REVIEWS

Magnetic topological insulators

Yoshinori Tokura1,2*, Kenji Yasuda 2 and Atsushi Tsukazaki 3 Abstract | The importance of global band topology is unequivocally recognized in condensed matter physics, and new states of matter, such as topological insulators, have been discovered. Owing to their bulk band topology, 3D topological insulators possess a massless Dirac dispersion with spin–momentum locking at the surface. Although 3D topological insulators were originally proposed in time-​reversal invariant systems, the onset of a spontaneous magnetization or, equivalently , a broken time-reversal​ symmetry leads to the formation of an exchange gap in the Dirac band dispersion. In such magnetic topological insulators, tuning of the Fermi level in the exchange gap results in the emergence of a quantum Hall effect at zero magnetic field, that is, of a quantum anomalous Hall effect. Here, we review the basic concepts of magnetic topological insulators and their experimental realization, together with the discovery and verification of their emergent properties. In particular, we discuss how the development of tailored materials through heterostructure engineering has made it possible to access the quantum anomalous Hall effect, the topological magnetoelectric effect, the physics related to the chiral edge states that appear in these materials and various spintronic phenomena. Further theoretical and experimental research on magnetic topological insulators will provide fertile ground for the development of new concepts for next-generation​ electronic devices for applications such as spintronics with low energy consumption, dissipationless topological electronics and topological quantum computation.

The dynamics of conduction electrons in magnetic The first example of a topological state observed solids, in particular the quantum transport of spin, in condensed matter was the integer quantum Hall has attracted much attention in the past few decades, effect (QHE) — an analogue of the ordinary Hall effect for example, in the context of heavy-​fermion systems1, in which the Hall conductance is quantized — meas- high-​temperature superconductors2, giant magnetore- ured in 2D electron systems in a magnetic field9. The sistance systems3, colossal magnetoresistance oxides4 experimental results were interpreted on the basis of and diluted magnetic semiconductors5. Some of these the concept of topology introduced in the Thouless– systems, in particular those exhibiting giant and colos- Kohmoto–Nightingale–den Nijs (TKNN)10,11 theory. sal magnetoresistance, have already found industrial The TKNN formula for a 2D electron system with bro- applications. More recently, another branch of work on ken time-reversal​ symmetry is characterized by the top- quantum transport based on the concept of topology has ological invariant called the TKNN number or Chern emerged. These two lines of research are now merging, number (C), which, in the integer QHE, corresponds to

producing new fields and directions; one example is the the occupancy of the Landau levels.2 The Hall conduc- study of magnetic topological insulators (TIs). tivity σ is expressed as Ce , where e is the electron xy σxy = h There is an increasing interest in topological quan- charge and h is Planck’s constant. Because C is an integer, 1RIKEN Center for Emergent tum materials, which exhibit electronic or magnetic there is no deformation of the material parameters that Matter Science (CEMS), states characterized by an integer topological invariant, can continuously connect it to a trivial insulator with Wako, Japan. such as the Chern number and the Z2 invariant for the C = 0 or to a vacuum (that also has C = 0). Owing to a 2Department of Applied topological electronic structure in momentum space6,7 phenomenon known as bulk–boundary correspondence, Physics, University of Tokyo, Tokyo, Japan. and the skyrmion number, defined by the winding num- this results in the appearance of a gapless state at the ber of the spin configuration, in real space8. One distin- sample edge that forms a number |C| of chiral edge chan- 3Institute for Materials Research, Tohoku University, guishing feature of topological materials is the presence nels. Charge transport in these edge channels is non- Sendai, Japan. of electronic or magnetic states that are robust against dissipative, and its direction (clockwise or anticlockwise) *e-mail:​ [email protected] external perturbations thanks to the topological pro- is uniquely determined by the sign of the charge (electron https://doi.org/10.1038/ tection afforded by the fact that the integer topological or hole) and by the direction of an applied magnetic field s42254-018-0011-5 number is invariant under continuous deformations. (up or down).

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Key points the discontinuity of ν2 at the surface of the topological material, a gapless 2D state appears at the surface of a • The chemical doping of topological insulators with transition metal elements induces 3D TI or at its interface with a trivial insulator (Fig. 1a,b). a spontaneous magnetization that interacts with the topological surface state to Therefore, if E lies within the bulk bandgap, the TI open a mass gap at the Dirac point. F shows metallic conduction at the surface (or interface) • The precise tuning of the Fermi level at the mass gap enables the observation of the but is insulating in the bulk. The 2D surface state is quantum anomalous Hall effect — a zero-​magnetic-field quantum Hall effect arising described by the Hamiltonian in the presence of a spontaneous magnetization — which is further stabilized by heterostructure engineering. • Chiral edge conduction associated with the quantum anomalous Hall effect is Hv=−Fy()kσxx+(kσy 1) manipulated by magnetic domain walls, and the edge modes can be turned into chiral Majorana edge modes via proximity coupling with a superconductor. • Heterostructure engineering and terahertz measurements enable the observation of where vF is the Fermi velocity of the linear dispersion the quantized topological magnetoelectric effect. and σx and σy are the Pauli matrices for spin. This • The spin–momentum-locked​ conduction electrons in the surface state lead to Hamiltonian implies the spin–momentum locking (Fig. 1a) versatile spintronic functionalities, such as an efficient generation of spin transfer of the massless Dirac electrons , which means torque, as a result of charge-​to-spin conversion. that electrons of opposite spins travel in opposite 23,24 • The further development of materials design and engineering will realize the quantum directions. Currently, many 3D TIs are known , anomalous Hall effect at higher temperatures, the control of this state with external including BixSb1-x, tetradymites such as Bi2Se3, Bi2Te 3 fields and exotic topological states of matter. and Sb2Te 3, and strained HgTe. For these materials, a massless Dirac dispersion with spin–momentum locking has been theoretically predicted21,25 and then experi­ From the analogy between the ordinary Hall effect mentally confirmed with surface-​sensitive probes, such and anomalous Hall effect (AHE), which are induced as angle-​resolved (and spin-​resolved) photoemission by an external magnetic field and a spontaneous magne­ spectroscopy and scanning tunnelling spectroscopy6,7. tization, respectively, it was natural to ask whether a Introducing a spontaneous magnetization into (or quantum AHE (QAHE) — an integer QHE at zero next to) the surface of a TI brings about important magnetic field — could be achieved. Initially, the QAHE modifications to the electronic structure of the surface, was theoretically predicted to arise in honeycomb and because the conduction electrons couple with the mag- kagome lattices with staggered magnetic fluxes12,13. netization via an exchange interaction, as described by These fluxes lead to the formation of a gapped state, the the model Hamiltonian6,17 quantum Hall insulator, giving a Berry curvature of the conduction electrons. Hv=−kσ++kσ mσ =(Rσ⋅ 2) Meanwhile, regarding the intrinsic mechanism F()yx xy z underlying the AHE, an interpretation in terms of the TKNN formula for an electronic band with relativistic with . The exchange interaction R =−()vkFFyx,,vk m spin–orbit coupling was established in 2D as well as 3D breaks time-​reversal symmetry, causing the opening of 14,29 (Fig. 1c) systems : σxy for the intrinsic AHE is given by the inte- a mass gap m in the Dirac surface state , which gration of the Berry curvature bnn()kk=×(∇ a ), where causes the Dirac fermions in the surface state to become ∂ is the Berry connection (u is the massive. In this 2D system with broken time-​reversal ann()k = iuk ∂k unk nk periodic part of the Bloch function of the nth band) symmetry, the Chern number can be defined10 using over the occupied states in the Brillouin zone, just as RR̂ = ∕ R as in the TKNN formula. Considering the close similar- ity between the QHE and AHE, it was predicted that    ∂RR̂̂∂  dkxydk the QAHE would be readily materialized in a 2D fer- C =2 R̂ ⋅  ×  =sgn()m (3) ∫  ∂kk∂4 π romagnet with spin–orbit coupling if gapping or full BZ  xy 15 carrier localization at the Fermi level (EF) was realized . However, the experimental discovery of the QAHE16 Here, the coefficient 2 counts the contributions from had to await the successful growth of a ferromagnetic the top and bottom surfaces; the integration (carried out TI following the theoretical design and prediction6,7,17,18. over the Brillouin zone, BZ) gives a winding number of TIs were theoretically discovered by extending the one-​half for the spin texture shown in Fig. 1c. Thus, the formalism for 2D (TKNN) topological phases with bro- Chern number is one, and its sign is that of the mass ken time-​reversal symmetry to 2D or 3D systems with term, which depends on the sign of the exchange cou- time-reversal​ symmetry19–22. The topological number in pling and on the magnetization direction. Therefore,

this case is the Z2 index, which has values ν2 = 0 or 1, when EF is located within the2 mass gap, the Hall con- unlike the TKNN number, ν, which can take any integer ductivity is quantized, e according to the TKNN σxy =± h value. ν2 = 1 corresponds to a TI, whereas ν2= 0 cor- formula. This implies the emergence of a QAHE, one of responds to a trivial insulator or vacuum. A necessary the hallmarks of magnetic TIs. The QAHE gives rise to a condition for the appearance of topologically protected chiral edge mode (CEM; Fig. 1d), the direction of which edge states is the presence of a band inversion between depends on the sign of the Chern number and on the the conduction and valence band by a spin−orbit inter- magnetization direction. action that leads to the opening of a bandgap, rather than According to the modern view of the intrinsic AHE to a semimetal state with overlapping bands. Reflecting in an itinerant ferromagnet14, spin–orbit coupling often

