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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

a r t i c l e i n f o

Article history:Received 1 September 2014Accepted 17 October 2014Communicated by “Xincheng Xie”

Keywords:Topological insulator MagnetismQuantum anomalous hallSpin-orbit torque

a b s t r a c t

When the magnetic order is introduced into topological insulators (TIs), the time-reversal symmetry(TRS) is broken, and the non-trivial topological surface is driven into a new massive Dirac fermions state.The study of such TRS-breaking systems is one of the most emerging frontiers in condensed-matterphysics. In this review, we outline the methods to break the TRS of the topological surface states. Withrobust out-of-plane magnetic order formed, we describe the intrinsic magnetisms in the magneticallydoped 3D TI materials and the approach to manipulate each contribution. Most importantly, wesummarize the theoretical developments and experimental observations of the scale-invariant quantumanomalous Hall effect (QAHE) in both the 2D and 3D Cr-doped (BiSb)2Te3 systems; at the same time, wealso discuss the correlations between QAHE and other quantum transport phenomena. Finally, wehighlight the use of TI/Cr-doped TI heterostructures to both manipulate the surface-related ferromag-netism and realize electrical manipulation of magnetization through the giant spin–orbit torques.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

When the spin–orbit coupling (SOC) is strong enough to invertthe conduction and valence bands in the material [1–8], it createsa new state of matter, known as the time-reversal-invariant Z2topological insulators (TIs). With the unique gapless Dirac-cone-like linear E�k dispersion relation and the spin-momentum-locking mechanism along the surface, TIs are regarded as one ofthe most intriguing 2D materials, which arises broad interestamong condensed-matter physics, material science, nano-electro-nics, spintronics, and energy harvesting applications [9–13]. Eversince this Z2 topological order was proposed in 2005 [1], enormousefforts have been dedicated toward the preparation [14–18],optimization [19–22], and utilization of the time-reversal-symmetry (TRS) protected topological non-trivial surface states[23–28]. In parallel with the pursuit of the massless Diracfermions, it is of equal significance to break the TRS of the TIsurfaces by introducing the perpendicular magnetic interaction[29–32]. In this TRS-breaking case, the intrinsic electrical [31–39],magnetic [8,40–46], and optical properties [47–50] of the surfacemassive Dirac fermions are subject to the interplay between theband topology (SOC strength) and the magnetic orders. Withadditional structural engineering (Fig. 1a), the functionalities ofassociated physics and applications can be further multiplied.

In this paper, we review the fundamental physics as wellas recent progress in the emerging magnetic TI research field. In

Part 1, we briefly introduce the methods used to break the TRS ofTIs; in Part 2, we discuss about the intrinsic magnetisms inside themagnetic TI systems and the driving forces behind the interplaybetween different magnetic orders; in Part 3, we present both thetheory models and experimental realizations of the quantumanomalous Hall effect (QAHE) in the magnetic TI thin films; andin Part 4, we briefly summarize the investigation of magneticTI-based heterostructures, the related new physics and potentialapplications.

1.1. Break the time-reversal-symmetry of topological insulators

Bermevig et al. proposed a four-band k�p model of the 2D TIs,which can be described by an effective Hamiltonian that isessentially a Taylor expansion in the wave vector k of theinteractions between the lowest conduction band and the highestvalence band that [3]

Heff ðkx; kyÞ ¼ EðkÞþ

mðkÞ Aðkxþ ikyÞ 0 0Aðkxþ ikyÞ �mðkÞ 0 0

0 0 mðkÞ �Aðkx� ikyÞ0 0 �Aðkxþ ikyÞ �mðkÞ

266664

377775

¼hðkÞ 00 hnð�kÞ

" #ð1Þ

where EðkÞ ¼ CþDðk2x þk2yÞ is the trivial band bending, mðkÞ ¼m0þBðk2x þk2y Þ is the finite mass or gap parameter (m0 is the energydifference between the inverted conduction and valence sub-bandsat the Γ point, and m0Bo0 is required for band inversion), and

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Solid State Communications

http://dx.doi.org/10.1016/j.ssc.2014.10.0220038-1098/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: [email protected] (K.L. Wang).

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

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hðkÞ ¼ EðkÞþmðkÞσzþAðkxσy�kyσxÞ. In the original undoped condi-tion, TRS is protected given that hðkÞ ¼ hnð�kÞ. However, onceadditional magnetic exchange Mz is introduced, the mass termchanges to mðk;MzÞ ¼m0þBðk2x þk2yÞþgMz . Consequently, Heff(kx,ky)no longer has the TRS-protected property, and the massless surfaceDirac fermions degrade to the massive case where the surface opens agap Δ, as shown in Fig. 1b [32].

Experimentally, the topological surface band gap was firstobserved by the Angle-resolved photoemission spectroscopy (ARPES).Chen et al. performed the measurements on the transition-metaldoped Bi2Se3 materials, and observed a small surface band gap(�7 meV) opening in the (Bi0.99Mn0.01)2Se3 sample [51]. In themeanwhile, Wray et al. also observed the surface gap opening whenthey deposited the Fe ions on top of the Bi2Se3 film [52]. Recently, Xuet al. used the more advanced spin-resolved ARPES to visualize thespin textures in the Mn-doped Bi2Se3 thin films [53]. The resolvedunique hedgehog-like spin reorientations around the surface electro-nic groundstate provided additional information about the mechanicsof TRS breaking on the TI surface.

1.2. Magnetically doped TI materials grown by MBE

In general, there are two methods to introduce the magneticexchange term Mz in order to break the TRS of the topologicalsurface states. First of all, by combing the TI material with atopologically trivial magnetic material, the magnetic proximityeffect at the interface is able to locally align the spin moments ofthe TI band/itinerant electrons out of plane, therefore breaking theTRS at the TI interface [54,55]. To date, such magnetic proximityeffects have been observed in both TI/YIG [56] and TI/EuS [57,58]heterostructures. However, since the interface qualities are usuallypoor in these systems, the magnetizations introduced by theshort-range magnetic exchange coupling are therefore quite weak.

Alternatively, incorporating magnetic ions into the host TI mate-rials has been proven to be the most effective way to generate robustmagnetism and open a gap of the surface states [45,51,53,59].Theoretically, Zhang et al. reported a detailed theoretical study ofthe formation energies, charge states, band structures, and magneticproperties of transition metal (V, Cr, Mn, and Fe) doped 3D TIs[29,60]. By using first-principles calculations based on the densityfunctional theory, they found that most transition metals favor thesubstitutional doping into the host TI cation-site, indicative of robustmagnetic moments formed in such magnetically doped TI systems.More importantly, Nunez et al. found that the exchange-inducedsingle ion magnetic anisotropy in such systems favored the perpen-dicular orientation [61]. As a result, the magnetic anisotropy easy-axis is expected to be out-of-plane in the magnetically doped 3D TIsregardless of the film thickness.

At the current stage, the transition-metal doped TI materials havebeen successfully grown by chemical vapor deposition (CVD) [40],bulk Bridgeman growth [44,62–64], and molecular beam epitaxy

(MBE) [39,42,45,46,53,59,65]. Among them, the MBE technique hasadvantages in terms of non-equilibrium physical deposition, accuratelayer thickness/doping profile control, wafer-scale growth capability,and potential integration of heterostructures and/or super-lattices formulti-functional device applications.

Here, we show one example of the MBE-grown Cr-doped(BixSb1�x)2Te3 thin films on the insulating GaAs substrate [46]. Withthe surface-sensitive reflection high-energy electron diffraction(RHEED) technique, in-situ growth dynamics were monitored [66].Fig. 2a shows the as-grown RHEED patterns taken at t¼0 s, 30 s, and65 s successively. Both the sharp 2D streaky lines and the bright zero-order specular spot persist during the entire growth process,indicative of the single-crystalline feature of the sample. The smoothsurface morphology was later confirmed by atomic force microscopy(AFM) as shown in Fig. 2b where typical TI triangular terraces werepreserved without any Cr aggregations or clusters. Meanwhile, byfitting the RHEED oscillation periods in Fig. 2c, the growth rate isextracted to be around 1 QL/min. More importantly, since the RHEEDpatterns directly reflect the in-plane atomic morphology in thereciprocal k-space [67], the as-grown surface configuration couldbe inspected by using the d-spacing evolution as shown in Fig. 2d. Itcan be seen clearly that the d-spacing drops quickly as soon as thegrowth starts, and the surface transition from the pristine GaAs tothe Cr-doped (BixSb1�x)2Te3 completes immediately after the forma-tion of the first quintuple layer (QL).

After the sample growth, an energy dispersive X-ray (EDX)spectrometer was employed to perform elemental mappings. Thecolored EDX maps collected from both the TI (Bi, Sb, Cr, and Te)and substrate (Ga and As) are illustrated individually in Fig. 3b–g.The distribution of every component (especially Cr) is uniform andthere is no substitutional or interstitial doping/diffusion from theGaAs substrate, therefore leaving the bottom TI surface intact. Toelaborate the detailed structural characteristics of the epitaxialfilm, high-resolution scanning transmission electron microscopy(HRSTEM) investigation was also performed. As shown in Fig. 3h,single-crystalline Cry(BixSb1�x)2Te3 film with a sharp interface canbe clearly seen on top of the GaAs substrate. At the same time, thecorresponding zoom-in HRTEM image in Fig. 3i manifeststhe typical TI quintuple-layered structure. It is also identified thatthe van der Waals interactions between adjacent quintuple layersindeed induce a larger gap compared with the neighboringBi/Sb–Te covalently-bonded sheets. Combined with Fig. 3d andthe theoretical anticipations [29], it may suggest that the Crdopant favors the stable substitution formation inside the host TImatrix without invoking any Cr interstitial defects or second phaseseparation.

