Progress and prospects in the quantum anomalous Hall effect

The quantum anomalous Hall (QAH) effect represents one of the triumphs in conceptualizing topological aspects of electronic states in condensed matter physics.^1–11 It constitutes an ever-evolving family of Hall effects.¹² In 1879, Edwin Herbert Hall discovered that the Lorentz force leads to a transverse voltage when the longitudinal current in a conductor is subjected to a perpendicular external magnetic field¹³—an effect that bears his name and inspires researchers to push the scientific and technological frontiers.¹⁴ This ordinary Hall (OH) effect offers a direct tool in assessing the charge carrier type and density in semiconductors, as well as acts as a practical probe in measuring the magnetic field. Hall later reported an unusual and stronger response in ferromagnets with qualitatively different field dependence.¹⁵ It hence came to be known as the anomalous Hall (AH) effect, correlated with the spontaneous magnetization M. Its deep roots in topology and geometry were appreciated in recent times,¹⁶ enabled by the discoveries of integer¹⁷ and fractional^18,19 quantum Hall (QH) effects in the 1980s.

For a two-dimensional electron gas (2DEG) under strong applied magnetic field, the Hall resistance R_yx develops well-defined plateaus, quantized to exact value of h/νe² at which the longitudinal resistance R_xx vanishes. Here, h is Planck’s constant, e is the elementary charge, and ν is the filling factor that is topological in nature and corresponds to the Chern numbers (by integrating the Berry curvature²⁰ over the first Brillouin zone) summed over the occupied bands. In the integer QH regime, the quasi-1D chiral edge channels, each contributing a quantized Hall conductance G_xy = e²/h, are immune to back scattering. For practical utilization of such dissipationless edge transport, e.g., in resistance metrology and low-energy-cost electronics, it is natural to desire a QAH system displaying a quantized version of the AH effect. It allows QH states to prevail at zero magnetic field, circumventing the necessity of forming Landau levels as well as the often stringent requirement on sample mobility.^21–23 

It became clear that, similar to the QH physics, the intrinsic AH conductivity σ_AH in a magnetic material is governed by the integral of the Berry curvature over occupied bands.^24–29 Effort toward realizing the QAH effect was nonetheless at an impasse for decades, as σ_AH is not quantized in metals with partially occupied bands and magnetic insulators with a non-zero Chern number are rare to come by. Since around 2005, however, new prospects emerged accompanying the discovery of topological insulators (TIs) with strong spin–orbit coupling (SOC) under the protection of time-reversal symmetry (TRS).^30,31 As shown in Fig. 1(a), a 2D TI characterized by a single Z₂ topological invariant hosts a pair of spin-polarized helical edge states (can be roughly understood as two copies of the chiral QH states with opposite spin³²) that leads to the quantum spin Hall (QSH) effect with quantized six-terminal R_xx of h/2e² and transverse spin-accumulation.^33–39 

Upon generalizing to 3D,⁴⁰ as exemplified by the Bi₂Te₃ family of materials,^41–44 TI features insulating bulk and gapless helical surface states (SSs) with Dirac-like linear dispersion and spin-momentum locking, see Fig. 1(b). Introducing magnetic order to break the TRS in 2D TI, will intuitively lead to a QAH state—when one spin block is driven out of the topologically nontrivial band inverted regime into the normal one with vanishing G_xy.⁴⁵ Despite early progress, this route was not pursued further, largely owing to the finite magnetic field needed to induce quantization as a result of the paramagnetic nature of Mn doping in HgTe.⁴⁶ As shown in Fig. 1(c), an alternative platform was brought forth to leverage 3D TI thin films instead, where each 2D SS of 3D TI contributes a half-quantized G_xy = ±e²/2h under broken TRS.^47–49 In 2013, this was realized in Cr-doped ternary (Bi,Sb)₂Te₃ thin films,⁵⁰ soon gaining wide acceptance with ideal performance.^51–61 More recently, the QAH effect was also demonstrated in one device of five-layer exfoliated intrinsic antiferromagnetic TI MnBi₂Te₄,⁶² as well as in Moiré materials, namely, magic-angle twisted bilayer graphene⁶³ and AB-stacked MoTe₂/WSe₂ heterostructures,⁶⁴ greatly expanding the available material platforms and physical mechanisms for investigating the QAH phenomena.

