Magnetic phase transition in Fe-doped topological insulator $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$

We study the effect of Fe impurities in $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ on the magnetic phase and topological insulating property using first-principles calculations. In particular, we investigate the ferromagnetic-antiferromagnetic phase transition and the energy gap variation of surface states in Fe-doped $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$. We find that Fe-doped $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ has a ferromagnetic phase at dilute doping regime by the interplay of the band inversion and intrinsic doping. For higher Fe concentration, $>1.7\mathrm{at}.\%, \mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ prefers the antiferromagnetic phase mediated by the superexchange interaction. We show that neighboring Fe impurities with antiferromagnetic ordering behave like nonmagnetic scattering centers that preserve the linear band dispersion and the in-plane spin texture of topological surface states. Our result indicates that interdependency of the magnetic phase and the band topology in transition-metal–doped topological insulators may tweak the electronic structure and topological surface states in peculiar ways.

Figure 1

(a)–(d) Calculated band structures of (a,c) pristine and (b,d) Fe-doped (1 $\mathrm{at}.\%$) $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$; calculations (a, b) without and (c, d) with SOC. The Fermi level is denoted by the dashed line at zero energy. (e) The first Brillouin zone for the bulk and the surface projection with the special $k$ points. (f) Calculated total energy for varying Fe distance relative to the case of a dilute regime (Fe distance of $10\AA$).

Figure 2

(a) Calculated spin gap (the energy difference between FM and AFM phase) as a function of inter-Fe distance ($D$); positive (negative) if AFM (FM) phase is the ground magnetic state. Blue dots (red triangles) for Fe-doped $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ without (with) $n$ doping. (b, c) Calculated band structure of Fe-doped $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ with inter-Fe distance of 28.5 and $4.6\AA$, respectively. (d, e) The states near the Fermi level for Fe distance of $4.6\AA$ calculated with the GGA and MBJ potentials, respectively. Red (Blue) dots denote Bi (Te) atomic character.

Figure 3

(a) Calculated total and formation energies of Fe-doped $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ for varying Fe positions from the top layer. The energy reference is the value for Fe at the topmost layer. Fe impurities prefer the inner layers in $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$. (b) Calculated energy gap of TSS for varying Fe position from the surface. The gap is the maximum when Fe impurities are located at the TSS-localized layer. (c) From the left to right panels, calculated band structures of $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ slab structure without defects, with arsenic doping, and with Fe doping. $Z$ is the distance of the impurity from the topmost Se layer. The shaded region denotes the bulk band projection into the surface Brillouin zone.

Figure 4

Band structures of $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ slab calculated with first-principles method for (a) a single atomic Fe defect, (b) AFM- and (c) FM-ordered Fe dimers with inter-Fe distance of $4.11\AA$, and (d) AFM-ordered Fe dimers with inter-Fe distance of $9.16\AA$. (e) In-plane spin texture of the upper Dirac cone in (b). (f, g) Band structures of the tight-binding Hamiltonian for a single atomic Fe defect and AFM Fe dimers, respectively. The parameters chosen from the first-principles results are $V=0.06\mathrm{eV}, {\lambda }_{so}=0.03\mathrm{eV}$, and ${\varepsilon }_{\tau }=\pm 0.3\mathrm{eV}$. (h) Schematics of TSS (blue and red arrows) interacting with an AFM-ordered Fe dimer (gray balls with spins in arrows).