High-Chern-number and high-temperature quantum Hall effect without Landau levels

Abstract The quantum Hall effect (QHE) with quantized Hall resistance of h/νe2 started the research on topological quantum states and laid the foundation of topology in physics. Since then, Haldane proposed the QHE without Landau levels, showing nonzero Chern number |C| = 1, which has been experimentally observed at relatively low temperatures. For emerging physics and low-power-consumption electronics, the key issues are how to increase the working temperature and realize high Chern numbers (C > 1). Here, we report the experimental discovery of high-Chern-number QHE (C = 2) without Landau levels and C = 1 Chern insulator state displaying a nearly quantized Hall resistance plateau above the Néel temperature in MnBi2Te4 devices. Our observations provide a new perspective on topological matter and open new avenues for exploration of exotic topological quantum states and topological phase transitions at higher temperatures.


INTRODUCTION
The quantum Hall effect (QHE) with quantized Hall resistance plateaus of height h/νe 2 was first observed in two-dimensional (2D) electron systems in 1980 [1]. Here, h is Planck's constant, ν is Landau filling factor and e is electron charge. The QHE in 2D electron systems with high mobility is originated from the formation of Landau levels (LLs) under strong external magnetic field. Subsequently, the exact quantization was explained by Laughlin based on gauge invariance and was later related to a topological invariance of the energy bands, which is characterized by Chern number C [2][3][4][5]. A nonzero Chern number distinguishes the QHE systems from vacuum with C = 0 [2,3]. The discovery of QHE introduces the concept of topology into condensed matter physics and is extremely important to physical sciences and technologies. However, the rigorous conditions of ultrahigh mobility, ultralow temperature and strong external magnetic field limit the deep exploration and wide applications of QHE.
Theoretical proposals based on the intrinsic band structure of 2D systems open up new opportunities.
In 1988, Haldane theoretically proposed a timereversal symmetry (TRS) breaking 2D condensedmatter lattice model with quantized Hall conductance of e 2 /h in the absence of an external magnetic field [6]. This indicates that QHE can be realized without the formation of LLs. The QHE induced by spontaneous magnetization in such insulators is called quantum anomalous Hall effect (QAHE), and such insulators are called Chern insulators. An alternative mechanism of realizing QAHE through localization of band electrons was later proposed in 2003 [7]. The emergence of topological insulators (TIs) in which strong spin-orbit coupling (SOC) gives rise to topological band structures provides a new system for the investigation of QHE without strong external magnetic field. The QAHE with quantized Hall conductance of e 2 /h was predicted to occur in magnetic TIs by doping transition metal elements (Cr or V) into time-reversal-invariant TIs Bi 2 Te 3 , Bi 2 Se 3 and Sb 2 Te 3 [8]. In 2013, the QAHE with quantized Hall conductance of e 2 /h was experimentally observed in thin films of C The Author(s) 2020. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Ge et al. 1281 chromium-doped (Bi, Sb) 2 Te 3 with the temperature down to 30 mK [9]. However, in the above-mentioned QHE systems without LLs, only a Hall resistance plateau with C = 1 can be obtained by coupling topological surface states with magnetism. High-Chern-number QHE without LLs has never been observed experimentally. Besides, the requirement of ultralow temperatures limits the study of QHE without LLs. Efforts on high-Chern-number and high-temperature QHE without LLs are still highly desired for exploring emergent physics and low-power-consumption electronics [10].

RESEARCH ARTICLE
Here we report the first experimental discovery of the high-Chern-number QHE without LLs above 10 K and C = 1 QHE without LLs above the Néel temperature (T N ) in MnBi 2 Te 4 devices. We show that when modulated into the insulating regime by a small back gate voltage, the nine-layer and ten-layer MnBi 2 Te 4 devices can be driven to Chern insulator with C = 2 at moderate perpendicular magnetic field. Quantized Hall resistance h/2e 2 accompanied by vanishing longitudinal resistance with the temperature as high as 13 K is observed in the ten-layer device. When reducing the thickness of the devices down to eight layer and seven layer, a quantized Hall resistance plateau h/e 2 is detected at a temperature much higher than the Néel temperature of the devices. This quantization temperature is the highest record in systems showing QHE without LLs. Our discoveries break new ground in the exploration of topological quantum states and provide a platform for potential applications in related low-consumption electronics.
MnBi 2 Te 4 is a layered material which can be viewed as a layer of Bi 2 Te 3 TI intercalated with an additional Mn-Te layer [11][12][13][14][15][16][17][18][19][20]. This material exhibits ferromagnetic (FM) order within septuple layer (SL) and anti-ferromagnetic (AFM) order between neighboring SLs with an out-of-plane easy axis [11], as displayed in Fig. 1a. By tuning the magnetic structure through thickness or magnetic field, exotic topological states, such as type-I topological Weyl semimetal (WSM) in 3D, Chern insulator in 2D and higher-order topological Möbius insulator, can be realized in MnBi 2 Te 4 [21,22]. In this work, the MnBi 2 Te 4 flakes were mechanically exfoliated from high-quality MnBi 2 Te 4 single crystals. These flakes were then transferred to 300 nm-thick SiO 2 /Si substrates and the standard e-beam lithography followed by e-beam evaporation was used to fabricate electrodes. The doped Si served as the back gate and a back gate voltage applied between Si and the sample could modulate the sample into insulating regime. The magnetic field is perpendicular to the samples throughout the text. Figure 1b shows an optical image of the MnBi 2 Te 4 device (s6) with Hall bar geometry. Atomic force microscope measurements were carried out to determine the thickness of s6 (Fig. S1f). The line profile reveals a thickness of 13.4 ± 0.4 nm, corresponding to 10-SL. The temperature dependence of longitudinal resistance R xx is shown in Fig. 1c, in which a sharp resistance peak gives the T N at around 22 K.

