Abstract
The intrinsic magnetic topological insulator, Mn(Bi_{1−x}Sb_{x})_{2}Te_{4}, has been identified as a Weyl semimetal with a single pair of Weyl nodes in its spinaligned strongfield configuration. A direct consequence of the Weyl state is the layer dependent Chern number, \(C\). Previous reports in MnBi_{2}Te_{4} thin films have shown higher \(C\) 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 surfaceband Landau levels under magnetic fields. Here, we report a tunable layerdependent \(C\) = 1 state with Sb substitution by performing a detailed analysis of the quantization states in Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} dualgated devices—consistent with calculations of the bulk Weyl point separation in the doped thin films. The observed Hall quantization plateaus for our thicker Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} films under strong magnetic fields can be interpreted by a theory of surface and bulk spinpolarised Landau level spectra in thin film magnetic topological insulators.
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Introduction
Magnetic topological insulators (MTIs) and Weyl semimetals have both received a great deal of attention in recent condensed matter physics research^{1,2,3}. Particularly, the intrinsic MTI MnBi_{2}Te_{4} has provided researchers with an ideal candidate to study the relationship between topological quantum states and magnetic phases^{4,5,6,7,8,9,10,11,12,13,14,15}. Theoretical predictions^{16,17,18,19} 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 MnBi_{2}Te_{4} have a single isolated pair of Weyl crossing points that are close to the Fermi level and therefore can be accessed by controlling the carrierdensity.
In this work, we report on thicknessdependent magnetotransport studies of the mechanicallyexfoliated Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} thin flakes with three different Sb concentrations. The Sb substitutions in the MnBi_{2}Te_{4} parent compound (i) move the Fermi level of the bulk bands closer to the 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 spinmoment aligned Mn(Bi_{1−x}Sb_{x})_{2}Te_{4}. We show that Sb substitution extends the surface gap regime to a wider thickness range by suppressing conduction from the trivial bulk bands. Thin films Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} provide a rich plethora of topologically distinct quasitwodimensional (2D) states that includes Chern insulator^{4,5,6,8,9,10} and axion insulator^{5,7,16,23} states. The application of external magnetic fields to Mn(Bi_{1−x}Sb_{x})_{2}Te_{4}, 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.
Results
Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} thin film Weyl semimetals
According to densityfunctionaltheory (DFT), the spinaligned magnetic configuration of Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} is a simple typeI (or II depending on lattice parameters^{16}) Weyl semimetal with Weyl points at \({\pm k}_{w}\) along the \(\Gamma Z\) line. As shown in Fig. 1a, the distance between Weyl points \((2{k}_{w})\) decreases with Sb fraction x. The bulk Hall conductivity normalized per layer can be expressed as \(\frac{{\sigma }_{{xy}}^{3D}}{d}=\frac{{e}^{2}}{h}\frac{{k}_{w}d}{\pi }\), where \({k}_{w}\) is the position of Weyl point and \(\pi /d\) is the size of Brillouin zone along the \(\Gamma Z\) line with d as the septuple layer (1SL ≈ 1.4 nm). The corresponding thin films can be viewed as quasi2D crystals and are expected to have quantized anomalous Hall conductivities with Chern numbers (\(C\)) that increase by one when the film thickness increases by \(\triangle t \sim \pi /{k}_{w}\)^{18}. For the case of Sb doping x ~ 25% as illustrated by the red curves in Fig. 1a, b, the position of the Weyl point (\({k}_{w}\, \approx \, 3\pi /25d\)) shifts closer to the \(\Gamma\) point than at x = 0 case, and thus corresponds to the expansion of the \(C\) = 1 state to larger film thicknesses. Figure 1b shows the theoretical thin film Chern gaps versus thickness obtained by fitting the bulk DFT bands to a simplified model^{18} as detailed in the Supplemental Note 1. The gaps close when a topological phase transition occurs between the different Chern numbers. As the Weyl semimetal state can exist only under the spinaligned condition, our focus is thus on the Chern insulator states in the magnetic field induced spinmoment alignment phase, which can be observed more consistently in Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} thin films.
