Strong single-ion anisotropy and anisotropic interactions of magnetic adatoms induced by topological surface states

The nature of the magnetism brought about by Fe adatoms on the surface of the topological insulator Bi${}_2$Se${}_3$ was examined in terms of density functional calculations. The Fe adatoms exhibit strong easy-axis magnetic anisotropy in the dilute adsorption limit due to the topological surface states (TSS). The spin-exchange J between the Fe adatoms follows a Ruderman-Kittel-Kasuya-Yosida behavior with substantial anisotropy, and the Dzyaloshinskii-Moriya interaction between them is quite strong with $|\mathrm{D}/\mathrm{J}|$ ≈ 0.3 under the mediation by the TSS and can be further raised to ∼0.6 by an external electric field. The apparent single-ion anisotropy of a Fe adatom is indispensable in determining the spin orientation.

Figure 1

(a) A top view of the pristine (111) surface of Bi${}_2$Se${}_3$. A and B represent the two possible adsorption sites for a magnetic Fe atom. The positions for various pairs of Fe atoms at the B sites are indicated by red segments with their optimized distances in units of angstroms. (b) A side view of a 1QL-slab with a single isolated Fe adatom showing the distorted local structure around the Fe atom. The spin of Fe (gray arrow) induces spin polarization to Bi${}^{*}$ and Se${}^{*}$, leading to small moments (red arrows/dark gray) on them. The spin-polarized spin moments of Bi${}^{*}$ and Se${}^{*}$ are antiparallel and parallel to that of Fe, respectively.

Figure 2

(a) The band structure calculated for the pristine 1QL-slab of Bi${}_2$Se${}_3$. The main surface components are marked in red/dark gray. (b)–(d) show the electronic structure of the 1QL-slab of Bi${}_2$Se${}_3$ with a single Fe atom adsorbed on the surface at the B site. (b) The band structure showing that the Fe 3d states interact strongly with the 6p states of the Bi${}^{*}$ atom and the 4p states of the Se${}^{*}$ atoms. (c) The PDOS plots for the 3d states of the adsorbed Fe atom. (d) An approximate crystal-field split pattern for the 3d states of the adsorbed Fe atom and their electron occupation.

Figure 3

The PDOS plots calculated for the p-states of Bi${}^{*}$ near the Fermi level before and after including SOC. In the absence of SOC, the $|{\mathrm{p}}_{\mathrm{z}},\uparrow \rangle$, $|{\mathrm{p}}_{\mathrm{x}},\downarrow \rangle$, and $|{\mathrm{p}}_{\mathrm{y}},\downarrow \rangle$ states located around the Fermi level are practically degenerate. These states are split in energy when SOC is included.

Figure 4

The split patterns of the three p-states of Bi${}^{*}$ near the Fermi level (namely, $|{\mathrm{p}}_{\mathrm{z}},\uparrow \rangle$, $|{\mathrm{p}}_{\mathrm{x}},\downarrow \rangle$, and $|{\mathrm{p}}_{\mathrm{y}},\downarrow \rangle$) under the effect of SOC when the spin orientation is along the $z$ axis (θ $=$ 0°) and perpendicular to the $z$ axis (θ $=$ $\frac{\pi }{2}$). (Note that the local $z$ axis is parallel to the $c$ axis.) With one electron to occupy the resulting three states, the SOC favors the $z$ axis orientation, i.e., //c SIA.

Figure 5

The SIA A vs ϕ curve calculated for the Fe adatom adsorbed on the B-site of Bi${}_2$Se${}_3$ (111), where the in-plane sweep angle ϕ is defined such that the direction for ϕ $=$ 0° is parallel to one Se${}^{*}$-Se${}^{*}$ edge of the Se${}^{*}$${}_3$ triangle.

Figure 6

The dependence of the spin-exchange J${}_{\mathrm{zz}}$ between two Fe atoms adsorbed on the (111) surface on the distance R between them, calculated with and without SOC effects.

Figure 7

The spin orientations predicted (a) in the absence and (b) in the presence of SIA by using the model Hamiltonian with the parameters determined from first-principles DFT calculations. (b) When considering SIA, a canted moment is left (red arrow) along the $x$ axis, and this canted moment could be delicately controlled by an external electric field.

Figure 8

The effect of an electric field $E$ along the $c$ direction on the magnetic interaction parameters and the difference charge density distribution of the (111) surface of Bi${}_2$Se${}_3$ with Fe atoms adsorbed at the nearest-neighbor B sites: (a) The dependence of the spin exchange, the DM vector, and the SIA term on $E$. (b) The difference charge density distribution, Δρ $=$ ρ($E$)$-$ρ(0), where ρ($E$) is the charge distribution under $E=-1.2V/\AA$ and ρ(0) is that without applied electric field. The negative field points down in this figure as shown with the red/dark gray arrow. The blue/medium gray area denotes charge loss, and the yellow/light gray area denotes charge accumulation.