Fig. 1. (Color online) Schematics of (a) a massless (m = 0) and (b) a massive (m ≠ 0) surface state of a 3D TI as for the time-reversal symmetry (TRS) broken by the introduction of effective magnetic interaction into the system.

Fig. 2. (Color online) STM image of Mn doped Bi₂Te₃ (x = 0.09). Adapted from Ref. [50]. (a) STM topograph of Bi_1.91Mn_0.09Te₃ (001) surface, +250 meV, 40 pA, 1000 × 1000 Ų. Substitutional Mn atoms appear as triangular suppressions of the LDOS. (b) and (c) Zoom-in topographies over Mn dopants of unoccupied (+500 mV, 30 pA) and filled states (−500 mV, 30 pA), 30 × 30 Ų. Reprinted with permission from Ref. [50].

Fig. 3. (Color online) Adapted from Ref. [42]. (a) X Ray Diffraction of undoped and Mn-doped Bi₂Te₃ thin films. (b) A high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) image of (Bi_1–xMn_x)₂Te₃ thin film (5% Mn concentration). Dotted yellow lines indicate QLs and unit layers composed of a Bi bilayer sandwiched between two QLs. (c) Atomic crystal structures of QL–Bi₂–QL. (d) Atomic crystal structures of Bi₂Te₃ QLs with Bi partially substituted by Mn. Reprinted with permission from Ref. [42].

Fig. 4. (Color online) A proposed process for the self-assembly of Bi₂Se₃ layers interspersed with septuple Bi₂MnSe₄. Adapted from Ref. [44]. (a) Bi, Se, and Mn atoms arrival at the growth surface. (b) Bi₂Se₃ forms thermodynamically while Mn atoms remain diffuse until pairing with Se atoms. (c) Self-assembling of Bi₂MnSe₄ SLs as interspersing between Bi₂Se₃ QLs. STEM image of 2.5% (d) and 4.2% Mn doped Bi₂Se₃ (e) showing the layered structure of Mn doped Bi₂Se₃. Reprinted with permission from Ref. [44].

Fig. 5. (Color online) HR-STEM images of Mn doped Bi₂Te₃ and Bi₂Se₃^[38], showing the layered heterostructure consisting of Bi₂MnTe₄ (Bi₂MnSe₄) SLs inserted between Bi₂Te₃ (Bi₂Se₃) QLs. Reprinted with permission from Ref. [38].

Fig. 6. (Color online) ARPES shows gap opening in the Dirac surface states of Mn doped Bi₂Se₃^[36]. (a) ARPES spectra of of (Bi_0.99Mn_0.01)₂Se₃ single crystal along K–Γ–K. Inset is a close-up of the dispersion in the vicinity of E_F, indicating a gap between the leading edge of the surface state band and E_F. (b) A leading-edge gap of 7 meV by comparison between the Γ point EDC and E_F. Reprinted with permission from Ref. [36].

Fig. 7. (Color online) Spin-resolved ARPES of Mn doped Bi₂Se₃. Adapted from Ref. [37]. (a) Spin-integrated data and (b) corresponding MDCs on film I (20 eV photons, MDC mode). (c) Spin-integrated dispersion and corresponding EDCs on film II (9 eV photons, EDC mode). Reprinted with permission from Ref. [37].

Fig. 8. (Color online) ARPES measurements of (Bi_1–xMn_x)₂Se₃ with different Mn doping and temperature. Adapted from Ref. [60]. (a–d) Mn doping-dependent ARPES for x values of (a) 0, (b) 0.02, (c) 0.04 and (d) 0.08, 50 eV photon energy, 12 K. The surface band gap increases with increasing Mn content. (e, f) ARPES dispersions of (Bi_1–xMn_x)₂Se₃ (x = 8%) at temperature of (e) 12 K and (f) 300 K. The surface band gap does not show a remarkable temperature dependence. Reprinted with permission from Ref. [60].

Fig. 9. (Color online) Magnetic gap of Mn-doped Bi₂Te₃ derived by ARPES. Adapted from Ref. [38]. (a–d) Measurements for Bi₂Te₃ with 6% Mn performed above and below the Curie temperature T_C ~ 10 K. Linear fits to the regions indicated in (c) yield shifts of 21 and 12 meV between these sections of the 20 and 1 K spectra. (d) Simulation showing that this corresponds to a magnetic gap = 90 ± 10 meV. (e–g) Same analysis for Mn doped Bi₂Se₃ with 6% Mn and a T_C of 6 K, revealing only a nonmagnetic gap of 220 ± 5 meV at 20 K and 205 ± 5 meV at 1 K, determined by least-square fit to the upper Dirac cone and to the lower Dirac cone at k_// = 0 Å^–1. Reprinted with permission from Ref. [38]

Fig. 10. (Color online) Magnetic-field-dependent magnetization of (Bi_1–xMn_x)₂Te₃ crystal (x = 0.045) with in-plane and out-of-plane fields, T = 1.8 K. Inset is the low field hysteresis MH loops of this Bi_1.91Mn_0.09Te₃ crystal. Reprinted with permission from Ref. [50].