High-Chern-number Chern insulator states
To get insight into the evolution of the Chern insulator states in the 10-SL MnBi 2 Te 4 device s6, we carried out magneto-transport measurements at various back gate voltages V bg . Figure 1d and e displays the gate-dependent magneto-transport properties of s6 under perpendicular magnetic field at T = 2 K. Two sharp transitions at around 3 T and 5 T can be clearly observed on both R xx and R yx in Fig. 1d and e. These two transitions may mark the beginning and ending of the spin-flipping process. With further application of a perpendicular magnetic field, the sample is supposed to enter the perfectly aligned FM state [19].
The well-quantized Hall resistance plateau with height of 0.99 h/2e 2 is detected at −15 T by applying a V bg = −17 V, accompanied by a longitudinal resistance as small as 0.004 h/2e 2 as shown in Fig. 1d and e. The quantized Hall resistance plateau almost does not change when further tuning V bg to −58 V (within the tolerance of the substrate), which can be clearly observed in Fig. 1f. Besides, the Hall resistance plateau deviates from the quantized value when V bg is above −5 V. The well-quantized Hall resistance plateau and nearly vanishing longitudinal resistance are characteristics of high-Chernnumber QHE without LLs contributed by dissipationless chiral edge states and indicate a well-defined Chern insulator state with C = 2.
In the absence of a magnetic field, MnBi 2 Te 4 bulk is an AFM TI, whose side surfaces are gapless and (111) surfaces are intrinsically gapped by exchange interactions [11,12,21]. The gapped surface states are characterized by a quantized Berry phase of π and can display the novel half-quantum Hall effect [23,24]. Due to the AFM nature of the bulk, Hall conductance or topological Chern number of MnBi 2 Te 4 (111) films is dictated by the surface states, which depend critically on the film thickness. For even-and odd-layer films, the two surfaces (on the top and bottom) display half-integer Hall conductance of opposite and identical signs, leading to C = 0 and 1, respectively [11]. Obviously, one would never obtain high Chern number C > 1 in AFM MnBi 2 Te 4 .  However, when MnBi 2 Te 4 is driven from AFM to FM states by external magnetic field, physical properties of the material change dramatically. While the interlayer coupling is restricted by the PT (combination of inversion and time-reversal) symmetry in AFM MnBi 2 Te 4 [11,21], it gets greatly enhanced in the FM state by PT symmetry breaking, which generates more dispersive bands along the − Z direction than the AFM state (Fig. S9). Remarkably, the magnetic transition results in a topological phase transition from an AFM TI to a ferromagnetic Weyl semimetal in the bulk [11,12], leading to a physical scenario in which Chern insulators with C > 1 are designed [21,[25][26][27]. Figure 1g shows the schematic FM order and electronic struc-ture of the C = 2 Chern insulator state with two chiral edge states across the band gap. Figure 2 shows the temperature evolution of the high-Chern-number QHE without LLs with the V bg = −19 V. As the temperature increases to 13 K, the height of the Hall resistance plateau stays above 0.97 h/2e 2 and R xx remains below 0.026 h/2e 2 . With the temperature further increasing to 15 K, the value of the Hall resistance plateau reduces to 0.964 h/2e 2 and R xx increases to 0.032 h/2e 2 . This working temperature of the high-Chernnumber QHE without LLs is much higher than liquid helium temperature, which shows potential application of QHE in low-dissipation electronics. Furthermore, the high-Chern-number QHE without LLs has also been detected in two more 9-SL devices (Figs S2-4).

