Rashba-splitting-induced topological flat band detected by anomalous resistance oscillations beyond the quantum limit in ZrTe5

Topological flat bands — where the kinetic energy of electrons is quenched — provide a platform for investigating the topological properties of correlated systems. Here, we report the observation of a topological flat band formed by polar-distortion-assisted Rashba splitting in the three-dimensional Dirac material ZrTe5. The polar distortion and resulting Rashba splitting on the band are directly detected by torque magnetometry and the anomalous Hall effect, respectively. The local symmetry breaking further flattens the band, on which we observe resistance oscillations beyond the quantum limit. These oscillations follow the temperature dependence of the Lifshitz–Kosevich formula but are evenly distributed in B instead of 1/B at high magnetic fields. Furthermore, the cyclotron mass gets anomalously enhanced about 102 times at fields ~ 20 T. Our results provide an intrinsic platform without invoking moiré or order-stacking engineering, which opens the door for studying topologically correlated phenomena beyond two dimensions.

Flat electronic bands harbor exotic quantum behaviors due to the quenched kinetic energy and subsequently dominated Coulomb interaction.The fractional quantum Hall effect 1,2 is an archetypical 2-dimensional (2D) flat band system.Recently developed moiréengineered 2D twisted bilayer graphene [3][4][5] and multilayer graphene in certain stacking order [6][7][8] are other examples of realizing topological flat bands.Flat bands are also theoretically predicted in some stoichiometric 3-dimensional (3D) materials forced by certain geometric lattices, like the Kagome or Lieb lattices 9,10 .Flat-band-induced correlation in 3D topological systems is crucial for realizing correlated 3D topological effects, like correlation on Weyl semimetals (WSM) and possible axionic dynamics [11][12][13] .Despite theoretical advancement in establishing the material database of topological flat bands 14 , experimental realization of an isolated topological flat band around Fermi level (FL) in 3D stoichiometric materials remains elusive.
In the twisted bilayer graphene, the two preconditions for realizing a 2D topological flat band are: 1. the pristine Dirac material graphene; 2. moiré superlattice as the method for flattening the band.We now ask a question: can we find a counterpart in 3D?If there is, then which material is the 3D counterpart of graphene?How do we flatten the energy band or enlarge the unit cell in 3D?Here, we report that ZrTe5 15 , as a typical Dirac material in 3D, can meet the first precondition; the polar distortion at low temperatures in ZrTe5 meets the second precondition without invoking van der Waals heterostructure-based engineering.The polar-distortion-assisted Rashba splitting in ZrTe5 is evidenced by torque magnetometry and the anomalous Hall effect (AHE), which shows the existence of a topological flat band in the magnetic field, on which anomalous resistance oscillations appear beyond the quantum limit.The cyclotron mass is enhanced by an order of 10 2 , consistent with the topological flat band.Our work also highlights the importance of local symmetry breaking in topological materials.
ZrTe5 was initially proposed as a typical candidate for a quantum spin Hall insulator in 2D limit 15 .As shown in Fig. 1a, 2D ZrTe5 layers stack along b axis (here, we use a, b, c and -x, z, y interchangeably) with a ZrTe3 chain that runs along a axis.The topological property of 3D ZrTe5 is sensitively dependent on the inter-and intra-layer coupling strengths.ZrTe5 is located near the boundary of weak topological (WTI) and strong topological insulators (STI) as typical Dirac material 15 , promoting many exotic phenomena [16][17][18][19][20][21][22] .Therefore, the properties of ZrTe5 are sensitively dependent on crystal growth methods, namely the chemical vapor transfer (CVT) and flux processes.Samples grown from Te-flux are more stoichiometric and closer to the phase boundary [23][24][25][26] .As shown in Fig. 1b, temperature-dependent resistivity measured on flux-grown single crystals exhibits semiconducting-like behavior, a typical profile in narrow-gapped semiconductors 27,28 .We focus on electrical transport within ac plane with current i along a axis as illustrated in the inset of Fig. 1b.Hall measurements (Fig. 1c) show hole is the only carrier down to 2 K. Hall conductivity   (Fig. 1d) fitted by Drude model , where p is the carrier density and  is the mobility, shows the existence of an anomalous term    , as indicated by the pink shadowed area.As shown in Fig. 1e, p is ultralow ~ 3×10 14 cm -3 and  is as high as 10 6 cm 2 V -1 s -1 at low temperatures, crucial for realizing the topological flat band and anomalous resistance oscillations.Therefore, ZrTe5 sample synthesized in this work is a 3D counterpart of graphene.