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Conduction band the study of CEMs, topological magnetoelectric effects a c and spintronic functionalities. We start by discussing the methods for the successful realization of magnetic TIs by transition metal elemental doping with a surface state exchange gap. Next, the experimental realization of the QAHE by tuning the Fermi level to the hybridization gap is explained, with a focus on the archetypal mat­ Edge state E erial system: (Bi,Sb)2Te 3 doped with transition metals. F This explanation is followed by a survey of the emer- EF Surface state gent phenomena related to the QAHE, such as CEMs and topological magnetoelectric and magneto-​optical effects. We also discuss various spintronic functional- Valence ities that leverage the spin–momentum locking of the band surface state. Finally, we overview the future perspectives for the field, including realization of the QAHE at higher temperatures, the external control of this state and the possible realization of exotic topological states of matter in magnetic TIs. b d Origins of magnetism in magnetic TIs For the emergent properties of magnetic TIs to appear, the formation of the exchange gap in the surface state is essential. The gap is induced by an interaction between the electrons in the surface state and the spontane- ous magnetization, as described by the mass term in 2 equation ; this term is mσJzS≡ − nSZZσ , where z ̂ is the unit vector normal to the surface, J is the exchange coup­

ling between the z component σZ of the spin of the Dirac

electrons and the localized spin S and nS is the areal den- sity of localized spins, which have average z component (Ref.18) SZ . The chemical doping with 3d transition metal elements is an effective approach to creating a magnetic Fig. 1 | The electronic structure of a topological insulator and of a magnetic | The massless Dirac-like​ dispersion of the surface state with interaction with the surface state, as was originally pro- topological insulator. a 17,34–37 spin–momentum locking in a topological insulator. The surface state band connects the posed theoretically . Transition metal doping in TIs bulk valence and the bulk conduction bands. b | Real-space​ picture of the surface state owes its success to the knowledge accumulated from in a topological insulator. Electrons with spins pointing up and down (red arrows) move in studies of diluted magnetic semiconductors5,38–40. opposite directions. c | The gapped Dirac-like​ dispersion of the surface state in a Two mechanisms have been considered as a possible magnetic topological insulator. d | The chiral edge mode that appears in a magnetic origin for ferromagnetism in magnetic TIs: the carrier-​ topological insulator when the Fermi level, EF, is located in the mass gap induced by the mediated Ruderman–Kittel–Kasuya–Yosida (RKKY) magnetic exchange interaction. The edge electrons conduct electricity without mechanism41,42 and the local valence-electron-mediated​ dissipation in one direction along the edge of the sample. Bloembergen–Rowland38,39,43, or Van Vleck17, mecha- nism. The role of the RKKY mechanism was experimen- (Ref.44) causes the opening of an anticrossing gap in the spin-​ tally verified in Mn-​doped Bi2Te 3 , as exemplified polarized band around a specific k-​point, while closing by measurements on an electric-​double-layer transis-

it at another k-​point, leading to the appearance of Weyl tor composed of MnxBi2-xSeyTe 3-y single crystal flakes fermions26–28. Many anticrossing gaps in momentum (x = 0.04 and y = 0.12; Fig. 2a,b)45. The Hall conductance space behave as magnetic monopoles in the context in this type of sample can be effectively controlled by

of the k-​space Berry curvature and hence contribute applying a gate bias VB; both the amplitude and hystere- to the AHE observed in itinerant ferromagnets14,29. In sis of the Hall conductance are enhanced as the electron

this context, the QAHE in magnetic TIs provides the density decreases. This behaviour of σxy reflects a carrier-​ ideal platform for the study of the AHE: a magnetic TI dependent ferromagnetism (Fig. 2a). The ferromagnetic

is characterized by a single magnetic monopole in the Curie temperature of the sample, TC, is dramatically 2D momentum space, located at k = 0 (the Γ point in enhanced with the reduction in the electron density, sig- the first Brillouin zone). The surface state band in mag- nalling that the ferromagnetism develops progressively (Fig. 2b) netic TIs maintains the spin–momentum locking when as EF approaches the Dirac point . This trend

EF is located away from the gap region while keeping the appears to be unexpected in the context of an intuitive spontaneous magnetization. This property of magnetic understanding of the RKKY mechanism, in which the TIs may bring about versatile spintronic functions, as effective carrier-​mediated interaction for the ferromag- discussed later. netism increases with the carrier density, at least in the Several excellent review papers on TIs4,6,23,24,30 and relatively low-​carrier-concentration region. However, 31–33 magnetic TIs are available. This Review is focused the observation of maximum TC at the Dirac point can on recent experimental developments, particularly in be rationally accounted for in terms of the maximization

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Mn-doped Bi2Te 3–ySey single crystal Cr-doped (BixSb2–x)Te3 thin film ab 13 –2 cdn2D (10 cm ) T=2K VT =0V 12 D 5 × 3 1 –1 –2 –6 C 60 n =3.3×1013 e– cm–2 10 x=0 2D B p-type n-type 8 VB =+100V 40 (K ) c

T 6 A 1 kΩ n =2.6×1013 e– cm–2 T c (K ) 20 2D 4 /h) 2 E (k Ω) (e

yx x=0.2 0 xy V =+20 2 σ B R 0.00.1 0.20.3 0.40.5 0 Bi content (x) 13 – –2 –1 0 123 4567 n2D =1.5×10 e cm 13 – –2 n2D (10 e cm )

×10 x=0.5 1.5K 10K 30K 3K 15K 40K 2 V =–100 0.04e /h B 5K 20K 60K –0.1 0.0 0.1 –1 0 1 B (T) B (T)

Fig. 2 | Two possible mechanisms for the ferromagnetism in magnetic topological insulators. a | The gate bias VB applied

to an electric-double-layer​ transistor based on MnxBi2-xSeyTe 3-y single crystal flakes (x = 0.04 and y = 0.12) controls the

anomalous Hall effect. The temperature T = 2 K , σxy is the Hall conductance, B is the external magnetic field and n2D is the

density of charge carriers in the surface state. b | Ferromagnetic Curie temperature TC as a function of n2D for Mn-​doped

MnxBi2-xSeyTe 3-y observed for five samples (A–E) with different carrier densities. c | Influence of the Bi content x on the anomalous

Hall effect for Cr0.22(BixSb1-x)1.78Te 3 thin films, measured at various temperatures. d | TC for Cr0.22(BixSb1-x)1.78Te 3 is plotted as a

function of x and charge n2D; the blue region corresponds to the charge neutral point. Ryx, Hall resistance. Panels a and b are adapted from Ref.45, Springer Nature Limited. Panels c and d are adapted with permission from Ref.47, Wiley-​VCH.

of the energy gain of the Dirac electronic system owing the constant TC, another interpretation based on the to the opening of the exchange gap induced by the fer- Bloembergen–Rowland mechanism38,39,43 was also put for- romagnetism41, analogously to the Peierls transition that ward. In particular, the ferromagnetic interaction medi- is accompanied by the opening of an electronic gap in ated by valence electrons in a narrow-gap​ semiconductor electron–lattice coupled systems. The related carrier-​ would explain the results without the need of invoking mediated ferromagnetism in the surface state is also a large concentration of itinerant conduction carriers38. observed in scanning tunnelling microscopy (STM) Thus, although the QAHE has been experimentally experiments46. An alternative interpretation is that the observed in both Cr-​doped and V-​doped systems, the

change in TC in gated MnxBi2-xSeyTe 3-y may originate microscopic origin of the ferromagnetism remains to from ferromagnetic spin–spin interactions mediated by be clarified with further theoretical and experimental the bulk valence band45. investigations49. The other mechanism underlying ferromagnetism, The opening of a gap in the surface state has been mediated by local valence electrons, is the Bloembergen– experimentally verified with surface-​sensitive tech- Rowland, or Van Vleck, mechanism. This kind of ferro- niques such as spectroscopic-​imaging STM (SI-​STM)50 magnetism, induced by doping with 3d transition metals, and angular-​resolved photoemission spectroscopy 51–53 is observed, for example, in Cr-doped​ (Bi,Sb)2Te 3. In 5-nm-​ (ARPES) . Because of the form of the exchange inter- 47 thick Cr0.22(BixSb1-x)1.78Te 3 thin films , the coercive field HC action, −JnSZSσZ, the size of the gap is governed by the in the hysteresis of the Hall resistance Ryx does not depend coupling strength as well as by the effective spin density. on the Bi content x (Fig. 2c). Nevertheless, the sign of The spatial distribution of the exchange gap in a single

