REVIEW National Science Review1: 3848, 2014doi: 10.1093/nsr/nwt029

Advance access publication 31 December 2013

PHYSICS

Special Topic: Topological Insulators

Quantum anomalous Hall effectKe He1,2, Yayu Wang1 and Qi-Kun Xue1,

1State Key Laboratoryof Low-DimensionalQuantum Physics,Department ofPhysics, TsinghuaUniversity, Beijing100084, China; and2Beijing NationalLaboratory forCondensed MatterPhysics, Institute ofPhysics, ChineseAcademy of Sciences,Beijing 100190, China

Correspondingauthor. E-mail:qkxue@mail.tsinghua.edu.cn

Received 23September 2013;Revised 13 October2013; Accepted 15October 2013

ABSTRACTHall effect is a well-known electromagnetic phenomenon that has been widely applied in the semiconductorindustry.The quantumHall effect discovered in two-dimensional electronic systems under a strongmagnetic field provided new insights into condensed matter physics, especially the topological aspect ofelectronic states.The quantum anomalous Hall effect is a special kind of the quantumHall effect that occurswithout a magnetic field. It has long been sought after because its realization will significantly facilitate thestudies and applications of the quantumHall physics. In this paper, we review how the idea of the quantumanomalous Hall effect was developed and how the effect was finally experimentally realized in thin films of amagnetically doped topological insulator.

Keywords: topological insulator, quantum anomalous Hall effect, quantumHall effect, ferromagneticinsulator, molecular beam epitaxy

THE HALL EFFECT FAMILYTheHall effect, first discovered by Hall in 1879 [1],indicates the voltage drop across a conductor trans-verse to the direction of the applied electrical cur-rent in thepresenceof aperpendicularmagnetic field(as depicted in Fig. 1a). In a non-magnetic mate-rial, the Hall voltage is proportional to the appliedmagnetic field because it results from the deflectionof charge carriers by the Lorentz force. The slope ofthe linear field dependenceof theHall resistance, de-fined as the transverse voltage divided by the cur-rent, is determinedby the type anddensity of carriers(Fig. 1b).This ordinary Hall effect (OHE) providesa probe for the properties of semiconductor materi-als and a direct measurement of the magnetic field,which have been widely used in industries.

Soon after the discovery of the OHE, Hall triedsimilar experiments on ferromagnetic materials. Heobserved that themagnetic-field-dependent Hall re-sistance shows an unusually large slope at a low field[2]. It was recognized that this unusually large Halleffect originates from the magnetization of ferro-magnetic materials, which was later known as theanomalous Hall effect (AHE) [3]. Since a ferromag-netic material keeps its spontaneous magnetizationeven when the external magnetic field is removed,

the AHE can be measured in the zero magnetic field(Fig. 1c).

Although it is generally believed that the spinorbit coupling (SOC) plays a fundamental role inthe AHE, its exact mechanism is a subject of de-bate for over one century. One group of thought as-cribes the AHE to impurity-induced skew-scatteringor side jump of carriers which are referred to as ex-trinsic mechanisms. The other believes that AHEresults from the property of the electronic energyband structures of ferromagnetic materials, which isknown as the intrinsic mechanism [3].

In 1980, about 100 years after Halls works, vonKlitzing discovered the integer quantum Hall effect(QHE) in a Si/SiO2 field effect transistor in a strongmagnetic field, which won him the Nobel Prize inphysics [4]. The discovery of the fractional QHEtwo years later gained another Nobel Prize for Tsui,Stormer, and Laughlin [5]. These milestone discov-eriesmade theQHEoneof themost important fieldsin modern condensed matter physics.

QHE occurs in a layer of two-dimensional elec-tron gas (2DEG) formed at the interface of asemiconductor heterostructure, such as Si/SiO2 orGaAs/AlGaAs, due to band bending.Without need-ing impurity doping, a 2DEG can have very high

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REVIEW He, Wang and Xue 39

Figure 1. (a) Measurement geometry of Hall effect. ((b), (c)) Magnetic field (B) depen-dence of Hall resistance (RH) in OHE (b) and AHE (c).

electronmobility, so that well-defined Landau levelscan appear under a strong magnetic field. With in-creasing magnetic field, the Hall resistance evolvesfrom a straight line into step-like behaviors withwell-defined plateaus. At the plateaus the Hall re-sistance has the exact value of h/e2, with h beingPlancks constant, e the electrons charge and aninteger or a certain fraction. At the same time, thefour-terminal longitudinal resistance drops to zero,suggesting dissipationless transport of electrons [6].

