Magnetic Impurities on the Surface of a Topological Insulator

The surface states of a topological insulator are described by an emergent relativistic massless Dirac equation in $2+1$ dimensions. In contrast with graphene, there is an odd number of Dirac points, and the electron spin is directly coupled to the momentum. We show that a magnetic impurity opens up a local gap and suppresses the local density of states. Furthermore, the Dirac electronic states mediate an RKKY interaction among the magnetic impurities which is always ferromagnetic, whenever the chemical potential lies near the Dirac point. Through this exchange mechanism, magnetic atoms uniformly deposited on the surface of a topological insulator could naturally form a ferromagnetically ordered film. These effects can be directly measured in STM experiments. We also study the case of quenched disorder through a renormalization group analysis.

Figure 1

(a) Charge local density of states (LDOS) as a function of the distance $r$ from the magnetic impurity for electron energies $E=30\mathrm{meV}$ (black line) and $E=70\mathrm{meV}$ (blue line). (b) Charge LDOS as a function of electron energy $E$ at positions $r=0$ (blue line) and $r=20\mathrm{nm}$ (red line). Here we assume a magnetic impurity strength $M_0=50\mathrm{meV}$ and a coupling range $r_0=13\mathrm{nm}$. The spin LDOS is plotted as a function of position at $E=10\mathrm{meV}$ for magnetization of the magnetic impurity placed at the origin (0,0) in (c) the $z$ direction and (d) the $y$ direction. Here the arrow indicates the in-plane spin LDOS and the color shows the $z$ direction spin LDOS.

Figure 2

RKKY interaction versus the distance $R$ between two magnetic impurities. Fermi momentum $k_F$ is chosen to be $0.5/a_0$ for the blue line, $1.0/a_0$ for the green line and $1.5/a_0$ for the red line, where $a_0$ is the lattice constant.