We then investigate the possible local modifications on the Dirac band, which might provide the clue for forming a topological flat band.In these flux-grown samples, preliminary evidence of polar distortion is reported by nonlinear transport 26 , while direct evidence is still lacking.We adopted torque magnetometry to measure the magnetic  As we will show, the flatness on the modified band is further supported by the observation of anomalous resistance oscillations beyond the quantum limit.The carrier density of sample S75 is 3×10 14 cm -3 at 2 K, corresponding to a quantum limit less than 0.05 T (B ∥ b axis), beyond which there are no quantum oscillations.However, as shown in Fig. 3a and 3b, we observe strong resistance oscillations in sample S75, and reproduced in another sample S74, where similar behaviors persist up to 18 T (Fig. 3c).As noted by blue vertical lines in Fig. 3c, we find evenly-distributed oscillations at fields higher than 3 T, which is clearly exhibited in Δ  (inset of Fig. 3c).This linear-in-B relation is against In the following, we will show that the kinetic energy of Dirac fermions in ZrTe5 is heavily quenched in magnetic fields.Figure 4a shows the temperature-dependent oscillatory component Δ  in sample S74.One prominent behavior is the high-field oscillations dampen quicker than the low-field ones.By extracting the amplitudes of peaks and valleys at characteristic fields, as shown in Fig. 4b, the temperature dependence of amplitudes Δ  can be well fitted by the temperature-damping prefactor  = Let's come to a picture based on the above experimental observations.As illustrated in Fig. 5a, the topmost Rashba-splitted band A flattens in a magnetic field, and the kinetic energy is heavily quenched.At the same time, the lower band is still dispersive with large kinetic energy and then goes up quickly in the magnetic field.The simplified band A, as a topological flat band, thus exhibits quenched kinetic energy and dominated Zeeman energy.
Then, the Landau levels will bend and appear to FL.We plot the kinetic energy  , = − � 2    ℏ (Fig. 5b) with fitted   , which quenches at small fields.As shown in Fig. 5c, we plot the total energy   = − � 2    ℏ +  � 2    (̅ ~15), and find the Landau levels cross the FL, distributing evenly in high magnetic fields.Therefore, the anomalous resistance oscillations observed beyond the quantum limit are consistent with forming a topological flat band in ZrTe5.
In conclusion, our work demonstrates that the 3D Dirac material ZrTe5 can be naturally transformed into a topological flat band system by symmetry breaking at low temperatures without invoking van der Waals heterostructure-based engineering.In this topological flat band, we observe anomalous resistance oscillations beyond the quantum limit, originating from the quenched kinetic energy of electrons on this flat band.Our work has two consequences: the first one is that the intrinsically formed topological flat band provides another route to create a topological flat band system beyond van der Waals heterostructures-based engineering; the second one is that the realization of a topological flat band in a 3D archetypical Dirac system opens the door for studying the specific exotic phenomena for topologically correlated effects uniquely existed in the dimensionality of three.

Crystal growth and characterizations
For single crystals of ZrTe5 grown by the Te-flux method: Starting materials Zr (Alfa, 99.95%) and Te (99.9999%) were sealed in a quartz tube with a ratio of 1:300 and placed in a box furnace.Then, the materials were heated up to 900 o C, followed by shaking the melt, and kept for two days.For single crystals of ZrTe5 grown by CVT method: ZrTe5 powder was synthesized by stoichiometric Zr (aladdin, 99.5%) and Te (99.999%).ZrTe5 powder and iodine were sealed in a quartz tube and then placed in a two-zone furnace with a temperature gradient of 540 o C and 445 o C for one month.
The crystal structure and composition were checked by powder x-ray diffraction (XRD, Rigaku) and energy dispersive x-ray (Hitachi-SU5000).