It is nowwell known that thequantizedHall resis-tance and vanishing longitudinal resistance are dueto the quantum transport of quasi-1D edge states[7]. In the QHE regime, all the bulk carriers in the2DEG are localized by impurities. However, elec-trons propagating along the sample edge remain ex-tendedover thewhole sample. In quantum transporttheory, each 1D edge channel contributes a quan-tized Hall conductance e2/h, so that at filling factor ( edge channels cross the Fermi level) the to-tal Hall conductance is e2/h. Moreover, the edgestates are chiral with the chirality determinedby thedirection of the externalmagnetic field (Fig. 2a). Be-cause the forwardandbackwardchannels are locatedat the opposite edges, the edge state electrons areimmune to back-scattering and keep dissipationlessover macroscopic length scale [6].

The unique chiral edge states responsible for theQHE originate from the magnetically induced Lan-dau levels. A QH system with the Fermi level lyingbetween two neighboring Landau levels can be con-sidered as a special insulator with a topologically dif-ferent electronic structure from that of the vacuum

and usual insulators [810]. Topology is a math-ematical concept which describes the fundamentalproperties of space that is insensitive to details [10].If one can find a topological character of a materialselectronic band structure, then the physical proper-ties related to the band topology will be insensitiveto the details of the material, such as sample size,shape, and degree of disorderness. Indeed, one candefine a topological character for a completely filledenergy band, which is known as the Chern number.The Chern number is proportional to the integralof the Berry curvature of the energy band over thewhole (magnetic) Brillouin zone, which is zero for ausual energy bandbut is unity for a Landau level.TheHall conductance is the sum of the Chern numbersof all the occupied bands in the unit of e2/h [9]. Atthe edge of a QH sample, the Chern number has toexperience a change from non-zero to zero.Thus, anenergy level is bound to cross theFermi level at somepoint around the edge, which results in the conduct-ing edge states.

The dissipationless chiral edge states of the QHEregime canbe used in lowpower consumption, high-speed electronic devices. However, well-definedLandau levels areonlypossible inhigh-mobility sam-ples under strong external magnetic fields. The de-manding requirements prevent the QHE from be-ing widely applied in industry.Therefore, it is highlydesirable to achieve the QHE without the need ofa strong magnetic field and an extraordinarily high-mobility sample. Since the discovery of the QHE,numerous theoretical models have been proposedto realize it in the zero magnetic field [3,11,12]. In1988,Haldaneproposed thefirstmodel for theQHEwithout Landau levels [11]. The model is basicallya graphene lattice with the time-reversal symmetry(TRS) broken by a periodic magnetic field (how-ever with zero net magnetic flux). Graphene has aDirac-cone-shaped gapless band structure. The bro-ken TRS opens a gap at the Dirac point, changinggraphene into an insulator with a Chern number ofunity.This model says yes to the possibility of realiz-ing the QHE in a system without Landau levels, butdoes not provide any particular materials for it.

As mentioned earlier, the AHE in a ferromagnetcan be induced by spontaneous magnetization with-out needing an external magnetic field [3]. There-fore, a quantized version of the AHE, namely thequantum anomalous Hall effect (QAHE), repre-sents the realization of the QHE in the zero mag-netic field [3,12]. After the discovery of the QHE,it was found that the theory for the QHE can alsobe used to explain the AHE in magnetic materials.The anomalous velocity that was used to introducethe intrinsic AHE is related to the Berry curvature ofthe energy band.TheAH conductivity of a magnetic

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Figure 2. (a) The chiral edge states in the QHE or QAHE ( = 1). (b) The helical edge states in the QSHE. Red and blue colorsof the lines in (a) and (b) indicate the spin-up and spin-down states, respectively, and the arrows indicate current direction.(c) The Dirac surface states of 3D TI. The red arrows indicate the spin direction.

material is also determined by the integral of theBerry curvature, but it is not quantized for a metal,since the valence band is partly filled and thus theintegral is not taken over the whole Brillouin zone[3]. Naturally, if one has a 2D ferromagnetic insu-lator with a non-zero Chern number, which can becalled a Chern insulator, the QAHE will be realized.However, finding a Chern insulator is very challeng-ing. The main difficulty lies in the fact that there arefew ferromagnetic insulators in nature. Thus, littleexperimental progress was made in this way to theQAHE.

