Anomalous Landau quantization in intrinsic magnetic topological insulators

The intrinsic magnetic topological insulator, Mn(Bi1−xSbx)2Te4, has been identified as a Weyl semimetal with a single pair of Weyl nodes in its spin-aligned strong-field configuration. A direct consequence of the Weyl state is the layer dependent Chern number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C$$\end{document}C. Previous reports in MnBi2Te4 thin films have shown higher \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C$$\end{document}C states either by increasing the film thickness or controlling the chemical potential. A clear picture of the higher Chern states is still lacking as data interpretation is further complicated by the emergence of surface-band Landau levels under magnetic fields. Here, we report a tunable layer-dependent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C$$\end{document}C = 1 state with Sb substitution by performing a detailed analysis of the quantization states in Mn(Bi1−xSbx)2Te4 dual-gated devices—consistent with calculations of the bulk Weyl point separation in the doped thin films. The observed Hall quantization plateaus for our thicker Mn(Bi1−xSbx)2Te4 films under strong magnetic fields can be interpreted by a theory of surface and bulk spin-polarised Landau level spectra in thin film magnetic topological insulators.


Introduction
Magnetic topological insulators (MTIs) and Weyl semimetals have both received a great deal of attention in recent condensed matter physics research [9][10][11] .Particularly, the intrinsic MTI MnBi2Te4 has provided researchers with an ideal candidate to study the relationship between topological quantum states and magnetic phases [5][6][7][8][12][13][14][15][16][17][18][19] . Theoreical predictions [1][2][3][4] and recent experimental results 20 show that when its Mn local moment spins are aligned by an external magnetic field, the energy bands of bulk MnBi2Te4 have a single isolated pair of Weyl crossing points that are close to the Fermi level and therefore can be accessed by controlling the carrier-density.
In this work, we report on thickness-dependent magneto-transport studies of the mechanicallyexfoliated Mn(Bi1-xSbx)2Te4 thin flakes with three different Sb concentrations.The Sb substitutions in the MnBi2Te4 parent compound (i) move the Fermi level of the bulk bands closer to charge neutrality point (CNP) [20][21][22] , and (ii) modulate the Weyl point separation in momentum space as illustrated in Fig. 1a.We focus here on the Chern insulator states of the spin-moment aligned Mn(Bi1-xSbx)2Te4.We show that Sb substitution extends the surface gap regime to a wider thickness range by suppressing conduction from the trivial bulk bands, enabling better access to their Weyl physics.Thin films MnBi2Te4 provide a rich plethora of topologically distinct quasi-two-dimensional (2D) states, that includes Chern insulator [5][6][7][8]12,13 and axion insulator 1,12,14,23 states.
The application of external magnetic fields to Mn(Bi1-xSbx)2Te4, therefore, generates an interplay between Chern gaps and Landau levels (LLs) quantization, allowing us to study rich quantum Hall physics that has not yet been fully explored.

Mn(Bi1-xSbx)2Te4 films at magnetic field B= 0
According to density-functional-theory (DFT), the spin-aligned magnetic configuration of Mn(Bi1-xSbx)2Te4 is a simple type-I (or II depending on lattice parameters 1 ) Weyl semimetal with Weyl points at ±  along the Γ −  line.As shown in Fig. 1a, the distance between Weyl points (2  ) decreases with Sb fraction x.The bulk Hall conductivity normalized per layer can be expressed as conditions on the sample quality required for observation of the quantum anomalous Hall effect in the absence of magnetic field.Thus, our focus in this article is on the Chern insulator state in the presence of a magnetic field, which can be observed more consistently than the quantum anomalous Hall effect but is intimately related to the zero magnetic field band topology.
We first examine the low-temperature transport properties of the Mn(Bi1-xSbx)2Te4 thin flakes at zero magnetic field.Fig. 1c plots the four-terminal resistivity (ρ xx CNP ) measured at the CNP as a function of Mn(Bi1-xSbx)2Te4 thickness for different Sb substitution levels (x= 0, 0.20, and 0.26).
We refer to a sample as surface-like when its conductivity is thermally activated at low temperatures, suggesting the presence of a bulk energy gap.Samples classified as bulk-like have weaker temperature dependence and are presumed to have disorder-induced bulk states at all energies.Representative temperature-dependent profiles for MnBi2Te4 are shown in Fig. S4, where the samples convert from being surface-like to being bulk-like as thickness increases., where   is the gate capacitance (≈80 nF/cm 2 for a ~30 nm mica dielectric), Gxx (= 1/xx) is the four-terminal conductance, and Vg is the voltage applied through the graphite gate-electrode, is plotted in Fig. 1d.We see that the electron mobility increases with gate voltage (electron density), while the hole mobility responds weakly and remains small in the low gate voltage (hole density) regime.The mobility is more than one order of magnitude higher mobility for electrons compared to hole carriers.By applying dualgate voltages, the top and bottom surface carrier densities can be modulated to achieve electron mobilities as high as 4000 cm 2 /Vs at a total carrier density of >510 11 cm 2 .