the ordinary Hall term (the slope of Ryx at a high magnetic crystal of Cr0.08(Bi0.1Sb0.9)1.92Te 3 (x = 0.08, TC = 18 K) was field) reverses from positive at x = 0 to negative at x = 0.5, characterized at T = 4.5 K by SI-​STM with quasiparticle 50 (Fig. 3a,b) indicating that EF can be controlled by varying x across the interference measurements . The conductance

charge neutral point and that EF is within the exchange gap spectra acquired at different positions on the surface

around x = 0.2. The TC in this system is rather insensitive indicate that the gap size ranges between 9 meV and to the carrier type or charge carrier density (Fig. 2d). This 48 meV (Fig. 3a), and the distribution of the gap sizes is result is in sharp contrast to the carrier-density-dependent​ centred around 30 meV (Fig. 3b). The width of the dis- (Fig. 2b) TC for the Mn-doped​ case . tribution indicates an inherent random distribution of In a theoretical study of the QAHE17, Van Vleck para­ Cr concentration or of the exchange interaction in the magnetism was proposed as a plausible origin for the crystal. Thus, the improvement in the spatial uniformity ferromagnetism with considerable gain of spin suscepti- of the strength of the exchange interaction plays a vital bility owing to the mixing of the conduction and valence role in the realization of a large gap in the surface state. bands. Experiments on the QAHE in Cr-​doped16,47 The gap formation has also been observed by ARPES 48 and V-​doped 3D TIs supported this idea. In light of in a Se-rich​ bulk crystal, (Bi0.84Fe0.16)2Se3.7. The exchange

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Cr-doped (Bi, Sb) Te single crystal ab2 3 c Fe-doped Bi2Se3 single crystal 1.0 4 ∆ (meV) KKГ 9 (2%) 0.0 0.8 18 (27%) 28 (48%) 3 )

3 2∆ 0.6 38 (19%) 48 (3%) gy (eV) SSB 2 0.4 0.2 Gap Count (x10 Conductance (nS) 0.2 1 Binding ener BVB 0.0 0.4 2∆ 0 –100 0 100 200 300 10 20 30 40 50 60 –0.05 00.05 E (meV) ∆ (meV) k (Å–1) Fig. 3 | Exchange gap formation by magnetization. a | Conductance spectra measured by scanning tunnelling

spectroscopy at different positions in a Cr0.08(Bi0.1Sb0.9)1.92Te 3 single crystal. The curves are shifted for clarity. The near-​ zero-conductance regions between the pairs of arrows correspond to the mass gap and enable evaluating its size, Δ.

b | Distribution of the gap sizes. c | Electronic band structure of a (Bi0.84Fe0.16)2Se3.7 crystal (Curie temperature of ~170 K) measured by angular-resolved​ photoemission spectroscopy. Γ and K , high-symmetry​ points in the first Brillouin zone; BVB, bulk valence band; E, electron energy ; k, electron momentum; SSB, surface state band. Panels a and b are adapted with permission from Ref.50, PNAS. Panel c is adapted with permission from Ref.51, AAAS.

gap in the crystal surface at T = 10 K is observed at the being insulating and the conduction occurring only at Dirac point of the surface state band51 (Fig. 3c). The size of the side edge of the surface. the gap, evaluated from the energy distribution curves, is In the theoretical prediction of the QAHE17, two tran- approximately 50 meV and decreases with decreasing Fe sition metal elements, Cr3+ and Fe3+, were proposed as concentration. Thus, the exchange interaction between candidates for magnetic-​ion doping. In the first exper- the surface states and the spontaneous magnetization imental demonstration of the QAHE (Fig. 4a–c), the 2 results in the formation of an exchange gap with a size quantization of the Hall resistance ρyx ≈ h/e was meas-

of a few tens of meV. ured in a uniformly Cr-​doped (Bi1-ySby)2Te 3 (CBST) 16 The experimental verification of the magnetic prop- 5-nm-​thick film on a SrTiO3 substrate . ρyx switches erties of the samples and the characterization of the between +h/e2 and –h/e2, accompanying the magnetiza- exchange gap are essential steps towards the materiali­ tion reversal resulting from the application of low posi- zation of the QAHE and of the topological magneto­ tive and negative magnetic fields. Two signatures of the electric effect in magnetic TIs. A careful consideration QAHE are clearly observed: one is the manifestation of of the inhomogeneity of magnetic dopants in terms of a QHE-​equivalent phenomenon even under zero mag- microscopic phase segregation, the coexistence of the netic field, and the other is the chirality reversal of the exchange gap and gapless bands54 and the impurity 1D edge conduction upon reversal of the magnetization 55 bands will be needed to improve the understanding direction. By applying a gate voltage Vg via the dielec-

and control over these systems. tric substrate to tune EF, the saturated ρyx value deviates from the quantized value h/e2, whereas the coercive field

Experimental observations of QAHEs remains constant. There is a narrow Vg region in which

As discussed in the introduction, the QAHE can be intu- the Hall conductance σxy is quantized and σxx is nearly (Fig. 4c) itively understood as the equivalent of the QHE at zero zero . The small range of values of Vg for which magnetic field and is characterized by the formation of this condition is realized indicates that precise tuning of

dissipationless 1D chiral edge conduction channels. The EF is crucial to realize the QAHE. number of quantized states is governed by the topologi- Two years after these measurements on CBST16, the

cal Chern number, C; in the case of the QAHE, there is a QAHE was also demonstrated for V-doped​ (Bi1-ySby)2Te 3 (Fig. 4d–f) one-to-one​ correspondence between the sign of C (= ±1) (VBST) 4-nm-thick​ films on SrTiO3 . VBST has and the magnetization direction. Two prerequisites for a larger coercive field (~1 T) than CBST and displays the observation of the QAHE are the formation of an better quantization characteristics, with values closer 2 (Ref.56) (Fig. 4e) exchange gap induced by the coupling with the mag- to ρyx ≈ h/e and ρxx ≈ 0 at T = 25 mK . (Fig. 4f) netization in or next to the TI and the precise tuning At T = 120 mK , however, ρxx maintains a finite

of EF into the gap. When EF is located in the exchange value, while Vg is tuned to maximize ρyx, meaning that

gap, the tangent of the Hall angle, σxy/σxx, diverges as a the residual in-​gap states remain at EF in the exchange

consequence of the quantization of the Hall conductiv- gap. Regardless of the relatively high TC of approximately 2 ity σxy ≈ e /h and the disappearance of the longitudinal 20 K for both CBST and VBST with the present transi-

conductivity σxx ≈ 0 owing to the Berry curvature at the tion metal doping level, the temperature at which the full exchange gap14,17. For the gap to form, the magnetization quantization is realized is lower than 100 mK. The tem- direction should be along the z direction (normal to the perature at which the QAHE can be observed is several

surface), which results in the top and bottom surfaces orders of magnitude lower than TC and the temperature

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CBST VBST Modulation-doped CBST 1nm a d g Crx(Bi1-ySby)2-xTe 3: 1nm

Crx(Bi1-ySby)2-xTe 3: 5nm Vx(Bi1-ySby)2-xTe 3: 4nm (Bi1-ySby)2Te 3: 4nm

1nm h 2 b 1.0 e 1.5 T=30mK ) 2

1.0 ) ) 0.5

2 1 2

(h/e 0.5 (h/e (h/e 0.0 yx ρ 0.0 0 –0.5 –0.5 R (0.5K) –1.0 –1 xx Resistance –1.0 Resistance R (0.5K) –0.4 –0.2 0.0 0.2 0.4 yx ρxx B (T) T=25mK V =V 0 V – V =0V R (4.2K) g g ρyx g CNP yx –1.5V–7.5V 28V 200V –2 –1 0 12 –1.0 –0.5 0.0 0.5 1.0 2.5V –14V –55V B (T) B (T) c f 0 i T=0.50K, B=0T T=30mK Vg =–1.5 V 1.0 σxx (modulation 1.0 ) /h ) 2 2 /h) doped) 2 0.8 0 1 Vg =–12V σxy (modulation 0.6 doped) 0.5 σ (uniformly 0.4 xx doped)

Resistance (h/e (uniformly σ 0.2 ρxx σxy

xx Conductance (e Conductance (e σ T=120mK ρyx doped) 0.0 xy 0.0 0 –10 –5 0 5 –200 –100 0 100 200––10 550 10 V –V (V) Vg (V) Vg (V) g CNP Fig. 4 | Experimental observations of the quantum anomalous hall effect. a | The first observation of the quantum

anomalous Hall effect was in a 5-nm-​thick film of uniformly Cr-​doped Crx(Bi1-ySby)2-xTe 3 (CBST) grown on a SrTiO3 substrate and measured at 30 mK. Yellow arrows indicate the locations of the surface states at the top and bottom of the film.

b | Magnetic field (B) dependence of the Hall resistance ρyx for CBST for various gate voltages (Vg). c | Vg dependence of the