In summary, the high-quality Cr-doped TI thin films with lowdefects, well-defined surfaces, and uniform Cr distribution havebeen prepared by MBE, and in turns they validate themselves asthe suitable system to investigate the magnetisms and QAHEphenomena, as we will present in the following parts.

Band topologySOC

Magnetic Exchange

MZ

Structure Eng.TI/Cr-TITI/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 inundoped TI (massless) and magnetically doped TI (massive), respectively. (Adopted from [51]).

X. Kou et al. / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





2. Magnetisms in magnetic TI materials

2.1. RKKY interaction and gate-controlled ferromagnetism

The first study on the transition metal (TM) doped Bi2Se3/Bi2Te3/Sb2Te3 materials was carried out even before the discoveryof topological insulators [68–75]. At that time, such systems weretreated as conventional dilute magnetic semiconductors (DMS).Compared with III–V based DMS counterparts [76–78], it wasfound that the solubility in the tetradymite-type thin film could beas high as 0.35 and the corresponding Curie temperature TC ashigh as 190 K was obtained [70].

In magnetically doped semiconductors, it is known that neighbor-ing magnetic dopants can be aligned through the Ruderman–Kittel–Kasuya–Yoshida (RKKY) interaction [79–81]. In the RKKY model, theitinerant carriers are polarized in the form of long-range oscillatorywaves, and the general interaction is described by [41,82]

HRKKY ¼ iπ

Z EF

�1dE∑

ia jTr � JðEÞ σ!� S

!i

h iGð R!ijÞ � JðEÞ σ!� S

!j

h iGð� R

!ijÞ

n oð2Þ

where J(E) is the exchange coupling coefficient, Si is the spin of localmagnetic ion, σ is the Pauli matrix of itinerant electrons, Rij is the

Inte

nsity

(a.u

)

65432Energy (keV)

10 nm Sb TeCr Ga AsBi

Pt

TI

GaAs

1 nm Cr0.2(Bi0.45Sb0.45)2Te3

Bi Sb

TeSb

Te Sb

Te

Bi

TeSb

Cr

Fig. 3. (a) HAADF image of the cross-section Cr-doped (Bi1�xSbx)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-crystallineCr0.2(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 be10%. (Adopted from [46]).

Fig. 2. (a) RHEED patterns along ½1 1 2 0� direction of the surface of Cr-doped (Bi1�xSbx)2Te3 thin film taken at 0 s, 30 s, and 65 successively during growth. (b) AFM imageof 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 fromthe oscillation period. (d) d-Spacing evolution of the surface lattice during growth. (Adopted from [46]).

X. Kou et al. / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





distance between the two localized ions, and the Green’s functionhas the form that [41,82].

Gð R!ijÞ ¼14π2

Z1

E�H0expð� i k

!� R!

ijÞd2k ð3Þ

In the case of Cr doped TI systems, particularly, the p–d couplingbetween the transition metal d-orbital (Cr3þ) and the itinerantcarrier p-orbital is found to be strongest due to the wavefunctionlocalization and symmetry argument [83], and the on-site p–dexchange coefficient J(E) can be estimated by using the empiricalmodel in which an itinerant carrier floats around the magnetic ion.It has the form [43]:

JðEÞ ¼∬ ψn

pð r!

1Þψn

dð r!

2Þðe2=Rpdþ3e2=RpCr3þ Þψdð r!1Þψpð r!2Þd r!1d r!2

ð4Þwhere ψp is the p-orbital wavefunction of the itinerant carrier, ψd isthe d-orbital wavefunction of the localized Cr3þ ion, Rpd is thedistance between the two mediating carriers, and RpCr3þ is thedistance between the carrier and the Cr3þ ion center. Uponcombing Eqs. (2)–(4), it is shown that the J(E) is determined bythe Cr 3d-orbital density of states (DOS) distribution Dd(E); in otherwords, the overall RKKY interaction strength scales with Dd(EF)2 (i.e.,it strongly depends on the Fermi level position and carrier density)[84].

In Zhang et al.’s work [15], first-principles DFT calculationsbased on the density functional theory were performed to inves-tigate the DOS distributions of Cr dopants inside the host 3D Bi2Se3material. It was revealed in Fig. 4a that there was strong hybridi-zation between the Cr 3d-orbit and Se 2p-orbit, and Dd(E) of the Crion was distributed majorly in the valence band, and reduced toalmost zero above the Dirac point. As a result, it would beexpected that holes were more favored in mediating the RKKYinteraction than electrons in the Cr-doped Bi2Se3 material. Inaddition, their DFT calculations also showed that such hole-mediated RKKY mechanism also dominated in Cr-doped Bi2Te3/Sb2Te3 systems (Fig. 4b) [46,60].

Experimentally, the most straightforward way to verify thecarrier-dependent RKKY interaction in the magnetic TI materials isto carry out the anomalous Hall effect (AHE) magneto-transportmeasurements [85,86]. It is known that in solids with a ferromag-netic (FM) phase, the total Hall resistance is expressed as [85]:

Rxy ¼ R0 � HþRA �M Hð Þ ð5Þwhere the ordinary Hall coefficient R0 is inversely proportional tothe Hall density, and the anomalous Hall coefficient RA is affected

by the intrinsic/extrinsic scatterings [87,88]. As a result, by tracingthe electric-field-controlled AHE curves, the intrinsic connectionbetween transport and magnetism can be investigated. Followingthe same method used in conventional DMS systems [89–91], wefabricated top-gated Hall bar devices on our MBE-grown 6 QL Cr-doped (BixSb1�x)2Te3 thin film, as shown in Fig. 5a. With a Crdoping concentration of 2%, the Bi/Sb ratio was chosen to bearound 1 so that the Fermi level (EF) of the as-grown film wasalready within the surface band gap. Accordingly, the electric fieldprovided by top gate (712 V) is used to effectively tune thesurface carrier type and EF position (i.e., 750 meV across thesurface band gap), and the ambipolar effect of the longitudinalresistance Rxx is also observed in Fig. 5c [20,92]. Fig. 5b shows thegate-dependent AHE results of the 6 QL Cr0.04(Bi0.49Sb0.49)2Te3 thinfilm. It can be clearly seen that the Hall resistance Rxy develops asquare-shaped hysteresis loop characteristic at 1.9 K, indicatingthe presence of a long-range FM order with out-of-plane magneticanisotropy in the film. More importantly, the magnetic hysteresisloops change notably as the sample is biased from p-type to n-type. Particularly, when the Fermi level is below the Dirac point(�12 V oVgo�2 V), the coercive field Hc (red hollow circles inFig. 5d) steadily reduces from 120 Oe down to 55 Oe; in contrast,once the dominant conduction holes are depleted when Vg40 V,Hc slowly stops decreasing, and finally approaches to its mini-mum value around 40 Oe when EF is far above the Dirac point.Similarly, the Curie temperature TC (i.e., blue solid squares inFig. 5d) also follows the reduction-and-saturation behavior bygate modulation, namely TC decreases from 7.5 K (Vg¼�12 V)down to 4.7 K (Vg45 V). As a result, the electric-field-controlledAHE results in the p-type region clearly manifest the hole-mediated RKKY coupling signature.

Beside the electric-field-controlled AHE curves, the RKKYcoupling can also be realized from the magnetic doping levelxM�TC relation [93–96]. In general, the Curie temperature underthe mean-field Zener theory is given by [94,96].

TC ¼SðSþ1Þ3kB

xMΩ0

J2mnkF4π2

¼ SðSþ1Þ32=34π4=3kB

xMΩ0

J2mnp1=3 ð6Þ

where mn is the effective mass, p is the free carrier (hole) density,and the Fermi wave vector kF ¼ ð3π2pÞ1=3. Accordingly, for a givenmagnetic system with RKKY interaction, TC is expected to scalewith xMp1=3. Recently, both Zhou et al. [70] and Li et al. [64]reported the experimental observations of such linear Tc�xMp1=3

relationship in both the Sb2�xCrxTe3 and the (CrxBiySb1�y�x)2Te3

Ene

rgy

(ev)

Ene

rgy

(ev)

Fig. 4. (a) Left: ab initio calculated energy band diagram and DOS of Cr d-orbital and Se p-orbital in Bi2�xCrxSe3, where x¼0.083. The red lines represent schematically thesurface 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. (Adoptedfrom [46]).






bulk samples, which again supported that the hole-mediatedRKKY interaction was responsible for the ferromagnetism inCr-doped magnetic TI materials.