C= 1 state in Mn(Bi1-xSbx)2Te4 films
The magnetic field-dependent transport properties of Mn(Bi1-xSbx)2Te4 films with the Sb concentrations x= 0, 0.20, and 0.26 for a variety of thicknesses were studied.When a perpendicular magnetic field is applied, Mn(Bi1-xSbx)2Te4 undergoes the magnetic transitions 20,22 from the antiferromagnetic (AFM), to the canted, and finally to the aligned spin-moment configurations.To compare the magnetic field dependence of the Mn(Bi1-xSbx)2Te4 film with different Sb substitutions, we plot the color maps of yx as functions of gate voltage and magnetic field for the 8-SL MnBi2Te4, 10-SL Mn(Bi0.8Sb0.2)2Te4,and 21-SL Mn(Bi0.74Sb0.26)2Te4,respectively, in Figs.2a-c.In general, the spin-flop and spin-flip transitions happen at magnetic fields of ~±2-3T and ~±7T, respectively, are observed in all the samples.The magnetic transition fields are identified by the color line marks depicted in Figs.2a-c, which agrees with the theoretically calculated values 25 .Line profiles of xx and yx curves for the 8-SL MnBi2Te4, 10-SL Mn(Bi0.8Sb0.2)2Te4,and and 10-SL Mn(Bi0.8Sb0.2)2Te4could be due to the uncompensated surface magnetization 15 or antiferromagnetic domain walls 26 , whereas no zero field hysteretic behavior observed in the 21-SL Mn(Bi0.74Sb0.26)2Te4film.The observed C= 1 states in all three Mn(Bi1-xSbx)2Te4 films with the different Sb substitutions confirm that their spin-alignment configuration is in the topological phase with a Chern insulator gap.
To further evaluate the thickness-dependence of the C= 1 state, we plot xx and yx values as a function of thickness, with the gate voltage tuned to the yx maximum for each thickness, at magnetic field of 9T.Our primary observation is that, despite their similar field-dependent quantization behavior, the C= 1 state prolongs to the larger thickness range with the Sb substitutions as shown in Figs.2g-i.This trend is consistent with our DFT calculations, which indicate the shift of the Weyl point position by Sb doping.In Fig. 2g, the observation of C= 1 quantum Hall states up to 8-SL MnBi2Te4 agrees with our calculations (Fig. 1b) and the literature 5 .
Although this can be somehow related to the extension of the surface-like regime in thin films by the optimal Sb substitutions, according to our calculations in Fig. 1b, the Mn(Bi0.74Sb0.26)2Te4films at the given film thickness range should lie in the higher Chern number states.