Transport measurements
Due to the reaction between silver paste and ZrTe5, the usual bonding method with silver paste directly applied on the sample surface is not applicable here, which causes huge contact resistance and contaminates the electrical transport.To make good contacts, the surface was cleaned first by Ar plasma, 5 nm Ti/50 nm Au was then deposited on the ac surface with a homemade Hall bar mask.Then, silver paste was used to bond the Au wires for typical four-probe electrical contacts.The contact resistance is around several Ohms, and no sizable increase in contact resistance was observed over several months.
Electrical transport measurements above 2 K were carried out in a Quantum Design Physical Property Measurement System 9 T (PPMS-9 T) with a sample rotator.
Measurements below 2 K were done in a top-loading dilution fridge (Oxford TLM, base temperature ~ 20 mK).High-field Measurements were carried out in a water-cooled magnet with steady fields up to 33 T in the Chinese High Magnetic Field Laboratory (CHMFL), Hefei.Lock-ins (SR830) were used to measure the resistance and the 2 resistance with a Keithley 6221 AC/DC current source.

Torque magnetometry
Magnetic torque measurements were carried out on a piezoresistive cantilever with a compensated Wheatstone bridge.The tiny sample cut from the pristine crystal was mounted on the tip of the cantilever.The resulted torque is roughly calculated by the relation: , where a is the leg width, t is the leg thickness, coefficient   = 4.5 × 10 −10  2 /, Δ is the voltage drop on the Wheatstone resistance bridge, i is the current fed into the bridge and   is the resistance of sample leg.

Torque data in CVT samples and related symmetry analyses
We also performed the same torque measurement on a sample grown by the chemical vapor transfer (CVT) method.As shown in Fig. S1b, the angular torque at 7T displays a negligible asymmetric amplitude reflecting a small A2 term.By adopting the same fitting process as outlined in the main text, as shown in Fig. S1c, the ratio  2 ()/ 2 (250 ) exhibits no enhancement at low temperatures and follows the ratio  1 ()/ 1 (250 ), which is distinct from the results of Fig. 2c in the main text and Fig. S1a.
We analyze possible crystal structures that flux-grown ZrTe5 samples adopt in low temperatures.Angular torque is powerful for systems with orthogonal axes, like cubic, tetragonal, or orthorhombic structures.Our torque results show the measured results are incompatible with an orthorhombic structure.For a monoclinic structure with a C2 operation, the  tensor adopts a form: Here x, y, z are used, and the magnetic torque  is: For a triclinic structure with no additional symmetry operations, the  tensor adopts a general form: and the magnetic torque  is: ZrTe5 adopts a crystal structure with lower symmetry, like monoclinic or triclinic with non-orthogonal axes, complicating the angular magnetic torque measurements and analyses.Nevertheless, in the above analyses, we can see there indeed appears sin 2  term when the magnetic field rotates in the yz (bc) plane, namely the A2 term, in the   tensor of the two crystal structures with lower symmetries.

Nonreciprocal transport measured under negligible Joule heating
We performed the nonreciprocal transport measurements on sample S75 with magnetic fields along b and c directions, respectively.In the measurements process, we found that the magnetic-field-symmetric Seebeck effect contributes to the raw signal even with a current as low as 0.1mA.The Seebeck effect is subtracted in the process of field antisymmetrization.While increasing the excitation current, a magnetic-fieldasymmetric signal arises at small fields, which superimposes on a linear background after antisymmetrization.The Nernst effect causes this anomaly at a small field due to an unevenly distributed thermal gradient caused by contact resistance Joule heating.
We have to set an excitation current of i = 0.1 mA to minimize this effect.As we know, nonreciprocal resistance 1 adopts a form ( 0 , ) ∝  0  0 , where  is the coefficient that characterizes the strength of nonreciprocal resistance.We adopted 2  measurements 2,3 , which produces  2 =

Angle-dependent 𝝆𝝆 𝒚𝒚𝒚𝒚 for ba and bc planes
As shown in Fig. 3a and 3b, we measured angle-dependent Hall resistivity   in both ba and bc planes, respectively.There is no in-plane anomalous Hall (AHE) signal in our flux-grown samples.Our samples are different from those used in ref 5 , but similar to those used in ref 6 .Our results are also consistent with the angular   reported in ref 6 .Therefore, the AHE appears when the magnetic field is along b axis, and suddenly disappears when the magnetic field is tilted along ac in-plane configuration, supporting the Rashba-splitting picture proposed in the main text.

