Magnetic Topological Insulators and Quantum Anomalous Hall Effect

Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Solid State Communications

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Magnetic topological insulators and quantum anomalous hall effect

Xufeng Kou, Yabin Fan, Murong Lang, Pramey Upadhyaya, Kang L. Wang n

Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, CA 90095, USA article info abstract

Article history: When the magnetic order is introduced into topological insulators (TIs), the time-reversal symmetry Received 1 September 2014 (TRS) is broken, and the non-trivial topological surface is driven into a new massive Dirac fermions state. Accepted 17 October 2014 The study of such TRS-breaking systems is one of the most emerging frontiers in condensed-matter “ ” Communicated by Xincheng Xie physics. In this review, we outline the methods to break the TRS of the topological surface states. With robust out-of-plane magnetic order formed, we describe the intrinsic magnetisms in the magnetically Keywords: doped 3D TI materials and the approach to manipulate each contribution. Most importantly, we Topological insulator Magnetism summarize the theoretical developments and experimental observations of the scale-invariant quantum Quantum anomalous hall anomalous Hall effect (QAHE) in both the 2D and 3D Cr-doped (BiSb)2Te3 systems; at the same time, we Spin-orbit torque also discuss the correlations between QAHE and other quantum transport phenomena. Finally, we highlight the use of TI/Cr-doped TI heterostructures to both manipulate the surface-related ferromagnetism and realize electrical manipulation of magnetization through the giant spin–orbit torques. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Part 1, we briefly introduce the methods used to break the TRS of TIs; in Part 2, we discuss about the intrinsic magnetisms inside the When the spin–orbit coupling (SOC) is strong enough to invert magnetic TI systems and the driving forces behind the interplay the conduction and valence bands in the material [1–8], it creates between different magnetic orders; in Part 3, we present both the a new state of matter, known as the time-reversal-invariant Z2 theory models and experimental realizations of the quantum topological insulators (TIs). With the unique gapless Dirac-cone- anomalous Hall effect (QAHE) in the magnetic TI thin films; and like linear Ek dispersion relation and the spin-momentum- in Part 4, we briefly summarize the investigation of magnetic locking mechanism along the surface, TIs are regarded as one of TI-based heterostructures, the related new physics and potential the most intriguing 2D materials, which arises broad interest applications. among condensed-matter physics, material science, nano-electro- – nics, spintronics, and energy harvesting applications [9 13]. Ever 1.1. Break the time-reversal-symmetry of topological insulators since this Z2 topological order was proposed in 2005 [1], enormous – efforts have been dedicated toward the preparation [14 18], Bermevig et al. proposed a four-band k p model of the 2D TIs, – optimization [19 22], and utilization of the time-reversal- which can be described by an effective Hamiltonian that is symmetry (TRS) protected topological non-trivial surface states essentially a Taylor expansion in the wave vector k of the – [23 28]. In parallel with the pursuit of the massless Dirac interactions between the lowest conduction band and the highest fi fermions, it is of equal signi cance to break the TRS of the TI valence band that [3] surfaces by introducing the perpendicular magnetic interaction 2 3 mðkÞ Aðkx þikyÞ 00 [29–32]. In this TRS-breaking case, the intrinsic electrical [31–39], 6 7 6 Aðk þik ÞmðkÞ 007 magnetic [8,40–46], and optical properties [47–50] of the surface ð ; Þ¼ ð Þþ6 x y 7 Heff kx ky E k 6 ð Þð Þ 7 massive Dirac fermions are subject to the interplay between the 4 00m k A kx iky 5 ð þ Þð Þ band topology (SOC strength) and the magnetic orders. With 00A kx iky m k additional structural engineering (Fig. 1a), the functionalities of "# hðkÞ 0 associated physics and applications can be further multiplied. ¼ ð Þ nð Þ 1 In this paper, we review the fundamental physics as well 0 h k as recent progress in the emerging magnetic TI research field. In ð Þ¼ þ ð 2 þ 2Þ ð Þ¼ where E k C D kx ky is the trivial band bending, m k þ ð 2 þ 2Þ fi m0 B kx ky is the nite mass or gap parameter (m0 is the energy n Corresponding author. difference between the inverted conduction and valence sub-bands E-mail address: [email protected] (K.L. Wang). at the Γ point, and m0Bo0 is required for band inversion), and http://dx.doi.org/10.1016/j.ssc.2014.10.022 0038-1098/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: X. Kou, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2014.10.022i 2 X. Kou et al. / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Band topology SOC

Magnetic Structure Eng. Exchange TI/Cr-TI

MZ TI/FM TI/NI ≠ Mz = 0 Mz 0

Fig. 1. (a) The interplay among band topology, magnetic exchange, and structure engineering in the TRS-breaking phenomena. (b) Topological surface band diagram in undoped TI (massless) and magnetically doped TI (massive), respectively. (Adopted from [51]). hðkÞ¼EðkÞþmðkÞσz þAðkxσy kyσxÞ. In the original undoped condi- (MBE) [39,42,45,46,53,59,65]. Among them, the MBE technique has n tion, TRS is protected given that hðkÞ¼h ðkÞ. However, once advantages in terms of non-equilibrium physical deposition, accurate additional magnetic exchange Mz is introduced, the mass term layer thickness/doping profile control, wafer-scale growth capability, ð ; Þ¼ þ ð 2 þ 2Þþ changes to m k Mz m0 B kx ky gMz.Consequently,Heff(kx,ky) and potential integration of heterostructures and/or super-lattices for no longer has the TRS-protected property, and the massless surface multi-functional device applications. Dirac fermions degrade to the massive case where the surface opens a Here, we show one example of the MBE-grown Cr-doped gap Δ,asshowninFig. 1b [32]. (BixSb1x)2Te3 thin films on the insulating GaAs substrate [46].With Experimentally, the topological surface band gap was first the surface-sensitive reflection high-energy electron diffraction observed by the Angle-resolved photoemission spectroscopy (ARPES). (RHEED) technique, in-situ growth dynamics were monitored [66]. Chen et al. performed the measurements on the transition-metal Fig. 2a shows the as-grown RHEED patterns taken at t¼0s,30s,and doped Bi2Se3 materials, and observed a small surface band gap 65 s successively. Both the sharp 2D streaky lines and the bright zero- (7 meV) opening in the (Bi0.99Mn0.01)2Se3 sample [51].Inthe order specular spot persist during the entire growth process, meanwhile, Wray et al. also observed the surface gap opening when indicative of the single-crystalline feature of the sample. The smooth they deposited the Fe ions on top of the Bi2Se3 film [52].Recently,Xu surface morphology was later confirmed by atomic force microscopy et al. used the more advanced spin-resolved ARPES to visualize the (AFM) as shown in Fig. 2b where typical TI triangular terraces were spin textures in the Mn-doped Bi2Se3 thin films [53].Theresolved preserved without any Cr aggregations or clusters. Meanwhile, by unique hedgehog-like spin reorientations around the surface electro- fitting the RHEED oscillation periods in Fig. 2c, the growth rate is nic groundstate provided additional information about the mechanics extracted to be around 1 QL/min. More importantly, since the RHEED of TRS breaking on the TI surface. patterns directly reflect the in-plane atomic morphology in the reciprocal k-space [67], the as-grown surface configuration could 1.2. Magnetically doped TI materials grown by MBE be inspected by using the d-spacing evolution as shown in Fig. 2d. It can be seen clearly that the d-spacing drops quickly as soon as the In general, there are two methods to introduce the magnetic growth starts, and the surface transition from the pristine GaAs to exchange term Mz in order to break the TRS of the topological the Cr-doped (BixSb1x)2Te3 completes immediately after the forma- surface states. First of all, by combing the TI material with a tion of the first quintuple layer (QL). topologically trivial magnetic material, the magnetic proximity After the sample growth, an energy dispersive X-ray (EDX) effect at the interface is able to locally align the spin moments of spectrometer was employed to perform elemental mappings. The the TI band/itinerant electrons out of plane, therefore breaking the colored EDX maps collected from both the TI (Bi, Sb, Cr, and Te) TRS at the TI interface [54,55]. To date, such magnetic proximity and substrate (Ga and As) are illustrated individually in Fig. 3b–g. effects have been observed in both TI/YIG [56] and TI/EuS [57,58] The distribution of every component (especially Cr) is uniform and heterostructures. However, since the interface qualities are usually there is no substitutional or interstitial doping/diffusion from the poor in these systems, the magnetizations introduced by the GaAs substrate, therefore leaving the bottom TI surface intact. To short-range magnetic exchange coupling are therefore quite weak. elaborate the detailed structural characteristics of the epitaxial Alternatively, incorporating magnetic ions into the host TI mate- film, high-resolution scanning transmission electron microscopy rials has been proven to be the most effective way to generate robust (HRSTEM) investigation was also performed. As shown in Fig. 3h, magnetism and open a gap of the surface states [45,51,53,59]. single-crystalline Cry(BixSb1x)2Te3 film with a sharp interface can Theoretically, Zhang et al. reported a detailed theoretical study of be clearly seen on top of the GaAs substrate. At the same time, the the formation energies, charge states, band structures, and magnetic corresponding zoom-in HRTEM image in Fig. 3i manifests properties of transition metal (V, Cr, Mn, and Fe) doped 3D TIs the typical TI quintuple-layered structure. It is also identified that [29,60].Byusingfirst-principles calculations based on the density the van der Waals interactions between adjacent quintuple layers functional theory, they found that most transition metals favor the indeed induce a larger gap compared with the neighboring substitutional doping into the host TI cation-site, indicative of robust Bi/Sb–Te covalently-bonded sheets. Combined with Fig. 3d and magnetic moments formed in such magnetically doped TI systems. the theoretical anticipations [29], it may suggest that the Cr More importantly, Nunez et al. found that the exchange-induced dopant favors the stable substitution formation inside the host TI single ion magnetic anisotropy in such systems favored the perpen- matrix without invoking any Cr interstitial defects or second phase dicular orientation [61].Asaresult,themagneticanisotropyeasy- separation. axis is expected to be out-of-plane in the magnetically doped 3D TIs In summary, the high-quality Cr-doped TI thin films with low regardless of the film thickness. defects, well-defined surfaces, and uniform Cr distribution have At the current stage, the transition-metal doped TI materials have been prepared by MBE, and in turns they validate themselves as been successfully grown by chemical vapor deposition (CVD) [40], the suitable system to investigate the magnetisms and QAHE bulk Bridgeman growth [44,62–64],andmolecularbeamepitaxy phenomena, as we will present in the following parts.