As shown in Fig. 2(a), the archetypical 3D TI of the Bi₂Te₃-based materials crystallize in a $R3̄m(D3d5,No.166)$ rhombohedral structure, featuring Te(1)–Bi–Te(2)–Bi–Te(1)-type quintuple layers (QLs) with weak van der Waals interlayer bonding.^65–67 The topologically nontrivial members of the family display a single surface Dirac cone⁴¹ and bandgap of 0.2–0.33, 0.21–0.3, and 0.13–0.2 eV, for Bi₂Se₃,^68–70 Sb₂Te₃,^68,71,72 and Bi₂Te₃,^{68,71,73–75} respectively. The chemical stoichiometry and prevalence of defects sensitively affect the transport properties.⁷⁶ When grown as thin films using molecular beam epitaxy (MBE), Bi₂Se₃ and Bi₂Te₃ are chiefly n type, owing to Se and Te vacancies, while Sb₂Te₃ is dominantly p type due to Sb_Te antisite defects. The isostructural nature of the compounds and availability of different carrier types offer the needed tunability for adjusting the chemical potential to zero-in on the surface transport.

Development of diluted magnetic semiconductors in the early 2000s has provided important insights into the induction of long-range magnetic order in TIs. Taking Sb₂Te₃ for example, Mn doping in Mn_xSb_2−xTe₃ bulk crystals does not stimulate long-range ordering for x up to 0.045,⁷⁷ although ferromagnetism can be stabilized in Mn_xBi_2−xTe₃ (x up to 0.09)⁷⁸ as well as in topological crystalline insulator Mn_xSn_1−xTe (x up to 0.12).⁷⁹ Doping with V and Cr, on the other hand, is effective in introducing magnetic order in V_xSb_2−xTe₃ (0.01 ≤ x ≤ 0.03)⁸⁰ and Cr_xSb_2−xTe₃ (x up to 0.095).⁸¹ With the incorporation of magnetic dopants (V/Cr) elevated to even higher levels in the out-of-equilibrium MBE growth, strong out-of-plane ferromagnetic ordering has been successfully demonstrated with Curie temperature T_C reaching 177 and 190 K in V_xSb_2−xTe₃ (x up to 0.35)⁸² and Cr_xSb_2−xTe₃ (x up to 0.59),⁸³ respectively. These early works on relatively thick films (hundreds of nm) have benchmarked the solubility limit, while setting expectation of the achievable exchange energy in films with thickness relevant to the QAH effect, considering that T_C generally decreases upon reducing the thickness.⁸⁴ The high doping, however, is expected to weaken and eventually destroy the topological nature of the band due to reduced SOC. At an intermediate level, V_xSb_2−xTe₃ (x up to 0.10) thin films can sustain high surface mobility, leading to the prominent Shubnikov–de Haas (SdH) quantum oscillations.⁸⁵ 

Despite its larger bandgap and more ideally positioned surface Dirac cone, the ferromagnetic response from Cr-doped Bi₂Se₃ is quite weak.⁸⁶ The effort toward realizing the QAH effect has since largely been focused on the ternary (Bi,Sb)₂Te₃ matrix, with systematically engineered thickness and Bi/Sb alloying ratio. The QAH state was first demonstrated by Cr doping in 5 QL Cr_0.15(Bi_0.1Sb_0.9)_1.85Te₃.⁵⁰ It was then discovered that V doping instead leads to a better reproducibility and a more robust QAH effect.⁵⁶ As shown in Fig. 2(b), a nearly ideal QAH state is present at the charge neutrality point in 4 QL V_0.11(Bi_0.29Sb_0.71)_1.89Te₃, manifesting zero magnetic field R_yx(0) = 1.000 19 ± 0.000 69 h/e², R_xx(0) = 0.000 13 ± 0.000 07 h/e² and an AH angle α of 89.993° ± 0.004° at 25 mK. As revealed by low energy muon spin rotation (LE-μSR) spectroscopy in Fig. 2(c), the full magnetic volume fraction is achieved in (Cr,V)_x(Bi,Sb)_2−xTe₃ only at doping levels with x ≥ 0.16. The evolution of the effective magnetic ordering is consistent with the formation and growth of ferromagnetic islands that eventually encompass full volume upon cooling.⁸⁷ 

