Fabrication-induced even-odd discrepancy of magnetotransport in few-layer MnBi2Te4

The van der Waals antiferromagnetic topological insulator MnBi2Te4 represents a promising platform for exploring the layer-dependent magnetism and topological states of matter. Recently observed discrepancies between magnetic and transport properties have aroused controversies concerning the topological nature of MnBi2Te4 in the ground state. In this article, we demonstrate that fabrication can induce mismatched even-odd layer dependent magnetotransport in few-layer MnBi2Te4. We perform a comprehensive study of the magnetotransport properties in 6- and 7-septuple-layer MnBi2Te4, and reveal that both even- and odd-number-layer device can show zero Hall plateau phenomena in zero magnetic field. Importantly, a statistical survey of the optical contrast in more than 200 MnBi2Te4 flakes reveals that the zero Hall plateau in odd-number-layer devices arises from the reduction of the effective thickness during the fabrication, a factor that was rarely noticed in previous studies of 2D materials. Our finding not only provides an explanation to the controversies regarding the discrepancy of the even-odd layer dependent magnetotransport in MnBi2Te4, but also highlights the critical issues concerning the fabrication and characterization of 2D material devices.


Introduction
The antiferromagnetic (AFM) topological insulator (TI) MnBi2Te4 provides promising opportunities for exploring various quantized topological phenomena [1][2][3][4][5][6] .As a layered A-type antiferromagnet, MnBi2Te4 bulk crystal is composed of septuple layers (SLs) stacked along the c-axis with intralayer ferromagnetic (FM) order and interlayer AFM order (Fig. 1a).The interplay between magnetic order and band topology gives rise to gapped surface states that exhibit half-quantized surface Hall conductivity σxy = 0.5 e 2 /h, where h represents the Plank constant and e denotes the elementary charge 7,8 .Therefore, depending on the magnetizations of the top and bottom surfaces, few-layer MnBi2Te4 with different SL-number-parity exhibits distinct topological quantum states 9 .In odd-number-SL MnBi2Te4, the parallel magnetization on the two surfaces gives rise to the quantum anomalous Hall (QAH) effect 3,10 characterized by quantized Hall resistivity (ρyx) and vanished longitudinal resistivity (ρxx) at zero magnetic field (H).In contrast, even-number-SL MnBi2Te4 displays a robust zero Hall plateau ρyx = 0 and large ρxx in a wide range of both μ0H and gate voltage (Vg), as the counter-propagating Hall currents in the two surfaces cancel out [11][12][13] .Because the zero Hall plateau with Chern number C = 0 is closely related to the topological magnetoelectric effect that stems from the axion electrodynamics [14][15][16] , magnetic TI with antiparallel magnetizations of two surfaces is widely believed as an ideal system for realizing the axion insulator state 5,[17][18][19] .Recently, using a circularly polarized light, the axion electrodynamics has been detected in a 6-SL MnBi2Te4 in the zero Hall plateau regime 20 .
Despite the experimental demonstration of the QAH effect and the axion insulator state, the fabrication of high-quality MnBi2Te4 devices with expected quantized properties remains a key challenge.In previous experiments, most odd-number-SL MnBi2Te4 exhibited a small AH effect that is far from quantization 3,21,22 , while even-number-SL devices usually exhibited linear normal Hall effect with negative slope in the AFM regime 4,23 .More puzzlingly, recent magnetic and transport measurements [24][25][26][27] found that the AH effect disappeared in some oddnumber-SL MnBi2Te4 with uncompensated AFM order, whereas a pronounced AH hysteresis occurred in some even-number-SL devices with fully compensated AFM order.Interestingly, the chirality of the AH hysteresis is opposite to the expected clockwise chirality for Mn-based TIs [28][29][30] .These counter-intuitive results have aroused widespread controversies regarding the topological nature of MnBi2Te4 in the AFM state, significantly impeding the explorations of other exotic topological quantum phenomena in topological antiferromagnets [14][15][16]31 . Sevral distinct scenarios have been proposed to account for these anomalies, such as the competition between intrinsic and extrinsic mechanisms of AH effect 32 , the magnetoelectric effect from the orbital magnetization 33 , and the layer-dependent hidden Berry curvature 34 .However, all the ideas assume MnBi2Te4 crystals with perfect sample qualities and electronic structure.As has been demonstrated by experiments [35][36][37] , even starting with the most optimized crystal, the electronic structure of a fabricated device may change dramatically, which is a critical issue in Bi2Te3 family TI materials.Theoretical calculations also suggested that the surface defects can result in redistribution of the surface charge from the first layer toward the second layer 38 , which will modify the magnetotransport performance of few-layer MnBi2Te4.Consequently, a promising yet unexplored research direction is to elucidate whether the fabrication process can lead to distinct transport behaviors in MnBi2Te4, which may offer a novel perspective for resolving the discrepancies in previous experiments.