2.2. Van Vleck mechanism and carrier-independent ferromagnetism

In a dilute magnetically doped semiconductor system, it wasfound possible that the magnetic exchange among local momentscan also be mediated by the band electrons other than the RKKYcoupling through itinerant carriers [97]. Yu et al. predicted thattetradymite TI materials (Bi2Se3/Bi2Te3/Sb2Te3) could form suchmagnetically ordered insulator when doped with Cr or Fe [32].They used the first-principles calculations to investigate the totalfree energy of the system that

Ftotal ¼12χ�1L M2

L þ12χ�1e M2

e � JeffMLMe�ðMLþMeÞH ð7Þ

where χL (χe) is the spin susceptibility of the local moments (bandelectrons), ML (Me) denotes the magnetization for the localmoment (electron) subsystem, and Jeff is the magnetic exchangecoupling between them. Accordingly, the onset of the FM phasecan be determined when the minimization procedure of the freeenergy gives a non-zero magnetization without the presence ofexternal magnetic field, which leads to J2eff �χ�1

L χ�1e 40, or

equivalently χL4 ðJ2eff � χeÞ�1; in other words, a larger χe gives riseto more prominent local coupling [32]. In fact, a considerable χevalue can be obtained through the Van Vleck mechanism that [32]

χe ¼∑4μ0μ2BonkjSzjmk4omkjSzjnk4

Emk�Enkð8Þ

where μ0 is the vacuum permeability, μB is the Bohr magneton, Szis the spin operator, |mk4 and |nk4 are the Bloch functions in theconduction and valence bands, respectively. In normal IV and III–VDMS systems where the conduction and valence bands are well-separated by the band gap Eg, there is no overlap between |mk4and |nk4 , χe is thus negligible. In contrast, the giant SOC in Bi2Se3-type materials causes the band inversion [7], and the mixing Bip1z-state (|nk4) and Se p2z-state (|mk4) therefore leads to asizable onk|Sz|mk4 [32]. Under such circumstances, large mag-netization can be generated through direct coupling between the

Cr d-orbit moment and the host TI valence electron without theassistance of any itinerant carriers.

Chang et al. carried out the systematic study to investigate theVan Vleck mechanism in magnetic TIs [42]. In their experiments,they prepared a set of Cr-doped (BixSb1�x)2Te3 thin films with afixed Cr doping level of 12% and same film thickness of 5 QL inorder to ensure that both χL and ML remain constant. They foundthat although the carrier type was tuned from n-type (x40.3) top-type (xo0.2) by varying the Bi/Sb ratio, the ferromagnetism ofthe Cr0.22(BixSb1�x)1.78Te3 films remained regardless of the sig-nificant change in carrier density. Most importantly, the Curietemperature TC (�30 K) showed little dependence on the carrierdensity and type (Fig. 6a and b). Together with gate-independentAHE data shown in Fig. 6c, their observations thus provided adirect evidence of the long-range FM order through the Van Vleckmechanism in the magnetic TI materials.

The large Van Vleck susceptibility resulting from the bulk bandinversion and the long-range exchange coupling through the bandelectrons distinguish the magnetic TIs from conventional DMSsystems. As will be discussed in Part 3, this unique magnetism isessential to form the ferromagnetic insulator with a non-zero firstChern number [85].

2.3. Interplay between different FM orders—Magnetic ionsdoping level

Since both the hole-mediated RKKY coupling and the carrier-independent Van Vleck magnetism are possible in the magneticTIs, it is thus important to understand the emergence of thesedifferent ferromagnetic orders, and the interplay/controllability ofeach contribution.

First of all, it is well-known that for any magnetic system, themagnetic ions doping level xM is essential to the magnetization.Specifically, in the dilute limit where the direct coupling betweenlocal moments is negligible, the ferromagnetic susceptibility (andthus the overall magnetization) is proportional to xM, following theCurie–Weiss form that [98]

χ ¼ CT�Tc

¼xMμ2J

3kBðT�TcÞð9Þ

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 effectat 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 ofcoercivity 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 thesample is biased from p-type to n-type, indicating the hole-mediated RKKY interaction signature. (Adopted from [46]).






where μJ ¼ gμBJðJþ1Þ is the magnetic moment, g is the Landég-factor, and J is the angular momentum quantum number. Inrecent years, several groups have contributed to this research field,and relevant xM-dependent experiments have been carried out inBi2�xCrxSe3 [45,59], Bi2�xMnxSe3 [65], Bi2�xMnxTe3 [62,99], andSb2�xCrxTe3 [70,71] systems, and all the results exhibited suchmonotonically increasing relation between the total magneticmoments and xM.

More importantly, the effect of the magnetic doping level onindividual magnetism can be further distinguished, given thatRKKY and Van Vleck mechanisms have different responses to theexternal electric field as elaborated in Sections 2.1 and 2.2.Accordingly, by controlling the Cr doping levels during MBEgrowth, we have prepared a set of Crx(Bi0.5Sb0.5)2Te3 samples withthe same Bi/Sb ratio (0.5/0.5) and film thickness (6 QL), butdifferent Cr doping levels ranging from 5% to 20%. Top-gated Hallbar devices were fabricated to investigate the electric-field-controlled magneto-transport properties [46]. From the extractedHc�Vg curves in Fig. 7e–h, it can be clearly observed that in themoderate doping region (5%, 10%, and 15%), the Cr-doped(Bi0.5Sb0.5)2Te3 thin films all exhibit the hole-mediated RKKYcoupling behaviors in the sense that the anomalous Hall resistanceRxy loops show a quick decrease of its coercivity field (Hc) whenthe majority holes are depleted. When the samples are furtherbiased deep into the n-type region (i.e., 450 meV above thesurface gap), Hc gradually saturates at finite values of 150 Oe(5%), 375 Oe (10%), and 685 Oe (15%), respectively. On the contrary,when the Cr doping concentration increases up to 20% in Fig. 7d,even though the surface Fermi level has been effectively adjustedby 50 meV (Inset of Fig. 7h), Hc remains a constant of 1000 Oe anddoes not show any change with respective to the gate bias.Consequently, from the detailed comparisons of the AHE resultsin Fig.7, it may be concluded that the bulk van Vleck mechanismand the hole-mediation both exist in the moderately doped system(i.e., o15%) and contribute to the net magnetizations.

Recall the DOS distribution of Cr3þ ions in Fig. 4b, the long-rangeRKKY coupling in the p-type region thus introduces muchmore robustferromagnetic moments over the weaker bulk van Vleck term; ittherefore becomes the dominant component in the magnetic TIsamples with moderate Cr doping. As more Cr atoms are incorporatedin the TI system, on the one hand, the bulk van Vleck-type magnetismis expected to become more pronounced due to the stronger out-of-plane magnetic moments Mz from the Cr ions; on the other hand,

however, since excessive Cr atoms promote the formation of n-typeBiTe anti-site defects [59], they therefore force the Fermi level EF“pinned” far above the surface band gap. Under such circumstances,the hole-mediated RKKY coupling becomes diminished or evencompletely suppressed, leaving only the bulk van Vleck responses inheavily Cr-doped TI samples. As a result, by designing the Cr dopingprofile and electric-gating strategy, we demonstrate an approach toeither enhance or suppress the magnetic contributions from differentmagnetisms.

2.4. Interplay between different FM orders—Band topology

Beside the magnetic ions doping level, the strength of ferro-magnetism is also determined by the band topology of the host TImaterials. In Section 2.2, it is argued that the strength of matrixelements onk|Sz|mk4 in Eq. (8) is determined by the mixingbetween the inverted conduction/valence bands (i.e., SOC) [32]. InBi2Se3 systems, it is found that the SOC is mostly contributed fromthe Bi atoms [7]. As a result, substitution of Bi atoms with muchlighter transition elements (Cr or Fe) will significantly lower theSOC, and the overall Van Vleck susceptibility (i.e., FM order) ofBi2Se3 is expected to get reduced by magnetic doping [100].

Zhang et al. calculated the band structures of the Bi2�xCrxSe3materials with different Cr contents by DFT [101]. In Fig. 8a, theircalculations showed that the inverted gap amplitude shrankdramatically with increased Cr content x. Finally, when the bandinversion between the conduction and valence bands disappearedat x40.17, the heavily-doped Bi2�xCrxSe3 would enter a topologi-cally trivial DMS regime without the presence of bulk Van Vleckmagnetism any more. (As the comparison, since Te atoms con-tribute giant SOC in Bi2Te3 and Sb2Te3 systems, the bulk bandinversion would always hold regardless of the magnetic dopinglevel [101]. Under such circumstances, robust Van Vleck order hasbeen observed in heavily doped Crx(BiSb)2Te3 materials [42,46]).

The topological phase transition due to magnetic doping hasbeen confirmed by experiments. Fig. 8b shows the surface bandE�k dispersion relations of the Bi2�xCrxSe3 films grown on the Si(1 1 1) substrate. The ARPES spectra reveal the systematic weak-ening of the surface states along with the enlarged surface gap Δ,as the Cr doping concentration gradually increases from 0 to 10%.With further reducing the SOC and modifying the Berry phaseaway from the nontrivial π state [102], the topological surface arealmost eliminated in the Bi1.8Cr0.2Se3 film.

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(BixSb1�x)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]).






In addition, Zhang et al. reported the quantitative study of thetopology-driven magnetic phase transition in the magnetic TImaterials [101]. In their study, the magneto-transport AHE resultson five Bi1.78Cr0.22(SexTe1�x)3 thin films with 0.22rxr0.86 weregiven in Fig. 9a. Three Te-rich samples with xr0.52 all showed ahysteretic response at low temperature, reflecting the FM phase. Incontrast, as the Se content was increased to x¼0.67, the hysteresisdisappeared, and the system became paramagnetic (PM). The signof the AHE also reversed to negative right at this doping level.With further increase of x to 0.86, the system remained asparamagnetic, and the negative AHE became more pronounced.Most importantly, by summarizing the intercept of Rxy as afunction of x and T, the critical Se content (x¼0.67) correspondingto the FM-to-PM transition was obtained in the magnetic phasediagram (Fig. 9b), and was found to be consistent with the criticalinverted-to-normal transition point xc�0.66 in the topologicalphase diagram. Consequently, their results provided strong evi-dence that the band topology of the host TI material would be oneof the fundamental driving forces for the FM order, and therelative robust Van Vleck component in Bi2Te3 and Sb2Te3 wereof great importance to further explore the quantum anomalousHall effect in the magnetic TI systems.

