Abstract
1. Introduction
In recent years, the research on topological insulators (TIs) has attracted much attention in the field of condensed matter physics. Conventional phases of materials are classified according to broken symmetries, while TIs are classified based on topological invariants determined by their band properties[
The history of topological matter began in the 1980, when the quantum Hall effect (QHE) was first observed by Von Klitzing[
The first 3D TI state was experimentally realized in the bismuth–antimony alloy system Bi1−xSbx[
The surface states of a 3D TI can be described with the Hamiltonian
Figure 1.(Color online) Schematics of (a) a massless (
When the TRS in 3D TIs is broken, for instance, by applying an out-of-plane magnetic field, introducing ferromagnetic order with perpendicular magnetization, or bring the sample in proximity to a magnetic insulator, a gap will open in the surface states[
in which the mass m is determined by Zeeman and/or magnetic exchange interaction. The corresponding surface states is shown in Fig. 1(b). Both the Berry phase and spin structure are modified due to the magnetic interaction. This leads to many interesting transport properties, such as the QAHE when the Fermi level is located in the mass gap[
Magnetic doping turned out to be an effective approach to break the TRS in 3D TIs to date. The seminal discovery of QAHE was first accomplished in Cr-doped (Bi,Sb)2Te3[
Concerning the properties of Mn doped TIs, the current studies are still in a fledgling stage. So far, even nominally identically prepared samples show a diversity of dopants sites, electronic states, magnetic transition temperatures, saturation magnetizations, and anisotropies. Mn dopants tend to enter the TI hosts not only substitutionally for Bi, which is the energetically most favorable site according to the calculation[
In this review paper, we focus on the experimental progress of Mn doped 3D TIs materials. The paper is organized as follows. Section 2 overviews the results of properties and characteristics on the aspects of structure, electronic, magnetic, and transport properties respectively. In Section 3, we show our recent findings of the two-component AHE in Mn-doped Bi2Se3 thin films[
2. Properties and characteristics of Mn doped TIs
2.1. Structure
(A1−xMnx)2B3 (where A = Bi, Sb and B = Se, Te) is the main chemical formula of Mn-doped TIs. Diverse experimental techniques revealed the incorporation of Mn dopants to the host matrix in various ways, from a substitutional position for Bi to interstitial sites within a quintuple layer (QL) and in the Van-der-Waals gap between QLs. Besides, the homogeneity of the Mn distribution has been questioned as a surface accumulation of the Mn dopant evidenced by the secondary ion mass spectroscopy measurements[
Especially, molecular beam epitaxy (MBE) growth is expected to produce a dilute magnetic alloy of Bi2B3, with Mn occupying Bi-substitutional sites randomly, if the doping level is not high[
Figure 2.(Color online) STM image of Mn doped Bi2Te3 (
X-ray absorption fine structure (XAFS) is usually used to discern the local electronic and structural environment of the dopants in an element-specific way. For (Bi1–xMnx)2Se3 thin films, analysis of the TM K-edge XAFS revealed that Mn occupies octahedral sites and possesses a divalent character (2+ oxidation state), in agreement with the Mn dopants substituting Bi3+ in the matrix[
Figure 3.(Color online) Adapted from Ref. [
Besides, there are reports about the formation of septuple-layer (SL) Bi2MnSe4 (Se–Bi–Se–Mn–Se–Bi–Se), which revises the assumption held by many that Mn arranging as randomly dispersed dopants in the Bi2Se3 lattice during MBE epitaxial growth of (Bi1–xMnx)2Se3. By EXAFS, STEM and DFT calculation, Hagmann et al. demonstrated that instead of Mn atoms incorporating randomly at Bi-substitutional sites, self-assembled layers of Bi2MnSe4 form as interspersing between layers of pure Bi2Se3[
Figure 4.(Color online) A proposed process for the self-assembly of Bi2Se3 layers interspersed with septuple Bi2MnSe4. Adapted from Ref. [
Similarly, the septuple layers Bi2MnTe4 can also form in MBE grown Mn-doped Bi2Te3 film. By high-resolution scanning transmission electron microscopy (HRSTEM), Rienks et al. confirmed this phenomenon[
Figure 5.(Color online) HR-STEM images of Mn doped Bi2Te3 and Bi2Se3[
