Bound state energy of a Coulomb impurity in gapped bilayer graphene

Application of a perpendicular electric field induces a band gap in bilayer graphene, and it also creates a “Mexican hat” structure in the dispersion relation. This structure has unusual implications for the hydrogen-like bound state of an electron to a Coulomb impurity. We calculate the ground state energy of this hydrogen-like state as a function of the applied interlayer voltage and the effective fine structure constant. Unlike in the conventional hydrogen atom, the resulting wave function has many nodes even in the ground state. Further, the electron state undergoes “atomic collapse” into the Dirac continuum both at small and large voltage.

Figure 1

(a)–(c) The dispersion relation $\varepsilon \left(k\right)$ of BLG, shown at (a) $V=0$, (b) $V<t$, and (c) $V>t$. Thick black lines show the valence and conduction bands, while thin gray lines show the outer bands, which are not relevant for this work. (d) The band edge $\Delta /2$ and the impurity state energy $\Delta /2+E$, plotted as a function of voltage for $Z\alpha =0.1$. (e) A schematic picture of the corresponding electron density ${\left|\psi \left(r\right)\right|}^2$, which has a strongly oscillating component with wave vector $2k_0$ and an exponentially decaying envelope.

Figure 2

The energy of Coulomb impurity states, measured relative to midgap, as a function of $V$ for different values of $Z\alpha$. Solid lines are calculated using a variational approach and the single-band approximation. Dashed lines are the result of Eq. (3). Different sets of curves are labeled by their corresponding value of $Z\alpha$. At $Z\alpha \gtrsim 0.26$ energy levels sink below midgap for all values of $V$.