Dualgate tuning of Chern states
To further identify the Chern insulator states in our thicker Mn(Bi0.74Sb0.26)2Te4,we perform a detailed analysis for these devices in a dualgating platform.(extended data Fig.E3).The existence of the C= 2 state in the CAFM phase is also captured by the xx and xy dualgate maps swept at magnetic field of 6T as shown in extended data Fig.E2.
Also, we note that the C= 2 state coincides with the ρ xx CNP at the zero magnetic field.
We further analyze the quantization states in the 14-SL Mn(Bi0.74Sb00.26)2Te4.In addition to the C= 1 state, the dualgate maps in Figs.4e and f show a series of oscillatory xx minima and xy plateaus, respectively, corresponding to the different quantum Hall states develop at 9T when tuning the dualgate voltages.The linecuts of xx and xy as a function of backgate voltage at magnetic field of 9T as depicted in Fig. 4i reveal the quantum Hall plateaus with C= 0, 1, 3, 5, etc.
The color maps of xx and yx as functions of magnetic field and backgate voltage (Fig. S8) resolve the fan diagram of Landau levels at odd integer fillings, where the xx minima for each filling factor can be traced down to a single gate voltage corresponding to the CNP.Such feature is similar to the surface states' LL fan diagram in non-magnetic topological insulators 27 .Different from the 18-SL and 21-SL Mn(Bi0.74Sb00.26)2Te4, the C= 1 state in the 14-SL Mn(Bi0.74Sb00.26)2Te4 at high magnetic field coincides with the CNP when traced down to zero magnetic field.The schematic diagrams inserted in Figs.3g-i illustrate the surface band structures with different Chern states resolved in the respective thicknesses of the Mn(Bi0.74Sb00.26)2Te4films.

Anomalous Landau levels in magnetic topological insulators
To interpret the rich Chern insulator states observed in our Mn(Bi0.74Sb0.26)2Te4films, we calculate their LLs spectra using a simplified model 3 , in which the 2D massive Dirac cones are coupled by tunneling within and between the compound's septuple-layer building blocks.For explanation purpose, we first present in Fig. 4a the LLs spectra for a non-magnetic TI thin film 28  A general picture of the relationship between the Chern numbers and exchange coupling is illustrated in Fig. 4b where the surface and bulk spin-splitting bands are plotted.In the case of   = 0, the number of subbands above and below the Fermi level equals, and thus the Chern number is 0. When the exchange field is turned on (  ≠ 0), the anomalous LLs are spin-polarized with two nearly degenerate surface anomalous LLs labeled with green and red curves in Fig. 4b, the filling factor is thus 1 since the two nearly degenerate surface anomalous LLs are either above or below the Fermi level.When the crossings between up (yellow curves) and down (blue curves) spins happen, the filling factors change by one.The quantization is therefore expected to be observable over a range of filling factors magnitudes centered on 1+ the number of  ⃗ ⊥ = 0 crossings that occur between up and down spins.When the magnetic field is reversed, the spin of the anomalous LLs is reversed, but the sign of the exchange coupling is also reversed, so the sign of the filling factor range at which strong quantum Hall effects will not change.The sign depends only on the sign of the exchange coupling between the local moments and the Dirac electrons.Our observations, therefore, provide evidence that the anomalous LLs are spin-polarized.
We also note that since the position of Weyl points depends on the exchange splitting 3 , which may be decreased by antisite defects leading to a further decrease of   , and thus may change the Chern numbers and the gaps in the absence of magnetic field 30 .Fig. 4c and 4d examine two cases, one is close to a transition (refer to Figs.S2 and S3 for the dependence of gaps versus various exchange splitting) between two different Chern numbers and hence has a small gap at B= 0 (with Js= 34 meV), and the other is a relatively large Chern gap at B= 0 (with Js= 30 meV).Although the size of the gap at B= 0 is sensitive to details that may vary from sample to sample, the strong magnetic field behavior in which well-spaced anomalous LLs dominate over a finite region of filling factor does not vary too much.
Finally, we compare our experimental results for the Mn(Bi0.74Sb0.26)2Te4films with the calculated LLs structure at the similar thickness and Sb doping level.We plot in the extended data Under a strong magnetic field, the calculated filling factor of C= 1 exhibits the largest LL gap in all three thicknesses of Mn(Bi0.75Sb0.25)2Te4film.This explains the experimental observation where the C= 1 state was observed over the wide thickness range in our Mn(Bi0.74Sb00.26)2Te4film.
One noticeable feature in the case of larger film thickness is that the C= 1 filling shifts below Fermi level and the surface gap at the Fermi level is at filling factor of higher Chern number (refer to Fig. S3 for details).This is consistent with the observation in our 18-SL Mn(Bi0.74Sb00.26)2Te4film, indicating that the 18-SL is in a higher Chern number state of C= 2.Moreover, the calculated gap size for the C= 3 state exceeds the C= 2 gap at high magnetic field, which explains the welldeveloped C= 3 state at 9T at this film thickness.This inevitably supports the development of the anomalous LLs near the Fermi level under strong magnetic field (red shades in Fig. 3g-i).Our results further imply that the Hall quantization states at larger film thickness provide an ideal platform for the realization of the spin-polarized anomalous LLs in thin film Weyl semimetals.