TIME-REVERSAL INVARIANTTOPOLOGICAL INSULATORSSince 2005, a new class of topological matters,namely topological insulators (TIs) were discov-ered, partly inspired byHaldaneswork in 1988 [1317]. The topological property of a TI is induced bythe SOC, which keeps the TRS, and is characterizedby the topological invariant Z2 number, instead ofthe Chern number [13]. A 2D TI is topologicallycharacterized by a single Z2 number and expectedto show the quantum spin Hall effect (QSHE) inwhich a pair of spin-polarized edge states counter-propagates at each edge (Fig. 2b). The helical edgestateswill lead toquantized spin-accumulation at thetwo edges transverse to the current flowing direc-tion and quantized longitudinal resistance (h/2e2,12.9 k for six-terminal measurements) [18,19].TheTR invariant TI can be generalized to 3D,whichis a bulk insulator topologically characterized by four

Z2 invariants and possesses 2D Dirac-type surfacestates at all surfaces [15]. The 2D Dirac-type sur-face states have similar band dispersion to that ofgraphene but are spin-polarized except for certainsymmetry points (TR invariant points) in momen-tum space (Fig. 2c).

TR invariant TIs attracted intense research inter-ests in short time because many existing materialswere soon found belonging to this class [16,17].TheQSHE was theoretically proposed and experimen-tally observed inHgTe/CdTeand InAs/GaSb/AlSbquantum wells, two semiconductor structures thathave been long studied and are now recognized as2D TIs [1821]. Well-known thermoelectric mate-rials such as BixSb1x alloys and chalcogenide com-poundsBi2Se3, Bi2Te3, andSb2Te3 were found tobe3DTIs [2228]. By far the Bi2Se3 family TIs are themost popular TI materials due to their stoichiomet-ric chemical composition, relatively large bulk gaps(up to 0.3 eV in Bi2Se3), and simple surface Dirac-cone structures [2528]. From thematerial point ofview, there is no essential difference between 2Dand3DTIs.One can obtain the 2DTI phase, though notalways, by reducing the thickness of a 3D TI to sev-eral nanometers [2931]. The 3D TI, on the otherhand, can result from stacking 2D TI quantum wellsunder certain conditions [32,33].

Breaking the TRS of a TI thin film with ferro-magnetism will lead to the QAHE [3438]. In a2D TI, if ferromagnetically induced exchange split-ting of the lowest order quantum well subbands islarge enough, so that one set of spin subbands aredriven back to the topologically trivial phase [35],

REVIEW He, Wang and Xue 41

there will be only one spin channel at each edgeremaining topologically protected, and thus the heli-cal edge states responsible for the QSHE will evolveinto chiral edge states exhibiting QAHE. In a 3DTI, ferromagnetism opens a gap in the Dirac surfaceband of the surface perpendicular to the magnetiza-tion vector and changes it into a QH system. Chi-ral edge states appear at each magnetic domain wallwhich is also the boundary between two topologi-cally different phases. In a uniformly perpendicularlymagnetized 3D TI film, the gapped surface bandsat the top and bottom surfaces have different topo-logical characters due to the opposite normal direc-tions. The chiral edge states reside at sample edgewhich acts as the topological boundary between thetop and bottom gapped surface bands. The QAHEcan thus be measured easily with the electrodes atthe sample edge [36].

When the thickness of a 3D TI is reduced toseveral nanometers, the hybridization between theDirac surface states of the top and bottom surfacesinduces gap-opening, which pushes the system intoa 2D TI or a 2D topologically trivial insulator phase[2931]. In both cases, the QAHE can be observed,as long as the hybridization gap is smaller than theferromagnetic exchange energy, because exchangesplitting can always lead to the situation that one setof spin subbands is topologically non-trivial, whereasthe other is topologically trivial [37].