Electrical transport in Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} films
We first examine the lowtemperature transport properties of the Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} films. Figure 1c plots the fourterminal resistivity (\({\uprho}_{{{{{\mathrm{xx}}}}}}^{{{{{\mathrm{CNP}}}}}}\)) measured at the CNP as a function of Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} thickness for different Sb substitution levels (x = 0, 0.20, and 0.26). We refer to a sample as surfacelike when its conductivity is thermallyactivated at low temperatures, suggesting the presence of a bulk energy gap. Samples classified as bulklike have weaker temperature dependence and are presumed to have disorderinduced bulk states at all energies. The representative resistivity versus temperature curves for the Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} are included in Supplementary Figs. 5, 6 to distinguish the surfacelike and bulklike behaviors. The surfacelike \({\uprho}_{{{{{\mathrm{xx}}}}}}^{{{{{\mathrm{CNP}}}}}}\) behavior persists to the largest thickness range for the Sb concentration x = 0.26 at T = 2 K while \({\uprho}_{{{{{\mathrm{xx}}}}}}^{{{{{\mathrm{CNP}}}}}}\) is bulklike for thickness above 21SLs. For x = 0.20, \({\uprho}_{{{{{\mathrm{xx}}}}}}^{{{{{\mathrm{CNP}}}}}}\) decreases abruptly at thicknesses above 12SLs. This trend persists for MnBi_{2}Te_{4}. This trend is expected since MnBi_{2}Te_{4} has bulk ntype doping^{5,24}, while the substitution of Sb on the Bi sites can shift the Fermi level of the bulk band toward ptype doping, with x = 0.26 being closest to CNP^{20}. Sb doping at higher concentrations x > 0.26 leads to excessive ptype doping and prevents access to CNP in thin flakes^{21}. Moving the Fermi level by the Sb to Bi ratio can thus maximize the surfacelike regime for probing their quantum transport properties.
Figure 1d shows a representative ρ_{xx} curve as a function of backgate voltage (V_{bg}) for an 18SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} device measured at a temperature of 2 K. The ambipolar gatedependent ρ_{xx} suggests a bulk gap with an intrinsic surface state at the CNP (V_{bg} ~ +3 V). The surface carrier density can be tuned to either hole or electron transport by controlling the gate voltage. The fieldeffect mobility \({\mu }_{{{{{{{\rm{FE}}}}}}}}=\frac{1}{{C}_{g}}\frac{{{{{{\rm{d}}}}}}{G}_{{xx}}}{{{{{{\rm{d}}}}}}{V}_{g}}\), where \({C}_{{g}}\) is the gate capacitance (≈80 nF/cm^{2} for a ~ 30 nm mica dielectric), G_{xx} ( = 1/ρ_{xx}) is the fourterminal conductance, and V_{g} is the voltage applied through the graphite gateelectrode, 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}.
\({{{{{\boldsymbol{C}}}}}}\) = 1 state in Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} films
The magnetic fielddependent transport properties of Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} 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(Bi_{1−x}Sb_{x})_{2}Te_{4} undergoes the spin–flop and spin–flip transitions^{20,22} from the antiferromagnetic (AFM) to the canted, and finally to the aligned spinmoment configurations. To compare the magnetic field dependence of the Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} film with different Sb substitutions, we plot the color maps of ρ_{yx} as functions of gate voltage and magnetic field for the 8SL MnBi_{2}Te_{4}, 10SL Mn(Bi_{0.8}Sb_{0.2})_{2}Te_{4}, and 21SL Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4}, respectively, in Fig. 2a–c. The spinflop and spinflip transition fields, as determined from the kinks in their ρ_{xx} and ρ_{yx} versus magnetic field curves, happen at magnetic fields of ~±2–3 T and ~±7 T, respectively, are observed in all the samples. The magnetic transition fields identified by the color line marks depicted in Fig. 2a–c agree with their parent bulk compounds^{20} and theoretically calculated values^{25}. Line profiles of ρ_{xx} and ρ_{yx} curves for the 8SL MnBi_{2}Te_{4}, 10SL Mn(Bi_{0.8}Sb_{0.2})_{2}Te_{4}, and 21SL Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4} are plotted in Fig. 2d–f. The ρ_{yx} increases sharply with magnetic field in the canted antiferromagnetic (CAFM) phase, and saturates at ~h/e^{2} as the thin film is driven into the FM phase by magnetic field. The suppression of ρ_{xx} in the FM phase further confirms the development of \(C\) = 1 state in all three samples. The interpretation is supported by the gatedependent ρ_{xx} and ρ_{yx} curves measured at magnetic field of 9 T for the three respective samples in Supplementary Figs. 7a, 8, and 10, respectively, where the ρ_{yx} plateau and ρ_{xx} minimum can be seen. Despite not being fully quantized, the 21SL Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4} film exhibits all the features of the \(C\) = 1 state. The anomalous Hall loop at low magnetic field for the 8SL MnBi_{2}Te_{4} and 10SL Mn(Bi_{0.8}Sb_{0.2})_{2}Te_{4} could be due to the uncompensated surface magnetization^{11} or antiferromagnetic domain walls^{26}, whereas no zero field hysteretic behavior observed in the 21SL Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4} film. The observed \(C\) = 1 states in all three Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} films with the different Sb substitutions confirm that their spinalignment configuration is in the topological phase with a Chern insulator gap.