The edge current–voltage I_i = (e²/h)∑_j(T_jiV_i − T_ijV_j), is determined by the transmission probability T_ji, connecting the ith electrode to the jth electrode in the Landauer–Büttiker theory.^88,89 As shown in Figs. 1(c) and 3, in the QAH regime, the chiral edge modes propagate clockwise (counter-clockwise) for M > 0 (M < 0), leading to T_i,i+1 = 1 (T_i+1,i = 1). The ideal dissipationless chiral edge transport at zero magnetic field has indeed been verified by comprehensive local and nonlocal magnetoresistance experiments.^52,90–92 In quantum phase transitions concerning a QAH insulator, the temperature dependence of the derivative of R_xx(H) at the critical field H_c follows a power law scaling behavior (dR_xx/dH)|_Hc ∝ T^−κ, with critical exponent κ in the range 0.22–0.62.^93–97 Utilizing a cryogenic current comparator, metrologically comprehensive low-current, high-precision measurements of the QAH states have demonstrated R_yx quantization within 1 part per million (ppm) of the von-Klitzing constant R_K,^98–100 and most recently down to 10 parts per billion (ppb), leveraging a permanent magnet design.¹⁰¹ It improves the prospects for zero-field quantum resistance standard and spintronics exploiting chiral edge transport, which favors a more robust QAH state at higher temperature, with even smaller R_xx and larger breakdown current density.^102,103

In early QAH studies, the critical temperature T_QAH reaching full quantization is rather low in the mK range. By Cr/V codoping, the T_QAH can be enhanced to 0.3 K with R_yx(0) = h/e² (within the experimental uncertainty) and R_xx(0) = 0.009 h/e², while at T = 1.5 K, R_yx(0) = 0.97 h/e² and R_xx(0) = 0.19 h/e².¹⁰⁴ The increase in the T_QAH via Cr/V codoping verifies to have originated from the improved magnetic homogeneity and favorably modulated surface band structure.

Further increase in the T_QAH would be beneficial in order to advance quantitative understanding of the QAH physics and to facilitate practical applications. There are apparent disadvantages in introducing ferromagnetic order via traditional doping though: (i) the foreign species act as defects that inevitably degrade the sample quality; (ii) the innate random distribution of magnetic dopants leads to undesirable disorder (including spin scattering) and fluctuation in the energy band (in addition to the more generic local variations of the Bi/Sb ratio and/or film thickness) that adversely affect the QAH state;^105–109 (iii) at the high level of doping, or rather substitution, needed for enhanced T_C, the band structure is prone to the development of impurity states in the bulk gap and its topological nature may also be affected, let alone the likelihood of secondary phase segregation.¹¹⁰ Thus, unfortunately, the doping route comes with inherent limitations.

The challenging yet highly desirable goal of raising the T_QAH is being actively pursued. A successful approach involves magnetic modulation doping. Specifically, as shown in Fig. 4(a), apart from the conventional single-layer Cr_x(Bi,Sb)_2−xTe₃ with uniform and modest Cr doping (x = 0.10), alternating combinations of heavily doped (x = 0.46) and pristine (x = 0) TI QLs are rationally designed to form tri- or penta-layer stacks. The penta-layer structure enables an impressively high T_QAH = 0.5 K, while R_yx(0) reaches 0.97 h/e² at 2 K, as shown in Fig. 4(b).¹¹¹ The greatly suppressed disorder and magnetic fluctuation are believed to be the key in enhancing the T_QAH. In the modulation doping case, as we shall see below, there is also the possibility that the internal interfacial exchange interaction becomes highly effective, leading to a better exchange gap opening in the “cleaner/pristine” layers of TI, thus enabling the achievement of a higher T_QAH.