High-temperature QHE without LLs
We further study the 7-SL and 8-SL MnBi 2 Te 4 devices (s2 and s3) and the results are displayed in Fig. 3 and Fig. S5. As shown in Fig. S5a, R yx of s2 reaches a well-quantized Hall resistance plateau with height of 0.98 h/e 2 by applying a small V bg = 6.5 V at T = 1.9 K, accompanied by R xx as low as 0.012 h/e 2 , which is a hallmark of Chern insulator state with C = 1. When further increasing V bg to 10 V, the quantized Hall resistance plateaus remain robust as shown in Fig. S5a and c. Temperature evolution of R yx and R xx in s2 with V bg = 6.5 V is shown in Fig. 3a and b. Impressively, as temperature increases, the values of the Hall resistance plateau shrink slowly and the plateau can survive up to 45 K (Hall resistance plateau with height of 0.904 h/e 2 ), much higher than the Néel temperature T N ∼ 21 K of s2 (Fig. S5b). The hightemperature QHE without LLs is also observed in the 8-SL device s3. As shown in Fig. 3d and e, R yx of s3 is 0.997 h/e 2 at 1.9 K (R xx ∼ 0.00006 h/e 2 ), 8 V, and even at 30 K (above Néel temperature T N = 22.5 K), R yx can reach 0.967 h/e 2 (R xx ∼ 0.0023 h/e 2 ). The quantized plateaus from 1.9 K to 30 K are very clear and overlapped. Figure 3c and f displays the color plot of R yx in s2 and s3 as a function of the temperature and magnetic field at V bg = 6.5 V and 8 V, respectively. Based on the experimental data, the B-T phase diagram can be summarized. The phase diagram is characterized by the phase boundaries, B AFM (T) and B QH (T). The B AFM (T) data points, as the boundary of the AFM states, are composed of the peak values of the R xx (B) curves ( Fig. 3b and e) at various temperatures (the cyan spheres) and the peak value of the R xx (T) curve ( Fig. S5b and Fig. S8) at zero magnetic field (the pink sphere). The B QH (T) curves, as the boundaries of the Chern insulator states (the yellow spheres), represent the magnetic fields required to reach 99% of the Hall resistance plateau at different temperatures, above which the device is driven to FM state and becomes a Chern insulator with C = 1. It is obvious that the AFM state disappears at T N . However, the Hall plateau shows nearly quantized resistance even at 45 K (0.904 h/e 2 ) in s2 and 30 K (0.967 h/e 2 ) in s3, which reveals that the Chern insulator state exists at a temperature much higher than T N, indicating a potential way to realize QHE without LLs above liquid nitrogen temperature.

DISCUSSION
A fundamental question is whether the observed quantized Hall resistance plateau is caused by Landau level quantization, as the ordinary QHE with LLs can also give rise to quantized Hall resistance plateaus and vanishing R xx . We estimate the mobility values of our devices according to the slope of Hall resistance near zero magnetic field [18]. The mobility values range from 100 to 300 cm 2 V −1 s −1 , which are typically below the critical value for formation of LLs up to 15 T [28]. To further exclude the possibility of QHE with LLs, we performed controlled measurements by changing the carrier type.
In general, the Chern number in ordinary QHE corresponds to the occupancy of LLs and the sign of the Chern number will change once the carrier type is switching. However, as shown in Fig. S6c and d, the carrier type in the device s4 (7-SL) with C = 1 is tuned from p to n when increasing the back gate voltage from 0 V to 99.5 V, while the sign of the Chern number does not change. Furthermore, for the C = 2 devices, the quantized R yx plateau in device s6 with n-type carriers (Fig. 1d) and s7 with ptype carriers (Fig. S4a) have the same sign. These observations unambiguously demonstrate that the observed quantized Hall resistance plateau has nothing to do with LLs and the quantized R yx originates from Chern insulator state. FM MnBi 2 Te 4 belongs to magnetic Weyl semimetals, and has one simple pair of Weyl points (WPs) along the − Z direction located at k W and −k W . The k z -dependent Chern number C(k z ) = σ xy (k z )h/e 2 defined for 2D momentum planes with specified k z must be quantized (except at the gapless WPs) and abruptly jumps at positions of Weyl points. C(k z ) equals one between the two Weyl points due to topological band inversion and zero elsewhere as illustrated in Fig. 4a. The averaged anomalous Hall conductance per unit layer is given by   σ xy = c 0 2π