Doping magnetic impurities is a convenient ap-proach to bring ferromagnetism in a TI. Manychoices in a TI material make magnetically dopedTIs a promising system to observe theQAHE.Thereare several challenges in material preparation to ob-tain a magnetically doped TI exhibiting the QAHE.First, the magnetically doped TI should have a long-range ferromagnetic order that can hold even inan insulating regime; the easy magnetization axisshould be perpendicular to the film plane. Second,the ferromagnetic TI film should have uniform and

well-controlled thickness; to localize the dissipativeconduction electrons, the film should be as thin aspossible, but should be thick enough to keep theenergy gap larger than the ferromagnetic exchangeenergy. Third, the Fermi level should be able to befinely tuned into the gap of the magnetically dopedTI thin film.

THIN FILMS OF MAGNETICALLY DOPEDTOPOLOGICAL INSULATORSWith molecular beam epitaxy (MBE), a standardtechnique to prepare high-quality semiconductorfilms, Bi2Se3 family TIs with well-controlled com-position and thickness can be grown on varioussubstrates [3942].The thickness-dependent angle-resolved photoemission spectroscopy (ARPES)clearly reveals gap-opening of Dirac surface statesdue to hybridization between top and bottomsurface states. For Bi2Se3, the hybridization gapreaches 40 meV in a 5 QL film, but cannotbe distinguished at 6 QL (Fig. 3) [41]. ForSb2Te3 and Bi2Te3, the crossover thickness is34 QL [39,42]. Considering that the typicalCurie temperature (TC) of a magnetically dopedsemiconductor/insulator is usually below 100K,corresponding to an exchange energy of the orderof 10 meV, a thickness near 3D2D crossoverthickness, i.e. 46 QL, should be chosen for Bi2Se3family TI films to observe the QAHE.

In most magnetically doped semiconductors/insulators, long-range ferromagnetic order cannotexist without itinerant charge carriers because theferromagnetic coupling between magnetic dopantsis of RKKY-type which requires mobile charge car-riers of bulk bands as a medium [43,44]. The bulkcarriers however will kill the QAHE by providinga parallel conduction channel. Hence, a differentlong-distance ferromagnetic coupling mechanism

Figure 3. ARPES bandmaps of the surface states of Bi2Se3 films with the thickness of 1 QL, 2 QL, 3 QL, 5 QL, and 6 QL, respectively. Reprinted withpermission from Zhang et al. [41], C 2010 NPG.

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independent of carriers is a prerequisite for the ob-servation of the QAHE in a magnetically doped TI.Fortunately, the unique band structure of Bi2Se3family TIs opens the possibility for such insulatingferromagnetism. Different from usual semiconduc-tors/insulators, Bi2Se3 family TIs have an invertedband structure, in which the conduction and va-lence bands are mixed together with the bulk gapopened by the SOC [25]. The special band struc-ture leads to a large Van Vleck susceptibility evenwhen the Fermi level lies in the bulk gap [37].Magnetic impurities dispersed in a Bi2Se3 family TIcan be ferromagnetically coupled by the strong VanVleck susceptibility without the need of free carriers.This mechanism can support ferromagnetism withTC up to tens of Kelvin in a magnetically dopedBi2Se3 family TI according to mean-field-basedcalculations [37].

Bi2Se3 is naturally the first choice of the threemembers of Bi2Se3 family TIs for its largest bulk gapand its Dirac point residing in the bulk gap. How-ever, long-range ferromagnetic order could not beenobserved in magnetically doped Bi2Se3 down to1.5 K. The absence of long-range ferromagnetic or-der is mainly due to the change in the band structureof Bi2Se3 induced by magnetic doping. The SOC ofBi2Se3 is mostly contributed by Bi atoms. Substi-tution of Bi atoms with much lighter magnetic ele-ments can significantly lower the SOC of Bi2Se3, sothat the band structure does no longer get invertedat a certain doping level. Van Vleck susceptibilitywill be greatly reduced as the result [45]. In Sb2Te3and Bi2Te3, because Te atoms also contribute sig-nificant SOC, the inverted band structure and VanVleck susceptibility are less influenced by magneticdoping. Clear long-range ferromagnetic order wasindeed observed in Cr- and V-doped Sb2Te3 andMn-doped Bi2Te3 [4649].