It is worth pointing out that, at such high level of Cr substitution on the Bi/Sb sites (exceeding 20%), the SOC is expected to be much weakened, and the band inversion may no longer sustain. It hence drives the Cr-rich layer away from the TI phase toward a magnetic insulator (MI) instead. The versatility of the MI-like heavily Cr-doped TI block has been recognized and extended to an architecture of [3 QL Cr:(Bi,Sb)₂Te₃/4 QL (Bi,Sb)₂Te₃]_n/3 QL Cr:(Bi,Sb)₂Te₃.¹¹² Similar to an earlier TI-normal insulator design of [6 QL (Cr,V):(Bi,Sb)₂Te₃/3.5 nm CdSe]_n/6 QL (Cr,V):(Bi,Sb)₂Te₃ multilayers,¹¹³ it successfully facilitates the demonstration of the high Chern number (adjustable by the stacking number n) QAH state.^114,115 This design also allows for probing the physics underlying the plateau-to-plateau phase transition connecting the Chern number C = 1 and C = 2 phases, by means of systematically engineering the strength of SOC, via tuning the Cr concentration, in the middle Cr-doped layer of a penta-layer device.¹¹⁶ The Chern number tunable QAH state attests to the capability and precise control of multichannel dissipationless chiral conduction. Recently, as illustrated in Fig. 4(c), asymmetric placement of the Cr-rich block in a semi-magnetic configuration allows selective opening of an exchange gap only in the top surface of TI.¹¹⁷ It enables condensed matter investigation of the relativistic parity anomaly in quantum field theory, offering long-sought experimental verification of the half-integer quantization of G_xy in Fig. 4(d).

The revelation of the role of MI/TI interfaces in modulation doped TI points to the significance of proximitized internal exchange coupling. Indeed, the QAH effect has been recently achieved in MI/TI/MI heterostructures.¹¹⁸ As shown in Fig. 5(a), Cr-doped ZnTe, a large gap MI with favorable lattice parameters in the (111) plane matching TI, can be grown with non-magnetic TI into high quality sandwiches. The stacks possess atomically sharp interfaces and minimal Cr interfacial migration, as confirmed by energy-dispersive x-ray spectroscopy (EDS) mapping and distribution profile analysis.^118–120 Full quantization has been observed at T = 30 mK [see Fig. 5(b)]. Despite the high growth temperature, the insignificant Cr diffusion across such an MI/TI interface is a salient feature, enabling reliable placement of alternating Cr-doped TI and pristine TI layers, instrumental in realizing the higher Chern number QAH state,¹¹² as well as spin–orbit-torque-based electrical switching of the chirality of edge current.¹²¹ 

Proximity-driven engineering offers a promising avenue for further enhancing the T_QAH beyond the presently available sub-Kelvin regime, as well as uncovering new physics. Higher temperature QAH states may be obtained by imprinting magnetism via high-quality heterostructures with compatible MIs possessing high T_C (or Néel temperature T_N for antiferromagnet) while preserving the structural integrity and salient surface electronic properties of TIs. The interface approach is a center piece at the forefront of the field.