Process of background subtraction for obtaining the oscillatory 𝚫𝚫𝝆𝝆 𝒚𝒚𝒚𝒚
As shown in Fig. S4, the magnetoresistance increases very quickly at small fields and tends to saturate at higher fields, which is found to be well fitted by exponential functions  *  −(− 0 )/ 1 , which is a smooth function shown by the black fitting line.
Topological flat band, on which the kinetic energy of topological electrons is quenched, represents a platform for investigating the topological properties of correlated systems.Recent experimental studies on flattened electronic bands have mainly concentrated on 2-dimensional materials created by van der Waals heterostructurebased engineering.Here, we report the observation of a topological flat band formed by polar-distortion-assisted Rashba splitting in a 3-dimensional Dirac material ZrTe5.The polar distortion and resulting Rashba splitting on the band are directly detected by torque magnetometry and the anomalous Hall effect, respectively.The local symmetry breaking further flattens the band, on which we observe resistance oscillations beyond the quantum limit.These oscillations follow the temperature dependence of the Lifshitz-Kosevich formula but are evenly distributed in B instead of 1/B in high magnetic fields.Furthermore, the cyclotron mass anomalously gets enhanced about 10 2 times at field ~20 T. These anomalous properties of oscillations originate from a topological flat band with quenched kinetic energy.The topological flat band, realized by polar-distortion-assisted Rashba splitting in the 3-dimensional Dirac system ZrTe5, signifies an intrinsic platform without invoking moiré or orderstacking engineering, and also opens the door for studying topologically correlated phenomena beyond the dimensionality of two.
enhanced and reaches a value of ~30 at 2 K, indicating the emergence of polar distortion at 150 K. Furthermore, as shown in Fig.2c,    shows up around 150 K, where the  2term suddenly gets enhanced.The concurrence of the polar distortion and    indicates that the AHE is locked to the polar-distortion-induced band modifications, which is also consistent with the fact that no AHE is observed in CVT samples 20 .ZrTe5 is a nonmagnetic material with time-reversal symmetry.Nevertheless, the observed AHE takes a profile like the AHE of ferromagnets with saturating plateaus.Based on our three observations: 1.The existence of polar distortion P//b; 2. The coincidence of the onsets of the AHE and polar distortions; 3. The absence of in-plane Hall effect in our samples (see Fig. S3 and SI section 3 for the in-plane Hall discussions).Considering the quasi-2D nature of ZrTe5, we interpret this behavior by invoking a Rashba model-based mechanism usually adopted for explaining the intrinsic AHE in magnetic materials 29 .With polar distortion P along b axis, the typical Rashba model appears as  =  ⋅ ( ×  ∥ ) + Δ()  with the time-reversal breaking term Δ(), here momentum  ∥ in the ac plane,  is the strength of Rashba splitting, and  is the Pauli matrix for real spins.Furthermore, we do not include   dispersion in the Rashba model due to the parabolic relation along   30 .As shown in Fig. 2d, the band splitting is very weak, and the crossing point is near the band edges due to tiny polar distortions, which means only samples with ultralow carrier density can access this region.When an out-of-plane magnetic field ( ⊥ ) is applied, the crossing point on the Rashba bands is gapped out due to a significant Zeeman effect Δ(), the region near the gap becomes Berry curvature hot spots.With increasing magnetic field, the FL located near the band edge falls into the Rashba gap at   , results in a saturated    in higher fields 29 .When we tilt the direction of a magnetic field to the ac plane, no in-plane Hall signal is detected (Fig. S3a & 3b), which is consistent with the Rashba model, because the in-plane magnetic field can only shift the Rashba crossing without opening a gap, then no AHE is expected to appear.The prominent feature of the formed Rashba band is that the extremum of the gapped band gets flatter with increasing magnetic field, creating an ideal, isolated topological flat band around FL, different from the proposal of NLSM-based flat band used to explain the results of nonlinear transport 31 .The origin of tiny polar distortions in the flux-grown ZrTe5 sample is not yet clear because no structural transition is observed by x-ray diffraction down to 10 K 26 , which might also indicate this weak polarity P is unable to drive the parent TIs to WSM phase via the Murakami's scheme 32 .As we know, the possible defects and disorder-induced polar distortion, as essential roles of local symmetry breakings, will lift the valley or spin degeneracies, and form a supercell 33-35 .As the main result of this work, we propose in this work that the effect of polar distortion together with Rashba splitting on the band edge, by forming the supercell, is similar to moiré engineering.As shown in the right column of Fig. 2d, the local symmetry breaking enlarges the unit cell, induces multiple Rashba splittings, and creates a topological flat band.