Bi2Te3 and Sb2Te3 have the same crystal struc-ture and close lattice constants, whereas the for-mer one is usually p-doped [50] and the latter oneis n-doped [51]. By simply mixing the two com-pounds into (Bi,Sb)2Te3 with a certain Bi:Sb ratio,one can tune the chemical potential between n- andp-types.The ARPES result has shown that the topo-logical surface states exist over the entire compo-sition range of (Bi,Sb)2Te3 with dominating carri-ers tunable between p- and n-types [52]. Moreover,near a charge neutral point, the carrier density esti-mated by ARPES and transport measurements arewell consistent. It suggests that (Bi,Sb)2Te3 filmshave negligible band bending and are little influ-enced by the ambient condition in transport mea-surements [53], both of which are important for theobservation of the QAHE.

Fig. 4af shows the magnetic-field-dependent Hall resistance (yx) of 5 QLCr0.22(BixSb1x)1.78Te3 films grown on sap-phire (0 0 0 1) substrates with the same Cr dopinglevel but different Bi:Sb ratio. At 1.5 K (the thickerlines) [46], for all the films the curves show nearlysquare-shaped hysteresis loops at a low field,suggesting good long-range ferromagnetic orderwith the easy magnetization axis perpendicular tothe sample plane. With increasing Bi content, theOHE (the slope of the linear background at highfield) evolves from positive to negative, indicatingthe change of the dominating carriers from p- ton-type. Therefore, the ferromagnetism of Cr-doped(BixSb1x)2Te3 always holds despite the significantchange in the carrier density and type induced byvariation in sample composition. The TC of thefilms shows little dependence on carrier density andtype, always3035K, even in the rather insulatingsamples around the pn crossover region (Fig. 4g).The carrier-independent ferromagnetism supportsthe existence of the ferromagnetic insulator phasepresumably induced by the Van Vleck mechanism.The AH resistance is significantly enhanced upto 3 k in the samples of lower carrier density.It is much larger than the AHE observed in mostferromagnetic metals, though still far from thequantized value (Fig. 4f).

Only by varying the Bi:Sb ratio, it is difficultto tune the Fermi level accurately in the magnet-ically induced gap. Electric field effect has to beapplied for fine-tuning of the chemical potential.SrTiO3 has a huge dielectric constant (20 000)at low temperature and is a commonly used sub-strate for MBE growth. By using the SrTiO3 sub-strate as a gate dielectric to tune the chemical po-tential of the TI thin film grown on it, one can real-ize the carrier density variation of 3 1013 cm2by changing the gate voltage (Vg) between 210V for a typical substrate thickness of 0.5 mm (seethe schematic drawing for the measurement setupin Fig. 5a) [54]. Fig. 5b shows the yx hysteresisloops of a 5 QL Cr0.22(Bi0.2Sb0.8)1.78Te3 film grownon SrTiO3 (1 1 1) measured with different gatevoltages (Vgs) (at 250mK). The coercivity and theshape of the hysteresis loops are nearly unchangedwith Vg, reconfirming the carrier-independent fer-romagnetism. At the same time, the AH resistancechanges dramatically with Vg, from 660 at 210V to 6.1 k (the maximum) at 35 V, 1/4 quan-tum Hall resistance, as a result of a lower carrierdensity (Fig. 5c). Hence, the ferromagnetic insula-tor phase, well-controlled thickness and tunable car-rier density are all satisfied in an MBE-grown Cr-doped (Bi,Sb)2Te3 thin film on the SrTiO3 (1 1 1)

REVIEW He, Wang and Xue 43

Figure 4. Transport properties of Cr-doped (BixSb1x)2Te3 films. (a)(f) Magnetic field dependent Hall resistance yx of the Cr0.22(BixSb1x)1.78Te3 filmswith x = 0 (a), x = 0.15 (b), x = 0.2 (c), x = 0.25 (d), x = 0.35 (e) and x = 0.5 (f) at different temperatures. (g) Dependence of Curie temperature (TC) onBi content (x) (bottom axis) and estimated carrier density (top axis). (h) Dependence of yx(0) (red solid squares) and xx(0) (blue solid circles) at 1.5 Kon Bi content (x) (bottom axis) and carrier density (top axis). Reprinted with permission from Chang et al. [46], C 2013 Wiley.

substrate, which validates the material as a perfectsystem to realize the QAHE.