RESEARCH ARTICLE
where c 0 is the out-of-plane thickness of each SL, andk W = |k W | c 0 /π . For an N-layer FM thin film, its electronic states can be viewed as quantum-well states and possess a finite band gap due to quantum confinement. Generally, σ xy of thin films would grow with film thickness, as its ideal bulk contribution is N|k W |e 2 / h. On the other hand, σ xy of 2D gapped films must take quantized values C (N)e 2 / h as topologically required. Therefore, for thick films with minor surface effects, the thickness-dependent Chern number C (N) would change discretely by 1 for every N = 1/|k W |, implying that high Chern number is feasible by increasing film thickness. The discrete increase of Chern number with increasing film thickness is a generic feature of ferromagnetic Weyl semimetals, which can also be understood by the topological band inversion picture as discussed in Methods. The above physical picture is confirmed by the first-principles study, which givesk W = 0.256 ≈ 1/4 for the bulk and shows that C (N) indeed increases by 1 for every N = 4 (Fig. 4b). Note that it is theoretically challenging to accurately predict C (N), since the predictedk W depends sensitively on the exchange-correlational functional and the lattice structure. Based on the mBJ functional [29], we systematically tested the influence of lattice parameter c 0 on band structure and C (N) (Fig. S9), and finally decided to use the experimental value c 0 = 13.6Å. As shown in Fig. 4b, the 9-SL film is a high-Chern-number band insulator with C = 2. Compared to the AFM films studied before [11], band structure of the FM film displays much more pronounced quantum confinement effects, as visualized by significant band splitting between quantum well states (Fig. 4c). A quantum confinement induced gap ∼5 meV is located at the point. The edge-state calculation reveals that there exist two chiral gapless edge channels within the gap (Fig. 4d), which confirms C = 2. Therefore, firstprinciples calculations indicate that high-Chernnumber band insulators can be realized in the FM Weyl semimetal MnBi 2 Te 4 by means of quantum confinement.
The theory suggests that the topological Chern number is tunable by controlling film thickness of FM MnBi 2 Te 4 . By reducing the film thickness to 7-SL, the Chern number decreases to C = 1, as found experimentally. The 8-SL is the marginal  case, which has C = 1 in experiment and C = 2 in theory. The discrepancy is possibly caused by the surface/interface effects that are not theoretically considered. Contrariwise, the increase of film thickness could lead to higher Chern numbers (C > 2), which is awaiting experimental confirmation. Moreover, since the Chern insulator phase appears in the FM state, the weak inter-SL antiferromagnetic exchange coupling is irrelevant to the topological physics. Thus, the working temperature of QHE without LLs will not be limited by the Néel temperature, and can be quite high due to the strong, ordered magnetism of MnBi 2 Te 4 .

CONCLUSION
In summary, we discovered high-Chern-number QHE (C = 2) without LLs showing two sets of dissipationless chiral edge states above 10 K and C = 1 Chern insulator state above the Néel temperature, which is also the highest temperature for QHE without LLs. Our findings open a new path for exploring the interaction between topology and magnetism, as well as the potential application of topo-logical quantum states in low-power-consumption electronics at higher temperatures.

Crystal growth
High-quality MnBi 2 Te 4 single crystals were grown by directly reacting a stoichiometric mixture of highpurity Bi 2 Te 3 and MnTe, which were prepared by reacting high-purity Bi (99.99%, Adamas) and Te (99.999%, Aladdin), and Mn (99.95%, Alfa Aesar) and Te (99.999%, Aladdin), respectively. The reactants were sealed in a silica ampoule under a dynamic vacuum, which was then heated to 973 K and slowly cooled down to 864 K, followed by the prolonged annealing at the same temperature over a month. The quality of mm-sized MnBi 2 Te 4 crystals was examined on a PANalytical Empyrean X-ray diffractometer with Cu Kα radiation.

Devices fabrication
The MnBi 2 Te 4 nanoflakes on 300 nm-thick SiO 2 /Si substrate were mechanically exfoliated from high quality single crystals using scotch tape. The substrates were pre-cleaned in oxygen plasma for five minutes with ∼60 mtorr pressure. To obtain flakes with thickness down to several nanometers, we heated the substrate after covering the scotch tape at 393 K (120 • C) for one minute. Standard electron beam lithography in a FEI Helios NanoLab 600i Dual Beam System was used to define electrodes after spin-coating PMMA resist. Then, metal electrodes (Ti/Au or Cr/Au, 65/180 nm) were deposited in a LJUHV E-400 L E-Beam Evaporator after Ar plasma cleaning.

Transport measurements
Electrical transport measurements were conducted in a 16T-Physical Property Measurement System (PPMS-16T) from Quantum Design with base temperature T = 1.9 K and magnetic field up to 16 T. Stanford Research Systems SR830 lock-in amplifiers were used to measure longitudinal resistance and Hall signals of the device with an AC bias current of 100 nA at a frequency of 3.777 Hz. The back gate voltages were applied by a Kethiley 2912A source meter.

First-principles calculations
First-principles calculations were performed in the framework of density functional theory (DFT) by