Defect bound states in the continuum of bilayer electronic materials without symmetry protection

Abstract

We analyze a class of bound defect states in the continuum electronic spectrum of bilayer materials, which emerge independent of symmetry protection or additional degrees of freedom. Taking graphene as a prototypical example, our comparative analysis of AA- and AB-stacked bilayer graphene demonstrates that these states originate from the intrinsic algebraic structure of the tight-binding Hamiltonian when trigonal warping is neglected rather than any underlying symmetry. Inclusion of trigonal warping and higher-order hoppings broaden the bound states into long-lived resonances. This discovery provides a pathway to previously unexplored approaches in defect and band-structure engineering. We conclude with a proposed protocol for observing these states in scanning tunneling microscopy experiments.

Figures (1)

• Figure 1: Bound state in the continuum of Bernal stacked graphene for $\gamma_1\!=\!0.103$, $\gamma_{3}\!=\!0$, $\gamma_{4}\!=\!0.041$, and $\Delta\!=\!0.034$. (a) Disperions relation for AB-stacked graphene near the $K$-point; all energy units are normalized by intra-layer hopping $\gamma_{0} = 2.9$ eV from Ref. MalardPimenta2007 and all momentum units are normalized by $a^{-1}$ for graphene lattice spacing $a$. The band in dashed blue (solid red) is associated to the factor $D_{1}$ ($D_{2}$). The horizontal line indicates an energy within a $D_{2}$-band and a $D_{1}$-gap for which a bound defect state exists. (b) The potential strengths on the AB-site needed to bind a defect state at the specified energy $E$; note that while the potential can be arbitrarily small, this happens exponentially close to the $D_{1}$ band edge ($E\approx 0.11$). How a pair of potentials on the AB-sites maps to a $(E,\Delta)$ pair is shown in the supplement supplement. (c) At $\gamma_{3} = 0$ with potentials $V(1B)\!=\!-18.7$ and $V(2A)\!=\!-7.3$, we bind an exponentially localized state at $E=0.07$ within the $D_{2}$-band. For $\gamma_{3}>0$, the defect state becomes a quasi-resonance as indicated by the widening of the density of states $\Delta\nu(E)$. Density of states is calculated with Lorentzian smearing $\eta = 0.00125$ as detailed in the supplement supplement. Dotted gray line indicates the experimental value in Ref. MalardPimenta2007.