2.5. Topological surface-related ferromagnetism in magnetic TIs

So far, we only focus on the bulk magnetic properties of themagnetic TIs. In fact, given the unique Dirac-cone-like linearenergy band of the topological surface states, it is thus equally

important to investigate the surface-related magnetism. Here, wewould like to summarize the work relate to the surface-carrier-mediated RKKY interaction [30,41,82].

The generic RKKY long-range exchange model discussed inSection 2.1 is still valid except the Green function, which is used todescribe the TI surface states, is now given by[41]

Gð R!ijÞ ¼ � E4ℏ2v2F

iHð1Þ0

RijEℏvF

� �8 zU ð σ!� nÞHð1Þ

1RijEℏvF

� �� ð10Þ

where Hð1Þν is the ν-order Hankel function of the first kind. After

combing Eq. (2) with Eq. (10), the surface-carrier-mediated RKKYinteraction can be written as [41]

HRKKY1;2 ¼ F1ðR; EF ÞS

,

1 U S!

2þF2ðR; EF Þð S!

1 � S!

2ÞyþF3ðR; EF ÞSy1Sy2 ð11Þ

where the range function F1 gives rise to the Heisenberg-like term,F2 represents the Dzyaloshinskii–Moriya-like term, and F3 standsfor the Ising-like term. According to Zhu et al.’s calculations, allthese range functions, F1–F3, still have the oscillation period of π/kF[93,95,103], where kF is the Fermi wave vector, and their magni-tudes scale with (J2/R3) [41]. When the surface Fermi energy levelmoves toward to the Dirac point, the oscillation period of the RKKYinteraction will diverge since the Fermi wavelength λF¼1/kFbecomes infinity as kF approaches zero [30], and thus neighboringmagnetic impurities can always be coupled through the itinerantcarriers when λF is much larger than the average distance betweenthem, as shown in Fig. 10a [30]. On the other hand, however, weshould point out that in addition to λF, the surface-mediated RKKY

Fig. 7. Gate-dependent AHE results for 6 QL Cr-doped (Bi1�xSbx)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 rangingfrom �12 V to þ12 V. (Adopted from [46]).






Fig. 8. (Color Online) (a) DFT-calculated bulk band structures of Bi2�xCrxSe3 with different Cr contents x¼0, 0.083, 0.167, and 0.25, respectively. The red and black brokenlines indicate the positions of Fermi level and the points. (Adopted from [101]) (b) ARPES intensity maps of MBE-grown 50 QL Bi2�xCrxSe3 thin films with x¼0, 0.02, 0.1, and0.2 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(SexTe1�x)3 thin films measured at 1.5 K. (b) The magnetic phase diagram of Bi1.78Cr0.22(SexTe1�x)3 summarizing the intercept of Rxy as afunction of x and T. (c) The topological phase diagram of Bi1.78Cr0.22(SexTe1�x)3 thin films. (Adopted from [101]).






interaction is also related to the DOS distributions of both themediating surface carriers and magnetic dopants, as we elaboratedin Section 2.1 [79,104,105]. Since the DOS of Dirac surface stateshave the form that DðEF Þ ¼ EF=ðπℏ2v2F Þ ¼ kF=ðπℏvF Þ, it is thus inver-sely proportional to λF. As a result, the overall strength of surfaceRKKY interaction is counterbalanced by the oscillation period(π/kF) and D(EF) (pkF). In particularly, as the Fermi level app-roaches the Dirac point, fewer Dirac fermions around the Fermilevel will take part in the surface-mediated RKKY interaction eventhough λF becomes larger, and the two effects combined lead to afinite surface RKKY coupling in the vicinity of the Dirac point [43].

In order to quantitatively study the surface-mediated RKKYinteraction in magnetic TIs, here, we show one example of amoderate-doped Cr0.16(Bi0.54Sb0.38)2Te3 sample. In particular, withthe Cr doping level of 8%, we find that the average distancebetween neighboring Cr3þ ions is R¼11 Å. If we further assumethat the on-site exchange parameter J is a constant of 7 eV/Å2 (thesame as Ref. [41]), we have the range function of surface-relatedRKKY interaction in Fig. 10b. Combined with the DOS distributionof the Cr3þ ions Dd(EF) which is majorly distributed below theDirac point (Fig. 4b), it is concluded that Dirac holes can effectivelycouple neighboring Cr ions together, and the surface-relatedferromagnetism is manifested in the p-type region, similar to thebulk RKKY case. Recently, Checkelsky et al. reported the experi-mental observation of such Dirac-hole-mediated RKKY interactionin the MnxBi2�xTe3�ySey materials where the highest Curietemperature was obtained when the samples were biased in thep-type region [44].

Nevertheless, due to similar hole-mediated behavior with thebulk states, it is difficult to distinguish the surface contribution inuniformly doped magnetic TI systems. As will be discussed inSection 4.1, modulation-doped magnetic TI heterostructures willbe served as a better platform for us to manipulate the surface-related ferromagnetic order [43].

3. Quantum anomalous hall effect in magnetic TI materials

It is known that both the ordinary Hall effect and anomalousHall effect in the solid-state transport are the result of the breakingof TRS. Specifically, in the former case, a perpendicular magneticfield is needed to deflect the conduction charge particles in orderto generate the transverse voltage Vxy [106]; in the latter case,spontaneous magnetization replace the external magnetic field,and Vxy arises from the spin–orbit interaction between chargecurrent and magnetic moments [85]. Since the discovery of thequantum Hall effect (QHE) 1980 [107], quantized dissipationlesschiral edge conduction have been observed in many high-mobility

two-dimensional electron gas (2DEG) systems under high mag-netic field. On the contrary, for decades, little progress was madeon the realization of quantum anomalous Hall (QAHE) effect afterthe first theoretical model proposed by Haldane [108]. It was notuntil the discovery of TI materials that the search for the suitablenon-zero Chern insulators and QAHE became practical. In thissection, we will briefly introduce both the theoretical predictionsand experimental observations of QAHE in the magnetic TImaterials, and we will also compare the physics among thesedifferent transport quantum phases.

3.1. Construct QAHE from QSHE in a 2D system

Since 2007, the realization of quantum spin Hall effect (QSHE)in the HgTe/CdTe 2D TI system [3,24] brought a new impetus toconstruct QAHE. Given the closely related topological relationbetween these two quantum phenomena, it is argued that thehelical edge state in the QSHE regime can be viewed as two copiesof the QAHE channels with opposite chirality [8,11]. Therefore, ifone spin block can be suppressed due to TRS-breaking, the singleQAHE conduction is thus realized. In 2008, based on the 2D TIfour-band model described by Eq. (1) [3], Liu et al. investigated thetopological surface band change by introducing additional Mn ionsinto the HgTe quantum wells (QW) [31]. In this case, the spinsplitting term induced by magnetization can be expanded into aphenomenological form as [31]

Hs ¼

GE 0 0 00 GH 0 00 0 �GE 00 0 0 �GH

26664

37775 ð12Þ

where the spin splitting energy difference is 2GE for the twoelectron sub-bands |E1, 74 , and 2GH for the two hole sub-bands|H1, 74 . Under such condition, if GE�GHo0 (i.e., assume GEo0and GH40), the spin splitting for |E1, 74 and |H1, 74 wouldhave opposite signs; with large enough splitting energy, the bandinversion of the |E1, þ4 and |H1, þ4 states still holds, while the|E1, �4 and |H1, �4 sets finally enter the normal regime, leavingonly spin-up edge states in the quantum transport (QAHE) regime,as illustrated in Fig. 11a. Following the same spin splitting (i.e.,GE�GHo0) scenario, Yu et al. later pointed out that even the four-band system was originally in the topological trivial (non-inverted) phase, proper exchange field would also induce a bandinversion in the spin-up sub-bands (|E1, þ4 and |H1, þ4) whilepushing the spin-down sub-bands (|E1, �4 and |H1, �4) evenfurther away from each other (Fig. 11b) [32]. Under such circum-stances, the single QAHE conduction could also be constructed.

-80 -40 0 40 80EF (meV)

80

40

0

-40

-80

-120

F1F2F3

EF = 100 meV

EF= 0 eV

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 showsthe 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. Inboth (a) and (b), the on-site exchange parameter J is assumed to be a constant of 7 eV/Å2.






In fact, although Liu et al.’s model sounds feasible, in reality, theMn moments in the Hg1�xMnxTe system do not order sponta-neously, therefore they fail to generate required spin splittingenergy. Alternatively, since pronounced magnetization can begenerated from the bulk Van Vleck susceptibility in the Cr-dopedTI films even if the bulk reaches in the insulating state (Section2.2), Yu et al. proposed an improved model to realize QAHE in theCr-doped tetradymite-type TI systems [32]. They found that whenthese Cr-doped 3D TI films reduced their thickness into a 2D

hybridization regime, quantum tunneling between the top andbottom surfaces became pronounced[8,16,92,109], giving rise to afinite mass term, mk ¼m0þBðk2x þk2yÞ, and the surface Hamiltoniancould be further modified from Eq. (1) by [32]

H¼hkþgMσz 0

0 hn

k�gMσz

" #ð13Þ

where hk ¼mkσzþνF ðkyσx�kxσyÞ. Under the spatial inversion sym-metry assumption (i.e., neglecting the band bending of the top/bottom surface states in the thin film), νF, g, and M were of samevalues for the top and bottom surfaces, and therefore, therequirement of GE�GHo0 was automatically achieved. In thesubsequent 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 areregarded as the promising candidate for QAHE: due to the giantSOC from the Te atoms, the bulk band is always inverted; largeout-of-plane magnetization is spontaneously established throughthe Van Vleck mechanism without the assistance of itinerantcarriers; and most importantly, the bulk carrier density canpractically be minimized by optimizing the Bi/Sb ratio, and theFermi level can be tuned across the Dirac point by gate modulation[42,46,110].