Hall and longitudinal conductance, σxy and σxx, respectively , for CBST. d | Measurements on a 4-nm-​thick uniformly V-​doped

Vx(Bi1-ySby)2-xTe 3 (VBST) film on SrTiO3 at 25 mK. e | Magnetic field dependence of the Hall and longitudinal resistance, ρyx and

ρxx, respectively , for VBST. f | Dependence of ρyx and ρxx on Vg for VBST at T = 120 mK. g | Schematic of the Cr-modulation-

doped heterostructure on which the measurements in panels h and i were recorded. h | Magnetic field dependence of Ryx at

0.5 K and 4.2 K and of Rxx at 0.5 K. i | Dependence of σxy and σxx on Vg for the uniformly doped and modulation-​doped CBST

at T = 50 mK. VCNP is the gate voltage used to tune the charge neutrality point (Dirac point). Panels b and c are reproduced with permission from Ref.16, AAAS. Panels e and f are reproduced from Ref.56, Springer Nature Limited. Panels h and i are adapted from Ref.61, Mogi, M. et al. Magnetic modulation doping in topological insulators toward higher-temperature​ quantum anomalous Hall effect. Appl. Phys. Lett. 107, 182401 (2015), with the permission of AIP Publishing.

corresponding to the energy of the spectroscopically even at T = 0.5 K (Fig. 4h), which indicates that disor- resolved magnetization gap of the surface state. The der is effectively suppressed by the insertion of BST as large discrepancy likely originates from the presence of a main body in the heterostructure. The gapped surface dissipative channels, such as nonchiral quasi-helical​ edge states are located only at the top and bottom surfaces modes57–59, residual carriers from the bulk valence band59 of the heterostructure, as in uniformly doped samples. and impurity channels formed by defects and magnetic The robustness of the QAHE in the heterostructure 60 dopants , which hinder the dissipationless nature of the is exemplified by the possibility of tuning EF using a

chiral edge states. fairly wide range of values of Vg through the underlying (Fig. 4i) To improve the observation temperature of the AlOx dielectric layer . A higher-​temperature QAHE, a magnetic modulation doping technique was QAHE was also realized in a Cr and V co-doped​ BST thin developed. As discussed, the suppression of the inhomo­ film thanks to the improvement in the homogeneity of geneity in the distribution of the gap size is a key issue. the ferromagnetism62. The QAHE was observed in a modulation-doped The Cr-​doped, V-​doped, (Cr, V)-doped and

heterostructure consisting of five alternating layers modulation-​doped (Bi,Sb)2Te 3 systems have thus been

of (Bi1-ySby)2Te 3 (BST) and CBST for a total thickness of used to realize the QAHE. The common universality in 8 nm, in which the film thickness and the dopant con- the QAHE and QHE is verified in terms of the scaling 61 59,63–65 centration are precisely controlled . Two magnetic lay- behaviour in the σxy–σxx plot , the quantum Hall ers with a high Cr concentration of ~20% induce a large breakdown at large current density66,67 and the criti- exchange coupling to the surface states at the top and cality of the quantum Hall plateau transition68, even bottom BST layers (Fig. 4g). The QAHE is well developed though the microscopic origins of the quantization

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are quite different: the gap opens at EF because of the field from the magnetized tip. Thus, the position of the do­ exchange coupling with the spontaneous magnetization main wall can be continuously changed by moving the tip for the QAHE, whereas it is the result of the formation (Fig. 5b). This technique was applied to study transport of Landau levels originating from the cyclotron motion of in a modulation-​doped CBST film at 0.5 K in a Hall bar 2D charge carriers for the QHE. The QAHE in other configuration77 (Fig. 5c). Initially, the magnetization points

candidate materials, such as (Bi,Sb)2Te 3 or Bi2Se3 doped downwards, and a single domain exists; moreover, the 17 with other 3d transition metals and Mn-​doped HgTe Hall (R13 and R24) and longitudinal (R12 and R34) resistance (Ref.69), although identified theoretically, has not yet values are −h/e2 and 0, respectively. When the magnetic been experimentally realized. However, the observation domain wall is in between the contacts, the Hall resistance 2 2 of the QAHE in different materials is highly desirable on the left side, R13, is ~+h/e , whereas R24 is ~−h/e , owing to further study its unconventional characteristics and to to a magnetization reversal only on the left-hand​ side, as

pursue its observation at higher temperatures. expected. Regarding the longitudinal resistance, R12 takes 2 One possible application of the QAHE would be the a high resistance value of ~2h/e , whereas R34 remains at electrical resistance standard, in analogy to the QHE70. almost zero. These resistance values are nothing but the Because the QAHE can be realized at zero or very low evidence of the presence of CEMs along the domain wall. magnetic fields, it would be easier to use as a reference As the two CEMs co-​propagate along the domain wall, compared with the QHE, which requires high magnetic they intermix and equilibrate with each other, and the fields. Moreover, the quantization of the conductance at electrons are finally equally distributed downstream zero magnetic field, combined with a Josephson device, of point D, which lies in between contacts 3 and 4. 71 may produce a new type of quantum metrology . Consequently, R34 becomes zero because the potentials

Presently, the uncertainty of the resistance in QAH sys- downstream are equal. By contrast, R12 assumes a value tems (V-doped​ 72 and Cr-doped​ 67 BST) has been reported that is twice the quantized resistance, 2h/e2, because only to be ~0.2–1 part per million, as characterized with a half of the electrons injected from current contact 5 would cryogenic current comparator bridge technique. These be ejected to the other current contact, that is, 6 (Ref.78). values are still three or four orders of magnitude worse Finally, when the magnetization direction is completely than those for GaAs/AlGaAs (Ref.73) and graphene74. reversed by the tip, the Hall and longitudinal resistance Further improvement of the uncertainty is linked to the values recover the quantized values for an upward-​ realization of the QAHE at higher temperatures and to pointing domain with C = +1. The presence of CEMs at its robustness against large current injection, which can the domain wall was also confirmed by another experi- be achieved through the development of less-disordered​ ment, in which the Meissner effect in a superconductor QAHE samples and improved device configurations. was used to shield the external magnetic field and obtain the desired magnetic domain structure79. Although the Chiral edge conduction basic characteristics of CEMs are expected to be almost As discussed, the QAH state is topologically characterized identical between the sample edge and domain walls, by a Chern number that is C = +1 or C = −1 when the the high controllability of domain walls enables the con- magnetization points upwards or downwards, respec- struction of dissipationless and reconfigurable circuits. In tively. This feature enables the control and manipulation fact, proof-​of-concept CEM circuits were demonstrated, of the physical properties of the chiral edge conduction exploiting the unique chiral nature of CEMs77,79. channels, for example, the edge conduction at magnetic An even more exotic CEM can be realized by lev- domain walls and the formation of chiral Majorana eraging the superconducting proximity effect on the edge modes, as argued in the following. At the interface QAHE. It was proposed that the superconducting prox- between the magnetic TI and a vacuum, the Chern num- imity effect on a TI results in the formation of a topo- ber changes from C = ±1 to C = 0 such that one CEM logical superconductor, accompanied by the appearance appears at the sample edge. By contrast, when the mag- of Majorana fermions at vortex cores or at the sample netization changes from downward to upward across a edge6,7,80–82. Majorana particles, which were originally magnetic domain wall, the Chern number changes from introduced in the context of high-​energy physics, are C = −1 to C = +1, and two CEMs co-​propagate along fermions that are their own antiparticles83. The reali- the domain wall6,7,18,75 (Fig. 5a). The presence of CEMs at zation of Majorana fermions as composite particles in domain walls is experimentally supported by transport condensed matter should enable non-​Abelian braid- measurements performed in the multidomain state at the ing operations in topological quantum computations, magnetization reversal45,76. In one experiment, increased which would then be robust against local perturba- conductivity was observed in the multidomain state and tions84–86. One experimental approach to the realization was attributed to the formation of dissipationless con- of Majorana quasiparticles exploited a junction between duction at the domain wall45. In another experiment, a a superconductor, Nb, and a magnetic TI87 (Fig. 5d). discrete jump in the longitudinal and Hall resistance, as The topological superconductor is characterized by a large as ~0.1 h/e2, was observed, accompanied by the for- Bogoliubov–de Gennes (BdG) Chern number N, which mation of a large magnetic domain wall and chiral edge is defined in the BdG Hamiltonian with particle–hole conduction76. Afterwards, CEMs were observed on a symmetry82. When the proximitized superconducting single domain wall in an experiment that used the tip of gap Δ at the top surface of the TI, the surface adja- a magnetic force microscope to write domains77. In this cent to Nb, is larger than the magnetization gap |m|, technique, the magnetization direction is locally reversed a topological superconducting state is stabilized, and by scanning over the sample owing to the stray magnetic the BdG Chern number becomes N = ±1. By contrast,

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a b d

1

2 Au

QAH Nb c Point D e 3 4 3 4 3 4 Δ<|m| Δ>|m|

QAH TSC QAH QAH TSC QAH 5 6 5 6 5 6 (C=–1) (N=–2) (C=–1) (C=–1) (N=–1) (C=–1)

