When a static electrical field is applied to a two-dimensional (2D) Dirac material, Landau-Zener transition (LZT) and Bloch-Zener oscillations can occur. Employing $\alpha \text{−}{\mathcal{T}}_3$ lattices as a paradigm for a broad class of 2D Dirac materials, we uncover two phenomena. First, due to the arbitrarily small energy gaps near a Dirac point that make it more likely for LZTs to occur than in other regions of the Brillouin zone, the distribution of differential LZT probability in the momentum space can form a complicated morphological pattern. Second, a change in the LZT morphology as induced by a mutual switching of the two distinct Dirac points can lead to irregular Bloch-Zener oscillations characterized by a nonsmooth behavior in the time evolution of the electrical current density associated with the oscillation. These phenomena are due to mixed interference of quantum states in multiple bands modulated by the geometric and dynamic phases. We demonstrate that the adiabatic-impulse model describing Landau-Zener-Stückelberg interferometry can be exploited to calculate the phases, due to the equivalence between the $\alpha \text{−}{\mathcal{T}}_3$ lattice subject to a constant electrical field and strongly periodically driven two- or three-level systems. The degree of irregularity of Bloch-Zener oscillations can be harnessed by selecting the morphology pattern, which is potentially experimentally realizable.

• Received 7 January 2023
• Revised 6 April 2023
• Accepted 7 April 2023

DOI:https://doi.org/10.1103/PhysRevB.107.165422