Summary
In summary, we studied the magnetoelectrical transport of the intrinsic MTI Mn(Bi1-xSbx)2Te4 for x= 0, 0.20, and 0.26 by probing Chern quantization states and their relationship with the flake thickness and Sb concentrations.We identified the thickness ranges for surface-like insulating and bulk-like metallic transport regime.The thickness-dependent Hall conductivities, particularly for the C= 1 Chern insulator state in the Mn(Bi1-xSbx)2Te4, show a correlation with the separation of Weyl points as described by our theoretical models, indicating that Mn(Bi1-xSbx)2Te4 behaves as thin film Weyl semimetals.Our transport results for different Sb concentrations highlight the importance of B= 0 Weyl point separation for the Chern state's quantization.Moreover, we showed that the interplay of Chern insulator states at B= 0and Landau quantization at high magnetic fields gives rise to the intriguing anomalous LLs in Mn(Bi1-xSbx)2Te4.Our work illustrates the complexity of the intertwined topological surface states and ferromagnetism in Landau quantization and thus can serve as a guide to bridge the gap between the 2D Chern insulators and 3D Weyl semimetals.

Methods
Materials.Mn(Bi1-xSbx)2Te4 bulk crystals at different Sb doping levels were grown by a self-flux growth method 20,24 .Variable thicknesses of Mn(Bi1-xSbx)2Te4 thin flakes were exfoliated from the parent bulk crystals and then transferred into the heterostructures of graphite/hexagonal boron nitride sandwiched layers using a micromanipulator transfer stage.The graphite/hexagonal boron nitride layers serve as the gate-electrode/dielectric layers.The exfoliation and transfer processes were performed in an argon gas-filled glovebox with O2 and H2O levels <1 ppm and <0.1 ppm, respectively, to prevent oxidation in thin flakes.We fabricated the Mn(Bi1-xSbx)2Te4 devices into the Hall bar configuration using a standard electron beam lithography process and metal deposition with Cr/Au (20 nm/60 nm) as the contact electrodes using a CHA Solution electron beam evaporator.The Mn(Bi1-xSbx)2Te4 flakes were protected by polymethyl methacrylate (PMMA) while transporting for lithography and metal deposition processes.Theoretical calculations.DFT calculations were performed using Vienna Ab initio Simulation Package (VASP) 31 in which Generalized gradient approximations (GGA) of Perdew-Burke-Ernzerhof (PBE) 32 have been adopted for exchange-correlation potential.On-site correlation on the Mn-3d states is treated by performing DFT + U calculations 33 with U-J as 5.34 eV.The global break condition for the electronic SC-loop is set to be 10 -7 eV and the cutoff energy for the plane wave basis set is 600 eV during the self-consistent (SC) calculations.A 9×9×6 Gamma-centered k-point integration grid was employed with Gaussian broadening factors as 50 meV.In the calculations of bulk Mn(Bi1-xSbx)2Te4, supercells of 2×2×1 unit cell were used to model the doping density of Sb atoms with densities of 0, 25%, 50%, 75%, and 100%.The calculations of Landau levels are based on the coupled Dirac cone model illustrated in the supplemental material, with the parameters estimated from the DFT calculations.