It needs to be emphasized here that the existence of multiple lattice sites would bring in complications for the associated charge and spin states. Concerning the valence state of the TM dopant in the TI lattice, it is generally believed that 3D TM substituting for Bi atoms would take the 3+ charge state. The divalent Mn replacing trivalent Bi should act as a strong acceptor. However, in the situation of Mn incorporated predominantly as interstitial in octahedral positions within the van der Waals (vdW) gaps, Mn turned out to little affect the Fermi level and carrier concentration[
2.2. Electronic structure
The electronic structures of Mn doped TIs have been mainly studied with angular resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). Large surface gap opening has been confirmed in Mn doped Bi2Se3 systems, yet comparisons to experiments with non-magnetic atoms doped TIs and strong impurity-induced resonance states observed around Dirac point[
Shen group was the first to observe the TRS protection lifted by magnetic dopants and the resulted gap opening in the Dirac surface states using ARPES[
Figure 6.(Color online) ARPES shows gap opening in the Dirac surface states of Mn doped Bi2Se3[
Figure 7.(Color online) Spin-resolved ARPES of Mn doped Bi2Se3. Adapted from Ref. [
However, Rader group[
Figure 8.(Color online) ARPES measurements of (Bi1–
In contrast to the nonmagnetic gap in Mn doped Bi2Se3, there is a pronounced magnetic exchange splitting at the Dirac point in Mn doped Bi2Te3 (Fig. 9), as reported in Rader group’s recent work[
Figure 9.(Color online) Magnetic gap of Mn-doped Bi2Te3 derived by ARPES. Adapted from Ref. [
Generally, the defect concentration in TIs plays an important role in determining the position of the Dirac point with respect to its Fermi energy. The evolution of local density of states (LDOS) with doping can be studied with STM. Hor et al. found Mn dopants act as electron acceptor in Mn-doped Bi2Te3 crystal[
Resonant photoemission is usually used to map the chemical and orbital character of the bands. This method provides a rather direct estimate of the 3d impurity DOS in the valence band by measuring the photoemission cross section close to the 2p-3d X-ray absorption (XAS) maximum. For Mn doped Bi2Se3 film, a localized, non-metallic Mn 3d5 ground state was inferred from X-ray photoemission spectroscopy characterization at the Mn 2p and Mn 3p core levels[
2.3. Magnetic properties
A variety of magnetic characterization techniques have been adopted to probe the magnetic properties of Mn doped TIs, including superconducting quantum interference device (SQUID), FM resonance (FMR), polarized neutron reflectivity (PNR), muon spin rotation (µSR), electron-spin resonance (ESR) spectroscopy and X-ray magnetic circular dichroism (XMCD). SQUID generally detects magnetization signal of the entire sample but not sensitive to samples with small magnetic moments, such as TI thin films. In contrast, XMCD is sensitive to the surface magnetization. The hitherto reported magnetic properties of Mn doped TIs are complicated. Signatures for multiple magnetic phases have been reported[
An early work with physical properties measurement system (PPMS) reported the crystal of Mn-doped Bi2Te3 and Sb2Te3 had ferromagnetic ordering at 10 and 17 K, while Mn-doped Bi2Se3 and Sb2Se3 showed spin glass and paramagnetic properties, respectively[
For (Bi1−xMnx)2Te3 and Bi2−xMnxTe3−ySey systems (x ~ 2%–10%), most studies show consistent results on the bulk magnetic characteristics[
Figure 10.(Color online) Magnetic-field-dependent magnetization of (Bi1–
However, there is not much consensus in the literature regarding the magnetic mechanism for (Bi1−xMnx)2Te3 and Bi2–xMnxTe3−ySey systems. First-principle calculations[
In contrast to (Bi1−xMnx)2Te3 and Bi2−xMnxTe3−ySey with an out-of-plane magnetization, the bulk-sensitive magnetometry measurements suggest a ferromagnetic ground phase in the bulk of (Bi1−xMnx)2Se3 (x = 1%−14%) with TC < 10 K and an in-plane magnetization [
Figure 11.(Color online) SQUID measurements of (Bi1–
Compared to the relatively consistent results on the bulk magnetism in (Bi1−xMnx)2Se3, there is little consensus regarding the surface magnetism. As mentioned in Section 2.2, using XMCD, Xu et al. found the surface TC of (Bi1−xMnx)2Se3 thin film (nominal concentration Mn = 2.5%) up to 45 K[
Collins-McIntyre et al.’s SQUID and XMCD study[
Rader group’s recent work compared Bi2−xMnxTe3 and Bi2−xMnxSe3 thin films[
Figure 12.(Color online) Magnetization
In Islam et al.’s comprehensive study on Cr, V, Fe, and Mn doped Sb2Te3 single crystal, Mn-doped sample showed the largest XMCD signal, indicative of a high-spin configuration of the dopants[
2.4. Transport
For most Mn doped TIs, the planar magnetoresistance (MR) signal exhibits the magnetic nature, of charge carriers with the evolution into weak localization behavior and hysteresis in a magnetic field along the easy axis below TC[
