Stability, electronic, and magnetic properties of the magnetically doped topological insulators ${\text{Bi}}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$

Magnetic interaction with the gapless surface states in a topological insulator (TI) has been predicted to give rise to a few exotic quantum phenomena. However, the effective magnetic doping of TI is still challenging in the experiment. Using first-principles calculations, the magnetic doping properties (V, Cr, Mn, and Fe) in three strong TIs (Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$) are investigated. We find that for all three TIs the cation-site substitutional doping is most energetically favorable with the anion-rich environment as the optimal growth condition. Further, our results show that under the nominal doping concentration of 4%, Cr- and Fe-doped Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Cr-doped Sb${}_2$Te${}_3$ remain as insulators, while all the V- and Mn-doped TIs, and Fe-doped Sb${}_2$Te${}_3$ become metal. We also show that the magnetic interaction of Cr-doped Bi${}_2$Se${}_3$ tends to be ferromagnetic, while Fe-doped Bi${}_2$Se${}_3$ is likely to be antiferromagnetic. Finally, we estimate the magnetic coupling and the Curie temperature for the promising ferromagnetic insulator (Cr-doped Bi${}_2$Se${}_3$) by Monte Carlo simulation. These findings may provide important guidance for the magnetism incorporation in TIs experimentally.

Figure 1

(a) Crystal structure illustration of a $2\times 2\times 1$ supercell for modeling a single dopant in bulk Bi${}_2$Se${}_3$, with (b) for side view and (c) for top view. (d) Brillouin zone and high symmetry points of the $2\times 2\times 1$ supercell. There are two types of Se atoms, including Se1 involving the van der Waals bonding and the other Se2.

Figure 2

The formation energy $▵H$ as a function of anion chemical potential for all the possible intrinsic defects in (a) Bi${}_2$Se${}_3$, (b) Bi${}_2$Te${}_3$, and (c) Sb${}_2$Te${}_3$. $V_{\mathrm{Bi}}$, $V_{\mathrm{Sb}}$, $V_{\mathrm{Se}}$, and $V_{\mathrm{Te}}$ stand for bismuth vacancy, antimony vacancy, selenium vacancy, and tellurium vacancy, respectively, while Bi${}_{\mathrm{Se}}$, Bi${}_{\mathrm{Te}}$, Sb${}_{\mathrm{Te}}$, Se${}_{\mathrm{Bi}}$, and Te${}_{\mathrm{Bi}}$ are antisites' defects. 1 and 2 are labeled to distinguish Se (Te) in different layers. $X$ rich ($X$ for Bi, Sb, Se, or Te) indicates the extreme growth condition with ${\mu }_X$ taking the maximum value in Eq. (1). Vertically dotted lines highlight the boundary of carrier types.

Figure 3

(a) Possible interstitial sites for Fe in Bi${}_2$Se${}_3$ after relaxation. (b) The formation energy $▵H$ as a function of anion chemical potential (${\mu }_{\mathrm{Se}}$) for interstitial Fe and Fe substitutional for Bi in Bi${}_2$Se${}_3$. Fe${}_{i1}$, Fe${}_{i2}$, and Fe${}_{i3}$ stand for different sites for interstitial Fe in (a), while Fe${}_{\mathrm{Bi}}$ denotes that the Bi atom is doped by the Fe atom.

Figure 4

Calculated formation energies of the most stable configurations of single V, Cr, Mn, and Fe impurities doped (a) Bi${}_2$Se${}_3$, (b) Bi${}_2$Te${}_3$, and (c) Sb${}_2$Te${}_3$ as a function of the host element chemical potentials.

Figure 5

Calculated defect formation energy $▵H$ as a function of Fermi energy $E_F$ for magnetically doped Bi${}_2$Se${}_3$ under (a) Bi-rich condition and (b) Se-rich condition; Bi${}_2$Te${}_3$ under (c) Bi-rich condition and (d) Te-rich condition and Sb${}_2$Te${}_3$ under (e) Sb-rich condition and (f) Te-rich condition. Here, $E_F$ is referenced to the valence-band maximum ($E_V$) in the bulk. The shaded areas highlight band gaps, where $E_F$ can range, as we mainly focus, from VBM ($E_F=0$) to CBM ($E_F=Eg$). Calculated values of band gaps with SOC for Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$ are 0.32, 0.15, and 0.12 eV, respectively. Solid dots denote the defect transition energies between different charge states.

Figure 6

Schematic representation of the thermodynamic transition levels for magnetically doped (a) Bi${}_2$Se${}_3$, (b) Bi${}_2$Te${}_3$, and (c) Sb${}_2$Te${}_3$, which correspond to the solid dots in Fig. 5. The transition levels shown are referenced to valence-band maximum (VBM) in the bulk. The distances of $\varepsilon (3+/2+)$ to VBM, for example, indicate the thermal ionization energies of simple acceptors. The shaded areas highlight band gaps of Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$, as shown in Fig. 5.

Figure 7

SOC band structures for relaxed V-, Cr-, Mn-, and Fe-doped Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$ with the nominal doping concentration of 4$\%$ ($x=0.083$). The size of the red dots denotes the contribution of TM-$d$ states.

Figure 8

Dynamics illustration of the magnetically doped Bi${}_2$Se${}_3$. Note that those Se atoms neighboring the dopants tend to be close to the dopants, while Bi atoms hardly move.

Figure 9

Monte Carlo simulation of Bi${}_{2-x}$Cr${}_x$Se${}_3$ with $x=0.074$. $M$ is the simulated magnetization and $U_L$, the Binder cumulant, is the normalized fourth-order cumulant of the magnetization. The Curie temperature is estimated at about 76 K.