1 2 1 2 1 2

QAH: (C=–1) (C=1) (C=–1) (C=1) T=0.5 K TSC: (N=–1) (N=1) (N=–1) (N=1) )

2 1 R h/ e ( 13 1.0 20mK

0 esistance r

R /h)

24 2 all (e H –1 12 0.5 σ ) 2 3 h/ e (

2 0.0 R 12 –300 –200 –100 0 100 200 300 esistanc e r B (mT)

inal 1 itu d R34 on g

L 0 0 10 20 30 40 48 Domain wall position (μm)

Fig. 5 | chiral edge conductions. a | Schematic illustration of chiral edge conduction along a domain wall of a magnetic topological insulator in the quantum anomalous Hall (QAH) state. Two chiral edge modes appear at the boundary between the domain with magnetization pointing upwards and that with magnetization pointing downwards. b | Schematic drawing of the measurement procedure for studying the dependence of the transport properties on the domain wall position using a magnetic force microscope (MFM). The domain wall position is continuously moved from

left to right, following the scan over the sample with the MFM tip. c | Variation of the Hall resistances, R13 and R24, and

longitudinal resistances, R12 and R34, measured during continuous MFM tip scanning from left to right on the Hall bar. The schematics in the upper panel illustrate the domain configurations of the Hall bar at the three relevant regions. Point D represents the downstream point on the domain wall. d | Schematic illustration of the measurement procedure for detecting chiral Majorana edge modes at the boundary between a QAH state and a superconductor, Nb, in a QAH–

topological superconductor (TSC) device. e | The dependence on the magnetic field (B) of the longitudinal conductivity σ12. A half-quantized​ conductance plateau is observed at the magnetization reversal point. The schematics in the upper panel illustrate the edge transport configurations for C = −1, N = −2 and C = −1, N = −1. Δ, superconducting gap; C, topological Chern number ; |m|, magnetization gap; N, Bogoliubov–de Gennes Chern number ; T, temperature. Panels a, b and c are reproduced with permission from Ref.77, AAAS. Panels d and e are adapted with permission from Ref.87, AAAS.

when Δ < |m|, a QAH state is realized, and the BdG numbers. Note that one CEM in the QAH state can Chern number is related to the Chern number, with the be interpreted as two co-​propagating chiral Majorana relation N = 2C = ±2. In this measurement, the chiral edge modes; in other words, one chiral Majorana edge Majorana edge mode, which can be interpreted as a 1D mode corresponds to half a CEM. The magnetic field

propagating Majorana fermion, is expected to appear at dependence of the two-​terminal conductivity σ12 of the the interface between phases with different BdG Chern sample is shown in Fig. 5e. The mass gap m is expected

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to continuously change from negative to positive as a electrodynamics in quantum chromodynamics95. result of the magnetization reversal64,88,89. As a result, Under time-​reversal symmetry, θ is equal to π (mod 2π) a sequential topological phase transition from a QAH within the TI and zero in a vacuum or in an ordinary state to a topological superconducting state is induced insulator. When time-reversal​ symmetry is broken under by the magnetic field. Initially, when the magnetization the presence of a magnetic field or magnetization, the points downwards, Δ is smaller than |m|; hence, C = −1 opening of the gap in the surface state allows a change

and N = −2, and the two-​terminal conductance σ12 is in θ, producing the exotic electromagnetic responses quantized to e2/h. When |m| becomes smaller than Δ at described below. the magnetization reversal point as the magnetic field is The conjunction of the axion term with the Maxwell swept, a single chiral Majorana edge mode is formed at term leads to the extended Maxwell equation, and the the boundary between the QAH state (C = −1, N = −2) following cross relation between the electric polarization and the topological superconductor (N = −1; Fig. 5e). P and magnetic field B and the magnetization M and In this situation, one of the two chiral Majorana edge electric field E can be derived96 modes is reflected back to the current terminal, while the other propagates along the edge of the sample. The 2 e θ ε0 θ splitting leads to a half quantization of the conductance, PB= = α B (6) 2 87 e /2h, which is observed as a half-​quantized plateau . 2h π μ0 π Although the interpretation of the half-​quantized pla- 90–92 teau appearing in σ12 is still controversial , the solid confirmation of chiral Majorana edge modes would 2 e θ ε0 θ enable non-​Abelian braiding and topological quantum ME= = α E (7) 93 computation in the future . 2h π μ0 π

Topological magnetoelectric effects This represents the magnetoelectric effect97. This kind of Aside from their topologically protected surface state, axion-type​ coupling, represented by the E ⋅ B term in the TIs also exhibit bulk magnetoelectricity94. Indeed, the Lagrangian, and the resultant diagonal magnetoelectric topological band inversion brings about an important susceptibility are observed broadly in magnetoelectric (Ref.98) modification of Maxwell’s equation. In TIs, the applica- materials, such as Cr2O3 and (Fe,Zn)2Mo3O8 tion of a magnetic field induces an electric polarization, (Ref.99). The uniqueness of TIs lies in the fact that the whereas an electric field induces a magnetization. This magnetoelectric susceptibility is quantized in units of is the magnetoelectric effect, which is characterized by the fine structure constant α.

a coefficient quantized to the fine structure constant. The axion term LTI causes an essential modification The experimental observation of the quantized magneto­ of Maxwell’s equations and a peculiar electromagnetic electric response is especially important from a funda- response called axion electrodynamics. The electro- mental point of view, because measuring the quantized magnetic response of a TI with a surface or interface in magnetoelectric response corresponds to a direct meas- contact with a vacuum or with a trivial material (θ = 0) is

urement of the Z2 invariant through a response func- determined by the boundary conditions. To be realistic, tion. Developments in the heterostructure engineering let us consider a magnetic TI thin film and its top and bot-

of magnetic TIs and the establishment of terahertz meas- tom surfaces. When EF lies within the exchange gap, the urement techniques enabled the direct observation of surface non-dissipative​ current is expressed as the sum of this effect. the polarization current (∂tP) and magnetization current 2 To describe the electromagnetic response in a TI, a (∇×M); jE= (eh∕⋅2 ) ∇∕(θπ) × . The rotation direction Lagrangian94 was derived that includes, in addition to the (clockwise or anticlockwise) of θ when going from the

conventional Maxwell term, the so-called​ axion term LTI, vacuum to the TI or vice versa depends on the sign of the mass gap or, equivalently, on the magnetization of the 2 top or bottom layer. Because the magnetizations of the top θe ε0 θ LTI = EB⋅⋅= α EB (4) and bottom surfaces are parallel, ∇θ can be additive (2π); 2πh μ π 0 hence, θ = 0→π (through the top surface) and θ = −π→0 (through the bottom surface). This results in the QAHE observed in a thin film with parallel magnetizations on the top and bottom surfaces and gives rise to the quan- 1 3  2  θ =− dkεaijkiTr  ∂−jkaiaaaijk (5) tized Faraday or Kerr rotation of incident electromagnetic 4π ∫  3  100–104 BZ waves , which corresponds to the measurement of the 105 2 magnetoelectric susceptibility of the TI . where 1 e is the fine structure cons­tant and A schematic representation of the magneto-​optical α(= 2ε hc ≈1∕137) ε0 and μ0 are0 the permittivity and permeability of free effects in the QAH state is shown in Fig. 6a. Magneto-​ μν ∂ space, respectively. ai (=iuμkkuν ) is the Berry optical effects include the Faraday rotation, the rotation ∂ki connection, and uνk is the lattice-​periodic part of the of the linear polarization of the transmitted light, and Bloch function for the occupied band v; the trace is the Kerr rotation, the rotation of the polarization of the taken over the occupied bands, and the integration reflected light. To probe quantized magneto-optical is carried out within the Brillouin zone. Interestingly, effects, the photon energy of the incident light has to LTI takes the same form as the term describing axion be tuned below the exchange gap of the surface state

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a b 0.5 (i)

Faraday rotation angle ) (ii) a.u. ( x

E 0 E(t): AC electric T=1.5K field of light 0.5 (i) (i) (ii) ) θ +θ θF F K a.u. 300 ( × y

(ii) E 0 AIO Propagation x direction of light TI film Substrate (InP) 0510 15 20 Time (ps) d Top/bottom e 1.0 surface surface 500mK 0.5 c Right-side V–V Cr–Cr /h) surface 2

