Localized States due to Expulsion of Resonant Impurity Levels from the Continuum in Bilayer Graphene

The Anderson impurity problem is considered for a graphene bilayer subject to a gap-opening bias. In-gap localized states are produced even when the impurity level overlaps with the continuum of band electrons. The effect depends strongly on the polarity of the applied bias as long as hybridization with the impurity occurs within a single layer. For an impurity level inside the conduction band a positive bias creates the new localized in-gap state. A negative bias does not produce the same result and leads to a simple broadening of the impurity level. The implications for transport are discussed including a possibility of the gate-controlled Kondo effect.

Figure 1

(a) Bilayer graphene subject to the gap-opening bias $2U$. The resonant impurity (black circle) is hybridized with $\pi$ electrons of a carbon atom in the upper layer. Tunneling ${\gamma }_1$ is indicated by the vertical dashed lines. (b) Band structure of the system for different impurity energies ${\epsilon }_0$. The left picture corresponds to a zero hybridization amplitude $V$. The right one illustrates the modification of the spectrum for finite couplings $V$ and positive bias $U>0$. The level with ${\epsilon }_0>U$ (blue) broadens due to coupling to the band electrons while generating an additional in-gap state. The level originally positioned inside the gap (shown with green for a specific case ${\epsilon }_0=0$) remains localized but is renormalized towards lower energies. States overlapping with the valence band ${\epsilon }_0<-U$ (red) are broadened without generating any additional in-gap states. The level at the top of the valence band (orange) leads to a square-root singularity in the density of states.

Figure 2

(a) Dimensionless density of states (scaled with the coupling strength $\Gamma$) of the $d$ electrons plotted in units of $1/\Gamma$ for $\Gamma ={\gamma }_1/2$, ${\epsilon }_0=0.7{\gamma }_1$, and three different values of bias $U$: zero bias (black dashed line), positive bias [red (dark grey)], negative bias [green (light grey)]. The bound in-gap state appears as a sharp peak for $U>0$, but not for $U<0$. Inset: development of a square-root singularity when the impurity level is approaching the top of the valence band from the inside of it.