2 𝜇𝜇
the normal Shubnikov-de Haas (SdH) oscillations evenly distributed in 1/B and also different from the logarithmic oscillations36  .As shown in Fig.3d, we index the oscillations by integers, which is found to be well fitted by ~ 0  +  1 *  , where the  0 term represents contribution from normal 1/B, and  1 term represents the additional contribution from linear-in-B.The same fitting is also employed to fit fan diagrams obtained from Δ (Fig.3e & 3f), where linear-in-B trend is highlighted by shadowed area.As shown in Fig.3g& 3h, we also tilt the magnetic field in ba and bc planes (defined as , ), and find the oscillations gradually shift towards high magnetic fields.As summarized in Fig.3i &3j, the angular dependence of a typical peak ( * = 0.7 T) shows anisotropies  * ()  * () ~9 and  * ()  * () ~7, respectively, infers a less anisotropic dispersion than anisotropic ratios 13, 8 in a previous report 17 .The two observations drive us to focus on the specific energy dispersion in the magnetic field.As we know, the Landau levels of the Dirac band, in ac plane of ZrTe5, are expressed as:   = � 2    ℏ, where e is the elemental charge.With fixed Fermi energy, we get a relation of ~1  .However, the Zeeman splitting  �   is large (for simplicity, we here use averaged g-factor ̅ ~15 20,37 ), results in a modified Landau level dispersion for the valence band:  (+) = − � 2    ℏ + , shows that the Zeeman effect is crucial for the specific -B relation observed in this work.Due to the high Fermi velocity   ~5 × 10 5 m/s 21 , the energy scale of the kinetic part ( � 2    ℏ) is 18 meV*√, and the Zeeman splitting part (  � 2   ) is 0.6 meV*, then the  1 is usually very small and overcome by a parabolic band mixing effect for non-ideal Dirac dispersion 38 .However, the appearance of oscillations beyond the quantum limit indicates that the kinetic part should be heavily quenched in the magnetic field.Otherwise, oscillations can only appear at unreal magnetic fields 39 .Quenched kinetic energy, or equivalently flattening the band, in Dirac dispersion is parameterized by the enhanced mass and reduced Fermi velocity, which is already indicated by the reduced anisotropy  * ()  * () .

1 3
Fig. 4c, the resulting cyclotron mass (  ) get heavily enhanced (ratio~10 2 ) in magnetic fields.The temperature-damping prefactor   of the L-K formula for Dirac fermion 41 is written as   = 2 2   || ℏ  2 , where the  is the chemical potential.Then, the cyclotron mass   fitted by  effectively reflects the quantity ||/  2 in the Dirac system, this is consistent with the universal definition   = ℏ 2 2 The temperature was then lowered to 660 o C within 7 hours and cooled to 460 o C in 200 hours.Single crystals of ZrTe5 were isolated from Te flux by centrifuging at 460 o C. Iterative temperature cycling was adopted to increase the size of ZrTe5.

Fig. 1|
Fig. 1| Electrical transport characterizations of flux-grown ZrTe5.a, Crystal structure of undistorted ZrTe5.b, Temperature-dependent resistivity   of typical flux-grown ZrTe5 samples measured in this work.c, Temperature-dependent Hall resistivity   of sample S75.Anomalous term develops when the temperature is low.d, Temperature-dependent Hall conductivity   converted from   .The anomalous term    , shadowed by the pink area, occurs after the Drude subtraction (dashed line).e, Temperature-dependent carrier density (p) and mobility () of sample S75.