Recently, polarized neutron reflectometry (PNR) experiments at the EuS/Bi₂Se₃ interface have demonstrated that long-range magnetic order can be induced at the surface of a TI without the complication of creating spin-scattering centers in the doping case.¹²² As shown in Fig. 5(c), the magnetic scattering length density (MSLD) profile reveals ferromagnetism that extends ∼2 nm into the Bi₂Se₃ at 5 K. The experimental spin asymmetry ratio, defined as SA = (R⁺ − R⁻)/(R⁺ + R⁻), with R⁺ or R⁻ being the reflectivity with neutron spin parallel (+) or antiparallel (−) to the external field, provides sensitive measurement of in-plane magnetism (SA = 0 designates no magnetic moment). The temperature evolution of SA profiles in EuS film and EuS/Bi₂Se₃ bilayer is drastically distinct¹²²—while SA becomes vanishingly small in EuS above T_C ∼ 17 K, the magnetism apparently survives up to 300 K at the EuS/Bi₂Se₃ interface [see Fig. 5(d) for MSLD depth profiles at selected temperatures]. This enhanced magnetic behavior has been further confirmed by superconducting quantum interference device (SQUID) magnetic measurements as well as x-ray magnetic circular dichroism (XMCD) studies. It is worth emphasizing that a particularly clean, sharp and controlled in situ interface between EuS and TI is required in achieving the needed effective magnetic proximity coupling due to the extreme short-range nature of the exchange interaction (≲0.5 nm).^123–126 Extraordinary care is warranted in the TI growth, and the magnetic interfacial heterostructuring thereafter, to properly mitigate the adverse influence from residual chalcogen atoms since the optimal TI demands a Se/Te-rich growth condition. This intriguing discovery has opened a vibrant arena for proximitized MI/TI heterostructures.^127–138 

Extensive investigations have since been devoted to interfacial exchange coupling and the magnetic proximity effect in MI/TI systems¹³⁹ using Y₃Fe₅O₁₂ (YIG),^140,141 Tm₃Fe₅O₁₂ (TIG),¹⁴² MnTe,¹⁴³ Cr₂Ge₂Te₆ (Refs. 119 and 144), and Fe₃GeTe₂ (Ref. 145), just to name a few. Convincing out-of-plane hysteretic AH data in platforms utilizing MIs with in-plane bulk magnetic anisotropy are not yet available and should be pursued. Initial studies leveraging TIG, a high-T_C (∼560 K) MI with perpendicular magnetic anisotropy, have shown that the ordering of TI can be increased to above 400 K.¹⁴² The R_yx at present is admittedly small though, on the order of 0.7–2 Ω at 2 K. Inspired by the high T_C, it warrants future effort to improve the interfacial coupling, for instance, by growing TIG and TI in situ without breaking vacuum during the interface fabrication. Sandwich heterostructures taking advantages of multiple interfaces^118,119 such as TIG/TI/TIG are also desirable, should one be able to control the magnetic anisotropy of TIG as out-of-plane oriented in subsequent layers. Furthermore, proximitizing magnetic TI with antiferromagnetic (Al:)Cr₂O₃ enables exchange that is biased at the interface,^146,147 allowing for further tunability in QAH-based devices.¹⁴⁸ 

Recognizing the critical importance of high-quality magnetic interfaces in QAH related physics, one may envision pushing the proximity-driven phenomena down to their ultimate limit— i.e., taking advantage of the layer-by-layer atomically ordered internal interfaces in intrinsic magnetic TIs. The promise and feasibility of such a strategy were perhaps well hinted by the large magnetic gap seen in the natural superlattice of QL Bi₂Te₃ and septuple layer (SL) MnBi₂Te₄ in Mn-doped Bi₂Te₃.¹⁴⁹ As shown in Fig. 6(a), MnBi₂Te₄ SLs feature intralayer ferromagnetic order and A-type interlayer antiferromagnetism. The interlayer coupling is tunable via intercalation of non-magnetic Bi₂Te₃ QLs, forming a family of MnBi₂Te₄(Bi₂Te₃)_m with rich topological and physical properties.^150–155 