In this work, we report systematic magnetotransport studies and the evolution of optical contrast (Oc) on 223 MnBi2Te4 devices with varied thickness.All the seven transport devices (from 5 SL to 8 SL) manifest quantized ρyx ~ h/e 2 in the field-polarized Chern insulator state, suggesting the high quality of our MnBi2Te4 devices.We demonstrate that fabrication process can result in mismatched even-odd layer dependent magnetotransport in few-layer MnBi2Te4.
A comprehensive study of the magnetotransport behaviors in a 6-and 7-SL device shows that both even-and odd-number-SL MnBi2Te4 can exhibit zero Hall plateau in zero magnetic field.
A statistical survey of the Oc in more than 200 MnBi2Te4 reveals that the effective thickness for magnetotransport could decrease by 1 SL after undergoing the electron-beam-lithography (EBL) method.Our finding not only provides an explanation to the controversies concerning the even-odd discrepancy of magnetotransport in few-layer MnBi2Te4, but also highlights the critical issues regarding the fabrication and characterization of 2D material devices.

Device Fabrication and Basic Calibration of Transport Properties
MnBi2Te4 few-layer flakes were prepared via mechanical exfoliation on 285 nm SiO2/Si substrates (see methods section).We then determined the thickness of the flakes using optical methods (Fig. 1b), atomic force microscopy (Fig. 1c) and scanning superconducting quantum interference device (SQUID) (see supplementary section A).The calibration of thickness was also examined by additional layer-dependent measurements on flakes exfoliated from crystal #1, including nonlocal transport, scanning microwave impedance microscopy (sMIM), ultrafast pump-probe reflectivity, and Raman spectroscopy 13,[39][40][41] .By conducting Oc measurement immediately after exfoliation in a glovebox, one can quickly determine the thickness without exposing the sample to the atmosphere.Figure 1d summarizes the one-to-one correspondence between Oc and thickness (SL number), which are highly consistent with the results measured in different crystals by another group 23 .For few-layer MnBi2Te4, a remarkable feature is that Oc changes its sign from negative to positive when the thickness increases from 6 SL to 7 SL, as guided by the dashed line.After the identification of thickness, the flakes were fabricated into field-effect-transistors by standard EBL method and coated with a layer of Polymethyl Methacrylate (PMMA) for protection (see supplementary section A for details).To study the layer-dependent transport properties, we first measured the temperature (T) dependent ρxx for a 6-SL and 7-SL device (S2 and S6) at μ0H = 0, with the Fermi levels (EFs) gated to the charge neutrality points (CNPs).Both the two flakes were derived from crystal #1.At the CNP, the transport is mainly conducted by the topological surface states or edge states.Therefore, both devices exhibit overall insulating behavior and display kink feature at their Néel temperatures (TNs).Compared to TN ~ 25 K for MnBi2Te4 bulk crystals 5 , the TNs for the few-layer devices are suppressed to 20.6 K and 21.6 K, respectively, possibly due to the enhanced fluctuations at lower dimensions.