EXPERIMENTAL OBSERVATIONOF THE QAHEThrough a careful optimization of sample growth pa-rameters and measurements at ultra-low tempera-ture, the QAHE was finally observed in Cr-doped(Bi,Sb)2Te3 thin films grown on SrTiO3 (1 1 1)substrates [55]. Figure 6a and c shows the mag-netic field dependence of yx and xx, of a 5 QLCr0.15(Bi0.1Sb0.9)1.85Te3 film, respectively, at differ-ent Vgs measured at T = 30mK. In the magnetizedstates,yx is nearly invariantwithmagnetic field, sug-gesting perfect ferromagnetic ordering and charge

neutrality of the sample. The AH resistance reachesa maximum value of h/e2 even at the zero magneticfield.Themagneto-resistance (MR) curves (Fig. 6c)exhibit the typical shape for a ferromagnetic mate-rial: two sharp symmetric peaks at the coercive fields.But theMR ratio ((xx(Hc)xx(0))/xx(0)) has asurprisingly large value of 2251% as yx reaches thequantized value (Fig. 6c).

The Vg dependences of zero field Hall and lon-gitudinal resistance (yx(0) and xx(0)) are plot-ted in Fig. 6b. The yx(0)Vg curve exhibits a dis-tinct plateau 1.5 V (indicated as Vg0) with thequantized value h/e2. Accompanying theyx plateauis a clear dip in the longitudinal resistance xx(0)down to 0.098 h/e2. For comparison with theory,we transform yx(0) and xx(0) into sheet conduc-tance via the relations xy = yx/(yx2 + xx2) and

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Figure 5. (a) A schematic drawing depicting the measurement geometry of the back-gated Cr-doped (Bi,Sb)2Te3 film with a SrTiO3 substrate as gatedielectrics. (b) Magnetic field dependent yx of a 5 QL Cr0.22(Bi0.2Sb0.8)1.78Te3 film grown on the SrTiO3 (1 1 1) substrate at different Vgs measured at250 mK. (c) Dependence of yx(0) (red solid squares) and xx(0) (blue solid line) of the 5 QL Cr0.22(Bi0.2Sb0.8)1.78Te3 film grown on the SrTiO3 (1 1 1)substrate on Vg at 250 mK. (b) and (c) are reprinted with permission from Chang et al. [46], C 2013 Wiley.

xx = xx/(yx2 + xx2) and plot them in Fig. 6d.Around Vg0, xy(0) has a notable plateau at 0.987e2/h, whereas xx(0) has a dip down to 0.096 e2/h.These observations unambiguously demonstrate therealization of the QAHE.

ThehugeMRcan also be attributed to theQAHEphenomenology. In the magnetized QAH state, theexistence of a dissipationless edge state leads to anearly vanishing xx. At the coercive field, the mag-netization reversal of a QAH system results in aquantum phase transition between two QH statesvia a highly dissipative phasewith a largexx [6].ThehugeMRthus reflects thedistinct difference in trans-port properties between an ordinary insulator and aQAH insulator.

For a QH system, if dissipative conduction chan-nels are not completely localized, xx has a non-zerovalue, whereas xy deviates slightly from the quan-tized plateau. The observations of the deviation of xy(0) from e2/h plateau and the non-zero xx(0)near the charge neutral point in Fig. 6d can thus beattributed to the residual dissipative channels, whichare expected to vanish completely at zero tempera-ture or in a strong magnetic field. Figure 7a and bdisplays the magnetic field dependence of yx and

xx of the same sample as in Fig. 6, respectively,with the magnetic field applied up to 18 T. Exceptfor the large MR peak at coercivity, increasing thefield suppresses xx toward zero. Above 10 T, xxvanishes completely, corresponding to a perfect QHstate. Since the increase in xx from zero (above 10T) to 0.098 h/e2 (at zero field) is very smooth andyx remains at the quantized value h/e2, no quantumphase transition occurs and the sample stays in thesameQHphase as the field sweeps from10T to zerofield.Therefore, the complete quantization above 10T can only be attributed to the same QAH state atthe zero field.