Fig. 2. (a) RHEED patterns along ½1120 direction of the surface of Cr-doped (Bi1xSbx)2Te3 thin ﬁlm taken at 0 s, 30 s, and 65 successively during growth. (b) AFM image of the Cr-doped TI thin ﬁlm with the size of 0.3 μm 0.3 μm. (c) RHEED oscillations of intensity of the specular beam. Inset: the growth rate of 1 QL/min is determined from the oscillation period. (d) d-Spacing evolution of the surface lattice during growth. (Adopted from [46]).

10 nm GaAs Bi SbCr Te Ga As

Te Sb

Bi Te Sb Bi Sb

Te Intensity (a.u) Te Sb Cr

2 3 4 5 6 1 nm Cr0.2(Bi0.45Sb0.45)2Te 3 Energy (keV)

Fig. 3. (a) HAADF image of the cross-section Cr-doped (Bi1xSbx)2Te3 ﬁlm grown on the GaAs substrate. (b)–(g) Distribution maps of each individual element: Bi (b), Te (c), Cr (d), Ta(e), Ga(f) and As (g). Cr dopants distribute uniformly inside the TI layer. (h) High-resolution cross-section STEM image, showing the epitaxial single-crystalline

Cr0.2(Bi0.45Sb0.45)2Te3 thin ﬁlm with sharp TI–GaAs interface. (i) Zoom-in STEM to demonstrate typical quintuple-layered crystalline structure of tetradymite-type materials. Neither Cr segregations nor interstitial defects are detected. (j) EDX spectrum to examine the chemical element compositions. The Cr doping concentration is estimated to be 10%. (Adopted from [46]).

2. Magnetisms in magnetic TI materials In magnetically doped semiconductors, it is known that neighboring magnetic dopants can be aligned through the Ruderman–Kittel– 2.1. RKKY interaction and gate-controlled ferromagnetism Kasuya–Yoshida (RKKY) interaction [79–81]. In the RKKY model, the itinerant carriers are polarized in the form of long-range oscillatory

The first study on the transition metal (TM) doped Bi2Se3/ waves, and the general interaction is described by [41,82] Bi Te /Sb Te materials was carried out even before the discovery 2 3 2 3 Z nohihi – EF ! ! ! ! of topological insulators [68 75]. At that time, such systems were RKKY ¼ i ∑ ð Þ!σ ð Þð Þ!σ ð Þ H π dE Tr J E S i G R ij J E S j G R ij treated as conventional dilute magnetic semiconductors (DMS). 1 i a j Compared with III–V based DMS counterparts [76–78],itwas ð2Þ found that the solubility in the tetradymite-type thin film could be as high as 0.35 and the corresponding Curie temperature TC as where J(E) is the exchange coupling coefficient, Si is the spin of local high as 190 K was obtained [70]. magnetic ion, σ is the Pauli matrix of itinerant electrons, Rij is the

Please cite this article as: X. Kou, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2014.10.022i 4 X. Kou et al. / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ distance between the two localized ions, and the Green’sfunction by the intrinsic/extrinsic scatterings [87,88]. As a result, by tracing has the form that [41,82]. the electric-field-controlled AHE curves, the intrinsic connection Z ! 1 1 ! ! between transport and magnetism can be investigated. Following ð Þ¼ ð Þ 2 ð Þ – G R ij 2 exp ik R ij d k 3 the same method used in conventional DMS systems [89 91],we 4π EH0 fabricated top-gated Hall bar devices on our MBE-grown 6 QL Cr- In the case of Cr doped TI systems, particularly, the p–d coupling doped (BixSb1x)2Te3 thin film, as shown in Fig. 5a. With a Cr 3þ between the transition metal d-orbital (Cr ) and the itinerant doping concentration of 2%, the Bi/Sb ratio was chosen to be carrier p-orbital is found to be strongest due to the wavefunction around 1 so that the Fermi level (EF) of the as-grown film was – localization and symmetry argument [83],andtheon-sitep d already within the surface band gap. Accordingly, the electric field fi exchange coef cient J(E)canbeestimatedbyusingtheempirical provided by top gate (712 V) is used to effectively tune the model in which an itinerant carrier floats around the magnetic ion. surface carrier type and EF position (i.e., 750 meV across the It has the form [43]: surface band gap), and the ambipolar effect of the longitudinal ! ! ! ! ! ! ð Þ¼∬ ψ nð Þψ nð Þð 2= þ 2= Þψ ð Þψ ð Þ resistance Rxx is also observed in Fig. 5c [20,92]. Fig. 5b shows the J E p r 1 d r 2 e Rpd 3e RpCr3 þ d r 1 p r 2 dr1dr2 gate-dependent AHE results of the 6 QL Cr (Bi Sb ) Te thin ð4Þ 0.04 0.49 0.49 2 3 film. It can be clearly seen that the Hall resistance Rxy develops a where ψp is the p-orbital wavefunction of the itinerant carrier, ψd is square-shaped hysteresis loop characteristic at 1.9 K, indicating 3þ the d-orbital wavefunction of the localized Cr ion, Rpd is the the presence of a long-range FM order with out-of-plane magnetic distance between the two mediating carriers, and RpCr3 þ is the anisotropy in the film. More importantly, the magnetic hysteresis þ distance between the carrier and the Cr3 ion center. Upon loops change notably as the sample is biased from p-type to n- combing Eqs. (2)–(4), it is shown that the J(E)isdeterminedby type. Particularly, when the Fermi level is below the Dirac point the Cr 3d-orbital density of states (DOS) distribution Dd(E); in other (12 V oVgo 2 V), the coercive field Hc (red hollow circles in 2 words, the overall RKKY interaction strength scales with Dd(EF) (i.e., Fig. 5d) steadily reduces from 120 Oe down to 55 Oe; in contrast, it strongly depends on the Fermi level position and carrier density) once the dominant conduction holes are depleted when Vg40V, [84]. Hc slowly stops decreasing, and finally approaches to its mini- In Zhang et al.’s work [15], first-principles DFT calculations mum value around 40 Oe when EF is far above the Dirac point. based on the density functional theory were performed to inves- Similarly, the Curie temperature TC (i.e., blue solid squares in tigate the DOS distributions of Cr dopants inside the host 3D Bi2Se3 Fig. 5d) also follows the reduction-and-saturation behavior by material. It was revealed in Fig. 4a that there was strong hybridi- gate modulation, namely TC decreases from 7.5 K (Vg¼12 V) zation between the Cr 3d-orbit and Se 2p-orbit, and Dd(E) of the Cr down to 4.7 K (Vg45 V). As a result, the electric-field-controlled ion was distributed majorly in the valence band, and reduced to AHE results in the p-type region clearly manifest the hole- almost zero above the Dirac point. As a result, it would be mediated RKKY coupling signature. expected that holes were more favored in mediating the RKKY Beside the electric-field-controlled AHE curves, the RKKY interaction than electrons in the Cr-doped Bi2Se3 material. In coupling can also be realized from the magnetic doping level addition, their DFT calculations also showed that such hole- xM TC relation [93–96]. In general, the Curie temperature under mediated RKKY mechanism also dominated in Cr-doped Bi2Te3/ the mean-field Zener theory is given by [94,96]. Sb2Te3 systems (Fig. 4b) [46,60]. ð þ Þ n ð þ Þ Experimentally, the most straightforward way to verify the S S 1 xM 2m kF S S 1 xM 2 n 1=3 TC ¼ J ¼ J m p ð6Þ Ω π2 2=3 4=3 Ω carrier-dependent RKKY interaction in the magnetic TI materials is 3kB 0 4 3 4π kB 0 to carry out the anomalous Hall effect (AHE) magneto-transport n measurements [85,86]. It is known that in solids with a ferromag- where m is the effective mass, p is the free carrier (hole) density, ¼ð π2 Þ1=3 netic (FM) phase, the total Hall resistance is expressed as [85]: and the Fermi wave vector kF 3 p . Accordingly, for a given ¼ þ ðÞ ðÞ magnetic system with RKKY interaction, TC is expected to scale Rxy R0 H RA MH 5 1=3 with xMp . Recently, both Zhou et al. [70] and Li et al. [64] 1=3 where the ordinary Hall coefficient R0 is inversely proportional to reported the experimental observations of such linear Tc xMp the Hall density, and the anomalous Hall coefficient RA is affected relationship in both the Sb2xCrxTe3 and the (CrxBiySb1yx)2Te3 Energy (ev) Energy (ev)

Fig. 4. (a) Left: ab initio calculated energy band diagram and DOS of Cr d-orbital and Se p-orbital in Bi2xCrxSe3, where x¼0.083. The red lines represent schematically the surface Dirac cone located at the Γ point. (Adopted from [29]) (b) energy band diagram and DOS of Cr d-orbital and Te p-orbital in Cr0.08(Bi0.48Sb0.48)2Te3 material. (Adopted from [46]).

Fig. 5. (Color Online) (a) Top-gated Hall bar device structure used for exploring the electric-field-controlled anomalous Hall effect. (b) Gate-dependent anomalous Hall effect at 1.9 K in 6 QL Cr0.04(Bi0.49Sb0.49)2Te3 thin films with the Cr doping levels of 2%. (c) Top-gate modulations in the 6 QL Cr0.04(Bi0.49Sb0.49)2Te3 sample. (d) The changes of coercivity field Hc at 1.9 K (red hollow circles) and Curie temperature TC (blue solid squares) with applied top-gate voltages. Both of them gradually decrease when the sample is biased from p-type to n-type, indicating the hole-mediated RKKY interaction signature. (Adopted from [46]).