Layer-dependent Magnetoelectric Transport Properties for Varied Vg
As a layered AFM TI, the most intriguing feature of MnBi2Te4 is the layer-dependent transport properties.We performed systematic μ0H dependent transport measurements on the two devices at different Vgs (see supplementary section B for transport data at various Ts), as presented in Figs.2a and 2b.With the application of Vg, EF is continuously tuned from the valence band towards the conduction band, manifested by the slope change of normal Hall effect from positive to negative.For the 6-SL MnBi2Te4, the most remarkable feature lies in the broad zero Hall plateau in the low-field AFM regime when its EF is tuned within the band gap.In the panels enclosed by thick magenta boundaries, the zero Hall plateau persists in a wide range of Vg from 36 to 49 V.Meanwhile, ρxx shows insulating behavior and reaches as high as 4 h/e 2 .These behaviors are indicative of the axion insulator state in even-number-SL MnBi2Te4, where the counter-propagating surface Hall currents give rise to a broad zero Hall plateau in ρyx and a large ρxx (ref. 5,12,13).An out-of-plane μ0H drives the system into a Chern insulator at the CNP (Vg = 42 V), where ρyx is quantized in h/e 2 and ρxx drops to zero for μ0H > 6 T.These behaviors are consistent with previous reports on the topological phase transition between axion insulator and Chern insulator in a 6-SL device 5,20 .
Figure 2b shows the μ0H-dependent ρyx and ρxx at various Vgs for the 7-SL device, which exhibit unexpected zero Hall plateau phenomenon rather than AH hysteresis in the AFM state.
At high field, the 7-SL device show transport behaviors very similar to the 6-SL device with quantized ρyx and vanished ρxx, as the Chern insulator quantization in the FM state does not depend on thickness.However, in the low-field AFM regime, some unexpected behaviors are observed.As guided by the black dashed lines, throughout the Vg range, ρyx displays overall linear behaviors and smoothly changes the slope from positive to negative.No discernable hysteresis is observed during the field sweep process.Remarkably, at Vg = 13 V, a wide zero Hall plateau appears between μ0H = ± 3 T.Meanwhile, ρxx reaches the maximum but with a smaller value than that of the 6-SL device.Theoretically, the zero Hall plateau phenomenon is unique to even-number-SL MnBi2Te4 with fully compensated AFM order, thus should be absent in odd-number-SL MnBi2Te4.These unexpected results strongly suggest the existence of some unknown mechanism that could modify the magnetotransport of few-layer MnBi2Te4.
In order to realize the QAH and axion insulator state in few-layer MnBi2Te4, EF must be tuned by Vg to lie in the Dirac point gap opened by FM order.To reveal the nature of the zero Hall phenomena in the two devices, we extract the value of ρxx and the slope of ρyx at μ0H = 0, and plot them as a function of Vg.As displayed in Fig. 3a, ρxx of the 6-SL device first goes up to a large value of 4 h/e 2 for Vg < 25 V and remains unchanged in a broad Vg window, and then decreases to a small value for Vg > 50 V.Meanwhile, dρyx/dH exhibits a clear three-stage transition with varying Vg.In the first stage with Vg < 30 V, dρyx/dH progressively decreases with increasing Vg, and is attributed to the depletion of hole-type carriers.For Vg from 25 to 30 V, dρyx/dH changes sign from positive to negative.As Vg is further increased, a broad zero plateau forms and persists within a Vg window of 13 V.Further application of Vg injects more electron-type carriers and ultimately leads to negative dρyx/dH.Such behaviors unequivocally suggest that the zero Hall plateau state in the 6-SL MnBi2Te4 is a genuine quantized Hall state (C = 0) with EF residing in the band gap, which is consistent with our previous report 5 .
Despite the superficially similar zero Hall plateau during μ0H sweep in the 7-SL device, it manifests different behavior in response to Vg.In contrast to the 6-SL device where dρyx/dH = 0 exists in a broad Vg window, for the 7-SL device, dρyx/dH = 0 only appears at a single Vg point corresponding to the sign change of ρyx slope.Meanwhile, we notice that for the 6-SL device, there is a broad Vg range where the zero Hall plateau and the Chern insulator coexist.
However, for the 7-SL device, the zero Hall plateau only occurs in a Vg smaller than the Chern insulator regime (see supplementary section C for colormaps of ρyx and ρxx).For longitudinal transport, the Vg range for large ρxx in the 7-SL device is also narrower than the 6-SL device.
To better visualize the different manifestations of the zero Hall plateaus, we summarize the variations of dρyx/dH with Vg and μ0H for the two devices to two colormaps, as shown in Figs.3c and 3d.The magenta dashed lines label the regimes for dρyx/dH = 0.It clearly shows that there is a well-defined zero Hall resistivity plateau regime in the parameter space for the 6-SL device.However, for the 7-SL device, the zero Hall plateau exists in a narrower regime.