(e 0.0

8 xy σ –0.5 f=α –1.0 6

) 1.0 –3 QAH insulator 0 60mK ) (1

K V–Cr ) θ

, 4 0.5 /h F 2 θ (

(e f σxx 0.0 2 –0.5 Conductance σxy 0 Axion insulator –1.0 0 0.2 0.4 0.6 0.8 1.0 –1.0 –0.5 0.00.5 1.0 d.c. (e2/h) σxy B (T) Fig. 6 | Topological magneto-​optical effect and magnetoelectric axion insulator. a | Schematic of a Faraday rotation measurement for a quantum anomalous Hall (QAH) state. The multiple reflection of terahertz light within the substrate (insulating InP) on which the magnetic topological insulator (TI) film is grown is shown in the right panel; the beam can either go directly through the sample (i) or be reflected from the substrate–vacuum and TI–substrate interfaces (ii). b | Temporal profiles of the terahertz light pulses. The upper panel shows the x component of the electric field E, which coincides with the polarization of the original incident light; the lower panel shows the y component of E, generated as a

result of the polarization rotation. Hence, the (i) and (ii) pulses in Ey contain the Faraday rotation (θF) and the | cotcθθFK− ot Faraday rotation plus Kerr rotation (θF + θK), respectively. c The values of the scaling function f()θθFK, = 2 cot2θθFF−−cotcotθK 1 obtained from the experimentally observed θF and θK values are plotted against the direct current values d.c. of the Hall conductivity , σxy . The scaling function value is expected to reach the fine structure constant α when quantization is realized in the QAH state. With decreasing temperature (T), σxy increases, approaching the quantized value of e2/h, and f approaches α. d | The upper schematic shows a QAH state with parallel magnetization directions on the top (V-​doped) and bottom (Cr-doped)​ magnetic layers; the lower schematic shows an axion insulator state with antiparallel

magnetization on those layers. e | The top panel shows the magnetic field dependence of σxy at 500 mK for symmetric

modulation-​doped heterostructures with V (pink) and Cr (green). The lower panel shows σxy and σxx at 60 mK for V-doped​ and Cr-​doped heterostructures. The insets show the magnetization configuration at each magnetic field (B). The antiparallel magnetization configuration realizes an axion insulator state. Panels a (right), b and c are adapted from Ref.102, CC-​BY-4.0. Panels d and e are adapted with permission from Ref.109, AAAS.

of the magnetic TI; otherwise, the surface state or bulk respect to the pulse going directly through the sample.

interband transitions would cause dissipation owing Results for a Cr-​modulation-doped (Bi,Sb)2Te 3 film to the real excitation of particles and holes. Thus, tera­ near the QAH state at 1.5 K are shown in Fig. 6b,c (Ref.102). hertz light, with a photon energy around a few meV, The comparison of the time profile of the transmitted

can be a good probe, and time-​domain spectroscopy light beam for the Ex and Ey polarizations with respect

on thin films grown on thick substrates is powerful for to the purely Ex-​polarized incident beam gives the rota- (Fig. 6b) deducing both the Faraday rotation (θF) and Kerr rota- tion angles (θF and θK) as the ratio of Ey /Ex . The

tion (θK) simultaneously by using the time delay of the pulse profiles (i) and (ii) represent the directly transmit- light pulse multiply reflected within the substrate with ted beam and the multiply reflected beam, respectively;

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106,107 therefore, the Ey profile contains contributions from (i) expected and could be measured using the non-​

of θF and from (ii) of θF + θK. Rotations on the order of reciprocal polarization rotation — termed gyrotropic several milliradian are readily visible despite the mag- birefringence — of terahertz light, as was demonstrated netic TI films being as thin as 8 nm, which indicates for multiferroics99. the very large magnitude of the magneto-​optical effect in the surface states. The transmission spectra in the Spintronic functionalities terahertz region, as obtained from the Fourier transfor- In addition to giving rise to QAH-​related phenomena, mation of the time-​domain profile, show no apprecia- the interaction between spontaneous magnetic moments ble electronic absorption, as expected from the presence and spin–momentum-locked​ conduction electrons in the of the large exchange gap in the QAH state. By using surface state leads to versatile spintronic functionalities.

only the θF and θK values, we can establish the relation Magnetic control of the electronic state or electric con-

f(θF, θK) = α in the QAH state. Here, f is a scaling func- trol of the magnetization is enabled via charge-​to-spin 111 tion of θF and θK that depends only on the experimental conversion by the Rashba–Edelstein effect . The configuration but contains no material parameters, such application of an in-​plane electric field shifts the Fermi as the thicknesses and dielectric constants (refractive surface in momentum space and causes the accumula- indices) of the magnetic TI film and substrate101,102; tion of net spins in the direction perpendicular to that for the present case, of the current because of the spin–momentum locking at the surface state of the TI. Studies on spin-torque​ ferro­

cotθFK−cotθ magnetic resonance and spin pumping highlighted the fθ( FK, θ ) = 2 ⋅ high efficiency of interfacial charge-​to-spin conversion cot−θcFF2−otθcotθK 1 3D 2D 3D qICS = JS /JC (with JS being the spin current density –2 2D (A m ) and JC being the surface charge current density –1 112 The variations of the experimentally obtained f(θF, θK) (A m )) , which corresponds to a spin Hall angle θCS values at various temperatures, plotted against the cor- of order unity, assuming an ~1-nm-​thick surface state 113–121 responding direct current values of Hall conductivity σxy, layer . The spin Hall angle is much larger for TIs than Fig. 6c are shown in . As σxy approaches the quantized for heavy metallic elements such as Pt, β-​Ta and β-​W, value (e2/h) with decreasing temperature (lowest at 1.5 K), which makes TIs an attractive arena to explore unprec- 123 f(θF, θK) approaches the value of α, thus confirming the edented spintronic functions . Furthermore, the Dirac near quantization of the topological magneto-​optical dispersion of the surface state provides an additional knob effect near the QAH state. to regulate the charge-​to-spin conversion efficiency by The topological magneto-​optical effect is one mani- controlling the Fermi level through composition tuning festation of axion electrodynamics. However, in the case and/or gate voltage112,124,125. of a QAH insulator, the θ term has opposite signs on One representative example of such spintronic func- the top and bottom surfaces. Hence, macroscopically, the tionalities is the nonlinear unidirectional magneto­ topological magnetoelectric effect is cancelled out. To resistance126–130. When the magnetization of the magnetic produce a topological (near-quantized)​ magnetoelectric TI points along the in-​plane direction (y direction), effect or an axion insulator, it is necessary to realize an perpendicular to the in-​plane current (x direction), the antiparallel magnetization (M) or an opposite sign of value of the resistance depends on the magnitude and the exchange gaps on the top and bottom surfaces106,107. direction (positive or negative) of the applied current. This is possible using the magnetic modulation doping This is phenomenologically understood in terms of technique to grow the TI film108–110 (Fig. 6d). The QAH the spin accumulation through the Rashba–Edelstein

effect observed for V-doped​ and Cr-doped​ (Bi,Sb)2Te 3 TI effect: the value of the resistance depends on the direc- Fig. 6e films is shown in (top2 panel); both samples show tion in which spin moments accumulate, either parallel the quantization of e , but they display different or antiparallel to the magnetization direction (Fig. 7a), σxy =± h coercive forces (Bc), as seen in the different hysteresis as is the case for the current-​in-plane giant magneto­ curves16,56. When a modulation-​doped film composed resistance in ferromagnetic/non-magnetic/ferromagnetic 131,132 of a large-​Bc V layer on top and a small-​Bc Cr layer on heterostructures . However, differently from giant the bottom is used, a state with M pointing up on the top magnetoresistance, unidirectional magnetoresistance surface and down on the bottom surface (or vice versa) is is a current-​nonlinear effect, because the magnitude realized within the wide hysteresis region109 (Fig. 6e, bot- of the spin accumulation is proportional to the current. tom panel). Remarkably, such a state is characterized by The in-​plane magnetic field dependence of the longi- nl a zero Hall plateau with zero conductance, which char- tudinal resistance Rxx measured under a DC current acterizes the axion insulator state, but is distinct from the of +2 μA and −2 μA for a heterostructure composed of

trivial insulator with θ = 0. In particular, the topological a magnetic TI/non-​magnetic TI, Crx(Bi,Sb)2-xTe 3/ Fig. 7a transition between a QAH insulator and an axion insu- (Bi,Sb)2Te 3, is shown in . In this heterostructure, lator is marked by the on–off switching of the chiral edge only the top surface interacts with magnetic moments;

channel. The two-terminal​ resistance R2T of the thin-film​ hence, cancellation due to opposite contributions from sample under discussion shows magnetic field switch- the top and bottom surfaces is avoided133. The magneti- ing between h/e2 ≈ 25.8 kΩ (irrespective of the sample zation points in the out-​of-plane direction at 0 T owing 8 109 length) and a high resistance value (R2T > 10 Ω) . to the magnetic anisotropy, but it points in the in-plane​ In this state, the topological magnetoelectric effect, direction at 1 T, causing a resistance drop resulting from ε0 ε0 manifested as Pz ≈ αBz and Mz ≈ αEz, can be the closing of the gap of the surface state. A difference μ0 μ0

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z a 15 b Bulk interband transition 2K M x y B Surface Dirac electron +J ) 20K Ω