bulk samples, which again supported that the hole-mediated Cr d-orbit moment and the host TI valence electron without the RKKY interaction was responsible for the ferromagnetism in assistance of any itinerant carriers. Cr-doped magnetic TI materials. Chang et al. carried out the systematic study to investigate the Van Vleck mechanism in magnetic TIs [42]. In their experiments,

they prepared a set of Cr-doped (BixSb1x)2Te3 thin films with a 2.2. Van Vleck mechanism and carrier-independent ferromagnetism fixed Cr doping level of 12% and same film thickness of 5 QL in

order to ensure that both χL and ML remain constant. They found In a dilute magnetically doped semiconductor system, it was that although the carrier type was tuned from n-type (x40.3) to found possible that the magnetic exchange among local moments p-type (xo0.2) by varying the Bi/Sb ratio, the ferromagnetism of can also be mediated by the band electrons other than the RKKY the Cr0.22(BixSb1x)1.78Te3 films remained regardless of the sig- coupling through itinerant carriers [97]. Yu et al. predicted that nificant change in carrier density. Most importantly, the Curie tetradymite TI materials (Bi2Se3/Bi2Te3/Sb2Te3) could form such temperature TC (30 K) showed little dependence on the carrier magnetically ordered insulator when doped with Cr or Fe [32]. density and type (Fig. 6a and b). Together with gate-independent fi They used the rst-principles calculations to investigate the total AHE data shown in Fig. 6c, their observations thus provided a free energy of the system that direct evidence of the long-range FM order through the Van Vleck 1 1 mechanism in the magnetic TI materials. F ¼ χ 1M2 þ χ 1M2 J M M ðM þM ÞH ð7Þ total 2 L L 2 e e eff L e L e The large Van Vleck susceptibility resulting from the bulk band inversion and the long-range exchange coupling through the band χ χ where L ( e) is the spin susceptibility of the local moments (band electrons distinguish the magnetic TIs from conventional DMS electrons), ML (Me) denotes the magnetization for the local systems. As will be discussed in Part 3, this unique magnetism is moment (electron) subsystem, and Jeff is the magnetic exchange essential to form the ferromagnetic insulator with a non-zero first coupling between them. Accordingly, the onset of the FM phase Chern number [85]. can be determined when the minimization procedure of the free energy gives a non-zero magnetization without the presence of fi 2 χ 1χ 1 4 external magnetic eld, which leads to Jeff L e 0, or 2.3. Interplay between different FM orders—Magnetic ions χ 4ð 2 χ Þ 1 χ equivalently L Jeff e ; in other words, a larger e gives rise doping level to more prominent local coupling [32]. In fact, a considerable χe value can be obtained through the Van Vleck mechanism that [32] Since both the hole-mediated RKKY coupling and the carrier- ^ ^ independent Van Vleck magnetism are possible in the magnetic onkjSzjmk4 omkjSzjnk4 χ ¼ ∑4μ μ2 ð8Þ TIs, it is thus important to understand the emergence of these e 0 B E E mk nk different ferromagnetic orders, and the interplay/controllability of where μ0 is the vacuum permeability, μB is the Bohr magneton, Sz each contribution. is the spin operator, |mk4 and |nk4 are the Bloch functions in the First of all, it is well-known that for any magnetic system, the conduction and valence bands, respectively. In normal IV and III–V magnetic ions doping level xM is essential to the magnetization. DMS systems where the conduction and valence bands are well- Specifically, in the dilute limit where the direct coupling between separated by the band gap Eg, there is no overlap between |mk4 local moments is negligible, the ferromagnetic susceptibility (and and |nk4, χe is thus negligible. In contrast, the giant SOC in Bi2Se3- thus the overall magnetization) is proportional to xM, following the type materials causes the band inversion [7], and the mixing Bi Curie–Weiss form that [98] p1z-state (|nk4) and Se p2z-state (|mk4) therefore leads to a μ2 o | | 4 C xM sizable nk Sz mk [32]. Under such circumstances, large mag- χ ¼ ¼ J ð Þ ð Þ 9 netization can be generated through direct coupling between the T Tc 3kB T Tc

Please cite this article as: X. Kou, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2014.10.022i 6 X. Kou et al. / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Fig. 6. (Color Online) (a) Dependence of Curie temperature (TC) on Bi content (x) (bottom axis) and carrier density (top axis) in the MBE grown Cr0.22(BixSb1x)1.78Te3 ﬁlms.

(b) Dependence of AHE resistance (RAH) (red solid squares) and longitudinal resistance (Rxx) (blue solid circles) at 1.5 K on Bi content x (bottom axis) and carrier density

(top axis). (c), Magnetic ﬁeld dependent Hall resistance Rxy of a 5QL Cr0.22(Bi0.2Sb0.8)1.78Te3 ﬁlm grown on SrTiO3 (1 1 1) at different back-gate voltages measured at 250 m K. (Adopted from [42]).

μ ¼ μ ð þ Þ where J g BJ J 1 is the magnetic moment, g is the Landé however, since excessive Cr atoms promote the formation of n-type g-factor, and J is the angular momentum quantum number. In BiTe anti-site defects [59], they therefore force the Fermi level EF recent years, several groups have contributed to this research field, “pinned” far above the surface band gap. Under such circumstances, and relevant xM-dependent experiments have been carried out in the hole-mediated RKKY coupling becomes diminished or even Bi2xCrxSe3 [45,59],Bi2xMnxSe3 [65],Bi2xMnxTe3 [62,99], and completely suppressed, leaving only the bulk van Vleck responses in Sb2xCrxTe3 [70,71] systems, and all the results exhibited such heavily Cr-doped TI samples. As a result, by designing the Cr doping monotonically increasing relation between the total magnetic profile and electric-gating strategy, we demonstrate an approach to moments and xM. either enhance or suppress the magnetic contributions from different More importantly, the effect of the magnetic doping level on magnetisms. individual magnetism can be further distinguished, given that RKKY and Van Vleck mechanisms have different responses to the 2.4. Interplay between different FM orders—Band topology external electric field as elaborated in Sections 2.1 and 2.2. Accordingly, by controlling the Cr doping levels during MBE Beside the magnetic ions doping level, the strength of ferro- growth, we have prepared a set of Crx(Bi0.5Sb0.5)2Te3 samples with magnetism is also determined by the band topology of the host TI the same Bi/Sb ratio (0.5/0.5) and film thickness (6 QL), but materials. In Section 2.2, it is argued that the strength of matrix different Cr doping levels ranging from 5% to 20%. Top-gated Hall elements onk|Sz|mk4 in Eq. (8) is determined by the mixing bar devices were fabricated to investigate the electric-field- between the inverted conduction/valence bands (i.e., SOC) [32].In controlled magneto-transport properties [46]. From the extracted Bi2Se3 systems, it is found that the SOC is mostly contributed from Hc Vg curves in Fig. 7e–h, it can be clearly observed that in the the Bi atoms [7]. As a result, substitution of Bi atoms with much moderate doping region (5%, 10%, and 15%), the Cr-doped lighter transition elements (Cr or Fe) will significantly lower the

(Bi0.5Sb0.5)2Te3 thin films all exhibit the hole-mediated RKKY SOC, and the overall Van Vleck susceptibility (i.e., FM order) of coupling behaviors in the sense that the anomalous Hall resistance Bi2Se3 is expected to get reduced by magnetic doping [100]. Rxy loops show a quick decrease of its coercivity field (Hc) when Zhang et al. calculated the band structures of the Bi2xCrxSe3 the majority holes are depleted. When the samples are further materials with different Cr contents by DFT [101].InFig. 8a, their biased deep into the n-type region (i.e., 450 meV above the calculations showed that the inverted gap amplitude shrank surface gap), Hc gradually saturates at finite values of 150 Oe dramatically with increased Cr content x. Finally, when the band (5%), 375 Oe (10%), and 685 Oe (15%), respectively. On the contrary, inversion between the conduction and valence bands disappeared when the Cr doping concentration increases up to 20% in Fig. 7d, at x40.17, the heavily-doped Bi2xCrxSe3 would enter a topologi- even though the surface Fermi level has been effectively adjusted cally trivial DMS regime without the presence of bulk Van Vleck by 50 meV (Inset of Fig. 7h), Hc remains a constant of 1000 Oe and magnetism any more. (As the comparison, since Te atoms con- does not show any change with respective to the gate bias. tribute giant SOC in Bi2Te3 and Sb2Te3 systems, the bulk band Consequently, from the detailed comparisons of the AHE results inversion would always hold regardless of the magnetic doping in Fig.7, it may be concluded that the bulk van Vleck mechanism level [101]. Under such circumstances, robust Van Vleck order has and the hole-mediation both exist in the moderately doped system been observed in heavily doped Crx(BiSb)2Te3 materials [42,46]). (i.e., o15%) and contribute to the net magnetizations. The topological phase transition due to magnetic doping has Recall the DOS distribution of Cr3þ ions in Fig. 4b, the long-range been confirmed by experiments. Fig. 8b shows the surface band

RKKY coupling in the p-type region thus introduces much more robust Ek dispersion relations of the Bi2xCrxSe3 ﬁlms grown on the Si ferromagnetic moments over the weaker bulk van Vleck term; it (1 1 1) substrate. The ARPES spectra reveal the systematic weak- therefore becomes the dominant component in the magnetic TI ening of the surface states along with the enlarged surface gap Δ, samples with moderate Cr doping. As more Cr atoms are incorporated as the Cr doping concentration gradually increases from 0 to 10%. in the TI system, on the one hand, the bulk van Vleck-type magnetism With further reducing the SOC and modifying the Berry phase is expected to become more pronounced due to the stronger out-of- away from the nontrivial π state [102], the topological surface are plane magnetic moments Mz from the Cr ions; on the other hand, almost eliminated in the Bi1.8Cr0.2Se3 ﬁlm.

Fig. 7. Gate-dependent AHE results for 6 QL Cr-doped (Bi1xSbx)2Te3 thin ﬁlms with different Cr doping concentrations (a) Cr¼5%, (b) Cr¼10%, (c) Cr¼15%, and (d) Cr¼20%. (e)–(h) Gate-modulated coercive ﬁeld changes for these four samples, respectively. Inset: Illustration of the Fermi level position as adjusted by the top-gate voltages ranging from 12 V to þ12 V. (Adopted from [46]).