By studying the AHE and magnetoconductance (MC) in Bi2−xMnxTe3−ySey single crystals (x = 0.04 and y = 0.12), Checkelsky et al. found robust ferromagnetism in this system as well as one-dimensional edge-state transport on the magnetic domain wall[
Figure 13.(Color online) AHE and
Figure 14.(Color online) Magnetoconductivity of Mn
For Mn doped Bi2Te3, several previous studies reported AHE. Lee et al. observed a strong AHE signal and a hysteretic magnetoresistance arising from domain-wall scattering, which indicated the presence of ferromagnetism in the system[
Figure 15.(Color online) Hall conductivity
Interestingly, Liu et al. reported a topological Hall effect (THE) in the Mn-doped Bi2Te3 thin films[
For Mn doped Bi2Se3, however, previous studies from other groups never report the observation of AHE. In an early work from Samarth group[
Figure 16.(Color online) Magneto-transport of Mn-Bi2Se3 thin films. Adapted from Ref. [
Point contact Andreev reflection (PCAR) spectroscopy can be used to detect the emerging magnetization induced effective transport spin polarization decreasing. In the TRS protected TIs, a current injected through the surface states becomes spin polarized and this transport spin-polarization leads to a proportionate suppression of Andreev reflection in superconductor/TI junctions. Kamboj et al.’s study showed that upon doping Bi2Se3 with Mn, the transport spin-polarization is monotonically suppressed[
3. Two-component AHE in Mn-doped Bi2Se3
From the above sections, we know that although ferromagnetism has been confirmed in both bulk and surface states, and also surface band gap has been observed with size varying from several tens to a hundred meV, no trace of AHE has been reported for Mn doped Bi2Se3 in previous transport measurements. Particularly, Rader group’s works[
Fig. 17(a) shows the Hall resistance Ryx for samples with different doping levels (x= 0–0.088) in the upper panels, and the corresponding AH resistances RAH in the lower panels. The nonlinear part of the Hall resistance is nearly zero for the entire field range in the undoped samples, while the AH resistances are clearly visible after Mn doping. At low doping levels (e.g. x = 1.8%), the sign of RAH above the (positive) magnetization saturation field is positive and opposite to that of the ordinary Hall resistance. In contrast, the samples with high Mn doping levels exhibit the negative AH resistances. The increase in the Mn concentration drives a crossover from the positive to negative RAH, and a kink appears at intermediate doping levels. This suggests coexistence of two component with opposite signs. Such a two-component AH effect can be observed for a wide range of Mn concentrations (x ≥ 2.4%), and the negative component becomes more pronounced relative to the positive component with increasing Mn doping level, as evidenced by the reversal of the sign of RAH in the high magnetic fields. The sign reversal in RAH and the two-component AH effect can also be obtained by gate-voltage tuning, as illustrated in Fig. 17(b). As the gate voltage is decreased from VG= +100 V to –210 V, the sheet electron density is reduced from ns= 0.91 × 1013 cm–2 to 0.35 × 1013 cm–2 and the sheet resistance ρxx increases from 2.7 to 6.0 kΩ. At high electron densities (VG ≥ 100 V), RAH only has the positive component. When the electron density is lowered by gating, a kink structure emerges at low magnetic fields and becomes more pronounced. At the lowest electron density, the magnitude of the negative component surpasses that of the positive one. The two-component AH effect by the gate-voltage tuning exist in all the samples with different Mn concentrations (x = 0.02–0.074).
Figure 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. [
The distinctively different characteristics of the two AH components strongly suggest that they originate from different electronic states. Fig. 18 displays the sheet longitudinal conductivity (σxx) dependences of
Figure 18.(Color online) Characteristics of the AH conductivity in a lightly doped (Bi1−
The chemical potential dependence of the negative AH component reveals an important role of the non-magnetic potential scatterings of the magnetic impurities in the transport properties of the surface states in the magnetically doped TIs. The interplay between the drastically different surface and bulk magnetizations, along with the competition from various impurity effects, could lead to novel spin structures, such as spin canting, noncolinear or topological spin textures.