Chang et al. from Tsinghua first reported the observation ofQAHE 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 waselectrically tuned into the surface gap (Vg��1.5 V), Rxy wasquantized at h/e2 (25.8 kΩ), and such quantized value was nearlyinvariant with the external magnetic field, suggesting perfect edgeconduction and charge neutrality of the film. In the meanwhile,

Fig. 11. Evolution of band structure and edge states upon increasing the spinsplitting for (a) inverted band and (b) non-inverted band. In both cases, GEo0 andGH40; the spin-up sub-bands (|E1, þ4 and |H1, þ4) remain inverted, while thespin-down sub-bands (|E1, �4 and |H1, �4) enter the normal insulating states.(Re-drawn from [31,32]).

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 bluesquares) 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 redcircles) on Vg. The vertical purple dashed-dotted lines in (b) and (d) indicate the Vg for Vg¼0. (Adopted from [39]).






the giant MR (2251%) was observed in Fig. 12c, and it related to thequantum phase transition between the two opposite QAHE statesvia a highly dissipative bulk channel. In addition, the gate-dependent Rxy and Rxx at zero field were provided in Fig. 12b,and the authors showed that Rxy exhibited a distinct plateau withthe quantized value h/e2; correspondingly, the longitudinal Rxyexperienced a sharp dip down to 0.098 h/e2. Recently, Lu et al.fitted these transport data with an effective conduction model, andthe agreement between the theory and the experiment confirmedthat the transport in the QAHE regime indeed originated from thetopological non-trivial conduction band, which had a concentratedBerry curvature and a local maximum in the group velocity [111].Besides, in Chang et al.’s experiments on the 5 QL film, a non-zerolongitudinal resistance Rxx was observed in the QAHE regime atzero field. By further applying a large perpendicular magnetic field(i.e., B410 T), the 5 QL film was finally driven into a perfect QHEregime, and Rxx diminished almost to zero [39]. The origins of suchphenomenon have been further analyzed, and both the variablerange hopping (VRH) [39] and the gapless quasi-helical edge statesin the 5 QL Cr-doped TI film [112] have been proposed to explainsuch field-dependent Rxx results.

3.3. Universal QAHE phase

Section 3.1 discussed the roadmap to construct the QAHE fromQSHE in the 2D hybridized magnetic TI system. In general, giventhe intrinsic relation between the Berry's phase and anomalousHall conductance [113–116], it has been suggested that thequantized Hall conductance without the presence of externalmagnetic field can be realized in a ferromagnetic insulator whichhas a non-zero first Chern number (C1) due to the non-zerointegral of the energy band Berry curvature [8,85]. To this regard,it was proposed by several groups that QAHE could also be derivedfrom the gapped top and bottom surfaces in a 3D magnetic TIsystems [8,36,117].

In fact, for an ideal 3D magnetic TI material with insulatingbulk state, the out-of-plane magnetization opens gaps on the topand bottom surfaces, and changes each surface into a non-zero C1phase with a half-quantized Hall conductance (e2/2h), as illu-strated in Fig. 13a [8,36]. Due to the opposite local basis definedwith respect to the normal direction _n (Fig. 13b), the effective

Hamiltonians of the top and bottom surfaces can be written as [36]

Heff ¼ 7νFℏðkxσy�kyσxÞþΔzσz ð14Þ

where7represents the top/bottom surfaces, and Δz¼gMz. There-fore, the massive Dirac fermions at the top and bottom surfaceshave opposite masses ð7Δz=v2F Þ [36]. In the meanwhile, since theside surfaces in the thick 3D magnetic TI system is parallel to themagnetic moments, their topological surface bands remain mass-less [36]. Under such circumstances, we can regard the 3Dmagnetic TI system as two massive Dirac fermions with oppositemasses separated by a massless Dirac fermions in between, orequivalently, the gapless side surface acts as the domain wallbetween the two topologically different phases, as shown inFig. 13b. Just like the case in a usual QHE system, the half-quantized chiral edge channels will be trapped at this interfacebetween the side surface and the top/bottom surface [8,36]. Moreimportantly, since the TRS is protected for the gapless side surfacestates, the side-surface conduction is non-chiral, and it does notcontribute to the Hall conductance; in other words, the stability ofthe quantized chiral edge state is well-maintained. As a result,QAHE can always be realized in the thick 3D magnetic TI samplesas long as the bulk remains insulating and the quantized chiralchannels are well-confined at the domain interface [8,36].

More recently, both Jiang et al. [33] and Wang et al. [34]proposed a more generic way to construct QAHE with tunableChern number in thick magnetic TIs. Following the same spinsplitting process as elaborated in Fig. 11, they found that the on-site band inversion depended on the critical film thickness dn as[34]:

dn4nπffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBz=ð7Δz�m0Þ

pð15Þ

where n represents the nth-order sub-bands, m0 is the energydifference between the lowest-order inverted conduction andvalence sub-bands, and Bz is the z-axis coefficient of the gapparameter m(k) defined in Eq. (1). Consequently, with a givenexchange field strength, QAHE with higher plateaus would bepossible in the thick magnetic TI films with insulating bulk states,and the QAHE was, in principle, a universal quantum phaseregardless of thickness (whereas in the QHE regime, the formationof the precise Landau level quantization requires the electrons tobe strictly confined in the 2D region [118]).

MZ

xy

Top Surface

Side Surface(Domain Wall)

Bottom Surface

n

nn

+ Δz

- 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 chiraledge 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-drawnfrom [8,36]).






3.4. Scale-invariant QAHE beyond 2D hybridization regime

To investigate the QAHE in 3D magnetic TIs, we prepared high-quality Cr-doped (BiSb)2Te3 films on the intrinsic GaAs (1 1 1)substrate by MBE. The film thickness was accurately controlled tobe 10 QL so as to prevent the top and bottom surfaces hybridizingwith each other. After growth, mm-size (2 mm�1 mm) six-terminalHall bar device was fabricated, as shown in Fig. 14a. Fig. 14c showsthe AHE results in our 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 film when thetemperature is far below TC. It can be seen that the film reaches theQAHE regime where the Hall resistance Rxy reaches the quantizedvalue of h/e2 at B¼0 T at the base temperature up to 85 m K. Toquantitatively understand the chiral edge transport in our sample,we thus apply the Landauer–Büttiker formalism that [119]

Ii ¼e2

h∑jðTjiV i�TijVjÞ ð16Þ

where Ii is the current flowing out of the ith contact into the sample,Vi is the voltage on the ith contact, and Tji is the transmissionprobability from the ith to the jth contacts [120,121]. In our Hall barstructure shown in Fig. 14d, the voltage bias is applied between the1st and 4th contacts (i.e., V1¼V, V4¼0, and I1¼� I4¼ I) and the otherfour contacts are used as the voltage probes such thatI2¼ I3¼ I5¼ I6¼0. Unlike QSHE helical states [24,122], in the QAHEregime, where the TRS is broken by the spontaneous magnetization,the electrons can only flow along one direction and such chiral edgeconduction route depends on the magnetization direction[43]. Spe-cifically, when the magnetic TI film is magnetized along þz direction(i.e., the upper panel of Fig. 14d), the non-zero transmission matrixelements for QAHE state are T61¼T56¼T45¼1 [43,122], and thecorresponding voltage distributions are given by V6¼V5¼V1¼(h/e2)I and V2¼V3¼V4¼0. On the other hand, when the magnetization isreversed (i.e., the lower panel of Fig. 14d), the edge current wouldflow through the 2nd and 3rd contacts, thus making V2¼V3¼V1¼(h/e2)I and V5¼V6¼V4¼0. Consequently, Rxy¼R14,62¼(V6�V2)/I isexpected to be positive ifMZ40 and change to be negative if MZo0.The observed polarities of the above QAHE results with respect to themagnetization directions are consistent with Eq. (16), which clearlydemonstrates the chiral edge conduction nature of QAHE.

Most important is the observation of the scale-invariant quan-tized edge conduction in the QAHE regime on the macroscopic

scale (i.e., from 200 mm to 2 mm). In fact, the chiral nature in theQAHE regime causes (Ti,iþ1, Tiþ1,i)¼(0, 1). Therefore, the backwardconduction is prohibited (unlike QSHE, in which the presence ofthe helical edge states would allow in-elastic backscattering andincoherent momentum-relaxation), and the de-coherence processfrom the lateral contacts does not lead to momentum and energyrelaxation [43,122]. Together with the thickness-dependentresults, we may conclude that when appropriate spin–orbit inter-action and perpendicular FM exchange strength are present in abulk insulating magnetic TI film where the Fermi level residesinside the surface gap [32,100,112], the QAHE resistance is alwaysquantized to be h/e2, regardless of the device dimensions and de-phasing process.

