Klein tunneling in deformed honeycomb-dice lattice: from massless to massive particles

TL;DR

This work studies Klein tunneling across sharp np junctions in a deformed α-T3 lattice, where uniaxial compression moves the Dirac cones toward each other and eventually opens a gap while the flat band remains intact. The authors derive a unified low-energy Hamiltonian that captures massless, semi-Dirac, and massive regimes through a deformation parameter Δ=(λ-2)t and an effective mass m, linking Dirac-point merging to transport properties. They show that KT persists in the Dirac phase for all α, while SKT persists in the dice lattice; as the system enters the gapped phase, all α exhibit anti-Klein tunneling (AKT), with α=1 also displaying anti-super-Klein tunneling (ASKT). Additionally, rotating the np junction relative to the deformation axis in the semi-Dirac phase induces KT→AKT transitions for all α, with abrupt AKT↔KT and ASKT↔SKT switches for α=1. Overall, the work reveals how deformation-induced mass generation and junction orientation jointly control tunneling regimes, offering tunable electron-optics functionalities in α-T3-like systems.

Abstract

We show that under compressive uniaxial deformation of the three-band $α-T_3$ lattice, the Dirac cones move toward each other, merge, and a gap opens, while the flat band remains unchanged. Consequently, the low-energy spectrum transitions from linear to quadratic dispersion, indicating the shift from massless to massive Dirac particles. Here, we theoretically investigate the tunneling properties of particles through a sharp $np$ junction in a deformed $α-T_3$ lattice, focusing on the case where the particle energy is half the junction height. We show that this transition from massless to massive particles leads to a change from omnidirectional total transmission, known as super-Klein tunneling, to omnidirectional total reflection, referred to as anti-super-Klein tunneling, in the case of the dice lattice ($α=1$). For all values of $α$, this transition manifests as a change from conventional Klein tunneling to anti-Klein tunneling.

Figures (7)

• Figure 1: (Color online) Schematic representation of the deformed $\alpha-T_3$ lattice. Each unit cell contains three sites, $A$, $B$, and $C$. The hopping amplitudes between $A$ and $B$ sites are $\lambda t \cos\varphi$ ($\lambda >1$) along the deformation direction (red thick lines) and $t \cos\varphi$ along the other directions (black thin lines). The hopping amplitude between $B$ and $C$ sites is $\lambda t \sin\varphi$ along the deformation direction (red dashed thick lines) and $t \sin\varphi$ (black dashed thin lines) along the other directions.
• Figure 2: (Color online) Energy spectrum of the deformed $\alpha-T_3$ model for different values of $\Delta$. The black lines represent the particle energy $E$. Dirac phase: $\Delta<\Delta_o<0$ and $E<<\left |\Delta \right |$, where the spectrum consists of two anisotropic Dirac cones and a flat band. Intermediate phase: $\Delta_o <\Delta \leq 0$ with $E\sim\left |\Delta \right |$, characterized by the coexistence of massless and massive particles along the $x$ direction. Semi-Dirac phase: $E > \Delta = 0$. The dispersion is linear along the $k_y$ direction (massless particles) and quadratic along the $k_x$ direction (massive particles), with a flat band at $E=0$. Gapped phase: $E > \Delta > 0$, where the spectrum is fully gapped, with a flat band at $E=0$ and the particles are massive. In all cases with $\Delta \leq 0$, particles remain massless along the $y$ direction.
• Figure 3: (Color online) Schematic of transmission and reflection probabilities for an $np$ junction along the deformation axis ($y$ direction) at $E = V_0/2$. (a) For $\Delta/E<-1$, the Fermi surfaces are disconnected. For a given parallel wave vector $k_y$ (horizontal dashed line), there are four propagating modes (red arrows) corresponding to four transmission and four reflection probabilities. The black arrows indicate the group velocities, showing the direction of propagation. (b) For $0<\Delta/E<1$, the Fermi surfaces are connected. For a given $k_y$, there is a single transmission and a single reflection probability. (c) For $-1 < \Delta/E < 0$, the Fermi surfaces are partially connected. For $k_{\mathrm{in}} < |k_y| < k_{\mathrm{max}}$, there are four transmission and four reflection probabilities as in (a), while for $|k_y| < k_{\mathrm{in}}$, there is only one transmission and one reflection probability, as in (b).
• Figure 4: (Color online) (a) The conductance in units of $G_x^{SKT}$ as a function of $\Delta/E$ and for the three values of $\alpha$. (b) Transmission probability at normal incidence, independent of $\alpha$. The three phases (DP, IP, GP, as in Fig. \ref{['figure2']}) are indicated. Here, $G_x^{SKT}$ is the conductance of the SKT regime in the Dirac phase [Eq. (\ref{['condskt']})] and the step height is set to $V_o=0.08 t$.
• Figure 5: (Color online) Schematic representation of the transmission probability in the semi-Dirac phase across an $np$ junction oriented at an angle $\beta$ relative to the deformation axis (magenta dashed line). Blue arrows indicate the group velocities, and the horizontal dashed lines show the conserved parallel wave vector, $k_\parallel$.