To further evaluate the thicknessdependence of the \(C\) = 1 state, we plot ρ_{xx} and ρ_{yx} values as a function of thickness, with the gate voltage tuned to the ρ_{yx} maximum for each thickness, at magnetic field of 9 T. Our primary observation is that, despite their similar fielddependent quantization behavior, the \(C\) = 1 state prolongs to the larger thickness range with the Sb substitutions as shown in Fig. 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 8SL MnBi_{2}Te_{4} agrees with our calculations (Fig. 1b) and the literature^{10}. The thickness limit for the \(C\) = 1 state extends to 11SLs for the Mn(Bi_{0.8}Sb_{0.2})_{2}Te_{4} as shown in Fig. 2h. The 16SL Mn(Bi_{0.8}Sb_{0.2})_{2}Te_{4} (Supplementary Fig. 9) shows the absence of \(C\) = 1 state at magnetic field up to 18 T, presumably due to the excessive bulk conduction channels in this sample. In Fig. 2i, we show the substantially wider thickness range of the \(C\) = 1 state resolved for the Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4}. Although this can be somehow related to the extension of the surfacelike regime in thin films by the optimal Sb substitutions, according to our calculations in Fig. 1b, the Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4} films 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(Bi_{0.74}Sb_{0.26})_{2}Te_{4}, we perform a detailed analysis for these devices in a dualgating platform. Figure 3a–f compare the dualgate maps of longitudinal conductivity (σ_{xx}) and Hall resistivity (ρ_{yx}) for the Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} devices at the thickness of 21, 18, and 14SLs, respectively, measured at 9 T. As shown in Fig. 3a and b, the 21SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} reveal a clear \(C\) = 1 plateau in the dualgate maps. This is also indicated by the σ_{xy} (σ_{xx}) versus V_{bg} 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 9 T. Careful tracking of the \(C\) = 1 state in magnetic field reveals its formation at Fermi energy slightly below the CNP as detailed in Supplementary Figs. 10 and 11. Similar behavior was also observed in the 18SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} (Supplementary Figs. 12, 13).
The 18SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} device with higher electron mobility (Fig. 1d) shows more quantization features at 9 T. 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 σ_{xx} versus dualgate voltages depicted in Fig. 3c shows multiple minima corresponding to the different quantized ρ_{yx} plateaus as indexed in the dualgate map in Fig. 3d, showing the different Chern numbers. The σ_{xy} (σ_{xx}) versus V_{bg} line profiles depicted in Fig. 3h reveal the welldeveloped \(C\) = 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 (Supplementary Fig. 14). The existence of the \(C\) = 2 state in the CAFM phase is also verified by the ρ_{xx} and ρ_{yx} dualgate maps and the flow diagram in (σ_{xx}, σ_{xy}) parameter space swept at magnetic field of 6 T as shown in Supplementary Figs. 12, 14, respectively. Also, we note that the \(C\) = 2 state coincides with the \({\uprho}_{{{{{\mathrm{xx}}}}}}^{{{{{\mathrm{CNP}}}}}}\) at the zero magnetic field.