3.5. Non-zero Rxx and non-local measurement in thick magneticTI film

In Section 3.2, Chang et al. reported a non-zero longitudinalresistance of the 5 QL Cr0.15(Bi0.1Sb0.9)1.85Te3 film in the QAHEregime [39]. Here, we also performed the Rxx�T and Rxy�T resultsof the 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 film at B¼3 T and 15 T in thelow-temperature region (To1 K), as shown in Fig. 15a. It is notedthat a similar non-zero Rxx is also observed when our 10 QLmagnetic TI film reaches the QAHE state below 85 m K. In contrastto the 5 QL film case, it is apparent from Fig. 15a that thelongitudinal resistance in our 10 QL (Cr0.12Bi0.26Sb0.62)2Te3 sampleremains at 3 kΩ even when the applied magnetic field reaches15 T. More importantly, unlike the bulk conduction case which hasa typical parabolic MR relation [59], Rxx at T¼85 m K exhibits littlefield dependence when the magnetic field is larger than 3 T(Fig. 15b). Consequently, it may be suggested that the non-zeroRxx in the thicker 10 QL magnetic TI film is more likely associatedwith a unique dissipative edge conduction, as illustrated inFig. 13b. In particular, with the out-of-plane magnetization in oursystem, the gapless Dirac point on the side surfaces is shifted awayfrom the symmetric point (Γ). Therefore, the backscattering is notsuppressed anymore, and the non-chiral side surface conductionthus becomes dissipative along the longitudinal direction [36].

Besides the conventional Hall experiments, another importantmethod to manifest the chiral/helical properties of quantum trans-port phenomena is through the non-local transport [112,122].To this regard, we carried out the non-local measurements on the

2 nm

5 mmIn

tens

ity (a

.u)

1086420# QLs

-20

-10

0

10

20

-0.2 0.0 0.2B (T)

< 85 mK

0.5 K

1 K

4.7 K

+ h/e2

- h/e2

MZ

MZ

R14

,62

(k)

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 is2 mm�1 mm. (b) Both the HRSTEM and RHEED oscillation confirms the film thickness. (c) Temperature-dependent hysteresis Rxy curves. When the base temperature isbelow 85 m K, the sample reaches the QAHE regime. (d) Schematics of the chiral edge conduction in the QAHE regime under different magnetization directions.






six-terminal Hall bar device. The top inset of Fig. 15c shows the non-local configuration: the current is passed through contacts 1(source) and 2 (drain) while the non-local resistance betweencontacts 5 and 4 (R12,54) is measured. In the QAHE regime(To85 m K), it can be clearly seen that R12,54 displays a square-shaped hysteresis windows (two-resistance states). In other words,R12,54 reaches the high-resistance state of 15 Ω when Bo�0.2 Twhile its magnitude vanishes close to zero once B40.2 T.

Here, we point out that the non-local resistances can be under-stood from the chirality of QAHE. From Eq. (16), it is known that thatthe transmission matrix element is given by (Ti,iþ1, Tiþ1,i)¼(0, 1) inthe ideal QAHE regime. If we solve the non-local transport config-uration shown in Fig. 15c using Eq. (16), the initial conditions thusbecome V1¼V, V2¼0, I1¼� I2¼ I, and I3¼ I4¼ I5¼ I6¼0. As a result, weshould obtain that V6¼V5¼V4¼V3¼V1¼(h/e2)I when MZ40, andV6¼V5¼V4¼V3¼V2¼0 when MZo0. In either case, there is novoltage drop among the four voltage probes, therefore leaving thenon-local resistance of R12,54 to be nearly zero regardless of themagnetization direction.

Now, if we consider the influence of the additional dissipative non-chiral edge channel as discovered in the 10 QL (Cr0.12Bi0.26Sb0.62)2Te3film, there will be two possible conditions for the non-local signals:the voltage probes measure the voltage drop along the same edgewhere the QAHE edge state is present, and the voltage probes areaway from the dissipationless chiral edge channel. With the samenon-local transport configuration as illustrated in Fig. 4(a), it can beseen that in the former case, the chirality forces the QAHE dissipation-less current flow from contact 1 to contact 2 through the 1-6-5-4-3-2 contacts successively, and in turns “shorts” the contacts sothat V6¼V5¼V4¼V3�V1¼V [119]. As a result, the voltage dropbetween these contacts is negligible, and R12,54 is driven into thelow-resistance state. On the other hand, when the magnetization isreversed (i.e., the latter condition), the 1st and 2nd contacts aredirectly connected by the QAHE channel in the upper edge, and the

voltage probes from V3 to V6 are now away from the dissipationlesschannel. Therefore, the non-local signal only relates to the voltagedrop caused by the dissipative edge channel, which thus gives riseto a larger value of R12,54. As a comparison, we also plot the field-dependent non-local results at 4.7 K (higher panel of Fig. 15c),where R12,54 is dominated by the larger bulk conduction component(i.e., the non-local resistances are more than 10 times larger thanthose probed at 85 m K). In such diffusive transport regime, thesquare-shaped hysteresis non-local signals are replaced by theordinary parabolic MR backgrounds, and the polarity of R12,54 alsodisappears. To summarize, both the field-independent Rxx and thenon-local resistances displayed in Fig. 15 confirm the coexistence ofQAHE chiral edge channel and the additional dissipative edgeconduction in the thick 10 QL Cr-doped TI sample.

3.6. Discussion on QHE, QSHE, and QAHE

As discussed in the above sections, although QHE, QSHE, andQAHE all belong to the quantized edge transport phenomena, theirintrinsic mechanisms are quite distinct. In this section, we sum-marize and compare both the relations and differences betweenthese quantum Hall trio, and show the advantages of QAHE forlow-power interconnect applications.

It is known that QHE can only be realized in a C1¼1 Cherninsulator where the bulk is insulating because of the discreteLandau levels (LLs) and the conduction channels only exist at theedges, as shown in Fig. 16a [119]. In the QHE regime, the externalmagnetic field acts as a gauge field which couples to the momen-tum of the electrons and forces the electrons to make cyclotronmotions in real space. In order to form the discrete LLs (i.e., tochange the band topology), the electrons need to complete thecyclotron motion circles before losing its momentum due toscattering, and this in turn requires the electrons to have a veryhigh mobility [119]. In addition, the precise LL quantization also

30

-10 0

85 mK

h/e2

Rxx

Rxy

25

20

15

10

5

0 5 6 7 80.1

2 3 4 5 6 7 81

T (K)

3 T 15 T

-0.2 0.0 0.20

5

10

15

B (T)

0.15

0.20

0.25

4.7 K

85 mK

1 42 3

6 5

1 42 3

6 51 4

2 3

6 5

V

R12

,54

(k)

R12

,54

(k)

R12

,54

(k)

R (k

)

50

40

20

10

-10 0 10B(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 fielddependence 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 indicatesthat the magnetic field is swept from positive (þz) to negative (�z), whereas the blue solid arrow corresponds to the opposite magnetic field sweeping direction.






requires the electrons to be confined in the 2D dimension so thatthe energy levels of these quantized orbitals take on discretevalues of En ¼ ℏωcðnþð1=2ÞÞ where ωc¼eB/m is the cyclotronfrequency. Therefore, once the Fermi level is located inside thegap between two neighboring LLs, the quantum chiral edge statereadily appears while the bulk carriers are localized by thecyclotron orbitals. In other words, the realization of QHE demandsa strong magnetic field and a high mobility material with a good2D confinement [118,119].

The QSHE [3], on the other hand, does not need any externalmagnetic field. It is realized in systems where the strong spin–orbit interaction leads to the band inversion at the Г point in theBrillouin zone. This band inversion makes the system into a C2¼1(C1¼0) Chern insulator with the quantized channel number M¼2[11]. Here the second Chern number is employed to describe thetopology in the QSHE state. Because the TRS is preserved in suchsystem (i.e., the external magnetic field is zero), the resultingquantum edge states are helical instead of chiral, as shown inFig. 16b. Accordingly, when the Fermi level is placed inside thebulk band gap, the longitudinal conductance exhibits a quantized2e2/h helical conduction. It is noted that since the bulk band gap,which is determined by the spin–orbit interaction strength, isusually small (i.e., up to 0.3 eV), the quantum helical edgeconduction is thus vulnerable to any intrinsic bulk impurity/defect-induced carrier states and band potential fluctuations.Consequently, the QSHE requires a very high material quality (i.e., low bulk carrier density and high mobility). To date, the QSHEhas only been realized experimentally in the HgTe/CdTe [24,122]and InAs/GaSb [123,124] quantum wells.

Compared with QHE and QSHE discussed above, the QAHEinvolves both the strong spin–orbit interaction and the magneticexchange interaction in a magnetic insulating system. In the QAHEregime with a robust magnetic moment, the strong spin–orbitinteraction as well leads to the band inversion. However, the largeexchange field, as large as 100 T, from the spontaneous magneti-zation couples with the spins of the band electrons, and splits thespin-up and spin-down sub-bands in the opposite directions. It inturn results in one pair of the spin-resolved bands un-invertedwhile keeping the other pair still remains within the non-trivialtopological state. As a result, in contrast to the QSHE state, the bulktopology is changed, the system becomes a C1¼1 Chern insulator,and the quantized QAHE chiral edge transport at zero externalmagnetic field can be achieved when the Fermi level is located inthe massive Dirac surface band gap, as displayed in Fig. 16c [39]. Inthe meanwhile, compared with QHE, the TRS in the QAHE state is

broken by the spontaneous magnetization rather than externalmagnetic field, and it does not require the formation of LLs.Accordingly, QAHE may have loose requirements on the filmthickness and quality, (i.e., our grown magnetic TI film has arelatively low mobility of 600 cm2/V s and the film is much thickerthan the 2D hybridization limit, but the quantized conduction ofe2/h is quite robust). In summary, the realization of QAHE requiresan appropriate spin–orbit interaction, a strong exchange interac-tion (i.e., in order to both splits the bands and localizes the bulkcarriers) and an out-of-plane magnetic anisotropy; all of which arefound to be satisfied in the Cr-doped (BiSb)2Te3 thin films.