The layer sensitive magnetism in MnBi₂Te₄ offers a versatile platform for exploring the magnetic field driven transitions connecting the (canted) antiferromagnetic and poled ferromagnetic phases in the few SL regime, as well as the rich QAH (odd) and/or axion (even) insulator physics with precise SL number control.^156–158 As illustrated in Fig. 6(b), the QAH effect has been realized in one optimal 5 SL MnBi₂Te₄ (device 5 SL A), displaying R_yx(0) = 0.97 h/e² and R_xx(0) < 0.061 h/e² at 1.4 K.⁶² While the T_QAH for full zero-field quantization was not reported, it is likely on par with the state-of-the-art thin film results using Cr/V codoping¹⁰⁴ and Cr-modulation doping.¹¹¹ A comparable QAH state has also been achieved in crystal flakes with the MnBi₂Te₄/Bi₂Te₃ natural superlattice.¹⁵⁹ It has stimulated intense investigations of intrinsic magnetic TIs, although reproducing the stunning QAH result⁶² in 5 SL MnBi₂Te₄ still appears to be challenging to date. It is likely owing to the complex interplay among crystal quality, magnetic/lattice defect, stacking/strain condition, fabrication processing, device morphology and substrate/film/capping interfaces, which leads to the often qualitatively different R_yx(H) profiles, as shown in Fig. 6(b), e.g., for devices from a different batch (device 5 SL B)⁶² or different group (device 5 SL C).¹⁵⁷ 

As depicted in Fig. 6(c), a zero Hall plateau has been observed in 6 SL MnBi₂Te₄ (device 6 SL A),¹⁶⁰ establishing another candidate system for solid state investigation of axion physics.^161–163 The characteristic critical fields, corresponding to the small but discernible kinks in R_yx and extrema in R_xx for device 6 SL A,¹⁶⁰ are generally consistent with other even SL devices (6 SL B¹⁵⁷ and 4 SL⁶²). By means of electric field tuning, Berry-curvature-induced layer Hall (LH) effect has been uncovered in this axion insulator like regime.¹⁶⁴ The capability of fusing magnetism and topology at the atomic level bodes particularly well for the future advancement of topological quantum effects. To further understand the key factors underlying the magnetic and topological nature of MnBi₂Te₄, and QAH insulators in general, multimodal investigations are of immediate interest, synergizing magnetotransport and various scanning probes, including magnetic force microscopy (MFM),¹⁶⁵ microwave impedance microscopy (MIM),¹⁶⁶ nano-superconducting quantum interference device (nanoSQUID),¹⁰⁷ magneto-optical Kerr effect (MOKE) and magnetic circular dichroism (MCD).¹⁶⁷ 

By interfacing 2D crystal layers, of either the same species at a small twist angle or a different kind possessing dissimilar lattice parameters, one creates artificial Moiré superlattices hosting intertwined topology and strong correlations.¹⁶⁸ Moiré graphene heterostructures with valley-spin-degenerate topological flat bands enable remarkable phenomena, including superconductivity,¹⁶⁹ correlated insulating states,¹⁷⁰ and when TRS is broken, orbital magnetism featuring Moiré-scale current loops.¹⁷¹ Significant AH effect with strong magnetic hysteresis has been observed in twisted bilayer graphene (tBLG)¹⁷² and ABC-trilayer graphene/hexagonal boron nitride (ABC-TLG/hBN) Moiré superlattices.¹⁷³ As shown in Fig. 7(a), a robust QAH state has been demonstrated at 1.6 K in a narrow range of density near band filling factor ν = 3 in a flat-band tBLG device aligned to hBN.⁶³  R_yx quantization, within 0.1% of R_K, has been found to survive up to 3 K, while the system displays a T_C of 7.5 K.

In electrically tunable semiconductor-based hetero-bilayers, the application of an out-of-plane gating electric field modulates the bandwidth as well as the band topology by intertwining Moiré bands from different layers. At ν = 1, the QAH effect was achieved at 0.3 K, upon spontaneous valley polarization in the MoTe₂/WSe₂ Moiré superlattice with AB configuration⁶⁴ [see Fig. 7(b)]. R_yx remains quantized up to about 2.5 K, while staying finite up to T_C ∼ 5–6 K. The two newly emerged Moiré platforms manifest R_yx > h/e² when approaching quantization, different from the TI-based phenomenology with R_yx < h/e², hinting that different disorder mechanisms might underpin the QAH effect owing to orbital magnetic states.