Measurements
is the position of Weyl point and / is the size of Brillouin zone along the Γ −  line with d as the septuple layer (1SL 1.4 nm).The corresponding thin films can be viewed as quasi-2D crystals and are expected to have quantized anomalous Hall conductivities with Chern numbers (C) that increase by one when the film thickness increases by ∆~/  3 .For the case of Sb doping x~25% as illustrated by the red curves in Figs.1a and b, the position of the Weyl point (  ≈ 3/25) shifts closer to the Γ point than at x= 0 case, and thus corresponds to the expansion of the C= 1 state to larger film thicknesses.Fig. 1b shows the theoretical thin film gaps versus thickness obtained by fitting the bulk DFT bands to a simplified model 3 as detailed in the supplemental material.The gaps close when a topological phase transition occurs between films with different Chern numbers.The small size of such size-quantization gaps places stringent

21 -
Figs. S5a, S6, and extended data Fig.E1, respectively, where the yx plateau and xx minimum can Figs. 3a-f compare the dualgate maps of longitudinal conductivity (xx) and Hall resistivity (yx) for the Mn(Bi0.74Sb00.26)2Te4devices at the thickness of 21, 18, and 14-SLs, respectively, measured at 9T.As shown in Figs.3a and b, the 21-SL Mn(Bi0.74Sb00.26)2Te4reveal a clear C= 1 plateau in the dualgate maps.This is also indicated by the xy (xx) versus Vbg line profiles as shown in Fig.3g.In this device, we observe no other Chern states develop near the CNP besides the C= 1 state at the highest accessible magnetic field of 9T.Careful tracking of the C= 1 state in magnetic field reveals its formation at Fermi energy slightly below the CNP as detailed in extended data Fig.1.Similar behavior was also observed in the 18-SL Mn(Bi0.74Sb00.26)2Te4(extended data Fig.E2).The 18-SL Mn(Bi0.74Sb00.26)2Te4device with higher electron mobility (Fig.1d) shows more quantization features at 9T.With the dualgating structure, we can access these quantum Hall states by tracing the boundaries of these states with dashed lines in the dualgate maps.The color map of xx versus dualgate voltages depicted in Fig.3cshows multiple minima corresponding to the different quantized yx plateaus as indexed in the dualgate map in Fig.3d, showing the different Chern numbers.The xy (xx) versus Vbg line profiles depicted in Fig. 3h reveal the well-developed C= 3 plateau and the developing C= 1 and 5 quantization states.Tracking the Chern state development at the lower magnetic field reveals an additional C= 2 state in the CAFM phase which has no exchange coupling.The degenerate n= 0 LLs (red curves) lie at the Fermi level due to the nontrivial Berry's phase29 in addition to the typical surface band LLs (black curves).When the exchange coupling is introduced, the quasi-2D films have discrete finite-length-chain hopping eigenstates, which have no spin-orbit coupling but are spin-split by exchange interactions with the aligned Mn spins.A series of spin-polarized anomalous LLs (red and blue lines) with magnetic field independent energies emerge from the  ⃗ ⊥ = 0 states as shown in Figs.4c and d calculatedfor an 18-SL Mn(Bi0.75Sb0.25)2Te4film with intralayer exchange coupling Js of 30 and 34 meV, respectively.The details of the LLs derivation can be referred to supplementary materials.The Chern numbers of the anomalous LLs are determined by ( <  −  >  )/2 where  <  ( >  ) is the total number of subbands below (above) Fermi level.The Chern numbers at B= 0 can be identified from the LLs plots in Figs.4c and dby determining the filling factor at Fermi level in the limit of zero magnetic field.The Chern number of C= 2 coincides well with the Hall conductivity calculations of the Chern number at B= 0 (Fig.1b).We see in Figs.4c and dthat the Mn(Bi0.75Sb0.25)2Te4films have an interval of carrier-density in which only anomalous LLs are present close to the Fermi level as indicated by the yellow arrows, and the gaps between these levels are large enough to support quantum Hall effects that are robust against disorder.The surface band LLs with n 0 indices (black curves) in conduction-valence pairs move further away from the Fermi level as magnetic fields strengthen and do not contribute to the B= 0 Chern numbers.