4. Physics and applications of magnetic TI heterostructures

As we addressed in Section 2.5 that the uniformly dopedmagnetic TI was not a proper platform to quantify the surface-related ferromagnetism since both surface and bulk holes arefavorable for mediating the RKKY interaction. In addition, theentanglement of surface carriers with bulk magnetic dopantsmade it more complicated to distinguish the surface contribution.Alternatively, if we design the TI/Cr-doped TI bilayer structure byintroducing an undoped TI layer one top, the top surface isseparated from the bulk magnetic ions. Under this condition, byvarying the thickness (d1) of the top undoped TI layer, we are ableto investigate and manipulate the surface-related ferromagnetismin a straightforward manner.

4.1. Manipulation of surface-related ferromagnetism in TI/Cr-TIbilayers

In principle, the proposed TI/Cr-doped TI bilayer structures canbe implemented by using the modulation-doped growth methodvia which the Cr distribution profile along the epitaxial growthdirection is accurately controlled [125–127]. The illustration of themodulation-doped growth methods are shown in Fig. 17a. Duringthe film growth, the feedback signal from the in-situ RHEEDoscillations was added to enable the effusion cell shutters beautomatically controlled by the computer. After growth, the Bi/Sbcomposition ratio (x, y) and the Cr doping level were determinedby an energy-dispersive X-ray (EDX) spectroscopy. Fig. 17b showsthe result of one 3 QL (Bi0.5Sb0.5)2Te3/6 QL Cr0.08(Bi0.57Sb0.39)2Te3film grown by the modulation-doped method. In contrast tothe lower figure, the absence of the Cr peak in the upper EDXspectrum provides strong evidence that the Cr atoms only

Quantum Hall effect Quantum Spin Hall effect Quantum Anomalous Hall effect

Bz Mz

C1 = 1M = 2

C2 = 1M = 2

C1 = 1M = 1

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 TRSbreaking, chiral edge conduction is dominant. (b) Quantum spin Hall case. Due to the protection of TRS and spin-momentum locking mechanism, opposite-spin electronsoccupy 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 alongþz-direction, only spin-up electrons flow through the edge clockwise.






distribute inside the bottom 6 QL Cr0.08(Bi0.57Sb0.39)2Te3 layer,while the top 3 QL layer is free of magnetic impurities.

In order to verify the surface-related magnetism in themodulation-doped TI hetero-structures, we first performed themagneto-optical Kerr effect (MOKE) measurements on three differ-ent samples as follows: Sample A has a 6 QL Cr0.16(Bi0.54Sb0.38)2Te3with a uniform Cr-doping profile (control sample); Sample B andSample C share the same 6 QL Cr0.16(Bi0.54Sb0.38)2Te3 designs in thebottom, but differ in the top undoped (Bi0.5Sb0.5)2Te3 layer thick-ness, namely 3QL for Sample B and 6 QL for Sample C. Fig. 18a–cshows the field-dependent MOKE results acquired at differenttemperatures. On the one hand, when the top layer thickness d1is 3 QL, the coercivity field Hc at T¼2.8 K has enlarged by more than5 times (i.e., from 4.5 mT to 30 mT) compared with the controlsample. On the other hand, Hc does not change monotonically withd1. Instead, when the top surface is 6 QL away from the bottomlayer, we observe the minimum Hc of only 2.5 mT. Here, since theidentical bottom Cr-doped TI layers of the three samples contributethe same “bulk”magnetization, the change of both Hc and TC amongthem hence can only be associated with the separation between thetop surface and the bulk Cr ions. Therefore, both the uniqued1-dependent magnetization behavior and the enhancement of TCin Sample B may provide us with the direct evidence about thepresence of the surface-related ferromagnetism in modulation-doped TI thin films.

To further investigate the magneto-electric responses from thetop surface states, gate-dependent AHE measurements werecarried out on five samples with fixed bottom magnetic TI layer(i.e., 6 QL Cr0.16(Bi0.54Sb0.38)2Te3) but different d1 in the topundoped layers (i.e., d1¼0, 2, 3, 4, 5 QL, respectively). The Hc�Vg

results of these five TI/Cr-doped TI bilayer samples are displayed inFig. 18d. Here, the unveiled monotonically decreasing relation ofHc�Vg in the p-type region possibly suggests the signature of thehole-mediated RKKY interaction, as discussed in Part 2. In themeanwhile, due to the effective Fermi level tuning, the coercivityfields Hc are found to achieve over 100% modulation via gate-voltage control in all samples; yet the most pronounced change isobtained in the 3 QL TI/6 QL Cr-doped TI sample, as highlighted inFig. 18e. It is important to point out that the Debye length of our

grown modulation-doped TI bilayer films is estimated to be lessthan 3 nm at 1.9 K when the samples are biased in the inversion/accumulation regions (i.e., n/p-type) [128,129]. Based on ourdevice geometry, it implies that the applied top-gate voltage isonly able to tune the Fermi level of the top surface states.Accordingly, we may conclude that the magneto-electric responsesamong the modulation-doped TI hetero-structures in the p-typeregions are predominantly due to the variations of the surface-related ferromagnetism, and the pronounced enhancement of ΔHc

may reflect the optimization of the magnetic RKKY couplingbetween neighboring Cr ions through top surface states.

To understand the optimization behavior of the surface-mediated magneto-electric effect in the TI/Cr-doped TI bilayerdevice, we can propose the realistic model based on two factors.First of all, it is known that both the magnetic dipole and exchangeinteractions from the bottom magnetic layer will force the topsurface open a surface gap Δ0. In our TI/Cr-doped TI bilayersystems, since the mediating carriers itinerate within the topsurface states, the resulting exchange coupling, and thus the gapopening, will thus decay exponentially with the separation dis-tance (d1). Because the RKKY coupling is a second-order-perturbation interaction, its exchange strength is thus inverselyproportional to the energy difference between the ground and theexcited states (i.e., Δ0) [130]. On the other hand, however, since thedensity of surface carriers follows the nsðdÞpe�2d=D0 distributionrelation and D0 characterizes the penetration depth of the surfacestates with a typical value of 2–3 QLs [131,132], it is expectedthat as d1 becomes larger, fewer surface Dirac fermions willparticipate into the RKKY mediating process, therefore weakeningthe surface-related RKKY coupling strength. Accordingly, if wetake these two competing factors into consideration, the overalld1-dependent surface-related RKKY exchange strength oΦ(d1)4follows [43]:

oΦðd1Þ4 ¼Φ0e�2d1=D0Δ0

Δ1e�d1=D1 þΔdipoleð17Þ

where Φ0 is the RKKY interaction strength in the uniformly Cr-doped TI film and Δ0¼Δ1þΔdipole is the total surface gap openingwhich takes into account both the direct magnetic exchange field

time

substrate

BiTe

Cr

Intensity

Cr-doped TI layer

TI layer

8.8

8.6

8.4

8.2

1 2 3 4 5Sample #

12

10

8

6

Cr-doped TI thickness

Cr (

%)

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 QAHEmeasurement. 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 andCr0.08(Bi0.57Sb0.39)2Te3 layers. (d) High repeatability of the MBE-grown TI film.






(Δ1) and the long-range magnetic dipolar field (Δdipole) from thedoped Cr ions. Based on Eq. (17), it is obvious to see that as thethickness d1 increases, the carrier density will drop rapidly whichfurther decreases the Φ0e�2d1=D0 part; on the contrary, the reducedsurface Dirac fermion gap Δ(d1) in the denominator enhances theRKKY interaction. For the d1¼0 case, Eq. (17) reduces to theuniformly doped case that oΦ(d1)4¼Φ0, while in the oppositeextreme condition where d1¼1, oΦ(d1)4 returns back to zero,corresponding to the surface-bulk decoupled condition. By fittingEq. (17) with the our experiment data, we obtain the d1-depen-dent oΦ(d1)4 behavior in Fig. 19b. In addition, the optimizedcondition d1¼3 QL (i.e., ∂ðoΦðd1Þ4 Þ=∂d1 ¼ 0) is consistent withFig. 18e, manifesting the fundamental surface-related RKKYmechanism.

4.2. Magnetization switching via SOTs in TI/Cr–TI bilayers

In Section 2.4, we discuss about the influence of the magneticions on the bulk band topology. In the meanwhile, the giant SOC inthe bulk TI matrix can also affect the magnetizations. In fact, it hasbeen observed in the heavy metals (Pt, Ta) with strong SOC,applied in-plane charge current can control the magnetizationdynamics in an adjacent ferromagnet layer (Co, CoFeB) due to thespin–orbit torques (SOTs) [133–144]. Compared with the heavymetals, in our conductive TI/Cr-doped TI bilayer heterostructures, adominant spin accumulation in the Cr-doped TI layer with spinpolarized in the transverse direction is expected when passing acharge current in the y-direction due to the SHE in the bulk andthe spin polarization arising from the Rashba-type interactions at

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

120

100

80

60

40

20

0

Hc

(mT)

Hc

(mT)

-10 -5 0 5 10Vg (V)

d1 = 0 QL d1 = 2 QL d1 = 3 QL d1 = 4 QL d1 = 5 QL

80

60

40

20

6543210d1 (QL)

60

50

40

30

20

10

0

K (a

.u)

K (a

.u)

K (a

.u)

Hc (m

T)

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-0.5

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-20 0 20 -40 -20 0 20 40 -40 -20 0 20 40

B (mT) B (mT) B (mT)

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 themodulation-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 coercivityfield 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 to6 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 withvarying 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]).