In Fig. 8a, we show Vg dependences of yx(0)and xx(0) measured at different temperatures. Theyx(0) maximum value is considerably suppressedby increasing temperatures, accompanied by dis-appearance of the dip in xx(0). The xx(0) ex-tracted from these measurements (on logarithmicscale, Fig. 8b) exhibits temperaturedependence sim-ilar to that in integer QH systems [56]: the drop of xx is at first rapid, resulting from the freezing of thethermal activation mechanism, and then becomesmuch slower when the temperature is below 1K. Itcan be attributed to variable range hopping (VRH),

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Figure 6. The QAHEmeasured at 30 mK. (a)Magnetic field dependence ofyx at different Vgs. (b) Dependence ofyx(0) (emptyblue squares) and xx(0) (empty red circles) on Vg. (c) Magnetic field dependence of xx at different Vgs. (d) Dependence of xy(0) (empty blue squares) and xx(0) (empty red circles) on Vg. The vertical purple dashed-dotted lines in (b) and (d) indicatethe Vg for Vg0. Reprinted with permission from Chang et al. [55], C 2013 AAAS.

but its exact mechanism remains unknown. Similarto the QHE, zero field xx is expected to decreaseto zero at sufficiently low temperature. In Fig. 8c,we plot the relation between xx(0) and xy(0)( xy = e2/h xy), which reflects the contribu-tion of dissipative channels). A power-law relation xy xx with 1.55 is obtained. For a ferro-magnetic insulator in the VRH regime, the AH con-ductivity is related to the longitudinal conductivitythrough AH = A xx (the power is 1.6, the

pre-factor A can be positive or negative dependingonmaterials) [3].The above result can thus be qual-itatively understood within the VRH framework.

CONCLUSION AND OUTLOOKThe experimental realization of the QAHE inmagnetically doped TI thin films not only con-cludes the searching of over 20 years for the QHE

Figure 7. The QAHE under strongmagnetic field measured at 30 mK. (a) Magnetic field dependence of yx at Vg0. (b) Magneticfield dependence of xx at Vg0. The blue and red lines in (a) and (b) indicate the data taken with increasing and decreasingfields, respectively. Reprinted with permission from Chang et al. [55], C 2013 AAAS.

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Figure 8. Temperature dependence of the QAHE. (a) Vg-dependent yx(0) and xx(0) measured at 90 mK, 400 mK, 1.5 K and 4 K, respectively. The verticalpurple dashed-dotted line indicates the Vg for Vg0. The variation in the position and width of the yx(0) peak at different temperatures results from thechange in substrate dielectric properties induced by temperature and charging cycles. (b) Dependences of logarithmically scaled xx(0) (empty redcircles) and xy(0) (empty blue squares) at Vg0 on inverse temperature. The dashed lines are a guide to the eye. (c) The relation between xy(0)and xx(0) at Vg0 on a double logarithmic scale. The red dashed line indicates the fit with a power law xy xx with 1.55. Reprinted withpermission from Chang et al. [55], C 2013 AAAS.

without magnetic field, but also confirms theexistence of intrinsic AHE that has been ques-tioned for over half century. The realization of theQAHE also brings new hopes for many other novelquantum phenomena predicted previously, such astopological magnetoelectric effect, image magneticmonopoles and Majorana states. Since the QAHEdoes not require high electronmobility, in principle,low-cost preparation techniques can be used tomake the QAH samples, which will significantlyreduce the barrier for the studies and applicationsbased on the QAHE.

The major challenge preventing further studieson the QAHE is the very low temperature that isneeded to reach the quantization plateau. So far it isstill not clear whatmainly contributes to the remain-ing non-localized channels. One may try to reducethe film thickness and artificially introduce disorderin a magnetically doped TI film to promote elec-tron localization, so that quantization canbe reachedat higher temperatures. There are two energy scalesthat determine at howhigh a temperature theQAHEcan be observed: one is the Curie temperature, andthe other is the gap of TI. The future endeavors will

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be focused on enhancing these two energy scalesby choosing novel TI materials, substrates and mag-netic dopants.

ACKNOWLEDGEMENTSThe authors would thank the collaborations and discussions withCui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, ZuochengZhang, Minghua Guo, Kang Li, Yunbo Ou, PangWei, Lili Wang,Li Lu, Zhongqing Ji, Jin-Feng Jia, Shuaihua Ji, Xi Chen, Xi Dai,Zhong Fang, Xincheng Xie, Jian Wang, Shengbai Zhang, Chao-Xing Liu, Bangfen Zhu, Wenhui Duan, Jian Wu, Peizhe Tang,Shun-Qing Shen, Qian Niu, Xiao-Liang Qi, Xu-Cun Ma, andShou-Cheng Zhang. This work was supported by the NationalNatural Science Foundation of China, the National Basic Re-search Program of China, and the Knowledge Innovation Pro-gram of Chinese Academy of Sciences.

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