In addition, Zhang et al. reported the quantitative study of the important to investigate the surface-related magnetism. Here, we topology-driven magnetic phase transition in the magnetic TI would like to summarize the work relate to the surface-carrier- materials [101]. In their study, the magneto-transport AHE results mediated RKKY interaction [30,41,82]. on five Bi1.78Cr0.22(SexTe1x)3 thin films with 0.22rxr0.86 were The generic RKKY long-range exchange model discussed in given in Fig. 9a. Three Te-rich samples with xr0.52 all showed a Section 2.1 is still valid except the Green function, which is used to hysteretic response at low temperature, reflecting the FM phase. In describe the TI surface states, is now given by[41] contrast, as the Se content was increased to x¼0.67, the hysteresis ! E ð Þ R E ! ð Þ R E disappeared, and the system became paramagnetic (PM). The sign Gð R Þ¼ iH 1 ij 8z^ Uð σ n^ÞH 1 ij ð10Þ ij 2 2 0 1 4ℏ v ℏvF ℏvF of the AHE also reversed to negative right at this doping level. F With further increase of x to 0.86, the system remained as ð1Þ where Hν is the ν-order Hankel function of the first kind. After paramagnetic, and the negative AHE became more pronounced. combing Eq. (2) with Eq. (10), the surface-carrier-mediated RKKY Most importantly, by summarizing the intercept of Rxy as a interaction can be written as [41] function of x and T, the critical Se content (x¼0.67) corresponding , ! ! ! to the FM-to-PM transition was obtained in the magnetic phase RKKY ¼ ð ; Þ U þ ð ; Þð Þ þ ð ; Þ y y ð Þ H1;2 F1 R EF S1 S 2 F2 R EF S 1 S 2 y F3 R EF S S 11 diagram (Fig. 9b), and was found to be consistent with the critical 1 2 inverted-to-normal transition point xc 0.66 in the topological where the range function F1 gives rise to the Heisenberg-like term, phase diagram. Consequently, their results provided strong evi- F2 represents the Dzyaloshinskii–Moriya-like term, and F3 stands dence that the band topology of the host TI material would be one for the Ising-like term. According to Zhu et al.’s calculations, all of the fundamental driving forces for the FM order, and the these range functions, F1–F3, still have the oscillation period of π/kF relative robust Van Vleck component in Bi2Te3 and Sb2Te3 were [93,95,103], where kF is the Fermi wave vector, and their magni- of great importance to further explore the quantum anomalous tudes scale with (J2/R3) [41]. When the surface Fermi energy level Hall effect in the magnetic TI systems. moves toward to the Dirac point, the oscillation period of the RKKY

interaction will diverge since the Fermi wavelength λF ¼1/kF 2.5. Topological surface-related ferromagnetism in magnetic TIs becomes inﬁnity as kF approaches zero [30], and thus neighboring magnetic impurities can always be coupled through the itinerant

So far, we only focus on the bulk magnetic properties of the carriers when λF is much larger than the average distance between magnetic TIs. In fact, given the unique Dirac-cone-like linear them, as shown in Fig. 10a [30]. On the other hand, however, we energy band of the topological surface states, it is thus equally should point out that in addition to λF, the surface-mediated RKKY

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Fig. 8. (Color Online) (a) DFT-calculated bulk band structures of Bi2xCrxSe3 with different Cr contents x¼0, 0.083, 0.167, and 0.25, respectively. The red and black broken lines indicate the positions of Fermi level and the points. (Adopted from [101]) (b) ARPES intensity maps of MBE-grown 50 QL Bi2xCrxSe3 thin ﬁlms with x¼0, 0.02, 0.1, and 0.2 on Si (1 1 1) along the Γ–K direction. All data are taken using 52 eV photons under the temperature of 10 K. (Adopted from [45]).

Fig. 9. (a) AHE results of Bi1.78Cr0.22(SexTe1x)3 thin ﬁlms measured at 1.5 K. (b) The magnetic phase diagram of Bi1.78Cr0.22(SexTe1x)3 summarizing the intercept of Rxy as a function of x and T. (c) The topological phase diagram of Bi1.78Cr0.22(SexTe1x)3 thin ﬁlms. (Adopted from [101]).

80 EF = 0 eV

0 F -40 1 F2 -80 F3 E = 100 meV F -120 -80 -40 0 40 80

EF (meV)

Fig. 10. (a) The range function of surface RKKY interaction as a function of the distance R between localized spin when EF is 100 meV above the Dirac point. The inset shows the intrinsic case (EF¼0). (Adopted from [41]) (b) The range function of surface RKKY interaction as a function of the Fermi energy EF for the Cr0.16(Bi0.54Sb0.38)2Te3 sample. In both (a) and (b), the on-site exchange parameter J is assumed to be a constant of 7 eV/Å2. interaction is also related to the DOS distributions of both the two-dimensional electron gas (2DEG) systems under high mag- mediating surface carriers and magnetic dopants, as we elaborated netic ﬁeld. On the contrary, for decades, little progress was made in Section 2.1 [79,104,105]. Since the DOS of Dirac surface states on the realization of quantum anomalous Hall (QAHE) effect after ð Þ¼ =ðπℏ2 2Þ¼ =ðπℏ Þ ﬁ have the form that D EF EF vF kF vF , it is thus inver- the rst theoretical model proposed by Haldane [108]. It was not sely proportional to λF. As a result, the overall strength of surface until the discovery of TI materials that the search for the suitable RKKY interaction is counterbalanced by the oscillation period non-zero Chern insulators and QAHE became practical. In this

(π/kF) and D(EF)(pkF). In particularly, as the Fermi level app- section, we will briefly introduce both the theoretical predictions roaches the Dirac point, fewer Dirac fermions around the Fermi and experimental observations of QAHE in the magnetic TI level will take part in the surface-mediated RKKY interaction even materials, and we will also compare the physics among these though λF becomes larger, and the two effects combined lead to a different transport quantum phases. finite surface RKKY coupling in the vicinity of the Dirac point [43]. In order to quantitatively study the surface-mediated RKKY 3.1. Construct QAHE from QSHE in a 2D system interaction in magnetic TIs, here, we show one example of a moderate-doped Cr (Bi Sb ) Te sample. In particular, with 0.16 0.54 0.38 2 3 Since 2007, the realization of quantum spin Hall effect (QSHE) the Cr doping level of 8%, we find that the average distance in the HgTe/CdTe 2D TI system [3,24] brought a new impetus to between neighboring Cr3 þ ions is R¼11 Å. If we further assume construct QAHE. Given the closely related topological relation that the on-site exchange parameter J is a constant of 7 eV/Å2 (the between these two quantum phenomena, it is argued that the same as Ref. [41]), we have the range function of surface-related helical edge state in the QSHE regime can be viewed as two copies RKKY interaction in Fig. 10b. Combined with the DOS distribution of the QAHE channels with opposite chirality [8,11]. Therefore, if of the Cr3 þ ions D (E ) which is majorly distributed below the d F one spin block can be suppressed due to TRS-breaking, the single Dirac point (Fig. 4b), it is concluded that Dirac holes can effectively QAHE conduction is thus realized. In 2008, based on the 2D TI couple neighboring Cr ions together, and the surface-related four-band model described by Eq. (1) [3], Liu et al. investigated the ferromagnetism is manifested in the p-type region, similar to the topological surface band change by introducing additional Mn ions bulk RKKY case. Recently, Checkelsky et al. reported the experi- into the HgTe quantum wells (QW) [31]. In this case, the spin mental observation of such Dirac-hole-mediated RKKY interaction splitting term induced by magnetization can be expanded into a in the Mn Bi Te Se materials where the highest Curie x 2 x 3 y y phenomenological form as [31] temperature was obtained when the samples were biased in the 2 3 p-type region [44]. GE 00 0 6 7 Nevertheless, due to similar hole-mediated behavior with the 6 0 GH 007 ¼ ð Þ fi Hs 6 7 12 bulk states, it is dif cult to distinguish the surface contribution in 4 00GE 0 5 uniformly doped magnetic TI systems. As will be discussed in 00 0 GH Section 4.1, modulation-doped magnetic TI heterostructures will be served as a better platform for us to manipulate the surface- where the spin splitting energy difference is 2GE for the two related ferromagnetic order [43]. electron sub-bands |E1, 7 4, and 2GH for the two hole sub-bands |H1, 7 4. Under such condition, if GE GH o0(i.e., assume GEo0 and GH 40), the spin splitting for |E1, 7 4 and |H1, 7 4 would 3. Quantum anomalous hall effect in magnetic TI materials have opposite signs; with large enough splitting energy, the band

inversion of the |E1, þ 4 and |H1, þ 4 states still holds, while the It is known that both the ordinary Hall effect and anomalous |E1, 4 and |H1, 4 sets ﬁnally enter the normal regime, leaving Hall effect in the solid-state transport are the result of the breaking only spin-up edge states in the quantum transport (QAHE) regime, of TRS. Speciﬁcally, in the former case, a perpendicular magnetic as illustrated in Fig. 11a. Following the same spin splitting (i.e.,

field is needed to deflect the conduction charge particles in order GE GH o0) scenario, Yu et al. later pointed out that even the four- to generate the transverse voltage Vxy [106]; in the latter case, band system was originally in the topological trivial (non- spontaneous magnetization replace the external magnetic field, inverted) phase, proper exchange field would also induce a band and Vxy arises from the spin–orbit interaction between charge inversion in the spin-up sub-bands (|E1, þ 4 and |H1, þ 4) while current and magnetic moments [85]. Since the discovery of the pushing the spin-down sub-bands (|E1, 4 and |H1, 4) even quantum Hall effect (QHE) 1980 [107], quantized dissipationless further away from each other (Fig. 11b) [32]. Under such circum- chiral edge conduction have been observed in many high-mobility stances, the single QAHE conduction could also be constructed.

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In fact, although Liu et al.’s model sounds feasible, in reality, the hybridization regime, quantum tunneling between the top and

Mn moments in the Hg1xMnxTe system do not order sponta- bottom surfaces became pronounced[8,16,92,109], giving rise to a fi ¼ þ ð 2 þ 2Þ neously, therefore they fail to generate required spin splitting nite mass term, mk m0 B kx ky , and the surface Hamiltonian energy. Alternatively, since pronounced magnetization can be could be further modified from Eq. (1) by [32] generated from the bulk Van Vleck susceptibility in the Cr-doped "# h þgMσ 0 TI films even if the bulk reaches in the insulating state (Section ¼ k z ð Þ H n σ 13 2.2), Yu et al. proposed an improved model to realize QAHE in the 0 hk gM z Cr-doped tetradymite-type TI systems [32]. They found that when where h ¼ m σ þν ðk σ k σ Þ. Under the spatial inversion sym- these Cr-doped 3D TI films reduced their thickness into a 2D k k z F y x x y metry assumption (i.e., neglecting the band bending of the top/

bottom surface states in the thin ﬁlm), νF, g, and M were of same values for the top and bottom surfaces, and therefore, the

requirement of GE GH o0 was automatically achieved. In the subsequent section, we will discuss the realization of QAHE in Cr- doped TI in the 2D hybridization regime based on Yu et al.’s model.