We further analyze the quantization states in the 14SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4}. In addition to the \(C\) = 1 state, the dualgate maps in Fig. 3e, f show a series of oscillatory σ_{xx} minima and ρ_{yx} plateaus, respectively, corresponding to the different quantum Hall states develop at 9 T when tuning the dualgate voltages. The linecuts of σ_{xx} and σ_{xy} as a function of backgate voltage at magnetic field of 9 T, as depicted in Fig. 3i, reveal the quantum Hall plateaus with \(C\) = 0, 1, 3, 5, etc. The color maps of ρ_{xx} and ρ_{yx} as functions of magnetic field and backgate voltage (Supplementary Fig. 15) resolve the fan diagram of Landau levels at odd integer fillings, where the ρ_{xx} minima for each filling factor can be traced down linearly to a gate voltage at zero magnetic field corresponding to the CNP. Such a feature is similar to the surface states’ LL fan diagram in nonmagnetic topological insulators^{27,28}. Different from the 18SL and 21SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4}, the \({\uprho}_{{{{{\mathrm{xx}}}}}}^{{{{{\mathrm{CNP}}}}}}\) obtained at zero magnetic field sits between \(C\) = 0 and 1 states at high magnetic field for the 14SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} (Supplementary Fig. 16). The schematic diagrams inserted in Fig. 3g–i illustrates the surface band structures with different Chern states resolved in the respective thicknesses of the Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} films.
Anomalous Landau levels in magnetic topological insulators
To interpret the rich Chern insulator states observed in our Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4} films, we calculate their LLs spectra using a simplified model^{18}, in which the 2D massive Dirac cones are coupled by tunneling within and between the compound’s septuplelayer building blocks. For explanation purposes, we first present in Fig. 4a the LLs spectrum for a nonmagnetic TI thin film^{29} which has no exchange coupling. The degenerate n = 0 LLs (red curves) lie at the Fermi level due to the nontrivial Berry’s phase^{30} in addition to the typical n ≠ 0 bulk LLs (black curves). When the exchange coupling is introduced, the quasi2D films have discrete finitelengthchain hopping eigenstates, which have no spinorbit coupling but are spinsplit by exchange interactions with the aligned Mn spins. A series of spinpolarized anomalous LLs (red and blue lines) with magnetic fieldindependent energies emerge from the \({\vec{k}}_{\perp }=0\) states, as shown in Fig. 4c, d. The derivation of anomalous LLs bands from a generalized Su–Schrieffer–Heeger (SSH) model (Supplementary Note 1) and their wavefunction distributions can be found in Supplementary Fig. 1. The Chern numbers of the anomalous LLs are determined by \(({N}_{E < {E}_{F}}{N}_{E > {E}_{F}})/2\) where \({N}_{E < {E}_{F}}\) (\({N}_{E > {E}_{F}}\)) is the total number of subbands below (above) Fermi level. Noted that for convenience in calculations, we ignore the magnetic transitions and assume the spinaligned phase for all magnetic field ranges. This assumption enables the determination of the actual \(C\) due to the band topology in zero magnetic field and the development of anomalous LLs under high magnetic field. For example, the \(C\) = 2 can be identified from the LLs spectra in Fig. 4c, d by tracing down to the zero magnetic field limit. The \(C\) = 2 can be further verified by the \(C\) calculated from the bulk Hall conductivity at the same layer thickness of the Mn(Bi_{0.75}Sb_{0.25})_{2}Te_{4} films (Fig. 1b). Under high magnetic field, the nonanomalous LLs with n ≠ 0 indices (black curves in Fig. 4c, d) in conductionvalence pairs move further away from the Fermi level as magnetic fields strengthen. This leaves an interval of carrier density in which only the n = 0 anomalous LLs (red and blue lines in Fig. 4c, d) are present close to the Fermi level as indicated by the yellow arrows, and the gaps between these levels are large enough to support Hall quantization that is robust against disorder. The energy gaps of the anomalous LLs depend on the n ≠ 0 nonanomalous LLs and thus are magnetic field dependence.
A general picture of the relationship between the Chern numbers and exchange coupling (\({J}_{S}\)) is illustrated in Fig. 4b where the surface and bulk spinsplitting bands are plotted. In the case of \({J}_{S}=0\), the number of subbands above and below the Fermi level equals (\({N}_{E < {E}_{F}}={N}_{E > {E}_{F}}\)), and thus the Chern number is 0. When the exchange field is turned on (\({J}_{S}\, \ne \, 0\)), the anomalous LLs are spinpolarized with two nearly degenerate surface anomalous LLs labeled represented by the green and red curves in Fig. 4b. The filling factor at the Fermi level becomes 1 (−1) for downspin (upspin) as the \({N}_{E < {E}_{F}}\) and \({N}_{E > {E}_{F}}\) are now differed by one. When the crossings between upspin (yellow curves) and downspin (blue curves) happen at the Fermi level by further increasing \({J}_{S}\), the filling factor will increase by one. Subsequently, the Chern quantization is therefore expected to be observable over a range of filling factors magnitudes centered on 1+ the number of \({\vec{k}}_{\perp }=0\) crossings that occur between up and down spins as indexed in Fig. 4b. When the magnetic field is reversed, both the spin of the anomalous LLs and the sign of the exchange coupling between the local moments and the Dirac electrons are reversed, therefore the sign of the filling factor will follow the sign of the magnetic field. The dependency between the spin of the anomalous LLs and the sign of the exchange coupling further infers that the anomalous LLs are spinpolarized.