The quantitative differences of the zero Hall plateaus in the Vg range, as well as that in the T range (see supplementary Fig. S4), indicate different manifestations of the zero Hall plateau associated with the axion insulator state of different energy gaps.
The observation of zero Hall plateau phenomenon in the 7-SL device bears resemblance to a recent observation of the discrepancies between magnetic order and transport properties in few-layer MnBi2Te4, where the absence of AH effect was observed in a 5-SL device with uncompensated AFM order, meanwhile a pronounced AH effect was found in a 6-SL device with fully compensated AFM order 24,25 .Previous magnetic measurements have demonstrated that the AFM order in MnBi2Te4 is highly robust and persist to the top surface 42,43 .In contrast, the surface electronic band structures have been found to be fragile and sensitive to the type and concentration of defects 38,[44][45][46][47] .We notice that most of the magnetization measurements in previous reports 24,25 were performed on MnBi2Te4 with fresh surface, whereas the transport measurements were conducted exclusively in devices after fabrication.It is highly possible that the even-odd discrepancy of magnetotransport in MnBi2Te4 arises from the influences of fabrication process.To verify our conjecture, we tracked the Oc values measured before and after fabrication for the two devices, as illustrated in Figs.3c and 3d.Surprisingly, we find a substantial Oc reduction from +12.5 to -0.2 % for the 7-SL device after fabrication, indicating that the thickness determined by Oc is significantly reduced by 1 SL.In contrast, the Oc value of the 6-SL device is less influenced, only changing slightly from -7.4 to -10.0 %.

Statistical Survey of Optical Properties and its Effects on Charge Transport
In order to figure out the reason for the color change and to exclude any artificial factor that may contribute to our observation, such as the transport electrodes, fabrication conditions, and imaging parameters etc., we conducted thorough control experiments on many few-layer flakes and compared Oc changes under different conditions (see supplementary section D for details).To mitigate the potential interferences from extrinsic effects, such as thermal cycling, environmental doping, and aging effect, Oc was obtained immediately after surface treatment in a glovebox 36,37,48,49 .Of the many relevant factors, we notice that the contact with PMMA plays the most crucial role on the reduction of Oc, a factor that was rarely noticed in previous studies of 2D materials.We have performed a statistical survey on more than 200 MnBi2Te4 exfoliated from four crystals grown by different groups, and the main results are summarized in Figs.4a and 4b.The most striking observation is that most of the studied MnBi2Te4 flakes exhibit Oc reduction, although to different extents, which is never reported in previous studies of MnBi2Te4.As presented in Fig. 4b

Discussion
Based on the above experimental observations, we discuss the possible explanations for the even-odd discrepancy of magnetotransport in few-layer MnBi2Te4.It may be suspected that one physical layer is unintentionally removed during the fabrication process, leading to an odd (even)-number-SL MnBi2Te4 to manifest transport behaviors that are characteristic of an even (odd)-number-SL MnBi2Te4 with 1 less SL 24,25 .However, such scenario can be safely excluded.We performed atomic force microscopy measurement on the flakes exfoliated from the most sensitive crystal (#4).All these samples exhibit pronounced Oc reduction during the fabrication process (see supplementary section E), however their physical heights determined by atomic force microscopy remain unchanged.In supplementary section F, we also compare the variations in the magneto-optical Kerr effect (MOKE) and the coherent interlayer phonon frequency of two MnBi2Te4 before and after PMMA contact.It further demonstrates that the fabrication mainly affects the effective thickness rather than the physical thickness.Therefore, a more plausible scenario is that the change of Oc arises from the modification of the magnetic or electronic structures 45,46,50 .In the experimental researches of MnBi2Te4, it is a widespread phenomenon that MnBi2Te4 exhibits sample-dependent behaviors, whether between different crystals or different flakes exfoliated from the same crystal 3,47,51 .A prevailing understanding attributes this to the various defects and the non-uniformity within MnBi2Te4 bulk crystal.It has been highlighted that the surface defects and the perturbations to the surface can result in instability of MnBi2Te4 (refs.38,47,50-56).Given the intricate physical and chemical process involved in the fabrication process, we attribute the Oc variation to the fabrication-catalyzed instability of MnBi2Te4 surface.