Fig. 3
Fig. 3 the calculated LL gaps at different filling factors as a function of magnetic field for the

.
Low-temperature magnetotransport measurements were performed in a Quantum Design Physical Properties Measurement System (PPMS) in helium-4 circulation (2K-300K) and magnetic field up to 9 T. Two synchronized Stanford Research SR830 lock-in amplifiers at a frequency of 5-8 Hz were used to measure the longitudinal and Hall resistances concurrently on the Mn(Bi1-xSbx)2Te4 devices.The devices were typically sourced with a small AC excitation current of 20-100 nA.Two Keithley 2400 source meters were utilized to source DC gate voltages separately to the top and bottom gate electrodes.Magnetotransport measurements at high magnetic field were carried out in a helium-3 variable temperature insert at a base temperature of 0.4 Kelvin and magnetic field up to 18 tesla based at the National High Magnetic Field laboratory.

Figure 2 .
Figure 2. Magnetic field induced quantization at different Sb substitution levels.Color maps of yx as a function of magnetic field and gate voltage for (a) MnBi2Te4, (b) Mn(Bi0.8Sb0.2)2Te4,and (c) Mn(Bi0.74Sb0.26)2Te4devices at flake thickness of 8-SLs, 10-SLs, and 21-SLs, respectively, measured at temperature of 2K.The red and blue lines mark the transition fields from AFM to CAFM, and CAFM to FM phases, respectively.The xx and yx line profiles in (d), (e), and (f) as a function of magnetic field are extracted from the color maps for the respective devices in (a), (b), and (c) at the gate voltages indicated by the blue arrows.The respective device images are inserted in (d), (e), and (f).The color maps are raw data, whereas the xx and yx line profiles are symmetrized and antisymmetrized, respectively, with respect to the magnetic field.The extracted xx (black rhombus) and yx (blue circle) values at the maximum yx as a function of flake thickness for (g) MnBi2Te4, (h) Mn(Bi0.8Sb0.2)2Te4,and (i) Mn(Bi0.74Sb0.26)2Te4devices measured at magnetic field of 9T.The blue color shades in (g)-(i) denote the thickness range where the C= 1 state is observed in the spin-aligned state.

Figure 3 .
Figure 3. Tunable Chern states by dualgating.Color maps of xx and yx as a function of dualgate voltages for Mn(Bi0.74Sb00.26)2Te4 at flake thicknesses of (a, b) 21-SLs, (c, d) 18-SLs, and (e, f) 14-SLs, respectively, measured at temperature of 2K and magnetic field of 9T.The white dashed lines in the color maps trace the boundaries of the quantization plateaus with the respective quantum states indexed in the yx maps.Line profiles of σxx and σxy versus backgate voltage curves swept across the charge neutrality as indicated by the black arrows in the color maps for the (g) 21-SLs, (h) 18-SLs, and (i) 14-SLs Mn(Bi0.74Sb00.26)2Te4.The σxy plateaus and the corresponding σxx minima are indexed to the Chern numbers in (g)-(i).Vertical dashed lines in (g)-(i) mark the backgate voltages corresponding to the ρ xx CNP at zero magnetic field.The surface band structures in (g)-(i) are sketched to illustrate the LL spectra observed for the respective thicknesses at 9T.The red shades in (g)-(i) denote the Chern states forming near the CNP.

Figure 4 .
Figure 4. Spectra of anomalous and non-anomalous Landau levels.Landau level fan diagrams and filling factors of an 18-SL Mn(Bi0.75Sb0.25)2Te4film.(a) The Landau level structures for the case of a non-magnetic thin film (Js= 0).The n≠ 0 non-anomalous Landau levels are plotted with black curves, while the n= 0 anomalous Landau levels whose energies are independent of magnetic field are plotted with blue and red curves.The red curves distinguish anomalous Landau levels that are localized at the surface.(b) Band at 2D wavevector  = 0 at magnetic field B= 0 versus the same-layer exchange splitting, Js.Spin up (down) states are labeled with orange (blue) color, and the bold green (red) curve is for the spin up (down) state localized at the thin film surfaces.(c) and (d) Landau level structures for thin film with an aligned moment spin configuration at Js= 30 and 34 meV, respectively, labeled in (b) with black vertical dashed lines.Strong quantum Hall states occur when only anomalous Landau levels are close the Fermi level.