Fig. 11. (Color online) SQUID measurements of (Bi_1–xMn_x)₂Se₃. Adapted from Ref. [45]. (a) Temperature-dependent magnetization curves (M–T) of the Bi/Mn = 12.5 sample. (b) Field-dependent magnetization plots (M–H) of the Bi/Mn = 12.5 sample with in-plane field at different temperatures. (c) M–H of the Bi/Mn = 12.5 sample with out-of-plane field at different temperatures. (d) M–T of the Bi/Mn = 23.6 sample. (e) M–H of the Bi/Mn = 23.6 sample with in-plane field at different temperatures. (f) M–H of the Bi/Mn = 23.6 sample with out-of-plane field at different temperatures. Reprinted with permission from Ref. [45].

Fig. 12. (Color online) Magnetization M(H) and Anomalous Hall effect (AHE) of Mn doped Bi₂Te₃ and Bi₂Se₃. Adapted from Ref. [38]. In-plane and out-of-plane M(H) of Bi₂Te₃ (a) and Bi₂Se₃ (b) films with Mn concentrations of 3 and 4% measured at 2 K by SQUID with the magnetic field either parallel or perpendicular to the surface, evidencing a perpendicular anisotropy (easy axis) for Bi₂Te₃ and an in-plane easy axis for Bi₂Se₃. The Curie temperature as a function of Mn concentration is depicted in the inserts, evidencing that T_C is significantly higher in the telluride system. (c, d) AHE measurements of the samples with the contribution of the ordinary Hall effect extracted from the high field data subtracted. Due to the perpendicular magnetic anisotropy, only Mn-doped Bi₂Te₃ displays a pronounced anomalous Hall effect appearing when the sample is cooled below T_C. Reprinted with permission from Ref. [38]

Fig. 13. (Color online) AHE and T_C of Mn_xBi_2–xTe_3–ySe_y single crystal (x = 0.04 and y = 0.12) at different carrier density tuned by gating. Adapted from Ref. [65]. (a) Hall conductivity σ_xy of device A at different back-gate voltages V_B. AHE increases with depleting carriers. (b) After application of a top-gate voltage V_T = –3 V, device B shows an enhanced σ_xy. The ordinary Hall conductivity changes from n-type to p-type at the most negative V_B. (c) Temperature-dependent σ_xy of device C. (d) T_C of devices A–E on the carrier density. Reprinted with permission from Ref. [65].

Fig. 14. (Color online) Magnetoconductivity of Mn_xBi_2–xTe_3–ySe_y (x = 0.04 and y = 0.12) with gating. Adapted from Ref. [65]. (a) σ_xx(B) at different V_B. (b) σ_xx(B) at V_B = –100 V. (c) Schematic of the domain structure in a magnetic topological insulator. A chiral mode appears in the domain walls across the opposite M domains. (d) σ_xx(B) at different temperatures at V_B = –100 V. (e) σ_xx(B) shows hysteresis at high carrier density. (f) The difference between the virgin and trained σ_xx. Reprinted with permission from Ref. [65].

Fig. 15. (Color online) Hall conductivity σ_xy and longitudinal conductivity σ_xx of Mn-Bi₂Te₃ films. Adapted from Ref. [42]. (a)–(c) Temperature dependence of σ_xy with Mn doping 2% (S3), 5% (S4), and 10% (S5). (d) Schematic of the Hall device. (e) Photo image of a Hall bar. (f) σ_xy with different Mn doping, T = 0.5 K. (g) Temperature dependence of σ_xy at zero magnetic field with different Mn concentrations. (h) Temperature dependence of σ_xx. (i) σ_xx (red) and σ_xy (black) at 0.5 K. Reprinted with permission from Ref. [42].

Fig. 16. (Color online) Magneto-transport of Mn-Bi₂Se₃ thin films. Adapted from Ref. [45]. (a) Measurement geometry. (b) –H curves at different field directions in xz plane. (c) R_xx–H plots under field (θ = 5° and ϕ = 0) at different temperatures. (d) R_xx–H in xy plane at different temperatures. Reprinted with permission from Ref. [45].

Fig. 17. (Color online) Evolution of the Hall effect and the corresponding AH resistances with Mn concentration (a) and gate-voltage tuning (b). Adapted from Ref. [61]. The magnetic field dependences of the Hall resistance R_yx and the nonlinear part of the Hall resistance (R_AH(B) = R_yx(B) – R_HB) are shown in the top panels and bottom panels of (a) and (b) respectively. The AH resistance R_AH is separated into a positive component (orange line) and a negative one (blue line). The right panels show the schematic band diagrams of the Fermi level changing with doping and gate-voltage tuning.

Fig. 18. (Color online) Characteristics of the AH conductivity in a lightly doped (Bi_1−xMn_x)₂Se₃ sample (x = 0.02). Adapted from Ref. [61]. (a) and (b) σ_xx dependences of the magnitudes of the positive and negative AH conductivities above the saturation fields, σ_AH,1 (panel a) and σ_AH,2 (panel b). (c) The ratio of the AH components, σ_AH,2/σ_AH,2, plotted as a function of σ_xx. Inset shows the schematic band diagrams for high and low Fermi levels.

Table 1. Magnetic characteristics of Mn doped TIs.

Table 2. The AHE sign in magnetically doped Tis.