We also note that since the position of Weyl points \({k}_{w}\) depends on the exchange splitting^{18}, which may be reduced by antisite defects^{31} leading to a change in the Chern numbers and the gaps in zero magnetic field spin alignment phase (Supplementary Fig. 2). To further elaborate on this effect, we examine two cases of \({J}_{S}\) = 30 and 34 meV for the 18SL Mn(Bi_{0.75}Sb_{0.25})_{2}Te_{4} film in Fig. 4c, d, respectively. As shown in their LL spectra, the \({J}_{S}\) = 30 and 34 meV exhibit very different \(C\) = 2 Chern insulator gaps at zero magnetic field. Nevertheless, the wellspaced anomalous LLs dominate over a finite region of filling factor at high magnetic field does not vary too much with the \({J}_{S}\).
Finally, we compare our experimental results for the Mn(Bi_{0.74}Sb_{0.26})_{2}Te_{4} films with the calculated LLs structure at a similar thickness and Sb doping level. We plot in Supplementary Fig. 17 the calculated LL gaps at different filling factors as a function of magnetic field for the Mn(Bi_{0.75}Sb_{0.25})_{2}Te_{4} films at different thicknesses. The LL gap size of each filling factor is determined either by gaps between anomalous LLs or by gaps between anomalous (n = 0) and nonanomalous (n ≠ 0) LLs. Under a strong magnetic field, the calculated filling factor of \(C\) = 1 exhibits the largest LL gap in all three thicknesses of Mn(Bi_{0.75}Sb_{0.25})_{2}Te_{4} film. This explains the experimental observation where the \(C\) = 1 state was observed over the wide thickness range in our Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} film. However, one noticeable feature in the case of larger film thickness is that the \(C\) = 1 filling shifts below the Fermi level and the surface gap at the Fermi level change to the filling factor of higher Chern number \(C\) = 2 (refer to Supplementary Figs. 4 and 17 for details). This is consistent with the observation in our 18SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} film, indicating that the 18SL is in a higher Chern number \(C\) = 2 state. 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 9 T at this film thickness.
Whereas for 14SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} film, both of our experimental observation and calculations suggest that the 14SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} film in the \(C\) = 1 state (Supplementary Figs. 3 and 17) with the higher filling factors of \(C\) = 3 and 5 states assigned to the nonanomalous band LLs. The calculations infer an anomalous LLs \(C\) = 2 state can develop at the higher magnetic field. To verify this, we performed measurements at a high magnetic field up to 18 T for the 11SL and 10SL Mn(Bi_{0.8}Sb0_{0.2})_{2}Te_{4} films, both with the \(C\) = 1 state at the Fermi level, as depicted in Supplementary Fig. 18. Similar to the 14SL Mn(Bi_{0.74}Sb0_{0.26})_{2}Te_{4} film, the quantization steps of \(C\) = 0, 1, and 3 states can be observed in the 11SL and 10SL Mn(Bi_{0.8}Sb_{0.2})_{2}Te_{4} at a magnetic field of 10 T. While the additional \(C\) = 2 plateau starts to develop at higher magnetic field of >14 T in both samples. The phase boundaries traced by the red dashed lines in Supplementary Fig. 18(b) and (e) covering the \(C\) = 1 and 2 plateaus are consistent with our theoretical picture of the Chern insulator \(C\) = 1 state and anomalous LLs \(C\) = 2 forming at high magnetic field as the ordinary n ≠ 0 bulk band LLs of \(C\) = 3 and above states move away from Fermi level. Our results can thus support the existence of the spinpolarized anomalous LLs in Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} thin film Weyl semimetals, which emerges at larger film thickness or strong magnetic field near the Fermi level.
