of $\epsilon_1 = \epsilon_2 = 0$, $t_1 = 1$, and $t_2 = 1.6$. The variation of hopping parameters can induce a gap opening for $t_2 \geq 2t_1$.
• Figure 2: (a) Bearded Su-Schrieffer-Heeger lattice and (b) its electronic band structure with the set of onsite energies $\epsilon_1 = \epsilon_2 = \epsilon_3 = 0$ and hopping parameters $t = h = 1$, and $t' = 0.5, 1, 1.5$.
• Figure 3: (a) Electronic band structure of a $pn$ junction for an electrostatic step potential. The curves correspond the energy bands in regions I and II. The dashed red line indicates the Fermi level, where gray regions are the occupied states. (b) Kinematical construction represents the conservation of energy (energy contours), linear momentum (dashed line), and current density (green semi-arcs). The arrows indicate the direction of linear momentum $\boldsymbol{k}_\textrm{in/r/t}$, pseudo-spin $\boldsymbol{s}_\textrm{in/r/t}$, and group velocity $\boldsymbol{v}_\textrm{in/r/t}$ for the scattering of electrons at the interface (red line).
• Figure 4: Electronic band structure of a bipolar $npn$ junction, which can be modeled through an electrostatic potential barrier. The curves correspond to energy bands; the blue area is the range of transmission from one of the conduction bands to the valence band.
• Figure 5: Arbitrary electrostatic potential modeled as a series of potential barriers, where the transfer matrix method can be used.