Electrically Controllable Surface Magnetism on the Surface of Topological Insulators

We study theoretically the RKKY interaction between magnetic impurities on the surface of three-dimensional topological insulators, mediated by the helical Dirac electrons. Exact analytical expression shows that the RKKY interaction consists of the Heisenberg-like, Ising-like, and Dzyaloshinskii-Moriya (DM)-like terms. It provides us a new way to control surface magnetism electrically. The gap opened by doped magnetic ions can lead to a short-range Bloembergen-Rowland interaction. The competition among the Heisenberg, Ising, and DM terms leads to rich spin configurations and an anomalous Hall effect on different lattices.

Figure 1

(a) Schematic of two local spins on the surface of TI, with electrically controllable RKKY interaction. The 3D topological thin film is epitaxially grown on ${\mathrm{SrTiO}}_3$ substrates [10]. (b) The massless and massive surface state with spin orientation. The $2\Delta$ denotes the gap caused by breaking of TRS. (c) The warping effect of ${\mathrm{Bi}}_2{\mathrm{X}}_3$ class TI with spin orientation.

Figure 2

The range function of RKKY interaction as a function of the distance between localized spins $R$ [(a),(c)] or the Fermi energy ${\epsilon }_F$ [(b),(d)]. The inset shows the intrinsic case (${\epsilon }_F=0$). (a),(b) depict the RKKY interaction with massless Dirac electrons, and (c),(d) the RKKY interaction with massive electrons ($\Delta =25\mathrm{meV}$) [13]. (e) is the range functions of the BR interaction, and (f) the absolute values of BR range functions with logarithmic coordinates. The blue (solid) lines are the range function $F_1(R,{\epsilon }_F)$, the red (long-dashed) lines are the $F_2(R,{\epsilon }_F)$, the green (short dashed) ones are $F_3(R,{\epsilon }_F)$, and the cyan (dash-dotted) ones are $F_4(R,{\epsilon }_F)$. We choose ${\epsilon }_F=100\mathrm{meV}$ for (a),(c), $R=30\mathrm{nm}$ for (b),(d), and ${\epsilon }_F=-18\mathrm{meV}$ for (e),(f). The zero point of energy is at the midpoint of the energy gap caused by breaking TRS in (c)–(f).

Figure 3

Schematic chiral spin configurations of spin systems in the square and triangular lattices.