Discussion
In summary, we studied the magnetoelectrical transport of the intrinsic MTI Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} 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 surfacelike insulating and bulklike metallic transport regimes. The thicknessdependent Hall conductivities, particularly for the \(C\) = 1 Chern insulator state in the Mn(Bi_{1−x}Sb_{x})_{2}Te_{4}, show a correlation with the separation of Weyl points as described by our theoretical models, indicating that Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} behaves as thin film Weyl semimetals. Our transport results for different Sb concentrations highlight the importance of the Weyl point separation in the spinaligned magnetic phase to the Hall quantization. Moreover, we showed that the Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} at larger film thickness and strong magnetic field can give rise to the unusual quantization sequence and the intriguing anomalous LLs near the Fermi level. 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(Bi_{1−x}Sb_{x})_{2}Te_{4} bulk crystals at different Sb doping levels were grown by a selfflux growth method^{20,24}. Variable thicknesses of Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} thin flakes were exfoliated from the parent bulk crystals and then transferred into the heterostructures of graphite/muscovite mica sandwiched layers using a micromanipulator transfer stage. The graphite and muscovite mica layers were subsequently transferred onto Si/SiO_{2} substrate using polypropylene carbonate, and followed by an annealing process in argon gas to clean the polymer residues. The graphite/muscovite mica layers serve as the gateelectrode/dielectric layers. The Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} exfoliation and transfer processes were performed in an argon gasfilled glovebox with O_{2} and H_{2}O levels <1 ppm and <0.1 ppm, respectively, to prevent oxidation in thin flakes. We fabricated the Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} 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(Bi_{1−x}Sb_{x})_{2}Te_{4} flakes were protected by polymethyl methacrylate (PMMA) while transporting for lithography and metal deposition processes.
Measurements
Lowtemperature magnetotransport measurements were performed in a Quantum Design Physical Properties Measurement System (PPMS) in helium4 circulation (2–300K) and magnetic field up to 9 T. Two synchronized Stanford Research SR830 lockin amplifiers at a frequency of 5–8 Hz were used to measure the longitudinal and Hall resistances concurrently on the Mn(Bi_{1−x}Sb_{x})_{2}Te_{4} 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 helium3 variable temperature insert at a base temperature of 0.4 K and magnetic field up to 18 T based at the National High Magnetic Field laboratory.
Theoretical calculations
DFT calculations were performed using Vienna Ab initio Simulation Package (VASP)^{32} in which Generalized gradient approximations (GGA) of PerdewBurkeErnzerhof (PBE)^{33} have been adopted for exchangecorrelation potential. Onsite correlation on the Mn3d states is treated by performing DFT + U calculations^{34} with U–J as 5.34 eV. The global break condition for the electronic SCloop is set to be 10^{−7 }eV and the cutoff energy for the plane wave basis set is 600 eV during the selfconsistent (SC) calculations. A 9 × 9 × 6 Gammacentered kpoint integration grid was employed with Gaussian broadening factors as 50 meV. In the calculations of bulk Mn(Bi_{1−x}Sb_{x})_{2}Te_{4,} 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.
Data availability
The data supporting the findings of this study are available within the article and supplementary information. The main data generated in this study are publicly available at https://doi.org/10.5068/D1097T. Additional data are available from the corresponding authors upon request.
Code availability
The codes used for the numerical simulation are available from the corresponding authors upon request.
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Acknowledgements
This work was supported by the National Science Foundation the Quantum Leap Big Idea under Grant No. 1936383 and the U.S. Army Research Office MURI program under Grants No. W911NF2020166 and No. W911NF1610472. Support for crystal growth and characterization was provided by the National Science Foundation through the Penn State 2D Crystal ConsortiumMaterials Innovation Platform (2DCCMIP) under NSF cooperative agreement DMR2039351. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement No. DMR1644779 and the State of Florida.
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S.K.C. and K.L.W. planned the experimental project. S.H.L. and Z.M. prepared the bulk crystals. S.K.C. fabricated the devices and conducted the transport measurements. J.J. helped with transport measurements conducted at National High Magnetic Field laboratory. C.L. and A.H.M. performed the theoretical calculations. S.K.C., L.C., A.H.M., and K.L.W. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Chong, S.K., Lei, C., Lee, S.H. et al. Anomalous Landau quantization in intrinsic magnetic topological insulators. Nat Commun 14, 4805 (2023). https://doi.org/10.1038/s4146702340383x
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DOI: https://doi.org/10.1038/s4146702340383x
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