0.8

0.6

0.4

0.2

0.0

<>

(a.u

)

1086420

d1 (QL)

1.0

Fig. 19. (a) Schematic representations of the surface-related magneto-electric effects in TI/Cr-doped TI bilayer structures with optimized top layer thickness d1¼3 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 spinaccumulation can be expected due to the spin-momentum lockingof the topological surface states [11,12,145–147]. The accumulatedspins’ angular momentum can be directly transferred to themagnetization M and therefore affect its dynamics. In particular,the complete form of the Landau–Lifshitz–Gilbert (LLG) equationwhich includes the SOT is given by [148,149]:

dm!dt

¼ �jγjμ0ðm!� H!

exÞþα

jjm!jjm!� dm!

dt

!þλ1I½m!� ðm!� xÞ�

þλ2I½x� m!� ð18Þ

where γ is the gyromagnetic ratio, I is the charge currentconducting along the longitudinal direction, m! is magnetizationvector, and α is the damping coefficient. In general, we have boththe spin-transfer like torque λ1I½m!� ðm!� xÞ� and the field liketorque λ2I½x� m!�. Accordingly, we illustrate the four stable statesin Fig. 20a where the applied DC current, Idc, conducts along thelongitudinal direction (i.e., 7y-axis), and the external magneticfield is also applied along the 7y-axis. It can be seen that in thepresence of a constant external magnetic field in the y-direction,the z-component magnetization Mz can be switched, dependingon the DC current conduction direction [133,135,140]; likewise,when the applied DC current is fixed, Mz can also be switched bychanging the in-plane external magnetic field.

Based on such a scenario, we carry out the (Idc-fixed, By-dependent)and the (By-fixed, Idc-dependent) experiments at 1.9 K; the results areshown in Fig. 20b and c, respectively. Specifically, when Idc¼þ10 μA

(blue squares in Fig. 20b), the AHE resistance RAHE goes from negativeto positive as the applied in-plane magnetic field By gradually changesfrom �3 T to 3 T, indicating the z-component magnetization Mz

switches from �z to þz. In contrast, when Idc¼�10 μA, the AHEresistance reverses sign (red circles in Fig. 20b) andMz varies from þzto �z as By is swept from �3 T to 3 T. It should be noted that in bothcases the AHE resistance hysteresis loops agree well with ourproposed scenario. At the same time, when we scan the DC currentIdc at a given fixed magnetic field, we also observe similar magnetiza-tion switching behavior: the AHE resistance RAHE changes fromnegative to positive for By¼þ0.6 T (blue squares in Fig. 20c), butreverses its evolution trend, i.e., changes from positive to negative, forBy¼þ0.6 T (red circles in Fig. 20c). For this case, the small hysteresiswindow in RAHE is clearly visible on expanded scale as shown in theinset of Fig. 20c. Consequently, both the (Idc-fixed, By-dependent) andthe (By-fixed, Idc-dependent) magnetization switching behaviorsclearly demonstrate that the magnetization can be effectively manipu-lated by the current-induced SOT in our TI/Cr-doped TI bilayerheterostructure. We summarize these switching behaviors in thephase diagram in Fig. 20d. For the four corner panels in Fig. 16dwhere the field value By and Idc are large, the magnetization state isdeterministic; however, in the central panel where By and Idc are small,both magnetization states, up and down, are possible; this behavioragrees with the hysteresis windows, as shown in Fig. 20b and c, wherein the low By and small Idc region the two magnetization states areboth allowed. Based on this phase diagram, it can be clearly seen thatthe magnetization can be easily switched with only tens of μA DCcurrent (i.e., below 8.9�104 A/cm2 in current density Jdc), suggesting

y

z

BK

M

ByBSO

M

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BSO

BSO

BSO

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By

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Idc

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3 4

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In-plane Field By (T)

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HE

(Ohm

)

Idc ( A)

By= +0.6T By= -0.6T

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RA

HE

(Ohm

)

Idc ( A)

Jdc (104 A/cm2)

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

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1.0

1.5

In-p

lane

Fie

ld B

y (T)

-12 -8 -4 0 4 8 12

Jdc (104 A/cm2)

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 magneticfield, 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 (bluesquares) 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 of aconstant in-plane magnetic field with By¼þ0.6 T (blue squares) and By¼�0.6 T (red circles), respectively. Inset: expanded scale to show the hysteresis windows. (d) Phasediagram 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]).






that the current-induced SOT in our TI/Cr-doped TI bilayer hetero-structure is quite efficient.

In order to quantitatively analyze the current-induced SOT inthe TI/Cr-doped TI bilayer system, we further carry out secondharmonic measurements of the AHE resistance to calibrate theeffective spin–orbit field (BSO) arising from the SOT. In particular,By sending an AC current, IacðtÞ ¼ I0 sin ðωtÞ, into the Hall bardevice, the alternating effective field, Beff ðtÞ ¼ BSO sin ðωtÞ, causesthe magnetization M to oscillate around its equilibrium position,which gives rise to a second harmonic AHE resistance R2ω

AHE ¼�1=2ðdRAHE=dIÞI0, where the second harmonic AHE resistanceR2ωAHE contains information of BSO, given by the simple formula as:

R2ωAHE ¼ �1

2RABSO

ðjBY �KjÞ ð19Þ

where RA is the out-of-plane saturation AHE resistance. In Fig. 21awe show the second harmonic AHE resistance R2ω

AHE as a function ofthe in-plane external magnetic field when the input AC current isgiven as IacðtÞ ¼ I0 sin ðωtÞ. When the in-plane external magneticfield is larger than the saturation field (i.e., |By|4K), the Cr-dopedTI layer will be in a single domain state and polarized in the samedirection as By, as illustrated in regions I and III in Fig. 21a.Following Eq. (19), it is noted that the magnetization M will bepolarized and the oscillation magnitude induced by the AC currentwill decrease if we further increase By, thus causing R2ω

AHE to scale as1/(|By|4K). By fitting the R2ω

AHE versus By curve in the large fieldregion with the above formula, we find the effective field value,By�726.2 mT, pointing along þz or �z-axis depending on thedirection of By, which is consistent with the definition of BSO.Consequently, the scaling relation of R2ω

AHE confirms that themeasured R2ω

AHE signal indeed comes from the SOT-induced mag-netization oscillation around its equilibrium position. In addition,we also investigate the strength of the field-like torque (Eq. (18))by rotating the applied magnetic field in the xz-plane (transverseto the current direction). The obtained results are plotted inFig. 21b. It can be seen that the effective field is negative atθB¼0 and changes to positive at θB¼π, which means the effectivefield is pointing along the –x direction for both cases. From Fig. 21bwe can get the effective field versus AC current density ratio is0.0005 mT/(A/cm2), which is one order of magnitude smaller thanthe one revealed from the spin transfer-like SOT.

The giant SOT and ultra-low magnetization-switching currentdensity revealed in our TI/Cr-doped TI bilayer heterostructures aresignificant. More importantly, since the large SOC is the intrinsic

property of the TI materials, it is expected that the TI-related SOTcan persist even at room-temperature. Recently, Melinik et al.reported a large SOT measured in the ferromagnetic permalloy(Ni81Fe19)–Bi2Se3 heterostructures at room temperature [150],again suggesting that TIs could enable efficient electrical manip-ulation of magnetic materials. In summary, the new progress onthe SOTs-related study in TIs and their heterostructures may openup a new route dedicate to the innovative spintronics applicationssuch as ultra-low power dissipation memory and logic devices.

5. Conclusion

The study of magnetic TIs is one of the most emerging frontiersto explore new physics and applications. In this review, we haveoutlined the methods to break the TRS of the topological surfacestates; with robust out-of-plane magnetic order formed in themagnetically doped 3D TI materials, we describe the intrinsicmagnetisms and the correlations between them; Most impor-tantly, by optimizing the carrier-independent Van Vleck suscept-ibility in the Cr-doped (BiSb)2Te3 system, we observe the scale-invariant QAHE in both the 2D hybridization regime and 10 QL3D limit, at the same time, the correlations between QHE, QSHE,and QAHE are elaborated; In addition, the TI/Cr-doped TI hetero-structures enable us to manipulate the surface-related ferromag-netism as well as to realize magnetization switching through thegiant SOTs.

Acknowledgements

We are grateful to the support from the DARPA Meso Programunder contract nos. N66001-12-1-4034 and N66001-11-1-4105.We also acknowledge the support from the FAME Center, one ofsix centers of STARnet, a Semiconductor Research CorporationProgram sponsored by MARCO and DARPA. K.L. W acknowledgesthe support of the Raytheon Endorsement. X.F. K acknowledgespartial support from the Qualcomm Innovation Fellowship.

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-12 -8 -4 0 4 8 12-0.4

-0.2

0.0

0.2

0.4

In-plane Field By (T)0 2 4 6 8 10

-10

-5

0

5

10

0.0 0.4 0.8 1.2 1.6 2.0

Jac (104 A/cm2)

R2 A

HE (O

hm)

Effe

ctiv

e Fi

eld

B (m

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 domainstate 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 rawdata. 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 figuresin 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 functionof 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 Tand the temperature is kept at 1.9 K. Straight lines in the figure are linear fittings (Re-drawn from [151]).



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