It is worth noting that some imaging experiment and theoretical calculations have clearly identified some physical mechanisms that can result in a decrease of effective thickness.For instance, a scanning transmission electron microscopy imaging experiment demonstrated that the synergistic effect of a high concentration of Mn-Bi site mixing and Te vacancy can trigger a surface reconstruction process from one SL of MnBi2Te4 to a quintuple layer of Mn-Bi2Te3 and an amorphous double layer of MnxBiyTe (ref.50).As a result, the effective thickness for the MnBi2Te4 structure is reduced by 1-SL.Theoretical calculations also reveal that a surface charge redistribution process can relocate the surface state from the first SL to the second SL, resulting in the decrease of effective thickness for magnetotransport 51 .Recently, a theoretical work demonstrates that a small expansion of the interlayer van der Waals gap can result in a noteworthy reduction in the surface gap 56 .Specifically, for a (7+1) SL MnBi2Te4, it triggers a topological phase transition with Chern number change by one.An odd (even)-number-SL MnBi2Te4 will naturally manifest magnetotransport properties akin to its even (odd)-number-SL counterpart with 1 less SL.Based on the sample-dependent defect type and concentration, as well as the susceptibility of MnBi2Te4 surface to perturbations 47,48,50,51,56 , we hypothesize that the sample dependent behaviors observed during the fabrication arise from the PMMAcatalyzed surface instability.Notably, prior researches on graphene, MoS2, and WSe2 indeed suggested that the PMMA residuals on the surface influence the intrinsic properties of the 2D materials [57][58][59] .It can not only increase the observed thickness in the atomic force microscopy measurement through absorption, but also act as charge source, prompting the surface charge redistribution.Our topography measurement has indeed shown island-like PMMA residuals on the MnBi2Te4 surface (see supplementary Fig. S8).In addition, various adsorbates trapped between layers during the fabrication can also expand the van der Waals gap 60 .Therefore, it is likely that the combined influences of non-uniformity, defects, and PMMA contribute to the sample dependent behaviors in response to fabrication.Further studies are needed to fully understand the underlying mechanisms.In Figs.4d and 4e, we display the process of effective thickness reduction with the magenta frame indicating the effective thickness for transport.
The reduced gap elucidates the narrower Vg and T range of the zero Hall plateau for the 7-SL sample (S6).
While the precise mechanism through which PMMA influences the quality of MnBi2Te4 samples remains incompletely understood, a potential solution to circumvent such fabrication issue involves isolating PMMA from the surface during the fabrication.Building upon recent advancements in low-damage lithography in the QAH system 35,37 , we suggest that depositing a thin layer of AlOx on the surface of MnBi2Te4 prior to fabrication may alleviate the damage of PMMA.In supplementary section G, we present our preliminary results obtained in crystal #5, which demonstrates the efficacy of the modified method in addressing the current issue.
In addition to the zero Hall plateau in the 7-SL MnBi2Te4 device, the fabrication-induced mismatched layer dependent magnetotransport behaviors are also evident in MnBi2Te4 flakes with other thicknesses, as displayed in Figs.4f and 4g.Among the seven samples, devices S1 and S5 were derived from crystals #3 and #2, respectively.All the other devices were derived from crystal #1.Notably, those PMMA-insensitive MnBi2Te4 with less-affected Oc (blue stars in Fig. 4b) exhibit the anticipated behaviors for both even-and odd-number-SL MnBi2Te4.In contrast, samples with pronounced Oc change (red stars in Fig. 4b) exhibit transport behaviors inconsistent with their nominal thickness.Specifically, as shown in Fig. 4g, odd-number-SL devices display vanished AH hysteresis in the AFM regime, while even-number-SL devices display hysteresis behaviors with counterclockwise chirality, as indicated by the black arrows.
The AH effect with reversed chirality may arise from the electric field due to gate or substrate, or the competition between various intrinsic and extrinsic mechanisms 23,32,33,48,61 .In addition to the Hall effect, since the transport of odd-and even-number-SL MnBi2Te4 are conducted by chiral and helical edge states 13,39 , the fabrication-induced mismatched even-odd dependent magnetotransport should also be manifested by the nonlocal transport measurements, which are observed in our experiment (see supplementary section H for details).