Fig. 2|
Fig. 2| Polar-distortion-induced Rashba band splittings in ZrTe5.a, Angle-dependent magnetic torque at 7 T measured at different temperatures.The inset illustrates the experimental setup.The solid black line shows the fitting by the formula   =  2 +  2 sin 2  .b, Temperature-dependent anomalous Hall conductivity    obtained after subtracting the Drude component.c, Temperature-dependent  1,2 ()/ 1,2 (270 ) and    in sample S75.The dashed vertical pink line indicates the concurrence of    and polar distortion.d, Illustration of band modifications in the presence of polar distortion.The lefthand column shows the magnetic field-induced gap in the original Rashba bands, and finally, a local topological flat band is formed at the top of the band in a higher magnetic field.The right-hand column shows the flatness is strongly enhanced by the local symmetry breaking promoted by polar distortions.

Fig. 3|
Fig. 3| Anomalous resistance oscillations beyond the quantum limit in ZrTe5.a&b, Magnetoresistance measured at different angles with standard and logarithmic scales in sample S75, respectively.The inset shows the experimental setup for rotation.c, Magnetoresistance measured up to 18 T at 0.2 K in sample S74.Inset shows conductivity oscillations Δ  obtained from background subtraction.d-f, Landau fan diagrams with arbitrary integers based on oscillatory components Δ  and Δ  , respectively.Solid blue and red lines are the fittings by the relation ~ 0  +  1 * .g&h, Oscillatory component Δ  of tilted angles  and , respectively. * denotes the characteristic peak ( * = 0.7 T). i&j, The  and  dependence of characteristic peak  * .Red lines are 1/cos,  relations.

Fig. 4| A
Fig. 4| A topological flat band evidenced by field-induced mass enhancement in ZrTe5.a, Temperature-dependent oscillatory component Δ  of sample S74.B1-B7 denote the seven characteristic peaks and valleys subjected for the L-K formula fitting.a, Temperature-dependent amplitudes of the component Δ  fitted by the L-K formula.The inset shows the L-K formula fitting at fields higher than 9 T measured in a 33 T watercooled magnet.c&d, Enhanced cyclotron mass (  ) and reduced Fermi velocity (  ) in the magnetic fields.

Fig. 5|
Fig. 5| Quenched kinetic energy of Landau levels.a, Formation of Landau levels on the topological flat band with dominated Zeeman effect.b, The kinetic energy,  , = − � 2    ℏ, of Landau levels obtained from experimental cyclotron mass, showing the quenched kinetic energy at high fields.c, The total energy of Landau levels, including the dominated Zeeman effect  � 2   , exhibiting the reappearance of Landau levels across the a linear-in-B secondharmonic voltage.As shown in Fig. S2a, We now can recover this linear-in-B secondharmonic voltage and find that the nonreciprocal signal   2 (B//c) is larger than   2 (B//b).This means there is a polar component (P) along the out-of-plane b axis in our samples, unlike the negligible polarity along the b axis in ref 4 .

FIG. S1 :
FIG. S1: Raw data of Fig. 2c and magnetic torque measurements in CVT-grown sample.a, Raw data of A1 and A2 for Fig. 2c in the main text, showing the A2 term suddenly appears around T =150 K, while A1 is smooth and almost unchanged in the whole temperature range.b, Angle-dependent magnetic torque measured on a CVT-grown sample.c, Ratios of  1,2 ()/ 1,2 (250 ) for CVT-grown sample, showing no anomaly on A2, which also follows the temperature dependence of A1.
susceptibility tensor   defined by   =     , where   is the magnetization.(−   ) .Therefore, we anticipate a pure 2 relation in angle-dependent   .As shown in Fig.2a,   at 7 T globally shows  periodicity consistent with 2 relation.However, the negative and positive amplitudes, noted as Amp+ and Ampin Fig.2a, show asymmetry against a pure 2 relation.Further, we find the torque signal at 4 K can be well-fitted by   =  2 +  2 sin 2  .The appearance of  2 term 2 () is almost negligible at temperatures higher than 150 K, below which the ratio of  2 ()/ 2 (270 ) suddenly gets