An interesting point we noted during our study is the sign of AHE when compared to the other magnetically doped TI systems. We summarized the AHE sign results of recent experimental works on Cr, V, Mn doped TIs in Table 2. Our (Bi1–xMnx)2Se3 (x = 0.01−0.09) shows negative AH sign for the surface state but positive sign for the bulk state[
4. Summary and perspective
In this review article, we have described recent experimental work on Mn doped TIs. In contrast to Cr and V-doped TIs in which the QAHE can be observed, the magnetism in Mn-doped TIs is far more complicated. In particular, the Mn doping in Bi2Se3 leads to strong non-magnetic resonant scatterings, which opens a large gap in the surface states, but on the other hand, suppresses the anomalous Hall effect arising from the magnetic ordering. Even though the competition between the magnetic and non-magnetic interactions is not good for realizing the QAHE in Mn-doped TIs, it renders a fertile ground for searching exotic magnetic orders or spin structures. Another promising direction is to explore the heterostructures based on MnTe or MnSe layers intercalated in quintuple layers of Bi2Se3, Bi2Te3 or their derivatives. Actually, recent theoretical works[
Acknowledgements
This work was supported by the National Key Research and Development Program (Project No. 2016YFA0300600), the National Science, Foundation of China (Projects No. 11604374 and No. 61425015), the National Basic Research Program of China (Project No. 2015CB921102), and the Strategic Priority Research Program of Chinese Academy of Sciences (Project No. XDB28000000).
References
[1] C L Kane, E J Mele. Z2 topological order and the quantum spin Hall effect. Phys Rev Lett, 95, 146802(2005).
[2] C L Kane, E J Mele. Quantum spin Hall effect in graphene. Phys Rev Lett, 95, 226801(2005).
[3] Y Zhang, Y W Tan, H L Stormer et al. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature, 438, 201(2005).
[4] B A Bernevig, S C Zhang. Quantum spin Hall effect. Phys Rev Lett, 96, 106802(2006).
[5]
[6] K V Klitzing, G Dorda, M Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys Rev Lett, 45, 494(1980).
[7] D J Thouless, M Kohmoto, M P Nightingale et al. Quantized Hall conductance in a two-dimensional periodic potential. Phys Rev Lett, 49, 405(1982).
[8] B Simon. Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys Rev Lett, 51, 2167(1983).
[9] B A Bernevig, T L Hughes, S C Zhang. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science, 314, 1757(2006).
[10] M Koenig, S Wiedmann, C Bruene et al. Quantum spin Hall insulator state in HgTe quantum wells. Science, 318, 766(2007).
[11] X L Qi, S C Zhang. Topological insulators and superconductors. Rev Mov Phys, 83, 1057(2011).
[12] Y Ando. Topological insulator materials. J Phys Soc Jpn, 82, 102001(2013).
[13] L Fu, C L Kane, E J Mele. Topological insulators in three dimensions. Phys Rev Lett, 98, 106803(2007).
[14] D Hsieh, D Qian, L Wray et al. A topological Dirac insulator in a quantum spin Hall phase. Nature, 452, 970(2008).
[15] Y Xia, D Qian, D Hsieh et al. Observation of a large-gap topological insulator class with a single Dirac cone on the surface. Nat Phys, 5, 398(2009).
[16] Y L Chen, J G Analytis, J H Chu et al. Experimental realization of a three-dimensional topological insulator, Bi2Te3. Science, 325, 178(2009).
[17] H Zhang, C X Liu, X L Qi et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat Phys, 5, 438(2009).
[18] J Chen, H J Qin, F Yang et al. Gate-voltage control of chemical potential and weak antilocalization in Bi2Se3. Phys Rev Lett, 105, 176602(2010).
[19] D X Qu, Y S Hor, J Xiong et al. Quantum oscillations and Hall anomaly of surface states in the topological insulator Bi2Te3. Science, 329, 821(2010).
[20] J G Analytis, J H Chu, Y Chen et al. Bulk Fermi surface coexistence with Dirac surface state in Bi2Se3: A comparison of photoemission and Shubnikov–de Haas measurements. Phys Rev B, 81, 205407(2010).
[21] R Yu, W Zhang, H J Zhang et al. Quantized anomalous Hall effect in magnetic topological insulators. Science, 329, 61(2010).
[22] C Z Chang, J Zhang, X Feng et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science, 340, 167(2013).
[23] A R Mellnik, J S Lee, A Richardella et al. Spin-transfer torque generated by a topological insulator. Nature, 511, 449(2014).
[24] X L Qi, T L Hughes, S C Zhang. Topological field theory of time-reversal invariant insulators. Phys Rev B, 78, 195424(2008).