3.2. Observation of QAHE in 5 QL Cr0.15(Bi0.1Sb0.9)1.85Te3 ﬁlm

As elaborated in Part 2, the Cr-doped Bi2Te3/Sb2Te3 systems are regarded as the promising candidate for QAHE: due to the giant SOC from the Te atoms, the bulk band is always inverted; large out-of-plane magnetization is spontaneously established through the Van Vleck mechanism without the assistance of itinerant carriers; and most importantly, the bulk carrier density can practically be minimized by optimizing the Bi/Sb ratio, and the Fermi level can be tuned across the Dirac point by gate modulation [42,46,110]. Chang et al. from Tsinghua ﬁrst reported the observation of

QAHE on the 5 QL Cr0.15(Bi0.1Sb0.9)1.85Te3 film grown on the SrTiO3 (1 1 1) substrate [39]. In their work, QAHE was achieved at 30 mK, as shown in Fig.12a and c. Specifically, when the Fermi level was Fig. 11. Evolution of band structure and edge states upon increasing the spin electrically tuned into the surface gap (V 1.5 V), R was o g xy splitting for (a) inverted band and (b) non-inverted band. In both cases, GE 0 and 2 Ω G 40; the spin-up sub-bands (|E , þ 4 and |H , þ 4) remain inverted, while the quantized at h/e (25.8 k ), and such quantized value was nearly H 1 1 fi spin-down sub-bands (|E1, 4 and |H1, 4) enter the normal insulating states. invariant with the external magnetic eld, suggesting perfect edge (Re-drawn from [31,32]). conduction and charge neutrality of the film. In the meanwhile,

Fig. 12. (Color Online) The QAH effect in the 5QL Cr0.15(Bi0.1Sb0.9)1.85Te3 film. (a) Magnetic field dependence of Rxy at different Vgs. (b) Dependence of Rxy(0) (empty blue squares) and Rxx(0) (empty red circles) on Vg. (c) Magnetic field dependence of Rxx at different Vgs. (d) Dependence of σxy(0) (empty blue squares) and σxx(0) (empty red circles) on Vg. The vertical purple dashed-dotted lines in (b) and (d) indicate the Vg for Vg¼0. (Adopted from [39]).

Please cite this article as: X. Kou, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2014.10.022i X. Kou et al. / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 11 the giant MR (2251%) was observed in Fig. 12c, and it related to the Hamiltonians of the top and bottom surfaces can be written as [36] quantum phase transition between the two opposite QAHE states ¼ 7ν ℏð σ σ ÞþΔ σ ð Þ via a highly dissipative bulk channel. In addition, the gate- Heff F kx y ky x z z 14 dependent R and R at zero field were provided in Fig. 12b, xy xx where7represents the top/bottom surfaces, and Δ ¼gM . There- and the authors showed that R exhibited a distinct plateau with z z xy fore, the massive Dirac fermions at the top and bottom surfaces the quantized value h/e2; correspondingly, the longitudinal R xy have opposite masses ð7Δ =v2Þ [36]. In the meanwhile, since the experienced a sharp dip down to 0.098 h/e2. Recently, Lu et al. z F side surfaces in the thick 3D magnetic TI system is parallel to the fitted these transport data with an effective conduction model, and magnetic moments, their topological surface bands remain mass- the agreement between the theory and the experiment confirmed less [36]. Under such circumstances, we can regard the 3D that the transport in the QAHE regime indeed originated from the magnetic TI system as two massive Dirac fermions with opposite topological non-trivial conduction band, which had a concentrated masses separated by a massless Dirac fermions in between, or Berry curvature and a local maximum in the group velocity [111]. equivalently, the gapless side surface acts as the domain wall Besides, in Chang et al.’s experiments on the 5 QL film, a non-zero between the two topologically different phases, as shown in longitudinal resistance R was observed in the QAHE regime at xx Fig. 13b. Just like the case in a usual QHE system, the half- zero field. By further applying a large perpendicular magnetic field quantized chiral edge channels will be trapped at this interface (i.e., B410 T), the 5 QL film was finally driven into a perfect QHE between the side surface and the top/bottom surface [8,36]. More regime, and R diminished almost to zero [39]. The origins of such xx importantly, since the TRS is protected for the gapless side surface phenomenon have been further analyzed, and both the variable states, the side-surface conduction is non-chiral, and it does not range hopping (VRH) [39] and the gapless quasi-helical edge states contribute to the Hall conductance; in other words, the stability of in the 5 QL Cr-doped TI film [112] have been proposed to explain the quantized chiral edge state is well-maintained. As a result, such field-dependent R results. xx QAHE can always be realized in the thick 3D magnetic TI samples as long as the bulk remains insulating and the quantized chiral channels are well-confined at the domain interface [8,36]. 3.3. Universal QAHE phase More recently, both Jiang et al. [33] and Wang et al. [34] proposed a more generic way to construct QAHE with tunable Section 3.1 discussed the roadmap to construct the QAHE from Chern number in thick magnetic TIs. Following the same spin QSHE in the 2D hybridized magnetic TI system. In general, given splitting process as elaborated in Fig. 11, they found that the on- the intrinsic relation between the Berry's phase and anomalous site band inversion depended on the critical film thickness dn as Hall conductance [113–116], it has been suggested that the [34]: quantized Hall conductance without the presence of external pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi magnetic field can be realized in a ferromagnetic insulator which dn 4nπ Bz=ð7Δz m0Þ ð15Þ has a non-zero first Chern number (C1) due to the non-zero integral of the energy band Berry curvature [8,85]. To this regard, where n represents the nth-order sub-bands, m0 is the energy it was proposed by several groups that QAHE could also be derived difference between the lowest-order inverted conduction and from the gapped top and bottom surfaces in a 3D magnetic TI valence sub-bands, and Bz is the z-axis coefficient of the gap systems [8,36,117]. parameter m(k)defined in Eq. (1). Consequently, with a given In fact, for an ideal 3D magnetic TI material with insulating exchange field strength, QAHE with higher plateaus would be bulk state, the out-of-plane magnetization opens gaps on the top possible in the thick magnetic TI films with insulating bulk states, and bottom surfaces, and changes each surface into a non-zero C1 and the QAHE was, in principle, a universal quantum phase phase with a half-quantized Hall conductance (e2/2h), as illu- regardless of thickness (whereas in the QHE regime, the formation strated in Fig. 13a [8,36]. Due to the opposite local basis defined of the precise Landau level quantization requires the electrons to _ with respect to the normal direction n (Fig. 13b), the effective be strictly confined in the 2D region [118]).

MZ y + Δz x

Top Surface nˆ

Side Surface (Domain Wall)

Bottom Surface nˆ

- Δz

Fig. 13. (a) Schematic of a 3D magnetic TI with an out-of-plane magnetization Mz, and the formation of chiral current on the top and bottom surface boundaries. (b) A chiral edge state will form around the domain wall between the 2D Dirac fermions with positive and negative masses. The arrows indicate the ﬂow of edge current. (Re-drawn from [8,36]).

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+ h/e2 < 85 mK MZ 0.5 K 20 1 K

10 5 mm 4.7 K

(k ) 0 14,62 R

-10 MZ

Intensity (a.u) -20 -h/e2

2 nm 0 2 4 6 8 10 -0.2 0.0 0.2 # QLs B (T)

Fig. 14. The QAH effect in the 10 QL Cr0.06(Bi0.26Sb0.62)2Te3 ﬁlm. (a) The image of the Hall bar-structure used for the QAHE measurement. The size of the Hall bar is

2mm 1 mm. (b) Both the HRSTEM and RHEED oscillation conﬁrms the ﬁlm thickness. (c) Temperature-dependent hysteresis Rxy curves. When the base temperature is below 85 m K, the sample reaches the QAHE regime. (d) Schematics of the chiral edge conduction in the QAHE regime under different magnetization directions.

3.4. Scale-invariant QAHE beyond 2D hybridization regime scale (i.e., from 200 mm to 2 mm). In fact, the chiral nature in the

QAHE regime causes (Ti,i þ1, Tiþ 1,i)¼(0, 1). Therefore, the backward To investigate the QAHE in 3D magnetic TIs, we prepared high- conduction is prohibited (unlike QSHE, in which the presence of quality Cr-doped (BiSb)2Te3 films on the intrinsic GaAs (1 1 1) the helical edge states would allow in-elastic backscattering and substrate by MBE. The film thickness was accurately controlled to incoherent momentum-relaxation), and the de-coherence process be 10 QL so as to prevent the top and bottom surfaces hybridizing from the lateral contacts does not lead to momentum and energy with each other. After growth, mm-size (2 mm 1 mm) six-terminal relaxation [43,122]. Together with the thickness-dependent Hall bar device was fabricated, as shown in Fig. 14a. Fig. 14cshows results, we may conclude that when appropriate spin–orbit inter- the AHE results in our 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 film when the action and perpendicular FM exchange strength are present in a temperature is far below TC. It can be seen that the film reaches the bulk insulating magnetic TI film where the Fermi level resides QAHE regime where the Hall resistance Rxy reaches the quantized inside the surface gap [32,100,112], the QAHE resistance is always value of h/e2 at B¼0Tatthebasetemperatureupto85mK.To quantized to be h/e2, regardless of the device dimensions and de- quantitatively understand the chiral edge transport in our sample, phasing process. we thus apply the Landauer–Büttiker formalism that [119]

3.5. Non-zero Rxx and non-local measurement in thick magnetic e2 TI film Ii ¼ ∑ðTjiV i TijV jÞð16Þ h j In Section 3.2, Chang et al. reported a non-zero longitudinal where Ii is the current flowing out of the ith contact into the sample, resistance of the 5 QL Cr0.15(Bi0.1Sb0.9)1.85Te3 film in the QAHE Vi is the voltage on the ith contact, and Tji is the transmission regime [39]. Here, we also performed the Rxx T and Rxy T results probability from the ith to the jth contacts [120,121].InourHallbar of the 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 film at B¼3 T and 15 T in the structure shown in Fig. 14d, the voltage bias is applied between the low-temperature region (To1 K), as shown in Fig. 15a. It is noted

1st and 4th contacts (i.e., V1¼V, V4¼0, and I1¼I4¼I) and the other that a similar non-zero Rxx is also observed when our 10 QL four contacts are used as the voltage probes such that magnetic TI ﬁlm reaches the QAHE state below 85 m K. In contrast

I2¼I3¼I5¼I6¼0. Unlike QSHE helical states [24,122],intheQAHE to the 5 QL film case, it is apparent from Fig. 15a that the regime, where the TRS is broken by the spontaneous magnetization, longitudinal resistance in our 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 sample the electrons can only flow along one direction and such chiral edge remains at 3 kΩ even when the applied magnetic field reaches conduction route depends on the magnetization direction[43].Spe- 15 T. More importantly, unlike the bulk conduction case which has cifically, when the magnetic TI film is magnetized along þz direction a typical parabolic MR relation [59], Rxx at T¼85 m K exhibits little (i.e., the upper panel of Fig. 14d), the non-zero transmission matrix field dependence when the magnetic field is larger than 3 T elements for QAHE state are T61 ¼T56¼T45¼1 [43,122],andthe (Fig. 15b). Consequently, it may be suggested that the non-zero 2 corresponding voltage distributions are given by V6¼V5¼V1¼(h/e ) Rxx in the thicker 10 QL magnetic TI film is more likely associated I and V2¼V3¼V4¼0. On the other hand, when the magnetization is with a unique dissipative edge conduction, as illustrated in reversed (i.e.,thelowerpanelofFig. 14d), the edge current would Fig. 13b. In particular, with the out-of-plane magnetization in our