) –5T 1 k 14 50 ( – l σ J n

x A x

M R +2μA M

M ent (n C CBST r 0 0T 13 x y BST –2μA InP hotocu r

InP(111) z CBST P –50 0.1 +5T

–J ) 0 200 400 600 800 1,000 Ω k ( Photon energy (meV) l

n 0.0 x x R ∆

–0.1 –2 –1 0 1 2 B (T) c j (1010 A m–2) d –5 pulse 0 5 Magnetic moment 1.2 B=0.1T M 2 Room temperature Jpulse Spin B BSOT B=0.02T accumulation B ∝σ ×M 0.6 SOT 1 2K CoTb ) Ω ) ( 0.0 H σ Ω k R ( 0 x y

R –0.6 –1 CBST Bi Se –1.2 BST 2 3 Charge –2 current –1 0 1 –4 –2 0 2 4 10 –2 Jpulse (mA) Je (10 A m )

Fig. 7 | spintronic functionalities. a | Schematic setup for the measurement of the nonlinear unidirectional magnetoresistance. +J and −J indicate the current direction; the red and blue arrows labelled σ and M represent the spin accumulation direction and magnetization direction, respectively. The black arrows represent the direction of movement nl of the surface electrons. The top graph shows the resistance Rxx of the magnetic topological insulator (TI)/non-magnetic​ TI heterostructure measured with positive (+2 μA) and negative (−2 μA) DC current under the in-plane​ magnetic field (B). nl The lower graph shows the difference between the two resistance curves, ΔRxx . b | Schematic experimental setup for measuring the photocurrent and zero-​bias photocurrent spectra under an in-plane​ magnetic field of 5 T, 0 T and −5 T. The red arrows represent the electron spin. c | Schematic illustration of current-​induced magnetization switching in a

magnetic TI/non-magnetic​ TI heterostructure. Jpulse is the current pulse, and BSOT is the effective field generated by the

spin–orbit torque. The dependence of Hall resistance Ryx on the current pulse amplitude Jpulse at T = 2 K under a small in-plane​ magnetic field of B = 0.02 T, which is required to determine the switching direction, is shown in the graph on the

right. The corresponding current pulse density jpulse is shown along the upper axis. d | Schematic illustration of the measurement procedure for the room temperature current-induced​ magnetization switching in a TI/ferromagnetic

heterostructure (Bi2Se3/CoTb). The dependence of the Hall resistance RH on the estimated current density Je inside the

Bi2Se3 layer is measured under a small in-plane​ magnetic field of 0.1 T. BST, (Bi,Sb)2-xTe 3; CBST, Crx(Bi,Sb)2-xTe 3. Panel a is adapted with permission from Ref.129, APS. Panel b is reproduced from Ref.134, CC-BY-4.0.​ Panel c is adapted with permission from Ref.148, APS. Panel d is adapted with permission from Ref.151, APS.

nl in resistance for positive and negative currents, ΔRxx , is surface state electrons, which would be forbidden in the clearly discerned (Fig. 7a), and it changes sign with the absence of magnetization. Because of the conservation magnetization reversal. Thus, this effect can be exploited of angular momentum in the spin–momentum-locked​ to perform an electrical reading of the magnetization state, electrons with momentum +k and −k are scattered direction using a device with a two-terminal​ geometry, through processes of magnon creation and annihilation, similar to giant magnetoresistance126–128. The value of respectively. This situation leads to a difference in the the unidirectional magnetoresistance in magnetic TIs lifetimes of right-moving​ and left-moving​ electrons and is 102–106 times larger than that for normal-​metal/fer- thus to the difference in the resistance for positive romagnetic-​metal heterostructures such as Pt/Co126–129. and negative currents: the electron–magnon scattering is Furthermore, gate tuning of the Fermi level around asymmetric129. The same mechanism likely results in the (Ref.129) the Dirac point resonantly enhances ΔRxx . The overestimation of the spin Hall angle in magnetic TIs in microscopic process underlying such a large unidirec- a second-​harmonic Hall measurement121,124,148. tional magnetoresistance in magnetic TIs is ascribable A related phenomenon is observed by optical excita- to the magnon-​mediated backward scattering of the tion. Under the in-​plane magnetization, mid-​infrared

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photoexcitation induces a zero-​bias photocurrent that asymmetric electron–magnon scattering, even without flows in plane, perpendicular to the magnetization magnetization switching121,124,148. direction134. The photocurrent direction depends on the For applications, it is desirable that these function- magnetization direction (positive or negative; Fig. 7b). alities are implemented at room temperature. However, The photocurrent spectra show a maximum at around at the moment, the Curie temperature of ferromag- 250 meV, below the bulk bandgap of the TI (~300 meV, netic TIs is a few tens of Kelvins for a moderate doping Fig. 7b), meaning that it is the surface Dirac electrons that level and limited to 250 K at the highest doping level149. mainly contribute to the observed photocurrent. The ori­ A platform that could overcome this limitation may be a gin of the photocurrent is attributed to the magnetization- TI/ferromagnetic heterostructure. In fact, nonlinear uni- induced modification of the energy dispersion resulting directional magnetoresistance150 and current-​induced from spin–momentum locking plus the parabolic k2-term. magnetization switching151–154 have been demonstrated Because of the asymmetric band dispersion for +k and for this type of composite structure or heterostructure. −k, both the excitation and relaxation of photoexcited The current-​induced magnetization switching in a electrons are asymmetric, which yields the zero-bias TI/ferromagnetic heterostructure at room tempera- photo­current. A shift current process or the presence of a ture, which is just like the switching in a magnetic TI, is photo-induced Floquet state may also contribute to the shown in Fig. 7d. The current required for magnetization photocurrent, as was recently found in polar semicon- switching is ~3 × 1010 A m–2, showing the potential of TIs ductors135,136. A similar phenomenon is observed in non- as practical spintronic materials151–154. magnetic TIs using obliquely incident circularly polari­ zed light instead of the magnetization: this is the circular Conclusions and perspectives photo­galvanic effect, which points to the possibility of In the past decade, pioneering theoretical studies have achieving the optical generation of spin currents in TIs137,138. advanced the field of topological quantum science and The presence of the topological surface state sup- motivated experimental studies of exotic phenomena in ports the formation of nontrivial magnetic structures topological materials. In this Review, we have discussed in ferromagnetic TIs and related heterostructures. The the emergent quantum phenomena in magnetic TIs Dzyaloshinskii–Moriya interaction or the asymmetric beyond the experimental confirmation of their topolog- exchange interaction between the localized magnetic ical electronic structures, focusing in particular on their moments is mediated by the surface state of the TI. electrical transport and optical properties. Magnetic TIs Similarly to the interface of non-​magnetic/ferromag- realize the massive Dirac gap and the materialization of netic heterostructures139, the Dzyaloshinskii–Moriya the QAHE and axion insulator state; moreover, they dis- interaction favours a noncolinear spin ordering and the play spintronic functionalities. From a materials science formation of magnetic skyrmions, which are detected perspective, the key issue regarding the massive Dirac through the topological Hall effect122,140,141. gap with broken time-reversal​ symmetry is the magnetic One other interesting spintronic phenomenon is the doping of the TI to induce the long-range​ ferromagnetic 142–145 current-​induced switching of the magnetization . order. In Cr-doped​ or V-doped​ (Bi,Sb)2Te 3, the interac- Efficient current-​induced magnetization switching is tion between surface state electrons and local magnetic highly desirable for the application of spintronic devices moments opens an exchange gap in the Dirac surface such as magnetic random access memories. As men- state. Heterostructure engineering, including the modu- tioned earlier, current injection in a TI induces a spin lation doping of magnetic ions, raises the temperature at accumulation at the surface by the Rashba–Edelstein which the QAHE can be observed, providing a concrete effect. The angular momentum of the accumulated platform for the exploration of exotic quantum effects spin is transferred to the local magnetization, m, which such as the topological magnetoelectric effect and the causes the damping-like​ spin–orbit torque (Slonczewski formation of axion insulators. The functionalization

torque) τSOT = m × (σ × m), where σ is the spin accu- of non-​dissipative charge transport will be material- mulation direction146,147. The torque leads to the estab- ized with the CEMs at magnetic domain walls and, in

lishment of an effective field Βeff = σ × m, leading to the superconducting heterostructures, chiral Majorana edge magnetization switching. The current-​induced magne­ modes. Moreover, the interaction between the surface tization switching for a magnetic TI/non-​magnetic TI state electrons and the magnetization induces rich spin- heterostructure under a small in-​plane magnetic field is tronic functions, such as current-induced​ magnetization shown in Fig. 7c (Ref.148). As current pulses of ~4 × 1010 A m–2 switching. Here, we discuss the future perspectives for are applied, the AHE signal changes from positive to research on magnetic TIs (Fig. 8). negative, which corresponds to a magnetization switch- ing from up to down. The switching current density High-​temperature realization of the QAHE. The real- is approximately one order of magnitude smaller than ization of a high-​temperature QAHE is necessary for that needed in ferromagnetic/non-​magnetic hetero- expanding the available experimental techniques to structures, owing to the large spin Hall angle of TIs. obtain further physical insights into QAH-​related phe- The measurement of the Hall effect using a DC cur- nomena, as well as for the application of dissipationless rent was reported to lead to an even higher efficiency channels based on the QAHE. Despite the improvements of current-​induced magnetization switching in a mag- in thin-film​ growth techniques, such as alloying and het- netic TI/non-​magnetic TI heterostructure121. However, erostructure engineering, the temperature at which the care should be taken, because the application of a large QAHE can be observed is still limited to 2 K (Refs61,62). DC current causes a spurious Hall effect owing to the This temperature is far below the Curie temperature