[25] A M Essin, J E Moore, D Vanderbilt. Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. Phys Rev Lett, 102, 146805(2009).
[26] W K Tse, A H MacDonald. Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators. Phys Rev Lett, 105, 057401(2010).
[27] R Li, J Wang, X L Qi et al. Dynamical axion field in topological magnetic insulators. Nat Phys, 6, 284(2010).
[28] L Fu, C L Kane. Superconducting proximity effect and Majorana Fermions at the surface of a topological insulator. Phys Rev Lett, 100, 096407(2008).
[29] X L Qi, R Li, J Zang et al. Inducing a magnetic monopole with topological surface states. Science, 323, 1184(2009).
[30] M Z Hasan, C L Kane. Colloquium: topological insulators. Rev Mod Phys, 82, 3045(2010).
[31] H Z Lu, J Shi, S Q Shen. Competition between weak localization and antilocalization in topological surface states. Phys Rev Lett, 107, 076801(2011).
[32] J G Checkelsky, R Yoshimi, A Tsukazaki et al. Trajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator. Nat Phys, 10, 731(2014).
[33] X Kou, S T Guo, Y Fan et al. Scale-invariant quantum anomalous Hall effect in magnetic topological insulators beyond the two-dimensional limit. Phys Rev Lett, 113, 137201(2014).
[34] A J Bestwick, E J Fox, X Kou et al. Precise quantization of the anomalous Hall effect near zero magnetic field. Phys Rev Lett, 114, 187201(2015).
[35] C Z Chang, W Zhao, D Y Kim et al. High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator. Nat Mater, 14, 473(2015).
[36] Y L Chen, J H Chu, J G Analytis, a et. Massive Dirac Fermion on the surface of a magnetically doped topological insulator. Science, 329, 659(2010).
[37] S Y Xu, M Neupane, C Liu et al. Hedgehog spin texture and Berry’s phase tuning in a magnetic topological insulator. Nat Phys, 8, 616(2012).
[38]
[39] J M Zhang, W Ming, Z Huang et al. Stability, electronic, and magnetic properties of the magnetically doped topological insulators Bi2Se3, Bi2Te3 and Sb2Te3. Phys Rev B, 88, 235131(2013).
[40] L B Abdalla, L Seixas, T M Schmidt et al. Topological insulator Bi2Se3(111) surface doped with transition metals: An ab initio investigation. Phys Rev B, 88, 045312(2013).
[41] J Růžička, O Caha, V Hol et al. Structural and electronic properties of manganese doped Bi2Te3 epitaxial layers. New J Phys, 17, 013028(2015).
[42] J S Lee, A Richardella, D W Rench et al. Ferromagnetism and spin-dependent transport in n-type Mn-doped bismuth telluride thin films. Phys Rev B, 89, 174425(2014).
[43] A I Figueroa, G van der Laan, L J Collins-McIntyre et al. Local structure and bonding of transition metal dopants in Bi2Se3 topological insulator thin films. J Phys Chem C, 119, 17344(2015).
[44] J A Hagmann, X Li, S Chowdbury et al. Molecular beam epitaxy growth and structure of self-assembled Bi2Se3/Bi2MnSe4 multilayer heterostructures. New J Phys, 19, 085002(2017).
[45] D Zhang, A Richardella, D W Rench et al. Interplay between ferromagnetism, surface states, and quantum corrections in a magnetically doped topological insulator. Phys Rev B, 86, 205127(2012).
[46] J Choi, S Choi, J Choi et al. Magnetic properties of Mn-doped Bi2Te3 and Sb2Te3. Phys Status Solidi B, 241, 1541(2004).
[47] J Choi, S Choi, J Choi et al. Mn-doped V2VI3 semiconductors: Single crystal growth and magnetic properties. J Appl Phys, 97, 10D(2005).
[48] J W G Bos, M Lee, E Morosan et al. Ferromagnetism below 10 K in Mn-doped BiTe. Phys Rev B, 74, 184429(2006).
[49] P Janíček, Č Drašar, P Lošt’ák et al. Transport, magnetic, optical and thermodynamic properties of Bi2–
[50] Y S Hor, P Roushan, H Beidenkopf et al. Development of ferromagnetism in the doped topological insulator Bi2–
[51] H J Von Bardeleben, J L Cantin, D M Zhang et al. Ferromagnetism in Bi2Se3:Mn epitaxial layers. Phys Rev B, 88, 075149(2013).