ﬂow through the 2nd and 3rd contacts, thus making V2¼V3¼V1¼ system, the gapless Dirac point on the side surfaces is shifted away 2 (h/e )I and V5¼V6¼V4¼0. Consequently, Rxy¼R14,62 ¼(V6V2)/I is from the symmetric point (Γ). Therefore, the backscattering is not expected to be positive if MZ40 and change to be negative if MZo0. suppressed anymore, and the non-chiral side surface conduction The observed polarities of the above QAHE results with respect to the thus becomes dissipative along the longitudinal direction [36]. magnetization directions are consistent with Eq. (16),whichclearly Besides the conventional Hall experiments, another important demonstrates the chiral edge conduction nature of QAHE. method to manifest the chiral/helical properties of quantum trans- Most important is the observation of the scale-invariant quan- port phenomena is through the non-local transport [112,122]. tized edge conduction in the QAHE regime on the macroscopic To this regard, we carried out the non-local measurements on the

h/e2 25 0.25 2 3 Rxy 20 1 4 15 0.20 (k ) 6 5 V R (k ) Rxx 10 3 T 12,54 15 T R 0.15 5 4.7 K 0 5 6 7 8 2 3 4 5 6 7 8 0.1 1 T (K) 15 2 3 2 3 1 4 1 4 50 85 mK (k ) 10 6 5 6 5

40 12,54 R

(k ) 30 5

12,54 20 85 mK R 0 -0.2 0.0 0.2 10 B (T)

-10 0 10 B(T)

Fig. 15. (Color Online) (a) Temperature-dependent Rxx and Rxy of the 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 film at B¼3 T and 15 T in the low-temperature region. (b) Magnetic field dependence of Rxx at 85 m K. Rxx in our 10 QL magnetic TI sample shows little field dependence when B43 T. (c) Non-local measurements of the six-terminal Hall bar device. the current is applied through 1st to 2nd contact and the non-local voltages are measured between 4th and 5th contacts at T¼85 m K and 4.7 K. The red solid arrow indicates that the magnetic field is swept from positive (þz) to negative (z), whereas the blue solid arrow corresponds to the opposite magnetic field sweeping direction.

six-terminal Hall bar device. The top inset of Fig. 15cshowsthenon- voltage probes from V3 to V6 are now away from the dissipationless local configuration: the current is passed through contacts 1 channel. Therefore, the non-local signal only relates to the voltage (source) and 2 (drain) while the non-local resistance between drop caused by the dissipative edge channel, which thus gives rise contacts 5 and 4 (R12,54) is measured. In the QAHE regime to a larger value of R12,54. As a comparison, we also plot the field- (To85 m K), it can be clearly seen that R12,54 displays a square- dependent non-local results at 4.7 K (higher panel of Fig. 15c), shaped hysteresis windows (two-resistance states). In other words, where R12,54 is dominated by the larger bulk conduction component R12,54 reaches the high-resistance state of 15 Ω when Bo 0.2 T (i.e., the non-local resistances are more than 10 times larger than while its magnitude vanishes close to zero once B40.2 T. those probed at 85 m K). In such diffusive transport regime, the Here, we point out that the non-local resistances can be under- square-shaped hysteresis non-local signals are replaced by the stood from the chirality of QAHE. From Eq. (16), it is known that that ordinary parabolic MR backgrounds, and the polarity of R12,54 also the transmission matrix element is given by (Ti,iþ 1, Tiþ1,i)¼(0, 1) in disappears. To summarize, both the field-independent Rxx and the the ideal QAHE regime. If we solve the non-local transport config- non-local resistances displayed in Fig. 15 confirm the coexistence of uration shown in Fig. 15cusingEq.(16), the initial conditions thus QAHE chiral edge channel and the additional dissipative edge become V1¼V, V2¼0, I1¼I2¼I,andI3¼I4¼I5¼I6¼0. As a result, we conduction in the thick 10 QL Cr-doped TI sample. 2 should obtain that V6¼V5¼V4¼V3¼V1¼(h/e )I when MZ40, and V6¼V5¼V4¼V3¼V2¼0whenMZo0. In either case, there is no 3.6. Discussion on QHE, QSHE, and QAHE voltage drop among the four voltage probes, therefore leaving the non-local resistance of R12,54 to be nearly zero regardless of the As discussed in the above sections, although QHE, QSHE, and magnetization direction. QAHE all belong to the quantized edge transport phenomena, their Now, if we consider the influence of the additional dissipative non- intrinsic mechanisms are quite distinct. In this section, we sum- chiral edge channel as discovered in the 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 marize and compare both the relations and differences between film, there will be two possible conditions for the non-local signals: these quantum Hall trio, and show the advantages of QAHE for thevoltageprobesmeasurethevoltagedropalongthesameedge low-power interconnect applications. where the QAHE edge state is present, and the voltage probes are It is known that QHE can only be realized in a C1¼1 Chern away from the dissipationless chiral edge channel. With the same insulator where the bulk is insulating because of the discrete non-local transport configuration as illustrated in Fig. 4(a), it can be Landau levels (LLs) and the conduction channels only exist at the seen that in the former case, the chirality forces the QAHE dissipation- edges, as shown in Fig. 16a [119]. In the QHE regime, the external less current flow from contact 1 to contact 2 through the 1-6-5- magnetic field acts as a gauge field which couples to the momen- 4-3-2 contacts successively, and in turns “shorts” the contacts so tum of the electrons and forces the electrons to make cyclotron that V6¼V5¼V4¼V3V1¼V [119].Asaresult,thevoltagedrop motions in real space. In order to form the discrete LLs (i.e.,to between these contacts is negligible, and R12,54 is driven into the change the band topology), the electrons need to complete the low-resistance state. On the other hand, when the magnetization is cyclotron motion circles before losing its momentum due to reversed (i.e., the latter condition), the 1st and 2nd contacts are scattering, and this in turn requires the electrons to have a very directly connected by the QAHE channel in the upper edge, and the high mobility [119]. In addition, the precise LL quantization also

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Bz Mz

C1 = 1 C2 = 1 C1 = 1 M = 2 M = 2 M = 1

Quantum Hall effect Quantum Spin Hall effect Quantum Anomalous Hall effect

Fig. 16. The quantum Hall trio. (a) Quantum Hall case. External perpendicular magnetic field is required to localize dissipative channels and quantize LLs. Due to the TRS breaking, chiral edge conduction is dominant. (b) Quantum spin Hall case. Due to the protection of TRS and spin-momentum locking mechanism, opposite-spin electrons occupy opposite sides in the quantum spin Hall system. (c) Quantum anomalous Hall case. The TRS is broken by magnetic doping. When the film is magnetized along þz-direction, only spin-up electrons flow through the edge clockwise. requires the electrons to be confined in the 2D dimension so that broken by the spontaneous magnetization rather than external the energy levels of these quantized orbitals take on discrete magnetic field, and it does not require the formation of LLs. values of En ¼ ℏωcðnþð1=2ÞÞ where ωc¼eB/m is the cyclotron Accordingly, QAHE may have loose requirements on the film frequency. Therefore, once the Fermi level is located inside the thickness and quality, (i.e., our grown magnetic TI film has a gap between two neighboring LLs, the quantum chiral edge state relatively low mobility of 600 cm2/V s and the film is much thicker readily appears while the bulk carriers are localized by the than the 2D hybridization limit, but the quantized conduction of cyclotron orbitals. In other words, the realization of QHE demands e2/h is quite robust). In summary, the realization of QAHE requires a strong magnetic field and a high mobility material with a good an appropriate spin–orbit interaction, a strong exchange interac- 2D confinement [118,119]. tion (i.e., in order to both splits the bands and localizes the bulk The QSHE [3], on the other hand, does not need any external carriers) and an out-of-plane magnetic anisotropy; all of which are magnetic field. It is realized in systems where the strong spin– found to be satisfied in the Cr-doped (BiSb)2Te3 thin films. orbit interaction leads to the band inversion at the Г point in the

Brillouin zone. This band inversion makes the system into a C2 ¼1 (C1 ¼0) Chern insulator with the quantized channel number M¼2 4. Physics and applications of magnetic TI heterostructures [11]. Here the second Chern number is employed to describe the topology in the QSHE state. Because the TRS is preserved in such As we addressed in Section 2.5 that the uniformly doped system (i.e., the external magnetic field is zero), the resulting magnetic TI was not a proper platform to quantify the surface- quantum edge states are helical instead of chiral, as shown in related ferromagnetism since both surface and bulk holes are Fig. 16b. Accordingly, when the Fermi level is placed inside the favorable for mediating the RKKY interaction. In addition, the bulk band gap, the longitudinal conductance exhibits a quantized entanglement of surface carriers with bulk magnetic dopants 2e2/h helical conduction. It is noted that since the bulk band gap, made it more complicated to distinguish the surface contribution. which is determined by the spin–orbit interaction strength, is Alternatively, if we design the TI/Cr-doped TI bilayer structure by usually small (i.e., up to 0.3 eV), the quantum helical edge introducing an undoped TI layer one top, the top surface is conduction is thus vulnerable to any intrinsic bulk impurity/ separated from the bulk magnetic ions. Under this condition, by defect-induced carrier states and band potential fluctuations. varying the thickness (d1) of the top undoped TI layer, we are able Consequently, the QSHE requires a very high material quality (i. to investigate and manipulate the surface-related ferromagnetism e., low bulk carrier density and high mobility). To date, the QSHE in a straightforward manner. has only been realized experimentally in the HgTe/CdTe [24,122] and InAs/GaSb [123,124] quantum wells. 4.1. Manipulation of surface-related ferromagnetism in TI/Cr-TI Compared with QHE and QSHE discussed above, the QAHE bilayers involves both the strong spin–orbit interaction and the magnetic exchange interaction in a magnetic insulating system. In the QAHE In principle, the proposed TI/Cr-doped TI bilayer structures can regime with a robust magnetic moment, the strong spin–orbit be implemented by using the modulation-doped growth method interaction as well leads to the band inversion. However, the large via which the Cr distribution profile along the epitaxial growth exchange field, as large as 100 T, from the spontaneous magneti- direction is accurately controlled [125–127]. The illustration of the zation couples with the spins of the band electrons, and splits the modulation-doped growth methods are shown in Fig. 17a. During spin-up and spin-down sub-bands in the opposite directions. It in the film growth, the feedback signal from the in-situ RHEED turn results in one pair of the spin-resolved bands un-inverted oscillations was added to enable the effusion cell shutters be while keeping the other pair still remains within the non-trivial automatically controlled by the computer. After growth, the Bi/Sb topological state. As a result, in contrast to the QSHE state, the bulk composition ratio (x, y) and the Cr doping level were determined topology is changed, the system becomes a C1 ¼1 Chern insulator, by an energy-dispersive X-ray (EDX) spectroscopy. Fig. 17b shows and the quantized QAHE chiral edge transport at zero external the result of one 3 QL (Bi0.5Sb0.5)2Te3/6QLCr0.08(Bi0.57Sb0.39)2Te3 magnetic field can be achieved when the Fermi level is located in film grown by the modulation-doped method. In contrast to the massive Dirac surface band gap, as displayed in Fig. 16c [39].In the lower figure, the absence of the Cr peak in the upper EDX the meanwhile, compared with QHE, the TRS in the QAHE state is spectrum provides strong evidence that the Cr atoms only

