Functionalized graphene in quantizing magnetic field: The case of bunched impurities

Resonant scattering at the atomic absorbates in graphene was investigated recently in relation with the transport and gap opening problems. Attaching an impurity atom to graphene is believed to lead to the creation of unusual zero-energy localized electron states. This paper aims to describe the behavior of the localized impurity-induced levels in graphene in a quantizing magnetic field. It is shown that in the magnetic field the impurity level effectively hybridizes with one of the $n=0$ Landau level states and splits into two opposite-energy states. The new hybridized state is doubly occupied, forming a spin singlet and reducing the polarization of a quantum Hall ferromagnet in undoped graphene. Taking into account the electron-electron interaction changes radically the spectrum of the electrons surrounding the impurity, which should be seen experimentally. While existing publications investigate graphene uniformly covered by adatoms, here we address a possibly even more experimentally relevant case of the clusterized impurity distribution. The limit of a dense bunch of the impurity atoms is considered, and it is shown how such a bunch changes the spectrum and spin polarization of a large dense electron droplet surrounding it. The droplet is encircled by an edge state carrying a persistent current.

Figure 1

Finding graphically the energy from the first equation (12). (Energy in units of ${\varepsilon }_B$ and $V_d=1$.) Straight solid line with small positive slope shows the left-hand side of the equation. Other solid lines show the right-hand side of Eq. (12), having a pole at $\varepsilon =0$ and other poles at each $\varepsilon =\pm \sqrt{n}{\varepsilon }_B$. Each crossing of two solid lines corresponds to an energy level. Impurity-induced localized states ${\Psi }_{S_{\pm }}$ correspond to two crossings closest to the zero energy. In addition, there are two crossings far to the right and far to the left from the energy segment shown in the figure (well outside the energy band of graphene). These crossings correspond to the bonding and antibonding states of the impurity atom and a single carbon atom closest to it. They are responsible for the creation of the true chemical bond between the two atoms. Thick dashed line with a single pole at $\varepsilon =0$ shows the approximate form of the energy equation (12) right-hand side described in Appendix pp1.

Figure 2

Radial dependence of the electron density at two sublattices ${\rho }_A$ and ${\rho }_B$ for the impurity-induced state ${\Psi }_{S_{-}}$ [Eq. (14)]. The density is doubled due to two spin components and normalized such that a single occupied Landau level has $\rho =1$. For drawing we choose $L=ln(l/d)=5$. The thick dashed line shows the density ${\rho }_{n=0}$ of the $n=0$ Landau level from sublattice $A$, valley $K$, with one spin orientation and having angular momenta $m=1,2,3,...$ (the states not affected by the impurity). Thin dashed lines show how this density is built by adding one by one electrons with $m=1,m=2$, etc. Together ${\rho }_A$ and ${\rho }_{n=0}$ give a constant charge density of a fully occupied Landau level. However, these constant density electrons have a very nontrivial exchange interaction, discussed in Sec. 3. The singular increase of density in state ${\Psi }_S$ at sublattice $B,{\rho }_B\sim 1/r^2$, is fully compensated by the decrease of density in Landau levels with $n>0$, as follows from the particle-hole symmetry.

Figure 3

Diagrammatic presentation of the exchange energy. Wavy line shows the Coulomb interaction $e^2/|\mathbf{r}-{\mathbf{r}}^{\prime }|$. Summation over the complete set of states in the first term in the right-hand side gives unity, or $\delta$ function.

Figure 4

Energies (schematic) of the $n=0$ Landau level states around the impurity with and without electrons interaction. Small rhombuses: show the energy levels for the noninteracting problem, filled red rhombuses for filled and empty blue rhombuses for empty states. For each $m>0$ there are two valley degenerate states $(K$ and $K^{\prime })$, each with two spin components, split due to the small Zeeman energy $\pm E_Z$. For $m=0$ there are total of six low-energy states. A pair of levels: spin split $n=0,m=0,K^{\prime }$-valley level behaves exactly the same way as its $m\ne 0$ companions. In addition, there are a pair of spin-split low-energy levels ${\Psi }_{S_{\pm }},{\varepsilon }_{S_{\pm }}$ [Eq. (14)]. Both spin components of ${\varepsilon }_{S_{-}}$ are occupied, forming a singlet. Big circles: show the energy levels for interacting electrons. Again, filled red circles show occupied states and empty blue circles show unoccupied states. States from the $K^{\prime }$ valley are not affected by the impurity independent on the value of $m$. Their energies are given by Eq. (20), which is shown by two dashed lines. Occasionally, the exchange interaction causes the same shift of the ${\varepsilon }_{S_{\pm }}$ levels, as it did for $n=0$ Landau level states in the absence of impurity. These states are shown by the two lowest red circles and by the two highest empty blue circles at $m=0$. The most interesting are the states with small, but finite values of the angular momentum number $m$ from the $K$ valley. Due to their reduced exchange interaction, these states fall inside the gap between the usual spin-up and -down $n=0$ Landau levels shown in Eq. (20).

Figure 5

Right: a large droplet of reduced spin polarization surrounding a bunch of impurities. Left: schematic angular momentum dependence of the energy of impurity-induced states (red circles) and the energy of the $n=0$ sublattice $A$ electron states surrounding the droplet (blue rhombuses). Due to their nonuniform exchange energy, these latter states carry an edge current around the droplet.