In recent years, tremendous progress has been made in advancing topological concepts in solid state. The realization and development of the QAH effect attests to the ever-more coherent synergy between theoretical prediction/interpretation and experimental exploration/discovery of new materials. We expect the fundamental interfacial magnetism in heterostructures to fuel future development in the field. Identifying new magnetic topological systems with suitable properties for implementing the QAH effect, or exploring the interplay between magnetism and topology in general, is of paramount interest.^174–176 The versatile selective tunability of the magnetic topological interfaces further enables the investigation of the high energy physics counterparts in materials laboratory, such as dyon particles¹⁷⁷ and Majorana bound states,^178,179 when additional superconductivity proximity is coupled.

From an application standpoint, the QAH effect bodes particularly well for future universal quantum standard units, combining the Josephson effect in one measurement setup, where uncertainty in the 1 ppb range is a prerequisite to rival that of existing QH systems.¹⁸⁰ The ideally dissipation-free nature of chiral edge transport in the QAH state inspires low-energy consumption spintronics or integration to existing computing architectures as chiral interconnects.¹⁸¹ Furthermore, when hybridized with superconductors, QAH insulators motivate topological quantum computing.¹⁸² These technological breakthroughs leverage on the capability of interfacial engineering structural, chemical, magnetic, and electronic properties at the atomic scale, ideally with the robust QAH state for practical operation temperature.

Despite impressive progress using interface-inspired approaches (heterostructure or intrinsic), the T_QAH manifesting quantized R_yx with negligible R_xx (≪1% h/e²) is still limited in the sub-Kelvin regime for all Bi₂Te₃-derived QAH systems. It implies that mechanisms intrinsic to tetradymites, e.g., spatial inhomogeneities in thickness and/or prevailing antisite defects, might be the main culprit limiting the operation temperature, in lieu of the magnetic ordering, as T_C/T_N is reasonably high of the order of tens of Kelvin. Indeed, recent analysis of non-local transport in a Corbino geometry has revealed QAH edge channels surviving up to T_C in V-doped TIs.¹⁸³ Further optimizing the bulk insulation in tetradymites,¹⁸⁴ or discovering an entirely new class of TIs beyond the current paradigm,¹⁸⁵ is beneficial toward increasing the practical relevance of the QAH effect.

In additional to MBE growth, recent development in alternative and industrial friendly routes such as sputtering¹⁸⁶ may bring about previously overlooked benefits, including quantum confinement and/or superior sample quality. The dynamic MBE deposition and equilibrium crystal synthesis bear dramatically different kinetics and thermodynamics. To realize the QAH state, the former route is instrumental in preparing doped films beyond the bulk solubility limit, while the latter is critical in ensuring the needed layer ordering and placement in intrinsic TIs, however not typically vice versa. Realizing intrinsic magnetic TI films capable of quantization is highly desirable, meaning material exploration. Prospects of advancing QAH physics also lie in the exploration of possibly fractional QAH by feasibly introducing the needed strong correlation, as well as the novel mechanisms exploiting, e.g., in-plane magnetization¹⁸⁷ and antiferromagnetism.¹⁸⁸ Further inspiration can also be drawn from the exciting development of twistronics, benefiting from magnetic states of orbital origin.¹⁸⁹ 

The work was supported by the Army Research Office (Grant No. W911NF-20-2-0061), the National Science Foundation (Grant Nos. NSF-DMR 1700137 and CIQM 1231319), and the Office of Naval Research (Grant No. N00014-20-1-2306). H.C. was sponsored by the Army Research Laboratory under Cooperative Agreement No. W911NF-19-2-0015.

The authors have no conflicts to disclose.

Hang Chi: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Jagadeesh S. Moodera: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.