Te Bi Cr

TI layer

Cr-doped TI layer

substrate Cr-doped TI thickness Intensity 8.8 12

8.6 10

Cr (%) 8.4 8

8.2 6 time 1 234 5 Sample #

Fig. 17. (a) Illustration of the modulation-doped growth method used to prepare the TI/Cr-doped TI bilayer ﬁlms. (b) The image of the Hall bar-structure used for the QAHE measurement. The size of the Hall bar is 2 mm 1 mm. (b) High-resolution cross-section STEM image of the bilayer thin ﬁlm. (c) EDX spectrum of (Bi0.5Sb0.5)2Te3 and

Cr0.08(Bi0.57Sb0.39)2Te3 layers. (d) High repeatability of the MBE-grown TI ﬁlm.

distribute inside the bottom 6 QL Cr0.08(Bi0.57Sb0.39)2Te3 layer, grown modulation-doped TI bilayer films is estimated to be less while the top 3 QL layer is free of magnetic impurities. than 3 nm at 1.9 K when the samples are biased in the inversion/ In order to verify the surface-related magnetism in the accumulation regions (i.e., n/p-type) [128,129]. Based on our modulation-doped TI hetero-structures, we first performed the device geometry, it implies that the applied top-gate voltage is magneto-optical Kerr effect (MOKE) measurements on three differ- only able to tune the Fermi level of the top surface states. ent samples as follows: Sample A has a 6 QL Cr0.16(Bi0.54Sb0.38)2Te3 Accordingly, we may conclude that the magneto-electric responses with a uniform Cr-doping profile (control sample); Sample B and among the modulation-doped TI hetero-structures in the p-type

Sample C share the same 6 QL Cr0.16(Bi0.54Sb0.38)2Te3 designs in the regions are predominantly due to the variations of the surface- bottom, but differ in the top undoped (Bi0.5Sb0.5)2Te3 layer thick- related ferromagnetism, and the pronounced enhancement of ΔHc ness, namely 3QL for Sample B and 6 QL for Sample C. Fig. 18a–c may reflect the optimization of the magnetic RKKY coupling shows the field-dependent MOKE results acquired at different between neighboring Cr ions through top surface states. temperatures. On the one hand, when the top layer thickness d1 To understand the optimization behavior of the surface- is 3 QL, the coercivity field Hc at T¼2.8 K has enlarged by more than mediated magneto-electric effect in the TI/Cr-doped TI bilayer 5 times (i.e., from 4.5 mT to 30 mT) compared with the control device, we can propose the realistic model based on two factors. sample. On the other hand, Hc does not change monotonically with First of all, it is known that both the magnetic dipole and exchange d1. Instead, when the top surface is 6 QL away from the bottom interactions from the bottom magnetic layer will force the top layer, we observe the minimum Hc of only 2.5 mT. Here, since the surface open a surface gap Δ0. In our TI/Cr-doped TI bilayer identical bottom Cr-doped TI layers of the three samples contribute systems, since the mediating carriers itinerate within the top the same “bulk” magnetization, the change of both Hc and TC among surface states, the resulting exchange coupling, and thus the gap them hence can only be associated with the separation between the opening, will thus decay exponentially with the separation dis- top surface and the bulk Cr ions. Therefore, both the unique tance (d1). Because the RKKY coupling is a second-order- d1-dependent magnetization behavior and the enhancement of TC perturbation interaction, its exchange strength is thus inversely in Sample B may provide us with the direct evidence about the proportional to the energy difference between the ground and the presence of the surface-related ferromagnetism in modulation- excited states (i.e., Δ0) [130]. On the other hand, however, since the 2d=D doped TI thin films. density of surface carriers follows the nsðdÞpe 0 distribution To further investigate the magneto-electric responses from the relation and D0 characterizes the penetration depth of the surface top surface states, gate-dependent AHE measurements were states with a typical value of 2–3 QLs [131,132], it is expected carried out on five samples with fixed bottom magnetic TI layer that as d1 becomes larger, fewer surface Dirac fermions will (i.e., 6 QL Cr0.16(Bi0.54Sb0.38)2Te3) but different d1 in the top participate into the RKKY mediating process, therefore weakening undoped layers (i.e., d1 ¼0, 2, 3, 4, 5 QL, respectively). The Hc Vg the surface-related RKKY coupling strength. Accordingly, if we results of these five TI/Cr-doped TI bilayer samples are displayed in take these two competing factors into consideration, the overall

Fig. 18d. Here, the unveiled monotonically decreasing relation of d1-dependent surface-related RKKY exchange strength oΦ(d1)4 HcVg in the p-type region possibly suggests the signature of the follows [43]: hole-mediated RKKY interaction, as discussed in Part 2. In the = Δ0 oΦð Þ4 ¼ Φ 2d1 D0 ð Þ meanwhile, due to the effective Fermi level tuning, the coercivity d1 0e = 17 Δ d1 D1 þΔ 1e dipole fields Hc are found to achieve over 100% modulation via gate- voltage control in all samples; yet the most pronounced change is where Φ0 is the RKKY interaction strength in the uniformly Cr- obtained in the 3 QL TI/6 QL Cr-doped TI sample, as highlighted in doped TI film and Δ0 ¼Δ1 þΔdipole is the total surface gap opening Fig. 18e. It is important to point out that the Debye length of our which takes into account both the direct magnetic exchange field

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(Δ1) and the long-range magnetic dipolar field (Δdipole) from the 4.2. Magnetization switching via SOTs in TI/Cr–TI bilayers doped Cr ions. Based on Eq. (17), it is obvious to see that as the thickness d1 increases, the carrier density will drop rapidly which In Section 2.4, we discuss about the influence of the magnetic 2d =D further decreases the Φ0e 1 0 part; on the contrary, the reduced ions on the bulk band topology. In the meanwhile, the giant SOC in surface Dirac fermion gap Δ(d1) in the denominator enhances the the bulk TI matrix can also affect the magnetizations. In fact, it has RKKY interaction. For the d1 ¼0 case, Eq. (17) reduces to the been observed in the heavy metals (Pt, Ta) with strong SOC, uniformly doped case that oΦ(d1)4 ¼Φ0, while in the opposite applied in-plane charge current can control the magnetization extreme condition where d1 ¼1, oΦ(d1)4 returns back to zero, dynamics in an adjacent ferromagnet layer (Co, CoFeB) due to the corresponding to the surface-bulk decoupled condition. By fitting spin–orbit torques (SOTs) [133–144]. Compared with the heavy

Eq. (17) with the our experiment data, we obtain the d1-depen- metals, in our conductive TI/Cr-doped TI bilayer heterostructures, a dent oΦ(d1)4 behavior in Fig. 19b. In addition, the optimized dominant spin accumulation in the Cr-doped TI layer with spin condition d1 ¼3QL(i.e., ∂ðoΦðd1Þ4Þ=∂d1 ¼ 0) is consistent with polarized in the transverse direction is expected when passing a Fig. 18e, manifesting the fundamental surface-related RKKY charge current in the y-direction due to the SHE in the bulk and mechanism. the spin polarization arising from the Rashba-type interactions at

1.0 1.0 1.0 d1 = 0 QL d1 = 3 QL d1 = 6 QL

0.5 0.5 0.5

0.0 0.0 0.0 (a.u) (a.u) (a.u) K K K

-0.5 -0.5 -0.5

-1.0 -1.0 -1.0 -20 0 20 -40 -20 0 20 40 -40 -20 0 20 40 B (mT) B (mT) B (mT)

d1 = 0 QL 120 80 60 d1 = 2 QL d = 3 QL 100 1 50 d1 = 4 QL 80 60 H d1 = 5 QL 40 c (mT) (mT) (mT) c c 60 30 H H 40 40 20

20 10 20 0 0 -10 -5 0 5 10 0 1 2 3 4 5 6

Vg (V) d1 (QL)

Fig. 18. Temperature-dependent magnetization MOKE measurement (a)–(c), Out-of-plane magnetizations (reﬂected in Hc) measured by a polar-mode MOKE setup. All of the modulation-doped samples have the same 6 QL Cr0.16(Bi0.54Sb0.38)2Te3 bottom layer, with different top layer thicknesses (0QL, 3QL, and 6QL). (d) Gate-controlled coercivity

ﬁeld Hc of modulation-doped structures. The Cr-doped TI layer is ﬁxed as 6 QL Cr0.16(Bi0.54Sb0.38)2Te3, whereas the thickness of the top undoped TI layer changes from 0 QL to

6 QL. All data are measured at 1.9 K. (e) Comparisons of the both the “neutral point” Hc and gate-controlled coercivity ﬁeld change ΔHc in modulation-doped samples with varying top surface thickness d1. The “neutral point” is deﬁned as the position where the surface Fermi levels are tuned close to the Dirac point. (Re-drawn from [43]).

1.0

0.8

0.6 > (a.u) 0.4 <

0.2

0.0 0 2 4 6 8 10

d1 (QL)

Fig. 19. (a) Schematic representations of the surface-related magneto-electric effects in TI/Cr-doped TI bilayer structures with optimized top layer thickness d1 ¼3 QL.