We have conducted a comprehensive investigation of the transport properties in a large number of few-layer MnBi2Te4 flakes.By tracking the quantized Hall plateau with respect to μ0H and Vg, and comparing the optical properties before and after the fabrication process, our study elucidates the relationship between transport behaviors and device fabrication process.
Our research has uncovered a condition in which the effective thickness for charge transport in MnBi2Te4 becomes decoupled from its pristine physical thickness, which is never reported in previous studies.Although the exact microscopic mechanism underlying the change of Oc remains to be determined, and we cannot exclude that those devices exhibiting unchanged Oc are not affected by fabrication because the AH effect (0.1 h/e 2 ) in odd-number-SL MnBi2Te4 is not quantized, our experiments still provide highly valuable insights for the fabrication of high-quality MnBi2Te4 toward realizing quantized phenomena.Our finding not only explains the controversies concerning the mismatched even-odd layer dependent magnetotransport in MnBi2Te4, but also highlights the critical issues regarding the fabrication and characterization of devices based on 2D materials.

Methods
Crystal growth High-quality MnBi2Te4 single crystals were synthesized independently by different methods.For crystal #1, it was grown by directly mixing Bi2Te3 and MnTe with the ratio of 1:1 in a vacuum-sealed silica ampoule.After heated to 973 K, the mixture was slowly cooled down to 864 K, followed by a long period of annealing process.The phase and crystal structure were examined by X-ray diffraction on a PANalytical Empyrean diffractometer with Cu Kα radiation.For crystal #2, it was grown by conventional flux method.Mn powders, Bi and Te were weighed with the ratio Mn:Bi:Te = 1:8:13 (MnTe:Bi2Te3 = 1:4) in an argon-filled glovebox.The mixtures were loaded into a corundum crucible which was sealed into a quartz tube.Then the tube was then put into a furnace and heated up to 1000 °C for 20 hours.After a quick cooling to 605 °C with the rate of 5 °C/h, the mixtures were then slowly cooled down to 590 with the rate of 0.5 °C/h and kept for 2 days.Finally, the crystals were obtained after centrifuging.For crystal #3, it was grown by the conventional high-temperature solution method.The Mn, Bi and Te blocks were weighed with a ratio of Mn:Bi:Te = 1:11.3:18,and placed in an alumina crucible, which were then sealed in a quartz tube in argon environment.
The assembly was first heated up in a box furnace to 950 °C and held for 10 hours, and then cooled down to 700 °C within 10 hours and further cooled down to 575 °C in about 100 hours.
After the heating procedure, the quartz tube was then taken out quickly and decanted into the centrifuge to remove the flux from the crystals.For crystal #4, it was grown by flux method using MnCl2 as the flux.The raw materials of Bi2Te3 powder, Mn lump, Te lump and MnCl2 powder were mixed with a molar ratio of 1:1:1:0.3 and then placed in a dry alumina crucible, which was sealed in a fused silica ampoule under vacuum.The ampoule was then placed in a furnace and heated up to 850 ℃ for over 20 hours, kept there for 24 hours, cooled down to 595 ℃ in over 5 hours, kept there for 150 hours, and finally cooled to room temperature in 5 hours.After the steps above, the yielded ingot was cleaved into millimeter-sized crystals with metallic luster.For crystal #5, it was grown by directly mixing Bi2Te3, MnTe and Te with the ratio of 1:1:0.2 in a vacuum-sealed silica ampoule.The ampoule was slowly heated to 900°C at a rate of 3°C/min and maintained at this temperature for 1 hour.Subsequently, the sample was cooled at a rate of 3°C/min to 700°C, held at this temperature for 1 hour.The temperature was then gradually decreased to 585°C at a rate of 0.5°C/min and maintained for annealing for 12 days.After the annealing process, the quartz ampoule was quenched in water to avoid phase impurities.Millimeter-sized MnBi2Te4 crystals were obtained after crushing the ingot.Scanning SUIID measurement Scanning SQUID measurements were carried in a different cryostat from the transport measurements.Scanning 2-junction SQUID susceptometers with two balanced pickup loops of 2 μm diameter in a gradiometric configuration were utilized as the SQUID sensors.Each of them was surrounded by a one-turn field coils of 10 μm diameter.

Device fabrication
The DC flux was measured through the pickup loop using a voltage meter (Zurich Instrument HF2LI) as a function of position and reflects the intrinsic magnetization of the sample.