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Majorana braiding Edge channel electronics High temperature QAHE

QAHI Manipulation of CEMs Magnetic TSC s-wave SC TSC TI MI proximity effect QAHI Current Voltage gate

Magnetic TI

Exotic quantum phases Fractional Chern High Chern number Topological Weyl semimetal insulator QAHE magnetoelectric effect

E M MTI M=αE

Fig. 8 | Future perspectives for magnetic topological insulators. Heterostructure engineering and the development of new measurement techniques will further broaden the field of magnetic topological insulators (TIs). Heterostructures based on magnetic insulators (MIs) and TIs may realize the quantum anomalous Hall effect (QAHE) at high temperatures through magnetic proximity effects. The stacking of MIs and TIs, or of trivial insulators and magnetic TIs, will realize Weyl semimetals through fine tuning of certain parameters, such as the film thickness. The axion term in the Lagrangian of TIs will establish cross relations between magnetism (M) and electricity (E), yielding quantized topological magnetoelectric effects. A QAHE characterized by high Chern numbers with multiple chiral edge modes (CEMs) will be enabled through appropriate material choices. Nearly flat bands with nontrivial topology will realize a fractional Chern insulator. The proximity to a superconductor, coupled with an elaborate device design, will realize non-Abelian​ braiding, an important step towards topological quantum computation. Finally , the manipulation of magnetic domain walls by electrical or optical means will enable fast control of CEMs. α, fine structure constant; QAHI, quantum anomalous Hall insulator ; SC, superconductor ; TSC, topological superconductor. Upper-left​ panel reproduced with permission from Ref.184, PNAS. Lower-​left image reproduced from Ref.93, Springer Nature Limited.

161 (a few tens of Kelvins) and the magnetization gap energy and Tm 3Fe5O12 (TIG) through measurements that of the surface state (~500 K). The most plausible reason used neutron scattering157, magnetic second-​harmonic for the suppression of the observation temperature is generation156, hysteretic magnetoresistance159 and the disorder in the exchange gap, which is inevitably intro- AHE155,158,160,161. However, the observed anomalous duced by the randomly distributed magnetic dopants50. Hall resistivity was far from the quantized value. This An alternative route towards the higher-​temperature is probably a result of the small magnetization gap realization of the QAHE is to leverage the magnetic resulting from the weak exchange coupling between proximity effect to ferromagnetic insulators with a the magnetic insulator and the TI. A suitable choice of the high Curie temperature94. The appropriate selection magnetic insulator yielding strong hybridization with of the ferromagnetic material and the synthesis of an the p orbitals of Bi, Sb and Te and the optimization of the abrupt interface activating the proximity effect are heterostructure growth will open new routes towards essential ingredients. The incorporation of the mag- the high-​temperature realization of the QAHE162. netic proximity effect into a TI has been exemplified Another possibility for the realization of a higher-​ in heterostructures with magnetic insulators such as temperature QAHE is to use an intrinsic magnetic (Refs155–157) (Ref.158) 159,160 EuS , Cr2Ge2Te 6 , Y3Fe5O12 (YIG) TI. One candidate is MnBi2Te 4, which is theoretically

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proposed as a layered antiferromagnet with perpen- chiral Majorana edge modes can be read out simply with dicular magnetic anisotropy163,164. The intrinsic mag- conductance measurements. From a materials science netization is expected to create a uniform and large perspective, the in situ fabrication of heterostructures by magnetization gap and hence to raise the observable molecular beam epitaxy171 and the strong hybridization temperature of the QAHE. In fact, the gap opening is of the superconducting state with the TI surface state172 observed by ARPES165, and a finite AHE appears in may realize a large superconducting proximity effect, transport measurements166. More interestingly, because similar to the magnetic proximity effect.

of its layered antiferromagnetic nature, MnBi2Te 4 is expected to be in a state that changes between the QAH Materials development towards exotic quantum state and an axion insulator state depending on the phases. More exotic quantum states may be imple- parity of the number of layers164. mented with heterostructure engineering and appropri- ate materials combinations. For example, it was proposed External control of the QAH state. One of the most that a magnetic Weyl semimetal would be realized in attractive features of the QAH state is the topologi- a superlattice composed of a magnetic TI and a trivial cal phase transition, which allows changing the Chern insulator through the tuning of the interlayer and intra- number in a non-​volatile way. Thanks to this feature, the layer tunnelling173,174. With a single pair of Weyl points magnetization reversal alters the direction of CEMs, in the Brillouin zone, the topologically protected Fermi the switching between a QAH state and an axion insu- arc, the nonlocal transport and the chiral anomaly are lator state controls their presence or absence108–110, and expected to be observed175. The in situ growth of such the domain wall motion changes their position77,79. To a heterostructure was realized by using CdSe as a trivial implement CEMs in devices as dissipationless channels, insulator176. However, because the trivial insulator was apart from the working-​temperature problem, an issue too thick, the superlattice became a simple stacking of that needs to be solved is the control of the magnetization QAH states176. The fine tuning of the thickness of the configuration in a local, rapid and reconfigurable manner. trivial insulator will produce a magnetic Weyl semimetal. One possibility is to use the spin–orbit torque magneti- QAH states with intrinsically high Chern numbers are zation switching, exploiting the spin accumulation at the proposed to emerge in a magnetic topological crystalline surface state148. Another option is to use magneto-optical​ insulator (TCI). TCIs possess four Dirac cones at the sur- recording, namely, optical writing of the magnetization face, protected by band inversion and mirror symmetry. through the modulation of the coercivity by heating167. The gapping of the four Dirac cones by magne­tization The effective control of the magnetic configuration will would lead to a QAH insulator with C = ±4 (Ref.177). The enable device functionalities based on the domain walls ferromagnetic order and the AHE are observed in mag- and on magnetoelectric effects, such as reconfigurable netically doped TCIs, such as Cr-​doped SnTe thin film, dissipationless circuits based on CEMs and reconfigurable and in magnetic-proximity-coupled​ SnTe (Refs178,179). 168 microwave circuits based on chiral edge plasmons . The fine tuning of EF by doping and the improvement of film quality are necessary for the realization of QAH Confirmation and control of Majorana modes. states. Beyond the single band model, the effect of correla- Although evidence for chiral Majorana edge modes tions on the topological order will constitute an intriguing has been experimentally reported87, the manipulation issue. The introduction of correlations in magnetic TIs or of Majorana fermions is still in its infancy. The obser- QAH states will realize fractional quantization without vation of the quantum coherence of chiral Majorana the need for an external magnetic field, in analogy to the fermions is the next step towards the realization of top- fractional QHE under a magnetic field. The choice of suit- ological quantum computation. Some theoretical works able materials will be crucial for the realization of nearly propose interferometry between two chiral Majorana flat bands with nontrivial topology180–184. edge modes, in which the conductance would be tuned All in all, magnetic TIs exhibit rich physics and by controlling the phase difference across the junc- emergent phenomena and functions. The integration tion through the parity of the number of vortices169,170. of strong electronic correlations and superconductiv- Another proposal is the realization of a Hadamard gate ity in topological quantum materials will open up new with a Corbino ring junction and a phase gate with phenomena and research avenues in the near future. the application of a gate voltage93. These proposals are Published online xx xx xxxx experimentally feasible because the quantum states of

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182. Neupert, T., Santos, L., Chamon, C. & Acknowledgements Author contributions Mudry, C. Fractional quantum Hall states at zero The authors thank R. Yoshimi, M. Mogi, M. Kawamura, All authors have read, discussed and contributed to the writing magnetic field. Phys. Rev. Lett. 106, 236804 N. Nagaosa, M. Kawasaki and T. Dietl for enlightening discus- of the manuscript. (2011). sions on magnetic topological insulators. This research was 183. Klinovaja, J., Tserkovnyak, Y. & Loss, D. Integer supported in part by the Japan Society for the Promotion of Competing interests statement and fractional quantum anomalous Hall effect in a Science (JSPS; no. 16J03476), the Ministry of Education, The authors declare no competing interests. strip of stripes model. Phys. Rev. B 91, 085426 Culture, Sports, Science and Technology (MEXT), Japan (2015). (no. JP15H05853), and Core Research for Evolutional Publisher’s note 184. Maciejko, J. & Fiete, G. A. Fractionalized topological Science and Technology (CREST), Japanese Science and Springer Nature remains neutral with regard to jurisdictional insulators. Nat. Phys. 11, 385–388 (2015). Technology (JST; no. JPMJCR16F1). claims in published maps and institutional affiliations.

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