(b) Simulation results of the surface-mediated RKKY coupling strength as the function of top layer thickness d1. (Re-drawn from [43]).

the interfaces. A strong enhancement of the interfacial spin (blue squares in Fig. 20b), the AHE resistance RAHE goes from negative accumulation can be expected due to the spin-momentum locking to positive as the applied in-plane magnetic field By gradually changes of the topological surface states [11,12,145–147]. The accumulated from 3 T to 3 T, indicating the z-component magnetization Mz spins’ angular momentum can be directly transferred to the switches from z to þz. In contrast, when Idc¼10 μA, the AHE magnetization M and therefore affect its dynamics. In particular, resistance reverses sign (red circles in Fig. 20b) and Mz varies from þz the complete form of the Landau–Lifshitz–Gilbert (LLG) equation to z as By is swept from 3Tto3T.Itshouldbenotedthatinboth which includes the SOT is given by [148,149]: cases the AHE resistance hysteresis loops agree well with our ! !! proposed scenario. At the same time, when we scan the DC current ! α dm ! ! dm ! ! ^ I at a given fixed magnetic field, we also observe similar magnetiza- ¼jγjμ ðm H exÞþ ! m þλ1I½m ðm xÞ dc dt 0 jj jj dt m tion switching behavior: the AHE resistance RAHE changes from ! negative to positive for B ¼þ0.6 T (blue squares in Fig. 20c), but þλ I½x^ m ð18Þ y 2 reverses its evolution trend, i.e., changes from positive to negative, for where γ is the gyromagnetic ratio, I is the charge current B ¼þ0.6 T (red circles in Fig. 20c).Forthiscase,thesmallhysteresis ! y conducting along the longitudinal direction, m is magnetization window in RAHE is clearly visible on expanded scale as shown in the vector, and α is the damping coefficient. In general, we have both inset of Fig. 20c. Consequently, both the (Idc-fixed, By-dependent) and ! ! ^ fi the spin-transfer like torque λ1I½m ðm xÞ and the field like the (By- xed, Idc-dependent) magnetization switching behaviors ^ ! torque λ2I½x m . Accordingly, we illustrate the four stable states clearly demonstrate that the magnetization can be effectively manipu- in Fig. 20a where the applied DC current, Idc, conducts along the lated by the current-induced SOT in our TI/Cr-doped TI bilayer longitudinal direction (i.e., 7y-axis), and the external magnetic heterostructure. We summarize these switching behaviors in the field is also applied along the 7y-axis. It can be seen that in the phase diagram in Fig. 20d. For the four corner panels in Fig. 16d presence of a constant external magnetic field in the y-direction, where the field value By and Idc are large, the magnetization state is the z-component magnetization Mz can be switched, depending deterministic; however, in the central panel where By and Idc are small, on the DC current conduction direction [133,135,140]; likewise, both magnetization states, up and down, are possible; this behavior when the applied DC current is fixed, Mz can also be switched by agrees with the hysteresis windows, as shown in Fig. 20bandc,where changing the in-plane external magnetic field. in the low By and small Idc region the two magnetization states are

Based on such a scenario, we carry out the (Idc-fixed, By-dependent) both allowed. Based on this phase diagram, it can be clearly seen that μ and the (By-fixed, Idc-dependent) experiments at 1.9 K; the results are the magnetization can be easily switched with only tens of ADC 4 2 shown in Fig. 20bandc,respectively.Specifically, when Idc ¼þ10 μA current (i.e.,below8.9 10 A/cm in current density Jdc), suggesting

z 8 Idc Idc

BK BK B BSO BSO B 4 y 2 1 y M M y (Ohm) 0 M M AHE

34 R By BSO BSO By -4

BK BK I I dc -8 dc -3 -2 -1 0 1 2 3

In-plane Field By (T)

4 2 J (104 A/cm2) Jdc (10 A/cm ) dc -12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12 6 1.5

1.0 3 (T) 2 y 0.5

By= +0.6T (Ohm) 0

(Ohm) 0 0.0 B = -0.6T AHE y R AHE -2 R -4 0 4 -0.5 -3 Idc (A) In-plane Field B -1.0

-6 -1.5 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60

Idc ( A) Idc (A)

Fig. 20. (Color Online) (a) Schematic of the four stable magnetization states (panels 1–4) when passing a large DC current, Idc, and applying an in-plane external magnetic

field, By, in the 7y directions. (b) The AHE resistance RAHE as a function of the in-plane external magnetic field when passing a constant DC current with Idc¼þ10 μA (blue squares) and Idc¼þ10 μA (red circles) along the Hall bar, respectively, at 1.9 K. (c) Current-induced magnetization switching in the Hall bar device at 1.9 K in the presence ofa constant in-plane magnetic field with By¼þ0.6 T (blue squares) and By¼0.6 T (red circles), respectively. Inset: expanded scale to show the hysteresis windows. (d) Phase diagram of the magnetization state in the presence of an in-plane external magnetic field By and a DC current Idc. The dashed lines and symbols (obtained from experiments) represent switching boundaries between the different states. In all panels, the symbol↑means Mz40 and↓means Mzo0. (Adopted from [151]).

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4 2 Jac (10 A/cm ) 0.0 0.4 0.8 1.2 1.6 2.0 0.4 10

0.2 5

(Ohm) 0.0 0 2 AHE R -5 -0.2 Effective Field B (mT) -10 -0.4 -12 -8 -4 0 4 8 12 0246810

In-plane Field By (T) AC current amplitude ( A)

Fig. 21. (Color Online) (a) Second harmonic AHE resistance as a function of the in-plane external magnetic field. The shaded regions I, II, and III represent a single domain state pointing in the y direction, magnetization reversal, and a single domain state pointing in the y direction, respectively. The solid black line is the experimental raw data. The dashed lines denote the fitting proportional to 1/(|By|K) in the negative field region (red line) and in the positive field region (blue line), respectively. Inset figures in regions I and III show the magnetization oscillation around its equilibrium position when passing an AC current. (b) The transverse effective spin–orbit field as a function of the AC current amplitude Iac for two different θB angles in the xz-plane, θB¼0 and π, respectively. In the experiments, the applied magnetic field magnitude is fixed at 2 T and the temperature is kept at 1.9 K. Straight lines in the figure are linear fittings (Re-drawn from [151]). that the current-induced SOT in our TI/Cr-doped TI bilayer hetero- property of the TI materials, it is expected that the TI-related SOT structure is quite efficient. can persist even at room-temperature. Recently, Melinik et al. In order to quantitatively analyze the current-induced SOT in reported a large SOT measured in the ferromagnetic permalloy the TI/Cr-doped TI bilayer system, we further carry out second (Ni81Fe19)–Bi2Se3 heterostructures at room temperature [150], harmonic measurements of the AHE resistance to calibrate the again suggesting that TIs could enable efficient electrical manip- effective spin–orbit field (BSO) arising from the SOT. In particular, ulation of magnetic materials. In summary, the new progress on By sending an AC current, IacðtÞ¼I0 sin ðωtÞ, into the Hall bar the SOTs-related study in TIs and their heterostructures may open fi ð Þ¼ ðω Þ device, the alternating effective eld, Beff t BSO sin t , causes up a new route dedicate to the innovative spintronics applications the magnetization M to oscillate around its equilibrium position, such as ultra-low power dissipation memory and logic devices. 2ω ¼ which gives rise to a second harmonic AHE resistance RAHE 1=2ðdRAHE=dIÞI0, where the second harmonic AHE resistance 2ω 5. Conclusion RAHE contains information of BSO, given by the simple formula as:

2ω ¼1 RABSO ð Þ The study of magnetic TIs is one of the most emerging frontiers RAHE ðj jÞ 19 2 BY K to explore new physics and applications. In this review, we have outlined the methods to break the TRS of the topological surface where RA is the out-of-plane saturation AHE resistance. In Fig. 21a 2ω states; with robust out-of-plane magnetic order formed in the we show the second harmonic AHE resistance RAHE as a function of the in-plane external magnetic field when the input AC current is magnetically doped 3D TI materials, we describe the intrinsic magnetisms and the correlations between them; Most impor- given as IacðtÞ¼I0 sin ðωtÞ. When the in-plane external magnetic tantly, by optimizing the carrier-independent Van Vleck suscept- field is larger than the saturation field (i.e., |By|4K), the Cr-doped TI layer will be in a single domain state and polarized in the same ibility in the Cr-doped (BiSb)2Te3 system, we observe the scale- invariant QAHE in both the 2D hybridization regime and 10 QL direction as By, as illustrated in regions I and III in Fig. 21a. Following Eq. (19), it is noted that the magnetization M will be 3D limit, at the same time, the correlations between QHE, QSHE, polarized and the oscillation magnitude induced by the AC current and QAHE are elaborated; In addition, the TI/Cr-doped TI hetero- 2ω structures enable us to manipulate the surface-related ferromag- will decrease if we further increase By, thus causing RAHE to scale as | |4 fi 2ω fi netism as well as to realize magnetization switching through the 1/( By K). By tting the RAHE versus By curve in the large eld region with the above formula, we find the effective field value, giant SOTs.

By 726.2 mT, pointing along þz or z-axis depending on the direction of B , which is consistent with the definition of B . y SO Acknowledgements 2ω fi Consequently, the scaling relation of RAHE con rms that the ω measured R2 signal indeed comes from the SOT-induced mag- AHE We are grateful to the support from the DARPA Meso Program netization oscillation around its equilibrium position. In addition, under contract nos. N66001-12-1-4034 and N66001-11-1-4105. we also investigate the strength of the field-like torque (Eq. (18)) We also acknowledge the support from the FAME Center, one of by rotating the applied magnetic field in the xz-plane (transverse six centers of STARnet, a Semiconductor Research Corporation to the current direction). The obtained results are plotted in Program sponsored by MARCO and DARPA. K.L. W acknowledges Fig. 21b. It can be seen that the effective field is negative at the support of the Raytheon Endorsement. X.F. K acknowledges θ ¼0 and changes to positive at θ ¼π, which means the effective B B partial support from the Qualcomm Innovation Fellowship. field is pointing along the –x direction for both cases. From Fig. 21b we can get the effective field versus AC current density ratio is 0.0005 mT/(A/cm2), which is one order of magnitude smaller than References the one revealed from the spin transfer-like SOT. The giant SOT and ultra-low magnetization-switching current [1] C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95 (2005) 146802. [2] C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95 (2005) 226801. density revealed in our TI/Cr-doped TI bilayer heterostructures are [3] B.A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314 (2006) 1757–1761. significant. More importantly, since the large SOC is the intrinsic [4] J.E. Moore, L. Balents, Phys. Rev. B: Condens. Matter 75 (2007) 121306.

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Please cite this article as: X. Kou, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2014.10.022i