[52] S Zimmermann, F Steckel, C Hess et al. Spin dynamics and magnetic interactions of Mn dopants in the topological insulator Bi2Te3. Phys Rev B, 94, 125205(2016).
[53] M F Islam, C M Canali, A Pertsova et al. Systematics of electronic and magnetic properties in the transition metal doped Sb2Te3 quantum anomalous Hall platform. Phys Rev B, 97, 155429(2018).
[54] C Niu, Y Dai, M Guo et al. Mn induced ferromagnetism and modulated topological surface states in Bi2Te3. Appl Phys Lett, 98, 252502(2011).
[55] Q Liu, C X Liu, C Xu et al. Magnetic impurities on the surface of a topological insulator. Phys Rev Lett, 102, 156603(2009).
[56] J J Zhu, D X Yao, S C Zhang et al. electrically controllable surface magnetism on the surface of topological insulators. Phys Rev Lett, 106, 097201(2011).
[57] P Sessi, F Reis, T Bathon et al. Signatures of Dirac fermion-mediated magnetic order. Nat Commun, 5, 5349(2014).
[58] B C Chapler, K W Post, A R Richardella et al. Infrared electrodynamics and ferromagnetism in the topological semiconductors Bi2Te3 and Mn-doped Bi2Te3. Phys Rev B, 89, 235308(2014).
[59] L J Collins-Mcintyre, M D Watson, A A Baker et al. X-ray magnetic spectroscopy of MBE-grown Mn-doped Bi2Se3 thin films. AIP Adv, 4, 127136(2014).
[60] J Sánchez-Barriga, A Varykhalov, G Springholz et al. Nonmagnetic band gap at the Dirac point of the magnetic topological insulator (Bi1−
[61] N Liu, J Teng, Y Li. Two-component anomalous Hall effect in a magnetically doped topological insulator. Nat Commun, 9, 1282(2018).
[62] R Tarasenko, M Vališka, M Vondráček et al. Magnetic and structural properties of Mn-doped Bi2Se3 topological insulators. Physica B, 481, 262(2016).
[63] M D Watson, L J Collins-McIntyre, L R Shelford et al. Study of the structural, electric and magnetic properties of Mn-doped Bi2Te3 single crystals. New J Phys, 15, 103016(2013).
[64] Y Li, X Zou, Li J, G Zhou et al. Ferromagnetism and topological surface states of manganese doped Bi2Te3: Insights from density-functional calculations. J Chem Phys, 140, 124704(2014).
[65] J G Checkelsky, J Ye, Y Onose et al. Dirac-fermion-mediated ferromagnetism in a topological insulator. Nat Phys, 8, 729(2012).
[66] G Rosenberg, M Franz. Surface magnetic ordering in topological insulators with bulk magnetic dopants. Phys Rev B, 85, 195119(2012).
[67] C Liu, Yunyi Zang, Wei Ruan et al. Dimensional crossover-induced topological Hall effect in a magnetic topological insulator. Phys Rev Lett, 119, 176809(2017).
[68] S Kamboj, S Das, A Sirohi et al. Suppression of transport spin-polarization of surface states with emergence of ferromagnetism in Mn-doped Bi2Se3. J Phys Cond Matt, 30, 355001(2018).
[69] I A Ado, I A Dmitriev, P M Ostrovsky et al. Anomalous Hall effect with massive Dirac fermions, Anomalous Hall effect with massive Dirac fermions. EPL, 111, 37004(2015).
[70] J Zhang, C Z Chang, P Tang et al. Topology-driven magnetic quantum phase transition in topological insulators. Science, 339, 1582(2013).
[71] Z Zhang, X Feng, M Guo et al. Electrically tuned magnetic order and magnetoresistance in a topological insulator. Nat Commun, 5, 4915(2014).
[72]
[73] M Liu, J Zhang, C Z Chang et al. Crossover between weak antilocalization and weak localization in a magnetically doped topological insulator. Phys Rev Lett, 108, 036805(2012).
[74] J Li, Y Li, S Du et al. Intrinsic magnetic topological insulators in van der Waals layered MnBi2Te4-family materials. Sci Adv, 5(2019).
[75] D Zhang, M Shi, T Zhu et al. Topological axion states in the magnetic insulator MnBi2Te4 with the quantized magnetoelectric effect. Phys Rev Lett, 122, 206401(2019).
[76]
[77]
[78]
[79]
[80] Y Gong, J Guo, J Li et al. Experimental realization of an intrinsic magnetic topological insulator. Chin Phys Lett, 36, 076801(2019).
[81]
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