Polar MOKE measurement Polar MOKE measurements were carried using a 633 nm HeNe laser.After transmitting through a linear polarizer, the light was focused to a 2µm spot on the sample by a reflective objective at normal incidence to avoid the large backgrounds that occur when a typical lens is used.The sample was mounted on a cold stage at 3 K within the vacuum chamber of an optical superconducting magnet system.The reflected beam is modulated at ~ 50 kHz by a PEM, split by a Wollaston prism, and detected using a balanced photodiode.The resulting 50 and 100 kHz modulations detected by lock-in amplifiers then correspond to the ellipticity and rotation angle of the beam respectively.We additionally modulate the intensity of the beam with a frequency of 2317 Hz chopper to measure the DC signal for normalization using a third lock-in.
, the blue and magenta dashed lines mark the area of Oc reduction of 0 and 20 %, respectively.The flakes situated close to the blue dashed line display little Oc change after device fabrication, whereas the flakes close to the magenta dashed line experience a pronounced Oc reduction, corresponding to an effective thickness decrease of 1 SL.The subtle increase of Oc in some certain samples is attributed to measurement error (see methods).In the top panel of Fig. 4a, we present the optical images of four typical MnBi2Te4 flakes, which clearly illustrate the pronounced color change caused in the fabrication process.In Fig. 4c, we further analyze the distribution of the Oc change for the different crystals.The leftward shift of the center of the blue lines clearly indicates that the impacts of the fabrication process on Oc are highly crystal-dependent.For most of the samples exfoliated from Crystal #1, their Oc values are only slightly affected.In contrast, almost all the flakes exfoliated from Crystal #4 exhibit significant reduction in Oc, corresponding to a thickness of 1 SL.
MnBi2Te4 flakes were exfoliated onto 285 nm-thick SiO2/Si substrates by using the Scotch tape method in an argon-filled glove box with O2 and H2O levels lower than 0.1 ppm.Before exfoliation, all SiO2/Si substrates were pre-cleaned by air plasma for 5 minutes at ~ 125 Pa pressure.To minimize the experimental errors due to the subtle difference in measurement conditions, such as the position of the flakes in the light fields, the uniformity of illumination, the size and shape of the sample, and the presence of electrode, the Oc shown in the main text were calculated by averaging the Oc of different parts across the sample.For the transport devices, thick flakes around the target sample were first scratched off by using a sharp needle in the glove box.A layer of 270 nm PMMA was spin-coated before EBL and heated at 60 °C for 5 minutes.After the EBL, 23 to 53 nm thick Cr/Au electrodes (3/20 to 3/50 nm) were deposited by a thermal evaporator connected with an argon-filled glove box.Before the fabrication and sample transfer process, the devices were always spin-coated with a PMMA layer to avoid contact with air.All the seven devices (S1-S7) shown in the text were fabricated through the same process.Transport measurement Four probe transport measurements were carried out in a cryostat with the lowest temperature 1.6 K and out-of-plane magnetic field up to 9 T. The longitudinal and Hall signals were acquired simultaneously via lock-in amplifiers with an AC current (200 nA, 13 Hz) generated by a Keithley 6221 current source meter.To correct for the geometrical misalignment, the longitudinal and Hall signals were symmetrized and antisymmetrized with magnetic field respectively.The back-gate voltages were applied by a Keithley 2400 source meter.

Fig. 4 |
Fig. 4 | Statistical analysis of Oc for more than two hundred flakes and distinct thickness dependent transport properties.a, Optical images of four representative samples taken in a glove box right after exfoliation (top panel) and after the removal of PMMA (bottom panel).b, Summary of the Oc values of 223 MnBi2Te4 flakes after exfoliation and after the removal of PMMA.The blue and magenta dashed lines mark the Oc reduction by 0 and 20 %.Different colored dots represent the data acquired from different crystals.c, Distribution of Oc change in the four different crystals.For the most PMMA-sensitive crystal (#4), fabrication can give rise to Oc change corresponding to a thickness of 1 SL.d-e, Illustrations of the influence of PMMA on the surface electronic structure for a 7-SL MnBi2Te4.f-g, Thickness dependent ρyx behaviors for MnBi2Te4 without (blue) and with (red) severe Oc change.